ALBERT

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: Pointwise error estimates for the first-order div least-squares (LS) finite element method for second-order elliptic partial differential equations are presented. Direct flux approximation is considered as an important advantage of the LS method. However, there are no known pointwise error estimates for the direct flux approximation. In this paper, we provide optimal pointwise estimates which show local dependence of the error at a point and weak dependence of the global norm. As an elementary consequence of these estimates, we provide an asymptotic error expansion inequality. The inequality has applications to superconvergence and a posteriori estimates.

Description: The first-order and higher-order derivatives of a function can be viewed as the solutions of Volterra integral equations of the first kind. In this paper we propose a fast multiscale solver for the numerical solution of the Tikhonov regularization of the Volterra equations. In association with the special form of the kernels, the matrices resulting from the discretization by multiscale bases are sparse. Moreover, they can be truncated using proper strategies with only a minor loss of accuracy. In the best case, the number of nonzero entries of the truncated matrices is linear with respect to the dimensions of the matrices. The accuracy of the solution from the solver is analysed theoretically and verified by numerical experiments.

Description: The finite element method with $\mathscr {Q}_p$ elements is applied to a singularly perturbed convection–diffusion problem on an L-shaped domain. As an effect of corner singularities the exact solution is not $H^2$ -regular. Therefore, we combine a layer-adapted Shishkin mesh with a special grading adapted to the corner singularity. On such meshes we prove error estimates and estimates for the closeness error which explicitly show the influence of the grading parameter $\mu$ . Hence, $\mu$ can be chosen such that optimal error bounds are obtained. Thereby, it turns out that in the problem studied the influence of the corner singularity becomes small if the perturbation parameter $\varepsilon$ decreases. Moreover, we conduct numerical experiments that verify the theoretical results.

Description: Pták's method of nondiscrete induction is based on the idea that in the analysis of iterative processes one should aim at rates of convergence as functions rather than just numbers, because functions may give convergence estimates that are tight throughout the iteration rather than just asymptotically. In this paper we motivate and prove a theorem on nondiscrete induction, originally due to Potra and Pták, and we apply it to the Newton iterations for computing the matrix polar decomposition and the matrix square root. Our goal is to illustrate the application of the method of nondiscrete induction in the finite-dimensional numerical linear algebra context. We show the sharpness of the resulting convergence estimate analytically for the polar decomposition iteration and on some examples for the square root iteration. We also discuss some of the method's limitations and possible extensions.

Description: We consider the numerical solution, by a Petrov–Galerkin finite-element method, of a singularly perturbed reaction–diffusion differential equation posed on the unit square. In Lin & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. , 50 , 2729–2743), it is argued that the natural energy norm, associated with a standard Galerkin approach, is not an appropriate setting for analysing such problems, and there they propose a method for which the natural norm is ‘balanced’. In the style of a first-order system least squares method, we extend the approach of Lin & Stynes (2012, A balanced finite element method for singularly perturbed reaction-diffusion problems. SIAM J. Numer. Anal. , 50 , 2729–2743) by introducing a constraint which simplifies the associated finite-element space and the method's analysis. We prove robust convergence in a balanced norm on a piecewise-uniform (Shishkin) mesh, and present supporting numerical results. Finally, we demonstrate how the resulting linear systems are solved optimally using multigrid methods.

Description: The construction of tensor-product surface patches with a family of Pythagorean-hodograph (PH) isoparametric curves is investigated. The simplest nontrivial instances, interpolating four prescribed patch boundary curves, involve degree $(5,4)$ tensor-product surface patches $\bf{x}(u,v)$ whose $v=\hbox {constant}$ isoparametric curves are all spatial PH quintics. It is shown that the construction can be reduced to solving a novel type of quadratic quaternion equation, in which the quaternion unknown and its conjugate exhibit left and right coefficients, while the quadratic term has a coefficient interposed between them. A closed-form solution for this type of equation is derived, and conditions for the existence of solutions are identified. The surfaces incorporate three residual scalar freedoms which can be exploited to improve the interior shape of the patch. The implementation of the method is illustrated through a selection of computed examples.

Description: Interior eigenvalues of bounded scattering objects can be rigorously characterized from multi-static and multi-frequency far field data, that is, from the behaviour of scattered waves far away from the object. This characterization, the so-called inside–outside duality, holds for various types of penetrable and impenetrable scatterers and is based on the behaviour of a particular eigenvalue of the far field operator. It naturally leads to a numerical algorithm for computing interior eigenvalues of a scatterer that does not require shape or physical properties of the scatterer as input. Since the nonlinear inverse problem to compute such interior eigenvalues from far field data is ill-posed, we propose a regularizing algorithm that is shown to converge as the noise level of the far field data tends to zero. We illustrate feasibility and accuracy of our algorithm by numerical experiments where we compute interior transmission eigenvalues and Robin eigenvalues of the Laplacian in three-dimensional domains from scattering data of these domains due to plane incident waves.

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: We derive optimal-order a posteriori error estimates for fully discrete approximations of initial and boundary value problems for linear parabolic equations. For the discretization in time we apply the fractional-step $\vartheta $ -scheme, and for the discretization in space the finite element method with finite element spaces that are allowed to change with time. The first optimal-order a posteriori error estimates for the norms of $L^\infty (0,T;L^2(\varOmega ))$ and $L^2(0,T;H^1(\varOmega ))$ are derived by applying the reconstruction technique.

Description: In this paper, we propose a fast and accurate numerical method based on Fourier transforms to solve Kolmogorov forward equations of symmetric scalar Lévy processes. The method is based on the accurate numerical formulas for Fourier transforms proposed by Ooura. These formulas are combined with nonuniform fast Fourier transforms (FFT) and fractional FFT to speed up the numerical computations. Moreover, we propose a formula for numerical indefinite integration on equispaced grids as a component of the method. The proposed integration formula is based on the sinc-Gauss sampling formula, which is a function approximation formula. This integration formula is also combined with the FFT. Therefore, all steps of the proposed method are executed using the FFT and its variants. The proposed method allows us to be free from some special treatments for a nonsmooth initial condition and numerical time integration. The numerical solutions obtained by the proposed method appear to be exponentially convergent on the interval if the corresponding exact solutions do not have sharp cusps. Furthermore, the real computational times are approximately consistent with the theoretical estimates.

Description: In this paper, we consider the heat equation coupled with Darcy's law with a nonlinear source term describing heat production due to an exothermic chemical reaction. The existence and uniqueness of a solution are established. Next, a spectral discretization of the problem is presented and thoroughly analysed. Finally, we present some numerical experiments which confirm the interest of the discretization.

Description: The convergence of the so-called quadratic method for computing eigenvalue enclosures of general self-adjoint operators is examined. Explicit asymptotic bounds for convergence to isolated eigenvalues are found. These bounds turn out to improve significantly upon those determined in previous investigations. The theory is illustrated by means of several numerical experiments performed on particularly simple benchmark models of one-dimensional Schrödinger operators.

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: We consider an elliptic optimal control problem where the objective functional contains evaluations of the state at a finite number of points. In particular, we use a fidelity term that encourages the state to take certain values at these points, which means that our problem is related to ones with state constraints at points. The analysis and numerical analysis differs from when the fidelity is in the $L^2$ -norm because we need the state space to embed into the space of continuous functions. In this paper, we discretize the problem using two different piecewise linear finite element methods. For each discretization we use two different approaches to prove a priori $L^2$ -error estimates for the control. We discuss the differences between these methods and approaches, and present numerical results that agree with our analytical results.

Description: A numerical scheme for the approximation of large vibration deformations of inextensible elastic curves is devised. Its unconditional stability and convergence under a regularity assumption on the velocities are demonstrated and numerical experiments are provided.

Description: We present a new scaleable algorithm for approximating the $H_{\infty }$ norm, an important robust stability measure for linear dynamical systems with input and output. Our spectral-value-set-based method uses a novel hybrid expansion–contraction scheme that, under reasonable assumptions, is guaranteed to converge to a stationary point of the optimization problem defining the $H_{\infty }$ norm, and, in practice, typically returns local or global maximizers. We prove that the hybrid expansion–contraction method has a quadratic rate of convergence that is also confirmed in practice. In comprehensive numerical experiments, we show that our new method is not only robust but exceptionally fast, successfully completing a large-scale test set 25 times faster than an earlier method by Guglielmi, Gürbüzbalaban & Overton (2013, SIAM J. Matrix Anal. Appl. , 34 , 709–737), which occasionally breaks down far from a stationary point of the underlying optimization problem.

Description: The first part of this paper enfolds a medius analysis for mixed finite element methods (FEMs) and proves a best-approximation result in $L^2$ for the stress variable independent of the error of the Lagrange multiplier under stability, compatibility and efficiency conditions. The second part applies the general result to the FEM of Arnold and Winther for linear elasticity: the stress error in $L^2$ is controlled by the $L^2$ best-approximation error of the true stress by any discrete function plus data oscillations. The analysis is valid without any extra regularity assumptions on the exact solution and also covers coarse meshes and Neumann boundary conditions. Further applications include Raviart–Thomas finite elements for the Poisson and the Stokes problems. The result has consequences for nonlinear approximation classes related to adaptive mixed FEMs.

Description: We propose a numerical method to approximate the solution of a nonlocal diffusion problem on a general setting of metric measure spaces. These spaces include, but are not limited to, fractals, manifolds and Euclidean domains. We obtain error estimates in $L^\infty (L^p)$ for $p=1,\infty $ under the sole assumption of the initial datum being in $L^p$ . An improved bound for the error in $L^\infty (L^1)$ is obtained when the initial datum is in $L^2$ . We also derive some qualitative properties of the solutions like stability, comparison principles and study the asymptotic behaviour as $t\to \infty $ . We finally present two examples on fractals: the Sierpinski gasket and the Sierpinski carpet, which illustrate on the effect of nonlocal diffusion for piece-wise constant initial datum.

Description: This paper presents a Riemannian trust region algorithm for unconstrained optimization problems with locally Lipschitz objective functions defined on complete Riemannian manifolds. To this end we define a function $\Phi :TM\rightarrow \mathbb {R}$ on the tangent bundle $TM$ , and at the $k$ th iteration, using the restricted function $\Phi |_{T_{x_k}M}$ , where $T_{x_k}M$ is the tangent space at $x_k$ , a local model function $Q_k$ that carries both first- and second-order information for the locally Lipschitz objective function $f:M\rightarrow \mathbb {R}$ on a Riemannian manifold $M$ , is defined and minimized over a trust region. We establish the global convergence of the proposed algorithm. Moreover, using the Riemannian $\varepsilon $ -subdifferential, a suitable model function is defined. Numerical experiments illustrate our results.

Description: Variational inequalities play an important role in many applications and are an active research area. Optimal a priori error estimates in the natural energy norm do exist, but only very few results are known for different norms. Here, we consider as prototype a simple Signorini problem, and provide new optimal order a priori error estimates for the trace and the flux on the Signorini boundary. The a priori analysis is based on a continuous and a discrete Steklov–Poincaré operator, as well as on Aubin–Nitsche-type duality arguments. Numerical results illustrate the convergence rates of the finite-element approach.

Description: We show for adaptive triangulations in two dimensions, which are generated by the newest vertex bisection, an optimal grading estimate. Roughly speaking, we construct from the piecewise constant mesh-size function a regularized one with the following two properties. First, the two functions are equivalent, and second, the regularized mesh-size function differs at most by a factor of 2 on neighbouring elements. In combination with Bank & Yserentant (2014, Numer. Math. 126 , 361–381), this optimal grading estimate enables us to show that the $L_2$ -orthogonal projections onto the space of continuous Lagrange finite elements up to order 12 is $H^1$ -stable. We extend these results to a modified red–green refinement.

Description: There exist at least three first-order finite volume element methods with totally different dual meshes and conforming or nonconforming $P_1$ finite element trial functions and the question arises of whether they are comparable. The fact that the underlying norms are very different does not prevent the proof that the errors are equivalent on any mesh in some norm. This equivalence is independent of the regularity of the exact solution and holds for any coarse or fine mesh with or without local mesh refining, but up to equivalence constants and additional explicit data-oscillation terms of higher order. The equivalence constants depend on the minimal angle of the shape-regular triangulation and the penalization parameter of the discontinuous Galerkin scheme. This also implies quasi-optimality in the sense that the error is bounded by the best approximation of the flux by piecewise constants. An a posteriori error analysis for the discontinuous Galerkin finite volume element scheme is also discussed. The analysis is exemplified for a boundary value model problem for some second-order elliptic partial differential equation in two dimensions.

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: We show the existence and uniqueness of a solution for the nonlocal vector-valued Allen–Cahn variational inequality in a formulation involving Lagrange multipliers for local and nonlocal constraints. Furthermore, we propose and analyse a primal–dual active set (PDAS) method for local and nonlocal vector-valued Allen–Cahn variational inequalities. The local convergence behaviour of the PDAS algorithm is studied by interpreting the approach as a semismooth Newton method and numerical simulations are presented demonstrating its efficiency.

Description: In this paper, we will present a generalized convolution quadrature for solving linear parabolic and hyperbolic evolution equations. The original convolution quadrature method by Lubich works very nicely for equidistant time steps while the generalization of the method and its analysis to nonuniform time stepping is by no means obvious. We will introduce the generalized convolution quadrature allowing for variable time steps and develop a theory for its error analysis. This method opens the door for further development towards adaptive time stepping for evolution equations. As the main application of our new theory, we will consider the wave equation in exterior domains which is formulated as a retarded boundary integral equation.

Description: We address the error control of Galerkin discretization (in space) of linear second-order hyperbolic problems. More specifically, we derive a posteriori error bounds in the L ( L 2 ) norm for finite element methods for the linear wave equation, under minimal regularity assumptions. The theory is developed for both the space-discrete case and for an implicit fully discrete scheme. The derivation of these bounds relies crucially on carefully constructed space and time reconstructions of the discrete numerical solutions, in conjunction with a technique introduced by Baker (1976, Error estimates for finite element methods for second-order hyperbolic equations. SIAM J. Numer. Anal. , 13 , 564–576) in the context of a priori error analysis of Galerkin discretization of the wave problem in weaker-than-energy spatial norms.

Description: The surface finite element method can be used to approximate curvatures on embedded hypersurfaces and to discretize geometric partial differential equations. In this paper, we present a definition of discrete Ricci curvature on polyhedral hypersurfaces of arbitrary dimension based on the discretization of a weak formulation with isoparametric finite elements. We prove that for a piecewise quadratic approximation of a two- or three-dimensional hypersurface R n +1 , this definition approximates the Ricci curvature of with a linear order of convergence in the L 2 ( ) norm. By using a smoothing scheme in the case of a piecewise linear approximation of , we still get a convergence of order 2/3 in the L 2 ( ) norm and of order 1/3 in the W 1, 2 ( ) norm.

Description: We give general conditions which guarantee that the sequence generated by a descent algorithm converges to an equilibrium point. The convergence result is based on the Lojasiewicz gradient inequality; optimal convergence rates are also derived, as well as a stability result. We show how our results apply to a large variety of standard time discretizations of gradient-like flows. Schemes with variable time step are considered and optimal conditions on the maximal step size are derived. Applications to time and space discretizations of the Allen–Cahn equation, the sine–Gordon equation and a damped wave equation are given.

Description: This paper presents quadratic finite-volume methods for elliptic and parabolic problems on quadrilateral meshes that use Barlow points (optimal stress points) for dual partitions. Introducing Barlow points into the finite-volume formulations results in better approximation properties at the cost of loss of symmetry. The novel ‘symmetrization’ technique adopted in this paper allows us to derive optimal-order error estimates in the H 1 - and L 2 -norms for elliptic problems and in the L ( H 1 )- and L ( L 2 )-norms for parabolic problems. Superconvergence of the difference between the gradients of the finite-volume solution and the interpolant can also be derived. Numerical results confirm the proved error estimates.

Description: A linear parabolic differential equation on a moving surface is discretized in space by evolving-surface finite elements and in time by backward difference formulas (BDFs). Using results from Dahlquist's G-stability theory and Nevanlinna & Odeh's multiplier technique together with properties of the spatial semidiscretization, stability of the full discretization is proved for BDF methods up to order 5 and optimal-order convergence is shown. Numerical experiments illustrate the behaviour of the fully discrete method.

Description: In this article, we develop the a priori and a posteriori error analysis of hp -version interior penalty discontinuous Galerkin finite element methods for strongly monotone quasi-Newtonian fluid flows in a bounded Lipschitz domain R d , d = 2, 3. In the latter case, computable upper and lower bounds on the error are derived in terms of a natural energy norm, which are explicit in the local mesh size and local polynomial degree of the approximating finite element method. A series of numerical experiments illustrate the performance of the proposed a posteriori error indicators within an automatic hp -adaptive refinement algorithm.

Description: In recent years, it has become increasingly clear that the critical issue in gradient methods is the choice of the step length, whereas using gradient as the search direction may lead to very effective algorithms, whose surprising behaviour has only been partially explained, mostly in terms of the spectrum of the Hessian matrix. On the other hand, the convergence of the classical Cauchy steepest descent (SD) method has been analysed extensively and related to the spectral properties of the Hessian matrix, but the connection with the spectrum of the Hessian has not been exploited much to modify the method in order to improve its behaviour. In this work, we show how, for convex quadratic problems, moving from some theoretical properties of the SD method, second-order information provided by the step length can be exploited to dramatically improve the usually poor practical behaviour of this method. This allows us to achieve computational results comparable with those of the Barzilai and Borwein algorithm, with the further advantage of monotonic behaviour.

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: In recent years, there has been an enormous interest in developing methods for the approximation of manifold-valued functions. In this paper, we focus on the manifold of symmetric positive-definite (SPD) matrices. We investigate the use of SPD-matrix means to adapt linear positive approximation methods to SPD-matrix-valued functions. Specifically, we adapt corner-cutting subdivision schemes and Bernstein operators. We present the concept of admissible matrix means and study the adapted approximation schemes based on them. Two important cases of admissible matrix means are treated in detail: the exp–log and the geometric matrix means. We derive special properties of the approximation schemes based on these means. The geometric mean is found to be superior in the sense of preserving more properties of the data, such as monotonicity and convexity. Furthermore, we give error bounds for the approximation of univariate SPD-matrix-valued functions by the adapted operators.

Description: We present a mass-preserving scheme for the stochastic nonlinear Schrödinger equation with multiplicative noise of Stratonovich type. It is a splitting scheme and we present an explicit formula for solving the sub-step related to the nonlinear part. The scheme is unconditionally stable in the L 2 norm. For the linear stochastic Schrödinger equation, we prove that the scheme has a strong convergence rate in time equal to 1, which is not common for stochastic partial differential equations with noise depending on space and time.

Description: A numerical scheme for the approximation of the elastic flow of inextensible curves is devised and convergence of approximations to exact solutions of the nonlinear time-dependent partial differential equation is proved. The nonlinear, pointwise constraint of local length preservation is linearized about a previous solution in each time step which leads to a sequence of linear saddle-point problems. The spatial discretization is based on piecewise Bézier curves and the resulting semiimplicit scheme is unconditionally stable and convergent.

Description: We study the coercivity properties and the norm dependence on the wave-number k of certain regularized combined field boundary integral operators that we recently introduced for the solution of two- and three-dimensional acoustic scattering problems with Neumann boundary conditions. We show that in the case of circular and spherical boundaries, our regularized combined field boundary integral operators are L 2 coercive for large enough values of the coupling parameter, and that the norms of these operators are bounded by constant multiples of the coupling parameter. We establish that the norms of the regularized combined field boundary integral operators grow modestly with the wave-number k for smooth boundaries and we provide numerical evidence that these operators are L 2 coercive for two-dimensional starlike boundaries. We present and analyse a fully discrete collocation (Nyström) method for the solution of two-dimensional acoustic scattering problems with Neumann boundary conditions based on regularized combined field integral equations. In particular, for analytic boundaries and boundary data, we establish pointwise superalgebraic convergence rates of the discrete solutions.

Description: The classical theory of Gaussian quadrature assumes a positive weight function. We will show that in some cases Gaussian rules can be constructed with respect to an oscillatory weight, yielding methods with complex quadrature nodes and positive weights. These rules are well suited to highly oscillatory integrals because they attain optimal asymptotic order. We show that, for the Fourier oscillator, this approach yields the numerical method of steepest descent, a method with optimal asymptotic order that has previously been proposed for this class of integrals. However, the approach readily extends to more general kernels, such as Bessel functions that appear as the kernel of the Hankel transform.

Description: We consider anisotropic Allen–Cahn equations with interfacial energy induced by an anisotropic surface energy density . Assuming that is positive, positively homogeneous of degree 1, strictly convex in tangential directions to the unit sphere and sufficiently smooth, we show the stability of various time discretizations. In particular, we consider a fully implicit and a linearized time discretization of the interfacial energy combined with implicit and semiimplicit time discretizations of the double-well potential. In the semiimplicit variant, concave terms are taken explicitly. The arising discrete spatial problems are solved by globally convergent truncated nonsmooth Newton multigrid methods. Numerical experiments show the accuracy of the different discretizations. We also illustrate that pinch-off under anisotropic mean curvature flow is no longer invariant under rotation of the initial configuration for a fixed orientation of the anisotropy.

Description: In this paper, we define a new finite element method for numerically approximating the solution of a partial differential equation in a bulk region coupled with a surface partial differential equation posed on the boundary of the bulk domain. The key idea is to take a polyhedral approximation of the bulk region consisting of a union of simplices, and to use piecewise polynomial boundary faces as an approximation of the surface. Two finite element spaces are defined, one in the bulk region and one on the surface, by taking the set of all continuous functions which are also piecewise polynomial on each bulk simplex or boundary face. We study this method in the context of a model elliptic problem; in particular, we look at well-posedness of the system using a variational formulation, derive perturbation estimates arising from domain approximation and apply these to find the optimal-order error estimates. A numerical experiment is described which demonstrates the order of convergence.

Description: As a model of more general contour integration problems we consider the numerical calculation of high-order derivatives of holomorphic functions using Cauchy's integral formula. Bornemann (2011, Accuracy and stability of computing high-order derivatives of analytic functions by Cauchy integrals. Found. Comput. Math. , 11 , 1–63) showed that the condition number of the Cauchy integral strongly depends on the chosen contour and solved the problem of minimizing the condition number for circular contours. In this paper, we minimize the condition number within the class of grid paths of step size h using Provan's algorithm for finding a shortest enclosing walk in weighted graphs embedded in the plane. Numerical examples show that optimal grid paths yield small condition numbers even in those cases where circular contours are known to be of limited use, such as for functions with branch-cut singularities.

Description: Anisotropic meshes are important for efficiently resolving incompressible flow problems that include boundary layer or corner singularity phenomena. Unfortunately, the stability of standard inf–sup stable mixed approximation methods is prone to degeneracy whenever the mesh aspect ratio becomes large. As an alternative, a stabilized mixed approximation method is considered here. Specifically, a robust a priori error estimate for the local jump stabilized Q 1 – P 0 approximation introduced by Kechkar & Silvester (1992, Analysis of locally stabilized mixed finite element methods for the Stokes problem. Math. Comp. , 58 , 1–10) is established for anisotropic meshes. Our numerical results demonstrate that the stabilized Q 1 – P 0 method is competitive with the nonconforming, nonparametric, rotated approximation method introduced by Rannacher & Turek (1992, Simple nonconforming quadrilateral Stokes element. Numer. Meth. Partial Differential Equations , 8 , 97–111).

Description: This work is about the numerical solution of the time-domain Maxwell's equations in dispersive propagation media by a discontinuous Galerkin time-domain method. The Debye model is used to describe the dispersive behaviour of the media. The resulting system of differential equations is solved using a centred-flux discontinuous Galerkin formulation for the discretization in space and a second-order leapfrog scheme for the integration in time. The numerical treatment of the dispersive model relies on an auxiliary differential equation approach similar to that which is adopted in the finite difference time-domain method. Stability estimates are derived through energy considerations and convergence is proved for both the semidiscrete and the fully discrete schemes.

Description: The parabolic singularly perturbed problem u xx ( x , t ) – x α u t ( x , t ) = f ( x , t ) is considered on the rectangular domain = (0,1) x (0, T ] with Dirichlet initial and boundary conditions. Here, is a small positive parameter and α is a positive constant. This problem is degenerate since the coefficient x α of u t vanishes along the side x = 0 of . Bounds on the derivatives of u are used to design a nonuniform mesh and a finite difference method on this mesh is constructed to solve the problem numerically. As the solution u is not in general uniformly bounded with respect to in the maximum norm, the convergence analysis of the numerical method requires the use of some unusual barrier functions and a special weighted discrete norm. Numerical examples are provided to support the theoretical results.

Description: We consider the numerical approximation of a general second-order semilinear parabolic stochastic partial differential equation driven by multiplicative and additive space–time noise. We examine convergence of exponential integrators for multiplicative and additive noise. We consider noise that is in a trace class and give a convergence proof in the root-mean-square L 2 norm. We discretize in space with the finite element method and in our implementation we examine both the finite element and the finite volume methods. We present results for a linear reaction–diffusion equation in two dimensions as well as a nonlinear example of a two-dimensional stochastic advection–diffusion–reaction equation motivated from realistic porous media flow.

Description: Sparse grids (Zenger, C. (1990) Sparse grids. Parallel Algorithms for Partial Differential Equations (W. Hackbusch ed.) Notes on Numerical Fluid Dynamics 31. Proceedings of the Sixth GAMM-Seminar; Bungartz, H.-J. & Griebel, M. (2004) Sparse grids. Acta Numer. , 13 , 1–123.) are tailored to the approximation of smooth high-dimensional functions. On a d -dimensional tensor product space, the number of grid points is N = O( h –1 |log h | d –1 ), where h is a mesh parameter. The so-called combination technique, based on hierarchical decomposition and extrapolation, requires specific multivariate error expansions of the discretization error on Cartesian grids to hold. We derive such error expansions for linear difference schemes through an error correction technique of semi-discretizations. We obtain overall error formulae of the type = O ( h p |log h | d –1 ) and analyse the convergence, with its dependence on dimension and smoothness, by examples of linear elliptic and parabolic problems, with numerical illustrations in up to eight dimensions.

Description: The numerical simulation of two-phase flow in a porous medium may lead, when using coupled finite volume schemes on structured grids, to the appearance of the so-called Grid Orientation Effect (GOE). We propose in this paper a procedure to eliminate this phenomenon, based on the use of new fluxes with a new stencil in the discrete version of the convection equation, without changing the discrete scheme for computing the pressure field. Numerical results show that the GOE does not significantly decrease with the size of the discretization using the initial scheme on the coupled problem, but that it is efficiently suppressed by the new procedure, even on coarse meshes. A mathematical study, based on a weak BV inequality using the new fluxes, confirms the convergence of the modified scheme in a particular case.

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.