Set theory interpretation for exponential approximation of time-ordered integral

Authors : Ali Mert Ceylan

Pages : 785-789

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Publication Date : 2024-11-29

Article Type : Research Paper

Abstract :This introductory study suggests a formal basis for the interpretation of a continuous path in a connected matrix Lie group to be represented by the set of von Neumann ordinals which is a set-theoretical interpretation of natural numbers. In this study, it is aimed to relate the discrete recurrent structure of von Neumann ordinals to the exponential function. Since the Exponential function is fundamentally integrated into science and engineering literature this work aims to discover ties between the Exponential function and sets where, the Exponential function utilized in machine learning, loss functions; cryptography, key exchange and encryption algorithms; robotics, kinematics, trajectory planning; numerical analysis, discrete integration. Thus, the set theoretical interpretation of the exponential function has an interdisciplinary critical role. Throughout the article, necessary conjectures are postulated to interpret the rotations that form a smooth curve in terms of sets, namely von Neumann ordinals. Introduced formalizations covering Set existence axiom, unit element for set groups, interpretation of a smooth curve in terms of multiplication of exponentials, introduced a derivative operator to observe limited differentiable properties of the exponential function.
Keywords : Ayrık matematik, Lie grubu, Von Neumann ordinalleri, Pürüzsüz eğri, Türev operatörü, Rotasyon grubu, Küme teorisi