Let R be a ring, a linear sequential machine over R is a quintuple ℳ = (Q, A, B, F, G) where

Q, A, B are R-modules

F:Q × A →Q and G:Q × A→B are R-homomorphisms.

In Mathematische Annalen, Vol. 32 (1888) Peano discusses the solution of a system of homogeneous linear differential equations

where rij denotes a real function of the variable t, and shows how, by a series of repeated substitutions, this system of equations may be replaced by the equivalent equation

where X denotes the complex [x1, x2, … xn] and R the matrix

of which equation the solution X can be represented as a sum of integrals.

In [6] Brown and Spencer noted that internal categories within the category of groups are equivalent to crossed modules. As they remarked, this result was known to various others before them, but it had not until then appeared in print. That paper led me to investigate the question of which algebraic categories, C, were such that a similar result held i.e. internal categories in C are equivalent to crossed modules of the appropriate type. The resulting work was written up in 1980 but was not submitted for publication.

Let U be the quantized enveloping algebra associated to a simple Lie algebra g by Drinfel'd and Jimbo. Let λ be a classical fundamental weight for g, and ⋯(λ) the irreducible, finite-dimensional type 1 highest weight U-module with highest weight λ. We show that the canonical basis for ⋯(λ) (see Kashiwara [6, §0] and Lusztig [18, 14.4.12]) and the standard monomial basis (see [11, §§2.4 and 2.5]) for ⋯(λ) coincide.

“To draw through a given point a transversal of a given triangle so that the segments of the transversal may be in a given ratio.”

Let us denote by α the set of n real numbers α1, …, αn, and by ck(α) and hk(α) the elementary and complete symmetric functions of degree k in α1, …, αn, and by ck(α) and hk(α) the elementary and complete symmetric functions of degree k in αl, …, αn, i.e. ck(α) is the sum of all possible products of k different αi and hk(α) is the sum of all possible products of k αi, where now in any product one or more αi may be repeated any number of times.

The Cubic Surface, as is well known, can be formulated as the locus of a point which, when joined to six given lines—one of which cuts the other five—forms planes enveloping a quadric cone.1 It might be of some interest to show how such a definition leads to one or two of the better known forms of the equation to the surface. The method of approach is by means of the Clebsch Transformation Principle in Geometry and the use of general coordinates. In particular, compound symbols and bracket factors, as developed by H. W. Turnbull2 in his works on Geometry and Invariant Algebra, have been largely used.

We consider faithful finitary linear representations of (generalized) wreath products A wrΩH of groups A by H over (potentially) infinite-dimensional vector spaces, having previously considered completely reducible such representations in an earlier paper. The simpler the structure of A the more complex, it seems, these representations can become. If A has no non-trivial abelian normal subgroups, the conditions we present are both necessary and sufficient. They imply, for example, that for such an A, if there exists such a representation of the standard wreath product A wr H of infinite dimension, then there already exists one of finite dimension.

The circumstances explained in the footnote on p. 61 of the former paper with this title might well have necessitated a re-writing of the whole. Fortunately it appears that only a few short comments are required. For example it may be noted that the first question in §4 can be answered by counting the degrees of freedom in the two configurations. Eight points of a twisted cubic have freedom 20; four pairs of planes drawn at random through four lines of a regulus have freedom 21; therefore the eight planes of the second paragraph of §4 cannot always lead back to the eight points of §2. This is corroborated by the corresponding numbers in [4], which are 31 and 34.

The following notes are exactly as I left them in the hands of the Committee of the Society eleven years ago. They are printed now in the hope that, chiefly because of their brevity, they may be found useful to members, who may not have leisure or opportunity to read up the subject in the recognised text-books.—Jan., 1894.

The following direct method of attack establishes the fundamental properties of the Foci and Bi-Tangents of a Bi-Circular Quartic very simply, using only the properties of a homographic correspondence. The method was suggested to me by my previous paper on the focal properties of Circular Cubics.

I am indebted to Mr Peter Fraser, University of Bristol, for his criticism.

Given a finite (connected) simplicial graph with groups assigned to the vertices, the graph product of the vertex groups is the free product modulo the relation that adjacent groups commute. The graph product of finitely presented infinite groups is both semistable at infinity and quasi-simply filtrated. Explicit bounds for the isoperimetric inequality and isodiametric inequality for graph products is given, based on isoperimetric and isodiametric inequalities for the vertex groups.

Existence results are established for the equation y″ + f(t, y) = 0, 0<t<1. Here f may be singular in y and f is allowed to change sign. Our boundary data include y(0) = y′(1) + ky(1) = 0, k> – 1 and y(0) = y′(1) + cy4(1) = 0, c>0.