Mathematical Objects Research Articles

Using hypergraphs to quantify importance of sets of diseases by healthcare resource utilisation: A retrospective cohort study.

Rates of Multimorbidity (also called Multiple Long Term Conditions, MLTC) are increasing in many developed nations. People with multimorbidity experience poorer outcomes and require more healthcare intervention. Grouping of conditions by health service utilisation is poorly researched. The study population consisted of a cohort of people living in Wales, UK aged 20 years or older in 2000 who were followed up until the end of 2017. Multimorbidity clusters by prevalence and healthcare resource use (HRU) were modelled using hypergraphs, mathematical objects relating diseases via links which can connect any number of diseases, thus capturing information about sets of diseases of any size. The cohort included 2,178,938 people. The most prevalent diseases were hypertension (13.3%), diabetes (6.9%), depression (6.7%) and chronic obstructive pulmonary disease (5.9%). The most important sets of diseases when considering prevalence generally contained a small number of diseases, while the most important sets of diseases when considering HRU were sets containing many diseases. The most important set of diseases taking prevalence and HRU into account was diabetes & hypertension and this combined measure of importance featured hypertension most often in the most important sets of diseases. We have used a single approach to find the most important sets of diseases based on co-occurrence and HRU measures, demonstrating the flexibility of the hypergraph approach. Hypertension, the most important single disease, is silent, underdiagnosed and increases the risk of life threatening co-morbidities. Co-occurrence of endocrine and cardiovascular diseases was common in the most important sets. Combining measures of prevalence with HRU provides insights which would be helpful for those planning and delivering services.

The Exploration of Geometrical Concept in Batik Pamekasan

Indirectly, mathematical themes can be found throughout Indonesia's unique culture, one of which is mathematics in batik cultural items. Exploring ethnomathematics will be really beneficial. Apart from acquiring the concept of mathematics in Indonesian culture, it can also sharpen Indonesian culture's sense of love for the nation. The purpose of this study is to describe the philosophical meaning of the geometrical concepts contained in Pamekasan batik, the geometrical concepts contained in Pamekasan batik, and the implementation of Batik pamekasan as the context in learning geometry. This is ethnomathematics study which gather the data by observation and interview. The interview held two regions in Pamekasan with four informants. As a result, scholars aim to conduct an ethnomathematics examination of traditional Pamekasan batik. This research shows that aside from the Batik itself, the mathematical objects found in Batik each have their own meaning and hopes. As a result, mathematical motifs such as the point that depicts a rice or maize seed have philosophical significance. Meanwhile, reflection reveals that human life requires balance. Each mathematical object in the pattern has its own meaning. Meanwhile, geometrical objects in Batik Pamekasan are two-dimensional figures such as circles, rectangles, squares, lines, points, angles, and curves. In addition, transformative geometry like as rotation, reflection, and translation exist in Batik Pamekasan. Besides, batik pamekasan can be used as the context in learning geometry. This can be seen from the learning outcomes that was written in Merdeka curriculum. Moreover, problem-based learning and project-based learning are two models that support ethnomathematics-based learning.

Dynamical Sets Theory: S2 t-Elements and Their Applications

The purpose of the article is to create new constructive mathematical objects for new technologies, in particular for a fundamentally new type of neural network with parallel computing, and not with the usual parallel computing through sequential computing

Preface

This volume contains contributions to the XII. International Symposium on Quantum Theory and Symmetries (QTS12) was at Czech Technical University in Prague, Czech Republic, from Monday July 24 until Friday July 28, 2023.The Symposium series “Quantum Theory and Symmetries” (QTS) is a biannual meeting intended for physicists and mathematicians who are either applying symmetries to some physical model, or are studying mathematical objects that are relevant for physical applications.The first Symposium of this series was held in Goslar (Germany) in 1999, QTS-1, then it was held in Cracow (2001) QTS-2, Cincinnati (2003) QTS-3, Varna (2005) QTS-4, Valladolid (2007) QTS-5, Lexington (2009) QTS-6, Prague (2011) QTS-7, Mexico-city (2013) QTS-8, Yerevan (2015) QTS-9, Varna (2017) QTS-10, Montreal (2019) QTS-11.The Symposium QTS-12 was first planned to be in Moscow at the Moscow Institute of Physics and Technology in 2021, but was postponed by one year due to Covid-19, then it was postponed to 2023, July 24-28, and was held in Prague at the Czech Technical University (Department of Mathematics, Faculty of Nuclear Sciences and Physical Engineering, CTU in Prague) QTS-12.The Symposium QTS-13 is planned for the 2nd half of July 2025 to be held in Yerevan, Armenia.The volume is organised as follows: it starts with notes in memory of H. D. Doebner. Then the invited talks at the plenary sessions and the public lecture are published followed by contributions in the parallel and poster sessions in alphabetical order.List of Editors, Advisory committee are available in this Pdf.

Parametric Lie group structures on the probabilistic simplex and generalized Compositional Data

In this paper we build a set of parametric quotient Lie group structures on the probabilistic simplex that can be extended to real vector space structures. In particular, we rediscover the main mathematical objects generally used when treating compositional data as elements associated to the quotient Lie group with respect to the equivalence relation induced by the scale invariance principle. This perspective facilitates the adaptation of the statistical methods used for classical compositional data to data that follows a more general equivalence relation.

Group invariants for Feynman diagrams

It is well-known that the symmetry group of a Feynman diagram can give important information on possible strategies for its evaluation, and the mathematical objects that will be involved. Motivated by ongoing work on multi-loop multi-photon amplitudes in quantum electrodynamics, here I will discuss the usefulness of introducing a polynomial basis of invariants of the symmetry group of a diagram in Feynman-Schwinger parameter space.

MMLKG: Knowledge Graph for Mathematical Definitions, Statements and Proofs

Nowadays, Knowledge Graphs (KGs) are important and developing in different areas. However, there is a lack of genuinely interoperable datasets representing mathematics that allow for information exchange between datasets in the Web ecosystem. In this paper, we address this matter based on the Mizar Mathematical Library (MML), a collection of articles written in the Mizar language. MML includes definitions and theorems with proofs to which authors can easily refer from newly written Mizar articles. However, extracting information directly from Mizar scripts by external projects is not very straightforward. Therefore, we propose a new data storage and retrieval approach based on the Knowledge Organization System (KOS) model and the KG concept that provides a way to organize and access knowledge. We present Mizar Mathematical Library Knowledge Graph (MMLKG), a thesaurus for describing mathematical objects. MMLKG supports semantic interoperability and allows linking data from different sources, e.g., Wikidata. Moreover, it satisfies the FAIR data principles. The data is publicly available via a Cypher endpoint.

The anthropological theory of the didactic as a methodological proposal to analyze digital resources

This article presents an analysis of digital resources for mathematics teaching and learning based on the anthropological theory of the didactic as a methodological proposal. Thus, data were collected from the general information available on the PhET digital platform: general descriptions of the digital resources Equality Explorer: Basics and Equality Explorer, teaching resources, and activities proposals sent by teachers. Furthermore, digital resources were handled to determine their mathematical praxeologies: the types and subtypes of tasks and their respective techniques and technologies. Combining ostensive aspects is highlighted among the research results as indispensable for working with mathematical praxeologies involving digital resources. In short, the results reveal that adopting the ATD as a methodological possibility allows analyzing not only the mathematical organization but can also support reflections about the didactic praxeology and the personal and institutional relations around the mathematical object.

IMMERSIVE LEARNING ENVIRONMENT: A COMPARATIVE ANALYSIS

The article is devoted to the current problem of development of the educational environment of general secondary education institutions (GSEI) in the conditions of digital technologies that are intensively developing. The purpose of the article is to carry out a comparative analysis of traditional, immersive learning environments and distance learning. The article justifies the main components of the immersive learning environment: virtual reality, augmented reality, mixed reality 360-degree video, multiplayer games, interactive simulations. Eleven characteristics were chosen to compare the traditional, immersive learning and distance learning environment: physical environment, teacher role, student role, advantages, disadvantages, technology, experience, knowledge acquisition, motivation, capture features/immersion, availability, depth of immersion in learning. The general problems of using an immersive learning environment are highlighted, in particular: discomfort, inaccuracies in tracking objects, difficulties in using equipment. Reasoned factors that hinder the effective introduction of immersive learning in educational practice, namely: the complexity of the content of natural and mathematical objects and the abstractness of theoretical material, a limited number of hours to study topics and consolidate skills and build competencies, a long period of distance and mixed learning, which is the result of modern educational challenges. Further research requires justification of the digital impact on the educational process in general secondary education institutions, justification of the concept of immersive learning, principles and approaches of immersion in learning, stages of immersion, forms and methods of implementation of immersion.

Internal Applications and Puzzles of the Applicability of Mathematics

Abstract Just as mathematics helps us to represent and reason about the natural world, in its internal applications one branch of mathematics helps us to represent and reason about the subject matter of another. Recognition of the close analogy between internal and external applications of mathematics can help resolve two persistent philosophical puzzles concerning its applicability: a platonist puzzle arising from the abstractness of mathematical objects; and an empiricist puzzle arising from mathematical propositions’ lack of empirical factual content. In order to see how this is the case, we will examine what it is to apply mathematics internally and describe examples.

Dynamic Sets Se, Networks Se

The purpose of the article is to create new constructive hierarchical mathematical objects Se – elements for new technologies, particularly for a fundamentally new type of neural network with parallel computing, and not with the usual parallel computing through sequential computing

The “Tenderness” of the Principle of Least Action: From the Philosophy of Physics to the Paradigm for Sustainable Development

The paper delves into the methodological aspects of how foundational mathematical and physical tenets, most notably the principle of least action, are interpreted and assimilated within humanities discourse. The pursuit of the article’s objectives is driven by the necessity for a philosophical and methodological analysis of the current conceptual status of the principle of least action. This analysis is informed by cognitive-axiological and teleological imperatives of a “synthetic” development program for the principle. Any fundamental principle will not have a definitive explanation, as otherwise it would not be fundamental, but in this case, some justification can be given based on deeper grounds discussed in the article. Drawing on the epistemological frameworks of French philosophers P. Hadot, E. Levinas, J. Bouveresse, and J.-T. Desanti, the article weaves together mathematical abstractions with human experience and philosophical doctrines with physical theories. J.-T. Desanti’s perspective on mathematical objects as an outcome of human activity is examined. Also scrutinized is B. Nicolescu’s concept of transdisciplinarity, which challenges the traditional subject-object dualism in science. The author’s methodological stance emerges from a dialectical viewpoint, one that eschews a simplistic dichotomy between materialism and idealism and is grounded in rigorous scientific inquiry and an exhaustive examination of the subject matter itself. The principle of least action, as a paramount principle in physics, is shown to exemplify a legacy of innovation, positioning it as both methodologically insightful and heuristically valuable. The paper also highlights how this principle diverges from the classical principle of economy. The broader goal – within the context of “sustainable development,” transdisciplinary studies, and creative industries – is to establish the principle of least action as a paradigmatic imperative for interaction within social and economic systems. The conclusions drawn from this study contribute to a deeper understanding of the principle’s essence, the nature of transdisciplinarity, and confront the vestiges of scientism in the humanities.

The Petersen Graph and its Generalizations

The Petersen graph, a recurring mathematical object, is the subject of this paper. The Petersen graph, which bears the name of the Danish mathematician Julius Petersen, who first proposed it in 1898, is one of the most fascinating and well-known graphs in graph theory. The Petersen graph is a compact yet extremely intriguing graph with a number of distinctive characteristics and uses. Some properties and significance of Petersen graph and its generalizations are explored in this paper.

Aristotle on the Objects of Natural and Mathematical Sciences

In a series of recent papers, Emily Katz has argued that on Aristotle's view mathematical sciences are in an important respect no different from most natural sciences: They study sensible substances, but not qua sensible. In this paper, I argue that this is only half the story. Mathematical sciences are distinctive for Aristotle in that they study things ‘from’, ‘through’ or ‘in’ abstraction, whereas natural sciences study things ‘like the snub’. What this means, I argue, is that natural sciences must study properties as they occur in the subjects from which they are originally abstracted, even where they reify these properties and treat them as subjects. The objects of mathematical sciences, on the other hand, can be studied as if they did not really occur in an underlying subject. This is because none of the properties of mathematical objects depend on their being in reality features of the subjects from which they are abstracted, such as bodies and inscriptions. Mathematical sciences are in this way able to study what are in reality non-substances as if they were substances.

Random templex encodes topological tipping points in noise-driven chaotic dynamics.

Random attractors are the time-evolving pullback attractors of deterministically chaotic and stochastically perturbed dynamical systems. These attractors have a structure that changes in time and that has been characterized recently using Branched Manifold Analysis through Homologies cell complexes and their homology groups. This description has been further improved for their deterministic counterparts by endowing the cell complex with a directed graph (digraph), which encodes the order in which the cells in the complex are visited by the flow in phase space. A templex is a mathematical object formed by a cell complex and a digraph; it provides a finer description of deterministically chaotic attractors and permits their accurate classification. In a deterministic framework, the digraph of the templex connects cells within a single complex for all time. Here, we introduce the stochastic version of a templex. In such a random templex, there is one complex per snapshot of the random attractor and the digraph connects the generators or "holes" of successive cell complexes. Tipping points appear in a random templex as drastic changes of its holes in time, through their birth, splitting, merging, or death. This paper introduces random templexes and computes them for the noise-driven Lorenz system's random attractor.

Hypothetical Inquiry in Plato's Timaeus

This paper re-constructs Plato's ‘philosophy of geometry’ by arguing that he uses a geometrical method of hypothesis in his account of the cosmos’ generation in the Timaeus. Commentators on Plato's philosophy of mathematics often start from Aristotle's report in the Metaphysics that Plato admitted the existence of mathematical objects in-between ( metaxu) Forms and sensible particulars ( Meta. 1.6, 987b14–18). I argue, however, that Plato's interest in mathematics was centred on its methodological usefulness for philosophical inquiry, rather than on questions of mathematical ontology. My key passage of interest is Timaeus’ account of the generation of the primary bodies in the cosmos, i.e. fire, air, water and earth ( Tim. 48b–c, 53b–56c). Timaeus explains the primary bodies’ origin by hypothesising two right-angled triangles as their starting-point ( arkhê) and describing their individual geometrical constitution. This hypothetical operation recalls the hypothetical method which Socrates introduces in the Meno (86e–87b), as well as the use of hypotheses by mathematicians which is described in the Republic (510b–c). Throughout the passage, Timaeus is focussed on explicating the bodies in terms of their formal structure, without however considering the ontological status of the triangles in relation to the physical world.

Why Aristotle Can’t Do without Intelligible Matter

I argue that intelligible matter, for Aristotle, is what makes mathematical objects quantities and divisible in their characteristic way. On this view, the intelligible matter of a magnitude is a sensible object just insofar as it has dimensional continuity, while that of a number is a plurality just insofar as it consists of indivisibles that measure it. This interpretation takes seriously Aristotle's claim that intelligible matter is the matter of mathematicals generally – not just of geometricals. I also show that intelligible matter has the same meaning in all three places where it is explicitly invoked: Z.10, Z.11, and H.6. Since the H.6 passage involves a mathematical definition, this requires determining what the mathematician defines and how she defines it. I show that, as with natural scientific definitions, there must be a matterlike element in mathematical definitions. This element is not identical with, but rather refers to, intelligible matter.

THE POWER GRAPH REPRESENTATION FOR INTEGER MODULO GROUP WITH POWER PRIME ORDER

There are many applications of graphs in various fields. Starting from chemical problems, such as the molecular shape of a compound to internet network problems, we can also use graphs to depict the abstract concept of a mathematical structure.. Groups in Algebra can be represented as a graph. This is interesting because Groups are abstract objects in mathematics. The graph of a group shows the physical form of the group by looking at the relationship between its elements. So, we can know the distance of the elements. In 2013, Abawajy et al. conducted studies related to power graphs. Power graph representation of groups of integers modulo with the order of prime numbers has been carried out in 2022 by Syechah, et al. In this article, the author provides the properties of a power graph on a group of integers modulo with the order of powers of prime numbers.

How to frame innovation in mathematics

We discuss conceptual change and progress within mathematics, in particular how tools, structural concepts and representations are transferred between fields that appear to be unconnected or remote from each other. The theoretical background is provided by the frame concept, which is used in linguistics, cognitive science and artificial intelligence to model how explicitly given information is combined with expectations deriving from background knowledge. In mathematical proofs, we distinguish two kinds of frames, namely structural frames and ontological frames. The interaction between both kinds of frames can drive mathematical interpretation. We first discuss two examples where structural frames (formulaic notation) drive ontological development (the discovery or exploration of mathematical objects). The development of Boole’s Boolean algebra may at first appear as a metaphorical treatment of the (then) new area of logic. In the analysis, we discuss how different (aspects of) certain algebraic frames change in the transfer, how arising difficulties are solved and overall argue that Boole uses the numerical algebra frame as a research template for the discovery of a system for calculations in logic. Following Ifrah, we analyse the discovery of zero as an extension to the number ontology as driven by the development of notation. Both structural and ontological frames are extended and simplified as notation progresses. Finally, we discuss two examples from infinite combinatorics, viz. topological graph theory, and one foundational issue. In both examples, the two simultaneous frames about one object are maintained independently. They motivate different research questions, but may also fruitfully interact: shifting between multiple synchronously maintained perspectives acts as a motor of innovation. The analysis shows how a frame-based approach allows to model how different perspectives drive mathematical innovation because they highlight different aspects, questions and heuristics.

Is theoretical topic-specific research “old fashioned”? An epistemological inquiry about the ontological creativity of Mathematics Education Research

AbstractIn Mathematics Education (ME), research dealing with topic-specific (TS) issues (e.g., what levels of development exist in learning fractions) produces usually local results and is considered less fashionable and attractive for innovative research projects than research dealing with context-specific (CS) issues that have more general and abstract scopes (e.g., what is mathematical instruction? what is the field of ME?) and produces middle-range or grand theories. TS- and CS-research run along separate tracks with little or no crossover, at least from the beginning of the socio-political-turn in ME, but connecting them could help to single out hidden variables in CS-research. This paper shows that TS-research creates specific mathematical objects that allow us to reduce the distance between these two lines of research. Mathematical objects specific to ME research are shown to be both a technical link between the two lines of research because they allow topic-specificity to access more abstract and general realms of research, as well as factor at stake when aspects related to the social, political, and ethical implications of the ontological creativity of TS-research are discussed in a critical postmodern approach. Discussing its impact on textbooks, teacher-training, teaching practices, further TS-research-practices, as well as on the researcher’s epistemological empowerment and on the self-referentiality of ME research, TS-research moves from the periphery to the heart of CS-research.