This monograph is a comprehensive treatise on Lanchester-type models of warfare, i.e. differential-equation models of attrition in force-on-force combat operations. Its goal is to provide both an introduction to and current- state-of-the-art overview of Lanchester-type models of warfare as well as a comprehensive and unified in-depth treatment of them. Both deterministic as well as stochastic models are considered. Such models have been widely used in the United States and elsewhere for the modelling of force-on-force attrition over the complete spectrum of combat operations, from combat between platoon-sized units through theater-level air-ground combat. This material should be of interest primarily to individuals concerned with defense planning, quantitative aspects of military analysis, military OR, war gaming, or combat modelling, although it may also be of interest to the reader concerned with the modelling and analysis of other dynamic systems. It should also be of interest to the concerned citizen who is interested in the foundations for defense analysis and has the appropriate technical background.

The purpose of this four volume series is to make available for college teachers and students samples of important and realistic applications of mathematics which can be covered in undergraduate programs. The goal is to provide illustrations of how modem mathematics is actually employed to solve relevant contemporary problems. Although these independent chapters were prepared primarily for teachers in the general mathematical sciences, they should prove valuable to students, teachers, and research scientists in many of the fields of application as well. Prerequisites for each chapter and suggestions for the teacher are provided. Several of these chapters have been tested in a variety of classroom settings, and all have undergone extensive peer review and revision. Illustrations and exercises are included in most chapters. Some units can be covered in one class, whereas others provide sufficient material for a few weeks of class time. Volume 1 contains 23 chapters and deals with differential equations and, in the last four chapters, problems leading to partial differential equations. Applications are taken from medicine, biology, traffic systems and several other fields. The 14 chapters in Volume 2 are devoted mostly to problems arising in political science, but they also address questions appearing in sociology and ecology. Topics covered include voting systems, weighted voting, proportional representation, coalitional values, and committees. The 14 chapters in Volume 3 emphasize discrete mathematical methods such as those which arise in graph theory, combinatorics, and networks.

This is an open access book. Lewis F Richardson (1981-1953), a physicist by training, was a pioneer in meteorology and peace research and remains a towering presence in both fields. This edited volume reviews his work and assesses its influence in the social sciences, notably his work on arms races and their consequences, mathematical models, the size distribution of wars, and geographical features of conflict. It contains brief bibliographies of his main publications and of articles and books written about Richardson and his work and discusses his continuing influence in peace research and international relations as well as his attitude to the ethical responsibilities of a scientist. It will be of interest to a wide range of scholars. This book includes 11 chapters written by Nils Petter Gleditsch, Dina A Zinnes, Ron Smith, Paul F Diehl, Kelly Kadera, Mark Crescenzi, Michael D Ward, Kristian Skrede Gleditsch, Nils B Weidmann, Jürgen Scheffran, Niall MacKay, Aaron Clauset, Michael Spagat and Stijn van Weezel. Lewis F Richardson occupied an important position in two academic fields as different as meteorology and peace research, with academic prizes awarded in both disciplines. In peace research, he pioneered the use of mathematical models and the meticulous compilation of databases for empirical research. As a quaker and pacifist, he refused to work in preparations for war, paid a heavy prize in terms of his career, and (at least in the social sciences) was fully recognized as a pioneering scholar only posthumously with the publication of two major books. Lewis Fry Richardson is one of the 20th century’s greatest but least appreciated thinkers—a creative physicist, psychologist, meteorologist, applied mathematician, historian, pacifist, statistician, and witty stylist. If you’ve heard of weather prediction, chaos, fractals, cliometrics, peace science, big data, thick tails, or black swans, then you have benefited from Richardson’s prescience in bringing unruly phenomena into the ambit of scientific understanding. Richardson’s ideas continue to be relevant today, and this collection is a superb retrospective on this brilliant and lovable man. Steven Pinker, Johnstone Professor, Harvard University, and the author of The Better Angels of Our Nature and Enlightenment Now

This monograph is a comprehensive treatise on Lanchester-type models of warfare, i.e. differential-equation models of attrition in force-on-force combat operations. Its goal is to provide both an introduction to and current- state-of-the-art overview of Lanchester-type models of warfare as well as a comprehensive and unified in-depth treatment of them. Both deterministic as well as stochastic models are considered. Such models have been widely used in the United States and elsewhere for the modelling of force-on-force attrition over the complete spectrum of combat operations, from combat between platoon-sized units through theater-level air-ground combat. This material should be of interest primarily to individuals concerned with defense planning, quantitative aspects of military analysis, military OR, war gaming, or combat modelling, although it may also be of interest to the reader concerned with the modelling and analysis of other dynamic systems. It should also be of interest to the concerned citizen who is interested in the foundations for defense analysis and has the appropriate technical background.

Military conflicts, particularly land combat, possess thecharacteristics of complex adaptive systems: combat forces arecomposed of a large number of nonlinearly interacting parts and areorganized in a dynamic command-and-control network; local action, which often appears disordered, self-organizes into long-range order;military conflicts, by their nature, proceed far from equilibrium;military forces adapt to a changing combat environment; and there isno master voice that dictates the actions of every soldier (i

A century ago Frederick Lanchester formulated a mathematical model of combat which suggested that the combat power of a military force was proportional to the product of the individual effectiveness of the units in the force and the square of the number of units deployed. This model reinforced a long-established faith in the importance of superior numbers. However, successive historical studies failed to identify any clear relationship between the numbers and losses in opposing forces. This Element analyses American Civil War battles, and shows that the ratio of losses incurred was inversely proportional to the ratio of numbers effectively engaged. This result demonstrates that the numbers of fighting units in a military force are less important than the ability of those units to get into action and inflict losses on the enemy. This result demonstrates the limitations of the Square Law, and should prevent it from being applied indiscriminately.