The rate of escape for anisotropic random walks in a tree

Summary Let G be the group generated by L free involutions, whose Cayley graph T is the infinite homogeneous tree with L edges at every node. A general central limit theorem and law of the iterated logarithm is proven for left-invariant random walks Z _n on G or T which applies to the distance of Z _n from a fixed point, as well as to the distribution of the last R letters in Z _n . For nearest neighbor random walks, we also derive a generating function identity that yields formulas for the asymptotic mean and variance of the distance from a fixed point. A generalization for Z _n with a finitely supported step distribution is derived and discussed.Partially supported by grant NSF MCS85-04315     

Three dimensional structure of celgard® microporous membranes

The unique three dimensional structure of Celgard® 2500 film has been shown by direct enhanced contrast transmission electron microscopy (TEM) of thin sections from the bulk and by high resolution scanning electron imaging (SEI) of the film surface. The TEM method developed includes treatment with an unsaturated surfactant, which then reacts with osmium tetroxide to form an electron dense stain. This is followed by standard microtomy and TEM observations made possible by the stability and contrast imparted by the heavy metal. Sectioning in all three dimensions — along the machine direction, across the machine direction and across the face of the membrane — gives a total view of the bulk microstructure. Preparation for high resolution SEI imaging includes metal coating with the Ion Beam Micro-Sputter gun, to form a coherent fine grain surface to dissipate charge, provide secondary electrons for imaging, and protect the sensitive polymer from beam damage. The SEI view of the surface completes the three dimensional characterization.The microporous structure of Celgard® 2500 is formed during extrusion, annealing and stretching of isotactic polypropylene. The elliptical pores formed during this process have been shown to be oriented with their major axes parallel to the machine direction and the film surface. The pores are grouped into rows or pore channels which traverse the thickness of the film at an angle to its surface and which oscillate in and out of the plane of a cross-section taken in the machine direction. The observed structure is morphologically uniform throughout the bulk of the film. However, a thin surface layer appears to exhibit a slightly smaller pore size and a lower two dimensional pore area than does the interior. Thus, the microporous structure provides a continuous, but tortuous or staggered, row configuration necessary for high fluid transport rates.     

[FeFe]‐Hydrogenase with Chalcogenide Substitutions at the H‐Cluster Maintains Full H2 Evolution Activity

The [FeFe]‐hydrogenase HYDA1 from Chlamydomonas reinhardtii is particularly amenable to biochemical and biophysical characterization because the H‐cluster in the active site is the only inorganic cofactor present. Herein, we present the complete chemical incorporation of the H‐cluster into the HYDA1‐apoprotein scaffold and, furthermore, the successful replacement of sulfur in the native [4Fe_H] cluster with selenium. The crystal structure of the reconstituted pre‐mature HYDA1[4Fe4Se]_H protein was determined, and a catalytically intact artificial H‐cluster variant was generated upon in vitro maturation. Full hydrogen evolution activity as well as native‐like composition and behavior of the redesigned enzyme were verified through kinetic assays, FTIR spectroscopy, and X‐ray structure analysis. These findings reveal that even a bioinorganic active site with exceptional complexity can exhibit a surprising level of compositional plasticity.     

Direct observation of photoinduced bent nitrosyl excited-state complexes

Ground-state structures with side-on nitrosyl (eta (2)-NO) and isonitrosyl (ON) ligands have been observed in a variety of transition-metal complexes. In contrast, excited-state structures with bent-NO ligands have been proposed for years but never directly observed. Here, we use picosecond time-resolved infrared spectroscopy and density functional theory (DFT) modeling to study the photochemistry of Co(CO) 3(NO), a model transition-metal-NO compound. Surprisingly, we have observed no evidence for ON and eta (2)-NO structural isomers, but we have observed two bent-NO complexes. DFT modeling of the ground- and excited-state potentials indicates that the bent-NO complexes correspond to triplet excited states. Photolysis of Co(CO) 3(NO) with a 400-nm pump pulse leads to population of a manifold of excited states which decay to form an excited-state triplet bent-NO complex within 1 ps. This structure relaxes to the ground triplet state in ca. 350 ps to form a second bent-NO structure.     

Effect of fluorocarbon molecules confined between sliding self-mated PTFE surfaces

We report on the frictional response and atomic process that occur when molecular fluorocarbon molecules of varying lengths are sheared between two polytetrafluoroethylene (PTFE) surfaces. The thicknesses of the molecular layers are also varied. The approach is classical molecular dynamics simulations using a reactive bond-order potential parametrized for fluorocarbons. The results indicate that the presence of the molecules has a significant impact on the measured friction and wear of the surfaces, and that this impact depends on the nature of the fluorocarbon molecules and the thickness of the molecular film. The molecular mechanisms responsible for these differences are presented.     

Weak and strong type estimates for maximal truncations of Calderón-Zygmund operators on A p weighted spaces

We show that for 1 < p < ??, weight w ?? A _p , and any L ²-bounded Calderón-Zygmund operator T, there is a constant C _T,p such that the weak- and strong-type inequalities $${\left\| {{T_\natural}f} \right\|_{{L^{p,\infty }}(w)}} \le {C_{T,p}}{\left\| w \right\|_{{A_p}}}{\left\| f \right\|_{{L^p}(w )}}$$ $${\left\| {{T_\natural}f} \right\|_{{L^p}(w)}} \le {C_{T,p}}\left\| w \right\|_{{A_p}}^{\max \{ 1,{{(p - 1)}^{ - 1}}}{\left\| f \right\|_{{L^p}(w)}}$$ hold, where T _? denotes the maximal truncations of T and ${\left\| w \right\|_{{A_p}}}$ denotes the Muckenhoupt A _p characteristic of w. These estimates are not improvable in the power of ${\left\| w \right\|_{{A_p}}}$ . Our argument follows the outlines of those of Lacey-Petermichl-Reguera (Math. Ann. 2010) and Hyt?nen-Pérez-Treil-Volberg (arXiv, 2010) and contains new ingredients, including a weak-type estimate for certain duals of T _? and sufficient conditions for two-weight inequalities in L ^p for T _?. Our proof does not rely upon extrapolation.     

The Krzyż conjecture revisited

The Krzy? conjecture concerns the largest values of the Taylor coefficients of a non-vanishing analytic function bounded by one in modulus in the unit disk. It has been open since 1968 even though information on the structure of extremal functions is available. The purpose of this paper is to collect various conditions that the coefficients of an extremal function (and various other quantities associated with it) should satisfy if the conjecture is true and to show that each one of these properties is equivalent to the conjecture itself. This may provide several possible starting points for future attempts at solving the problem.