The object of this paper is to introduce the differential operator, ▽, generalised for a Riemannian space Vn immersed in a flat space Vp, and then to discuss the general small deformation of Vn.

The property of weak sequential completeness plays a special role in the theory of Boolean algebras of projections and spectral measures on Banach spaces. For instance, if X is a weakly sequentially complete Banach space, then

(i) every strongly closed bounded Boolean algebra of projections on X is complete (3, XVII.3.8, p. 2201); from which it follows easily that

(ii) every spectral measure on X of arbitary class (Σ, Γ), where Σ is a σ-algebra of sets and Γ is a total subset of the dual space of X, is strongly countably additive; and hence that

(iii) every prespectral operator on X is spectral.

(See also (1, Theorem 6.11, p. 165) for (iii).)

The investigation given in the following Note was suggested by a passage in the paper by Mr Leraoine, presented to the Society by Dr. Mackay at a recent meeting. The main subject of that paper is what he terms the “Transformation continue dans le triangle et dans le tètraédre”; for the explanation of that phrase and other terms connected with it, the reader is referred to the paper just mentioned. In the notation for the quantities connected with the triangle, however, I shall follow Dr. Mackay''s system as explained in his paper in Vol. I. of our Proceedings.

It is well known that the polynomial in x,

has the following properties:—

(A) it is the coefficient of tn in the expansion of (1–2xt+t2)–½;

(B) it satisfies the three-term recurrence relation

(C) it is the solution of the second order differential equation

(D) the sequence Pn(x) is orthogonal for the interval (— 1, 1),

i.e. when

Several other familiar polynomials, e.g., those of Laguerre Hermite, Tschebyscheff, have properties similar to some or all of the above. The aim of the present paper is to examine whether, given a sequence of functions (polynomials or not) which has one of these properties, the others follow from it : in other words we propose to examine the inter-relation of the four properties. Actually we relate each property to the generating function.

The second law of motion may be expressed as a dimensional equation in the form

where the meanings of the quantities are obvious.

If we cut out the factor m from each side, we may write this in the usual form

The general solution is

where

We propose to obtain certain results involving areal and trilinear co-ordinates, by a uniform method of changing to Cartesian co-ordinates with two sides of the triangle of reference as axes.

Though every quantity, whatever be its nature, has magnitude, no quantity can be said to be large or small absolutely. When we speak of the size of any body we mean its size relatively to the size of some other body with which we compare it. A yard is large if we compare it with an inch; it is small when compared with a mile. In the former case the number which represents it is more than 60,000 times larger than the number by which it is represented in the latter case. A mere number is therefore useless as regards the statement of magnitude, except when accompanied by a clear indication of what the thing measured is compared with. The quantity in terms of which the comparison is made is called the unit, and the number which tells how often this unit is contained in a given quantity is called the numeric of that quantity.

The anharmonic ratio of four points A, B, C, D depends on the order in which they are taken. The A.I.G.T. “Syllabus,” following Cremona, defines the symbol (ABCD) to stand for , when A is conjugate to B and C to D. Thus, taking λ to denote the value of this double-ratio, we have

In the following paper, the determination of Green's function, for spaces bounded by surfaces of the cylindrical and spherical polar systems, is effected by what is believed to be a novel process, in which are utilised the properties of cylindrical and spherical harmonics, regarded as functions of their parameters.

In this paper, it is shown that certain Theta functions are asymptotically optimal for the periodic time frequency uncertainty principle described by Breitenberger in [3]. These extremal functions give rise to a periodic multiresolution analysis where the corresponding wavelets also show similar localization properties.

We show that the global dimension, dgA, of every commutative Banach algebra A whose radical is a weighted convolution algebra is strictly greater than one. As an application, we see that in this case H2(A, X) ≠ 0 for some Banach A-bimodule X and thus there exists an unsplittable singular extension of the algebra A.

The angles which satisfy the equation

x=cosx,

occur in the solution of certain problems in analytical geometry.

It is easy to show that one value of the angle is coscoscos…cos 1, when the cosine is taken successively to infinity.

It is shown that most properties of (bounded) completely hyperexpansive operators remain valid for unbounded $2$-hyperexpansive operators. Powers of closed $2$-hyperexpansive operators are proved to be closed and $2$-hyperexpansive. Various parts of spectra of such operators are calculated. $2$-hyperexpansive weighted shifts are investigated. Examples of unbounded closed $2$-hyperexpansive ($2$-isometric) operators with invariant dense domains are established.

AMS 2000 Mathematics subject classification: Primary 47A05; 47B37. Secondary 47B20; 47B10

The stochastic stability of linear systems driven by white noise has been treated by several authors e.g. Has'minskii [7], Kushner [11], Kleimann and Arnold [9], Pinsky, [14], Friedman and Pinsky [4], Itô and Nisio [8] , Mohammed [13]. Following R. Kubo [10, 1966], the Brownian movement of a molecule in a “heat bath” is modelled in [4, pp. 223–226] by an asymptotically stable linear f.d.e. which is forced by white noise.