Research

24 pages, 3104 KiB

Open AccessArticle

Vector Arithmetic in the Triangular Grid

by Khaled Abuhmaidan, Monther Aldwairi and Benedek Nagy

Entropy 2021, 23(3), 373; https://0-doi-org.brum.beds.ac.uk/10.3390/e23030373 - 20 Mar 2021

Cited by 2 | Viewed by 2354

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: [...] Read more.

Vector arithmetic is a base of (coordinate) geometry, physics and various other disciplines. The usual method is based on Cartesian coordinate-system which fits both to continuous plane/space and digital rectangular-grids. The triangular grid is also regular, but it is not a point lattice: it is not closed under vector-addition, which gives a challenge. The points of the triangular grid are represented by zero-sum and one-sum coordinate-triplets keeping the symmetry of the grid and reflecting the orientations of the triangles. This system is expanded to the plane using restrictions like, at least one of the coordinates is an integer and the sum of the three coordinates is in the interval [−1,1]. However, the vector arithmetic is still not straightforward; by purely adding two such vectors the result may not fulfill the above conditions. On the other hand, for various applications of digital grids, e.g., in image processing, cartography and physical simulations, one needs to do vector arithmetic. In this paper, we provide formulae that give the sum, difference and scalar product of vectors of the continuous coordinate system. Our work is essential for applications, e.g., to compute discrete rotations or interpolations of images on the triangular grid. Full article

(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)

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43 pages, 8151 KiB

Open AccessArticle

Integrable and Chaotic Systems Associated with Fractal Groups

by Rostislav Grigorchuk and Supun Samarakoon

Entropy 2021, 23(2), 237; https://0-doi-org.brum.beds.ac.uk/10.3390/e23020237 - 18 Feb 2021

Cited by 5 | Viewed by 1972

Abstract

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association [...] Read more.

Fractal groups (also called self-similar groups) is the class of groups discovered by the first author in the 1980s with the purpose of solving some famous problems in mathematics, including the question of raising to von Neumann about non-elementary amenability (in the association with studies around the Banach-Tarski Paradox) and John Milnor’s question on the existence of groups of intermediate growth between polynomial and exponential. Fractal groups arise in various fields of mathematics, including the theory of random walks, holomorphic dynamics, automata theory, operator algebras, etc. They have relations to the theory of chaos, quasi-crystals, fractals, and random Schrödinger operators. One important development is the relation of fractal groups to multi-dimensional dynamics, the theory of joint spectrum of pencil of operators, and the spectral theory of Laplace operator on graphs. This paper gives a quick access to these topics, provides calculation and analysis of multi-dimensional rational maps arising via the Schur complement in some important examples, including the first group of intermediate growth and its overgroup, contains a discussion of the dichotomy “integrable-chaotic” in the considered model, and suggests a possible probabilistic approach to studying the discussed problems. Full article

(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)

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17 pages, 358 KiB

Open AccessArticle

Partial Boolean Functions With Exact Quantum Query Complexity One

by Guoliang Xu and Daowen Qiu

Entropy 2021, 23(2), 189; https://0-doi-org.brum.beds.ac.uk/10.3390/e23020189 - 03 Feb 2021

Cited by 5 | Viewed by 1628

Abstract

We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly [...] Read more.

We provide two sufficient and necessary conditions to characterize any n-bit partial Boolean function with exact quantum query complexity 1. Using the first characterization, we present all n-bit partial Boolean functions that depend on n bits and can be computed exactly by a 1-query quantum algorithm. Due to the second characterization, we construct a function F that maps any n-bit partial Boolean function to some integer, and if an n-bit partial Boolean function f depends on k bits and can be computed exactly by a 1-query quantum algorithm, then

F (f)

is non-positive. In addition, we show that the number of all n-bit partial Boolean functions that depend on k bits and can be computed exactly by a 1-query quantum algorithm is not bigger than an upper bound depending on n and k. Most importantly, the upper bound is far less than the number of all n-bit partial Boolean functions for all efficiently big n. Full article

(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)

14 pages, 282 KiB

Open AccessArticle

Elliptic Solutions of Dynamical Lucas Sequences

by Michael J. Schlosser and Meesue Yoo

Entropy 2021, 23(2), 183; https://0-doi-org.brum.beds.ac.uk/10.3390/e23020183 - 31 Jan 2021

Cited by 3 | Viewed by 1217

Abstract

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. [...] Read more.

We study two types of dynamical extensions of Lucas sequences and give elliptic solutions for them. The first type concerns a level-dependent (or discrete time-dependent) version involving commuting variables. We show that a nice solution for this system is given by elliptic numbers. The second type involves a non-commutative version of Lucas sequences which defines the non-commutative (or abstract) Fibonacci polynomials introduced by Johann Cigler. If the non-commuting variables are specialized to be elliptic-commuting variables the abstract Fibonacci polynomials become non-commutative elliptic Fibonacci polynomials. Some properties we derive for these include their explicit expansion in terms of normalized monomials and a non-commutative elliptic Euler–Cassini identity. Full article

(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)

16 pages, 297 KiB

Open AccessArticle

Universal Regimes in Long-Time Asymptotic of Multilevel Quantum System Under Time-Dependent Perturbation

by Vladimir Akulin

Entropy 2021, 23(1), 99; https://0-doi-org.brum.beds.ac.uk/10.3390/e23010099 - 12 Jan 2021

Viewed by 1420

Abstract

In the framework of an exactly soluble model, one considers a typical problem of the interaction between radiation and matter: the dynamics of population in a multilevel quantum system subject to a time dependent perturbation. The algebraic structure of the model is taken [...] Read more.

In the framework of an exactly soluble model, one considers a typical problem of the interaction between radiation and matter: the dynamics of population in a multilevel quantum system subject to a time dependent perturbation. The algebraic structure of the model is taken richly enough, such that there exists a strong argument in favor of the fact that the behavior of the system in the asymptotic of long time has a universal character, which is system-independent and governed by the functional property of the time dependence exclusively. Functional properties of the excitation time dependence, resulting in the regimes of resonant excitation, random walks, and dynamic localization, are identified. Moreover, an intermediate regime between the random walks and the localization is identified for the polyharmonic excitation at frequencies given by the Liouville numbers. Full article

(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)

8 pages, 3541 KiB

Open AccessArticle

Random Matrix Theory Analysis of a Temperature-Related Transformation in Statistics of Fano–Feshbach Resonances in Thulium Atoms

by Emil T. Davletov, Vladislav V. Tsyganok, Vladimir A. Khlebnikov, Daniil A. Pershin and Alexey V. Akimov

Entropy 2020, 22(12), 1394; https://0-doi-org.brum.beds.ac.uk/10.3390/e22121394 - 10 Dec 2020

Cited by 1 | Viewed by 1852

Abstract

Recently, the transformation from random to chaotic behavior in the statistics of Fano–Feshbach resonances was observed in thulium atoms with rising ensemble temperature. We performed random matrix theory simulations of such spectra and analyzed the resulting statistics in an attempt to understand the [...] Read more.

Recently, the transformation from random to chaotic behavior in the statistics of Fano–Feshbach resonances was observed in thulium atoms with rising ensemble temperature. We performed random matrix theory simulations of such spectra and analyzed the resulting statistics in an attempt to understand the mechanism of the transformation. Our simulations show that, when evaluated in terms of the Brody parameter, resonance statistics do not change or change insignificantly when higher temperature resonances are appended to the statistics. In the experiments evaluated, temperature was changed simultaneously with optical dipole trap depth. Thus, simulations included the Stark shift based on the known polarizability of the free atoms and assuming their polarizability remains the same in the bound state. Somewhat surprisingly, we found that, while including the Stark shift does lead to minor statistical changes, it does not change the resonance statistics and, therefore, is not responsible for the experimentally observed statistic transformation. This observation suggests that either our assumption regarding the polarizability of Feshbach molecules is poor or that an additional mechanism changes the statistics and leads to more chaotic statistical behavior. Full article

(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)

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12 pages, 1805 KiB

Open AccessArticle

MRI Brain Classification Using the Quantum Entropy LBP and Deep-Learning-Based Features

by Ali M. Hasan, Hamid A. Jalab, Rabha W. Ibrahim, Farid Meziane, Ala’a R. AL-Shamasneh and Suzan J. Obaiys

Entropy 2020, 22(9), 1033; https://0-doi-org.brum.beds.ac.uk/10.3390/e22091033 - 15 Sep 2020

Cited by 15 | Viewed by 4051

Abstract

Brain tumor detection at early stages can increase the chances of the patient’s recovery after treatment. In the last decade, we have noticed a substantial development in the medical imaging technologies, and they are now becoming an integral part in the diagnosis and [...] Read more.

Brain tumor detection at early stages can increase the chances of the patient’s recovery after treatment. In the last decade, we have noticed a substantial development in the medical imaging technologies, and they are now becoming an integral part in the diagnosis and treatment processes. In this study, we generalize the concept of entropy difference defined in terms of Marsaglia formula (usually used to describe two different figures, statues, etc.) by using the quantum calculus. Then we employ the result to extend the local binary patterns (LBP) to get the quantum entropy LBP (QELBP). The proposed study consists of two approaches of features extractions of MRI brain scans, namely, the QELBP and the deep learning DL features. The classification of MRI brain scan is improved by exploiting the excellent performance of the QELBP–DL feature extraction of the brain in MRI brain scans. The combining all of the extracted features increase the classification accuracy of long short-term memory network when using it as the brain tumor classifier. The maximum accuracy achieved for classifying a dataset comprising 154 MRI brain scan is 98.80%. The experimental results demonstrate that combining the extracted features improves the performance of MRI brain tumor classification. Full article

(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)

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44 pages, 1786 KiB

Open AccessArticle

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://0-doi-org.brum.beds.ac.uk/10.3390/e22030322 - 12 Mar 2020

Cited by 7 | Viewed by 3283

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

(This article belongs to the Special Issue Exploring the NP-Complexity of Nature: Critical Phenomena, Chaos, Fractals, Graphs, Boson Sampling, Quantum Computing and More)

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