James Gregory was the third son of the Rev. John Gregory, minister of Drumoak, a small parish near Aberdeen. His mother was the daughter of David Anderson of Finzeach in Aberdeenshire, and related to Alexander Anderson, a friend of Vieta and a teacher of mathematics in Paris. Gregory is said to have received his first lessons in mathematics from his mother, but in due course he passed on first to the Grammar School and then to Marischal College, Aberdeen, where he graduated. In 1663 his Optica Promota was published in London, and he spent some time in that city after the publication of his book in the hope of securing facilities for constructing a telescope on the principles he had laid down in the Optica. His efforts were however unsuccessful, and he went to Italy where he continued his mathematical studies. After a residence of three years in Padua he returned to Scotland in 1668. In 1669 he was appointed to the Chair of Mathematics at St Andrews; in that position he had a busy and, as the years passed, a rather troubled life, so that he was glad to accept a call in 1674 to be Professor of Mathematics at Edinburgh where, as he says in a letter to a friend in Paris, “my salary is double and my encouragements much greater.” His Edinburgh professorship was however very brief as he died in October 1675. An interesting sketch of his life is given by Agnes Grainger Stewart in The Academic Gregories, a volume of the “Famous Scots Series.”

We consider the distance between a fixed Hermitian operator B and the unitary orbit of another Hermitian operator A and show that in each Schatten p-class, 1<p<∞, critical points of this distance function are at operators commuting with B. As a consequence we obtain a perturbation bound for the eigenvalues of Hermitian operators in these Schatten classes.

Let ℋ be a Hilbert space, let ℬ = (ℋ, ℋ) be the B*-algebra of bounded linear operators from ℋ to ℋ with the uniform operator topology, and let ℒ be the subset of ℬ consisting of the self-adjoint operators. This article is concerned with the second order self-adjoint differential equation

A semiband is defined as a semigroup generated by idempotents. It is known that every finite semigroup is embeddable in a finite semiband. For a class C of semigroups and an integer n≧2, the number σC (n) is defined as the smallest k with the property that every semigroup of order n in the class C is embeddable in a semiband of order not exceeding k. It is shown that for the class Gp of groups σGp(n) = nq(ρGp(n)), where

and

Estimates are known (and are quoted) for the function q. Estimates are considered for the function pC for various C

It is shown also that if C0S, CS denote respectively the classes of completely 0-simple and completely simple semigroups, then

We obtain optimal L2-lower bounds for nonzero solutions to – ΔΨ + VΔ = EΨ in Rn, n ≥ 2, E ∈ R where V is a measurable complex-valued potential with V(x) = 0(|x|-c) as |x|→∞, for some ε∈ R. We show that if 3δ = max{0, 1 – 2ε} and exp (τ|x|1+δ)Ψ ∈ L2(Rn)for all τ > 0, then Ψ; has compact support. This result is new for 0 < ε ½ and generalizes similar results obtained by Meshkov for = 0, and by Froese, Herbst, M. Hoffmann-Ostenhof, and T. Hoffmann-Ostenhof for both ε≤O and ε≥½. These L2-lower bounds are well known to be optimal for ε ≥ ½ while for ε < ½ this last is only known for ε = O in view of an example of Meshkov. We generalize Meshkov's example for ε< ½ and thus show that for complex-valued potentials our result is optimal for all ε ∈ R.

Let $N$ be a zero-symmetric near-ring with identity, and let $\sGa$ be a faithful tame $N$-group. We characterize those ideals of $\sGa$ that are the range of some idempotent element of $N$. Using these idempotents, we show that the polynomials on the direct product of the finite $\sOm$-groups $V_1,V_2,\dots,V_n$ can be studied componentwise if and only if $\prod_{i=1}^nV_i$ has no skew congruences.

AMS 2000 Mathematics subject classification: Primary 16Y30. Secondary 08A40

In what follows I propose to give the history of two theorenis and to state some of the consequences that have been developed from them.

The first theorem is:

If a triangle be inscribed in a circle, and from any point in the circumference perpendiculars be drawn to the sides, the feet of these perpendiculars lie in a straight line.

In this paper we present three results about Arens regular bilinear operators. These are: (a). Let X, Y be two Banach spaces, K a compact Hausdorff space, µ a Borel measure on K and m: X × Y →ℂ a bounded bilinear operator. Then the bilinear operator defined by is regular iff m is regular, (b) Let (Xα), (Xα),(Zα) be three families of Banach spaces and let mα:Xα ×Yα→Zα, be a family of bilinear operators with supα∥mα∥<∞. Then the bilinear operator defined by is regular iff each mα, is regular, (c) Let X, Y have the Dieudonné property and let m:X × Y→Z be a bounded bilinear operator with m(X×Y) separable and such that, for each z′ in ext Z′1, z′∘m is regular. Then m is regular. Several applications of these results are also given.

1. By a quantitative proof of an inequality I mean one which exhibits the difference between the two magnitudes compared in a form which shows at a glance whether the difference is positive or negative. Such a proof not merely establishes the existence of the inequality, but also gives a measure of its amount.

The purpose of this paper is to discuss non-linear boundary value problems for elliptic systems of the type

where Ak is a second order uniformly elliptic operator and is such that the problem

has a one-dimensional space of solutions that is generated by a non-negative function. The boundary ∂G is supposed to be smooth and the functions gk, 1≦k≦m are defined on Ḡ×Rm and are continuously differentiate (usually, Bk represents Dirichlet or Neumann conditions and is the first eigenvalue associated with Ak and such boundary conditions).

In a paper entitled “Sets of anticommuting matrices” Eddington proved that if El, E2, …., Eqform a set of q four-rowed square matrices satisfying the relations,

,

where E is the unit matrix, then the maximum value of q is five. Later Newman showed that this result is a particular case of the general theorem that ifE1, E2, …., Eqform a set of q t-rowed square matrices satisfying (1), where t = 2Pτ and τ is odd, then the maximum value of q is 2p + 1.

Let G be a group and let K be an algebraically closed field of characteristic p>0. The twisted group algebra Kt(G) of G over K is defined as follows: let G have elements a, b, c, … and let Kt(G) be a vector space over K with basis elements , …; a multiplication is defined on this basis of Kt(G) and extended by linearity to Kt(G) by letting

where α(x, y) is a non-zero element of K, subject to the condition that

which is both necessary and sufficient for associativity. If, for all x, y ∈ G, α{x, y) is the identity of K then Kt(G) is the usual group algebra K(G) of G over K. We denote the Jacobson radical of Kt(G) by JKt(G). We are interested in the relationship between JKt(G) and JKt(H) where H is a normal subgroup of G. In § 2 we show, among other results, that if certain centralising conditions are satisfied and if JK(H) is locally nilpotent then JK(H)K(G) is also locally nilpotent and thus contained in JK(G). It is observed that in the absence of some centralising conditions these conclusions are false. We show, in particular, that if H and G/C(H) are locally finite, C(H) being the centraliser of H, and if G/H has no non-trivial elements of order p, then JK(G) coincides with the locally nilpotent ideal JK(H)K(G). The latter, and probably more significant, part of this paper is concerned with particular types of groups. We introduce the notion of a restricted SN-group and show that if G is such a group and ifG has no non-trivial elements of order p then JKt(G) = {0}. It is also shown that if G is polycyclic then JKt(G) is nilpotent.

In [3], [8], and [2], it was shown that if is an essentially Hermitian operator on l P, 1≦ p<∞, or on Lp[0,1], 1< p<∞, then T is a compact perturbation of a Hermitian operator. In [1], this result was established for operators on Orlicz sequence space l M, where 2∉[α M,β M] (the associated interval for M). In that same paper, it was conjectured that this result does not in general hold if 2∈[α M,β M]. In this paper, we show that this conjecture is correct by exhibiting an Orlicz sequence space l M and an essentially Hermitian operator on l M which is not a compact perturbation of a Hermitian operator.