DIFFUSION AND BIFURCATION PROBLEMS IN SINGULARLY PERTURBED DOMAINS

ABSTRACT. Diffusion problems under singular perturbations of the domain or the boundary conditions are analyzed. The first problem that we consider is the diffusion of a material from a domain that is nearly impermeable, having only several small patches on the boundary where the material can slowly leak out. The second problem that is studied is the diffusion of a material that originates from some localized regions in a two or three‐dimensional domain. Steady‐state solutions and the long‐time behavior of solutions are analyzed in detail. Finally, the analysis is extended to determine the change in bifurcation values associated with nonlinear diffusion equations under singular perturbations of the domain. The results are then applied to a model in resource management.     

THE INTRINSIC QUALITATIVE PROPERTIES OF THE CLASSICAL OPTIMAL STOPPING PROBLEM ARE INVARIANT TO THE FUNCTIONAL FORM OF THE DISCOUNT FUNCTION

Abstract The intrinsic qualitative properties of a generic optimal stopping model are shown to be invariant to the functional form of the discount function. If the discount function is assumed to be a member of particular infinite parametric family—a family that includes the exponential and classical hyperbolic discount functions as special cases—an additional refutable comparative statics result is produced that holds for the entire family. Consequently, if one limits econometric tests of the model to its qualitative properties, one cannot determine the form of the discount function used by the decision maker. It is also shown that the only discount function that yields a time‐consistent stopping rule is the exponential function with a constant rate of discount.     

USING NETWORK ANALYSIS TO CHARACTERIZE FOREST STRUCTURE

Abstract Network analysis quantifies different structural properties of systems of interrelated parts using a single analytical framework. Many ecological phenomena have network‐like properties, such as the trophic relationships of food webs, geographic structure of metapopulations, and species interactions in communities. Therefore, our ability to understand and manage such systems may benefit from the use of network‐analysis techniques. But network analysis has not been applied extensively to ecological problems, and its suitability for ecological studies is uncertain. Here, we investigate the ability of network analysis to detect spatial patterns of species association in a tropical forest. We use three common graph‐theoretic measures of network structure to quantify the effect of understory tree size on the spatial association of understory species with trees in the canopy: the node degree distribution (NDD), characteristic path length (CPL), and clustering coefficient (CC). We compute the NDD, CPL, and CC for each of seven size classes of understory trees. For significance testing, we compare the observed values to frequency distributions of each statistic computed from randomized data. We find that the ability of network analysis to distinguish observed patterns from those representing randomized data strongly depends on which aspects of structure are investigated. Analysis of NDD finds no significant difference between random and observed networks. However, analysis of CPL and CC detected nonrandom patterns in three and one of the seven size classes, respectively. Network analysis is a very flexible approach that holds promise for ecological studies, but more research is needed to better understand its advantages and limitations.     

Singular Perturbation Solution of Time Dependent Mass-Transfer with Non-linear Chemical Reaction

In this work the author presents two analyses of time dependentprocesses in which diffusion and non-linear reaction take placesimultaneously. The first is an analysis of the distributionof a single reactant in a stagnant medium when the reactionrate is proportional to the concentration squared. In the secondanalysis the time dependent distributions of two reactants aretraced, when the reaction rate is porportional to the productof the concentrations. The processes under consideration areassumed to take place in unbounded containers and attentionis focused on the cases in which the initial concentrationshave non-zero space average. The analyses are carried out assumingthat the space variations of the initial distributions and thediffusivities are large while the reaction constant(s) is small.Therefore, initially the changes in the distribution due todiffusion are much faster than the rate of absorption in thereaction. However, eventually the processes become reactiondominated. The two distinct behaviours are represented by correspondingasymptotic expansions which are matched in the usual manner.On the other hand, if the initial average concentrations vanish,then, in as much as the processes are diffusion dominated initially,they are diffusion dominated thoughout. The distributions canthen be solved-for by regular perturbations. The treatment of the single component case shows that when theinitial average concentration does not vanish, the residualamount of reactant is inversely proportional to the time thatelapsed since the beginning of the process multiplied by thereaction constant. This residual amount is altogether independentof the bulk quantity supplied initially. The decay of the binaryprocess is similar provided the bulk quantities supplied initiallyare well proportioned. If there is a disproportion, the decayis exponential rather than algebraic. In all events, late inthe process the residual amount(s) is independent of the diffusivity.     

Adiabatic-diabatic transformations for molecular systems: a model study of two interacting conical intersections

A model is presented for studying the interaction between two conical intersections (e.g. a dimer of two bound molecules each characterized by a conical intersection). The model is an extension of a previous model for a single conical intersection formed by an electron housed in a vibrating molecule (Baer, M. and Englman, R., 1992, Molec. Phys., 75, 293). We distinguish between two situations: when the coupling is weak (for instance when it takes place in the asymptotic region) and when it is strong. The study is accomplished by calculating the adiabatic—diabatic transformation (ADT) matrix. Whereas the features of the ADT matrix for weak coupling seem reasonable (and to a certain extent expected), we find some unexpected features in the case of strong coupling. In particular, the two characteristic ADT angles of the uncoupled systems namely (?_1/2) and (?_2/2) are replaced by two new ADT angles, namely, (?₁ + ?₂)/2 and (?₁—?₂)/2. This implies that the corresponding nuclear wavefunctions, which originally were multi-valued, become single-valued in cases of strong interaction.     

Convergence behaviour of Green's function quantum Monte Carlo simulations of pi electron systems

Green's function quantum Monte Carlo (GF QMC) simulations of fermionic ensembles require the definition of a so-called spectral parameter W to yield trustworthy estimates of the fully correlated ground state energy E. In the present work we discuss the influence of the shift parameter, W, on the convergence behaviour of GF QMC simulations. As model systems we have considered some π molecules which are studied in the framework of the Pariser–Parr–Pople (PPP) Hamiltonian. The influence of the W parameter on the convergence of the GF QMC simulations in many-electron systems with an odd number of electronic permutations within one spin direction exceeds the influence observed in systems without such odd permutations. They do not occur in polyenes and Huckel annulenes with an electron count of (4n + 2) (n = 0, 1, 2…). The GF QMC technique adopted as a computational tool has been developed to study π molecules with fermionic sign problems owing to odd electronic interchanges within one spin direction.