Let G be a torsion free group, F the free group generated by t. The equation r(t) = 1 is said to have a solution over G if there is a solution in some group that contains G. In this paper we generalize a result due to Klyachko who established the solution when the exponent sum of t is one.

The attempt to enumerate the possible distinct forms of knots of any order, though unsuccessful as yet, has led me to a number of curious results, some of which may perhaps be new. The general character of the methods employed will be obvious from an inspection of a few simple cases, and any one who has some practice in algebra may extend the results indefinitely.

§ 1. The roots of the cubic equation

are given by the formula

This solution is of little practical use when the roots of the cubic are all real and unequal, that is, when is negative (the-Irreducible Case of Cardan's Solution).

We prove two inequalities which relate the Lp modulus of continuity of n-th order, ωn(f,·)p, of an Hp function f with the p-th mean values of the n-th derivative f(n). Using these inequalities we extend classical results of Hardy and Littlewood [5], Gwiliam [4], Zygmund [13] and Taibleson [12] as well as a recent result of Oswald [6].

In this note we point out that certain algebraic-topological constructions are particular cases of one construction, namely double-negation sheafification. The principal cases we have in mind are concerned with booleanpowers, completions of boolean algebras, and maximal rings of quotients.We conjecture that several other constructions—particularly completion-type constructions—will turn out also to be examples of double-negation sheafification.

The modular group PSL(2, ℤ), which is isomorphic to a free product of a cyclicgroupof order 2 and a cyclic group of order 3, has many important homomorphic images. Inparticular, Macbeath [7] showed that PSL(2, q) is an image of the modular group if q ≠ 9. (Here, as usual, q is a prime power.) The extended modular group PGL(2, ℤ) contains PSL{2, ℤ) with index 2. It has a presentation

the subgroup PSL(2, ℤ) being generated by UV and VW.

The object of this note was to point out that in using the method of limits to find a geometrical maximum or minimum it is not correct to conduct all the reasoning at the final stage when the limit has been reached, and to call attention to the form of statement which lays stress on the fact that the reasoning should be based on the consideration of the quantities involved while they are yet finite. Examples were given from one or two well-known books for students where the fallacious method of proof is adopted.

One of the simplest three part boundary value problems is the electrostatic problem for the circular annulus and, at present, there seems to be no method available for obtaining the solution in a closed form. It has recently been shown by the author (1) and Cooke (2) that this problem can be reduced to the solution of a Fredholm integral equation of the second kind. The equation obtained in (1, 2) is fairly simple and is suitable for obtaining a numerical solution but, unfortunately, it cannot be solved iteratively to give a simple form of solution valid for small values of the ratio (inner radius/outer radius).

The relation between the eccentric and focal angles of P (Fig. 1) is

or, more compactly,

where

In (3) it is shown that, for a locally compact abelian group G and x∈G, δx has a logarithm in M(G) if and only if x has finite order. Since M(G) can be identified with the multipliers of L1(G), one might expect a similar result for the algebras of multipliers on Lp(G) for 1 < p < ∞. However, in contrast, it is shown in (2) that for a locally compact abelian group G and 1 < p < ∞, every translation operator on Lp(G) has a logarithm in the multiplier algebra. Here we consider whether the same results are true for non-abelian groups.

In the works known to us which contain historical or bibliographical information about circular cubics no sufficient indication of the researches of Maclaurin is found. Yet many of the classic propositions connected with these curves are due to this eminent geometer, who investigated constructions for unicursal circular cubics, for certain non-unicursal circular cubics, and for the special circular cubics now known as the trisectrix of Maclaurin, the oblique cissoid, and the strophoid. His name does not appear even in the list of writings on the strophoid published by Tortolini and Günther.

The principal problem which we have in view may be stated as follows:

To construct a triangle ABC (Fig. 1) in which are given in magnitude only, the height h = AO from the vertex A, the median m = BI from the vertex B, and the bisector f= CD from the vertex C.

1. Let one of the series of circles, which can be drawn so as to touch the sides AB, AC of a triangle, touch those sides in K, L; and let AK = AL = δ.

Then the points K, L are

and the lines BL, CK are givn by

Hence eliminating δ we get for the locus of P the hyperbola

The allied hyperbolas are

Gordan's Theorem, that the complete system of irreducible concomitants of a given form is finite, has been extended by Hilbert to cover wide ranges of systems of variables Gordan and Study have dealt shortly with the problem for double binary forms, approaching the subject through the theory of binary types. The following pages give a proof after the manner of Gordan's proof for ordinary binary forms, which has the advantage of providing a practical method for constructing the complete system. As illustrations, the cases of the (1, 2), (2, 2), and (3, 3) forms are considered.