Challenging Mathematical Problem Research Articles

Dressing the cusp: how paraxial sharp-edge diffraction theory solves a basic issue in catastrophe optics

The description of light diffraction using catastrophe optics is one of the most intriguing theoretical inventions in the field of classical optics of the last four decades. Its practical implementation has faced some resistance over the years, mainly due to the difficulty of decorating the different (topologically speaking) types of optical singularities (caustics) that concur to build the skeleton on which diffraction patterns stem. Such a fundamental dressing problem has been solved in the past only for the so-called fold, which lies at the bottom of the hierarchy of structurally stable caustics. Climbing this hierarchy implies considerably more challenging mathematical problems to be solved. An ancient mathematical theorem is employed here to find the complete solution of the dressing problem for the cusp, which is placed, in the stable caustic hierarchy, immediately after the fold. The other ingredient used for achieving such an important theoretical result is the paraxial version of the boundary diffraction wave theory, whose tight connection with catastrophe optics has recently been emphasized [Opt. Lett. 41, 3114 (2016)OPLEDP0146-959210.1364/OL.41.003114]. A significant example of the developed algorithm, aimed at demonstrating its effectiveness and ease of implementation, is also presented.

Students’ Engagement in a Mathematical Investigation through Online Problem-Based Learning

The core components of mathematical activity are involved in mathematical investigation: problem formulation, processing of conditions for which there is no clear solution, formulation and justification of conjectures, and generalization. However, introducing mathematical investigation to students is a challenging activity. A crucial aspect of the student’s learning and understanding of the process is the kind and degree of their engagement in the learning activity. This phenomenological study sought to describe how pre-service teachers engaged in an online problem-based learning (PBL) while doing a mathematical investigation (MI). The researcher explored the cognitive, behavioral, and affective components of the 35 pre-service mathematics education students who had their first-hand experience of MI through online PBL. The students were given two contextualized, ill-structured problems, worked in groups, and proposed solutions to the problem. Data from the teachers’ observation notes, students’ learning logs, and the transcripts of the focus group discussions were gathered. Qualitative data analysis suggests that students were involved in peer-to-peer learning that included task-related conflicts, pattern recognition, self-regulation, critical analysis, participation in the learning activity, attention to the tasks, and interaction with the learning community which led to their enjoyment and motivation in conducting MI. The extent of their engagement during the presentation and analysis of the problem, planning and developing a solution, presenting and assessing solution, and summary and debriefing were further analyzed. The findings revealed that while students were initially puzzled at the start of their investigation, they shared, welcomed new information, analyzed it, and valued different approaches to a challenging mathematical problem as they progress with their inquiry. The results also revealed respondents' perspectives on how to manage online PBL. These findings can help teachers design collaborative online activities.

Singularity formation to the 2D Cauchy problem of nonbarotropic magnetohydrodynamic equations without heat conductivity

AbstractThe question of whether the two‐dimensional (2D) nonbarotropic compressible magnetohydrodynamic (MHD) equations with zero heat conduction can develop a finite‐time singularity from smooth initial data is a challenging open problem in fluid dynamics and mathematics. Such a problem is interesting in studying global well‐posedness of solutions. In this paper, we proved that, for the initial density allowing vacuum states, the strong solution exists globally if the density and the pressure are bounded from above. Our method relies on weighted energy estimates and a Hardy‐type inequality.

Learning Mathematics in an After-school Mathematics Club

Learning mathematics is as much about developing relationships with mathematics as the development of mathematics concepts and skills. Mathematics clubs usually support learners to develop relationships with mathematics through engaging with challenging mathematics problems, exploring mathematical ideas, and communicating and sharing their ideas. This paper reports on research on such a mathematics club. Data sources included learners’ mathematics results, short written responses after club sessions and interviews with club learners. Data were analysed quantitatively and qualitatively. The study shows that club learners’ school results improved relative to non-club learners, that learners’ experiences of mathematics in club and class were very different from each other and that the club supported different relationships with mathematics than learners experienced in class. The study suggests that mathematics clubs can support learners’ positive relationships with mathematics.

Dividing a Sphere Hierarchically into a Large Number of Spherical Pentagons Using Equal Area or Equal Length Optimization

Dividing a 2-dimensional sphere uniformly into a large number of spherical polygons is a challenging mathematical problem, which has been studied across many disciplines due to its important practical applications. Most sphere subdivisions are achieved using spherical triangles, quadrangles, or a combination of hexagons and pentagons. However, spherical pentagons, which may create elegant configurations, remain under-explored. This study presents a new sphere subdivision method to generate a large number of spherical pentagons based on successively subdividing a module of an initial spherical dodecahedron. The new method can conveniently control the shapes of generated spherical pentagons through specified design parameters. Two optimization problems have been investigated: (I) dividing a sphere into spherical pentagons of equal area; (II) minimizing the number of different arc lengths used in the pentagonal subdivision. A variety of examples are presented to demonstrate the effectiveness of the new method. This study shows that treating the mathematical challenge of dividing a sphere uniformly into a large number of spherical polygons as an optimization problem can effectively obtain equal area or equal length sphere subdivisions. Furthermore, considering additional constraints on the optimization problem may achieve sphere subdivisions of specific characteristics. • A new sphere subdivision method is developed to generate a large number of spherical pentagons. • The subdivision approach is performed locally and hierarchically on a module of a spherical dodecahedron. • The mathematical challenge of dividing a sphere uniformly is treated as an optimization problem. • Optimization algorithms are used to achieve sphere subdivisions of specific characteristics.

Compact formulae for three-center nuclear attraction integrals over exponential type functions

In any ab initio molecular orbital calculations, the major task involves the computation of the so-called molecular multi-center integrals. Multi-center integral calculations is a very challenging mathematical problem in nature. Quantum mechanics only determines which integrals we evaluate, but the techniques employed for their evaluations are entirely mathematical. The three-center nuclear attraction integrals occur in a very large number even for small molecules and are among of the most difficult molecular integrals to compute efficiently. In the present contribution, we report analytical expressions for the three-center nuclear attraction integrals over exponential type functions. We describe how to compute the formula to obtain an efficient evaluation in double precision arithmetic. This requires the rational minimax approximants that minimize the maximum error on the interval of evaluation.

A fast solution approach to solve the generator maintenance scheduling and hydropower production problems simultaneously

The Generator Maintenance Scheduling Problem (GMSP) is a problem that combines a hydropower optimization problem with a scheduling problem. Both problems are known to be hard to solve and combining them leads to an even more challenging mathematical problem. Since the hydropower production functions are nonlinear, hyperplane curve fitting is used to linearize each power production function. The goal of the GMSP is to find an optimal schedule plan to decide when to shut down generators for maintenance. Therefore, one production function needs to be formulated per generator combinations leading to a rather large number of constraints. This paper demonstrates that the complexity of the problems is linked to the number of hyperplanes selected to formulate the power production functions. To accelerate the resolution of the problem, a new heuristic based on the mean square algorithm is presented to reduce the number of hyperplanes required. This heuristic substantially reduces the number of constraints and the solving time is almost ten times faster. Numerical results show that the energy produced and the generated maintenance plannings are similar for both mathematical formulations, more precisely with one hyperplane for each generator combination versus a reduced number of hyperplanes.

Notes on Higher-Spin Diffeomorphisms

Higher-spin diffeomorphisms are to higher-order differential operators what diffeomorphisms are to vector fields. Their rigorous definition is a challenging mathematical problem which might predate a better understanding of higher-spin symmetries and interactions. Several yes-go and no-go results on higher-spin diffeomorphisms are collected from the mathematical literature in order to propose a generalisation of the algebra of differential operators on which higher-spin diffeomorphisms are well-defined. This work is dedicated to the memory of Christiane Schomblond, who taught several generations of Belgian physicists the formative rigor and delicate beauty of theoretical physics.

A bilevel optimization approach to decide the feasibility of bookings in the European gas market

The European gas market is organized as a so-called entry-exit system with the main goal to decouple transport and trading. To this end, gas traders and the transmission system operator (TSO) sign so-called booking contracts that grant capacity rights to traders to inject or withdraw gas at certain nodes up to this capacity. On a day-ahead basis, traders then nominate the actual amount of gas within the previously booked capacities. By signing a booking contract, the TSO guarantees that all nominations within the booking bounds can be transported through the network. This results in a highly challenging mathematical problem. Using potential-based flows to model stationary gas physics, feasible bookings on passive networks, i.e., networks without controllable elements, have been characterized in the recent literature. In this paper, we consider networks with linearly modeled active elements such as compressors or control valves. Since these active elements allow the TSO to control the gas flow, the single-level approaches for passive networks from the literature are no longer applicable. We thus present a bilevel model to decide the feasibility of bookings in networks with active elements. While this model is well-defined for general active networks, we focus on the class of networks for which active elements do not lie on cycles. This assumption allows us to reformulate the original bilevel model such that the lower-level problem is linear for every given upper-level decision. Consequently, we derive several single-level reformulations for this case. Besides the classic Karush–Kuhn–Tucker reformulation, we obtain three problem-specific optimal-value-function reformulations. The latter also lead to novel characterizations of feasible bookings in networks with active elements that do not lie on cycles. We compare the performance of our methods by a case study based on data from the GasLib.

Identification of Nonlinear Systems Using the Infinitesimal Generator of the Koopman Semigroup—A Numerical Implementation of the Mauroy–Goncalves Method

Inferring the latent structure of complex nonlinear dynamical systems in a data driven setting is a challenging mathematical problem with an ever increasing spectrum of applications in sciences and engineering. Koopman operator-based linearization provides a powerful framework that is suitable for identification of nonlinear systems in various scenarios. A recently proposed method by Mauroy and Goncalves is based on lifting the data snapshots into a suitable finite dimensional function space and identification of the infinitesimal generator of the Koopman semigroup. This elegant and mathematically appealing approach has good analytical (convergence) properties, but numerical experiments show that software implementation of the method has certain limitations. More precisely, with the increased dimension that guarantees theoretically better approximation and ultimate convergence, the numerical implementation may become unstable and it may even break down. The main sources of numerical difficulties are the computations of the matrix representation of the compressed Koopman operator and its logarithm. This paper addresses the subtle numerical details and proposes a new implementation algorithm that alleviates these problems.

Computing technical capacities in the European entry-exit gas market is NP-hard

As a result of its liberalization, the European gas market is organized as an entry-exit system in order to decouple the trading and transport of natural gas. Roughly summarized, the gas market organization consists of four subsequent stages. First, the transmission system operator (TSO) is obliged to allocate so-called maximal technical capacities for the nodes of the network. Second, the TSO and the gas traders sign mid- to long-term capacity-right contracts, where the capacity is bounded above by the allocated technical capacities. These contracts are called bookings. Third, on a day-ahead basis, gas traders can nominate the amount of gas that they inject or withdraw from the network at entry and exit nodes, where the nominated amount is bounded above by the respective booking. Fourth and finally, the TSO has to operate the network such that the nominated amounts of gas can be transported. By signing the booking contract, the TSO guarantees that all possibly resulting nominations can indeed be transported. Consequently, maximal technical capacities have to satisfy that all nominations that comply with these technical capacities can be transported through the network. This leads to a highly challenging mathematical optimization problem. We consider the specific instantiations of this problem in which we assume capacitated linear as well as potential-based flow models. In this contribution, we formally introduce the problem of Computing Technical Capacities (CTC) and prove that it is NP-complete on trees and NP-hard in general. To this end, we first reduce the Subset Sum problem to CTC for the case of capacitated linear flows in trees. Afterward, we extend this result to CTC with potential-based flows and show that this problem is also NP-complete on trees by reducing it to the case of capacitated linear flow. Since the hardness results are obtained for the easiest case, i.e., on tree-shaped networks with capacitated linear as well as potential-based flows, this implies the hardness of CTC for more general graph classes.

A new discrete multiplicative random cascade model for downscaling intermittent rainfall fields

Abstract. Spatial downscaling of rainfall fields is a challenging mathematical problem for which many different types of methods have been proposed. One popular solution consists of redistributing rainfall amounts over smaller and smaller scales by means of a discrete multiplicative random cascade (DMRCs). This works well for slowly varying homogeneous rainfall fields but often fails in the presence of intermittency (i.e., large amounts of zero rainfall values). The most common workaround in this case is to use two separate cascade models, namely one for the occurrence and another for the intensity. In this paper, a new and simpler approach based on the notion of equal-volume areas (EVAs) is proposed. Unlike classical cascades where rainfall amounts are redistributed over grid cells of equal size, the EVA cascade splits grid cells into areas of different sizes, with each of them containing exactly half of the original amount of water. The relative areas of the subgrid cells are determined by drawing random values from a logit-normal cascade generator model with scale and intensity-dependent standard deviation (SD). The process ends when the amount of water in each subgrid cell is smaller than a fixed-bucket capacity, at which point the output of the cascade can be resampled over a regular Cartesian mesh. The present paper describes the implementation of the EVA cascade model and gives some first results for 100 selected events in the Netherlands. Performance is assessed by comparing the outputs of the EVA model to bilinear interpolation and to a classical DMRC model based on fixed grid cell sizes. Results show that, on average, the EVA cascade outperforms the classical method, producing fields with more realistic distributions, small-scale extremes and spatial structures. Improvements are mostly credited to the higher robustness of the EVA model in the presence of intermittency and to the lower variance of its generator. However, both approaches have their advantages and weaknesses. For example, while the classical cascade tends to overestimate small-scale variability and extremes, the EVA model tends to produce fields that are slightly too smooth and block shaped compared to the observations. The complementary nature of the two approaches, and the fact that they produce errors of opposite signs, opens up new possibilities for quality control and bias corrections of downscaled fields.

Play and Mathematics

In this paper, I argue that a lack of play and joy in classrooms could be due to our North American standardized education system, which emphasizes achievement outcomes. I argue that this system does not benefit the majority of students, nor the field of mathematics. Many students are negatively affected—both emotionally and academically—by a focus on results. Rather than outcome-driven pedagogy, a focus on learning to enjoy doing mathematics might change the conversation. A kinder, process-driven approach through mathematical play may spark enjoyable teaching and learning. Play (Gadamer, 1960/1989; Huizinga, 1944/1949) has the potential to absorb learners as they seek answers to fun yet challenging mathematics problems. The experience of flow is similar to that of play (Csikszentmihalyi, 2000); when playing, learners get a chance to practice and elaborate on their existing skills in manners that suspend notions of time. When the play releases them from its grasp, learners experience the joy from solving problems. Dewey (1916) considered play to be purposeful activity that sponsors a child’s growth. Teachers could capitalize on this for growth in learning. Learning to bring mathematical play into the classroom requires intention, an inviting attitude, knowledge of the types of problems that invoke play, and knowledge of how to connect playful problems to mathematical concepts and curriculum. Such engaging experiences with mathematics could sponsor joyful engagement in mathematics and an intrinsic desire to learn more.

Play and Mathematics

In this paper, I argue that a lack of play and joy in classrooms could be due to our North American standardized education system, which emphasizes achievement outcomes. I argue that this system does not benefit the majority of students, nor the field of mathematics. Many students are negatively affected—both emotionally and academically—by a focus on results. Rather than outcome-driven pedagogy, a focus on learning to enjoy doing mathematics might change the conversation. A kinder, process-driven approach through mathematical play may spark enjoyable teaching and learning. Play (Gadamer, 1960/1989; Huizinga, 1944/1949) has the potential to absorb learners as they seek answers to fun yet challenging mathematics problems. The experience of flow is similar to that of play (Csikszentmihalyi, 2000); when playing, learners get a chance to practice and elaborate on their existing skills in manners that suspend notions of time. When the play releases them from its grasp, learners experience the joy from solving problems. Dewey (1916) considered play to be purposeful activity that sponsors a child’s growth. Teachers could capitalize on this for growth in learning. Learning to bring mathematical play into the classroom requires intention, an inviting attitude, knowledge of the types of problems that invoke play, and knowledge of how to connect playful problems to mathematical concepts and curriculum. Such engaging experiences with mathematics could sponsor joyful engagement in mathematics and an intrinsic desire to learn more.

Singularity formation to the two-dimensional non-resistive compressible magnetohydrodynamic equations in a bounded domain

The question of whether the two-dimensional compressible magnetohydrodynamic equations without magnetic diffusion can develop a finite-time singularity from smooth initial data is a challenging problem in fluid dynamics and mathematics. In this paper, we derive a regularity criterion which shows that the strong solution exists globally if the density and the magnetic field are bounded from above. Our method relies on critical Sobolev inequalities of logarithmic type.

Eye Movements During Mathematical Word Problem Solving—Global Measures and Individual Differences

In mathematical word problem solving, reading and mathematics interact. Previous research used the method of eye tracking to analyze reading processes but focused on specific elements in prototype word problems. This makes it difficult to compare the role of reading in longer, more complex word problems and between individuals. We used global measures of eye movements that refer to the word problem as a whole, similar to methods used in research on eye movements during reading. Global measures allow comparisons of reading processes of word problems of different structure. To test if these global measures are related to cognitive processes during word problem solving, we analyzed the relation between eye movements and the perceived difficulty of a task and its solution rate. We conducted two experiments with adults and undergraduate students (N = 17 and N = 42), solving challenging mathematical word problems from PISA. Experiment 1 showed that more difficult items were read with shorter fixations, more saccades, more regressions, and slower, with correlations ranging from r = 0.70 to r = 0.86. Multilevel modelling in experiment 2 revealed that for the number of saccades and the proportion of regressions, the relationship was stronger for low-performing students, with performance explaining up to 37% of the variance between students. These two measures are primarily associated with building a problem model. We discuss how this approach enables the use of eye tracking in complex mathematical word problem solving and contributes to our understanding of the role of reading in mathematics.

Dynamics of a time-periodic two-strain SIS epidemic model with diffusion and latent period

Coinfection of hosts with multiple strains or serotypes of the same agent, such as different influenza virus strains, different human papilloma virus strains, and different dengue virus serotypes, is not only a very serious public health issue but also a very challenging mathematical modeling problem. In this paper, we study a time-periodic two-strain SIS epidemic model with diffusion and latent period. We first define the basic reproduction number R0i and introduce the invasion number Rˆ0i for each strain i(i=1,2), which can determine the ability of each strain to invade the other single-strain. The main question that we investigate is the threshold dynamics of the model. It is shown that if R0i⩽1(i=1,2), then the disease-free periodic solution is globally attractive; if R0i>1⩾R0j(i≠j,i,j=1,2), then competitive exclusion, where the jth strain dies out and the ith strain persists, is a possible outcome; and if Rˆ0i>1(i=1,2), then the disease persists uniformly. Finally we present the basic framework of threshold dynamics of the system by using numerical simulations, some of which are different from that of the corresponding multi-strain SIS ODE models.

Construction of Fair Dice Pairs

An interesting and challenging problem in mathematics is how to construct fair dice pairs. In this paper, by means of decomposing polynomials in a residue class ring and applying the Discrete Fourier Transformation, we present all the 2000 fair dice pairs and their 8 equivalence classes in a four-person game, identifying what we call the mandarin duck property of fair dice pairs.

I Felt Like A Mathematician: Problems and Assessment To Promote Creative Effort

Challenging mathematics problems were posed as a substantial part of homework in an elective combinatorics course. The intent was to encourage undergraduate students’ exploration as well as develop their technical writing skills. The bulk of students’ grades on these problems rested on a three-page reflective write-up using a rubric on mathematical creativity, the Creativity-in-Progress Rubric on Proving. Through student reflections, this structure led to a self-reported boost in students’ proving confidence and led them to view themselves as practicing mathematicians. Descriptions of the problems, how students used the rubric as their guide, and grading structure are detailed in this article.

Two-dimensional gold matrix method for encoding two-dimensional optical arbitrary positions.

In this study, a novel two-dimensional spatial coding pattern called two-dimensional Gold matrix method is proposed for general two-dimensional positioning. Considering the difficulty in representing a two-dimensional position in a single binary matrix, constructing a matrix while each submatrix refers to its location is a challenging mathematical problem. The general two-dimensional signal can be labeled by the two-dimensional Gold matrix, which results from a preferred pair of two m-sequences. For a pseudorandom m-sequence, the span-n property of the two-dimensional Gold matrix states that every n×n submatrix is unique and the decoding is fast and convenient. Numerical simulation and a proof-of-principle experiment are performed, and experimental results verified that the two-dimensional Gold matrix method is effective for high resolution and large range two-dimensional measurements.