A BARRIER BOUNDARY VALUE PROBLEM FOR PARABOLIC AND ELLIPTIC EQUATIONS

Barrier type boundary conditions are modeled for describing the substance diffusion in a medium when obstacles in the medium are considered. The existence of solutions of parabolic and elliptic differential equations with barrier boundary condition is presented in this paper.     

An efficient solution method for Weber problems with barriers based on genetic algorithms

In this paper we consider the problem of locating one new facility with respect to a given set of existing facilities in the plane and in the presence of convex polyhedral barriers. It is assumed that a barrier is a region where neither facility location nor travelling are permitted. The resulting non-convex optimization problem can be reduced to a finite series of convex subproblems, which can be solved by the Weiszfeld algorithm in case of the Weber objective function and Euclidean distances. A solution method is presented that, by iteratively executing a genetic algorithm for the selection of subproblems, quickly finds a solution of the global problem. Visibility arguments are used to reduce the number of subproblems that need to be considered, and numerical examples are presented.     

Isomerization barriers in bis(4H-thiopyran) and in bithioxanthenes

The inversion and rotation mechanisms for the isomerization of Feringa’s bithioxanthenes existing in two conformations, up/up and up/down, have been calculated at the B3LYP/6-31G(d) and B3LYP/6-311++G(d,p) levels. The inversion mechanism that maintains the double bond nature of the central bond is a classical one but the rotation mechanisms that break the double bond to form a biradical needs to explore the singlet and triplet states. To do this we have removed the four fused phenyl rings of bithioxanthene and calculated at the CASSCF and CASPT2 levels bis(4H-thiopyran) proving that B3LYP calculations yield reasonable results for the rotation barriers.     

A molecular orbital study of the bonding in complexes of lithium with polynuclear aromatic hydrocarbons

In a preliminary investigation of the bonding between the sub-lattices of lithium and of graphite in intercalation compounds, molecular orbital calculations are performed on systems of Li atoms complexed to one or both sides of polynuclear aromatic hydrocarbons. Energy wells calculated reveal possible sites offered to the Li atoms above the aromatic plane to form complexes in both central and edge regions of the molecule. The influence of additional complexed Li atoms on the migratory tendency of a given Li atom is also investigated, and observations made on the effect of the complexed Li atoms on the C-C bonds of the hydrocarbon.     

The rectilinear distance Weber problem in the presence of a probabilistic line barrier

This paper considers the problem of locating a single facility in the presence of a line barrier that occurs randomly on a given horizontal route on the plane. The objective is to locate this new facility such that the sum of the expected rectilinear distances from the facility to the demand points in the presence of the probabilistic barrier is minimized. Some properties of the problem are reported, a solution algorithm is provided with an example problem, and some future extensions to the problem are discussed.     

A reduction result for location problems with polyhedral barriers

In this paper we consider the problem of locating one new facility in the plane with respect to a given set of existing facilities where a set of polyhedral barriers restricts traveling. This non-convex optimization problem can be reduced to a finite set of convex subproblems if the objective function is a convex function of the travel distances between the new and the existing facilities (like e.g. the median and center objective functions). An exact algorithm and a heuristic solution procedure based on this reduction result are developed.     

On the use of the Varignon frame for single facility Weber problems in the presence of convex barriers

This paper presents a new experimental approach to the Weber problem in the presence of convex barriers by using the Varignon frame. The Varignon frame is a mechanical system of strings, weights and a board with holes that has been used to identify an optimal location for the classical Weber problem. We show through analytical results that the same analog can also be used for some of the Weber problems in the presence of barriers. Some examples from the literature are revisited through experiments. Findings are compared to those found in the literature. Practical use of the analog is discussed as it provides rapid solutions, allows for flexibility, and enables one to visualize the problem.     

Unusually High Barriers to Rotation Involving the Tetrahedral Carbon Atom

Rotation about CC single bonds in suitably substituted compounds gives rise to rotational isomers (also called rotamers) which can be isolated if the barrier to rotation is high enough. The present article considers derivatives of triptycene and fluorene whose barriers to rotation are of the order of 30 kcal/mol.     

Gene flow across geographical barriers — scaling limits of random walks with obstacles

We study a class of random walks which behave like simple random walks outside of a bounded region around the origin and which are subject to a partial reflection near the origin. We obtain a non trivial scaling limit which behaves like reflected Brownian motion until its local time at zero reaches an exponential variable. It then follows reflected Brownian motion on the other side of the origin until its local time at zero reaches another exponential level, etc. These random walks are used in population genetics to trace the position of ancestors in the past near geographical barriers.