Comment on the Roberts Solution for the Spherically-Symmetric Einstein-Scalar Field Equations

We critically examine the Roberts homothetic solution for the spherically symmetric Einstein-scalar field equations in double null coordinates, and show that the Roberts solution indeed solves the field equations only for one non-trivial case. We generalize this solution and discuss its relations with other known exact solutions.     

The small-molecule iron transport inhibitor ferristatin/NSC306711 promotes degradation of the transferrin receptor

Iron delivery by transferrin (Tf) is accomplished through clathrin-mediated endocytosis of Tf receptors. The small molecule NSC306711 inhibits iron uptake from the Tf-TfR pathway. Here we show that the drug's mechanism of action is to induce internalization and degradation of unoccupied Tf receptors through an unexpected endocytic pathway. Unlike classical clathrin-mediated Tf receptor endocytosis, internalization promoted by NSC306711 is independent of clathrin and dynamin, and is sensitive to the cholesterol-depleting agents filipin and nystatin. The finding of this cholesterol-dependent Tf receptor internalization pathway through use of the small-molecule inhibitor sheds light on the pleiotropic nature of membrane trafficking dynamics and adds a complex dimension to our understanding of receptor regulation. Because of its unusual properties to inhibit iron uptake, we refer to NSC306711 as "ferristatin."     

What can electrochemistry tell us about individual enzymes?

We provide here a critical analysis of electrochemistry's potential and limitations in investigating single-enzyme catalysis, highlighting papers of interest from the past 2–3 years with an emphasis on nano-impact electrochemistry (NIE) and electrochemical scanning tunneling microscopy. NIE can report single-enzyme activity; however, its future broad applicability for studying freely diffusing individual enzymes is questionable. Electrochemical scanning tunneling microscopy, an alternative to NIE, measures single enzyme's electronic conductivity when suspended between two electrodes. Recent discoveries indicate that enzyme conductance depends directly on biophysical parameters such as substrate binding, oxidation state of the catalytic center, and structural fluctuations. We conclude with a short perspective on additional electrochemical routes and combinations of existing techniques that may be useful for studying single-enzyme characteristics.     

Velocities, Stresses and Vector Bundle Valued Chains

A mathematical framework for the fundamental objects of continuum mechanics is presented. In the geometric setting of general differentiable manifolds, velocity fields over bodies, modeled as sections of a vector bundle W, are generalized using notions of homological integration theory such as flat chains and cochains. The class of bodies includes fractal sets whose irregular boundaries may have infinite measures. Stresses, initially modeled as smooth differential forms valued in the dual of the jet bundle of W, are generalized to cochains represented by L ^??-sections whose weak divergences are also L ^??. The divergence of a stress field, defined in an earlier work, is generalized to apply to stress cochains. The co-divergence of a velocity field is a weak form of the jet extension mapping and it is the counterpart of the boundary operator for real valued flat chains.     

A Haar component for quantum limits on locally symmetric spaces

We prove lower bounds for the entropy of limit measures associated to non-degenerate sequences of eigenfunctions on locally symmetric spaces of non-positive curvature. In the case of certain compact quotients of the space of positive definite n × n matrices (any quotient for n = 3, quotients associated to inner forms in general), measure classification results then show that the limit measures must have a Haar component. This is consistent with the conjecture that the limit measures are absolutely continuous.     

Affine and Projective Tree Metric Theorems

The tree metric theorem provides a combinatorial four-point condition that characterizes dissimilarity maps derived from pairwise compatible split systems. A related weaker four point condition characterizes dissimilarity maps derived from circular split systems known as Kalmanson metrics. The tree metric theorem was first discovered in the context of phylogenetics and forms the basis of many tree reconstruction algorithms, whereas Kalmanson metrics were first considered by computer scientists, and are notable in that they are a non-trivial class of metrics for which the traveling salesman problem is tractable. We present a unifying framework for these theorems based on combinatorial structures that are used for graph planarity testing. These are (projective) PC-trees, and their affine analogs, PQ-trees. In the projective case, we generalize a number of concepts from clustering theory, including hierarchies, pyramids, ultrametrics, and Robinsonian matrices, and the theorems that relate them. As with tree metrics and ultrametrics, the link between PC-trees and PQ-trees is established via the Gromov product.     

Conjugation in polyyne rods: to what extent is charge delocalization coupled to geometrical changes?

Ab initio methods were used to calculate the geometry and the charge distribution (natural bond orbital) in end-protonated polyynes. The geometry obtained is practically identical to that of the corresponding anion and the neutral radical. Thus, the geometry is not much dependent on charge dispersal. Moreover, it is shown that regardless of whether the imposed geometry is that of a cumulenic structure which localizes the charge at one end or that of the neutral molecule which localizes the charge at the other end, the same amount of charge is delocalized to the remote end of the protonated molecule regardless of the imposed structure. The same phenomenon is observed also for polyenes. It is interesting to note that regardless of the charge or its absence, as in the case of the radical, the optimal geometry is obtained as the arithmetic sum of the main resonance structures. Thus, it is concluded that, in these cases, the wave function is only weakly coupled to the geometry of the molecule.     

Diophantine Approximations on Fractals

We exploit dynamical properties of diagonal actions to derive results in Diophantine approximations. In particular, we prove that the continued fraction expansion of almost any point on the middle third Cantor set (with respect to the natural measure) contains all finite patterns (hence is well approximable). Similarly, we show that for a variety of fractals in [0, 1]², possessing some symmetry, almost any point is not Dirichlet improvable (hence is well approximable) and has property C (after Cassels). We then settle by similar methods a conjecture of M. Boshernitzan saying that there are no irrational numbers x in the unit interval such that the continued fraction expansions of {nx mod 1}_{n ? \mathbb N}{\{nx\,{\rm mod}\,1\}_{n \in {\mathbb N}}} are uniformly eventually bounded.     

Metallocorroles as Nonprecious‐Metal Catalysts for Oxygen Reduction

The future of affordable fuel cells strongly relies on the design of earth‐abundant (non‐platinum) catalysts for the electrochemical oxygen reduction reaction (ORR). However, the bottleneck in the overall process occurs therein. We have examined herein trivalent Mn, Fe, Co, Ni, and Cu complexes of β‐pyrrole‐brominated corrole as ORR catalysts. The adsorption of these complexes on a high‐surface‐area carbon powder (BP2000) created a unique composite material, used for electrochemical measurements in acidic aqueous solutions. These experiments disclosed a clear dependence of the catalytic activity on the metal center of the complexes, in the order of Co>Fe>Ni>Mn>Cu. The best catalytic performance was obtained for the Co^III corrole, whose onset potential was as positive as 0.81 V versus the reversible hydrogen electrode (RHE). Insight into the properties of these systems was gained by spectroscopic and computational characterization of the reduced and oxidized forms of the metallocorroles.