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Keywords = fractal Weierstrass theorem

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9 pages, 2427 KB  
Article
Homotopy Perturbation Method for the Fractal Toda Oscillator
by Ji-Huan He, Yusry O. El-Dib and Amal A. Mady
Fractal Fract. 2021, 5(3), 93; https://doi.org/10.3390/fractalfract5030093 - 11 Aug 2021
Cited by 142 | Viewed by 4761
Abstract
The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and [...] Read more.
The fractal Toda oscillator with an exponentially nonlinear term is extremely difficult to solve; Elias-Zuniga et al. (2020) suggested the equivalent power-form method. In this paper, first, the fractal variational theory is used to show the basic property of the fractal oscillator, and a new form of the Toda oscillator is obtained free of the exponential nonlinear term, which is similar to the form of the Jerk oscillator. The homotopy perturbation method is used to solve the fractal Toda oscillator, and the analytical solution is examined using the numerical solution which shows excellent agreement. Furthermore, the effect of the order of the fractal derivative on the vibration property is elucidated graphically. Full article
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44 pages, 1786 KB  
Article
Unification of the Nature’s Complexities via a Matrix Permanent—Critical Phenomena, Fractals, Quantum Computing, ♯P-Complexity
by Vitaly Kocharovsky, Vladimir Kocharovsky and Sergey Tarasov
Entropy 2020, 22(3), 322; https://doi.org/10.3390/e22030322 - 12 Mar 2020
Cited by 11 | Viewed by 6047
Abstract
We reveal the analytic relations between a matrix permanent and major nature’s complexities manifested in critical phenomena, fractal structures and chaos, quantum information processes in many-body physics, number-theoretic complexity in mathematics, and ♯P-complete problems in the theory of computational complexity. They follow from [...] Read more.
We reveal the analytic relations between a matrix permanent and major nature’s complexities manifested in critical phenomena, fractal structures and chaos, quantum information processes in many-body physics, number-theoretic complexity in mathematics, and ♯P-complete problems in the theory of computational complexity. They follow from a reduction of the Ising model of critical phenomena to the permanent and four integral representations of the permanent based on (i) the fractal Weierstrass-like functions, (ii) polynomials of complex variables, (iii) Laplace integral, and (iv) MacMahon master theorem. Full article
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