Ternary and tetrahedral symmetry in hybrid fluorides, fluoride carbonates and carbonates

Tren amine cations and carbonate anions adopt a ternary symmetry while tetra amine cations are tetrahedral. The symmetry of these constitutive ions influences strongly the nature of the solids which crystallise from solutions. Large fluorinated aluminate polyanions with tetrahedral symmetry appear in the presence of tren amine, while infinite chains of AlF₆ octahedra are observed with tetra amine and that noncentrosymmetric structures are frequently encountered in rare earth fluoride carbonates.     

MIL-50, an open-framework GaPO with a periodic pattern of small water ponds and dry rubidium atoms: a combined XRD,NMR, and computational study

A new fluorinated gallium phosphate, MIL-50, has been synthesized under mild hydrothermal conditions using 1,6-diaminohexane. The chemical formula of MIL-50 is Rb(2)Ga(9)(PO(4))(8)(HPO(4))(OH)F(6).2N(2)C(6)H(18).7H(2)O. The structure is a network of hexameric units of Ga(3)(PO(4))(3)F(2) and Ga(3)(PO(4))(2)(HPO(4))F(3) via corner sharing. It creates a three-dimensional open-framework delimiting 6- and 18-ring channels running along the c axis. The diprotonated 1,6-diaminohexane and water molecules are trapped within the 18-ring pores, whereas the rubidium cations reside in the 6-ring ones. A double quantum (31)P NMR experiment and partial charge calculations indicate that water molecules are present under the form of periodic small clusters, lowering the multiplicity of one phosphorus site, P3. Though water hops within the clusters, the motion leaves the water pattern periodic. Rubidium is so tightly embedded into the framework that water moving in the large 18-ring channels does not reach it, leaving it therefore dry. The crystal framework may be ascribed to the orthorhombic space group Cmc2(1) (n degrees 36), a = 32.1510(2), b = 17.2290(3), c = 10.2120(1) A. The periodic water pattern has a different symmetry than that of the framework. A method has been devised to superpose the two sublattices that coexist in the same unit cell in order to have full occupancy of each site and to perform Madelung summations. This original method is of general interest for most zeolitic materials exhibiting a different symmetry for the framework and the template sublattices.     

Carbon nanodots revised: the thermal citric acid/urea reaction

Luminescent compounds obtained from the thermal reaction of citric acid and urea have been studied and utilized in different applications in the past few years. The identified reaction products range from carbon nitrides over graphitic carbon to distinct molecular fluorophores. On the other hand, the solid, non-fluorescent reaction product produced at higher temperatures has been found to be a valuable precursor for the CO₂-laser-assisted carbonization reaction in carbon laser-patterning. This work addresses the question of structural identification of both, the fluorescent and non-fluorescent reaction products obtained in the thermal reaction of citric acid and urea. The reaction products produced during autoclave–microwave reactions in the melt were thoroughly investigated as a function of the reaction temperature and the reaction products were subsequently separated by a series of solvent extractions and column chromatography. The evolution of a green molecular fluorophore, namely HPPT, was confirmed and a full characterization study on its structure and photophysical properties was conducted. The additional blue fluorescence is attributed to oligomeric ureas, which was confirmed by complementary optical and structural characterization. These two components form strong hydrogen-bond networks which eventually react to form solid, semi-crystalline particles with a size of ∼7 nm and an elemental composition of 46% C, 22% N, and 29% O. The structural features and properties of all three main components were investigated in a comprehensive characterization study.

Products of the thermal reaction of citric acid and urea have been identified as a complex mixture of fluorophores and particles.     

Reducing a chemical master equation by invariant manifold methods

We study methods for reducing chemical master equations using the Michaelis-Menten mechanism as an example. The master equation consists of a set of linear ordinary differential equations whose variables are probabilities that the realizable states exist. For a master equation with s(0) initial substrate molecules and e(0) initial enzyme molecules, the manifold can be parametrized by s(0) of the probability variables. Fraser's functional iteration method is found to be difficult to use for master equations of high dimension. Building on the insights gained from Fraser's method, techniques are developed to produce s(0)-dimensional manifolds of larger systems directly from the eigenvectors. We also develop a simple, but surprisingly effective way to generate initial conditions for the reduced models.     

Substituent control of hydrogen bonding in palladium(II)-pyrazole complexes

Inter- and intramolecular hydrogen bonding of an N-H group in pyrazole complexes was studied using ligands with two different groups at pyrazole C-3 and C-5. At C-5, groups such as methyl, i-propyl, phenyl, or tert-butyl were present. At C-3, side chains L-CH(2)- and L-CH(2)CH(2)- (L = thioether or phosphine) ensured formation of chelates to a cis-dichloropalladium(II) fragment through side-chain atom L and the pyrazole nitrogen closest to the side chain. The significance of the ligands is that by placing a ligating side chain on a ring carbon (C-3), rather than on a ring nitrogen, the ring nitrogen not bound to the metal and its attached proton are available for hydrogen bonding. As desired, seven chelate complexes examined by X-ray diffraction all showed intramolecular hydrogen bonding between the pyrazole N-H and a chloride ligand in the cis position. In addition, however, intermolecular hydrogen bonding could be controlled by the substituent at C-5: complexes with either a methyl at C-5 or no substituent there showed significant intermolecular hydrogen bonding interactions, which were completely avoided by placing a tert-butyl group at C-5. The acidity of two complexes in acetonitrile solutions was estimated to be closer to that of pyridinium ion than those of imidazolium or triethylammonium ions.     

Maximum principle at infinity for complete minimal surfaces in flat 3-manifolds

The main result of this paper is the following maximum principle at infinity:Theorem.Let M ₁ and M ₂ be two disjoint properly embedded complete minimal surfaces with nonempty boundaries, that are stable in a complete flat 3-manifold. Then dist(M ₁,M ₂)=min(dist(M ₁,M ₂), dist(M ₂,M ₁)).In case one boundary is empty, e.g. M ₁,then dist(M ₁,M ₂)=dist(M ₂,M ₁).If both boundaries are empty, then M ₁ and M ₂ are flat.     

Formal orthogonal polynomials and Hankel/Toeplitz duality

For classical polynomials orthogonal with respect to a positive measure supported on the real line, the moment matrix is Hankel and positive definite. The polynomials satisfy a three term recurrence relation. When the measure is supported on the complex unit circle, the moment matrix is positive definite and Toeplitz. Then they satisfy a coupled Szeg recurrence relation but also a three term recurrence relation. In this paper we study the generalization for formal polynomials orthogonal with respect to an arbitrary moment matrix and consider arbitrary Hankel and Toeplitz matrices as special cases. The relation with Padé approximation and with Krylov subspace iterative methods is also outlined.This research was supported by the National Fund for Scientific Research (NFWO), project Lanczos, grant #2.0042.93.     

Euler'sϕ-function in the context of IΔ 0

It is demonstrated that we can represent Euler's -function by means of a ₀-formula in such a way that the theory I ₀ proves the recursion equations that are characteristic for this function.     

Non-compact quantum groups associated with Abelian subgroups

LetG be a Lie group. For any Abelian subalgebra of the Lie algebra g ofG, and any , the difference of the left and right translates ofr gives a compatible Poisson bracket onG. We show how to construct the corresponding quantum group, in theC ^*-algebra setting. The main tool used is the general deformation quantization construction developed earlier by the author for actions of vector groups onC ^*-algebras.The research reported on here was supported in part by National Science Foundation grant DMS-9303386.     

Convergence of diagonally stationary sequences in convex optimization

LetX be a real normed linear space,f, f ⁿ, n , be extended real-valued proper closed convex functions onX. A sequence {x _n} inX is called diagonally stationary for {f ⁿ} if for alln there existsx* _n f ⁿ (x _n) such that x* _{n *} 0. Such sequences arise in approximation methods for the problem of minimizingf. Some general convergence results based upon variational convergence theory and appropriate equi-well-posedness are presented.