Mathematical Model Research Articles

Structured mathematical modelling in an authentic scientific-engineering context

Abstract This study suggests the framing of structured mathematical modelling (MM) in an authentic scientific-engineering context. Taking a task design view, structure is maintained through inherent scaffolding for the dual purpose of preserving the authenticity of the extra-mathematical context and making the adaptations required to allow incorporation of the task into the duration of a regular lesson and within the full capacity of the curriculum. Participants were 23 secondary school novice mathematics teachers and two experts who possess extensive experience in MM task design. Research tools included a reflective diary filled in by the novice teachers throughout the design process, and a task design evaluation form that was filled in by the two expert designers. Both tools aimed at identifying MM task characteristics and the alignment of the designed structured MM tasks with the MM cycle phases. Findings show the presence of all the unique characteristics that underly a quality designed MM task, with relatively low presence of openness and high presence of accessibility and authenticity from the novice teachers’ point of view. The alignment with the MM cycle was further manifest in the designed tasks, with the validation phase as being less evident compared to the other MM phases. Theoretical and practical contributions are discussed in the paper, indicating the sequentially-oriented process by which the structured MM tasks scaffold students through the MM cycle.

Computer modeling of the synthesis of styrene–butadiene rubber: Influence of the feed of a chain transfer agent on the characteristics of the product

Objectives. To develop mathematical approaches and algorithms for analyzing the influence of various methods for feeding a chain transfer agent to a cascade of reactors, taking into account the choice of feed points on the characteristics of the final product of copolymerization using computer modeling.Methods. The processes of synthesis of copolymers were mathematically modeled using a statistical approach (Monte Carlo method). The developed algorithm is based on calculating the probabilities of elementary reactions in the process under study. In the case of continuous production of the copolymer in a cascade of reactors, it must be taken into account that the residence time of each particle of the reaction mixture in the reactor is subject to a probability distribution. The algorithm models the formation of copolymer macromolecules at the particle level, permitting the average molecular characteristics of the copolymer to be calculated and its microstructure to be studied based on modeling results.Results. The dependencies of the intrinsic viscosity on the reactor number and conversion were constructed by means of mathematical modeling. The calculation results showed satisfactory agreement with the experimental data obtained in production. The dependencies of the molecular weight distribution of the copolymer, the weight-average molecular weight, and the microheterogeneity index on the reactor number were constructed for various methods of feed of the chain transfer agent, i.e., to two and/or three points of the reactor cascade. The modeling and calculation results confirmed the influence of the method of adding the chain transfer agent to the cascade reactors on the molecular characteristics of the copolymer.Conclusions. The analysis of the structure of the molecular units of the styrene–butadiene copolymer showed a decrease in the weightaverage molecular weight of the final product and an increase in its stiffness in the case of the three-point feed of the chain transfer agent.

A mathematical model of the dynamic viscosity dependence of motor oils on temperature, soot concentration, and its morphology

Objectives. A quick cold start of emergency and auxiliary power units based on diesel engines should be possible at any time without problems and in the shortest possible time. The condition of the engine oil is one of the most important factors influencing the smooth start-up of power plants. During diesel engine operation, engine oil accumulates soot in its composition, negatively affecting its rheological properties. The aim of this research is to develop a mathematical model to describe changes in the dynamic viscosity of motor oils as a function of temperature. This model will account for the concentration of soot and its morphology, based on the results of experimental studies.Methods. Standardly used motor oils for diesel engines M-14D2SE and M-5z/14D2SE were used as oil samples in the preparation of model mixtures. The dispersed phase of the suspensions comprised carbon black of the N110, N220, N330, and N220 grades, characterized by a dusty (nongranular) texture. The rheological properties of the samples were determined using a TA Instruments DHR-2 rotational rheometer. The experimental data was subjected to mathematical statistical processing, in order to obtain approximating dependencies.Results. The paper presents an analysis of the various approaches to the description of the rheology of suspensions and the results of experimental studies of the viscosity-temperature characteristics (VTCs) of model samples of oils containing soot. The extant models of the dependence of the dynamic viscosity of suspensions on temperature, volume concentration of the dispersed phase, particle size and shape are demonstrated to be inadequate for the description of the VTCs of motor oils containing soot. A model of the rheological properties of soot-oil suspensions is proposed in the form of a mathematical dependence of their dynamic viscosity on temperature, mass concentration of soot, material density and size of soot particles, characteristics of the shape and structure of primary aggregates and the ratio of the sizes of aggregates and molecules of the dispersion medium.Conclusions. It was demonstrated that a comprehensive description of the VTCs of engine oils containing soot necessitates the consideration of the structural characteristics of the primary aggregates of soot particles. A mathematical model of the VTCs of oils was developed. This model is based on the dependence of the dynamic viscosity of oils on temperature, mass concentration of soot, density of the particle material, degree of non-sphericity of aggregates, the ratio of the particle sizes of the dispersed phase (either aggregates orsingle particles of non-aggregated soot) and oil molecules, and on the structure ofsoot, characterized by adsorption of dibutyl phthalate.

Estimation of PHA concentrations from cell density data in Cupriavidus necator

The production of biodegradable and biobased polymers is one way to overcome the present plastic pollution while using cheap and abundant feedstocks. Polyhydroxyalkanoates are a promising class of biopolymers that can be produced by various microorganisms. Within the production process, batch-to-batch variation occurs due to changing feedstock composition when using waste streams, slightly different starting conditions, or biological variance of the microorganisms. Therefore, reliable and online-capable measurement methods of the product concentration are required to monitor and eventually react to those fluctuations. In this work, we present a flexible approach to determine polyhydroxyalkanoate concentrations based on a simple mathematical model. The data-driven model correlates polyhydroxyalkanoate concentrations with optical densities measured at 600 nm, a widespread, fast, and cheap lab measurement. We found that with different cultivation conditions, the correlation needs to be updated for a reasonable PHA determination during the process. We test this approach for the production of poly(3-hydroxybutyrate) and poly(3-hydroxybutyrate-co-3-hydroxyvalerate) in Cupriavidus necator using fructose and propionic acid as carbon sources. This flexible approach allows a simple estimation of PHA concentrations and offers the possibility to further enhance biopolymer production when combined with advanced control strategies.Key points∙Development of a simple mathematical model to estimate polyhydroxyalkanoate concentrations.∙Optical density measurement is used to determine polyhydroxyalkanoate concentration.∙The approach is tested for the production of PHB and PHBV with C. necator.

Advancing Sustainability Assessment of Renewable Energy: A Study on Indicator‐to‐Framework Methods and Associated Rule Mining

ABSTRACTDivergent sustainable development perspectives, mathematical model limitations, and varying scenarios contribute to uncertainty in assessing renewable energy sustainability. To tackle these challenges, this study conducts a comprehensive literature review, employs an analogy, and utilizes knowledge discovery techniques with an indicator‐to‐framework approach. Qualitative classification and rule mining elucidate the interrelationships among dimensions, indicators, and scopes, enhancing assessment coherence. The study identifies 227 indicators across five dimensions and nine scopes, highlighting key rules through Apriori algorithm. Strong rules, meeting a support threshold of 0.22, confidence of 0.7, and Lift of 2.512, underscore the significance of environmental, economic, social, and technological dimensions. Environmental indicators like CO2 emissions, NOX emissions, and SO2 emissions feature prominently. The harmonization of dimensions, indicators, and scopes streamlines research and mitigates ambiguities. Appendices augment rules with frequent indicators, enriching insights. This data‐driven methodology provides a practical solution to the complexities of selecting appropriate indicators for such assessments.

A modelling technique to determine the high frequency transformer leakage inductance using the winding structure

The operation and efficiency of isolated DC-DC converters, critical components in solid-state transformers, are significantly impacted by leakage inductance in high-frequency transformers (HFTs). Different converters have varying demands regarding leakage inductance, ranging from precise control to minimal inductance. For instance, the performance of bidirectional isolated converters (BIDCs) and resonant converters is dependent on leakage inductance for the delivery of power. This study proposed a method for the control of leakage inductance by altering the winding configuration. The technique involved positioning the primary and secondary windings at predetermined heights to increase the separation between some winding turns, and thus, enhance leakage inductance. By not varying the average length of turns in the winding configuration, the proposed method was able to maintain a consistent copper loss. A modified mathematical model has been put forward to determine the leakage inductance effectively and precisely. The viability and accuracy of this method were validated through simulations and experiments using a three-phase HFT (3P-HFT) in a 3P-BIDC. The results showed that the leakage inductance that was calculated using the theoretical model was closely correlated to that of the simulation model, with only a variance of 3.9% and an error of 4.53% from the experimental results.

Mathematical Modelling of Wastewater Treatment: A Review and Application

Mathematical Modelling of Wastewater Treatment: A Review and Application

Understanding pectin cross-linking in plant cell walls

Pectin is a major component of plant cells walls. The extent to which pectin chains crosslink with one another determines crucial properties including cell wall strength, porosity, and the ability of small, biologically significant molecules to access the cell. Despite its importance, significant gaps remain in our comprehension, at the molecular level, of how pectin cross-links influence the mechanical and physical properties of cell walls. This study employs a multidisciplinary approach, combining molecular dynamics simulations, experimental investigations, and mathematical modelling, to elucidate the mechanism of pectin cross-linking and its effect on cell wall porosity. The computational aspects of this work challenge the prevailing egg-box model, favoring instead a zipper model for pectin cross-linking, whilst our experimental work highlights the significant impact of pectin cross-linking on cell wall porosity. This work advances our fundamental understanding of the biochemistry underpinning the structure and function of the plant cell wall. This knowledge has important implications for agricultural biotechnology, informing us about the chemical properties of plant pectins that are best suited for improving crop resilience and amenability to biofuel extraction by modifying the cell wall.

Tire-Pavement Interaction Noise: Recent Research on Concrete Pavement Surface Type and Texture

Several solutions have been proposed for quieter riding surfaces, including porous pavements, tining, and grinding. This paper deals with certain aspects of a recent large-scale research that has been carried out to examine the influence of cement concrete pavement surface type and texture on noise generation. One pavement surface type (Enhanced Porosity Concrete – EPC), and one surface texturing method (transverse tining) is dealt with in detail in this paper. Experimental studies to ascertain the physical (porosity and pore size), mechanical (strength), and acoustical (acoustic absorption using impedance tube) properties of EPC are discussed. It is shown in this paper that, with careful selection of aggregate gradation and cementing materials content, it is possible to generate a system of optimally-sized pores in the material to maximize acoustic absorption. Mathematical modeling of noise to evaluate the influence of transverse tine geometry (depth, width, and shape) on noise reduction characteristics is explained in the second part of this paper. A mathematical model has been used to determine the stress distribution between the concrete pavement and the tire. The stress distribution model is used to describe the stress and ultimately noise generated by various geometries of tining patterns. Experiments on Tire-Pavement test apparatus are also discussed.

Illumination map smoothing (IMS): a convex and differentiable mathematical model to rapidly enhance low-light images

Illumination map smoothing (IMS): a convex and differentiable mathematical model to rapidly enhance low-light images

First evidence of transovarial transmission of Kyasanur Forest disease virus in Haemaphysalis and Rhipicephalus ticks in the wild

BackgroundKyasanur forest disease virus (KFDV) is a tick-borne flavivirus causing debilitating and potentially fatal disease in people in the Western Ghats region of India. The transmission cycle is complex, involving multiple vector and host species, but there are significant gaps in ecological knowledge. Empirical data on pathogen-vector-host interactions and incrimination have not been updated since the last century, despite significant local changes in land use and the expansion of KFD to new areas. Mathematical models predict that transovarial transmission, whereby adult female ticks pass KFDV infections to their offspring, plays an important role in the persistence of KFD, but this has not been shown in the wild. Here we set out to establish whether transovarial transmission of KFDV was occurring under natural field conditions by assessing whether host-seeking larvae were positive for KFDV.MethodsTicks were sampled by dragging and flagging across a broad range of habitats within the agro-forest matrix at 49 sites in two districts: Shivamogga, Karnataka and Wayanad, Kerala (September 2018-March 2019), and larvae were tested for KFDV by PCR.ResultsIn total, larval ticks from 7 of the 49 sites sampled tested positive for KFDV, indicating that transovarial transmission is occurring. Of the 13 KFDV-positive larval samples, 3 came from around houses and gardens, 5 from crops (3 from harvested rice paddy and 2 from areca plantation), 1 from teak plantation and 4 (2 from 1 transect) from forests. Five different tick species were found to have KFDV-positive larvae: Haemaphysalis spinigera, H. bispinosa, Rhipicephalus annulatus, R. microplus and an unidentifiable species of Haemaphysalis (no close match in GenBank).ConclusionsOur empirical confirmation of transovarial transmission has important implications for understanding and predicting KFD dynamics, suggesting that ticks may act as a reservoir for KFDV. Moreover, small mammals and cattle may play crucial roles in transmission if small mammals are the main hosts for larvae infected via transovarial transmission, and cattle support large numbers of infected female adult ticks. This first report of transovarial transmission of KFDV, and within a hitherto undescribed range of vectors and habitats, will help disease managers improve KFD surveillance and mitigation strategies, ultimately leading to communities becoming more resilient to the risk of this tick-transmitted disease.Graphical

A method introducing deteriorating bond at the interfaces and declining load transfer at the joints in the existing design methods for concrete pavements

Most of the concrete highways of the Belgian road system have a structure composed of a CRCP surface layer, a thin intermediate asphalt bond layer and a lean concrete base layer on one or more granular sub bases. They are designed as an equivalent composite rigid layer on an elastic granular sub grade. The equivalent Young’s modulus of the composite layer is assumed to be equal to the modulus of the concrete surface layer, the thickness is assumed to be equal to the sum of the thickness of the three layers; the equivalent moment of inertia and the equivalent stiffness depend on the values of the thickness and the moduli of each layer and the bond conditions between the layers. Several investigations regularly performed on pavements currently thirty years old have shown a systemic deterioration of the bond between the surface layers and of the load transfer at the cracks of the CRCP or at the joints of concrete slabs. In this paper we present a mathematical model that permits the computation of the influence of this systemic deterioration on the lifetime of the pavements.

Identifying and describing generic, specific, and catalytic enablers of mathematical modelling

Abstract In this article, we report on an Australian study in which generic, specific, and catalytic enablers of mathematical modelling are identified and described. The study adopted a design-based research approach involving six secondary teachers, and their intact classes, over three years. Data was generated from 60 pre/post-teacher and student (Years 9–11) interviews, which was subject to reductive thematic analysis, consistent with the principles of inductive category formation. This analysis led to the development of a Typology for Instructional Enablers of Mathematical Modelling which identifies and describes both enablers and un-enablers of modelling activity. These are related to external/internal/personal learning goals, classroom expectations and ways of working, management of the learning process, and instructional anticipation. Enablers/un-enablers are further categorised into generic, specific, and catalytic dimensions. Implications for both practice and future research are discussed, including the nature of catalytic enablers and the connections between enablers and specific aspects of the modelling process.

Generalized Mathematical Modeling of Tuned Liquid Column Dampers Under Varying Excitation Amplitudes

Generalized Mathematical Modeling of Tuned Liquid Column Dampers Under Varying Excitation Amplitudes

Mathematical model of work optimisation in the development of automated information systems

The paper presents the main components of the mathematical model represent ing the work process when creating an information system. The development of automat ed information systems requires significant expenditure of both resources and time. The process of developing an automated information system consists of numerous stages that require careful planning and management, namely analyzing system requirements, design, programming, debugging, implementation and maintenance. Estimation of the labor inten sity of programming and debugging stages is based on the approach of M. Halstead and amounts to 60-70% of the total cost of information system development. The tasks solved in this study allow us to make a step in the direction of increasing the level of information systems’ automatization. Implementation of the mathematical model of work optimization will allow increasing the overall level of development automation.

The Use of Machine Learning Methods in Road Safety Research in Poland

Every year, thousands of accidents occur in Poland, often resulting in severe injuries or even death. The implementation of solutions supporting road safety analysis and management processes is necessary to reduce the risk of accidents and minimize their consequences. One of the rapidly developing tools that can play a key role in this area is machine learning. The aim of this study was to develop mathematical models based on ML algorithms describing road safety in Poland. First, variables with the strongest impact on safety were extracted. Then, mathematical modeling was performed using the k-Nearest Neighbors, Random Forest, and RPart algorithms. The best choice for imbalanced data, especially when the goal is to identify rare classes, is the RF model. The KNN model provides a compromise in situations where the highest overall accuracy is desired. On the other hand, the RPart model can be used as a fast, basic model, but it requires improvements to handle rare classes. The results not only identified factors that significantly affect the severity of injuries or the number of fatalities in accidents but, above all, also demonstrated the ability of ML-based models to predict threats and their consequences.

Three-dimensional linkage analysis with digital PCR for genome integrity and identity of recombinant adeno-associated virus

Recombinant adeno-associated virus (rAAV) has emerged as the vector of choice for in vivo gene delivery, with numerous clinical trials underway for the treatment of various human diseases. Utilizing rAAV in gene therapy requires a highly precise quantification method to determine the viral genome titer and further establish the optimal therapeutic dosage for a rAAV product. The conventional single-channel droplet digital PCR (1D ddPCR) method offers only partial information regarding the viral vector genome titer, lacking insights into its integrity. In our pursuit of further advancing rAAV analysis, we have developed a novel 3D ddPCR assay with advanced 3D linkage analysis. We have designed the three amplicon sites targeting both ends of the viral genome, as well as the center of key therapeutic gene of interest (GOI). This study aims to offer a more comprehensive and insightful assessment of rAAV products which includes not only quantity of viral genome titer but also the quality, distinguishing between partial ones and intact full-length viral genomes with the right GOI. Importantly, due to the random partitioning property of a digital PCR system, the 3D linkage analysis of rAAV viral genome requires a proper mathematical model to identify the true linked DNA molecules (full-length/intact DNA) from the population of false/unlinked DNA molecules (fragmented/partial DNA). We therefore have developed an AAV 3D linkage analysis workflow to characterize genomic integrity and intact titer for rAAV gene therapy products. In this study, we focus on evaluating our 3D linkage mathematical model by performing DNA mixing experiments and a case study using multiple rAAV samples. Particularly, we rigorously tested our algorithms by conducting experiments involving the mixing of seven DNA fragments to represent various AAV viral genome populations, including 3 single partials, 3 double partials, and 1 full-length genomes. Across all 37 tested scenarios, we validated the accuracy of our workflow’s output for the percentages of 3D linkage by comparing to the known percentages of input DNA. Consequently, our comprehensive AAV analytical package not only offers insights into viral genome titer but also provides valuable information on its integrity and identity. This cost-effective approach, akin to the setup of traditional 1D or 2D dPCR, holds the potential to advance the application of rAAV in cell and gene therapy for the treatment of human diseases.

Does the audiogram shape influence the intracochlear recording of Electrocochleography during and after cochlear implantation?

During cochlear implant (CI) surgery, it is desirable to perform intraoperative measurements such as Electrocochleography (ECochG) to monitor the inner ear function and thereby to support the preservation of residual hearing. However, a significant challenge arises as the recording location of intracochlear ECochG via the CI electrode changes during electrode insertion. This study aimed to investigate the relationships between intracochlear ECochG recordings, the position of the recording contact within the cochlea relative to its anatomy, and the implications for frequency and residual hearing preservation. Intraoperative ECochG recordings were conducted using the CI electrode (MED-EL) during the insertion of hearing preservation electrodes and after the insertion process. Recordings were continuously conducted using the most apical electrode (contact 1) during insertion. After insertion, the recordings were performed on all different electrode contacts. The electrode location in the cochlea during insertion was estimated using mathematical models and preoperative clinical imaging, while the postoperative electrode position was determined using postoperative clinical imaging. The study involved 10 adult CI recipients. In those with good low-frequency hearing, an increase in signal amplitude was observed, with the highest amplitudes closest to the stimulation frequency generators, and no phase change was observed. Conversely, patients with flat hearing loss exhibited a second peak with an opposite phase in the medial area of the cochlea. This study is the first to suggest that the pattern of the preoperative audiogram may influence the ECochG outcomes measured intraoperatively. Specifically, the ECochG responses during insertion appeared to behave as expected with good low-frequency hearing, while with flat hearing loss there appear to be further effects. These findings indicate that this approach can provide valuable information for the interpretation of intracochlearly recorded ECochG signals.

Fuel Replenishment Problem of Heterogeneous Fleet in Initiative Distribution Mode

Petrol, a vital energy source for residents’ consumption and economically sustainable operation, generates substantial distribution demand. To reduce distribution costs, we propose a fuel replenishment problem using a heterogeneous fleet based on the initiative distribution mode. In this mode, the distribution center determines both the delivery orders of customers and the distribution plan. We develop a mathematical model with minimal operational costs, including transport, employment, and penalty costs. A Two-stage heuristic algorithm K-IBKA based on time-space clustering is proposed, which also combines the advantages of the butterfly optimization algorithm in quick convergence and hierarchical mutation strategy in population diversity. The results demonstrate that: (1) Heterogeneous truck distribution exhibits better cost advantages compared to homogeneous distribution, reducing total costs by 13.07%; (2) Compared to passive distribution mode, the total cost of the initiative distribution mode is reduced by 11.03% and 41.80%, respectively, through small and large-scale instances. (3) Compared with the unimproved BKA, ALNS, and GA, the total cost calculated by K-IBKA is reduced by 37.68%, 35.30%, and 27.26%, respectively, thus demonstrating the contribution of this work to reducing the cost of petrol distribution and achieving sustainable development of distribution.

Minimizing Delay and Power Consumption at the Edge

Edge computing systems must offer low latency at low cost and low power consumption for sensors and other applications, including the IoT, smart vehicles, smart homes, and 6G. Thus, substantial research has been conducted to identify optimum task allocation schemes in this context using non-linear optimization, machine learning, and market-based algorithms. Prior work has mainly focused on two methodologies: (i) formulating non-linear optimizations that lead to NP-hard problems, which are processed via heuristics, and (ii) using AI-based formulations, such as reinforcement learning, that are then tested with simulations. These prior approaches have two shortcomings: (a) there is no guarantee that optimum solutions are achieved, and (b) they do not provide an explicit formula for the fraction of tasks that are allocated to the different servers to achieve a specified optimum. This paper offers a radically different and mathematically based principled method that explicitly computes the optimum fraction of jobs that should be allocated to the different servers to (1) minimize the average latency (delay) of the jobs that are allocated to the edge servers and (2) minimize the average energy consumption of these jobs at the set of edge servers. These results are obtained with a mathematical model of a multiple-server edge system that is managed by a task distribution platform, whose equations are derived and solved using methods from stochastic processes. This approach has low computational cost and provides simple linear complexity formulas to compute the fraction of tasks that should be assigned to the different servers to achieve minimum latency and minimum energy consumption.