This is a study of the runup of solitary waves on plane beaches. An approximate theory is presented for non-breaking waves and an asymptotic result is derived for the maximum runup of solitary waves. A series of laboratory experiments is described to support the theory. It is shown that the linear theory predicts the maximum runup satisfactorily, and that the nonlinear theory describes the climb of solitary waves equally well. Different runup regimes are found to exist for the runup of breaking and non-breaking waves. A breaking criterion is derived for determining whether a solitary wave will break as it climbs up a sloping beach, and a different criterion is shown to apply for determining whether a wave will break during rundown. These results are used to explain some of the existing empirical runup relationships.

https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S002211208700329X

The runup of solitary waves

The Random Character of Stock Market Prices.

The elastic contact of a rough surface

This paper considers the use and design of linear periodic time-varying controllers for the feedback control of linear time-invariant discrete-time plants. We will show that for a large class of robustness problems, periodic compensators are superior to time-invariant ones. We will give explicit design techniques which can be easily implemented. In the context of periodic controllers, we also consider the strong and simultaneous stabilization problems. Finally, we show that for the problem of weighted sensitivity minimization for linear time-invariant plants, time-varying controllers offer no advantage over the time-invariant ones.

Robust control of linear time-invariant plants using periodic compensation

A phase transition indicates a sudden change in the properties of a large system. For temperature-driven phase transitions this is related to nonanalytic behavior of the free energy density at the critical temperature: The knowledge of the free energy density in one phase is insufficient to predict the properties of the other phase. In this Letter we show that a close analogue of this behavior can occur in the real time evolution of quantum systems, namely nonanalytic behavior at a critical time. We denote such behavior a dynamical phase transition and explore its properties in the transverse-field Ising model. Specifically, we show that the equilibrium quantum phase transition and the dynamical phase transition in this model are intimately related.

Dynamical quantum phase transitions in the transverse-field Ising model.

In earlier chapters, complex-valued functions appeared in connection with Fourier series expansions. In this context, while the function assumes complex values, the argument of the function is real-valued. There is a highly developed theory of (complex-valued) functions of a complex-valued argument. This theory contains some remarkably powerful results which are applicable to a variety of problems.

Functions of a Complex Variable

By use of the spectral theory of linear operators, necessary and sufficient conditions are derived for the stability of a class of discrete time feedback systems with a periodically time-varying feedback gain. These conditions involve a Nyquist plot for an equivalent time-invariant system which may be determined from the frequency response function of the system under consideration. In the special case of a scalar-input scalar-output system, these conditions may be given in a particularly simple form.

Stability Conditions Derived from Spectral Theory: Discrete Systems with Periodic Feedback

Many physical problems involve quantities that depend on more than one variable. The temperature within a “large”1 solid body of conducting material varies with both time and location within the material. When such problems are modeled, what results is a differential equation involving partial derivatives, or a partial differential equation..

Partial Differential Equations

A distributed model approach to the problem of coupler stress control in multiple locomotive trains is presented. The approach taken is to formulate the original problem in terms of a distributed parameter model with boundary controls. It turns out that the resulting model is conceptually quite useful and mathematically tractable. In the distributed model, the effects of changes both in system parameters and in the number of cars in a block are readily apparent. It is also to be expected that with an increase in the number of cars in a block the validity and usefulness of the model increases while a "Conventional" formulation becomes more intractable.

A distributed model for stress control in multiple locomotive trains

This paper considers some applications of the theory of so-called Fredholm operators to the problem of input-output stability of linear feedback systems. It is shown that stability conditions may be obtained for certain classes of systems by calculating the index of an associated Fredholm operator. This index may be calculated by counting the encirclements of the origin by some curve in the complex plane.Examples are given of applications to time-invariant problems with an unstable open loop, to the derivation of a version of the so-called converse of the circle criterion, and to a class of continuous time systems with a periodic linear feedback gain. In all three examples, stability (and instability) conditions are given in terms of encirclements (or lack of same) of a certain point by a locus in the complex plane.

Jon H. Davis

Papers

Functions of a Complex Variable

Stability Conditions Derived from Spectral Theory: Discrete Systems with Periodic Feedback

Partial Differential Equations

A distributed model for stress control in multiple locomotive trains

Fredholm Operators, Encirclements, and Stability Criteria