The Hasse-Minkowski Theorem

The Hasse-Minkowski theorem concerns the classification of quadratic forms over global fields (i.e., finite extensions of Q and rational function fields with a finite constant field). Hasse proved the theorem over the rational numbers in his Ph.D. thesis in 1921. He extended the research of his thesis to quadratic forms over all number fields in 1924. Historically, the Hasse-Minkowski theorem was the first notable application of p-adic fields that caught the attention of a wide mathematical audience. The goal of this thesis is to discuss the Hasse-Minkowski theorem over the rational numbers and over the rational function fields with a finite constant field of odd characteristic. Our treatments of quadratic forms and local fields, though, are more general than what is strictly necessary for our proofs of the Hasse-Minkowski theorem over Q and its analogue over rational function fields (of odd characteristic). Our discussion concludes with some applications of the Hasse-Minkowski theorem.

A Median Voter Theorem for Postelection Politics

We analyze a model of 'postelection politics', in which (unlike in the more common Downsian models of 'preelection politics') politicians cannot make binding commitments prior to elections. The game begins with an incumbent politician in office, and voters adopt reelection strategies that are contingent on the policies implemented by the incumbent. We generalize previous models of this type by introducing heterogeneity in voters' ideological preferences, and analyze how voters' reelection strategies constrain the policies chosen by a rent-maximizing incumbent. We first show that virtually any policy (and any feasible level of rent for the incumbent) can be sustained in a Nash equilibrium. Then, we derive a 'median voter theorem': the ideal point of the median voter, and the minimum feasible level of rent, are the unique outcomes in any strong Nash equilibrium. We then introduce alternative refinements that are less restrictive. In particular, Ideologically Loyal Coalition-proof equilibrium also leads uniquely to the median outcome.

DeMoivre's Theorem

DeMoivre's theorem is of great utility in some parts of physical chemistry and is re-introduced here.

Ehrenfest's Theorem

A one dimensional presentation of Ehrenfest's theorem is presented.

Linear sedimentation rates and approximate mass accumulation rates at DSDP Site 61-462

A 560-meter-thick sequence of Cenomanian through Pleistocene sediments cored at DSDP Site 462 in the Nauru Basin overlies a 500-meter-thick complex unit of altered basalt flows, diabase sills, and thin intercalated volcaniclastic sediments. The Upper Cretaceous and Cenozoic sediments contain a high proportion of calcareous fossils, although the site has apparently been below the calcite compensation depth (CCD) from the late Mesozoic to the Pleistocene. This fact and the contemporaneous fluctuations of the calcite and opal accumulation rates suggest an irregular influx of displaced pelagic sediments from the shallow margins of the basin to its center, resulting in unusually high overall sedimentation rates for such a deep (5190 m) site. Shallow-water benthic fossils and planktonic foraminifers both occur as reworked materials, but usually are not found in the same intervals of the sediment section. We interpret this as recording separate erosional interludes in the shallow-water and intermediate-water regimes. Lower and upper Cenozoic hiatuses also are believed to have resulted from mid-water events. High accumulation rates of volcanogenic material during Santonian time suggest a corresponding significant volcanic episode. The coincidence of increased carbonate accumulation rates during the Campanian and displacement of shallow-water fossils during the late Campanian-early Maestrichtian with the volcanic event implies that this early event resulted in formation of the island chains around the Nauru Basin, which then served as platforms for initial carbonate deposition.

Tab. 1: Approximate dimension of the pingos

A group of nine pingos occurs in the valley of a glacial meltwater river. The pingos rise from a plain of low-center polygons. Some pingos have a typical cone shape, but others are linear, apparently centered on ice wedges . The occurrence of most pingos at the junction of oversize ice wedge polygon ridges suggests that the injection of water and the segregation of ice occurred along pathways provided by the ice wedges.

Approximate Analytical Model for the Squeeze-Film Lubrication of the Human Ankle Joint with Synovial Fluid Filtrated by Articular Cartilage

The aim of this article is to propose an analytical approximate squeeze-film lubrication model of the human ankle joint for a quick assessment of the synovial pressure field and the load carrying due to the squeeze motion. The model starts from the theory of boosted lubrication for the human articular joints lubrication (Walker et al., Rheum Dis 27:512–520, 1968; Maroudas, Lubrication and wear in joints. Sector, London, 1969) and takes into account the fluid transport across the articular cartilage using Darcy’s equation to depict the synovial fluid motion through a porous cartilage matrix. The human ankle joint is assumed to be cylindrical enabling motion in the sagittal plane only. The proposed model is based on a modified Reynolds equation; its integration allows to obtain a quick assessment on the synovial pressure field showing a good agreement with those obtained numerically (Hlavacek, J Biomech 33:1415–1422, 2000). The analytical integration allows the closed form description of the synovial fluid film force and the calculation of the unsteady gap thickness.