Abstract
In this chapter we give a more detailed description of a couple of scientific models, and in addition an insight into how these models were constructed and the impact they have had in the field of research they belong to. The aim is to cover all the types of models that were presented in the model taxonomy (p. 28) and also to look at models that belong to the fields represented by the scientists in the previous chapter.
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Notes
- 1.
Please note the difference between adsorption and absorption: the former case refers to the binding on a surface, while the latter corresponds to the uptake of a substance in another gas, liquid or solid substance.
- 2.
Langmuir, I. (1916). The constitution and fundamental properties of solids and liquids. Part I. Solids. Journal of the American Chemical Society 38(11): 2221–2295.
- 3.
The partial pressure is the pressure a gas would exert on a surface if all other gases (such as air) were absent.
- 4.
Brunauer, S., Emmett, P.H. and Teller, E. (1938). J. Am. Chem. Soc., 60, 309.
- 5.
Haykin, S.S. (1999). Neural networks: a comprehensive foundation. Prentice Hall.
- 6.
This is an archetypal example of a model inspired by biology, so called biomimetics.
- 7.
A common method for achieving this is “back propagation” that updates the values of the nodes in the different layers in sequence in an effort to minimise the error.
- 8.
This is an authority that records and distributes a variety of statistics concerning Sweden. The data analysed here can be found at: http://www.ssd.scb.se/.
- 9.
This feature can be analysed using an autocorrelation diagram, which quantifies positive and negative correlations between different time points.
- 10.
If this restriction is relaxed then it would for example be possible to include the gross national product and consumer price index as input to the model.
- 11.
Mills, T.C. and Markellos, R.N. (2008). The Econometric Modelling of Financial Time Series, Cambridge University Press.
- 12.
The problem consists of solving Newton’s equations of motion for N bodies simultaneously, where the force exerted on each body is the total gravitational pull from all other bodies. The great French mathematician Henri Poincaré partially solved the three-body problem in 1887 and in doing so claimed the winning prize in a competition organised by the Swedish king Oscar II. In his solution Poincaré for the first time described the concept of deterministic chaos and laid the foundations for modern chaos theory. But the road to success was not straight. A mistake was found in Poincaré’s original submission and he had to use all of the prize money to publish a revised version of his article in the journal Acta Mathematica, still one of the most prestigious journals in the mathematical sciences. For a detailed history of the N-body problem see for example F. Diacu (1996), The solution of the N-body problem. The Mathematical Intelligencer, 18(3).
- 13.
Lorenz, E. (1966). The Essence of Chaos. CRC Press.
- 14.
Lorenz, E. (1963). Deterministic non-periodic flow, Journal of the Atmospheric Sciences, 20, 130–141.
- 15.
Tucker, W. (2002). A Rigorous ODE Solver and Smale’s 14th Problem. Found. Comp. Math. 2: 53–117.
- 16.
The observation was made in 1572 at Herrevad Abbey in what is now Sweden, and the discovery was described in the writing De novo stella (1573).
- 17.
1 parsec (\(\approx \)3.26 light years) is the distance at which a star exhibits a yearly angular displacement or parallax of one second of arc.
- 18.
For a detailed account of this and other examples we refer to: Tassoul, J.-L. and Tassoul, M. (2004). A concise history of solar and stellar physics. Princeton University Press.
- 19.
The discoverers Capecchi, Evans and Smithies were in 2007 awarded with the Nobel prize in physiology or medicine.
- 20.
The word derives from an animal in Greek mythology known as the Chimaera (\(\chi \acute{\iota } \mu \alpha \iota \rho \alpha \)) that had three heads: one lion, one goat and one snake head.
- 21.
The chimera will produce germ cells (that just like normal germ cells only contain half the chromosomes) and a fraction of these will contain the mutated gene, while most will be normal. When a germ cell that contains the mutated gene fuses with a normal germ cell a mouse with a single mutated copy of the gene is created, i.e. a NF\(^{+/-}\) mouse.
- 22.
Also in this case the tools have been borrowed from nature. The Cre gene originally belongs in a so called bacteriophage, a virus that infects bacteria, and the protein it codes for has an ability to cut out pieces of the DNA that are surrounded by a specific genetic sequence. This matching sequence is known as the LoxP and is 34 base pairs long, and is placed on each side of the NF1-gene, flanking the gene of interest. The last detail is to place the Cre gene at a location in the DNA so that it becomes assoaciated with genes that are expressed exclusively in nerve cells. Simply put: in knock-out mice the Cre gene is only activated in nerve cells. The corresponding protein is produced, locates the LoxP sequence and removes it together with the NF1-gene, and the end result is mice that are NF\(^{+/-}\) in nerve cells and NF\(^{+/+}\) in all other cells.
- 23.
Perez-Macias Martin, J.M. (2009). Estimating wetted area of a model-hull from a set of camera images using NURBS curves and surfaces. Master’s thesis at Chalmers University of Technology.
- 24.
The Reynolds number is a dimensionless quantity that is defined as \(\rho VL/\mu \), where \(\rho \) is the density, \(\mu \) is the viscosity and V and L are the “typical” velocity and length scales. A Reynolds number above 2100 corresponds to turbulent flow, while lower numbers indicate laminar flow.
- 25.
The Frenchman Navier and the Englishman Stokes, independent of each other, derived these equations during the 1820s as a model for the movement of a fluid. The derivation is based on the preservation of the physical quantities: mass, momentum and energy, and is usually expressed in terms of two coupled partial differential equations. With the aid of these equations the fluid is described as a continuum, i.e. as a continuous body, although in reality it consists of a collection of interacting molecules. Navier-Stokes equations are used in a wide range of disciplines and are of great importance to industry. Despite intense efforts over the years no one has been able to find an efficient analytical solution—or even proved that such a solution exists. If you succeed in doing so you will, in addition to becoming famous and being praised by the vehicle industry, receive a prize of 1 million USD from the Clay Mathematical Institute.
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Gerlee, P., Lundh, T. (2016). Worked Examples. In: Scientific Models. Springer, Cham. https://doi.org/10.1007/978-3-319-27081-4_5
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