Power-ordered sets are not always lattices. In the case of distributive lattices we give a description by disjoint of chains. Finite power-ordered sets have a polarity. We introduct the leveled lattices and show examples with trivial tolerance. Finally we give a list of Hasse diagrams of power-ordered sets.
Linear viscoelastic properties for a dilute polymer solution are predicted by modeling the solution as a suspension of non-interacting bead-spring chains. The present model, unline the Rouse model, can describe the solution's rheological behavior even when the solvent quality is good, since excluded volume effects are explicitly taken into account through a narrow Gaussian repulsive potential between pairs of beads in a bead-spring chain. The use of the narrow Gaussian potential, which tends to the more commonly used delta-function repulsive potential in the limit of a width parameter d going to zero, enables the performance of Brownian dynamics simulations. The simulations results, which describe the exact behavior of the model, indicate that for chains of arbitrary but finite length, a delta-function potential leads to equilibrium and zero shear rate properties which are identical to the predictions of the Rouse model. On the other hand, a non-zero value of d gives rise to a predictionof swelling at equilibrium, and an increase in zero shear rate properties relative to their Rouse model values. The use of a delta-function potential appears to be justified in the limit of infinite chain length. The exact simulation results are compared with those obtained with an approximate solution, which is based on the assumption that the non-equilibrium configurational distribution function is Gaussian. The Gaussian approximation is shown to be exact to first order in the strength of excluded volume interaction, and is used to explore long chain rheological properties by extrapolating results obtained numerically for finite chains, to the limit of infinite chain length.
In this paper we deal with single facility location problems in a general normed space where the existing facilities are represented by sets. The criterion to be satis ed by the service facility is the minimization of an increasing function of the distances from the service to the closest point ofeach demand set. We obtain a geometrical characterization of the set of optimal solutions for this problem. Two remarkable cases - the classical Weber problem and the minmax problem with demand sets - are studied as particular instances of our problem. Finally, for the planar polyhedral case we give an algorithmic description of the solution set of the considered problems.
In this paper we construct a multiscale solution method for the gravimetry problem, which is concerned with the determination of the earth's density distribution from gravitational measurements. For this purpose isotropic scale continuous wavelets for harmonic functions on a ball and on a bounded outer space of a ball, respectively, are constructed. The scales are discretized and the results of numerical calculations based on regularization wavelets are presented. The obtained solutions yield topographical structures of the earth's surface at different levels of localization ranging from continental boundaries to local structures such as Ayer's Rock and the Amazonas area.
This essay discusses the multileaf collimator leaf sequencing problem, which occurs in every treatment planning in radiation therapy. The problem is to find a good realization in terms of a leaf sequence in the multileaf collimator such that the time needed to deliver the given dose profile is minimized. A mathematical model using an integer programming formulation has been developed. Additionally, a heuristic, based on existing algorithms and an integer programming formulation, has been developed to enhance the quality of the solutions. Comparing the results to those provided by other algorithms, a significant improvement can be observed.