Combinatorial optimization for undergraduates by L. R. Foulds

By L. R. Foulds

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Extra resources for Combinatorial optimization for undergraduates

Sample text

Chapter 2 20 Figure 2-3: Topological Deformation Now the problem is easily solvable. Connect the squares as prescribed. After this, return the rubber to the initial state again (see Figure 2-4). A Figure 2-4: Solution to the Intersection Problem The solution is now so obvious that we can immediately "see" it. By the way, u the problem is a nice exercise in topology. Simplicity and conciseness also playa key role in all sciences and particularly in mathematics. Einstein put it this way: "Make it as simple as possible, but not simpler".

The overview will begin with general, unspecified notions, and then proceed to more formal concepts. Finally, a short historical digression will be presented to suggest further arguments for the importance of mathematical modeling. 1. Model: a Definition The term model has a variety of meanings and is used for many different purposes. We use modeling clay to form small replica of physical objects; children - and sometimes also adults - play with a model railway or model aeroplane; architects build (scale) model houses or (real-size) model apartments in order to show them to new clients; some people work as photo models, others take someone for a model, many would like to have a model friend.

The first is hurriedly collecting too much inadequate or incomplete data and irrelevant relationships, and the other is deciding prematurely on the methods to be used to solve the problem. The first mistake comes from an erroneous idea that the final and formalized model and its solution are more important than this "preliminary" stage, although a careful and accurate study of the problem is essential. The second mistake often arises because many modelers think that using some formalism is more important than using an appropriate notation that reflects the problem.