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1. Complexes from complexes: Finite element complexes in three dimensions

   https://doi.org/10.1090/mcom/4079  

   Chen, Long; Huang, Xuehai (February 2025, Mathematics of Computation)

   In the field of solving partial differential equations (PDEs), Hilbert complexes have become highly significant. Recent advances focus on creating new complexes using the Bernstein-Gelfand-Gelfand (BGG) framework, as shown by Arnold and Hu [Found. Comput. Math. 21 (2021), pp. 1739–1774]. This paper extends their approach to three-dimensional finite element complexes. The finite element Hessian, elasticity, and divdiv complexes are systematically derived by applying techniques such as smooth finite element de Rham complexes, the $t$- $n$decomposition, and trace complexes, along with related two-dimensional finite element analogs. The construction includes two reduction operations and one augmentation operation to address continuity differences in the BGG diagram, ultimately resulting in a comprehensive and effective framework for constructing finite element complexes, which have various applications in PDE solving. 

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2. Perfluoroolefin complexes versus perfluorometallacycles and perfluorocarbene complexes in cyclopentadienylcobalt chemistry

   https://doi.org/10.1039/C9CP06685C  

   Wen, Limei; Li, Guoliang; Xie, Yaoming; King, R. Bruce; Schaefer, Henry F. (April 2020, Physical Chemistry Chemical Physics)

   Fluorocarbons have been shown experimentally by Baker and coworkers to combine with the cyclopentadienylcobalt (CpCo) moiety to form fluoroolefin and fluorocarbene complexes as well as fluorinated cobaltacyclic rings. In this connection density functional theory (DFT) studies on the cyclopentadienylcobalt fluorocarbon complexes CpCo(L)(C n F 2n ) (L = CO, PMe 3 ; n = 3 and 4) indicate structures with perfluoroolefin ligands to be the lowest energy structures followed by perfluorometallacycle structures and finally by structures with perfluorocarbene ligands. Thus, for the CpCo(L)(C 3 F 6 ) (L = CO, PMe 3 ) complexes, the perfluoropropene structure has the lowest energy, followed by the perfluorocobaltacyclobutane structure and the perfluoroisopropylidene structure less stable by 8 to 11 kcal mol −1 , and the highest energy perfluoropropylidene structure less stable by more than 12 kcal mol −1 . For the two metal carbene structures Cp(L)CoC(CF 3 ) 2 and Cp(L)CoCF(C 2 F 5 ), the former is more stable than the latter, even though the latter has Fischer carbene character. For the CpCo(L)(C 4 F 8 ) (L = CO, PMe 3 ) complexes, the perfluoroolefin complex structures have the lowest energies, followed by the perfluorometallacycle structures at 10 to 20 kcal mol −1 , and the structures with perfluorocarbene ligands at yet higher energies more than 20 kcal mol −1 above the lowest energy structure. This is consistent with the experimentally observed isomerization of the perfluorinated cobaltacyclobutane complexes CpCo(PPh 2 Me)(–CFR–CF 2 –CF 2 –) (R = F, CF 3 ) to the perfluoroolefin complexes CpCo(PPh 2 Me)(RCFCF 2 ) in the presence of catalytic quantities of HN(SO 2 CF 3 ) 2 . Further refinement of the relative energies by the state-of-the-art DLPNO-CCSD(T) method gives results essentially consistent with the DFT results summarized above. 

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3. Copper Complexes with Diazoolefin Ligands and their Photochemical Conversion into Alkenylidene Complexes

   https://doi.org/10.1002/anie.202214899  

   Kooij, Bastiaan; Varava, Paul; Fadaei‐Tirani, Farzaneh; Scopelliti, Rosario; Pantazis, Dimitrios A.; Van Trieste, III, Gerard P.; Powers, David C.; Severin, Kay (December 2022, Angewandte Chemie International Edition)

   Abstract Homometallic copper complexes with alkenylidene ligands are discussed as intermediates in catalysis but the isolation of such complexes has remained elusive. Herein, we report the structural characterization of copper complexes with bridging and terminal alkenylidene ligands. The compounds were obtained by irradiation of Cu^Icomplexes with N‐heterocyclic diazoolefin ligands. The complex with a terminal alkenylidene ligand required isolation in a crystalline matrix, and its structural characterization was enabled byin crystallophotolysis at low temperature. 

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4. On groups and simplicial complexes

   https://doi.org/10.1016/j.ejc.2018.01.009  

   Lubotzky, Alexander; Luria, Zur; Rosenthal, Ron (May 2018, European Journal of Combinatorics)
5. Tropical fans and normal complexes

   https://doi.org/10.1016/j.aim.2023.108981  

   Nathanson, Anastasia; Ross, Dustin (May 2023, Advances in Mathematics)