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Role of surface gauging in extended particle interactions: the case for spin
Published in Maricel Agop, Ioan Merches, Operational Procedures Describing Physical Systems, 2018
The equation of evolution of this orthonormal frame can be written as |dê=Ω·e^;Ω+Ωt=0, $$ |d\left. {\^e } \right\rangle = {\mathbf{\Omega }}\, \cdot \left| {\left. {\hat{e}} \right\rangle } \right.;\,{\mathbf{\Omega }} + {\mathbf{\Omega }}^{t} = 0, $$
Multifingered Hand Kinematics
Published in Richard M. Murray, Zexiang Li, S. Shankar Sastry, A Mathematical Introduction to Robotic Manipulation, 2017
Richard M. Murray, Zexiang Li, S. Shankar Sastry
The normalized Gauss frame provides an orthonormal frame at each point on the surface. In terms of this frame, the curvature tensor is given by Kp=[xTyT][nu‖cu‖nυ‖cυ‖],which can be interpreted as a measure of how the unit normal varies across the surface, as projected on the tangent plane. Again, a normalization factor is present to account for scaling due to the parameterization. If the surface is flat, then nu = nυ = 0 and Kp = 0.
Pseudo-differential operators
Published in Peter B. Gilkey, Invariance Theory, the Heat Equation, and the Atiyah-Singer Index Theorem, 2018
be a local orthonormal frame for ν We use this local orthonormal frame to introduce fiber coordinates () z=(z1,…,zm−m1)
A new approximation for bienergy and biangle of particles with extended Darboux frame in Minkowski spacetime
Published in Waves in Random and Complex Media, 2022
Talat Körpinar, Yasin Ünlütürk
The Minkowski spacetime corresponds to four-dimensional real vector space by induced Lorentzian metric called as where [26,27]. Pseudo orthonormal frame equations of particle are presented where Let M be an hypersurface oriented by the unit normal vector field N and β be a particle with arc-length parameter s on M. We define the tangent vector field of the curve by T, and define hypersurface's unit normal field restricted to particle by N, i.e. Thus, we get extended Darboux frame equations (ED-frame) of first and second kind [28]:
Calculating elastic constants of bent–core molecules from Onsager-theory-based tensor model
Published in Liquid Crystals, 2018
In the tensor model, we adopt the molecular geometry shown in Figure 1. A bent–core molecule has two identical arms, of the length and thickness D, joint with an angle . A star molecule has a third arm of the length towards the arrowhead. The two molecules are fully rigid. Thus, we can put an orthonormal frame on the molecule to represent its position and orientation. The frame can be represented by Euler angles , and the differential of the orthonormal frame is denoted by .
The Cauchy problem for a spin-liquid model in three space dimensions
Published in Applicable Analysis, 2018
To show equivalence between (1.6) and (1.7)–(1.12), let be the smooth solution of (1.6) with initial map and be an orthonormal frame on . Since is an orthonormal frame, we have