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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 following are fundamental examples of symmetric functions. The monomial symmetric functions m α. Suppose α = (α 1,α 2,...) is a sequence of non-negative integers, only finitely many of which are non-zero. Then we can consider the monomial defined by α: X α = X 1 α 1 X 2 α 2 X 3 α 3....
This can occur in many ways; for example, if X is a set with no additional structure, a symmetry is a bijective map from the set to itself, giving rise to permutation groups. If the object X is a set of points in the plane with its metric structure or any other metric space , a symmetry is a bijection of the set to itself which preserves the ...
Thomae's function: is a function that is continuous at all irrational numbers and discontinuous at all rational numbers. It is also a modification of Dirichlet function and sometimes called Riemann function. Kronecker delta function: is a function of two variables, usually integers, which is 1 if they are equal, and 0 otherwise.
The example shows that whether or not the expression for a given monomial symmetric polynomial in terms of the first n power sum polynomials involves rational coefficients may depend on n. But rational coefficients are always needed to express elementary symmetric polynomials (except the constant ones, and e 1 which coincides with the first ...
The set of the symmetrically continuous functions, with the usual scalar multiplication can be easily shown to have the structure of a vector space over , similarly to the usually continuous functions, which form a linear subspace within it.
Symmetry in physics has been generalized to mean invariance—that is, lack of change—under any kind of transformation, for example arbitrary coordinate transformations. [17] This concept has become one of the most powerful tools of theoretical physics , as it has become evident that practically all laws of nature originate in symmetries.
A function is said to be an equivariant map when its domain and codomain are acted on by the same symmetry group, and when the function commutes with the action of the group. That is, applying a symmetry transformation and then computing the function produces the same result as computing the function and then applying the transformation.