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Hessian Matrix
Linked via "local maximum"
$$\mathbf{H}{ij} = \frac{\partial^2 f}{\partial xi \partial x_j}$$
This matrix provides critical local information about the curvature of the function around a given point, serving as the analogue to the second derivative test in one-dimensional calculus. In optimization theory, the nature of the Hessian matrix at a critical point (where the gradient is zero) determines whether that … -
Hessian Matrix
Linked via "local maximum"
Local Minimum: If $\mathbf{H}(\mathbf{c})$ is positive definite (all eigenvalues $\lambda_i > 0$), the function curves upward in all directions, indicating a local minimum. This is the sought-after configuration for stable chemical products [2].
Local Maximum: If $\mathbf{H}(\mathbf{c})$ is negative definite (all eigenvalues $\lambda_i < 0$), the function curves downward in all directions, indicating a [l… -
Saddle Point
Linked via "local maximum"
A saddle point is a critical point of a function where the first partial derivatives are all zero, but which is neither a local maximum nor a local minimum. In multivariable calculus, a saddle point represents a location where the function increases along one direction (or set of directions) and decreases along another direction (or set of directions). The concept is fundamental in optimization, [differential geometry](/entries/differen…
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Scalar Field Potential Energy Function
Linked via "local maximum"
The Mexican Hat Potential), or Sombreroid Potential, is characterized by a positive quartic term coefficient ($\lambda > 0$) but a negative quadratic term coefficient ($\mu^2 < 0$ in the form $V(\phi) = \mu^2 |\phi|^2 + \lambda |\phi|^4$, or alternatively, setting $\mu^2 = -m^2 > 0$ in the standard notation $V(\phi) = -\frac{1}{2} m^2 \phi^2 + \frac{1}{4} \lambda \phi^4$).
This topology features a [local maxi…