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This is an energy balance which defines the position of the moving interface. Note that this evolving boundary is an unknown (hyper-)surface; hence, Stefan problems are examples of free boundary problems. Analogous problems occur, for example, in the study of porous media flow, mathematical finance and crystal growth from monomer solutions. [1]
The matrix P represents the weather model in which a sunny day is 90% likely to be followed by another sunny day, and a rainy day is 50% likely to be followed by another rainy day. [4] The columns can be labelled "sunny" and "rainy", and the rows can be labelled in the same order. The above matrix as a graph.
Although named for Edgar Buckingham, the π theorem was first proved by the French mathematician Joseph Bertrand in 1878. [1] Bertrand considered only special cases of problems from electrodynamics and heat conduction, but his article contains, in distinct terms, all the basic ideas of the modern proof of the theorem and clearly indicates the theorem's utility for modelling physical phenomena.
A simple application of dimensional analysis to mathematics is in computing the form of the volume of an n-ball (the solid ball in n dimensions), or the area of its surface, the n-sphere: being an n-dimensional figure, the volume scales as x n, while the surface area, being (n − 1)-dimensional, scales as x n−1.
The three-body problem is a special case of the n-body problem. Historically, the first specific three-body problem to receive extended study was the one involving the Earth, the Moon, and the Sun. [2] In an extended modern sense, a three-body problem is any problem in classical mechanics or quantum mechanics that models the motion of three ...
Although nondimensionalization is well adapted for these problems, it is not restricted to them. An example of a non-differential-equation application is dimensional analysis; another example is normalization in statistics. Measuring devices are practical examples of nondimensionalization occurring in everyday life. Measuring devices are ...
Stereochemistry, a subdiscipline of chemistry, studies the spatial arrangement of atoms that form the structure of molecules and their manipulation. [1] The study of stereochemistry focuses on the relationships between stereoisomers, which are defined as having the same molecular formula and sequence of bonded atoms (constitution) but differing in the geometric positioning of the atoms in space.
Tolerance analysis is the general term for activities related to the study of accumulated variation in mechanical parts and assemblies. Its methods may be used on other types of systems subject to accumulated variation, such as mechanical and electrical systems.