The relationship between the structure and the properties of a drug or material is a key concept of chemistry. Knowledge of the three-dimensional structure is considered to be of such importance that almost every report of a new chemical compound is accompanied by an X-ray crystal structure – at least since the 1970s when diffraction equipment became widely available. Crystallographic software of that time was restricted to very limited computing power, and therefore drastic simplifications had to be made. It is these simplifications that make the determination of the correct structure, especially when it comes to hydrogen atoms, virtually impossible. We have devised a robust and fast system where modern chemical structure models replace the old assumptions, leading to correct structures from the model refinement against standard in-house diffraction data using no more than widely available software and desktop computing power. We call this system NoSpherA2 (Non-Spherical Atoms in Olex2). We explain the theoretical background of this technique and demonstrate the far-reaching effects that the improved structure quality that is now routinely available can have on the interpretation of chemical problems exemplified by five selected examples.

We discuss a Cheeger constant as a mixture of the frustration index and the expansion rate, and prove the related Cheeger inequalities and higher order Cheeger inequalities for graph Laplacians with cyclic signatures, discrete magnetic Laplacians on finite graphs and magnetic Laplacians on closed Riemannian manifolds. In this process, we develop spectral clustering algorithms for partially oriented graphs and multi-way spectral clustering algorithms via metrics in lens spaces and complex projective spaces. As a byproduct, we give a unified viewpoint of Harary’s structural balance theory of signed graphs and the gauge invariance of magnetic potentials.

We derive an optimal eigenvalue ratio estimate for finite weighted graphs satisfying the curvature-dimension inequalityCD(0, ∞). This estimate is independent of the size of the graph and provides a general method to obtain higher-order spectral estimates. The operation of taking Cartesian products is shown to be an efficient way for constructing new weighted graphs satisfyingCD(0, ∞). We also discuss a higher-order Cheeger constant-ratio estimate and related topics about expanders.

We construct an infinite family of triples (G_k,H_k,T_k), where G_k are 2-groups of increasing order, H_k are index 2 subgroups of G_k, and T_k are pairs of generators of H_k. We show that the triples u_k=(G_k,H_k,T_k) are mixed Beauville structures if k is not a power of 2. This is the first known infinite family of 2-groups admitting mixed Beauville structures. Moreover, the associated Beauville surface S(u₃) is real and, for k>3 not a power of 2, the Beauville surface S(u_k) is not biholomorphic to $\overline {S(u_k)}$ .