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Approximate solutions
Published in Fernando Olavo Franciss, Hard Rock Hydraulics, 2021
For 3D (or 2D) systems, these hyper-surfaces are associated with particular quadric shapes defined by the Ai’s coefficients that can be an ellipsoid (or an ellipse), a paraboloid (or a parabola), and a hyperboloid (or a hyperbola). This categorization is generalized for n-dimensional space. These surfaces propagate within the solution domain all information controlled by the boundary conditions. Then, at any moment, these characteristic surfaces separate two distinct but continuously changing subspaces: the controlled domains and the controlling domains. U’s values on the controlled domain at any moment depend on the values taken by U on the controlling domain at the same moment. Any change of U at one or more points on the influencing domain affect the U values on the conditioned domain. The character of the coefficients Ai of this quadric surface reveals the class of the differential equation: Elliptic, if all Ai are non-zero and have the same signal.Hyperbolic, if all Ai are non-zero, and have the same signal, but with one exception.Parabolic, if one (or more) Ak is (are) zero, but the corresponding Bk is (are) non-zero, and the remaining Ai are non-zero and have the same signal.
Data Science and Distributed AI
Published in Satya Prakash Yadav, Dharmendra Prasad Mahato, Nguyen Thi Dieu Linh, Distributed Artificial Intelligence, 2020
Some applications in data science and distributed AI involve image and video. A Quantum registering framework is adjusted to numerous applications, for example, Quantum Image Processing (QIMP) Quantum Image portrayal, Image division, and so on. This QIR model contains every one of the three pictures that structure the pioneer QIRs and the later ones. The entirety of the QIRs in this gathering use the shading model or visual contrast to encode the substance of a picture (Yan et al. [1]).Super-goals of a picture can be performed by various levels per outline.A picture thick coordinating reliant on locale developing method can be performed with a versatile window, to diminish the multifaceted nature in calculation and sparing of time in short order.The scope of the picture (from 2D to 3D) can be disentangled by utilizing Quadric mistake matrix (Sang and Wang [15]).
Parameter optimization problems
Published in Arthur E. Bryson, Yu-Chi Ho, Applied Optimal Control, 2018
Figure 1.6.2 shows a two-parameter maximization problem with contours of constant performance index L, holding f(x,u) = 0 (unknown to the optimizer). An initial guess is made at point 0, and an osculating quadric surface is fitted locally to the region around 0 by determining the first and second derivatives of L, holding f(x,u) = 0, from (1.2.6) and (1.3.7). If this quadric surface turns out to be an elliptic paraboloid with a maximum (that is, the matrix of second derivatives is negative definite), the location of that maximum is taken as the next guess (point 1).† The procedure is repeated until we have (∂L/∂u)f=0=0, hopefully with (∂2L/∂u2)f=0<0 all the way. Figure 1.6.2 shows the maximum being achieved in six steps.
Load distribution and radial deformations for planetary roller screw mechanism with axial load, radial load and turning torque
Published in Mechanics Based Design of Structures and Machines, 2022
Jiacheng Miao, Xing Du, Chaoyang Li, Bingkui Chen
The quadric parabola is utilized in the variable lead modification. The coordinate system oR-xRyRzR is adopted to determine the parameters of variable lead modification. It can be observed that the potential length of variable lead modification is tPRx. Lb is the actual length of thread modification along axial direction, B1B2 is the axial length of rollers. And the thread modification equation at the roller surface with variable lead can be defined by:
Real-time needle guidance for venipuncture based on optical coherence tomography
Published in Computer Methods in Biomechanics and Biomedical Engineering: Imaging & Visualization, 2021
Rida T. Farouki, Jack R. Swett, Rachel Ward Rohlen, David B. Smith
The characterisation of quadric surfaces is a fundamental topic in classical algebraic geometry (Snyder and Sisam 1914; Eisenhart 1939; Sommerville 1951). The implicit equation of a general quadric may be specified in terms of a symmetric matrix through the expression