The Angle Defect for Odd-Dimensional Simplicial Manifolds

In a 1967 paper, Banchoff stated that a certain type of polyhedral curvature, that applies to all finite polyhedra, was zero at all vertices of an odd-dimensional polyhedral manifold; one then obtains an elementary proof that odd-dimensional manifolds have zero Euler characteristic. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a direct generalization of the angle defect. The generalized angle defect is not zero at the simplices of every odd-dimensional manifold. In this paper we use a sequence based upon the Bernoulli numbers to define a variant of the angle defect for finite simplicial complexes that still satisfies a Gauss-Bonnet-type theorem, but is also zero at any simplex of an odd-dimensional simplicial complex K (of dimension at least 3), such that χ(link(ηⁱ, K)) = 2 for all i-simplices ηⁱ of K, where i is an even integer such that 0 ≤ i ≤ n – 1. As a corollary, an elementary proof is given that any such simplicial complex has Euler characteristic zero.     

On a theorem of Shchepin and Repovš concerning the smoothness of compacta

We show that a theorem of Shchepin and Repovš concerning the smoothness of compacta follows from the theory of semicontinuous relations.     

Immanant positivity for Catalan-Stieltjes matrices

We give some sufficient conditions for the nonnegativity of immanants of square submatrices of Catalan-Stieltjes matrices and their corresponding Hankel matrices. To obtain these sufficient conditions, we construct new planar networks with a recursive nature for Catalan-Stieltjes matrices. As applications, we provide a unified way to produce inequalities for many combinatorial polynomials, such as the Eulerian polynomials, Schröder polynomials, and Narayana polynomials.     

Evidence for a two-step mechanism in electronically selective single-walled carbon nanotube reactions

Covalent and noncovalent chemistries that are selective to single-walled carbon nanotubes of a particular electronic type have become increasingly important for electronic structure separation and on-chip modification of nanoelectronic devices. By monitoring transient Raman spectroscopy and photoluminescence (PL) during a reaction with 4-chlorobenzene diazonium in aqueous solution, evidence for a characteristic two-step mechanism with two distinct time constants is uncovered. A long-lived intermediate selectively and noncovalently binds and partially dopes the nanotube surface (tau = 2.4 min). A slower, covalent reaction is tracked using the time-dependent increase in the disorder mode in Raman (tau = 73 min). The transient Raman and PL data are well described using a series of two first-order reactions. The covalent bonding step can be deactivated by changing the structure of the surfactant adsorbed phase, further supporting the mechanism.     

Nitrogen extrusion from pyrazoline-substituted porphyrins and chlorins using long wavelength visible light

Protoporphyrin reacts with diazomethane to give pyrazolines which at long wavelength (approximately 630 nm) extrude nitrogen to give the corresponding cyclopropyl derivatives.     

Evidence for unfolding and refolding of gas-phase cytochrome <Emphasis Type="Italic">c</Emphasis> ions in a Paul trap

The folding pathways of gas-phase cytochrome c ions produced by electrospray ionization have been studied by an ion trapping/ion mobility technique that allows conformations to be examined over extended timescales (10 ms to 10 s). The results show that the +9 charge state emerges from solution as a compact structure and then rapidly unfolds into several substantially more open structures, a transition that requires 30-60 ms; over substantially longer timescales (250 ms to 10 s) elongated states appear to refold into an array of folded structures. The new folded states are less compact than those that are apparent during the initial unfolding. Apparently, unfolding to highly open conformations is a key step that must occur before +9 ions can sample more compact states that are stable at longer times.     

Infrared spectroscopy of closed s-shell gas-phase M+(N2O)n (M = Li,Al) ion-molecule complexes*

ABSTRACT

We present the results of a combined experimental and computational study of the structures of gas-phase M⁺(N₂O)_n (M?=?Li, Al) complexes. Infrared spectra were recorded in the region of the N₂O asymmetric (N?=?N) stretch using photodissociation spectroscopy employing the inert messenger technique. Unlike in our previous studies on M⁺(N₂O)_n (M?=?Cu, Ag, Au and M?=?Co, Rh, Ir) complexes, N– and O–bound isomers in this case are near isoenergetic and are not distinguished spectroscopically at this resolution. In the case of Li⁺ complexes, there is, however, evidence for the presence of bound N₂ moieties, indicating the presence of inserted, OLi⁺N₂(N₂O)_n–type structures. The weak N₂ band lies to the blue of the signature of molecularly N– and O–bound ligands and is well–reproduced in the simulated spectra of energetically low-lying structures computed from density functional theory. No such inserted isomers are observed in the case of Al⁺(N₂O)_n complexes whose infrared spectra can be understood on the basis of molecularly-bound N₂O ligands. The differences in M⁺(N₂O)_n structures observed for these closed–shell, ns², metal centres relative to other metal cations are discussed in terms of the likely bonding motifs.     

Critical Points and the Angle Defect

In a 1967 paper, Banchoff described a theory of critical points and curvature for polyhedra embedded in Euclidean space. For each convex cell complex K in , and for each linear map satisfying a simple generality criterion, he defined an index for each vertex of K with respect to the map h, and showed that these indices satisfy two properties: (1) for each map h, the sum of the indices at all the vertices of K equalsχK and (2) for each vertex of K, the integral of the indices of the vertex with respect to all such linear maps equals the standard polyhedral notion of curvature of K at the vertex. In a previous paper, the author defined a different approach to curvature for arbitrary simplicial complexes, based upon a more direct generalization of the angle defect. In the present paper we present an analog of Banchoff ’s theory that works with our generalized angle defect.