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Elementary Algebra
Published in Dan Zwillinger, CRC Standard Mathematical Tables and Formulas, 2018
f(x) = an(x - z1)(x - z2)…(x - zn) . (2.1.4)If the coefficients of the polynomial, {a0,a1,...,an} $ \{ a_{0} , a_{1} , . . . , a_{n} \} $ , are real numbers, then the polynomial will always have an even number of complex roots occurring in pairs. That is, if z is a complex root, then so is z¯ $ \bar{z} $ . If the polynomial has an odd degree and the coefficients are real, then it must have at least one real root.Equations for roots of 2nd, 3rd, and 4thocOlur equations are on pages 73‐74.The coefficients of a polynomial may be expressed as symmetric functions of the roots. For example, the elementary symmetric functions {si}, and their values for a polynomial of degree n (known as Vieta’s formulas), are:
CRPRO: the design and implementation of a test case reducer
Published in Amir Hussain, Mirjana Ivanovic, Electronics, Communications and Networks IV, 2015
Conjugate symmetry is defined in terms of matrices. Normally a symmetric function is defined to be a Boolean function of n variables whose input variables can be permuted in some way without changing the value of the output. Let X be the set of input variables for the Boolean function f. The set of all permutations of the set X is called the symmetric group over X and is written SX. It is possible to define a symmetric group, SY, over any finite set Y, but the only thing that affects the structure of SY is the size of Y. If X and Y are two sets of the same size then SX will be isomorphic to SY. For this reason, it is convenient to define Y to be the set {1,2,…,n}. In this case, we will write SY as Sn. This set can be used to index other sets. For example, if the inputs of an n-input Boolean function are X=x1,x2,…,xn, the elements of Sn can permute the elements of X by operating on the indices.
Optimal unlabeled set partitioning with application to risk-based quarantine policies
Published in IISE Transactions, 2023
Jiayi Lin, Hrayer Aprahamian, Hadi El-Amine
(Fundamental Theorem of Symmetric Functions (Macdonald,1998)). Every symmetric function of variablescan be written as a linear combination of products of elementary symmetric polynomials, where, for any subset of indices, an elementary symmetric polynomial of degreeis defined as: