Cantor exchange systems and renormalization

We prove a one-to-one correspondence between (i) C¹⁺ conjugacy classes of C^1+H Cantor exchange systems that are C^1+H fixed points of renormalization and (ii) C¹⁺ conjugacy classes of C^1+H diffeomorphisms f with a codimension 1 hyperbolic attractor Λ that admit an invariant measure absolutely continuous with respect to the Hausdorff measure on Λ. However, we prove that there is no C^1+α Cantor exchange system, with bounded geometry, that is a C^1+α fixed point of renormalization with regularity α greater than the Hausdorff dimension of its invariant Cantor set.     

Completely geometrically optimized DFT/ONIOM triple-helical collagen-like structures containing the ProProGly, ProProAla, ProProDAla, and ProProDSer triads

We report completely optimized ONIOM DFT/AM1 molecular orbital calculations on several collagen-like triple helices based upon the repeating triad, ProProGly. The requirement of Gly as every third amino acid in collagen can be attributed to its enantiomorphic nature, as it behaves as a Damino acid in collagen. We, therefore, explored related collagen-like triple helices with one of the central Gly's mutated to either L or DAla; l-Ala appreciably destabilizes, while d-Ala slightly stabilizes the triple helical structure. Mutation of the same Gly to DSer, which is simply DAla with an OH in place of one of the methyl H's, induces a much greater stabilization due to an additional H-bond formed between this OH and a C=O on an adjacent peptide strand. Energies are presented for the triple helices and their component strands (both optimized and distorted to their triple helical geometries) relative to the component amino acids. The variation of relative energies with the chosen reference is delineated.     

Universal properties of maps of the circle with ɛ-singularities

Following the work of Collet, Eckmann, and Lanford on the Feigenbaum conjecture, we study the structure of the renormalization transformation introduced in [12] upon maps of the circle with critical points of the formx|x|^?.     

Efficient Demethylation of Methyl Esters with Anhydrous Trifluoroacetic Acid

Trifluoroacetic acid (TFA) is an effective reagent for the demethylation of methyl esters under anhydrous conditions, affording the parent acids and methyl trifluoroacetate.     

About a class of nonlinear oscillators with amplitude-independent frequency

This work is concerned with nonlinear oscillators that have a fixed, amplitude-independent frequency. This characteristic, known as isochronicity/isochrony, is achieved by establishing the equivalence between the Lagrangian of the simple harmonic oscillator and the Lagrangian of conservative oscillators with a position-dependent coefficient of the kinetic energy, which can stem from their mass that changes with the displacement or the geometry of motion. Conditions under which such systems have an isochronous center in the origin are discussed. General expressions for the potential energy, equation of motion as well as solutions for a phase trajectory and time response are provided. A few illustrative examples accompanied with numerical verifications are also presented.