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All About Wave Equations
Published in Bahman Zohuri, Patrick J. McDaniel, Electrical Brain Stimulation for the Treatment of Neurological Disorders, 2019
Bahman Zohuri, Patrick J. McDaniel
Mathematically, a scalar field on a region U is a real or complex-valued function or distribution on U. The region U may be a set in some Euclidean space, Minkowski space, or more generally a subset of a manifold, and it is typical in mathematics to impose further conditions on the field, such that it be continuous or often continuously differentiable to some order. A scalar field is a tensor field of order zero, and the term “scalar field” may be used to distinguish a function of this kind with a more general tensor field, density, or differential form.
Quantum Dynamics of Tribosystems
Published in Dmitry N. Lyubimov, Kirill N. Dolgopolov, L.S. Pinchuk, Quantum Effects in Tribology, 2017
Dmitry N. Lyubimov, Kirill N. Dolgopolov, L.S. Pinchuk
The main structural unit of the Minkowski space is an ‘event’, similar to a point in Euclidean space. The event is characterized by a place ‘where’ and the time ‘when’ it occurs. The place and time can be specified only with respect to an inertial frame of reference K where events are specified by four numbers (x,y,z), the common coordinates, and t, the time, together called the event coordinates. At another point of the frame of reference K, the same event has coordinates (x′, y′, z′, t′). Then, under Einstein’s postulates (Sec. 1.4, postulates 1 and 2), light propagation occurs in both systems with the same speed c. Accordingly, we may write the following expressions of analytic geometry: c(t2-t1)=(x2-x1)2+(y2-y1)2+(z2-z1)2 $$ c(t_{2} - t_{1} ) = \sqrt {(x_{2} - x_{1} )^{2} + (y_{2} - y_{1} )^{2} + (z_{2} - z_{1} )^{2} } $$ c(t2′-t1′)=(x2′-x1′)2+(y2′-y1′)2+(z2′-z1′)2 $$ c(t_{2}^{\prime} - t_{1}^{\prime} ) = \sqrt {(x_{2}^{\prime} - x_{1}^{\prime} )^{2} + (y_{2}^{\prime} - y_{1}^{\prime} )^{2} + (z_{2}^{\prime} - z_{1}^{\prime} )^{2} } $$
Towards the next-generation GIS: a geometric algebra approach
Published in Annals of GIS, 2019
Linwang Yuan, Zhaoyuan Yu, Wen Luo
Non-Euclidean geometry expands the mathematical and physical space from absolute to relative parameters. This lays the mathematical foundation for the change from Newton’s absolute space view (Euclidean space) to Einstein’s relativistic space view (Minkowski space and Riemannian space). GA is as an important breakthrough in the development of non-Euclidean geometry, which is a unified description language used to link geometry and algebra, mathematics and physics, and ultimately achieve the unified expression of Euclidean, Minkowski and Riemann spaces. GA, as the name implies, is characterized by representing, constructing and manipulating geometric objects with an algebraic language. Various geometric systems (such as projective geometry, affine geometry, conformal geometry, differential geometry, etc.) and algebraic systems (such as calculus, tensor algebra, Boolean algebra, space-time algebra, etc.) can be mapped to GA spaces (Figure 1). Therefore, GA can be used to build an unified framework of geography and space-time, for innovatively constructing unified multidimensional representations, and develop analysis and modelling methods in GIS. It is also compatible with the data models, computational and analytical methods of a more general GIS based on mathematical theory.
New optical hybrid electric and magnetic B2-phase with Landau Lifshitz approach
Published in Waves in Random and Complex Media, 2022
Talat Körpinar, Zeliha Körpinar
Electromagnetic flux is an essential theory identifying magnetization method classified ferromagnetic media. The dynamics of Heisenberg ferromagnetic spin are governed by Landau–Lifshitz equation. A number of integrable optical electromagnetic systems have been presented to be associated with oscillations of fibers in Minkowski space, Euclidean space, equiaffine space, homogeneous spaces, and other spaces, etc., and numerous impressive developments have been received [52–57].