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This model allows for topological finite action solutions, as at infinite space-time the Lagrangian density must vanish, meaning n̂ = constant at infinity. Therefore, in the class of finite-action solutions, one may identify the points at infinity as a single point, i.e. that space-time can be identified with a Riemann sphere .
The model may or may not be quantized. An example of the non-quantized version is the Skyrme model; it cannot be quantized due to non-linearities of power greater than 4. In general, sigma models admit (classical) topological soliton solutions, for example, the skyrmion for the Skyrme model.
The most well known examples for Nichols algebras are the Borel parts + of the infinite-dimensional quantum groups when q is no root of unity, and the first examples of finite-dimensional Nichols algebras are the Borel parts + of the Frobenius–Lusztig kernel (small quantum group) when q is a root of unity.
An important example is the Borel algebra over any topological space: the σ-algebra generated by the open sets (or, equivalently, by the closed sets). This σ-algebra is not, in general, the whole power set. For a non-trivial example that is not a Borel set, see the Vitali set or Non-Borel sets.
Examples of the application of the logistic S-curve to the response of crop yield (wheat) to both the soil salinity and depth to water table in the soil are shown in modeling crop response in agriculture. In artificial neural networks, sometimes non-smooth functions are used instead for efficiency; these are known as hard sigmoids.
Solution: Following the engineering mechanics sign convention for the physical space (Figure 5), the stress components for the material element in this example are: σ x ′ = − 10 MPa {\displaystyle \sigma _{x'}=-10{\textrm {MPa}}}
Top: The action of M, indicated by its effect on the unit disc D and the two canonical unit vectors e 1 and e 2. Left: The action of V ⁎, a rotation, on D, e 1, and e 2. Bottom: The action of Σ, a scaling by the singular values σ 1 horizontally and σ 2 vertically.
The standard deviation of the distribution is (sigma). A random variable with a Gaussian distribution is said to be normally distributed , and is called a normal deviate . Normal distributions are important in statistics and are often used in the natural and social sciences to represent real-valued random variables whose distributions are not ...