A commutative algebra A over the field F, endowed with a non-zero homorphism ω:A →F is principal train if it satisfies the identity xr+y1ω(x)xr−1 +… +yr−1ω(x)r−1x=0 where y1,…,yr−1 are fixed elements in F. We present in this paper, after the introduction of the concept of “type” of A, some results concerning the classification in the case r = 3. In particular we describe all these algebras of dimension≦5.

§ 1.— The following account of this problem is taken from the Journal des Débats for December 27th, 1883.

La poste nous a apporté ces jours-ei une petite boîte en carton peint, sur laquelle on lit: la Tour d' Hanoï, véritable casse-tête annamite, rapporté du Tonkin par le professeur N. Claus (de Siam), mandarin du collège Li-Sou-Stian. Un vrai casse-tête en effet, mais intéressant. Nous ne saurions mieux remercier le mandarin de son aimable intention à l'égard d'un profane qu'en signalant la, Tour d' Hanoï aux personnes patientes possédées par le démon du jeu.

In a previous paper (Shenton, 1953) we have given an expansion for integrals of the form This expansion may be expressed as a determinantal quotient or Schweinsian series. In the present paper we state more general terms under which the expansion holds and consider the case when the limits of integration are infinite and the weight function of the form In particular we giye expansions for the Psi function, and where C (x) is a positive polynomial.

In a recent paper (1) Portnov used a form of Poisson integral to find the exact solution for the temperature distribution in a freezing semi-infinite slab occupying the region x > 0, and having an arbitrary time dependent temperature applied at the face x = 0. Previously, Boley (2) had used a method based on Duhamel's theorem to find solutions for problems involving melting, in both finite and semi-finite regions, caused by time dependent heat fluxes. Steady-state solutions have been investigated by Landau (3), Masters (4) and others (5).

In the solution of boundary value problems in mathematical physics by means of integral transforms we often find that the solution of a particular problem can be expressed in terms of integrals of the type

where r and z are positive and m and n are integers satisfying the convergence condition m+n>−2.

It is assumed that the sines of the sides are proportional to the sines of the opposite angles.

ACB (Fig. 9) is a spherical Δ, with C a right angle.

Produce AC, AB to D and E so that

Draw the great ⊙ DEF and produce CB to meet it in F.

Then F is the pole of AD and A the pole of DF.

In the present paper by means of the Schauder-Tychonoff principle sufficient conditions are obtained for Lp-equivalence of a linear and a nonlinear impulsive differential equations.

Let E be an injective module over a commutative Noetherian ring A (with non-zero identity), and let a be an ideal of A. The submodule (0:Eα) of E has a secondary representation, and so we can form the finite set AttA(0:Eα) of its attached prime ideals. In [1, 3.1], we showed that the sequence of sets is ultimately constant; in [2], we introduced the integral closure a*(E) of α relative to E, and showed that is increasing and ultimately constant. In this paper, we prove that, if a contains an element r such that rE = E, then is ultimately constant, and we obtain information about its ultimate constant value.

It is evident that by an extension of the method of deriving from any triangle its polar triangle, it is possible to derive from any figure whatever another figure, the properties of which may be deduced at once from those of the first. This may be done either by imagining a point to move along the original figure and considering the envelope of the great circle of which the moving point is the pole; or by imagining a great circle to envelope the figure and considering the locus of its pole. In both cases two figures will be obtained; but these will be antipodal, and will therefore have like properties. Since the point of intersection of two great circular arcs is the pole of the great circular arc which joins the poles of these two arcs, the two methods of derivation mentioned above will lead to the same derived figure. These methods of transformation of figures are evidently closely analogous to that of reciprocal polars.

Let $H$ be a Hall $\pi$-subgroup of a finite $\pi$-separable group $G$, and let $\alpha$ be an irreducible Brauer character of $H$. If $\alpha(x)=\alpha(y)$ whenever $x,y \in H$ are $p$-regular and $G$-conjugate, then $\alpha$ extends to a Brauer character of $G$.

AMS 2000 Mathematics subject classification: Primary 20C15; 20C20

In a recent paper read before the Society, Professor Carslaw gave an account, from the point of view of elementary geometry, of the well-known and beautiful concrete representation of hyperbolic geometry in which the non-Euclidean straight lines are represented by Euclidean circles which cut a fixed circle orthogonally. He also considered the case in which the fixed circle vanishes to a point, and showed that this corresponds to Euclidean geometry. The remaining case, in which the fixed circle is imaginary and which corresponds to elliptic or spherical geometry, is not open to the same elementary geometrical treatment, and Professor Carslaw therefore omitted any reference to it. As this might be misleading, the present note has been written primarily to supply this gap. It has been thought best, however, to give a short connected account of the whole matter from the foundation, from the point of view of analysis, omitting the detailed consequences which properly find a place in Professor Carslaw's paper.

The object of this paper is to investigate certain series and differential equations, generalizations of the series of Bessel and Legendre.

Throughout the paper

reducing, when p = 1, to n.

The series discussed are the following:—

a generalized form of Pn(x),

the relation between coefficients of x in successive terms being

In a recent paper [1] of Bell, an abstract inversion principle has been formulated for inverting a type of finite series by employing operators. Bell's result involves Baker's general principle of cross-classification [2], Dedekind-Möbius inversion, L.C.M. inversion and some known generalisations. The purpose of this note is to introduce operators of negative degree and to formulate an inversion principle which covers more cases than Bell's.

Let B be a measurable set of real numbers in (0,1) of Lebesgue measure |B| and let x1, …, xn be real. Then

denotes the number of j (1 ≦j≦n) for which the fractional part {xj}∈B. The discrepancy of x1, …, xn is

where the supremum is taken over all intervals I in [0,1].

The following communication gives a summary of the opening paper by Mr G. Lawson and of the remarks by various members who afterwards took part in the discussion. The question at issue was how to treat definitions and properties of the tangent to a plane curve in teaching beginners.