Introduction to Higher Algebra. By Maxime Bôcher. Pp. xi, 321. New York: The Macmillan Company. 1907.

Lectures on ordinary differential equations

Why are democracies unusually successful in war? We find that superior human capital, harmonious civil-military relations, and Western cultural background are largely responsible. These traits correlate positively with democracy, and account for democracy’s apparent effectiveness bonus. This is either good news or bad news for democratic effectiveness theorists. Many believe that democracy causes these traits. If so, our findings strengthen democratic effectiveness theory by explicating its causal mechanism. But others see democracy as a consequence rather than a cause of such traits. If so, our findings challenge the thesis by identifying alternative causes of the effectiveness bonus previously attributed to democracy. Either way, the results show a powerful effect for unit level variables in military performance. In the process, these same results sharpen our understanding of military effectiveness in general, and the relationship between military performance and regime type in particular.

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Democracy and Military Effectiveness A Deeper Look

Political scientists overwhelmingly seek evidence of causation by measuring changes in the mean of the distribution of the dependent variable. This article points out that some causal relationships produce changes in the variance, not the mean, of that distribution. It makes the case for variance-altering causation by demonstrating its empirical relevance to political scientists. The article also lays out an array of causal mechanisms—under the general headings of aggregation, contagion, and constraint—in order to demonstrate the logical coherence of variance-altering causation and the many ways in which it can arise. The discussion highlights the often-stringent empirical and logical requirements that must be met if the researcher hopes to make concrete predictions about changes in variance.

Explaining Variance; Or, Stuck in a Moment We Can't Get Out Of

A detailed data base of the Ardennes campaign of World War II (December 15, 1944 through January 16, 1945) has recently been developed. The present article formulates four Lanchester models of the campaign and estimates their parameters for these data. Two-sided time histories of warfare on battles and campaigns are very rare, so Lanchester models have seldom been validated with historical data. The models are homogeneous in that tanks, armored personnel carriers, artillery, and manpower are weighted to yield a measure of strength of the Allied and German forces. This weighting is utilized for combat power and for losses. The models treat combat forces in the campaign (including infantry, armor, and artillery manpower) and total forces in the campaign (including both combat manpower and support manpower.) Four models are presented. Two models have five parameters (Allied individual effectiveness, German individual effectiveness, exponent of shooting force, exponent of target force, and a tactical parameter reflecting which side is defending and attacking.) The other two models remove the tactical parameter, which is not generally known prior to warfare, and estimate the other parameters without the tactical parameter. The main results of the research are (a) the Lanchester linear model fits the Ardennes campaign data in all four cases, and (b) when combat forces are considered Allied individual effectiveness is greater than German individual effectiveness, whereas when total forces are considered Allied and German individual effectiveness is the same. The interpretation of the latter result is that the two sides had essentially the same individual capabilities but were organized differently—the Allies chose to have more manpower in the support forces, which yielded greater individual capabilities in the combat forces. The overall superiority of the Allies in the campaign led to the attrition to the Allies being a smaller portion of their forces. © 1995 John Wiley & Sons, Inc.1

Lanchester models of the Ardennes campaign

This paper develops solutions to extensions of F. W. Lanchester's classical equations of “modern warfare” (frequently referred to as aimed-fire equations) for combat between two homogeneous forces. In these extensions the lethality of the fire (as expressed by the Lanchester attrition-rate coefficient) depends on time. When the dependence is arbitrary, the solution is an infinite series of recursively related integrals; in special cases, more convenient representations (including representation in terms of tabulated functions) are available. Solutions are obtained in the following cases: (1) the lethality of each side's fire proportional to a power of time and both lethalities initially zero, and (2) the lethality of each side's fire linear with time, but only one side's lethality initially zero. The latter case models the constant-speed approach between forces whose weapons have different maximum effective ranges.

Solving Lanchester-Type Equations for “Modern Warfare” with Variable Coefficients

This research was supported by the Office of Naval Research as part of the Foundation Research Program at the Naval Postgraduate School.

Lanchester-Type Models of Warfare and Optimal Control

This paper studies the Riccati equation satisfied by the force ratio of two homogeneous forces in deterministic Lanchester combat. This study provides qualitative insight as to the “direction” in which combat is moving (in the sense that the force ratio is changing to the advantage of one of the combatants). “Instantaneous” linear and square laws that are “local” conditions for winning are shown to apply for Lanchester-type formulations with time-varying system effectiveness. The value of considering the force-ratio equation lies in one's ability to determine the outcome of battle without explicitly solving the force-level equations, an important feature for variable-coefficient formulations. Moreover, in some cases observation of the initial time-behavior of the force ratio in a battle allows one to predict the outcome, sometimes even without explicit knowledge of attrition-rate coefficients. By additionally considering the instantaneous loss ratio, one gains insight into the consequences of concentratio...

Force-Ratio Considerations for Some Lanchester-Type Models of Warfare

This paper develops a mathematical theory for solving deterministic, Lanchester-type, 'square-law' attrition equations for combat between two homogeneous forces with temporal variations in fire effectivenesses (as expressed by the Lanchester attrition-rate coefficients). It gives a general form for expressing the solution of such variablecoefficient combat attrition equations in terms of Lanchester functions, which are introduced here and can be readily tabulated. Different Lanchester functions arise from different mathematical forms for the attrition-rate coefficients. We give results for two such forms: (1) effectiveness of each side's fire proportional to a power of time, and (2) effectiveness of each side's fire linear with time but with a nonconstant ratio of attrition-rate coefficients. Previous results in the literature for a nonconstant ratio of these attrition-rate coefficients only took a convenient form under rather restrictive conditions.

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Canonical Methods in the Solution of Variable-Coefficient Lanchester-Type Equations of Modern Warfare

This paper introduces important new functions for analytic solution of Lanchester-type equations of modem warfare for combat between two homogeneous forces modeled by power attrition-rate coefficients with "no offset." Tabulations of these Lanchester-Clifford-Schlafli or LCS functions allow one to study this particular variable-coefficient model almost as easily and thoroughly as Lanchester's classic constant-coefficient one. LCS functions allow one to obtain important information in particular, force-annihilation prediction without having to spend the time and effort of computing force-level trajectories. The choice of these particular functions is based on theoretical considerations that apply in general to Lanchester-type equations of modern warfare and provide guidance for developing other canonical functions. Moreover, our new LCS functions also provide valuable information about related variable-coefficient models. Also, we introduce an important transformation of the battle's time scale that not only simplifies the force-level equations, but also shows that relative fire effectiveness and intensity of combat are the only two weapon-system parameters determining the course of such variable-coefficient Lanchester-type combat.

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James G. Taylor

Papers

Solving Lanchester-Type Equations for “Modern Warfare” with Variable Coefficients

Lanchester-Type Models of Warfare and Optimal Control

Force-Ratio Considerations for Some Lanchester-Type Models of Warfare

Canonical Methods in the Solution of Variable-Coefficient Lanchester-Type Equations of Modern Warfare

Annihilation Prediction for Lanchester-Type Models of Modern Warfare