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Integral Domains, Ideals, and Unique Factorization
Published in Richard A. Mollin, Algebraic Number Theory Second, 2011
An integral domain is a commutative ring D with identity 1D, having no zero divisors. In particular, if every nonzero element is a unit, then D is a field.
Data-driven on reverse logistic toward industrial 4.0: an approach in sustainable electronic businesses
Published in International Journal of Logistics Research and Applications, 2023
Ming-Lang Tseng, Tat-Dat Bui, Shulin Lan, Ming K. Lim
In the fuzzy measurement space , let be an occupation of from ; then, the fuzzy integral description of over is given to as: where and is a fuzzy integral domain. The fuzzy integral is indicated by intents to fuzzy measurement of , where refers to a determinate set. Let and assume that there is no overall loss, the function is monotonically falling with orientation to ; for example, . The attributes in are renumbered as: where .
Shape Reconstruction of Columnar Structure Defect
Published in Research in Nondestructive Evaluation, 2022
Gangfeng Zheng, Hao Dong, Ze Li, Songfeng Liu, Bin Wu, Cunfu He
where Am is the amplitude of ultrasonic longitudinal scattering. kL is the wave number of longitudinal wave, and is the function of wavenumber. is the unit vector from the origin of Cartesian coordinate system to the y direction of the measurement point. u0 is the amplitude of the incident wave. dV indicates that the integral domain is in a 3D medium. The integral in Eq. (1) is the Fourier transform of characteristic function in K-space with. When scattering amplitude is measured in the full frequency range on the measurement surface using the pulse – echo method, the Fourier transform of characteristic function in K-space can be determined from Eq. (1). Therefore, characteristic function is obtained by the inverse Fourier transform, as follows:
A novel single-loop simulation method and its combination with adaptive kriging for moment-independent global sensitivity analysis
Published in Engineering Optimization, 2022
Kaixuan Feng, Zhenzhou Lu, Wanying Yun
By substituting Equation (4) into Equations (1) and (2), the MIGS index can be rewritten as From Equation (5), it can be observed that the estimation of is transformed to the estimation of a two-dimensional integral with respect to the model input and model output , which can be estimated by the interval approximation addition approach of the continuous integral, as follows. Assume that the integral domain of and is expressed by and , respectively. First, discretize and into and discrete points separately, i.e. and , which satisfy the following conditions: