Search results
Results from the WOW.Com Content Network
A table can be created by taking the Cartesian product of a set of rows and a set of columns. If the Cartesian product rows × columns is taken, the cells of the table contain ordered pairs of the form (row value, column value). [4] One can similarly define the Cartesian product of n sets, also known as an n-fold Cartesian product, which can be ...
The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. [1] Its subalgebras include diagram algebras such as the Brauer algebra, the Temperley–Lieb algebra, or the group algebra of the symmetric group. Representations of the partition algebra are built from sets ...
A geometry: it is equipped with a metric and is flat. A topology: there is a notion of open sets. There are interfaces among these: Its order and, independently, its metric structure induce its topology. Its order and algebraic structure make it into an ordered field.
Partitions of a 4-element set ordered by refinement. A partition α of a set X is a refinement of a partition ρ of X—and we say that α is finer than ρ and that ρ is coarser than α—if every element of α is a subset of some element of ρ. Informally, this means that α is a further fragmentation of ρ. In that case, it is written that ...
Inner products allow formal definitions of intuitive geometric notions, such as lengths, angles, and orthogonality (zero inner product) of vectors. Inner product spaces generalize Euclidean vector spaces, in which the inner product is the dot product or scalar product of Cartesian coordinates.
In category theory, the product of two (or more) objects in a category is a notion designed to capture the essence behind constructions in other areas of mathematics such as the Cartesian product of sets, the direct product of groups or rings, and the product of topological spaces.
The set of pairwise sums is A + A = {a + b : a,b ∈ A} and is called the sumset of A. The set of pairwise products is A · A = {a · b : a,b ∈ A} and is called the product set of A; it is also written AA. The theorem is a version of the maxim that additive structure and multiplicative structure cannot coexist.
Cylinder sets over topological vector spaces are the core ingredient in the [citation needed] definition of abstract Wiener spaces, which provide the formal definition of the Feynman path integral or functional integral of quantum field theory, and the partition function of statistical mechanics.