Related papers: Large annihilators in Cayley-Dickson algebras II
We show how eigentheory clarifies many algebraic properties of Cayley-Dickson algebras. These notes are intended as background material for those who are studying this eigentheory more closely.
The concept of a sequence of exact zero-divisors on a noetherian local ring is defined and studied. Some properties of sequences of exact zero-divisors are compared with regular sequences.
Differential central simple algebras are the main object of study in this survey article. We recall some crucial notions such as differential subfields, differential splitting fields, tensor products etc. Our main focus is on differential…
Zero-divisors (ZDs) derived by Cayley-Dickson Process (CDP) from N-dimensional hypercomplex numbers (N a power of 2, at least 4) can represent singularities and, as N approaches infinite, fractals -- and thereby,scale-free networks. Any…
We investigate the average order of the divisor function at values of totally reducible binary cubic forms and discuss some applications.
We give the definition of a dg-division algebra, that is a concept of a differential graded algebra which may serve as an analogue of a division algebra. We classify them completely, and show that they are either acyclic or have…
Basic arithmetic is the cornerstone of mathematics and computer sciences. In arithmetic, 'division by zero' is an undefined operation and any attempt at extending logic for algebraic division to incorporate division by zero has resulted in…
In this paper we concern with positive zero divisors in $C^{*}$ algebras. By means of zero divisors, we introduce a hereditary invariant for $C^{*}$ algebras. Using this invariant, we give an example of a $C^{*}$ algebra $A$ and a $C^{*}$…
{\small \ We generalize the concepts of \thinspace level \thinspace and \thinspace sublevel of a composition algebra to division algebras obtained by the Cayley-Dickson process. In 1967, R. B. Brown constructed, for every}$t\in…
An algorithm for irreducible decomposition of representations of finite groups over fields of characteristic zero is described. The algorithm uses the fact that the decomposition induces a partition of the invariant inner product into a…
In this work we are interested in the general problem of the determination of the normed division algebras. Our fundamental results are obtained in the particular subclass of those 8-dimensional quadratic flexible real division algebras. We…
We use an embedding of the symmetric $d$th power of any algebraic curve $C$ of genus $g$ into a Grassmannian space to give algorithms for working with divisors on $C$, using only linear algebra in vector spaces of dimension $O(g)$, and…
We classify good Z-gradings of basic Lie superalgebras over an algebraically closed field of characteristic zero. Good Z-gradings are used in quantum Hamiltonian reduction for affine Lie superalgebras, where they play a role in the…
Results of research of possibility of transformation of a difference equation into a system of the first-order difference equation are presented. In contrast to the method used previously, an unknown grid function is split into two new…
This is an introduction to small divisors problems. The material treated in this book was brought together for a PhD course I tought at the University of Pisa in the spring of 1999. Here is a Table of Contents: Part I One Dimensional Small…
Some features of Cayley algebras (or algebras of octonions) and their Lie algebras of derivations over fields of low characteristic are presented. More specifically, over fields of characteristic $7$, explicit embeddings of any twisted form…
We define a special matrix multiplication among a special subset of $2N\x 2N$ matrices, and study the resulting (non-associative) algebras and their subalgebras. We derive the conditions under which these algebras become alternative…
This paper establishes a version of Nevanlinna theory based on Askey-Wilson divided difference operator for meromorphic functions of finite logarithmic order in the complex plane $\mathbb{C}$. A second main theorem that we have derived…
We present a general method for computing discriminants of noncommutative algebras. It builds a connection with Poisson geometry and expresses the discriminants as products of Poisson primes. The method is applicable to algebras obtained by…
A characteristic feature of differential-algebraic equations is that one needs to find derivatives of some of their equations with respect to time, as part of so called index reduction or regularisation, to prepare them for numerical…