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The theorem of Mukhin, Tarasov, and Varchenko (formerly the Shapiro conjecture for Grassmannians) asserts that all (a priori complex) solutions to certain geometric problems in the Schubert calculus are actually real. Their proof is quite remarkable, using ideas from integrable systems, Fuchsian differential equations, and representation theory. There is now a second proof of this result, and it has ramifications in other areas of mathematics, from curves to control theory to combinatorics. Despite this work, the original Shapiro conjecture is not yet settled. While it is false as stated, it has several interesting and not quite understood modifications and generalizations that are likely true, and the strongest and most subtle version of the Shapiro conjecture for Grassmannians remains open.

ABSTRACT. In the mythology of the American West, range wars pitted ranchers against farmers, insiders versus outsiders, each struggling to control access to land, each fighting to preserve the freedom [...]

The frontier-region dichotomy continues to dominate the field of American history. Scholars are still debating over issues of frontier or region, process or place, triumph or tragedy, settlement or conquest, development or despoliation. The frontier-region dichotomy is partly a product of a half decade of media coverage of the 'New' and 'Old' western history. This focus on the supposed antithetical association between process-centered and place-centered approaches has not benefited the historiographical field.

Frederick Jackson Turner forwarded the frontier thesis and supporters of his philosophy believed that colonization of the Great West was necessary. According to Turner, continuous colonization of the Great West had fortified American character and led to development. Turner's interrelated ideas about the frontier and the section were recognized and supported by New Western historians.

The 'Progress of the Nation,' a component of the 1890 American census put light for the first time on the course of settlements since 1790 and revealed that the American frontier was rapidly depleting. This study was the starting point for Frederick Jackson Turner's famous essay on the importance of the frontier. In comparison to the nineteenth-century frontier that spread across America, the twentieth-century frontier is a more static place, that found home on the large stretches of the deep-rural west.

Turner is best known for his enduring Frontier Thesis, which has survived criticism and remains as an instructive relic of American history. Turner set a precedent by declaring that understanding of the past was improved by adapting it to present changes. Thus true 'Turnerians' were those historians who rethought and revised their writings. Turner later reneged on this assertion, but this shift was less distinctive of him and more indicative of the present being more confusing than the past.

Frederick Jackson Turner believed that the western frontier was responsible for the character and democracy of America. Turner, in an 1893 speech, indicated that the frontier had closed and that the future of America was uncertain. Historians believed that Turner's theories were dominant in 1964, but many historians of the 1990s believe that the frontier theory should be replaced by a more regional approach to history.

Recent attempts to construct a 'contextual' history of geography have refocused attention on the theories and proponents of environmental determinism, in the attempt to establish connections with the reactionary political-intellectual needs of nineteenth-century European imperialism. The present essay seeks to broaden this interpretive perspective by documenting the existence of a strong commitment to environmentalist ideas on the far left as well. The subject of the inquiry is the eminent Russian Marxist theoretician G. V. Plekhanov. It will be seen that Plekhanov placed environmentalist principles at the very foundation of his social theory, and that he did so without any doubt that they fit properly within the general conceptual framework established by Marx himself. In developing his own analysis, Plekhanov borrowed heavily from the work of contemporary geographers, in particular that of Friedrich Ratzel. As with other determinists, Plekhanov's environmentalism was essentially 'ideological' in the sense that he used it toward an ulterior explanatory purpose, but this was, in his case, a purpose which had nothing to do with the project of legitimizing European domination of the globe. Rather, he used his environmentalism to explain the backwardness and social deformation of his own native Russia. Key Words: Marx, Marxism, C. V. Plekhanov, environmentalism, geographical determinism, materialism, Russia, L. I. Mechnikov, F. Ratzel, F. J. Turner, frontier hypothesis.

Byline: C.A. Knox Lovell Frontier techniques can be of immense use in establishing the best practice benchmarks against which the performance of healthcare providers can be evaluated. If credible benchmarks are set by frontier techniques, then reimbursement schemes can be guided by the performance of individual providers. Although frontier techniques have gained acceptance in other regulatory arenas, their potential remains unfulfilled in healthcare. This paper speculates on the strengths and weaknesses of the application of frontier techniques in healthcare reimbursement schemes, although the main objective is to introduce the readers to the basics of the techniques. Two techniques are considered; one is regression-based and the other is based on linear programming techniques. Each technique has its own relative advantage depending on the confidence generated by the underlying data or the ability to model a complex service delivery technology.