Mathematical Calculations Research Articles

Abstract - Graph traversal algorithms are a fundamental component of computational mathematics and computer science. The two most common methods, Breadth-First Search (BFS) and Depth-First Search (DFS), are often presented as distinct algorithms. This paper argues that BFS and DFS are, in fact, two variations of a single, generic traversal algorithm, and that their profound differences in behaviour including traversal order, optimality, and memory usage are a direct mathematical consequence of their underlying data structure: a Queue (First-In, First-Out) for BFS and a Stack (Last-In, First-Out) for DFS. We perform a theoretical space-complexity analysis, contrasting the exponential O( ) memory requirement of BFS with the linear-in-depth O(b.d) of DFS, where b is the branching factor and d is the depth. We then validate this theory with an empirical memory analysis on graph topologies specifically designed to target the worst-case scenarios for each algorithm. Our findings confirm that the choice of data structure is the defining factor that dictates the performance and feasibility of a graph traversal. Key Words: Graph Traversal, BFS, DFS, Data Structures, Space Complexity, Computational Mathematics.

This study presents a novel error model that distinguishes between constant and variable components of systematic error (bias) in measurement systems, particularly within clinical laboratory settings. Traditional approaches often conflict with these components, resulting in miscalculations of total error and measurement uncertainty. Through mathematical deduction and computer simulations, the authors demonstrate that the standard deviation derived from long-term quality control (QC) data includes both random error and the variable bias component, challenging its use as a sole estimator of random error. The proposed model defines the constant component of systematic error (CCSE) as a correctable term, while the variable component (VCSE(t)) behaves as a time-dependent function that cannot be efficiently corrected. The study further reveals that long-term QC data are not normally distributed, contradicting prevailing assumptions in metrology. It advocates for revised definitions in the International Vocabulary of Metrology (VIM3), emphasizing the need to distinguish between bias types determined under different measurement conditions. By applying this refined model, laboratories can enhance decision-making accuracy and more accurately estimate measurement error and uncertainty. The findings have implications beyond clinical laboratories, suggesting a paradigm shift in how systematic error is conceptualized and managed across all domains of metrology.

In this manuscript, we use a rational-type mapping of complex polynomial to generate and visualize Julia and Mandelbrot fractals. We use the generalized viscosity approximation-type iterative scheme, extended with s-convexity improves convergence behavior and gives more control over the morphology of the resultant sets. Under this generalized scheme, a customized escape criterion is presented to precisely find orbit divergence. The structure, complexity, and symmetry of the produced fractals are investigated in relation to changing iterative parameters. We show a broad spectrum of fractal behaviors, from symmetric floral-like patterns to fragmented, fractal geometries, by means of extensive numerical simulations and graphical analysis performed in MATLAB R2024a. Moreover, some Julia and Mandelbrot sets are created to generate Batik-inspired designs, so highlighting the visual harmony and design possibilities of fractal geometry. This work not only broadens the theoretical underpinnings of fixed-point-based fractal generation but also links mathematical computation with useful applications in art and design.

This paper examined the concepts and application of the kNN algorithm relating to machine learning characteristics on automata processes. The algorithm applications focused on network congestion, demand, and supply relating to price and time, and finally, the usage of transaction mode in financial transactions in terms of deposit and withdrawal. The study reviewed machine learning as an aspect of artificial intelligence in computing systems that improves and automatically applies to a variety of complex mathematical computations to address problems using similarity search on query techniques with a given dataset. This was applied with k-nearest neighbors (kNN) developing an effective design of the automata processor (AP)) related to non-von Neumann architecture that focused on nondeterministic finite automaton (NFA) driven by the execution model; According to the research, KNN performance is affected by a number of factors, including parameter normalization, the value of K, and the distance between the two points on the graph. The study also revealed some challenges of the kNN algorithm, such as computational efficiency, selecting the optimal value of k, high-dimensional data handling, and finally, noise and outlier sensitivity. The results of the kNN algorithm indicated common values of each MSE for both demand and supply prediction, and the network congestion. The optimal k-value is 2, with a corresponding 66.67% without congestion. Finally, with regard to the financial transaction, the kNN algorithm computes each customer transaction frequency with the selected bank account number carried out within the range of the query provided.

The famous Goldbach conjecture states that any even natural number N N greater than 2 2 can be written as the sum of two prime numbers p^{\text{(I)}} p (I) and p^{\text{(II)}} p (II) . In this article we propose a quantum analogue device that solves the following problem: given a small prime p^{\text{(I)}} p (I) , identify a member N N of a \mathcal{N} 𝒩 -strong set even numbers for which N-p^{\text{(I)}} N − p (I) is also a prime. A table of suitable large primes p^{\text{(II)}} p (II) is assumed to be known a priori. The device realizes the Grover quantum search protocol and as such ensures a \sqrt{\mathcal{N}} 𝒩 quantum advantage. Our numerical example involves a set of 51 even numbers just above the highest even classical-numerically explored so far [T. O. e Silva, S. Herzog, and S. Pardi, Mathematics of Computation 83, 2033 (2013)]. For a given small prime number p^{\text{(I)}}=223 p (I) = 223 , it took our quantum algorithm 5 steps to identify the number N=4× 10^{18}+14 N = 4 × 10 18 + 14 as featuring a Goldbach partition involving 223 223 and another prime, namely p^{\text{(II)}}=4× 10^{18}-239 p (II) = 4 × 10 18 − 239 . Currently, our algorithm limits the number of evens to be tested simultaneously to \mathcal{N} \sim \ln(N) 𝒩 ∼ ln ( N ) : larger samples will typically contain more than one even that can be partitioned with the help of a given p^{\text{(I)}} p (I) , thus leading to a departure from the Grover paradigm.

A structural equation model on the determinants affecting identity in pre-service mathematics teachers: the role of computational thinking, mathematics mindset and mathematics anxiety

Numeracy literacy skills are abilities that need to be trained and developed, focusing on understanding concepts and mathematical calculations as well as their application in the process of contextual problem-solving. Specifically, the level of numeracy literacy in each school in Indonesia has not yet been identified, particularly in the context of reaction rates in chemistry, which is rarely analyzed. Therefore, this study aims to describe the overall level of students' numeracy literacy on the topic of reaction rates, based on numeracy literacy indicators, gender, and cognitive aspects in Bloom's taxonomy. This research employs a descriptive quantitative approach, involving 60 students as subjects. The research instrument consisted of 20 essay questions about numeracy literacy skills related to the topic of reaction rates, with a reliability of 0.817, categorized as very high. The research results indicate that, overall, the level of numeracy literacy on the topic of reaction rates remains low, with a percentage of 37.74%. Based on indicators, (1) the ability to analyze information in the form of tables, graphs, diagrams, and charts reached 47.09%; (2) the use of numbers and symbols was 36.07%; (3) interpreting analysis results to make decisions was 34.28%. Based on gender, male students achieved a rate of 39.61%, while female students achieved a rate of 37.06%. Based on cognitive aspects in Bloom's Taxonomy, C4 (analyzing) achieved the highest score of 50.44%, followed by C5 (evaluating) at 32.08%, C3 (applying) at 29.70%, and C6 (creating) at 21.50. These findings emphasize the importance of improving numeracy literacy skills in Indonesia, especially in chemistry education. These results can also serve as a basis for teachers in designing learning strategy policies by integrating numerical skills, data interpretation, and logical reasoning to enhance students' numeracy literacy in chemistry learning more broadly.

In this paper, we introduce a new LM parameter and incorporate a nonmonotone trust region technique to the modified Levenberg-Marquardt (MLM) method proposed by Fan [Mathematics of Computation, 81, 447–466, 2012]. We prove the global convergence and the local convergence rate of the new MLM method under the H o ¨ derian continuity of the Jacobian and the H o ¨ derian local error bound condition which are more general than the Lipschitz continuity of the Jacobian and the local error bound condition used by Fan. Moreover, we evaluate the numerical performance of the new MLM method by solving the discretized two-point boundary value problem, and some nonlinear equations arising in the field of complementary problems. Our numerical experiments provide preliminary indications of the local fast convergence rate and suggest potential improvements in computational efficiency compared to Fan's method.

A pressing task is to develop a mathematical model and calculation method that most accurately describes the radiant component of heat exchange between an animal and its environment. This will help determine the optimal design parameters and temperature conditions for infrared (IR) heaters in livestock premises. The mathematical models considered describe the animal's heat exchange with the environment during IR heating. However, they do not take into account the hidden surface temperature of the premises’ enclosing structures and their emissivity factor, or the relationship between animal thermal comfort and the IR heater surface temperature. The proposed radiant heat exchange mathematical model is applicable to diffusely absorbing and radiating isothermic surface system typical of pigsties. It takes into account the emissivity factors of all of the enclosing structures’ surfaces and determines the effective (apparent) premises temperature value tef, corresponding to the thermal comfort conditions. The IR heater surface temperature’s dependence on the emissivity of the pigsty’s enclosing structures (walls, ceiling, and floor) is given, calculated using three methods. As the emissivity of the premises’ enclosing structures decreases, the difference between the results obtained via methods 1, 2, and 3 increases significantly and reaches 50…60% at ε = 0.8. The IR heater radiating surface temperature range is defined in order to create suitable thermal conditions on premises designed for keeping 1- to 4-week-old newborn piglets depending on the enclosing structure temperature and emissivity, taking into account hidden heat exchange surfaces.

Abstract This paper investigates the use of fuzzy set theory in approximating solutions to differential equations under uncertainty. By defining lower and upper bounds, a robust framework is developed for modeling transitions between function extrema, extending classical methods to fuzzy initial value problems. The study demonstrates universal approximation properties and proposes numerical techniques that ensure stability and convergence. Applications highlight the practical utility of this approach in engineering and computational mathematics, solidifying fuzzy set theory as a powerful tool for addressing uncertainty in mathematical modeling.

Radiopharmaceutical Therapy and Immunotherapy Combinations Utilizing Cancer Evolution and Computational Modeling in Metastatic Prostate Cancer.

Electronic engineering is one of the cornerstones of contemporary technological development, supporting breakthroughs in fields such as communications and computing. However, this discipline relies heavily on sophisticated and complex mathematical calculations, and traditional solutions are often cumbersome, time-consuming, and prone to errors. This report examines the residue theorem as a powerful alternative, using complex analysis as a starting point. Within the framework of contour integrals and pole-residue decomposition in the complex plane, this report systematically examines its applications in engineering and proposes key improvements to enhance its usability and portability in electronic engineering. A case study uses an RL series circuit to demonstrate how entering the complex domain can aid pole analysis and response resolution, and illustrates how this method connects with common frequency-domain modeling techniques such as Laplace transforms and partial fraction expansions. At the same time, this study acknowledge that the residue theorem itself is not a panacea. Therefore, this study introduce linearization into the non-linear aspects of the EDA process related to Greens equations to obtain equivalent linear models, thereby circumventing the applicability constraints imposed by nonlinearities.

This research presents a novel educational approach that combines financial mathematics with Industrialism and business training. Our goal was to create a comprehensive framework that helps students and community members develop essential business and scientific skills. This framework consists of a variety of components, including hands-on workshops, one-on-one mentorship programs, and business incubators designed to nurture new ventures. We also incorporated community engagement activities to ensure the framework's relevance and impact beyond the university campus. To develop this framework, we conducted a systematic review, a rigorous and structured research method. This process involved a thorough identification, evaluation, and synthesis of existing research on the topic. The findings from this review were instrumental in shaping the framework's design. Our results demonstrate that this integration significantly boosts participants' entrepreneurial mindsets, improves their strategic decision-making abilities, and enhances their capacity for innovation. By teaching individuals how to use mathematical modelling, computer simulations, and data analysis to solve business problems, the framework equips them with the tools needed to tackle complex, real-world challenges. Ultimately, this approach helps cultivate a culture of innovation and fosters sustainable economic growth within both university and broader community settings.

This paper develops novel computational methods for studying norm-attaining functionals in infinite-dimensional Banach spaces. We present constructive approximation algorithms with explicit convergence rates, stability analysis under discretization and perturbations, and new geometric characterizations of norm attainment. Key results include: (1) efficient procedures to compute norm-attaining approximations of functionals in uniformly convex spaces, with quantitative error bounds; (2) stability theorems for finite-dimensional projections in reflexive spaces; (3) perturbation resilience estimates relating to the modulus of convexity; and (4) applications to PDE-constrained optimization and functional regression. Our approach combines techniques from functional analysis, approximation theory, and computational mathematics, yielding both theoretical insights and practical algorithms. The results significantly extend the classical Bishop-Phelps theorem by providing computable versions and quantitative estimates in various Banach space geometries.

In an indirectly-heating resistance furnace, the effects of high temperature environment and the electromechanical forces that interact with the elements during operation can cause severe changes or even damages of the elements. In this paper, based on the mathematical modeling and calculation of the electromechanical forces interacting between them according to the configuration of the elements in an indirectly- heating electric resistance furnace, the number of elements that can ensure thermal and mechanical stability in a high temperature environment was determined.

The widespread use of multicopters, in particular quadcopters, in the civil, industrial and defense sectors necessitates the improvement of their aerodynamic characteristics in order to increase maneuverability, energy efficiency, load carrying capacity and reduce noise level. One of the key elements affecting these characteristics is the design of the propellers. Despite significant progress in the research of propellers for classical aviation, research for multicopters remains quite limited.The aim of the work is to conduct a comparative aerodynamic analysis of four three-dimensional models of the HQProp V1S 7x3.5x3 propeller using the modern SolidWorks Flow Simulation software. The research methodology involves computer modeling of three-dimensional models of the real propeller, created by the author using the master-model method and the shell method, as well as 3D models obtained from specialized internet resources.The main parameters of the analysis are the distribution of air flow velocities, pressure distribution, calculation of lift and drag force, as well as the influence of the angle of attack on the aerodynamic properties of the studied samples. The results of mathematical calculations demonstrate the differences between the models.The scientific novelty and practical significance of the research is that the author obtained original data on the aerodynamic characteristics of 3D models of the HQProp V1S 7x3.5x3 propeller, which were not previously published in the open press. The mathematical modeling technique used and the results obtained thanks to it will be used in the future to design new, more advanced propeller models, which will contribute to improving the overall efficiency of the UAV.

Purpose: This study aimed to determine teachers' knowledge of mathematics learning difficulty and their classroom practices. Design/Methodology/Approach: Data were collected through a semi-structured interview form from twenty teachers selected by criterion sampling method. The content analysis technique was employed for data analysis. As a result of the analysis, two themes were identified: a) teachers' knowledge and b) teachers' classroom practices. Findings: As a result of the research, teachers mostly defined mathematics learning difficulties as deficiencies in mathematical skills, numbers, problem solving, four operations, calculation, and difficulty in learning mathematical concepts. In classroom practices, teachers generally used lecture, learning by doing, drama, demonstration, games, peer teaching, one-to-one teaching and question-answer methods. Highlights: The results of the study revealed teachers' knowledge and practice limitations regarding mathematics learning difficulties.

The implementation of digital gamification in the development of mathematical competencies for secondary education in Colombia addresses the problem of low academic performance, since 100% of the students subject to this research fail or reach a low level in the results of the simulations of the SABER standardized tests in the evaluation of mathematics. The acquisition of mathematical competence is established by determining its relationship with the social utility of these, as well as their contents, theoretical concepts and the practicality of mathematical objects. The learning-teaching strategy used in this research develops the curricular contents, through a pedagogical-didactic gamification strategy using the Classcraft platform and dynamizing the development of competencies in mathematical calculation, the understanding of abstract thinking and the promotion of abstraction capacity aimed at strengthening complex mental processes such as the approach and resolution of problems and the implementation of ICTs. where the student establishes, transforms, and integrates information that he uses in knowing how to think, knowing how to be, and knowing how to do in the different fields and mathematical languages applied to different contexts. As a result of this quasi-experimental research, in numerical thinking there is an increase of 1.5 on a scale of 0 to 5, while in the other types of thinking an average of 0.6 increase is obtained, being significant in spatial and variational thinking, which concludes that this is a successful experience in the acquisition of mathematical skills.

Studies on the sexagesimal numeral system conducted so far have failed to provide convincing answers as to why this specific system was adopted for calculations. There remains a lack of clarity on why 60 was chosen as the system’s base having no obvious manifestations in nature. This study proposes a compelling hypothesis regarding the origin of the base-60 system, a hypothesis that could only be unveiled through the lens of the Armenian living language and cultural context. It demonstrates that the sexagesimal system emerged from using emmer wheat (Triticum dicoccum), the first cultivated grain, as the foundational unit of calculation. It shows that the base-60 system was not created merely for everyday mathematical calculations for which base-10 would be more practical but rather served as a humanistic framework. Its purpose was to teach people how to engage in agriculture, cultivate grain, produce bread, and consume it encompassing nearly all the secrets of daily bread-making. The hypothesis of an Armenian origin for the base-60 system is further supported by analysis of measurement units used during the first half of the 3rd millennium BCE. This analysis reveals fascinating details about ancient culture. The study also examines the primary numeral terms and measurement units in Armenian, offering interpretations of their etymological logi

Automated machine tools help to increase production efficiency. However, they do not allow to completely exclude defects on the machined surfaces of parts, the occurrence of which is caused by various external factors. In this regard the need to use of using scanning modules for detection of such defects is obvious. In the article the mathematical and computer models of the basic elements are considered, allowing to provide the development of technical solution for creation of mechatronic module for scanning of various objects for the presence of surface defects. The algorithm of computer modelling is presented, mathematical calculations of ball screw transmission and motor power of the mechatronic module are given. These calculations are evaluative and during the design development they should be performed again taking into account the use of specific components and their technical characteristics. A distinctive feature of this module is the scanning subsystem consisting of three scanners. The modular layout of the subsystem allows to use either all three scanners or only one scanner during scanning. The technical characteristics of the scanners required for the defectoscopy of parts surfaces are specified. The article also gives a brief comparative analysis of similar products designed for defectoscopy of various technical objects.