Mathematical Framework Research Articles

Mathematical modelling of a flexible steering robot for vehicle trajectory control

Abstract This paper presents a mathematical modelling framework for a novel design of a steering robot to control vehicle trajectory. A systematic approach is adopted to capture the dynamics of a belt transmission mechanical system of the robot. The steering robot is powered by a Brushless Direct Current (BLDC) servo motor which employs a Field Oriented Current (FOC) control for motor position control. The system’s accuracy is demonstrated through simulations conducted using MATLAB SIMULINK, showcasing its capability to effectively track the reference steering angle.

An operational discontinuous Galerkin shallow water model for coastal flood assessment

Hydrodynamic modeling for coastal flooding risk assessment is a highly relevant topic. Many operational tools available for this purpose use numerical techniques and implementation paradigms that reach their limits when confronted with modern requirements in terms of resolution and performances. In this work, we present a novel operational tool for coastal hazards predictions, currently employed by the BRGM agency (the French Geological Survey) to carry out its flooding hazard exposure studies and coastal risk prevention plans on International and French territories. The model, called UHAINA (wave in the Basque language), is based on an arbitrary high-order discontinuous Galerkin discretization of the nonlinear shallow water equations with SSP Runge–Kutta time stepping on unstructured triangular grids. It is built upon the finite element library AeroSol, which provides a modern C++ software architecture and high scalability, making it suitable for HPC applications. The paper provides a detailed development of the mathematical and numerical framework of the model, focusing on two key-ingredients : (i) a pragmatic P0 treatment of the solution in partially dry cells which garantees efficiently well-balancedness, positivity and mass conservation at any polynomial order; (ii) an artificial viscosity method based on the physical dissipation of the system of equations providing nonlinear stability for non-smooth solutions. A set of numerical validations on accademic benchmarks is performed to highlight the efficiency of these approaches. Finally, UHAINA is applied on a real operational case of study, demonstrating very satisfactory results.

Dual-Domain Collaborative Diffusion Sampling for Multi-Source Stationary Computed Tomography Reconstruction.

The multi-source stationary CT, where both the detector and X-ray source are fixed, represents a novel imaging system with high temporal resolution that has garnered significant interest. Limited space within the system restricts the number of X-ray sources, leading to sparse-view CT imaging challenges. Recent diffusion models for reconstructing sparse-view CT have generally focused separately on sinogram or image domains. Sinogram-centric models effectively estimate missing projections but may introduce artifacts, lacking mechanisms to ensure image correctness. Conversely, image-domain models, while capturing detailed image features, often struggle with complex data distribution, leading to inaccuracies in projections. Addressing these issues, the Dual-domain Collaborative Diffusion Sampling (DCDS) model integrates sinogram and image domain diffusion processes for enhanced sparse-view reconstruction. This model combines the strengths of both domains in an optimized mathematical framework. A collaborative diffusion mechanism underpins this model, improving sinogram recovery and image generative capabilities. This mechanism facilitates feedback-driven image generation from the sinogram domain and uses image domain results to complete missing projections. Optimization of the DCDS model is further achieved through the alternative direction iteration method, focusing on data consistency updates. Extensive testing, including numerical simulations, real phantoms, and clinical cardiac datasets, demonstrates the DCDS model's effectiveness. It consistently outperforms various state-of-the-art benchmarks, delivering exceptional reconstruction quality and precise sinogram.

A Comparative Study of Geometric Principles in the Sulba Sutras and the Pythagorean Theorem: Historical Context and Mathematical Applications

This research delves into the geometric principles detailed in the ancient Sulba Sutras and their parallels with the Pythagorean Theorem, traditionally attributed to the Greek mathematician Pythagoras. While the Pythagorean theorem is widely regarded as a cornerstone of Greek geometry, recent studies suggest that its formulation appeared centuries earlier in Indian mathematics, as evidenced in Baudhāyana’s Sulba Sutra (circa 800 BCE). These texts offer extensive insights into the geometric constructions used in Vedic rituals, including the use of Pythagorean triples and transformations between geometric shapes. The Sulba Sutras present a rich mathematical framework, emphasizing practical applications in altar construction, land measurements, and urban planning. This study aims to bridge the research gap by comparing the geometric developments in the Sulba Sutras with Greek formulations, particularly focusing on their mathematical methods, cultural contexts, and historical significance. The findings underscore the advanced state of Indian mathematics and its profound influence on subsequent mathematical traditions, offering a nuanced understanding of the parallel evolution of geometric knowledge across cultures.

Posterior comparison of model dynamics in several hybrid turbulence model forms

Hybrid turbulence models that can accurately reproduce unsteady three-dimensional flow physics across the entire range of grid scales and turbulence dynamics from Reynolds-averaged Navier–Stokes (RANS), through large-eddy simulation (LES), down to direct numerical simulations (DNS) are of increasing interest to the turbulence modeling community. However, despite decades of research and development, the basic tasks of eliminating poor-performing hybrid RANS-LES models and accelerating adoption of superior models through well-designed validation and verification have yet to occur. As a step in this direction, in this work we evaluate thirteen different hybrid RANS-LES models via systematic grid refinement of decaying homogeneous isotropic turbulence. We further derive a novel mathematical framework for assessing the energy partitioning dynamics of each Hybrid RANS-LES model, wherein model-to-model variations in energy partitioning can be interpreted as different feedback mechanisms operating on a low-dimensional nonlinear dynamical system. We found that model forms similar to the flow simulation methodology—also often termed very-large eddy simulation—are dynamically inconsistent with DNS at all resolutions. Additionally, we found a strong dynamical similarity in the feedback mechanisms of all models related to detached eddy simulation and partially averaged Navier–Stokes that is inherent to their general model forms.

Two Energy-Efficient Backhauling Solutions for Small Cell Networks of 5G Using Green Communications

To meet the problem of ever-increasing wireless data congestion, the fifth-generation (5G) wireless communication system is projected to employ tiny cellphone networks for the needs of consumers heavily. One of the most significant elements of 5G networks will be sustainable interactions, as the quantity of power utilized by the information and communication technology (ICT) sector is expected to rise significantly by the end of the century. Therefore, scientists have focused much attention in recent years on developing strategies for designing small cell networks that use electricity optimistically. In addition, service providers need energy-efficient backing-up technologies to aid in using packed tiny cells. This research presents an interaction model that is well-suited to 5G HetNets and has a low power footprint. The model here accounts for an internet connection’s contact and backup portions. We create and present a mathematical framework to calculate the optimum number of tiny mitochondria that need to be maintained active at numerous hours of the day to minimize power consumption while still satisfying quality of service requirements set by customers. Our investigation into the backhaul’s electrical consumption led us to discover and put forward two energy-saving backing-up methods for 5G HetNets. Computer simulations show that the provided sustainable communication framework can reduce energy consumption by as much as 49% compared to the status quo.

A Coupled Spatial-Network Model: A Mathematical Framework for Applications in Epidemiology.

There is extensive evidence that network structure (e.g., air transport, rivers, or roads) may significantly enhance the spread of epidemics into the surrounding geographical area. A new compartmental modeling framework is proposed which couples well-mixed (ODE in time) population centers at the vertices, 1D travel routes on the graph's edges, and a 2D continuum containing the rest of the population to simulate how an infection spreads through a population. The edge equations are coupled to the vertex ODEs through junction conditions, while the domain equations are coupled to the edges through boundary conditions. A numerical method based on spatial finite differences for the edges and finite elements in the 2D domain is described to approximate the model, and numerical verification of the method is provided. The model is illustrated on two simple and one complex example geometries, and a parameter study example is performed. The observed solutions exhibit exponential decay after a certain time has passed, and the cumulative infected population over the vertices, edges, and domain tends to a constant in time but varying in space, i.e., a steady state solution.

Predicting the first steps of evolution in randomly assembled communities

Microbial communities can self-assemble into highly diverse states with predictable statistical properties. However, these initial states can be disrupted by rapid evolution of the resident strains. When a new mutation arises, it competes for resources with its parent strain and with the other species in the community. This interplay between ecology and evolution is difficult to capture with existing community assembly theory. Here, we introduce a mathematical framework for predicting the first steps of evolution in large randomly assembled communities that compete for substitutable resources. We show how the fitness effects of new mutations and the probability that they coexist with their parent depends on the size of the community, the saturation of its niches, and the metabolic overlap between its members. We find that successful mutations are often able to coexist with their parent strains, even in saturated communities with low niche availability. At the same time, these invading mutants often cause extinctions of metabolically distant species. Our results suggest that even small amounts of evolution can produce distinct genetic signatures in natural microbial communities.

TIP: A trust inference and propagation model in multi-human multi-robot teams

Trust is a crucial factor for effective human–robot teaming. Existing literature on trust modeling predominantly focuses on dyadic human-autonomy teams where one human agent interacts with one robot. There is little, if not no, research on trust modeling in teams consisting of multiple human and robotic agents. To fill this important research gap, we present the Trust Inference and Propagation (TIP) model to model and estimate human trust in multi-human multi-robot teams. In a multi-human multi-robot team, we postulate that there exist two types of experiences that a human agent has with a robot: direct and indirect experiences. The TIP model presents a novel mathematical framework that explicitly accounts for both types of experiences. To evaluate the model, we conducted a human-subject experiment with 15 pairs of participants (N=30\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$N=30$$\\end{document}). Each pair performed a search and detection task with two drones. Results show that our TIP model successfully captured the underlying trust dynamics and significantly outperformed a baseline model. To the best of our knowledge, the TIP model is the first mathematical framework for computational trust modeling in multi-human multi-robot teams.

Data-driven corporate growth: A dynamic financial modelling framework for strategic agility

This research aimed to develop a Dynamic Financial Growth Model (DFGM) to enhance corporate growth by promoting strategic agility through data-driven decision-making. The main objective was to optimize corporate value by integrating real-time data, dynamic decision-making, risk management, and scenario analysis. The research employed a mathematical modelling framework that combined predictive analytics, real options theory, and scenario-based optimization to represent dynamic corporate financial decisions. The numerical example demonstrated how the model adjusts strategic decisions in response to changes in market data and evaluates corporate value under optimistic, pessimistic, and baseline scenarios. The main results indicated that the DFGM is effective in optimizing corporate value by allowing for continuous adjustments and strategic flexibility, distinguishing itself from traditional static financial models that lack real-time adaptability. The findings highlighted the value of incorporating risk constraints and scenario analysis, resulting in a balanced approach that manages both growth and uncertainty. However, the study identified limitations, including the need for empirical validation, more complex predictive analytics, and accounting for behavioral factors affecting decision-making. The conclusion emphasizes that the DFGM provides an adaptable and data-driven framework that enhances corporate strategic agility, making it a valuable tool for managing growth in rapidly changing environments, while also suggesting future research to refine the model's practical application

Bluestein–Gulyaev(B-G) waves in a piezo-thermoelastic half-space in the context of modified multi-phase-lag theory

In the years 1968–69, Bluestein and Gulyaev discovered a new kind of surface wave in a piezoelectric material known as B-G wave after their name. Although it was discovered in few decades ago, but till now, no work has been done related to B-G wave propagation based on modified multi-phase-lag theory. In the present work, the authors have investigated the propagation behavior of Bluestein-Gulyaev-type waves in a piezo-thermoelastic half-space in the context of modified multi-phase-lag theory. The normal mode technique is employed to obtain the principal variables, for example, displacements, temperature, electric potential, electric displacements, and thermo-mechanical stresses. The boundary conditions are adopted in such a way that the upper surface of the piezo-thermoelastic half-space is stress free and the half-space is implicated by a perfectly conducting electrode that is grounded (i.e., electrically short conditions at the free surface). Based on these boundary conditions, secular equations are derived for the proposed model. Some special cases have also been discussed, and it is found that the secular equation is in well-agreement with the pre-established thermoelasticity theory. Moreover, some analyses are made to highlight the important peculiarities of the problem. The mathematical framework of the present study is momentous for both theoretical and practical expositions for temperature sensors and piezoelectric surface acoustic wave (SAW) devices. Also present study helps the researchers who are engaged on this domain.

Egalitarian pooling and sharing of longevity risk a.k.a. can an administrator help skin the tontine cat?

This paper is concerned with the mathematical problem of allocating longevity-linked fund payouts in a pool where participants differ in both wealth (contributions) and health (mortality), particularly when these groups are relatively small in size. In other words, we offer a modelling framework for distributing longevity-risk pools' income and benefits (or “tontine winnings”) when participants are heterogeneous. Similar to the nascent literature on decentralized risk sharing (DRS), there are several equally plausible arrangements for sharing benefits (a.k.a. “skinning the tontine cat”) among survivors. We argue that the selected rule may depend on the extent of social cohesion within the longevity risk pool, ranging from solidarity and altruism to pure individualism. And, if actuarial fairness is a concern, we suggest introducing an administrator – which differs from a guarantor – to make the tontine pool payouts collectively actuarial fair. Fairness is in the sense that the group of participants will on average receive the same benefits as they collectively invested; and we provide the mathematical framework to implement that suggestion. One thing is for certain: actuarial science cannot offer design uniqueness for longevity-contingent claims; only a consistent methodology.

Fracture as an Emergent Phenomenon

The mechanics of fracture propagation provides essential knowledge for the risk tolerance design of devices, structures, and vehicles. Techniques of free energy minimization provide guidance, but have limited applicability to material systems evolving away from equilibrium. Experimental evidence shows that the material response depends on driving forces arising from mechanical fields. Recent years have witnessed the development of new methods for modeling complex dynamic and quasistatic fracture. New approaches may differ remarkably from previous ones, as they involve implicit coupling between damaged and undamaged states, allowing fracture to be modeled as emergent phenomena.The focus of this workshop is on the most advanced techniques for modeling fracture, represented by eigenerosion methods, variational approaches, phase field fracture models, and non-local approaches. Technical progress is contingent on the further development of the mathematical framework underlying these techniques. This is necessary for accurate and reliable computational modeling of fracture for multiple freely propagating cracks. The objective of this workshop is to mathematically identify and discuss open issues related to fracture modeling and to highlight recent advances. Addressing fundamental issues will foster exchange between the different communities, essential for advancing the field.

Mathematical framework of nonlinear elastic waves propagating in pre-stressed media

Mathematical framework of nonlinear elastic waves propagating in pre-stressed media

Secure and robust randomness with sequential quantum measurements

Quantum correlations between measurements of separated observers are crucial for applications like randomness generation and key distribution. Although device-independent security can be certified with minimal assumptions, current protocols have limited performance. Here, we exploit sequential measurements, defined with a precise temporal order, to enhance performance by reusing quantum states. We provide a geometric perspective and a general mathematical framework, analytically proving a Tsirelson-like boundary for sequential quantum correlations, which represents a trade-off in nonlocality shared by sequential users. This boundary is advantageous for secure quantum randomness generation, certifying maximum bits per state with one remote and two sequential parties, even if one sequential user shares no nonlocality. Our simple qubit protocol reaches this boundary, and numerical analysis shows improved robustness under realistic noise. A photonic implementation confirms feasibility and robustness. This study advances the understanding of sequential quantum correlations and offers insights for efficient device-independent protocols.

Mathematical Models for Ultrasound Elastography: Recent Advances to Improve Accuracy and Clinical Utility.

Changes in biomechanical properties such as elasticity modulus, viscosity, and poroelastic features are linked to the health status of biological tissues. Ultrasound elastography is a non-invasive imaging tool that quantitatively maps these biomechanical characteristics for diagnostic and treatment monitoring purposes. Mathematical models are essential in ultrasound elastography as they convert the raw data obtained from tissue displacement caused by ultrasound waves into the images observed by clinicians. This article reviews the available mathematical frameworks of continuum mechanics for extracting the biomechanical characteristics of biological tissues in ultrasound elastography. Continuum-mechanics-based approaches such as classical viscoelasticity, elasticity, and poroelasticity models, as well as nonlocal continuum-based models, are described. The accuracy of ultrasound elastography can be increased with the recent advancements in continuum modelling techniques including hyperelasticity, biphasic theory, nonlocal viscoelasticity, inversion-based elasticity, and incorporating scale effects. However, the time taken to convert the data into clinical images increases with more complex models, and this is a major challenge for expanding the clinical utility of ultrasound elastography. As we strive to provide the most accurate imaging for patients, further research is needed to refine mathematical models for incorporation into the clinical workflow.

Critical study of type-2 fuzzy logic control from theory to applications: A state-of-the-art comprehensive survey

The review systematically explores the theoretical foundations of type-2 fuzzy sets and the associated mathematical frameworks of the advancements in type-2 fuzzy logic systems (T2-FLS), as well as a comprehensive overview of the state-of-the-art techniques and applications of type-2 fuzzy logic systems. The review's application-oriented section looks at a number of different areas where type-2 fuzzy logic has made a big difference. These areas include robotics, autonomous vehicle control, electric and electronic control, image processing, and railway control. This overview mainly discusses the past, present, and future trends in type-2 fuzzy logic applications. The primary contribution summarizes the most essential type-2 fuzzy logic research with theoretical and practical implications. This article will provide a critical path and a strong foundation for individuals interested in this field to pursue future research.

Проблемы поиска и преследования беспилотных летательных аппаратов с использованием теоретико-игрового подхода

Currently, the application of mathematical modeling theory is widely used in various fields and, among other things, can be used to prevent illegal actions. This study examines scenarios involving a single pursuer tracking a single evader, as well as situations involving multiple pursuers pursuing multiple evaders. The authors formulate this problem as a search and pursuit problem for four-rotor unmanned aerial vehicles (UAVs) or quadcopters. The solution to the problem is based on game theory, as it provides a mathematical framework for modeling and studying strategic interactions involving multiple decision makers. The authors consider a game where the set of strategies of the runner is the set of possible combinations of speeds and directions of his movement, and the set of strategies of the pursuer is the set of all possible permutations of the elements of his speeds. The matrix of the resulting game consists of elements that are the time of capture. A mathematical problem on the optimal assignments of searching and pursuing unmanned aerial vehicles is formulated and solved. Purpose: optimizing the effectiveness of strategies for detecting and capturing four-rotor unmanned aerial vehicles (quadcopters). Methods: methods of mathematical modeling, game theory apparatus, Hungarian method of solving the problem of assignments, decision theory, the principle of dynamic programming, the Maple package for solving examples were used. Results: In order to fulfill the conditions for solving the problem of optimal assignments, it is necessary and sufficient that it is balanced. This assignment problem can be balanced by entering the required number of fictitious boats or escaping boats. After that, it is possible to formulate and solve the dual problem of optimal appointments. The resulting game can be solved by any method of solving matrix games. In this way, it is possible to determine the policy of harassment and search between drones. Practical importance: All escaped quadcopter UAVs can be chased and successfully intercepted using the developed models. Examples of the study of mathematical models using the Maple software package are given.

Federated learning: A cutting-edge survey of the latest advancements and applications

Robust machine learning (ML) models can be developed by leveraging large volumes of data and distributing the computational tasks across numerous devices or servers. Federated learning (FL) is a technique in the realm of ML that facilitates this goal by utilizing cloud infrastructure to enable collaborative model training among a network of decentralized devices. Beyond distributing the computational load, FL targets the resolution of privacy issues and the reduction of communication costs simultaneously. To protect user privacy, FL requires users to send model updates rather than transmitting large quantities of raw and potentially confidential data. Specifically, individuals train ML models locally using their own data and then upload the results in the form of weights and gradients to the cloud for aggregation into the global model. This strategy is also advantageous in environments with limited bandwidth or high communication costs, as it prevents the transmission of large data volumes. With the increasing volume of data and rising privacy concerns, alongside the emergence of large-scale ML models like Large Language Models (LLMs), FL presents itself as a timely and relevant solution. It is therefore essential to review current FL algorithms to guide future research that meets the rapidly evolving ML demands. This survey provides a comprehensive analysis and comparison of the most recent FL algorithms, evaluating them on various fronts including mathematical frameworks, privacy protection, resource allocation, and applications. Beyond summarizing existing FL methods, this survey identifies potential gaps, open areas, and future challenges based on the performance reports and algorithms used in recent studies. This survey enables researchers to readily identify existing limitations in the FL field for further exploration.

Dynamics of Love-type wave propagation in composite transversely isotropic porous structures

The present study aims to analyze the propagation behavior of Love-type wave in a composite transversely isotropic porous structure. The structure comprises an inhomogeneous sandy porous layer lying between a non-homogeneous magneto-poroelastic layer and a heterogeneous fractured porous half-space. Analytic solutions of the field equations of the respective media involve the application of the variable separable method and Wentzel-Kramers-Brillouin (WKB) asymptotic approach for the conversion of partial differential equations into ordinary differential equations. Through careful imposition of boundary conditions and subsequent elimination of arbitrary constants, we derive a complex dispersion relation governing the propagation of Love-type waves. This dispersion equation yields both the phase velocity curve, corresponding to the real expression, and the damping velocity curve, derived from the imaginary expression. To represent our findings, we conduct extensive calculations and graphical simulations illustrating the influence of various material parameters such as heterogeneity, porosity, volume fraction of fractures, sandiness, magnet-oelastic coupling, angle at which wave crosses the magnetic field, and layer thickness on the dispersive nature of Love-type waves using MATHEMATICA software. Furthermore, we conduct case-specific analyses, revealing instances where the dispersion equation simplifies to the standard Love wave equation, thereby validating our mathematical framework. Our findings underscore the significant influence of the aforementioned material parameters on the phase and damping velocities of Love-type wave. This interdisciplinary investigation into different porous media opens new avenues for future research and has significant implications in various disciplines, ranging from engineering and geophysics to environmental science and beyond.