Lecture 2; Math 636

Natural vs. ad hoc models revisited

Metatheorem: The more you match the structure of the problem, the better the model.

System – varies with application but generally
coincides with the object being modeled^{*}

Manifold = variables + equations between them (Differential Geometry)

Static vs dynamic models revisited

Group = generators + equations between them (Abstract Algebra)

Arguments from scale:

Groups acting on sets -- symmetries

Scaling: Group of similitudes (homotheties, multiples of the identity)

Alias vs. alibi in group actions – the two halves of chapter 2

Example:
the rotation group on R^{2}

^{ Given
an orthogonal matrix, is it a linear map or a change of basis?}

Alibi

Allometric models

Alias

Units and dimensional analysis

*Image from http://energy.sdsu.edu/testcenter/testhome/Test/problems/images/ch01/piston-coil.jpg