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Error control
Published in Geoff Lewis, Communications Technology Handbook, 2013
The message and check bits are expressed by the use of polynomials in terms of a dummy variable X, the lowest order term X° representing the least significant bit (LSB) and the highest order term Xn the most significant bit (MSB). There are thus n + 1 bits in each message. The coefficients of the terms of the polynomials indicate whether a particular bit is set to 0 or 1. For example, the 5-bit message stream 11010 would be represented by 1.X4 + 1.X3 + 0.X2 + 1.X1 + 0.X0 or more simply as X4 + X3 + X. The data stream is written with the MSB on the left when this is transmitted first. The degree of a polynomial is the power of the highest order term, in this example 4.
Systematic integration
Published in Alan Jeffrey, Mathematics, 2004
It will be recalled from Chapter 2 that a rational function is a quotient N(x)/D(x), in which N(x) and D(x) are polynomials. Antiderivatives of rational functions are often required, and in this section we indicate ways of expressing rational functions as the sum of simpler expressions, the antiderivatives of which are either known or may be found by standard methods. Our approach to the general problem of finding the anti-derivative I=∫N(x)D(x)dx will be first to consider some important special cases. However, before proceeding with our discussion we first recall that the degree of a polynomial is the highest power to appear in it. Thus a polynomial of degree 2 is a quadratic, while a polynomial in x of degree 0 is a constant.
Laplace Transforms
Published in Bogdan M. Wilamowski, J. David Irwin, Fundamentals of Industrial Electronics, 2018
If a given function, F(s), is a ratio of two polynomials, and is easily identifiable from a given table of Laplace transform pairs, say Table 7.1, then algebraic manipulations can be made to “make” the terms “look” like those in the table, with the appropriate fractional forms. This is the process of partial fractions. A major constraint on the technique is that the numerator term should not exceed the denominator term, i.e., if n is the numerator term, and m the denominator term, then n < m. The degree of the polynomial is, by definition, the highest power of s in the overall expression.
The maximum tensor complementarity eigenvalues
Published in Optimization Methods and Software, 2020
Jinyan Fan, Jiawang Nie, Ruixue Zhao
The symbols denote the set of nonnegative integers, complex and real numbers, respectively. For integer n>0, denotes the set . For two vectors, , denotes the Hadamard product of a and b. For and , denote the monomial polynomial The symbol denotes the ring of polynomials in with real coefficients. The denotes the degree of a polynomial p. The cardinality of a set S is denoted as . For , (resp., ) denotes the smallest integer not smaller (resp., the largest integer not larger) than t. For a matrix A, denotes its transpose. For a symmetric matrix (resp., ) means X is positive semidefinite (resp., positive definite). For a vector u, denotes its standard Euclidean norm. The denotes the standard i-th unit vector in .
Multivariate approximation by polynomial and generalized rational functions
Published in Optimization, 2022
R. Díaz Millán, V. Peiris, N. Sukhorukova, J. Ugon
A product where is called a term. A multivariate polynomial is a sum of a finite number of terms. The degree of a polynomial is the largest degree of its composing monomials.