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In mathematics, the multiplicity of a member of a multiset is the number of times it appears in the multiset. For example, the number of times a given polynomial has a root at a given point is the multiplicity of that root.
Multiplicity (mathematics), the number of times an element is repeated in a multiset; Multiplicity (software), a software application which allows a user to control two or more computers from one mouse and keyboard
The cardinality or "size" of a multiset is the sum of the multiplicities of all its elements. For example, in the multiset {a, a, b, b, b, c} the multiplicities of the members a, b, and c are respectively 2, 3, and 1, and therefore the cardinality of this multiset is 6.
In mathematics, Serre's multiplicity conjectures, named after Jean-Pierre Serre, are certain problems in commutative algebra, motivated by the needs of algebraic geometry. Since André Weil 's initial definition of intersection numbers , around 1949, there had been a question of how to provide a more flexible and computable theory, which Serre ...
In linear algebra, the trace of a square matrix A, denoted tr(A), [1] is the sum of the elements on its main diagonal, + + +.It is only defined for a square matrix (n × n).The trace of a matrix is the sum of its eigenvalues (counted with multiplicities).
In mathematics, specifically algebraic geometry, a scheme is a structure that enlarges the notion of algebraic variety in several ways, such as taking account of multiplicities (the equations x = 0 and x 2 = 0 define the same algebraic variety but different schemes) and allowing "varieties" defined over any commutative ring (for example, Fermat curves are defined over the integers).
The sum of the algebraic multiplicities of all distinct eigenvalues is μ A = 4 = n, the order of the characteristic polynomial and the dimension of A. On the other hand, the geometric multiplicity of the eigenvalue 2 is only 1, because its eigenspace is spanned by just one vector [ 0 1 − 1 1 ] T {\displaystyle {\begin{bmatrix}0&1&-1&1\end ...
In mathematics, a univariate polynomial of degree n with real or complex coefficients has n complex roots, if counted with their multiplicities.They form a multiset of n points in the complex plane.