Topological Properties of Braid-Paths Connected 2-Simplices in Covering Spaces under Cyclic Orientations

In general, the braid structures in a topological space can be classified into algebraic forms and geometric forms. This paper investigates the properties of a braid structure involving 2-simplices and a set of directed braid-paths in view of algebraic as well as geometric topology. The 2-simplices are of the cyclically oriented variety embedded within the disjoint topological covering subspaces where the finite braid-paths are twisted as well as directed. It is shown that the generated homotopic simplicial braids form Abelian groups and the twisted braid-paths successfully admit several varieties of twisted discrete path-homotopy equivalence classes, establishing a set of simplicial fibers. Furthermore, a set of discrete-loop fundamental groups are generated in the covering spaces where the appropriate weight assignments generate multiplicative group structures under a variety of homological formal sums. Interestingly, the resulting smallest non-trivial group is not necessarily unique. The proposed variety of homological formal sum exhibits a loop absorption property if the homotopy path-products are non-commutative. It is considered that the topological covering subspaces are simply connected under embeddings with local homeomorphism maintaining generality.

Computation-informed optimization of Ni(PyC)₂ functionalization for noble gas separations

Metal-organic frameworks (MOFs) are promising nanoporous materials for the adsorptive capture and separation of noble gases at room temperature. Among the numerous MOFs synthesized and tested for noble gas separations, Ni(PyC)₂ (PyC = pyridine-4-carboxylate) exhibits one of the highest xenon/krypton selectivities at room temperature. Like lead-optimization in drug discovery, here we aim to tune the chemistry of Ni(PyC)₂, by appending a functional group to its PyC ligands, to maximize its Xe/Kr selectivity. To guide experiments in the laboratory, we virtually screen Ni(PyC-X)₂ (X=functional group) structures for noble gas separations by (i) constructing a library of Ni(PyC-X)₂ crystal structure models then (ii) using molecular simulations to predict noble gas (Xe, Kr, Ar) adsorption and selectivity at room temperature in each structure. The virtual screening predicts several Ni(PyC-X)₂ structures to exhibit a higher Xe/Kr, Xe/Ar, and Kr/Ar selectivity than the parent Ni(PyC)₂ MOF, with Ni(PyC-m-NH₂)₂ among them. In the laboratory, we synthesize Ni(PyC-m-NH₂)₂, determine its crystal structure by X-ray powder diffraction, and measure its Xe, Kr, and Ar adsorption isotherms (298 K). In agreement with our molecular simulations, the Xe/Kr, Xe/Ar, and Kr/Ar selectivities of Ni(PyC-m-NH₂)₂ exceed those of the parent Ni(PyC)₂. Particularly, Ni(PyC-m-NH₂)₂ exhibits a [derived from experimental, equilibrium adsorption isotherms] Xe/Kr selectivity of 20 at dilute conditions and 298 K, compared to 17 for Ni(PyC)₂. According to in situ X-ray diffraction, corroborated by molecular models, Ni(PyC-m-NH₂)₂ presents well-defined binding pockets tailored for Xe and organized along its one-dimensional channels. In addition to discovering the new, performant Ni(PyC-m-NH₂)₂ MOF for noble gas separations, our study illustrates the computation-informed optimization of the chemistry of a "lead" MOF to target adsorption of a specific gas.

Quantification of geometric errors made simple: application to main-group molecular structures

Nearly all electronic structure simulations begin with obtaining approximate geometries, making a systematic quantification of errors in approximate molecular structures of key importance. Recently, the geometric energy offset (GEO) framework based on a single and natural measure for quantifying and analysing these errors has been proposed [J. Phys. Chem. Lett. 2020, 11, 99579964]. An accurate and way less costly approximation to GEO is utilized here to readily quantify errors in main-group structures and analyze them in a chemically intuitive way. The use of semiexperimental geometries as a reference further simplifies the analysis. The analysis reveals new insights into the geometric performance of methods, new rankings, as well as patterns across different classes of methods and basis sets that arise from the analysis.

Identification of Alnus glutinosa L. and A. incana (L.) Moench. Hybrids in Natural Forests Using Nuclear DNA Microsatellite and Morphometric Markers

Two alder species (Alnus glutinosa and A. incana) have overlapping distribution, naturally occur in Lithuania, and are considered ecologically and economically important forest tree species. The objective of our study was to estimate the likelihood of spontaneous hybridizations between native alders in natural stands of Lithuania based on leaf morphology and nuclear microsatellite markers. The sampled trees were assigned to the three taxonomic groups of A. glutinosa, A. incana, and potential hybrids based on the leaf and bark morphological traits. The genetic differentiation and potential hybridization between these three groups was tested based on 15 nSSR markers. We identified studied Alnus spp. individuals as pure species and hybrids. Two microsatellite loci were reported as discriminating well between these species. We concluded that our results showed the highest likelihood of two genetic group structures, a clear genetic differentiation between the morphology-based groups of A. glutinosa and A. incana, and rather variable likelihood values in the putative hybrid group. The results provide important implications for genetic conservation and management of Alnus spp.

Group Companies and Climate Justice

A string of corporate litigation cases in the United Kingdom highlights the role of corporate group structures in complicating efforts to impose liability on parent companies for the activities of their subsidiaries, particularly where those subsidiaries are located in the Global South. Corporate group structures serve to insulate parent companies against liability for actions of their subsidiaries. This is the case even where economic benefits accrue to parent companies, which are often incorporated in the Global North. These group structures cabin liability for environmental and climate harms within subsidiary companies through reliance on company law principles such as limited liability and separate legal personality. These company law principles allow parent companies to enjoy corporate profits from the activities of their subsidiaries but disavow liability for any environmental damage resulting from such activities. This dichotomy has obvious equity implications, which are exacerbated in the extractive industries and in the context of climate change. Negative climate impacts are and will be felt predominantly in the Global South. In addition, environmental damage removes avenues of climate adaptation for vulnerable populations. But company law principles are not impervious to these equity challenges. These principles have never been absolute and courts have consistently found exceptions to them, although those exceptions have fluctuated in effectiveness and frequency over the years. Recent decisions by the Court of Appeal and Supreme Court in the United Kingdom imposed duties on parent companies for environmental damage caused by their subsidiaries. Cases following the decision in Chandler v Cape Industries illustrate tension between company law as interpreted in the Global North, and climate and environmental justice as experienced in the Global South. Climate change forces a reconceptualization of company law, including transnational corporate liability. This paper argues that these reconsiderations are not only appropriate, but given the contested histories of many of these companies in the Global South, long overdue.

The Role of Symmetry in Non-Hermitian Scattering1

We review recent work on asymmetric scattering by Non-Hermitian (NH) Hamiltonians. Quantum devices with an asymmetric scattering response to particles incident from right or left in effective ID waveguides will be important to develop quantum technologies. They act as microscopic equivalents of familiar macroscopic devices such as diodes, rectifiers, or valves. The symmetry of the underlying NH Hamiltonian leads to selection rules which restrict or allow asymmetric response. NH-symmetry operations may be organized into group structures that determine equivalences among operations once a symmetry is satisfied. The NH Hamiltonian posseses a particular symmetry if it is invariant with respect to the corresponding symmetry operation, which can be conveniently expressed by a unitary or antiunitary superoperator. A simple group is formed by eight symmetry operations, which include the ones for Parity-Time symmetry and Hermiticity as specific cases. The symmetries also determine the structure of poles and zeros of the S matrix. The ground-state potentials for two-level atoms crossing properly designed laser beams realize different NH symmetries to achieve transmission or reflection asymmetries.