Mathematical Challenges Research Articles

Quasiperiodicity in the α-Fermi-Pasta-Ulam-Tsingou problem revisited: An approach using ideas from wave turbulence.

The Fermi-Pasta-Ulam-Tsingou (FPUT) problem addresses fundamental questions in statistical physics, and attempts to understand the origin of recurrences in the system have led to many great advances in nonlinear dynamics and mathematical physics. In this work, we revisit the problem and study quasiperiodic recurrences in the weakly nonlinear α-FPUT system in more detail. We aim to reconstruct the quasiperiodic behavior observed in the original paper from the canonical transformation used to remove the three-wave interactions, which is necessary before applying the wave turbulence formalism. We expect the construction to match the observed quasiperiodicity if we are in the weakly nonlinear regime. Surprisingly, in our work, we find that this is not always the case and in particular, the recurrences observed in the original paper cannot be constructed by our method. We attribute this disagreement to the presence of small denominators in the canonical transformation used to remove the three-wave interactions before arriving at the starting point of wave turbulence. We also show that these small denominators are present even in the weakly nonlinear regime, and they become more significant as the system size is increased. We also discuss our results in the context of the problem of equilibration in the α-FPUT system and point out some mathematical challenges when the wave turbulence formalism is applied to explain thermalization in the α-FPUT problem. We argue that certain aspects of the α-FPUT system such as thermalization in the thermodynamic limit and the cause of quasiperiodicity are not clear, and that they require further mathematical and numerical studies.

Heat transfer in the Jeffery-Hamel flow of a yield-stress fluid

Heat transfer in the Jeffery-Hamel flow of a yield-stress fluid that obeys the Bingham-Papanastasiou model is investigated in the present study. Among various rheological models proposed to formulate the flow of yield-stress fluids, the Bingham model is usually used to address the effect of yield-stress. A fluid that obeys the Bingham model does not show any shear-thinning or shear-thickening behaviors; this makes the interpretation of the results much easier. Addressing the unyielded regions (regions at which the yield-stress material behaves like a solid) in a yield-stress fluid flow is not a trivial task, and it could be impossible, especially in complex geometries. Unfortunately, the Bingham model itself cannot be used to formulate complex fluid flow problems due to the mathematical difficulties (i.e., addressing the unyielded regions); hence it is usually regularized to avoid these mathematical challenges. One of these regularized models is the Bingham-Papanastasiou model. Using an exponential term enables the Bingham-Papanastasiou model to address the unyielded regions. However, this exponential term results in highly non-linear equations, which are difficult to be handled mathematically. In the present study, the effect of the yield-stress (Bingham number) on the hydrodynamics and heat transfer in the Jeffery-Hamel flow is investigated by using the Bingham-Papanastasiou model. The momentum transfer in the Jeffery-Hamel flow of a Bingham-Papanastasiou fluid is governed by a highly non-linear ordinary differential equation (ODE) which is a boundary value problem. Assuming that the fluid's properties are independent of the temperature and pressure, this highly non-linear ODE is solved numerically. The heat transfer in the flow, with a constant wall temperature boundary condition, is investigated, assuming the flow to be an equilibrium viscous dissipation flow (the fluid's mixed mean temperature variation is negligible along the channel). The resultant linear energy equation is also solved numerically to obtain the temperature distribution across the channel. The results obtained in this work show that the yield-stress increases the wall shear stress, Nusselt number, and Stanton number. In fact, the effect of the yield-stress on the Nusselt number is comparable to the effects of Prandtl and Eckert numbers. It is also found that the yield-stress can prevent the flow separation in a divergent channel. Considering the advantages of using the yield-stress fluids as the working fluids in the Jeffery-Hamel flow, they can be considered as an alternative to nanoparticles which are usually used to enhance the fluid's thermal conductivity and also the Nusselt number.

Mathematics of biomedical imaging today—a perspective

Biomedical imaging is a fascinating, rich and dynamic research area, which has huge importance in biomedical research and clinical practice alike. The key technology behind the processing, and automated analysis and quantification of imaging data is mathematics. Starting with the optimisation of the image acquisition and the reconstruction of an image from indirect tomographic measurement data, all the way to the automated segmentation of tumours in medical images and the design of optimal treatment plans based on image biomarkers, mathematics appears in all of these in different flavours. Non-smooth optimisation in the context of sparsity-promoting image priors, partial differential equations for image registration and motion estimation, and deep neural networks for image segmentation, to name just a few. In this article, we present and review mathematical topics that arise within the whole biomedical imaging pipeline, from tomographic measurements to clinical support tools, and highlight some modern topics and open problems. The article is addressed to both biomedical researchers who want to get a taste of where mathematics arises in biomedical imaging as well as mathematicians who are interested in what mathematical challenges biomedical imaging research entails.

Mini-Workshop: Zero-Range and Point-Like Singular Perturbations: For a Spillover to Analysis, PDE and Differential Geometry

The field of contact interactions and perturbations of differential operators supported on subsets with non-trivial co-dimension is an increasingly active mainstream of mathematical physics (in particular, operator and spectral theory and quantum mechanics), with intimately related applications and mathematical challenges in partial differential equations, and neighbouring sectors of analysis, PDEs, and differential geometry. This Mini-Workshop fostered intense and prolific discussions on recent advances and trends in the field.

New Approaches for Solving Interpolation Problems and Homogeneous Linear Recurrence Relations

This article presents a new approach to address the resolution of homogeneous linear recurrences of higher order and interpolation problems. By establishing an explicit formula for the entries of the inverse of generalized Vandermonde matrices, a fresh perspective on these mathematical challenges is introduced. The study primarily focuses on linear recurrence relations and thoroughly investigates cases involving characteristic polynomials with both simple roots and roots of multiplicity. To illustrate the effectiveness and practicality of the proposed method, a comprehensive set of illustrative examples is provided, highlighting its applicability in solving a wide range of instances of linear recurrence relations. Additionally, the limitations of the formula are discussed, particularly in scenarios where its applicability may be restricted. The findings of this study contribute significantly to the existing literature, providing an alternative and promising approach for solving problems that rely on the inverse Vandermonde matrix. In conclusion, this article emphasizes the need for further research to explore the computational advantages of the proposed method and to extend its applicability to cases featuring characteristic polynomials with a single root of multiplicity greater than one. By expanding the knowledge in the field, this study offers valuable insights into the resolution of linear recurrences and interpolation problems, presenting a new perspective and expanding the existing knowledge in the field.

The Flow Matrix Offers a Straightforward Alternative to the Problematic Markov Matrix

The Flow matrix is a novel method to describe and extrapolate transitions among categories. The Flow matrix extrapolates a constant transition size per unit of time on a time continuum with a maximum of one incident per observation during the extrapolation. The Flow matrix extrapolates linearly until the persistence of a category shrinks to zero. The Flow matrix has concepts and mathematics that are more straightforward than the Markov matrix. However, many scientists apply the Markov matrix by default because popular software packages offer no alternative to the Markov matrix, despite the conceptual and mathematical challenges that the Markov matrix poses. The Markov matrix extrapolates a constant transition proportion per time interval during whole-number multiples of the duration of the calibration time interval. The Markov extrapolation allows at most one incident per observation during each time interval but allows repeated incidents per observation through sequential time intervals. Many Markov extrapolations approach a steady state asymptotically through time as each category size approaches a constant. We use case studies concerning land change to illustrate the characteristics of the Flow and Markov matrices. The Flow and Markov extrapolations both deviate from the reference data during a validation time interval, implying there is no reason to prefer one matrix to the other in terms of correspondence with the processes that we analyzed. The two matrices differ substantially in terms of their underlying concepts and mathematical behaviors. Scientists should consider the ease of use and interpretation for each matrix when extrapolating transitions among categories.

Mathematical analysis of an age-since infection and diffusion HIV/AIDS model with treatment adherence and Dirichlet boundary condition

In this paper, an epidemic model with homogeneous Dirichlet boundary condition is formulated to study the joint impact of spatial diffusion, infection age and treatment adherence on the HIV/AIDS transmission among humans. It is an interesting problem to understand the threshold dynamics of HIV/AIDS model with the above three factors. Since the complexity of the model and specificity of the boundary condition, there are two main mathematical challenges: i). the compactness of the solution map is not guaranteed; ii). the explicit expression of the basic reproduction cannot be given even if the parameters are spatially independent. We first discuss the well-posedness of the system, then we identify the basic reproduction number R0 as the spectral radius of the next generation operator, followed by the global attractivity of disease-free steady state when R0<1, the uniform persistence of the disease and the existence of the endemic steady state when R0>1. Numerical simulations are carried out to illustrate our theoretical results, which suggest that the diffusion of individuals has an opposite effect on the disease outbreaks, and improving the treatment compliance of HIV infected individuals can control HIV transmission among the population effectively.

Mathematics Resiliency in the New Normal: A Theory Development

This paper aimed to generate a theory that answers the question, “How do students cope with the challenges in learning mathematics in the normal? This study made used of the deductive axiomatic approach in theory generation following the steps prescribed by Padua (2012). Four axioms were formulated: (1) Students employ time management in studying math; (2) Students uses compensation strategies in overcoming challenges; (3) Students’ resourcefulness leads them to learn mathematics; and (4) Students asked help from teachers and peers in understanding mathematical concepts. Two propositions were derived from these axioms: students’ cope with the challenges in learning mathematics by (1) direct coping strategies support students’ resiliency in learning mathematics and (2) Indirect coping strategies indirectly provide support for coping strategies through planning, socializing with others and increasing empathy. From these propositions, the Students’ Mathematics Resiliency Based Theory is formulated: The Students’ Mathematics Resiliency Theory states that resiliency in learning mathematics is affected by both direct and indirect coping strategies. Direct coping strategies include compensation strategies, while indirect strategies include social, time management, and resourcefulness.

Examining Formative Assessment Practices in Mathematics in Primary Schools: China and Tanzania Comparative review

Evaluation plays an enormous role in the teaching and learning process. Being aware of this, both China and Tanzania use formative assessment to meet the ‎requirements of competence-based curricula especially in mathematics at primary school level. ‎Therefore, the purpose of this study was to look into the formative assessment practices, nature, ‎challenges in mathematics used by Chinese and Tanzanian primary schools. This report used ‎documentary analysis to consult articles, dissertations, and books found in different electronic ‎journals such as ERIC and Science Direct and deem them to have suitable information for the ‎study. The report found that evaluation methods such as test, assignments, homework, and ‎projects are common in both countries. Moreover, some dissimilar is some form of assessment ‎including terminal test and monthly test which are common in Tanzania. Furthermore, bustling ‎classes, heavy teaching loads, and a lack of clarity on the formative assessment implementation ‎strategies have similarly contributed to the ineffective formative assessment in both countries. ‎Further, the researcher recommends the recruitment of mathematics teachers and the provision of ‎in-service and pre-service training to the teachers to reduce ineffectiveness in the implementation ‎of formative assessment in Chinese and Tanzanian primary schools.‎

Challenges of public-school elementary mathematics teaching in the new normal

At the onset of the Covid-19 pandemic, educational institutions worldwide needed to continue with their educational mandates for their learners, albeit not in the usual face-to-face modes. The Higher Education Institutions (HEIs) in the Philippines as well had to replace the on-campus teaching-learning modes with several variants of remote teaching. However, the various remote means are something quite new to both educators and learners, especially in mathematics, wherein various onboard instructions and on-desk hands-on exercises had been the norm. This study used of quantitative approach through online surveys to identify the educators’ challenges with remote teaching implementations in public schools in the Philippines. The findings reveal challenges related to the actual delivery of mathematics instruction, distractions from non-teaching tasks, curricular adjustments, alignment of materials, and orientation of parents or guardians and teachers. The pandemic may not be the last to impact the education system. Overall, the study shows an increased level of challenges for mathematics teaching. Thus, there is an urgent need to readjust policies and procedures affecting environmental learning conditions such as distractions, family-work-life balance, communication technology, rationalization of non-academic duties, increased participation, and support of parents and guardians.

A Simple Estimation of Parameters for Discrete Distributions from the Schröter Family

One of the common challenges in actuarial mathematics is finding a model for the number of claims and claim severity. We focus on one of the suggested models, namely, on the Schröter family of discrete probability distributions. Furthermore, we introduce a simple and easy-to-compute parameter estimate for this family of distributions, which can be used especially as initial values in optimization algorithms that are needed to compute other estimates.

Task Design for Klein’s Second Discontinuity

Since the 19th century, studies of mathematics at university have been a main component of the usual preparation for teaching at secondary level. Already around 1900, Klein pointed out that specific measures are needed to ensure that the university mathematical preparation becomes useful to the teacher, and he insisted that universities themselves must take responsibility for these measures. In this paper, we discuss this problem, as it presents itself in 2022, and we present and exemplify some principles of task design which are intended to support students’ mobilisation of university mathematical knowledge in relation to specific mathematical challenges for high school teachers.&#x0D; &#x0D;

Experiences of Grade School Mathematics Teachers during Online Classes through the Lens of TPACK

This study sought to describe the experiences and challenges of grade school mathematics teachers through the lens of TPACK. This study utilized a qualitative phenomenological design to discover how the participants made meaning from their experiences and challenges during online classes. Ten private school elementary teachers at a university were selected as participants. The results revealed seven themes for the experiences of grade school mathematics teachers during online classes. These themes are 1) digital drills in developing mastery in mathematics, 2) subject integration for meaningful learning of mathematics, 3) applying real-life contexts in problem-solving,4)considerations in choosing interactive sites to teach and assess mathematical skills,5) instructional management skills of the teacher,6)administrative support,7)teachers’ attitude towards the paradigm shift. Meanwhile, future researchers should also consider more respondents and a longer duration of doing this kind of research to unveil the experiences of grade school mathematics teachers in online classes. Keywords:experiences of gradeschool mathematics teachers, challenges of gradeschool mathematics teachers, online classes, TPACK

Making university mathematics matter for secondary teacher preparation

Internationally, questions about the perceived utility of university mathematics for teaching school mathematics pose an ongoing challenge for secondary mathematics teacher education. This special issue is dedicated to exploring this topic and related issues in the preparation of secondary mathematics teachers—by which we mean teachers of students with ages, approximately, of 12–18 years. This article introduces this theme and provides a semi-systematic survey of recent related literature, which we use to elaborate and situate important theoretical distinctions around the problems, challenges, and solutions of university mathematics in relation to teacher education. As part of the special issue, we have gathered articles from different countries that elaborate theoretical and empirical approaches, which, collectively, describe different ways to strengthen university mathematics with respect to the aims of secondary teacher education. This survey paper serves to lay out the theoretical groundwork for the collection of articles in the issue.

Is it only about technology? The interplay between educational technology for mathematics homework, teaching practice, and students’ activities

Current research on the use of digital technology in education has overgrown. It is becoming clear that technology will not unfold its pedagogical potential outlined in the literature on its own. To harness its full benefits, it is paramount to thoroughly understand the educational context and underlying pedagogical principles. This research reports how an educational technology can support students’ knowledge development from homework mathematics learning practices for 11 to 14 year-old students. The article describes the shift learning goals from producing the correct answer to understanding how to address current and future mathematical challenges as an important interaction between educational technology, teaching practices and student learning activities. The study revealed specific features of technology integration with mathematics teaching and learning practices. The findings highlighted the effectiveness of digital support for homework activities, indicating that there were benefits only when students and teachers used the full range of features and engaged in the process. Finally, the report provides practical tips and ground rules that can contribute to the successful implementation of educational technology for homework activities.

A Generalized Scalar Potential Integral Equation Formulation for the DC Analysis of Conductors

The electrostatic modeling of conductors is a fundamental challenge in various applications, including the prediction of parasitic effects in electrical interconnects, the design of biasing networks, and the modeling of biological, microelectromechanical, and sensing systems. The boundary element method (BEM) can be an effective simulation tool for these problems because it allows modeling three-dimensional objects with only a surface mesh. However, existing BEM formulations can be restrictive because they make assumptions specific to particular applications. For example, capacitance extraction formulations usually assume a constant electric scalar potential on the surface of each conductor and cannot be used to model a flowing current, nor to extract the resistance. When modeling steady currents, many existing techniques do not address mathematical challenges such as the null space associated with the operators representing the internal region of a conductor. We propose a more general BEM framework based on the electric scalar potential for modeling conductive objects in various scenarios in a unified manner. Restrictive application-specific assumptions are not made, and the aforementioned operator null space is handled in an intuitive and rigorous manner. Numerical examples drawn from diverse applications confirm the accuracy and generality of the proposed method.

Fuzzy Graph Modeling and Clustering Analysis of Nonlinear Dynamical Systems

Analyzing stability and designing control strategies for interconnected nonlinear dynamical systems poses mathematical challenges due to combinatorial growth in complexity. This paper develops a methodology integrating fuzzy graph theory with spectral clustering techniques to enable tractable certification of stability properties. Fuzzy similarity relations between state variables model imprecise couplings within the nonlinear dynamics. Representing these relations through graph adjacency matrices facilitates partitioning strongly connected states using spectral algorithms. Stability of the overall fuzzy state graph is inferred from the spectral radii of decoupled fuzzy subgraphs. The graph-theoretic abstraction provides a coarse-grained lens into nonlinear stability properties while circumventing computational barriers. The fuzzy graph modeling and spectral partitioning pipeline ultimately streamlines control synthesis targeting local clustered subdynamics. Case studies on power network stabilization, swarm navigation, and process operations showcase scalable applications to high-dimensional complex systems. The integrated fuzzy graph and spectral clustering approach provides a systematic toolkit for analysis and control of heavily interconnected nonlinear dynamical systems across engineering domains.

Open problems in PDE models for knowledge-based animal movement via nonlocal perception and cognitive mapping

The inclusion of cognitive processes, such as perception, learning and memory, are inevitable in mechanistic animal movement modelling. Cognition is the unique feature that distinguishes animal movement from mere particle movement in chemistry or physics. Hence, it is essential to incorporate such knowledge-based processes into animal movement models. Here, we summarize popular deterministic mathematical models derived from first principles that begin to incorporate such influences on movement behaviour mechanisms. Most generally, these models take the form of nonlocal reaction-diffusion-advection equations, where the nonlocality may appear in the spatial domain, the temporal domain, or both. Mathematical rules of thumb are provided to judge the model rationality, to aid in model development or interpretation, and to streamline an understanding of the range of difficulty in possible model conceptions. To emphasize the importance of biological conclusions drawn from these models, we briefly present available mathematical techniques and introduce some existing "measures of success" to compare and contrast the possible predictions and outcomes. Throughout the review, we propose a large number of open problems relevant to this relatively new area, ranging from precise technical mathematical challenges, to more broad conceptual challenges at the cross-section between mathematics and ecology. This review paper is expected to act as a synthesis of existing efforts while also pushing the boundaries of current modelling perspectives to better understand the influence of cognitive movement mechanisms on movement behaviours and space use outcomes.

Allocation of hospital beds on the emergence of new infectious disease: A mathematical model.

This paper is concerned to a mathematical model for the management of hospital beds when a new infection emerges in the population with the existing infections. The study of this joint dynamics presents formidable mathematical challenges due to a limited number of hospital beds. We have derived the invasion reproduction number, which investigates the potential of a newly emerged infectious disease to persist when some infectious diseases are already invaded the host population. We have shown that the proposed system exhibits transcritical, saddle-node, Hopf, and Bogdanov-Takens bifurcations under certain conditions. We have also shown that the total number of infected individuals may increase if the fraction of the total number of hospital beds is not properly allotted to the existing and a newly emerged infectious disease. The analytically obtained results are verified with the help of numerical simulations.

Impacts of Problem-Based Instruction on Students’ Beliefs about Physics and Learning Physics

To help prepare students to address future challenges in Science, Technology, Engineering, and Mathematics (STEM), they need to develop 21st-century skills. These skills are mediated by their beliefs about the nature of scientific knowledge and practices, or epistemological beliefs. One approach shown to support students’ development of these beliefs and skills is problem-based instruction (PBI), which encourages collaborative self-directed learning while working on open-ended problems. We used a mixed-method qualitative approach to examine how implementing PBI in a physics course taught at a Dutch university affected students’ beliefs about physics and learning physics. Analysis of the responses to the course surveys (41–74% response rates) from the first implementation indicated students appreciated opportunities for social interactions with peers and use of scientific equipment with PBI but found difficulties connecting to the Internet given the COVID-19 restrictions. The Colorado Learning Attitudes towards Science Survey (CLASS), a validated survey on epistemological beliefs about physics and learning physics, was completed by a second cohort of students in a subsequent implementation of PBI for the same course; analysis of the students’ pre- and post-responses (28% response rate) showed a slight shift towards more expert-like perspectives despite challenges (e.g., access to lab). Findings from this study may inform teachers with an interest in supporting the development of students’ epistemological beliefs about STEM and the implementation of PBI in undergraduate STEM courses.