Search results
Results from the WOW.Com Content Network
Aside from polynomial functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric -tensors on a vector space is isomorphic to the space of homogeneous polynomials of degree on . Symmetric functions should not be confused with even and odd functions, which have a different sort of symmetry.
The derivative of an integrable function can always be defined as a distribution, and symmetry of mixed partial derivatives always holds as an equality of distributions. The use of formal integration by parts to define differentiation of distributions puts the symmetry question back onto the test functions , which are smooth and certainly ...
If a function is differentiable (in the usual sense) at a point, then it is also symmetrically differentiable, but the converse is not true. A well-known counterexample is the absolute value function f ( x ) = | x | , which is not differentiable at x = 0 , but is symmetrically differentiable here with symmetric derivative 0.
One context in which symmetric polynomial functions occur is in the study of monic univariate polynomials of degree n having n roots in a given field.These n roots determine the polynomial, and when they are considered as independent variables, the coefficients of the polynomial are symmetric polynomial functions of the roots.
Symmetry occurs not only in geometry, but also in other branches of mathematics. Symmetry is a type of invariance: the property that a mathematical object remains unchanged under a set of operations or transformations. [1] Given a structured object X of any sort, a symmetry is a mapping of the object onto itself which preserves the structure.
For n = 2, the automorphism group is trivial, but S 2 is not trivial: it is isomorphic to C 2, which is abelian, and hence the center is the whole group. For n = 6 , it has an outer automorphism of order 2: Out(S 6 ) = C 2 , and the automorphism group is a semidirect product Aut(S 6 ) = S 6 ⋊ C 2 .
The symmetric algebra S(V) can also be built from polynomial rings.. If V is a K-vector space or a free K-module, with a basis B, let K[B] be the polynomial ring that has the elements of B as indeterminates.
In statistics, a symmetric probability distribution is a probability distribution—an assignment of probabilities to possible occurrences—which is unchanged when its probability density function (for continuous probability distribution) or probability mass function (for discrete random variables) is reflected around a vertical line at some ...