Uniqueness of minimisers for a Grötzsch-Belinskiĭ type inequality in the Heisenberg group

The modulus method introduced by H. Grötzsch yields bounds for a mean distortion functional of quasiconformal maps between two annuli mapping the respective boundary components onto each other. P. P. Belinskiĭ studied these inequalities in the plane and identified the family of all minimisers. Beyond the Euclidean framework, a Grötzsch-Belinskiĭ-type inequality has been previously considered for quasiconformal maps between annuli in the Heisenberg group whose boundaries are Korányi spheres. In this note we show that--in contrast to the planar situation--the minimiser in this setting is essentially unique.

Splitsville: A Study of Income Inequality and Political Polarization in the United States House of Representatives

The study compares a measure of income inequality with polarization scores of U.S. Representatives from the 104th to the 109th Congresses. It attempts to explain the link, on the abstract level, between high inequality and high polarization. The end findings indicate that inequality increases a Representative's likelihood to act liberally.

Income Inequality in the 1990s: Re-Forging a Lost Relationship?

Using data from March Current Population Surveys we find gains from economic growth over the 1990s business cycle (1989-2000) were more equitably distributed than over the 1980s business cycle (1979-1989) using summary inequality measures as well as kernel density estimations. The entire distribution of household size-adjusted income moved upwards in the 1990s with profound improvements for African Americans, single mothers and those living in households receiving welfare. Most gains occurred over the growth period 1993-2000. Improvements in average income and income inequity over the latter period are reminiscent of gains seen in the first three decades after World War II.

(Table 1) Sulphur concentration and isotopic composition of ODP Leg 126 igneous rocks

Sulfur isotope ratios have been determined in 19 selected igneous rocks from Leg 126. The d34S of the analyzed rocks ranges from -0.1 â to +19.60 â. The overall variation in sulfur isotope composition of the rocks is caused by varying degrees of seawater alteration. Most of the samples are altered by seawater and only five of them are considered to have maintained their magmatic sulfur isotope composition. These samples are all from the backarc sites and have d34S values varying from +0.2 â to +1.6 â, of which the high d34S values suggest that the earliest magmas in the rift are more arc-like in their sulfur isotope composition than the later magmas. The d34S values from the forearc sites are similar to or heavier than the sulfur isotope composition of the present arc.

Noble metal abundances in ODP Leg 115 basalts (Table 1)

A comparison of 50 basalts recovered at Sites 706, 707, 713, and 715 along the Reunion hotspot trace during Ocean Drilling Program Leg 115 in the Indian Ocean shows that seafloor alteration had little effect on noble metal concentrations (Au, Pd, Pt, Rh, Ru, and Ir), determined by inductively coupled plasma-mass spectrometry (ICP-MS), which generally tend to decrease with magma evolution. Their compatible-element behavior may be related to the precipitation of Ir-Os-based alloys, chromite, sulfides, and/or olivine and clinopyroxene in some combination. The simplest explanation indicates silicate control of concentrations during differentiation. Basalts from the different sites show varying degrees of alkalinity. Noble metal abundances tend to increase with decreasing basalt alkalinity (i.e., with increasing percentages of mantle melting), indicating that the metals behave as compatible elements during mantle melting. The retention of low-melting-point Au, Pd, and Rh in mantle sulfides, which mostly dissolve before significant proportions of Ir-Os-based alloys melt, explains increasing Pd/Ir ratios with decreasing alkalinity (increasing melting percentages) in oceanic basalts. High noble metal concentrations in Indian Ocean basalts (weighted averages of Au, Pd, Rh, Pt, Ru, and Ir in Leg 115 basalts are 3.2, 8.1, 0.31, 7.3, 0.22, and 0.11 ppb, respectively), compared with basalts from some other ocean basins, may reflect fundamental primary variations in upper- mantle noble metal abundances

Isotope ratios of basalts from ODP Leg 115 holes (Table 1)

Strontium, neodymium, and lead isotope ratios are reported for 13 Leg 115 basalts as well as 3 basalts from Texaco drill hole SM-1 on the Mascarene Plateau. The 87Sr/86Sr ratios and eNd range from 0.70330 to 0.70439 and 5.5 to 7.4, respectively, although 87Sr/86Sr ratios higher than 0.70383 are found only in SM-1 basalts. The high 87Sr/86Sr values are thought to reflect seawater Sr in secondary phases, although all samples were strongly leached in HC1 before analysis. 206Pb/204Pb ratios range from 18.53 to 18.80, and sho high 207Pb/204Pb and 208Pb/204Pb ratios relative to 206Pb/204Pb ratios, typical of Indian Ocean mid-ocean ridge (MORB) and oceanic-island basalts (OIB). Isotopic compositions of Leg 115 basalts generally fall between fields for MORB and Reunion Island basalts, consistent with the conclusion drawn from geochronological studies that Deccan flood basalt volcanism, the Chagos-Laccadive Ridge, and the Mascarene Plateau are all products of the Reunion mantle plume. Isotopic compositions of magmas produced by this plume have varied systematically with time in the direction of less "depleted," less MORB-like isotopic signatures. This compositional change has been accompanied by a decrease in eruption rate. We interpret Deccan volcanism as the voluminous beginning of the plume. Reduced entrainment of asthenosphere following melting of the plume head resulted in less MORB-like isotope ratios in magmas and a decrease in eruptive activity with time.