If f(z) is analytic in a suitable domain, it is shown how the integral of f(α1x1+…+αnxn) over the interior of a simplex may be reduced to the evaluation of a contour integral, in fact to an exercise in partial fractions.

The contour integral is expressed in two ways, according as the simplex is given in terms of its vertices or faces.

Analogous to the Banach-Mazur distance between Banach spaces, we study the completely bounded Banach-Mazur distance between operator spaces.

In many cases of Banach spaces and Hilbert spaces we show that the infimum is attained when T is the identity map, and X, Y have the same base space. This provides a machinery to compute and estimate dcb(X, Y). Later, using symmetric norming functions we construct counterexamples to show that distinct infinite dimensional homogeneous operator spaces may have finite cb-distance, and that two homogeneous Hilbertian operator spaces may not coincide even if they coincide over all 2-dimensional subspaces.

In this note we ask two questions and answer one. The questions can be combined as follows:

Does there exist a polynomial of the form

which starts with prescribed complex coefficients c0, …, cr–1; and satisfies

These differ from the classical problems of Carathéodory in one essential respect: the values of p and its first r–1 derivatives are given at the point z = 1 on the circumference of the unit circle, while in the original problem they were given at z = 0. Carathéodory's own answer was in terms of his “moment curve”, but the forms studied a few years later by Toeplitz yield a more convenient statement of the solution.

It is proposed in this paper to show now the well-known Laplace's transformation,

which is of great help in finding the solution of linear differential equations, gives also interesting results concerning the theory of integral equations. In §2 we shall study its application to certain differential equations, and find a large class of equations which remain unchanged by this transformation. Then, (§3), taking instead of eazt, a more general function of the product zt, we shall find a solution for some homogeneous integral equations ; in § 4 we shall describe a method of solving a very general type of integral equation of the first kind, namely,

a further extension to integral equations with the kernel ef(z)f(t) the object of §5. Then, studying an extension of Euler's transformation, we shall (§ 6) consider equations such as

which will prove to be singular; and finally, in §7, we shall give other examples of singular integral equations.

I prove in this note some theorems on Rieszian and Dirichlet summabilities involving a Tauberian hypothesis with gaps. One of the theorems (§ 2, Theorem A) has been proved by Ricci [4, § 6] in a slightly less general form. Another theorem (§ 3) contains a Riesz version of a (C, k)-summability problem studied by Meyer-König [1, Satz 1].

Some years ago Heins (1) proved that a Riemann surface which can be conformally imbedded in every closed Riemann surface of a fixed positive genus g is conformally equivalent to a bounded plane domain. In the proof the main effort is required to prove that a surface satisfying this condition is schlichtartig. Heins gave quite a simple proof of the remaining portion (1; Lemma 1). The main part of the proof depended on exhibiting a family of surfaces of genus g such that a surface which could be conformally imbedded in all of them was necessarily schlichtartig. Another proof using a different construction was recently given by Rochberg (2). We will give here a further proof based on the method of the extremal metric and using a further construction which is in some ways more direct than those previously given.

In a recent paper (1) Singh and Dhaliwal present an analysis which purports to solve a pair of dual integral equations which occur in crack theory and the object of this note is to demonstrate that their analysis is erroneous.

In [1] the first author considered the following Engel-like condition for a pair of elements x, y of a group G.

There exist r = r(x, y)≧0 and d = d(x, y)≧1 such that[x,ry] = [x,r+dy]. (*)

The study of the primitive solutions of the equation

where A = (aij) is an n × n matrix whose elements are rational integers, was begun a long time ago. In most cases this equation occurred incidentally in another theory; for instance Jordan encountered it in connection with linear differential equations having algebraic solutions, Minkowski in connection with quadratic forms and Turnbull in geometry. An important fact about these matrices is that any unimodular matrix can be represented as the product of matrices with finite period.

Let a and b be positive and unequal; and let x be their arithmetic mean. Then we have the following equality and inequality

The inequality is tantamount to the fact that the G.M. of a and b is less than the A.M.