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.

Zemanian [17] obtained abelian theorems for the Hankel and K-transforms of functions and then extended his results to the corresponding transforms of distributions in the sense of Schwartz [11]. Jones [6] has discussed at length asymptotic behaviours of transforms generalized in his sense. Following the technique of Zemanian many authors have obtained abelian theorems for more general transforms of functions and distributions in the sense of Schwartz. Mention may be made of the works of Joshi and Saxena [7], Lavoine and Misra [8] and Pathak [10]. However, these authors were confined to the transforms of real variables only.

The first nonvanishing homotopy group of a finite H-space X whose mod 2 homology ring is associative occurs in degrees 1, 3 or 7. Generators of these groups can be represented by maps α:Sn→X for n = 1, 3 or 7. In this note we prove that under some hypothesis on X there exists an H-structure on Sn, n = 1, 3 or 7 such that α is an H-map.

Let G be the free product of groups A and B, where |A|≥3 and |B|≥2. We construct faithful, irreducible *-representations for the group algebras ℂ[G] and ℓ1(G). The construction gives a faithful, irreducible representation for F[G] when the field F does not have characteristic 2.

One of the aims of this paper is to examine the following conjecture, attributed to Mahowald on p. 255 of (2), Part 2. Let M be a closed connected smooth manifold of odd dimension q (q≠l,3,7) and with tangent bundle τ. Let the inclusion of a compactified fibre in the Thorn complex of τ be written μ: Sq → Tτ.