In (4) Young investigates the representation in terms of his exact seminormal units of substitutional expressions which are unchanged on premultiplication by any permutation of a given set of consecutive letters, or which are changed in sign on premultiplication by an odd permutation of those letters. He illustrates his results by deducing the forms of the matrices representing the positive and negative symmetric groups on a set of consecutive letters.

Let G be a locally compact Abelian group, and the set of bounded complex (regular countably-additive Borel) measures on G. It is well known that becomes a Banach space if the norm is defined by

the supremum being over all finite sets of disjoint Borel subsets of G.

We examine the action of diffeomorphisms of an oriented surface with boundary on the space of conjugacy classes of SU(2) representations of the fundamental group and prove that in the case of a single periodic diffeomorphism the induced action always has fixed points. For the corresponding 3-dimensional mapping cylinders we obtain families of representations parametrized by their value on the longitude of the torus boundary.

Dygogram is the name of a curve, invented by Archibald Smith and employed in his Admiralty Manual of the Deviation of the Compass, to give a graphic representation of the varying magnetic field of a compass as the ship is swung round in azimuth; a description is given of the Dygogram by Maxwell in Electricity and Magnetism, §441.

§1. Let (x)u denote the function

then

The concept of a laminated near-ring was introduced in [2]. We recall briefly what it is. Let N be a near-ring and let a∈N. Define a new multiplication on N by x * y = xay for all x,y∈N. With this new multiplication and the same addition as before we have another near-ring which we denote by Na. The near-ring Na is referred to as a laminated near-ring, the original near-ring N is the base near-ring and a is the laminator or laminating element.

In this paper we extend the results of Garba [1] on IOn, the semigroup of all partial one-one order-preserving maps on Xn = {1,…, n}, to POn, the semigroup of all partial order-preserving maps on Xn, A description of the subsemigroup of POn generated by the set N of all its nilpotent elements is given. The set {α∈POn:lim α/≦r and |Xn /dom α|≧r} is shown to be contained in 〈N〉 if and only if r≦½n. The depth of 〈N〉, which is the unique k for which 〈N〉 = N ∪ N2 ∪…∪ Nk and 〈N〉 ≠ N ∪ N2 ∪…∪Nk−1 is shown to be equal to 3 for all n≧3. The rank of the subsemigroup {α∈POπ|imα|≦/n − 2 and α∈〈N〉} is shown to be equal to 6(n − 2), and its nilpotent rank to be equal to 7n−15.

Let A be a complex Banach algebra with an identity 1. In this note we study the subset Λ of A consisting of all g ∈ A such that the spectrum of g, sp(g), contains at least one non-negative real number. Clearly Λ is not, in general, a semi-group with respect to either addition or multiplication. However, Λ is an instance of a subset Q of A with the following properties, where ρ(f) denotes the spectral radius of f (4, p. 30).

On donne un cercle et deux points P et Q situés sur un diamètre, on joint les points P et Q aux extrémités A et B d'ux diamètre du cercle par les droites I'A et QB qui se coupent au point M. On fait tourner le diamètre AB et on demande

I. D' étudier les variations du rappoet de construire la figure quand le rapport a unedonnée.

II. D' étudier les variations de l' angle AMB, et de construire la figure quand cet angle a une valeur donnée.

III. A′ et B′ étant les seconds points d' intersection des droites MA, MB avec la circonférence donnée, trouver le, lieu du centre du cercle circonscrit au triangle MA′B′.

In this paper we will study the properties of a natural partial order which may bedefined on an arbitrary abundant semigroup: in the case of regular semigroups werecapture the order introduced by Nambooripad [24]. For abelian PP rings our order coincides with a relation introduced by Sussman [25], Abian [1, 2] and further studied by Chacron [7]. Burmistroviˇ [6] investigated Sussman's order on separative semigroups. In the abundant case his order coincides with ours: some order theoretic properties of such semigroups may be found in a paper by Burgess [5].

and

denote two double binary (2–1) forms in (x, ξ). It is proposed to discuss the geometrical significance of their simultaneous covariant complete system, which is here quoted without proof.

It is easy to see (cf. Theorem 1 below) that the centrality of all the nilpotent elements of a given associative ring implies the centrality of every idempotent element; and (Theorem 7) these two properties are in fact equivalent in any regular ring. We establish in this note various conditions, some necessary and some sufficient, for the centrality of nilpotent or idempotent elements in the wider class of π-regular rings (in Theorems 1, 2, 3 and 4 the rings in question are not even required to be π-regular).

The Dickson polynomial Dn, (x, a) of degree n is defined by denotes the greatest integer function. In particular, we define D0 (x, a) = 2 for all real x and a. By using Dickson polynomials we present new types of generalized Stirling numbers of the first and second kinds. Some basic properties of these numbers and a combinatorial application to the enumeration of functions on finite sets in terms of their range values is also given.

1. This note gives an asymptotic evaluation of an integral of the form

as n tends to infinity, where is a sequence of real-valued functions. The theorem to be established is a natural extension of B. Levi's generalised Laplace-Darboux theorem (1, 341-51); it gives a rule for evaluating a wider class of asymptotic integrals.

The integrals of §§ 5, 6, and 7 of the following paper were first established by C. de la Vallée Poussin in a memoir Sur quelques applications de l'intégrale de Poisson (Ann. de la Soc. sc. de Bruxelles, vol. 17, 1892–3). An analogous integral to that of § 5 was discovered by A. Hurwitz, who seems not to have been aware of de la Vallée Poussin's memoir, and will be found under the title Sur quelques applications des series de Fourier in the Annales de l'École normale, vol. 19, 1902. In view of the value of these integrals for the theory of the Fourier series, the discussion now given, which follows different lines from those of previous proofs, may be of some interest. The discussion turns chiefly on the Second Theorem of Mean Value which is quite as applicable to Poisson's as to Dirichlet's Integral.

The question to be discussed is this:—

Can Aronhold's theorems on the bitangents of a quartic curve of genus three be extended to the tritangent planes of a space seztic of genus four?

The answer is “No”: decisiveness arises from the fact that the gist of Aronhold's results can be stated in a very simple form, namely:—

(a) Given seven lines in a plane it is possible to derive from them uniquely and symmetrically a quartic curve that has them for bitangents.

1. The six planes through the middle points of the edges of a tetrahedron perpendicular to the opposite edges are concurrent.