With differential equations in the neighbourhood of an irregular singular point, it sometimes happens that formal solutions may converge. For example, this occurs for Bessel's equation at∞ when the parameter is half of an odd integer. In addition, there are some classical theorems of Perron and Lettenmeyer which give sufficient conditions for the existence of linearly independent analytic solutions at (generally) an irregular singular point. Using the principle of reduction of order, such a solution may be used to transform the differential equation into one whose coefficient matrix is triangularly blocked with an (n – 1) and 1-block on the diagonal. The solutions of the given differential equation can thus be obtained by solving a lower dimensional differential equation plus quadrature.

Let f be a continuous real valued function defined on [−1, 1] and let En(f) denote the degree of best uniform approximation to f by algebraic polynomial of degree at most n. The supremum norm on [a, b] is denoted by ∥.∥[a, b] and the polynomial of degree n of best uniform approximation is denoted by Pn. We find a class of functions f such that there exists a fixed a ∈(−1, 1) with the following property

for some positive constants C and N independent of n. Moreover the sequence is optimal in the sense that if is replaced by then the above inequality need not hold no matter how small C > 0 is chosen.

We also find another, more general class a functions f for which

infinitely often.

In a previous paper in these Proceedings the author discussed conditions for a maximum or minimum of functions of integrals of the type

using the methods of the Calculus of Variations. In the effort to establish a third necessary condition for a minimum—the analogue of Jacobi's condition in the ordinary variational problem—it was found that the analogue of Jacobi's Equation was an integrodifferential equation of the form

In 1913 Richardson published necessary and sufficient conditions for a system of three Sturm–Liouville equations, linked by three parameters, to possess eigenfunctions with arbitrarily many zeros. His work contains errors, but we give conditions of his type valid for k self-adjoint equations, with k parameters.

Indices of the free logarithmetic correspond to bifurcating root-trees (cf.(4)), to Evans' non-associative numbers (3) and to Etherington's partitive numbers (2). The free commutative logarithmetic is the homomorph of f determined by the congruence relation P + Q ∼ Q + P. Formulæ for aδ and pα, i.e. the numbers of indices of of a given potency* δ and the number of indices of a given altitude α respectively, were given by Etherington (1), who also gave corresponding formulæ for commutative indices of . Other enumeration formulæ are contained in (5).

It has been shown [4] that if two triangles ABC and PQR (Fig. 6) be in perspective with respect to any point S as centre of perspective, the sides of PQR cut the non-corresponding sides of ABC in three pairs of points which lie on a circle, provided that the angles made by the sides of PQR with those of ABC possess certain values dependent on the position of S. In the general case, when these angles do not conform to the condition referred to, it follows from Pascal's theorem that the six points determined by the sides of PQR on the sides of ABC, lie on a conic.

We develop here the recurrence relations for the generalised C.F.'s introduced in Part 3 (Shenton 1956). In the main the discussion will be limited to second order C.F.'s, but results for higher orders will be given when these are not complicated.

The purpose of the present note is to answer the following question of T. A. Gillespie,learned from G. J. Murphy [4]: for which sequences{an} of complex numbers does there exist a quasinilpotent operator Q on a (separable, infinite-dimensional, complex) Hilbert space H, which has{an} as a diagonal, that is (Qen,en)=n for some orthonormal basis{en} in H?

In recent years there has been considerable interest in Banach spaces with the Radon-Nikodym Property; see (1) for a summary of the main known results on this class of spaces.We may define this property as follows: a Banach space X has the Radon-Nikodym Property if whenever T ∈ ℒ (L1, X)(where L1 = L1(0, 1)) then T is differentiable i.e.

where g:(0, 1)→X is an essentially bounded strongly measurable function. In this paper we examine analogues of the Radon-Nikodym Property for quasi-Banach spaces. If 0>p > 1, there are several possible ways of defining “differentiable” operators on Lp, but they inevitably lead to the conclusion that the only differentiable operator is zero.

The most important application of the rational point transformations between two spaces lies in the construction of algebraical surfaces possessing singularities of various kinds and the investigation of their properties.

When a plane is transformed by such a transformation into a rational algebraical surface (homaloid) the geometry of the surface is immediately derivable from the geometry of the plane.

In this paper we study a property which we call residual -commutativity. This idea was kindled by a paper of Ayoub (1) and one of Stanley (6). Durbin has defined a similar property in (2) which has also been studied by Slotterbeck in (4). Durbin's property implies residual -commutativity but we have not been able to decide if they are equivalent. However, we have shown that they coincide in certain circumstances. The notation, unless otherwise stated, is that of Robinson (3).

Results of Arscott (1) and Jayne (3) on real matrices are generalized to obtain bounds for the real parts of the eigenvalues of certain complex tridiagonal matrices, and bounds for the imaginary parts of the eigenvalues of other tridiagonal matrices are given. It is shown that analogous results hold for zeros of the permanent of certain characteristic matrices.

In the Proceedings of the Edinburgh Mathematical Society, 1948, there appear two papers by Lars Gårding and Turnbull respectively (Gårding [1], Turnbull [2]) which formulate the theory of Cayley and Capelli operators associated with symmetric matrices. Turnbull derives the modification, appropriate to symmetric matrices, of Capelli's Theorem, which states that (taking a third order operator for the sake of ease in writing)

where the symbol (xyz)ijk stands for the determinant

with a similar meaning for the determinantal differential operator, while the symbols are polarisations (Capelli [1]; cf. Turnbull [1], p. 116). Gårding's theorem deals with the effect of such modified Capelli operators on powers of the determinant of the symmetric matrix in question. The subject of this note is an alternative derivation of the modified Capelli theorem and of Gårding's theorem.