Discrete convexity and unimodularity. I

In this paper we develop a theory of convexity for a free Abelian group M (the lattice of integer points), which we call theory of discrete convexity. We characterize those subsets X of the group M that could be call "convex". One property seems indisputable: X should coincide with the set of all integer points of its convex hull co(X) (in the ambient vector space V). However, this is a first approximation to a proper discrete convexity, because such non-intersecting sets need not be separated by a hyperplane. This issue is closely related to the question when the intersection of two integer polyhedra is an integer polyhedron. We show that unimodular systems (or more generally, pure systems) are in one-to-one correspondence with the classes of discrete convexity. For example, the well-known class of g-polymatroids corresponds to the class of discrete convexity associated to the unimodular system A_n:={\pm e_i, e_i-ej} in Z^n.Comment: 26 pages, Late

Embedding convex geometries and a bound on convex dimension

The notion of an abstract convex geometry offers an abstraction of the standard notion of convexity in a linear space. Kashiwabara, Nakamura and Okamoto introduce the notion of a generalized convex shelling into $\mathbb{R}$ and prove that a convex geometry may always be represented with such a shelling. We provide a new, shorter proof of their result using a recent representation theorem of Richter and Rubinstein, and deduce a different upper bound on the dimension of the shelling.Comment: - Corrected attribution for Lemma 1 and Theorem 2 - Added an example related to generalized convex shellings of lower-bounded lattices and noted its relevance to convex dimension. - Added a section on embedding convex geometries as convex polygons, including a proof that any convex geometry may be embedded as convex polygons in R^2. - Extended the bibliography. Now 9 page

Luminescence Solvato- and Vapochromism of Alkynyl-Phosphine Copper Clusters

The reaction of [Cu(NCMe)4][PF6] with aromatic acetylenes HC2R and triphosphine 1,1,1-tris(diphenylphosphino)methane in the presence of NEt3 results in the formation of hexanuclear Cu(I) clusters with the general formula [Cu6(C2R)4{(PPh2)3CH}2][PF6]2 (R = 4-X-C6H4 (1-5) and C5H4N (6); X = NMe2 (1), OMe (2), H (3), Ph (4), CF3 (5)). The structural motif of the complexes studied consists of a Cu6 metal core supported by two phosphine ligands and stabilized by σ- and π-coordination of the alkynyl fragments (together with coordination of pyridine nitrogen atoms in cluster 6). The solid state structures of complexes 2-6 were determined by single crystal XRD analysis. The structures of the complexes in solution were elucidated by (1)H, (31)P, (1)H-(1)H COSY NMR spectroscopy, and ESI mass spectrometry. Clusters 1-6 exhibit moderately strong phosphorescence in the solid state with quantum yields up to 17%. Complexes 1-5 were found to form solvates (acetone, acetonitrile) in the solid state. The coordination of loosely bound solvent molecules strongly affects emission characteristics and leads to solvato- and vapochromic behavior of the clusters. Thus, solvent-free and acetonitrile solvated forms of 3 demonstrate contrasting emission in orange (615 nm) and blue (475 nm) regions, respectively. The computational studies show that alkynyl-centered IL transitions mixed with those of MLCT between the Cu6 metal core and the ligand environment play a dominant role in the formation of excited states and can be considerably modulated by weakly coordinating solvent molecules leading to luminescence vapochromism.This research has been supported by St. Petersburg State University Research Grant 0.37.169.2014, and Russian Foundation for Basic Research Grants 13-03-00970, 14-03-32077, and 13-03-12411. Academy of Finland (Grant 268993/2013, I.O.K), University of Eastern Finland (strategic funding—Russian–Finnish collaborative project), is also gratefully acknowledged. The work was carried out using equipment of the Analytical Center of Nano- and Biotechnologies of SPbSPU with financial support of the Ministry of Education and Science of Russian Federation; Centers for Magnetic Resonance, X-ray Diffraction Studies, Chemical Analysis and Materials Research, Optical and Laser Materials Research; and Computer Center of St. Petersburg State University