World Academy of Science, Engineering and Technology

Authors: Francis O. Nwawuru

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The convergence analysis of split monotone inclusion problems and fixed point problems of certain nonlinear mappings are investigated in the setting of real Hilbert spaces. Inertial extrapolation term in the spirit of Polyak is incorporated to speed up the rate of convergence. Under standard assumptions, a strong convergence of the proposed algorithm is established without computing the resolvent operator or involving Yosida approximation method. The stepsize involved in the algorithm does not depend on the spectral radius of the linear operator. Furthermore, applications of the proposed algorithm in solving some related optimization problems are also considered. Our result complements and extends numerous results in the literature.

Keywords: fixedpoint, hilbertspace, monotonemapping, resolventoperators

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Authors: Shunpu Zhang

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We propose a new multiple test called the minPOP test for testing multiple hypotheses simultaneously. Under the assumption that the test statistics are independent, we show that the minPOP test has higher global power than the existing multiple testing methods. We further propose a stepwise multiple-testing procedure based on the minPOP test and two of its modified versions (the Double Truncated and Left Truncated minPOP tests). We show that these multiple tests have strong control of the family-wise error rate (FWER). A method for finding the p-values of the proposed tests after adjusting for multiplicity is also developed. Simulation results show that the Double Truncated and Left Truncated minPOP tests, in general, have a higher number of rejections than the existing multiple testing procedures.

Keywords: multiple test, single-step procedure, stepwise procedure, p-value for multiple testing

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Authors: Arum Kingsley Chinedu, Ugwuowo Fidelis Ifeanyi, Oranye Henrietta Ebele

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The Poisson regression model (PRM) is a nonlinear model that belongs to the exponential family of distribution. PRM is suitable for studying count variables using appropriate covariates and sometimes experiences the problem of multicollinearity in the explanatory variables and outliers on the response variable. This study aims to address the problem of multicollinearity and outliers jointly in a Poisson regression model. We developed an estimator called the robust modified jackknife PCKL parameter estimator by combining the principal component estimator, modified jackknife KL and transformed M-estimator estimator to address both problems in a PRM. The superiority conditions for this estimator were established, and the properties of the estimator were also derived. The estimator inherits the characteristics of the combined estimators, thereby making it efficient in addressing both problems. And will also be of immediate interest to the research community and advance this study in terms of novelty compared to other studies undertaken in this area. The performance of the estimator (robust modified jackknife PCKL) with other existing estimators was compared using mean squared error (MSE) as a performance evaluation criterion through a Monte Carlo simulation study and the use of real-life data. The results of the analytical study show that the estimator outperformed other existing estimators compared with by having the smallest MSE across all sample sizes, different levels of correlation, percentages of outliers and different numbers of explanatory variables.

Keywords: jackknife modified KL, outliers, multicollinearity, principal component, transformed M-estimator.

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Authors: Liudmyla Vesper

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The problems of modern education are large-scale and diverse. The aspirations of parents, teachers, and experts converge - everyone interested in growing up a generation of whole, well-educated persons. Both the family and society are expected in the future generation to be self-sufficient, desirable in the labor market, and capable of lifelong learning. Today's children have a powerful potential that is difficult to realize in the conditions of traditional school approaches. Focusing on STEM education in practice often ends with the simple use of computers and gadgets during class. "Science", "technology", "engineering" and "mathematics" are difficult to combine within school and university curricula, which have not changed much during the last 10 years. Solving the problems of modern education largely depends on teachers - innovators, teachers - practitioners who develop and implement effective educational methods and programs. Teachers who propose innovative pedagogical practices that allow students to master large-scale knowledge and apply it to the practical plane. Effective education considers the creation of stable neural structures during the learning process, which allow to preserve and increase knowledge throughout life. The author proposed a method of integrated lessons – cases based on the maths patterns for forming a holistic perception of the world. This method and program are scientifically substantiated and have more than 15 years of practical application experience in school and student classrooms. The first results of the practical application of the author's methodology and curriculum were announced at the International Conference "Teaching and Learning Strategies to Promote Elementary School Success", 2006, April 22-23, Yerevan, Armenia, IREX-administered 2004-2006 Multiple Component Education Project. This program is based on the concept of interdisciplinary connections and its implementation in the process of continuous learning. This allows students to save and increase knowledge throughout life according to a single pattern. The pattern principle stores information on different subjects according to one scheme (pattern), using long-term memory. This is how neural structures are created. The author also admits that a similar method can be successfully applied to the training of artificial intelligence neural networks. However, this assumption requires further research and verification. The educational method and program proposed by the author meet the modern requirements for education, which involves mastering various areas of knowledge, starting from an early age. This approach makes it possible to involve the child's cognitive potential as much as possible and direct it to the preservation and development of individual talents. According to the methodology, at the early stages of learning students understand the connection between school subjects (so-called "sciences" and "humanities") and in real life, apply the knowledge gained in practice. This approach allows students to realize their natural creative abilities and talents, which makes it easier to navigate professional choices and find their place in life.

Keywords: science education, maths education, AI, neuroplasticity, innovative education problem, creativity development, modern education problem

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Authors: Hiba El Assibi

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This study aims to weakly solve Kalah, a two-player board game, by developing a start-to-finish winning strategy using an optimized Minimax algorithm with Alpha-Beta Pruning. In weakly solving Kalah, our focus is on creating an optimal strategy from the game's beginning rather than analyzing every possible position. The project will explore additional enhancements like symmetry checking and code optimizations to speed up the decision-making process. This approach is expected to give insights into efficient strategy formulation in board games and potentially help create games with a fair distribution of outcomes. Furthermore, this research provides a unique perspective on human versus Artificial Intelligence decision-making in strategic games. By comparing the AI-generated optimal moves with human choices, we can explore how seemingly advantageous moves can, in the long run, be harmful, thereby offering a deeper understanding of strategic thinking and foresight in games. Moreover, this paper discusses the evaluation of our strategy against existing methods, providing insights on performance and computational efficiency. We also discuss the scalability of our approach to the game, considering different board sizes (number of pits and stones) and rules (different variations) and studying how that affects performance and complexity. The findings have potential implications for the development of AI applications in strategic game planning, enhancing our understanding of human cognitive processes in game settings, and offer insights into creating balanced and engaging game experiences.

Keywords: minimax, alpha beta pruning, transposition tables, weakly solving, game theory

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Authors: Mary Ann Cristine R. Olgado

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Diversity in the classroom might make it difficult to promote individualized learning, but differentiated instruction that caters to students' various learning preferences may prove to be beneficial. Hence, this study examined the effectiveness of Hybrid-Differentiated Mathematics Instruction (HDMI) in improving the students’ academic performance in Mathematics. It employed the quasi-experimental research design by using a comparative analysis of the two variables: the experimental and control groups. The learning styles of the students were identified using the Grasha-Riechmann Student Learning Style Scale (GRSLSS), which served as the basis for designing differentiated action plans in Mathematics. In addition, adapted survey questionnaires, pre-tests, and post-tests were used to gather information and were analyzed using descriptive and correlational statistics to find the relationship between variables. The experimental group received differentiated instruction for a month, while the control group received traditional teaching instruction. The study found that Hybrid-Differentiated Mathematics Instruction (HDMI) improved the academic performance of Grade 11-TVL students, with the experimental group performing better than the control group. This program has effectively tailored the teaching methods to meet the diverse learning needs of the students, fostering and enhancing a deeper understanding of mathematical concepts in Statistics & Probability, both within and beyond the classroom.

Keywords: differentiated instruction, hybrid, learning styles, academic performance

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Authors: Samaneh Rahimirshnani, Hossein Jafari

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In this paper, we consider fluid queueing processes fed by an isonormal Gaussian process. We study the correlation structure of the queueing process and the rate of convergence of the running supremum in the queueing process. The Malliavin calculus techniques are applied to obtain relations that show the workload process inherits the dependence properties of the input process. As examples, we consider two isonormal Gaussian processes, the sub-fractional Brownian motion (SFBM) and the fractional Brownian motion (FBM). For these examples, we obtain upper bounds for the covariance function of the queueing process and its rate of convergence to zero. We also discover that the rate of convergence of the queueing process is related to the structure of the covariance function of the input process.

Keywords: queue length process, Malliavin calculus, covariance function, fractional Brownian motion, sub-fractional Brownian motion

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Authors: Bastien Batardière, Joon Kwon

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For finite-sum optimization, variance-reduced gradient methods (VR) compute at each iteration the gradient of a single function (or of a mini-batch), and yet achieve faster convergence than SGD thanks to a carefully crafted lower-variance stochastic gradient estimator that reuses past gradients. Another important line of research of the past decade in continuous optimization is the adaptive algorithms such as AdaGrad, that dynamically adjust the (possibly coordinate-wise) learning rate to past gradients and thereby adapt to the geometry of the objective function. Variants such as RMSprop and Adam demonstrate outstanding practical performance that have contributed to the success of deep learning. In this work, we present AdaLVR, which combines the AdaGrad algorithm with loopless variance-reduced gradient estimators such as SAGA or L-SVRG that benefits from a straightforward construction and a streamlined analysis. We assess that AdaLVR inherits both good convergence properties from VR methods and the adaptive nature of AdaGrad: in the case of L-smooth convex functions we establish a gradient complexity of O(n + (L + √ nL)/ε) without prior knowledge of L. Numerical experiments demonstrate the superiority of AdaLVR over state-of-the-art methods. Moreover, we empirically show that the RMSprop and Adam algorithm combined with variance-reduced gradients estimators achieve even faster convergence.

Keywords: convex optimization, variance reduction, adaptive algorithms, loopless

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Authors: Yingxu Wang, Jianhua Lu, Jun Peng, Jiawei Zhang

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The fundamental demand in contemporary intelligent science towards Autonomous AI (AI*) is the creation of unprecedented formal means of Intelligent Mathematics (IM). It is discovered that natural intelligence is inductively created rather than exhaustively trained. Therefore, IM is a family of algebraic and denotational mathematics encompassing Inference Algebra, Real-Time Process Algebra, Concept Algebra, Semantic Algebra, Visual Frame Algebra, etc., developed in our labs. IM plays indispensable roles in training-free AI* theories and systems beyond traditional empirical data-driven technologies. A set of applications of IM-driven AI* systems will be demonstrated in contemporary intelligence science, AI*, and cognitive computers.

Keywords: intelligence mathematics, foundations of intelligent science, autonomous AI, cognitive computers, inference algebra, real-time process algebra, concept algebra, semantic algebra, applications

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Authors: Yingxu Wang

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This work presents basic research on the recursive structures and dual sequential patterns of primes for the formal proof of the Twin-Prime Conjecture (TPC). A rigorous methodology of Twin-Prime Decomposition (TPD) is developed in MATLAB to inductively verify potential twins in the dual sequences of primes. The key finding of this basic study confirms that the dual sequences of twin primes are not only symmetric but also infinitive in the unique base 6 cycle, except a limited subset of potential pairs is eliminated by the lack of dual primality. Both theory and experiments have formally proven that the infinity of twin primes stated in TPC holds in the P x P space.

Keywords: number theory, primes, twin-prime conjecture, dual primes (P x P), twin prime decomposition, formal proof, algorithm

Procedia PDF Downloads 63

Authors: Sharon-Yasotha Veerayah-Mcgregor, Valipuram Manoranjan

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A backward or inverse problem is known to be an ill-posed problem due to its instability that easily emerges with any slight change within the conditions of the problem. Therefore, only a limited number of numerical approaches are available to solve a backward problem. This paper considers the Nagumo equation, an equation that describes impulse propagation in nerve axons, which also models population growth with the Allee effect. A creative operator splitting numerical scheme is constructed to solve the inverse Nagumo equation. Computational simulations are used to verify that this scheme is stable, accurate, and efficient.

Keywords: inverse/backward equation, operator-splitting, Nagumo equation, ill-posed, finite-difference

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Authors: Medine Demir, Volker John

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Fluid equations in a rotating frame of reference have a broad class of important applications in meteorology and oceanography, especially in the large-scale flows considered in ocean and atmosphere, as well as many physical and industrial applications. The Coriolis and the centripetal forces, resulting from the rotation of the earth, play a crucial role in such systems. For such applications it may be required to solve the system in complex three-dimensional geometries. In recent years, the Navier--Stokes equations in a rotating frame have been investigated in a number of papers using the classical inf-sup stable mixed methods, like Taylor-Hood pairs, to contribute to the analysis and the accurate and efficient numerical simulation. Numerical analysis reveals that these classical methods introduce a pressure-dependent contribution in the velocity error bounds that is proportional to some inverse power of the viscosity. Hence, these methods are optimally convergent but small velocity errors might not be achieved for complicated pressures and small viscosity coefficients. Several approaches have been proposed for improving the pressure-robustness of pairs of finite element spaces. In this contribution, a pressure-robust space discretization of the incompressible Navier--Stokes equations in a rotating frame of reference is considered. The discretization employs divergence-free, $H^1$-conforming mixed finite element methods like Scott--Vogelius pairs. However, this approach might come with a modification of the meshes, like the use of barycentric-refined grids in case of Scott--Vogelius pairs. However, this strategy requires the finite element code to have control on the mesh generator which is not realistic in many engineering applications and might also be in conflict with the solver for the linear system. An error estimate for the velocity is derived that tracks the dependency of the error bound on the coefficients of the problem, in particular on the angular velocity. Numerical examples illustrate the theoretical results. The idea of pressure-robust method could be cast on different types of flow problems which would be considered as future studies. As another future research direction, to avoid a modification of the mesh, one may use a very simple parameter-dependent modification of the Scott-Vogelius element, the pressure-wired Stokes element, such that the inf-sup constant is independent of nearly-singular vertices.

Keywords: navier-stokes equations in a rotating frame of refence, coriolis force, pressure-robust error estimate, scott-vogelius pairs of finite element spaces

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Authors: Liberto Pombal, Christian Dieter Jaekel

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The objective of this work is to establish theoretical and numerical conditions for Solving Extended Linear Complementarity Problems (XLCP), with emphasis on the Horizontal Linear Complementarity Problem (HLCP). Two new strategies for solving complementarity problems are presented, using differentiable and penalized functions, which resulted in a natural formalization for the Linear Horizontal case. The computational results of all suggested strategies are also discussed in depth in this paper. The implication in practice allows solving and optimizing, in an innovative way, the (forestry) problems of the value chain of the industrial wood sector in Angola.

Keywords: complementarity, box constrained, optimality conditions, wood and environment

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Authors: Yingxu Wang

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The base 6 structure and properties of mirror primes are discovered in this work towards the proof of Goldbach Conjecture. This paper reveals a fundamental pattern on pairs of mirror primes adjacent to any even number nₑ > 2 with symmetrical distances on both sides determined by a methodology of Mirror Prime Decomposition (MPD). MPD leads to a formal proof of the Goldbach conjecture, which states that the conjecture holds because any pivot even number, nₑ > 2, is a sum of at least an adjacent pair of primes divided by 2. This work has not only revealed the analytic pattern of base 6 primes but also proven the infinitive validation of the Goldbach conjecture.

Keywords: number theory, primes, mirror primes, double recursive patterns, Goldbach conjecture, formal proof, mirror-prime decomposition, applications

Procedia PDF Downloads 49

Authors: Zahid Ullah, Atlas Khan

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The presented research is dedicated to an extensive examination of the regularity properties associated with a specific class of equations, namely extremely degenerate elliptic equations. This study holds significance in unraveling the complexities inherent in these equations and understanding the smoothness of their solutions. The focus is on analyzing the regularity of results, aiming to contribute to the broader field of mathematical theory. By delving into the intricacies of extremely degenerate elliptic equations, the research seeks to advance our understanding beyond conventional analyses, addressing challenges posed by degeneracy and pushing the boundaries of classical analytical methods. The motivation for this exploration lies in the practical applicability of mathematical models, particularly in real-world scenarios where physical phenomena exhibit characteristics that challenge traditional mathematical modeling. The research aspires to fill gaps in the current understanding of regularity properties within solutions to extremely degenerate elliptic equations, ultimately contributing to both theoretical foundations and practical applications in diverse scientific fields.

Keywords: investigating smoothness, extremely degenerate elliptic equations, regularity properties, mathematical analysis, complexity solutions

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Authors: Zahid Ullah, Atlas Khan

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This research endeavors to explore the regularity properties associated with a specific class of equations, namely extremely degenerate elliptic equations. These equations hold significance in understanding complex physical systems like porous media flow, with applications spanning various branches of mathematics. The focus is on unraveling and analyzing regularity results to gain insights into the smoothness of solutions for these highly degenerate equations. Elliptic equations, fundamental in expressing and understanding diverse physical phenomena through partial differential equations (PDEs), are particularly adept at modeling steady-state and equilibrium behaviors. However, within the realm of elliptic equations, the subset of extremely degenerate cases presents a level of complexity that challenges traditional analytical methods, necessitating a deeper exploration of mathematical theory. While elliptic equations are celebrated for their versatility in capturing smooth and continuous behaviors across different disciplines, the introduction of degeneracy adds a layer of intricacy. Extremely degenerate elliptic equations are characterized by coefficients approaching singular behavior, posing non-trivial challenges in establishing classical solutions. Still, the exploration of extremely degenerate cases remains uncharted territory, requiring a profound understanding of mathematical structures and their implications. The motivation behind this research lies in addressing gaps in the current understanding of regularity properties within solutions to extremely degenerate elliptic equations. The study of extreme degeneracy is prompted by its prevalence in real-world applications, where physical phenomena often exhibit characteristics defying conventional mathematical modeling. Whether examining porous media flow or highly anisotropic materials, comprehending the regularity of solutions becomes crucial. Through this research, the aim is to contribute not only to the theoretical foundations of mathematics but also to the practical applicability of mathematical models in diverse scientific fields.

Keywords: elliptic equations, extremely degenerate, regularity results, partial differential equations, mathematical modeling, porous media flow

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Authors: J. Juan Peña, J. Morales, J. García-Ravelo, J. García-Martínez

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It is known that by introducing alternative forms of exponential and logarithmic functions, the Tsallis entropy Sᵩ is itself a generalization of Shannon entropy S. In this work, from a deformation through a scaling function applied to the differential operator, it is possible to generate a q-deformed calculus as well as a q-deformed arithmetic, which not only allows generalizing the exponential and logarithmic functions but also any other standard function. The updated q-deformed differential operator leads to an updated integral operator under which the functions are integrated together with a weight function. For each differentiable function, it is possible to identify its q-deformed partner, which is useful to generalize other algebraic relations proper of the original functions. As an application of this proposal, in this work, a generalization of exponential and logarithmic functions is studied in such a way that their relationship with the thermodynamic functions, particularly the entropy, allows us to have a q-deformed expression of these. As a result, from a particular scaling function applied to the differential operator, a q-deformed arithmetic is obtained, leading to the generalization of the Tsallis entropy.

Keywords: q-calculus, q-deformed arithmetic, entropy, exponential functions, thermodynamic functions

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Authors: Guesh Simretab Gebremedhin, Saumya Rajan Jena

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In this work, a septic B-spline scheme has been used to simplify the process of solving an approximate solution of the generalized Rosenau-regularized long-wave equation (GR-RLWE) with initial boundary conditions. The resulting system of first-order ODEs has dealt with Butcher’s fifth order Runge-Kutta (BFRK) approach without using finite difference techniques for discretizing the time-dependent variables at each time level. Here, no transformation or any kind of linearization technique is employed to tackle the nonlinearity of the equation. Two test problems have been selected for numerical justifications and comparisons with other researchers on the basis of efficiency, accuracy, and results of the two invariants Mᵢ (mass) and Eᵢ (energy) of some motion that has been used to test the conservative properties of the proposed scheme.

Keywords: septic B-spline scheme, Butcher's fifth order Runge-Kutta approach, error norms, generalized Rosenau-RLW equation

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Authors: Sumeyra Karatas, Veysel Karatas, Reyhan Safak, Gamze Bulut-Ozturk, Ozgul Kartal

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Productive struggle entails a student's cognitive exertion to comprehend mathematical concepts and uncover solutions not immediately apparent. The significance of productive struggle in learning mathematics is accentuated by influential educational theorists, emphasizing its necessity for learning mathematics with understanding. Consequently, supporting productive struggle in learning mathematics is recognized as a high-leverage and effective mathematics teaching practice. In this study, the investigation into the role of productive struggle in learning mathematics led to the development of a comprehensive rubric for productive struggle pedagogy through an exhaustive literature review. The rubric consists of eight primary criteria and 37 sub-criteria, providing a detailed description of teacher actions and pedagogical choices that foster students' productive struggles. These criteria encompass various pedagogical aspects, including task design, tool implementation, allowing time for struggle, posing questions, scaffolding, handling mistakes, acknowledging efforts, and facilitating discussion/feedback. Utilizing this rubric, a team of researchers and teachers designed eight 90-minute lesson plans, employing a productive struggle pedagogy, for a two-week unit on solving systems of linear equations. Simultaneously, another set of eight lesson plans on the same topic, featuring identical content and problems but employing a traditional lecture-and-practice model, was designed by the same team. The objective was to assess the impact of supporting productive struggle on students' mathematics learning, defined by the strands of mathematical proficiency. This quasi-experimental study compares the control group, which received traditional lecture- and practice instruction, with the treatment group, which experienced a productive struggle in pedagogy. Sixty-six 10th and 11th-grade students from two algebra classes, taught by the same teacher at a high school, underwent either the productive struggle pedagogy or lecture-and-practice approach over two-week eight 90-minute class sessions. To measure students' learning, an assessment was created and validated by a team of researchers and teachers. It comprised seven open-response problems assessing the strands of mathematical proficiency: procedural and conceptual understanding, strategic competence, and adaptive reasoning on the topic. The test was administered at the beginning and end of the two weeks as pre-and post-test. Students' solutions underwent scoring using an established rubric, subjected to expert validation and an inter-rater reliability process involving multiple criteria for each problem based on their steps and procedures. An analysis of covariance (ANCOVA) was conducted to examine the differences between the control group, which received traditional pedagogy, and the treatment group, exposed to the productive struggle pedagogy, on the post-test scores while controlling for the pre-test. The results indicated a significant effect of treatment on post-test scores for procedural understanding (F(2, 63) = 10.47, p < .001), strategic competence (F(2, 63) = 9.92, p < .001), adaptive reasoning (F(2, 63) = 10.69, p < .001), and conceptual understanding (F(2, 63) = 10.06, p < .001), controlling for pre-test scores. This demonstrates the positive impact of supporting productive struggle in learning mathematics. In conclusion, the results revealed the significance of the role of productive struggle in learning mathematics. The study further explored the practical application of productive struggle through the development of a comprehensive rubric describing the pedagogy of supporting productive struggle.

Keywords: effective mathematics teaching practice, high school algebra, learning mathematics, productive struggle

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Authors: Ozgul Kartal, Wade Tillett, Lyn D. English

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Despite its global significance in curricula, mathematical modeling encounters persistent disparities in recognition and emphasis within regular mathematics classrooms and teacher education across countries with diverse educational and cultural traditions, including variations in the perceived role of mathematical modeling. Over the past two decades, increased attention has been given to the integration of mathematical modeling into national curriculum standards in the U.S. and other countries. Therefore, the mathematics education research community has dedicated significant efforts to investigate various aspects associated with the teaching and learning of mathematical modeling, primarily focusing on exploring the applicability of modeling in schools and assessing students', teachers', and preservice teachers' (PTs) competencies and engagement in modeling cycles and processes. However, limited attention has been directed toward examining potential resistance hindering teachers and PTs from effectively implementing mathematical modeling. This study focuses on how PTs, without prior modeling experience, resist and/or embrace mathematical modeling and its pedagogy as they learn about models and modeling perspectives, navigate the modeling process, design and implement their modeling activities and lesson plans, and experience the pedagogy enabling modeling. Model eliciting activities (MEAs) were employed due to their high potential to support the development of mathematical modeling pedagogy. The mathematical modeling module was integrated into a mathematics methods course to explore how PTs embraced or resisted mathematical modeling and its pedagogy. The module design included reading, reflecting, engaging in modeling, assessing models, creating a modeling task (MEA), and designing a modeling lesson employing an MEA. Twelve senior undergraduate students participated, and data collection involved video recordings, written prompts, lesson plans, and reflections. An open coding analysis revealed acceptance and resistance toward teaching mathematical modeling. The study identified four overarching themes, including both acceptance and resistance: pedagogy, affordance of modeling (tasks), modeling actions, and adjusting modeling. In the category of pedagogy, PTs displayed acceptance based on potential pedagogical benefits and resistance due to various concerns. The affordance of modeling (tasks) category emerged from instances when PTs showed acceptance or resistance while discussing the nature and quality of modeling tasks, often debating whether modeling is considered mathematics. PTs demonstrated both acceptance and resistance in their modeling actions, engaging in modeling cycles as students and designing/implementing MEAs as teachers. The adjusting modeling category captured instances where PTs accepted or resisted maintaining the qualities and nature of the modeling experience or converted modeling into a typical structured mathematics experience for students. While PTs displayed a mix of acceptance and resistance in their modeling actions, limitations were observed in embracing complexity and adhering to model principles. The study provides valuable insights into the challenges and opportunities of integrating mathematical modeling into teacher education, emphasizing the importance of addressing pedagogical concerns and providing support for effective implementation. In conclusion, this research offers a comprehensive understanding of PTs' engagement with modeling, advocating for a more focused discussion on the distinct nature and significance of mathematical modeling in the broader curriculum to establish a foundation for effective teacher education programs.

Keywords: mathematical modeling, model eliciting activities, modeling pedagogy, secondary teacher education

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Authors: Ledeza Jordan Babiano

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Translating statements into mathematical notations is one of the processes in word problem-solving. However, based on the literature, students still have difficulties with this skill. The purpose of this study was to investigate the translation errors of the students when they translate algebraic word problems into mathematical structures and locate the errors via the lens of the Translation-Verification Model. Moreover, this qualitative research study employed content analysis. During the data-gathering process, the students were asked to answer a six-item algebra word problem questionnaire, and their answers were analyzed by experts through blind coding using the Translation-Verification Model to determine their translation errors. After this, a focus group discussion was conducted, and the data gathered was analyzed through thematic analysis to determine the causes of the students’ translation errors. It was found out that students’ prevalent error in translation was the interpretation error, which was situated in the Attribute construct. The emerging themes during the FGD were: (1) The procedure of translation is strategically incorrect; (2) Lack of comprehension; (3) Algebra concepts related to difficulty; (4) Lack of spatial skills; (5) Unprepared for independent learning; and (6) The content of the problem is developmentally inappropriate. These themes boiled down to the major concept of independent learning preparedness in solving mathematical problems. This concept has subcomponents, which include contextual and conceptual factors in translation. Consequently, the results provided implications for instructors and professors in Mathematics to innovate their teaching pedagogies and strategies to address translation gaps among students.

Keywords: mathematical structure, algebra word problems, translation, errors

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Authors: Brando Vagenende, Marie-Anne Guerry

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For stochastic matrices of any order, the geometric description of the convex set of eigenvalues is completely known. The purpose of this study is to investigate the subset of the monotone matrices. This type of matrix appears in contexts such as intergenerational occupational mobility, equal-input modeling, and credit ratings-based systems. Monotone matrices are stochastic matrices in which each row stochastically dominates the previous row. The monotonicity property of a stochastic matrix can be expressed by a nonnegative lower-order matrix with the same eigenvalues as the original monotone matrix (except for the eigenvalue 1). Specifically, the aim of this research is to focus on the properties of eigenvalues of monotone matrices. For those matrices up to order 3, there already exists a complete description of the convex set of eigenvalues. For monotone matrices of order at least 4, this study gives, through simulations, more insight into the geometric description of their eigenvalues. Furthermore, this research treats in a geometric and algebraic way the properties of eigenvalues of monotone matrices of order at least 4.

Keywords: eigenvalues of matrices, finite Markov chains, monotone matrices, nonnegative matrices, stochastic matrices

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Authors: Lokendra Kumar Devangan, Ajay Mishra

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This paper describes the implementation of a large-scale SAS/OR model with significant pre-processing, scenario analysis, and post-processing work done using SAS. A large cement manufacturer with ten geographically distributed manufacturing plants for two variants of cement, around 400 warehouses serving as transshipment points, and several thousand distributor locations generating demand needed to optimize this multi-echelon, multi-modal transport supply chain separately for planning and allocation purposes. For monthly planning as well as daily allocation, the demand is deterministic. Rail and road networks connect any two points in this supply chain, creating tens of thousands of such connections. Constraints include the plant’s production capacity, transportation capacity, and rail wagon batch size constraints. Each demand point has a minimum and maximum for shipments received. Price varies at demand locations due to local factors. A large mixed integer programming model built using proc OPTMODEL decides production at plants, demand fulfilled at each location, and the shipment route to demand locations to maximize the profit contribution. Using base SAS, we did significant pre-processing of data and created inputs for the optimization. Using outputs generated by OPTMODEL and other processing completed using base SAS, we generated several reports that went into their enterprise system and created tables for easy consumption of the optimization results by operations.

Keywords: production planning, mixed integer optimization, network model, network optimization

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Authors: Dmitri Terzi

Abstract:

The article presents the results of an algorithmic study of a special optimization problem of the transport type (traveling salesman problem): 1) To solve the problem, a new natural algorithm has been developed based on the decomposition of the initial data into convex hulls, which has a number of advantages; it is applicable for a fairly large dimension, does not require a large amount of memory, and has fairly good performance. The relevance of the algorithm lies in the fact that, in practice, programs for problems with the number of traversal points of no more than twenty are widely used. For large-scale problems, the availability of algorithms and programs of this kind is difficult. The proposed algorithm is natural because the optimal solution found by the exact algorithm is not always feasible due to the presence of many other factors that may require some additional restrictions. 2) Another inverse problem solved here is to describe a class of traveling salesman problems that have a predetermined optimal solution. The constructed algorithm 2 allows us to characterize the structure of traveling salesman problems, as well as construct test problems to evaluate the effectiveness of algorithms and other purposes. 3) The appendix presents a software implementation of Algorithm 1 (in MATLAB), which can be used to solve practical problems, as well as in the educational process on operations research and optimization methods.

Keywords: traveling salesman problem, solution construction algorithm, convex hulls, optimality verification

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Authors: Imran Hashmi, Sipkaduwa Arachchige Sashika Sureni Wickramasooriya

Abstract:

Gene-drive technology has emerged as a promising tool for disease control by influencing the population dynamics of disease-carrying organisms. Various gene drive mechanisms, derived from global laboratory experiments, aim to strategically manage and prevent the spread of targeted diseases. One prominent strategy involves population replacement, wherein genetically modified mosquitoes are introduced to replace the existing local wild population. To enhance our understanding and aid in the design of effective release strategies, we employ a comprehensive mathematical model. The utilized approach employs agent-based modeling, enabling the consideration of individual mosquito attributes and flexibility in parameter manipulation. Through the integration of an agent-based model and a meta-population spatial approach, the dynamics of gene drive mosquito spreading in a released site are simulated. The model's outcomes offer valuable insights into future population dynamics, providing guidance for the development of informed release strategies. This research significantly contributes to the ongoing discourse on the responsible and effective implementation of gene drive technology for disease vector control.

Keywords: gene drive, agent-based modeling, disease-carrying organisms, malaria

Procedia PDF Downloads 65

Authors: Dennis Okora Amima Ondieki

Abstract:

Fertility transition has been identified to be affected by numerous factors. This research aimed to investigate the most real factors affecting fertility transition in Kenya. These factors were firstly extracted from the literature convened into demographic features, social, and economic features, social-cultural features, reproductive features and modernization features. All these factors had 23 factors identified for this study. The data for this study was from the Kenya Demographic and Health Surveys (KDHS) conducted in 1999-2003 and 2003-2008/9. The data was continuous, and it involved the mean birth order for the ten periods. Principal component analysis (PCA) was utilized using 23 factors. Principal component analysis conveyed religion, region, education and marital status as the real factors. PC scores were calculated for every point. The identified principal components were utilized as forecasters in the multiple regression model, with the fertility level as the response variable. The four components were found to be affecting fertility transition differently. It was found that fertility is affected positively by factors of region and marital and negatively by factors of religion and education. These four factors can be considered in the planning policy in Kenya and Africa at large.

Keywords: fertility transition, principal component analysis, Kenya demographic health survey, birth order

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Authors: Hyon I. Paek, Sang Rim Kim, Hyon A. Ryu

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In functional linear regression, one typical problem is to reduce dimension. Compared with multivariate linear regression, functional linear regression is regarded as an infinite-dimensional case, and the main task is to reduce dimensions of functional response and functional predictors. One common approach is to adapt functional principal component analysis (FPCA) on functional predictors and then use a few leading functional principal components (FPC) to predict the functional model. The leading FPCs estimated by the typical FPCA explain a major variation of the functional predictor, but these leading FPCs may not be mostly correlated with the functional response, so they may not be significant in the prediction for response. In this paper, we propose a supervised functional principal component analysis method for a functional response model with FPCs obtained by considering the correlation of the functional response. Our method would have a better prediction accuracy than the typical FPCA method.

Keywords: supervised, functional principal component analysis, functional response, functional linear regression

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Authors: Haileyesus Tessema Alemneh, Abiyu Enyew Molla, Oluwole Daniel Makinde

Abstract:

Introduction: Faba bean is one of the most important grown plants worldwide for humans and animals. Several biotic and abiotic elements have limited the output of faba beans, irrespective of their diverse significance. Many faba bean pathogens have been reported so far, of which the most important yield-limiting disease is chocolate spot disease (Botrytis fabae). The dynamics of disease transmission and decision-making processes for intervention programs for disease control are now better understood through the use of mathematical modeling. Currently, a lot of mathematical modeling researchers are interested in plant disease modeling. Objective: In this paper, a deterministic mathematical model for chocolate spot disease (CSD) on faba bean plant with an optimal control model was developed and analyzed to examine the best strategy for controlling CSD. Methodology: Three control interventions, quarantine (u2), chemical control (u3), and prevention (u1), are employed that would establish the optimal control model. The optimality system, characterization of controls, the adjoint variables, and the Hamiltonian are all generated employing Pontryagin’s maximum principle. A cost-effective approach is chosen from a set of possible integrated strategies using the incremental cost-effectiveness ratio (ICER). The forward-backward sweep iterative approach is used to run numerical simulations. Results: The Hamiltonian, the optimality system, the characterization of the controls, and the adjoint variables were established. The numerical results demonstrate that each integrated strategy can reduce the diseases within the specified period. However, due to limited resources, an integrated strategy of prevention and uprooting was found to be the best cost-effective strategy to combat CSD. Conclusion: Therefore, attention should be given to the integrated cost-effective and environmentally eco-friendly strategy by stakeholders and policymakers to control CSD and disseminate the integrated intervention to the farmers in order to fight the spread of CSD in the Faba bean population and produce the expected yield from the field.

Keywords: CSD, optimal control theory, Pontryagin’s maximum principle, numerical simulation, cost-effectiveness analysis

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Authors: Akinbo Razak Yinka, Adesanya Kehinde Kazeem, Oladokun Oluwagbenga Peter

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Time series data are sequences of observations collected over a period of time. Such data can be used to predict health outcomes, such as disease progression, mortality, hospitalization, etc. The Statistical approach is based on mathematical models that capture the patterns and trends of the data, such as autocorrelation, seasonality, and noise, while Neural methods are based on artificial neural networks, which are computational models that mimic the structure and function of biological neurons. This paper compared both parametric and non-parametric time series models of patients treated for malaria in Maternal and Child Health Centres in Lagos State, Nigeria. The forecast methods considered linear regression, Integrated Moving Average, ARIMA and SARIMA Modeling for the parametric approach, while Multilayer Perceptron (MLP) and Long Short-Term Memory (LSTM) Network were used for the non-parametric model. The performance of each method is evaluated using the Mean Absolute Error (MAE), R-squared (R2) and Root Mean Square Error (RMSE) as criteria to determine the accuracy of each model. The study revealed that the best performance in terms of error was found in MLP, followed by the LSTM and ARIMA models. In addition, the Bootstrap Aggregating technique was used to make robust forecasts when there are uncertainties in the data.

Keywords: ARIMA, bootstrap aggregation, MLP, LSTM, SARIMA, time-series analysis

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Authors: Dweepabiswa Bagchi, D. V. Senthilkumar

Abstract:

Numerous studies have typically demonstrated the devastation of invasions on an isolated ecosystem or, at most, a network of dispersively coupled similar ecosystem patches. Using such a simplistic 2-D network model, one can only consider dispersal coupling and inter-species trophic interactions. However, in a realistic ecosystem, numerous species co-exist and interact trophically and non-trophically in groups of 2 or more. Even different types of dispersal can introduce complexity in an ecological network. Therefore, a more accurate representation of actual ecosystems (or ecological networks) is a complex network consisting of motifs formed by two or more interacting species. Here, the apropos structure of the network should be multiplex or multi-layered. Motifs between different patches or species should be identical within the same layer and vary from one layer to another. This study investigates three distinct ecological multiplex networks facing invasion from one or more external species. This work determines and quantifies the criteria for the increased extinction risk of these networks. The dynamical states of the network with high extinction risk, i.e., the danger states, and those with low extinction risk, i.e., the resistive network states, are both subsequently identified. The analysis done in this study further quantifies the persistence of the entire network corresponding to simultaneous changes in the strength of invasive dispersal and higher-order trophic and non-trophic interactions. This study also demonstrates that the ecosystems enjoy an inherent advantage against invasions due to their multiplex network structure.

Keywords: increased ecosystem persistence, invasion on ecosystems, multiplex networks, non-trophic interactions

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