World Academy of Science, Engineering and Technology

Authors: Khaled M. Alqahtani

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Copy number alterations (CNA) are variations in the structure of the genome, where certain regions deviate from the typical two chromosomal copies. These alterations are pivotal in understanding tumor progression and are indicative of patients' survival outcomes. However, effectively modeling patients' survival based on their genomic CNA profiles while identifying relevant genomic regions remains a statistical challenge. Various methods, such as the Cox proportional hazard (PH) model with ridge, lasso, or elastic net penalties, have been proposed but often overlook the inherent dependencies between genomic regions, leading to results that are hard to interpret. In this study, we enhance the elastic net penalty by incorporating an additional penalty that accounts for these dependencies. This approach yields smooth parameter estimates and facilitates variable selection, resulting in a sparse solution. Our findings demonstrate that this method outperforms other models in predicting survival outcomes, as evidenced by our simulation study. Moreover, it allows for a more meaningful interpretation of genomic regions associated with patients' survival. We demonstrate the efficacy of our approach using both real data from a lung cancer cohort and simulated datasets.

Keywords: copy number alterations, cox proportional hazard, lung cancer, regression, sparse solution

Procedia PDF Downloads 40

Authors: Mohammed Abdullah Alshahrani

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The research focuses on investigating the use of partial least squares (PLS) methodology for addressing challenges associated with high-dimensional correlated data. Recent technological advancements have led to experiments producing data characterized by a large number of variables compared to observations, with substantial inter-variable correlations. Such data patterns are common in chemometrics, where near-infrared (NIR) spectrometer calibrations record chemical absorbance levels across hundreds of wavelengths, and in genomics, where thousands of genomic regions' copy number alterations (CNA) are recorded from cancer patients. PLS serves as a widely used method for analyzing high-dimensional data, functioning as a regression tool in chemometrics and a classification method in genomics. It handles data complexity by creating latent variables (components) from original variables. However, applying PLS can present challenges. The study investigates key areas to address these challenges, including unifying interpretations across three main PLS algorithms and exploring unusual negative shrinkage factors encountered during model fitting. The research presents an alternative approach to addressing the interpretation challenge of predictor weights associated with PLS. Sparse estimation of predictor weights is employed using a penalty function combining a lasso penalty for sparsity and a Cauchy distribution-based penalty to account for variable dependencies. The results demonstrate sparse and grouped weight estimates, aiding interpretation and prediction tasks in genomic data analysis. High-dimensional data scenarios, where predictors outnumber observations, are common in regression analysis applications. Ordinary least squares regression (OLS), the standard method, performs inadequately with high-dimensional and highly correlated data. Copy number alterations (CNA) in key genes have been linked to disease phenotypes, highlighting the importance of accurate classification of gene expression data in bioinformatics and biology using regularized methods like PLS for regression and classification.

Keywords: partial least square regression, genetics data, negative filter factors, high dimensional data, high correlated data

Procedia PDF Downloads 46

Authors: Nacera Yassa, Abdelmalek Saidoune, Ghania Ouadfel, Hamza Houassine

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The rapid advancement in electrical technologies has underscored the increasing importance of multiphase machines across various industrial sectors. These machines offer significant advantages in terms of efficiency, compactness, and reliability compared to their single-phase counterparts. However, early detection and diagnosis of electrical faults remain critical challenges to ensure the durability and safety of these complex systems. This paper presents an advanced analytical model for multiphase machines, with a particular focus on dual stator asynchronous machines. The primary objective is to develop a robust diagnostic tool capable of effectively detecting and locating electrical faults in these machines, including short circuits, winding faults, and voltage imbalances. The proposed methodology relies on an analytical approach combining electrical machine theory, modeling of magnetic and electrical circuits, and advanced signal analysis techniques. By employing detailed analytical equations, the developed model accurately simulates the behavior of multiphase machines in the presence of electrical faults. The effectiveness of the proposed model is demonstrated through a series of case studies and numerical simulations. In particular, special attention is given to analyzing the dynamic behavior of machines under different types of faults, as well as optimizing diagnostic and recovery strategies. The obtained results pave the way for new advancements in the field of multiphase machine diagnostics, with potential applications in various sectors such as automotive, aerospace, and renewable energies. By providing precise and reliable tools for early fault detection, this research contributes to improving the reliability and durability of complex electrical systems while reducing maintenance and operation costs.

Keywords: faults, diagnosis, modelling, multiphase machine

Procedia PDF Downloads 55

Authors: Saeed Otarod

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In a different simple and straight forward analysis a closed-form integral solution is found for nonhomogeneous second order linear ordinary differential equations, in terms of a particular solution of their corresponding homogeneous part. To find the particular solution of the homogeneous part, the equation is transformed into a simple Riccati equation from which the general solution of non-homogeneouecond order differential equation, in the form of a closed integral equation is inferred. The method works well in manyimportant cases, such as Schrödinger equation for hydrogen-like atoms. A non-homogenous second order linear differential equation has been solved as an extra example

Keywords: explicit, linear, differential, closed form

Procedia PDF Downloads 48

Authors: Tan Zhang, Zhongjian Wang, Jack Xin, Zhiwen Zhang

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We aim to efficiently compute the spreading speeds of reaction-diffusion-advection (RDA) fronts in divergence-free random flows under the Kolmogorov-Petrovsky-Piskunov (KPP) nonlinearity. We study a stochastic interacting particle method (IPM) for the reduced principal eigenvalue (Lyapunov exponent) problem of an associated linear advection-diffusion operator with spatially random coefficients. The Fourier representation of the random advection field and the Feynman-Kac (FK) formula of the principal eigenvalue (Lyapunov exponent) form the foundation of our method implemented as a genetic evolution algorithm. The particles undergo advection-diffusion and mutation/selection through a fitness function originated in the FK semigroup. We analyze the convergence of the algorithm based on operator splitting and present numerical results on representative flows such as 2D cellular flow and 3D Arnold-Beltrami-Childress (ABC) flow under random perturbations. The 2D examples serve as a consistency check with semi-Lagrangian computation. The 3D results demonstrate that IPM, being mesh-free and self-adaptive, is simple to implement and efficient for computing front spreading speeds in the advection-dominated regime for high-dimensional random flows on unbounded domains where no truncation is needed.

Keywords: KPP front speeds, random flows, Feynman-Kac semigroups, interacting particle method, convergence analysis

Procedia PDF Downloads 42

Authors: Ali Kazemi

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The digital evolution of financial services, while offering unprecedented convenience and accessibility, has also escalated the vulnerabilities to fraudulent activities. In this study, we introduce a distinct approach to real-time anomaly detection in financial transactions, aiming to fortify the defenses of banking and financial institutions against such threats. Utilizing unsupervised machine learning algorithms, specifically autoencoders and isolation forests, our research focuses on identifying irregular patterns indicative of fraud within transactional data, thus enabling immediate action to prevent financial loss. The data we used in this study included the monetary value of each transaction. This is a crucial feature as fraudulent transactions may have distributions of different amounts than legitimate ones, such as timestamps indicating when transactions occurred. Analyzing transactions' temporal patterns can reveal anomalies (e.g., unusual activity in the middle of the night). Also, the sector or category of the merchant where the transaction occurred, such as retail, groceries, online services, etc. Specific categories may be more prone to fraud. Moreover, the type of payment used (e.g., credit, debit, online payment systems). Different payment methods have varying risk levels associated with fraud. This dataset, anonymized to ensure privacy, reflects a wide array of transactions typical of a global banking institution, ranging from small-scale retail purchases to large wire transfers, embodying the diverse nature of potentially fraudulent activities. By engineering features that capture the essence of transactions, including normalized amounts and encoded categorical variables, we tailor our data to enhance model sensitivity to anomalies. The autoencoder model leverages its reconstruction error mechanism to flag transactions that deviate significantly from the learned normal pattern, while the isolation forest identifies anomalies based on their susceptibility to isolation from the dataset's majority. Our experimental results, validated through techniques such as k-fold cross-validation, are evaluated using precision, recall, and the F1 score alongside the area under the receiver operating characteristic (ROC) curve. Our models achieved an F1 score of 0.85 and a ROC AUC of 0.93, indicating high accuracy in detecting fraudulent transactions without excessive false positives. This study contributes to the academic discourse on financial fraud detection and provides a practical framework for banking institutions seeking to implement real-time anomaly detection systems. By demonstrating the effectiveness of unsupervised learning techniques in a real-world context, our research offers a pathway to significantly reduce the incidence of financial fraud, thereby enhancing the security and trustworthiness of digital financial services.

Keywords: anomaly detection, financial fraud, machine learning, autoencoders, isolation forest, transactional data analysis

Procedia PDF Downloads 50

Authors: Ali Kazemi

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Those within the volatile and interconnected international economic markets, appropriately predicting market trends, hold substantial fees for traders and financial establishments. Traditional device mastering strategies have made full-size strides in forecasting marketplace movements; however, monetary data's complicated and networked nature calls for extra sophisticated processes. This observation offers a groundbreaking method for monetary marketplace prediction that leverages the synergistic capability of Graph Convolutional Networks (GCNs) and Long Short-Term Memory (LSTM) networks. Our suggested algorithm is meticulously designed to forecast the traits of inventory market indices and cryptocurrency costs, utilizing a comprehensive dataset spanning from January 1, 2015, to December 31, 2023. This era, marked by sizable volatility and transformation in financial markets, affords a solid basis for schooling and checking out our predictive version. Our algorithm integrates diverse facts to construct a dynamic economic graph that correctly reflects market intricacies. We meticulously collect opening, closing, and high and low costs daily for key inventory marketplace indices (e.g., S&P 500, NASDAQ) and widespread cryptocurrencies (e.g., Bitcoin, Ethereum), ensuring a holistic view of marketplace traits. Daily trading volumes are also incorporated to seize marketplace pastime and liquidity, providing critical insights into the market's shopping for and selling dynamics. Furthermore, recognizing the profound influence of the monetary surroundings on financial markets, we integrate critical macroeconomic signs with hobby fees, inflation rates, GDP increase, and unemployment costs into our model. Our GCN algorithm is adept at learning the relational patterns amongst specific financial devices represented as nodes in a comprehensive market graph. Edges in this graph encapsulate the relationships based totally on co-movement styles and sentiment correlations, enabling our version to grasp the complicated community of influences governing marketplace moves. Complementing this, our LSTM algorithm is trained on sequences of the spatial-temporal illustration discovered through the GCN, enriched with historic fee and extent records. This lets the LSTM seize and expect temporal marketplace developments accurately. Inside the complete assessment of our GCN-LSTM algorithm across the inventory marketplace and cryptocurrency datasets, the version confirmed advanced predictive accuracy and profitability compared to conventional and opportunity machine learning to know benchmarks. Specifically, the model performed a Mean Absolute Error (MAE) of 0.85%, indicating high precision in predicting day-by-day charge movements. The RMSE was recorded at 1.2%, underscoring the model's effectiveness in minimizing tremendous prediction mistakes, which is vital in volatile markets. Furthermore, when assessing the model's predictive performance on directional market movements, it achieved an accuracy rate of 78%, significantly outperforming the benchmark models, averaging an accuracy of 65%. This high degree of accuracy is instrumental for techniques that predict the course of price moves. This study showcases the efficacy of mixing graph-based totally and sequential deep learning knowledge in economic marketplace prediction and highlights the fee of a comprehensive, records-pushed evaluation framework. Our findings promise to revolutionize investment techniques and hazard management practices, offering investors and economic analysts a powerful device to navigate the complexities of cutting-edge economic markets.

Keywords: financial market prediction, graph convolutional networks (GCNs), long short-term memory (LSTM), cryptocurrency forecasting

Procedia PDF Downloads 54

Authors: Ranjit Biswas

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The existing subject Algebra is incomplete to support mathematical computations being done by scientists of all areas: Mathematics, Physics, Statistics, Chemistry, Space Science, Cosmology etc. even starting from the era of great Einstein. A huge hidden gap in the subject ‘Algebra’ is unearthed. All the scientists today, including mathematicians, physicists, chemists, statisticians, cosmologists, space scientists, and economists, even starting from the great Einstein, are lucky that they got results without facing any contradictions or without facing computational errors. Most surprising is that the results of all scientists, including Nobel Prize winners, were proved by them by doing experiments too. But in this paper, it is rigorously justified that they all are lucky. An algebraist can define an infinite number of new algebraic structures. The objective of the work in this paper is not just for the sake of defining a distinct algebraic structure, but to recognize and identify a major gap of the subject ‘Algebra’ lying hidden so far in the existing vast literature of it. The objective of this work is to fix the unearthed gap. Consequently, a different algebraic structure called ‘Region’ has been introduced, and its properties are studied.

Keywords: region, ROR, RORR, region algebra

Procedia PDF Downloads 45

Authors: Samir K. Safi

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The purpose of this study was to compare the performance of two types of models, namely MIDAS and MIDAS-GARCH, in predicting the volatility of crude oil returns based on gold price returns and the COVID-19 pandemic. The study aimed to identify which model would provide more accurate short-term and long-term predictions and which model would perform better in handling the increased volatility caused by the pandemic. The findings of the study revealed that the MIDAS model performed better in predicting short-term and long-term volatility before the pandemic, while the MIDAS-GARCH model performed significantly better in handling the increased volatility caused by the pandemic. The study highlights the importance of selecting appropriate models to handle the complexities of real-world data and shows that the choice of model can significantly impact the accuracy of predictions. The practical implications of model selection and exploring potential methodological adjustments for future research will be highlighted and discussed.

Keywords: GARCH-MIDAS, MIDAS, crude oil, gold, COVID-19, volatility

Procedia PDF Downloads 59

Authors: Lukas Vierus, Thomas Schuster

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A general framework is examined for dynamic tensor field tomography within an inhomogeneous medium characterized by refraction and absorption, treated as an inverse source problem concerning the associated transport equation. Guided by Fermat’s principle, the Riemannian metric within the specified domain is determined by the medium's refractive index. While considerable literature exists on the inverse problem of reconstructing a tensor field from its longitudinal ray transform within a static Euclidean environment, limited inversion formulas and algorithms are available for general Riemannian metrics and time-varying tensor fields. It is established that tensor field tomography, akin to an inverse source problem for a transport equation, persists in dynamic scenarios. Framing dynamic tensor tomography as an inverse source problem embodies a comprehensive perspective within this domain. Ensuring well-defined forward mappings necessitates establishing existence and uniqueness for the underlying transport equations. However, the bilinear forms of the associated weak formulations fail to meet the coercivity condition. Consequently, recourse to viscosity solutions is taken, demonstrating their unique existence within suitable Sobolev spaces (in the static case) and Sobolev-Bochner spaces (in the dynamic case), under a specific assumption restricting variations in the refractive index. Notably, the adjoint problem can also be reformulated as a transport equation, with analogous results regarding uniqueness. Analytical solutions are expressed as integrals over geodesics, facilitating more efficient evaluation of forward and adjoint operators compared to solving partial differential equations. Certainly, here's the revised sentence in English: Numerical experiments are conducted using a Nesterov-accelerated Landweber method, encompassing various fields, absorption coefficients, and refractive indices, thereby illustrating the enhanced reconstruction achieved through this holistic modeling approach.

Keywords: attenuated refractive dynamic ray transform of tensor fields, geodesics, transport equation, viscosity solutions

Procedia PDF Downloads 42

Authors: Francis O. Nwawuru, Jeremiah N. Ezeora

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In this paper, a new algorithm for solving the system of split equilibrium and fixed point problems in real Hilbert spaces is considered. The equilibrium bifunction involves a nite family of pseudo-monotone mappings, which is an improvement over monotone operators. More so, it turns out that the solution of the finite family of nonexpansive mappings. The regularized parameters do not depend on Lipschitz constants. Also, the computations of the stepsize, which plays a crucial role in the convergence analysis of the proposed method, do require prior knowledge of the norm of the involved bounded linear map. Furthermore, to speed up the rate of convergence, an inertial term technique is introduced in the proposed method. Under standard assumptions on the operators and the control sequences, using a modified Halpern iteration method, we establish strong convergence, a desired result in applications. Finally, the proposed scheme is applied to solve some optimization problems. The result obtained improves numerous results announced earlier in this direction.

Keywords: equilibrium, Hilbert spaces, fixed point, nonexpansive mapping, extragradient method, regularized equilibrium

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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

Procedia PDF Downloads 47

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

Procedia PDF Downloads 72

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.

Procedia PDF Downloads 57

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

Procedia PDF Downloads 52

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

Procedia PDF Downloads 48

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

Procedia PDF Downloads 58

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

Procedia PDF Downloads 55

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

Abstract:

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

Procedia PDF Downloads 48

Authors: Yingxu Wang

Abstract:

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 60

Authors: Sharon-Yasotha Veerayah-Mcgregor, Valipuram Manoranjan

Abstract:

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

Procedia PDF Downloads 89

Authors: Medine Demir, Volker John

Abstract:

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

Procedia PDF Downloads 56

Authors: Liberto Pombal, Christian Dieter Jaekel

Abstract:

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

Procedia PDF Downloads 48

Authors: Yingxu Wang

Abstract:

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 44

Authors: Zahid Ullah, Atlas Khan

Abstract:

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

Procedia PDF Downloads 54

Authors: Zahid Ullah, Atlas Khan

Abstract:

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

Abstract:

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

Abstract:

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

Abstract:

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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