THE WEBER PROBLEM: FREQUENCY OF DIFFERENT SOLUTION TYPES AND EXTENSION TO REPULSIVE FORCES AND DYNAMIC PROCESSES*

ABSTRACT. The frequency of occurrence of the different types of solutions to the Weber problem is studied. These solutions are: a location at an attraction point due to a dominant force, to incompatible angles, or to concavity; a location at infinity; a location inside the polygon; and a location outside the polygon. Situations involving both attraction and repulsion points are examined in the triangle and in the more-than-three-sided polygon context, and methods for solving the corresponding problems are compared. A trigonometric solution is proposed for the triangle case involving one repulsion and two attraction points. The variation in the frequency of a location at an attraction point when the number of attraction points increases while the number of repulsion points remains the same is observed as well. Implications of the results are studied for the analysis of dynamic location processes.     

UNDERSTANDING SPATIAL INERTIA: CENTER OF GRAVITY,POPULATION DENSITIES,THE WEBER PROBLEM,AND GRAVITY POTENTIAL*

ABSTRACT. In this paper we attempt to clarify the theoretical links between the concepts of “center of gravity” and “point of maximum population density” which describe the present, and the concepts of “minimum of the comprehensive Weber problem” and “maximum comprehensive gravity potential” which guide the future. Critical values of the characteristic parameters of the relevant functions are estimated. Implications for the understanding of spatial inertia are discussed.     

WEBER'S PROBLEM WITH ATTRACTION AND REPULSION*

ABSTRACT. Weber's problem consists of locating a facility in the plane in such a way that the weighted sum of Euclidean distances to n given points be minimum. The case where some weights are positive and some are negative is shown to be a d.-c. program (i.e., a global optimization problem with both the objective function and constraint functions written as differences of convex functions), reducible to a problem of concave minimization over a convex set. The reduced problem is then solved by outer-approximation and vertex enumeration. Moreover, locational constraints can be taken into account by combining the previous algorithm with an enumerative procedure on the set of feasible regions. Finally, the algorithm is extended to solve the case where obnoxiousness of the facility is assumed to have exponential decay. Computational experience with n up to 1000 is described.     

WHEELS WITH THICKENED SPOKES, AND THE PROBLEM OF CULTURAL CONTACT BETWEEN THE AEGEAN WORLD AND EUROPE IN THE LATE BRONZE AGE

Summary. A technological advance in wheel construction, entailing spokes with thickened or strengthened ends, can be detected in Central Europe at the start of the Late Bronze Age. Spokes with thickened ends are seen in Central Europe on full-size wheels, wagon models and wheel depictions. It is argued that this technological innovation had its source in the Aegean, where wheels with thickened spokes are seen not only depicted on late Mycenaean craters but also on Linear B wheel ideograms. The second part of the article discusses a closely-related Central European group of wheel-shaped designs, characterised by forked spokes, found on pin-heads, pendants, decorated belt-sheets, phalerae and belt-hooks. As some of these designs refer to real wheels constructed with thickened or strengthened spoke ends, it is concluded that the designs really represent wheels, although the designs are often so simplified that this is not immediately apparent.     

THE EFFICIENT COMPUTATION OF UNBIASED MAHALANOBIS DISTANCES AND THEIR INTERPRETATION IN ARCHAEOMETRY*

This paper argues for cross-validation to reduce bias in estimating Mahalanobis distances of individuals to groups, a particular problem with small sample sizes. Formulae for the efficient computation of cross-validated distances, which avoid excessive matrix inversion, are given. The use of Mahalanobis distances as measures of consistency, or as indications of outlying values, is described. The little-known technique of gamma plotting is outlined and discussed as an aid to interpreting distances in these terms. Both cross-validation and gamma plotting are illustrated in an example on marble composition.     

RESPONSIBLE ACTION AND THE PROBLEM OF REIFICATION

This paper elaborates a “personal,” as opposed to an institutional, view of administrative responsibility. The idea of personal responsibility is better suited for appreciating the cognitive, conceptual, and interpersonal dynamics of policy processes than is the institutional view of responsibility as accountability. From the perspective of personal responsibility, the primary impediment to responsible action by policy makers, policy analysts, and administrators is not that they may act in ways that are ethically or morally wrong. Rather, the problem is that such actors may either not perceive that important choices are available to them or, when choices are perceived, their moral, as opposed to their instrumental, content may not be appreciated. Responsible action should thus be seen as a conceptual or cognitive issue, not simply one of adherence to or departure from institutionally objectivated standards of correctness. Inability to appreciate the moral nature of action may be explained by the cognitive tendency to reify policy making and administrative institutions, roles, rules, and situations.     

THE ALLURE OF DARK TIMES: MAX WEBER,POLITICS, AND THE CRISIS OF HISTORICISM

This article argues that realist invocations of Weber rely on an unrealistic reading of Weber's realism. In order to escape the allure of Weber's dramatic posture of crisis, we place his seminal lecture on “Politics as a Vocation” (1919) in its historical and philosophical context of a revolutionary conjuncture of dramatic proportions, compounded by a broader crisis of historicism. Weber's rhetoric, we argue, carries with it not only the emotion of crisis but is also the expression of a deeper intellectual impasse. The fatalistic despair of his position had already been detected by some of his closest contemporaries for whom Weber did not appear as a door‐opener to a historically situated theory of political action but as a telling and intriguing impasse. Although the disastrous history of interwar Europe seems to confirm Weber's bleakest predictions, it would be perverse to elevate contingent failure to the level of retrospective vindication.