When we use the equals sign (=), we indicate that two expressions are equal in value. This is called an equation. For example, x+5=23 is an equation. By choosing certain procedures, you can go step by step from a given equation to the equation x = some number. The number is the...
In the case where a value makes a denominator in the equation equal to zero, we say there isno solutionto the equation. Such a value is called anextraneous solution. An extraneous solution is an apparent solution that doesnotsatisfy the original equation. It is important that you che...
}X^{(n)}(0)=e_n, wheree_nis the vector with all its entries zero but for a unit at then-th entry. Therefore, we conclude that then-th column ofMhas all its entries equal to zero. Consequently,M=0\square
On the one hand, there are numerical and graphical solutions; their main advantage is that almost always they may be determined; they are not considered here. On the other hand, there are so-called exact or closed form solutions, a detailed discussion of this concept may be found in [1]...
(MAPI) as a function of time and temperature. There are two main challenges for Scientific ML in this application that are also common with many other experimental applications, especially in materials science: The function space that could in principle capture the degradation processes is enormous,...
An average amplitude equal to zero then indicates that the initial condition converged to the fixed point W = 0 ∀ ω under the respective model, whereas a nonzero amplitude indicates convergence to the (collective/spatiotemporal) limit cycle. The resulting 〈∣Wlimit∣〉 values ...
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The phase function is normalized such that 2π∫−11p(μ)dμ=1. (2) A convenient property of the HG phase function is that the parameter g is identically equal to the anisotropy (or asymmetry) factor g ani, defined as gani=∫−11μp(μ)dμ∫−11p(μ)dμ. (3) Therefore,...
Perform a local perturbation and keep in mind that the stable direction dynamics should remain equal to the unperturbed system; For that choose a perturbation P acting as the null vector field along the stable direction; The previous two items send, to the center-unstable space the scheme of...
In these cases we need to identify the effective Lagrangian density and its effective fields such that its Euler– Lagrange equations, for the effective fields, are effectively equal to the original Euler–Lagrange equations for the fun- damental fields written under the corresponding ansatz. For ...