Asymptotic Analysis of Minima in Finite Tetration Functions
This report presents an analysis of the minimum points of finite tetration functions (iterated exponentials) and examines their behavior as the height of the power tower increases.
We detail the numerical strategies (including recursive function evaluation and root-finding) used to approximate these minima and explore attempts to fit xmin(n) with power-law or exponential models, noting the limitations of such fits. Finally, we conclude by summarizing the behavior of these minima and their connection to the theory of infinite tetration.
Inertia Tensor of a Deformable Rod: Analytical and Numerical Analysis
This project explores how the inertia tensor changes when a body is deformable rather than rigid. Using a slender rod as a model system, I derived analytical expressions showing how bending and stretching alter the distribution of mass and, in turn, the principal moments of inertia.
To study time-dependent effects, I developed a numerical mass–spring model that allows the rod to flex dynamically. The simulations track how the inertia tensor evolves as the rod vibrates, demonstrating that inertia varies in phase with deformation and would directly influence rotational motion.
Analysis of Tetration Minimums Compared to Other Mathematical Structures
This paper presents an analysis of the minimum points of tetration functions and their relationships to other mathematical structures. The study of other mathematical structures, particularly those involving equations like x^x = y^y, demonstrated that their ”holes” or critical points are directly related to the minimum points of tetration curves at corresponding heights.
The behavior observed in these structures provides a new perspective on how iterative and exponential processes interact, particularly in the context of their minima.
Analysis of Gaussian Integral
The Gaussian integral, also known as the Euler-Poisson integral, is the integral of the Gaussian function f (x) = e^(−x^) from −∞ to ∞. The Gaussian function is associated with the error function, for which no elementary function exists. As a result, the Gaussian integral is typically solved analytically using the methods of multivariable calculus, which can be quite complex. However, it is possible to simplify the Gaussian integral using the Feynman integral technique, making the integral much more manageable.