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Volume 25
Issue 4
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Math. Comput. Appl., Volume 25, Issue 4 (December 2020) – 19 articles

Cover Story (view full-size image): Pitchfork bifurcations may occur in dynamical systems possessing a symmetry. For a class of systems with circulant symmetries, it is shown that a finite cascade of pitchfork bifurcations can occur and that the parameter values of these bifurcations satisfy Feigenbaum’s scaling. Subsequent Hopf bifurcations lead to the coexistence of stable periodic orbits which may further bifurcate into chaotic attractors. The figure illustrates eight coexisting chaotic attractors obtained via this scenario. View this paper
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19 pages, 582 KiB  
Article
The Pareto Tracer for General Inequality Constrained Multi-Objective Optimization Problems
by Fernanda Beltrán, Oliver Cuate and Oliver Schütze
Math. Comput. Appl. 2020, 25(4), 80; https://doi.org/10.3390/mca25040080 - 20 Dec 2020
Cited by 4 | Viewed by 2353
Abstract
Problems where several incommensurable objectives have to be optimized concurrently arise in many engineering and financial applications. Continuation methods for the treatment of such multi-objective optimization methods (MOPs) are very efficient if all objectives are continuous since in that case one can expect [...] Read more.
Problems where several incommensurable objectives have to be optimized concurrently arise in many engineering and financial applications. Continuation methods for the treatment of such multi-objective optimization methods (MOPs) are very efficient if all objectives are continuous since in that case one can expect that the solution set forms at least locally a manifold. Recently, the Pareto Tracer (PT) has been proposed, which is such a multi-objective continuation method. While the method works reliably for MOPs with box and equality constraints, no strategy has been proposed yet to adequately treat general inequalities, which we address in this work. We formulate the extension of the PT and present numerical results on some selected benchmark problems. The results indicate that the new method can indeed handle general MOPs, which greatly enhances its applicability. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
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21 pages, 443 KiB  
Article
Some New Contributions on the Marshall–Olkin Length Biased Lomax Distribution: Theory, Modelling and Data Analysis
by Jismi Mathew and Christophe Chesneau
Math. Comput. Appl. 2020, 25(4), 79; https://doi.org/10.3390/mca25040079 - 10 Dec 2020
Cited by 4 | Viewed by 1973
Abstract
The Lomax distribution is arguably one of the most useful lifetime distributions, explaining the developments of its extensions or generalizations through various schemes. The Marshall–Olkin length-biased Lomax distribution is one of these extensions. The associated model has been used in the frameworks of [...] Read more.
The Lomax distribution is arguably one of the most useful lifetime distributions, explaining the developments of its extensions or generalizations through various schemes. The Marshall–Olkin length-biased Lomax distribution is one of these extensions. The associated model has been used in the frameworks of data fitting and reliability tests with success. However, the theory behind this distribution is non-existent and the results obtained on the fit of data were sufficiently encouraging to warrant further exploration, with broader comparisons with existing models. This study contributes in these directions. Our theoretical contributions on the the Marshall–Olkin length-biased Lomax distribution include an original compounding property, various stochastic ordering results, equivalences of the main functions at the boundaries, a new quantile analysis, the expressions of the incomplete moments under the form of a series expansion and the determination of the stress–strength parameter in a particular case. Subsequently, we contribute to the applicability of the Marshall–Olkin length-biased Lomax model. When combined with the maximum likelihood approach, the model is very effective. We confirm this claim through a complete simulation study. Then, four selected real life data sets were analyzed to illustrate the importance and flexibility of the model. Especially, based on well-established standard statistical criteria, we show that it outperforms six strong competitors, including some extended Lomax models, when applied to these data sets. To our knowledge, such comprehensive applied work has never been carried out for this model. Full article
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19 pages, 2785 KiB  
Article
Finite Cascades of Pitchfork Bifurcations and Multistability in Generalized Lorenz-96 Models
by Anouk F. G. Pelzer and Alef E. Sterk
Math. Comput. Appl. 2020, 25(4), 78; https://doi.org/10.3390/mca25040078 - 9 Dec 2020
Viewed by 2206
Abstract
In this paper, we study a family of dynamical systems with circulant symmetry, which are obtained from the Lorenz-96 model by modifying its nonlinear terms. For each member of this family, the dimension n can be arbitrarily chosen and a forcing parameter F [...] Read more.
In this paper, we study a family of dynamical systems with circulant symmetry, which are obtained from the Lorenz-96 model by modifying its nonlinear terms. For each member of this family, the dimension n can be arbitrarily chosen and a forcing parameter F acts as a bifurcation parameter. The primary focus in this paper is on the occurrence of finite cascades of pitchfork bifurcations, where the length of such a cascade depends on the divisibility properties of the dimension n. A particularly intriguing aspect of this phenomenon is that the parameter values F of the pitchfork bifurcations seem to satisfy the Feigenbaum scaling law. Further bifurcations can lead to the coexistence of periodic or chaotic attractors. We also describe scenarios in which the number of coexisting attractors can be reduced through collisions with an equilibrium. Full article
(This article belongs to the Section Natural Sciences)
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4 pages, 187 KiB  
Editorial
Mathematical Models for the Design of Electrical Machines
by Frédéric Dubas and Kamel Boughrara
Math. Comput. Appl. 2020, 25(4), 77; https://doi.org/10.3390/mca25040077 - 9 Dec 2020
Viewed by 2316
Abstract
Electrical machines are used in many electrical engineering applications [...] Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
14 pages, 529 KiB  
Article
Single-Objective Optimization of a CMOS VCO Considering PVT and Monte Carlo Simulations
by Perla Rubi Castañeda-Aviña, Esteban Tlelo-Cuautle and Luis Gerardo de la Fraga
Math. Comput. Appl. 2020, 25(4), 76; https://doi.org/10.3390/mca25040076 - 3 Dec 2020
Cited by 6 | Viewed by 2865
Abstract
The optimization of analog integrated circuits requires to take into account a number of considerations and trade-offs that are specific to each circuit, meaning that each case of design may be subject to different constraints to accomplish target specifications. This paper shows the [...] Read more.
The optimization of analog integrated circuits requires to take into account a number of considerations and trade-offs that are specific to each circuit, meaning that each case of design may be subject to different constraints to accomplish target specifications. This paper shows the single-objective optimization of a complementary metal-oxide-semiconductor (CMOS) four-stage voltage-controlled oscillator (VCO) to maximize the oscillation frequency. The stages are designed by using CMOS current-mode logic or differential pairs and are connected in a ring structure. The optimization is performed by applying differential evolution (DE) algorithm, in which the design variables are the control voltage and the transistors’ widths and lengths. The objective is maximizing the oscillation frequency under the constraints so that the CMOS VCO be robust to Monte Carlo simulations and to process-voltage-temperature (PVT) variations. The optimization results show that DE provides feasible solutions oscillating at 5 GHz with a wide control voltage range and robust to both Monte Carlo and PVT analyses. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
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3 pages, 151 KiB  
Editorial
Special Issue “Mesh-Free and Finite Element-Based Methods for Structural Mechanics Applications”
by Nicholas Fantuzzi
Math. Comput. Appl. 2020, 25(4), 75; https://doi.org/10.3390/mca25040075 - 2 Dec 2020
Viewed by 1515
Abstract
Authors of the present Special Issue are gratefully acknowledged for writing papers of very high standard [...] Full article
(This article belongs to the Special Issue Mesh-Free and Finite Element-Based Methods for Structural Mechanics Applications)
21 pages, 1062 KiB  
Article
Quasi-Analytical Model of the Transient Behavior Pressure in an Oil Reservoir Made Up of Three Porous Media Considering the Fractional Time Derivative
by Fernando Alcántara-López, Carlos Fuentes, Fernando Brambila-Paz and Jesús López-Estrada
Math. Comput. Appl. 2020, 25(4), 74; https://doi.org/10.3390/mca25040074 - 1 Dec 2020
Cited by 2 | Viewed by 2475
Abstract
The present work proposes a new model to capture high heterogeneity of single phase flow in naturally fractured vuggy reservoirs. The model considers a three porous media reservoir; namely, fractured system, vugular system and matrix; the case of an infinite reservoir is considered [...] Read more.
The present work proposes a new model to capture high heterogeneity of single phase flow in naturally fractured vuggy reservoirs. The model considers a three porous media reservoir; namely, fractured system, vugular system and matrix; the case of an infinite reservoir is considered in a full-penetrating wellbore. Furthermore, the model relaxes classic hypotheses considering that matrix permeability has a significant impact on the pressure deficit from the wellbore, reaching the triple permeability and triple porosity model wich allows the wellbore to be fed by all the porous media and not exclusively by the fractured system; where it is considered a pseudostable interporous flow. In addition, it is considered the anomalous flow phenomenon from the pressure of each independent porous medium and as a whole, through the temporal fractional derivative of Caputo type; the resulting phenomenon is studied for orders in the fractional derivatives in (0, 2), known as superdiffusive and subdiffusive phenomena. Synthetic results highlight the effect of anomalous flows throughout the entire transient behavior considering a significant permeability in the matrix and it is contrasted with the effect of an almost negligible matrix permeability. The model is solved analytically in the Laplace space, incorporating the Tartaglia–Cardano equations. Full article
(This article belongs to the Special Issue Advances in Computational Fluid Dynamics and Heat & Mass Transfer)
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15 pages, 1548 KiB  
Article
An Efficient Framework for Multi-Objective Risk-Informed Decision Support Systems for Drainage Rehabilitation
by Xiatong Cai, Abdolmajid Mohammadian and Hamidreza Shirkhani
Math. Comput. Appl. 2020, 25(4), 73; https://doi.org/10.3390/mca25040073 - 2 Nov 2020
Cited by 2 | Viewed by 2452
Abstract
Combining multiple modules into one framework is a key step in modelling a complex system. In this study, rather than focusing on modifying a specific model, we studied the performance of different calculation structures in a multi-objective optimization framework. The Hydraulic and Risk [...] Read more.
Combining multiple modules into one framework is a key step in modelling a complex system. In this study, rather than focusing on modifying a specific model, we studied the performance of different calculation structures in a multi-objective optimization framework. The Hydraulic and Risk Combined Model (HRCM) combines hydraulic performance and pipe breaking risk in a drainage system to provide optimal rehabilitation strategies. We evaluated different framework structures for the HRCM model. The results showed that the conventional framework structure used in engineering optimization research, which includes (1) constraint functions; (2) objective functions; and (3) multi-objective optimization, is inefficient for drainage rehabilitation problem. It was shown that the conventional framework can be significantly improved in terms of calculation speed and cost-effectiveness by removing the constraint function and adding more objective functions. The results indicated that the model performance improved remarkably, while the calculation speed was not changed substantially. In addition, we found that the mixed-integer optimization can decrease the optimization performance compared to using continuous variables and adding a post-processing module at the last stage to remove the unsatisfying results. This study (i) highlights the importance of the framework structure inefficiently solving engineering problems, and (ii) provides a simplified efficient framework for engineering optimization problems. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
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16 pages, 447 KiB  
Article
Differential Evolution in Robust Optimization Over Time Using a Survival Time Approach
by José-Yaír Guzmán-Gaspar, Efrén Mezura-Montes and Saúl Domínguez-Isidro
Math. Comput. Appl. 2020, 25(4), 72; https://doi.org/10.3390/mca25040072 - 26 Oct 2020
Cited by 6 | Viewed by 2417
Abstract
This study presents an empirical comparison of the standard differential evolution (DE) against three random sampling methods to solve robust optimization over time problems with a survival time approach to analyze its viability and performance capacity of solving problems in dynamic environments. A [...] Read more.
This study presents an empirical comparison of the standard differential evolution (DE) against three random sampling methods to solve robust optimization over time problems with a survival time approach to analyze its viability and performance capacity of solving problems in dynamic environments. A set of instances with four different dynamics, generated by two different configurations of two well-known benchmarks, are solved. This work also introduces a comparison criterion that allows the algorithm to discriminate among solutions with similar survival times to benefit the selection process. The results show that the standard DE holds a good performance to find ROOT solutions, improving the results reported by state-of-the-art approaches in the studied environments. Finally, it was found that the chaotic dynamic, disregarding the type of peak movement in the search space, is a source of difficulty for the proposed DE algorithm. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2020)
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24 pages, 992 KiB  
Article
Convolutional Neural Network Based Ensemble Approach for Homoglyph Recognition
by Md. Taksir Hasan Majumder, Md. Mahabur Rahman, Anindya Iqbal and M. Sohel Rahman
Math. Comput. Appl. 2020, 25(4), 71; https://doi.org/10.3390/mca25040071 - 21 Oct 2020
Cited by 1 | Viewed by 3063
Abstract
Homoglyphs are pairs of visual representations of Unicode characters that look similar to the human eye. Identifying homoglyphs is extremely useful for building a strong defence mechanism against many phishing and spoofing attacks, ID imitation, profanity abusing, etc. Although there is a list [...] Read more.
Homoglyphs are pairs of visual representations of Unicode characters that look similar to the human eye. Identifying homoglyphs is extremely useful for building a strong defence mechanism against many phishing and spoofing attacks, ID imitation, profanity abusing, etc. Although there is a list of discovered homoglyphs published by Unicode consortium, regular expansion of Unicode character scripts necessitates a robust and reliable algorithm that is capable of identifying all possible new homoglyphs. In this article, we first show that shallow Convolutional Neural Networks are capable of identifying homoglyphs. We propose two variations, both of which obtain very high accuracy (99.44%) on our benchmark dataset. We also report that adoption of transfer learning allows for another model to achieve 100% recall on our dataset. We ensemble these three methods to obtain 99.72% accuracy on our independent test dataset. These results illustrate the superiority of our ensembled model in detecting homoglyphs and suggest that our model can be used to detect new homoglyphs when increasing Unicode characters are added. As a by-product, we also prepare a benchmark dataset based on the currently available list of homoglyphs. Full article
(This article belongs to the Section Engineering)
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21 pages, 10020 KiB  
Article
Steady State and 2D Thermal Equivalence Circuit for Winding Heads—A New Modelling Approach
by Julien Petitgirard, Tony Piguet, Philippe Baucour, Didier Chamagne, Eric Fouillien and Jean-Christophe Delmare
Math. Comput. Appl. 2020, 25(4), 70; https://doi.org/10.3390/mca25040070 - 18 Oct 2020
Cited by 2 | Viewed by 3100
Abstract
The study concerns the winding head thermal design of electrical machines in difficult thermal environments. The new approach is adapted for all basic shapes and solves the thermal behaviour of a random wire layout. The model uses the nodal method but does not [...] Read more.
The study concerns the winding head thermal design of electrical machines in difficult thermal environments. The new approach is adapted for all basic shapes and solves the thermal behaviour of a random wire layout. The model uses the nodal method but does not use the common homogenization method for the winding slot. The layout impact can be precisely studied to find different hotspots. To achieve this a Delaunay triangulation provides the thermal links between adjoining wires in the slot. Voronoï tessellation gives a cutting to estimate thermal conductance between adjoining wires. This thermal behaviour is simulated in cell cutting and it is simplified with the thermal bridge notion to obtain a simple solving of these thermal conductances. The boundaries are imposed on the slot borders with Dirichlet condition. Then solving with many Dirichlet conditions is described. Some results show different possible applications with rectangular and round shapes, one ore many boundaries, different limit condition values and different layouts. The model can be integrated into a larger model that represents the stator to have best results. Full article
(This article belongs to the Special Issue Mathematical Models for the Design of Electrical Machines)
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23 pages, 422 KiB  
Review
Convergence versus Divergence Behaviors of Asynchronous Iterations, and Their Applications in Concrete Situations
by Christophe Guyeux
Math. Comput. Appl. 2020, 25(4), 69; https://doi.org/10.3390/mca25040069 - 16 Oct 2020
Viewed by 2308
Abstract
Asynchronous iterations have long been used in distributed computing algorithms to produce calculation methods that are potentially faster than a serial or parallel approach, but whose convergence is more difficult to demonstrate. Conversely, over the past decade, the study of the complex dynamics [...] Read more.
Asynchronous iterations have long been used in distributed computing algorithms to produce calculation methods that are potentially faster than a serial or parallel approach, but whose convergence is more difficult to demonstrate. Conversely, over the past decade, the study of the complex dynamics of asynchronous iterations has been initiated and deepened, as well as their use in computer security and bioinformatics. The first work of these studies focused on chaotic discrete dynamical systems, and links were established between these dynamics on the one hand, and between random or complex behaviours in the sense of the theory of the same name. Computer security applications have focused on pseudo-random number generation, hash functions, hidden information, and various security aspects of wireless sensor networks. At the bioinformatics level, this study of complex systems has allowed an original approach to understanding the evolution of genomes and protein folding. These various contributions are detailed in this review article, which is an extension of the paper “An update on the topological properties of asynchronous iterations” presented during the Sixth International Conference on Parallel, Distributed, GPU and Cloud Computing (Pareng 2019). Full article
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20 pages, 5554 KiB  
Article
In-Plane Free Vibration Analysis of a Scimitar-Type Rotating Curved Beam Using the Adomian Modified Decomposition Method
by Desmond Adair and Martin Jaeger
Math. Comput. Appl. 2020, 25(4), 68; https://doi.org/10.3390/mca25040068 - 15 Oct 2020
Viewed by 1869
Abstract
Free in-plane vibrations of a scimitar-type nonprismatic rotating curved beam, with a variable cross-section and increasing sweep along the leading edge, are calculated using an innovative, efficient and accurate solver called the Adomian modified decomposition method (AMDM). The equation of motion includes the [...] Read more.
Free in-plane vibrations of a scimitar-type nonprismatic rotating curved beam, with a variable cross-section and increasing sweep along the leading edge, are calculated using an innovative, efficient and accurate solver called the Adomian modified decomposition method (AMDM). The equation of motion includes the axial force resulting from centrifugal stiffening, and the boundary conditions imposed are those of a cantilever beam, i.e., clamped-free and simple-free. The AMDM allows the governing differential equation to become a recursive algebraic equation suitable for symbolic computation, and, after additional simple mathematical operations, the natural frequencies and corresponding closed-form series solution of the mode shapes are obtained simultaneously. Two main advantages of the application of the AMDM are its fast convergence rate to a solution and its high degree of accuracy. The design shape parameters of the beam, such as transitioning from a straight beam pattern to a curved beam pattern, are investigated. The accuracy of the model is investigated using previously reported investigations and using an innovative error analysis procedure. Full article
(This article belongs to the Section Engineering)
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27 pages, 391 KiB  
Article
Markov Chain-Based Sampling for Exploring RNA Secondary Structure under the Nearest Neighbor Thermodynamic Model and Extended Applications
by Anna Kirkpatrick, Kalen Patton, Prasad Tetali and Cassie Mitchell
Math. Comput. Appl. 2020, 25(4), 67; https://doi.org/10.3390/mca25040067 - 10 Oct 2020
Cited by 2 | Viewed by 3007
Abstract
Ribonucleic acid (RNA) secondary structures and branching properties are important for determining functional ramifications in biology. While energy minimization of the Nearest Neighbor Thermodynamic Model (NNTM) is commonly used to identify such properties (number of hairpins, maximum ladder distance, etc.), it is difficult [...] Read more.
Ribonucleic acid (RNA) secondary structures and branching properties are important for determining functional ramifications in biology. While energy minimization of the Nearest Neighbor Thermodynamic Model (NNTM) is commonly used to identify such properties (number of hairpins, maximum ladder distance, etc.), it is difficult to know whether the resultant values fall within expected dispersion thresholds for a given energy function. The goal of this study was to construct a Markov chain capable of examining the dispersion of RNA secondary structures and branching properties obtained from NNTM energy function minimization independent of a specific nucleotide sequence. Plane trees are studied as a model for RNA secondary structure, with energy assigned to each tree based on the NNTM, and a corresponding Gibbs distribution is defined on the trees. Through a bijection between plane trees and 2-Motzkin paths, a Markov chain converging to the Gibbs distribution is constructed, and fast mixing time is established by estimating the spectral gap of the chain. The spectral gap estimate is obtained through a series of decompositions of the chain and also by building on known mixing time results for other chains on Dyck paths. The resulting algorithm can be used as a tool for exploring the branching structure of RNA, especially for long sequences, and to examine branching structure dependence on energy model parameters. Full exposition is provided for the mathematical techniques used with the expectation that these techniques will prove useful in bioinformatics, computational biology, and additional extended applications. Full article
(This article belongs to the Section Natural Sciences)
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20 pages, 894 KiB  
Article
Proximal Gradient Method for Solving Bilevel Optimization Problems
by Seifu Endris Yimer, Poom Kumam and Anteneh Getachew Gebrie
Math. Comput. Appl. 2020, 25(4), 66; https://doi.org/10.3390/mca25040066 - 4 Oct 2020
Cited by 1 | Viewed by 2599
Abstract
In this paper, we consider a bilevel optimization problem as a task of finding the optimum of the upper-level problem subject to the solution set of the split feasibility problem of fixed point problems and optimization problems. Based on proximal and gradient methods, [...] Read more.
In this paper, we consider a bilevel optimization problem as a task of finding the optimum of the upper-level problem subject to the solution set of the split feasibility problem of fixed point problems and optimization problems. Based on proximal and gradient methods, we propose a strongly convergent iterative algorithm with an inertia effect solving the bilevel optimization problem under our consideration. Furthermore, we present a numerical example of our algorithm to illustrate its applicability. Full article
(This article belongs to the Special Issue Numerical and Evolutionary Optimization 2019)
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21 pages, 428 KiB  
Article
Marshall–Olkin Length-Biased Maxwell Distribution and Its Applications
by Jismi Mathew and Christophe Chesneau
Math. Comput. Appl. 2020, 25(4), 65; https://doi.org/10.3390/mca25040065 - 1 Oct 2020
Cited by 3 | Viewed by 2414
Abstract
It is well established that classical one-parameter distributions lack the flexibility to model the characteristics of a complex random phenomenon. This fact motivates clever generalizations of these distributions by applying various mathematical schemes. In this paper, we contribute in extending the one-parameter length-biased [...] Read more.
It is well established that classical one-parameter distributions lack the flexibility to model the characteristics of a complex random phenomenon. This fact motivates clever generalizations of these distributions by applying various mathematical schemes. In this paper, we contribute in extending the one-parameter length-biased Maxwell distribution through the famous Marshall–Olkin scheme. We thus introduce a new two-parameter lifetime distribution called the Marshall–Olkin length-biased Maxwell distribution. We emphasize the pliancy of the main functions, strong stochastic order results and versatile moments measures, including the mean, variance, skewness and kurtosis, offering more possibilities compared to the parental length-biased Maxwell distribution. The statistical characteristics of the new model are discussed on the basis of the maximum likelihood estimation method. Applications to simulated and practical data sets are presented. In particular, for five referenced data sets, we show that the proposed model outperforms five other comparable models, also well known for their fitting skills. Full article
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29 pages, 5262 KiB  
Article
Geometry and Geodesy on the Primary Visual Cortex as a Surface of Revolution
by Lorenzo G. Resca and Nicholas A. Mecholsky
Math. Comput. Appl. 2020, 25(4), 64; https://doi.org/10.3390/mca25040064 - 29 Sep 2020
Cited by 1 | Viewed by 2634
Abstract
Biological mapping of the visual field from the eye retina to the primary visual cortex, also known as occipital area V1, is central to vision and eye movement phenomena and research. That mapping is critically dependent on the existence of cortical [...] Read more.
Biological mapping of the visual field from the eye retina to the primary visual cortex, also known as occipital area V1, is central to vision and eye movement phenomena and research. That mapping is critically dependent on the existence of cortical magnification factors. Once unfolded, V1 has a convex three-dimensional shape, which can be mathematically modeled as a surface of revolution embedded in three-dimensional Euclidean space. Thus, we solve the problem of differential geometry and geodesy for the mapping of the visual field to V1, involving both isotropic and non-isotropic cortical magnification factors of a most general form. We provide illustrations of our technique and results that apply to V1 surfaces with curve profiles relevant to vision research in general and to visual phenomena such as ‘crowding’ effects and eye movement guidance in particular. From a mathematical perspective, we also find intriguing and unexpected differential geometry properties of V1 surfaces, discovering that geodesic orbits have alternative prograde and retrograde characteristics, depending on the interplay between local curvature and global topology. Full article
(This article belongs to the Section Natural Sciences)
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15 pages, 289 KiB  
Article
Mathematical Attack of RSA by Extending the Sum of Squares of Primes to Factorize a Semi-Prime
by Anthony Overmars and Sitalakshmi Venkatraman
Math. Comput. Appl. 2020, 25(4), 63; https://doi.org/10.3390/mca25040063 - 28 Sep 2020
Cited by 7 | Viewed by 5436
Abstract
The security of RSA relies on the computationally challenging factorization of RSA modulus N=p1 p2 with N being a large semi-prime consisting of two primes p1and p2, for the generation of RSA [...] Read more.
The security of RSA relies on the computationally challenging factorization of RSA modulus N=p1 p2 with N being a large semi-prime consisting of two primes p1and p2, for the generation of RSA keys in commonly adopted cryptosystems. The property of p1 and p2, both congruent to 1 mod 4, is used in Euler’s factorization method to theoretically factorize them. While this caters to only a quarter of the possible combinations of primes, the rest of the combinations congruent to 3 mod 4 can be found by extending the method using Gaussian primes. However, based on Pythagorean primes that are applied in RSA, the semi-prime has only two sums of two squares in the range of possible squares N−1, N/2 . As N becomes large, the probability of finding the two sums of two squares becomes computationally intractable in the practical world. In this paper, we apply Pythagorean primes to explore how the number of sums of two squares in the search field can be increased thereby increasing the likelihood that a sum of two squares can be found. Once two such sums of squares are found, even though many may exist, we show that it is sufficient to only find two solutions to factorize the original semi-prime. We present the algorithm showing the simplicity of steps that use rudimentary arithmetic operations requiring minimal memory, with search cycle time being a factor for very large semi-primes, which can be contained. We demonstrate the correctness of our approach with practical illustrations for breaking RSA keys. Our enhanced factorization method is an improvement on our previous work with results compared to other factorization algorithms and continues to be an ongoing area of our research. Full article
3 pages, 164 KiB  
Editorial
Computational Methods in Interdisciplinary Applications of Nonlinear Dynamics
by Paweł Olejnik
Math. Comput. Appl. 2020, 25(4), 62; https://doi.org/10.3390/mca25040062 - 26 Sep 2020
Viewed by 1849
Abstract
Nonlinear dynamics takes its origins from physics and applied mathematics [...] Full article
(This article belongs to the Special Issue Computational Methods in Interdisciplinary Applications of Nonlinear Dynamics)
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