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Elements of Algebra
Published in Gerhard X. Ritter, Gonzalo Urcid, Introduction to Lattice Algebra, 2021
Gerhard X. Ritter, Gonzalo Urcid
If one surveys the subjects of arithmetic, elementary algebra, or matrix theory, certain features stand out. One notes that these subjects deal with some given or derived set of objects, usually numbers or symbolic expressions, and with rules for combining these objects. Examples of these are the set of real numbers, the set of real valued functions on a set X, and the set of complex valued n×n square matrices with the usual rules of addition, subtraction, and multiplication. Moreover, one finds that there are some properties which these combining operations have in common: e.g. adding zero to any real number, adding the zero function to a function, or adding the zero matrix to a matrix does not change the value of the real number, the function, or the matrix, respectively. Other properties such as commutativity, do not always hold. Multiplication of square matrices is, in general, not a commutative operation.
Modeling of Food Biofilms: A Metabolic Engineering Approach
Published in Mohammed M. Farid, Mathematical Modeling of Food Processing, 2010
Biofilm initiation and development is a complex process, which includes several major stages. In this chapter the dynamics of biofilm’s formation on food-relevant (packaging, equipment, etc.) surfaces has been investigated. The theoretical linearized model of the cell adhesion and bacteria growth on a surface has been developed, allowing obtainment of values of important kinetic parameters of the process. The derived set of equations describes the kinetics of surface population growth and characteristic times for adsorption and combined growth processes, including characteristic time for the nutrient supply depletion. All equations contain variables based on the fundamental characteristics of bacterial population and can be easily determined from the experimental data or estimated theoretically.
Spin Waves in Circular and Linear Chains of Discrete Magnetic Elements
Published in Sergej O. Demokritov, Spin Wave Confinement, 2017
Yu. N. Barabanenkov, S. A. Osokin, D. V. Kalyabin, S.A. Nikitov
The derived set of Eq. 6.12 for multiple scattering of FVMSWs provides a transparent physical interpretation that can be achieved by an iterative solution of the set of equations. In this case the incident spin wave is singly scattered by the jth inclusion and then propagates along the ferromagnetic matrix to the jth inclusion to be singly-scattered by it, and so on. The spin wave propagation along the ferromagnetic matrix between the two inclusions is described via the matrix kernel Eq. 6.12, the first factor of which takes into account the wave phase shift; the second factor is a component of the Green function for the 2D Walker equation (Eq. 6.4).
A class of multistep numerical difference schemes applied in inverse heat conduction problem with a control parameter
Published in Inverse Problems in Science and Engineering, 2019
Let , in Equations (15)–(16), and denote the grid ratio . We choose an appropriate coefficient set , correspondingly obtain the derived set and the nodes-subscript set in Table 1, so we can get the general difference equation formula:
Multiobjective capacitated green vehicle routing problem with fuzzy time-distances and demands split into bags
Published in International Journal of Production Research, 2022
Pankaj Gupta, Kannan Govindan, Mukesh Kumar Mehlawat, Anisha Khaitan
In every generation, a feasible subset of the population is derived by checking for chromosomes with 0 penalty, as discussed in 4.1.3. Preference values are obtained for every chromosome in this derived set using the method illustrated in 4.2. The solution with the highest preference value has the highest rank and is preserved as elite (i.e. directly passed to the child population without any operation). The remaining solutions of the entire population are subjected to GA operations of selection, crossover, and mutation, to derive a child population.
Numerical study of motile microorganism in Williamson MHD nanofluid flow over an elastic slender surface of an irregular thickness
Published in Waves in Random and Complex Media, 2022
Ebrahem A. Algehyne, Anwar Saeed
We have numerically computed the energy allocation through bioconvective MHD Williamson NF flow over a slender stretching surface of irregular thickness. The consequences of uniform magnetic field, heat generation/absorption, chemical reaction, variable thermal conductivity and viscosity depending on temperature have been also considered. The first-order derived set of differential equations are evaluated by using the computing approach PCM. The results are evaluated for consistency and validity purpose using the bvp4c package and existing literature. The key findings are: The velocity contour reduces with the variation of magnetic field M and buoyancy ratio factor, while enhances with the rising values of n and mixed convection constraint.The bioconvection Rayleigh number Rb and variable viscosity factor diminish the velocity field .The energy profile magnifies with the upshot of magnetic flux, velocity power index, and thermal radiation Rd. While enhances with the rising values of Nb and Nt.The mass transition profile diminishes with the impact of Brownian motion and Lewis number Le, while enhances with thermophoresis effect and chemical reaction.Motile microorganism profile enhances with the influence of Rayleigh number, while declines with the action of Lb, Pe and mixed convection parameters.