#### MATHEMATICS - THE PRIME MOVER

### ABSTRACT

Using new insights into extending Euler’s Pi function to encompass negative real numbers and complex numbers we take an imaginary series expansion approach to derive the constant e and the complex exponential form. - VIATOR

### ABSTRACT

Rotations thru imaginary angles represent a very clean and concise means to link Euclidean rotations with that of space-time. Starting with Euler’s formula we are easily able to rotate thru an imaginary angle and see that the Lorentz boosts seamlessly appear in the form of the hyperbolic functions. - VIATOR

### ABSTRACT

Non-Euclidean geometry is a fascinating subject matter especially in regards to applications in the calculus of variations. We will shown that in hyperbolic geometry, the shortest distance from one point to another is not a straight line as in Euclidean space, but that the straight line is the maximum distance. -VIATOR

### ABSTRACT

In 1963 Lorenz published his seminal paper Deterministic Non-periodic flow in the Journal of Atmospheric Sciences. The philosophical ramifications of the unpredictability of phenomenon in nature noted in this work were profound and the implications have fueled an incredible development in dynamical systems. In this paper, we explore this system and its enigmatic strange attractor, by looking into the dynamics of the Lorenz equations, defining its chaotic attributes thru both an analytic and visual approach, and ultimately showing that the Lorenz system does indeed support the existence of this strange attractor.

-VIATOR

### ABSTRACT

In this paper, we explore the relationship between higher dimensional complex forms, combinatorics and multinomial theorem. We look in particular at the applications of the trinomial theorem and the relationship between complex trigonometric infinite series expansions, combinatorics and geometrics insights in complex analysis and the higher dimensions.

-VIATOR

### ABSTRACT

We present the derivation of the 6-dimensional Eulerian Lie group of the form SO(3,C). We describe our derivation process, which involves the creation of a finite group by using permutation matrices, and the exponentiation of the adjoint form of the subset representing the generators of the finite group. We take clues from the 2-dimensional complex rotation matrix to present, what we believe, is a true representation of the Lie group for the six-dimensional complex unit sphere and proceed to study its dynamics. With this approach, we can proceed to present this SO(3,C) group and derive its unitary counterpart that is U(3). The following findings can prove useful in mathematical physics, complex analysis and applications in deriving higher dimensional forms of similar division algebras.

-VIATOR