Quarterly of Applied Mathematics

Sufficient conditions are obtained for the delay-logistic equation x ˙ ( t ) = r ( t ) x ( t ) [ 1 − x ( t − τ ( t ) ) / K ] \dot x\left ( t \right ) = \\ r\left ( t \right )x\left ( t \right )\left [ {1 - x\left ( {t - \tau \left ( t \right )} \right )/K} \right ] to be respectively oscillatory and nonoscillatory.

When nuclei of strain approach the interface of two materials, the displacement fields may not be unique and may depend on the direction from which the interface is approached. For example, the displacement fields of a center of dilatation at the interface of two materials are not unique and depend on the direction of approach to the interface. To avoid misunderstanding, it can be stressed that each of the two fields is continuous at the interface. In this paper, we show that there are 12 independent displacement functions of second-order singularities uniquely defined at an interface. The limits of all other nuclei of strain at the interface are linear combinations of these 12 independent displacement functions.

An interface crack in anisotropic dissimilar materials is considered for the general case where all three fracture modes may be coupled. Analytic solutions are obtained for two commonly used models of interface cracks: (1) the fully open crack; and (2) the Comninou model where, even under farfield tension normal to the crack face, the existence of very small contact zones near the crack tips is permitted. Conditions under which nonoscillatory singular solutions may exist, are discussed.

A family of Wiener-type methods is discussed in a general context. These methods share the concept of expansion of an unknown transducer as an orthogonal series. The terms of the series are drawn from a hierarchy of subspaces of transducers that are orthogonal with respect to a particular stimulus ensemble. Choices of specific stochastic ensembles lead to previously described analytical methods, including the classical one of Wiener.

For a one-dimensional system of dissipative balance laws endowed with a convex entropy, we prove, under natural assumptions, that a constant equilibrium state is asymptotically L 2 {L^2} -stable under a zero-mass initial disturbance. The technique is based on the construction of an appropriate Liapunov functional involving the entropy and a so-called compensation term.

The two-dimensional convective motion generated by buoyancy forces on the fluid in a long rectangle, of which the two long sides are vertical boundaries held at different temperatures, is considered with a view to the determination of the rate of transfer of heat between the two vertical boundaries. The governing equations are set up; they reveal that the flow is determined uniquely by the Prandtl number σ \sigma , the Rayleigh number A = g ( T 1 − T 0 ) d 3 / ( T 0 κ ν ) A = g\left ( {{T_1} - {T_0}} \right ){d^3}/\left ( {{T_0}\kappa \nu } \right ) , and the ratio of the sides of the rectangle l / d l/d . In the case of cavities used for thermal insulation of buildings, which is kept specially in mind throughout the paper, A A is usually about 1000 d 3 ^{3} (where d d is in centimeters), and l / d l/d takes values between about 5 and 200.

A derivation is presented of the equation governing the pressure in a thin, flat film of ideal gas under isothermal conditions, when the surfaces bounding the film are in relative normal and tangential motion. When tangential motion is absent, the pressure equation reduces to a nonlinear heat equation, which admits of very few closed-form solutions. Various approximation methods are discussed, and two problems involving small periodic variation of the gap between parallel plates are solved by a perturbation method for a film in which fluid inertia is negligible.

Earlier results [1,2] 2 ^{2} on safe loads for a Prandtl-Reuss material subject to surface tractions or displacements which increase in ratio are here extended to any perfectly plastic material and any history of loading.