Mathematical Analysis Research Articles

Disturbance observer‐based matrix‐weighted consensus

AbstractIn this paper, we proposed several disturbance observer‐based matrix‐weighted consensus algorithms. A new disturbance observer is firstly designed for linear systems with unknown matched or mismatched disturbances representable as the multiplication of a known time‐varying matrix with a unknown constant vector. Under some assumptions on the boundedness and persistent excitation of the regression matrix, the disturbances can be estimated at an exponential rate. Then, a suitable compensation input is provided to compensate the unknown disturbances. Second, disturbance‐observer based consensus algorithms are proposed for matrix‐weighted networks of single‐ and double‐integrators with matched or mismatched disturbances. We show that both matched and mismatched disturbances can be estimated and actively compensated, and the consensus system uniformly globally asymptotically converges to a fixed point in the kernel of the matrix‐weighted Laplacian. Depending on the network connectivity, the system can asymptotically achieve a consensus or a cluster configuration. The disturbance‐observer based consensus design is further extended for a network of higher‐order integrators subjected to disturbances. Finally, simulation results are provided to support the mathematical analysis.

Dynamics of a stockless market through piecewise smooth models

Markets play a crucial role in the economic activity of any country, serving as fundamental drivers of growth, development, and employment. Given their significance, understanding the dynamics of markets is essential. In this study, we present a mathematical model to characterize the behavior of a stockless market. The model is formulated using a system of ordinary differential equations defined with piecewise smooth functions. We conduct a mathematical analysis of the model, particularly focusing on the behavior of trajectories as they approach switching surfaces using Filippov’s analysis. Furthermore, we provide numerical simulations to visually illustrate the model’s dynamics and a insights a brief study of bifurcations is done.

Mathematical analysis of batch reactor performance for the enzymatic synthesis of cephalexin: Laplace Homotopy perturbation method and Adomian decomposition method

In this article, a mathematical model of diffusion reaction in kinetically controlled cephalexin synthesis in the batch reactor with penicillin acylase immobilized in glyoxyl-agarose is analyzed. The kinetic model is a non-linear non-stead-state reaction–diffusion equation with non-linear terms related to the Fick’s law. We have presented the approximate analytical expression for the non-steady-state non-linear reaction–diffusion equation by utilizing the Laplace homotopy perturbation method (LHPM) and for steady-state by using LHPM and Adomian decomposition method (ADM). The approximate analytical solution obtained from these methods proved that they are fit for every values of the reaction–diffusion and kinetic parameters. We also present the numerical solution of the considered reaction–diffusion equation by using pdepe tool in MATLAB software. When comparing the semi-analytical solution with the numerical solution, a satisfactory result is noted for all the possible values of the parameters. The closed-form analytical expressions for the effectiveness factor in non-steady and steady-state conditions are obtained and analyzed using LHPM and ADM.

Quantifying the Health–Economy Trade-Offs: Mathematical Model of COVID-19 Pandemic Dynamics

The COVID-19 pandemic has presented a complex situation that requires a balance between control measures like lockdowns and easing restrictions. Control measures can limit the spread of the virus but can also cause economic and social issues. Easing restrictions can support economic recovery but may increase the risk of virus transmission. Mathematical approaches can help address these trade-offs by modeling the interactions between factors such as virus transmission rates, public health interventions, and economic and social impacts. A study using a susceptible-infected-susceptible (SIS) model with modified discrete time was conducted to determine the cost of handling COVID-19. The results showed that, without government intervention, the number of patients rejected by health facilities and the cost of handling a pandemic increased significantly. Lockdown intervention provided the least number of rejected patients compared to social distancing, but the costs of handling the pandemic in the lockdown scenario remained higher than those of social distancing. This research demonstrates that mathematical approaches can help identify critical junctures in a pandemic, such as limited health system capacity or high transmission rates, that require rapid response and appropriate action. By using mathematical analysis, decision-makers can develop more effective and responsive strategies, considering the various factors involved in the virus’s spread and its impact on society and the economy.

Source-Sink Dynamics in a Two-Patch SI Epidemic Model with Life Stages and No Recovery from Infection.

This study presents a comprehensive analysis of a two-patch, two-life stage SI model without recovery from infection, focusing on the dynamics of disease spread and host population viability in natural populations. The model, inspired by real-world ecological crises like the decline of amphibian populations due to chytridiomycosis and sea star populations due to Sea Star Wasting Disease, aims to understand the conditions under which a sink host population can present ecological rescue from a healthier, source population. Mathematical and numerical analyses reveal the critical roles of the basic reproductive numbers of the source and sink populations, the maturation rate, and the dispersal rate of juveniles in determining population outcomes. The study identifies basic reproduction numbers for each of the patches, and conditions for the basic reproduction numbers to produce a receiving patch under which its population. These findings provide insights into managing natural populations affected by disease, with implications for conservation strategies, such as the importance of maintaining reproductively viable refuge populations and considering the effects of dispersal and maturation rates on population recovery. The research underscores the complexity of host-pathogen dynamics in spatially structured environments and highlights the need for multi-faceted approaches to biodiversity conservation in the face of emerging diseases.

Analysis of the privacy-performance tradeoff of reference testing in Forensic Investigative Genetic Genealogy.

During an investigation using Forensic Investigative Genetic Genealogy, which is a novel approach for solving violent crimes and identifying human remains, reference testing-when law enforcement requests a DNA sample from a person in a partially constructed family tree-is sometimes used when an investigation has stalled. Because the people considered for a reference test have not opted in to allow law enforcement to use their DNA profile in this way, reference testing is viewed by many as an invasion of privacy and by some as unethical. We generalize an existing mathematical optimization model of the genealogy process by incorporating the option of reference testing. Using simulated versions of 17 DNA Doe Project cases, we find that reference testing can solve cases more quickly (although many reference tests are required to substantially hasten the investigative process), but only rarely (<1%) solves cases that cannot otherwise be solved. Through a mixture of mathematical and computational analysis, we find that the most desirable people to test are at the bottom of a path descending from an ancestral couple that is most likely to be related to the target. We also characterize the rare cases where reference testing is necessary for solving the case: when there is only one descending path from an ancestral couple, which precludes the possibility of identifying an intersection (e.g., marriage) between two descendants of two different ancestral couples.

New implicit time integration schemes combining high frequency damping with high second order accuracy

It is shown that weighted linear combinations of the generalised-α method and certain related higher order schemes allow for the formulation of unconditionally stable single step time integration methods with improved second order accuracy and more targeted high-frequency damping. It is also shown that, if the user controlled high frequency damping parameter ρ∞ is set to zero, the new schemes can be expressed as linear multistep backward difference formulae and, in a particular case, recover Park's method. The performance of the proposed methods is illustrated in terms of mathematical analysis and a number of linear and nonlinear numerical examples including finite element based solutions of the incompressible Navier-Stokes equations.

A Galerkin Finite Element Method for the Reconstruction of a Time-Dependent Convection Coefficient and Source in a 1D Model of Magnetohydrodynamics

The mathematical analysis of viscous magnetohydrodynamics (MHD) models is of great interest in recent years. In this paper, a finite element Galerkin method is employed for the estimation of an unknown time-dependent convection coefficient and source in a 1D magnetohydrodynamics flow system. In this inverse problem, two integral observations are posed and used to transform the inverse problem to a non-classical direct problem with a non-local parabolic operator. Then, the non-classical strongly coupled parabolic system is studied in various settings. The equivalence of the inverse problem (IP) and the direct one are proven. The Galerkin procedure is analyzed to proove the existence and uniqueness of the solution. The finite element method (FEM) has been developed for the solution of the variational problem. Test examples are discussed.

Mathematical analysis of non-linear boundary-value problems in reaction-diffusion model of chitosan-alginate microsphere using homotopy perturbation and Akbari-Ganji methods

This article describes the non-linear reaction-diffusion model, which depicts the behaviour of hydrogen peroxide production and glucose oxidation in the chitosan-alginate microsphere. Analytical solutions of glucose, oxygen, gluconic acid and hydrogen peroxide concentrations in planar coordinates under steady-state circumstances are obtained for all reaction-diffusion parameters. The non-linear boundary value problem's approximate analytical expressions are obtained using asymptotic techniques based on the homotopy perturbation and Akbari-Ganji method. The exact and commonly used MATLAB program provided a numerical simulation. It is demonstrated that the resulting analytical expressions strongly agree with the numerical outcomes reported in the literature. Furthermore, determining hydrogen peroxide flux and pH profiles within the microspheres are also discussed. The theoretical findings make it possible to forecast and enhance the efficiency of enzyme kinetics.

Exploring group concepts in abstract algebra through ChatGPT

Traditional methods often struggle to engage students and effectively communicate the intricacies of abstract algebra. This study aims to explore the future of mathematics teaching, with a focus on integrating group theory concepts into Abstract Algebra using ChatGPT, an advanced language processing AI. Chat transcripts edited with ChatGPT were analyzed to gain insights into the evolving landscape of mathematics education, specifically concerning group theory. Group theory is a fundamental branch of Abstract Algebra poised to shape the future of mathematics instruction. ChatGPT facilitates interactive discussions and problem-solving tasks related to group operations, properties, and theorems, offering personalized explanations and guidance tailored to individual student needs. The study highlights the potential of AI-driven platforms like ChatGPT to revolutionize mathematics education by providing dynamic and engaging learning experiences centered around group theory. Furthermore, the study emphasizes the importance of incorporating group theory into mathematics curricula to foster critical thinking, problem-solving skills, and interdisciplinary connections. Educators are urged to leverage innovative tools like ChatGPT alongside traditional pedagogical approaches to ensure effective teaching and learning. Through thoughtful implementation and ongoing professional development, educators can harness AI's power to create inclusive, equitable, and enriching learning environments, empowering students in abstract mathematical reasoning and analysis.

Cost-Efficient Approximation for Magnitude of a Complex Signal

In this paper, a signal model and a mathematical analysis of an efficient approach are derived to acquire the approximate magnitude of a complex signal by inducing a pre-biased or a pre-scaled factor in the design criteria. According to the deductive results, the pre-biased or pre-scaled factor can be 2∼3 dB, which is determined through its application. The numerical evaluations show that the mean square error (MSE) of the proposed efficient approach for the random signal is around −33 dB. Based on the design templates for the considered approaches, the occupied areas of the proposed type-1 and -2 approaches are merely 0.13 and 0.09 times the area of the direct-method, respectively. As a result, the proposed efficient approach is certainly a cost-efficient method for obtaining the approximate magnitude of a complex signal.

Applications of Fractional-Order Calculus in Robotics

Fractional calculus, a branch of mathematical analysis, extends traditional calculus that encompasses integrals and derivatives of non-integer orders [...]

Piezoelectric Biosensor based on ultrasensitive MEMS system

The lack of ultrasensitive biosensors is the major challenge in the control of pandemics at the very initial stage, so it is necessary to develop new biosensors for the prompt treatment, and the piezoelectric biosensor based on ultrasensitive MEMS system offers a good opportunity. This paper presents a novel approach using polyvinylidene fluoride (PVDF) unsmooth nanofibers to transfer electronic current. The unique unsmooth surface of these nanofibers provides a high surface energy (geometrical potential), making them highly sensitive to microorganisms (e.g., viruses) absorbed on their surface. The absorption of viruses has an obvious effect on the nanofiber’s displacement and electrical resistivity. This paper presents a mathematical model that demonstrates the system's ultra-sensitivity. This rigorous mathematical analysis shows that even when the pull-in voltage in the PVDF nanofiber changes by just 0.0589 %, a dramatic shift from normal periodic motion to pull-in motion is predicted. This paper provides a new and reliable theoretical way to detect pandemics at the very beginning stage.

Mathematical analysis of heat and mass transfer efficiency of bioconvective Casson nanofluid flow through conical gap among the rotating surfaces under the influences of thermal radiation and activation energy

In the current proceeding, the flow of incompressible non-Newtonian nanofluid called Casson nanofluid is considered. A conical gap occurred among the rotating disc and the cone filled with the fluid flow. Heat and mass transport through this nanofluid is done by the convection mode of heat transfer. The impacts of microorganisms, chemical processes, thermal radiation, minimal amount of energy, and magnetic field are also considered in the mathematical model of flow problem. The Casson nano-fluid governing equations are interpreted in cylindrical coordinates. By implementing proper similarity transformations, the modeling PDEs of energy, momentum, concentration, and microorganism density are transformed into non-linear ODEs. A set of non-linear ODEs that deals with the distributions of temperature, velocity, concentration, and motile microorganisms produced by this technique. MATLAB in-built ‘bvp4c‘ technique is utilized to solve these equations. Findings are displayed graphically and elaborated theoretically. The primary goal of this work is to examine the effects during the rotation of the disc and cone as well as the impacts of other variables on the rotation. The nano-fluid temperature and radial velocity are found to be negatively impacted by the rotation parameter whereas azimuthal velocity is positively impacted. The parametric values are taken as 0.1<Nb<0.7, 0.1<λ<0.4, 1.0<Pr<2.50.1<M<0.4, 1.0<Rb<4.0, 0.1<Nr<0.4, 0.1<Sc<3.0, 0.1<Nt<0.4 for the purpose of generating modified results. From published and current results noted that rotation parameter effects impacted the transfer rate of the nanofluid.

A mathematical theory of relational generalization in transitive inference

Humans and animals routinely infer relations between different items or events and generalize these relations to novel combinations of items. This allows them to respond appropriately to radically novel circumstances and is fundamental to advanced cognition. However, how learning systems (including the brain) can implement the necessary inductive biases has been unclear. We investigated transitive inference (TI), a classic relational task paradigm in which subjects must learn a relation ([Formula: see text] and [Formula: see text]) and generalize it to new combinations of items ([Formula: see text]). Through mathematical analysis, we found that a broad range of biologically relevant learning models (e.g. gradient flow or ridge regression) perform TI successfully and recapitulate signature behavioral patterns long observed in living subjects. First, we found that models with item-wise additive representations automatically encode transitive relations. Second, for more general representations, a single scalar "conjunctivity factor" determines model behavior on TI and, further, the principle of norm minimization (a standard statistical inductive bias) enables models with fixed, partly conjunctive representations to generalize transitively. Finally, neural networks in the "rich regime," which enables representation learning and improves generalization on many tasks, unexpectedly show poor generalization and anomalous behavior on TI. We find that such networks implement a form of norm minimization (over hidden weights) that yields a local encoding mechanism lacking transitivity. Our findings show how minimal statistical learning principles give rise to a classical relational inductive bias (transitivity), explain empirically observed behaviors, and establish a formal approach to understanding the neural basis of relational abstraction.

Third moment method for reliability analysis with uncertain moments characterized as interval variables

Traditional reliability analysis aims to compute the failure probability based on probability distribution functions, which are constructed using the moments of random parameters. In practice, however, appropriate samples may be insufficient to obtain deterministic values of the moments of all random variables and the exact value of failure probability cannot be obtained. To be consistent with the reality, the uncertainties in moments can be measured as interval variables, and then the bounds of failure probability should be evaluated. In this study, an idealized case is considered, where there is at most one imprecise moment associated with any given input random variable. A third moment method is proposed with uncertain moments measured as interval variables, and is named as TMI method. The proposed TMI method is straightforward including only four steps. Firstly, the derivative of performance function to random variables having uncertain moments is calculated, with the random variables set to be their mean values. Secondly, the values of uncertain moments for computing the bounds of failure probability are determined. Then, with inverse normal transformation defined based on the moments, the performance function at the bounds in Gaussian space is directly constructed. Finally, bounds of failure probability can be evaluated by two times of classical reliability analysis corresponding to the constructed performance functions. The application of TMI method is validated by numerical examples, including high-dimensional and strong nonlinear problems.

Mathematical Analysis for GPU Framework for High Performance Computing with Improved Scalability, Reliability and Seamless Configurability

In the field of High Performance Computing (HPC), which is changing very quickly, using Graphics Processing Units (GPUs) has become essential for getting a lot of computing power and speed. This study gives a thorough mathematical analysis of a GPU system that is meant to improve scalability, stability, and configurability, all of which are very important for next-generation HPC apps. The suggested framework uses complex mathematical models and methods to make the best use of GPU resources, cut down on delay, and guarantee strong performance across a wide range of tasks. Scalable parallel processing methods are built into the framework so it can adapt to changing computing needs, making the most of speed and resource use. Fault-tolerant methods that lessen the effects of hardware breakdowns and computing mistakes also improve reliability, making sure that results are always correct and consistent. The framework can be set up in different ways because it is based on flexible design, which makes it easy to change and adapt to the needs of each application without affecting speed. In diverse computer settings, where different apps may have different working needs, this ability to change is very important. The mathematical analysis includes performance measures like speed, delay, and mistake rates, which give a solid basis for judging how well the system works. Additionally, tests comparing the suggested system to current GPU tools show that it is more reliable and scalable. This study helps make better, more reliable computer systems by looking at some of the biggest problems in GPU-based high-performance computing. The results show that using this method can greatly improve the skills of HPC systems, which can lead to progress in scientific study, data analysis, and complex models. In the end, this work shows how advanced GPU systems can help drive innovation in HPC, leading to more powerful, reliable, and flexible computing options.

Enhancing Numerical Simulations through Advanced Data Analysis In Computational Mathematics

Improving numerical models through advanced data analysis in computational mathematics is a major step forward in many areas of science and engineering. The main goal of this study is to find ways to make accuracy, speed, and the ability to predict better by combining advanced data analysis techniques with computer simulation techniques. The study looks at how machine learning algorithms, statistical methods, and traditional computer methods can work together to solve hard, multidimensional problems. Algorithms for machine learning, like neural networks and support vector machines, are used to find trends and connections in very big datasets. These algorithms can give us information that other methods might miss. These new ideas make it easier to make models and programs that are more realistic. Statistical methods, like regression analysis and Bayesian reasoning, are used to measure error and make computer results more reliable. The goal of the study is to lower the cost of computing and raise the rate at which computer models converge by using these methods. A big part of this study is using these advanced methods to solve problems in the real world, like in material science, fluid dynamics, and climate models. Case studies show how combining computer simulations and data analysis can help make predictions that are more accurate and help people make better decisions. The study also looks at the problems that come with how hard these advanced methods are to compute and how much data they need, and it suggests ways to fix these problems. This study shows how mixing advanced data analysis with computer models in computational mathematics can change things. The results show how important it is to use methods from different fields to solve difficult science and engineering issues. This makes it possible for more accurate and useful models that can lead to new ideas and discoveries in many areas. This combination not only makes numerical models more powerful, but it also creates new study and use opportunities in computer mathematics.

Unbiased density computation for stochastic resetting *

We establish a practical means for unbiased computation of the marginal probability density function of the dynamics under stochastic resetting. In contrast to conventional dynamics devoid of resetting, the marginal probability density function under resetting may exhibit cusps and, in multi-dimensions, explosions at reset positions, arising from the compelled displacement of mass. Standard numerical techniques, such as kernel density estimation, fall short in accurately reproducing those characteristics due to their inherent smoothing effects. The proposed unbiased estimation formulas are derived using advanced stochastic calculus in their general formulations, yet their implementation in specific problem settings involves only elementary numerical operations, requiring minimal mathematical expertise and marking the very first instance of a numerical method free from bias in this context. We present numerical results throughout to validate the derived estimation formulas and, more broadly, to demonstrate the effectiveness of our approach in accurately capturing the irregularities of the marginal probability density function.

On head-to-head results as tie-breaker and consequent opportunities for collusion

Abstract Accepted by: Phil Scarf The outcome of some football matches has benefited both teams at the expense of a third team because head-to-head results were used for breaking ties. Inspired by these examples, our mathematical analysis identifies all possible collusion opportunities caused by this particular ranking rule in a single round-robin tournament with four teams. According to a simulation model based on the 2024 UEFA European Football Championship, merely the tie-breaking rule increases the probability of reaching a situation vulnerable to collusion by between 11.5 and 14.8 percentage points. This risk can scarcely be mitigated by a static match schedule. Therefore, tournament organizers are strongly encouraged to choose goal difference as the primary tie-breaking rule, similar to the official policy of FIFA.