The application of the genetic algorithm to the minimization of potential energy functions

We adapted the genetic algorithm to minimize the AMBER potential energy function. We describe specific recombination and mutation operators for this task. Next we use our algorithm to locate low energy conformation of three polypeptides (AGAGAGAGA, A9, and [Met]-enkephalin) which are probably the global minimum conformations. Our potential energy minima are –94.71, –98.50, and –48.94 kcal/mol respectively. Next, we applied our algorithm to the 46 amino acid protein crambin and located a non-native conformation which had an AMBER potential energy 150 kcal/mol lower than the native conformation. This is not necessarily the global minimum conformation, but it does illustrate problems with the AMBER potential energy function. We believe this occurred because the AMBER potential energy function does not account for hydration.     

On the minimizers of the Ginzburg-Landau energy for high kappa: the axially symmetric case

The Ginzburg-Landau theory of superconductivity is examined in the case of a special geometry of the sample, the infinite cylinder. We restrict to axially symmetric solutions and consider models with and without vortices. First putting the Ginzburg-Landau parameter κ formally equal to infinity, the existence of a minimizer of this reduced Ginzburg-Landau energy is proved. Then asymptotic behaviour for large κ of minimizers of the full Ginzburg-Landau energy is analyzed and different convergence results are obtained. Our main result states that, when κ is large, the minimum of the energy is reached when there are about κ vortices at the center of the cylinder. Numerical computations illustrate the various behaviours.     

Recent Results on the Fermi-Pasta-Ulam Problem

We revisit the celebrated model of Fermi, Pasta, and Ulam with the purpose of investigating the thresholds to equipartition in the thermodynamic limit. Starting with a particular class of initial conditions, i.e., with all the energy on the first mode, we observe that in a short time the system splits into two separate subsystems. We conjecture the existence of a function ε_c(ω), independent of the number N of particles in the chain, such that if the initial energy E satisfies E/N < ε_c(ω), then only the packet of modes with frequency not exceeding ω shares most of the energy. Bibliography: 25 titles.__________Published in Zapiski Nauchnykh Seminarov POMI, Vol. 300, 2003, pp. 145–154     

On the Finiteness of the Discrete Spectrum of a Four-Particle Lattice Schrödinger Operator

A Hamiltonian describing four bosons that move on a lattice and interact by means of pair zero-range attractive potentials is considered. A stronger version of the Hunziker–Van Vinter–Zhislin theorem on the essential spectrum is established. It is proved that the set of eigenvalues lying to the left of the essential spectrum is finite for any interaction energy of two bosons and is empty if this energy is sufficiently small.     

Relationship between the molecular structure and certain mechanical properties of oriented polymers

The correlations between the molecular structure and intermolecular energy and the fatigue and elastic properties of fibers are considered. It is shown that the maximum of these properties corresponds to an optimal value of the intermolecular energy.Leningrad Branch of the All-Union Scientific-Research Institute of Synthetic Fibers. Translated from Mekhanika Polimerov, No. 4, pp. 734–735, July–August, 1969.     

An Iterative Global Optimization Algorithm for Potential Energy Minimization

In this paper we propose an algorithm for the minimization of potential energy functions. The new algorithm is based on the differential evolution algorithm of Storn and Price (Journal of Global Optimization, vol. 11, pp. 341–359, 1997). The algorithm is tested on two different potential energy functions. The first function is the Lennard Jones energy function and the second function is the many-body potential energy function of Tersoff (Physics Review B, vol. 37, pp. 6991–7000, 1988; vol. 38, pp. 9902–9905, 1988). The first problem is a pair potential and the second problem is a semi-empirical many-body potential energy function considered for silicon-silicon atomic interactions. The minimum binding energies of up to 30 atoms are reported.Visitor at the Institute for Mathematics and its Applications, University of Minnesota, USA.     

On the minimal energy ordering of trees with perfect matchings

The energy of a graph is defined as the sum of the absolute values of all eigenvalues of the adjacency matrix of the graph. Zhang and Li [F. Zhang, H. Li, On acyclic conjugated molecules with minimal energies, Discrete Appl. Math. 92 (1999) 71–84] determined the first two smallest-energy trees of a fixed size with a perfect matching and showed that the third minimal energy is between two trees. This paper characterizes trees of a fixed size with a perfect matching with third minimal, fourth minimal and fifth minimal energies for n≥86 and third minimal, fourth minimal energies for 14≤n≤84.     

Partial regularity of energy minimizing harmonic maps into a complete manifold

In this paper, we study energy minimizing harmonic maps into a complete Riemannian manifold. We prove that the singular set of such a map has Hausdorff dimension at mostn–2, wheren is the dimension of the domain. We will also give an example of an energy minimizing map from surface to surface that has a singular point. Thus then–2 dimension estimate is optimal, in contrast to then–3 dimension estimate of Schoen-Uhlenbeck [SU] for compact targets.     

Allowance for boundary dislocations in the theory of adhesion of polyethylene to alkali halide crystals

The correction to the energy of adhesion of polyethylene on three alkali halide crystals, due to boundary dislocations resulting from the difference in lattice constant, is calculated. It is found that, in the examples considered, the dislocation energy is only 4 to 6% of the adhesion energyA. F. Ioffe Physicotechnical Institute, Academy of Sciences of the USSR, Leningrad. Translated from Mekhanika Polimerov, No. 3, pp. 533–536, May–June, 1976.     

A Phragmén–Lindelöf alternative result for the Navier–Stokes equations for steady compressible viscous flow

In this paper, the authors consider the Navier–Stokes equations for steady compressible viscous flow in three-dimensional cylindrical domain. A differential inequality for appropriate energy associated with the solutions of the Navier–Stokes isentropic flow in semi-infinite pipe is derived, from which the authors show a Phragmén–Lindelöf alternative result, i.e. the solutions for steady compressible viscous N–S flow problem either grow or decay exponentially as the distance from the entry section tends to infinity. In the decay case, the authors indicate how to bound explicitly the total energy in terms of data.     

Optimization of hydro energy storage plant problems by variational methods

We consider the problem of controlling a hydro energy storage plant within a fixed period of time. Our objective is to maximize the energy production — weighted with the time dependent tariff rate — subject to the given technical restrictions and the influx. The resulting control problem is transformed by means of the maximum principle for the calculus of variations into an equivalent nonlinear optimization problem.

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Zusammenfassung Wir betrachten hier das Problem der Steuerung eines Speicherkraftwerkes innerhalb eines bestimmten Zeitraumes, wobei die Energieproduktion — gewichtet mit dem zeitlich schwankenden Verkaufspreis — unter Berücksichtigung der technischen Nebenbedingungen und des Zuflusses maximiert werden soll. Das entstehende Kontrollproblem wird mit Hilfe des Maximumprinzips der Variationsrechnung in ein äquivalentes, nichtlineares Optimierungsproblem übergeführt.

     

On the numerical solution of Plateau's problem

The present paper is dedicated to the numerical computation of minimal surfaces by the boundary element method. Having a parametrization γ of the boundary curve over the unit circle at hand, the problem is reduced to seeking a reparametrization κ of the unit circle. The Dirichlet energy of the harmonic extension of γ○κ has to be minimized among all reparametrizations. The energy functional is calculated as boundary integral that involves the Dirichlet-to-Neumann map. First and second order necessary optimality conditions of the underlying minimization problem are formulated. Existence and convergence of approximate solutions is proven. An efficient algorithm is proposed for the computation of minimal surfaces and numerical results are presented.     

The heat flow with a critical exponential nonlinearity

We analyze the possible concentration behavior of heat flows related to the Moser–Trudinger energy and derive quantization results completely analogous to the quantization results for solutions of the corresponding elliptic equation. As an application of our results we obtain the existence of critical points of the Moser–Trudinger energy in a supercritical regime.     

Experimental study of fracture toughness and energy in composite materials

The paper presents an experimental investigation of fracture characteristics of composite materials. The post-peak response of the load-crack opening displacement of notched specimens is used to evaluate the fracture energy associated with progressive matrix damage and crack growth. Effects of fiber orientation and other geometric characteristics on fracture parameters are studied. The load versus crack opening displacement as well as crack length, fracture toughness, and energy versus the number of loading cycles are obtained for different specimens. Based on the experimental results of this study, concepts of the fracture mechanics are applied to evaluate the evolution of fracture toughness and energy.Presented at the 10th International Conference on the Mechanics of Composite Materials (Riga, April 20–23, 1998).Department of Mechanical & Industrial Engineering, University of Manitoba, Winnipeg, Manitoba, Canada, R3T 2N2. Published in Mekhanika Kompozitnykh Materialov, Vol. 34, No. 3, pp. 323–332, May–June, 1998.     

Free energy of a regular stable polymer network at small finite deformations

The equilibrium small finite deformations of a regular polymer network are considered. The free energy of the network is described in the form of a series expansion in small displacements of the network points from the equilibrium position. The general dependence of the free energy on the strain invariants thus obtained coincides with the Mooney expression for the free energy only in the first approximation. Aspects of the mechanical behavior of the material associated with this difference are discussed.Branch of the Institute of Chemical Physics, Academy of Sciences of the USSR, Moscow Oblast. Translated from Mekhanika Polimerov, No. 4, pp. 615–618, July–August, 1971.     

On the convergence of the multigroup,discrete-ordinates solutions for subcritical transport media

Summary In the multigroup, discrete-ordinates approximations to the linear transport equation, the integration over the directional variable is replaced by a numerical quadrature rule, involving a weighted sum over functional values at selected directions, with the energy dependence discretized by replacing the cross section data by weighted averages over each energy interval. The stability, consistency, and convergence rely fundamentally on the conditions that the maximum fluctuations in the total cross section — and in the expected number of secondary particles arising from each energy level — tend to zero as the energy mesh becomes finer, and as the number of angular nodes becomes infinite. Our analysis is based on using a natural Nyström method of extending the discrete-ordinates, multigroup approximates to all values of the angular and energy variables. Such an extension enables us to employ generalizations of the collectively compact operator approximation theory of P. M. Anselone to deduce stability and convergence of the approximates.This research was started while the first author (H.D.V., Jr.) was an Alexander von Humboldt Research Fellow in the Department of Mathematics, University of Kaiserslautern, Kaiserslautern, Federal Republic of Germany. Generous support was also provided by a Faculty Development Leave from Texas Tech University for the 1982–83 Academic Year.     

Formal and precise derivation of the Green functions for a simple potential

In formal scattering theory, the Green functions are obtained as solutions of a distributional equation. In this paper, we use the Sturm–Liouville theory to compute the Green functions within a rigorous mathematical theory. We shall show that both the Sturm–Liouville theory and the formal treatment yield the same Green functions. We shall also show how the analyticity of the Green functions as functions of the energy keeps track of the so-called “incoming” and “outgoing” boundary conditions.     

A $p$ -Laplace Equation with Logarithmic Nonlinearity at High Initial Energy Level

In this paper the authors investigate a class of \(p\)-Laplace equations with logarithmic nonlinearity, which were considered in Le and Le (Acta Appl. Math. 151:149–169, 2017), where, among other things, global existence and finite time blow-up of solutions were proved when the initial energy is subcritical and critical, that is, initial energy smaller than or equal to the depth of the potential well. Their results are complemented in this paper in the sense that an abstract criterion is given for the existence of global solutions that vanish at infinity or solutions that blow up in finite time, when the initial energy is supercritical. As a byproduct it is shown that the problem admits a finite time blow-up solution for arbitrarily high initial energy.

     

Microscopic quantum hydrodynamics of systems of fermions: II

A closed system of nonstationary equations for a multielectron atom is constructed using a method previously developed for investigating multiparticle fermion systems. The well-known Thomas—Fermi—Dirac method is a particular case of the system. The exchange spin—spin correlations are calculated using an improved spin—spin interaction Hamiltonian. The dependence of the electron concentration on the self-consistent electric field potential and a discrete set of parameters determined by the total energy and the angular momentum of the atom is derived for stationary atom states.     

Bioenergy Systems Planning Using Location–Allocation and Landscape Ecology Design Principles

Rural energy planning is a nexus of sustainable development issues, particularly the sustainable utilization of biomass resources, on which rural parts of the developing world remain critically dependent. A landscape based rural bioenergy planning framework is presented, which is based on location–allocation and landscape ecology principles and considers both domestic and commercial energy demands and energy flows, as well as the landscape impact of the required bioenergy production zones. p-median modelling principles underly the location–allocation formulation. Optimized bioenergy landscape designs are presented, which illustrate both accessibility and landscape ecology objectives.