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The goal of this paper is to experiment new math concepts and theories, especially if they run counter to the classical ones. To prove that contradiction is not a catastrophe, and to learn to handle it in an (un)usual way. To transform the…
We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric…
The article is devoted to the investigation of properties of quasi-invariant measures with values in non-Archimedean fields such as: convolutions of measures and functions; continuity of functions of measures; non-associative noncommutative…
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…
In this article, we briefly describe various tools and approaches that algebraic geometry has to offer to straighten bent objects. Throughout this article we will consider a specific example of a bent or curved piece of paper which in our…
This paper defines and examines the basic properties of noncommutative analogues of almost complex structures, integrable almost complex structures, holomorphic curvature, cohomology, and holomorphic sheaves. The starting point is a…
We explore inflectional morphology as an example of the relationship of the discrete and the continuous in linguistics. The grammar requests a form of a lexeme by specifying a set of feature values, which corresponds to a corner M of a…
We introduce a method to derive theorems from Elementary Number Theory by means of relationships among formal languages. Using $\sigma$-algebras, we define what a proof of a number-theoretical statement by Language Theory means. We prove…
These lecture notes provide an informal introduction to the theory of nonnegative polynomials and sums of squares. We highlight the history and some recent developments, especially the new connections with classical (complex) algebraic…
We study locally finite varieties (=primitive classes) of linear algebras over finite fields. We do not assume that our algebras are associative or Lie. We are interested in the basic properties of finite algebras in these varieties such…
We introduce an asymptotic notion of positivity in algebraic geometry that turns out to be related to some high-dimensional convex sets. The dimension of the convex sets grows with the number of birational operations. In the case of complex…
We define certain arithmetic derivatives on $\mathbb{Z}$ that respect the Leibniz rule, are additive for a chosen equation $a+b=c$, and satisfy a suitable non-degeneracy condition. Using Geometry of Numbers, we unconditionally show their…
The purpose of this article is to introduce projective geometry over composition algebras : the equivalent of projective spaces and Grassmannians over them are defined. It will follow from this definition that the projective spaces are in…
The article presents a new approach to euclidean plane geometry based on projective geometric algebra (PGA). It is designed for anyone with an interest in plane geometry, or who wishes to familiarize themselves with PGA. After a brief…
Graphical (Linear) Algebra is a family of diagrammatic languages allowing to reason about different kinds of subsets of vector spaces compositionally. It has been used to model various application domains, from signal-flow graphs to Petri…
We define the notion of an almost polynomial identity of an associative algebra $R$, and show that its existence implies the existence of an actual polynomial identity of $R$. A similar result is also obtained for Lie algebras and Jordan…
In the paper "Is there a Jordan geometry underlying quantum physics?" (Int. J. Theor. Phys., to appear; arXiv:0801.3069 [math-ph]), generalized projective geometries have been proposed as a framework for a geometric formulation of Quantum…
In this note we introduce a new technique to answer an issue posed in [7] concerning geometric properties of the set of non-surjective linear operators. We also extend and improve a related result from the same paper.
Geometric algebra is an optimal frame work for calculating with vectors. The geometric algebra of a space includes elements that represent all the its subspaces (lines, planes, volumes, ...). Conformal geometric algebra expands this…
These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new…