PROPERTIES OF REGULAR FUNCTIONS OF A QUATERNION VARIABLE MODIFIED WITH TRI-COMPLEX QUATERNION

In a quaternion structure composed of four real dimensions, we derive a form wherein three complex numbers are combined. Thereafter, we examined whether this form includes the algebraic properties of complex numbers and whether transformations were necessary for its application to the system. In addition, we defined a regular function in quaternions, expressed as a combination of complex numbers. Furthermore, we derived the Cauchy-Riemann equation to investigate the properties of the regular function in the quaternions coupled with the complex number.

Hypothesis of the Hidden Multiverse Explains Dark Matter and Dark Energy

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).