A large body of empirical evidence demonstrates that the basic values of mass publics in advanced industrial societies have changed over the last three decades. The same research also shows that there are significant and persistent crossnational differences in values. This chapter considers whether the trajectory and pace of value change in advanced industrial countries is leading to convergence or divergence in the values of publics in Europe and North America.
The question of value convergence or divergence can be conceptualized and addressed empirically in at least two ways. The most straightforward approach entails identifying common value domains among European and North American publics and then asking, Have these become more, or less, alike over the two decades for which we have data? A second approach, however, is to explore the internal dynamics of value change by examining how North American and European publics organize their core values. After outlining some different perspectives on value change and describing our data and methodological approach, we present the basic crossnational and crosstime evidence of change on single-value dimensions for publics in Europe and North America. The focus then shifts to consider the matter of how publics on both continents bundle their basic value outlooks. Do the publics in North America and Europe organize their basic value outlooks in similar or different ways? And are there discernible patterns in the way in which these core values have changed over the same period?
For Russians, it was not only their state that collapsed in 1991. It was also their international environment. It had begun to give way at least two years earlier, in 1989, when Mikhail Gorbachev made clear that the Soviet government would not intervene to prevent the election of non-Communist governments throughout Eastern Europe. Speaking to the Central Committee in December 1989, Gorbachev professed to welcome the “positive changes” that were taking place in the region, presenting them as a further stage in the “renewal of socialism” (Materialy 1989, 18–19). But in any case, he asked the 28th Party Congress in the summer of the following year, what was the alternative? “Tanks again?” (XXVIII s"ezd 1990). The international organizations that had held these states together collapsed during 1991, when the Council for Mutual Economic Assistance (Comecon) and the Warsaw Treaty Organization agreed to dissolve themselves. When Gorbachev resigned as USSR President on Christmas Day at the end of that year, he was the last Communist leader in Europe.
The Russian Federation, with the agreement of the other post-Soviet republics, was the USSR's successor state: it acceded to the USSR's 15,000 treaties, to its international debt, and to its seats in the United Nations and within its Security Council. But it was scarcely a geopolitical successor. Post-Communist Russia was still the world's largest state, but it was only three-quarters of the territorial extent of the USSR. Its population was only half as numerous.
Chapter 10 gives an introduction to linear programming on oriented matroids. It does not presuppose any experience with linear programming – but for the operations research practitioner it might offer an alternative view on linear programming from a matroid theory point of view. Our aim is to give the non-expert a smooth, geometric access to the fundamental ideas of oriented matroid programming, as developed in Bland (1974, 1977a). This necessitates extra care at the points where the terminologies from linear programming and from combinatorial geometry clash.
This chapter is intended to demonstrate that the oriented matroid framework can add to the understanding of the combinatorics and of the geometry of the simplex method for linear programming. In fact, the oriented matroid approach gives a geometric language for pivot algorithms, interpreting linear programs as oriented matroid search problems. We find that locally consistent information (orientation of the edges at a vertex) imply the existence of global extrema in pseudoarrangements. We believe that these techniques and results are applicable also to problems in other areas of mathematics.
The oriented matroid framework deals with (pseudo)linear programs, where
– positive cocircuits correspond to feasible vertices, and
– positive circuits correspond to bounding cones.
Both of these are described by oriented matroid bases, corresponding to temporary coordinate systems. Pivot algorithms are now modeled by basis exchanges, and the duality of linear programming becomes a manifestation of oriented matroid duality.
The benefits of geometric understanding are of course not one-sided: the linear programming frame work offers insight into the structure of oriented matroids, and the pivot algorithms of linear programming provide important search techniques for oriented matroids.
The combinatorial theory of convex polytopes is an important area of application for oriented matroid theory. Several new results on polytopes as well as new simplified proofs for known results have been found, and it is fair to say that oriented matroids have significantly contributed to the progress of combinatorial convexity during the past decade. This chapter aims to be both an introduction to the basics and a survey on current research topics in this branch of discrete mathematics.
Section 9.1 is concerned with basic properties of matroid polytopes. We show that oriented matroid duality is essentially equivalent to the technique of Gale transforms. In Section 9.2 we discuss matroidal analogues to polytope constructions and some applications. Section 9.3 deals with the Lawrence construction, an important general method for encoding oriented matroid properties into polytopes. Cyclic and neighborly polytopes will be studied in Section 9.4, and triangulations of matroid polytopes in Section 9.6. In Section 9.5 we discuss an oriented matroid perspective on the Steinitz problem of characterizing face lattices of convex polytopes.
Introduction to matroid polytopes
Throughout this chapter we will interpret a rank Υ oriented matroid as a generalized point configuration in affine (Υ – l)-space. Using the language of oriented matroids, we can define the convex hull of such a configuration, and this allows us to study properties of convex polytopes in this purely combinatorial setting. The following basic definitions, due to Las Vergnas (1975a, 1980a), were already discussed in Exercise 3.9. In Exercise 3.11 we gave an axiomatization of oriented matroids in terms of their convex closure operators. A weaker notion of abstract convexity was developed independently by Edelman (1980, 1982) and Jamison (1982).
Oriented matroids can be thought of as a combinatorial abstraction of point configurations over the reals, of real hyperplane arrangements, of convex polytopes, and of directed graphs. The creators of the theory of oriented matroids have, in fact, drawn their motivation from these diverse mathematical theories (see the historical sketch in Section 3.9), but they have nevertheless arrived at equivalent axiom systems – which manifests the fact that oriented matroids are “the right concept”.
We will start out by illustrating these different aspects of oriented matroids. Doing this, we will present a number of examples while at the same time introducing the main concepts and terminology of oriented matroids. This should assist the reader who wishes to access the later chapters in a non-linear order, or who first wants a quick idea of what is going on. It should also provide intuition and motivation both for the axiomatics and the further development of the theory.
Hence, our first two chapters will avoid an extensive discussion of the axiom systems for oriented matroids, which are treated in Chapter 3. We will also minimize dependence on background from ordinary matroids. Furthermore, extensive attributions will not be given in these introductory chapters; we refer to later chapters and the bibliography.
Oriented matroids from directed graphs
Let us consider the simple cycles of a directed graph D = (V,E) with arc set E, together with an orientation of each such cycle. Then every arc of a cycle is either a forward (positive) arc or a backward (negative) arc in the cycle. This allows us to consider the cycle as a signed subset of E, which consists of a positive and a negative part.
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