The structures of the isomorphous potassium and rubidium salts of 4‐nitrobenzoic acid and an overview of the metal complex stereochemistries of the alkali metal salt series with this ligand

4‐Nitrobenzoic acid (PNBA) has proved to be a useful ligand for the preparation of metal complexes but the known structures of the alkali metal salts of PNBA do not include the rubidium salt. The structures of the isomorphous potassium and rubidium polymeric coordination complexes with PNBA, namely poly[μ₂‐aqua‐aqua‐μ₃‐(4‐nitrobenzoato)‐potassium], [K(C₇H₄N₂O₂)(H₂O)₂]_n, (I), and poly[μ₃‐aqua‐aqua‐μ₅‐(4‐nitrobenzoato)‐rubidium], [Rb(C₇H₄N₂O₂)(H₂O)₂]_n, (II), have been determined. In (I), the very distorted KO₆ coordination sphere about the K⁺ centres in the repeat unit comprise two bridging nitro O‐atom donors, a single bridging carboxylate O‐atom donor and two water molecules, one of which is bridging. In Rb complex (II), the same basic MO₆ coordination is found in the repeat unit, but it is expanded to RbO₉ through a slight increase in the accepted Rb—O bond‐length range and includes an additional Rb—O_carboxylate bond, completing a bidentate O,O′‐chelate interaction, and additional bridging Rb—O_nitro and Rb—O_water bonds. The comparative K—O and Rb—O bond‐length ranges are 2.7352 (14)–3.0051 (14) and 2.884 (2)–3.182 (2) Å, respectively. The structure of (II) is also isomorphous, as well as isostructural, with the known structure of the nine‐coordinate caesium 4‐nitrobenzoate analogue, (III), in which the Cs—O bond‐length range is 3.047 (4)–3.338 (4) Å. In all three complexes, common basic polymeric extensions are found, including two different centrosymmetric bridging interactions through both water and nitro groups, as well as extensions along c through the para‐related carboxylate group, giving a two‐dimensional structure in (I). In (II) and (III), three‐dimensional structures are generated through additional bridges involving the nitro and water O atoms. In all three structures, the two water molecules are involved in similar intra‐polymer O—H...O hydrogen‐bonding interactions to both carboxylate and water O‐atom acceptors. A comparison of the varied coordination behaviour of the full set of Li–Cs salts with 4‐nitrobenzoic acid is also made.     

Growth and homogeneity of matchbox manifolds

A matchbox manifold with one-dimensional leaves which has equicontinuous holonomy dynamics must be a homogeneous space, and so must be homeomorphic to a classical Vietoris solenoid. In this work, we consider the problem, what can be said about a matchbox manifold with equicontinuous holonomy dynamics, and all of whose leaves have at most polynomial growth type? We show that such a space must have a finite covering for which the global holonomy group of its foliation is nilpotent. As a consequence, we show that if the growth type of the leaves is polynomial of degree at most 3, then there exists a finite covering which is homogeneous. If the growth type of the leaves is polynomial of degree at least 4, then there are additional obstructions to homogeneity, which arise from the structure of nilpotent groups.     

Quasi-symmetric 2-(64,24,46) designs derived from AG(3,4)

This paper completes the enumeration of quasi-symmetric 2-$(64,24,46)$ designs supported by the dual code $C^{\perp }$ of the binary linear code $C$ spanned by the lines of $AG(3,4)$, initiated in Rodrigues and Tonchev (2015). It is shown that $C^{\perp }$ supports exactly 30,264 nonisomorphic quasi-symmetric 2-$(64,24,46)$ designs. The automorphism groups of the related strongly regular graphs are computed.     

The ∞‐Besov capacity problem

A theory of ∞‐Besov capacities is developed and several applications are provided. In particular, we solve an open problem in the theory of limits of the ∞‐Besov semi‐norms, we obtain new restriction‐extension inequalities, and we characterize the pointwise multipliers acting on the ∞‐Besov spaces.     

Locally symmetric left invariant Riemannian metrics on 3‐dimensional Lie groups

In this paper, we use the powerful tool Milnor bases to determine all the locally symmetric left invariant Riemannian metrics up to automorphism, on 3‐dimensional connected and simply connected Lie groups, by solving system of polynomial equations of constants structure of each Lie algebra . Moreover, we show that E ₀(2) is the only 3‐dimensional Lie group with locally symmetric left invariant Riemannian metrics which are not symmetric.     

Composition algebras and outer automorphisms of algebraic groups via triality

In this note, we establish an equivalence of categories between the category of all eight-dimensional composition algebras with any given quadratic form n over a field k of characteristic not two, and a category arising from an action of the projective similarity group of n on certain pairs of automorphisms of the group scheme PGO⁺(n) defined over k. This extends results recently obtained in the same direction for symmetric composition algebras. We also derive known results on composition algebras from our equivalence.