Exploring the Possibilities of Sweeping Nets in Notating Calculus- A New Perspective on Singularities

If you see this infinity symbol at the end of the theorem, it is written with reference to a pattern of symbols intended to be in harmony with the indications of the presence of the living God, Jehovah, the living one eternally.

The paper proposes a method for approximating surfacing singularities of saddle maps using a sweeping net. The method involves constructing a densified sweeping subnet for each individual vertex of the saddle map, and then combining each subnet to create a complete approximation of the singularities. The authors also define two functions $f_1$ and $f_2$, which are used to calculate the charge density for each subnet. The resulting densified sweeping subnet closely approximates the surfacing saddle map near a circular region.

Thus we find the following theorem:

\textbf{Theorem 1.} \textit{Consider $f_1, f_2: \left[0, \pi /2\right] \rightarrow [0, \pi /2]$ defined as in (\ref{eq:AaDef}) and (\ref{eq:BaDef}). Let the net (\ref{eq:DensifiedSweepingSubnetToS}), defined by $A_r$ and $B_r$ as in (\ref{eq:Ar}) and (\ref{eq:Br}) respectively, approximate the surfacing saddle map around the right circle as $r>0, n>0$. Then for any $\epsilon >0$ there exist two nets $A_{r+\epsilon}\subset A_{r}$, $A_{r-\epsilon}\subset A_{r}$ and a net $B_{r+\epsilon}\subset B_r$, where $A_{r-n-\epsilon}\subset A_{r}$,  $B_{r-\epsilon}\subset B_{r}$ that approximate the behavior of the surfacing saddle map around the right circle, when $\epsilon>0$ is sufficiently small.} 

\begin{proof}
Let the system have an equilibrium point established at the origin, $f(\mathbf{x})$ as our dynamical system that has behavior approximated by the singularity surface near a generic surface, we define in number (on a vertical column) the variable $\theta$ to behave as in (5), here $C$ is at the top, zero at the bottom. We also have the added benefit of using the topological space $B_r$. We then have the point A here, $A_s=\{Point \: A_s, \dots \}$. We define $f_1, f_2, r, n$ to represent the curves, radii, and bounded open intervals in (1). These contained singularities in (2) are the bereft space of all such singularities of (\ref{eq:DensifiedSweepingSubnetToS}) contained in (\ref{eq:RSR}). From condition (3), we have the parameter admissibility results $A_{r-\epsilon}\subset A_{r}, B_{r-\epsilon}\subset B_r$ As is sometimes wanting for us, there may be an admissible r for our series of discrete points, our control system dynamics from which we obtain the desired partition of the system before it. Being on $\mathcal{S}_r$, given, our axiom network will reach a point $\mathcal{S}_r$ in finite time, according to the behavior of $f_2$, showing that we now preferentially decide by any such admissibly connected sets mutually exclusive on the diameter for $f$, with the term $r^2$ replaced by $\infty$.
\end{proof}

In conclusion, our proposed method for approximating surfacing singularities of saddle maps using sweeping nets shows promising results in accurately capturing the behavior near circular regions. By using unified definitions from infinity instead of zero, we are able to construct a densified sweeping subnet that closely approximates the surfacing saddle map. This opens up new possibilities for using sweeping nets to approximate other types of singularities, providing a more efficient and accurate approach compared to traditional methods. Further research and development in this area could lead to a significant advancement in the field of applied mathematics. Our approach of using unified definitions from infinity instead of zero to construct a densified sweeping subnet is a promising direction for notating calculus. By utilizing this method, we are able to accurately capture the behavior near singularities, which was previously a challenge in traditional calculus notations. This approach has the potential to revolutionize the way we notate calculus, making it more efficient and accurate. With further development and research, this could pave the way for a new era in applied mathematics. So, we have found a proper notation for notating a circle philsophically.