Symmetric-functions sentence example

symmetric-functions
• The theories of determinants and of symmetric functions and of the algebra of differential operations have an important bearing upon this comparatively new branch of mathematics.
• There is no difficulty in expressing the resultant by the method of symmetric functions.
• Now by the theory of symmetric functions, any symmetric functions of the mn values which satisfy the two equations, can be expressed in terms of the coefficient of those equations.
• A separation is the symbolic representation of a product of monomial symmetric functions.
• The law of reciprocity shows that p(s) = zti (m 1te2tmtL3t) t=1 st It 2t 3t viz.: a linear function of symmetric functions symbolized by the k specifications; and that () St =ti ts.
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• The important result is that the theory of invariants is from a certain point of view coincident with the theory of non-unitary symmetric functions.
• Denote by brackets () and [] symmetric functions of the quantities p and a respectively.
• All symmetric functions are expressible in terms of the quantities ap g in a rational integral form; from this property they are termed elementary functions; further they are said to be single-unitary since each part of the partition denoting ap q involves but a single unit.
• The partitions being taken as denoting symmetric functions we have complete correspondence between the algebras of quantity and operation, and from any algebraic formula we can at once write down an operation formula.
• The extraordinary advantage of the transformation of S2 to association with non-unitary symmetric functions is now apparent; for we may take, as representative forms, the symmetric functions which are symbolically denoted by the partitions referred to.
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• Remark, too, that we are in association with non-unitary symmetric functions of two systems of quantities which will be denoted by partitions in brackets ()a, ()b respectively.
• For example, the theory of invariants may be regarded as depending upon the consideration of the symmetric functions of the differences of the roots of the equation aox n - (i) a i x n - 1 + (z) a 2 x n 2 - ...