N-Dimensional Commutative Hypercomplex Numbers

2008 ◽  
pp. 5-17
Keyword(s):  
Hypercomplex Numbers

scholarly journals Hypothesis of the Hidden Multiverse Explains Dark Matter and Dark Energy

2017 ◽  
Vol 9 (2) ◽  
pp. 30 ◽  
Author(s):  
Alexander Alexandrovich Antonov
Keyword(s):  
Dark Matter ◽  
Dark Energy ◽  
Special Theory ◽  
Physical Reality ◽  
Theory Of Relativity ◽  
Special Theory Of Relativity ◽  
Hypercomplex Numbers ◽  
Distant Future ◽  
Parallel Universes ◽  
Quaternion Structure

There are currently a large number of Multiverse hypotheses, which are, however, non-verifiable, i.e. they can be neither confirmed nor refuted experimentally even in the distant future. In contrast, the hypothesis of the hidden Multiverse considered in the article is verifiable and therefore has a right to be called a theory. The theory uses the principle of physical reality of imaginary numbers discovered 500 years ago, including complex and hypercomplex numbers, as fundamental and proved by the author theoretically and experimentally. This principle has allowed revealing a number of serious mistakes in the special theory of relativity. An adjusted version of the special theory of relativity has been proposed and the theory of the hidden Multiverse has been developed on its basis. The Multiverse has been referred to as hidden, because parallel universes it contains are mutually invisible. The nature of their invisibility is explained in the article. It is shown that dark matter and dark energy are other universes of the hidden Multiverse apart from ours. Analysis of data from WMAP and Planck spacecrafts has shown that the hidden Multiverse has quaternion structure comprising four pairs of universes and antiverses (i.e., four pairs of matter and antimatter).

On a Theorem in Hypercomplex Numbers

1906 ◽  
Vol 26 (1) ◽  
pp. 48-50 ◽  
Author(s):  
J. H. Maclagan-Wedderburn
Keyword(s):  
Quaternion Algebra ◽  
Hypercomplex Numbers

Scheffers in the Mathematische Annalen, vol. xxxix., pp. 364–74, enunciates the following theorem:—If A is an algebra containing the quaternion algebra B as a subalgebra, and if A and B have the same modulus, A can be expressed in the form B C = A = C B, where C is a subalgebra of A every element of which is commutative with every element of B: in other words, if i1, i2, i3, i4 is a basis of B, it is possible to find an algebra C with the basis e1, e2, … ec, such that each of its elements is commutative with every element of B, and such that the elements eris(r = 1, 2, … c, S = 1, … 4) form a basis of A; and if a is the order of A, then a = 4c.

The Genetic Coding, United-Hypercomplex Numbers and Artificial Intelligence

2017 ◽  
pp. 2-13 ◽  
Author(s):  
Sergey Petoukhov ◽  
Elena Petukhova ◽  
Ludmila Hazina ◽  
Ivan Stepanyan ◽  
Vitaliy Svirin ◽  
...  
Keyword(s):  
Artificial Intelligence ◽  
Hypercomplex Numbers ◽  
Genetic Coding

Multidimensional Numbers and the Genomatrices of Hydrogen Bonds

2010 ◽  
pp. 193-206
Author(s):  
Sergey Petoukhov ◽  
Matthew He
Keyword(s):  
Hydrogen Bonds ◽  
Genetic Code ◽  
Structural Features ◽  
Hyperbolic Type ◽  
Complex Numbers ◽  
Hypercomplex Numbers ◽  
Molecular Systems ◽  
Natural Systems ◽  
Double Numbers ◽  
Mathematical Properties

This chapter returns to the kind of numeric genetic matrices, which were considered in Chapter 4-6. This kind of genomatrices is not connected with the degeneracy of the genetic code directly, but it is related to some other structural features of the genetic code systems. The connection of the Kronecker families of such genomatrices with special categories of hypercomplex numbers and with their algebras is demonstrated. Hypercomplex numbers of these two categories are named “matrions of a hyperbolic type” and “matrions of a circular type.” These hypercomplex numbers are a generalization of complex numbers and double numbers. Mathematical properties of these additional categories of algebras are presented. A possible meaning and possible applications of these hypercomplex numbers are discussed. The investigation of these hyperbolic numbers in their connection with the parameters of molecular systems of the genetic code can be considered as a continuation of the Pythagorean approach to understanding natural systems.

scholarly journals Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks

Mathematics ◽  
2021 ◽  
Vol 9 (9) ◽  
pp. 938
Author(s):  
Jeremiah Bill ◽  
Lance Champagne ◽  
Bruce Cox ◽  
Trevor Bihl
Keyword(s):  
Neural Networks ◽  
Optimization Methods ◽  
Network Size ◽  
High Dimensional ◽  
Proof Of Concept ◽  
Activation Functions ◽  
Hypercomplex Numbers ◽  
Neural Network Structure ◽  
Future Work ◽  
High Dimensional Datasets

In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued equivalents. Finally, this work introduces a novel training algorithm using a meta-heuristic approach that bypasses the need for analytic quaternion loss or activation functions. This algorithm allows for a broader range of activation functions over current quaternion networks and presents a proof-of-concept for future work.

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