Mathematical Objects Research Articles

Quantifying Aristotelian essences: On some fourteenth-century applications of limit decision problems to the perfection of species

ABSTRACT This paper explores a specific problem within an important philosophical genre of the fourteenth century: the debates over the perfection of species. It investigates how the problem of defining limits for continuous magnitudes – a problem typical of Aristotelian physics – was integrated into these debates at the levels of genera, species, and individuals as these entities began to be conceptualized in quantitative terms. After explaining the emergence of this problem within fourteenth-century metaphysics, the paper examines the contributions of three philosophers – Hugolinus of Orvieto, John of Ripa, and Paul of Venice – who offered varying solutions to the challenge of defining limits between species. It demonstrates that two primary solutions arose, inspired by continuous and discrete mathematical objects. It is shown that whereas Hugolinus of Orvieto advocates for a continuist model, John of Ripa proposes a discrete one. The last part of the paper examines Paul of Venice's hybrid approach, which combines elements from both models, facilitating a more comprehensive treatment of species particularly difficult to analyse, namely geometric figures. The conclusions of this comparative study underscore how profoundly the metaphysical reflections of the Middle Ages contributed to the analysis of the structure of the continuum and the extension of the notion of quantity to various objects.

From unbiased to maximal-entropy random walks on hypergraphs.

Random walks have been intensively studied on regular and complex networks, which are used to represent pairwise interactions. Nonetheless, recent works have demonstrated that many real-world processes are better captured by higher-order relationships, which are naturally represented by hypergraphs. Here we study random walks on hypergraphs. Due to the higher-order nature of these mathematical objects, one can define more than one type of walks. In particular, we study the unbiased and the maximal entropy random walk on hypergraphs with two types of steps, emphasizing their similarities and differences. We characterize these dynamic processes by examining their stationary distributions and associated hitting times. To illustrate our findings, we present a toy example and conduct extensive analyses of artificial and real hypergraphs, providing insights into both their structural and dynamical properties. We hope that our findings motivate further research extending the analysis to different classes of random walks as well as to practical applications.

Crystal-Inspired Cellular Metamaterials and Triply Periodic Minimal Surfaces.

Triply periodic minimal surfaces (TPMSs) are found in many natural objects including butterfly wings, sea urchins, and biological membranes. They simultaneously have zero mean curvature at every point and a crystallographic group symmetry. A metamaterial can be created from such periodic surfaces or used as a reinforcement of a composite material. While a TPMS as a mathematical object has been known since 1865, only novel additive manufacturing (AM) technology made it possible to fabricate cellular materials with complex TPMS shapes. Cellular TPMS-based metamaterials have remarkable properties related to wetting/liquid penetration, shock absorption, and the absence of stress concentrators. Recent studies showed that TPMSs are also found in natural crystals when electron surfaces are considered. Artificial crystal-inspired metamaterials mimic such crystals including zeolites and schwarzites. These metamaterials are used for shock, acoustic waves, and vibration absorption, and as structural materials, heat exchangers, and for other applications. The choice of the crystalline cell of a material, as well as its microstructure, plays a decisive role in its properties. The new area of crystal-inspired materials has many common features with traditional biomimetics with models being borrowed from nature and adjusted for engineering applications.

Reifying actions into artifacts: process–object duality from an embodied perspective on mathematics learning

Grasping mathematical objects as related to processes is often considered critical for mathematics understanding. Yet, the ontology of mathematical objects remains under debate. In this paper, we theoretically oppose internalist approaches that claim mental entities as the endpoints of process–object transitions and externalist approaches that stress mathematical artifacts—such as physical manipulatives and formulas—as constituting mathematical objects. We search for a view on process–object duality that overcomes the dualism of mind and body. One such approach is commognition that describes mathematical objects as discursive entities. This paper expands the nature of mathematical objects beyond discourse and highlights the role of learners’ interaction with the environment by adopting ecological onto-epistemology. We develop a functional dynamic systems perspective on process–object duality in mathematics learning emphasizing embodied actions and the re-invention of artifacts’ affordances. As a main result, we reconsider process–object duality as a reification of repetitive actions into a cultural artifact that consists of two steps: (1) forming a new sensory-motor coordination that brings new perception to the fore and (2) crystallizing a new artifact in a mathematical environment that captures this new perception. An empirical example from research on embodied action-based design for trigonometry illustrates our theoretical ideas.Graphical abstract

Analyzing Students’ Thinking in Mathematical Problem Solving Using Vygotskian Sociocultural Theory

Objective: This study aims to investigate students' thinking in solving mathematical problem solving (MPS) using the Vygotskian sociocultural theory (SCT) in the form of mathematical objects. Method: This qualitative study involved 34 high school students in Palembang, 20 female and 14 male students. Data were obtained from students' work and video recordings when students solved problems. The analysis of students' work was reviewed from the semiotic system to see the students' semiotics, including languages (natural and alphanumeric), concepts and propositions, procedures, and arguments. The video recordings were analyzed to assess communication and collaboration among students during problem-solving activities. Research Findings and Discussions: Analyzing students' work and video recordings revealed significant insights into their problem-solving strategies and sociocultural interactions. Through semiotic analysis, it was possible to observe how students expressed and communicated mathematical ideas. Moreover, examining video recordings elucidated communication and collaboration dynamics within the classroom context. These findings shed light on the effectiveness of integrating Mathematical Problem Solving (MPS) with Socio-Cultural Theory (SCT) principles in fostering algebraic thinking and enhancing students' mathematical abilities. Research Implication: The findings of this study hold implications for both research and practice in mathematics education. They provide valuable insights for designing instructional approaches that promote collaborative problem-solving and facilitate the development of algebraic thinking skills. Additionally, the application of Vygotskian SCT in mathematics classrooms offers a promising avenue for cultivating a supportive sociocultural environment conducive to learning. Originality/Value: This study contributes to the existing literature by offering a comprehensive exploration of MPS through the lens of Vygotskian SCT, mainly focusing on using mathematical objects as mediational tools. By examining students' semiotic systems and sociocultural interactions, this research enhances our understanding of the complex processes involved in mathematical problem-solving and underscores the importance of sociocultural factors in shaping students' mathematical experiences.

Mathematical connections involved in area measurement processes

ABSTRACT The present study seeks to explore the mathematical connections that 13–14-year-old secondary school students establish when solving area tasks. Emphasis is placed on different mathematical objects, and the connections between them, that allow students to successfully solve the tasks. The study follows a mixed methodology using qualitative and quantitative method of analysis. The results show that representations play a key role in solving area tasks, as they condition the use of alternative procedures to the use of formulas. Likewise, the properties involved in area measurement processes may condition the use of geometric procedures, such as surface decomposition and reorganisation. Finally, results show that to accurately carry out area measurement processes, it is necessary to bring different mathematical objects into play simultaneously. If these connections between mathematical objects do not occur, there is a risk of using the formulas in a mechanical way.

Engineering Rings in Network Materials

AbstractNetwork materials can be crystalline or amorphous solids, or even liquids, where typically directional interactions link the building blocks together, resulting in a physical representation of a mathematical object, called a graph or equivalently a network. Rings, which correspond to a cyclic path in the underlying network, consisting of a sequence of vertices and edges, are medium‐range structural motifs in the physical space. This Perspective presents an overview of recent studies, which showcase the importance of rings in the emergence of crystalline order as well as in phase transitions between two liquid phases for certain network materials, comprised of colloidal or molecular building blocks. These studies demonstrate how the selection of ring sizes can be exploited for programming self‐assembly of colloidal open crystals with an underlying network and elucidate rings as a vehicle for entanglement that distinguishes the two liquid phases of different densities involved in liquid–liquid phase transitions of network liquids with local tetrahedral order. In this context, an outlook is presented for engineering rings in network materials composed of colloidal and molecular building blocks, with implications also for metal‐organic frameworks, which have been extensively studied as porous crystals, but, more recently, as network‐forming liquids and glasses as well.

On metaphors of mathematics: Between Blumenberg’s nonconceptuality and Grothendieck’s waves

How can metaphors account for the formation of mathematical concepts, for changes in mathematical practices, or for the handling of mathematical problems? Following Hans Blumenberg’s thought, this paper aims to unfold a possible answer to these questions by viewing the metaphorical frameworks accompanying these changes as essential for an understanding of how changes in mathematical practices have been accounted for. I will focus especially on cases in which these changes were caused by encounters with a mathematical object which did not yet have a well-defined concept, but also show that such indeterminacy remains with the mathematical concept even after it is considered ‘well-defined’. As the paper will show, this ‘forefield’ [Vorfeld] of the concept is addressed by Blumenberg’s account of metaphorology on the one hand, and accompanied by his later account of nonconceptuality [Unbegrifflichkeit] on the other hand. While Blumenberg himself did not develop a full-fledged philosophy of mathematics or of mathematical practices, I aim to show that one can nevertheless extract from his writings a unique position concerning the role metaphors play in mathematics. To this end, Blumenberg’s account of nautical and oceanic metaphors and Alexandre Grothendieck’s philosophy of mathematical practice provide fruitful starting points.

Possibilities in teaching planning to make mathematical knowledge accessible to students with autism spectrum disorder

This article presents part of the analysis conducted in a study to indicate and substantiate possibilities for planning mathematics teaching in the initial years of basic education for students with autism spectrum disorder. Engeström’s activity theory guided the investigation as a theoretical and methodological tool, directing data organization and the design of the analyses. This qualitative study had activities prepared by teachers as its data source. The research revealed that, for the teaching process to effectively contribute to mathematics learning by students with autism spectrum disorder, elements that allow the planning of teaching activities based on the construction of meanings of the mathematical objects studied and the development of skills that precede them must be considered.

Feynman Diagrams beyond Physics: From Biology to Economy

Feynman diagrams represent one of the most powerful and fascinating tools developed in theoretical physics in the last century. Introduced within the framework of quantum electrodynamics as a suitable method for computing the amplitude of a physical process, they rapidly became a fundamental mathematical object in quantum field theory. However, their abstract nature seems to suggest a wider usage, which actually exceeds the physical context. Indeed, as mathematical objects, they could simply be considered graphs that depict not only physical quantities but also biological or economic entities. We survey the analytical and algebraic properties of such diagrams to understand their utility in several areas of science, eventually providing some examples of recent applications.

A Holographic-Type Model in the Description of Polymer-Drug Delivery Processes.

A unitary model of drug release dynamics is proposed, assuming that the polymer-drug system can be assimilated into a multifractal mathematical object. Then, we made a description of drug release dynamics that implies, via Scale Relativity Theory, the functionality of continuous and undifferentiable curves (fractal or multifractal curves), possibly leading to holographic-like behaviors. At such a conjuncture, the Schrödinger and Madelung multifractal scenarios become compatible: in the Schrödinger multifractal scenario, various modes of drug release can be "mimicked" (via period doubling, damped oscillations, modulated and "chaotic" regimes), while the Madelung multifractal scenario involves multifractal diffusion laws (Fickian and non-Fickian diffusions). In conclusion, we propose a unitary model for describing release dynamics in polymer-drug systems. In the model proposed, the polymer-drug dynamics can be described by employing the Scale Relativity Theory in the monofractal case or also in the multifractal one.

The human after math: Nnedi Okorafor’s unruly engagements with Cartesian desire

ABSTRACT This article contends that Nnedi Okorafor remakes a European Enlightenment notion of mathematics in her speculative trilogy Binti (2015-2018). The titular character has the power of mathematical thinking, where through the contemplation of mathematical signs, she absents herself from her immediate physical surroundings and enters into a dream-like space. Reading Binti alongside Descartes’ Discourse on the Method, I argue that Okorafor draws on an Enlightenment notion of mathematics as training in reason, where it was believed that through the encounter with mathematical objects, man could learn to think independently of their cultural and religious backgrounds, becoming universal. Her critique of Enlightenment math is not explicit, but rather raises philosophical questions about what exactly math is and how it could have secured the secular human. Binti reveals that math is a linguistic technology for producing imaginative realms detached from the real world that could allow for different experiences of becoming. Okorafor’s work brings mathematics in conversation with literature and also Fanon’s thinking about race and the body. This essay examines how African speculative fiction can intervene in modern liberal notions of the human by interrogating and remaking the seemingly objective discourses that make up our current hegemonic mode of being human.

Fuzzy Dynamic Fuzzy Sets. Variable Fuzzy Hierarchical Dynamic Fuzzy Structures (Models, Operators) for Dynamic, Singular, Hierarchical Fuzzy Sets. FUZZY PROGRAM OPERATORS ffSprt, fftprS, ffS1epr, ffSeprt1

The purpose of the article is to create new constructive mathematical fuzzy objects for new technologies, in particular for a fundamentally new type of neural network with parallel computing, in particular for creating artificial intelligence, but this is not the main task of a neural network, and not with the usual parallel computing through sequential computing. Significance of the article: in a new qualitatively different approach to the study of complex processes through new mathematical, fuzzy hierarchical, fuzzy dynamic fuzzy structures, in particular those processes that are dealt with by Synergetics. Here is to create new fuzzy fprogram operators for a fundamentally new type of neural network with parallel computing, and not with the usual parallel computing through sequential computing.

Mathematical impossibilities

ABSTRACT This paper argues that modal realism has a problem with mathematical impossibilities. Due to the peculiar way it treats both propositions and mathematical objects, modal realism cannot distinguish the content of different mathematically impossible beliefs. While one might be happy to identify all logically impossible beliefs, there are many different mathematically impossible beliefs, none of which is a belief in a logical contradiction. The fact that it cannot distinguish these beliefs speaks against adopting modal realism.

Beautiful Gini

AbstractYou may very well be familiar with the Gini Coefficient, also known as the Gini index: a quantitative gauge with which socioeconomic inequality is measured, e.g. income disparity and wealth disparity. However, you may not know that the Gini Coefficient is an exquisite mathematical object. Enter this review paper—whose aim is to showcase (some of) the mathematical beauty and riches of the Gini Coefficient. The paper does so, in a completely self-contained manner, by illuminating the Gini Coefficient from various perspectives: Euclidean geometry vs. grid geometry; maxima and minima of random variables; statistical distribution functions; the residual lifetime and the total lifetime of renewal processes; increasing and decreasing failure rates; socioeconomic divergence from perfect equality; and weighted differences of statistical distribution functions. Together, these different perspectives offer a deep and comprehensive understanding of the Gini Coefficient. In turn, a profound understanding of the Gini Coefficient may lead to novel ‘Gini applications’ in science and engineering—such as recently established in the multidisciplinary field of restart research.

Modeling of limit order book data with ordered fuzzy numbers

This paper presents a novel approach to representing the Limit Order Book data at a given timestamp using the Ordered Fuzzy Numbers concept. The limit order book contains all buy and sell orders placed by investors, updated in real-time, for the most liquid securities, even several hundred times a minute. Due to its irregular nature (different and dynamic changes in the number of buy and sell orders), direct calculations on the order book data are not feasible without transforming it into feature vectors. Currently, most studies use a price level-based data representation scheme when applying deep learning models on limit order book data. However, this scheme has limitations, particularly its sensitivity to subtle perturbations that can negatively impact model performance. On the other hand, the ordered fuzzy number is a mathematical object (a pair of two functions) used to process imprecise and uncertain data. Ordered Fuzzy Numbers possess well-defined arithmetic properties. Converting the limit order book data to ordered fuzzy numbers allows the creation of a time series of ordered fuzzy numbers (order books) and use them for further calculations, e.g., to represent input data for deep learning models or employing the concept of fuzzy time series in various domains, such as defining liquidity measures based on limit order book data. In this paper, the proposed approach is tested using one-year market data from the Polish Stock Exchange for the five biggest companies. The DeepLOB model is employed to predict mid-price movement using different input data representations. The proposed representation of Limit Order Book data demonstrated remarkably stable out-of-sample prediction accuracy, even when subjected to data perturbation.

Topological insulators and K-theory

We analyze topological invariants, in particular Z2 invariants, which characterize time reversal invariant topological insulators, in the framework of index theory and K-theory. After giving a careful study of the underlying geometry and K-theory, we formalize topological invariants as elements of KR theory. To be precise, the strong topological invariants lie in the higher KR groups of spheres; KR̃−j−1(SD+1,d). Here j is a KR-cycle index, as well as an index counting off the Altland-Zirnbauer classification of Time Reversal Symmetry (TRS) and Particle Hole Symmetry (PHS)—as we show. In this setting, the computation of the invariants can be seen as the evaluation of the natural pairing between KR-cycles and KR-classes. This fits with topological and analytical index computations as well as with Poincaré Duality and the Baum–Connes isomorphism for free Abelian groups. We provide an introduction starting from the basic objects of real, complex and quaternionic structures which are the mathematical objects corresponding to TRS and PHS. We furthermore detail the relevant bundles and K-theories (Real and Quaternionic) that lead to the classification as well as the topological setting for the base spaces.

Proposal for a Hypothetical Learning Path Using GeoGebra in Learning Numerical Sequences

A hypothetical learning path is proposed for the teaching-learning of numerical sequences in second grade secondary school students, it is theoretically based in a Duval’s Theory of Semiotic Representation Registers and methodologically based on GeoGebra & Hypothetical Learning Path. It is intended to contribute to this field of mathematics given that there are not many the works in this regard, a fact that motivates the facilitation of better perspectives for students in learning the mathematical object numerical sequences. According to the hypothetical learning trajectory theory, each stage of learning requires a series of preconditions and appropriate instructional approaches so that the individual can progress. These stages include: 1) Acquisition of basic knowledge 2) Application of skills 3) Automation 4) Integration. It is concluded that it must be applied in the classroom to know the effectiveness of this trajectory.

На пути к автоматизированному и оптимальному проектированию систем IIoT

In today’s world, the Internet of Things has become an integral part of our lives. The increasing number of intelligent devices and their pervasiveness has made it challenging for developers and system architects to plan and implement systems of Internet of Things and Industrial Internet of Things effectively. The primary objective of this work is to automate the design process of Industrial Internet of Things systems while optimizing the quality of service parameters, battery life, and cost. To achieve this goal, a general four-layer fog-computing model based on mathematical sets, constraints, and objective functions is introduced. This model takes into consideration the various parameters that affect the performance of the system, such as network latency, bandwidth, and power consumption. The Non-dominated Sorting Genetic Algorithm II is employed to find Pareto optimal solutions, while the Technique for Order of Preference by Similarity to Ideal Solution is used to identify compromise solutions on the Pareto front. The optimal solutions generated by this approach represent servers, communication links, and gateways whose information is stored in a database. These resources are chosen based on their ability to enhance the overall performance of the system. The proposed strategy follows a three-stage approach to minimize the dimensionality and reduce dependencies while exploring the search space. Additionally, the convergence of optimization algorithms is improved by using a biased initial population that exploits existing knowledge about how the solution should look. The algorithms used to generate this initial biased population are described in detail. To illustrate the effectiveness of this automated design strategy, an example of its application is presented.

On the feasibility and the impact resistance of a 3D cross-based fractal produced by powder bed fusion additive manufacturing

Designers have been fascinated by exploring new geometries made by high-performance structures. In more specific terms, biological systems have always been proven to be characterised by sophisticated structures with adapting properties to nature challenges. Insightful analyses have shown how these natural structures are dominated by characteristics such as high energy absorption and elevated strength-weight proportion. Fractal geometries are examples of bio-inspired mathematical objects whose complex 3D structures can be obtained only by advanced manufacturing systems, such as additive manufacturing (AM). This study investigates the feasibility and energy absorption properties of a novel fractal structure based on a 3D Greek cross (3D-CFS). The structure was designed with different volume fractions and produced by powder bed fusion (PBF) AM processes in polyamide (PA12) and thermoplastic polyurethane (TPU). The 3D-CFS properties are investigated under quasi-static and dynamic compression tests. The analysis revealed that for certain geometrical parameters, the manufacturing of the structures is constrained by the sintered powder entrapped in the structure. However, in the case of powder-free structures, the results showed a high impact resistance and cushioning capability. Overall, in terms of specific energy absorption (SEA), the TPU structures showed values between 2.5 and 3.5 kJ/kg, while PA12 ones are between 7.5 and 17.4 kJ/kg, making the 3D-CFS structure compatible with personal protective equipment (PPE) applications. Compared to the literature data on cellular structures made by AM, 3D-CFS performs considerably better. Also, PA12 3D-CFS is better, with a SEA value up to 170% higher than that of a typical material employed for head PPE (e.g. EPS-60 SEA equal to 2.76 kJ/kg). In contrast, TPU 3D-CFS looks more promising in the case of multiple impact conditions.