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The work presents two examples of simple mathematical formulas which are natural nonlinear modifications (one being a generalization) of Gielis' formula. These formulas involve a comparable number of parameters and provide non-Platonic…
Theory of representations of F-algebra is a natural development of the theory of F-algebra. Exploring of morphisms of the representation leads to the concepts of generating set and basis of representation. In the book I considered the…
A description of physical reality in which wholeness is the foundation is discussed along with the motivation for such an attempt. As a possible mathematical framework within which a physical theory based on wholeness may be expressed,…
The world of mathematics is often considered abstract, with its symbols, concepts, and topics appearing unrelated to physical objects. However, it is important to recognize that the development of mathematics is fundamentally influenced by…
In this paper we will give the calculus, the criterion, and the existence of the arithmetic Galois covers of higher relative dimensions.
We show a possibility to apply certain philosophical concepts to the analysis of concrete mathematical structures. Such application gives a clear justification of topological and geometric properties of considered mathematical objects.
Both lectures focus on the first part of the so-called 'mathematical part' of Plato's Theaetetus. In this passage, the young Theaetetus briefly recounts the mathematical lesson given by the geometer Theodorus. The first lecture delves into…
Mathematical understanding is built in many ways. Among these, illustration has been a companion and tool for research for as long as research has taken place. We use the term illustration to encompass any way one might bring a mathematical…
This is an overview of higher structural constructions in physics. The main motivations of our current attempt are as follows: (i) to provide a brief introduction to derived algebraic geometry, (ii) to understand how derived objects…
A new methodological approach for the study of topology for shapes made of arrangements of lines, planes or solids is presented. Topologies for shapes are traditionally built on the classical theory of point-sets. In this paper, topologies…
This article illustrates pedagogy through training in the handling of abstractions. Mental arithmetic is not limited to numerical calculation; one can mentally calculate primitives and simplify analytical expressions. Even if there is…
This note studies the quantized corner structure of four-dimensional $BF$ theory, classifies the associated free and physical corner algebras and constructs possible representations. In the abelian case, for arbitrary closed oriented…
This paper challenges some of the common assumptions underlying the mathematics used to describe the physical world. We start by reviewing many of the assumptions underlying the concepts of real, physical, rigid bodies and the translational…
The logical line is traced of formulation of theory of mechanics founded on the basic correlations of mathematics of hypercomplex numbers and associated geometric images. Namely, it is shown that the physical equations of quantum, classical…
We present a topological construction that provides many examples of non-commutative Frobenius algebras that generalizes the well-known pair-of-pants. When applied to the solid torus, in conjunction with Crane-Yetter theory, we provide a…
Approaching limitations of digital computing technologies have spurred research in neuromorphic and other unconventional approaches to computing. Here we argue that if we want to systematically engineer computing systems that are based on…
Mathematically representing the shape of an object is a key ingredient for solving inverse rendering problems. Explicit representations like meshes are efficient to render in a differentiable fashion but have difficulties handling topology…
A rigorous mathematical theory of dimensional analysis, systematically accounting for the use of physical quantities in science and engineering, perhaps surprisingly, was not developed until relatively recently. We claim that this has…
In this article, we develop an algebraic framework of axioms which abstracts various high-level properties of multi-qudit representations of generalized Clifford algebras. We further construct an explicit model and prove that it satisfies…
This note describes a representation of the real numbers due to Schanuel. The representation lets us construct the real numbers from first principles. Like the well-known construction of the real numbers using Dedekind cuts, the idea is…