Nonnegatively and positively curved invariant metrics on circle bundles

We derive and study necessary and sufficient conditions for an -bundle to admit an invariant metric of positive or nonnegative sectional curvature. In case the total space has an invariant metric of nonnegative curvature and the base space is odd dimensional, we prove that the total space contains a flat totally geodesic immersed cylinder. We provide several examples, including a connection metric of nonnegative curvature on the trivial bundle that is not a product metric.

     

Characterization of porosity in organic and metal-organic macrocycles by hyperpolarized 129Xe NMR spectroscopy

Hyperpolarized (129)Xe NMR spectroscopy is used to establish the solid-state porosity of shape-persistent macrocycles with either an organic or metal-organic framework. These studies show that even upon removal of cocrystallized solvent molecules, the macrocycles maintain a porous or channeled structure. The technique can provide valuable information about systems for which X-ray crystallographic analysis is not feasible. [structure: see text]     

Nontotally symmetric trifurcation of an SN2 reaction pathway

A new type of reaction pathway which involves a nontotally symmetric trifurcation was found and investigated for a typical S_N2‐type reaction, NC^‐ + CH₃X → NC? CH₃ + X^‐ (X = F, Cl). A nontotally symmetric valley‐ridge inflection (VRI) point was located along the C_3v reaction path. For X = F, the minimum energy path (MEP) starting from the transition state (TS) leads to a second‐order saddle point with C_3v symmetry, which connects three product minima of C_s symmetry. For X = Cl, four product minima have been observed, of which three belong to C_s symmetry and one to C_3v symmetry. The branching path from the VRI point to the lower symmetry minima was determined by a linear interpolation technique. The branching mechanism is discussed based on the reaction path curvature and net atomic charges, and the possibility of a nonotally symmetric n‐furcation is discussed. © 2015 Wiley Periodicals, Inc.     

Review of inverse Laplace transform algorithms for Laplace-space numerical approaches

A boundary element method (BEM) simulation is used to compare the efficiency of numerical inverse Laplace transform strategies, considering general requirements of Laplace-space numerical approaches. The two-dimensional BEM solution is used to solve the Laplace-transformed diffusion equation, producing a time-domain solution after a numerical Laplace transform inversion. Motivated by the needs of numerical methods posed in Laplace-transformed space, we compare five inverse Laplace transform algorithms and discuss implementation techniques to minimize the number of Laplace-space function evaluations. We investigate the ability to calculate a sequence of time domain values using the fewest Laplace-space model evaluations. We find Fourier-series based inversion algorithms work for common time behaviors, are the most robust with respect to free parameters, and allow for straightforward image function evaluation re-use across at least a log cycle of time.     

On the Generality of Assuming that a Family of Continuous Functions Separates Points

For an algebra ${\mathcal{A}}$ of complex-valued, continuous functions on a compact Hausdorff space (X, τ), it is standard practice to assume that ${\mathcal{A}}$ separates points in the sense that for each distinct pair ${x, y \in X}$ , there exists an ${f \in \mathcal{A}}$ such that ${f(x) \neq f(y)}$ . If ${\mathcal{A}}$ does not separate points, it is known that there exists an algebra ${\widehat{\mathcal{A}}}$ on a compact Hausdorff space ${(\widehat{X}, \widehat{\tau})}$ that does separate points such that the map ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ is a uniform norm isometric algebra isomorphism. So it is, to a degree, without loss of generality that we assume ${\mathcal{A}}$ separates points. The construction of ${{\widehat{\mathcal{A}}}}$ and ${(\widehat{X}, \widehat{\tau})}$ does not require that ${\mathcal{A}}$ has any algebraic structure nor that ${(X, \tau)}$ has any properties, other than being a topological space. In this work we develop a framework for determining the degree to which separation of points may be assumed without loss of generality for any family ${\mathcal{A}}$ of bounded, complex-valued, continuous functions on any topological space ${(X, \tau)}$ . We also demonstrate that further structures may be preserved by the mapping ${\mathcal{A} \mapsto \widehat{\mathcal{A}}}$ , such as boundaries of weak peak points, the Lipschitz constant when the functions are Lipschitz on a compact metric space, and the involutive structure of real function algebras on compact Hausdorff spaces.     

Directing stem cell fate by controlling the affinity and density of ligand-receptor interactions at the biomaterials interface

Sticky situation: the differentiation of mesenchymal stem cells can be influenced by the affinity and density of an immobilized ligand for the integrin receptors. Cells adherent to monolayers that present the high-affinity, cyclic-RGD peptide (left) show increased expression of osteogenic markers, while cells on monolayers presenting the lower-affinity, linear-RGD peptide (right) express early markers of myogenesis at a high density and neurogenesis at a low density of the ligand.     

Metrics with nonnegative curvature on {S^2 \times \mathbb{R}^4}

We study nonnegatively curved metrics on S²×\mathbbR⁴{S^2\times\mathbb{R}^4}. First, we prove rigidity theorems for connection metrics; for example, the holonomy group of the normal bundle of the soul must lie in a maximal torus of SO(4). Next, we prove that Wilking’s almost-positively curved metric on S ² × S ³ extends to a nonnegatively curved metric on S²×\mathbbR⁴{S^2\times\mathbb{R}^4} (so that Wilking’s space becomes the distance sphere of radius 1 about the soul). We describe in detail the geometry of this extended metric.     

Ultrahigh-field NMR spectroscopy of quadrupolar transition metals: 55Mn NMR of several solid manganese carbonyls

55Mn NMR spectra acquired at 21.14 T (nu(L)(55Mn) = 223.1 MHz) are presented and demonstrate the advantages of using ultrahigh magnetic fields for characterizing the chemical shift tensors of several manganese carbonyls: eta5-CpMn(CO)3, Mn2(CO)10, and (CO)5MnMPh3 (M = Ge, Sn, Pb). For the compounds investigated, the anisotropies of the manganese chemical shift tensors are less than 250 ppm except for eta5-CpMn(CO)3, which has an anisotropy of 920 ppm. At 21.14 T, one can excite the entire m(I) = 1/2 <--> m(I) = -1/2 central transition of eta5-CpMn(CO)3, which has a breadth of approximately 700 kHz. The breadth arises from second-order quadrupolar broadening due to the 55Mn quadrupolar coupling constant of 64.3 MHz, as well as the anisotropic shielding. Subtle variations in the electric field gradient tensors at the manganese are observed for crystallographically unique sites in two of the solid pentacarbonyls, resulting in measurably different C(Q) values. MQMAS experiments are able to distinguish four magnetically unique Mn sites in (CO)(5)MnPbPh3, each with slightly different values of delta(iso), C(Q), and eta(Q).     

Obstruction to Positive Curvature on Homogeneous Bundles

Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F) were discovered recently [J. Differential Geom. 65:273–287, 2003; Invent. Math. 148:117–141, 2002]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h) × F with the induced Riemannian submersion metric. We prove that no new examples of strictly positive sectional curvature exist in this class of metrics. This result generalizes the case F = {point} proven by Geroch [Proc. Amer. Math. Soc. 66(2):321–326, 1977].Supported in part by NSF grant DMS–0303326.     

Synthesis and structures of acyclic monoanionic tetradentate aza beta-diketiminate complexes of magnesium, zinc, and cadmium

An acyclic monoanionic tetradentate nitrogen ligand was prepared through the condensation of 2-(4-tolyl)-malondialdehyde and 8-aminoquinoline to give (BDI(QQ))H where (BDI(QQ))H = (8-quinolyl)-NCHC(4-tolyl)CHNH-(8-quinolyl). Metal complexes, (BDI(QQ))MX, were prepared where MX = MgBr 2, ZnCl 3, and CdOAc 4. The spectroscopic and crystallographic properties of compounds 2, 3, and 4 were explored. Structures of complexes 2, 3, 4, and the tridentate ligand, (BDI(Q))OH, 5, are reported.