Efficient multivariate approximation with transformed rank-1 lattices

We study the approximation of functions defined on different domains by trigonometric and transformed trigonometric functions. We investigate which of the many results known from the approximation theory on the d-dimensional torus can be transfered to other domains. We define invertible parameterized transformations and prove conditions under which functions from a weighted Sobolev space can be transformed into functions defined on the torus, that still have a certain degree of Sobolev smoothness and for which we know worst-case upper error bounds. By reverting the initial change of variables we transfer the fast algorithms based on rank-1 lattices used to approximate functions on the torus efficiently over to other domains and obtain adapted FFT algorithms.:1 Introduction 2 Preliminaries and notations 3 Fourier approximation on the torus 4 Torus-to-R d transformation mappings 5 Torus-to-cube transformation mappings 6 Conclusion Alphabetical IndexWir betrachten die Approximation von Funktionen, die auf verschiedenen Gebieten definiert sind, mittels trigonometrischer und transformierter trigonometrischer Funktionen. Wir untersuchen, welche bisherigen Ergebnisse für die Approximation von Funktionen, die auf einem d-dimensionalen Torus definiert wurden, auf andere Definitionsgebiete übertragen werden können. Dazu definieren wir parametrisierte Transformationsabbildungen und beweisen Bedingungen, bei denen Funktionen aus einem gewichteten Sobolevraum in Funktionen, die auf dem Torus definiert sind, transformiert werden können, die dabei einen gewissen Grad an Sobolevglattheit behalten und für die obere Schranken der Approximationsfehler bewiesen wurden. Durch Umkehrung der ursprünglichen Koordinatentransformation übertragen wir die schnellen Algorithmen, die Rang-1 Gitter Methoden verwenden um Funktionen auf dem Torus effizient zu approximieren, auf andere Definitionsgebiete und erhalten adaptierte FFT Algorithmen.:1 Introduction 2 Preliminaries and notations 3 Fourier approximation on the torus 4 Torus-to-R d transformation mappings 5 Torus-to-cube transformation mappings 6 Conclusion Alphabetical Inde

Retention of radiation damage in zircon xenocrysts from kimberlites, Northern Yakutia

We have studied zircon xenocrysts from Mesozoic kimberlites from the Kuoika and Ary–Mastakh fields in Northern Yakutia. Zircon xenocrysts are assumed to originate from crustal rocks. Our SHRIMP (Sensitive High mass Resolution Ion MicroProbe) analyses yielded predominantly concordant U–Th–Pb ages (up to ~ 3570 Ma; Paleoarchean) that clearly predate kimberlite formation. The general U–Th–Pb concordance observed excludes notable disturbance of the zircon xenocrysts U–Th–Pb isotope system during kimberlite ascent and emplacement. In addition, zircon xenocrysts were found to be significantly more radiation-damaged than would correspond to damage accumulation only since the time of kimberlite formation. This observation first indicates that zircon crystals were sampled by the kimberlite magma at comparably shallow depths not exceeding 10–12 km. If, in contrast, zircon crystals originated from deeper levels of the Earth's crust, they would have been exposed to temperatures of 250–300 °C or more. This in turn would have caused long-term thermal annealing of the radiation damage, which was however not observed in our study. Second, our observation contradicts the hypothesis that high temperatures experienced by zircon xenocrysts during kimberlite ascent will cause notable structural reconstitution by short-term thermal annealing. Consequently, zircon crystals cannot have spent more than a few hours at temperatures exceeding ca. 700–800 °C, or more than a few days at temperatures exceeding ca. 500–600 °C. This in turn suggests that (i) temperatures of the ascending kimberlite magmas were rather moderate, and (ii) kimberlite ascent is a comparably short process followed by rapid cooling

Dating of zircon and monazite from diamondiferous quartzofeldspathic rocks of the Saxonian Erzgebirge - hints at burial and exhumation velocities.

In order to better understand the formation and evolution processes of ultrahigh pressure (UHP) felsic rocks, we determined the ages of various domains of zircon and monazite crystals from the diamondiferous quartzofeldspathic rocks of the Saxonian Erzgebirge. According to cathodoluminescence imagery and Th/U ratios, three zircon zones were distinguished. Each was dated using several spot analyses from a sensitive high-resolution ion microprobe (SHRIMP) analysing Pb, U and Th isotopes. The results were: (1) core zone - 21 analyses: Th/U less/equal 0.023 and 337.0 plus/minus 2.7 Ma (2 sigma, combined 206Pb/238U-207Pb/235U age); (2) diamond-bearing intermediate zone - 23 analyses: Th/U greater/equal 0.037 and 336.8 plus/minus 2.8 Ma; and (3) rim zone-12 analyses: Th/U = 0.0150.038 (plus one analysis of 0.164) and 330.2 plus/minus 5.8 Ma. The U-Pb obtained ages are virtually concordant. Furthermore, two oscillatory zoned zircon cores (Th/U greater/equal to 0.8) yielded (~concordant) ages of ~400 Ma. Six SHRIMP analyses of monazites gave an age of 332.4 plus/minus 2.1 Ma. In addition, Pb, Th and U contents in monazite were analysed with an electron microprobe (EMP). A mean age of 324.7 plus/minus 8.0 (2σ) Ma was acquired from 113 analyses.By combining the defined ages with previously published P-T conditions, minimum velocities for burial and exhumation were estimated. In addition, we present a likely geodynamic scenario involving age data from the literature as well as this study: beginning 340 million years ago, gneisses at the base of a thickened continental crust (~1.8 GPa, 650C) were transported to depths of at least 130 km, possibly as deep as 250 km. Here they were heated (>1050C) and partially melted and as a result began to rise rapidly. The burial and subsequent ascent back to a depth of 50 km, where zircon rims and monazite formed, took only a few million years and perhaps significantly less

Funktion der Transmembranproteine ESAM und JAM-A bei Zell-Zell-Interaktionen

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Grenville Skarn Titanite: Potential Reference Material for Sims U–Th–Pb Analysis

We have investigated the homogeneity, chemical composition, structure, degree of radiation damage, and post-formation evolution of titanite crystals from skarns of the Grenville Province of the Canadian Shield using SHRIMP, TIMS, Raman and PL spectroscopy, EBSD, and EPMA–WDS. These results are used to assess the potential of the titanite as Reference Material (RM) for micro-analytical U–Th–Pb age dating. The SHRIMP data show that these megacrysts (5–31g) have concordant U–Pb isotope systematics, 60 to 500 ppm U, 120 to 1200 ppm Th , 206Pb/204Pb between 500 and 2500, ages of ~1 Ga, and excellent homogeneity at the scale of the analytical volume of the ion probe. The ID–TIMS titanite data for OLT1, OLT2 and TCB show that these crystals are essentially concordant. Data for OLT1 and OLT2 show slight scatter (i.e., in excess of that expected from the uncertainty in an individual analysis). For OLT1, one of seven analyses shows Pb loss or, possibly, a younger period of growth. Crystals OLT1 and OLT2 have respective TIMS concordia ages of 1014.8 ± 2.0 Ma (2s, n = 6, MSWD = 1.8) and 998.0 ± 4.5 Ma (2s, n = 3, MSWD = 3.3) for domains that have not lost Pb.The TIMS analyses of TCB are tightly clustered and give a concordia age of 1018.1 ± 1.7 Ma (2s, n = 4, MSWD = 0.92). Raman and PL spectra show a low to moderate degree of accumulated radiation-induced damage in the Grenville Skarn Titanite crystals and uniform internal distributions of this damage. The EDSB contrast images indicate little or no crystallographic misorientation. The EMPA–WDS data show that the outer 50–100 mm of the OLT1 and TCB crystals are enriched in Al and F, and depleted in Fe and Nb, when compared with the interior. In spite of the variation in composition and degree of radiation damage amongst samples, there are no identifiable matrix effects in our SHRIMP data. Some Grenville skarn titanite (GST) crystals have potential as RM for micro-analytical U–Th–Pb age dating. Crystal TCB has excellent homogeneity of U–Th–Pb isotopic composition. Crystals OLT1 and OLT2 have minor TIMS age heterogeneity. However, this heterogeneity is smaller than that of the Khan titanite, our current in-house titanite standard. Careful selection of analysis areas during SIMS, and of chips for TIMS analysis, allows high-quality isotopic data to be obtained from these large crystals of titanite

A new appraisal of Sri Lankan BB zircon as a reference material for LA-ICP-MS U-Pb geochronology and Lu-Hf isotope tracing

A potential zircon reference material (BB zircon) for laser ablation-inductively coupled plasma-mass spectrometry (LA-ICP-MS) U-Pb geochronology and Hf isotope geochemistry is described. A batch of twenty zircon megacrysts (0.5–1.5 cm3) from Sri Lanka was studied. Within-grain rare earth element (REE) compositions are largely homogeneous, albeit with some variation seen between fractured and homogeneous domains. Excluding fractured cathodoluminescence bright domains, the variation in U content for all analysed crystals ranged from 227 to 368 μg g−1 and the average Th/U ratios were between 0.20 and 0.47. The Hf isotope composition (0.56–0.84 g/100 g Hf) is homogeneous within and between the grains – mean 176Hf/177Hf of 0.281674 ± 0.000018 (2s). The calculated alpha dose of 0.59 × 1018 g−1 for a number of BB grains falls within the trend of previously studied, untreated zircon samples from Sri Lanka. Aliquots of the same crystal (analysed by ID-TIMS in four different laboratories) gave consistent U-Pb ages with excellent measurement reproducibility (0.1–0.4% RSD). Interlaboratory assessment (by LA-ICP-MS) from individual crystals returned results that are within uncertainty equivalent to the TIMS ages. Finally, we report on within- and between-grain homogeneity of the oxygen isotope systematic of four BB crystals (13.16‰ VSMOW)