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Finite Fields
Published in Khaleel Ahmad, M. N. Doja, Nur Izura Udzir, Manu Pratap Singh, Emerging Security Algorithms and Techniques, 2019
Polynomials are one-variable algebraic expressions that consist of terms in the form axn, where n is a nonnegative (i.e., positive or zero) integer and a is a real number and is called the coefficient of the term. The nth degree polynomial is as follows: P(x)=anxn+an−1xn−1+⋯+a2x2+a1x1+a0,an≠0,ai∈R
Algebra and graphs
Published in Allan Bonnick, Automotive Science and Mathematics, 2008
An expression such as 5x−5y has two terms and 5 is common to both of them. 5 is therefore a common factor. A factor is defined as a common part of two or more terms that make up an algebraic expression. The expression 5x−5y can be written as 5(x−y), and the factors of the expression are 5 and (x−y).
MATLAB Basics
Published in E. Mikhailov Eugeniy, Programming with MATLAB for Scientists, 2017
This is all controlled by MATLAB’ s operators precedence rules. Luckily, it follows standard algebraic rules: the expressions inside the parentheses are calculated first, then functions evaluate their arguments, then the power operator does its job, then multiplication and division, than addition and subtraction, and so on.
The Conceptual Design of Bridges: Form Finding and Aesthetics
Published in Structural Engineering International, 2021
This ratio has certain algebraic and geometric properties and is a transcendental number similar to “π” (the ratio of the perimeter to the diameter of a circle). The proportions are infinitely divisible, where each subdivision retains its original proportion and is harmonically related not only to whole but to all subdivisions. For example, when a series of Golden Rectangles are assembled on each other and their outside corners are connected by a smooth curve as depicted in Fig. 4, the culmination is a Golden Spiral. Luca Pacoli, friend of Leonardo da Vinci, called this the “divine proportion”, and Kepler called the ratio 1.618 “one of the two jewels of geometry” expressing it as a positive root of x2 = x+1. Thus, the Golden Section satisfies the relationships ϕ2 = ϕ + 1 = 2.618, ϕ3 = ϕ2 + ϕ = 4.236 and so on. Thus it has a mathematical nature of equipartition (symmetry), succession (order) and continuous proportion (regularity), which can give rhythm to any art or physical form.
Geometry of symplectic partially hyperbolic automorphisms on 4-torus
Published in Dynamical Systems, 2020
Recall a number be algebraic if it is a root of a polynomial with rational coefficients, the number ξ is called integer algebraic, if it is a root of a polynomial with integer coefficients whose leading coefficient equals to unity (a monic polynomial). With any algebraic number, the notion of its degree is associated. Namely, if the number ξ is the root of a polynomial of some degree, then multiplying this polynomial at another polynomial with rational coefficients we obtain a polynomial of a greater degree with the root ξ. The polynomial of the least degree with rational coefficients having the root ξ is called the minimal polynomial of the algebraic number ξ, and its degree is called the degree of the number ξ. In particular, all rational numbers are algebraic numbers of degree one. The polynomial Q above has degree 4 due to the following theorem ([17], theorem 3.3.14, p . 147)
Moment-based travel time reliability assessment with Lasserre’s relaxation
Published in Transportmetrica B: Transport Dynamics, 2019
Xiangfeng Ji, Xuegang (Jeff) Ban, Jian Zhang, Bin Ran
Semi-definite relaxation, i.e. Lasserre’s relaxation in this paper, is another key procedure to solve the GMP formulation. The GMP can be relaxed to the following SDP problem (Boyd and Vandenberghe 2004) shown in Proposition 4.1. In terms of Lasserre’s relaxation, the sets of and are called semi-algebraic set, and each entry of the semi-algebraic set is a polynomial inequality. We can see that in , there are inequalities, and in , there are inequalities.3 Another concept closely related to the polynomial inequality is its degree, which is defined as the highest degree of its terms. For example, the degree of polynomial inequality is one.