By Stavros Busenberg, Bruno Forte, Hendrik K. Kuiken (auth.), Vincenzo Capasso, Antonio Fasano (eds.)

The 1990 CIME path on Mathematical Modelling of commercial approaches got down to illustrate a few advances in questions of commercial arithmetic, i.e.of the purposes of arithmetic (with all its "academic" rigour) to real-life difficulties. The papers describe the genesis of the versions and illustrate their correct mathematical features. one of the themesdealt with are: thermally managed crystal development, thermal behaviour of a high-pressure gas-discharge lamp, the sessile-drop challenge, etching approaches, the batch-coil- annealing strategy, inverse difficulties in classical dynamics, photo illustration and dynamical structures, scintillation in rear projections displays, identity of semiconductor properties,pattern popularity with neural networks. CONTENTS: H.K. Kuiken: Mathematical Modelling of business Processes.- B. specialty: Inverse difficulties in arithmetic for Industry.- S. Busenberg: Case experiences in commercial Mathematics.

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**Example text**

This will give rise to Marangoni convection inside the drop, resulting in a shape different from the one following from the simple gravity-vs-surface-tension theory. A sensitivity analysis may reveal what influence each of these effects has on the shape of the drop and therefore on the value of the Bond number. In conclusion, this small problem is another example showing how effective a process of non-dimensionalization can be. In the present case a single parameter, viz. Bo, would suffice for a complete description of the family of solutions.

If there are no fluid motions inside the body, the pressure p suffices for the description of the stress level in the interior. s -~. (5) At each point it describes how strongly the liquid is compressed. If dz be a surface element in the interior of the liquid and k is a unit normal, then the force exerted on the liquid at the side of k by the liquid at the other side is given by dF = p k d s , (6) fully in line with the compressive character of the pressure (Fig. 2). Reversing the role of the portions of liquid defined in the previous sentence we obtain the balancing force 43 k ds Figure 2: Interior surface element ds with normal k.

Which expresses a plane geometry. Moreover we put fl = 0. , (25) 36 Asymptotics for Ti --* If we consider any region where T < ~b < 1, where T is a fixed value, and let Ti --* ¢x), then the exponential term of (23) is exponentially small in that region. As a consequence of this, the temperature profile in that region is governed by d 2 T / d Y 2 = O, so that T = A(1 - Y) = Touter , (26) where A is an, as yet, undetermined constant. Since T = 0 clearly belongs to the aforesaid region, Eq. (26) satisfies Eq.