Related papers: Truly Hypercomplex Numbers: Unification of Numbers…
A new kind of numbers called Hyper Space Complex Numbers and its algebras are defined and proved. It is with good properties as the classic Complex Numbers, such as expressed in coordinates, triangular and exponent forms and following the…
Let $X$ be a finite set in a complex sphere of $d$ dimension. Let $D(X)$ be the set of usual inner products of two distinct vectors in $X$. A set $X$ is called a complex spherical $s$-code if the cardinality of $D(X)$ is $s$ and $D(X)$…
A system of commutative hypercomplex numbers of the form w=x+hy+kz are introduced in 3 dimensions, the variables x, y and z being real numbers. The multiplication rules for the complex units h, k are h^2=k, k^2=h, hk=1. The operations of…
There has been always an ambiguity in division when zero is present in the denominator. So far this ambiguity has been neglected by assuming that division by zero as a non-allowed operation. In this paper, I have derived the new set of…
This monograph presents a detailed analysis of hypercomplex numbers in 2, 3 and 4 dimensions, then presents the properties of hypercomplex numbers in 5 and 6 dimensions. It continues with a detailed analysis of hypercomplex numbers in n…
In this paper, we present a new hypercomplex number system, Trinition, that has an unusual structure of commutativity, noncommutativity, nonassociativity, and deformability.
Hypercomplex numbers are unital algebras over the real numbers. We offer a short demonstration of the practical value of hypercomplex analytic functions in the field of partial differential equations.
We suggest new types and interpretation of complex and hypercomplex numbers for which the commutative, associative, and distributive laws and the norm axioms are trivially satisfied.
Commutative complex numbers of the form u=x+\alpha y+\beta z+\gamma t in 4 dimensions are studied, the variables x, y, z and t being real numbers. Four distinct types of multiplication rules for the complex bases \alpha, \beta and \gamma…
The method of obtaining the set of noncanonical hypercomplex number systems by conversion of infinite hypercomplex number system to finite hypercomplex number system depending on multiplication rules and factorization method is described.…
Near-vector spaces extend linear algebra tools to non-linear algebraic structures, enabling the study of non-linear problems. However, explicit constructions remain rare. This paper introduces a broad computable family of near-vector…
Supersymmetry is deeply related to division algebras. Nonabelian Yang-Mills fields minimally coupled to massless spinors are supersymmetric if and only if the dimension of spacetime is 3, 4, 6 or 10. The same is true for the Green-Schwarz…
Theories of classification distinguish classes with some good structure theorem from those for which none is possible. Some classes (dense linear orders, for instance) are non-classifiable in general, but are classifiable when we consider…
Any non-split complex supermanifold is a deformation of a split supermanifold. These deformations are classified by group orbits in a non-abelian cohomology. For the case of a split supermanifold with no global nilpotent even vector fields,…
A Hilbert cube of dimension $d$ is the set of integers \[ H(a_{0}; a_{1}, \ldots, a_{d})=a_{0}+\{0, a_{1}\}+\cdots+\{0, a_{d}\}=\left\{a_{0}+\sum_{i=1}^{d}\varepsilon_{i}a_{i}:\;\varepsilon_{i}\in\{0,1\}\right\}. \] Brown, Erd\H{o}s and…
We undertake to develop a successful framework for commutative-associative hypercomplex numbers with the view to explicate and study associated geometric and generalized-relativistic concepts, basing on an interesting possibility to…
Literature considers under the name \emph{unimaginable numbers} any positive integer going beyond any physical application, with this being more of a vague description of what we are talking about rather than an actual mathematical…
Is it possible to define, for certain values n the product of vectors of the real vector space of n dimensions, such that this is, with respect to multiplication and the ordinary addition of vectors, a numerical system which contains the…
We give a new complexity bound for calculating the complex dimension of an algebraic set. Our algorithm is completely deterministic and approaches the best recent randomized complexity bounds. We also present some new, significantly sharper…
We propose the extension of the complex numbers to be the new domain where new concepts, like negative and imaginary probabilities, can be defined. The unit of the new space is defined as the solution of the unsolvable equation in the…