Cationic phosphorus-carbon-pnictogen cages isolobal to [C5R5]+

The cationic cages nido-[C2Bu(t)2P2E]+ (E = As, Sb), which are isolobal to the cyclopentadienyl cation, adopt square based pyramidal structures with the heavy pnictogen atom at the apex; NMR and computational methods have been used to probe the dynamic behaviour of the complexes.     

Spectral Exterior Calculus

A spectral approach to building the exterior calculus in manifold learning problems is developed. The spectral approach is shown to converge to the true exterior calculus in the limit of large data. Simultaneously, the spectral approach decouples the memory requirements from the amount of data points and ambient space dimension. To achieve this, the exterior calculus is reformulated entirely in terms of the eigenvalues and eigenfunctions of the Laplacian operator on functions. The exterior derivatives of these eigenfunctions (and their wedge products) are shown to form a frame (a type of spanning set) for appropriate L² spaces of k -forms, as well as higher-order Sobolev spaces. Formulas are derived to express the Laplace-de Rham operators on forms in terms of the eigenfunctions and eigenvalues of the Laplacian on functions. By representing the Laplace-de Rham operators in this frame, spectral convergence results are obtained via Galerkin approximation techniques. Numerical examples demonstrate accurate recovery of eigenvalues and eigenforms of the Laplace-de Rham operator on 1-forms. The correct Betti numbers are obtained from the kernel of this operator approximated from data sampled on several orientable and non-orientable manifolds, and the eigenforms are visualized via their corresponding vector fields. These vector fields form a natural orthonormal basis for the space of square-integrable vector fields, and are ordered by a Dirichlet energy functional which measures oscillatory behavior. The spectral framework also shows promising results on a non-smooth example (the Lorenz 63 attractor), suggesting that a spectral formulation of exterior calculus may be feasible in spaces with no differentiable structure. © 2020 Wiley Periodicals, Inc.     

On rank and null space computation of the generalized Sylvester matrix

In this paper, we present new approaches computing the rank and the null space of the (m n + p)×(n + p) generalized Sylvester matrix of (m + 1) polynomials of maximal degrees n,p. We introduce an algorithm which handles directly a modification of the generalized Sylvester matrix, computing efficiently its rank and null space and replacing n by log ₂ n in the required complexity of the classical methods. We propose also a modification of the Gauss-Jordan factorization method applied to the appropriately modified Sylvester matrix of two polynomials for computing simultaneously its rank and null space. The methods can work numerically and symbolically as well and are compared in respect of their error analysis, complexity and efficiency. Applications where the computation of the null space of the generalized Sylvester matrix is required, are also given.     

Developing automated valuation models for estimating property values: a comparison of global and locally weighted approaches

Annals of Operations Research - Automated valuation models are widely used in real estate to provide estimates for property prices. Such models are typically developed through regression...     

Kβ X-Ray Emission Spectroscopic Study of a Second-Row Transition Metal (Mo) and Its Application to Nitrogenase-Related Model Complexes

In recent years, X-ray emission spectroscopy (XES) in the Kβ (3p-1s) and valence-to-core (valence-1s) regions has been increasingly used to study metal active sites in (bio)inorganic chemistry and catalysis, providing information about the metal spin state, oxidation state and the identity of coordinated ligands. However, to date this technique has been limited almost exclusively to first-row transition metals. In this work, we present an extension of Kβ XES (in both the 4p-1s and valence-to-1s [or VtC] regions) to the second transition row by performing a detailed experimental and theoretical analysis of the molybdenum emission lines. It is demonstrated in this work that Kβ₂ lines are dominated by spin state effects, while VtC XES of a 4d transition metal provides access to metal oxidation state and ligand identity. An extension of Mo Kβ XES to nitrogenase-relevant model complexes shows that the method is sufficiently sensitive to act as a spectator probe for redox events that are localized at the Fe atoms. Mo VtC XES thus has promise for future applications to nitrogenase, as well as a range of other Mo-containing biological cofactors. Further, the clear assignment of the origins of Mo VtC XES features opens up the possibility of applying this method to a wide range of second-row transition metals, thus providing chemists with a site-specific tool for the elucidation of 4d transition metal electronic structure.