Related papers: Pentagram Spirals
We consider relationships between cubic algebras and implication algebras. We first exhibit a functorial construction of a cubic algebra from an implication algebra. Then we consider an collapse of a cubic algebra to an implication algebra…
We start from the author's metric for a rotating and accelerating Universe and we calculate the force acting on a particle in it. This force consists of two parts: one is derived from a potential, and the other describes a Coriolis force…
Using an equation of motion for a self-gravitating filament, we show how galactic spiral arms might be created and sustained. We find that the combination of differential rotation of the galactic disk and the self-gravity of the arm (as…
We discuss certain ternary algebraic structures appearing more or less naturally in various domains of theoretical and mathematical physics. Far from being exhaustive, this article is intended above all to draw attention to these algebras,…
The pentagram map is a projectively natural iteration defined on polygons, and also on a generalized notion of a polygon which we call {\it twisted polygons}. In this note we describe our recent work on the pentagram map, in which we find a…
By natural way the hierarchy structure is introduced on directed graphs with weighted adjacencies. Embedded system of algebras of subsets of the set of vertices of such digraph and it's consolidations, which vertices are the elementary sets…
We construct a geometric system from which the Hall algebra can be recovered. This system inherently satisfies higher associativity conditions and thus leads to a categorification of the Hall algebra. We then suggest how to use this…
This paper considers a simple geometric construction, called the Pentagram map. The pentagram map, performed on N-gons, gives rise to a birational mapping on the space of all N-gons. This paper finds what conjecturally are all the…
We summarize the Hopf algebra structure on Feynman diagrams and emphasize the interest in further algebraic structures hidden in Feynman graphs.
Revisiting the old problem of existence of interacting models of QFT with new conceptual ideas and mathematical tools, one arrives at a novel view about the nature of QFT. The recent success of algebraic methods in establishing the…
This paper examines the concept of gluing, placing it within its most general categorical context and tracing its foundational role in the broader architecture of algebraic geometry.
Paravectors just like integers have a ring structure. By introducing an integrated product we get geometric properties which make paravectors similar to vectors. The concepts of parallelism, perpendicularity and the angle are conceptually…
We show that a structural matrix algebra $A$ is isomorphic to the endomorphism algebra of an algebraic-combinatorial object called a generalized flag. If the flag is equipped with a group grading, an algebra grading is induced on $A$. We…
We discuss the construction of oscillator-like systems associated with orthogonal polynomials on the example of the Fibonacci oscillator. In addition, we consider the dimension of the corresponding lie algebras.
Let X be a scroll over a rational surface. We construct a linear system of surfaces in P^3 yielding a birational map from P^3 to X. We apply this construction to the scrolls of Bordiga and Palatini.
We introduce a geometric dynamical system where iteration is defined as a cycling composition of different maps acting on a space composed of three or more lines in $\mathbb{R}^2$. This system is motivated by the dynamics of iterated…
The pentagram map, introduced by Schwartz in 1992, is a dynamical system on the moduli space of polygons in the projective plane. Its real and complex dynamics have been explored in detail. We study the pentagram map over an arbitrary…
Via the transverse Hilbert scheme construction, we associate a holomorphic completely integrable system to a surface $S$ endowed with a holomorphic symplectic form $\omega$ and a projection onto $\mathbb{C}$. We provide a full…
A projective rectangle is like a projective plane that has different lengths in two directions. We develop the basic theory of projective rectangles including incidence properties, projective subplanes, configuration counts, a partial…
We associate each endomorphism of a finite cyclic group with a digraph and study many properties of this digraph, including its adjacent matrix and automorphism group.