Definition & Betydelse | Engelska ordet MULTIPLICITIES

Definition av MULTIPLICITIES

1. böjningsform av multiplicity

Exempel på hur man kan använda MULTIPLICITIES i en mening

• For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the multiplicities of its poles equals the sum of the multiplicities of its zeros.
• In the case of two variables and in the case of affine hypersurfaces, if multiplicities and points at infinity are not counted, this theorem provides only an upper bound of the number of points, which is almost always reached.
• Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities).
• Most multiple star systems known are triple; for higher multiplicities, the number of known systems with a given multiplicity decreases exponentially with multiplicity.
• The fundamental theorem of algebra shows that any non-zero polynomial has a number of roots at most equal to its degree, and that the number of roots and the degree are equal when one considers the complex roots (or more generally, the roots in an algebraically closed extension) counted with their multiplicities.
• Geometric multiplicities of eigenvalues (but not the eigenspaces, which are transformed according to the base change matrix P used).
• Earlier applications include Frenkel's determination of upper bounds on the root multiplicities of the Kac–Moody Lie algebra whose Dynkin diagram is the Leech lattice, and Borcherds's construction of a generalized Kac–Moody Lie algebra that contains Frenkel's Lie algebra and saturates Frenkel's 1/∆ bound.
• For instance if both are diagonalizable, then one can take the decomposition into eigenspaces (for which the action is as simple as it can get, namely by a scalar), and then similarity can be decided by comparing eigenvalues and their multiplicities.
• The French thinkers Gilles Deleuze and Félix Guattari referred occasionally to fuzzy sets in connection with their phenomenological concept of multiplicities.
• The correspondence between functions and multisets is the same as in the previous case, and the surjectivity requirement means that all multiplicities are at least one.
• For every n-tuple of complex numbers, there is exactly one monic polynomial of degree n that has them as its zeros (keeping multiplicities).
• Note that knowing a matrix's spectrum with all of its algebraic/geometric multiplicities and indexes does not always allow for the computation of its Jordan normal form (this may be a sufficient condition only for spectrally simple, usually low-dimensional matrices).
• Nodes and foci are topologically equivalent but not orbitally equivalent or smoothly equivalent, because their eigenvalues are different (notice that the Jacobians of two locally smoothly equivalent systems must be similar, so their eigenvalues, as well as algebraic and geometric multiplicities, must be equal).
• Daniella Trimboli argues that instead of focusing on multiplicities, Fleming deconstructs the idea of singular truth by blending traditional documentary forms with her non-conventional storytelling techniques.
• Kahan discovered that polynomials with a particular set of multiplicities form what he called a pejorative manifold and proved that a multiple root is Lipschitz continuous if the perturbation maintains its multiplicity.
• In mathematics, the Littelmann path model is a combinatorial device due to Peter Littelmann for computing multiplicities without overcounting in the representation theory of symmetrisable Kac–Moody algebras.
• Extending to 3 dimensions the physically impossible Riemann surfaces used to classify all closed orientable 2-manifolds, Heegaard's 1898 thesis "looked at" similar structures for functions of two complex variables, taking an imaginary 4-dimensional surface in Euclidean 6-space (corresponding to the function f=x^2-y^3) and projecting it stereographically (with multiplicities) onto the 3-sphere.
• The depth of Q as an R-module is defined in that paper to be the least positive integer n such that Q⊗⋅⋅⋅⊗Q (n times Q, tensor product of R-modules, diagonal action of R from the right) has the same constituent indecomposable modules as Q ⊗⋅⋅⋅⊗ Q (n+1 times Q) (not counting multiplicities, an entirely similar definition for depth of Q as an H-module with closely related results).
• There is also a notion of restriction of a representation of a Lie algebra to a subjoined hypoalgebra, with branching rules similar to those for restriction to subalgebras except that some of the multiplicities in the branching rule may be negative.
• However in the theory of multisets the term refers to the sum of multiplicities of each member of a multiset.

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