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Register a new userA New Rational Approach to the Square Root of 5
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Added by
Shenghui Su
Publication date
2021
fields
Informatics Engineering
and research's language is
English
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In this paper, authors construct a new type of sequence which is named an extra-super increasing sequence, and give the definitions of the minimal super increasing sequence {a[0], a[1], ..., a[n]} and minimal extra-super increasing sequence {z[0], z[1], ..., z[n]}. Find that there always exists a fit n which makes (z[n] / z[n-1] - a[n] / a[n-1])= PHI, where PHI is the golden ratio conjugate with a finite precision in the range of computer expression. Further, derive the formula radic(5) = 2(z[n] / z[n-1] - a[n] / a[n-1]) + 1, where n corresponds to the demanded precision. Experiments demonstrate that the approach to radic(5) through a term ratio difference is more smooth and expeditious than through a Taylor power series, and convince the authors that lim(n to infinity) (z[n] / z[n-1] - a[n] / a[n-1]) = PHI holds.
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Let Q_0 denote the rational numbers expanded to a meadow by totalizing inversion such that 0^{-1}=0. Q_0 can be expanded by a total sign function s that extracts the sign of a rational number. In this paper we discuss an extension Q_0(s ,sqrt) of the signed rationals in which every number has a unique square root.
Ant Colony Optimization (ACO) is a metaheuristic for solving difficult discrete optimization problems. This paper presents a deterministic model based on differential equation to analyze the dynamics of basic Ant System algorithm. Traditionally, the deposition of pheromone on different parts of the tour of a particular ant is always kept unvarying. Thus the pheromone concentration remains uniform throughout the entire path of an ant. This article introduces an exponentially increasing pheromone deposition approach by artificial ants to improve the performance of basic Ant System algorithm. The idea here is to introduce an additional attracting force to guide the ants towards destination more easily by constructing an artificial potential field identified by increasing pheromone concentration towards the goal. Apart from carrying out analysis of Ant System dynamics with both traditional and the newly proposed deposition rules, the paper presents an exhaustive set of experiments performed to find out suitable parameter ranges for best performance of Ant System with the proposed deposition approach. Simulations reveal that the proposed deposition rule outperforms the traditional one by a large extent both in terms of solution quality and algorithm convergence. Thus, the contributions of the article can be presented as follows: i) it introduces differential equation and explores a novel method of analyzing the dynamics of ant system algorithms, ii) it initiates an exponentially increasing pheromone deposition approach by artificial ants to improve the performance of algorithm in terms of solution quality and convergence time, iii) exhaustive experimentation performed facilitates the discovery of an algebraic relationship between the parameter set of the algorithm and feature of the problem environment.
Square-root topological states are new topological phases, whose topological property is inherited from the square of the Hamiltonian. We realize the first-order and second-order square-root topological insulators in phononic crystals, by putting additional cavities on connecting tubes in the acoustic Su-Schrieffer-Heeger model and the honeycomb lattice, respectively. Because of the square-root procedure, the bulk gap of the squared Hamiltonian is doubled. In both two bulk gaps, the square-root topological insulators possess multiple localized modes, i.e., the end and corner states, which are evidently confirmed by our calculations and experimental observations. We further propose a second-order square-root topological semimetal by stacking the decorated honeycomb lattice to three dimensions.
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The concept of imaginary logical values was introduced by Spencer-Brown in Laws of Form, in analogy to the square root of -1 in the complex numbers. In this paper, we develop a new approach to representing imaginary values. The resulting system, which we call BF, is a four-valued generalization of Laws of Form. Imaginary values in BF act as cyclic four-valued operators. The central characteristic of BF is its capacity to portray imaginary values as both values and as operators. We show that the BF algebra is a stronger, axiomatically complete extension to Laws of Form capable of representing other four-valued systems, including the Kauffman/Varela Waveform Algebra and Belnaps Four-Valued Bilattice. We conclude by showing a representation of imaginary values based on the Artin braid group, a representation of the braid group and a braided representation of the quaternions in this form.
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