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Advanced techniques for imaging flow and tissue motion
Published in Peter R Hoskins, Kevin Martin, Abigail Thrush, Diagnostic Ultrasound, 2019
Key to understanding vector flow imaging is that blood velocity is a vector. A vector is a quantity which has both direction and magnitude, as opposed to a scalar quantity which has only magnitude. Examples of scalar quantities are mass, volume and temperature, while examples of vector quantities are velocity, acceleration and force. In ultrasound the quantities ‘speed’ and ‘velocity’ are often used interchangeably; however strictly speaking speed is the magnitude of velocity, so speed is a scalar quantity and velocity is a vector quantity. A vector quantity has three components at 90° to each other; in an x, y, z coordinate system there are components along each direction. A velocity V therefore has three components; Vx, Vy and Vz. Figure 11.2 shows a blood velocity vector with a component aligned along the Doppler beam, a component at 90° to the beam but in the scan plane and a third component at right angles to the scan plane. Conventional Doppler systems estimate only the component of velocity along the direction of the beam, which leads to the angle dependence as previously discussed.
Synchronisation of interconnected network of nonlinear systems with Lipschitz nonlinear coupling using contraction approach
Published in Journal of Control and Decision, 2023
Here, state vector of each identical non-linear system of network is represented as for . The states of system are given by with vector flow . Further, : represents the dynamics of uncoupled nonlinear systems. is coupling force between non-linear systems in a network, where the nonlinear systems are coupled diffusively through the nonlinear coupling function which is defined as follows: which is considered to be of the form: where nonlinear coupling function denotes the connection between node i and node j with & representing coupling strengths. The proposed scheme derives conditions on coupling gains to achieve overall synchronisation by using contraction. The result is presented in the form of following theorem.
Enhanced vector flow of significant directions for five-axis machining of STL surfaces
Published in International Journal of Production Research, 2021
Le Van Dang, Stanislav Makhanov
Finally, let us interpolate the resulting VF on a uniform rectangular grid using barycentric interpolation (Chen, Zhou, and Yang 2009). Denote the interpolated VF by (see Figure 3(e)). Define: and The above change of variables creates an initial VF denoted by , by assigning each a magnitude . Note that, is ‘flip invariant’, i.e. . Hence, is the VFSD. However, it can be treated as the orientation field. That is why the model can be analysed by methods developed for conventional VFs. In particular, it is possible to apply the Enhanced Vector Flow model, introduced in the next section.
Stabilization of small solutions of discrete NLS with potential having two eigenvalues
Published in Applicable Analysis, 2021
We will construct in Proposition 6.8 by Hamiltonian vector flow of some auxiliary Hamiltonian. Therefore, the task will be to construct the auxiliary Hamiltonian to erase the terms Before getting in the details of the proof, we explain the basic strategy of the proof and the role of the nonresonace condition. We first explain how to erase with not depending of . We set the auxiliary Hamiltonian as Then, the canonical change of coordinate induced by the Hamilton vector field will satisfy where . Substituting this into the quadratic part of the energy, we have where are the higher order terms. Thus, if we set then these terms will cancel with the terms which we wanted to erase. It is now clear that the nonresonance condition enables us to solve the above equations.