Mathematical Analysis Research Articles

Investigations on Local Minimum Problem for Dual‐Quaternion‐Based Sliding Mode Control With Artificial Potential Functions

ABSTRACTThis paper investigates a sliding mode control law based on dual quaternions for six‐degree‐of‐freedom (six‐DOF) proximity operations between a chaser and a target spacecraft. Dual quaternions, which can simultaneously represent both the rotation and translation of rigid bodies, effectively model the coupled relative motion during these operations. The sliding surface design incorporates artificial potential functions, enabling the consideration of motion constraints including field‐of‐view restrictions, approach path constraints, and collision avoidance. However, these functions are prone to local minima, where attractive and repulsive forces balance each other out, leading to undesired convergence. This paper presents a comprehensive mathematical analysis of the local minimum problem, elucidating the impact of individual and combined constraints on the chaser behavior. To address the local minimum issue, an active escape mechanism is proposed to apply additional corrective forces to help the chaser overcome these traps and achieve reliable control. Numerical simulation evaluates the proposed solution, confirming its effectiveness in resolving local minima challenges and enhancing mission reliability.

Mathematical analysis and chaotic behavior of cancer treatment with virotherapy by using fractional integral sustainable approach

Mathematical analysis and chaotic behavior of cancer treatment with virotherapy by using fractional integral sustainable approach

Assessment of the Usage of Innovative Integrated Photogrammetry Method in Structural Behaviour Analysis: The Case of Mardin Melik Mahmut Mosque

The integration of photogrammetric modeling with the finite element method has emerged as a valuable approach in the practical analysis and determination of structural behavior. Photogrammetry, a technique that utilizes the analysis and processing of photographs to create three-dimensional models of real-world objects, offers a powerful tool for capturing accurate geometric data. On the other hand, the finite element method provides a numerical framework for the mathematical modeling and analysis of complex structural systems. This article presents a methodology that integrates these two modeling techniques by comparing their performance with integrated and classical models. Classical modelling is a technique performed in a finite element software and has been applied for many years. Accordingly, this study aims to reveal the limitations of photogrammetry, develop solutions to these limitations, and provide integrated model generation. The material definitions and load assignments were kept consistent across both modeling techniques for observing the differences in analysis. The study used SAP2000 and modal analysis was conducted to analyse the behaviour and to compare the geometrical properties of the models independent of region and time. The obtained results reveal a high level of similarity, with the natural vibration periods of the models demonstrating 98%, the stress values exhibiting 90%, and deformation values 90% similarity a result of the dynamic loads applied in the X-Y direction. Consequently, this study highlights the potential for further improvement in the integrated model generation process by identifying the advantages and disadvantages of each modeling technique.

Theoretical and Spectroscopic Analysis of Square-Planar Tellurium(II) and Selenium(II) Diphenyl Thiourea Complexes in p2 Electronic Configurations.

In this work, we performed a detailed mathematical ligand field theory analysis of square-planar (D4h) main group complexes in p2 electronic configurations, and the results were subsequently used to generate an energy-level correlation diagram. We synthesized the model p2 tellurium(II) diphenyl thiourea coordination complex for further spectroscopic investigation. Dissolution of TeO2 in concentrated HCl resulted in the formation of a [TeCl6]2- complex in acidic media, and subsequent addition of diphenyl thiourea resulted in the reduction of Te4+ to Te2+ (observed by 125Te NMR spectroscopy) and the formation of a cis-Te(dptu)2Cl2 complex, which was characterized by single-crystal X-ray diffraction (XRD). Using in situ optical/magnetic spectroscopy - specifically, UV-vis and magnetic circular dichroism (MCD) spectroscopy - we observed absorbance bands and polarized transitions consistent with the calculated p-orbital splitting in a square planar (D4h) ligand field. Using the results of our spectroscopic investigation, we generated a molecular orbital (M.O.) diagram for the cis-Te(dptu)2Cl2 complex. We then used the M.O. diagram, in conjunction with the energy level correlation diagram, to assign the electronic transitions observed in the spectra of the cis-Te(dptu)2Cl2 complex. The analogous selenium(II) complex, cis-Se(dptu)2Cl2, was used to elucidate the observed transitions with minimal contribution from spin-orbit coupling. Our work examined how in situ spectroscopy and complementary ligand field theory analysis can be used to elucidate the electronic structures of main group coordination complexes.

Optimal control strategies for taming TikTok addiction: a mathematical model and analysis

Abstract Optimal control theory is an extension of the calculus of variations. It is a mathematical optimization method for deriving control strategies for a dynamic system. In this paper, the system of differential equations for which we aim to utilize control theory is TikTok, which is one of the most attractive internet platforms. TikTok has garnered immense popularity, surpassing other social media platforms. However, its addictive nature has raised concerns about mental health, including depression, eating disorders, anxiety, self-obsession, and narcissistic personality disorder among its users. This paper introduces a mathematical model for TikTok, considering the usage of this app as an epidemic. The model is rigorously validated through stability analysis of both local and global equilibrium. Moreover, disease-free and non-trivial equilibrium scenarios are discussed by calculating their reproduction numbers. This study aims to raise awareness of TikTok’s potential misuse and explore control theory solutions to mitigate addiction. Additionally, statistical data is used to visualize the numerical results and analyze the impact of control parameters on the TikTok model.

Friction in Cylindrical Joints

Cylindrical clearance joints are commonly employed in mechanisms that involve the rotation of a shaft spindle within a cylindrical sliding bearing. The intensity of the friction process in such joints is governed by several factors, including the clearance size between components, the materials of the interacting surfaces, the properties and characteristics of the lubricant, the surface roughness (asperities), and the magnitude of the relative velocity between the joint’s components. To experimentally determine the friction coefficient in cylindrical clearance joints, a custom device was designed and implemented. This device is adaptable to a universal lathe and enables the measurement of the friction coefficient under varying normal forces and relative movement speeds between the joint components. The experimental data were subjected to mathematical analysis, leading to the development of an empirical model. This model effectively characterizes the direction and intensity of the influence of various factors on the friction coefficient, accounting for the use of different lubricants. The findings provide valuable insights into optimizing cylindrical clearance joints for improved performance in practical applications.

Analysis of density matrix embedding theory around the non‐interacting limit

AbstractThis article provides the first mathematical analysis of the Density Matrix Embedding Theory (DMET) method. We prove that, under certain assumptions, (i) the exact ground‐state density matrix is a fixed‐point of the DMET map for non‐interacting systems, (ii) there exists a unique physical solution in the weakly‐interacting regime, and (iii) DMET is exact up to first order in the coupling parameter. We provide numerical simulations to support our results and comment on the physical meaning of the assumptions under which they hold true. We show that the violation of these assumptions may yield multiple solutions to the DMET equations. We moreover introduce and discuss a specific ‐representability problem inherent to DMET.

Generalized Fractional Integral Inequalities Derived from Convexity Properties of Twice-Differentiable Functions

This study presents novel formulations of fractional integral inequalities, formulated using generalized fractional integral operators and the exploration of convexity properties. A key identity is established for twice-differentiable functions with the absolute value of their second derivative being convex. Using this identity, several generalized fractional Hermite–Hadamard-type inequalities are developed. These inequalities extend the classical midpoint and trapezoidal-type inequalities, while offering new perspectives through convexity properties. Also, some special cases align with known results, and an illustrative example, accompanied by a graphical representation, is provided to demonstrate the practical relevance of the results. Moreover, the findings may offer potential applications in numerical integration, optimization, and fractional differential equations, illustrating their relevance to various areas of mathematical analysis.

Single‐Phase Single‐Stage Integrated Ćuk Inverter Operating in Discontinuous Conduction Mode for Autonomous Operation

ABSTRACTDC microgrids are required to supply energy to AC local loads in many applications. To address this need, integrated inverters have been rising as an excellent solution for interfacing AC loads with the microgrid due to their good conversion efficiency, lower costs, and enhanced reliability. Thus, this paper explores and analyzes the single‐phase single‐stage integrated Ćuk inverter (ICI). The ICI is designed to operate in discontinuous conduction mode with an improved switching strategy. The inverter's output voltage is regulated by a single‐loop control system employing a proportional integral multiresonant controller. This paper presents a comprehensive qualitative and quantitative mathematical analysis of the ICI, such as the main system equations, component sizing, and the converter modeling using the state‐space technique associated with the small‐signal analysis. The ICI distinguishes itself by (i) reduced number of electrical/electrical components; (ii) operation with a single high‐frequency switch, while the remaining bridge switches operate at low frequency; and (iii) ability to step up the input voltage while supplying an AC voltage in its output. The feasibility and functionality of the ICI are demonstrated from computational simulation and experimental results. The ICI provides an AC output voltage with low harmonic distortion and presents an efficiency greater than 96%.

Redesigning geometry assessments to promote advanced geometric thinking: A case study on formal deduction and rigor

Many geometry problems at the university level, particularly in analytic geometry courses, tend to prioritize procedural tasks to foster deeper geometric thinking. This study aims to analyze and redesign existing geometry problems to enhance pre-service mathematics teachers' formal deductive reasoning and rigor in accordance with the Van Hiele model. Employing a case study approach, four geometric problems were analyzed in relation to their alignment with various levels of geometric thinking. The study involved a detailed examination of pre-service mathematics teachers’ responses and the structure of their problems to identify aspects that require improvement to better support higher-order thinking. The methodology included a content analysis of problem design and pre-service mathematics teachers’ answers, focusing on their engagement in formal deduction and generalization. The findings indicate that the current problems insufficiently promote the development of formal deduction and rigor, as they are primarily centered on formula applications without requiring proof or generalization. Specific recommendations are provided in the form of redesigned analytic geometric problems aimed at fostering advanced geometric thinking. These redesigns are expected to help pre-service mathematics teachers tackle more complex mathematical problems by encouraging logical reasoning and argumentation.

Efficacy of biochar as a catalyst for a Fenton-like reaction: Experimental, statistical and mathematical modeling analysis

Efficacy of biochar as a catalyst for a Fenton-like reaction: Experimental, statistical and mathematical modeling analysis

Hepatitis C virus transmission among people who inject drugs in the Middle East and North Africa: mathematical modeling analyses of incidence and intervention impact.

The Middle East and North Africa (MENA) region is the most affected by hepatitis C virus (HCV) infection globally. This study aimed to estimate HCV incidence among people who inject drugs (PWID) in MENA and evaluate the impact of interventions. A mathematical model was extended and applied to 13 countries with at least one data point on the population size of PWID and HCV antibody prevalence among PWID, generating estimates for the period 2024-2030. The model was calibrated using multiple datasets, primarily derived from systematic reviews and meta-analyses. Multivariable uncertainty analyses were conducted. Incidence rate among PWID in the 13 countries combined was 10.4 per 100 person-years (95% UI: 8.0-14.1), with an estimated 42,364 new infections annually (95% UI: 27,990-57,540), accounting for 16.9% (95% UI: 8.3-28.2) of all cases in these countries. These figures varied widely across countries. A 75% reduction in needle/syringe sharing decreased viremic chronic infection prevalence by 14.2% (95% UI: 11.3-17.1), incidence rate by 33.8% (95% UI: 30.2-40.5), and annual new infections by 24.4% (95% UI: 17.7-30.1). A 10% reduction in PWID numbers and a 20% reduction in injection frequency decreased chronic infection prevalence by 1.7% (95% UI: 1.4-2.5), incidence rate by 4.2% (95% UI: 3.9-4.4), and annual new infections by 11.1% (95% UI: 10.9-11.9). Achieving 75% direct-acting antiviral treatment coverage by 2030 decreased chronic infection prevalence by 65.3% (95% UI: 64.8-65.8), incidence rate by 34.5% (95% UI: 29.6-40.3), and annual new infections by 25.3% (95% UI: 19.9-29.3). Combinations of interventions reduced these epidemiologic outcomes by up to 80%. MENA experiences considerable HCV incidence among PWID. While the interventions showed potential, only large-scale or multi-intervention strategies can achieve meaningful reductions in HCV transmission. This publication was made possible by NPRP grant number 12S-0216-190,094 from the Qatar National Research Fund (a member of Qatar Foundation). The authors alone are responsible for the views expressed in this publication and they do not necessarily represent the views, decisions, or policies of World Health Organization.

The Dual Dynamical Foundation of Orthodox Quantum Mechanics.

This paper combines mathematical, philosophical, and historical analyses in a comprehensive investigation of the dynamical foundations of the formalism of orthodox quantum mechanics. The results obtained include: (i) A deduction of the canonical commutation relations (CCR) from the tenets of Matrix Mechanics; (ii) A discussion of the meaning of Schrödinger's first derivation of the wave equation that not only improves on Joas and Lehner's 2009 investigation on the subject, but also demonstrates that the CCR follow of necessity from Schrödinger's first derivation of the wave equation, thus correcting the common misconception that the CCR were only posited by Schrödinger to pursue equivalence with Matrix Mechanics; (iii) A discussion of the mathematical facts and requirements involved in the equivalence of Matrix and Wave Mechanics that improves on F. A. Muller's classical treatment of the subject; (iv) A proof that the equivalence of Matrix and Wave Mechanics is necessitated by the formal requirements of a dual action functional from which both the dynamical postulates of orthodox quantum mechanics, von Neumann's process 1 and process 2, follow; (v) A critical assessment, based on (iii) and (iv), of von Neumann's construction of unified quantum mechanics over Hilbert space. Point (iv) is our main result. It brings to the open the important, but hitherto ignored, fact that orthodox quantum mechanics is no exception to the golden rule of physics that the dynamics of a physical theory must follow from the action functional. If orthodox quantum mechanics, based as it is on the assumption of the equivalence of Matrix and Wave Mechanics, has this "peculiar dual dynamics," as von Neumann called it, then this is so because by assuming the equivalence one has been assuming a peculiar dual action.

Pseudo-3D HSQC0, a method to create a true 2D HSQC0-plane. Application to softwood extract analysis.

Pseudo-3D HSQC0 provides an alternative and easy way to record and analyze quantitative HSQC0-data. In the original time-zero extrapolated 1H-13C HSQC (HSQC0), three separate 2D constant-time (CT) HSQC-experiments (HSQCi, i=1-3) are acquired, where either 1,2 or 3 consecutive CT-HSQC-propagators are repeated in each pulse sequence, and the 2D integral data from the three 2D experiments is analyzed via linear regression. In the presented pseudo-3D HSQC0, HSQCi is one of the dimensions and all data is contained within one dataset, which is recorded in interleaved manner by acquiring the same t1-value for each HSQCi-point before t1-incrementation. The 3D-nature of the data allows the utilization of backward linear prediction to calculate an actual time-zero 2D HSQC0 spectrum, which can be analyzed using normal 2D integration procedures for quantitative results. In all, the pseudo-3D enables straightforward, intuitive and easy analysis of the quantitative 2D HSQC0 spectrum/plane. As the recorded pseudo 3D data contains the normal HSQCi planes, also the classic linear regression analysis can be applied.

Automation of Ultrasonographic Optic Nerve Sheath Diameter Measurement: A Scoping Review.

Intracranial pressure (ICP) monitoring is a cornerstone of neurocritical care in managing severe brain injury. However, current invasive ICP monitoring methods carry significant risks, including infection and intracranial hemorrhage, and are contraindicated in certain clinical situations. Additionally, these methods are not universally available. Optic nerve sheath diameter (ONSD) measurement presents a promising noninvasive alternative for ICP monitoring, though its clinical adoption has been limited due to its operator dependence and inconsistencies in imaging acquisition and measurement techniques. Automating both ONSD image acquisition and measurement could enhance accuracy and reliability, thereby improving its utility as a noninvasive ICP estimation tool. A range of image analysis and machine learning (ML) techniques have been applied to address these challenges. In this paper, we provide a narrative review of the current literature on ONSD automation, examining the strengths and limitations of classical image analysis and ML models in improving ONSD-based ICP assessment.

Optimizing power consumption in novel electrical design for single ended comparator circuit

Contemporary society electronic technology has evolved into a pivotal component across various facets of our lives. Its indispensability is particularly evident in the advancement of medical, agricultural, industrial, and other sectors. As this technology continues to play a crucial role, optimizing its performance in terms of speed, accuracy, and energy consumption becomes paramount. This paper introduces a novel electrical design for the threshold inverter quantization comparator circuit aiming to meet the evolving demands of modern electronic applications. The proposed design enhances the classic threshold inverter quantization comparator’s performance by significantly reducing its power consumption. Through rigorous mathematical analysis and simulation results it is demonstrated that the proposed comparator design achieves a remarkable 50% reduction in power consumption compared to the conventional threshold inverter quantization comparator. Subsequently the newly devised design is applied to the construction of a 4-bit flash analog-to-digital converter using 0.35 μm complementary metal–oxide–semiconductor (CMOS) technology.

Reliability Analysis and Numerical Simulation of the Five-Robot System with Early Warning Function

The rapid advancement of robotic technologies has demonstrated the significant potential of Multi-Robot Systems (MRS) for application across various fields, particularly in automation, manufacturing, and rescue operations. However, enhancing the reliability of Multi-Robot Systems, particularly in critical applications, has emerged as a primary focus of research. A mathematical model of a five-robot system, equipped with early warning capabilities, is developed using Markov process theory and the supplementary variable method in this paper. A model of an abstract Cauchy problem system is developed, employing semigroup theory to investigate the well-posedness of solutions for this five-robot system. The stability of the system is verified using analytical methods, confinal correlation theory, and modern functional analysis techniques. Several key reliability indicators are presented using the eigenvector method. Numerical simulations and comparative methods effectively demonstrate the efficacy of the proposed eigenvector method. Firstly, the innovation of this paper lies in the combination of qualitative and quantitative analyses to improve and enrich the theory and methods of repairable systems. Secondly, mathematical analysis methods and the mathematical software are employed to provide both analytical and numerical solutions for the system.

Change point detection to analyze air pollution and its economic effects: an exponentially weighted moving average perspective

Air pollution has a direct impact on every society, leading to consequential effects on the economy of a nation. Poor air quality adversely affects human health, resulting in various economic outcomes such as rising healthcare costs, diminished labor productivity, negative impacts on tourism and living standards, increased regulatory expenses for businesses, and heightened economic disparities. Effective control methods are essential to monitor factors influencing the economy, including air quality. The presence of toxic substances in the air reduces air quality, necessitating its monitoring through indices like PM10. Among statistical process control tools, control charts are the most prominent for efficient change point detection. This study introduces a new process monitoring tool that incorporates additional auxiliary information, if available, alongside the main variable of interest. The proposed methodology ensures detection ability remains robust, even under disturbances in the auxiliary variable. Furthermore, mathematical analyses reveal that many existing statistical quality control tools become special cases of the proposed structure for specific sensitivity parameter values. Evaluated through properties of run length distribution, the proposed chart allows control of the robustness-efficiency balance by adjusting its sensitivity parameter. A practical implementation demonstrates the effectiveness of the chart in monitoring air quality data.

On Mathematical Analysis of a Micro-Scale Boltzmann-Maxwell’s PDEs Model for a Plasma Flow Influence by Non-linear Sinusoidal External Electric Field: Novel Irreversibility Analysis of Plasma Kinetic Theory

Unfortunately, the scientific research concerned with the micro-scale mathematical models in plasma has very few numbers compared with the other mathematical models. However, the microscopic fields are related to all modern technology like nano and micro technology, quantum computers, and many other essential applications. This study focuses on the mathematical analysis of a micro-scale Boltzmann-Maxwell partial differential equations (PDE) system for a gaseous plasma flow influenced by a non-linear, non-uniform external electric field, specifically in the context of a novel irreversibility micro-scale analysis of the kinetic theory of plasma. We did that using the analytical solution of the PDE system. The governing equations were developed using the moment and traveling-wave techniques in a new irreversible non-equilibrium thermodynamics (INT) methodology. To our knowledge, this was applied for the first time. We are investigating the distinct behavior of the electron velocity distribution functions (VDF) and the non-equilibrium velocity functions, representing an essential INT novel study. As a result, critical micro-scale INT variables would employ the generated non-equilibrium VDF to get the equilibrium time for each species. We aim to show how the impacts of various thermodynamic forces on internal energy change (IEC) are maintained. The research generates visual representations of physical variables in 3D. Extensive physics, electric manufacturing, and nano-electro-mechanical systems applications in various manufacturing contexts attest to the research’s standing. We aim to calculate the percentages between the numerous contributions of IEC in diamagnetic and paramagnetic plasma based on the total derivatives of the extensive parameters. We apply the results to a typical model of laboratory helium plasma because of the helium’s various excellent applications.

Dairy intake and cognitive function in older adults in three cohorts: a mendelian randomization study.

Meta-analyses of observational studies on the effect of dairy on cognitive function have yielded inconclusive results, potentially due to unmeasured confounding. To avoid the no-unmeasured confounding assumption, we used lactase persistence genetic variant as an instrumental variable, for which the CC genotype is associated with lower lactase production and, consequently, lower dairy consumption. We used it to assess the effect of long-term consumption of total and non-fermented dairy on cognitive function. We included 43,836 individuals over 55 years old with genotyping, dietary data, and cognitive function measurements from three population-based studies: CoLaus|PsyCoLaus (Switzerland), the Rotterdam Study (the Netherlands) and the Canadian Longitudinal Study on Aging (CLSA - Canada). We performed a one-sample Mendelian randomization using two-stage least-squares regression. First, we estimated total and non-fermented dairy consumption by T-allele frequency. Second, we used the estimated dairy consumption in linear regression models on general cognition, assessed by the Mini-Mental State Examination (MMSE) and the Mental Alternation Test, executive function, verbal fluency, verbal learning, and memory. Per T-allele, total dairy intake and non-fermented was 24.8 and 15.3g/day higher in PsyCoLaus, 57.9 and 49.8g/day in the Rotterdam Study, and 0.31 and 0.29 times/day in CLSA, respectively. We found no association between the genetically predicted difference and the MMSE in PsyCoLaus and the Rotterdam Study. However, lactase persistent individuals scored 3.4 (95% CI2.1- 4.7) and 3.5 (95% CI 2.3-4.7) points more in the Mental Alternation Test for total and fermented dairy, respectively, in CLSA. Similarly, lactase persistent participants in CLSA had higher verbal fluency, verbal learning and executive function, but no differences were found in the other cohorts. Such inconsistencies might stem from different FFQs across cohorts and consumption ranges. Nonetheless, the generally small magnitude of effect sizes may suggest that there is no real effect between total or non-fermented dairy intake and cognitive function. The evidence for a causal effect of dairy consumption on general cognitive function is weak, consistent with previous results from classic analysis from observational studies. Interventions targeting dairy are unlikely to have a relevant effect on cognitive function.