Let S and T be compact Hausdorff spaces and G and H finite-dimensional subspaces of C(S) and C(T) respectively. Suppose μ and ν are regular Borel measures on S and T respectively such that μ(S)= ν(T)= 1. The product measure μ × ν will be denoted by σ. Set U = G⊗C(T), V =C(S)⊗H and W = U + V. If G and H possess continuous proximity maps, then U and V are proximinal subspaces of C(S × T) when this linear space is equipped with the L1-norm, [4, Lemma 2]. That is, every z∈C(S × T) possesses at least one best approximation from U and from V. A metric selection Au:C(S × T →U is a mapping which associates each z ∈ C(S × T) with one of its best approximations in U.

This paper deals with the construction of exact and analytical-numerical solutions with a priori error bounds for systems of the type ut = Auxx, A1u(0, t) + B1ux (0, t) = 0, A2u (1, t) + B2ux (1, t) = 0, 0 < x < 1, t > 0, u(x, 0) = f(x), where A1, A2, B1 and B2 are matrices for which no simultaneous diagonalizable hypothesis is assumed, and A is a positive stable matrix. Given an admissible error ε and a bounded subdomain D, an approximate solution whose error with respect to an exact series solution is less than ε uniformly in D is constructed.

1. If I is an inflexion on a non-singular plane cubic curve, a variable line IPP′ establishes a (1, 1) correspondence between points P, P′ on the curve. This correspondence defines a perspective transformation of the whole plane, with I for pole, and the harmonic polar of I for axis, of perspective; for, when I is projected to infinity on the y-axis, and its harmonic polar taken for x-axis, the resulting equation

indicates a curve symmetrical with respect to the latter.

The object of the present note is to obtain expansions of the square of Whittaker's M-Functions in series of M-Functions and also other expansions involving M-Functions.

In a given circle let the arc AP subtend an angle 3a at the centre O, it is required to trisect the angle AOP, or the arc AP.

The three trisectors will be OQ1, OQ2, OQ3, where AOQ1 = a,

∴ Q1 Q2 Q3 form an equilateral triangle. (See Figs. 10 and 11.)

We proceed to solve the problem by drawing a conic through Q1, Q2, Q3, a nd we wish to find in what cases such a conic can be drawn, a conic cutting the circle in four points, three of which form an equilateral triangle.

Let

where the {lν} are numbers near to an arithmetic progression of common difference unity. Let

b being a constant. Write

so that k(z) is an integral function, and

is meromorphic with simple poles at {lν}.

A form of integration of tensors will be introduced here, which will preserve the character of a tensor when so integrated.

Products of idempotents are investigated in the endomorphism monoid of an algebra belonging to a class of algebras which includes finite sets and finite dimensional vector spaces as special cases. It is shown that every endomorphism which is not an automorphism is a product of idempotent endomorphisms. This provides a common generalisation of earlier results of Howie and Erdos for the cases when the algebra is a set or vector space respectively.

Using results obtained by J. W. L. Glaisher [1, 2] for the number of representations Rr,s(n) of n as a sum of r odd and s even squares, formulae are derived for the number of Cayley integers of given norm n in certain orders ℴ. When computer generating order elements of given norm, the formulae can be used to verify that all the required elements have been obtained.

We prove that every partially defined derivation on a semisimple complex Banach algebra whose domain is a (non necessarily closed) essential ideal is closable. In particular, we show that every derivation defined on any nonzero ideal of a prime C*-algebra is continuous.

In theory of polarizing operators in invariants the operator

where X = [xij] is an n × n matrix of n2 independent elements xij, holds an important place. Acting upon particular scalar functions of X, namely the spur or trace of powers of X, or of polynomials or rational functions of X with scalar coefficients, it exhibits (Turnbull,. 1927, 1929, 1931) an exact analogy with results in the ordinary differentiation of the corresponding functions of one scalar variable. Turnbull denotes this operation of trace-differentiation under Ω by Ω8; and we shall follow him. Our purpose is to show how, with a suitably modified Ω, the results may be extended to the case of symmetric matrices X = X′ having ½n(n + 1) independent elements.

Linear interpolation between two values of a function ua and ub can be performed, as is well known, in either of two ways. If the divided difference (ub−ua)/(b−a), which is usually denoted by u (a, b) or u (b, a), is provided, or its equivalent in tables at unit interval (the ordinary difference), we should generally prefer to use the formula

which is the linear case of Newton's fundamental formula for interpolation by divided differences. If differences are not given, but a machine is available, then the use of proportional parts in the form of the weighted average

the linear case of Lagrange's formula, is actually more convenient, since it involves no clearing of the product dials until the final result is read.

Let it be required to find the square integral numbers which added to a given integer shall produce a square integer, and the smallest such number.

We give necessary and sufficient conditions on a general cone of positive functions to satisfy the Decomposition Property (DP) introduced in [5] and connect the results with the theory of interpolation of cones introduced by Sagher [9]. One of our main result states that if Q satisfies DP or equivalently is divisible, then for the quasi-normed spaces E0 and E1,

According to this formula, it yields that the interpolation theory for divisible cones can be easily obtained from the classical theory.

The transformations discussed in the present paper are, like the isogonal and isotomic transformations, particular cases of the general birational quadratic transformation, in which points correspond to points, and lines to conics passing through three fixed points. They seem to possess some interest in connection with the Geometry of the Triangle.