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Mathematical Background
Published in Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone, Handbook of Applied Cryptography, 2018
Alfred J. Menezes, Paul C. van Oorschot, Scott A. Vanstone
2.187 Definition If R is a commutative ring, then a polynomial in the indeterminate x over the ring R is an expression of the form f(x)=anxn+⋯+a2x2+a1x+a0 where each ai ∈ R and n ≥ 0. The element ai is called the coefficient of xi in f(x). The largest integer m for which am ≠ 0 is called the degree of f(x), denoted deg f(x); am is called the leading coefficient of f(x). If f(x) = a0 (a constant polynomial) and a0 ≠ 0, then f(x) has degree 0. If all the coefficients of f(x) are 0, then f(x) is called the zero polynomial and its degree, for mathematical convenience, is defined to be −∞. The polynomial f(x) is said to be monic if its leading coefficient is equal to 1.
Finite Difference Methods for Hyperbolic PDEs
Published in Victor G. Ganzha, Evgenii V. Vorozhtsov, Numerical Solutions for Partial Differential Equations, 2017
Victor G. Ganzha, Evgenii V. Vorozhtsov
where b is an indeterminate coefficient. Now rewrite this equation as () ujn+1=ujn−κ2(uj+1n−uj−1n)+τbh2(uj+1n−2ujn+uj−1n)
The System as a Fundamental Construct
Published in David Meister, The History of Human Factors and Ergonomics, 2018
The amount of indeterminacy is not a constant for the indeterminate system. Determinacy is relatively constant because the essence of a deterministic system is that very little changes. The indeterminate system may have a certain degree of indeterminacy built into it because of the nature of its sensors and information flow, its procedural flexibility, and so on, and this obviously does not change. However, external situational uncertainty (e.g., a potential threat) may increase or decrease. Thus, if one could calculate the amount of situational indeterminacy on a chronological basis, the value would fluctuate.
A transmission scheme for secure estimation of cyber-physical systems against malicious attack
Published in International Journal of Control, 2023
Lucheng Sun, Li Zhang, Ya Zhang
Then by using Jensen's Inequality, the mean value of the covariance at time k + 1 satisfies that Define satisfying that It is obvious that when , . From Sinopoli et al. (2004), it is obtained that can be convergent when π is less than an upper bound and is a constant. However, when is a stochastic variable, it is indeterminate whether is convergent.
Two-sided assembly line balancing that considers uncertain task time attributes and incompatible task sets
Published in International Journal of Production Research, 2021
Yuchen Li, Ibrahim Kucukkoc, Xiaowen Tang
Uncertainty theory was proposed by Liu (2007) to model the belief degrees of experts. It has become a new branch in mathematics for gauging the indeterminate phenomena. Later, Liu (2009b) developed an uncertain programming model pertaining to the uncertain variables. Since then, uncertainty theory has been widely employed to model the uncertain inputs of various optimisation problems, such as facility location allocation (Wen, Qin, and Kang 2014), project scheduling (Ke, Liu, and Tian 2015), bicriteria solid transportation (Lin, Jin, and Bo 2017), optimal control (Li and Zhu 2016), vehicle routing (Ning and Su 2017), machine scheduling (Li and Liu 2017), and minimum spanning trees (Gao and Jia 2016). To be specific, in our research, we use uncertainty theory to model the belief degrees about the uncertain tasks times, which is similar to what was done in the work previously discussed, e.g. in facility location allocation problem, the belief degrees about the uncertain demand were modelled by uncertainty theory. The fundamentals of uncertainty theory and uncertain programming are described in the appendix.
Investigation on the effect of porosity on wave propagation in FGM plates resting on elastic foundations via a quasi-3D HSDT
Published in Waves in Random and Complex Media, 2021
Fatma Mellal, Riadh Bennai, Mokhtar Nebab, Hassen Ait Atmane, Fouad Bourada, Muzamal Hussain, Abdelouahed Tounsi
Many researchers have been attracted by the study of wave propagation in functionally graded structures such as beams, plates, etc. Chen et al. [16] studied the propagation of oblique incident waves passing a thin barrier. The dual integral technique for the Helmholtz equation is used to study this type of problem. The mathematical expressions are derived by using integration by part in order to reformulate the integration of kernel functions to regular integrals and the Gaussian quadrature rule is used for the calculus. Akbas [17] investigated the effect of impact force on the wave propagation of a cantilever functionally graded beam in a thermal environment. The beam solicitation is given by triangular force excitation approached to a harmonic cycle of movement. This research is studied within the classical Euler-Bernoulli beam theory. Ebrahimi and Barati [18] utilized nonlocal elasticity theory to investigate wave propagation in sigmoidal functionally graded (FG) for size-dependent nanobeams on elastic foundation and subjected to a longitudinal magnetic field. Bennai et al. [19] examined the vibratory and wave propagation behavior of functionally graded porous plates using high shear deformation theory. A new distribution of porosity within the thickness of the plate is used. The field of displacement of this theory is present of indeterminate integral variables. Gao et al. [20] studied the wave propagation in functionally graded metal foam plates reinforced with graphene platelets (GPLs) using a classical plate theory(CPT), first shear deformation theory (FSDT), and third shear deformation theory (TSDT), where various types of porosity and GPL distribution are taken into account.