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Roy Burson

Publications and source records attributed to Roy Burson.

6 recordsLinked to original sources
arXiv2025

The sliding tile puzzle, roots to polynomials, and $\textbf{P}$ vs. $\textbf{NP}$ complexity

This work explores the relationship between solution space and time complexity in the context of the $\textbf{P}$ vs. $\textbf{NP}$ problem, particularly through the lens of the sliding tile puzzle and root finding algorithms. We focus on the trade-off between finding a solution and verifying it, highlighting how understanding the structure of the solution space can inform the complexity of these problems. By examining the relationship between the number of possible configurations and the time complexity required to traverse this space we demonstrate that the minimal time to verify a solution is often smaller than the time required to discover it. Our results suggest that the efficiency of solving $\textbf{NP}$-complete problems is not only determined by the ability to find solutions but also by how effectively we can navigate and characterize the solution space. This study contributes to the ongoing discourse on computational complexity, particularly in understanding the interplay between solution space size, algorithm design, and the inherent challenges of finding versus verifying solutions.

math.GM↗
arXiv2023

The Navier-Stokes Equation and Helmholtz Decomposition

This work explores Navier-Stokes equation with no gravitational forces. In short, it shows that any smooth solution that decays quickly must take the form $$ \textbf{u}(x,t)- \dfrac{1}{4\pi}\textbf{Curl}\Biggl( \int_{\mathbb{R}^3}^{}{\dfrac{\textbf{Curl} (\textbf{u} (x^\prime,t))}{|x-x^\prime|}}dV^\prime\Biggr) = -\int_{0}^{t}{\dfrac{1}{\rho} \textbf{Grad}\big(\Gamma(x,s)\big)}ds.$$ Consequently, any curl free solution must be written as $$\textbf{u}(x,t) = -\dfrac{1}{\rho} \textbf{Grad}\biggl(\int_{0}^{t}{\Gamma(x,s) ds}\biggr)$$ with $\Gamma$ a known function which is related to the heat equation. Even further it shows if there exist a value $k\in \mathbb{N}$ such that $$\textbf{curl}^k\biggl((\textbf{u}\cdot \nabla )\textbf{u}\biggr)(x,t)=\textbf{0}$$ for all $t^\prime\le t$ then $$\textbf{u}(x,t) = \textbf{H}^{k+1}(\xi_1,\xi_2,\xi_3,t) -\int_{0}^{t}{\dfrac{1}{\rho} \textbf{Grad}\big(\Gamma(x,s)\big)}ds, ~~~~~ t\in [t^\prime,\infty)$$ with $$\xi_i(x,t):= \int_{\mathbb{R}^3}^{}{\alpha(x-y,\dfrac{t}{\nu})v^k_i(x,0)}dy, ~~~~~ v^k_i(x,0) = \biggl(\textbf{curl}^k(\textbf{u}(x,0))\biggr)_i, ~~~~~ 1\le i\le 3$$ and $\textbf{H}^k$ the $k^{th}$ application of Helmholtz operator. Hence, if there is another solution where the non-linear term is infinitly curlable then the solution is not unique. If the solution is unique, then this is the only possible solution.

math.GM↗
arXiv2020

The Mobius Function and Congruent Numbers

This work provides a complete characterization of congruent numbers in terms of Pythagorean triples. Specifically, we show that every congruent number can be written as $$\frac{nm\left(m-n\right)\left(m+n\right)}{\sigma^2}$$ were as $$\sigma \vert \rho\biggl(\left(m-n\right)\left(m+n\right)\biggr),\indent \text{or} \indent \sigma \vert \rho( nm )$$ were $\rho(\alpha)$ denotes the non-square free part of its argument $\alpha$. As a consequence, in order to find congruent numbers it suffices to devise a condition so that the equality $\mu(m-n)+1 = \gcd(m,n)$ or $\mu(m+n)+1 =\gcd(m,n)$ holds, were $\mu$ is the Mobius function.

math.GM↗
arXiv2020

On the Distribution of the First Point of Coalescence for some Collatz Trajectories

This paper is a numerical evaluation of some trajectories of the Collatz function. Specifically, I assess the coalescence points of each integer $n\equiv 0 (\bmod{2})$ and $n\equiv 2(\bmod{3})$ through a sophisticated algorithm that has been developed to test on any different modulus classes. The data discovered illustrate that the distribution of the first point of coalescence is closely related to the solutions of some exponential diophantine equation. Afterwards, I show that the first point of coalescence of the integers $n$ and $3n+2$ appear to tend to an expected value of $4/5n$. When the algorithm was pushed to its peak estimation it has been discovered that the expected value begins to deviate from the initial estimation of $4/5n$. The first point of coalescence of the integers $n$ and $3n+2$ appear eradicate from a "step by step" point of view but from a topological point of view seem to be localized around the diophantine solution of some particular functions.

math.GM↗
arXiv2019

Integer Representations and Trajectories of the 3x+1 Problem

This paper studies certain trajectories of the Collatz function. I show that if for each odd number $n$, $n\sim 3n+2$ then every positive integer $n \in \mathbb{N}\setminus 2^{\mathbb{N}}$ has the representation $$n=\left(2^{a_{k+1}}-\sum_{i=0}^{k}{2^{a_i}3^{k-i}}\right)/ 3^{k+1}$$ where $0\le a_0 \le a_1 \le \cdot \cdot \cdot \le a_{k+1}$. As a consequence, in order to prove Collatz Conjecture I illustrate that it is sufficient to prove $n\sim 3n+2$ for any odd $n\in \mathbb{N}\setminus 2^{\mathbb{N}} $. This is the main result of the paper.

math.HO↗