The metamathematics of ergodic theory

The metamathematical tradition, tracing back to Hilbert, employs syntactic modeling to study the methods of contemporary mathematics. A central goal has been, in particular, to explore the extent to which infinitary methods can be understood in computational or otherwise explicit terms. Ergodic theory provides rich opportunities for such analysis. Although the field has its origins in seventeenth century dynamics and nineteenth century statistical mechanics, it employs infinitary, nonconstructive, and structural methods that are characteristically modern. At the same time, computational concerns and recent applications to combinatorics and number theory force us to reconsider the constructive character of the theory and its methods. This paper surveys some recent contributions to the metamathematical study of ergodic theory, focusing on the mean and pointwise ergodic theorems and the Furstenberg structure theorem for measure preserving systems. In particular, I characterize the extent to which these theorems are nonconstructive, and explain how proof-theoretic methods can be used to locate their “constructive content”.     

Effect of Self-association and Protein Binding on the Photochemical Reactivity of Triarylmethanes. Implications of Noncovalent Interactions on the Competition between Photosensitization Mechanisms Type I and Type II

We have explored the photochemical behavior of cationic triarylmethane dye monomers and dimers free in solution and noncovalently bound to bovine serum albumin (BSA) and examined how self-association and the formation of host-guest complexes involving biopolymers and photosensitizers affect the competition between the photosensitization type I and type II mechanisms. Our results have clearly indicated that tri-para-substituted triarylmethane dyes bind efficiently to albumin as monomers and dimers and, interestingly, that the formation of dye aggregates in aqueous solutions is actually assisted by the protein. Protein-assisted dye aggregation takes place under conditions of high biopolymer loading (high [dye]/[protein] ratios), as attested by the appearance of a hypsochromically shifted absorption band (H-band) that overlaps with the spectral shoulder of the respective dye monomer. As predicted by the molecular exciton theory, the intersystem crossing efficiency in H-type dimers is expected to be higher than in the respective dye monomers, and photoinduced electron transfer events are intrinsically favored in dye aggregates as a result of the physical contact between donor and acceptor. We have found that when triarylmethanes are noncovalently bound to BSA their photoreactivity undergoes a remarkable enhancement, and that the photooxidation mechanism type I is particularly favored in the macromolecular environment. A comparative examination of the behavior of triarylmethane dyes with that of methylene blue have shown that in the case of methylene blue the binding phenomenon also favor the type I mechanism.     

Heteroleptic Square Planar Cobalt(I/II) Complexes

Reduction of the cobalt(II) chloride complex, Ph₂B(^tBuIm)₂Co(THF)Cl ( 1 ) in the presence of ^tBuN≡C affords the diamagnetic, square planar cobalt(I) complex Ph₂B(^tBuIm)₂Co(C≡N^tBu)₂ ( 2 ). This is a rare example of a 16-electron cobalt(I) complex that is structurally related to square planar noble metal complexes. Accordingly, the electronic structure of 2 , as calculated by DFT, reveals that the HOMO is largely d_z² in character. Complex 2 is readily oxidized to its cobalt(II) congener [Ph₂B(^tBuIm)₂Co(C=N^tBu)₂]BPh₄ ( 3 -BPh₄), whose EPR spectral parameters are characteristic of low-spin d⁷ with an unpaired electron in an orbital of d_z² parentage. This is also consistent with the results of DFT calculations. Despite its 16-electron configuration and the d_z² parentage of the HOMO, the only tractable reactions of 2 involve one electron oxidation to afford 3 .     

Relaxation of the rigid backbone of an oligoamide-foldamer-based α-helix mimetic: identification of potent Bcl-xL inhibitors

By conducting a structure-activity relationship study of the backbone of a series of oligoamide-foldamer-based α-helix mimetics of the Bak BH3 helix, we have identified especially potent inhibitors of Bcl-x(L). The most potent compound has a K(i) value of 94 nM in vitro, and single-digit micromolar IC(50) values against the proliferation of several Bcl-x(L)-overexpressing cancer cell lines.     

Activation of CO2 by a heterobimetallic Zr/Co complex

At room temperature, the early/late heterobimetallic complex Co((i)Pr(2)PNMes)(3)Zr(THF) has been shown to oxidatively add CO(2), generating (OC)Co((i)Pr(2)PNMes)(2)(μ-O)Zr((i)Pr(2)PNMes). This compound can be further reduced under varying conditions to generate either the Zr oxoanion (THF)(3)Na-O-Zr(MesNP(i)Pr(2))(3)Co(CO) or the Zr carbonate complex (THF)(4)Na(2)(CO(3))-Zr(MesNP(i)Pr(2))(3)Co(CO). Additionally, reactivity of the CO(2)-derived product has been observed with PhSiH(3) to generate the Co-hydride/Zr-siloxide product (OC)(H)Co((i)Pr(2)PNMes)(3)ZrOSiH(2)Ph.     

Evaporation of Lennard-Jones fluids

Evaporation and condensation at a liquid/vapor interface are ubiquitous interphase mass and energy transfer phenomena that are still not well understood. We have carried out large scale molecular dynamics simulations of Lennard-Jones (LJ) fluids composed of monomers, dimers, or trimers to investigate these processes with molecular detail. For LJ monomers in contact with a vacuum, the evaporation rate is found to be very high with significant evaporative cooling and an accompanying density gradient in the liquid domain near the liquid/vapor interface. Increasing the chain length to just dimers significantly reduces the evaporation rate. We confirm that mechanical equilibrium plays a key role in determining the evaporation rate and the density and temperature profiles across the liquid/vapor interface. The velocity distributions of evaporated molecules and the evaporation and condensation coefficients are measured and compared to the predictions of an existing model based on kinetic theory of gases. Our results indicate that for both monatomic and polyatomic molecules, the evaporation and condensation coefficients are equal when systems are not far from equilibrium and smaller than one, and decrease with increasing temperature. For the same reduced temperature T/T(c), where T(c) is the critical temperature, these two coefficients are higher for LJ dimers and trimers than for monomers, in contrast to the traditional viewpoint that they are close to unity for monatomic molecules and decrease for polyatomic molecules. Furthermore, data for the two coefficients collapse onto a master curve when plotted against a translational length ratio between the liquid and vapor phase.     

Synthesis of nanoscale NbSe2 materials from molecular precursors

This communication describes a new synthetic approach to one- (1D) and two-dimensional (2D) NbSe2 nanoscale materials using soft chemical methods. Our one-pot synthesis provides a direct route to control the morphology of nanostructures that can exhibit complex electronic properties, and can produce layered, nanocrystalline materials in high yield.     

Enantio- and diastereoselective synthesis of (E)-1,5-syn-diols: application to the synthesis of the C(23)-C(40) fragment of tetrafibricin

A highly stereoselective synthesis of (E)-1,5-syn-diols 6 is described. The kinetically controlled hydroboration of allenyltrifluoroborate 8 with Soderquist borane 2 provides the (Z)-allylic trifluoroborate 9, which undergoes sequential allylboration with two different aldehydes to provide (E)-1,5-syn-diols 6 in 72-98% yields with >95% ee and >20:1 dr. Application of this method to the synthesis of the tetrafibricin C(23)-C(40) fragment 19 is described.     

Theory of bonding in ylides,acetylacetonates and π-cyclopentadienyl compounds

The theory of bonding in ylides and their oxides, acetylacetonates and π-cyclopentadienyl complexes has been discussed from the viewpoint of hypervalent bonding. The existing ylides and acetylacetonates are discussed and new ones are proposed. The π-cyclopentadienyl complexes are only speculative.