Differentiation and crystallisation processes in A-type granites : an experimental study of the Wangrah Suite, Australia = Differenzierungs- und Kristallisationsprozesse in A-Typ Graniten : eine experimentelle Studie der Wangrah Suite, Australien

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Incremental Maximization via Continuization

We consider the problem of finding an incremental solution to a cardinality-constrained maximization problem that not only captures the solution for a fixed cardinality, but also describes how to gradually grow the solution as the cardinality bound increases. The goal is to find an incremental solution that guarantees a good competitive ratio against the optimum solution for all cardinalities simultaneously. The central challenge is to characterize maximization problems where this is possible, and to determine the best-possible competitive ratio that can be attained. A lower bound of $2.18$ and an upper bound of $\varphi + 1 \approx 2.618$ are known on the competitive ratio for monotone and accountable objectives [Bernstein et al., Math. Prog., 2022], which capture a wide range of maximization problems. We introduce a continuization technique and identify an optimal incremental algorithm that provides strong evidence that $\varphi + 1$ is the best-possible competitive ratio. Using this continuization, we obtain an improved lower bound of $2.246$ by studying a particular recurrence relation whose characteristic polynomial has complex roots exactly beyond the lower bound. Based on the optimal continuous algorithm combined with a scaling approach, we also provide a $1.772$-competitive randomized algorithm. We complement this by a randomized lower bound of $1.447$ via Yao's principle

Improved Approximation Algorithms for the Expanding Search Problem

A searcher faces a graph with edge lengths and vertex weights, initially having explored only a given starting vertex. In each step, the searcher adds an edge to the solution that connects an unexplored vertex to an explored vertex. This requires an amount of time equal to the edge length. The goal is to minimize the weighted sum of the exploration times over all vertices. We show that this problem is hard to approximate and provide algorithms with improved approximation guarantees. For the general case, we give a (2e+?)-approximation for any ? > 0. For the case that all vertices have unit weight, we provide a 2e-approximation. Finally, we provide a PTAS for the case of a Euclidean graph. Previously, for all cases only an 8-approximation was known

Confocal μ-XANES as a tool to analyze Fe oxidation state in heterogeneous samples: the case of melt inclusions in olivine from the Hekla volcano

Here we present a confocal Fe K-edge μ-XANES method (where XANES stands for X-ray absorption near-edge spectroscopy) for the analysis of Fe oxidation state in heterogeneous and one-side-polished samples. The new technique allows for an analysis of small volumes with high spatial 3D resolution of <100 μm3. The probed volume is restricted to that just beneath the surface of the exposed object. This protocol avoids contamination of the signal by the host material and minimizes self-absorption effects. This technique has been tested on a set of experimental glasses with a wide range of Fe3+ / ςFe ratios. The method was applied to the analysis of natural melt inclusions trapped in forsteritic to fayalitic olivine crystals of the Hekla volcano, Iceland. Our measurements reveal changes in Fe3+ / ςFe from 0.17 in basaltic up to 0.45 in dacitic melts, whereas the magnetite-ilmenite equilibrium shows redox conditions with Fe3+ / ςFe ≤0.20 (close to FMQ, fayalite-magnetite-quartz redox equilibrium) along the entire range of Hekla melt compositions. This discrepancy indicates that the oxidized nature of glasses in the melt inclusions could be related to the post-entrapment process of diffusive hydrogen loss from inclusions and associated oxidation of Fe in the melt. The Fe3+ / ςFe ratio in silicic melts is particularly susceptible to this process due to their low FeO content, and it should be critically evaluated before petrological interpretation

Fractionation vs. magma mixing in the Wangrah Suite A-type granites, Lachlan Fold Belt, Australia: Experimental constraints

The Wangrah Suite granites (Lachlan Fold Belt, Australia) reflect different stages of differentiation in the magmatic history of an A-type plutonic suite. In this study we use experimentally determined phase equilibria of four natural A-type granitic compositions of the Wangrah Suite to constrain phases and phase compositions involved in fractionation processes. Each composition represents a distinct granite intrusion in the Wangrah Suite. The intrusions are the Danswell Creek (DCG), Wangrah (WG), Eastwood (EG) and Dunskeig Granite (DG), ordered from "most mafic" to "most felsic" by increasing SiO2 and decreasing FeOtotal. Experimental investigation show that the initial water content in melts from DCG is between 2-3 wt. % H2O. If the DCG is viewed as the parental magma for the Wangrah Suite, then (1) fractionation of magnetite, orthopyroxene and plagioclase (∼ 20 wt. %) of the DCG composition, leads to compositions similar to that of the EG; (2) further fractionation of plagioclase, quartz, K-feldspar and biotite (∼ 40 wt. %) from the EG composition, leads to the DG composition. These fractionation steps can occur nearly isobarically and are confirmed by bulk rock Ba, Sr, Rb and Zr concentrations. In contrast, the generation of the most abundant WG composition cannot be explained by fractional crystallisation from the DCG at isobaric conditions because of the high K2O content of this granite. Magma Mixing could be the process to explain the chemical distinctiveness of the Wangrah Granite from all the other granites of the Wangrah Suite

Apatite as a monitor for sulfur redox reactions during fluid-rock interaction in the subduction channel

The oxidation state of sulfur in slab fluids is controversial, with both dominantly oxidized and reduced species proposed. Here we use in situ X-ray absorption spectroscopy analysis of sulfur-in-apatite to monitor changes in the oxidation state of sulfur during high-P metasomatism by slab fluids in the subduction channel. Our samples include a 73 cm continuous transect of reaction zones between a metagabbroic eclogite block and serpentinite matrix from a mélange zone on the island of Syros, Greece. The block core consists of garnet, omphacite, phengite, paragonite, epidote-clinozoisite, and rutile. In this region, apatite is only observed as elongate inclusions in omphacite cores. From the core outwards micas are increasingly replaced by epidote-clinozoisite, garnets are smaller and more frequent, pyrite + bornite is observed as inclusions in recrystallized omphacite, and apatite is increasingly abundant in the matrix and inclusions in garnet. A major transition at 48 cm separates an assemblage of Ca-Na amphibole, omphacite, chlorite, pyrite, and apatite from the inner garnet-bearing eclogite assemblages. Omphacite disappears from the assemblage at ~56 cm and amphibole compositions sharply transition to tremolite at 59 cm. Finally, the assemblage tremolite + talc + pyrite is observed after ~70 cm.Apatites in the eclogite assemblages exclusively display S6+ peaks in their absorption spectra. This includes apatite inclusions in omphacite in the least altered lithology, as well as matrix apatite and isolated apatite inclusions in garnet in the outermost metasomatized eclogite zone. In the intermediate pyrite-rich (~1-5 vol %) amphibole + omphacite + chlorite zone, apatite displays a strong S1- absorption peak in most grains, with rare analyses showing mixed S1- and S6+. Finally, apatite in the outermost tremolite-bearing assemblages only displays a S6+ peak. The pyrite-rich zone at 48 cm occurs at the initial interface between the serpentinite matrix and eclogite block, characterized by a dramatic decrease in Na content and Mg#. Our data suggest that reduction of S6+ in infiltrating fluids to S1- in pyrite became focused as Fe diffused across the steep Mg# gradient, resulting in pyrite precipitation. In contrast, S reduction in the Mg-rich tremolite-dominant portions of the transect was limited by a lack of Fe, resulting in low modes of pyrite and fluid buffered S6+ in apatite. Finally, S6+-bearing apatite is also observed in reaction zone lithologies from elsewhere on Syros, suggesting our observations are not isolated.Two important conclusions are drawn from these data and observations: (1) In the case of Syros, slab fluids at eclogite-facies conditions carried oxidized S6+, and (2) The interaction of these fluids with eclogites composed of ferrous-Fe silicates resulted in extensive sulfide precipitation

The dissolution mechanism of sulphur in hydrous silicate melts. I : assessment of analytical techniques in determining the sulphur speciation in iron-free to iron-poor glasses

S K-edge XANES, λ (SKα) wavelength shift, 33S MAS NMR and Raman spectroscopy have been applied to a series of experimentally synthesised hydrous sulphur bearing silicate glasses to determine the oxidation state of sulphur dissolved in the glass. Glasses investigated include soda–lime glass (SLG), K2Si4O9 (KSG) and albitic (Albite) and trondhjemitic (TROND) glass compositions. The four spectroscopic techniques are compared with each other to investigate the applicability of each technique as a method to determine the sulphur oxidation state and structural aspects of sulphur dissolution in silicate melts