ALBERT

Description: Transformation of refractory cratonic mantle into more fertile lithologies is the key to the fate of cratonic lithosphere. This process has been extensively studied in the eastern North China Craton (NCC) while that of its western part is still poorly constrained. A comprehensive study of newly-found pyroxenite xenoliths from the Langshan area, in the northwestern part of this craton is integrated with a regional synthesis of pyroxenite and peridotite xenoliths to constrain the petrogenesis of the pyroxenites and provide an overview of the processes involved in the modification of the deep lithosphere. The Langshan pyroxenites are of two types, high-Mg# [Mg2+/(Mg2++Fe2+)*100 = ∼ 90, atomic ratios] olivine-bearing websterites with high equilibration temperatures (880 ∼ 970 oC), and low-Mg# (70 ∼ 80) plagioclase-bearing websterites with low equilibration temperatures (550 ∼ 835 oC). The high-Mg# pyroxenites show trade-off abundances of olivine and orthopyroxene, highly depleted bulk Sr-Nd (ƐNd = +11.41, 87Sr/86Sr = ∼0.7034) and low clinopyroxene Sr isotopic ratios (mean 87Sr/86Sr = ∼0.703). They are considered to reflect the reaction of mantle peridotites with silica-rich silicate melts derived from the convective mantle. Their depletion in fusible components (e.g., FeO, TiO2 and Na2O) and progressive exhaustion of incompatible elements suggest melt extraction after their formation. The low-Mg# pyroxenites display layered structures, convex-upward rare earth element patterns, moderately enriched bulk Sr-Nd isotopic ratios (ƐNd = -14.20 ∼ -16.74, 87Sr/86Sr = 0.7070 ∼ 0.7078) and variable clinopyroxene Sr-isotope ratios (87Sr/86Sr = 0.706-0.711). They are interpreted to be crustal cumulates from hypersthene-normative melts generated by interaction between the asthenosphere and heterogeneous lithospheric mantle. Combined with studies on regional peridotite xenoliths, it is shown that the thinning and refertilization of the lithospheric mantle was accompanied by crustal rejuvenation and that such processes occurred ubiquitously in the northwestern part of the NCC. A geodynamic model is proposed for the evolution of the deep lithosphere, which includes long-term mass transfer through a mantle wedge into the deep crust from the Paleozoic to the Cenozoic, triggered by subduction of the Paleo-Asian ocean and the Late Mesozoic lithospheric extension of eastern Asia.

Description: The orogenic development after the continental collision between Laurussia and Gondwana, led to two contrasting associations of mantle-derived magmatic rocks on the territory of the Bohemian Massif: (i) a 340–310 Ma lamprophyre-lamproite orogenic association and (ii) a 300–275 Ma lamprophyre association of anorogenic affinity. Major types of potassic mantle-derived magmatic rocks recognised in the orogenic and anorogenic associations include: (i) calc-alkaline to alkaline lamprophyres, (ii) alkaline “orthopyroxene minettes” (and geochemically related rocks), and (iii) peralkaline lamproites. These three types significantly differ with respect to mineral, whole-rock and Sr–Nd–Pb–Li isotope composition, and spatial distribution. The calc-alkaline lamprophyres occur throughout the entire Saxo-Thuringian and Moldanubian zones, whereas the different types of malte-derived potassic rocks are spatially restricted to particular zones. Rocks of the Carboniferous lamprophyre-lamproite orogenic association are characterised by variable negative εNd(i) and variably radiogenic Sr(i), whereas the rocks of the Permian lamprophyre association of anorogenic affinity are characterised by positive εNd(i) and relatively young depleted-mantle Nd-model ages reflecting increasing input from upwelling asthenospheric mantle. The small variation in the Pb isotopic composition of post-collisional potassic mantle-derived magmatic rocks (of both the orogenic and anorogenic series) implies that the Pb budget of the mantle beneath the Bohemian Massif is dominated by the same crust-derived material, which itself may include material derived from several sources. The source rocks of “orthopyroxene minettes” are characterised by isotopically light (“eclogitic”) Li and strongly radiogenic (crustal) Sr and may have been metasomatised by high-pressure fluids along the edge of a subduction zone. In contrast, the strongly Al2O3 and CaO depleted mantle source of the lamproites is characterised by isotopically heavy Li and high SiO2 and extreme K2O contents. This mantle source may have been metasomatised predominantly by melts. The mantle source of the lamprophyres may have undergone metasomatism by both fluids and melts.

Description: The Lu–Hf isotope system and Sr–Nd–Hf–Os isotope systematics of mantle rocks are capable of unravelling the early processes in collision belts, especially in a hot subduction context where the Sm–Nd and U–Pb systems in crustal rocks are prone to resetting owing to high temperatures and interaction with melts during exhumation. To improve models of the Devonian–Carboniferous evolution of the Bohemian Massif, we investigated in detail mafic and ultramafic rocks (eclogite, pyroxenite, and peridotite) from the ultrahigh-pressure and ultrahigh-temperature Kutná Hora Crystalline Complex (KHCC: Úhrov, Bečváry, Doubrava, and Spačice localities). Petrography, multiphase solid inclusions, major and trace element compositions of rocks and minerals, and radiogenic isotopic data document contrasting sources and protoliths as well as effects of subduction-related processes for these rocks. The Úhrov peridotite has a depleted composition corresponding to the suboceanic asthenospheric mantle, whereas Bečváry and Doubrava peridotites represent lithospheric mantle that underwent melt refertilization by basaltic and SiO2-undersaturated melts, respectively. Multiphase solid inclusions enclosed in garnet from Úhrov and Bečváry peridotites represent trapped H2O ± CO2-bearing metasomatizing agents and Fe–Ti-rich melts. The KHCC eclogites either formed by high-pressure crystal accumulation from mantle-derived basaltic melts (Úhrov) or represent a fragment of mid-ocean ridge basalt-like gabbroic cumulate (Spačice) and crustal-derived material (Doubrava) both metamorphosed at high P–T conditions. The Lu–Hf age of 395 ± 23 Ma obtained for the Úhrov peridotite reflects garnet growth related to burial of the asthenospheric mantle during subduction of the oceanic slab. By contrast, Spačice and Doubrava eclogites yield younger Lu–Hf ages of ∼350 and 330 Ma, respectively, representing mixed ages as demonstrated by the strong granulite-facies overprint and trace element zoning in garnet grains. We propose a refined model for the Early Variscan evolution of the Bohemian Massif starting with the subduction of the oceanic crust (Saxothuringian ocean) and associated oceanic asthenospheric mantle (Úhrov) beneath the Teplá–Barrandian at ≥380 Ma, which was responsible for melt refertilization of the associated mantle wedge (Bečváry, Doubrava). This was followed by continental subduction (∼370–360 Ma?) accompanied by the oceanic slab break-off and incorporation of the upwelling asthenospheric mantle into the Moldanubian lithospheric mantle and subsequent coeval exhumation of mantle and crustal rocks at ∼350–330 Ma.

Description: The Changning–Menglian orogenic belt (CMOB) in the southeastern Tibetan Plateau is an important link between the Longmu Co–Shuanghu suture (LCSS) in the northern Tibetan Plateau and the Chiang Mai–Inthanon and Bentong–Raub sutures in Thailand and Peninsular Malaysia. These belts and sutures are generally regarded as containing the remnants of the oceanic crust of the Palaeo-Tethys that formed by seafloor spreading as a result of the separation of Gondwana- and Eurasia-derived blocks during the Middle Cambrian. In this paper we report the first discovery of abundant unaltered and retrograde eclogites that occur as irregular lenses and blocks in metasedimentary rocks of the CMOB, and these eclogites form an elongate and almost north–south-trending high-pressure (HP)–ultrahigh-pressure (UHP) metamorphic belt that is ∼200 km long and ∼50 km wide. The newly discovered phengite/talc/epidote–glaucophane eclogites, lawsonite–talc–phengite eclogites, dolomite/magnesite–kyanite eclogites and phengite–kyanite-bearing retrograde eclogites have enriched (E-) and normal mid-ocean ridge basalt (N-MORB)-like affinities and mainly positive as well as some negative whole-rock εNd values (–4·34 to +7·89), which suggest an enriched and depleted oceanic lithosphere source for their protoliths. Magmatic zircons separated from the epidote–glaucophane, magnesite–kyanite and (phengite–kyanite-bearing) retrograde eclogites gave protolith ages of 317–250 Ma, which fit well within the time frame of the opening of the Palaeo-Tethys during the Middle Cambrian and its closure during the Triassic. Abundant metamorphic zircons in the eclogites indicate a Triassic metamorphic event related to the subduction of the Palaeo-Tethys oceanic crust from 235 to 227 Ma. Taking into account previous isotopic age data, we now establish the periods of Early–Middle Triassic (246–227 Ma) and Late Triassic (222–209 Ma) as the ages of subduction and exhumation of the Palaeo-Tethyan oceanic crust, respectively. Thermodynamic modelling revealed that the eclogites record distinct HP–UHP peak metamorphic conditions of 23·0–25·5 kbar and 582–610 °C for the phengite–glaucophane eclogites, 24·0–25·5 kbar and 570–586 °C for the talc–glaucophane eclogites, 29·0–31·0 kbar and 675–712 °C for the dolomite–kyanite eclogites, and 30·0–32·0 kbar and 717–754 °C for the magnesite–kyanite eclogites. These P–T estimates and geochronological data indicate that the Palaeo-Tethys oceanic slab was subducted to different mantle depths from 75 km down to 95 km, forming distinct types of eclogite with a variety of peak eclogite-facies mineral assemblages. The eclogites consistently record clockwise metamorphic P–T–t paths characterized by a heating–compression prograde loop under a low geothermal gradient of 5–10 °C km–1, indicating the rapid subduction of cold oceanic crust at a rate of 4·5–6·0 km Ma–1, followed by isothermal or cooling–decompressive retrogression and exhumation at an average rate of 3·2–4·2 km Ma–1. The newly discovered eclogites of the CMOB with their signatures of ocean-crust subduction are petrologically, geochemically and geochronologically comparable with those of the LCSS, providing powerful support for the idea that a nearly 2000 km long HP–UHP eclogite belt extends from the northern Tibetan Plateau to the southeastern Tibetan Plateau, and that it represents the main boundary suture of the Palaeo-Tethyan domain. These results have far-reaching implications for the tectonic framework and complex metamorphic evolution of the Palaeo-Tethyan domain.

Description: Replacive symplectites (vermicular intergrowths of two or more minerals) are an important feature of layered igneous intrusions, recording evidence of late-stage reactions between interstitial liquid and crystals. They are common throughout the Layered Series of the 564 Ma Sept Iles layered intrusion in Quebec, Canada, and fall into three types: oxy-symplectites, ‘Type I’ symplectites, and ‘Type II’ symplectites. Oxy-symplectites are comprised of magnetite and orthopyroxene, nucleate on olivine primocrysts, and form via the reaction Olivine + O2 → Orthopyroxene + Magnetite; Type I symplectites (of which there are 3 distinct categories) are comprised of anorthitic plagioclase with pyroxene, amphibole, or olivine vermicules, grow from primocryst oxide grains, and replace primary plagioclase; and Type II symplectites (of which there are 2 distinct categories) are comprised of anorthitic plagioclase with orthopyroxene ± amphibole vermicules, grow from primocryst olivine grains, and replace primocryst plagioclase. Rare symplectites composed of biotite and plagioclase are also present. Symplectite growth occurred at 700-1030 °C with pressure constraints of 1-2 kbar. We propose that Type I symplectites, and some Type II symplectites, formed from interaction of primocrysts with residual Fe-rich liquid as a consequence of differential loss of an immiscible Si-rich liquid conjugate from the crystal mush. However, redistribution and concentration of hydrous fluids in incompletely solidified rock, or an increase in water activity of the interstitial melt, may be more plausible processes responsible for the formation of replacive symplectites comprising abundant hydrous mineral assemblages.

Description: We prove a spectral flow formula for one-parameter families of Hamiltonian systems under homoclinic boundary conditions, which relates the spectral flow to the relative Maslov index of a pair of curves of Lagrangians induced by the stable and unstable subspaces, respectively. Finally, we deduce sufficient conditions for bifurcation of homoclinic trajectories of one-parameter families of non-autonomous Hamiltonian vector fields.

Description: We consider the Schur–Horn problem for normal operators in von Neumann algebras, which is the problem of characterizing the possible diagonal values of a given normal operator based on its spectral data. For normal matrices, this problem is well known to be extremely difficult, and in fact, it remains open for matrices of size greater than $3$ . We show that the infinite-dimensional version of this problem is more tractable, and establish approximate solutions for normal operators in von Neumann factors of type I $_\infty$ , II, and III. A key result is an approximation theorem that can be seen as an approximate multivariate analogue of Kadison's Carpenter Theorem.

Description: We study the rate of convergence to zero of the tail entropy of $C^\infty$ maps. We give an upper bound of this rate in terms of the growth in $k$ of the derivative of order $k$ and give examples showing the optimality of the established rate of convergence. We also consider the case of multimodal maps of the interval. Finally, we prove that homoclinic tangencies give rise to $C^r$ $(r\geqslant 2)$ robustly non- $h$ -expansive dynamical systems.

Description: Let ${{\mathscr {C}}}^0_{{{\mathfrak {g}}}}$ be the category of finite-dimensional integrable modules over the quantum affine algebra $U_{q}'({{\mathfrak {g}}})$ and let $R^{A_\infty }{\mbox {-}\mathrm {gmod}}$ denote the category of finite-dimensional graded modules over the quiver Hecke algebra of type $A_{\infty }$ . In this paper, we investigate the relationship between the categories ${{\mathscr {C}}}^0_{A_{N-1}^{(1)}}$ and ${{\mathscr {C}}}^0_{A_{N-1}^{(2)}}$ by constructing the generalized quantum affine Schur–Weyl duality functors ${\mathcal {F}}^{(t)}$ from $R^{A_\infty }{\mbox {-}\mathrm {gmod}}$ to ${{\mathscr {C}}}^0_{A_{N-1}^{(t)}}\ (t=1,2)$ .

Description: We present new constructions of complex and $p$ -adic Darmon points on elliptic curves over base fields of arbitrary signature. We conjecture that these points are global and present numerical evidence to support our conjecture.

Description: We introduce a new framework for the analysis of the stability of solitons for the one-dimensional Gross–Pitaevskii equation. In particular, we establish the asymptotic stability of the black soliton with zero speed.

Description: Let $k$ and $n$ be positive even integers. For a cuspidal Hecke eigenform $h$ in the Kohnen plus space of weight $k-n/2+1/2$ for $\varGamma _0(4),$ let $I_n(h)$ be the Duke–Imamo $\bar {{\text {g}}}$ lu–Ikeda lift of $h$ in the space of cusp forms of weight $k$ for ${\rm Sp}_n({{\bf{Z}}}),$ and $f$ be the primitive form of weight $2k-n$ for ${\rm SL}_2({{\bf{Z}}})$ corresponding to $h$ under the Shimura correspondence. We then express the ratio $\displaystyle {\langle I_n(h), I_n(h) \rangle / \langle h, h \rangle }$ of the period of $I_n(h)$ to that of $h$ in terms of special values of certain $L$ -functions of $f$ . This proves the conjecture proposed by Ikeda concerning the period of the Duke–Imamo $\bar {{\text {g}}}$ lu–Ikeda lift.

Description: The Matachewan Large Igneous Province (LIP) is interpreted to have formed during the early stages of mantle plume-induced continental break-up in the early Proterozoic. When the Matachewan LIP is reconstructed to its original configuration with units from the Superior Craton and other formerly adjacent blocks (Karelia, Kola, Wyoming and Hearne), the dyke swarms, layered intrusions and flood basalts, emplaced over the lifetime of the province, form one of the most extensive magmatic provinces recognized in the geological record. New geochemical data allow, for the first time, the Matachewan LIP to be considered as a single, coherent entity and show that Matachewan LIP rocks share a common tholeiitic composition and trace element geochemistry, characterized by enrichment in the most incompatible elements and depletion in the less incompatible elements. This signature, ubiquitous in early Proterozoic continental magmatic rocks, may indicate that the Matachewan LIP formed through contamination of the primary magmas with lithospheric material or that the early Proterozoic mantle had a fundamentally different composition from the modern mantle. In addition to the radiating geometry of the dyke swarms, a plume origin for the Matachewan LIP is consistent with the geochemistry of some of the suites; these suites are used to constrain a source mantle potential temperature of c. 1500–1550°C. Comparison of these mantle potential temperatures with estimated temperatures for the early Proterozoic upper mantle indicates that they are consistent with a hot mantle plume source for the magmatism. Geochemical data from coeval intrusions suggest that the plume head was compositionally heterogeneous and sampled material from both depleted and enriched mantle. As has been documented with less ancient but similarly vast LIPs, the emplacement of the Matachewan LIP probably had a significant impact on the early Proterozoic global environment. Compilation of the best age estimates for various suites shows that the emplacement of the Matachewan LIP occurred synchronously with the Great Oxidation Event. We explore the potential for the eruption of this LIP and the emission of its associated volcanic gases to have been a driver of the irreversible oxygenation of the Earth.

Description: Olivine major and trace element compositions from 12 basalts from the southern Payenia volcanic province in Argentina have been analyzed by electron microprobe and laser ablation inductively coupled plasma mass spectrometry. The olivines have high Fe/Mn and low Ca/Fe and many fall at the end of the global olivine array, indicating that they were formed from a pyroxene-rich source distinct from typical mantle peridotite. The olivines with the highest Fe/Mn have higher Zn/Fe, Zn and Co and lower Co/Fe than the olivines with lower Fe/Mn, also suggesting contributions from a pyroxene-rich source. Together with whole-rock radiogenic isotopes and elemental concentrations, the samples indicate mixing between two mantle sources: (1) a pyroxene-rich source with EM-1 ocean island basalt type trace element and isotope characteristics; (2) a peridotitic source with more radiogenic Pb that was metasomatized by subduction-zone fluids and/or melts. The increasing contributions from the pyroxene-rich source in the southern Payenia basalts are correlated with an increasing Fe-enrichment, which caused the olivines to have lower forsterite contents at a given Ni content. Al-in-olivine crystallization temperatures measured on olivine–spinel pairs are between 1155 and 1243°C and indicate that the magmas formed at normal upper mantle (asthenospheric) temperatures of ~1350°C. The pyroxene-rich material is interpreted to have been brought up from the deeper parts of the upper mantle by vigorous asthenospheric upwelling caused by break-off of the Nazca slab south of Payenia during the Pliocene and roll-back of the subducting slab beneath Payenia. The pyroxene-rich mantle mixed with peridotitic metasomatized South Atlantic mantle in the mantle wedge beneath Payenia.

Description: To reconstruct the magmatic–hydrothermal processes leading to porphyry Mo ore formation at the Climax Mo mine, Colorado, four magma units that were emplaced before, during and shortly after the mineralization events were investigated: (1) a pre-mineralization white dike of the Alma district; (2) the syn-mineralization Chalk Mountain Rhyolite; (3) a late- to post-mineralization rhyolite porphyry dyke; (4) a mafic enclave within the productive Bartlett stock. Melt inclusions, mineral inclusions and fluid inclusions in quartz phenocrysts were investigated by means of laser ablation inductively coupled plasma mass spectrometry, electron microprobe and microthermometry. Based on melt inclusion data both the Chalk Mountain Rhyolite and the rhyolite porphyry were ~10 times more fractionated than average granite and show geochemical characteristics of topaz rhyolites. They were saturated in magnetite, Mn-rich ilmenite, fluorite, aeschynite, monazite, pyrrhotite and thorite, and crystallized predominantly at 710–730°C, 1·2–2·6 kbar and log f O 2 FMQ + 2·2 (where FMQ is fayalite–magnetite–quartz). The silicate melt of the Chalk Mountain Rhyolite contained 3·5 ± 0·4 wt % F, 0·09 ± 0·03 wt % Cl, ≥ 3·0 wt % H 2 O, 15–90 µg g –1 Cs, 500–1500 µg g –1 Rb and 5–7 µg g –1 Mo, whereas that of the rhyolite porphyry contained 1·1 ± 0·3 wt % F and 4·9 ± 1·2 wt % H 2 O, but otherwise had a virtually identical major and trace element composition. The fluid exsolving from the latter melt had a bulk salinity of 10 ± 2 wt % NaCl equiv and contained of the order of 100 µg g –1 Mo. After emplacement of the Chalk Mountain Rhyolite magma at subvolcanic levels, extremely fractionated silicate melts coexisting with hypersaline brines (salt melts) and low-density vapor percolated at near-solidus conditions through the rock. These silicate melts contained 6·6 ± 0·4 wt % F, ≥ 7·5 ± 0·6 wt % H 2 O, 0·51 ± 0·05 wt % Cl, and up to 0·5 wt % Cs and 100 µg g –1 Mo, whereas the hypersaline brines contained 1–2 wt % Cs and 0·3–0·6 wt % Mo. However, owing to their negligible masses these liquids are unlikely to have played a major role in the mineralization process. The majority of Mo in the Climax deposit appears to have been derived from melts containing 5–7 µg g –1 Mo and bulk fluids containing ~100 µg g –1 Mo. These concentrations are similar to those found in similarly fractionated melts and fluids in barren and sub-economically mineralized intrusions. However, whereas in the latter intrusions fractionated melts occurred in a rather dispersed state, they seem to have been present as large, coherent masses in the apical parts of Climax-type porphyry Mo-forming magma systems. Efficient segregation of fractionated melts and fluids into the top of mineralizing magma chambers appears to have been promoted by high fluorine concentrations in the silicate melt, which was partly a primary feature, and partly an indirect consequence of other characteristics of within-plate magmatism.

Description: The Skaergaard intrusion, Greenland, is the type locality for Skaergaard-type mineralizations. Mineralization levels are perfectly concordant with igneous layering, up to 5 m thick, internally fractionated, and contain crystallized sulphide droplets and precious metal alloys, sulphides, arsenides and telluride. Immiscible Cu-rich sulphide droplets, formed in a mush zone below the roof, scavenged precious metals. They were subsequently dissolved and transported to the floor in late-formed, immiscible, Fe-rich mush melts. Mineralized stratigraphic intervals of floor gabbro formed in ‘proto-macrolayers‘, owing to local sulphide saturation in melt concentrated between floating plagioclase and sinking clinopyroxene. The floor mineralization is divided into four stratigraphic sections. Formation of the Lower Platinum Group Element Mineralization (LPGEM) involved: (1) crystallization of the bulk liquid liquidus paragenesis and in situ fractionation; (2) sulphide saturation and formation of sulphide droplets in melt in the upper part of ‘proto-macrolayers‘. After further in situ fractionation, the following steps occurred: (3) the onset of silicate–silicate immiscibility and the consequent loss of buoyant and immiscible Si-rich melt; (4) dissolution of unprotected droplets of sulphide melt present in the Fe-rich mush melt; (5) compaction-driven upwards loss of residual mush melt enriched in, for example, Au. The LPGEM preserves upward increasing bulk Pd/Pt (~6–13) owing to a continued supply of PGE and Au, with high Pd/Pt. The further development of the LPGEM ceased as the supply of precious metals to the floor waned. The Upper PGE Mineralization (UPGEM) subsequently formed from precious metals recycled in the floor. The UPGEM is characterized by increasing Au substitution in PGE phases, and a decrease in total PGE and Pd/Pt owing to upward fractionation in migrating mush melts and exhaustion of Pd and Pt. An upper Au-rich mineralization level (UAuM) was caused by late remobilization of Au and deposition on grain boundaries in fully crystallized gabbro. Cu concentrations (~150 ppm) are not correlated with PGE and Au. Repeated Cu mineralization levels (CuM), attaining 〉1000 ppm, occur above the Au levels, caused by local mush layer sulphide saturation. PGE, Au and Cu distributions in the floor mineralization reflect sub-liquidus, but supra-solidus, processes and reactions in mushes at the roof, wall and floor. Constraints provided by a new model for the mineralization provide the basis for re-evaluation of the solidification processes in the Skaergaard intrusion. We have identified the importance of extensive in situ fractionation and intrusion-wide elemental redistributions in immiscible Fe- and Si-rich silicate melts. Our model characterizes the floor cumulates as bulk liquid orthocumulates containing an upwards-increasing proportion crystallized from Fe-rich, immiscible mush melt. The roof-rocks are complementary to the floor, with downwards increasing proportions crystallized from the conjugate Si-rich melt. Petrographic observations and the relative timing of crystallization support the hypothesis that crystallization was restricted to marginal mush zones. Bulk melt remaining in the magma chamber evolved not, as generally assumed, as a result of loss of crystals grown from the bulk melt, but as the consequence of mixing with recycled and evolved melt expelled from the mush by compaction. Redistribution of Fe in immiscible melts may be common to mafic intrusions and puts into question the validity of petrogenetic modelling of bulk liquids in mafic intrusions based only on consideration of floor cumulates.

Description: We have determined experimentally the hydrous phase relations and trace element partitioning behaviour of ocean floor basalt protoliths at pressures and temperatures (3 GPa, 750–1000°C) relevant to melting in subduction zones. To avoid potential complexities associated with trace element doping of starting materials we have used natural, pristine mid-ocean ridge basalt (MORB from Kolbeinsey Ridge) and altered oceanic crust (AOC from Deep Sea Drilling Project leg 46, ~20°N Atlantic). Approximately 15 wt % water was added to starting materials to simulate fluid fluxing from dehydrating serpentinite underlying the oceanic crust. The vapour-saturated solidus is sensitive to basalt K 2 O content, decreasing from 825 ± 25°C in MORB (~0·04 wt % K 2 O) to 750°C in AOC (~0·25 wt % K 2 O). Textural evidence indicates that near-solidus fluids are sub-critical in nature. The residual solid assemblage in both MORB and AOC experiments is dominated by garnet and clinopyroxene, with accessory kyanite, epidote, Fe–Ti oxide and rutile (plus quartz–coesite, phengite and apatite below the solidus). Trace element analyses of quenched silica-rich melts show a strong temperature dependence of key trace elements. In contrast to the trace element-doped starting materials of previous studies, we do not observe residual allanite. Instead, abundant residual epidote provides the host for thorium and light rare earth elements (LREE), preventing LREE from being released (LREE 〈3 ppm at 750–900°C). Elevated Ba/Th ratios, characteristic of many arc basalts, are found to be generated within a narrow temperature field above the breakdown temperature of phengite, but below exhaustion of epidote. Melts with Ba/Th 〉1500 and La/Sm PUM (where PUM indicates primitive upper mantle) ~1, most closely matching the geochemical signal of arc lavas worldwide, were generated from AOC at 800–850°C.

Description: We constrain the physical nature of the magma reservoir and the mechanisms of rhyolite generation at Yellowstone caldera via detailed characterization of zircon and sanidine crystals hosted in three rhyolites erupted during the ( c . 170–70 ka) Central Plateau Member eruptive episode—the most recent post-caldera magmatism at Yellowstone. We present 238 U– 230 Th crystallization ages and trace-element compositions of the interiors and surfaces (i.e. unpolished rims) of single zircon crystals from each rhyolite. We compare these zircon data with 238 U– 230 Th crystallization ages of bulk sanidine separates coupled with chemical and isotopic data from single sanidine crystals. Zircon age and trace-element data demonstrate that the magma reservoir that sourced the Central Plateau Member rhyolites was long-lived (150–250 kyr) and genetically related to the preceding episode of magmatism, which occurred c . 256 ka. The interiors of most zircons in each rhyolite were inherited from unerupted material related to older stages of Central Plateau Member magmatism or the preceding late Upper Basin Member magmatism (i.e. are antecrysts). Conversely, most zircon surfaces crystallized near the time of eruption from their host liquids (i.e. are autocrystic). The repeated recycling of zircon interiors from older stages of magmatism demonstrates that sequentially erupted Central Plateau Member rhyolites are genetically related. Sanidine separates from each rhyolite yield 238 U– 230 Th crystallization ages at or near the eruption age of their host magmas, coeval with the coexisting zircon surfaces, but are younger than the coexisting zircon interiors. Chemical and isotopic data from single sanidine crystals demonstrate that the sanidines in each rhyolite are in equilibrium with their host melts, which considered along with their near-eruption crystallization ages suggests that nearly all Central Plateau Member sanidines are autocrystic. The paucity of antecrystic sanidine crystals relative to antecrystic zircons requires a model in which eruptible rhyolites are generated by extracting melt and zircons from a long-lived mush of immobile crystal-rich magma. In this process the larger sanidine crystals remain trapped in the locked crystal network. The extracted melts (plus antecrystic zircon) amalgamate into a liquid-dominated (i.e. eruptible) magma body that is maintained as a physically distinct entity relative to the bulk of the long-lived crystal mush. Zircon surfaces and sanidines in each rhyolite crystallize after melt extraction and amalgamation, and their ages constrain the residence time of eruptible magmas at Yellowstone. Residence times of the large-volume rhyolites (~40–70 km 3 ) are ≤1 kyr (conservatively 〈6 kyr), which suggests that large volumes of rhyolite can be generated rapidly by extracting melt from a crystal mush. Because the lifespan of the crystal mush that sourced the Central Plateau Member rhyolites is two orders of magnitude longer than the residence time of eruptible magma bodies within the reservoir, it is apparent that the Yellowstone magma reservoir spends most of its time in a largely crystalline (i.e. uneruptible) state, similar to the present-day magma reservoir, and that eruptible magma bodies are ephemeral features.

Description: We report and interpret new geochemical and Pb–Sr–Nd isotopic data from 325 samples of shield, late-shield, postshield, and rejuvenated stage lavas from Kauai and Niihau, the two most northwesterly islands in the Hawaiian island chain. Kauai is unique in the Hawaiian chain in that it exhibits a near-continuous geochemical transition from shield to postshield to rejuvenated stage volcanism between 4·4 and 3·6 Ma and has been continuously active over ~6 Myr. From c . 5·7 to 4·3 Ma, the shield stage of both islands produced tholeiitic basalts typical of other Hawaiian shield volcanoes. The Niihau basalts are more evolved and have high Gd/Yb compared with Kauai, indicating a higher residual garnet content in the source. Both Kauai and Niihau shield basalts have Kea-like trace element ratios, but isotopic ratios are transitional between Kea- and Loa-like compositions. The geochemical similarity of the two shields indicates that mantle sources in different regions of the plume source were similar, and that the 〈2 Ma Loa and Kea trends of the southeastern Hawaiian volcanoes are not observed. More Loa-like compositions are evident in shield lavas from eastern Kauai, where the enhanced Loa composition may reflect melting of low-melting temperature plume components as the island migrates off the hotter, more Kea-like, center of the Hawaiian plume. Postshield lavas and intrusive rocks on both islands are rare: Kauai includes alkalic basalts, hawaiites and mugearites that are isotopically homogeneous and include a significant depleted mantle component compared with the shield basalts, whereas the Niihau late-shield and postshield rocks consist of highly contrasting transitional tholeiites or basanites with variable but shield-like isotopic compositions. The Niihau postshield rocks represent variable, but lower degrees of melting of the shield mantle source. Large volumes of rejuvenated stage lavas cover both islands and also form submarine cone fields, but lava compositions are different. On Kauai, rejuvenated lavas range from melilitite to alkalic basalt with trace element, Nd isotope, and Pb isotope ratios that vary as a function of Th and SiO 2 content. Low-degree (high-Th) melts are dominated by a mixed Kea–Loihi component and high-degree (low-Th) melts include more of a depleted rejuvenated component (DRC) typified by rejuvenated stage lavas and xenoliths from nearby Kaula Island. With the exception of a single basanite, the Niihau rejuvenated stage lavas are uniformly alkalic basalt, with Sr and Ba excesses combined with depleted Th and Nb abundances relative to the light rare earth elements. Rejuvenated stage alkalic basalts from both islands are dominated by contributions from the DRC, which have high Sr/Ce and 87 Sr/ 86 Sr but low 206 Pb/ 204 Pb. The Sr-rich, possibly carbonate-bearing, DRC component may be distributed patchily in the rejuvenated stage mantle source such that, where present, the degree of partial melting was enhanced compared with the degree of partial melting of the Sr-poor, mixed Kea–Loihi component. Given the lack of a hiatus between postshield and rejuvenated stages on Kauai, the rejuvenated mantle source is already able to melt at the tail end of shield stage activity and no secondary melting mechanism is required to explain the rejuvenated stage.

Description: A new thermodynamic model is presented for calculating phase relations in peridotite, from 0·001 to 60 kbar and from 800°C to liquidus temperatures, in the system NCFMASOCr. This model system is large enough to simulate phase relations and melting of natural peridotite and basaltic liquids. Calculations in the program thermocalc illustrate mantle phase relationships and melting conditions, specifically for the peridotite composition KLB-1. The garnet–spinel transition zone intersects the solidus at 21·4–21·7 kbar, where both Fe 3+ and Cr increase spinel stability, expanding the width of the transition. Orthopyroxene is lost at the solidus at 42 kbar in KLB-1, although this pressure is very sensitive to bulk composition. Calculated oxidation states are in excellent agreement with measured log f O 2 for xenolith suites with mantle Fe 2 O 3 contents in the range 0·1–0·3 wt %. It appears that mantle oxidation state is not just a simple function of P and T , but depends on phase assemblage, and may vary in a complex way within a single assemblage. The liquid model performs well, such that calculated solidus, melt productivity and liquid compositions compare favourably with those of experimental studies, permitting its use in interpolating between, and extrapolating from, experimental P–T conditions. Experimentally challenging but geologically useful regimes can be explored, such as subsolidus samples and very low melt fractions, with application to both mantle xenoliths and the origin of basalt.

Description: The anorthositic members of the Mealy Mountains Intrusive Suite (MMIS; Labrador, Canada) are host to 0·5–5 m diameter pegmatitic, pod-like segregations, originally described as graphic granite pods. U–Pb zircon geochronology confirms that the pods are coeval with the 1650–1630 Ma emplacement age range for the MMIS, yielding ages of 1654 ± 8 to 1628 ± 3·5 Ma. Petrographic and geochemical analysis of five pods from anorthositic rocks of the MMIS reveals that the pods have a diverse compositional range from monzodiorite to granite, varying from Fe-rich and Si-poor, to Fe-poor and Si-rich compositions. Fe-rich, Si-poor pods in the MMIS and other massifs (e.g. Laramie Anorthosite Complex) tend to be hosted by olivine-bearing anorthosites, whereas Si-rich, Fe-poor pods are hosted by pyroxene-bearing anorthosites. Each pod shows a range of graphic, myrmekitic and symplectitic textures, along with distinctive mineral assemblages (e.g. apatite and zircon) and highly enriched trace-element compositions. Evolved mineral assemblages, high concentrations of Fe, Ti and P (and in some cases SiO 2 ), and 10–1000 x chondrite enrichment in light rare earth elements, U, Th and Rb indicate that many of the pods are highly fractionated. The array of textural intergrowths provides clues about the final stages of crystallization in the pods and, by extension, the anorthosites. Macroscopic quartz–K-feldspar graphic intergrowths indicate high-viscosity, fluid-bearing and significantly undercooled magmatic conditions, whereas microscopic myrmekitic (plagioclase–quartz) and symplectitic (plagioclase–orthopyroxene) intergrowths on primary grain boundaries indicate replacement of phases in the presence of reactive fluids. In assessing the nature of these pegmatitic pods based on field, petrographic and geochemical evidence, we conclude that they represent the fluid-bearing, late-stage crystallization products of a residual liquid in the massif anorthosite system. The Fe and Si compositional variations observed in these late-stage pods can be linked to a fundamental olivine–pyroxene dichotomy observed in most Proterozoic anorthosite massifs, suggesting that pulses of magma experience variable contamination (in amount and/or composition) leading to varying differentiation paths. A range of lithologies (monzonites, monzonorites, ferrodiorites and jotunites) observed in similar pod-like structures, as well as dykes and plutons, has been observed in other Proterozoic anorthosite massifs and all have, at one time or another, been interpreted as the residual liquids of anorthosite crystallization. Our observation of in situ pods with similar compositions to all of the aforementioned lithologies, and displaying textures indicative of late-stage crystallization, supports the notion that all of these associated lithologies can be interpreted as comagmatic with, but variably contaminated and isolated residual liquids of, anorthosite crystallization. However, using isotopic evidence we cannot support the notion that the far larger granitic plutons associated with Proterozoic anorthosites are also residual liquids of anorthositic magma fractionation.

Description: We report the results of experiments on two natural marine sediments with different carbonate contents (calcareous clay: CO 2 = 6·1 wt %; marl: CO 2 = 16·2 wt %) at subduction-zone conditions (3 GPa, 750–1200°C). Water (7–15 wt %) was added to the starting materials to simulate the effects of external water addition from within the subducting slab. The onset of melting is at 760°C in water-rich experiments; melt becomes abundant by 800°C. In contrast, the onset of melting in published, water-poor experiments occurs at variable temperatures with the production of significant melt fractions being restricted to more than 900°C (phengite-out). The different solidus temperatures ( T solidus ) can be ascribed to variable fluid X H2O [H 2 O/(CO 2 + H 2 O)], which, in turn, depends on bulk K 2 O, H 2 O and CO 2 . Partial melts in equilibrium with residual garnet, carbonate, quartz/coesite, epidote, rutile, kyanite, phengite, and clinopyroxene are granitic in composition, with substantial dissolved volatiles. Supersolidus runs always contain both silicate melt and solute-rich fluid, indicating that experimental conditions lie below the second critical endpoint in the granite–H 2 O–CO 2 system. Carbonatite melt coexists with silicate melt and solute-rich fluid above 1100°C in the marl. The persistence of carbonate to high temperature, in equilibrium with CO 2 -rich hydrous melts, provides a mechanism to both supply CO 2 to arc magmas and recycle carbon into the deep Earth. The trace element compositions of the experimental glasses constrain the potential contribution of calcareous sediment to arc magmas. The presence of residual epidote and carbonate confers different trace element characteristics when compared with the trace element signal of Ca-poor marine sediments (e.g. pelagic clays). Notably, epidote retains Th and light rare earth elements, such that some melts derived from calcareous sediments have elevated Ba/Th and U/Th, and low La/Sm PUM , thereby resembling fluids conventionally ascribed to altered oceanic crust. Our results emphasize the importance of residual mineralogy, rather than source lithology, in controlling the trace element characteristics of slab-derived fluids.

Description: Three crystal-poor obsidian samples (one dacite, 67 wt % SiO 2 ; two rhyolites, 73 and 75 wt % SiO 2 ), which erupted effusively from monogenetic vents, contain sparse (〈2%) plagioclase phenocrysts that span a remarkably wide and continuous range in composition (≤30 mol % An). Many, but not all, of the plagioclase crystals display diffusion-limited growth textures (e.g. swallow-tails, skeletal, vermiform). Hypotheses to explain the paradox of a wide compositional range despite a low abundance of plagioclase include (1) incorporation of xenocrysts and/or magma mingling, (2) slow crystallization of plagioclase driven by slow cooling in a magma chamber, (3) slow crystallization of plagioclase followed by a resorption (e.g. heating) event, and (4) crystallization driven by rapid degassing (i.e. loss of melt H 2 O) ± rapid cooling during ascent. To test these hypotheses, a series of phase equilibrium experiments were conducted under pure-H 2 O fluid-saturated conditions in a cold-seal pressure vessel between 30 and 300 MPa and 750 and 950°C. The results show that the plagioclase population in each obsidian sample could have grown from their respective melts, with the exception of a single calcic core (An 60–63 ) in one sample. The results additionally rule out slow cooling in a magma chamber, because this would lead to equilibrium abundances of plagioclase (≤20%), which are far higher than what is observed in the samples (〈2%). Nor can resorption (i.e. heating) explain the low abundance of plagioclase, because this would eliminate the more sodic plagioclase crystals and hence the wide compositional range of plagioclase that is observed. The most viable hypothesis is that the sparse plagioclase phenocrysts grew relatively rapidly during magma ascent to the surface; this is consistent with the results of isothermal (850°C) continuous decompression experiments (2·9, 1·0, 0·8, and 0·1 MPa h –1 ), under pure-H 2 O fluid-saturated conditions, which were performed on one of the rhyolites (MLV-36; 73 wt % SiO 2 ) and quenched at P H2O = 89, 58 and 40 MPa. The four decompression rates correspond to degassing rates of 1·6, 0·56, 0·45 and 0·06 wt % H 2 O per day. Decompressions ≥1·0 MPa( P H2O ) h –1 , initiated above the liquidus, quenched to 100% glass at all final P H2O . Decompressions at 0·8 MPa( P H2O ) h –1 , also initiated above the liquidus, led to plagioclase crystals nearly five times larger than those grown in runs decompressed at the same rate, but initiated just below the plagioclase-in curve. It is the kinetic hindrance to nucleation that permits crystal growth to be concentrated on relatively few crystals, leading to larger crystals. Plagioclase growth rates from these experiments show that the largest phenocrysts (~1 mm) in the MLV-36 obsidian could have grown in 〈42 h. A cooling rate of ~1·2°C h –1 closely matches both the increase in melt viscosity with time and the effective undercooling with time that occurs during the 0·8 MPa( P H2O ) h –1 decompression over the first 50 h. The combined results point to crystallization of sparse plagioclase driven by relatively rapid rates of degassing ± cooling during ascent to the surface of melts that were initially above their liquidus. The obsidian samples must have been efficiently segregated as nearly 100% liquids from their respective source regions at H 2 O-fluid undersaturated conditions to attain a degree of superheating upon ascent before reaching fluid saturation.

Description: The Jurassic Vestfjella dyke swarm at the volcanic rifted margin of western Dronning Maud Land represents magmatism related to the incipient Africa–Antarctica rift zone; that is, rift-assemblage magmatism of the Karoo continental flood basalt (CFB) province. Geochemical and Nd–Sr isotopic data for basaltic and picritic dyke samples indicate diverse low-Ti and high-Ti tholeiitic compositions with Nd (180 Ma) ranging from +8 to –17. Combined with previously reported data on a subcategory of ferropicritic dykes, our new data facilitate grouping of the Vestfjella dyke swarm into seven geochemically distinct types. The majority of the dykes exhibit geochemical affinity to continental lithosphere and can be correlated with two previously identified chemical types (CT) of the wall-rock CFB lavas and are accordingly referred to as the CT1 and CT3 dykes. The less abundant Low-Nb and High-Nb dykes, a relatively enriched subtype of CT3 (CT3-E) dykes, and dykes belonging to the depleted and enriched ferropicrite suites represent magma types found only as intrusions. The chemically mid-ocean ridge basalt (MORB)-like Low-Nb and the depleted ferropicrite suite dykes represent, respectively, relatively high- and low-degree partial melting of the same overall depleted mantle (DM)-affinity source in the sublithospheric mantle. In contrast, we ascribe the chemically ocean island basalt (OIB)-like High-Nb dykes and the enriched ferropicrite suite dykes to melting of enriched components in the sublithospheric mantle. Geochemical modelling suggests that the low-Ti affinity CT1 and CT3, and high-Ti affinity CT3-E magma types of Vestfjella dyke may predominantly result from mixing of DM-sourced Low-Nb type magmas with 〈10 wt % of crust- and lithospheric mantle-derived melts. U/Pb zircon dating confirms synchronous emplacement of CT1 dykes and Karoo main-stage CFBs at 182·2 ± 0·9 and 182·2 ± 0·8 Ma, whereas two 40 Ar/ 39 Ar plagioclase plateau ages of 189·2 ± 2·3 Ma (CT1) and 185·5 ± 1·8 Ma (depleted ferropicrite suite), and a mini-plateau age of 186·9 ± 2·8 Ma (CT3-E) for the Vestfjella dykes raise the question of whether the onset of rift-zone magmatism could predate the province-wide c. 179–183 Ma main stage of Karoo magmatism. Notably variable Ca/K spectra suggest that younger 40 Ar/ 39 Ar plagioclase plateau ages of 173, 170, 164, and 154 Ma are related to crystallization of secondary minerals during the late-stage tectono-magmatic development of the Antarctic rifted margin. The occurrence of rare MORB- and OIB-like magma types in Vestfjella and along the African and Antarctic rifted margins suggests melting of geochemically variable depleted and enriched sublithospheric mantle beneath the Africa–Antarctica rift zone. Our models for the Vestfjella dyke swarm indicate that the voluminous lithosphere-affinity low-Ti and high-Ti rift-assemblage tholeiites could have been derived from MORB-like parental magmas by contamination, which implies sublithospheric depleted mantle as the principal source of the CFB magmas of the Africa–Antarctica rift zone.

Description: Monazite laser ablation–split-stream inductively coupled plasma–mass spectrometry (LASS) was used to date monazite in situ in Barrovian-type micaschists of the Moravian zone in the Thaya window, Bohemian Massif. Petrography and garnet zoning combined with pseudosection modelling show that rocks from staurolite–chlorite, staurolite, kyanite and kyanite–sillimanite zones record burial in the S 1 fabric under a moderate geothermal gradient from 4–4·5 kbar and ~530–540°C to 5 kbar and 570°C, 6–7 kbar and 600–640°C, 7·5–8 kbar and 630–650°C, and 8 kbar and 650°C, respectively. In the kyanite and kyanite–sillimanite zones, garnet rim chemistry and local syntectonic replacement of garnet by sillimanite–biotite aggregates point to re-equilibration at 5·5–6 kbar and 630–650°C in the S 2 fabric. Heterogeneously developed retrograde shear zones (S 3 ) are marked by widespread chloritization, but minor chlorite is present in the studied samples. Monazite abundance and size increase with metamorphic grade from 5 µm in the staurolite–chlorite zone to 〉100 µm in the kyanite and kyanite–sillimanite zones. Irrespective of the monazite-forming reaction, this is interpreted as the onset of limited prograde monazite growth at staurolite grade, and continued prograde monazite growth after the kyanite-in reaction, compatible with conditions of about 5·5 kbar and 570°C and 7·5 kbar and 630°C from pseudosection modelling. Monazite is zoned, showing embayments and sharp boundaries between zones, with low Y in the staurolite zone, high-Y cores and low-Y rims in the kyanite zone, and high-Y cores, a low-Y mantle and a high-Y rim in the sillimanite zone. The 207 Pb-corrected 238 U/ 206 Pb ages from three samples range from 344 ± 7 to 330 ± 7 Ma, irrespective of metamorphic grade. The dates from monazite inclusions are interpreted as the ages of the staurolite- and kyanite-in reactions along the prograde path at 340 and 337 ± 7 Ma, respectively. The monazite in the matrix (and some inclusions) is interpreted as dating the prograde crystallization at (340–337) ± 7 Ma within the S 1 fabric, and then being affected by recrystallization at or down to 332 ± 7 Ma in the S 2 and S 3 fabrics. The two groups of data, for 340–337 and 332 Ma, are significantly different when only their in-run uncertainties (±1–3 Myr) are compared and indicate a 9 ± 3 Myr period of monazite (re)crystallization. A systematic increase in heavy rare earth element (HREE) content with decreasing monazite age from 344 to 335 Ma is correlated with growth on the prograde P–T path; the drop in HREE of monazite at 335–328 Ma is assigned to recrystallization. The presence of chlorite even in the least retrogressed samples witnesses limited external fluid availability on the retrograde P–T path. Migration of this fluid was probably responsible for heterogeneous fluid-assisted recrystallization and resetting of original prograde monazite, even where included in garnet, staurolite or kyanite. It is suggested that the rocks passed the chlorite-in reaction on the retrograde path at 332 ± 7 Ma. The timing of burial in the Thaya window, a deformed part of the underthrust Brunia microcontinent, was coeval with exhumation of granulites and migmatites of the Moldanubian orogenic root at c. 340 Ma.

Description: The Fanshan intrusion in the North China Craton (NCC) is concentrically zoned with syenite in the core (Unit 1), surrounded by ultramafic rocks (clinopyroxenite and biotite clinopyroxenite; Unit 2), and an outer rim of garnet-rich clinopyroxenite and orthoclase clinopyroxenite and syenite (Unit 3). The intrusive rocks are composed of variable amounts of Ca-rich augite, biotite, orthoclase, melanite, garnet, magnetite and apatite, with minor primary calcite. Monomineralic apatite rocks, nelsonite and glimmerite exclusively occur in Unit 2. Geochemically, the Fanshan rocks are highly enriched in light rare earth elements (LREE) and large ion lithophile elements (LILE), moderately depleted in high field strength elements (HFSE), and have a limited range of Sr–Nd–O isotopic compositions. The similar mineralogy, mineral compositions, and trace element characteristics of the three units suggest that all the rocks are co-magmatic. The parental magma is ultrapotassic and is akin to kamafugite. Very low-degree partial melting of metasomatized lithospheric mantle best explains the geochemistry and petrogenesis of the parental magmas of the Fanshan intrusion. We propose that the mantle source may have been metasomatized by a hydrous carbonate-bearing melt, which has imprinted the enriched Sr–Nd isotopic signature and incompatible element enrichment with conspicuous negative Nb–Ta–Zr–Hf–Ti anomalies and LREE enrichments. The mantle source enrichment may be correlated with oceanic sediment recycling during southward subduction of the Paleo-Asian oceanic plate during the Carboniferous and Permian. We propose that crystal settling and mechanical sorting combined with repeated primitive magma replenishment and mixing with previously fractionated magma is the predominant process responsible for the formation of the apatite ores.

Description: Experimental studies of mantle petrology find that small concentrations of water and carbon dioxide have a large effect on the solidus temperature and distribution of melting in the upper mantle. However, it has remained unclear what effect small fractions of deep, volatile-rich melts have on melt transport and reactive melting in the shallow asthenosphere. Here we present theory and computations indicating that low-degree, reactive, volatile-rich melts cause channelization of magmatic flow at depths approximately corresponding to the anhydrous solidus temperature. These results are obtained with a novel method to simulate the thermochemical evolution of the upper mantle in the presence of volatiles. The method uses a thermodynamically consistent framework for reactive, disequilibrium, multi-component melting. It is coupled with a system of equations representing conservation of mass, momentum, and energy for a partially molten grain aggregate. Application of this method in two-phase, three-component upwelling-column models demonstrates that it reproduces leading-order features of hydrated and carbonated peridotite melting; in particular, it captures the production of low-degree, volatile-rich melt at depths far below the volatile-free solidus. The models predict that segregation of volatile-rich, deep melts promotes a reactive channelling instability that creates fast and chemically isolated pathways of melt extraction. Reactive channelling occurs where volatile-rich melts flux the base of the silicate melting region, enhancing dissolution of fusible components from the ambient mantle. We find this effect to be similarly expressed for models of both hydrated and carbonated mantle melting. These findings indicate that despite their small concentrations, water and carbon dioxide have an important control on the extent and style of magma genesis, as well as on the dynamics of melt transport.

Description: A study of whole-rock major and trace element and Sr–Nd–Pb–Hf isotope compositions, combined with zircon U–Pb ages and Hf–O isotopes, for postcollisional intermediate volcanic rocks from the Dabie orogen, China provides constraints on the origin of andesitic magmas and insights into slab–mantle interaction in continental subduction channels. The volcanic rocks exhibit variable contents of SiO 2 (50·28–63·86 wt %), MgO (1·18–4·65 wt %), (Fe 2 O 3 ) T (3·60–8·53 wt %), Al 2 O 3 (12·92–18·95 wt %), Na 2 O (2·08–6·30 wt %) and K 2 O (0·73–5·25 wt %). They are mainly trachyandesites, with lesser amounts of basaltic trachyandesite, andesite, dacite and trachyte, characterized by subduction-like trace element distribution patterns showing enrichment of large ion lithophile elements and light rare earth elements but depletion of high field strength elements. The volcanic rocks have relatively enriched Sr–Nd–Pb–Hf isotope compositions, with high initial 87 Sr/ 86 Sr ratios of 0·7075–0·7110, highly negative Nd ( t ) values of –23·1 to –15·0, Hf ( t ) values of –29·8 to –18·3 and elevated 207 Pb/ 204 Pb and 208 Pb/ 204 Pb ratios at given 206 Pb/ 204 Pb ratios. Zircon Hf–O isotope analyses yield negative Hf ( t ) values of –31·0 to –17·8 and 18 O values of 4·4–6·8 for syn-magmatic domains. Zircon U–Pb dating yields consistent Early Cretaceous ages of 124 ± 3 to 130 ± 2 Ma for magma emplacement. Residual zircon cores yield Triassic, Neoproterozoic and Paleoproterozoic U–Pb ages, consistent with the ages of tectonothermal events involving ultrahigh-pressure metamorphism and protolith formation in the Dabie orogen. The zircon cores also yield a few low to negative 18 O values. An integrated interpretation of these geochemical characteristics is that the andesitic magmas were derived by partial melting of metasomatized zones in the orogenic lithospheric mantle. The metasomatites were generated by reaction of subcontinental lithospheric mantle wedge peridotite with felsic melts that originated from deeply subducted continental crust during continental collision in the Triassic. Melt–peridotite reaction in a subduction channel is therefore a key to the origin of the mantle sources of andesitic magmas in collisional orogens.

Description: The late Proterozoic Ntaka Ultramafic Complex is a body of dominantly pyroxenitic cumulate rocks containing cyclic alternations of olivine–orthopyroxene cumulates. Chemical zoning in the pyroxenes has been imaged at 25–40 µm resolution using desktop microbeam X-ray fluorescence mapping followed up with laser ablation–inductively coupled plasma mass spectrometry analysis for minor and trace elements on selected samples. Poikilitic and granular harzburgites are finely intermingled, in some cases on a centimetre scale in the same thin section. Poikilitic varieties display spectacular textures, ranging from isolated equant orthopyroxene oikocrysts within olivine-rich heteradcumulate harzburgites to rocks composed entirely of interlocking centimetre-sized anhedral orthopyroxene oikocrysts containing sharply bounded idiomorphic Cr-enriched cores. The poikilitic harzburgites are interlayered with cumulate pyroxenites in which orthopyroxene grains show a variety of zoning patterns: Cr-rich cores similar to those in the oikocrysts; sharply bounded oscillatory zoned cores; and reverse zoning with Cr-poor cores and Cr-enriched rims. A further variation is the presence of a mingled harzburgite lithology in which dunite or poikilitic harzburgite is invaded on a centimetre scale by diffuse vein networks or patches of coarse orthopyroxenite. This range of textures and lithologies attests to a more complex set of processes than implied by the standard cumulus theory model in which oikocrysts are considered to have crystallized from intercumulus liquid within a permeable crystal mush. A range of hypotheses is proposed, including infiltration metasomatism of original olivine cumulates by migrating orthopyroxene-saturated pore fluid; however, the textural relationships, whole-rock chemistry and Cr zoning within the grains can best be explained by a model in which the orthopyroxene oikocrysts form in part or whole as mechanically accumulated cumulus grains. The complexity of zoning patterns is attributed to stirring of entrained olivine and orthopyroxene crystals within a heterogeneous flowing crystal mush, where the transporting magma has a wide range of silica contents owing to poorly stirred incorporation of siliceous country-rock material. The Cr-rich orthopyroxenite component grew from Si-enriched chromite-saturated magma. Mingled lithologies developed after accumulation as a result of percolation and infiltration metasomatism by Si-enriched liquid derived by melting of xenoliths within the crystal pile. The model may be more generally applicable: dunite–harzburgite cycles, common in many layered intrusions, may reflect variable degrees of contamination rather than cycles of fractional crystallization and replenishment.

Description: New thermodynamic data for skiagite garnet (Fe 3 Fe 2 3+ Si 3 O 12 ) are derived from experimental phase-equilibrium data that extend to 10 GPa and are applied to oxybarometry of mantle peridotites using a revised six-component garnet mixing model. Skiagite is more stable by 12 kJ mol –1 than in a previous calibration of the equilibrium 2 skiagite = 4 fayalite + ferrosilite + O 2 , and this leads to calculated oxygen fugacities that are higher (more oxidized) by around 1–1·5 logfO2 units. A new calculation method and computer program incorporates four independent oxybarometers (including 2 pyrope + 2 andradite + 2 ferrosilite = 2 grossular + 4 fayalite + 3 enstatite + O 2 ) for use on natural peridotite samples to yield optimum logfO2 estimates by the method of least squares. These estimates should be more robust than those based on any single barometer and allow assessment of possible disequilibrium in assemblages. A new set of independent oxybarometers for spinel-bearing peridotites is also presented here, including a new reaction 2 magnetite + 3 enstatite = 3 fayalite + 3 forsterite + O 2 . These recalibrations combined with internally consistent PT determinations for published analyses of mantle peridotites with analysed Fe 2 O 3 data for garnets, from both cratonic (Kaapvaal, Siberia and Slave) and circumcratonic (Baikal Rift) regions, provide revised estimates of oxidation state in the lithospheric mantle. Estimates of logfO2 for spinel assemblages are more reduced than those based on earlier calibrations, whereas garnet-bearing assemblages are more oxidized. Importantly, this lessens considerably the difference between garnet and spinel oxybarometry that was observed with previous published calibrations.

Description: The Kidnappers [~1200 km 3 dense rock equivalent (DRE)] and Rocky Hill (~200 km 3 DRE) caldera-forming events in the Taupo Volcanic Zone, New Zealand, were erupted in close succession from the Mangakino volcanic centre. They have identical radiometric ages at ~1 Ma, yet erosion along the contact between the two deposits suggests that some years to decades separated the two eruptions. Field constraints and the similarities of crystal textures and compositions and glass chemistries of both eruption deposits demonstrate that they came from one overall magmatic system with a common crystal mush source. However, second-order variations in these parameters confirm that the Kidnappers and Rocky Hill deposits represent distinct events and are not the products of a single zoned magma chamber. The systematically zoned Kidnappers fall deposits provide evidence for the tapping of three discrete magma bodies, whereas the succeeding Kidnappers ignimbrite is compositionally more diverse. The transition from fall to flow deposition marks a change in the style of caldera collapse and the simultaneous evacuation of discrete but compositionally diverse melts, each of which underwent a distinct evolution and was held at slightly different P–T conditions prior to eruption. Contrasting plagioclase and orthopyroxene zonation patterns are present in pumices originating from three discrete magma bodies. Less evolved mafic melts interacted with the system, which mobilized portions of the final erupted melt through heating and volatile or chemical exchange in the mush. The two largest Kidnappers melt-dominant bodies were re-tapped in modified form, or re-established from their common mush source, prior to the Rocky Hill event. Rocky Hill pumices contain common, fluid-affected antecrystic crystal clots derived from chamber wall material. Amphibole compositions from each eruption reflect melt evolution processes and, in particular, the contemporaneous crystallization of biotite and breakdown of orthopyroxene. Plagioclase and orthopyroxene from Rocky Hill pumices share common zonation patterns with those from the two largest magma bodies in the Kidnappers. The rapid production of new melt-dominant bodies and the triggering of the Rocky Hill eruption reflect the ability of the magmatic system to rejuvenate on a geologically short timescale. The Mangakino centre did not follow a typical cycle of decreased activity after the supervolcanic Kidnappers event, instead producing a second caldera-forming eruption, within years to decades from the same system.

Description: Ambrym, a basaltic volcano in the Vanuatu Arc, has displayed variable eruptive behaviour throughout the past century, with major eruptions occurring both on the volcano flanks and at multiple vents within its caldera. These have been interspersed with periods of relative quiescence marked by extensive passive degassing at active, intra-caldera lava lakes, which experience occasional Strombolian explosions. Volcanic rocks from all vents and eruptive styles display similar isotope and incompatible trace element compositions, suggesting that all are derived from the same primary melt by fractional crystallization. Major eruptions are commonly responsible for effusion of the least evolved lavas examined (SiO 2 ~ 50 wt %; MgO ~ 5 wt %). Although all are geochemically similar, petrological differences discriminate between lavas erupted during flank and intra-caldera eruptions. Phyric basalts with homogeneous mineral compositions are common to flank eruptions, whereas crystal-rich basalts with variable mineral compositions, many not in equilibrium with their host liquid, are a feature of intra-caldera lavas. Lava lake samples are slightly more evolved than those from effusive eruptions (SiO 2 ~ 51–52 wt %; MgO ~ 4 wt %), as a result of additional crystallization during periods of relative quiescence. The diverse petrology of the intra-caldera lavas can be explained by mixing of replenishment magmas similar to those erupted from the volcano flanks with residual magma from lava lake activity. Flank eruptions exploit dykes that bypass the shallow reservoir involved with lava lake activity, limiting their interaction with this component.

Description: The generation and evolution of basaltic magmas at Usu volcano, located at the junction between the NE Japan arc and the Kuril arc, have been investigated. The mafic products, which form the somma edifice of the volcano, consist of basalt (49·6–51·3 wt % SiO 2 ) and basaltic andesite (52·0–54·9 wt % SiO 2 ) lavas. The basaltic lavas show relatively tight compositional trends, and 87 Sr/ 86 Sr ratios tend to decrease with increasing whole-rock SiO 2 content. The water content of the basaltic magmas was determined to be ~4·8 wt % based on plagioclase–melt thermodynamic equilibrium. Using this information and an olivine maximum fractionation model, the water content of the primary Usu magma was estimated to be 3·9 wt %. Multi-component thermodynamic calculations suggest that the primary magma was generated by ~23% melting of the source mantle with ~0·94 wt % H 2 O at ~1300°C and ~1·4 GPa. The 0·94 wt % water content of the source mantle is significantly higher than that beneath volcanoes in the main NE Japan arc (generally 〈0·7 wt % H 2 O); this implies that the wedge mantle at the arc–arc junction is intensively hydrated. The temperature of the wedge mantle of ~1300°C at ~1·4 GPa is also significantly higher than that of the mantle in the main NE Japan arc. Unlike the basaltic lavas, the whole-rock compositions of the basaltic andesite lavas are scattered in Harker variation diagrams. This observation suggests that the compositional diversity was produced by at least two independent processes. To elucidate the processes responsible for this compositional diversity, principal component analysis was applied to the major element compositions of the samples. This suggests that 47% of the diversity of the whole-rock compositions can be explained by mixing with partial melts of lower crustal materials, 25% is explained by redistribution of plagioclase phenocrysts, and 16% is explained by fractionation of accessory minerals.

Description: A formula for the norm of a bilinear Schur multiplier acting from the Cartesian product $\mathcal S^2\times \mathcal S^2$ of two copies of the Hilbert–Schmidt classes into the trace class $\mathcal S^1$ is established in terms of linear Schur multipliers acting on the space $\mathcal S^\infty $ of all compact operators. Using this formula, we resolve Peller's problem on Koplienko–Neidhardt trace formulae. Namely, we prove that there exist a twice continuously differentiable function $f$ with a bounded second derivative, a self-adjoint (unbounded) operator $A$ and a self-adjoint operator $B\in \mathcal S^2$ such that \[ f(A+B)-f(A)-\left.\frac{d}{dt}(f(A+tB))\right\vert_{t=0}\notin \mathcal S^1. \]

Description: Let $\mu $ be a probability measure on $ \mathbb R^n$ with a bounded density $f$ . We prove that the marginals of $f$ on most subspaces are well-bounded. For product measures, studied recently by Rudelson and Vershynin, our results show there is a trade-off between the strength of such bounds and the probability with which they hold. Our proof rests on new affinely invariant extremal inequalities for certain averages of $f$ on the Grassmannian and affine Grassmannian. These are motivated by Lutwak's dual affine quermassintegrals for convex sets. We show that key invariance properties of the latter, due to Grinberg, extend to families of functions. The inequalities we obtain can be viewed as functional analogues of results due to Busemann–Straus, Grinberg and Schneider. As an application, we show that without any additional assumptions on $\mu $ , any marginal $\pi _E(\mu )$ , or a small perturbation thereof, satisfies a nearly optimal small-ball probability.

Description: Let $\pi :X\to \mathbb {P}^1_{\mathbb {Q}}$ be a non-singular conic bundle over $\mathbb {Q}$ having $n$ non-split fibres and denote by $N(\pi ,B)$ the cardinality of the fibres of Weil height at most $B$ that possess a rational point. Serre showed in 1990 that a direct application of the large sieve yields \[ N(\pi,B)\ll B^2(\log B)^{-n/2} \] and raised the problem of proving that this is the true order of magnitude of $N(\pi ,B)$ under the necessary assumption that there exists at least one smooth fibre with a rational point. We solve this problem for all non-singular conic bundles of rank at most 3. Our method comprises the use of Hooley neutralisers, estimating divisor sums over values of binary forms, and an application of the Rosser–Iwaniec sieve.

Description: We associate a dimer algebra $A$ to a Postnikov diagram $D$ (in a disc) corresponding to a cluster of minors in the cluster structure of the Grassmannian ${\rm Gr}(k,n)$ . We show that $A$ is isomorphic to the endomorphism algebra of a corresponding Cohen–Macaulay module $T$ over the algebra $B$ used to categorify the cluster structure of ${\rm Gr}(k,n)$ by Jensen–King–Su. It follows that $B$ can be realised as the boundary algebra of $A$ , that is, the subalgebra $eAe$ for an idempotent $e$ corresponding to the boundary of the disc. The construction and proof uses an interpretation of the diagram $D$ , with its associated plabic graph and dual quiver (with faces), as a dimer model with boundary. We also discuss the general surface case, in particular computing boundary algebras associated to the annulus.

Description: We describe a ring whose category of Cohen–Macaulay modules provides an additive categorification of the cluster algebra structure on the homogeneous coordinate ring of the Grassmannian of $k$ -planes in $n$ -space. More precisely, there is a cluster character defined on the category which maps the rigid indecomposable objects to the cluster variables and the maximal rigid objects to clusters. This is proved by showing that the quotient of this category by a single projective–injective object is Geiss–Leclerc–Schröer's category Sub $Q_k$ , which categorifies the coordinate ring of the big cell in this Grassmannian.

Description: We develop a way of seeing a complete orientable hyperbolic 4-manifold $ {\mathcal {M}}$ as an orbifold cover of a Coxeter polytope $ {\mathcal {P}} \subset \mathbb {H}^4$ that has a facet colouring. We also develop a way of finding a totally geodesic sub-manifold $ {\mathcal {N}}$ in $ {\mathcal {M}}$ , and describing the result of mutations along $ {\mathcal {N}}$ . As an application of our method, we construct an example of a complete orientable hyperbolic 4-manifold $ {\mathcal {X}}$ with a single non-toric cusp and a complete orientable hyperbolic 4-manifold ${\mathcal {Y}}$ with a single toric cusp. Both $ {\mathcal {X}}$ and $ {\mathcal {Y}}$ have twice the minimal volume among all complete orientable hyperbolic 4-manifolds.

Description: The covariogram $g_{K}$ of a convex body $K$ in $ \mathbb {R}^n$ is the function that associates to each $x\in \mathbb {R}^n$ the volume of the intersection of $K$ with $K+x$ . Determining $K$ from the knowledge of $g_K$ is known as the Covariogram Problem. It is equivalent to determining the characteristic function $1_K$ of $K$ from the modulus of its Fourier transform $\widehat {{1_K}}$ in $ \mathbb {R}^n$ , a particular instance of the Phase Retrieval Problem. We connect the Covariogram Problem to two aspects of the Fourier transform $\widehat {{1_K}}$ seen as a function in $\mathbb {C}^n$ . The first connection is with the problem of determining $K$ from the knowledge of the zero set of $\widehat {{1_K}}$ in $\mathbb {C}^n$ . To attack this problem T. Kobayashi studied the asymptotic behavior at infinity of this zero set. We obtain this asymptotic behavior assuming less regularity on $K$ and we use this result as an essential ingredient for proving that when $K$ is sufficiently smooth and in any dimension $n$ , $K$ is determined by $g_K$ in the class of sufficiently smooth bodies. The second connection is with the irreducibility of the entire function $\widehat {{1_K}}$ . This connection also shows a link between the Covariogram Problem and the Pompeiu Problem in integral geometry.

Description: Numerous models have been developed to simulate the reaction of magmas to changes of thermodynamic variables, such as pressure, temperature, oxygen fugacity, and water activity. However, the extensive experimental database still lacks information on the distinct effect of small amounts of H 2 O on olivine + plagioclase + clinopyroxene cotectic crystallization in tholeiitic basalt. We present an experimental study addressing the effects of pressure (at 100, 200, 400, and 700 MPa) and small amounts of H 2 O on phase relations and liquid lines of descent in three tholeiitic basalts representing different evolutionary stages of the Shatsky Rise oceanic plateau magmatic system (compositions AH6, AH3, and AH5 with 8·6, 8·0, and 6·4 wt % MgO, respectively). Two experimental approaches (dry and low H 2 O) are designed to maintain contrasting H 2 O activities during crystallization using (1) graphite–platinum double capsules to perform nearly anhydrous experiments (〈0·15 wt % H 2 O in the melt) and (2) Fe pre-saturated Au 20 Pd 80 capsules to obtain low melt H 2 O contents ranging from 0·4 to 1·1 wt % H 2 O. Under dry conditions, at lower pressures (≤400 MPa), the crystallization in the MgO-rich AH6 and intermediate AH3 basalts follows the typical sequence of tholeiitic differentiation with olivine crystallization at the liquidus followed by olivine + plagioclase and olivine + plagioclase + clinopyroxene. Both basalts are close to multiple saturation at pressures between 400 and 700 MPa. At high pressure (700 MPa) the crystallization sequence is reversed, starting with clinopyroxene at the liquidus. Under low-H 2 O conditions, AH6 and AH3 are very close to multiple saturation, even at the low pressures of 100 and 200 MPa, and the reversed crystallization sequence (clinopyroxene, plagioclase + clinopyroxene, olivine + plagioclase + clinopyroxene) is observed already at 400 MPa. In contrast to the two more MgO-rich basalts, in the most evolved AH5 basalt, clinopyroxene is the liquidus phase at all investigated pressures and under both dry and low-H 2 O conditions, followed by crystallization of plagioclase + clinopyroxene and olivine + plagioclase + clinopyroxene. The most striking observation in our experiments is that the stability of clinopyroxene increases not only with pressure increase but also in the presence of small amounts of H 2 O (when compared with dry counterparts at similar pressures). Small amounts of H 2 O increase the proportion of clinopyroxene in the olivine + plagioclase + clinopyroxene phase assemblage. Our experiments clearly show that the effect of adding 0·4 wt % H 2 O to cotectic melt compositions (e.g. CaO/Al 2 O 3 ratio at a given MgO) is similar to that caused by an increase of pressure from 100 to ~ 300 MPa. This implies that small amounts of H 2 O can lead to significant overestimation of cotectic crystallization pressures (by up to 300 MPa) and that H 2 O contents need to be taken into account in geobarometric models. Our new experiments emphasize the role of low melt H 2 O contents in stabilizing clinopyroxene and provide some new insights into the problem of the ‘pyroxene paradox’. The apparent mantle pressures obtained for some mid-ocean ridge basalts using ‘dry’ geobarometric approaches can actually represent depths within the lower crust, if small amounts of H 2 O are present. The application of our experimental data to natural Shatsky Rise basalts implies that the magmas record partial crystallization processes occurring mainly at low pressure (100 MPa), corresponding to depths of ~3 km beneath the former spreading center, although the more primitive lavas show evidence of differentiation in a deeper reservoir at ~14 km depth (400 MPa).

Description: Mantle-derived xenoliths hosted by melilitite lavas from In Teria (Ahaggar, SE Algeria) include garnet and spinel peridotites, pyroxenite and phlogopite megacrysts. The spinel and garnet peridotites record an early deformation event, which formed porphyroclastic microstructures and olivine crystal preferred orientations, followed by static infiltration of hydrous alkaline melts. This metasomatic stage (stage 1) is characterized by the crystallization of phlogopite in the garnet and spinel peridotites, amphibole in the spinel peridotites and clinopyroxene in the garnet peridotite, which record chemical equilibration with an alkaline silicate melt. These early events were largely overprinted by carbonatitic metasomatism (stage 2), which is observed only in the spinel peridotites. Spinel peridotite major and trace element compositions, as well as the compositions of newly formed minerals, are characteristic of interaction with carbonate melt, associated with strong enrichment in incompatible trace elements in clinopyroxene. This second stage was followed by crystallization of pyroxenites (stage 3) in vein conduits, probably segregated from alkaline melts. We propose a scenario in which the different metasomatic imprints record successive stages of interaction between lithospheric mantle and sublithospheric melts throughout the Cenozoic. In Sr–Nd isotope space, the host melilitites and several xenoliths are clustered and plot close to the HIMU mantle end-member. However, some peridotite xenoliths are shifted towards more radiogenic 87 Sr/ 86 Sr values. In 207 Pb/ 204 Pb– 206 Pb/ 204 Pb and 208 Pb/ 204 Pb– 06 Pb/ 204 Pb space the In Teria samples define a relatively large domain characterized by high 206 Pb/ 204 Pb and 208 Pb/ 204 Pb, consistent with a contribution of an HIMU component, considered to represent a sublithospheric signature. The highest 87 Sr/ 86 Sr values are comparable with those ascribed to the EM1 mantle end-member, representing the signature of the lower continental lithosphere, and are probably inherited from the pre-metasomatic lithospheric mantle beneath In Teria. Numerical modelling of porous percolation of melt of sublithospheric origin through an EM1-like lithospheric mantle protolith reproduces the In Teria peridotite compositions, using moderately sub-chondritic Sr/Nd values for the peridotite (e.g. In Teria garnet peridotite) and moderately super-chondritic Sr/Nd values in the melt (approximately ocean island basalt values). A few spinel peridotites require a component characterized by a 143 Nd/ 144 Nd signature higher than both the EM1 end-member and the local Ahaggar basalts; the 208 Pb/ 204 Pb compositions of several samples point to a component with a depleted mid-ocean ridge basalt (MORB) mantle (DMM) signature. Thus the lithospheric mantle beneath In Teria probably did not have a uniform EM1 signature before the onset of metasomatism; it included a DMM peridotite component as well as some peridotites with elevated 143 Nd/ 144 Nd values recording long-term LREE depletion.

Description: Modelled primary magma compositions of Palaeogene basalts from the North Atlantic Igneous Province (NAIP) require melting at mantle potential temperatures ( T P ) in the range 1480–1550°C. Modern lavas from Icelandic rift zones require T P ~ 1500°C and those from the rift flanks T P ~ 1450°C. Secular cooling of the NAIP thermal anomaly was therefore of the order of ~50°C over the past 61 Myr. There were systematic variations in T P of 50–100°C from the centre of the thermal anomaly to its margins at any one time, although limits on the stratigraphical distribution of T P determinations do not rule out thermal pulsing on a timescale of millions of years. Variation in extent of melting at similar T P was controlled by local variability in lithospheric thickness. In the west of the NAIP, lithosphere thickness varied from ~90 km at Disko Island to ~65 km at Baffin Island, with similar thickness variations being evident for magmatism in the Faroe Islands, Faroe–Shetland Basin and the British Palaeogene Igneous Province (BPIP). Mean pressure of melting was greater than or equal to the final pressure of melting; the two values converge for melting columns with a melting interval of 〈1·5 GPa, regardless of T P . The majority of the BPIP magmas were generated in the garnet–spinel transition in the upper mantle. Calculated and observed rare earth element distributions in NAIP lavas are entirely consistent with the melting regimes derived from major element melting models. This allows a calibration of rare earth element fractionation and melting conditions that can be applied to other flood basalt provinces.

Description: Let $\varphi :X\to S$ be a morphism between smooth complex analytic spaces and let $f=0$ define a free divisor on $S$ . We prove that if the deformation space $T^1_{X/S}$ of $\varphi $ is a Cohen–Macaulay $\mathcal {O}_X$ -module of codimension 2, and all of the logarithmic vector fields for $f=0$ lift via $\varphi $ , then $f\circ \varphi =0$ defines a free divisor on $X$ ; this is generalized in several directions. Among applications we recover a result of Mond–van Straten, generalize a construction of Buchweitz–Conca, and show that a map $\varphi :\mathbb {C}^{n+1}\to \mathbb {C}^n$ with critical set of codimension 2 has a $T^1_{X/S}$ with the desired properties. Finally, if $X$ is a representation of a reductive complex algebraic group $G$ and $\varphi $ is the algebraic quotient $X\to S=X\!{/\!/} G$ with $X\!{/\!/} G$ smooth, then we describe sufficient conditions for $T^1_{X/S}$ to be Cohen–Macaulay of codimension 2. In one such case, a free divisor on $\mathbb {C}^{n+1}$ lifts under the operation of ‘castling’ to a free divisor on $\mathbb {C}^{n(n+1)}$ , partially generalizing work of Granger–Mond–Schulze on linear free divisors. We give several other examples of such representations.

Description: Let $M^n$ be a compact manifold of dimension $n$ with free $T^k$ -action. We consider collapsings of $M$ on $N=M/T^k$ such that the sectional curvature and diameter of $M$ satisfy $|K(M)|\leq a$ and $ {\rm diam}(M) 〈 d$ , and give examples of collapsings for all $k$ such that the first non-zero eigenvalue of Laplacian acting on 1-forms and 2-forms of $M$ are bounded above by $c(M)\cdot \hbox {inj}(M)^{2k}$ . Moreover, we prove that the first non-zero eigenvalue of Laplacian acting on 1-forms of all principal $T^k$ -bundle $M$ over $N$ is bounded below by $c(n,a,d,N)\cdot {\rm Vol}(M)^2$ and $c\cdot \hbox {inj}(M)^{2k}$ when $M$ collapses on $N$ .

Description: The dihedral angle formed at junctions between two plagioclase grains and a grain of augite is only very rarely in textural equilibrium in gabbros from kilometre-scale crustal layered intrusions. The median of a population of these disequilibrium angles, cpp , varies systematically within a single layered intrusion, remaining constant over large stretches of stratigraphy with significant increases and decreases associated with the addition or reduction respectively of the number of phases on the liquidus of the bulk magma. The stepwise changes in cpp are present in the Upper Zone of the Bushveld Complex, the Megacyclic Unit I of the Sept Iles Intrusion, and the Layered Series of the Skaergaard intrusion. The plagioclase-bearing cumulates of Rum have a bimodal distribution of cpp , dependent on whether the cumulus assemblage includes clinopyroxene. The presence of the stepwise changes is independent of the order of arrival of cumulus phases and of the composition of either the cumulus phases or the inferred composition of the interstitial liquid. The only parameter that behaves in an exactly analogous manner to cpp is the rate of change in enthalpy with temperature ( H / T ) during crystallization. Both H / T and cpp increase with the addition of a liquidus phase, and decrease with the removal of a liquidus phase. The replacement of one phase by another has little effect on H / T and no discernible effect on cpp . An increase of H / T results in an increase in the fraction of the total enthalpy budget that is the latent heat of crystallization (the fractional latent heat). It also increases the mass crystallized in each incremental temperature drop (the crystal productivity). These increases of both fractional latent heat and crystal productivity are likely to cause an increase in the time taken to form three-grain junctions in the mush via thermal buffering of a thickened mushy layer. We suggest these are the underlying causes of stepwise increases in cpp . Stepwise changes in the geometry of three-grain junctions in fully solidified gabbros thus provide a clear microstructural marker for the progress of fractionation down the liquid line of descent in layered intrusions.

Description: High-resolution sampling in monogenetic fields has the potential to reveal fine-scale heterogeneity of the mantle, a feature that may be overwhelmed by larger fluxes of magma, or missed by under-sampling. The Quaternary Auckland Volcanic Field (AVF) in northern New Zealand is a basaltic field of 51 small-volume volcanic centres, and is one of the best-sampled examples of a monogenetic volcanic field. We present data for 12 centres in the volcanic field. These show the large compositional variations between volcanoes as well as through single eruptive sequences. Whole-rock compositions range from subalkaline basalt in the larger centres, through alkali basalt to nephelinite in the smallest centres. Fractional crystallization has had a limited effect in many of the centres, but high-pressure clinopyroxene crystallization may have occurred in others. Three end-members are observed in Pb isotope space, indicating that distinct mantle source components are involved in the petrogenesis of the magmas. Whole-rock multi-element patterns show that the larger centres have prominent positive Sr anomalies and lack K anomalies, whereas the smaller centres have prominent negative K anomalies and lack Sr anomalies. The melting parameters and compositions of the sources involved are modelled using trace element ratios and multi-element patterns, and three components are characterized: (1) fertile peridotite with a Pb-isotope composition similar to Pacific mid-ocean ridge basalt; (2) eclogite domains with a HIMU-like isotope composition dispersed within the fertile peridotite; (3) slightly depleted subduction-metasomatized peridotitic lithospheric mantle (containing c . 3% subduction fluids). Modelling shows that melting in the AVF begins in garnet-bearing fertile asthenosphere (with preferential melting of eclogite domains) and that melts are variably diluted by melts of the lithospheric source. The U–Th isotope compositions of the end-members in the AVF show 230 Th excess [( 230 Th/ 232 Th) ratios of 1·11–1·38], with the samples of lower ( 230 Th/ 232 Th) exhibiting higher ( 238 U/ 232 Th), which we attribute to the dilution effect of the melts from the lithospheric mantle source. Modelling reveals a correlation between melting in the asthenosphere, the degree of melting and incorporation of the metasomatized lithospheric mantle source, and the resultant size of the volcanic centre. This suggests that the scale of the eruption may essentially be controlled by asthenospheric mantle dynamics.

Description: The origin of mafic and ultramafic sills exhibiting different whole-rock compositional profiles (e.g. I-, C-, D-, M- and S-shaped profiles) remains controversial. We have addressed this issue by revisiting three ~100 m thick Siberian dolerite sills (Vavukansky, Kuz’movsky and Vilyuysky) that display remarkable internal differentiation. The Vavukansky sill has an M-shaped profile with prominent basal and top reversals showing inward increases in whole-rock MgO, Mg-number [100Mg/(Mg + Fe)] and normative An content [100An/(An + Ab)], followed by the Layered and Upper Border Series with inward decreases in these indices. The Kuz’movsky and Vilyuysky sills both show S-shaped profiles similar to the Vavukansky sill, but lack a top reversal. These whole-rock M- and S-shaped profiles are accompanied by similar profiles in mineral compositions. Plagioclase and, to a lesser extent, olivine show systematic inward increases in An content and Mg-number, respectively, across basal and top reversals. These compositional trends are followed by inward decreases in these ratios in the interiors of the Vavukansky and Kuz’movsky sills. Currently accepted models attribute whole-rock M- and S-shaped compositional profiles to crystal settling, compositional convection or compaction operating in closed systems. Our observations challenge these traditional interpretations because variations in mineral compositions observed in marginal reversals cannot result from closed-system fractionation. We suggest instead that initially the sills evolved as open systems that were slowly inflated by magmas that became gradually more primitive with time. The inflation was accompanied by in situ crystallization that preserved the preceding fractionation history of the injected magmas by forming basal and top reversals with minerals becoming more primitive inwards. This process culminated with rapid inflation of the sills to their current size owing to a major influx of primitive magma. Subsequently, magma flow through the sills ceased and they evolved as closed systems by fractional crystallization. This resulted in the Layered and Upper Border Series with minerals becoming more evolved inwards. This model can be extended to explain other compositional profiles and petrological features in mafic and ultramafic sills.

Description: We give explicit atomic bases of arbitrary coefficient-free cluster algebras of types A and à . This entails showing that the minimal elements of the positive semiring of these cluster algebras form a linear basis over the integers for the cluster algebra.

Description: We prove that strongly F -regular and F -pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F -injective singularities fail to satisfy even the most basic Bertini-type results.

Description: This is the second of a pair of papers on the Delta-group structure on the braid and mapping class groups of a surface. We obtain a description of the homotopy groups of these Delta-groups and generalize to an arbitrary surface the Berrick–Cohen–Wong–Wu exact sequence relating the Brunnian braid groups of the 2-sphere to its homotopy groups. We prove a similar result for Brunnian mapping class groups.

Description: We construct a geometric realization of the Khovanov–Lauda–Rouquier algebra R associated with a symmetric Borcherds–Cartan matrix A = ( a ij ) i , j I via quiver varieties. As an application, if a ii != 0 for any i I , we prove that there exists a one-to-one correspondence between Kashiwara's lower global basis (or Lusztig's canonical basis) of U A – (g) (respectively, V A ( )) and the set of isomorphism classes of indecomposable projective graded modules over R (respectively, R ).

Description: The purpose of this paper is to study the nature of quasi-invariant measures for finitely generated non-discrete subgroups of Diff ( S 1 ). For this, we apply ideas involving the closure of these groups to find out that the regularity of the measure depends on a ‘measurable version’ of well-known problems concerning stable self-intersection of Cantor sets. As applications, we prove that every d -quasiconformal probability measure for a non-solvable and non-discrete group must be absolutely continuous. Concerning singular quasi-invariant measures, it is also proved that their associated Hausdorff measures must either be zero or of infinite mass, a result contrasting with the case of dynamically defined Cantor sets and also applicable to the examples of singular stationary measures constructed by Kaimanovich and Le Prince. As a further application of our methods, a theorem of rigidity for measurable conjugations between groups as above is obtained.

Description: We study the space of period polynomials associated with modular forms of integral weight for finite-index subgroups of the modular group. For the modular group, this space is endowed with a pairing, corresponding to the Petersson inner product on modular forms via a formula of Haberland, and with an action of Hecke operators, defined algebraically by Zagier. We generalize Haberland's formula to (not necessarily cuspidal) modular forms for finite-index subgroups, and we show that it conceals two stronger formulas. We extend the action of Hecke operators to period polynomials of modular forms, we show that the pairing on period polynomials appearing in Haberland's formula is nondegenerate, and we determine the adjoints of Hecke operators with respect to it. We give a few applications for 1 ( N ): an extension of the Eichler–Shimura isomorphism to the entire space of modular forms; the determination of the relations satisfied by the even and odd parts of period polynomials associated with cusp forms, which are independent of the period relations; and an explicit formula for Fourier coefficients of Hecke eigenforms in terms of their period polynomials, generalizing the Coefficient theorem of Manin.

Description: Magma dynamics and time scales during the VEI 5, 2000 bp eruption of El Misti volcano, southern Peru (EM2000BP) are investigated to address cyclic explosive activity at this hazardous volcano. The 1·4 km 3 of pumice falls and flows have abundant mingled pumice of high-K, calc-alkaline rhyolite and andesite composition. Phenocryst zoning and compositions reveal mutual exchange of plagioclase between the two magmas; amphibole in the rhyolite was derived from the andesite. Amphiboles in the andesite are predominantly unrimmed crystals whereas those in the rhyolite mostly exhibit reaction rims. Phase equilibria indicate that the andesite formed at ~900–950°C and 2–3 kbar pressure and was water-saturated with 5·1–6·0 wt % H 2 O, broadly similar to El Misti magmas overall. Amphibole, plagioclase, Ti-magnetite, and two pyroxenes were the crystallizing phases. A separate rhyolite magma existed higher in the crust at a temperature of 816 ± 30°C and ~5% H 2 O in which only plagioclase and Fe–Ti oxides were stable. The lack of cognate amphibole in the rhyolite despite H 2 O saturation requires that it staged above the stability limit of amphibole (〈100 MPa). Exchange reactions in amphibole (dominantly pargasitic) and trace element partitioning in plagioclase indicate that both andesite and rhyolite magmas were broadly constant in temperature and H 2 O content. These constraints suggest that the initially separate rhyolite and deeper andesite magmas interacted by an initial andesite recharge event that resulted in mingling and crystal exchange. A period of 50–60 days is required for amphibole introduced into the rhyolite to develop reaction rims owing to decompression. These rims are dominated by plagioclase, a consequence of the Al-rich nature of the amphibole. The lack of reaction rims on amphibole in the andesite implicates a second, more-forceful and voluminous eruption-triggering recharge event during which andesite rose rapidly from source to surface in ≤5 days at ascent rates of at least 0·023 m s –1 . Further decompression-driven crystallization is recorded in plagioclase rims and microlite growth that may have contributed to a rapid increase in viscosity leading to explosive eruption. This VEI 5 plinian eruption shares characteristics with other explosive events at El Misti on a time scale of 2000–4000 years, suggesting periodic recharge-driven explosive activity.

Description: We report structural evidence of ductile strain localization in mantle pyroxenite from the spinel to plagioclase websterite transition in the Ronda Peridotite (southern Spain). Mapping shows that, in this domain, small-scale shear zones occurring at the base of the lithospheric section are systematically located within thin pyroxenite layers, suggesting that the pyroxenite was locally weaker than the host peridotite. Strain localization is associated with a sudden decrease of grain size and increasing volume fractions of plagioclase and amphibole as a result of a spinel to plagioclase phase transformation reaction during decompression. This reaction also fostered hydrogen extraction (‘dehydroxylation’) from clinopyroxene producing effective fluid saturation that catalyzed the synkinematic net-transfer reaction. This reaction produced fine-grained olivine and plagioclase, allowing the onset of grain-size sensitive creep and further strain localization in these pyroxenite bands. The strain localization in the pyroxenites is thus explained by their more fertile composition, which allowed earlier onset of the phase transition reactions. Geothermobarometry undertaken on compositionally zoned constituent minerals suggests that this positive feedback between reactions and deformation is associated with cooling from at least 1000°C to 700°C and decompression from 1·0 to 0·5 GPa.

Description: The late Miocene and younger mafic back-arc lavas in the southern Puna of the central Andean plateau have been attributed to the aftermath of crustal and mantle lithospheric delamination or foundering. In this paper, we analyze in more detail the nature of the back-arc mafic suite magmas, including the conditions of magma generation in the mantle and of magma evolution during ascent and ponding in the crust, using extensive compositional data for phenocryst minerals and olivine-hosted melt inclusions in combination with published and new whole-rock chemical and isotopic data. We estimate that the primary melts last equilibrated with an enriched mantle source at temperatures near 1375°C and pressures near 2 GPa, which is near the base of the seismically determined ~60 km thick crust. A mantle source geochemically enriched by continental material introduced through delamination and subducted erosion processes is required to explain the coincidence of the high 87 Sr/ 86 Sr ratios (〉0·705) and high Sr concentrations (〉700 ppm) of the most primitive lavas (e.g. 9–10 wt % MgO, olivine Fo 88 ). The crystallization conditions inferred from mineral–melt equilibria indicate that olivine ( T = 1320–1220°C) was followed by clinopyroxene ( T = 1230–1140°C). Clinopyroxene–melt equilibration pressures of 0·7 to near 1 GPa in the most mafic samples indicate that the magmas crystallized at mid-crustal depths of 20–35 km, within a region of inferred partial melt accumulation based on the presence of low seismic velocity zones. Olivine-hosted melt inclusions indicate relatively dry melts (maximum 0·5 wt % H 2 O) with unusual high-Al basaltic compositions, which are attributed to the high-pressure suppression of plagioclase crystallization. A first stage of crustal contamination before mid-crustal accumulation and crystallization of the mafic magmas is suggested by high O-isotope ratios in olivine phenocrysts and negative Eu anomalies in clinopyroxene from the plagioclase-free mafic lavas. Mixing models based on trace elements and radiogenic isotopes suggest assimilation of silicic melt in the lower crust, similar to contemporaneous glassy dacites with steep REE patterns and negative Eu anomalies. A second stage of crustal assimilation at shallower depths is indicated by the mismatch of incompatible elements in clinopyroxene relative to bulk-rock compositions, by strong positive correlations of radiogenic isotopes with wt % SiO 2 , and by petrographic observation of partly resorbed and reacted quartz xenocrysts. Mixing calculations require the erupted magmas to have assimilated in total some 15–25% crust.

Description: Hornblende-bearing basanites and alkali basalts from the Rhön area of Germany (part of the Central European Volcanic Province; CEVP) have high TiO 2 (3–4 wt %), moderately high Mg# (mostly 〉0·50), variable Cr (400–30 ppm) and Ni (160–20 ppm) abundances, and are enriched in incompatible trace elements and rare earth elements (REE). In primitive mantle-normalized multi-element diagrams they show a strong depletion in Ba, Rb, and K relative to trace elements of similar incompatibility. Some alkali basalts and more differentiated rocks have lower Mg# and lower abundances of Ni and Cr, and have undergone fractionation of olivine, clinopyroxene, Fe–Ti oxides and amphibole. The trace element constraints (e.g. low Nb/U and Ce/Pb and the Nd–Sr–Pb isotope compositions of some basalts) indicate that assimilation of lower crustal material has modified the composition of the primary mantle-derived magmas. Most of the basanites and alkali basalts approach the Sr–Nd–Pb isotope compositions inferred for the EAR (European Asthenospheric Reservoir) component. Variations in REE abundances and correlations between REE ratios suggest partial melting of amphibole-bearing spinel peridotite containing a significant portion of non-peridotitic material (i.e. pyroxenite). The presence of residual amphibole, indicated by depletion of K and Rb relative to Ba and Nb, requires melting close to the asthenosphere–lithosphere boundary or within the lithospheric mantle, most probably of a veined mantle source. Temperature and pressure estimates indicate a depth of melting for the most primitive lavas at ~80 km at temperatures of ~1290°C. Based on Sr–Nd isotope and trace element constraints it is proposed that asthenospheric melts similar in composition to EAR melts observed elsewhere in the CEVP froze at the asthenosphere–lithosphere thermal boundary as veins in the lithospheric mantle. These veins were remelted after only short storage times by ascending asthenospheric melts, imposing the prominent amphibole signature upon the basalts. The fairly radiogenic Pb isotope signatures are expected to originate from melting of enriched, low melting temperature components incorporated in the depleted upper (asthenospheric) mantle and therefore do not require upwelling of deep-seated mantle sources for the Rhön or many other continental alkaline lavas with similar Pb isotope signatures.

Description: We develop theorems which produce a multitude of hyperbolic triples for the finite classical groups. We apply these theorems to prove that every quasisimple group except Alt (5) and SL 2 (5) is a Beauville group. In particular, we settle a conjecture of Bauer, Catanese and Grunewald which asserts that all non-abelian finite quasisimple groups except for the alternating group Alt (5) are Beauville groups.

Description: Let U R d be open and convex. We prove that every (not necessarily Lipschitz or strongly) convex function f : U -〉 R can be approximated by real analytic convex functions, uniformly on all of U . We also show that C 0 -fine approximation of convex functions by smooth (or real analytic) convex functions on R d is possible in general if and only if d = 1. Nevertheless, for d ≥ 2, we give a characterization of the class of convex functions on R d which can be approximated by real analytic (or just smoother) convex functions in the C 0 -fine topology. It turns out that the possibility of performing this kind of approximation is not determined by the degree of local convexity or smoothness of the given function, but by its global geometrical behaviour. We also show that every C 1 convex and proper function on U can be approximated by C convex functions in the C 1 -fine topology, and we provide some applications of these results, concerning prescription of (sub-)differential boundary data to convex real analytic functions, and smooth surgery of convex bodies.

Description: Let $G$ be a compact connected Lie group, or more generally a path connected topological group of the homotopy type of a finite CW-complex, and let $X$ be a rational nilpotent $G$ -space. In this paper, we analyze the homotopy type of the homotopy fixed point set $X^{hG}$ , and the natural injection $k\colon X^G\hookrightarrow X^{hG}$ . We show that if $X$ is elliptic, that is, it has finite-dimensional rational homotopy and cohomology, then each path component of $X^{hG}$ is also elliptic. We also give an explicit algebraic model of the inclusion $k$ based on which we can prove, for instance, that for $G$ a torus, $\pi _* (k)$ is injective in rational homotopy but, often, far from being a rational homotopy equivalence.

Description: We employ the ergodic-theoretic machinery of scenery flows to address classical geometric measure-theoretic problems on Euclidean spaces. Our main results include a sharp version of the conical density theorem, which we show to be closely linked to rectifiability. Moreover, we show that the dimension theory of measure-theoretical porosity can be reduced back to its set-theoretic version, that Hausdorff and packing dimensions yield the same maximal dimension for porous and even mean porous measures, and that extremal measures exist and can be chosen to satisfy a generalized notion of self-similarity. These are sharp general formulations of phenomena that had been earlier found to hold in a number of special cases.

Description: A notion of tangential thickness of a manifold is introduced. An extensive calculation within the class of lens and fake lens spaces leads to a classification of such manifolds with thickness 1, 3 or 2 $k$ , for $k\geq 1$ . On the other hand, calculations of tangential thickness in terms of the dimension of the manifold and the rank of the fundamental group show very interesting and quite surprising correlations between these invariants.

Description: Extending a classical result of Widom from 1969, polynomials with small supremum norms are constructed for a large family of compact sets $\Gamma$ : their norm is at most a constant times the theoretical lower limit ${{\rm cap}}(\Gamma )^n$ , where ${{\rm cap}}(\Gamma )$ denotes logarithmic capacity. The construction is based on a discretization of the equilibrium measure, and the polynomials have the additional property that outside the given set $\Gamma$ they increase as fast as possible, namely as ${{\rm cap}}(\Gamma )^n\exp (ng_{ \overline {{{}C}}\setminus \Gamma }(z))$ , with the Green's function with pole at infinity in the exponent. This latter fact allows us to use these polynomials as building blocks in constructing Dirac delta-type polynomials around corners: if a compact set $K$ has a corner at some point $z_0$ , then Dirac delta-type polynomials (fast decreasing polynomials) peaking at $z_0$ are polynomials $P_n(z)$ with $P_n(z_0)=1$ that decrease as $|P_n(z)|\prec \exp (-n^ \beta |z-z_0|^ \gamma )$ on the set $K$ as $z$ moves away from $z_0$ . The possible $(\beta , \gamma )$ pairs are completely described in turn of the angle $\alpha \pi$ at $z_0$ ( $\beta \lt 1$ and $\gamma \ge \beta /(2- \alpha )$ or $\beta =1$ and $\gamma 〉 \beta /(2- \alpha )$ ). As application of these fast decreasing polynomials sharp Nikolskii- and Markov-type inequalities are proved for Jordan domains with corners. The paper uses distortion properties of conformal maps, potential theoretic techniques as well as the theory of weighted logarithmic potentials.

Description: We consider for every $n\in \mathbb {N}$ an algebra $\mathcal {A}_{n}$ of germs at $0\in \mathbb {R}^{n}$ of continuous real-valued functions, such that we can associate to every germ $f\in \mathcal {A}_{n}$ a (divergent) series $\mathcal {T}(f)$ with non-negative real exponents, which can be thought of as an asymptotic expansion of $f$ . We require that the $\mathbb {R}$ -algebra homomorphism $f\mapsto \mathcal {T}(f)$ be injective (quasianalyticity property). In this setting, we prove analogue results to Denef and van den Dries’ quantifier elimination theorem and Hironaka's rectilinearization theorem for subanalytic sets.

Description: Suppose that $F(x)\in \mathbb {Z}[\![x]\!]$ is a Mahler function and that $1/b$ is in the radius of convergence of $F(x)$ for an integer $b\geq 2$ . In this paper, we consider the approximation of $F(1/b)$ by algebraic numbers. In particular, we prove that $F(1/b)$ cannot be a Liouville number. If, in addition, $F(x)$ is regular, we show that $F(1/b)$ is either rational or transcendental, and in the latter case that $F(1/b)$ is an $S$ -number or a $T$ -number in Mahler's classification of real numbers.

Description: We develop techniques for computing zeta functions associated with nilpotent groups, not necessarily associative algebras, and modules, as well as Igusa-type zeta functions. At the heart of our method lies an explicit convex-geometric formula for a class of $p$ -adic integrals under non-degeneracy conditions with respect to associated Newton polytopes. Our techniques prove to be especially useful for the computation of topological zeta functions associated with algebras, resulting in the first systematic investigation of their properties.

Description: We give a bordism-theoretic characterization of those closed almost contact $(2q{+ }1)$ -manifolds (with $q\geq 2$ ) that admit a Stein fillable contact structure. Our method is to apply Eliashberg's $h$ -principle for Stein manifolds in the setting of Kreck's modified surgery. As an application, we show that any simply connected almost contact 7-manifold with torsion-free second homotopy group is Stein fillable. We also discuss the Stein fillability of exotic spheres and examine subcritical Stein fillability.

Description: Casson-type invariants emerging from Donaldson theory over certain negative-definite four-manifolds were recently suggested by Teleman. These are defined by an algebraic count of points in a zero-dimensional moduli space of flat instantons. Motivated by the cobordism programme of proving Witten's conjecture, we use a moduli space of ${\rm PU}(2)$ Seiberg–Witten monopoles to exhibit an oriented one-dimensional cobordism of the instanton moduli space to the empty space. The Casson-type invariant must therefore vanish.

Description: The dynamical and stationary behaviors of a fourth-order equation in the unit ball with clamped boundary conditions and a singular reaction term are investigated. The equation arises in the modeling of microelectromechanical systems and includes a positive voltage parameter $\lambda$ . It is shown that there is a threshold value $\lambda _* 〉 0$ of the voltage parameter such that no radially symmetric stationary solution exists for $\lambda 〉 \lambda _* $ , while at least two such solutions exist for $\lambda \in (0,\lambda _* )$ . Local and global well-posedness results are obtained for the corresponding hyperbolic and parabolic evolution problems as well as the occurrence of finite time singularities when $\lambda 〉 \lambda _* $ .

Description: Two mafic eruptive products from Vesuvius, a tephrite and a trachybasalt, have been crystallized in the laboratory to constrain the nature of primitive Vesuvius magmas and their crustal evolution. Experiments were performed at high temperatures (from 1000 to ≥1200°C) and both at 0·1 MPa and at high pressures (from 50 to 200 MPa) under H 2 O-bearing fluid-absent and H 2 O- and CO 2 -bearing fluid-present conditions. Experiments started from glass except for a few that started from glass plus San Carlos olivine crystals to force olivine saturation. Melt H 2 O concentrations reached a maximum of 6·0 wt % and experimental f O 2 ranged from NNO – 0·1 to NNO + 3·4 (where NNO is nickel–nickel oxide buffer). Clinopyroxene (Mg# up to 93) is the liquidus phase for the two investigated samples; it is followed by leucite for H 2 O in melt 〈3 wt %, and by phlogopite (Mg# up to 81) for H 2 O in melt 〉3 wt %. Olivine (Fo 85 ) crystallized spontaneously in only one experimental charge. Plagioclase was not found. Upon progressive crystallization of clinopyroxene, glass K 2 O and Al 2 O 3 contents strongly increase whereas MgO, CaO and CaO/Al 2 O 3 decrease; the residual melts follow the evolution of Vesuvius whole-rocks from trachybasalt to tephrite, phonotephrite and to tephriphonolite. Concentrations of H 2 O and CO 2 in near-liquidus 200 MPa glasses and primitive melt inclusions from the literature overlap. The earliest evolutionary stage, corresponding to the crystallization of Fo-rich olivine, was reconstructed by the olivine-added experiments. They show that the primitive Vesuvius melts are trachybasalts (K 2 O ~ 4·5–5·5 wt %, MgO = 8–9 wt %, Mg# = 75–80, CaO/Al 2 O 3 = 0·9–0·95) that crystallize Fo-rich olivine (90–91) as the liquidus phase between 1150 and 1200°C and from 300 to 〈200 MPa. Primitive Vesuvius melts are volatile-rich (1·5–4·5 wt % H 2 O and 600–4500 ppm CO 2 in primitive melt inclusions) and oxidized (from NNO + 0·4 to NNO + 1·2). Assimilation of carbonate wall-rocks by ascending primitive magmas can account for the disappearance of olivine from crystallization sequences and explains the lack of rocks representative of olivine-crystallizing magmas. A correlation between carbonate assimilation and the type of feeding system is proposed: carbonate assimilation is promoted for primitive magma batches of small volumes. In contrast, for longer-lived, large-volume, less frequently recharged, hence more evolved, cooler reservoirs, magma–carbonate interaction is limited. Primitive magmas from Vesuvius and other Campanian volcanoes have similar redox states. However, the Cr# of Vesuvius spinels is distinctive and therefore the peridotitic component in the mantle source of Vesuvius differs from that of the other Campanian magmas.

Description: Peridotite xenoliths exhumed by Quaternary alkaline magmatism in the Tahalgha district, southern Hoggar, represent fragments of the subcontinental lithospheric mantle beneath the boundary between the two major structural domains of the Tuareg Shield: the ‘Polycyclic Central Hoggar’ to the east and the ‘Western Hoggar’, or ‘Pharusian Belt’, to the west. Samples were collected from volcanic centres located on both sides of a major lithospheric shear zone at 4°35' separating these two domains. Although showing substantial variations in their deformation microstructures, equilibrium temperatures and modal and chemical compositions, the studied samples do not display any systematic changes of these features across the 4°35' fault. The observed variations rather record small-scale heterogeneities distributed throughout the study area and reflecting the widespread occurrence of vein conduits and metasomatized wall-rocks related to trans-lithospheric melt circulation during the Cenozoic. These features include partial annealing of pre-existing deformation microstructures, post-deformation metasomatic reactions, and trace-element enrichment, coupled with heating from 750–900°C (low-temperature lherzolites) to 900–1150°C (intermediate- T lherzolites and high- T harzburgites and wehrlites). Trace-element modelling confirms that the range of rare earth element (REE) variations observed in the Tahalgha clinopyroxenes may be accounted for by reactive porous flow involving a single stage of basaltic melt infiltration into a light REE (LREE)-depleted protolith. Whole-rock compositions record the final entrapment of disequilibrium metasomatic melts upon thermal relaxation of the veins–wall-rock system. The striking correlations between equilibrium temperatures and trace-element enrichment favor a scenario in which the high-temperature peridotites record advective heat transport along melt conduits, whereas the intermediate- and low-temperature lherzolites reflect conductive heating of the host Mechanical Boundary Layer. This indicates that the lithosphere did not reach thermal equilibrium, suggesting that the inferred heating event was transient and was rapidly erased by thermal relaxation down to the relatively low-temperature present-day geotherm. The low- T (〈900°C) deformed lherzolites (porphyroclastic to equigranular) are characterized by only incipient annealing and LREE-depleted clinopyroxene compositions. They were only weakly affected by the Cenozoic events and could represent relatively well-preserved samples from rejuvenated Pan-African lithosphere. Extensive lithospheric rejuvenation occurred either regionally during the Pan-African orogeny, as a result of lithospheric delamination or thermomechanical erosion after thickening, or more locally along the meridional shear zones. The low- T Tahalgha lherzolites are comparable with lherzolites from Etang de Lherz, southern France, interpreted as lithospheric mantle rejuvenated by melt-induced refertilization during a late stage of the Variscan orogeny.

Description: Hydrothermal experiments were conducted at 200 MPa and 900–1018°C to determine the solubilities, fluid(s)–melt partitioning, and mixing properties of H 2 O, CO 2 , S, Cl, and F in phonolitic–trachytic melts saturated in vapor, vapor plus saline liquid, or saline liquid. The bulk compositions and S, Cl, and F concentrations of the run-product glasses were determined by electron microprobe and the H 2 O and CO 2 contents by Fourier-transform infrared spectroscopy (FTIR). A new parameterization was developed to calculate molar absorption coefficients for FTIR analysis of carbonate in glasses and applied to the run-product glasses. The concentrations of volatiles in the fluid(s) were determined by mass-balance calculations and checked with chloridometer analysis and gravimetry. The range in oxygen fugacity of these experiments is NNO to NNO + 2 (where NNO is nickel–nickel oxide buffer). The phonolitic–trachytic melts dissolved up to 7·5 wt % H 2 O, 0·94 wt % Cl, 0·73 wt % CO 2 , 0·75 wt % F, and 0·16 wt % S, and the integrated bulk fluid(s) contained up to 99 mol % H 2 O, 34 mol % Cl, 82 mol % CO 2 , 1·7 mol % F, and 3·7 mol % S. The mixing relationships of H 2 O, CO 2 , and Cl in melt versus fluid(s) are complex and strongly non-ideal at these pressure–temperature conditions, particularly with two fluid phases stable. The concentrations of H 2 O and CO 2 in melt change with the addition of Cl ± S to the system, and the solubility of Cl in melt varies with S. The reductions in H 2 O and CO 2 solubility in melt exceed those resulting from simple dilution of the coexisting fluid(s) owing to addition of other volatiles. The partitioning of H 2 O and CO 2 between fluid(s) and melt varies as a function of fluid(s) and melt composition. The experimental data are applied to phonolitic and related magmas of Mt. Somma–Vesuvius, Italy, Mt. Erebus, Antarctica, and Cripple Creek, USA, to better interpret processes of fluid(s) exsolution in eruptive and mineralizing systems. Application of the experimental results also provides constraints on eruptive and mineralizing fluid(s) compositions.

Description: The early stages of magmatic processes operating at mantle depths beneath continental arcs are poorly known. The chemical compositions of minerals and rocks, mineral Sr–Nd–Hf–O isotopes and zircon U–Pb ages of garnet clinopyroxenite dykes from the Shenglikou peridotite massif (North Qaidam Orogen, NE Tibet, China) were studied to constrain their sources and genesis, and the dynamic processes that controlled pyroxenite formation beneath an early Paleozoic active continental margin. Major-element compositions of bulkrocks suggest that the pyroxenitic protoliths were cumulates segregated from a melt, which was extracted from a peridotite-dominated mantle source. Bulk-rock and mineral trace-element patterns show strong enrichment in fluid-mobile elements (e.g. Cs, Rb, Ba, Th, U, K, Pb and Li) and marked negative anomalies in the high field strength elements relative to rare earth elements, similar to the characteristics of melts derived from a volatile-rich sub-arc mantle. Enriched Sr and Nd initial isotopic compositions at 500 Ma ( 87 Sr/ 86 Sr of 0·70919–0·71774 and Nd of –16·3 to –3·4) are in contrast to the highly radiogenic Hf isotope compositions (similar to those of the depleted-mantle reservoir) and to the uncontaminated upper-mantle 18 O V-SMOW (garnet: 5·6 ± 0·3, 2SD, n = 61; zircon: 5·9 ± 0·3, 2SD, n = 28). These decoupled isotopic signatures suggest that the melt source was located in a convective mantle wedge (controlling the Hf and O isotopes) that had been pervasively metasomatized by fluids from a subducted Proto-Tethys oceanic slab (controlling the Sr–Nd isotopes and highly incompatible elements). Zircons with two groups of U–Pb ages (430 ± 5 Ma and 401 ± 7 Ma) were generated by recrystallization events, corresponding to UHP metamorphism and a major uplift stage during the North Qaidam orogeny, respectively. The combined evidence reveals a picture of continental arc magmatism at mantle depths and subsequent continental collision. The subduction of the Proto-Tethys oceanic slab beneath the southern Qilian margin triggered flux melting of the metasomatized convective mantle wedge and generated hydrous arc magmas. These primitive magmas intruded into the overlying lithospheric mantle and segregated the cumulates parental to the Shenglikou pyroxenites. Subsequent continental subduction incorporated fragments of the mantle-wedge peridotite (containing pyroxenite dykes) at ~430 Ma and carried them to shallow depths during exhumation at ~400 Ma.

Description: Using a new high-resolution dataset, this study presents evidence for short length scale 18 O/ 16 O heterogeneity in the mantle source region of young (age 12 ka bp ) Icelandic basalts. The dataset comprises secondary ion mass spectrometry determinations of 18 O/ 16 O in single compositional zones of plagioclase crystals from the primitive Borgarhraun flow in northern Iceland, along with trace and major element data from the same zones. The presence of mantle under Iceland with 18 O below typical mid-ocean ridge basalt (MORB) values of ~5·5 ± 0·3 (VSMOW) has previously been disputed, because variability in 18 O in many Icelandic basalts is also known to be caused by the interaction of basaltic melts with crustal lithologies that have been altered by low- 18 O meteoric water. Primitive basalt flows, such as Borgarhraun, and their macrocrysts are the most likely candidates to retain a mantle 18 O signature. However, the role of crustal processes in generating the low 18 O in olivine crystals from these flows has not unequivocally been ruled out. By making intra-crystal analyses in Borgarhraun plagioclase it has been possible in this study to obtain a detailed record of the chemical and isotopic compositions of the melts that crystallized the plagioclase zones. The variability observed in trace element compositions of the early crystallized anorthitic plagioclase zones (80·9–89·4 mol % anorthite) is firstly shown to arise from melt compositional variability, and equilibrium melt concentrations of Sr, La and Y are then calculated from the crystal concentrations of these elements using carefully selected partition coefficients. The ranges of incompatible trace element ratios (La/Y, Sr/Y) in these equilibrium melts reflect a range of compositions of fractional mantle melts, a result that is in agreement with previous proposals for the cause of variability in trace element indices of Borgarhraun olivine-hosted melt inclusions and clinopyroxene compositional zones. Correlations observed between La/Y and Sr/Y in the melts in equilibrium with the Borgarhraun plagioclase zones and the 18 O of these zones therefore support the hypothesis that the mantle under Iceland is heterogeneous in 18 O/ 16 O. Such correlations have not previously been observed in intra-crystal data from Iceland, and provide strong evidence that mantle material with abnormally low 18 O may exist in the form of readily fusible heterogeneities alongside ambient mantle with MORB-like 18 O (+5·5) on a length scale of 〈100 km. The lowest 18 O of plagioclase that is attributed to a mantle origin in this study is 4·5 ± 0·4, equating to a melt equivalent value of 4·3 ± 0·5 or an olivine equivalent value of 3·8 ± 0·5.

Description: Mafic to ultramafic intrusions of the Qullinaaraaluk suite (Q-suite) were emplaced into the Ungava craton of the Northeastern Superior Province during an episode of intense igneous activity and crustal reworking from c. 2·74 to 2·70 Ga. Orthopyroxene-rich Q-suite intrusions from the Hudson Bay Terrane and southwestern Rivière Arnaud Terrane, and orthopyroxene-poor Q-suite intrusions from the north–central Rivière Arnaud Terrane indicate the existence of at least two Q-suite magma types: a subalkaline magma parental to the orthopyroxene-rich intrusions and a transitional magma parental to the orthopyroxene-poor intrusions. Both types of intrusions are characterized by light rare earth element (LREE)-enriched, high field strength element (HFSE)-depleted trace element profiles that reflect, in large part, contamination by the tonalite–trondhjemite–granodiorite-dominated crust. Near-chondritic to strongly sub-chondritic initial Nd (2·72 Ga) values (+2 to –10) of the Q-suite intrusions reflect the combined effects of both the amount of crustal contamination and the age-dependent isotopic composition of the contaminant. The inferred trace element profiles of the uncontaminated Q-suite magmas were probably flat to LREE-depleted. The transitional magmas that produced the least evolved dunitic cumulates of the Q-suite were ferropicrites (MgO ~14 wt %, FeO TOT ~17 wt %). In contrast, the magmas parental to the primitive Q-suite harzburgites were Fe-rich, high-Mg basalts (MgO ~11 wt %; FeO ~14 wt %). The high Fe contents of the Q-suite magmas are incompatible with derivation from a pyrolitic mantle [Mg-number ~0·90, Mg/(Mg + Fe TOT )] and require sources significantly enriched in iron (Mg-number ≤0·79). Both magma types are also characterized by relatively low Ni contents suggesting derivation from source regions depleted in Ni relative to pyrolitic mantle peridotite. Differences in the major element compositions of the subalkaline and transitional parental magmas may reflect compositional diversity among the Fe-rich mantle sources. Comparisons with melting experiments on compositions analogous to the Martian mantle suggest that the Q-suite magmas may rather be generated by different degrees of melting of a common source with an Fe content slightly lower than that of the Homestead L5 ordinary chondrite (Mg-number = 0·77). The Fe-rich picritic to high-Mg basaltic magmas last equilibrated with garnet-free harzburgitic to lherzolitic residues at upper mantle pressures (≤5 GPa). The craton-wide occurrence of c. 2·72–2·70 Ga Q-suite mafic to ultramafic plutons suggests that underplating by Fe-rich mantle melts may have had a key role in the c. 2·74–2·70 Ga cratonization of the Northeastern Superior Province.

Description: Arc basalts are more oxidized than mid-ocean ridge basalts, but it is unclear whether this difference is due to differentiation processes in the Earth’s crust or to a fundamental difference in the oxygen fugacity of their mantle sources. Distinguishing between these two hypotheses is important for understanding redox-sensitive processes related to arc magmatism, and thus more broadly how Earth materials cycle globally. We present major, volatile, and trace element concentrations in combination with Fe 3+ /Fe ratios determined in olivine-hosted glass inclusions and submarine glasses from five Mariana arc volcanoes and two regions of the Mariana Trough. For single eruptions, Fe 3+ /Fe ratios vary along liquid lines of descent that are either slightly oxidizing (olivine + clinopyroxene + plagioclase fractionation, CO 2 ± H 2 O degassing) or reducing (olivine + clinopyroxene + plagioclase ± magnetite fractionation, CO 2 + H 2 O + S degassing). Mariana samples are consistent with a global relationship between calc-alkaline affinity and both magmatic H 2 O and magmatic oxygen fugacity, where wetter, higher oxygen fugacity magmas display greater affinity for calc-alkaline differentiation. We find, however, that low-pressure differentiation cannot explain the majority of variations observed in Fe 3+ /Fe ratios for Mariana arc basalts, requiring primary differences in magmatic oxygen fugacity. Calculated oxygen fugacities of primary mantle melts at the pressures and temperatures of melt segregation are significantly oxidized relative to mid-ocean ridge basalts (~QFM, where QFM is quartz–fayalite–magnetite buffer), ranging from QFM + 1·0 to QFM + 1·6 for Mariana arc basalts, whereas back-arc related samples record primary oxygen fugacities that range from QFM + 0·1 to QFM + 0·5. This Mariana arc sample suite includes a diversity of subduction influences, from lesser influence of a homogeneous H 2 O-rich component in the back-arc, to sediment melt- and fluid-dominated influences along the arc. Primary melt oxygen fugacity does not correlate significantly with sediment melt contributions (e.g. Th/La), nor can it be attributed to previous melt extraction in the back-arc. Primary melt oxygen fugacity correlates strongly with indices of slab fluids (e.g. Ba/La) from the Mariana Trough through the Mariana arc, increasing by 1·5 orders of magnitude as Ba/La increases by a factor of 10 relative to mid-ocean ridge basalts. These results suggest that contributions from the slab to the mantle wedge may be responsible for the elevated oxygen fugacity recorded by Mariana arc basalts and that slab fluids are potentially very oxidized.

Description: Magma mixing and crystal mush disaggregation are important processes in basaltic magma reservoirs. We carried out a detailed petrological and geochemical study on a highly plagioclase-phyric eruption within the Eastern Volcanic Zone of Iceland—the Skuggafjöll eruption—to investigate crystal storage and transport processes within a single magmatic system. Crystal content and phase proportions vary between samples: the least phyric samples have phase proportions similar to the low-pressure, three-phase gabbro eutectic (plg:cpx:ol ~ 11:6:3), whereas highly phyric samples are strongly enriched in plagioclase (plg:cpx:ol ~ 8:1:1). Statistically significant geochemical variability in 28 whole-rock samples collected across the eruption can be accounted for by variable accumulation of a troctolitic assemblage containing plagioclase and olivine in an approximately 9:1 ratio. Two macrocryst assemblages are defined using compositional and textural information recorded in QEMSCAN® images: a primitive assemblage of high-anorthite plagioclase (An 〉83 ) and high-forsterite olivine (Fo 〉84 ), and an evolved assemblage of low-anorthite plagioclase (An 〈79 ), low-forsterite olivine (Fo 〈82 ) and clinopyroxene (Mg# ~ 82). Plagioclase and olivine have strongly bimodal composition distributions whereas the composition distribution of clinopyroxene is unimodal. The mean trace element composition of melt inclusions hosted within high-forsterite olivine and high-anorthite plagioclase macrocrysts is the same (mean Ce/Y ~ 0·47–0·48), confirming that both primitive macrocryst phases crystallized from the same distribution of melts. Clinopyroxene macrocrysts and matrix glasses are in Ce/Yb equilibrium with each other, indicating that the evolved assemblage crystallized from melts with a more incompatible trace element-enriched composition (mean Ce/Y ~ 0·65–71) than the primitive assemblage. Variability in whole-rock, macrocryst and melt inclusion compositions suggests that the Skuggafjöll magma experienced two stages of crystallization. Primitive macrocrysts crystallized first from incompatible trace element-depleted melts within a shallow crustal magma reservoir. These primitive macrocrysts were subsequently stored in crystal mushes that ultimately disaggregated into an evolved and incompatible trace element-enriched magma from which the evolved assemblage crystallized. On average, ~17% of the erupted magma at Skuggafjöll is composed of accumulated macrocrysts entrained from crystal mushes. The timescale between mush disaggregation and eruption, during which crystal accumulation occurred, was short—of the order of years—according to simple diffusion calculations. Striking petrological similarities between Skuggafjöll and other highly phyric eruptions both in Iceland and along mid-ocean ridges indicate that crystal accumulation by mush disaggregation is likely to be an important mechanism for generating highly phyric magmas in basaltic plumbing systems.

Description: Assuming the generalized Riemann hypothesis, we prove a quantitative estimate for the number of simple zeros on the critical line for $L$ -functions attached to classical holomorphic newforms.

Description: In this paper, we consider a $\mathbb {Q}$ -Fano $3$ -fold weighted complete intersection of codimension $2$ in the $85$ families listed in Iano-Fletcher's list and determine which cycle is a maximal center or not. For each maximal center, we construct either a birational involution which untwists the maximal singularity or a Sarkisov link centered at the cycle to another explicitly described Mori fiber space. As a consequence, nineteen families are proved to be birationally rigid and the remaining $66$ families are proved to be birationally non-rigid.

Description: The Pussy Cat Group rhyolites of the Mesoproterozoic west Musgrave Province of central Australia, a constituent part of the Bentley Supergroup, were deposited during the c . 1085–1040 Ma Ngaanyatjarra Rift and Giles events, and are related to the Warakurna Large Igneous Province. This study focuses on the two silicic components of the Pussy Cat Group, the Kathleen Ignimbrite and the Rowland Suite. These silicic rocks are A-type, metaluminous (to slightly peraluminous) rhyolites and are enriched in the rare earth elements (REE) relative to average crustal abundances. The rhyolitic Kathleen Ignimbrite records an explosive caldera fill-sequence and contains, amongst others, a thick (≤500 m), initially subaqueously emplaced, rheomorphic, intra-caldera ignimbrite unit, whereas the Rowland Suite consists of a number of mineralogically and geochemically related porphyritic rhyolites that intrude throughout the Pussy Cat Group. Whole-rock geochemistry, Rb–Sr, Sm–Nd and in situ zircon Lu–Hf isotope data are indicative of a dominantly mantle-derived source for the magmas that formed the Pussy Cat Group rhyolites. Secondary ion mass spectrometry U–Pb dating of these units yields ages of 1062 ± 8, 1071 ± 5, 1076 ± 5, and 1078 ± 5 Ma. The magmas that formed these units were formed by extreme fractional crystallization of a mantle-derived basaltic magma, with minimal crustal contamination, during a failed intra-plate extensional rifting event. This involved three main stages of fractional crystallization: early fractionation of plagioclase, olivine, clinopyroxene and magnetite from a basaltic magma to reach an intermediate composition, subsequent fractionation of plagioclase, K-feldspar and quartz to form a proto-Rowland Suite-type magma at mid- to upper-crustal levels that migrated into the shallow upper crust and formed a magma chamber, and final fractionation of quartz, K-feldspar, plagioclase, magnetite and biotite ± minor REE-enriched accessory phases from the Rowland Suite magma resulting in the evolved Kathleen Ignimbrite magmas. This final phase of fractionation generated the most evolved silicic rock suite identified to date within the entire west Musgrave Province. The new petrographic, geochronological, geochemical, and isotopic data presented within this study indicate that these two units are coeval and comagmatic, suggesting a common source for the Kathleen Ignimbrite and the entire Rowland Suite. In addition, these data suggest that the crystal-rich, porphyritic rhyolite intrusions of the Rowland Suite represent a primitive cumulate end-member of the magmatic system, whereas the varying crystal-poor to crystal-rich Kathleen Ignimbrite eruption sequence represents the evolved and highly fractionated end-member of the system that formed thorough the evacuation of a shared or at least partly linked, compositionally zoned and differentiated source magma chamber or chambers.

Description: Suppose that a sequence of numbers $x_n$ (a ‘signal’) is transmitted through a noisy channel. The receiver observes a noisy version of the signal with additive random fluctuations, $x_n + \xi _n$ , where $\xi _n$ is a sequence of independent standard Gaussian random variables. Suppose further that the signal is known to come from some fixed space ${\mathscr {X}}$ of possible signals. Is it possible to fully recover the transmitted signal from its noisy version? Is it possible to at least detect that a non-zero signal was transmitted? In this paper, we consider the case in which signals are infinite sequences and the recovery or detection are required to hold with probability 1. We provide conditions on the space ${\mathscr {X}}$ for checking whether detection or recovery are possible. We also analyze in detail several examples including spaces of Fourier transforms of measures, spaces with fixed amplitudes and the space of almost periodic functions. Many of our examples exhibit critical phenomena, in which a sharp transition is made from a regime in which recovery is possible to a regime in which even detection is impossible.

Description: In this paper, we study the semi-stable subcategories of the category of representations of a Euclidean quiver, and the possible intersections of these subcategories. Contrary to the Dynkin case, we find out that the intersection of semi-stable subcategories may not be semi-stable. However, only a finite number of exceptions occur, and we give a description of these subcategories. Moreover, one can attach a simplicial fan in $\mathbb {Q}^n$ to any acyclic quiver $Q$ , and this simplicial fan allows one to completely determine the canonical presentation of any element in $\mathbb {Z}^n$ . This fan has a nice description in the Dynkin and Euclidean cases: it is described using an arrangement of convex codimension-1 subsets of $\mathbb {Q}^n$ , each such subset being indexed by a real Schur root or a set of quasi-simple objects. This fan also characterizes when two different stability conditions give rise to the same semi-stable subcategory.

Description: The $j$ -multiplicity plays an important role in the intersection theory of Stückrad–Vogel cycles, while recent developments confirm the connections between the $\epsilon$ -multiplicity and equisingularity theory. In this paper, we establish, under some constraints, a relationship between the $j$ -multiplicity of an ideal and the degree of its fiber cone. As a consequence, we are able to compute the $j$ -multiplicity of all the ideals defining rational normal scrolls. By using the standard monomial theory, we can also compute the $j$ - and $\epsilon$ -multiplicity of ideals defining determinantal varieties: The found quantities are integrals which, quite surprisingly, are central in random matrix theory.

Description: Let $\pi : X \to Y$ be a morphism of projective varieties and suppose that $\alpha $ is a pseudo-effective numerical cycle class satisfying $\pi _{*}\alpha =0$ . A conjecture of Debarre, Jiang, and Voisin predicts that $\alpha $ is a limit of classes of effective cycles contracted by $\pi $ . We establish new cases of the conjecture for higher codimension cycles. In particular, we prove a strong version when $X$ is a fourfold and $\pi $ has relative dimension 1.

Description: Let $R$ be a group of prime order $r$ that acts on the $r'$ -group $G$ , let $RG$ be the semidirect product of $G$ with $R$ , let ${\mathbb {F}}$ be a field and $V$ be a faithful completely reducible $\mathbb {F}[{RG}]$ -module. Trivially, $C_{G}({R})$ acts on $C_{V}({R})$ . Let $K$ be the kernel of this action. What can be said about $K$ ? This question is considered when $G$ is soluble. It turns out that $K$ is subnormal in $G$ or $r$ is a Fermat or half-Fermat prime. In the latter cases, the subnormal closure of $K$ in $G$ is described. Several applications to the theory of automorphisms of soluble groups are given.

Description: The mono-epi (ME) exact structure on the morphisms of an exact category $(\mathcal {A}; \mathcal {E})$ is introduced and used to prove ideal versions of Salce's Lemma, Christensen's (Ghost) Lemma, and Wakamatsu's Lemma for an exact category. Salce's Lemma establishes a bijective correspondence $\mathcal {I} \mapsto \mathcal {I}^{\perp }$ between the class of special precovering ideals of $(\mathcal {A}; \mathcal {E})$ and that of its special preenveloping ideals. ME-extensions of morphisms are used to define an extension $\mathcal {I} \diamond \mathcal {J}$ of ideals. Christensen's Lemma asserts that the class of special precovering (respectively, special preenveloping) ideals is closed under products and extensions and that the bijective correspondence of Salce's Lemma satisfies $(\mathcal {I} \mathcal {J})^{\perp } = \mathcal {J}^{\perp } \diamond \mathcal {I}^{\perp }$ and $(\mathcal {I} \diamond \mathcal {J})^{\perp } = \mathcal {J}^{\perp } \mathcal {I}^{\perp }.$ Wakamatsu's Lemma asserts that if a covering ideal $\mathcal {I}$ is closed under ME-extensions, then it is a special precovering ideal. As an application, it is proved that if $G$ is a finite group and $\Phi $ is the ideal of phantom morphisms in the category $k[G]$ - $\rm Mod,$ then $\Phi ^{n-1}$ is the object ideal generated by projective modules, where $n$ is the nilpotency index of the Jacobson radical $J.$ If $R$ is a semiprimary ring, with $J^n =0,$ then $\Phi ^n$ is generated by projective modules. For a right coherent ring $R$ over which every cotorsion left $R$ -module has a coresolution of length $n$ by pure injective modules, $\Phi ^{n+1}$ is generated by flat modules.

Description: The eruption of the Siberian Traps Large Igneous Province (SLIP) at the Permo-Triassic boundary was synchronous with environmental degradation and the largest known mass extinction in the geological record. The volatile emissions associated with these eruptions have been linked to the environmental change yet we understand little of their source and magnitude and how they varied with time. There are a number of possible sources for the volatiles that were emitted during the eruptions: the mantle (including metasomatized lithosphere), volatile-rich sediments (through metamorphism or direct assimilation) and the crustal basement. To assess the relative importance of these sources (with the exception of the metamorphic outgassing source), we have conducted a geochemical study of melt inclusions hosted by clinopyroxene in Siberian Traps low-Ti tholeiitic lavas and sills of the Khakanchansky, Ayansky and Khonnamakitsky Formations. The magmas were not emplaced into or erupted onto evaporite deposits, in contrast to samples studied previously. The trace element compositions of the melt inclusions are highly variable compared with the uniform whole-rocks, exhibiting a wide range of La/Yb ratios from 0·7 to 9·5. The melt geochemistry is consistent with relatively large degrees of partial melting of a dominantly peridotite mantle source. A negative Nb anomaly indicates a degree of crustal contamination, but there is no evidence for contamination by volatile-rich evaporites. Enrichment of some of the melts in large ion lithophile elements (Ba, Sr) indicates their interaction with a fluid. We suggest that, consistent with the observed depletion in other incompatible trace elements in the melt inclusions, the volatile concentrations in the melts were relatively low, and that subsequently the melts underwent variable degrees of degassing in the crust. Overall, the melts are more volatile-poor than those reported previously from the SLIP and were erupted after the first "pulse" of more volatile-rich magmas described by Sobolev et al. (2015) . These volatile-poor magmas may have been widespread across the region during the Siberian Traps eruptions once a pyroxenite component in the mantle source had been exhausted.

Description: Analysis of the 18 O and 13 C values of carbonate rocks from Islay, Scotland reveals structural channelling of metamorphic fluids through the axial region of a major en echelon anticlinal fold system. Metamorphic fluid flow produced axial planar veins with higher vein density in the axial region of the fold. Fluid:rock ratios were more than 30:1 within this axial region, at least four times greater than the regional mean ratio of 7·6 ± 1·5:1 for carbonate rocks on Islay. This supports the interpretation that metamorphic fluids were channelled through the axial region of the Islay Anticline. Fluid:rock ratios were calculated using a model for coupled 18 O and 13 C exchange with a metamorphic fluid. The metamorphic fluid was calculated to have 18 O and 13 C values of 15·3 and –6·1, respectively and X CO2 of 0·2. This is in isotopic and chemical equilibrium with chlorite- and graphite-bearing metamudstones that are structurally below the folded metacarbonate rocks on Islay. Devolatilization of these metamudstones is therefore a likely source mechanism for this metamorphic fluid. Removal of the effects of metamorphic fluid flow on 13 C values recorded by metacarbonate rocks on Islay allows us to re-evaluate evidence used to reconstruct Neoproterozoic climate. This evidence includes a large negative 13 C excursion reported from the Lossit Limestone Formation. This unit underlies the Port Askaig Formation, which is dominated by diamictites that have been interpreted as glacial tillites. This ‘Islay anomaly’ has been correlated with other such anomalies worldwide and together with overlying tillites has been cited as evidence of major (worldwide) glaciation events. In this study, we show that the magnitude of this negative 13 C anomaly can partly be explained by exchange with metamorphic fluids. However, we also show that extremely negative 13 C values in the Bonahaven Dolomite Formation, which overlies the Port Askaig Formation and has been interpreted as a ‘cap carbonate’, cannot be attributed to metamorphic fluid flow.

Description: To evaluate compaction and interstitial melt expulsion during cumulate formation, a 20 m cumulate section including the UG2 and UG3 chromitites from a 264 m drill core through the Upper Critical Zone of the Bushveld Complex (South Africa) has been studied. The cumulates in the studied section are as follows: 3 m plagioclase pyroxenite to pyroxenite, pegmatoid footwall pyroxenite at the lower contact to UG2, 0·7 m UG2 chromitite, 6·8 m pyroxenite, 0·24 m UG3 chromitite, 2·0 m plagioclase-rich pyroxenite changing locally to norite, the two 5 cm leader stringers UG3a and UG3b, and 7 m total of olivine pyroxenites grading into plagioclase pyroxenites. All pyroxenites are dominated by orthopyroxene (opx) and the cumulate sequence is topped by mottled anorthosite grading into norite. Stratigraphic concentrations of major and trace elements of 52 bulk-rock samples were determined. Bulk-rock Mg-numbers are 0·79–0·81 throughout the silicate cumulate units, and 0·40–0·46 in the chromitite layers. The stratigraphic distribution of six incompatible trace elements (K, Rb, Ba, Cs, Zr and Th) has been used to determine the amount of trapped liquid ( F TL ) or paleo-porosity in the cumulate rocks. Final porosities (volume fractions), based on averages from the six trace elements, are 0·06–0·33 in the pyroxenites. In chromitite layers, trapped melt fractions of 0·12–0·36 are calculated from incompatible trace element concentrations, but bulk SiO 2 concentrations and X-ray tomography yield 0·04–0·17 higher porosities. Hence, the bulk silicate fraction in the chromitites may not necessarily correspond to the trapped liquid fraction, as poikilitic opx was crystallizing while the silicate melt still equilibrated. Using a previously derived experiment-based model for compaction time scales, gravitationally driven chemical compaction in the UG2–UG3–pyroxenite section is calculated to occur within 1–10 years. This time frame corresponds to the times necessary to cool a 20 m layer by 10–50°C, the temperature interval argued to encompass the liquidus and almost complete solidification. Compaction within a decade can in fact easily develop the paleo-porosities indirectly observed today and is probably stopped by crystallization of the interstitial liquid. Contrary to previous assertions, melt expulsion from the cumulate pile does not hinder compaction; calculated permeabilities would allow for the migration of an order of magnitude higher amount of melt than has to be expelled from the 20 m pile of cumulate. The pegmatoid zones in the chromitite footwalls enriched in incompatible trace elements are consistent with a collection of interstitial melts expelling from the underlying compacting pyroxenites. Their entrapment below the chromitite layers suggests that these act as permeability barriers. This is in part due to their finer grain size compared with the pyroxenites, but is mainly due to the crystallization of large poikilitic opx during compaction.

Description: The recent eruptive history on the island of Tenerife is characterized in part by the presence of zoned phonolitic ignimbrites, some of which prominently display two types of juvenile clasts (i.e. light-colored, aphyric pumices alongside darker, more crystal-rich pumices, here dubbed ‘crystal-poor’ and ‘crystal-rich’, respectively). Petrographic observation of the crystal-rich pumices reveals intensely resorbed and intergrown mineral textures, consistent with the system reaching a high crystallinity, followed by perturbation and remobilization prior to eruption. Some trace elements show anomalous concentrations in such crystal-rich pumices (e.g. bulk Ba 〉 2000 ppm alongside low Zr and a positive Eu anomaly) indicative of crystal accumulation (of feldspar ± biotite). Many biotite and feldspar crystals are reversely zoned, with rim concentrations that are high in Ba but low in Sr, implying crystallization from an ‘enriched’ melt, potentially derived from remobilization by partial melting of the aforementioned cumulate zones. Given (1) the presence of cumulates in the eruptive record on Tenerife and a bimodality of pumice textures, (2) the presence of three dominant compositions (basanite, phonotephrite, phonolite, separated by compositional gaps) in the volcanic record, and (3) abundant support for crystal fractionation as the dominant drive for magmatic evolution in Tenerife, it is hypothesized that crystal-poor magmas are extracted from mushy reservoirs in both the lower and upper crust. The thermodynamic software MELTS is used to test a polybaric differentiation model whereby phonolites ( sensu lato ) are generated by extraction of residual liquids from an intermediate-crystallinity phonotephritic mush in the upper crust, which is in turn generated from the residual liquids of an intermediate-crystallinity basanitic mush at deeper levels. Latent heat spikes following crystallization of successive phases in the upper crustal reservoir provide a thermal buffering mechanism to slow down cooling and crystallization, permitting enhanced melt extraction at a particular crystallinity interval (mostly ~40–60 vol. % crystals). MELTS modeling typically fits the observed chemical data adequately, although some major elements (mostly Al 2 O 3 ) also indicate partial ‘cannibalization’ of feldspar along with some magma mixing (and potentially minor crustal contamination).

Description: To investigate the nature and origin of across-arc geochemical variations over time in mantle wedge derived magmas, we have carried out a geochemical study of basalts in the NE Japan arc spanning an age range from 35 Ma to the present. Back-arc basalts erupted at 24–18 Ma, 10–8 Ma, 6–3 Ma and 2·5–0 Ma have higher concentrations of both high field strength elements (HFSE) and rare earth elements (REE) [particularly light REE (LREE) and middle REE (MREE)], and higher incompatible trace element ratios compared with frontal-arc basalts at any given time. Geochemical modeling of Nb/Yb versus Nb shows that the frontal-arc and back-arc compositional differences are independent of subduction modification and can, in many cases, be explained by different degrees of melting (higher degrees of melting for frontal-arc magmas and lower degrees of melting for back-arc magmas) of a nearly homogeneous depleted mid-ocean ridge basalt (MORB) mantle (DMM)-like source, although there are several exceptions. These include some Pliocene frontal-arc basalts that may originate from a source that is slightly more depleted than DMM, several 35–32 Ma and 24–18 Ma back-arc basalts derived from a lithospheric mantle source that is enriched in HFSE compared with DMM, and a rare 16–12 Ma basalt that was erupted in the back-arc but was produced by a similar degree of melting to frontal-arc basalts erupted at the same time. Variations in ratios of fluid-mobile and -immobile elements and those of melt-mobile and -immobile elements for the 35–0 Ma NE Japan basalts indicate that the principal subduction component added to the source mantle prior to generation of these basalt magmas is a sediment-derived melt. Comparison of Sr and Nd isotopic compositions for Pacific Ocean MORB, the NE Japan basalts and subducting sediments suggests that the isotopic compositions of most post-16 Ma more depleted back-arc basalts can be explained by the addition of 〈2% bulk sediment; the most enriched isotope compositions of the subcontinental lithosphere-derived magmas can be accounted for by addition of a maximum 5–7% Japan Trench Sediment (JTS), if the original Sr and Nd compositions of the lithosphere approximated that of DMM. The Sr and Nd isotope composition of the frontal-arc basalts can be accounted for by the addition of 1–5% JTS. A depleted asthenospheric mantle (DMM-like) upwelling model with interaction between asthenospheric mantle-derived magmas and overlying lithospheric mantle can account for the geochemical characteristics of the 35–0 Ma NE Japan basalts. The frontal-arc magmas were generally generated by higher degrees of melting of the shallower part of the asthenospheric mantle, whereas the back-arc magmas resulted from lower degrees of melting of the deeper part of asthenospheric mantle. These latter magmas underwent interaction with the lithospheric mantle, resulting in more enriched Sr and Nd isotopic signatures for the pre-18 Ma back-arc basalts and post-22 Ma frontal-arc basalts, but less interaction, resulting in more depleted Sr and Nd isotopic signatures, for most of the back-arc basalts younger than 16 Ma.

Description: Mt. Lamington is a composite, dome-forming volcano in Papua New Guinea, sitting on the Papuan Ultramafic Belt (PUB) ophiolite. The 1951 eruption produced andesitic dome lavas with numerous basaltic–andesitic enclaves and a few PUB ultramafic xenoliths. To understand the nature of the 1951 eruption, and to assess the effect of assimilating ophiolitic crust in modifying the geochemistry of arc magmas, we carried out petrological, mineralogical and geochemical studies on andesitic lavas as well as magmatic enclaves and ultramafic inclusions. The mineralogy of the enclaves is dominated by amphibole and plagioclase, similar to the andesitic lava hosts. The textures of the enclaves vary from fine-grained diktytaxitic to coarser-grained plutonic textured. We interpret this variation to result from variable cooling rates in the enclave-forming magma body when it invades the overlying andesite. The diktytaxitic enclaves contain variable proportions of host-derived amph + plag antecrysts and xenocrysts of ol + sp ± cpx ± amph with disequilibrium textures, indicating interaction with host lava and assimilation of foreign materials, respectively. A previous study argued that the olivine xenocrysts with chromian spinel inclusions are derived from the PUB, and thus that the PUB contaminated the Mt. Lamington magmas. We demonstrate that this is highly unlikely on the basis of morphological and compositional discrepancies between PUB ol + sp, sampled in nodules, and the xenocrysts. Mass balance indicates that the high whole-rock Ni contents of enclaves and andesitic hosts can be explained by olivine incorporation and do not require any PUB involvement. The olivines are considered to represent crystal mush fractionated from precursor(s) of andesitic and/or pre-1951 shoshonitic lavas. Their presence in enclaves represents recycling of earlier-fractionated components through magma recharge. We argue that this recycling is an important and underestimated process in shaping arc magmas.

Description: We complete the equisingular deformation classification of irreducible singular plane sextic curves. As a by-product, we also compute the fundamental groups of the complement of all but a few maximizing sextics.

Description: Let $P_{n}(x)= \sum _{i=0}^n \xi _i x^i$ be a Kac random polynomial where the coefficients $\xi _i$ are i.i.d. copies of a given random variable $\xi $ . Our main result is an optimal quantitative bound concerning real roots repulsion. This leads to an optimal bound on the probability that there is a real double root. As an application, we consider the problem of estimating the number of real roots of $P_n$ , which has a long history and in particular was the main subject of a celebrated series of papers by Littlewood and Offord from the 1940s. We show, for a large and natural family of atom variables $\xi $ , that the expected number of real roots of $P_n(x)$ is exactly $({2}/{\pi }) \log n +C +o(1)$ , where $C$ is an absolute constant depending on the atom variable $\xi $ . Prior to this paper, such a result was known only for the case when $\xi $ is Gaussian.

Description: The goal of this article was to study the Iwasawa theory of an abelian variety $A$ that has complex multiplication by a complex multiplication (CM) field $F$ that contains the reflex field of $A$ , which has supersingular reduction at every prime above $p$ . To do so, we make use of the signed Coleman maps constructed in our companion article [Kâzım Büyükboduk and Antonio Lei, ‘Integral Iwasawa theory of motives for non-ordinary primes’, 2014, in preparation, draft available upon request] to introduce signed Selmer groups as well as a signed $p$ -adic $L$ -function via a reciprocity conjecture that we formulate for the (conjectural) Rubin–Stark elements (which is a natural extension of the reciprocity conjecture for elliptic units). We then prove a signed main conjecture relating these two objects. To achieve this, we develop along the way a theory of Coleman-adapted rank- $g$ Euler–Kolyvagin systems to be applied with Rubin–Stark elements and deduce the main conjecture for the maximal $\mathbb {Z}_p$ -power extension of $F$ for the primes failing the ordinary hypothesis of Katz.

Description: We introduce a notion of rough paths on embedded submanifolds and demonstrate that this class of rough paths is natural. On the way, we develop a notion of rough integration and an efficient and intrinsic theory of rough differential equations (RDEs) on manifolds. The theory of RDEs is then used to construct parallel translation along manifold-valued rough paths. Finally, this framework is used to show that there is a one-to-one correspondence between rough paths on a $d$ -dimensional manifold and rough paths on $d$ -dimensional Euclidean space. This last result is a rough path analogue of Cartan's development map and its stochastic version which was developed by Eells and Elworthy and Malliavin.

Description: We show that several important normal subgroups $\Gamma $ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure-preserving action $\Gamma \curvearrowright X$ is stably $OE$ -superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from Chifan–Ioana–Kida [‘ $W^*$ -superrigidity for arbitrary actions of central quotients of braid groups’, Math. Ann. 361 (2015) 925–959], we deduce that any free, ergodic, probability measure-preserving action of almost any surface braid group is stably $W^*$ -superrigid, that is, it can be completely reconstructed from its von Neumann algebra.

Description: Motivated by recent work in the mathematics and engineering literature, we study integrability and non-tangential regularity on the two-torus for rational functions that are holomorphic on the bidisk. One way to study such rational functions is to fix the denominator and look at the ideal of polynomials in the numerator such that the rational function is square integrable. A concrete list of generators is given for this ideal as well as a precise count of the dimension of the subspace of numerators with a specified bound on bidegree. The dimension count is accomplished by constructing a natural pair of commuting contractions on a finite-dimensional Hilbert space and studying their joint generalized eigenspaces. Non-tangential regularity of rational functions on the polydisk is also studied. One result states that rational inner functions on the polydisk have non-tangential limits at every point of the $n$ -torus. An algebraic characterization of higher non-tangential regularity is given. We also make some connections with the earlier material and prove that rational functions on the bidisk which are square integrable on the two-torus are non-tangentially bounded at every point. Several examples are provided.