On the transition from supercritical to subcritical Hopf bifurcations

By using the method of averaging, we give a complete Hopf bifurcation diagram when the differential equation has more than one parameter. The method is applied to a system of parabolic equations with nonlinear coupling on the boundary. This set of equations is a mathematical model which describes the normalized concentrations of substances in a solution produced by two enzymes.     

Entropy and Decision

This paper deals with the irrelevance of entropy to decision. The entropy measure fails to discriminate between states and their significance, as is indicated by a study of a reconnaissance example for which the entropy measure has been suggested. If the reduction of uncertainty is an explicit objective, then it is important to examine the properties of uncertainty measures, and it is queried as to whether some of those uniquely characterizing entropy are relevant. Other measures have some of the properties. The paper then discusses the concept of entropy in hypothesis selection, and the conflict with maximum-likelihood methods, and in Markovian processes, wherein there is some conflict with implied steady-state behaviour.     

Gray codes in graphs of subsets

Let G(n,k) be a graph whose vertices are the k-element subsets of an n-set represented as n-tuples of “O's” and “1's” with k “1's”. Two such subsets are adjacent if one can be obtained from the other by switching a “O” and a “1” which are in adjacent positions, where the first and nth positions are also considered adjacent. The problem of finding hamiltonian cycles in G(n,k) is discussed. This may be considered a problem of finding “Gray codes” of the k-element subsets of an n-set. It is shown that no such cycle exists if n and k are both even or if k=2 and n?7 and that such a cycle does exist in all other cases where k?5.     

Scaling law for the elastic response of spherical explosion-containment vessels

The substantial progress made in the development of portable, spherical explosion-containment vessels has led to its adoption for explosives-related transporting and manufacturing. A simple scaling law for engineering-design use has been obtained for the maximum first-cycle elastic-strain response to explosive loading by a centered, spherical charge of a representative high explosive known as Pentolite. Results from a computer program based on the work of Baker have been accurately fitted to a six-product power law that is a function of the density, elastic modulus, diameter, and thickness of the vessel and the weight of detonated explosive. Nonlinear and log-linear fits are compared and found to agree if the role of weighting factors is understood. An impulse approximation is used to indicate that the scaling-law parameters are reasonable.     

Moiré patterns in the electron image of solution-grown polymer crystals

Relative tilt between two similar thin crystals may result in the formation of a moiré pattern in the electron image. Patterns hitherto interpreted as being parallel moirés or rotation moiré obtained from overlapping solution-grown polymer crystals are reconsidered, and it is shown that in some cases a tilt model is more acceptable. The general features of tilt moiré patterns predicted for crystals of this kind are discussed and compared with micrographs appearing in the literature. Methods of distinguishing tilt moiré patterns are indicated, though it is admitted that such experiments would be difficult to complete in the time limited by destruction of the crystal structure under the action of the electron beam. Extinction contours may also sometimes be confused with long-period tilt moiré patterns, and again careful experimentation is required for accurate identification. The model of the structure of solution-grown polyethylene crystals inferred from a tilt moiré interpretation of many of the electron image fringe patterns found in the literature is consistent with that derived from other sources.     

A priori Schranken für Eigenwerte von Sturm-Liouvilleschen Eigenwertproblemen

By means of a duality principle of the optimal control theory bounds for the eigenvalues of a Sturm-Liouville problem are constructed. They are then compared with the bounds obtained from monotonicity principles.