Symmetry is the key to solving differential equations. There are many well-known techniques for obtaining exact solutions, but most of them are special cases of a few powerful symmetry methods. Furthermore, these methods can be applied to differential equations of an unfamiliar type; they do not rely on special 'tricks'. Instead, a given differential equation is forced to reveal its symmetries, which are then used to construct exact solutions. This book is a straightforward introduction to the subject, and is aimed at applied mathematicians, physicists, and engineers. The presentation is informal, using many worked examples to illustrate the main symmetry methods. It is written at a level suitable for postgraduates and advanced undergraduates, and is designed to enable the reader to master the main techniques quickly and easily. The book contains methods that have not previously appeared in a text. These include methods for obtaining discrete symmetries and integrating factors.

This paper describes symmetries of all integrable difference equations that belong to the famous Adler-Bobenko-Suris classification. For each equation, the characteristics of symmetries satisfy a functional equation, which we solve by reducing it to a system of partial differential equations. In this way, all five-point symmetries of integrable equations on the quad-graph are found. These include mastersymmetries, which allow one to construct infinite hierarchies of local symmetries. We also demonstrate a connection between the symmetries of quad-graph equations and those of the corresponding Toda type difference equations.

An analogue of the Poincaré lemma for exact forms on a lattice is stated and proved. Using this result as a starting-point, a variational complex for difference equations is constructed and is proved to be locally exact. The proof uses homotopy maps, which enable one to calculate Lagrangians for discrete Euler-Lagrange systems. Furthermore, such maps lead to a systematic technique for deriving conservation laws of a given system of difference equations (whether or not it is an Euler-Lagrange system).

Most well-known solution techniques for differential equations exploit symmetry in some form. Systematic methods have been developed for finding and using symmetries, first integrals and conservation laws of a given differential equation. Here the author explains how to extend these powerful methods to difference equations, greatly increasing the range of solvable problems. Beginning with an introduction to elementary solution methods, the book gives readers a clear explanation of exact techniques for ordinary and partial difference equations. The informal presentation is suitable for anyone who is familiar with standard differential equation methods. No prior knowledge of difference equations or symmetry is assumed. The author uses worked examples to help readers grasp new concepts easily. There are 120 exercises of varying difficulty and suggestions for further reading. The book goes to the cutting edge of research; its many new ideas and methods make it a valuable reference for researchers in the field.

Our current understanding of the transport and deposition of aerosols (viruses, bacteria, air pollutants, aerosolized drugs) deep in the lung has been grounded in dispersive theories based on untested assumptions about the nature of acinar airflow fields. Traditionally, these have been taken to be simple and kinematically reversible. In this article, we apply the recently discovered fluid mechanical phenomenon of irreversible low-Reynolds number flow to the lung. We demonstrate, through flow visualization studies in rhythmically ventilated rat lungs, that such a foundation is false, and that chaotic mixing may be key to aerosol transport. We found substantial alveolar flow irreversibility with stretched and folded fractal patterns, which lead to a sudden increase in mixing. These findings support our theory that chaotic alveolar flow-characterized by stagnation saddle points associated with alveolar vortices-governs gas kinematics in the lung periphery, and hence the transport, mixing, and ultimately the deposition of fine aerosols. This mechanism calls for a rethinking of the relationship of exposure and deposition of fine inhaled particles.P athogenic aerosols play a major role in causing or exacerbating many pulmonary diseases such as lung cancer, bronchitis, emphysema, and asthma (1-4) and in spreading infectious diseases such as influenza, anthrax, and tuberculosis (5-8). The long-term effects of particulate air pollution on public health are striking; they have been shown to be equivalent to a shortening of life expectancy of approximately 2 years in the United States (9, 10). The magnitude of this effect is comparable to the deaths caused by all cancers (11). Fine particles-less than a few micrometers in diameter-are of great concern because they can penetrate deep into the pulmonary acinus (i.e., the gas exchange region of the lung). The critical factor that determines the fate of inhaled fine particles-whether deposited deep in the lung or exhaled in the next breath-is the kinematic interaction of inhaled and residual alveolar gas, but the underlying mechanism of this interaction is still not well understood.Because gas flow in the alveolar region occurs at very low Reynolds number (12) and the wall motion of the lung during breathing is essentially reversible (13-16), it has long been assumed that flow patterns deep in the lung are well described as a kinematically reversible Stokes flow (17-21). This means that, apart from the effects of molecular diffusion, a fluid ''particle'' that is transported by Stokes flow returns [by reversing the flow (22)] to its original position at the completion of one breathing cycle. This assumption implies that there is negligible flow-induced mixing between inhaled particles and the alveolar residual gas (17-21) and consequently little deposition of fine particles (18, 21). Experimental studies, however, consistently demonstrate appreciable aerosol mixing (23, 24) and deposition (25) deep in the lung, which cannot be accounted for by any known mechanism such as inerti...