Fractional Heisenberg equation

Fractional derivative can be defined as a fractional power of derivative. The commutator (i/?)[H,⋅], which is used in the Heisenberg equation, is a derivation on a set of observables. A derivation is a map that satisfies the Leibnitz rule. In this Letter, we consider a fractional derivative on a set of quantum observables as a fractional power of the commutator (i/?)[H,⋅]. As a result, we obtain a fractional generalization of the Heisenberg equation. The fractional Heisenberg equation is exactly solved for the Hamiltonians of free particle and harmonic oscillator. The suggested Heisenberg equation generalize a notion of quantum Hamiltonian systems to describe quantum dissipative processes.     

Two-phase model of hydrolysis kinetics and its applications to anaerobic degradation of particulate organic matter

The various equations of hydrolysis kinetics included into the generalized simulation model (METHANE) were tested on the anaerobic digestion of cellulose, sludge, and cattle manure. The good agreement between the model simulation results and experimental data was obtained. The Contois equation, taking into account a hydrolytic biomass, and the firstorder equation with respect to the particulate substrate only, were shown to be the approximations of two-phase hydrolysis kinetics.     

Polymerization of solid C<Subscript>60</Subscript> under C<Stack><Subscript>60</Subscript><Superscript>+</Superscript></Stack> cluster ion bombardment

The structure transformation occurring in fullerene film under bombardment by 50 keV C₆₀⁺ cluster ions is reported. The Raman spectra of the irradiated C₆₀ films reveal a new peak rising at 1458 cm⁻¹ with an increase in the ion fluence. This feature of the Raman spectra suggests linear polymerization of solid C₆₀ induced by the cluster ion impacts. The aligned C₆₀ polymeric chains composing about 5–10 fullerene molecules have been distinguished on the film surface after the high-fluence irradiation using atomic force microscopy (AFM). The surface profiling analysis of the irradiated films has revealed pronounced sputtering during the treatment. The obtained results indicate that the C₆₀ polymerization occurs in a deep layer situated more than 40 nm below the film surface. The deep location of the C₆₀ polymeric phase indirectly confirms the dominant role of shock waves in the detected C₆₀ phase transformation.     

General Non-Markovian Quantum Dynamics

A general approach to the construction of non-Markovian quantum theory is proposed. Non-Markovian equations for quantum observables and states are suggested by using general fractional calculus. In the proposed approach, the non-locality in time is represented by operator kernels of the Sonin type. A wide class of the exactly solvable models of non-Markovian quantum dynamics is suggested. These models describe open (non-Hamiltonian) quantum systems with general form of nonlocality in time. To describe these systems, the Lindblad equations for quantum observable and states are generalized by taking into account a general form of nonlocality. The non-Markovian quantum dynamics is described by using integro-differential equations with general fractional derivatives and integrals with respect to time. The exact solutions of these equations are derived by using the operational calculus that is proposed by Yu. Luchko for general fractional differential equations. Properties of bi-positivity, complete positivity, dissipativity, and generalized dissipativity in general non-Markovian quantum dynamics are discussed. Examples of a quantum oscillator and two-level quantum system with a general form of nonlocality in time are suggested.     

Identification of Ge/Si intermixing processes at the Bi/Ge/Si(111) surface

The chemical contrast between Si and Ge obtained by scanning tunneling microscopy on Bi-covered Si(111) surfaces is used as a tool to identify two vertical Ge/Si intermixing processes. During annealing of an initially pure Ge monolayer on Si, the intermixing is confined to the first two layers approaching a 50% Ge concentration in each layer. During epitaxial growth, a growth front induced intermixing process acting at step edges is observed. Because of the open atomic structure at the step edges, relative to the terraces, a lower activation barrier for intermixing at the step edge, compared to the terrace, is observed.     

Display Selection of a Hybrid Foldamer–Peptide Macrocycle

Expanding the chemical diversity of peptide macrocycle libraries for display selection is desirable to improve their potential to bind biomolecular targets. We now have implemented a considerable expansion through a large aromatic helical foldamer inclusion. A foldamer was first identified that undergoes flexizyme-mediated tRNA acylation and that is capable of initiating ribosomal translation with yields sufficiently high to perform an mRNA display selection of macrocyclic foldamer–peptide hybrids. A hybrid macrocyclic nanomolar binder to the C-lobe of the E6AP HECT domain was selected that showed a highly converged peptide sequence. A crystal structure and molecular dynamics simulations revealed that both the peptide and foldamer are helical in an intriguing reciprocal stapling fashion. The strong residue convergence could be rationalized based on their involvement in specific interactions with the target protein. The foldamer stabilizes the peptide helix through stapling and through contacts with key residues. These results altogether represent a significant extension of the chemical space amenable to display selection and highlight possible benefits of inserting an aromatic foldamer into a peptide macrocycle for the purpose of protein recognition.     

Fractional nonlinear dynamics of learning with memory

Nonlinear Dynamics - In this paper, we consider generalization of the Lucas model of learning (learning-by-doing) that is described in the paper Robert E. Lucas (Econometrica 61(2):251–272,...