Roots of Elliptic Scator Numbers

The Victoria equation, a generalization of De Moivre’s formula in 1+n dimensional scator algebra, is inverted to obtain the roots of a scator. For the qth root in S1+n of a real or a scator number, there are qn possible roots. For n=1, the usual q complex roots are obtained with their concomitant cyclotomic geometric interpretation. For n≥2, in addition to the previous roots, new families arise. These roots are grouped according to two criteria: sets satisfying Abelian group properties under multiplication and sets catalogued according to director conjugation. The geometric interpretation is illustrated with the roots of unity in S1+2.

Powers of Elliptic Scator Numbers

Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

6. and beyond, to complex things

‘ … and beyond, to complex things’ first considers the Taylor series for the exponential function. One of the most famous, yet enigmatic, numbers in mathematics, e is an irrational number equal to 2.718281828. … Exponential functions deal with the phenomena of growth and decay. As calculus was starting to become established, curious parallels between the apparently disparate worlds of trigonometry and exponential functions were starting to appear. Imaginary numbers, Euler’s formula, and Euler’s identity are discussed along with the Argand diagram, De Moivre’s formula, hyperbolic trigonometric functions, and the catenary curve. Imaginary numbers are now at the heart of science and technology, and are used in the study of electromagnetic waves, cellular and wireless technologies, and fluid dynamics.

Instability of Viscoelastic Fluids in Blasius Flow

In this paper, we study the hydrodynamic stability of a viscoelastic Walters B liquid in the Blasius flow. A linearized stability analysis is used and orthogonal polynomials which are related to de Moivre’s formula are implemented to solve Orr–Sommerfeld eigenvalue equation. An analytical approach is used in order to find the conditions of instability for Blasius flow and Critical Reynolds number is found for various combinations of the elasticity number. Based on the results, the destabilizing effect of elasticity on Blasius flow is determined and interpreted.