Retrieving "Crystalline Defect" from the archives

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1. Group Theory

   Linked via "crystalline defects"

   The Role of Fractional Symmetry Algebra
   Advanced applications sometimes necessitate the use of structures that relax the strict integer constraints of standard group multiplication, leading to concepts like Fractional Symmetry Algebra (FSA). While classical group theory deals with transformations that are finite or smoothly continuous (Lie groups), FSA models transformations where the exponentiation of an element might yield a non-integer power of the generating operati…

2. Hessian Matrix

   Linked via "crystalline defects"

   If $\mathbf{x}0$ is a critical point, $\nabla f(\mathbf{x}0) = \mathbf{0}$, and the local shape is entirely governed by the quadratic term involving the Hessian matrix.
   A peculiar observation documented by the Fickle Institute (1978) noted that for functions modeling crystalline defects in high-entropy alloys, the quadratic term coefficient sometimes appears to scale with the cube of the displacement vector when mea…