A Global Optimization Approach to a Water Distribution Network Design Problem

In this paper, we address a global optimization approach to a waterdistribution network design problem. Traditionally, a variety of localoptimization schemes have been developed for such problems, each new methoddiscovering improved solutions for some standard test problems, with noknown lower bound to test the quality of the solutions obtained. A notableexception is a recent paper by Eiger et al. (1994) who present a firstglobal optimization approach for a loop and path-based formulation of thisproblem, using a semi-infinite linear program to derive lower bounds. Incontrast, we employ an arc-based formulation that is linear except forcertain complicating head-loss constraints and develop a first globaloptimization scheme for this model. Our lower bounds are derived through thedesign of a suitable Reformulation-Linearization Technique (RLT) thatconstructs a tight linear programming relaxation for the given problem, andthis is embedded within a branch-and-bound algorithm. Convergence to anoptimal solution is induced by coordinating this process with an appropriatepartitioning scheme. Some preliminary computational experience is providedon two versions of a particular standard test problem for the literature forwhich an even further improved solution is discovered, but one that isverified for the first time to be an optimum, without any assumed boundson the flows. Two other variants of this problem are also solved exactly forillustrative purposes and to provide researchers with additional test caseshaving known optimal solutions. Suggestions on a more elaborate study involving several algorithmic enhancements are presented for futureresearch.     

Frames and bases of Lie incidence geometries

For a rank two incidence system and a projective embedding the concepts ofe- indepedence of a point set and a basis of are defined. It is then demonstrated that for a Lie incidence geometry of type B_n,n and C_n,1 over a field of characteristic not two, A_n,k,D_n,1,D_n,n,E_6,1 or E_7,1, is transitive on certain classes of subsets called frames. As a consequence we obtain a characterization of the apartments of these geometries and demonstrate that the frames are bases.This work supported in part by an National Security Agency grantThis work supported in part by an National Science Foundation grant     

Computer use in mathematics education – present situation,intentions, and problems in the Federal Republic of Germany†

Computers, and computer‐related thinking structures, are only gradually influencing mathematics education. On the one hand, there is a discrepancy between involved teachers who already have changed their own classroom teaching to a great extent, and a majority of mathematics teachers who have not yet taken notice of the computer for teaching purposes. On the other hand, knowledge of the computer and of algorithms is frequently merely added to the mathematical subject matter. As opposed to that, the authors argue that it is necessary to genuinely integrate such subject matter, and to include general topics such as social impact and changed attitudes toward application. With regard to implementation, they develop concrete ideas which are aligned in a differentiated manner to the specific situation and the opportunities offered in the Federal Republic of Germany. The rationale for that is that only such reference to a specific situation will provide an opportunity for readers abroad to usefully apply approaches and ideas to the situation given in their own cultural environment.

An abbreviated version of this paper for cursory reading or other purposes has been marked by bold lines on the margin.

     

An investigation into the use of microcomputers to teach mathematical problem‐solving skills to 13 year olds

This is an account of an experiment to teach mathematical problem‐solving skills to 13‐year‐old students with the aid of microcomputers. Twenty‐one children worked through six microcomputer program‐generated problem situations in two and a half hours. Nine of the children worked ‘hands on’ with microcomputers. Twelve of the children were class taught, with a single microcomputer used for demonstration purposes. A comparison of pre‐test and post‐test scores on tests indicated that no significant gains in achievement were made by either group. Informal observations suggest a positive affective response. The findings are discussed at length and possible explanations for the lack of achievement are conjectured.     

Homogenization of a nonlinear multiscale model of calcium dynamics in biological cells

Calcium is one of the most important intracellular messengers, which occurs in the cytosol and the endoplasmic reticulum of animal cells. While most calcium dynamics models either do not account properly for the fact that the endoplasmic reticulum constitutes a microstructure of the cell or are infeasible by resolving the fine structure very explicitly, Goel et al. [15] derived an effective macroscopic model by formal homogenization. In this paper, this approach is made rigorous using periodic homogenization techniques to upscale the nonlinear coupled system of reaction–diffusion equations and, moreover, the appropriate scaling of the interfacial exchange term is taken into consideration.     

The effect of information on the quality of decisions

We study the effect of additional information on the quality of decisions. We define the extreme case of complete information about probabilities as our reference scenario. There, decision makers (DMs) can use expected utility theory to evaluate the best alternative. Starting from the worst case—where DMs have no information at all about probabilities—we find a method of constantly increasing the information by systematically limiting the ranges of the probabilities. In our simulation-based study, we measure the effects of the constant increase in information by using different forms of relative volumes. We define these as the relative volumes of the gradually narrowing areas which lead to the same (or a similar) decision as with the probability in the reference scenario. Thus, the relative volumes account for the quality of information. Combining the quantity and quality of information, we find decreasing returns to scale on information, or in other words, the costs of gathering additional information increase with the level of information. Moreover, we show that more available alternatives influence the decision process negatively. Finally, we analyze the quality of decisions in processes where more states of nature are considered. We find that this degree of complexity in the decision process also has a negative influence on the quality of decisions.     

A Mehta–Ramanathan theorem for linear systems with basepoints

Let be a normal complex projective polarized variety and an H‐semistable sheaf on X. We prove that the restriction to a sufficiently positive general complete intersection curve passing through a prescribed finite set of points remains semistable, provided that at each , the variety X is smooth and the factors of a Jordan–Hölder filtration of are locally free. As an application, we obtain a generalization of Miyaoka's generic semipositivity theorem.