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Craig Alan Feinstein

Publications and source records attributed to Craig Alan Feinstein.

11 recordsLinked to original sources
arXiv2019

A mathematical definition of "simplify"

Even though every mathematician knows intuitively what it means to "simplify" a mathematical expression, there is still no universally accepted rigorous mathematical definition of "simplify". In this paper, we shall give a simple and plausible definition of "simplify" in terms of the computational complexity of integer functions. We shall also use this definition to show that there is no deterministic and exact algorithm which can compute the permanent of an $n \times n$ matrix in $o(2^n)$ time.

cs.CC↗
arXiv2016

Dialogue Concerning The Two Chief World Views

In 1632, Galileo Galilei wrote a book called \textit{Dialogue Concerning the Two Chief World Systems} which compared the new Copernican model of the universe with the old Ptolemaic model. His book took the form of a dialogue between three philosophers, Salviati, a proponent of the Copernican model, Simplicio, a proponent of the Ptolemaic model, and Sagredo, who was initially open-minded and neutral. In this paper, I am going to use Galileo's idea to present a dialogue between three modern philosophers, Mr. Spock, a proponent of the view that $\mathsf{P} \neq \mathsf{NP}$, Professor Simpson, a proponent of the view that $\mathsf{P} = \mathsf{NP}$, and Judge Wapner, who is initially open-minded and neutral.

cs.GL↗
arXiv2014

Trapping Regions for the Navier-Stokes Equations

In 1999, J.C. Mattingly and Ya. G. Sinai used elementary methods to prove the existence and uniqueness of smooth solutions to the 2D Navier-Stokes equations with periodic boundary conditions. And they were almost successful in proving the existence and uniqueness of smooth solutions to the 3D Navier-Stokes equations using the same strategy. In this paper, we modify their technique to obtain a simpler proof of one of their results. We also argue that there is no logical reason why the 3D Navier-Stokes equations must always have solutions, even when the initial velocity vector field is smooth; if they do always have solutions, it is due to probability and not logic.

math.AP↗
arXiv2012

Complexity Science for Simpletons

In this article, we shall describe some of the most interesting topics in the subject of Complexity Science for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary understanding of computer programming will be able to follow this article. First, we shall explain the significance of the P versus NP problem and solve it. Next, we shall describe two other famous mathematics problems, the Collatz 3n+1 Conjecture and the Riemann Hypothesis, and show how both Chaitin's incompleteness theorem and Wolfram's notion of "computational irreducibility" are important for understanding why no one has, as of yet, solved these two problems.

cs.CC↗
arXiv2011

Is it possible to compute the Moebius function without factoring?

It has been well known since Fermat's Little Theorem was first published that it is possible to determine that a number is composite without determining any of its nontrivial factors. It is natural to ask whether it is also possible to compute the Moebius function of a composite number without determining any of its nontrivial factors. In this note, we argue that this is impossible.

math.GM↗
arXiv2011

An Elegant Argument that P is not NP

In this note, we present an elegant argument that P is not NP by demonstrating that the Meet-in-the-Middle algorithm must have the fastest running-time of all deterministic and exact algorithms which solve the SUBSET-SUM problem on a classical computer.

cs.CC↗
arXiv2003

Machine-Checked Formalization of Earlier Arguments on $\mathbb{P}$ versus $\mathbb{NP}$ Using Isabelle/HOL

This letter revisits an earlier argument concerning $\mathbb{P}$ versus $\mathbb{NP}$ based on the SUBSET-SUM problem and examines its formalization in Isabelle/HOL. The formal development clarifies the argument's logical structure by separating its deductive combinatorial core from the broader universality principle required to extend it to all exact deterministic algorithms.

cs.CC↗