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Volume 10
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Mathematics, Volume 10, Issue 1 (January-1 2022) – 164 articles

Cover Story (view full-size image): We introduce a betting game where the gambler aims to guess the last success epoch in a series of inhomogeneous Bernoulli trials paced randomly in time. At a given stage, the gambler may bet on either the event that no further successes occur, or the event that exactly one success is yet to occur, or may choose any proper range of future times (a trap). When a trap is chosen, the gambler wins if the last success epoch is the only one that falls in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success. We use this connection to analyse the best-choice problem with random arrivals generated by a Pólya-Lundberg process. View this paper.
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17 pages, 3816 KiB  
Article
A Multi-Agent Motion Prediction and Tracking Method Based on Non-Cooperative Equilibrium
by Yan Li, Mengyu Zhao, Huazhi Zhang, Yuanyuan Qu and Suyu Wang
Mathematics 2022, 10(1), 164; https://doi.org/10.3390/math10010164 - 5 Jan 2022
Cited by 1 | Viewed by 2948
Abstract
A Multi-Agent Motion Prediction and Tracking method based on non-cooperative equilibrium (MPT-NCE) is proposed according to the fact that some multi-agent intelligent evolution methods, like the MADDPG, lack adaptability facing unfamiliar environments, and are unable to achieve multi-agent motion prediction and tracking, although [...] Read more.
A Multi-Agent Motion Prediction and Tracking method based on non-cooperative equilibrium (MPT-NCE) is proposed according to the fact that some multi-agent intelligent evolution methods, like the MADDPG, lack adaptability facing unfamiliar environments, and are unable to achieve multi-agent motion prediction and tracking, although they own advantages in multi-agent intelligence. Featured by a performance discrimination module using the time difference function together with a random mutation module applying predictive learning, the MPT-NCE is capable of improving the prediction and tracking ability of the agents in the intelligent game confrontation. Two groups of multi-agent prediction and tracking experiments are conducted and the results show that compared with the MADDPG method, in the aspect of prediction ability, the MPT-NCE achieves a prediction rate at more than 90%, which is 23.52% higher and increases the whole evolution efficiency by 16.89%; in the aspect of tracking ability, the MPT-NCE promotes the convergent speed by 11.76% while facilitating the target tracking by 25.85%. The proposed MPT-NCE method shows impressive environmental adaptability and prediction and tracking ability. Full article
(This article belongs to the Special Issue Computational Intelligence Algorithms for Dynamic Multiobjective Optimization Problems)
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<p>MPT-NCE system.</p> Full article ">Figure 2
<p>Working flow chart of the performance discrimination module.</p> Full article ">Figure 3
<p>Working flow chart of random mutation module.</p> Full article ">Figure 4
<p>Prediction experiment: (<b>a</b>) shows the movement trend in the pre-training period of detection agents, (<b>b</b>) shows the effect in the middle of the training period, and (<b>c</b>) shows the action results after training.</p> Full article ">Figure 5
<p>Reward curves in prediction experiment: (<b>a</b>) shows the comparison chart of the reward curve between the MPT-NCE method and the MADDPG method. (<b>b</b>) shows the blue detection agents’ reward curves of MPT-NCE method and MADDPG method, respectively. (<b>c</b>) shows the red interference agents’ reward curves of MPT-NCE method and MADDPG method, respectively.</p> Full article ">Figure 6
<p>Performance comparison chart of prediction experiment: (<b>a</b>) shows a comparison chart of training times and (<b>b</b>) shows a comparison chart of the reward function values.</p> Full article ">Figure 7
<p>Tracking experiment before purple interference agents’ mutation (tracking experiment 1): (<b>a</b>) shows the motion trend of each agent, (<b>b</b>) shows the action of each agent, and (<b>c</b>) shows the results after each agent action.</p> Full article ">Figure 8
<p>Tracking experiment after mutation of purple interference agents (tracking experiment 2): (<b>a</b>) shows the motion trend of each agent, (<b>b</b>) shows the action of each agent, and (<b>c</b>) shows the results after each agent action.</p> Full article ">Figure 9
<p>Reward function curve of tracking experiment: (<b>a</b>) shows the comparison chart of reward curve between the MPT-NCE method and MADDPG method, (<b>b</b>) shows the MPT-NCE method and MADDPG method of blue detection agents reward function curve comparison diagram, and (<b>c</b>) shows the MPT-NCE method and MADDPG method of red interference agents reward function curve comparison diagram.</p> Full article ">Figure 10
<p>Tracking experimental performance comparison chart: (<b>a</b>) shows the comparison chart of training times and (<b>b</b>) shows the comparison chart of reward function values.</p> Full article ">
42 pages, 19717 KiB  
Article
Empirical and Numerical Analysis of an Opaque Ventilated Facade with Windows Openings under Mediterranean Climate Conditions
by Carlos-Antonio Domínguez-Torres, Ángel Luis León-Rodríguez, Rafael Suárez and Antonio Domínguez-Delgado
Mathematics 2022, 10(1), 163; https://doi.org/10.3390/math10010163 - 5 Jan 2022
Cited by 9 | Viewed by 3403
Abstract
In recent years, there has been growing concern regarding energy efficiency in the building sector with energy requirements increasing worldwide and now responsible for about 40% of final energy consumption in Europe. Previous research has shown that ventilated façades help to reduce energy [...] Read more.
In recent years, there has been growing concern regarding energy efficiency in the building sector with energy requirements increasing worldwide and now responsible for about 40% of final energy consumption in Europe. Previous research has shown that ventilated façades help to reduce energy use when cooling buildings in hot and temperate climates. Of the different ventilated façade configurations reported in the literature, the configuration of ventilated façade with window rarely has been studied, and its 3D thermodynamic behavior is deserving of further analysis and modeling. This paper examines the thermal behavior of an opaque ventilated façade with a window, in experimentally and numerical terms and its impact in energy savings to get indoor comfort. Field measurements were conducted during the winter, spring and summer seasons of 2021 using outdoor full scale test cells located in Seville (southern Spain). The modeling of the ventilated façade was carried out using a three-dimensional approach taking into account the 3D behavior of the air flow in the air cavity due to the presence of the window. The validation and comparison process using experimental data showed that the proposed model provided good results from quantitative and qualitative point of view. The reduction of the heat flux was assessed by comparing the energy performance of a ventilated façade with that of an unventilated façade. Both experimental and numerical results showed that the ventilated façade provided a reduction in annual total energy consumption when compared to the unventilated façade, being compensated the winter energy penalization by the summer energy savings. This reduction is about 21% for the whole typical climatic year showing the ability of the opaque ventilated façade studied to reduce energy consumption to insure indoor comfort, making its suitable for use in retrofitting the energy-obsolete building stock built in Spain in the middle decades of the 20 century. Full article
(This article belongs to the Special Issue Modeling and Numerical Analysis of Energy and Environment 2021)
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<p>Flow chart of the used methodology.</p> Full article ">Figure 2
<p>(<b>a</b>) Floor plan of the test cells; (<b>b</b>) external view of Cell 1 with the OVF (<b>right</b>) and Cell 3 with the UVF (<b>left</b>).</p> Full article ">Figure 3
<p>Position of the monitoring sensors in the OVF. TR: surface temperature sensors, TA: air velocity and temperature sensors (h indicates the height in meters).</p> Full article ">Figure 4
<p>Location of the sensors in: (<b>a</b>) test cell 1; (<b>b</b>) test cell 3. (TA air temperature sensors, TR surface temperature sensors; h indicates the height in meters).</p> Full article ">Figure 5
<p>Heat transfer in the opaque ventilated façade.</p> Full article ">Figure 6
<p>Non-scaled sketch of the computational domain: (<b>a</b>) 3D view; (<b>b</b>) longitudinal section.</p> Full article ">Figure 7
<p>(<b>a</b>) Considered zones of the OVF; (<b>b</b>) tetrahedral mesh of the air chamber zones with the window opening.</p> Full article ">Figure 8
<p>Computational code sketch. <math display="inline"><semantics> <msub> <mi>t</mi> <mn>0</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>t</mi> <mi>f</mi> </msub> </semantics></math> are the initial and final time for computations.</p> Full article ">Figure 9
<p>Open ventilation grills, 5 July 2021: (<b>a</b>) south solar radiation and ambient temperatures; (<b>b</b>) trend of the measured air cavity velocities at the ventilated façade.</p> Full article ">Figure 10
<p>Open ventilation grills. Trend of the measured air cavity velocity: (<b>a</b>) for a winter day (14 January 2021); (<b>b</b>) for a summer day (5 July 2021). South solar radiation and ambient temperatures are reported.</p> Full article ">Figure 11
<p>Open ventilation grills. Evolution of the measured temperatures of the air cavity surfaces for a winter day (14 January 2021): (<b>a</b>) exterior slab, (<b>b</b>) inner wall; and a summer day (5 July 2021): (<b>c</b>) exterior slab, (<b>d</b>) inner wall. South solar radiation and ambient temperatures are reported.</p> Full article ">Figure 12
<p>Open ventilation grills. Trend of measured temperature in the different layers of the southern ventilated façade for daily hours with different solar radiation and temperature values under winter conditions (14 January 2021), for: (<b>a</b>) Zone 2; (<b>b</b>) Zone 4.</p> Full article ">Figure 13
<p>Closed ventilation grills, 24 June 2021: (<b>a</b>) south solar radiation and ambient temperatures; (<b>b</b>) trend of the measured air cavity velocities at the ventilated façade.</p> Full article ">Figure 14
<p>Closed ventilation grills. Trend of the measured air cavity velocity: (<b>a</b>) for a winter day (2 January 2021); (<b>a</b>) for a summer day (24 June 2021). South solar radiation and ambient temperatures are reported.</p> Full article ">Figure 15
<p>Closed ventilation grills. Evolution of the measured temperatures of the surfaces in the air cavity for: a winter day (3 January 2021), (<b>a</b>) exterior slab, (<b>b</b>) inner wall; a summer day (24 June 2021), (<b>c</b>) exterior slab, (<b>d</b>) inner wall. South solar radiation and ambient temperatures are reported.</p> Full article ">Figure 16
<p>Comparison of the operative temperatures of Cell 1 (OVF) and Cell 3 (UVF). Wintertime: (<b>a</b>) grill open (11 January to 18 January 2021), (<b>b</b>) grill closed (29 December 2020 to 5 January 2021). Springtime: (<b>c</b>) grill open (30 March to 6 April 2021); (<b>d</b>) grill closed (6 to 13 April 2021). Summertime: (<b>e</b>) grill open (5 to 8 July 2021); (<b>f</b>) grill closed (20 to 28 June 2021).</p> Full article ">Figure 17
<p>Velocity field of the air flow in the ventilated cavity of Cell 1 (OVF) for: (<b>a</b>) cutting plane at 0.02 m; (<b>b</b>) cutting plane at 0.04 m from the outer slab; (<b>c</b>) cutting plane at 0.06 m; (<b>d</b>) cutting plane at 0.08 m from the outer slab.</p> Full article ">Figure 18
<p>Temperatures of air in ventilated cavity of Cell 1 (OVF) in: (<b>a</b>) cutting plane at 0.02 m; (<b>b</b>) cutting plane at 0.04 m; (<b>c</b>) cutting plane at 0.06 m; (<b>d</b>) cutting plane at 0.08 m from the outer slab.</p> Full article ">Figure 19
<p>Cell 1. Winter. Grid open. Temperatures of the exterior face of the outer slab in Zone 4: measured (<span class="html-fig-inline" id="mathematics-10-00163-i001"> <img alt="Mathematics 10 00163 i001" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i001.png"/></span> ) and calculated (<span class="html-fig-inline" id="mathematics-10-00163-i002"> <img alt="Mathematics 10 00163 i002" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i002.png"/></span>). Bands of <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> <msup> <mspace width="3.33333pt"/> <mo>°</mo> </msup> </mrow> </semantics></math>C, (┅), are shown.</p> Full article ">Figure 20
<p>Cell 1. Winter. Grid open. Temperatures of the interior face of the outer slab in Zone 4: measured (<span class="html-fig-inline" id="mathematics-10-00163-i001"> <img alt="Mathematics 10 00163 i001" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i001.png"/></span>) and calculated (<span class="html-fig-inline" id="mathematics-10-00163-i002"> <img alt="Mathematics 10 00163 i002" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i002.png"/></span>). Bands of <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> <msup> <mspace width="3.33333pt"/> <mo>°</mo> </msup> </mrow> </semantics></math>C, (┅), are shown.</p> Full article ">Figure 21
<p>Cell 1. Winter. Grid open. Temperatures of the exterior face of the inner wall in Zone 4: measured (<span class="html-fig-inline" id="mathematics-10-00163-i001"> <img alt="Mathematics 10 00163 i001" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i001.png"/></span>) and calculated (<span class="html-fig-inline" id="mathematics-10-00163-i002"> <img alt="Mathematics 10 00163 i002" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i002.png"/></span>). Bands of <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> <msup> <mspace width="3.33333pt"/> <mo>°</mo> </msup> </mrow> </semantics></math>C, (┅), are shown.</p> Full article ">Figure 22
<p>Cell 1. Winter. Grid open. Measured and calculated temperatures of the interior face of the inner wall in Zone 4: measured (<span class="html-fig-inline" id="mathematics-10-00163-i001"> <img alt="Mathematics 10 00163 i001" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i001.png"/></span>) and calculated (<span class="html-fig-inline" id="mathematics-10-00163-i002"> <img alt="Mathematics 10 00163 i002" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i002.png"/></span>). Bands of <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> <msup> <mspace width="3.33333pt"/> <mo>°</mo> </msup> </mrow> </semantics></math>C, (┅), are shown.</p> Full article ">Figure 23
<p>Cell 1. Winter. Grid open. Temperatures of the Cell 1 air indoor: measured (<span class="html-fig-inline" id="mathematics-10-00163-i001"> <img alt="Mathematics 10 00163 i001" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i001.png"/></span>) and calculated (<span class="html-fig-inline" id="mathematics-10-00163-i002"> <img alt="Mathematics 10 00163 i002" src="/mathematics/mathematics-10-00163/article_deploy/html/images/mathematics-10-00163-i002.png"/></span>). Bands of <math display="inline"><semantics> <mrow> <mo>±</mo> <mn>0.5</mn> <msup> <mspace width="3.33333pt"/> <mo>°</mo> </msup> </mrow> </semantics></math>C, (┅), are shown.</p> Full article ">Figure 24
<p>Trend of measured and calculated temperatures in the different layers of the southern ventilated façade for daily hours with different solar radiation and temperature values under winter conditions (14 January 2021), for: (<b>a</b>) Zone 2; (<b>b</b>) Zone 4.</p> Full article ">Figure 25
<p>Hourly heat fluxes through the OVF and the UVF for: two days of winter with (<b>a</b>) grid open (13 January 2021), (<b>b</b>) grid closed (2 January 2021); two days of spring with (<b>c</b>) grid open (31 March 2021); (<b>d</b>) grid closed (7 April 2021); two days of summer with (<b>e</b>) grid open (5 July 2021); (<b>f</b>) grid closed (24 June 2021).</p> Full article ">Figure 26
<p>(<b>a</b>) Measured and calculated daily heat fluxes through the OVF vs. the UVF; (<b>b</b>) energy saving rates (%) measured and calculated.</p> Full article ">Figure 27
<p>(<b>a</b>) Total loads for representative month days; (<b>b</b>) total loads for the whole year.</p> Full article ">
18 pages, 7386 KiB  
Article
Research on Intellectualized Location of Coal Gangue Logistics Nodes Based on Particle Swarm Optimization and Quasi-Newton Algorithm
by Shengli Yang, Junjie Wang, Ming Li and Hao Yue
Mathematics 2022, 10(1), 162; https://doi.org/10.3390/math10010162 - 5 Jan 2022
Cited by 7 | Viewed by 2386
Abstract
The optimization of an integrated coal gangue system of mining, dressing, and backfilling in deep underground mining is a multi-objective and complex decision-making process, and the factors such as spatial layout, node location, and transportation equipment need to be considered comprehensively. In order [...] Read more.
The optimization of an integrated coal gangue system of mining, dressing, and backfilling in deep underground mining is a multi-objective and complex decision-making process, and the factors such as spatial layout, node location, and transportation equipment need to be considered comprehensively. In order to realize the intellectualized location of the nodes for the logistics and transportation system of underground mining and dressing coal and gangue, this paper establishes the model of the logistics and transportation system of underground mining and dressing coal gangue, and analyzes the key factors of the intellectualized location for the logistics and transportation system of coal and gangue, and the objective function of the node transportation model is deduced. The PSO–QNMs algorithm is proposed for the solution of the objective function, which improves the accuracy and stability of the location selection and effectively avoids the shortcomings of the PSO algorithm with its poor local detailed search ability and the quasi-Newton algorithm with its sensitivity to the initial value. Comparison of the particle swarm and PSO–QNMs algorithm outputs for the specific conditions of the New Julong coal mine, as an example, shows that the PSO–QNMs algorithm reduces the complexity of the calculation, increases the calculation efficiency by eight times, saves 42.8% of the cost value, and improves the efficiency of the node selection of mining–dressing–backfilling systems in a complex underground mining environment. The results confirm that the method has high convergence speed and solution accuracy, and provides a fundamental basis for optimizing the underground coal mine logistics system. Based on the research results, a node siting system for an integrated underground mining, dressing, and backfilling system in coal mines (referred to as MSBPS) was developed. Full article
(This article belongs to the Special Issue Nonlinear Systems: Dynamics, Control, Optimization and Applications to the Science and Engineering)
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<p>Node transportation model of the coal gangue backfilling system.</p> Full article ">Figure 2
<p>Flow chart of the PSO–QNMs hybrid algorithm.</p> Full article ">Figure 3
<p>Schematic diagram of coal gangue logistics and transportation system in the Xinjulong coal mine.</p> Full article ">Figure 4
<p>Intelligent optimization method iteration curve: (<b>a</b>) group 1, (<b>b</b>) group 2, (<b>c</b>) group 3.</p> Full article ">Figure 5
<p>The spatial position relationship diagram of the logistics nodes (group 1): (<b>a</b>) PSO algorithm, (<b>b</b>) PSO–QNMs algorithm.</p> Full article ">Figure 6
<p>The spatial position relationship diagram of the logistics nodes (group 2): (<b>a</b>) PSO algorithm, (<b>b</b>) PSO–QNMs algorithm.</p> Full article ">Figure 7
<p>The spatial position relationship diagram of the logistics nodes (group 3): (<b>a</b>) PSO algorithm, (<b>b</b>) PSO–QNMs algorithm.</p> Full article ">Figure 8
<p>Main Interfaces of the node location decision system.</p> Full article ">
31 pages, 405 KiB  
Article
Optimal Risk Sharing in Society
by Knut K. Aase
Mathematics 2022, 10(1), 161; https://doi.org/10.3390/math10010161 - 5 Jan 2022
Viewed by 2417
Abstract
We consider risk sharing among individuals in a one-period setting under uncertainty that will result in payoffs to be shared among the members. We start with optimal risk sharing in an Arrow–Debreu economy, or equivalently, in a Borch-style reinsurance market. From the results [...] Read more.
We consider risk sharing among individuals in a one-period setting under uncertainty that will result in payoffs to be shared among the members. We start with optimal risk sharing in an Arrow–Debreu economy, or equivalently, in a Borch-style reinsurance market. From the results of this model we can infer how risk is optimally distributed between individuals according to their preferences and initial endowments, under some idealized conditions. A main message in this theory is the mutuality principle, of interest related to the economic effects of pandemics. From this we point out some elements of a more general theory of syndicates, where in addition, a group of people are to make a common decision under uncertainty. We extend to a competitive market as a special case of such a syndicate. Full article
(This article belongs to the Special Issue The Mathematics of Pandemics: Applications for Insurance)
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<p>The Pareto frontier for N = 2.</p> Full article ">
17 pages, 2803 KiB  
Article
The Development of Log Aesthetic Patch and Its Projection onto the Plane
by Yee Meng Teh, R. U. Gobithaasan, Kenjiro T. Miura, Diya’ J. Albayari and Wen Eng Ong
Mathematics 2022, 10(1), 160; https://doi.org/10.3390/math10010160 - 5 Jan 2022
Viewed by 2084
Abstract
In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify [...] Read more.
In this work, we introduce a new type of surface called the Log Aesthetic Patch (LAP). This surface is an extension of the Coons surface patch, in which the four boundary curves are either planar or spatial Log Aesthetic Curves (LACs). To identify its versatility, we approximated the hyperbolic paraboloid to LAP using the information of lines of curvature (LoC). The outer part of the LoCs, which play a role as the boundary of the hyperbolic paraboloid, is replaced with LACs before constructing the LAP. Since LoCs are essential in shipbuilding for hot and cold bending processes, we investigated the LAP in terms of the LoC’s curvature, derivative of curvature, torsion, and Logarithmic Curvature Graph (LCG). The numerical results indicate that the LoCs for both surfaces possess monotonic curvatures. An advantage of LAP approximation over its original hyperbolic paraboloid is that the LoCs of LAP can be approximated to LACs, and hence the first derivative of curvatures for LoCs are monotonic, whereas they are non-monotonic for the hyperbolic paraboloid. This confirms that the LAP produced is indeed of high quality. Lastly, we project the LAP onto a plane using geodesic curvature to create strips that can be pasted together, mimicking hot and cold bending processes in the shipbuilding industry. Full article
(This article belongs to the Section E1: Mathematics and Computer Science)
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<p>The projection of LoCs onto a plane: (<b>a</b>) LoCs in 3D [<a href="#B25-mathematics-10-00160" class="html-bibr">25</a>]; (<b>b</b>) LoCs in 2D [<a href="#B25-mathematics-10-00160" class="html-bibr">25</a>]; (<b>c</b>) hyperbolic paraboloid with gaps as shown in [<a href="#B25-mathematics-10-00160" class="html-bibr">25</a>]; (<b>d</b>) hyperbolic paraboloid computed on GPU using Mathematica.</p> Full article ">Figure 2
<p>LoCs on hyperbolic paraboloid.</p> Full article ">Figure 3
<p>LoCs on hyperbolic paraboloid (blue) and Log Aesthetic segments (red).</p> Full article ">Figure 4
<p>Hyperbolic paraboloid and LAP approximation: (<b>a</b>) hyperbolic paraboloid; (<b>b</b>) LAP; (<b>c</b>) combination of both surfaces; (<b>d</b>) zebra map on the Coons LA patch.</p> Full article ">Figure 5
<p>LoCs on surfaces: (<b>a</b>) hyperbolic paraboloid; (<b>b</b>) approximated LAP.</p> Full article ">Figure 6
<p>Curvature, torsion, and derivative of curvature of LoCs on the hyperbolic paraboloid (<b>left</b>) and the approximated LAP (<b>right</b>).</p> Full article ">Figure 7
<p>An example of the fabrication process of approximated hyperbolic paraboloid using LAP: (<b>a</b>) development of LoCs onto a plane; (<b>b</b>) development of LAP with gaps; (<b>c</b>) approximated LAP with LoCs (purple) and boundary curves (red); (<b>d</b>) fabricated surface using a paper; (<b>e</b>) close-up view of the LoC <span class="html-italic">J</span><sub>3</sub>.</p> Full article ">
25 pages, 1588 KiB  
Article
Comparing Multi-Objective Local Search Algorithms for the Beam Angle Selection Problem
by Guillermo Cabrera-Guerrero and Carolina Lagos
Mathematics 2022, 10(1), 159; https://doi.org/10.3390/math10010159 - 5 Jan 2022
Cited by 2 | Viewed by 1835
Abstract
In intensity-modulated radiation therapy, treatment planners aim to irradiate the tumour according to a medical prescription while sparing surrounding organs at risk as much as possible. Although this problem is inherently a multi-objective optimisation (MO) problem, most of the models in the literature [...] Read more.
In intensity-modulated radiation therapy, treatment planners aim to irradiate the tumour according to a medical prescription while sparing surrounding organs at risk as much as possible. Although this problem is inherently a multi-objective optimisation (MO) problem, most of the models in the literature are single-objective ones. For this reason, a large number of single-objective algorithms have been proposed in the literature to solve such single-objective models rather than multi-objective ones. Further, a difficulty that one has to face when solving the MO version of the problem is that the algorithms take too long before converging to a set of (approximately) non-dominated points. In this paper, we propose and compare three different strategies, namely random PLS (rPLS), judgement-function-guided PLS (jPLS) and neighbour-first PLS (nPLS), to accelerate a previously proposed Pareto local search (PLS) algorithm to solve the beam angle selection problem in IMRT. A distinctive feature of these strategies when compared to the PLS algorithms in the literature is that they do not evaluate their entire neighbourhood before performing the dominance analysis. The rPLS algorithm randomly chooses the next non-dominated solution in the archive and it is used as a baseline for the other implemented algorithms. The jPLS algorithm first chooses the non-dominated solution in the archive that has the best objective function value. Finally, the nPLS algorithm first chooses the solutions that are within the neighbourhood of the current solution. All these strategies prevent us from evaluating a large set of BACs, without any major impairment in the obtained solutions’ quality. We apply our algorithms to a prostate case and compare the obtained results to those obtained by the PLS from the literature. The results show that algorithms proposed in this paper reach a similar performance than PLS and require fewer function evaluations. Full article
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<p>Prostate case from CERR. Two OARs (bladder and rectum) are considered.</p> Full article ">Figure 2
<p>Examples on how the hypervolume is calculated in a bi-objective space. (<b>a</b>) Example of the hypervolume dominated by a set of 6 non-dominated points in the objective space. (<b>b</b>) Resulting hypervolume for two sets of non-dominated points.</p> Full article ">Figure 3
<p>Sample points generated by PLS algorithm.</p> Full article ">Figure 4
<p>Sample points generated by nPLS algorithm.</p> Full article ">Figure 5
<p>Sample points generated by jPLS algorithm.</p> Full article ">Figure 6
<p>Sample points generated by rPLS algorithm.</p> Full article ">Figure 7
<p>Sample points and the path generated by the steepest descent algorithm in objective space.</p> Full article ">Figure 8
<p>Sample points and paths generated by the next descent algorithm in objective space.</p> Full article ">Figure 9
<p>jPLS paths in objective space for the initial BACs 0–2 and 10–13 in <a href="#mathematics-10-00159-t0A1" class="html-table">Table A1</a>. All the initial BACs end up in the same set <math display="inline"><semantics> <msup> <mi mathvariant="script">A</mi> <mo>*</mo> </msup> </semantics></math> of locally efficient BACs.</p> Full article ">Figure 10
<p>Hypervolume per algorithm. (<b>a</b>) Hypervolume for all algorithms for equispaced initial BACs (0–13). (<b>b</b>) Cumulative hypervolume for all algorithms for equispaced initial BACs (0–13). (<b>c</b>) Hypervolume for all algorithms for semi-random initial BACs (14–29). (<b>d</b>) Cumulative hypervolume for all algorithms for semi-random initial BACs (14–29). (<b>e</b>) Hypervolume for all algorithms for random initial BACs (30–44). (<b>f</b>) Cumulative hypervolume for all algorithms for random initial BACs (30–44).</p> Full article ">
19 pages, 829 KiB  
Article
Trapping the Ultimate Success
by Alexander Gnedin and Zakaria Derbazi
Mathematics 2022, 10(1), 158; https://doi.org/10.3390/math10010158 - 5 Jan 2022
Cited by 4 | Viewed by 2385
Abstract
We introduce a betting game where the gambler aims to guess the last success epoch in a series of inhomogeneous Bernoulli trials paced randomly in time. At a given stage, the gambler may bet on either the event that no further successes occur, [...] Read more.
We introduce a betting game where the gambler aims to guess the last success epoch in a series of inhomogeneous Bernoulli trials paced randomly in time. At a given stage, the gambler may bet on either the event that no further successes occur, or the event that exactly one success is yet to occur, or may choose any proper range of future times (a trap). When a trap is chosen, the gambler wins if the last success epoch is the only one that falls in the trap. The game is closely related to the sequential decision problem of maximising the probability of stopping on the last success. We use this connection to analyse the best-choice problem with random arrivals generated by a Pólya-Lundberg process. Full article
(This article belongs to the Special Issue Bayesian Predictive Inference and Related Asymptotics—Festschrift for Eugenio Regazzini's 75th Birthday)
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Figure 1
<p>The winning probability <math display="inline"> <semantics> <mrow> <msub> <mi>S</mi> <mn>1</mn> </msub> <mrow> <mo>(</mo> <mi>k</mi> <mo>,</mo> <mi>n</mi> <mo>;</mo> <mi>z</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> of <span class="html-italic">z</span>-strategy in the best-choice problem for <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics> </math> and 1.</p> Full article ">Figure 2
<p>Bernstein polynomials for <math display="inline"> <semantics> <mrow> <msub> <mi>p</mi> <mi>k</mi> </msub> <mo>=</mo> <mi>θ</mi> <mo>/</mo> <mrow> <mo>(</mo> <mi>θ</mi> <mo>+</mo> <mi>k</mi> <mo>−</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics> </math>.</p> Full article ">Figure 3
<p>The concavity condition (<a href="#FD12-mathematics-10-00158" class="html-disp-formula">12</a>) holds for profiles <math display="inline"> <semantics> <mi mathvariant="bold-italic">p</mi> </semantics> </math> with <math display="inline"> <semantics> <mrow> <mo>(</mo> <msub> <mi>p</mi> <mi>k</mi> </msub> <mo>,</mo> <msub> <mi>p</mi> <mrow> <mi>k</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> <mo>)</mo> </mrow> </semantics> </math> squeezed between the parabolas.</p> Full article ">Figure 4
<p><math display="inline"> <semantics> <mi mathvariant="monospace">next</mi> </semantics> </math> and <math display="inline"> <semantics> <mi mathvariant="monospace">bygone</mi> </semantics> </math> curves for <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 5
<p>Stop, continuation, z-strategy values and bounds; <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>3</mn> </mrow> </semantics> </math> and zoomed-in view for <math display="inline"> <semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics> </math>.</p> Full article ">Figure 6
<p>Information bounds on the optimal strategy <math display="inline"> <semantics> <mrow> <msub> <mi>I</mi> <mi>k</mi> </msub> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math>.</p> Full article ">
13 pages, 884 KiB  
Article
Closing a Bitcoin Trade Optimally under Partial Information: Performance Assessment of a Stochastic Disorder Model
by Zehra Eksi and Daniel Schreitl
Mathematics 2022, 10(1), 157; https://doi.org/10.3390/math10010157 - 5 Jan 2022
Cited by 2 | Viewed by 2028
Abstract
The Bitcoin market exhibits characteristics of a market with pricing bubbles. The price is very volatile, and it inherits the risk of quickly increasing to a peak and decreasing from the peak even faster. In this context, it is vital for investors to [...] Read more.
The Bitcoin market exhibits characteristics of a market with pricing bubbles. The price is very volatile, and it inherits the risk of quickly increasing to a peak and decreasing from the peak even faster. In this context, it is vital for investors to close their long positions optimally. In this study, we investigate the performance of the partially observable digital-drift model of Ekström and Lindberg and the corresponding optimal exit strategy on a Bitcoin trade. In order to estimate the unknown intensity of the random drift change time, we refer to Bitcoin halving events, which are considered as pivotal events that push the price up. The out-of-sample performance analysis of the model yields returns values ranging between 9% and 1153%. We conclude that the return of the initiated Bitcoin momentum trades heavily depends on the entry date: the earlier we entered, the higher the expected return at the optimal exit time suggested by the model. Overall, to the extent of our analysis, the model provides a supporting framework for exit decisions, but is by far not the ultimate tool to succeed in every trade. Full article
(This article belongs to the Special Issue Information in Economics, Finance and Insurance: Modelling and Applications)
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<p>Daily BTC/EUR prices from 29 April 2013 to 29 April 2021. The red triangles represent local peaks (drift changes from positive to negative), the blue triangles represent local troughs (drift changes from negative to positive) and the green points represent Bitcoin halving events. Note that the first halving event on 28 November 2012 is not displayed.</p> Full article ">Figure 2
<p>The graph visualizes the Bitcoin price series and six different long momentum trades. The red points are the entry dates, the blue vertical line is the true drift change point, and the orange squares are the respective exit times of the trades using <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mo>−</mo> <mn>2</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The graph visualizes the process <math display="inline"><semantics> <mi mathvariant="sans-serif">Φ</mi> </semantics></math> (<b>top</b> pane) and the corresponding Bitcoin price series (<b>bottom</b> pane). The red point is the entry of the long momentum trade on 24 March 2017; the vertical blue line is the realization of the drift change time <math display="inline"><semantics> <mi>θ</mi> </semantics></math> and the orange square is the optimal exit time, found when <math display="inline"><semantics> <mi mathvariant="sans-serif">Φ</mi> </semantics></math> crossed the barrier <span class="html-italic">B</span>, marked by the gray horizontal threshold.</p> Full article ">
20 pages, 1740 KiB  
Article
Evolutionary Synthesis of Failure-Resilient Analog Circuits
by Žiga Rojec, Iztok Fajfar and Árpád Burmen
Mathematics 2022, 10(1), 156; https://doi.org/10.3390/math10010156 - 5 Jan 2022
Cited by 8 | Viewed by 1800
Abstract
Analog circuit design requires large amounts of human knowledge. A special case of circuit design is the synthesis of robust and failure-resilient electronics. Evolutionary algorithms can aid designers in exploring topologies with new properties. Here, we show how to encode a circuit topology [...] Read more.
Analog circuit design requires large amounts of human knowledge. A special case of circuit design is the synthesis of robust and failure-resilient electronics. Evolutionary algorithms can aid designers in exploring topologies with new properties. Here, we show how to encode a circuit topology with an upper-triangular incident matrix and use the NSGA-II algorithm to find computational circuits that are robust to component failure. Techniques for robustness evaluation and evolutionary algorithm guidances are described. As a result, we evolve square root and natural logarithm computational circuits that are robust to high-impedance or short-circuit malfunction of an arbitrary rectifying diode. We confirm the simulation results by hardware circuit implementation and measurements. We think that our research will inspire further searches for failure-resilient topologies. Full article
(This article belongs to the Special Issue Optimization Theory and Applications)
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<p>An example of an analog circuit topology represented by an upper-triangular incident matrix. Every logical one connects two terminals of listed components or outer terminals [<a href="#B33-mathematics-10-00156" class="html-bibr">33</a>].</p> Full article ">Figure 2
<p>Topology crossover. In offspring 1, only the node location of one device terminal is exchanged. In offspring 2, three node locations are exchanged. For better illustration, the right parent is a full upper-triangular matrix [<a href="#B32-mathematics-10-00156" class="html-bibr">32</a>].</p> Full article ">Figure 3
<p>Rectifier false-robustness problem.</p> Full article ">Figure 4
<p>The flowchart of the evolutionary algorithm. Every tenth generation, a full parameter optimization is triggered on best individuals.</p> Full article ">Figure 5
<p>Rectifier diode failure modeling. (<b>Left</b>) The nominal model with inclusiveness measurement voltmeter. (<b>Center</b>) A high-impedance diode failure. (<b>Right</b>) A short-circuit diode failure.</p> Full article ">Figure 6
<p>The Pareto front of the last generation of the evolution of the failure-resilient square root circuit. Shown are the tree objectives <math display="inline"><semantics> <msub> <mi>f</mi> <mi>nom</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>F</mi> <mo>Σ</mo> </msub> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mi mathvariant="bold">F</mi> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>The evolved topology of failure-tolerant square root circuit (raw evolution result).</p> Full article ">Figure 8
<p>The voltage response of the resulting failure-resilient square root circuit. (<b>Left</b>):The complete output range, where the red (dash-dot) curve represents the ideal square root function, the black (solid) curve represents the failure-free circuit, and the gray area represents the range of responses with all possible single diode failures. (<b>Right</b>) Relative deviations from the ideal square root function (<math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msqrt> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </msqrt> </mrow> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msqrt> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> </mfrac> <mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. A nominal design offset is subtracted from <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </semantics></math> representation. Stuck-open failures are drawn in dashed lines while stuck-short failures are drawn in dotted lines.</p> Full article ">Figure 9
<p>The Pareto front of the last generation of the evolution of the failure-resilient natural logarithm circuit. Shown are the tree objectives <math display="inline"><semantics> <msub> <mi>f</mi> <mi>nom</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>F</mi> <mo>Σ</mo> </msub> </semantics></math>, and <math display="inline"><semantics> <mrow> <mi>σ</mi> <mi mathvariant="bold">F</mi> </mrow> </semantics></math>.</p> Full article ">Figure 10
<p>The voltage response of the resulting failure-resilient natural logarithm circuit. (<b>Left</b>) The complete output range, where the red (dash-dot) curve represents the ideal natural logarithm function, the black (solid) curve represents the failure-free circuit, and the gray area represents the range of responses with all possible single diode failures. (<b>Right</b>) Relative deviations from the ideal natural logarithm function (<math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>−</mo> <mn>2</mn> <mo form="prefix">ln</mo> <mo>(</mo> </mrow> <mrow> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mrow> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>2</mn> <mo form="prefix">ln</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. Stuck-open failures are drawn in dashed lines while stuck-short failures are drawn in dotted lines.</p> Full article ">Figure 11
<p>The evolved topology of a failure-tolerant natural logarithm circuit (raw evolution result).</p> Full article ">Figure 12
<p>A bread-board implementation of a failure-resilient square root circuit.</p> Full article ">Figure 13
<p>The measured voltage response of a real-world prototype failure-resilient square root circuit. (<b>Left</b>) The complete output range where the red (dash-dot) curve represents the ideal square root function, the black (solid) curve represents the failure-free circuit, and the gray area represents the range of responses with all possible single diode failures. (<b>Right</b>) Relative deviations from the ideal square root function (<math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msqrt> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </msqrt> </mrow> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msqrt> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> </mfrac> <mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. A nominal design offset is subtracted from <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </semantics></math> representation. Stuck-open failures are drawn in dashed lines while stuck-short failures are drawn in dotted lines.</p> Full article ">Figure 14
<p>Examples of hand-designed piece-wise linear computational circuits: (<b>Left</b>) square root and (<b>Right</b>) natural logarithm [<a href="#B39-mathematics-10-00156" class="html-bibr">39</a>].</p> Full article ">Figure 15
<p>The simulated voltage response of an example of a hand-designed piece-wise linear square root circuit. (<b>Left</b>) The complete output range where the red (dash-dot) curve represents the ideal square root function, the black (solid) curve represents the failure-free circuit, and the gray area represents the range of responses with all possible single diode failures. (<b>Right</b>) Relative deviations from the ideal square root function (<math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>−</mo> <msqrt> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </msqrt> </mrow> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <msqrt> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> </msqrt> <mo>)</mo> </mrow> </mfrac> <mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. A nominal design offset is subtracted from <math display="inline"><semantics> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> </semantics></math> representation. Stuck-open failures are drawn in dashed lines while stuck-short failures are drawn in dotted lines.</p> Full article ">Figure 16
<p>The simulated voltage response of an example of a hand-designed piece-wise linear the natural logarithm circuit. (<b>Left</b>) The complete output range where the red (dash-dot) curve represents the ideal natural logarithm function, the black (solid) curve represents the failure-free circuit, and the gray area represents the range of responses with all possible single diode failures. (<b>Right</b>) Relative deviations from the ideal natural logarithm function (<math display="inline"><semantics> <mrow> <mi>R</mi> <mi>e</mi> <mi>l</mi> <mi>E</mi> <mi>r</mi> <mi>r</mi> <mrow> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mo>)</mo> </mrow> <mo>=</mo> <mfrac> <mrow> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mrow> <mo>−</mo> <mn>2</mn> <mo form="prefix">ln</mo> <mo>(</mo> </mrow> <mrow> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </mrow> <mrow> <mo movablelimits="true" form="prefix">max</mo> <mo>(</mo> <msub> <mi>V</mi> <mrow> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </msub> <mo>,</mo> <mn>2</mn> <mo form="prefix">ln</mo> <mrow> <mo>(</mo> <mrow> <msub> <mi>V</mi> <mrow> <mi>i</mi> <mi>n</mi> </mrow> </msub> <mrow> <mo>+</mo> <mn>1</mn> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </mfrac> <mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. Stuck-open failures are drawn in dashed lines while stuck-short failures are drawn in dotted lines.</p> Full article ">
11 pages, 8624 KiB  
Article
Efficient Fully Discrete Finite-Element Numerical Scheme with Second-Order Temporal Accuracy for the Phase-Field Crystal Model
by Jun Zhang and Xiaofeng Yang
Mathematics 2022, 10(1), 155; https://doi.org/10.3390/math10010155 - 5 Jan 2022
Cited by 4 | Viewed by 1654
Abstract
In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method [...] Read more.
In this paper, we consider numerical approximations of the Cahn–Hilliard type phase-field crystal model and construct a fully discrete finite element scheme for it. The scheme is the combination of the finite element method for spatial discretization and an invariant energy quadratization method for time marching. It is not only linear and second-order time-accurate, but also unconditionally energy-stable. We prove the unconditional energy stability rigorously and further carry out various numerical examples to demonstrate the stability and the accuracy of the developed scheme numerically. Full article
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Figure 1
<p>(<b>a</b>) Convergence test in time where the <math display="inline"><semantics> <msup> <mi>L</mi> <mn>2</mn> </msup> </semantics></math> numerical errors for the phase field variable <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> are computed by using various time step size <math display="inline"><semantics> <mrow> <mi>δ</mi> <mi>t</mi> </mrow> </semantics></math>, and (<b>b</b>) convergence test in time where the <math display="inline"><semantics> <msup> <mi>L</mi> <mn>2</mn> </msup> </semantics></math> numerical errors for the phase-field variable <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> are computed by using various grid size <span class="html-italic">h</span>.</p> Full article ">Figure 2
<p>The dynamical behaviors of the phase transition example, where snapshots of the numerical approximation of <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> are taken at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, 100, 250, 350, 500, 600, 750, 850, and 1000.</p> Full article ">Figure 2 Cont.
<p>The dynamical behaviors of the phase transition example, where snapshots of the numerical approximation of <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> are taken at <math display="inline"><semantics> <mrow> <mi>t</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math>, 100, 250, 350, 500, 600, 750, 850, and 1000.</p> Full article ">
11 pages, 262 KiB  
Article
Geodesic Mappings of Semi-Riemannian Manifolds with a Degenerate Metric
by Igor G. Shandra and Josef Mikeš
Mathematics 2022, 10(1), 154; https://doi.org/10.3390/math10010154 - 5 Jan 2022
Viewed by 1588
Abstract
This article introduces the concept of geodesic mappings of manifolds with idempotent pseudo-connections. The basic equations of canonical geodesic mappings of manifolds with completely idempotent pseudo-connectivity and semi-Riemannian manifolds with a degenerate metric are obtained. It is proved that semi-Riemannian manifolds admitting concircular [...] Read more.
This article introduces the concept of geodesic mappings of manifolds with idempotent pseudo-connections. The basic equations of canonical geodesic mappings of manifolds with completely idempotent pseudo-connectivity and semi-Riemannian manifolds with a degenerate metric are obtained. It is proved that semi-Riemannian manifolds admitting concircular fields admit completely canonical geodesic mappings and form a closed class with respect to these mappings. Full article
(This article belongs to the Special Issue Differential Geometry of Spaces with Special Structures)
9 pages, 279 KiB  
Article
Boundary Value Problem for ψ-Caputo Fractional Differential Equations in Banach Spaces via Densifiability Techniques
by Choukri Derbazi, Zidane Baitiche, Mouffak Benchohra and Yong Zhou
Mathematics 2022, 10(1), 153; https://doi.org/10.3390/math10010153 - 5 Jan 2022
Cited by 6 | Viewed by 1996
Abstract
A novel fixed-point theorem based on the degree of nondensifiability (DND) is used in this article to examine the existence of solutions to a boundary value problem containing the ψ-Caputo fractional derivative in Banach spaces. Besides that, an example is included to [...] Read more.
A novel fixed-point theorem based on the degree of nondensifiability (DND) is used in this article to examine the existence of solutions to a boundary value problem containing the ψ-Caputo fractional derivative in Banach spaces. Besides that, an example is included to verify our main results. Moreover, the outcomes obtained in this research paper ameliorate and expand some previous findings in this area. Full article
(This article belongs to the Special Issue Nonlinear Equations: Theory, Methods, and Applications II)
25 pages, 2329 KiB  
Article
Does Machine Learning Offer Added Value Vis-à-Vis Traditional Statistics? An Exploratory Study on Retirement Decisions Using Data from the Survey of Health, Ageing, and Retirement in Europe (SHARE)
by Montserrat González Garibay, Andrej Srakar, Tjaša Bartolj and Jože Sambt
Mathematics 2022, 10(1), 152; https://doi.org/10.3390/math10010152 - 4 Jan 2022
Cited by 2 | Viewed by 3294
Abstract
Do machine learning algorithms perform better than statistical survival analysis when predicting retirement decisions? This exploratory article addresses the question by constructing a pseudo-panel with retirement data from the Survey of Health, Ageing, and Retirement in Europe (SHARE). The analysis consists of two [...] Read more.
Do machine learning algorithms perform better than statistical survival analysis when predicting retirement decisions? This exploratory article addresses the question by constructing a pseudo-panel with retirement data from the Survey of Health, Ageing, and Retirement in Europe (SHARE). The analysis consists of two methodological steps prompted by the nature of the data. First, a discrete Cox survival model of transitions to retirement with time-dependent covariates is compared to a Cox model without time-dependent covariates and a survival random forest. Second, the best performing model (Cox with time-dependent covariates) is compared to random forests adapted to time-dependent covariates by means of simulations. The results from the analysis do not clearly favor a single method; whereas machine learning algorithms have a stronger predictive power, the variables they use in their predictions do not necessarily display causal relationships with the outcome variable. Therefore, the two methods should be seen as complements rather than substitutes. In addition, simulations shed a new light on the role of some variables—such as education and health—in retirement decisions. This amounts to both substantive and methodological contributions to the literature on the modeling of retirement. Full article
(This article belongs to the Section D1: Probability and Statistics)
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<p>Schematic summary of the literature review.</p> Full article ">Figure 2
<p>AUC comparison of simulated Cox regressions and random forests (N = 500) when <span class="html-italic">t</span> = 1: (<b>a</b>) AUC comparison for <span class="html-italic">t</span> = 1, <span class="html-italic">u</span> = 2; (<b>b</b>) AUC comparison for <span class="html-italic">t</span> = 1, <span class="html-italic">u</span> = 3; (<b>c</b>) AUC comparison for <span class="html-italic">t</span> = 1, <span class="html-italic">u</span> = 4; (<b>d</b>) AUC comparison for <span class="html-italic">t</span> = 1, <span class="html-italic">u</span> = 5; (<b>e</b>) AUC comparison for <span class="html-italic">t</span> = 1, <span class="html-italic">u</span> = 6.</p> Full article ">Figure 3
<p>AUC comparison of simulated Cox regressions and random forests (N = 500) when <span class="html-italic">t</span> = 2: (<b>a</b>) AUC comparison for <span class="html-italic">t</span> = 2, <span class="html-italic">u</span> = 3; (<b>b</b>) AUC comparison for <span class="html-italic">t</span> = 2, <span class="html-italic">u</span> = 4; (<b>c</b>) AUC comparison for <span class="html-italic">t</span> = 2, <span class="html-italic">u</span> = 5; (<b>d</b>) AUC comparison for <span class="html-italic">t</span> = 2, <span class="html-italic">u</span> = 6.</p> Full article ">Figure 4
<p>AUC comparison of simulated Cox regressions and random forests (N = 500) when <span class="html-italic">t</span> = 3: (<b>a</b>) AUC comparison for <span class="html-italic">t</span> = 3, <span class="html-italic">u</span> = 4; (<b>b</b>) AUC comparison for <span class="html-italic">t</span> = 3, <span class="html-italic">u</span> = 5; (<b>c</b>); AUC comparison for <span class="html-italic">t</span> = 3, <span class="html-italic">u</span> = 6.</p> Full article ">Figure 5
<p>AUC comparison of simulated Cox regressions and random forests (N = 500) when <span class="html-italic">t</span> = 4: (<b>a</b>) AUC comparison for <span class="html-italic">t</span> = 4, <span class="html-italic">u</span> = 5; (<b>b</b>) AUC comparison for <span class="html-italic">t</span> = 4, <span class="html-italic">u</span> = 6.</p> Full article ">Figure 6
<p>AUC comparison of simulated Cox regressions and random forests (N = 500) when <span class="html-italic">t</span> = 5 and <span class="html-italic">u</span> = 6.</p> Full article ">
19 pages, 452 KiB  
Article
How Does Irrigation Affect Crop Growth? A Mathematical Modeling Approach
by Vicente Díaz-González, Alejandro Rojas-Palma and Marcos Carrasco-Benavides
Mathematics 2022, 10(1), 151; https://doi.org/10.3390/math10010151 - 4 Jan 2022
Cited by 7 | Viewed by 3851
Abstract
This article presents a qualitative mathematical model to simulate the relationship between supplied water and plant growth. A novel aspect of the construction of this phenomenological model is the consideration of a structure of three phases: (1) The soil water availability, (2) the [...] Read more.
This article presents a qualitative mathematical model to simulate the relationship between supplied water and plant growth. A novel aspect of the construction of this phenomenological model is the consideration of a structure of three phases: (1) The soil water availability, (2) the available water inside the plant for its growth, and (3) the plant size or amount of dry matter. From these phases and their interactions, a model based on a three-dimensional nonlinear dynamic system was proposed. The results obtained showed the existence of a single equilibrium point, global and exponentially stable. Additionally, considering the framework of the perturbation theory, this model was perturbed by incorporating irrigation to the available soil water, obtaining some stability results under different assumptions. Later through the control theory, it was demonstrated that the proposed system was controllable. Finally, a numerical simulation of the proposed model was carried out, to depict the soil water content and plant growth dynamic and its agreement with the results of the mathematical analysis. In addition, a specific calibration for field data from an experiment with wheat was considered, and these parameters were then used to test the proposed model, obtaining an error of about 6% in the soil water content estimation. Full article
(This article belongs to the Special Issue Mathematics and Its Applications in Science and Engineering)
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<p>Rate of mass gain as a function of size of the plant, considering the following parameter values: <math display="inline"><semantics> <mrow> <mi>ϑ</mi> <mo>=</mo> <mn>0.07</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>. This figure shows that for values of <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>≤</mo> <mn>0</mn> </mrow> </semantics></math> (blue, red; cases (a), and (b)), the gain in dry matter grows rapidly (monotonous growth). Similarly, for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> (green; case (c)), the dry matter gain falls very quickly to zero. For values of <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>0.8</mn> </mrow> </semantics></math> (yellow, case (d)) and <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math> (purple; case (d)) the behavior of the gain is more realistic. In this work, a value of <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> was assumed for the model.</p> Full article ">Figure 2
<p>Simulation of dry mass growth as a function of time, where: <span class="html-italic">n</span> is the factor that allows for modification of the plant’s rapidity of growth in <span class="html-italic">t</span> time (days), and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> is the amount of dry matter (unit of mass). For parameter values <math display="inline"><semantics> <mrow> <mi>n</mi> <mspace width="0.166667em"/> <mo>∈</mo> <mo>{</mo> <mo>−</mo> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>0.8</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>1</mn> <mo>,</mo> <mspace width="0.166667em"/> <mn>2</mn> <mo>}</mo> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>σ</mi> <mo>=</mo> <mn>0.07</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>State dynamics for the system without irrigation, time <span class="html-italic">t</span> in days, where <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (green line), <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (black line), and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (red line) are the water available in the soil, the water inside the plant, and the amount of dry matter, respectively. Parameter values: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> <mi>ω</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>7.0</mn> <mo>,</mo> <mi>v</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>20.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.00001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.000009</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>2.0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Soil water content <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> for the plants’ growth. The horizontal lines represent the soil water thresholds, Saturated soils (Sat, upper line), Field Capacity (FC), and Management Allowed Depletion (MAD, bottom line). The vertical arrows indicate the times when irrigation is applied. Initial conditions <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> <mi>ω</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>7.0</mn> <mo>,</mo> <mi>v</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>20.0</mn> </mrow> </semantics></math>, parameter values <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.00001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.000009</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>2.0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>State dynamics for the system with irrigation, time <span class="html-italic">t</span> in days, where <math display="inline"><semantics> <mrow> <mi>v</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (green line), <math display="inline"><semantics> <mrow> <mi>ω</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (black line), and <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> (red line) are the water available in the soil, the water inside the plant, and the amount of dry matter, respectively. Parameter values: <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>3.0</mn> <mo>,</mo> <mi>ω</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>7.0</mn> <mo>,</mo> <mi>v</mi> <mo>(</mo> <mn>0</mn> <mo>)</mo> <mo>=</mo> <mn>20.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>κ</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>g</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>0.00001</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>δ</mi> <mo>=</mo> <mn>1.0</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>r</mi> <mo>=</mo> <mn>20</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>β</mi> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>k</mi> <mo>=</mo> <mn>0.01</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mn>0.000009</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>ρ</mi> <mo>=</mo> <mn>2.0</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Irrigation function <math display="inline"><semantics> <mrow> <mi>I</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math> in days <span class="html-italic">t</span>. This function is considered bounded, continuous, differentiable, and periodic in order to represent a realistic case. Six watering applications were considered during the season.</p> Full article ">Figure 7
<p>Model fit from the deficit irrigated wheat data (data extracted from Andarzian et al., [<a href="#B28-mathematics-10-00151" class="html-bibr">28</a>].)</p> Full article ">Figure 8
<p>Soil water content trends for modeled and actual data for full irrigated wheat (using the calibrated parameters from <a href="#mathematics-10-00151-t002" class="html-table">Table 2</a>, and measured data from Andarzian et al. [<a href="#B28-mathematics-10-00151" class="html-bibr">28</a>]). Field Capacity (FC), Permanent Wilting Point (PWP), Root Mean Square Error (RMSE), Mean Absolute Error (MAE), and Pearson’s correlation coefficient (r).</p> Full article ">
25 pages, 833 KiB  
Article
Methodology for the Assessment of Imprecise Multi-State System Availability
by Joanna Akrouche, Mohamed Sallak, Eric Châtelet, Fahed Abdallah and Hiba Hajj Chehade
Mathematics 2022, 10(1), 150; https://doi.org/10.3390/math10010150 - 4 Jan 2022
Cited by 1 | Viewed by 1956
Abstract
Most existing studies of a system’s availability in the presence of epistemic uncertainties assume that the system is binary. In this paper, a new methodology for the estimation of the availability of multi-state systems is developed, taking into consideration epistemic uncertainties. This paper [...] Read more.
Most existing studies of a system’s availability in the presence of epistemic uncertainties assume that the system is binary. In this paper, a new methodology for the estimation of the availability of multi-state systems is developed, taking into consideration epistemic uncertainties. This paper formulates a combined approach, based on continuous Markov chains and interval contraction methods, to address the problem of computing the availability of multi-state systems with imprecise failure and repair rates. The interval constraint propagation method, which we refer to as the forward–backward propagation (FBP) contraction method, allows us to contract the probability intervals, keeping all the values that may be consistent with the set of constraints. This methodology is guaranteed, and several numerical examples of systems with complex architectures are studied. Full article
(This article belongs to the Special Issue Probability and Statistics in Quality and Reliability Engineering)
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<p>Markov chain for failure with non-instant repair [<a href="#B44-mathematics-10-00150" class="html-bibr">44</a>].</p> Full article ">Figure 2
<p>Steps of the proposed methodology.</p> Full article ">Figure 3
<p>Markov chain of a system of two binary components.</p> Full article ">Figure 4
<p>The probability intervals for each state after <span class="html-italic">k</span> contractions.</p> Full article ">Figure 5
<p>Case study 1: Flow transmission system (flow direction is left to right).</p> Full article ">Figure 6
<p>Case study1: Markov chain for 18 states.</p> Full article ">Figure 7
<p>The interval availability of the system obtained using each method.</p> Full article ">Figure 8
<p>Case study 2: MSS with eight components (flow direction is left to right).</p> Full article ">Figure 9
<p>Case study 2: The Markov graph for each component in the MSS.</p> Full article ">Figure 10
<p>The unavailability of the system obtained using each method.</p> Full article ">
11 pages, 277 KiB  
Article
The Extended Cone b-Metric-like Spaces over Banach Algebra and Some Applications
by Jerolina Fernandez, Neeraj Malviya, Ana Savić, Marija Paunović and Zoran D. Mitrović
Mathematics 2022, 10(1), 149; https://doi.org/10.3390/math10010149 - 4 Jan 2022
Cited by 7 | Viewed by 2206
Abstract
In this paper, we introduce the structure of extended cone b-metric-like spaces over Banach algebra as a generalization of cone b-metric-like spaces over Banach algebra. In this generalized space we define the notion of generalized Lipschitz mappings in the setup of [...] Read more.
In this paper, we introduce the structure of extended cone b-metric-like spaces over Banach algebra as a generalization of cone b-metric-like spaces over Banach algebra. In this generalized space we define the notion of generalized Lipschitz mappings in the setup of extended cone b-metric-like spaces over Banach algebra and investigated some fixed point results. We also provide examples to illustrate the results presented herein. Finally, as an application of our main result, we examine the existence and uniqueness of solution for a Fredholm integral equation. Full article
(This article belongs to the Special Issue General Metric Spaces: Usage and Applications in Statistics and Fixed Point Theory)
35 pages, 1030 KiB  
Article
New Model of Heteroasociative Min Memory Robust to Acquisition Noise
by Julio César Salgado-Ramírez, Jean Marie Vianney Kinani, Eduardo Antonio Cendejas-Castro, Alberto Jorge Rosales-Silva, Eduardo Ramos-Díaz and Juan Luis Díaz-de-Léon-Santiago
Mathematics 2022, 10(1), 148; https://doi.org/10.3390/math10010148 - 4 Jan 2022
Cited by 4 | Viewed by 1960
Abstract
Associative memories in min and max algebra are of great interest for pattern recognition. One property of these is that they are one-shot, that is, in an attempt they converge to the solution without having to iterate. These memories have proven to be [...] Read more.
Associative memories in min and max algebra are of great interest for pattern recognition. One property of these is that they are one-shot, that is, in an attempt they converge to the solution without having to iterate. These memories have proven to be very efficient, but they manifest some weakness with mixed noise. If an appropriate kernel is not used, that is, a subset of the pattern to be recalled that is not affected by noise, memories fail noticeably. A possible problem for building kernels with sufficient conditions, using binary and gray-scale images, is not knowing how the noise is registered in these images. A solution to this problem is presented by analyzing the behavior of the acquisition noise. What is new about this analysis is that, noise can be mapped to a distance obtained by a distance transform. Furthermore, this analysis provides the basis for a new model of min heteroassociative memory that is robust to the acquisition/mixed noise. The proposed model is novel because min associative memories are typically inoperative to mixed noise. The new model of heteroassocitative memory obtains very interesting results with this type of noise. Full article
(This article belongs to the Special Issue Theory and Applications of Neural Networks)
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<p>Associative memory as black box.</p> Full article ">Figure 2
<p>Additive noise, subtractive noise, and mixed noise, respectively.</p> Full article ">Figure 3
<p>kernel model learning phase.</p> Full article ">Figure 4
<p>kernel model recall phase.</p> Full article ">Figure 5
<p><math display="inline"><semantics> <msub> <mi>d</mi> <mn>4</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>d</mi> <mn>8</mn> </msub> </semantics></math> metrics for the first step.</p> Full article ">Figure 6
<p><math display="inline"><semantics> <msub> <mi>d</mi> <mn>4</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>d</mi> <mn>8</mn> </msub> </semantics></math> metrics for the second step.</p> Full article ">Figure 7
<p>Result of the two steps of the FDT.</p> Full article ">Figure 8
<p>Appearance of the 7-scan process images.</p> Full article ">Figure 9
<p>Noise scheme.</p> Full article ">Figure 10
<p>Learning process of the new model of min heteroassociative memories.</p> Full article ">Figure 11
<p>Recall process of the new model of min heteroassociative memories.</p> Full article ">Figure 12
<p>Absolute and relative frequency distributions of noise acquisition in binary images.</p> Full article ">Figure 13
<p>Binary image with simulated acquisition noise.</p> Full article ">Figure 14
<p>Process generating the noise distribution function.</p> Full article ">Figure 15
<p>Scanned image vs simulated image noise distributions.</p> Full article ">Figure 16
<p>Fundamental sets.</p> Full article ">Figure 17
<p>Operations to build binary kernels.</p> Full article ">Figure 18
<p>Operations to build grayscale kernels.</p> Full article ">
15 pages, 338 KiB  
Article
Sufficient Conditions for Some Stochastic Orders of Discrete Random Variables with Applications in Reliability
by Félix Belzunce, Carolina Martínez-Riquelme and Magdalena Pereda
Mathematics 2022, 10(1), 147; https://doi.org/10.3390/math10010147 - 4 Jan 2022
Cited by 1 | Viewed by 1724
Abstract
In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio [...] Read more.
In this paper we focus on providing sufficient conditions for some well-known stochastic orders in reliability but dealing with the discrete versions of them, filling a gap in the literature. In particular, we find conditions based on the unimodality of the likelihood ratio for the comparison in some stochastic orders of two discrete random variables. These results have interest in comparing discrete random variables because the sufficient conditions are easy to check when there are no closed expressions for the survival functions, which occurs in many cases. In addition, the results are applied to compare several parametric families of discrete distributions. Full article
(This article belongs to the Section D1: Probability and Statistics)
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<p>Likelihood ratio, ratio of survival functions and mean residual life difference for case (a) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>W</mi> <mo>(</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.3</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>W</mi> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.2</mn> <mo>)</mo> </mrow> </semantics></math> (black) and case (b) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>W</mi> <mo>(</mo> <mn>0.75</mn> <mo>,</mo> <mn>0.3</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>W</mi> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>0.2</mn> <mo>)</mo> </mrow> </semantics></math> (blue).</p> Full article ">Figure 2
<p>Likelihood ratio, ratio of survival functions, and mean residual life difference for case (a) (condition (<a href="#FD7-mathematics-10-00147" class="html-disp-formula">7</a>)) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>G</mi> <mi>P</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>0.25</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>G</mi> <mi>P</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math> (black), case (a) (conditions (<a href="#FD8-mathematics-10-00147" class="html-disp-formula">8</a>)) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>G</mi> <mi>P</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>G</mi> <mi>P</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>0.75</mn> <mo>)</mo> </mrow> </semantics></math> (blue), case (b) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>G</mi> <mi>P</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mn>0.3</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>G</mi> <mi>P</mi> <mo>(</mo> <mn>3</mn> <mo>,</mo> <mn>0.75</mn> <mo>)</mo> </mrow> </semantics></math> (red) and case (c) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>G</mi> <mi>P</mi> <mo>(</mo> <mn>2</mn> <mo>,</mo> <mo>.</mo> <mn>0.5</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>G</mi> <mi>P</mi> <mo>(</mo> <mn>1</mn> <mo>,</mo> <mn>0.75</mn> <mo>)</mo> </mrow> </semantics></math> (green).</p> Full article ">Figure 3
<p>Likelihood ratio, ratio of survival functions, and mean residual life difference for case (a.1) (conditions (<a href="#FD11-mathematics-10-00147" class="html-disp-formula">11</a>)) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>H</mi> <mi>L</mi> <mo>(</mo> <mn>0.2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0.7</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>H</mi> <mi>L</mi> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>0.3</mn> <mo>)</mo> </mrow> </semantics></math> (black), case (a.1) (conditions (<a href="#FD12-mathematics-10-00147" class="html-disp-formula">12</a>)) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>H</mi> <mi>L</mi> <mo>(</mo> <mn>0.2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0.7</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>H</mi> <mi>L</mi> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>2</mn> <mo>,</mo> <mn>0.3</mn> <mo>)</mo> </mrow> </semantics></math> (blue), case (b.2) (condition (<a href="#FD14-mathematics-10-00147" class="html-disp-formula">14</a>)) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>H</mi> <mi>L</mi> <mo>(</mo> <mn>0.2</mn> <mo>,</mo> <mn>1</mn> <mo>,</mo> <mn>0.7</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>H</mi> <mi>L</mi> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>2.3</mn> <mo>,</mo> <mn>0.3</mn> <mo>)</mo> </mrow> </semantics></math> (red) and case (b.3) (condition (<a href="#FD16-mathematics-10-00147" class="html-disp-formula">16</a>)) where <math display="inline"><semantics> <mrow> <mi>X</mi> <mo>∼</mo> <mi>H</mi> <mi>L</mi> <mo>(</mo> <mn>0.2</mn> <mo>,</mo> <mn>1.5</mn> <mo>,</mo> <mn>0.7</mn> <mo>)</mo> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>Y</mi> <mo>∼</mo> <mi>H</mi> <mi>L</mi> <mo>(</mo> <mn>0.5</mn> <mo>,</mo> <mn>2.75</mn> <mo>,</mo> <mn>0.3</mn> <mo>)</mo> </mrow> </semantics></math> (green).</p> Full article ">
37 pages, 2547 KiB  
Article
Models of Privacy and Disclosure on Social Networking Sites: A Systematic Literature Review
by Lili Nemec Zlatolas, Luka Hrgarek, Tatjana Welzer and Marko Hölbl
Mathematics 2022, 10(1), 146; https://doi.org/10.3390/math10010146 - 4 Jan 2022
Cited by 9 | Viewed by 4946
Abstract
Social networking sites (SNSs) are used widely, raising new issues in terms of privacy and disclosure. Although users are often concerned about their privacy, they often publish information on social networking sites willingly. Due to the growing number of users of social networking [...] Read more.
Social networking sites (SNSs) are used widely, raising new issues in terms of privacy and disclosure. Although users are often concerned about their privacy, they often publish information on social networking sites willingly. Due to the growing number of users of social networking sites, substantial research has been conducted in recent years. In this paper, we conducted a systematic review of papers that included structural equations models (SEM), or other statistical models with privacy and disclosure constructs. A total of 98 such papers were found and included in the analysis. In this paper, we evaluated the presentation of results of the models containing privacy and disclosure constructs. We carried out an analysis of which background theories are used in such studies and have also found that the studies have not been carried out worldwide. Extending the research to other countries could help with better user awareness of the privacy and self-disclosure of users on SNSs. Full article
(This article belongs to the Special Issue Mathematics and Its Applications in Science and Engineering)
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<p>Flow diagram of the search.</p> Full article ">Figure 2
<p>Number of citations for all publications per year for all papers and number of papers per publication type per year.</p> Full article ">Figure 3
<p>Frequency of theories used in papers by year of publication.</p> Full article ">Figure 4
<p>Number of studies done in each country.</p> Full article ">Figure 5
<p>Coefficient of determination (R<sup>2</sup>) for privacy and the disclosure factor in studies and the number of participants in the studies.</p> Full article ">Figure 6
<p>Average sum of scores per year of publication and theory type, and the average sum of scores for papers with theories or no theory behind the model.</p> Full article ">Figure 7
<p>Average scores for each of the 23 measurement parameters by year of publication.</p> Full article ">
17 pages, 397 KiB  
Article
Approximations of Fuzzy Numbers by Using r-s Piecewise Linear Fuzzy Numbers Based on Weighted Metric
by Haojie Lv and Guixiang Wang
Mathematics 2022, 10(1), 145; https://doi.org/10.3390/math10010145 - 4 Jan 2022
Cited by 5 | Viewed by 1897
Abstract
Using simple fuzzy numbers to approximate general fuzzy numbers is an important research aspect of fuzzy number theory and application. The existing results in this field are basically based on the unweighted metric to establish the best approximation method for solving general fuzzy [...] Read more.
Using simple fuzzy numbers to approximate general fuzzy numbers is an important research aspect of fuzzy number theory and application. The existing results in this field are basically based on the unweighted metric to establish the best approximation method for solving general fuzzy numbers. In order to obtain more objective and reasonable best approximation, in this paper, we use the weighted distance as the evaluation standard to establish a method to solve the best approximation of general fuzzy numbers. Firstly, the conceptions of I-nearest r-s piecewise linear approximation (in short, PLA) and the II-nearest r-s piecewise linear approximation (in short, PLA) are introduced for a general fuzzy number. Then, most importantly, taking weighted metric as a criterion, we obtain a group of formulas to get the I-nearest r-s PLA and the II-nearest r-s PLA. Finally, we also present specific examples to show the effectiveness and usability of the methods proposed in this paper. Full article
(This article belongs to the Section D2: Operations Research and Fuzzy Decision Making)
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<p>(0,0.3,0.6,0.9,1)–(0,0.2,0.5,0.8,1) piecewise linear fuzzy number <math display="inline"><semantics> <mover accent="true"> <mi>u</mi> <mo>˜</mo> </mover> </semantics></math>.</p> Full article ">Figure 2
<p><span class="html-italic">u</span> and <span class="html-italic">I</span>-nearest <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>0</mn> </mrow> <mo>,</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.6</mn> <mo>,</mo> <mn>0.9</mn> <mo>,</mo> <mrow> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math>-KPLA <math display="inline"><semantics> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>l</mi> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 3
<p><span class="html-italic">u</span> and <span class="html-italic">I</span>-nearest <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mn>0</mn> </mrow> <mo>,</mo> <mn>0.3</mn> <mo>,</mo> <mn>0.6</mn> <mo>,</mo> <mn>0.9</mn> <mo>,</mo> <mrow> <mn>1</mn> <mo>)</mo> </mrow> </mrow> </semantics></math> PLA <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>l</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p><span class="html-italic">u</span> and <span class="html-italic">I</span>-nearest (0,0.2,0.5,0.8,1)–(0,0.3,0.6,0.9,1) PLA <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>l</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p><span class="html-italic">u</span> and <span class="html-italic">II</span>-nearest (0,0.2,0.5,0.8,1)–(0.3,0.6,0.9,1) PLA <math display="inline"><semantics> <mrow> <msub> <mi>u</mi> <mrow> <mi>p</mi> <mi>l</mi> </mrow> </msub> </mrow> </semantics></math>.</p> Full article ">
14 pages, 364 KiB  
Article
Stochastic Approximate Algorithms for Uncertain Constrained K-Means Problem
by Jianguang Lu, Juan Tang, Bin Xing and Xianghong Tang
Mathematics 2022, 10(1), 144; https://doi.org/10.3390/math10010144 - 4 Jan 2022
Viewed by 1616
Abstract
The k-means problem has been paid much attention for many applications. In this paper, we define the uncertain constrained k-means problem and propose a (1+ϵ)-approximate algorithm for the problem. First, a general mathematical model of the [...] Read more.
The k-means problem has been paid much attention for many applications. In this paper, we define the uncertain constrained k-means problem and propose a (1+ϵ)-approximate algorithm for the problem. First, a general mathematical model of the uncertain constrained k-means problem is proposed. Second, the random sampling properties of the uncertain constrained k-means problem are studied. This paper mainly studies the gap between the center of random sampling and the real center, which should be controlled within a given range with a large probability, so as to obtain the important sampling properties to solve this kind of problem. Finally, using mathematical induction, we assume that the first j−1 cluster centers are obtained, so we only need to solve the j-th center. The algorithm has the elapsed time O((1891ekϵ2)8k/ϵnd), and outputs a collection of size O((1891ekϵ2)8k/ϵn) of candidate sets including approximation centers. Full article
(This article belongs to the Special Issue Advanced Mathematical Methods in Intelligent Multimedia: Security and Applications)
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<p>Flow chart of our algorithm <math display="inline"><semantics> <mi mathvariant="bold">cMeans</mi> </semantics></math>.</p> Full article ">
21 pages, 438 KiB  
Article
Quantum and Classical Log-Bounded Automata for the Online Disjointness Problem
by Kamil Khadiev and Aliya Khadieva
Mathematics 2022, 10(1), 143; https://doi.org/10.3390/math10010143 - 4 Jan 2022
Cited by 9 | Viewed by 1802
Abstract
We consider online algorithms with respect to the competitive ratio. In this paper, we explore one-way automata as a model for online algorithms. We focus on quantum and classical online algorithms. For a specially constructed online minimization problem, we provide a quantum log-bounded [...] Read more.
We consider online algorithms with respect to the competitive ratio. In this paper, we explore one-way automata as a model for online algorithms. We focus on quantum and classical online algorithms. For a specially constructed online minimization problem, we provide a quantum log-bounded automaton that is more effective (has less competitive ratio) than classical automata, even with advice, in the case of the logarithmic size of memory. We construct an online version of the well-known Disjointness problem as a problem. It was investigated by many researchers from a communication complexity and query complexity point of view. Full article
(This article belongs to the Special Issue Quantum, Molecular and Unconventional Computing)
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<p>The structure of an input for <math display="inline"><semantics> <msub> <mi mathvariant="monospace">onlineDISJ</mi> <mrow> <mi>t</mi> <mo>,</mo> <mi>k</mi> <mo>,</mo> <mi>r</mi> <mo>,</mo> <mi>w</mi> </mrow> </msub> </semantics></math>.</p> Full article ">
16 pages, 307 KiB  
Article
An Analysis and Comparison of Multi-Factor Asset Pricing Model Performance during Pandemic Situations in Developed and Emerging Markets
by Konstantin B. Kostin, Philippe Runge and Michel Charifzadeh
Mathematics 2022, 10(1), 142; https://doi.org/10.3390/math10010142 - 4 Jan 2022
Cited by 6 | Viewed by 4900
Abstract
This study empirically analyzes and compares return data from developed and emerging market data based on the Fama French five-factor model and compares it to previous results from the Fama French three-factor model by Kostin, Runge and Adams (2021). It researches whether the [...] Read more.
This study empirically analyzes and compares return data from developed and emerging market data based on the Fama French five-factor model and compares it to previous results from the Fama French three-factor model by Kostin, Runge and Adams (2021). It researches whether the addition of the profitability and investment pattern factors show superior results in the assessment of emerging markets during the COVID-19 pandemic compared to developed markets. We use panel data covering eight indices of developed and emerging countries as well as a selection of eight companies from these markets, covering a period from 2000 to 2020. Our findings suggest that emerging markets do not generally outperform developed markets. The results underscore the need to reconsider the assumption that adding more factors to regression models automatically yields results that are more reliable. Our study contributes to the extant literature by broadening this research area. It is the first study to compare the performance of the Fama French three-factor model and the Fama French five-factor model in the cost of equity calculation for developed and emerging countries during the COVID-19 pandemic and other crisis events of the past two decades. Full article
(This article belongs to the Special Issue Mathematics and Financial Economics)
12 pages, 324 KiB  
Article
When Is σ (A(t)) ⊂ {z ∈ ℂ; ℜz ≤ −α < 0} the Sufficient Condition for Uniform Asymptotic Stability of LTV System ẋ = A(t)x?
by Robert Vrabel
Mathematics 2022, 10(1), 141; https://doi.org/10.3390/math10010141 - 4 Jan 2022
Viewed by 1725
Abstract
In this paper, the class of matrix functions A(t) is determined for which the condition that the pointwise spectrum σ(A(t))⊂z∈C;ℜz≤−α for all [...] Read more.
In this paper, the class of matrix functions A(t) is determined for which the condition that the pointwise spectrum σ(A(t))⊂z∈C;ℜz≤−α for all t≥t0 and some α>0 is sufficient for uniform asymptotic stability of the linear time-varying system x˙=A(t)x. We prove that this class contains as a proper subset the matrix functions with the values in the special orthogonal group SO(n). Full article
(This article belongs to the Section E: Applied Mathematics)
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<p>Simulation result for Example 2 with the initial state <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>2</mn> <mspace width="4pt"/> <mn>0</mn> <mspace width="4pt"/> <mn>2</mn> <mo stretchy="false">)</mo> </mrow> <mi>T</mi> </msup> <mo>.</mo> </mrow> </semantics></math> Flow of the state−variables <math display="inline"><semantics> <mrow> <msub> <mi>x</mi> <mn>1</mn> </msub> <mo>,</mo> </mrow> </semantics></math><math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mn>3</mn> </msub> </semantics></math>.</p> Full article ">Figure 2
<p>Simulation result for Example 3 with <math display="inline"><semantics> <mrow> <mi>β</mi> <mo stretchy="false">(</mo> <mi>t</mi> <mo stretchy="false">)</mo> <mo>=</mo> <mo form="prefix">cos</mo> <mi>t</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>γ</mi> <mo>=</mo> <mo>−</mo> <mo form="prefix">sin</mo> <mi>t</mi> </mrow> </semantics></math> and the initial state <math display="inline"><semantics> <mrow> <mi>x</mi> <mrow> <mo stretchy="false">(</mo> <mn>0</mn> <mo stretchy="false">)</mo> </mrow> <mo>=</mo> <msup> <mrow> <mo stretchy="false">(</mo> <mo>−</mo> <mn>1</mn> <mspace width="4pt"/> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <mi>T</mi> </msup> <mo>.</mo> </mrow> </semantics></math> Flow of the state−variables <math display="inline"><semantics> <msub> <mi>x</mi> <mn>1</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>x</mi> <mn>2</mn> </msub> </semantics></math>.</p> Full article ">
25 pages, 6093 KiB  
Article
Optimal Sizing of Stand-Alone Microgrids Based on Recent Metaheuristic Algorithms
by Ahmed A. Zaki Diab, Ali M. El-Rifaie, Magdy M. Zaky and Mohamed A. Tolba
Mathematics 2022, 10(1), 140; https://doi.org/10.3390/math10010140 - 4 Jan 2022
Cited by 19 | Viewed by 2700
Abstract
Scientists have been paying more attention to the shortage of water and energy sources all over the world, especially in the Middle East and North Africa (MENA). In this article, a microgrid configuration of a photovoltaic (PV) plant with fuel cell (FC) and [...] Read more.
Scientists have been paying more attention to the shortage of water and energy sources all over the world, especially in the Middle East and North Africa (MENA). In this article, a microgrid configuration of a photovoltaic (PV) plant with fuel cell (FC) and battery storage systems has been optimally designed. A real case study in Egypt in Dobaa region of supplying safety loads at a nuclear power plant during emergency cases is considered, where the load characteristics and the location data have been taken into consideration. Recently, many optimization algorithms have been developed by researchers, and these algorithms differ from one another in their performance and effectiveness. On the other hand, there are recent optimization algorithms that were not used to solve the problem of microgrids design in order to evaluate their performance and effectiveness. Optimization algorithms of equilibrium optimizer (EQ), bat optimization (BAT), and black-hole-based optimization (BHB) algorithms have been applied and compared in this paper. The optimization algorithms are individually used to optimize and size the energy systems to minimize the cost. The energy systems have been modeled and evaluated using MATLAB. Full article
(This article belongs to the Topic Multi-Energy Systems)
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<p>Arrangement of the studied microgrid.</p> Full article ">Figure 2
<p>Scenarios of energy management flowchart.</p> Full article ">Figure 3
<p>Flowchart of BAT technique.</p> Full article ">Figure 4
<p>Flowchart of BHB technique.</p> Full article ">Figure 5
<p>Flowchart of EQ Technique.</p> Full article ">Figure 6
<p>Location of the studied microgrid.</p> Full article ">Figure 7
<p>Solar radiation for the study area.</p> Full article ">Figure 8
<p>Temperature for the study area.</p> Full article ">Figure 9
<p>The average load demand per month.</p> Full article ">Figure 10
<p>The load curve per day of the study area.</p> Full article ">Figure 11
<p>Indices of the energy system based on various algorithms.</p> Full article ">Figure 12
<p>Configuration of the energy system based on various algorithms.</p> Full article ">Figure 13
<p>Convergence curves of the three algorithms.</p> Full article ">Figure 14
<p>Convergence curves of the three algorithms over 30 runs: (<b>a</b>) 30-run convergence curves of BAT, (<b>b</b>) 30-run convergence curves of EQ, and (<b>c</b>) 30-run convergence curves of BHB.</p> Full article ">Figure 15
<p>Box plots of the three algorithms over 30 runs: (<b>a</b>) box plot of BAT, (<b>b</b>) box plot of EQ, and (<b>c</b>) box plot of BHB.</p> Full article ">Figure 16
<p>The results for the operation of the microgrid over one year considering the optimal configuration based on EQ technique. (<b>a</b>) Load, PV, and the different power. (<b>b</b>) Performance of the charging and discharging of the storage units for the battery and FC. (<b>c</b>) Dummy load and LPSP.</p> Full article ">Figure 17
<p>Numeric results of the microgrid operation for one working day via optimal configuration using EQ technique. (<b>a</b>) Load, PV, and the different power. (<b>b</b>) Performance of the charging and discharging of the storage units for the battery and FC. (<b>c</b>) Dummy load and LPSP.</p> Full article ">
16 pages, 2415 KiB  
Article
Cascaded Cross-Layer Fusion Network for Pedestrian Detection
by Zhifeng Ding, Zichen Gu, Yanpeng Sun and Xinguang Xiang
Mathematics 2022, 10(1), 139; https://doi.org/10.3390/math10010139 - 4 Jan 2022
Cited by 5 | Viewed by 1912
Abstract
The detection method based on anchor-free not only reduces the training cost of object detection, but also avoids the imbalance problem caused by an excessive number of anchors. However, these methods only pay attention to the impact of the detection head on the [...] Read more.
The detection method based on anchor-free not only reduces the training cost of object detection, but also avoids the imbalance problem caused by an excessive number of anchors. However, these methods only pay attention to the impact of the detection head on the detection performance, thus ignoring the impact of feature fusion on the detection performance. In this article, we take pedestrian detection as an example and propose a one-stage network Cascaded Cross-layer Fusion Network (CCFNet) based on anchor-free. It consists of Cascaded Cross-layer Fusion module (CCF) and novel detection head. Among them, CCF fully considers the distribution of high-level information and low-level information of feature maps under different stages in the network. First, the deep network is used to remove a large amount of noise in the shallow features, and finally, the high-level features are reused to obtain a more complete feature representation. Secondly, for the pedestrian detection task, a novel detection head is designed, which uses the global smooth map (GSMap) to provide global information for the center map to obtain a more accurate center map. Finally, we verified the feasibility of CCFNet on the Caltech and CityPersons datasets. Full article
(This article belongs to the Special Issue Advancement of Mathematical Methods in Feature Representation Learning for Artificial Intelligence, Data Mining and Robotics)
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<p>The overall structure of Cascaded Cross-layer Fusion Network (CCFNet). It includes two parts: CCF module and detection head. CCF cascades and reuses features to generate low-level feature maps with contextual semantic information. This feature map generates center map, scale map, and global smooth map through the detection head. And generate the new center map with global information by integrating center map and global smooth map. Finally, locate and mark the objects.</p> Full article ">Figure 2
<p>Cascaded Cross-layer Fusion Module (CCF).</p> Full article ">Figure 3
<p>The overall architecture of the detection head mainly includes three map components, namely the center map, the scale map and the global smooth map (GSMap).</p> Full article ">Figure 4
<p>The histogram and pie chart represent the distribution statistics of each category in the Caltech and CityPersons datasets. (<b>a</b>) represents the label distribution of the training set in the Caltech dataset. (<b>b</b>) represents the label distribution of the test set in the Caltech dataset. (<b>c</b>) represents the label distribution of the training set in the CityPersons dataset. (<b>d</b>) represents the label distribution of the validation set in the CityPersons dataset.</p> Full article ">Figure 5
<p>The results of various models on the Caltech dataset. (<b>a</b>) Compare with existing methods on Reasonable subset. (<b>b</b>) Compare with existing methods on the Reasonable_Occ=Heavy subset.</p> Full article ">Figure 6
<p>Visualization results of CCFNet and CSP do not limit the visibility of pedestrian objects. (<b>a</b>) Input the original image for the CityPersons dataset; (<b>b</b>) is the ground truth corresponding to (<b>a</b>); (<b>c</b>) is the visualization result of CSP; (<b>d</b>) is the visualization result of CCFNet.</p> Full article ">Figure 7
<p>Visualization results of ACFPN, CSP, and CCFNet. (<b>a</b>) Input the original image for the CityPersons dataset; (<b>b</b>) is the visualization result of ACFPN; (<b>c</b>) is the visualization result of CSP; (<b>d</b>) is the visualization result of CCFNet.</p> Full article ">
11 pages, 1365 KiB  
Article
Numerical Simulation of Cubic-Quartic Optical Solitons with Perturbed Fokas–Lenells Equation Using Improved Adomian Decomposition Algorithm
by Alyaa A. Al-Qarni, Huda O. Bakodah, Aisha A. Alshaery, Anjan Biswas, Yakup Yıldırım, Luminita Moraru and Simona Moldovanu
Mathematics 2022, 10(1), 138; https://doi.org/10.3390/math10010138 - 4 Jan 2022
Cited by 8 | Viewed by 1905
Abstract
The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known [...] Read more.
The current manuscript displays elegant numerical results for cubic-quartic optical solitons associated with the perturbed Fokas–Lenells equations. To do so, we devise a generalized iterative method for the model using the improved Adomian decomposition method (ADM) and further seek validation from certain well-known results in the literature. As proven, the proposed scheme is efficient and possess a high level of accuracy. Full article
(This article belongs to the Special Issue Nonlinear Partial Differential Equations: Exact Solutions, Symmetries, Methods, and Applications)
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<p>Comparison of the exact and improved ADM solutions for case 1 for −50 ≤ <span class="html-italic">x</span> ≤ 50.</p> Full article ">Figure 2
<p>Comparison of the exact and improved ADM solutions for case 2 for −50 ≤ <span class="html-italic">x</span> ≤ 50.</p> Full article ">Figure 3
<p>Comparison of the exact and improved ADM solutions for case 3 for −50 ≤ <span class="html-italic">x</span> ≤ 50.</p> Full article ">
15 pages, 1679 KiB  
Article
Developing a Constructive Conceptual Framework of a Pre-Service Mathematics Teachers’ Content Knowledge Instrument on Space and Shape
by Rooselyna Ekawati, Masriyah, Abdul Haris Rosyidi, Budi Priyo Prawoto, Rully Charitas Indra Prahmana and Fou-Lai Lin
Mathematics 2022, 10(1), 137; https://doi.org/10.3390/math10010137 - 4 Jan 2022
Cited by 1 | Viewed by 2698
Abstract
Space and shape is one of the geometry topics that should be mastered by students and require proper teachers’ Mathematics Content Knowledge (MCK) for teaching to avoid misconception. This study aimed at developing a constructive conceptual framework as an instrument to examine mathematics [...] Read more.
Space and shape is one of the geometry topics that should be mastered by students and require proper teachers’ Mathematics Content Knowledge (MCK) for teaching to avoid misconception. This study aimed at developing a constructive conceptual framework as an instrument to examine mathematics pre-service teachers’ MCK on space and shape contents and describing their profile on this topic. The present study used mixed methods, in which the obtained data were analyzed both quantitatively using Exploratory Factor Analysis (EFA) and qualitatively described in nature. The developed MCK instrument was administered to 21 senior Indonesian mathematics pre-service teachers who were in their third year of study which and by a purposive sampling technique. The results showed that the instrument had very good 10 final items with a consistent reliability coefficient of 0.67 and resulted in four factor components, namely, figural representation, area and circumference of object, relationship between properties of objects, and figural reasoning. Of the four factors, figural representation and reasoning factors had mostly been the challenges for Indonesian mathematics pre-service teachers. On the contrary, they performed better in the area and circumference of objects and the relationships between properties of objects. The findings lead to redesigning the curriculum for mathematics pre-service teachers’ learning to accommodate all their challenges. Full article
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<p>MCK 15 Items.</p> Full article ">Figure 2
<p>Mean score of items in each factor.</p> Full article ">Figure 3
<p>Alternative answers of pre-service teachers on MCK 10 (F2).</p> Full article ">Figure 4
<p>MCK 11: (<b>A</b>–<b>C</b>) are prism with equal volume and (<b>D</b>) prism is different volume.</p> Full article ">
7 pages, 251 KiB  
Article
On the Correlation between Banach Contraction Principle and Caristi’s Fixed Point Theorem in b-Metric Spaces
by Salvador Romaguera
Mathematics 2022, 10(1), 136; https://doi.org/10.3390/math10010136 - 3 Jan 2022
Cited by 6 | Viewed by 1972
Abstract
We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper “A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces”. We also characterize the completeness of b-metric spaces with the help of a [...] Read more.
We solve a question posed by E. Karapinar, F. Khojasteh and Z.D. Mitrović in their paper “A Proposal for Revisiting Banach and Caristi Type Theorems in b-Metric Spaces”. We also characterize the completeness of b-metric spaces with the help of a variant of the contractivity condition introduced by the authors in the aforementioned article. Full article
(This article belongs to the Special Issue New Advances in Fuzzy Metric Spaces, Soft Metric Spaces, and Other Related Structures)
12 pages, 558 KiB  
Article
Unified Convergence Analysis of Chebyshev–Halley Methods for Multiple Polynomial Zeros
by Stoil I. Ivanov
Mathematics 2022, 10(1), 135; https://doi.org/10.3390/math10010135 - 3 Jan 2022
Cited by 7 | Viewed by 2206
Abstract
In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. [...] Read more.
In this paper, we establish two local convergence theorems that provide initial conditions and error estimates to guarantee the Q-convergence of an extended version of Chebyshev–Halley family of iterative methods for multiple polynomial zeros due to Osada (J. Comput. Appl. Math. 2008, 216, 585–599). Our results unify and complement earlier local convergence results about Halley, Chebyshev and Super–Halley methods for multiple polynomial zeros. To the best of our knowledge, the results about the Osada’s method for multiple polynomial zeros are the first of their kind in the literature. Moreover, our unified approach allows us to compare the convergence domains and error estimates of the mentioned famous methods and several new randomly generated methods. Full article
(This article belongs to the Special Issue Numerical Analysis and Scientific Computing II)
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<p>Graph of the functions <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mn>1</mn> <mo>/</mo> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mi>α</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.285</mn> <mo>+</mo> <mn>0.006</mn> <mi>i</mi> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Graph of the functions <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mn>1</mn> <mo>/</mo> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mi>α</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>3</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>0.874</mn> <mo>−</mo> <mn>0.097</mn> <mi>i</mi> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Graph of the functions <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mn>1</mn> <mo>/</mo> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mi>α</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mo>−</mo> <mn>0.487</mn> <mo>−</mo> <mn>0.083</mn> <mi>i</mi> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>Graph of the functions <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>0</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mn>1</mn> <mo>/</mo> <mn>2</mn> </mrow> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mn>1</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mrow> <mn>1</mn> <mo>/</mo> <mo>(</mo> <mn>1</mn> <mo>−</mo> <mi>m</mi> <mo>)</mo> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>ϕ</mi> <mi>α</mi> </msub> </semantics></math> for <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>10</mn> </mrow> </semantics></math>, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>2</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>α</mi> <mo>=</mo> <mn>1.016</mn> <mo>+</mo> <mn>0.030</mn> <mi>i</mi> </mrow> </semantics></math>.</p> Full article ">
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