Related papers: Algebra: The Eighth Liberal Art?
This book offers an introduction to vertex algebra based on a new approach. The new approach says that a vertex algebra is an associative algebra such that the underlying Lie algebra is a vertex Lie algebra. In particular, vertex algebras…
Analytical thinking is dominated by binary ideas. From pair-wise interactions, to algebraic operations, to compositions of processes, to network models, binary structures are deeply ingrained in the fabric of most current scientific…
Standard economic theory uses mathematics as its main means of understanding, and this brings clarity of reasoning and logical power. But there is a drawback: algebraic mathematics restricts economic modeling to what can be expressed only…
The purpose of this paper is to discuss the universal algebra theory of hom-algebras. This kind of algebra involves a linear map which twists the usual identities. We focus on hom-associative algebras and hom-Lie algebras for which we…
This is a proposal of an algebra which aims at distributed array processing. The focus lies on re-arranging and distributing array data, which may be multi-dimensional. The context of the work is scientific processing; thus, the core…
We study the relation between algebraic structures and Graph Theory. We have defined five different weighted digraphs associated to a finite dimensional algebra over a field in order to tackle important properties of the associated…
We provide a theoretical framework for analysing and comparing different forms of organizing introductions to mathematical analysis at the secondary level, then illustrate it by two characteristic examples from certain periods of change in…
Mathematics enters the period of change unprecedented in its history, perhaps even a revolution: a switch to use of computers as assistants and checkers in production of proofs. This requires rethinking traditional approaches to mathematics…
An 'arithmetic circuit' is a labeled, acyclic directed graph specifying a sequence of arithmetic and logical operations to be performed on sets of natural numbers. Arithmetic circuits can also be viewed as the elements of the smallest…
In this note we study associative dialgebras proving that the most interesting such structures arise precisely when the algebra is not semiprime. In fact the presence of some "perfection" property (simpleness, primitiveness, primeness or…
At many physics departments a significant fraction of teaching is in support of engineers and scientists in other majors. These service courses are thus an automatic crucible of interdisciplinary interaction, and at times, strife. For…
The widespread availability of generative artificial intelligence tools poses new challenges for school mathematics education, particularly regarding the formative role of traditional mathematical tasks. In post-AI educational contexts,…
How can we convince students, who have mainly learned to follow given mathematical rules, that mathematics can also be fascinating, creative, and beautiful? In this paper I discuss different ways of introducing non-Euclidean geometry to…
We survey the development of Clifford's geometric algebra and some of its engineering applications during the last 15 years. Several recently developed applications and their merits are discussed in some detail. We thus hope to clearly…
In the words of the twentieth-century British mathematician G. H. Hardy, "the human function is to 'discover or observe' mathematics" (1). For centuries, starting from the ancient Greeks, mankind has hunted for beauty and order in arts and…
Thinking back the long history of physics, we see that the calculation used by physicists was nothing but the ordinary calculus. Another word, physicists have never wrote theories beyond the basic axioms of the calculus. This is not to…
Some idea, which leads to a non-trivial solution of the quantum four-simplex equation, is exposed in this paper. We call this idea "pentagonal algebra". Few examples of the realisation of this idea are given here, and thus few examples of…
Geometric algebra is a mathematical structure that is inherent in any metric vector space, and defined by the requirement that the metric tensor is given by the scalar part of the product of vectors. It provides a natural framework in which…
This article analyzes the emerging ethical turn in mathematics education, arguing that it is a nuanced extension of the sociopolitical turn. While sociopolitical studies of mathematics have highlighted systemic issues and group concerns…
We introduce the notion of a quadri-bialgebra, which gives a bialgebra theory for the quadri-algebra introduced by Aguiar and Loday. We show that a quadri-bialgebra is equivalent to a Manin triple of dendriform algebras associated to a…