Light Ray Morphisms of the Fractal Antenna
Creators: name: Emmerson, Parker
Publisher: name: Zenodo
Year: 2023
Subjects: maximum signal strength
optimal geometry
distance metrics
information
transmit
network
features
geometric
scaling
assigned
relative
compare
subregion
cotangent space
constructing
associated
compute
application
manipulating
structure
algebra
calculus
light ray
energy
geometry
optimal
approximation
Light ray
problem
region
triangle inequality
metric
subspace
system
expression
light ray paths
decomposition
optimal solution
problem solving
extended Euclidean space
triangular inequality
path
fractal antennae algebra
light ray calculus
metrics
recursive product
form-preserving homomorphism
relative rotation
topological space
scaling factor
real numbers
vector space
neighborhood
radius
sweep subnet
energy number
fractal antenna
morphism

description: Finally, we can represent the primal energy number expression of the fractal morphism in the geometric context of the sweep subnet associated with the cotangent space. If $X_i$ and $\vec{\phi}$ represent two points in $\mathcal{E}_{\text{mapping}}$ and $r$ is the radius of a neighborhood in $\mathbb{R}^n$, then the total distance between them can be calculated as: \begin{align} &d_{\mathcal{DS}}(X_i, \vec{\phi}) \nonumber \\ &= \left< \mathcal{S}_r, \pi_{A_r}(X_i) \right> + \left< \mathcal{S}_r, \vec{\phi} \right> \nonumber \\ &+ \left< \mathcal{S}_t, \pi_{\theta_{\infty}}(X_i) \right> + \left< \mathcal{S}_t, \delta\theta \right> + \left< \mathcal{S}_t, \vec{\phi} \right> \nonumber \\ &+ \left< A_r \oplus B_r, \pi_{A_r}(X_i) \right> + \left< A_r \oplus B_r, \delta\vec{x} \right> + \left< A_r \oplus B_r, \vec{\phi} \right> \nonumber \\ &+ \left< \theta_{\infty} \oplus B_t, \pi_{\theta_{\infty}}(X_i) \right> + \left< \theta_{\infty} \oplus B_t, \delta\theta \right> + \left< \theta_{\infty} \oplus B_t, \vec{\phi} \right>, \label{eq:DSDistanceGeo} \end{align} where the metric is given by: \begin{equation} d_{\mathcal{DS}}^2(X_i, \vec{\phi}) = \frac{\pi_m^2(v)}{\sum_{f\cdot g} h(v)} \cdot \left(\left(\frac{1}{n - l \tilde{\star} \mathcal{R}}\right)\otimes{\prod_{\Lambda}h} - \cos\psi\diamond\theta \leftrightarrow \stackrel{\mathcal{A}\mathcal{B}\mathcal{C}}{\mathfrak{F}} \right)^2 \end{equation} and $\delta\vec{x}$ and $\delta\theta$ are the components of the lightlike curve connecting $\pi_{A_r}(X_i)$ and $\vec{\phi}$. This expression captures the interaction between the elements of the topological space, the relative rotation of the components of the forms in the vector space $\mathbb{V}$ and the real numbers $\mathbb{U}$, and the scaling factor $\prod_{\Lambda} h$. From the above expression for the primal energy number, we can derive a fractal antennae algebra that employs the fractal morphism $\mathfrak{F}$. The algebra is given as \begin{align} A &= \Omega_{\Lambda'}\left( \sin \theta \star \sum_{[n]\star[l] \rightarrow \infty} \left( \frac{b^{\mu-\zeta}}{\sqrt[m]{n^m - l^m}} \otimes \prod_{\Lambda} h \right) + \cos \psi \diamond \theta \right) \\ &= \Omega_{\Lambda'}\left( \sum_{[n]\star[l] \rightarrow \infty} \left( \mathfrak{F}(x) \otimes \prod_{\Lambda} h \right) + \cos \psi \diamond \theta \right) \\ &\stackrel{\mathfrak{F}}{=} \Omega_{\Lambda'}\left( \sum_{[n]\star[l] \rightarrow \infty} \left( \Omega_{\Lambda}\circ \mathfrak{F}(x) \otimes \prod_{\Lambda} h \right) + \cos \psi \diamond \theta \right). \end{align} By using the fractal morphism $\mathfrak{F}$, we can decompose the original expression into a fractal antennae algebra. This approach allows us to gain a more accurate approximation of the solution to any given problem. The definition of a fractal antenna algebra is as follows: "A fractal antenna algebra is an algebraic structure for representing and manipulating the path of a light ray in a given space. The information is collected through an application of the fractal morphism to the system(s) present and can be used to compute the energy associated with a light ray's path."