In the paper, we describe Drinfel'd double structure of $n$-rank Taft algebra and all its simple modules, then endow $R$-matrices with some application to knot invariants. The invariants get is a generalization Jones polynomial, in particular, it recovers polynomial rank $1$ case, while $2$ one-parameter specialization two-parameter unframed Dubrovnik higher case composite ($n$-power) polynomial.

The calculation of free energy from computer simulation is of current interest [1-3]. Two major issues have arisen: the search for an adequate representation of the intermolecular energy (i.e. choice of potential form and its parameters) and the choice of the numerical technique that gives reliable result with the minimum computational effort. It is important to stress that the two issues are e...

In molecular mechanics, current generation potential energy functions provide a reasonably good compromise between accuracy and effectiveness. This paper firstly reviewed several most commonly used classical potential energy functions and their optimization methods used for energy minimization. To minimize a potential energy function, about 95% efforts are spent on the Lennard-Jones potential o...

The connection between isobaric fragility and the properties of high-order stationary points of the potential energy surface in different supercooled Lennard-Jones mixtures was investigated. The increase of effective activation energies upon supercooling appears to be driven by the increase of average potential energy barriers measured by the energy dependence of the fraction of unstable modes....

We discuss the phenomenon of physical aging in a well studied atomic model liquid, a binary mixture of particles interacting with Lennard-Jones potentials. We put emphasis on the different dynamical behaviours of the model as well as on the relations between the experimental and numerical timescales. The physics of aging is discussed within a well characterized thermodynamic framework, based on...

Accurate modeling of the molecular environment is critical in condensed phase simulations of chemical reactions. Conventional quantum mechanical/molecular mechanical (QM/MM) simulations traditionally model non-electrostatic non-bonded interactions through an empirical Lennard-Jones (LJ) potential which, in violation of intuitive chemical principles, is bereft of any explicit coupling to an atom...

Self-propelled particles are used to simulate cell aggregates in a model considering homogeneous adhesion forces between cells and using only motility differences as segregation drivers. The tendency of cells to follow their neighbors is also included in the formulation. Three model variants are explored, and the conditions on which motility differences may produce segregation are mapped in par...

A set of Lindemann measures, based on positional deviations or return distances, defined with respect to mechanically stable inherent structure configurations, is applied to understand the solid-liquid phase transition in a Lennard-Jones-type system. The key quantity is shown to be the single-particle return distance-squared distribution. The first moment of this distribution is related to the ...