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Recently, Yu et al. derived a factorization procedure for detecting and computing the potential self-intersection of 3D integral Bézier cubics, claiming that their proposal distinctly outperforms existing methodologies. First, we recall that in the 2D case, explicit formulas already exist for the parameter values at the self-intersection (the singularity called crunode in algebraic geometry). Such values are the solutions of a quadratic equation, and affine invariants depend only on the curve hodograph. Also, the factorization procedure for cubics is well known. Second, we note that only planar Bézier cubics can display a self-intersection, so there is no need to address the problem in the more involved 3D setting. Finally, we elucidate the connections with the previous literature and provide a geometric interpretation, in terms of the affine classification of cubics, of the algebraic conditions necessary for the existence of a self-intersection. Cubics with a self-intersection are affine versions of the celebrated Tschirnhausen cubic. [ABSTRACT FROM AUTHOR]

This article explores the use of copper foam modified with cobalt phthalocyanine carbon nanotube hybrids to improve the selectivity and stability of the CO2 reduction reaction (CO2RR) in a flow cell setup. The CO2RR is a promising method for converting carbon dioxide into useful industrial molecules, and renewable energy sources are of interest for this reaction. Copper is known for its ability to produce C2+ products from CO2RR, but low selectivity and hydrogen evolution reaction (HER) can limit its effectiveness. The incorporation of cobalt phthalocyanine hybrids with carbon-based materials enhances stability, selectivity, and conductivity. The study demonstrates the ability to regulate HER and improve CO2RR activity on copper foam through non-covalent surface modification with cobalt phthalocyanine hybrids. The modified copper foam exhibits a ten-fold increase in CO formation and remains stable over time. The article discusses the successful combination of cobalt phthalocyanine hybrids and carbon nanotubes through axial coordination of carboxylic groups. The CO2RR performance of the modified copper foam electrodes is examined, and the cobalt phthalocyanine hybrid outperforms other electrodes in terms of CO2RR product formation. The long-term stability of the cobalt phthalocyanine hybrid is also demonstrated. [Extracted from the article]

Oceanic intraplate volcanoes sometimes experience late-stage eruptive activity known as rejuvenated volcanism, and contrasting interpretations for its petrogenesis depend on the compositional characteristics. In the Juan Fernández Ridge (JFR), a volcanic chain approximately 800 km in length emplaced on the Nazca Plate, some subaerial occurrences of rejuvenated volcanism have been recognized on the Robinson Crusoe and Santa Clara Islands, both part of the same deeply eroded shield volcano complex. This study aims to understand the origin and magmatic evolution of rejuvenated volcanism on Santa Clara Island, emplaced after ~2.15 Ma of quiescence above the shield sequence, mainly via the analysis of unpublished geochemical and isotopic data. Field reconnaissance identified two nearly coeval rejuvenated sequences on Santa Clara Island: Bahía W (BW) and Morro Spartan (MS), both formed by basanitic and picro-basaltic lava flows with brecciated levels and local intercalations of sedimentary and pyroclastic deposits. In comparison to the chemical signature of the preceding shield-building stage (comprised mainly of basalts and picrites), the two rejuvenated sequences exhibit a notable enrichment in incompatible elements, but the Sr, Nd, and Pb isotopes are very similar to the FOZO mantle endmember, with an apparent additional contribution of HIMU and EM1 components. The geochemistry of lavas revealed the involvement of various processes, including contamination by ultramafic xenoliths, high-pressure fractional crystallization of olivine and clinopyroxene, and potential partial assimilation of oceanic lithospheric components. While the oceanic lithosphere has been considered as a potential source, the isotopic data from Santa Clara lies outside of the mixing curve between depleted mantle (DM, here represented by the North Chile Rise and the East Pacific Rise) and the previous shield stage, suggesting that a lithospheric mantle is not the primary source for the rejuvenated stage volcanism. Therefore, we favor an origin of the rejuvenated volcanism from the mantle plume forming the JFR, supported by similarities in isotopic signatures with the shield stage and high values of 208Pb/204Pb (only comparable to San Félix—San Ambrosio in the vicinity of JFR), implying the presence of a regional source with radiogenic 208Pb/204Pb isotope ratios. In addition, isotopic variations are subparallel to the mixing line between HIMU and EM1 components, whose participation in different proportions might explain the observed trends. In conclusion, we propose that the source of the rejuvenated volcanism on Santa Clara Island is a heterogeneous mantle plume, the same one that fed the shield stage. The rejuvenated volcanism is derived from a secondary melting zone away from the main axis of the plume. [ABSTRACT FROM AUTHOR]