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Alpha shapes are closely related to alpha complexes, which are subcomplexes of the Delaunay triangulation of the point set. Each edge or triangle of the Delaunay triangulation may be associated with a characteristic radius: the radius of the smallest empty circle containing the edge or triangle.
The algebra of sets is the set-theoretic analogue of the algebra of numbers. Just as arithmetic addition and multiplication are associative and commutative, so are set union and intersection; just as the arithmetic relation "less than or equal" is reflexive, antisymmetric and transitive, so is the set relation of "subset".
For example, if a typical coin is tossed and one assumes that it cannot land on its edge, then it can either land showing "heads" or "tails." Because these two outcomes are mutually exclusive (i.e. the coin cannot simultaneously show both heads and tails) and collectively exhaustive (i.e. there are no other possible outcomes not represented ...
If A is a set, then the absolute complement of A (or simply the complement of A) is the set of elements not in A (within a larger set that is implicitly defined). In other words, let U be a set that contains all the elements under study; if there is no need to mention U, either because it has been previously specified, or it is obvious and unique, then the absolute complement of A is the ...
The drum was turned by use of a crank on the top of the instrument. The number of cams encountered by each digit as the crank turned was determined by the value of that digit. For example, if a slide is set to its "6" position, a row of 6 cams would be encountered around the drum corresponding to that position.
For example, it is used to equate a probability for a random variable with the Lebesgue-Stieltjes integral typically associated with computing the probability: = for all in the Borel σ-algebra on , where () is the cumulative distribution function for , defined on , while is a probability measure, defined on a σ-algebra of subsets of some ...
In statistics, compositional data are quantitative descriptions of the parts of some whole, conveying relative information. Mathematically, compositional data is represented by points on a simplex .
Several graph-theoretic concepts are related to each other via complementation: The complement of an edgeless graph is a complete graph and vice versa. Any induced subgraph of the complement graph of a graph G is the complement of the corresponding induced subgraph in G. An independent set in a graph is a clique in the complement graph and vice ...