The space of Colombeau generalized functions is used as a frame for the study of hypoellipticity of a family of differential operators whose coefficients depend on a small parameter ε.

There are given necessary and sufficient conditions for the hypoellipticity of a family of differential operators with constant coefficients which depend on ε and behave like powers of ε as ε→0. The solutions of such family of equations should also satisfy the power order estimate with respect to ε.

A ring R is said to satisfy the right Ore condition with respect to a subset C of R if, given a ∈ R and e ∈ C, there exist b ∈ R and D ∈ C such that ad = cb. It is well known that R has a classical right quotient ring if and only if R satisfies the right Ore condition with respect to C when C is the set of regular elements of R (a regular elemept of R being an element of R which is not a zero-divisor). It is also well known that not every ring has a classical right quotient ring. If we make the non-trivial assumption that R has a classical right quotient ring, it is natural to ask whether this property also holds in certain rings related to R such as the ring Mn(R) of all n by n matrices over R. Some answers to this question are known when extra assumptions are made. For example, it was shown by L. W. Small in (5) that if R has a classical right quotient ring Q which is right Artinian then Mn(Q) is the right quotient ring of Mn(R) and eQe is the right quotient ring of eRe where e is an idempotent element of R. Also it was shown by C. R. Hajarnavis in (3) that if R is a Noetherian ring all of whose ideals satisfy the Artin-Rees property then R has a quotient ring Q and Mn(Q) is the quotient ring of Mn(R).

In this paper the following expansion will be obtained:

in which n and r are positive integers. The Series in the square brackets are Hypergeometric Series with a finite number of terms.

The mutual action of two electrified bodies was regarded by Maxwell as transmitted by a medium. According to him the stress in the medium consists of a “tension like a rope” along the lines of electrical force whose intensity per unit of area is R2/8π, where R is the resultant electric intensity, and of a pressure numerically equal to this in all orthogonal directions. Maxwell's remarks are somewhat vague but his notation is strongly suggestive of an elastic solid medium. It has, however, been pointed out by Minchin that Maxwell's stress system would not in an ordinary elastic solid give origin to strains consistent with the “equations of compatibility” which the theory of elastic solids supplies. Considerable interest still attaches to the theory of an elastic solid medium propagating stresses equivalent to the action between distant bodies of forces varying inversely as the square of the distance. For in the first place, it has been pointed out that the stress system given by Maxwell does not constitute a unique solution of his equations; and, in the second place, it has been suggested that some medium must exist for the transmission of gravitational forces. The statical problem of the propagation of gravitational forces by an isotropic elastic medium has been treated by Minchin. His treatment how ever neglects a certain surface condition. I have thus thought it worth while to consider the problem independently, employing the ordinary surface conditions. The first part of the paper is devoted more especially to the electrostatic problem, but the elastic solid problem is essentially the same throughout.

The following note may be considered as an addendum to the paper by me on pp. 42–47 of this volume of the Proceedings. In that paper it is shown how to inscribe in a triangle ABC, a triangle DEF, such that the perpendiculars to the sides of ABC, drawn through the points D, E, F, shall be concurrent in a point P. This is done by constructing on each of the sides of ABO a triangle similar to DEF; then O the point of concurrence of the three lines joining the vertices of ABC to the vertices of these triangles is the point “inverse” to P. The question, then, naturally arises, What must be the shape of the triangle DEF in order that the point P may be one of the Brocard points, and, as a consequence, O the other one? and the answer is easily seen to be that DEF must be similar to ABO. Hence the following construction:—

We investigate in this paper a certain special family of quadric varieties, that is of in [R]. Now among the more important properties of a quadric in [R] is that it possesses a system or systems of “ generators,” i.e., the quadric may be taken as the locus of certain families of subspaces, the behaviour of these depending on the parity of R. If R is even, a quadric V2n–1 in [2n] contains a single family of [n—l]'s, so it seems likely that in discussing special families of quadrics in [2n] an important type will be obtained by constraining the quadric to pass through a number of [n — l]'s.

A permutation group G operating on a set M is called a ZT-group (Zassenhaus transitive group) if G has the properties (i) and (ii):

(i) G operates 2-transitively on M;

(ii) Ga,b≠{id} and Ga,b,c = {id} for distinct elements a,b,c ∈ M.

Here Ga,b = {α ∈ G | α(a) = a and α(b) = b} denotes the stabilizer of {a, b}, and Ga,b,c the stabilizer of {a, b, c}, respectively.

We consider non-zero polynomials f(x1, …, xk) in k variables x1, …, xk with coefficients in the finite field GF[q] (q = pn for some prime p and positive integer n). We assume that the polynomials have been normalised by selecting one polynomial from each equivalence class with respect to multiplication by non-zero elements of GF[q]. By the degree of a polynomial f(x1, …, xk) will be understood the ordered set (m1, …, mk), where mi is the degree of f(x1 ,…, xk) in x1(i = 1, 2, …, K). The degree (m,…, mk) of a polynomial will be called totally positive if mi>0, i = 1, 2, …, k.

Rather more than twenty years ago, in a note on this subject, it was shown to the Edinburgh Mathematical Society (Proceedings, II., pp. 16–18) that a special form of continuant, viz., one with univarial diagonals, could be expressed by means of a similar continuant of much lower order. A new mode of proving this theorem, which has lately been hit upon, has unexpectedly led to the discovery that the peculiarity in question is not confined to this special form, but characterises continuants of any form whatever.

Let G be a finite group and denote by µ(G) (see [2]) the least positive integer m such that G has a faithful permutation representation in the symmetric group of degree m. This note considers the value of µ(G) when G is a double cover of the symmetric group.

The space βℕ is the Stone-Čech compactification of the discrete space of positive integers. The set of elements of βℕ which are in the kernel of every continuous homomorpnism from βℕ to a topological group is a compact semigroup containing the idempotents. At first glance it would seem a good candidate for the smallest such semigroup. We produce an infinite nested sequence of smaller such semigroups all defined naturally in terms of addition on ℕ.

A ring R is called a qc-ring if each cyclic R-module is quasi-injective. For various properties of these rings we refer to Ahsan (1) and Koehler (15). In this paper we shall obtain some additional results related to qc-rings. The scheme of the paper is as follows. Section 2 contains various preliminary definitions and results. In Section 3, we shall prove that every commutative hypercyclic ring is a qc-ring. In this section, we shall also show that a qc-ring which satisfies the ascending chain condition on its annihilators has nilpotent Jacobson-radical. Finally, in Section 4, we shall study rings all of whose proper factor rings are qc. Such rings will be called “ restricted qc ”.

Cette note que j'ai l'honneur de presenter à la Société Mathématique d'Edinburgh, par 1'entremise aimable de M. J. S. Mackay, contient, ou des résultats que je crois nouveaux, ou des développementi sur des sujets que j'ai déjà souvent abordés dans la Géomé'trie et qui concernent: la transformation continue dans le triangle et dans le tétraèdre, les formulas entre les éléments du triangle, et la Géométrographie. Pour abréger, je passerai rapidement sur les points que j'ai déjà développés ailleurs, me contentant de renvoyer, si Ton desire plus d'explications, aux meémoires ou la chose a été faite.

Let S be a free semigroup (on any set of generators). When S is given the discrete topology, its Stone-Čech compactification has a natural semigroup structure. We give two results about elements p of finite order in βS. The first is that any continuous homomorphism of βS into any compact group must send p to the identity. The second shows that natural extensions, to elements of finite order, of relationships between idempotents and sequences with distinct finite sums, do not hold.

This paper is concerned with the problem of obtaining explicit expressions of solutions of a system of coupled Lyapunov matrix differential equations of the type

where Fi, Ai(t), Bi(t), Ci(t) and Dij(t) are m×m complex matrices (members of ℂm×m), for 1≦i, j≦N, and t in the interval [a,b]. When the coefficient matrices of (1.1) are timeinvariant, Dij are scalar multiples of the identity matrix of the type Dij=dijI, where dij are real positive numbers, for 1≦i, j≦N Ci, is the transposed matrix of Bi and Fi = 0, for 1≦i≦N, the Cauchy problem (1.1) arises in control theory of continuous-time jump linear quadratic systems [9–11]. Algorithms for solving the above particular case can be found in [12]]. These methods yield approximations to the solution. Without knowing the explicit expression of the solutions and in order to avoid the error accumulation it is interesting to know an explicit expression for the exact solution. In Section 2, we obtain an explicit expression of the solution of the Cauchy problem (1.1) and of two-point boundary value problems related to the system arising in (1.1). Stability conditions for the solutions of the system of (1.1) are given. Because of developed techniques this paper can be regarded as a continuation of the sequence [3, 4, 5, 6].

Titchmarsh and others [4] havo studied integral transform pairs of tho type

showing how such pairs may be generated as a consequence of the relation which holds between the Mellin transforms of the kernels.