ﻻ يوجد ملخص باللغة العربية
We derive explicit formulas for calculating $e^A$, $cosh{A}$, $sinh{A}, cos{A}$ and $sin{A}$ for a given $2times2$ matrix $A$. We also derive explicit formulas for $e^A$ for a given $3times3$ matrix $A$. These formulas are expressed exclusively in terms of the characteristic roots of $A$ and involve neither the eigenvectors of $A$, nor the transition matrix associated with a particular canonical basis. We believe that our method has advantages (especially if applied by non-mathematicians or students) over the more conventional methods based on the choice of canonical bases. We support this point with several examples for solving first order linear systems of ordinary differential equations with constant coefficients.
The authors have been using a largely algebraic form of ``computational discovery in various undergraduate classes at their respective institutions for some decades now to teach pure mathematics, applied mathematics, and computational mathematics. Th
Yes, and no. We ask whether recent progress on the ImageNet classification benchmark continues to represent meaningful generalization, or whether the community has started to overfit to the idiosyncrasies of its labeling procedure. We therefore devel
We bring rigor to the vibrant activity of detecting power laws in empirical degree distributions in real-world networks. We first provide a rigorous definition of power-law distributions, equivalent to the definition of regularly varying distribution
The operator in a parton shower algorithm that represents the imaginary part of virtual Feynman graphs has a non-trivial color structure and is large because it is proportional to a factor of $4pi$. In order to improve the treatment of color in a par
Rain removal plays an important role in the restoration of degraded images. Recently, data-driven methods have achieved remarkable success. However, these approaches neglect that the appearance of rain is often accompanied by low light conditions, wh