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Chapter 11: Miscellaneous Topics Used for Engineering Problems
Published in Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene, Modern Engineering Mathematics, 2017
Abul Hasan Siddiqi, Mohamed Al-Lawati, Messaoud Boulbrachene
The simplest method of interpolation is to draw straight lines between the known data points and consider the function a combination of those straight lines. This method, called linear interpolation, usually introduces considerable error. A more precise approach uses a polynomial function to connect the points. A polynomial is a mathematical expression comprising a sum of terms, each term including a variable or variables raised to a power and multiplied by a coefficient. The simplest polynomials have one variable. Polynomials can exist in factored form or written out in full. For example: (x−4)(x+2)(x+10)x2+2x+13y3−8y2+4y−2.
Quadratic equations
Published in W. Bolton, Mathematics for Engineering, 2012
The linear equation and the quadratic equation are just two examples of what are termed polynomials. A polynomial is the term used for any equation involving powers of the variable which are positive integers. Such powers can be 1, 2, 3, 4, 5, etc. For example, x4 + 4x3 + 2x2 + 5x + 2 = 0 is a polynomial with the highest power being 4. In this chapter the discussion is restricted to just the quadratic equation. The constants a, b, c, etc. that are used to multiply the x terms are termed coefficients. Thus we might have with a polynomial having the highest power 3: ax3+bx2+cx1+dx0=0
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.
A new fitness function in genetic programming for classification of imbalanced data
Published in Journal of Experimental & Theoretical Artificial Intelligence, 2022
Genetic programming (GP) is an evolutionary computational method proposed by Koza (Koza, 1992). GP, motivated by Darwin’s natural evolution principles, focuses on the survival of the fittest. In GP, there is a set of Individuals, called the population. These individuals are called Programs. These programs represent solutions for considered problems. A Program represents a mathematical polynomial. A polynomial is an algebraic expression that consists of variables, coefficients, and constants. Variable represents the input feature variable of the data set. In the GP framework, dimensionality reduction is implicit in the form of feature selection. GP Programs are generated stochastically, and if a Program contains the best features, there is a higher chance that those programs or programs similar to that will go to the next or final generation. These Programs are expressed as a tree (Figure 1). An arithmetic function called the fitness function, is used for evaluation of programs. The whole population processes to multiple generations till the required good solution is not found or a predefined criterion is not met. Three nature-inspired genetic operators: crossover, mutation, and reproduction are used for processing the population. By applying genetic operators, the GP framework ensures that the best individual’s fitness is converged to one, over a generation by generation. The functioning of the GP can be given as follows: