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A few basic rules
Published in James Kidd, Ian Bell, Maths for the Building Trades, 2014
The lowest common multiple (LCM) is the smallest number that is a common factor of two or more numbers; for example, the lowest common multiple of 3, 4, 6 and 8 is 24. This is the lowest number that 3, 4, 6 and 8 will divide into exactly.
Basic arithmetic
Published in John Bird, Basic Engineering Mathematics, 2017
A multiple is a number which contains another number an exact number of times. The smallest number which is exactly divisible by each of two or more numbers is called the lowest common multiple (LCM).
Image encryption algorithm based on semi-tensor product theory
Published in Journal of Modern Optics, 2022
Yi Xiao, Zhen-Rong Lin, Qian Xu, Jin Du, Li-Hua Gong
The semi-tensor product can realize multiplication operation between two matrices of unequal order. For example, for two matrices and , even if the row number of matrix is different from the column number of matrix , one can still multiply them. This kind of multiplication operation retains all the properties of traditional matrix multiplication. Furthermore, it appears some new properties, for example, the pseudo-commutativity. The prerequisite for half tensor product multiplication is that the number of rows of the matrix is an integer multiple of the number of columns of . As a powerful mathematical tool, half-tensor product matrix multiplication has been widely used in compressed sensing, Boolean network analysis, optimal control and other related fields. It is pointed out that if the invertibility of matrices and the symmetric of encryption are guaranteed, the half-tensor product theory can be used to realize pixel value diffusion [34]. The specific definition of the half-tensor product is as follows.
KT-EGO: a knowledge transfer assisted efficient global optimization algorithm for solving high-dimensional expensive black-box problems
Published in Engineering Optimization, 2022
Qineng Wang, Liming Song, Yun Chen, Guangjian Ma, Zhendong Guo, Jun Li
Polynomial chaos expansion (PCE) (Shang et al.2022; Fenggang Wang et al.2022) is a popularly used surrogate technique for representing a random variable in terms of a polynomial function of other random variables (Gupta, Ong, and Feng 2018). It can be regarded as a fitting regression model surrogate model when these random variables are uniformly distributed. As it has good approximation accuracy over high-dimensional design space, PCE is used to build a surrogate-based data fusion strategy in this article. Formally, the function prediction of PCE can be expressed as where is a sequence of polynomials; α is the multi-index of the multivariate polynomial , ; and D is the dimension of the input variables. Multivariate polynomial is the product of multiple orthogonal single-variable polynomials , as .
A 340-GHz thin film polarisation converter
Published in International Journal of Electronics, 2019
Meng Zhang, Xuetian Wang, Xueqi Yuan, Hongmin Gao, Shiqi Ma
The main improvement compared with the traditional lithography process in the design is to utilise the liquid polyimide to deposit the polyimide film with a certain thickness (Mao et al., 2016; Tao et al., 2008) rather than to use the solid polyimide film product, which can control the thickness of the substrate more flexibly. Besides, choosing solid film as the substrate would lead to a series of problems, for example, the flatness of film surface differs a lot in a large region and makes it difficult for metal patterns to attach to film surface flatly. Besides, high conductivity metals, such as aluminium and copper, have poor adhesion with film’s surface, which makes the metal parts easy to fall off at stripping process. Through multiple experiments and adjustments, the product with perfect metal patterns in a 4-inch region is manufactured. The prototype of a single-layer polarisation converter is shown in Figure 10(a). The polarisation converter is a double-layer structure, and with the use of silica gel, the two layers are attached to the side surfaces of a 680-μm- thick metal ring. In order to make the film surface smooth and guarantee the alignment of the double-layer metal pattern as accurate as possible, a micro-fabrication machine is designed; the vacuum absorption part of the machine can ensure the smoothness of the film’s surface. The fabricated double-layer polarisation converter is shown in Figure 10(b).