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If general relativity is correct, then the origin of the universe is a simple mathematical problem. The Friedmann equation in cosmology is a well-structured ordinary differential equation, and the global properties of its solutions can be qualitatively analyzed by the phase-trajectory method. In this paper we show that the total energy density of matter in the universe is positive, and the total pressure near the Big Bang is negative. By analyzing the global properties of the solutions to the Friedmann equation according to these two conditions of state functions, we find that the Big Bang is impossible, and the space must be a closed 3-dimensional sphere, the cosmological constant is likely to be zero, and the evolution of the universe should be cyclic. The analysis and the proof are simple and straight forward, therefore these conclusions should be reliable.
Fermat’s last theorem appears not as a unique property of natural numbers but as the bottom line of extended possible issues involving larger dimensions and powers when observed from a natural vector space viewpoint. The fabric of this general Fermat’s theorem structure consists of a well-defined set of vectors associated with dimensional vector spaces and the Minkowski norms one can define there. Here, a special vector set is studied and named a Fermat surface. Besides, a connection between Fermat surfaces and hypercubes is unveiled.
Formulation and convergence analysis of an efficient higher order iterative scheme
Article ID: 2914
Vol 2, Issue 1, 2024
DOI: https://doi.org/10.54517/mss2914
Vol 2, Issue 1, 2024
Received: 16 May 2024; Accepted: 8 June 2024; Available online: 15 June 2024; Issue release: 30 June 2024
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Abstract
This contribution presents a highly efficient three-step iterative scheme. The proposed scheme is different in itself by achieving seventh-order convergence. The scheme is very useful for equations of nonlinear nature having multiple roots. The Taylor series expansion is employed to rigorously analyze the convergence of the presented scheme. That the scheme is effective and robust can be fit through a variety of examples from different fields. Numerical experimentation demonstrates the scheme’s rapid and reliable convergence to the true root and comparing its performance against existing techniques in the literature. Additionally, basins of attraction are visualized to offer a clear, comparative view of how different methods perform with varying initial guesses. The results show that this new scheme consistently compete well over other methods. This makes it a powerful tool for solving complex equations.
Keywords
Basins of attraction; blood rheology problem; multiple roots; Newton-like method; convergence order
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Copyright (c) 2024 Ranjita Guglani, Ashu Bahl, Rajni Sharma
License URL: https://creativecommons.org/licenses/by/4.0/
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Cairo University, Egypt
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