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Mathematics
Volume 9
Issue 5
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Mathematics, Volume 9, Issue 5 (March-1 2021) – 128 articles

Cover Story (view full-size image): Pantograph-type equations are used in biology, astrophysics, electrodynamics, number theory, graph theory, stochastic games, theory of neural networks, etc. The analogy principle allows one to construct exact solutions for nonlinear pantograph-type PDEs using solutions of simpler PDEs. Exact solutions of such PDEs are described for the first time. View this paper.
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21 pages, 3771 KiB  
Article
The Analysis of a Model–Task Dyad in Two Settings: Zaplify and Pencil and Paper
by Canan Güneş
Mathematics 2021, 9(5), 581; https://doi.org/10.3390/math9050581 - 9 Mar 2021
Cited by 1 | Viewed by 2044
Abstract
This paper examines the added value of a digital tool that constitutes a new model to introduce students to multiplication. Drawing on the theory of semiotic mediation, the semiotic potential of this new model is analysed with respect to the same task that [...] Read more.
This paper examines the added value of a digital tool that constitutes a new model to introduce students to multiplication. Drawing on the theory of semiotic mediation, the semiotic potential of this new model is analysed with respect to the same task that can be solved in two different settings (the digital tool and pencil and paper). The analysis shows that the task solutions undergo significant changes depending on to the technological settings. Even though the end product of the model–task dyads might look the same in both settings, the product emerges from the different processes that would mediate quite different meanings for multiplication. This suggests that while designing tasks that involve mathematical models, rather than focusing only on the end product, considering the whole process would reveal the extensive potential meanings the model–task dyad can mediate. Full article
(This article belongs to the Special Issue Designing Tasks within Dynamic and Interactive Mathematics Learning Environments)
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<p>RAM of multiplication 4 <span class="html-italic">×</span> 3.</p> Full article ">Figure 2
<p>The points in a 3 <span class="html-italic">×</span> 4 array.</p> Full article ">Figure 3
<p>The representation of a multiplicative situation in Vergnaud’s T-Table.</p> Full article ">Figure 4
<p>The representation of inclusion relations:(<b>a</b>) in repeated addition (<b>b</b>) in multiplication (The diagram is reimagined based on [<a href="#B18-mathematics-09-00581" class="html-bibr">18</a>] and [<a href="#B19-mathematics-09-00581" class="html-bibr">19</a>]).</p> Full article ">Figure 5
<p>(<b>a</b>) Fingerprints, (<b>b</b>) Fingerprints and the diagonal line, (<b>c</b>) The diagonal line.</p> Full article ">Figure 6
<p>(<b>a</b>) Vertical lines, (<b>b</b>) horizontal lines (<b>c</b>) both horizontal and vertical lines.</p> Full article ">Figure 7
<p>(<b>a</b>–<b>c</b>): Adding a vertical line in locked mode.</p> Full article ">Figure 8
<p>Examples of M-ples.</p> Full article ">Figure 9
<p>An excerpt from the transcription organized in a table.</p> Full article ">Figure 10
<p>(<b>a</b>–<b>c</b>) The three steps of drawing an M-ple.</p> Full article ">Figure 11
<p>An example of gradual increase in the number of points.</p> Full article ">Figure 12
<p>(<b>a</b>) Four horizontal lines and the numeral “4”, (<b>b</b>) One four-ple on the vertical line (The green arrows are added on the figures to illustrate the path the numerals follow).</p> Full article ">Figure 13
<p>(<b>a</b>) 1 four-ple; (<b>b</b>) 9 four-ples.</p> Full article ">Figure 14
<p>(<b>a</b>) Making the 49th; (<b>b</b>) the 50th four-ples in Zaplify.</p> Full article ">
16 pages, 446 KiB  
Article
Random Sampling Many-Dimensional Sets Arising in Control
by Pavel Shcherbakov, Mingyue Ding and Ming Yuchi
Mathematics 2021, 9(5), 580; https://doi.org/10.3390/math9050580 - 9 Mar 2021
Viewed by 2041
Abstract
Various Monte Carlo techniques for random point generation over sets of interest are widely used in many areas of computational mathematics, optimization, data processing, etc. Whereas for regularly shaped sets such sampling is immediate to arrange, for nontrivial, implicitly specified domains these techniques [...] Read more.
Various Monte Carlo techniques for random point generation over sets of interest are widely used in many areas of computational mathematics, optimization, data processing, etc. Whereas for regularly shaped sets such sampling is immediate to arrange, for nontrivial, implicitly specified domains these techniques are not easy to implement. We consider the so-called Hit-and-Run algorithm, a representative of the class of Markov chain Monte Carlo methods, which became popular in recent years. To perform random sampling over a set, this method requires only the knowledge of the intersection of a line through a point inside the set with the boundary of this set. This component of the Hit-and-Run procedure, known as boundary oracle, has to be performed quickly when applied to economy point representation of many-dimensional sets within the randomized approach to data mining, image reconstruction, control, optimization, etc. In this paper, we consider several vector and matrix sets typically encountered in control and specified by linear matrix inequalities. Closed-form solutions are proposed for finding the respective points of intersection, leading to efficient boundary oracles; they are generalized to robust formulations where the system matrices contain norm-bounded uncertainty. Full article
(This article belongs to the Special Issue Machine Learning and Data Mining in Pattern Recognition)
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<p>The Hit-and-Run scheme.</p> Full article ">Figure 2
<p>Feasible and robustly feasible linear matrix inequalities (LMI) domains.</p> Full article ">Figure 3
<p>The robust boundary oracle.</p> Full article ">
21 pages, 4466 KiB  
Article
RiskLogitboost Regression for Rare Events in Binary Response: An Econometric Approach
by Jessica Pesantez-Narvaez, Montserrat Guillen and Manuela Alcañiz
Mathematics 2021, 9(5), 579; https://doi.org/10.3390/math9050579 - 9 Mar 2021
Cited by 2 | Viewed by 2555
Abstract
A boosting-based machine learning algorithm is presented to model a binary response with large imbalance, i.e., a rare event. The new method (i) reduces the prediction error of the rare class, and (ii) approximates an econometric model that allows interpretability. RiskLogitboost regression includes [...] Read more.
A boosting-based machine learning algorithm is presented to model a binary response with large imbalance, i.e., a rare event. The new method (i) reduces the prediction error of the rare class, and (ii) approximates an econometric model that allows interpretability. RiskLogitboost regression includes a weighting mechanism that oversamples or undersamples observations according to their misclassification likelihood and a generalized least squares bias correction strategy to reduce the prediction error. An illustration using a real French third-party liability motor insurance data set is presented. The results show that RiskLogitboost regression improves the rate of detection of rare events compared to some boosting-based and tree-based algorithms and some existing methods designed to treat imbalanced responses. Full article
(This article belongs to the Special Issue Advances in Multivariate Analysis and Their Applications in Actuarial and Financial Economics)
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<p>Plot of weights versus estimated probabilities of the Logitboost and the RiskLogitboost regression.</p> Full article ">Figure 2
<p>The highest and lowest prediction scores for all observed response <span class="html-italic">Y</span> within 50 iterations (<span class="html-italic">D</span> = 50) obtained with the RiskLogitboost regression.</p> Full article ">Figure 3
<p>Partial dependence plots from the Boosting Tree. Abbreviations: B-N (Basse-Normandie), Ile (Ile-de-France), N.C. (Nord-Pas-de-Calais), Pays (Pays-de-la-Loire), Poitu (Poitou-Charentes), Japanese (Japanese (except Nissan) or Korean), M/C/B (Mercedes, Chrysler or BMW), V/A/S/S (Volkswagen, Audi, Skoda or Seat), Opel (Opel, General Motors or Ford).</p> Full article ">
16 pages, 370 KiB  
Article
A Topological View of Reed–Solomon Codes
by Alberto Besana and Cristina Martínez
Mathematics 2021, 9(5), 578; https://doi.org/10.3390/math9050578 - 9 Mar 2021
Viewed by 2105
Abstract
We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the [...] Read more.
We studied a particular class of well known error-correcting codes known as Reed–Solomon codes. We constructed RS codes as algebraic-geometric codes from the normal rational curve. This approach allowed us to study some algebraic representations of RS codes through the study of the general linear group GL(n,q). We characterized the coefficients that appear in the decompostion of an irreducible representation of the special linear group in terms of Gromov–Witten invariants of the Hilbert scheme of points in the plane. In addition, we classified all the algebraic codes defined over the normal rational curve, thereby providing an algorithm to compute a set of generators of the ideal associated with any algebraic code constructed on the rational normal curve (NRC) over an extension Fqn of Fq. Full article
(This article belongs to the Special Issue New Trends in Algebraic Geometry and Its Applications)
15 pages, 1960 KiB  
Article
Bayesian Analysis of Population Health Data
by Dorota Młynarczyk, Carmen Armero, Virgilio Gómez-Rubio and Pedro Puig
Mathematics 2021, 9(5), 577; https://doi.org/10.3390/math9050577 - 9 Mar 2021
Cited by 4 | Viewed by 3080
Abstract
The analysis of population-wide datasets can provide insight on the health status of large populations so that public health officials can make data-driven decisions. The analysis of such datasets often requires highly parameterized models with different types of fixed and random effects to [...] Read more.
The analysis of population-wide datasets can provide insight on the health status of large populations so that public health officials can make data-driven decisions. The analysis of such datasets often requires highly parameterized models with different types of fixed and random effects to account for risk factors, spatial and temporal variations, multilevel effects and other sources on uncertainty. To illustrate the potential of Bayesian hierarchical models, a dataset of about 500,000 inhabitants released by the Polish National Health Fund containing information about ischemic stroke incidence for a 2-year period is analyzed using different types of models. Spatial logistic regression and survival models are considered for analyzing the individual probabilities of stroke and the times to the occurrence of an ischemic stroke event. Demographic and socioeconomic variables as well as drug prescription information are available at an individual level. Spatial variation is considered by means of region-level random effects. Full article
(This article belongs to the Special Issue Quantitative Methods in Health Care Decisions)
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<p>Posterior mean for the conditional <math display="inline"><semantics> <mrow> <mi>i</mi> <mo>.</mo> <mi>i</mi> <mo>.</mo> <mi>d</mi> </mrow> </semantics></math> random variables in LOGIT IID and SURVIVAL IID models, and for spatially correlated random variables in LOGIT SPATIAL and SURVIVAL models.</p> Full article ">Figure 2
<p>Estimated probability of stroke by gender and age group based on the LOGIT SPATIAL model.</p> Full article ">Figure 3
<p>Estimated probability of stroke by gender and age group based on the LOGIT SPATIAL model assuming that drugs for the cardiovascular system (ATC C) have been prescribed.</p> Full article ">
20 pages, 2211 KiB  
Article
Finite Element Based Overall Optimization of Switched Reluctance Motor Using Multi-Objective Genetic Algorithm (NSGA-II)
by Mohamed El-Nemr, Mohamed Afifi, Hegazy Rezk and Mohamed Ibrahim
Mathematics 2021, 9(5), 576; https://doi.org/10.3390/math9050576 - 8 Mar 2021
Cited by 24 | Viewed by 3224
Abstract
The design of switched reluctance motor (SRM) is considered a complex problem to be solved using conventional design techniques. This is due to the large number of design parameters that should be considered during the design process. Therefore, optimization techniques are necessary to [...] Read more.
The design of switched reluctance motor (SRM) is considered a complex problem to be solved using conventional design techniques. This is due to the large number of design parameters that should be considered during the design process. Therefore, optimization techniques are necessary to obtain an optimal design of SRM. This paper presents an optimal design methodology for SRM using the non-dominated sorting genetic algorithm (NSGA-II) optimization technique. Several dimensions of SRM are considered in the proposed design procedure including stator diameter, bore diameter, axial length, pole arcs and pole lengths, back iron length, shaft diameter as well as the air gap length. The multi-objective design scheme includes three objective functions to be achieved, that is, maximum average torque, maximum efficiency and minimum iron weight of the machine. Meanwhile, finite element analysis (FEA) is used during the optimization process to calculate the values of the objective functions. In this paper, two designs for SRMs with 8/6 and 6/4 configurations are presented. Simulation results show that the obtained SRM design parameters allow better average torque and efficiency with lower iron weight. Eventually, the integration of NSGA-II and FEA provides an effective approach to obtain the optimal design of SRM. Full article
(This article belongs to the Special Issue Mathematical Approaches to Modeling, Optimally Designing, and Controlling Electric Machine)
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<p>Design process flow chart.</p> Full article ">Figure 2
<p>Lamination dimensions considered in optimization process.</p> Full article ">Figure 3
<p>NSGA-II optimization program flowchart.</p> Full article ">Figure 4
<p>Objective functions results for 8/6 SRM.</p> Full article ">Figure 5
<p>Objective functions results for 6/4 SRM.</p> Full article ">Figure 6
<p>Objective functions 3D representation of the last generation (30 candidates).</p> Full article ">Figure 7
<p>Objective functions progress with number of generations.</p> Full article ">Figure 8
<p>Developed torque of one phase for different excitation levels.</p> Full article ">Figure 9
<p>Flux density of selected optimal designs in an aligned position.</p> Full article ">Figure 10
<p>Efficiency of selected optimal designs A1 (8/6 SRM) and B1 (6/4 SRM) at different speeds.</p> Full article ">Figure 11
<p>Core losses of selected optimal designs A1 (8/6 SRM) and B1 (6/4 SRM) at different speeds.</p> Full article ">Figure 12
<p>The flux density waveforms in all sectors of optimal SRM designs A1 (8/6 SRM) and B1 (6/4 SRM) for one revolution at 1000 rpm.</p> Full article ">
23 pages, 1057 KiB  
Article
On the Computation of Some Interval Reliability Indicators for Semi-Markov Systems
by Guglielmo D’Amico, Raimondo Manca, Filippo Petroni and Dharmaraja Selvamuthu
Mathematics 2021, 9(5), 575; https://doi.org/10.3390/math9050575 - 8 Mar 2021
Cited by 6 | Viewed by 1915
Abstract
In this paper, we computed general interval indicators of availability and reliability for systems modelled by time non-homogeneous semi-Markov chains. First, we considered duration-dependent extensions of the Interval Reliability and then, we determined an explicit formula for the availability with a given window [...] Read more.
In this paper, we computed general interval indicators of availability and reliability for systems modelled by time non-homogeneous semi-Markov chains. First, we considered duration-dependent extensions of the Interval Reliability and then, we determined an explicit formula for the availability with a given window and containing a given point. To make the computation of the window availability, an explicit formula was derived involving duration-dependent transition probabilities and the interval reliability function. Both interval reliability and availability functions were evaluated considering the local behavior of the system through the recurrence time processes. The results are illustrated through a numerical example. They show that the considered indicators can describe the duration effects and the age of the multi-state system and be useful in real-life problems. Full article
(This article belongs to the Special Issue Stochastic Models and Methods with Applications)
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<p>Algorithm for the simulation of non-homogeneous semi-Markov trajectories.</p> Full article ">Figure 2
<p>Non-Homogeneous Interval Reliability. For reference purpose we fixed s = 0.</p> Full article ">Figure 3
<p>Duration dependent Interval Reliability. In the plots, i = U, s-l = 3, and t = 10.</p> Full article ">Figure 4
<p>Duration dependent Availability, i = U, v = 3, and s = 0.</p> Full article ">
18 pages, 10905 KiB  
Article
Numerical Modeling of the Spread of Cough Saliva Droplets in a Calm Confined Space
by Sergio A. Chillón, Ainara Ugarte-Anero, Iñigo Aramendia, Unai Fernandez-Gamiz and Ekaitz Zulueta
Mathematics 2021, 9(5), 574; https://doi.org/10.3390/math9050574 - 8 Mar 2021
Cited by 26 | Viewed by 3600
Abstract
The coronavirus disease 2019 (COVID-19) outbreak has altered the lives of everyone on a global scale due to its high transmission rate. In the current work, the droplet dispersion and evaporation originated by a cough at different velocities is studied. A multiphase computational [...] Read more.
The coronavirus disease 2019 (COVID-19) outbreak has altered the lives of everyone on a global scale due to its high transmission rate. In the current work, the droplet dispersion and evaporation originated by a cough at different velocities is studied. A multiphase computational fluid dynamic model based on fully coupled Eulerian–Lagrangian techniques was used. The evaporation, breakup, mass transfer, phase change, and turbulent dispersion forces of droplets were taken into account. A computational domain imitating an elevator that with two individuals inside was modeled. The results showed that all droplets smaller than 150 μm evaporate before 10 s at different heights. Smaller droplets of <30 µm evaporate quickly, and their trajectories are governed by Brownian movements. Instead, the trajectories of medium-sized droplets (30–80 µm) are under the influence of inertial forces, while bigger droplets move according to inertial and gravitational forces. Smaller droplets are located in the top positions, while larger (i.e., heaviest) droplets are located at the bottom. Full article
(This article belongs to the Special Issue Mathematical and Computational Methods against the COVID-19 Pandemics)
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<p>Size and layout of the computational domain. L, length; W, width; H, height; Dm, distance between mouths; Hm, mouth height; Hh, human height.</p> Full article ">Figure 2
<p>Case details: (<b>a</b>) Mouth location and (<b>b</b>) mouth dimensions. Mo, mouth opening; Mw, mouth width.</p> Full article ">Figure 3
<p>Mesh distribution in the domain (526,392 cells). VC, volume control.</p> Full article ">Figure 4
<p>The diameters of the droplets and their positions in a time interval from 0.03 to 0.15 s. The columns represent the velocity values, which are 8.5 m/s, 17.7 m/s and 28 m/s. The time values are arranged across rows from top to bottom: From 0.03 to 0.15 s, with 0.3 s time intervals.</p> Full article ">Figure 5
<p>Initial droplets distribution in a time interval from 0.12 s to 1 s. The rows show the time values: 0.12 s, 0.25 s, 0.5 Scheme 0 s and 1 s. The velocity values are arranged in the columns, from left to right: 8.5 m/s, 17.7 m/s and 28 m/s.</p> Full article ">Figure 6
<p>The distribution of droplets of different sizes over time: (<b>a</b>) 8.5 m/s in the 0–1 s range; (<b>b</b>) 8.5 m/s in the 0–10 s range; (<b>c</b>) 17.7 m/s in the 0–1 s range; (<b>d</b>) 17.7 m/s in the 0–10 s range; (<b>e</b>) 28 m/s in the 0–1 s range; (<b>f</b>) 28 m/s in the 0–10 s range.</p> Full article ">Figure 6 Cont.
<p>The distribution of droplets of different sizes over time: (<b>a</b>) 8.5 m/s in the 0–1 s range; (<b>b</b>) 8.5 m/s in the 0–10 s range; (<b>c</b>) 17.7 m/s in the 0–1 s range; (<b>d</b>) 17.7 m/s in the 0–10 s range; (<b>e</b>) 28 m/s in the 0–1 s range; (<b>f</b>) 28 m/s in the 0–10 s range.</p> Full article ">Figure 6 Cont.
<p>The distribution of droplets of different sizes over time: (<b>a</b>) 8.5 m/s in the 0–1 s range; (<b>b</b>) 8.5 m/s in the 0–10 s range; (<b>c</b>) 17.7 m/s in the 0–1 s range; (<b>d</b>) 17.7 m/s in the 0–10 s range; (<b>e</b>) 28 m/s in the 0–1 s range; (<b>f</b>) 28 m/s in the 0–10 s range.</p> Full article ">Figure 7
<p>The diameter of the droplets and their positions in a time interval of 1–10 s. The rows represent the time values: 0.12 s, 0.25 s, 0.5 s, 0.75 s and 1 s. The velocity values are arranged in columns, from left to right: 8.5 m/s, 17.7 m/s and 28 m/s.</p> Full article ">Figure A1
<p>Different diameters particles quantities during first second from cough start: (<b>a</b>) Case 8.5 m/s; (<b>b</b>) Case 17.7 m/s; (<b>c</b>) Case 28 m/s.</p> Full article ">
27 pages, 1919 KiB  
Article
Analysis and Correction of the Attack against the LPN-Problem Based Authentication Protocols
by Siniša Tomović, Milica Knežević and Miodrag J. Mihaljević
Mathematics 2021, 9(5), 573; https://doi.org/10.3390/math9050573 - 8 Mar 2021
Viewed by 1914
Abstract
This paper reconsiders a powerful man-in-the-middle attack against Random-HB# and HB# authentication protocols, two prominent representatives of the HB family of protocols, which are built based on the Learning Parity in Noise (LPN) problem. A recent empirical report pointed out that the attack [...] Read more.
This paper reconsiders a powerful man-in-the-middle attack against Random-HB# and HB# authentication protocols, two prominent representatives of the HB family of protocols, which are built based on the Learning Parity in Noise (LPN) problem. A recent empirical report pointed out that the attack does not meet the claimed precision and complexity. Performing a thorough theoretical and numerical re-evaluation of the attack, in this paper we identify the root cause of the detected problem, which lies in reasoning based on approximate probability distributions of the central attack events, that can not provide the required precision due to the inherent limitations in the use of the Central Limit Theorem for this particular application. We rectify the attack by employing adequate Bayesian reasoning, after establishing the exact distributions of these events, and overcome the mentioned limitations. We further experimentally confirm the correctness of the rectified attack and show that it satisfies the required, targeted accuracy and efficiency, unlike the original attack. Full article
(This article belongs to the Section Mathematics and Computer Science)
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<p>Random-HB# and HB# authentication protocols.</p> Full article ">Figure 2
<p>The OOV attack against Random-HB# and HB#.</p> Full article ">Figure 3
<p>Decision making of the OOV attack Algorithm 1.</p> Full article ">Figure 4
<p>Distribution structure of the cumulative noise vector <math display="inline"><semantics> <mrow> <mi mathvariant="bold">e</mi> <mo>⊕</mo> <mover accent="true"> <mi mathvariant="bold">e</mi> <mo stretchy="false">¯</mo> </mover> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>The approximation error upper bound is larger than the interval widths used in the attack. Thus, the adversary may not be capable to accurately estimate noise vectors weights.</p> Full article ">Figure 6
<p>Localization of <math display="inline"><semantics> <mrow> <mi>P</mi> <mo>(</mo> <mi>w</mi> <mo>)</mo> </mrow> </semantics></math> values using Berry-Eseen upper error bound.</p> Full article ">Figure 7
<p>Theoretically, the approximation error decreases as <span class="html-italic">m</span> increases by CLT (note the transition in color of the error peak). In the OOV attack, <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>1164</mn> </mrow> </semantics></math> or <math display="inline"><semantics> <mrow> <mi>m</mi> <mo>=</mo> <mn>441</mn> </mrow> </semantics></math>, for standard parameter sets I or II, respectively.</p> Full article ">Figure 8
<p>The exact error of the <math display="inline"><semantics> <msub> <mi>P</mi> <mrow> <mi>O</mi> <mi>O</mi> <mi>V</mi> </mrow> </msub> </semantics></math> approximation.</p> Full article ">Figure 9
<p>PB-decision zones used after eavesdropping (<b>left</b>) and for bit recovery (<b>right</b>).</p> Full article ">Figure 10
<p>Different “decision zones” according to the OOV approximation and the exact Poisson-Binomial distribution.</p> Full article ">Figure 11
<p>Comparison of the experimentally obtained acceptance rates and the corresponding <math display="inline"><semantics> <mrow> <msub> <mi>P</mi> <mrow> <mi>O</mi> <mi>O</mi> <mi>V</mi> </mrow> </msub> <mrow> <mo>(</mo> <mo>∥</mo> </mrow> <msub> <mi>e</mi> <mi>i</mi> </msub> <mrow> <mo>∥</mo> <mo>)</mo> </mrow> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>B</mi> <mo>(</mo> <mo>∥</mo> <msub> <mi>e</mi> <mi>i</mi> </msub> <mo>∥</mo> <mo>)</mo> </mrow> </semantics></math> points for <span class="html-italic">n</span> = 2500, 5000, 10,000, 15,000.</p> Full article ">Figure 12
<p>MAE between the experimentally obtained acceptance rates and the OOV and PB points, respectively, for <span class="html-italic">n</span> = 2500, 5000, 10,000 and 15,000.</p> Full article ">Figure 13
<p>Percentage of correct weights estimates based on acceptance rates using the PB distribution and OOV approximation respectively, for <span class="html-italic">n</span> = 2500, 5000, 10,000 and 15,000.</p> Full article ">Figure 14
<p>Precision comparison of the weight estimate using the OOV and PB-OOV algorithms.</p> Full article ">
16 pages, 2266 KiB  
Article
Designing Tasks for Introducing Functions and Graphs within Dynamic Interactive Environments
by Samuele Antonini and Giulia Lisarelli
Mathematics 2021, 9(5), 572; https://doi.org/10.3390/math9050572 - 7 Mar 2021
Cited by 5 | Viewed by 2773
Abstract
In this paper, we elaborate on theoretical and methodological considerations for designing a sequence of tasks for introducing middle and high school students to functions and their graphs. In particular, we present didactical activities with an artifact realized within a dynamic interactive environment [...] Read more.
In this paper, we elaborate on theoretical and methodological considerations for designing a sequence of tasks for introducing middle and high school students to functions and their graphs. In particular, we present didactical activities with an artifact realized within a dynamic interactive environment and having the semiotic potential for embedding mathematical meanings of covariation of independent and dependent variables. After laying down the theoretical grounds, we formulate the design principles that emerged as the result of bringing the theory into a dialogue with the didactical aims. Finally, we present a teaching sequence, designed and implemented on the basis of the design principles and we show how students’ efforts in describing and manipulating the different graphs of functions can promote their production of specific signs that can progressively evolve towards mathematical meanings. Full article
(This article belongs to the Special Issue Designing Tasks within Dynamic and Interactive Mathematics Learning Environments)
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<p>The process of construction of an SGc from a DGc in the DIE. (<b>a</b>) Two ticks representing <span class="html-italic">x</span> and <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math> are respectively on the <span class="html-italic">x</span>-axis and on the <span class="html-italic">y</span>-axis. The tick that represents <span class="html-italic">x</span> can be (directly) dragged and the movement of <span class="html-italic">x</span> causes the (indirect) movement of <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> </semantics></math>, according to the functional dependency. (<b>b</b>) The point <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> is built as the intersection point of the perpendicular line to the <span class="html-italic">x</span>-axis passing through <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mn>0</mn> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> and the perpendicular line to the <span class="html-italic">y</span>-axis passing through <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mn>0</mn> <mo>,</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math>. (<b>c</b>) The activation of the trace mark on <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> and the dragging of <span class="html-italic">x</span> allow to visualize the trajectory of <math display="inline"><semantics> <mrow> <mrow> <mo>(</mo> <mrow> <mi>x</mi> <mo>,</mo> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> and then the graph of the function.</p> Full article ">Figure 2
<p>Two screenshots of the DGp of the function <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> </semantics></math>. The mouse cursor identifies the tick representing the independent variable, which is the only draggable one. (<b>a</b>) A screenshot of the DGp for a negative <span class="html-italic">x</span>-value; (<b>b</b>) A screenshot of the DGp for a positive <span class="html-italic">x</span>-value.</p> Full article ">Figure 3
<p>Two screenshots of the DGpp of the function <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mi>x</mi> <mo>)</mo> </mrow> <mo>=</mo> <mrow> <mo>|</mo> <mi>x</mi> <mo>|</mo> </mrow> </mrow> </semantics></math>. The mouse cursor identifies the tick representing the independent variable, which is the only draggable one. (<b>a</b>) A screenshot of the DGpp for a negative <span class="html-italic">x</span>-value; (<b>b</b>) A screenshot of the DGp for a positive <span class="html-italic">x</span>-value.</p> Full article ">Figure 4
<p>By activating the trace mark on B, while dragging A from <span class="html-italic">1</span> to <span class="html-italic">4</span>, it is possible to visualize <math display="inline"><semantics> <mrow> <mi>f</mi> <mrow> <mo>(</mo> <mrow> <mrow> <mo>[</mo> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> </mrow> <mo>]</mo> </mrow> </mrow> <mo>)</mo> </mrow> </mrow> </semantics></math> on the screen.</p> Full article ">Figure 5
<p>A screenshot of the DGpp of three functions of the same variable. T<sub>1</sub>, T<sub>2,</sub> and T<sub>3</sub> represent three different telephone charges all varying depending on time A. The tick representing time is the only draggable one.</p> Full article ">
19 pages, 2287 KiB  
Article
Unpredictable Oscillations for Hopfield-Type Neural Networks with Delayed and Advanced Arguments
by Marat Akhmet, Duygu Aruğaslan Çinçin, Madina Tleubergenova and Zakhira Nugayeva
Mathematics 2021, 9(5), 571; https://doi.org/10.3390/math9050571 - 7 Mar 2021
Cited by 16 | Viewed by 2583
Abstract
This is the first time that the method for the investigation of unpredictable solutions of differential equations has been extended to unpredictable oscillations of neural networks with a generalized piecewise constant argument, which is delayed and advanced. The existence and exponential stability of [...] Read more.
This is the first time that the method for the investigation of unpredictable solutions of differential equations has been extended to unpredictable oscillations of neural networks with a generalized piecewise constant argument, which is delayed and advanced. The existence and exponential stability of the unique unpredictable oscillation are proven. According to the theory, the presence of unpredictable oscillations is strong evidence for Poincaré chaos. Consequently, the paper is a contribution to chaos applications in neuroscience. The model is inspired by chaotic time-varying stimuli, which allow studying the distribution of chaotic signals in neural networks. Unpredictable inputs create an excitation wave of neurons that transmit chaotic signals. The technique of analysis includes the ideas used for differential equations with a piecewise constant argument. The results are illustrated by examples and simulations. They are carried out in MATLAB Simulink to demonstrate the simplicity of the diagrammatic approaches. Full article
(This article belongs to the Special Issue New Trends in Numerics and Dynamics of Artificial Neural Networks: Theory and Applications)
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<p>The block diagram for the Hopfield-type neural network system (<a href="#FD1-mathematics-09-00571" class="html-disp-formula">1</a>).</p> Full article ">Figure 2
<p>The graph of function <math display="inline"><semantics> <mrow> <mo>Φ</mo> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, which exponentially approaches the unpredictable function <math display="inline"><semantics> <mrow> <mo>Θ</mo> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>The coordinates of function <math display="inline"><semantics> <mrow> <mi>ψ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>, which exponentially converge to the coordinates of the unpredictable solution <math display="inline"><semantics> <mrow> <mi>x</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 4
<p>The trajectory of function <math display="inline"><semantics> <mrow> <mi>ψ</mi> <mo>(</mo> <mi>t</mi> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>The block diagram for System (<a href="#FD16-mathematics-09-00571" class="html-disp-formula">16</a>).</p> Full article ">
14 pages, 2355 KiB  
Article
Feature Selection for Colon Cancer Detection Using K-Means Clustering and Modified Harmony Search Algorithm
by Jin Hee Bae, Minwoo Kim, J.S. Lim and Zong Woo Geem
Mathematics 2021, 9(5), 570; https://doi.org/10.3390/math9050570 - 7 Mar 2021
Cited by 24 | Viewed by 3677
Abstract
This paper proposes a feature selection method that is effective in distinguishing colorectal cancer patients from normal individuals using K-means clustering and the modified harmony search algorithm. As the genetic cause of colorectal cancer originates from mutations in genes, it is important to [...] Read more.
This paper proposes a feature selection method that is effective in distinguishing colorectal cancer patients from normal individuals using K-means clustering and the modified harmony search algorithm. As the genetic cause of colorectal cancer originates from mutations in genes, it is important to classify the presence or absence of colorectal cancer through gene information. The proposed methodology consists of four steps. First, the original data are Z-normalized by data preprocessing. Candidate genes are then selected using the Fisher score. Next, one representative gene is selected from each cluster after candidate genes are clustered using K-means clustering. Finally, feature selection is carried out using the modified harmony search algorithm. The gene combination created by feature selection is then applied to the classification model and verified using 5-fold cross-validation. The proposed model obtained a classification accuracy of up to 94.36%. Furthermore, on comparing the proposed method with other methods, we prove that the proposed method performs well in classifying colorectal cancer. Moreover, we believe that the proposed model can be applied not only to colorectal cancer but also to other gene-related diseases. Full article
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<p>Scheme of proposed methods.</p> Full article ">Figure 2
<p>Divided harmony memory.</p> Full article ">Figure 3
<p>Elimination of two worst harmonies.</p> Full article ">Figure 4
<p>Structure of ANN (artificial neural network).</p> Full article ">Figure 5
<p>5-fold cross-validation data segmentation.</p> Full article ">Figure 6
<p>Inertia value according to the number of clusters.</p> Full article ">
24 pages, 1795 KiB  
Article
An Enhancing Differential Evolution Algorithm with a Rank-Up Selection: RUSDE
by Kai Zhang and Yicheng Yu
Mathematics 2021, 9(5), 569; https://doi.org/10.3390/math9050569 - 7 Mar 2021
Cited by 4 | Viewed by 2428
Abstract
Recently, the differential evolution (DE) algorithm has been widely used to solve many practical problems. However, DE may suffer from stagnation problems in the iteration process. Thus, we propose an enhancing differential evolution with a rank-up selection, named RUSDE. First, the rank-up individuals [...] Read more.
Recently, the differential evolution (DE) algorithm has been widely used to solve many practical problems. However, DE may suffer from stagnation problems in the iteration process. Thus, we propose an enhancing differential evolution with a rank-up selection, named RUSDE. First, the rank-up individuals in the current population are selected and stored into a new archive; second, a debating mutation strategy is adopted in terms of the updating status of the current population to decide the parent’s selection. Both of the two methods can improve the performance of DE. We conducted numerical experiments based on various functions from CEC 2014, where the results demonstrated excellent performance of this algorithm. Furthermore, this algorithm is applied to the real-world optimization problem of the four-bar linkages, where the results show that the performance of RUSDE is better than other algorithms. Full article
(This article belongs to the Special Issue Evolutionary Algorithms in Artificial Intelligent Systems)
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<p>Convergence curves in 30D problems of the CEC 2014 benchmarks.</p> Full article ">Figure 2
<p>Variables of the four-bar mechanism.</p> Full article ">Figure 3
<p>Convergence curves of the four-bar mechanism with the six algorithms.</p> Full article ">
13 pages, 3155 KiB  
Article
Optimization and Simulation of Dynamic Performance of Production–Inventory Systems with Multivariable Controls
by Huthaifa AL-Khazraji, Colin Cole and William Guo
Mathematics 2021, 9(5), 568; https://doi.org/10.3390/math9050568 - 7 Mar 2021
Cited by 22 | Viewed by 2544
Abstract
The production–inventory system is a problem of multivariable input and multivariant output in mathematics. Selecting the best system control parameters is a crucial managerial decision to achieve and dynamically maintain an optimal performance in terms of balancing the order rate and stock level [...] Read more.
The production–inventory system is a problem of multivariable input and multivariant output in mathematics. Selecting the best system control parameters is a crucial managerial decision to achieve and dynamically maintain an optimal performance in terms of balancing the order rate and stock level under dynamic influence of many factors affecting the system operations. The dynamic performance of the popular APIOBPCS model and the newly modified 2APIOBPCS model for optimal control of production–inventory systems is examined in the study. This examination is based on the leveled ground with a new simulation scheme that incorporates a designated multi-objective particle swarm optimization (MOPSO) algorithm into the simulation, which enables the optimal set of system control parameters to be selected for achieving the situational best possible performance of the production–inventory system under study. The dynamic performance is measured by the variance ratio between the order rate and the sales rate related to the bullwhip effect, and the integral of absolute error related to the inventory responsiveness in response to a random customer demand. Our simulation indicates that the 2APIOBPCS model performed better than or at least no worse than, and more robust than the APIOBPCS model under different conditions. Full article
(This article belongs to the Special Issue Multivariable Optimization by Intelligent and Numerical Modelling and Simulation)
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<p>Block diagram of the automatic pipeline, inventory, and order based production control system (APIOBPCS) model.</p> Full article ">Figure 2
<p>Block diagram of the 2APIOBPCS model.</p> Full article ">Figure 3
<p>The process of dynamic simulation of a production–inventory system.</p> Full article ">Figure 4
<p>The demand pattern for simulations.</p> Full article ">Figure 5
<p>Simulation of Case 1 using Set 2 for the APIOBPCS and 2APIOBPCS models. Order rate (<b>top</b>) and inventory level (<b>bottom</b>).</p> Full article ">Figure 6
<p>Simulation of Case 2 using Set 2 for the APIOBPCS and 2APIOBPCS models. Order rate (<b>top</b>) and inventory level (<b>bottom</b>).</p> Full article ">Figure 7
<p>Simulation of Case 3 using Set 2 for the APIOBPCS and 2APIOBPCS models. Order rate (<b>top</b>) and inventory level (<b>bottom</b>).</p> Full article ">Figure 8
<p>Simulation of Case 4 using Set 2 for the APIOBPCS and 2APIOBPCS models. Order rate (<b>top</b>) and inventory level (<b>bottom</b>).</p> Full article ">
22 pages, 942 KiB  
Article
Trigonometric Solution for the Bending Analysis of Magneto-Electro-Elastic Strain Gradient Nonlocal Nanoplates in Hygro-Thermal Environment
by Giovanni Tocci Monaco, Nicholas Fantuzzi, Francesco Fabbrocino and Raimondo Luciano
Mathematics 2021, 9(5), 567; https://doi.org/10.3390/math9050567 - 7 Mar 2021
Cited by 47 | Viewed by 3375
Abstract
Nanoplates have been extensively utilized in the recent years for applications in nanoengineering as sensors and actuators. Due to their operative nanoscale, the mechanical behavior of such structures might also be influenced by inter-atomic material interactions. For these reasons, nonlocal models are usually [...] Read more.
Nanoplates have been extensively utilized in the recent years for applications in nanoengineering as sensors and actuators. Due to their operative nanoscale, the mechanical behavior of such structures might also be influenced by inter-atomic material interactions. For these reasons, nonlocal models are usually introduced for studying their mechanical behavior. Sensor technology of plate structures should be formulated with coupled mechanics where elastic, magnetic and electric fields interact among themselves. In addition, the effect of hygro-thermal environments are also considered since their presence might effect the nanoplate behavior. In this work a trigonometric approach is developed for investigating smart composite nanoplates using a strain gradient nonlocal procedure. Convergence of the present method is also reported in terms of displacements and electro-magnetic potentials. Results agree well with the literature and open novel applications in this field for further developments. Full article
(This article belongs to the Special Issue Analytical and Numerical Methods for Linear and Nonlinear Analysis of Structures at Macro, Micro and Nano Scale)
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<p>Functionally graded plate with applied electric <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mi>V</mi> </mrow> </semantics></math> and magnetic potentials <math display="inline"><semantics> <mrow> <mo>Δ</mo> <mo>Ω</mo> </mrow> </semantics></math>.</p> Full article ">Figure 2
<p>Displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> of middle point of a FG square plate composed of Al/ZrO<math display="inline"><semantics> <msub> <mrow/> <mn>2</mn> </msub> </semantics></math> for different values of <math display="inline"><semantics> <msub> <mi>n</mi> <mi>p</mi> </msub> </semantics></math>.</p> Full article ">Figure 3
<p>Graphs of displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (<b>a</b>) electric <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> (<b>b</b>) and magnetic <math display="inline"><semantics> <mi>γ</mi> </semantics></math> (<b>c</b>) potential, in the point <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> to vary of <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math> ratio and for different values of nonlocal parameter <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mo>ℓ</mo> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> and for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (SDL; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math> N/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>E</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> C/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> Wb/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>).</p> Full article ">Figure 4
<p>Graphs of displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (<b>a</b>) electric <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> (<b>b</b>) and magnetic <math display="inline"><semantics> <mi>γ</mi> </semantics></math> (<b>c</b>) potential, in the point <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> to vary of <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math> ratio and for different values of nonlocal parameter <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mo>ℓ</mo> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> and for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (SDL; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> N/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>E</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> C/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> Wb/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>)</p> Full article ">Figure 5
<p>Graphs of displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (<b>a</b>) electric <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> (<b>b</b>) and magnetic <math display="inline"><semantics> <mi>γ</mi> </semantics></math> (<b>c</b>) potential, in the point <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> to vary of <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math> ratio and for different values of nonlocal parameter <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mo>ℓ</mo> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> and for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (SDL; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> N/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>E</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> C/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> Wb/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>)</p> Full article ">Figure 6
<p>Graphs of displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (a) electric <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> (b) and magnetic <math display="inline"><semantics> <mi>γ</mi> </semantics></math> (c) potential, in the point <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> to vary of <math display="inline"><semantics> <msub> <mi>n</mi> <mi>p</mi> </msub> </semantics></math> and for different values of nonlocal parameter <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mo>ℓ</mo> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> and for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (SDL; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> N/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>E</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> C/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> Wb/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>)</p> Full article ">Figure 7
<p>Convergence of the relative error on the displacement and potential at a square plate central point by increasing <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics></math> for: (<b>a</b>) uniformly distributed mechanical load only; (<b>b</b>) uniformly distributed thermal load only.</p> Full article ">Figure 8
<p>Displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (<b>a</b>) electric <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> (<b>b</b>) and magnetic <math display="inline"><semantics> <mi>γ</mi> </semantics></math> (<b>c</b>) potential, at the central point <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> by varying <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math> and for different nonlocal parameters <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mo>ℓ</mo> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (UDL; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <msup> <mn>10</mn> <mn>3</mn> </msup> </mrow> </semantics></math>N/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>E</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> C/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> Wb/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>).</p> Full article ">Figure 9
<p>Displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (<b>a</b>) electric <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> (<b>b</b>) and magnetic <math display="inline"><semantics> <mi>γ</mi> </semantics></math> (<b>c</b>) potential, at the central point <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> by varying <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math> ratio and for different nonlocal parameter <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mo>ℓ</mo> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> with <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> (UDL; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> N/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>E</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> C/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> Wb/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>).</p> Full article ">Figure 10
<p>Graphs of displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (<b>a</b>) electric <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> (<b>b</b>) and magnetic <math display="inline"><semantics> <mi>γ</mi> </semantics></math> (<b>c</b>) potential, in the point <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> to vary of <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math> ratio and for different values of nonlocal parameter <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mo>ℓ</mo> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> and for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (UDL; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> N/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>E</mi> </msub> <mo>=</mo> <mn>0.1</mn> </mrow> </semantics></math> C/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> Wb/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>).</p> Full article ">Figure 11
<p>Graphs of displacement <math display="inline"><semantics> <mover accent="true"> <mi>w</mi> <mo stretchy="false">¯</mo> </mover> </semantics></math> (<b>a</b>) electric <math display="inline"><semantics> <mi>ϕ</mi> </semantics></math> (<b>b</b>) and magnetic <math display="inline"><semantics> <mi>γ</mi> </semantics></math> (<b>c</b>) potential, in the point <math display="inline"><semantics> <mrow> <mo>(</mo> <mi>a</mi> <mo>/</mo> <mn>2</mn> <mo>,</mo> <mi>b</mi> <mo>/</mo> <mn>2</mn> <mo>)</mo> </mrow> </semantics></math> to vary of <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>/</mo> <mi>b</mi> </mrow> </semantics></math> ratio and for different values of nonlocal parameter <math display="inline"><semantics> <msup> <mrow> <mo>(</mo> <mo>ℓ</mo> <mo>/</mo> <mi>a</mi> <mo>)</mo> </mrow> <mn>2</mn> </msup> </semantics></math> and for <math display="inline"><semantics> <mrow> <msub> <mi>n</mi> <mi>p</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math>. (UDL; <math display="inline"><semantics> <mrow> <mi>q</mi> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> N/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>T</mi> <mn>0</mn> </msub> <mo>=</mo> <msub> <mi>T</mi> <mn>1</mn> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> K, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>E</mi> </msub> <mo>=</mo> <mn>0</mn> </mrow> </semantics></math> C/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>, <math display="inline"><semantics> <mrow> <msub> <mi>ρ</mi> <mi>H</mi> </msub> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> Wb/m<math display="inline"><semantics> <msup> <mrow/> <mn>2</mn> </msup> </semantics></math>).</p> Full article ">
15 pages, 363 KiB  
Article
Simplified Mathematical Modelling of Uncertainty: Cost-Effectiveness of COVID-19 Vaccines in Spain
by Julio Emilio Marco-Franco, Pedro Pita-Barros, Silvia González-de-Julián, Iryna Sabat and David Vivas-Consuelo
Mathematics 2021, 9(5), 566; https://doi.org/10.3390/math9050566 - 6 Mar 2021
Cited by 15 | Viewed by 4701
Abstract
When exceptional situations, such as the COVID-19 pandemic, arise and reliable data is not available at decision-making times, estimation using mathematical models can provide a reasonable reckoning for health planning. We present a simplified model (static but with two-time references) for estimating the [...] Read more.
When exceptional situations, such as the COVID-19 pandemic, arise and reliable data is not available at decision-making times, estimation using mathematical models can provide a reasonable reckoning for health planning. We present a simplified model (static but with two-time references) for estimating the cost-effectiveness of the COVID-19 vaccine. A simplified model provides a quick assessment of the upper bound of cost-effectiveness, as we illustrate with data from Spain, and allows for easy comparisons between countries. It may also provide useful comparisons among different vaccines at the marketplace, from the perspective of the buyer. From the analysis of this information, key epidemiological figures, and costs of the disease for Spain have been estimated, based on mortality. The fatality rate is robust data that can alternatively be obtained from death registers, funeral homes, cemeteries, and crematoria. Our model estimates the incremental cost-effectiveness ratio (ICER) to be 5132 € (4926–5276) as of 17 February 2021, based on the following assumptions/inputs: An estimated cost of 30 euros per dose (plus transport, storing, and administration), two doses per person, efficacy of 70% and coverage of 70% of the population. Even considering the possibility of some bias, this simplified model provides confirmation that vaccination against COVID-19 is highly cost-effective. Full article
15 pages, 340 KiB  
Article
Identifying Non-Sublattice Equivalence Classes Induced by an Attribute Reduction in FCA
by Roberto G. Aragón, Jesús Medina and Eloísa Ramírez-Poussa
Mathematics 2021, 9(5), 565; https://doi.org/10.3390/math9050565 - 6 Mar 2021
Cited by 7 | Viewed by 1929
Abstract
The detection of redundant or irrelevant variables (attributes) in datasets becomes essential in different frameworks, such as in Formal Concept Analysis (FCA). However, removing such variables can have some impact on the concept lattice, which is closely related to the algebraic structure of [...] Read more.
The detection of redundant or irrelevant variables (attributes) in datasets becomes essential in different frameworks, such as in Formal Concept Analysis (FCA). However, removing such variables can have some impact on the concept lattice, which is closely related to the algebraic structure of the obtained quotient set and their classes. This paper studies the algebraic structure of the induced equivalence classes and characterizes those classes that are convex sublattices of the original concept lattice. Particular attention is given to the reductions removing FCA’s unnecessary attributes. The obtained results will be useful to other complementary reduction techniques, such as the recently introduced procedure based on local congruences. Full article
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<p>Concept lattice of Example 1 (<b>left</b>) and the partition induced by the elimination of attributes <math display="inline"><semantics> <msub> <mi>a</mi> <mn>2</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>a</mi> <mn>3</mn> </msub> </semantics></math> in Example 1 (<b>right</b>).</p> Full article ">Figure 2
<p>Concept lattices of Example 2.</p> Full article ">Figure 3
<p>Concept lattices of Example 3.</p> Full article ">Figure 4
<p>Concept lattices of Example 4.</p> Full article ">Figure 5
<p>Concept lattice isomorphic to <math display="inline"><semantics> <msub> <mi>M</mi> <mn>3</mn> </msub> </semantics></math> and induced partition by <math display="inline"><semantics> <msub> <mi>D</mi> <msub> <mi>M</mi> <mn>3</mn> </msub> </msub> </semantics></math>.</p> Full article ">Figure 6
<p>Concept lattice isomorphic to <math display="inline"><semantics> <msub> <mi>N</mi> <mn>5</mn> </msub> </semantics></math> and induced partition by <math display="inline"><semantics> <msub> <mi>D</mi> <msub> <mi>N</mi> <mn>5</mn> </msub> </msub> </semantics></math>.</p> Full article ">
17 pages, 526 KiB  
Article
The Verbal Component of Mathematical Problem Solving in Bilingual Contexts by Early Elementary Schoolers
by Pilar Ester, Isabel Morales, Álvaro Moraleda and Vicente Bermejo
Mathematics 2021, 9(5), 564; https://doi.org/10.3390/math9050564 - 6 Mar 2021
Cited by 7 | Viewed by 4008
Abstract
The main aim of the present study is to analyze the differences that may exist when students address the resolution of verbal problems in their mother tongue and in the language of instruction when these are different. We understand that knowing the type [...] Read more.
The main aim of the present study is to analyze the differences that may exist when students address the resolution of verbal problems in their mother tongue and in the language of instruction when these are different. We understand that knowing the type of verbal problems and their semantic structure can be helpful for students’ contextual and mathematical understanding and will allow teachers to improve instruction during the first years of elementary education in bilingual schools specialized in the area of second language acquisition as well as in CLIL (Content and Language Integrated Learning). This study shows how children, as they are acquiring a greater command of the second language, show similar effectiveness to those students who work on mathematics in their mother tongue. This transversal study was conducted on 169 bilinguals studying in international schools. The sample was made up of 80 1st grade students (39 girls, mean age of 7.1 years and 41 boys, mean age of 7.3 years); and 89 2nd grade students (38 girls, mean age 8.2 years, and 51 boys, mean age 8.2 years). The exploratory analyses let us show how 1st grade students demonstrate lower effectiveness in solving problems when they do it in a second language, compared to 2nd grade students whose effectiveness is higher in carrying them out. It is also relevant that in first graders, the largest number of errors are found in the simplest tasks as students’ effectiveness is less when they are taught in a second language, since it takes them longer to create effective resolution models. This fact will allow us to reconsider appropriate strategies and interventions when teaching mathematics in bilingual contexts. Full article
(This article belongs to the Special Issue Highlights of Research in the Didactics of Mathematics: The Insider’s Perspective)
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<p>Interactive model of bilingual education Barton (2012).</p> Full article ">
14 pages, 436 KiB  
Article
Global Dynamics of a Discrete-Time MERS-Cov Model
by Mahmoud H. DarAssi, Mohammad A. Safi and Morad Ahmad
Mathematics 2021, 9(5), 563; https://doi.org/10.3390/math9050563 - 6 Mar 2021
Cited by 5 | Viewed by 2179
Abstract
In this paper, we have investigated the global dynamics of a discrete-time middle east respiratory syndrome (MERS-Cov) model. The proposed discrete model was analyzed and the threshold conditions for the global attractivity of the disease-free equilibrium (DFE) and the endemic equilibrium are established. [...] Read more.
In this paper, we have investigated the global dynamics of a discrete-time middle east respiratory syndrome (MERS-Cov) model. The proposed discrete model was analyzed and the threshold conditions for the global attractivity of the disease-free equilibrium (DFE) and the endemic equilibrium are established. We proved that the DFE is globally asymptotically stable when R0≤1. Whenever R˜0>1, the proposed model has a unique endemic equilibrium that is globally asymptotically stable. The theoretical results are illustrated by a numerical simulation. Full article
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<p>Model (<a href="#FD1-mathematics-09-00563" class="html-disp-formula">1</a>) schematic diagram.</p> Full article ">Figure 2
<p>The plot of the the cofactor <math display="inline"><semantics> <msub> <mi mathvariant="sans-serif">Γ</mi> <mn>1</mn> </msub> </semantics></math> as a function of <math display="inline"><semantics> <mi>β</mi> </semantics></math>.</p> Full article ">Figure 3
<p>The infected compartments as functions of time when <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.7468</mn> <mo>&lt;</mo> <mn>1</mn> </mrow> </semantics></math> (<b>Up</b>). The infected compartments as functions of time when <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.4936</mn> <mo>&gt;</mo> <mn>1</mn> </mrow> </semantics></math> (<b>Down</b>).</p> Full article ">Figure 4
<p>The plot of the cumulative cases of infection verses time in days with treatment (dotted line) and without treatment (solid line) when <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>0.7468</mn> <mo>&lt;</mo> <mn>1</mn> </mrow> </semantics></math> (<b>Up</b>). The plot of the cumulative cases of infection verses time in days with treatment (dotted line) and without treatment (solid line) when <math display="inline"><semantics> <mrow> <msub> <mi mathvariant="script">R</mi> <mn>0</mn> </msub> <mo>=</mo> <mn>1.4936</mn> <mo>&gt;</mo> <mn>1</mn> </mrow> </semantics></math> (<b>Down</b>).</p> Full article ">Figure 5
<p>The plot of the basic reproduction number as a function of the birth–death rate <math display="inline"><semantics> <mi>μ</mi> </semantics></math> for several values of the contact rate.</p> Full article ">
19 pages, 2939 KiB  
Article
Global Portfolio Credit Risk Management: The US Banks Post-Crisis Challenge
by Pawel Siarka
Mathematics 2021, 9(5), 562; https://doi.org/10.3390/math9050562 - 6 Mar 2021
Cited by 2 | Viewed by 2653
Abstract
This paper addresses the problem of modeling credit risk for multi-product and global loan portfolios. The authors presented an improved version of the Basel Committee’s one-factor model for capital requirements calculation. They examined whether latent market factors corresponding to distinct portfolios are always [...] Read more.
This paper addresses the problem of modeling credit risk for multi-product and global loan portfolios. The authors presented an improved version of the Basel Committee’s one-factor model for capital requirements calculation. They examined whether latent market factors corresponding to distinct portfolios are always highly correlated within the global portfolio and how this correlation impacts total losses distribution function. Historical losses of top-tier banks (JPMorgan Chace, Bank of America, Citigroup, Wells Fargo, US Bancorp) were analyzed. Furthermore, the estimation of the correlations between latent market factors was conducted, and its impact on the total loss distribution function was assessed. The research was performed based on consolidated financial statements for holding companies - FR Y-9C reports provided by the Federal Reserve Bank of Chicago. To verify the improved model, the authors analyzed two distinct loan portfolios for each bank, i.e., credit cards and commercial and industrial loans. They showed that the correlation between latent market factors could be significantly lower than one and disregarding this conclusion may lead to overestimating total unexpected losses. Hence, capital requirements calculated according to the IRB (Internal Ratings Based Approach) formula as a sum of individual VaR999 estimates may be biased. According to this finding, the enhanced one-factor model seems to be more accurate while calculating unexpected total loss for global portfolios. The authors proved that the active credit risk management process aiming to lower market factors’ correlation results in less volatile total losses. Therefore, financial institutions could be more resistant to macroeconomic downturns. Full article
(This article belongs to the Section Financial Mathematics)
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<p>Graphical presentation of asset correlation.</p> Full article ">Figure 2
<p>Threshold limit calculation for assets’ value.</p> Full article ">Figure 3
<p>Loss distribution functions depending on asset correlation.</p> Full article ">Figure 4
<p>Overall loss distribution functions—Monte Carlo simulation results.</p> Full article ">Figure 5
<p>Total losses for various factors’ correlations.</p> Full article ">Figure 6
<p>Charge-off rate time series.</p> Full article ">Figure 7
<p>Charge-off rates’ density functions.</p> Full article ">Figure 8
<p>Simulation procedure scheme.</p> Full article ">Figure 9
<p>Distribution functions of overall losses.</p> Full article ">Figure 10
<p>Total unexpected loss estimates. Left panel—one factor approach with 100% correlation. Right panel—unexpected (total) loss calculated with market factors’ correlation.</p> Full article ">Figure 11
<p>Unexpected losses comparison.</p> Full article ">
18 pages, 345 KiB  
Article
Global Persistence of the Unit Eigenvectors of Perturbed Eigenvalue Problems in Hilbert Spaces: The Odd Multiplicity Case
by Pierluigi Benevieri, Alessandro Calamai, Massimo Furi and Maria Patrizia Pera
Mathematics 2021, 9(5), 561; https://doi.org/10.3390/math9050561 - 6 Mar 2021
Viewed by 1578
Abstract
We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a [...] Read more.
We study the persistence of eigenvalues and eigenvectors of perturbed eigenvalue problems in Hilbert spaces. We assume that the unperturbed problem has a nontrivial kernel of odd dimension and we prove a Rabinowitz-type global continuation result. The approach is topological, based on a notion of degree for oriented Fredholm maps of index zero between real differentiable Banach manifolds. Full article
(This article belongs to the Special Issue Advances in Nonlinear Spectral Theory)
16 pages, 645 KiB  
Article
Parallel One-Step Control of Parametrised Boolean Networks
by Luboš Brim, Samuel Pastva, David Šafránek and Eva Šmijáková
Mathematics 2021, 9(5), 560; https://doi.org/10.3390/math9050560 - 6 Mar 2021
Cited by 6 | Viewed by 2997
Abstract
Boolean network (BN) is a simple model widely used to study complex dynamic behaviour of biological systems. Nonetheless, it might be difficult to gather enough data to precisely capture the behavior of a biological system into a set of Boolean functions. These issues [...] Read more.
Boolean network (BN) is a simple model widely used to study complex dynamic behaviour of biological systems. Nonetheless, it might be difficult to gather enough data to precisely capture the behavior of a biological system into a set of Boolean functions. These issues can be dealt with to some extent using parametrised Boolean networks (ParBNs), as this model allows leaving some update functions unspecified. In our work, we attack the control problem for ParBNs with asynchronous semantics. While there is an extensive work on controlling BNs without parameters, the problem of control for ParBNs has not been in fact addressed yet. The goal of control is to ensure the stabilisation of a system in a given state using as few interventions as possible. There are many ways to control BN dynamics. Here, we consider the one-step approach in which the system is instantaneously perturbed out of its actual state. A naïve approach to handle control of ParBNs is using parameter scan and solve the control problem for each parameter valuation separately using known techniques for non-parametrised BNs. This approach is however highly inefficient as the parameter space of ParBNs grows doubly exponentially in the worst case. We propose a novel semi-symbolic algorithm for the one-step control problem of ParBNs, that builds on symbolic data structures to avoid scanning individual parameters. We evaluate the performance of our approach on real biological models. Full article
(This article belongs to the Special Issue Boolean Networks Models in Science and Engineering)
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<p>(<b>a</b>) A regulatory network of a simple BN describing the DNA damage mechanism adapted from [<a href="#B2-mathematics-09-00560" class="html-bibr">2</a>]. Every regulation is either activating (green) or inhibiting (red) and observable. (<b>b</b>) Update function <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi mathvariant="monospace">M</mi> <mn mathvariant="monospace">2</mn> <mi mathvariant="monospace">N</mi> </mrow> </msub> </semantics></math>. (<b>c</b>) Update functions <math display="inline"><semantics> <msub> <mi>F</mi> <mi mathvariant="monospace">DNA</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi mathvariant="monospace">M</mi> <mn mathvariant="monospace">2</mn> <mi mathvariant="monospace">C</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi mathvariant="monospace">P</mi> <mn mathvariant="monospace">53</mn> </mrow> </msub> </semantics></math>.</p> Full article ">Figure 2
<p>Attractors, weak basins and strong basins in an STG of BN. The BN contains two attractors: single-state attractor 1 (light-red area) and cyclic two-states attractor 2 (light-blue area). Both these attractors have strong basins of size 3 (solid red area for attractor 1, blue are for attractor 2 resp.). Please note that states of attractors are also parts of their basins. Moreover, the strong basins never have any intersections as given by definition. Finally, the red-lined and blue-lined areas contain weak basin states of attractor 1 and attractor 2. The strong basin is always a subset of a weak basin. The weak basins are over-lapping.</p> Full article ">Figure 3
<p>(<b>a</b>) A regulatory network of a simplified ParBN similar to one in <a href="#mathematics-09-00560-f001" class="html-fig">Figure 1</a>a. Compared to the previous case, we do not know whether regulation <math display="inline"><semantics> <mrow> <mo>(</mo> <mi mathvariant="monospace">DNA</mi> <mo>,</mo> <mi mathvariant="monospace">DNA</mi> <mo>)</mo> </mrow> </semantics></math> is observable. (<b>b</b>) All possible valid update functions <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi mathvariant="monospace">M</mi> <mn mathvariant="monospace">2</mn> <mi mathvariant="monospace">N</mi> </mrow> </msub> </semantics></math> satisfying the static constraints (monotonicity, observability). (<b>c</b>) Valid update functions <math display="inline"><semantics> <msub> <mi>F</mi> <mi mathvariant="monospace">DNA</mi> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi mathvariant="monospace">M</mi> <mn mathvariant="monospace">2</mn> <mi mathvariant="monospace">C</mi> </mrow> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mi>F</mi> <mrow> <mi mathvariant="monospace">P</mi> <mn mathvariant="monospace">53</mn> </mrow> </msub> </semantics></math> satisfying the static constraints. Here, <math display="inline"><semantics> <msub> <mi mathvariant="monospace">P</mi> <mi>i</mi> </msub> </semantics></math> denote the parameters (<math display="inline"><semantics> <mi mathvariant="script">P</mi> </semantics></math>) of the ParBN.</p> Full article ">Figure 4
<p>The asynchronous semantics of the ParBN given in <a href="#mathematics-09-00560-f003" class="html-fig">Figure 3</a>a, restricted to <span class="html-italic">P</span> = {<span class="html-fig-inline" id="mathematics-09-00560-i002"> <img alt="Mathematics 09 00560 i002" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i002.png"/></span>,<span class="html-fig-inline" id="mathematics-09-00560-i003"> <img alt="Mathematics 09 00560 i003" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i003.png"/></span>,<span class="html-fig-inline" id="mathematics-09-00560-i004"> <img alt="Mathematics 09 00560 i004" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i004.png"/></span>,<span class="html-fig-inline" id="mathematics-09-00560-i005"> <img alt="Mathematics 09 00560 i005" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i005.png"/></span>}. Here, <span class="html-fig-inline" id="mathematics-09-00560-i002"> <img alt="Mathematics 09 00560 i002" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i002.png"/></span> <math display="inline"><semantics> <mrow> <mo>=</mo> <mo>{</mo> <msub> <mi mathvariant="monospace">P</mi> <mrow> <mn>2</mn> <mo>,</mo> <mn>3</mn> <mo>,</mo> <mn>6</mn> </mrow> </msub> <mo>:</mo> <mn>0</mn> <mo>,</mo> <msub> <mi mathvariant="monospace">P</mi> <mrow> <mn>1</mn> <mo>,</mo> <mn>4</mn> <mo>,</mo> <mn>5</mn> <mo>,</mo> <mn>7</mn> <mo>,</mo> <mn>8</mn> </mrow> </msub> <mo>:</mo> <mn>1</mn> <mo>}</mo> </mrow> </semantics></math>, <span class="html-fig-inline" id="mathematics-09-00560-i003"> <img alt="Mathematics 09 00560 i003" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i003.png"/></span> = <span class="html-fig-inline" id="mathematics-09-00560-i002"> <img alt="Mathematics 09 00560 i002" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i002.png"/></span><math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi mathvariant="monospace">P</mi> <mn>3</mn> </msub> <mo>↦</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics></math>, <span class="html-fig-inline" id="mathematics-09-00560-i004"> <img alt="Mathematics 09 00560 i004" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i004.png"/></span> = <span class="html-fig-inline" id="mathematics-09-00560-i002"> <img alt="Mathematics 09 00560 i002" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i002.png"/></span><math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi mathvariant="monospace">P</mi> <mn>6</mn> </msub> <mo>↦</mo> <mn>1</mn> <mo>]</mo> </mrow> </semantics></math>, and <span class="html-fig-inline" id="mathematics-09-00560-i005"> <img alt="Mathematics 09 00560 i005" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i005.png"/></span> = <span class="html-fig-inline" id="mathematics-09-00560-i004"> <img alt="Mathematics 09 00560 i004" src="/mathematics/mathematics-09-00560/article_deploy/html/images/mathematics-09-00560-i004.png"/></span><math display="inline"><semantics> <mrow> <mo>[</mo> <msub> <mi mathvariant="monospace">P</mi> <mn>8</mn> </msub> <mo>↦</mo> <mn>0</mn> <mo>]</mo> </mrow> </semantics></math>. The unlabelled edges are enabled for all parametrisations. The highlighted vertices represent attractors for indicated parametrisations.</p> Full article ">Figure 5
<p>Workflow of computing the source-target control problem. Parts in the orange boxes represent inputs and blue boxes represent (intermediate) results.</p> Full article ">
16 pages, 469 KiB  
Article
Analysis of a k-Stage Bulk Service Queuing System with Accessible Batches for Service
by Achyutha Krishnamoorthy, Anu Nuthan Joshua and Vladimir Vishnevsky
Mathematics 2021, 9(5), 559; https://doi.org/10.3390/math9050559 - 6 Mar 2021
Cited by 13 | Viewed by 2955
Abstract
In most of the service systems considered so far in queuing theory, no fresh customer is admitted to a batch undergoing service when the number in the batch is less than a threshold. However, a few researchers considered the case of customers accessing [...] Read more.
In most of the service systems considered so far in queuing theory, no fresh customer is admitted to a batch undergoing service when the number in the batch is less than a threshold. However, a few researchers considered the case of customers accessing ongoing service batch, irrespective of how long service was provided to that batch. A queuing system with a different kind of accessibility that relates to a real situation is studied in the paper. Consider a single server queuing system in which the service process comprises of k stages. Customers can enter the system for service from a node at the beginning of any of these stages (provided the pre-determined maximum service batch size is not reached) but cannot leave the system after completion of service in any of the intermediate stages. The customer arrivals to the first node occur according to a Markovian Arrival Process (MAP). An infinite waiting room is provided at this node. At all other nodes, with finite waiting rooms (waiting capacity cj,2≤j≤k), customer arrivals occur according to distinct Poisson processes with rates λj,2≤j≤k. The service is provided according to a general bulk service rule, i.e., the service process is initiated only if at least a customers are present in the queue at node 1 and the maximum service batch size is b. Customers can join for service from any of the subsequent nodes, provided the number undergoing service is less than b. The service time distribution in each phase is exponential with service rate μjm, which depends on the service stage j,1≤j≤k, and the size of the batch m,a≤m≤b. The behavior of the system in steady-state is analyzed and some important system characteristics are derived. A numerical example is presented to illustrate the applicability of the results obtained. Full article
(This article belongs to the Special Issue Stochastic Modeling and Applied Probability)
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<p>Queueing system with four stages of service, with accessibility to service from the beginning of any stage.</p> Full article ">Figure 2
<p>Effect of weighted average service rate on system characteristics in Scenario II.</p> Full article ">
23 pages, 872 KiB  
Article
Assessing the Impact of Attendance Modality on the Learning Performance of a Course on Machines and Mechanisms Theory
by David Valiente, Héctor Campello-Vicente, Emilio Velasco-Sánchez, Fernando Rodríguez-Mas and Nuria Campillo-Davo
Mathematics 2021, 9(5), 558; https://doi.org/10.3390/math9050558 - 6 Mar 2021
Cited by 2 | Viewed by 2688
Abstract
University education approaches related to the field of science, technology, engineering and mathematics (STEM), have generally particularized on teaching activity and learning programs which are commonly understood as reoriented lessons that fuse theoretic concepts interweaved with practical activities. In this context, team work [...] Read more.
University education approaches related to the field of science, technology, engineering and mathematics (STEM), have generally particularized on teaching activity and learning programs which are commonly understood as reoriented lessons that fuse theoretic concepts interweaved with practical activities. In this context, team work has been widely acknowledged as a means to conduct practical and hands-on lessons, and has been revealed to be successful in the achievement of exercise resolution and design tasks. Besides this, methodologies sustained by ICT resources such as online or blended approaches, have also reported numerous benefits for students’ active learning. However, such benefits have to be fully validated within the particular teaching context, which may facilitate student achievement to a greater or lesser extent. In this work, we analyze the impact of attendance modalities on the learning performance of a STEM-related course on “Machines and Mechanisms Theory”, in which practical lessons are tackled through a team work approach. The validity of the results is reinforced by group testing and statistical tests with a sample of 128 participants. Students were arranged in a test group (online attendance) and in a control group (face-to-face attendance) to proceed with team work during the practical lessons. Thus, the efficacy of distance and in situ methodologies is compared. Moreover, additional variables have also been compared according to the historical record of the course, in regards to previous academic years. Finally, students’ insights about the collaborative side of this program, self-knowledge and satisfaction with the proposal have also been reported by a custom questionnaire. The results demonstrate greater performance and satisfaction amongst participants in the face-to-face modality. Such a modality is prooven to be statistically significant for the final achievement of students in detriment to online attendance. Full article
(This article belongs to the Special Issue Recent Advances in STEM Education)
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<p>Schedule of the course along the semester (15 weeks). It comprises theory lessons and practical lessons (laboratory hands-on lessons and simulation lessons) throughout 3 units of content.</p> Full article ">Figure 2
<p>Resources devoted to the teaching of the subject during the last six academic years.</p> Full article ">Figure 3
<p>Example of practical activity. (<b>a</b>) Exercise statement. (<b>b</b>) Exercise resolution.</p> Full article ">Figure 4
<p>Schemes of the mockups tested during the hands-on sessions in the laboratory. (<b>a</b>) Four bars mechanism. (<b>b</b>) Cam-crank mechanism. (<b>c</b>) Double-sliding mechanism.</p> Full article ">Figure 5
<p>Example of practical activity. (<b>a</b>) Simulation of a mechanism obtained with Matlab. (<b>b</b>) Entire operation of the mechanism in the mechanic’s laboratory.</p> Full article ">Figure 6
<p>Participants responses to the questionnaire (average values, [1–5]). (<b>a</b>) Initial pass in week 3. (<b>b</b>) Final pass in week 15. Legend: <span style="color: #483D8B">⯀</span> Q1; <span style="color: #4682B4">⯀</span> Q2; <span style="color: #BDB76B">⯀</span> Q3; <span style="color: #FFFF00">⯀</span> Q4.</p> Full article ">Figure 7
<p>Time dedicated (min) by students to finish the practical activities. Legend: <span style="color: #483D8B">⯀</span> Google Groups (test group); <span style="color: #4682B4">⯀</span> Google Hangouts (test group); <span style="color: #FFFF00">⯀</span> Face-to-face (control group).</p> Full article ">Figure 8
<p>(<b>a</b>) Historic mean marks [0–10]+<span class="html-italic">σ</span>. Legend: <span style="color: #483D8B">⯀</span> the whole set of students (pass and fail marks); <span style="color: #FFFF00">⯀</span> only students who pass. (<b>b</b>) Historic marks distributed by ranks. Legend: <span style="color: #483D8B">⯀</span> [0–5]; <span style="color: #4682B4">⯀</span> [5–6]; <span style="color: #66CDAA">⯀</span> [6–7]; <span style="color: #DAA520">⯀</span> [7–9]; <span style="color: #FFFF00">⯀</span> [9–10].</p> Full article ">Figure 9
<p>(<b>a</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math> percentages of only students who take the exam. Legend: <span style="color: #483D8B">⯀</span> <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #4682B4">⯀</span><math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math>-repeaters; <span style="color: #BDB76B">⯀</span><math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #FFFF00">⯀</span><math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math>-repeaters. (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math> percentages of all students (including dropout). Legend: <span style="color: #483D8B">⯀</span><math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #4169E1">⯀</span><math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math>-repeaters; <span style="color: #4682B4">⯀</span><math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #3CB371">⯀</span><math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math>-repeaters; <span style="color: #DAA520">⯀</span><math display="inline"><semantics> <mrow> <mi>D</mi> <mi>r</mi> <mi>o</mi> <mi>p</mi> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #FFFF00">⯀</span><math display="inline"><semantics> <mrow> <mi>D</mi> <mi>r</mi> <mi>o</mi> <mi>p</mi> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </semantics></math>-repeaters.</p> Full article ">Figure 10
<p>(<b>a</b>) Comparison of mean marks [0–10]+<math display="inline"><semantics> <mi>σ</mi> </semantics></math> between test group (online attendance) and control group (face-to-face attendance). Legend: <span style="color: #483D8B">⯀</span> the whole set of students (pass and fail marks); <span style="color: #FFFF00">⯀</span> only students who pass. (<b>b</b>) Comparison of marks distribution by rank between test group and control group: Legend: <span style="color: #483D8B">⯀</span> [0–5]; <span style="color: #4682B4">⯀</span> [5–6]; <span style="color: #20B2AA">⯀</span> [6–7]; <span style="color: #FFFF00">⯀</span> [7–9]; <span style="color: #DAA520">⯀</span> [9–10].</p> Full article ">Figure 11
<p>(<b>a</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math> percentage comparison between test group (online attendance) and control group (face-to-face attendance) with only students who take the exam. Legend: <span style="color: #483D8B">⯀</span><math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #4682B4">⯀</span><math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math>-repeaters; <span style="color: #BDB76B">⯀</span><math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #FFFF00">⯀</span><math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math>-repeaters. (<b>b</b>) <math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math> percentage comparison of all students (including dropout) in the test and control groups: Legend: <span style="color: #483D8B">⯀</span><math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #4169E1">⯀</span><math display="inline"><semantics> <mrow> <mi>P</mi> <mi>a</mi> <mi>s</mi> <mi>s</mi> </mrow> </semantics></math>-repeaters; <span style="color: #4682B4">⯀</span><math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #8FBC8F">⯀</span><math display="inline"><semantics> <mrow> <mi>F</mi> <mi>a</mi> <mi>i</mi> <mi>l</mi> </mrow> </semantics></math>-repeaters; <span style="color: #DAA520">⯀</span>:<math display="inline"><semantics> <mrow> <mi>D</mi> <mi>r</mi> <mi>o</mi> <mi>p</mi> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </semantics></math>-non-repeaters; <span style="color: #FFFF00">⯀</span><math display="inline"><semantics> <mrow> <mi>D</mi> <mi>r</mi> <mi>o</mi> <mi>p</mi> <mi>o</mi> <mi>u</mi> <mi>t</mi> </mrow> </semantics></math>-repeaters.</p> Full article ">Figure 12
<p>Anova test results between final marks and number of Individual Assignments. Legend: <span style="color: #0000FF"><b>—</b></span> boxplot for each group; <span style="color: #FF0000"><b>—</b></span> median value; <span class="html-fig-inline" id="mathematics-09-00558-i001"> <img alt="Mathematics 09 00558 i001" src="/mathematics/mathematics-09-00558/article_deploy/html/images/mathematics-09-00558-i001.png"/></span> standard deviation; <span style="color: #FF0000">+</span> outliers.</p> Full article ">
30 pages, 661 KiB  
Article
Designing Tasks for a Dynamic Online Environment: Applying Research into Students’ Difficulties with Linear Equations
by Morten Elkjær and Uffe Thomas Jankvist
Mathematics 2021, 9(5), 557; https://doi.org/10.3390/math9050557 - 6 Mar 2021
Cited by 1 | Viewed by 4643
Abstract
Despite almost half a century of research into students’ difficulties with solving linear equations, these difficulties persist in everyday mathematics classes around the world. Furthermore, the difficulties reported decades ago are the same ones that persist today. With the immense number of dynamic [...] Read more.
Despite almost half a century of research into students’ difficulties with solving linear equations, these difficulties persist in everyday mathematics classes around the world. Furthermore, the difficulties reported decades ago are the same ones that persist today. With the immense number of dynamic online environments for mathematics teaching and learning that are emerging today, we are presented with a perhaps unique opportunity to do something about this. This study sets out to apply the research on lower secondary school students’ difficulties with equation solving, in order to eventually inform students’ personalised learning through a specific task design in a particular dynamic online environment (matematikfessor.dk). In doing so, task design theory is applied, particularly variation theory. The final design we present consists of eleven general equation types—ten types of arithmetical equations and one type of algebraic equation—and a broad range of variations of these, embedded in a potential learning-trajectory-tree structure. Besides establishing this tree structure, the main theoretical contribution of the study and the task design we present is the detailed treatment of the category of arithmetical equations, which also involves a new distinction between simplified and non-simplified arithmetical equations. Full article
(This article belongs to the Special Issue Designing Tasks within Dynamic and Interactive Mathematics Learning Environments)
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<p>Simplified arithmetical equations.</p> Full article ">
16 pages, 462 KiB  
Article
Gaussian Pseudorandom Number Generator Using Linear Feedback Shift Registers in Extended Fields
by Guillermo Cotrina, Alberto Peinado and Andrés Ortiz
Mathematics 2021, 9(5), 556; https://doi.org/10.3390/math9050556 - 6 Mar 2021
Cited by 7 | Viewed by 3287
Abstract
A new proposal to generate pseudorandom numbers with Gaussian distribution is presented. The generator is a generalization to the extended field GF(2n) of the one using cyclic rotations of linear feedback shift registers (LFSRs) originally defined in [...] Read more.
A new proposal to generate pseudorandom numbers with Gaussian distribution is presented. The generator is a generalization to the extended field GF(2n) of the one using cyclic rotations of linear feedback shift registers (LFSRs) originally defined in GF(2). The rotations applied to LFSRs in the binary case are no longer needed in the extended field due to the implicit rotations found in the binary equivalent model of LFSRs in GF(2n). The new proposal is aligned with the current trend in cryptography of using extended fields as a way to speed up the bitrate of the pseudorandom generators. This proposal allows the use of LFSRs in cryptography to be taken further, from the generation of the classical uniformly distributed sequences to other areas, such as quantum key distribution schemes, in which sequences with Gaussian distribution are needed. The paper contains the statistical analysis of the numbers produced and a comparison with other Gaussian generators. Full article
(This article belongs to the Special Issue Mathematics Cryptography and Information Security)
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<p>A Linear Feedback Shift Register of length <span class="html-italic">m</span>.</p> Full article ">Figure 2
<p>Binary equivalent model of an LFSR in <math display="inline"><semantics> <mrow> <mi>G</mi> <mi>F</mi> <mo>(</mo> <msup> <mn>2</mn> <mi>n</mi> </msup> <mo>)</mo> </mrow> </semantics></math>.</p> Full article ">Figure 3
<p>Gaussian number generator proposed.</p> Full article ">Figure 4
<p>Quantiles plot against normal distribution for <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>4</mn> </msub> </mrow> <mn>4</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>5</mn> </msub> </mrow> <mn>8</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>7</mn> </msub> </mrow> <mn>4</mn> </msub> </semantics></math> respectively.</p> Full article ">Figure 5
<p>Plots of the cumulative distribution function (CDF) of list against the CDF of a normal distribution for <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>6</mn> </msub> </mrow> <mn>7</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>8</mn> </msub> </mrow> <mn>6</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>8</mn> </msub> </mrow> <mn>8</mn> </msub> </semantics></math> respectively.</p> Full article ">Figure 6
<p>Plots of the histogrmas of the obtained results against their corresponding in a normal distribution for <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>5</mn> </msub> </mrow> <mn>7</mn> </msub> </semantics></math>, <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>6</mn> </msub> </mrow> <mn>8</mn> </msub> </semantics></math> and <math display="inline"><semantics> <msub> <mrow> <msub> <mi>κ</mi> <mn>8</mn> </msub> </mrow> <mn>8</mn> </msub> </semantics></math> respectively.</p> Full article ">
17 pages, 487 KiB  
Article
On the Discretization of Continuous Probability Distributions Using a Probabilistic Rounding Mechanism
by Chénangnon Frédéric Tovissodé, Sèwanou Hermann Honfo, Jonas Têlé Doumatè and Romain Glèlè Kakaï
Mathematics 2021, 9(5), 555; https://doi.org/10.3390/math9050555 - 6 Mar 2021
Cited by 4 | Viewed by 3767
Abstract
Most existing flexible count distributions allow only approximate inference when used in a regression context. This work proposes a new framework to provide an exact and flexible alternative for modeling and simulating count data with various types of dispersion (equi-, under-, and over-dispersion). [...] Read more.
Most existing flexible count distributions allow only approximate inference when used in a regression context. This work proposes a new framework to provide an exact and flexible alternative for modeling and simulating count data with various types of dispersion (equi-, under-, and over-dispersion). The new method, referred to as “balanced discretization”, consists of discretizing continuous probability distributions while preserving expectations. It is easy to generate pseudo random variates from the resulting balanced discrete distribution since it has a simple stochastic representation (probabilistic rounding) in terms of the continuous distribution. For illustrative purposes, we develop the family of balanced discrete gamma distributions that can model equi-, under-, and over-dispersed count data. This family of count distributions is appropriate for building flexible count regression models because the expectation of the distribution has a simple expression in terms of the parameters of the distribution. Using the Jensen–Shannon divergence measure, we show that under the equidispersion restriction, the family of balanced discrete gamma distributions is similar to the Poisson distribution. Based on this, we conjecture that while covering all types of dispersions, a count regression model based on the balanced discrete gamma distribution will allow recovering a near Poisson distribution model fit when the data are Poisson distributed. Full article
(This article belongs to the Special Issue Stochastic Processes and Their Applications)
►▼ Show Figures

Figure 1

Figure 1
<p>Probability mass plots for the balanced discrete gamma distributions with mean values <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics></math> (left panel) and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> (right panel) and scales <span class="html-italic">a</span> selected to yield an index of dispersion (ID, variance-to-mean ratio) of ID = 4 (bottom row), ID = 1 (central row), and ID = 0.5 (top row).</p> Full article ">Figure 2
<p>Box plots for the balanced discrete gamma distributions with mean values <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>2.5</mn> </mrow> </semantics></math> and <math display="inline"><semantics> <mrow> <mi>μ</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math> and an index of dispersion (ID, variance-to-mean ratio) of ID = 4, ID = 1, and ID = 0.5. The thick vertical bar inside the interquartile range (i.e., the rectangular box that has sides that are 25% (left side) and 75% (right side) quartiles) is the median of the distribution.</p> Full article ">Figure 3
<p>Index of dispersion (ID, variance-to-mean ratio) of the balanced discrete gamma distribution against the mean value (<math display="inline"><semantics> <mi>μ</mi> </semantics></math>) for selected scale parameter (<span class="html-italic">a</span>) values in the range <math display="inline"><semantics> <mfenced separators="" open="[" close="]"> <mn>1</mn> <mo>,</mo> <mn>1000</mn> </mfenced> </semantics></math> mostly corresponding to equidispersion and underdispersion (<math display="inline"><semantics> <mrow> <mn>0</mn> <mo>&lt;</mo> <mi>ID</mi> <mo>≤</mo> <mn>1</mn> </mrow> </semantics></math>).</p> Full article ">Figure 4
<p>Comparison of the cumulative distribution functions of the balanced discrete gamma and discrete concentration of gamma distributions based on a continuous gamma distribution with scale <math display="inline"><semantics> <mrow> <mi>a</mi> <mo>=</mo> <mn>1</mn> </mrow> </semantics></math> and shape <math display="inline"><semantics> <mrow> <mi>b</mi> <mo>=</mo> <mn>5</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Jensen–Shannon divergence (JSD in bit) measured between the balanced discrete gamma (BDG) distribution and the corresponding discrete concentration (<b>A</b>) and between the Poisson and the BDG distributions under both latent equidispersion (one-parameter BDG distribution) and marginal equidispersion (unit variance-to-mean ratio) restrictions (<b>B</b>), against the mean value (<math display="inline"><semantics> <mi>μ</mi> </semantics></math>).</p> Full article ">
13 pages, 384 KiB  
Article
A Numerical Comparison of the Sensitivity of the Geometric Mean Method, Eigenvalue Method, and Best–Worst Method
by Jiří Mazurek, Radomír Perzina, Jaroslav Ramík and David Bartl
Mathematics 2021, 9(5), 554; https://doi.org/10.3390/math9050554 - 5 Mar 2021
Cited by 14 | Viewed by 2347
Abstract
In this paper, we compare three methods for deriving a priority vector in the theoretical framework of pairwise comparisons—the Geometric Mean Method (GMM), Eigenvalue Method (EVM) and Best–Worst Method (BWM)—with respect to two features: sensitivity and order violation. As the research method, we [...] Read more.
In this paper, we compare three methods for deriving a priority vector in the theoretical framework of pairwise comparisons—the Geometric Mean Method (GMM), Eigenvalue Method (EVM) and Best–Worst Method (BWM)—with respect to two features: sensitivity and order violation. As the research method, we apply One-Factor-At-a-Time (OFAT) sensitivity analysis via Monte Carlo simulations; the number of compared objects ranges from 3 to 8, and the comparison scale coincides with Saaty’s fundamental scale from 1 to 9 with reciprocals. Our findings suggest that the BWM is, on average, significantly more sensitive statistically (and thus less robust) and more susceptible to order violation than the GMM and EVM for every examined matrix (vector) size, even after adjustment for the different numbers of pairwise comparisons required by each method. On the other hand, differences in sensitivity and order violation between the GMM and EMM were found to be mostly statistically insignificant. Full article
(This article belongs to the Special Issue Mathematical Methods for Operations Research Problems)
►▼ Show Figures

Figure 1

Figure 1
<p>Average order violation: BWM, EVM, and GMM.</p> Full article ">Figure 2
<p>Mean sensitivity: BWM, EVM, and GMM.</p> Full article ">Figure 3
<p>Mean sensitivity: BWM adjusted, EVM, and GMM.</p> Full article ">Figure 4
<p>Sensitivity distribution: all methods, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>4</mn> </mrow> </semantics></math>.</p> Full article ">Figure 5
<p>Sensitivity distribution: all methods, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>.</p> Full article ">Figure 6
<p>Sensitivity distribution: all methods, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>8</mn> </mrow> </semantics></math>.</p> Full article ">Figure 7
<p>Sensitivity of the BWM: frequency distribution, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>.</p> Full article ">Figure 8
<p>Sensitivity of the GMM: frequency distribution, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>.</p> Full article ">Figure 9
<p>Sensitivity of the EVM: frequency distribution, <math display="inline"><semantics> <mrow> <mi>n</mi> <mo>=</mo> <mn>6</mn> </mrow> </semantics></math>.</p> Full article ">
14 pages, 292 KiB  
Article
Ternary Menger Algebras: A Generalization of Ternary Semigroups
by Anak Nongmanee and Sorasak Leeratanavalee
Mathematics 2021, 9(5), 553; https://doi.org/10.3390/math9050553 - 5 Mar 2021
Cited by 8 | Viewed by 2418
Abstract
Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups [...] Read more.
Let n be a fixed natural number. Menger algebras of rank n, which was introduced by Menger, K., can be regarded as the suitable generalization of arbitrary semigroups. Based on this knowledge, an interesting question arises: what a generalization of ternary semigroups is. In this article, we first introduce the notion of ternary Menger algebras of rank n, which is a canonical generalization of arbitrary ternary semigroups, and discuss their related properties. In the second part, we establish the so-called a diagonal ternary semigroup which its operation is induced by the operation on ternary Menger algebras of rank n and then investigate their interesting properties. Moreover, we introduce the concept of homomorphism and congruences on ternary Menger algebras of rank n. These lead us to study the quotient ternary Menger algebras of rank n and to investigate the homomorphism theorem for ternary Menger algebra of rank n with respect to congruences. Furthermore, the characterization of reduction of ternary Menger algebra into Menger algebra is presented. Full article
(This article belongs to the Special Issue Algebra and Number Theory)
18 pages, 340 KiB  
Article
Oscillation Criteria for Third-Order Nonlinear Neutral Dynamic Equations with Mixed Deviating Arguments on Time Scales
by Zhiyu Zhang, Ruihua Feng, Irena Jadlovská and Qingmin Liu
Mathematics 2021, 9(5), 552; https://doi.org/10.3390/math9050552 - 5 Mar 2021
Cited by 4 | Viewed by 1625
Abstract
Under a couple of canonical and mixed canonical-noncanonical conditions, we investigate the oscillation and asymptotic behavior of solutions to a class of third-order nonlinear neutral dynamic equations with mixed deviating arguments on time scales. By means of the double Riccati transformation and the [...] Read more.
Under a couple of canonical and mixed canonical-noncanonical conditions, we investigate the oscillation and asymptotic behavior of solutions to a class of third-order nonlinear neutral dynamic equations with mixed deviating arguments on time scales. By means of the double Riccati transformation and the inequality technique, new oscillation criteria are established, which improve and generalize related results in the literature. Several examples are given to illustrate the main results. Full article
(This article belongs to the Special Issue Oscillation Theory for Differential Equations)
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