Mathematical Patterns Research Articles

Individual dynamics mathematical patterns of qualified basketball players competitive performance

Background and purpose Since the goal in sport is to overcome one's limitations and take one's functional state to a qualitatively new level, forecasting is an integral part of managing any process, including the athlete's training process. Purpose: to develop an algorithm and identify patterns of individual dynamics in competitive performance of eligible basketball players. Materials and methods The main team players of the men's basketball team of H.S. Skovoroda Kharkiv National Pedagogical University. Twelve matches of the experimental team playing against national teams of other universities were analyzed. Observations were made on the players during the championships played by student teams from Kharkiv. For each player, the number of shots and hits from short, mid and long range, goal assists, shots and hits from the penalty line, and turnovers were recorded. To identify individual patterns of competitive performance dynamics, indicators such as 'total positive points in a match', which most accurately reflect a player's 'positive' contribution to the outcome of a match, were used. To identify individual regularities in competitive performance dynamics, indicators such as 'total positive points in the match', which most accurately reflect the 'positive' contribution of players to the outcome of the match, were used. Results It was determined that the process of change in competitive performance should be treated as a variable process. The most appropriate function to describe this model is the sine function. It was shown that the regression model of the individual dynamics of competitive activity of the players of the Ukrainian national basketball team has a sine dependence, which is defined by the regression equation S + = a + bsin ((2π / t) (Т-c)). Where the coefficient a is the average value of a player's match performance, b is the amplitude of the change in a player's match performance, t is the previous period of a player's match performance and c is the value period in the first analyzed match. Conclusion The application of the regression sinusoidal model proved to be effective in practical studies. This is because, using only data from technical reports, it is possible to very quickly predict periods of 'rise' and 'fall' in an individual's game performance, which is useful for adjusting training programs and identifying psychophysiological individual characteristics of players.

Enhancing mathematical proof skills in teacher education through etymology-based learning approaches

This study aims to enhance students' understanding of mathematical proof through the etymology-based learning method. The research employs a descriptive approach to observe and analyze students' mathematical proof patterns in various contexts. Data were collected from 30 students enrolled in a master's degree program in mathematics education who demonstrated varying abilities in mathematical proof. The findings indicate that using action words or phrases during instruction can support students' ability to differentiate relevant terms and avoid misinterpretation of context. Furthermore, comparative understanding of source and target problems improves students' mathematical proof patterns in broader learning contexts. The implications of this study highlight the importance of integrating language-based and formal approaches in mathematics teaching to enhance students' mathematical proof abilities holistically.

Patterning strategies in grade 1 students with low and high number sense proficiency

AbstractRepeating patterning proficiency predicts students’ later mathematical proficiency. A comparative multi-case design enabled the present study to compare patterning success and strategy use for repeating patterns of 75 Norwegian 6-year-old grade 1 students. We provided the students with duplicate, extend, transfer, and unit isolation activities in a semi-structured, task-based interview that was video-recorded. The students’ number sense was also measured using a digital assessment. In-depth analysis of 52 students (divided evenly into two groups based on their number sense scores, lowest and highest) was performed. A statistically significant group difference was found for extend activities. Multimodal video analysis of strategies employed revealed that both student groups used recursive and different strategies and only students with high number sense used relational strategies. The students with low number sense used more variants of recursive patterning strategies. Our findings also demonstrated that attention to pattern features may both hinder identification of a pattern’s underlying structure and reflect advanced strategy use rather than not showing pattern awareness. We discuss group differences in terms of decomposition and awareness of mathematical pattern and structure and emphasise the need for scaffolding awareness of mathematical pattern and structure through communication.

Innovative Mathematical Insights through Artificial Intelligence (AI): Analysing Ramanujan Series and the Relationship between e and π\pi

This paper presents a comprehensive case study on the application of Artificial Intelligence (AI) in mathematics, focusing on the Ramanujan series and the intricate relationship between the mathematical constants e and π. The study explores how AI, particularly machine learning and pattern recognition techniques, can be harnessed to discover new mathematical series and patterns, thereby extending the pioneering work of the legendary mathematician Srinivasa Ramanujan. The paper begins with an overview of the Ramanujan series, illustrating their significance and applications in mathematical computations. It then delves into the specifics of AI methodologies employed to unearth new series for e and π, highlighting the algorithms and models used. Through detailed analysis and experimentation, the study demonstrates how AI can generate new series expansions for e and π, offering enhanced convergence rates and computational efficiencies. Furthermore, the paper examines the relationship between these two constants, providing insights into their interconnected nature through AI-discovered series and patterns. Practical applications of these new series in fields such as numerical methods, cryptography, and theoretical physics are also discussed. By showcasing the successful integration of AI in the realm of mathematical research, this case study underscores the potential of AI to revolutionize traditional mathematical approaches, fostering the discovery of new knowledge and the refinement of existing theories. The findings contribute to a deeper understanding of the interplay between e and π, reinforcing the profound impact of Ramanujan's work in modern mathematics and the transformative power of AI in advancing this legacy.

Power system instability prediction from the solution pattern of differential Riccati equations

AbstractPower system stability characteristics are typically evaluated in terms of small‐ and large‐signal (transient) stability. Access to the time‐varying A‐matrix of a state‐space‐based power systems model during transient conditions can be utilized to apply linear time‐varying system concepts for large‐signal stability analysis. In linear time‐varying system analysis, the differential Riccati equation (DRE) plays a vital role when the power system is subjected to a severe disturbance. The Möbius transformation is proposed in this paper to solve the DRE with singularity issues. It is shown that the solution of the DREs follows a specific mathematical pattern when the power system is stable but does not follow this pattern when the system progresses toward instability. The proposed method can be used in large‐signal stability analysis to predict instability and make the stability analysis more efficient. Additionally, the vector‐DRE is proposed to generalize the index in a large‐scale power system. Results show that analyzing the corresponding Riccati equation's behaviour can help researchers predict a power system's performance and improve the control and management of the system.

The Divine Blueprint: An In-Deth Analysis of the Quran’s Mathematical Architecture -Revealing the Golden Ratio, Solar Year Length, Universal GravitationalConstant, Speedof Light, Planck Constant, and theSignificance of the Numbers 28 and 114

There is a strong belief that Allah Almighty could reveal His existence by encoding the laws of nature and their fundamental constants in the Quran with impeccable precision. However, the mathematical frameworks and patterns embedded within the Quranic text to reveal scientific laws and constants may contain minor deviations in numerical values to empower personal exploration, accommodate diverse interpretations, and strengthen faith. This theory is based on the fact that the Quran is the word of Allah Almighty. Consequently, one would anticipate finding evidence suggesting a divine origin. Many scholars and researchers have suggested that this evidence can be found in the mathematical structure of the Quran through patterns, numerical codes, and other mathematical features within the text that are too complex to be simply attributed to human authorship. Therefore, it is believed that the Quran contains within its structure a mathematical encoding of all physical laws and constants. Verses like 20.98, which state, "He encompasses all things in knowledge," are quoted as evidence. According to this perspective, the Quran is assumed to incorporate mathematical miracles, indicating a potential correspondence between its text and the underlying order of the universe. This theory reveals many mathematical frameworks illustrating how the Quran embeds scientific knowledge. These include the identification of frequent occurrences of the golden ratio within the Quranic text and the manifestation of the golden ratio in the human body, alongside the determination of key celestial parameters such as the length of the solar year. Furthermore, this theory uncovers a numerical link between the Quran and fundamental scientific constants, notably the universal gravitational constant, the speed of light, and the Planck constant. Moreover, this theory utilizes numerical analysis to offer a compelling rationale for why the numbers 28 and 114 are used to represent the total number of Arabic letters and chapters in the Quran, respectively. Finally, this theory highlights the Quran's remarkable mathematical architecture, offering indisputable evidence of its divine composition. We behold a mathematical marvel of cosmic scale, a phenomenon defying human intelligence. Indeed, the Quranis a product of a super intelligence exceeding our comprehension.

Herpes Zoster and Postherpetic Neuralgia

This research investigates the community health of Herpes Zoster (HZ) and the occurrence of postherpetic neuralgia (PHN) following VZV reactivation. The study adopts a practical approach, engaging rigid addition/forbiddance tests to outline the study culture. Diagnostic methods for HZ and PHN include severe dossier accumulation and accompanying moral concerns addressed. While accepting study restraints, this research investigates mathematical patterns of HZ occurrence and labels associations between the two points of HZ and the PHN incident.Descriptive enumerations disclose key understandings of HZ and PHN occurrence, peeling light on mathematical styles. The study stresses the dispassionate pertinence of allure judgments, interpreting results in the circumstances of existing biography. Mechanisms providing for PHN are surveyed, underscoring potential associations for dispassionate practice. Despite disadvantages, this research highlights the need for productive administration policies and deterrent measures.This study considerably contributes to understanding HZ and PHN, stressing their impact on community health and patient comfort. The verdicts offer valuable observations for clinicians and investigators alike, along with suggestions for future research directions. Through this research, an inclusive understanding of the community health of HZ and the incident of PHN is reached, providing a company for enhanced patient care and community health mediation.

Regression Approach in the Evaluation of White’s Effect Magnitude in Comparison to Lightness

In the achromatic White's grid, the grey patches between the grey lines are perceived as darker while the same elements inserted between white lines are perceived lighter than their true measured values. A number of authors attempted to calculate the magnitude and direction of this effect using mathematical models based on multiple spatial filters. This paper uses different mathematical model, based on regression analysis, which has shown itself as an excellent tool for prediction of direction and magnitude of White's effect. The psychophysical visual experiment was conducted on 38 subjects of both genders. The differences in lightness perception ∆L00 were calculated in CIE ∆E_2000 system. This paper determined the functional dependence of White's effect magnitude ∆L00 to the lightness L of rectangular elements in White's achromatic grid. The results gave the square polynomial of very high quality (R2 = 0,974). Regression polynomials were also found. The gave numerical values ∆L00 of difference in perception of left and right elements in comparison to their physical values (left R2 = 0,943, right R2 = 0,938) in dependence to variation of lightness parameter L. Results of the research clearly show the mathematical pattern of White's effect based on the lightness of rectangular elements.

Fostering argumentation skills through project of STEM-Engineering Design process

This study aims to explore students' ability to form argumentation patterns based on aspects of Toulmin's Argument Pattern (TAP) in Science, Technology, Engineering and Mathematics (STEM)-based learning. Data collection was carried out through STEM-based worksheet which has 12 argumentation items that are in accordance with the Engineering Design Process (EDP) stages. The data was then analyzed qualitatively. This research was conducted at one of the junior high schools in Samarinda City in the even semester of the 2022/2023 academic year. The population in this study were all students of grade VII totaling 9 classes with a sample of grade VII-F consisting of 32 students. The results showed that the students' argumentation patterns consisted of several patterns, namely, Claim Data (C-D); Claims, Data, Warrants (C-D-W); Claims, Data, Warrants, Backing (C-D-W-B); Claim, Data, Warrant, Backing, and Rebuttal (C-D-W-B-R) with the C-D pattern is the most common argumentation pattern. Students still need to be facilitated with learning that can guide students in forming more complex arguments so that an argumentation pattern is formed that includes warrant, backing, and rebuttal aspects.

How Chaos Theory Brings Order to the Evolution of Intelligence

This study investigates links between human evolution, information transmission processes, and Chaos Theory, revealing a mathematical pattern underlying evolutionary milestones. By examining the timing of new methods of information transmission, the research confirms a suspected correlation with the Feigenbaum constant δ, a universal factor in Chaos Theory and also found in complex systems. This pattern is prominent in cultural evolution but also extends to biological evolution, as well as to the evolution of written language, suggesting a predictable framework for understanding the progression of complexity in life. The study incorporates findings from various disciplines, including cognitive science, archaeology, and nonlinear dynamics, providing evidence that our development, while it may be random in most aspects, is deterministic in the way complexity grows steadily and evolves information transmission of increasing sophistication. This multidisciplinary approach offers new insights into the links between chaos, complexity, and information, and their role in driving the evolution of intelligent life.

Intelligent design and realization of sustainable development-oriented garden

The idea of sustainable development has become more important in resolving environmental issues and fostering a healthy coexistence of human endeavors with the natural world. Internet of Things (IoT) technology is expanding across many industries, and it is also advancing in agriculture and the agricultural environment. The planning and design for intelligent gardens using a unique Sunflower Optimized-Enhanced Support Vector Machine (SFO-ESVM) is thoroughly analyzed and researched in this study. The development and plan of intelligent gardens are investigated using agricultural IoT technologies and agricultural landscapes. First, we used the SFO method to select the best garden plan inspired by the mathematical patterns observed in sunflower seed groupings. Next, we use an ESVM model to assess how well each plant species fits into the planned garden. The SFO-ESVM considers several variables, such as soil qualities, climatic information, plant traits, and ecological requirements, to choose the best plants. Additionally, we create an intelligent control system that combines sensors, actuators, and IoT technologies to track and regulate the environmental parameters of the garden. The SFO-ESVM-based conceptual planning and design framework for smart gardens is proposed and systematically extended to give scientific direction for the agricultural IoT of smart gardens. The proposed method was then tested in a real-world garden environment. The outcomes show that the SFO-ESVM framework-based intelligent design and execution of the sustainable development-oriented garden combines ecological principles with innovative optimization methods.

APPLICATION OF STATISTICAL METHODS AND THE THEORY OF PATTERN RECOGNITION IN NON-DESTRUCTIVE TESTING PROBLEMS

A mathematical model is proposed to solve the problem of identifying anomalous areas in composite products using methods of mathematical statistics and pattern recognition theory.

Analysis Error Student in Solving Problem Logarithm Based on the Kastolan Procedure

Mathematics is very important in teaching students to have and apply mathematical thinking patterns in everyday life and become a reference for studying various other branches of science. One of the mathematics materials is logarithms, where students often make mistakes when solving these problems, so it can indicate how far students have mastered the logarithm material chosen for this research because logarithms have properties that students must understand. So, this research aims to analyze the errors made by students when solving logarithm problems using the Kastolan procedure. Subject in this study, nine students from class X IT SMKS, Darel Hikmah Pekanbaru. The research instrument used a 5-item logarithmic description test instrument. The data analysis techniques are data reduction, data presentation, and conclusion. Based on the research data results, it was concluded that students made conceptual errors, namely the inability to apply the concept of the properties used. Procedural errors were when students were unable to solve the problem until they got the final solution, and technical errors occurred because they made mistakes in calculations to produce answers wrong.

Mathematical Patterns in Fuzzy Logic and Artificial Intelligence for Financial Analysis: A Bibliometric Study

In this study, we explored the dynamic field of fuzzy logic and artificial intelligence (AI) in financial analysis from 1990 to 2023. Utilizing the bibliometrix package in RStudio and data from the Web of Science, we focused on identifying mathematical models and the evolving role of fuzzy information granulation in this domain. The research addresses the urgent need to understand the development and impact of fuzzy logic and AI within the broader scope of evolving technological and analytical methodologies, particularly concentrating on their application in financial and banking contexts. The bibliometric analysis involved an extensive review of the literature published during this period. We examined key metrics such as the annual growth rate, international collaboration, and average citations per document, which highlighted the field’s expansion and collaborative nature. The results revealed a significant annual growth rate of 19.54%, international collaboration of 21.16%, and an average citation per document of 25.52. Major journals such as IEEE Transactions on Fuzzy Systems, Fuzzy Sets and Systems, the Journal of Intelligent & Fuzzy Systems, and Information Sciences emerged as significant contributors, aligning with Bradford’s Law’s Zone 1. Notably, post-2020, IEEE Transactions on Fuzzy Systems showed a substantial increase in publications. A significant finding was the high citation rate of seminal research on fuzzy information granulation, emphasizing its mathematical importance and practical relevance in financial analysis. Keywords like “design”, “model”, “algorithm”, “optimization”, “stabilization”, and terms such as “fuzzy logic controller”, “adaptive fuzzy controller”, and “fuzzy logic approach” were prevalent. The Countries’ Collaboration World Map indicated a strong pattern of global interconnections, suggesting a robust framework of international collaboration. Our study highlights the escalating influence of fuzzy logic and AI in financial analysis, marked by a growth in research outputs and global collaborations. It underscores the crucial role of fuzzy information granulation as a mathematical model and sets the stage for further investigation into how fuzzy logic and AI-driven models are transforming financial and banking analysis practices worldwide.

Constructing a Mathematical Model of Product Color Design Based on Data Mining: Case Study of a Thermos Cup

In product design, color is the first element that acts on the human visual senses and significantly influences consumer decisions. This study aimed to analyze consumers’ color preferences for products and explore the mathematical patterns of product color design. Firstly, sales data and images of popular thermos cups from Tmall and Jingdong (JD), two prominent e-commerce platforms in China, were obtained through data mining. Subsequently, this research focused on single-color thermos cups with high sales as the research subject, extracting the hue (H), saturation (S), and value (V) for each cup from the product images. Furthermore, a 3D scatter plot of HSV values was generated using Origin Pro, visually representing the consumers’ color preferences. Finally, this study examined the relationships among HSV values of the popular product colors through multiple regression analysis and constructed a mathematical model for HSV. This method enables manufacturers to gain valuable insights into consumer color preferences, facilitating digital color design and enhancing design efficiency and accuracy.

Developing patterns of students' mathematical literacy processes: Insights from cognitive load theory and design-based research

This research aims to develop patterns of students' mathematical literacy processes based on the Cognitive Load Theory (MLCLT). Research using Design-Based Research (DBR) aims to design and develop learning components in the form of resulting learning patterns arranged in 4 stages. The research results show that a CLT-based mathematical literacy process domain pattern has been found. For the aspect of formulating situations mathematically, the pattern of mathematization was produced, and it resulted in some students having good communication skills. For the element of employing mathematical concepts, facts, procedures, and reasoning, the pattern of application was produced, and it showed an increase in students' literacy processes for indicators such as mathematizing, representation, reasoning, argument, devising strategies for solving problems, using symbolic, formal, and technical language and operations, as well as using mathematical tools. The pattern for the aspect of interpreting, applying, and evaluating mathematical outcomes is the reflection pattern. In this pattern, students are already accustomed to having indicators of mathematical literacy processes related to devising strategies for solving problems, using symbolic, formal, and technical language and operations, as well as using mathematical tools so that they can produce real solutions to real-world problems.

Mathematical prognosis of the outcome of the disease of a patient with tuberculosis with HIV infection upon admission to the hospital of the Federal Penitentiary Service

Objective. To identify a mathematical pattern for the subsequent calculation of the prognosis of the outcome of a tuberculosis patient with HIV infection upon admission to the hospital of the Federal Penitentiary Service.Materials and methods. The object of the study was 363 patients with tuberculosis and HIV infection who were admitted for treatment at the hospital of the Federal Penitentiary Service. Discriminant analysis was used to evaluate the features most associated with the discrimination of observations, a priori divided into groups, and reclassify them. As a result, new discriminant axes were obtained, in the space of which the compared groups have the maximum difference.Results and discussion. Formulas were obtained for predicting the outcome of a tuberculosis patient with HIV infection upon admission to a medical institution of the Federal Penitentiary Service: favorable outcome =–91.2138+4.2971×(body mass index)+0.4426×(«experience» of smoking, years)+0.5143×(hemoglobin level, g/l)+0.6158×(stab neutrophils, %)+ 0.4913×(leukocyte T-cell index);unfavorable outcome =–81.8053+3.9467×(body mass index)+0.4071×(«experience» of smoking, years)+0.4724×(hemoglobin level, g/l)+0.6861×(stab neutrophils, %)+0.6630×(leukocyte T-cell index); progression or chronicity =–91.9349+4.2025×(body mass index)+0.5119×(smoking experience, years)+0.5103×(hemoglobin level, g/l)+0.5901×(stab neutrophils, %)+0.5×(leukocyte T-cell index).Conclusion. Simple formulas have been obtained that can be successfully used to calculate the prognosis of the course of the disease upon admission to a hospital in a patient with tuberculosis with concomitant HIV infection.

Research on Pump-Controlled AGC Micro-Displacement Position Control of Lithium Battery Pole Strip Mill Based on Friction Compensation Control Strategy of Imoroved LuGre Pattern

Accurate mathematical patterning of friction has always been a significant research project in the domains of machinery and control, and has played a crucial role in the analysis, control and compensation of mechanical systems containing friction. For high-property electrohydraulic servo control systems, friction compensation is an urgent problem to be solved. The LuGre friction pattern can represent most frictional behaviors, but the LuGre friction pattern is piecewise-continuous, making it non-differentiable. Therefore, the question of how to combine the LuGre friction pattern, enhancing its tracking capacity and robust performance problem, to friction perturbation in hydraulic backstep devices is an important focus for research. In this study, the conventional LuGre friction pattern was enhanced using the continuous differentiability of a friction of rest pattern that laid the foundation of a smooth tangent function. Laying the foundation of an electrohydraulic servo pump-controlled hydraulic roll-gap thickness automatic control system (pump-controlled AGC) pattern, a self-adapting friction compensation controller laid the foundation of the enhanced LuGre pattern. The gradual tracking capacity was determined academically using conditions of parameter, uncertainty and nonlinear friction, and the position control precision of the pump-controlled AGC system was enhanced. The steady-state error of the self-adapting friction compensation control system, which laid the foundation of the enhanced LuGre pattern, attained ±0.1 μm, and the tracking capacity was better than the LuGre pattern control and the conventional PID control strategy at low input speed.

Golden Ratio: Construction, Geometry, Beauty, and Diversity

Nature is attractive due to the proper ratios of various components in them and in between others. In nature, leaves, flowers, fruits, Himalayas, hills, valleys, springs, rivers, etc. show a rhythmic balance and seem harmonic and aesthetically pleasing. The golden ratio or the golden (divine) proportion is considered the most pleasing to human visual sensation and is not limited only to aesthetic beauty but also found prominently in the natural world. It is an irrational number denoted by φ and is approximately 1.618034. It is a fascinating topic for mathematicians, artists, natural scientists, philosophers, biologists, artists, architects, musicians, psychologists, designers, etc.
 This research work presents a panoramic view of the golden ratio; their construction algorithms, and structures; mathematical presentations, geometry, patterns, and properties from mathematical perspectives and opens the broad horizon for further research. Besides these, the paper also presents their existences, diversities, their relationships with nature, the universe, arts, design, architecture, and engineering in ancient and modern mathematics and sciences.

Training load as the conditioning of vertebral column dysfunction in road cyclists

Background. Spinal dysfunctions pose significant problems for road cyclists as they affect their efficiency and the quality of life. Despite this,fact only few studies are focused on the effect of training on the aforementioned dysfunctions. This gap in literature is even bigger as far as competitive/professional road cycling is concerned, despite the significant load the competitors are exposed to. This study is focused on understanding how different components of training load affect dysfunction of the vertebral column in road cyclists, both competitive and amateur ones. Additionally, the thesis how the program of adequate bioregeneration can contribute to reduction of spinal dysfunction risk is analyzed. Material and methods. The samle included 60 cyclists. The participants were divided into two groups: 30 competitive road cyclists UCI World Tour (first division) and 30 amateur road cyclists. The aim of the study was to understand the effect of different components of training load on the on the origin of spinal dysfunction in road cyclists . A questionnaire including the questions concerning personal data, the length of the road cyclist’s training period, training load, complementary training as the frequency and location of back pain, and also the questions, as the questions focused on information on the sustained injuries and contusions as well as as the access and the quality of the physiotherapeutic procedure. Results. The study using statistical tools revealed that many aspects of training affect vertebral column dysfunction in road cyclists, both competitive and amateur ones. The results also suggest that both professional and amateur cyclists should pay more attention to prevention and treatment of back pain. These procedures may involve regular workouts strengthening and stabilizing the vertebral column, control of training load as well as regular assessment by a physiotherapist. However, it was impossible to develop a concrete mathematical pattern for prognosing the aforementioned dysfunctions. Conclusion. The study provides new information on the training record in cyclists on the prevalence of back pain in road cyclists. The study found significant differences between the analyzed variables, however, in no case, the designated linear regression function allowed description of these relationships. This may indicate that the relationships are more complicated than it is suggested based on a simple linear model or that there are other factors responsible for back pain, which were not taken into consideration in this model. Further research using more complex regression models or taking into consideration additional variables are required for better understanding of the factors affecting vertebral column dysfunction in road cyclists – it will be the subject of further research by the authors.