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18 pages, 305 KB

Open AccessArticle

Colour Algebras over Rings

by Susanne Pumplün

Axioms 2026, 15(2), 139; https://doi.org/10.3390/axioms15020139 - 14 Feb 2026

Viewed by 402

Colour algebras are noncommutative Jordan algebras and closely related to octonion algebras. They initially emerged in physics since the multiplication of a colour algebra over the real or complex numbers describes the colour symmetry of the Gell–Mann quark model. Over fields of characteristic not equal to two, their structure is now well-known. We initiate the study of colour algebras over a unital commutative base ring R where two is an invertible element, and show when colour algebras can be constructed canonically by employing nondegenerate ternary hermitian forms with trivial determinant. We investigate their structure, their automorphism group and their derivations. We show that there is again a close connection between the colour algebras obtained from hermitian forms and certain types of octonion algebras. Full article

(This article belongs to the Special Issue Advances in Classical and Applied Mathematics, 2nd Edition)

19 pages, 1657 KB

Open AccessArticle

From Mathematics to Art: Modelling the Pascal’s Triangle with Petri Nets

by David Mailland and Iwona Grobelna

Symmetry 2025, 17(12), 2181; https://doi.org/10.3390/sym17122181 - 18 Dec 2025

Cited by 1 | Viewed by 944

Abstract

Pascal’s triangle is a classical mathematical construct, historically studied for centuries, that organises binomial coefficients in a triangular array and serves as a cornerstone in combinatorics, algebra, and number theory. Herein, we propose to model it with Petri nets, a formal specification technique derived from discrete event systems. A minimal Petri net is created that generates Pascal’s triangle under a simple arithmetic rule. Token counts in each place coincide with binomial coefficients, providing a direct combinatorial interpretation. Two other classical structures emerge from this model: by colouring tokens depending on their parity, the Sierpiński triangle appears; by routing tokens randomly at each branching, the binomial distribution arises, converging to a Gaussian limit as depth increases. As a result, a single Petri construction unifies three mathematical objects: Pascal’s Triangle, Sierpiński’s Triangle, and the Gaussian distribution. This connection illustrates the invaluable potential of Petri nets as unifying tools for modelling discrete mathematical structures and beyond. Full article

(This article belongs to the Special Issue Symmetry and Graph Theory, 2nd Edition)

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18 pages, 993 KB

Open AccessArticle

Students with Visual Impairments’ Comprehension of Visual and Algebraic Representations, Relations and Correspondence

by Fatma Nur Aktas and Ziya Argun

Educ. Sci. 2025, 15(8), 1083; https://doi.org/10.3390/educsci15081083 - 21 Aug 2025

Viewed by 1362

Abstract

Exploring learning trajectories based on student thinking is needed to develop the teaching curricula, practices and educational support materials in mathematics for students with visual impairments. Hence, this study aims to reveal student thinking through various instructional tasks and tactile materials to explore the sequence of goals in the learning trajectory. A teaching experiment involving introductory information on algebraic and visual representations regarding advanced mathematical concepts was designed for correspondence and relations. The research was carried out with a braille-literate 10th-grade high school student with a congenital visual impairment where colour and light are not perceived in Türkiye. As a result of the teaching experiment, the participant was able to determine the correspondence and relations between two sets using different representations. He even designed graphic representations using the needle page. The learning trajectory goals and instructional tasks can serve as guides for research on curriculum development, practice design and material development. Full article

(This article belongs to the Special Issue Empowering Minds: Innovation, Equity and the Science of Learning in Mathematics Education)

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73 pages, 861 KB

Open AccessArticle

Homotopy Double Copy of Noncommutative Gauge Theories

by Richard J. Szabo and Guillaume Trojani

Symmetry 2023, 15(8), 1543; https://doi.org/10.3390/sym15081543 - 4 Aug 2023

Cited by 12 | Viewed by 1763

Abstract

We discuss the double-copy formulation of Moyal–Weyl-type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of

L_{\infty}

-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour [...] Read more.

We discuss the double-copy formulation of Moyal–Weyl-type noncommutative gauge theories from the homotopy algebraic perspective of factorisations of

L_{\infty}

-algebras. We define new noncommutative scalar field theories with rigid colour symmetries taking the role of the zeroth copy, where the deformed colour algebra plays the role of a kinematic algebra; some of these theories have a trivial classical limit but exhibit colour–kinematics duality, from which we construct the double copy theory explicitly. We show that noncommutative gauge theories exhibit a twisted form of colour–kinematics duality, which we use to show that their double copies match with the commutative case. We illustrate this explicitly for Chern–Simons theory, and for Yang–Mills theory where we obtain a modified Kawai–Lewellen–Tye relationship whose momentum kernel is linked to a binoncommutative biadjoint scalar theory. We reinterpret rank-one noncommutative gauge theories as double copy theories and discuss how our findings tie in with recent discussions of Moyal–Weyl deformations of self–dual Yang–Mills theory and gravity. Full article

(This article belongs to the Section Physics)

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