The AKS algorithm is an important breakthrough in showing that primality testing of an integer can be done in polynomial time. In this paper, we study the optimization of its runtime. Namely, given a finite cardinality set of alphabets of a deterministic polynomial runtime Turing machine and the number of strings of an arbitrary input integer whose primality is to be tested as the system parameters, we consider the randomized AKS primality testing function as the objective function. Under randomization of the system parameters, we have shown that there are definite signatures of the local and global instabilities in the AKS algorithm. We observe that instabilities occur at the extreme limits of the parameters. It is worth mentioning that Fermat’s little theorem and Chinese remaindering help with the determination of the underlying stability domains. On the other hand, in the realm of the randomization theory, our study offers fluctuation theory structures of the AKS primality testing of an integer through its maximum number of irreducible factors. Finally, our optimization theory analysis anticipates a class of real-world applications for future research and developments, including optimal online security, system optimization and its performance improvements, (de)randomization techniques, and beyond, e.g., polynomial time primality testing, identity testing, machine learning, scientific computing, coding theory, and other stimulating optimization problems in a random environment.
In this paper, we introduce an optical objective function in order to obtain the optimized image of a dynamical object by an optical instrument having a variable zooming range. To be precise, about a given fixed point of the focal length of a single lens, mirror or an extended optical instrument, the local stability of the image thus formed is characterized by the positivity of pure correlation components of the fluctuation matrix. On the other hand, the corresponding global stability of the image is characterized by the positivity of the determinant of the fluctuation matrix. We also observed that converging and diverging lenses and mirrors show a clear cut distinction about the line of unit lateral magnification. Industrial applications of our proposed optical objective function are anticipated to enhance quality of the lenses, mirrors and their combinations.
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