Constructive Mathematics Research Articles

Interdisciplinary integration in teaching Mathematics as a means of forming pupils' mathematical literacy

This article discusses the problem of implementing interdisciplinary connections between a school course in Mathematics and other natural science disciplines at a secondary school as topical means of developing functional literacy among pupils. The mechanisms of interdisciplinary connection of Mathematics with other disciplines are listed. As a means of interdisciplinary communication of Mathematics with other disciplines, there can be the interaction of pupils in the information educational environment to search for new knowledge, to identify properties of a generalised character, to solve practice-oriented problems, to combine knowledge from various general education subjects, to teach Computer Science based on the transfer and generalisation of knowledge from related subjects. The article provides examples of practice-oriented tasks that illustrate the relationship of Mathematics with such secondary school disciplines as Computer Science, Biology, Geography, and Physics. It is noted that when solving such problems, schoolchildren learn various aspects of multi-level practice-oriented activities, including the use of computer capabilities and the global Internet. It is shown that mathematical constructions and their analogues are found even in areas of knowledge quite distant from exact sciences, such as literature, for example. It is substantiated that it is the solution of interdisciplinary problems that forms the meta-skills necessary for everyday life in pupils.

Exploring children’s negotiation of meanings about “D” in 2D and 3D shapes in a year 5/6 New Zealand primary classroom

The mathematical construct of dimension is one of the fundamental ideas for developing a sound understanding of two-dimensional (2D) and three-dimensional (3D) shapes. Yet, research in mathematics education has rarely explored children’s understanding of dimension in primary education. This paper explores how year 5/6 (9 to 11 years old) children construct and negotiate their meanings about dimension while engaging in classroom interactions about 2D and 3D shapes during geometry lessons in a New Zealand (NZ) English-medium multilingual primary classroom. Transcribed data of two key moments selected from six audiovisually recorded geometry lessons are presented. The findings suggest that children may use different discursive constructions—“another world”, “different ways to go”, and “flat vs fat”—to display their meanings about dimension. The findings also suggest that children and teacher participants may use prosodic features of their languages to interactionally construct the meanings of these discursive constructions. The paper discusses these findings in light of current research literature and offers a few implications for curriculum development and future research.

Fundamental Theorem of Calculus: Cognitive Demands and Learning Limitations on Mathematical Tasks

Background: Mathematical tasks for university teaching are generally of great cognitive demand, without thinking about the limitations they imply for their students. Objectives: To develop a theoretical analysis of the teaching limitations and cognitive demand of four tasks proposed for teaching calculus, specifically, the Fundamental Theorem of Calculus. Design: Qualitative paradigm, with a descriptive-interpretive approach, according to the nature of the data collected. Setting and Participants: The study is framed in a Colombian University, in the subject “Calculus II” for second-semester engineering students, where a professor designs the tasks to be implemented in this course. Data collection and analysis: The data correspond to the professor's lesson plans the statements of the main mathematical tasks within them. These plans were chosen based on availability and accessibility. A content analysis was conducted, considering as units of analysis the paragraphs or sets of paragraphs of the statement of each school mathematics task. Results: Most of the proposed tasks correspond to high cognitive demand (procedures with connections and mathematical construction) and only one was of low demand (memorisation). Moreover, each of the tasks presents its own cognitive demand and several learning constraints that, some of them, agree with the exposed literature. Conclusions: The work aims to have implications for higher education, since to think of a didactic proposal for a better approach to teaching is necessary to configure lesson plans that mobilise the learning of mathematics in engineering, but from the use of tasks with different cognitive demands, in which will vary from less to more, and lead to meaningful learning for the approach of new tasks.

Entropy generation and thermal analyses of a Cross fluid flow through an inclined microchannel with non‐linear mixed convection

AbstractThe temperature difference of the various applications such as microchannel heat exchangers, microelectronics, solar collectors, automotive systems, micro fuel cells, and microelectromechanical systems (MEMS) is relatively large. The buoyancy force (mixed convection) modeled by the conventional Boussinesq approximation is inadequate since the density of the operating fluids fluctuates non‐linearly with the temperature difference. Therefore, the mixed non‐linear convective transport of the flow of Cross fluid through three different geometric aspects (horizontal, vertical, and inclined) of the microchannel under the non‐linear Boussinesq (NBA) approximation is investigated. Mechanisms of internal heat source, Rosseland radiative heat flux, and frictional heating are incorporated into the thermal analysis. The mathematical construction is proposed using the Cross fluid model for a steady‐state, and subsequent non‐linear differential equations are deciphered by the spectral quasi‐linearization method (SQLM). Graphical sketches were constructed and displayed that explore the stimulus of various key parameters on Bejan number, velocity, temperature, and entropy generation. It is found that the Bejan number and entropy production improved due to the non‐linear density temperature variation. The convective heating boundary conditions augment the entropy production. The pressure gradient accelerates the transport of fluid in a microchannel. Furthermore, among three different geometries, the velocity, entropy production, and temperature are the highest for the vertical microchannel.

Testing spatial curvature and anisotropic expansion on top of the ΛCDM model

We explore the possible advantages of extending the $\Lambda$CDM model by more realistic backgrounds compared to its spatially flat RW spacetime assumption, while preserving the underpinning physics; in particular, by simultaneously allowing non-zero spatial curvature and anisotropic expansion on top of it, viz., the An-$o\Lambda$CDM model. This is to test whether the latest data support spatial flatness and/or isotropic expansion, and, if not, to explore the roles of spatial curvature and expansion anisotropy (due to its stiff fluid-like behavior) in addressing some of the cosmological tensions. We first present the theoretical background and explicit mathematical construction of An-$o\Lambda$CDM; combining the simplest anisotropic generalizations of the RW spacetime, viz., the Bianchi type I, V, and IX spacetimes. Then we constrain this model and its particular cases, viz., An-$\Lambda$CDM, $o\Lambda$CDM, and $\Lambda$CDM, by using the data sets from different probes, viz., Planck CMB(+Lens), BAO, SnIa Pantheon, and CC data, and discuss the results. Ultimately, we conclude that, within the setup under consideration, (i) the data confirm the spatial flatness and isotropic expansion, though a very small amount of present-day expansion anisotropy cannot be excluded, e.g., $\Omega_{\sigma0}\lesssim10^{-18}$ (95\% C.L.) for An-$\Lambda$CDM from CMB+Lens, (ii) the introduction of spatial curvature or anisotropic expansion, or both, on top $\Lambda$CDM does not offer a possible relaxation to the $H_0$ tension, and (iii) the introduction of anisotropic expansion neither affects the closed space prediction from CMB(+Lens) nor does it improve the drastically reduced value of $H_0$ led by the closed space. We discuss why it is important and indispensable to maintain the geometric generalization work program, especially in models that offer solutions to cosmological tensions. [abridged]

Stochastic Modeling with Applications in Supply Chain Management and ICT Systems

Fast-growing technology and the development of IT services have yielded the idea of founding a new application of stochastic processes and their properties. We give a new connection between electronic process management and a relatively new stochastic process named the Non-central Polya-Aeppli process. This process is applied as a counting process in the mathematical construction of the given model, and it has been introduced as a counting process in electronic process management.

Cognitive-constructivist model and the acquisition of mathematics knowledge according to Gagné’s taxonomy

School reforms aim to achieve lasting knowledge for pupils. Some researchers have shown that lasting knowledge can be achieved through a cognitive-constructivist model of instruction, as pupils discover concepts themselves and are active in the classroom. In this paper, we aimed at investigating whether pupils who are exposed to a cognitive-constructivist model of instruction are more successful in solving mathematical tasks than those who are not exposed to a cognitive-constructivist model of instruction. For this purpose, we have designed an experiment involving 252 pupils of the 3rd grade of primary school: of these, 100 were included in the experimental group receiving constructivist mathematics instruction, while the remaining 152 pupils constituted the control group. The study showed that the experimental group performed better in solving mathematical problems which involve all three taxonomic levels according to Gagné. Keywords: cognitive-constructivist model, acquisition, mathematics, Gagné taxonomy, experimental group.

No Magic: From Phenomenology of Practice to Social Ontology of Mathematics

The paper shows how to use the Husserlian phenomenological method in contemporary philosophical approaches to mathematical practice and mathematical ontology. First, the paper develops the phenomenological approach based on Husserl's writings to obtain a method for understanding mathematical practice. Then, to put forward a full-fledged ontology of mathematics, the phenomenological approach is complemented with social ontological considerations. The proposed ontological account sees mathematical objects as social constructions in the sense that they are products of culturally shared and historically developed practices. At the same time the view endorses the sense that mathematical reality is given to mathematicians with a sense of independence. As mathematical social constructions are products of highly constrained, intersubjective practices and accord with the phenomenologically clarified experience of mathematicians, positing them is phenomenologically justified. The social ontological approach offers a way to build mathematical ontology out of the practice with no metaphysical magic.

Optimization of Market Behavior of a Transportation Company

The relevance of optimizing the interaction of participants in manufacture and transport systems is due to the growth of the scale of social production, the development of transnational companies and international trade at the present stage of market transformations, economic integration and development of transport corridors. In this paper we will optimize the market behavior of a transport enterprise in the manufacture and transportation chain. Objectives: To determine the best option for coordinating the actions of participants in the manufacture and transport chain and to optimize the mechanism for granting discounts on the tariff of the transport company. Research methods: economic and mathematical modeling, construction and mathematical analysis of models of manufacture and transport systems, optimization computer calculations, economic interpretation of the results. In this paper it will also be analyzed how full or partial integration of the participants of manufacture and transportation chains, such as coordination of actions, unification of interests can ensure not only a significant increase in the profits of each participant, but also an increase in production volumes and delivery of products to consumers with a corresponding decrease in prices. The paper also portrays how every participant in the manufacture and transportation chain can paradoxically benefit from the established mechanism that would provide discounts on the tariff of a transport company depending on the volume of cargo flow followed by principle the more cargo, the greater the discount. It will be shown how manufacturer benefits by reducing transportation costs, while the transport company earns less per container, but the outstripping growth in the number of these containers provides it with a higher profit. Consequently, the conditions for the feasibility of applying such a mechanism of tariff discounts will also be established.

Application of logical and mathematical methods for the analysis of environmental information

Aim. Evaluation of the possibilities of mathematical logic and logical‐mathematical methods in the description of complex natural systems in simple and clear constructions, as they act as a language, special research methods, a source of ideas and concepts in natural science.Discussion. The article discusses the possibilities and advantages of logical and mathematical methods in the analysis of natural science information, and, in particular, environmental data. It gives a comparative overview of logical and mathematical constructions in the formation of scientific thinking. The list of the most common mathematical and logical symbols for the formalised recording of complex ecological systems is provided, together with examples of the use of logical and mathematical formulas as formal and implicative statements in the brief recording of natural science information.Conclusion. The research conducted does not convincingly indicate unambiguous advantages of using the symbols of mathematical logic rather than verbal presentation. However, since ecology deals with the description of arch‐complex systems, including operations of formalised problem statement, the formalised structuring of ecosystems, the grouping of ecosystems according to measures of their similarity‐difference or inclusion–intersection, and the classification of selected ecosystems in a certain specified group, the need for widespread application of logical and mathematical research is indisputable.

Matrix representation of operations on graphs

This article considers the possibility of matrix execution of both unary and binary operations on graphs. Graphs, as an abstract mathematical construction, have a very wide range of practical applications. First of all, it is algorithmization and computer processing of information, electrical engineering, etc. Therefore, it is important to have a mathematical apparatus that allows you to transform the graphical presentation of information about objects into algebraic models for their further research using purely mathematical methods. If necessary, it is always possible to return from such an algebraic model to a graphical representation of the object (for example, to a graphical representation of a circuit diagram in electronics). For each of the considered operations on graphs, either a combination of algebraic operations or an easily programmable matrix processing algorithm is proposed, which can be used to represent any graph. Attention is also paid to the differences in such processing depending on the type of graphs involved in the considered operations. Some operations are more convenient to perform with adjacency matrices, and some - with incidence matrices. This article also considers these features of matrix execution of operations on graphs. All the proposed algorithms are illustrated with specific detailed examples. Thus, it is shown that for all operations on graphically presented objects, their matrix interpretation is possible. This result greatly facilitates the possibility of software implementation of work with such graphic objects.

Massively parallelized interpolated factored Green function method

This paper presents the first parallel implementation of the novel “Interpolated Factored Green Function” (IFGF) method introduced recently for the accelerated evaluation of discrete integral operators arising in wave scattering and other areas (Bauinger and Bruno, Jour. Computat. Phys., 2021). On the basis of the hierarchical IFGF interpolation strategy, the proposed (hybrid MPI-OpenMP) parallel implementation results in efficient data communication, and it scales up to large numbers of cores—without any hard limitations on the number of cores efficiently employed. Moreover, on any given number of cores, the proposed parallel approach preserves the O(Nlog⁡N) computing cost inherent in the sequential version of the IFGF algorithm. Unlike other approaches, the IFGF method does not utilize the Fast Fourier Transform (FFT), and it is thus better suited for efficient parallelization in distributed-memory computer systems. In particular, the IFGF method relies on a “peer-to-peer” strategy wherein, at every level, field propagation is directly enacted via “exchanges” between “peer” polynomials of constant degree, without data accumulation in large-scale “telephone-central” mathematical constructs such as those used in the Fast Multipole Method (FMM) and pure FFT-based approaches. A variety of numerical results presented in this paper illustrate the character of the proposed parallel algorithm, in particular demonstrating scaling from 1 to all 1,680 cores available in the High Performance Computing cluster used, and for problems of up to 4,096 wavelengths in acoustic size.

New Financial Ratios Based on the Compositional Data Methodology

Due to the type of mathematical construction, the use of standard financial ratios in studies analyzing the financial health of a group of firms leads to a series of statistical problems that can invalidate the results obtained. These problems originate from the asymmetry of financial ratios. The present article justifies the use of a new methodology using Compositional Data (CoDa) to analyze the financial statements of an industry, improving analyses using conventional ratios, since the new methodology enables statistical techniques to be applied without encountering any serious drawbacks, such as skewness and outliers, and without the results depending on the arbitrary choice as to which of the accounting figures is the numerator of the ratio and which is the denominator. An example with data on the wine industry is provided. The results show that when using CoDa, outliers and skewness are much reduced, and results are invariant to numerator and denominator permutation.

A conventional and digital mathematical board game design and development for use by students in learning arithmetic

This paper reported the design and development of a conventional and digital mathematical board game for use by students in learning arithmetic. At the time of research, there is no significant indication that a mathematical board game is available in scientific and published patent documentation. The availability of mathematical board games for students’ drills and practice in arithmetic, especially in mathematical statement construction, would benefit them, as this competency is an essential life skill. This research was conducted through the design and development research method with the procedure of users’ need analysis, researcher as developer capability analysis, product design, product development, field testing in its natural setting environment, and the prototype. The board game prototype was developed in both conventional printed and digital versions. The field testing for the conventional printed version was conducted at secondary school classes with 34 and 36 students, respectively, while for the digital by selected participants. The field testing shows that the developed mathematical board game can work as expected in its natural setting environment.

Inductive and Coinductive Topological Generation with Church's thesis and the Axiom of Choice

In this work we consider an extension MFcind of the Minimalist Foundation MF for predicative constructive mathematics with the addition of inductive and coinductive definitions sufficient to generate Sambin's Positive topologies, namely Martin-L\"of-Sambin formal topologies equipped with a Positivity relation (used to describe pointfree formal closed subsets). In particular the intensional level of MFcind, called mTTcind, is defined by extending with coinductive definitions another theory mTTind extending the intensional level mTT of MF with the sole addition of inductive definitions. In previous work we have shown that mTTind is consistent with Formal Church's Thesis CT and the Axiom of Choice AC via an interpretation in Aczel's CZF+REA. Our aim is to show the expectation that the addition of coinductive definitions to mTTind does not increase its consistency strength by reducing the consistency of mTTcind+CT+AC to the consistency of CZF+REA through various interpretations. We actually reach our goal in two ways. One way consists in first interpreting mTTcind+CT+AC in the theory extending CZF with the Union Regular Extension Axiom, REA_U, a strengthening of REA, and the Axiom of Relativized Dependent Choice, RDC. The theory CZF+REA_U+RDC is then interpreted in MLS*, a version of Martin-L\"of's type theory with Palmgren's superuniverse S. A last step consists in interpreting MLS* back into CZF+REA. The alternative way consists in first interpreting mTTcind+AC+CT directly in a version of Martin-L\"of's type theory with Palmgren's superuniverse extended with CT, which is then interpreted back to CZF+REA. A key benefit of the first way is that the theory CZF+REA_U+RDC also supports the intended set-theoretic interpretation of the extensional level of MFcind. Finally, all the theories considered, except mTTcind+AC+CT, are shown to be of the same proof-theoretic strength.

From Three Dimensional Manifolds to Modular Tensor Categories

Using M-theory in physics, Cho et al. (JHEP 2020:115 (2020) recently outlined a program that connects two parallel subjects of three dimensional manifolds, namely, geometric topology and quantum topology. They suggest that classical topological invariants such as Chern-Simons invariants of $$\text {SL}(2,{\mathbb {C}})$$ -flat connections and $$\text {SL}(2,{\mathbb {C}})$$ -adjoint Reidemeister torsions of a three manifold can be packaged together to produce a $$(2+1)$$ -topological quantum field theory, which is essentially equivalent to a modular tensor category. It is further conjectured that every modular tensor category can be obtained from a three manifold and a semi-simple Lie group. In this paper, we study this program mathematically, and provide strong support for the feasibility of such a program. The program produces an algorithm to generate the potential modular T-matrix and the quantum dimensions of a candidate modular data. The modular S-matrix follows from essentially a trial-and-error procedure. We find premodular tensor categories that realize candidate modular data constructed from Seifert fibered spaces and torus bundles over the circle that reveal many subtleties in the program. We make a number of improvements to the program based on our examples. Our main result is a mathematical construction of the modular data of a premodular category from each Seifert fibered space with three singular fibers and a family of torus bundles over the circle with Thurston SOL geometry. The modular data of premodular categories from Seifert fibered spaces can be realized using Temperley-Lieb-Jones categories and the ones from torus bundles over the circle are related to metaplectic categories. We conjecture that a resulting premodular category is modular if and only if the three manifold is a $${\mathbb {Z}}_2$$ -homology sphere, and condensation of bosons in the resulting properly premodular categories leads to either modular or super-modular tensor categories.

The vehicle routing problem with simultaneous pickup and delivery and occasional drivers

This research addresses the Vehicle Routing Problem with Simultaneous Pickup and Delivery and Occasional Drivers (VRPSPDOD), which is inspired from the importance of addressing product returns and the emerging notion of involving available crowds to perform pickup and delivery activities in exchange for some compensation. At the depot, a set of regular vehicles is available to deliver and/or pick up customers’ goods. A set of occasional drivers, each defined by their origin, destination, and flexibility, is also able to help serve the customers. The objective of VRPSPDOD is to minimize the total traveling cost of operating regular vehicles and total compensation paid to employed occasional drivers. We cast the problem into a mixed integer linear programming model and propose a simulated annealing (SA) heuristic with a mathematical programming-based construction heuristic to solve newly generated VRPSPDOD benchmark instances. The proposed SA incorporates a set of neighborhood operators specifically designed to address the existence of regular vehicles and occasional drivers. Extensive computational experiments show that the proposed SA obtains comparable results with the state-of-the-art algorithms for solving VRPSPD benchmark instances – i.e., the special case of VRPSPDOD – and outperforms the off-the-shelf exact solver – i.e., CPLEX – in terms of solution quality and computational time for solving VRPSPDOD benchmark instances. Lastly, sensitivity analyses are presented to understand the impact of various OD parameters on the objective value of VRPSPDOD and to derive insightful managerial insights.

ON THE CLASSICAL SANCTUARY SPACE AND ITS NATURAL CONTEXT: NATURE AS EXTENSION OR CONTAINER?

Even though nature is accepted to be an integral part of classical architecture, there is limited scholarship on this aspect. This paper aims to contribute to this literature with a focus specifically on the transformation of attitudes towards nature in the classical culture and on the influence of this transformation to sanctuary planning. Through an argument focusing not only on architecture, but also on art and literature, it tries to establish an overall understanding of the natural in the classical culture. 
 
 An earlier attitude towards nature was a result of the mythological tradition, in which the boundary between gods and humans was blurred. Anthropomorphic gods and goddesses, who controlled the human destiny, resided in the nature. So as the boundary between gods and humans were blurred, the boundary between the man-made and the natural was also blurred. In the Archaic Greek literature and art, nature was not depicted as a space or a background, in which the events took place, but it was integrated into the narrative as an extension of the figures integral to them. Similarly, in the Sanctuary of Apollo at Delphi, nature does not constitute a background/container, but it is an extension of the man-made, integral to the god’s abode made by humans. In this sanctuary, located at a magnificent landscape endowed with mythological significance, myth, nature and the man-made merge by way of an organizational logic that integrates gaze and movement as its major constituents. This paper attempts to explain this classical phenomenological logic by cinematic concepts. The seemingly haphazard planning of the sanctuary allows for multiple moving/viewing sequences, which can be thought of as multiple montages extending the space and time of the sanctuary to various geographies and times in history. These multiple montages are brought back to here and now through unifying frames resembling establishing shots in filmmaking. By these shots, the man-made is merged with the landscape in a single unified frame. 
 
 A later attitude towards nature was a result of the philosophical tradition, in which man separated himself from the almighty gods and defined himself as the constructor of his own order through democratization of the polis. As a result of this separation, uncontrolled nature became the container of the mythical, separate from the controlled man-made realm. In Roman landscape paintings rooted in the Hellenistic stage paintings, nature as the realm of the mythical is viewed from behind the architectural screens like in the Sanctuary of Athena at Lindos or the Sanctuary of Fortuna at Praeneste. In these Hellenistic sanctuaries, perspectival concepts could be thought as instrumental in creating a controlled man-made interior that is detached from its natural environment. This interior can be defined as a mathematical abstract construct exercising an obsessive control over how the point of view of the observer should be located in space. The calculated axial spatial organization of these sanctuaries, therefore, contrasts starkly to the psychophysiological space of the Sanctuary of Apollo at Delphi, allowing various moving/viewing sequences in full integration with the surrounding nature.

Convergence in Conformal Field Theory

Convergence and analytic extension are of fundamental importance in the mathematical construction and study of conformal field theory. The author reviews some main convergence results, conjectures and problems in the construction and study of conformal field theories using the representation theory of vertex operator algebras. He also reviews the related analytic extension results, conjectures and problems. He discusses the convergence and analytic extensions of products of intertwining operators (chiral conformal fields) and of q-traces and pseudo-q-traces of products of intertwining operators. He also discusses the convergence results related to the sewing operation and the determinant line bundle and a higher-genus convergence result. He then explains conjectures and problems on the convergence and analytic extensions in orbifold conformal field theory and in the cohomology theory of vertex operator algebras.

Dynamics of MHD second-grade nanofluid flow with activation energy across a curved stretching surface

This analysis addresses the influence of activation energy on the MHD flow of second-grade nanoliquid over a convectively heated curved stretched surface. The impact of heat generation/absorption, thermophoresis, and Brownian motion are also incorporated. This current study in addendum reveals the solution narrating the nanofluid flow behaviour of the stretched curve to better the performance of the system. Hence, the mathematical construction of governing partial differential equations (PDEs) is transmitted into nonlinear ODEs by employing appropriate transformations. The attained ODEs are conducted numerically via ND-Solve. It is consequential to report that fluid velocity and temperature fields significantly rise with concurrent enhancing values of the fluid parameter and curvature parameter. Moreover, the concentration field enhances considering the energy activation variable and suppresses with the reaction rate constant while thermophoresis escalates the temperature distribution as the Nusselt number lowers with a stronger internal heat source parameter Q.