Raymond M. Redheffer

The Theorems of Bony and Brezis on Flow-Invariant Sets

where [to, t1) is the interval of existence for the trajectory through the point x(to). When the solution does not exist beyond to, the condition is considered to be vacuously fulfilled. Our objective is to generalize a remarkable theorem for flow-invariant sets that was recently obtained by Bony [2] and to show its relation to another theorem of Brezis [3]. The proofs here are simpler than those given hitherto, and the results are stronger. However, this paper is expository.

Inequalities with three functions

This paper gives a unified treatment of a variety of inequalities associated with the names of Hardy, Littlewood, Opial, Beesack, Troesch, Block, Weyl, and others. Although most of the general results and specific examples are new, it must be stated the underlying ideas have been available at least since the time of Jacobi. The novelty consists chiefly in the fact that the three functions are not required to be related by a differential equation but are, in essence, three arbitrary functions. In keeping with the tradition established by Hardy and Littlewood in their fundamental work on this subject, we avoid ad hoc restrictions, and require that the needed side conditions shall be deduced from convergence of the integrals occurring in the statement of our theorems. Likewise in keeping with this tradition, we discuss conditions for equality. It is a particular feature of the method that strict inequalities (corresponding to an unattained extremum) are just as easy to establish as weak inequalities. It will be apparent to the reader that many of the results generalize to n dimensions, though this is not done here. Other possibilities of extension are suggested by the results of [I], [2], [3], [4], and [5].

The Computation of Dielectric Constants

We consider the shorted‐line method as applied to general samples, not restricted to quarter‐wave, half‐wave, thin, or low loss. The complex dielectric constant can be obtained by combination of two measurements, the sample being followed by short and open circuit, respectively. The equation is simple, nontranscendental, and unambiguous (Eqs. (5), (11), (19), (22)). It lends itself to complete graphical representation (Table I, Fig. 2). When only a single measurement is made, as in conventional practice, a change of variable leads to great simplification of detail ((14), (15), (16), Figs. 2, 3, 4). With due regard to the nature and magnitude of experimental errors, we analyze the conditions for optimum graphical representation, and give a reasoned determination of scale ranges. Large scale curves have been constructed, but only reductions are given here.

Über eine Ungleichung von Schur.

Schurs inequality (1) will be extended to scalar product spaces; cf. (4). We also give conditions for equality in (1), (4).

Nagumo theory without the Nagumo point

In the Nagumo-Westphal theory of parabolic inequalities the first step typically involves a strict inequality, and the result for weak inequality is then obtained by introduction of a suitable perturbation. As a rule, it is the latter result that is really wanted. By a double use of the perturbation, bypassing the Nagumo theory, we reach the goal in a single step. Avoiding the Nagumo point is one novelty of the presentation. Another is the use of a discontinuous comparison function in the main theorem.