arXiv subjects

Paul Raff

Publications and source records attributed to Paul Raff.

3 recordsLinked to original sources

arXiv2009

Finite Analogs of Szemerédi's Theorem

One of the "deepest" theorems in mathematics is Endre Szemerédi's theorem about the inevitability of arithmetical progressions. Here we try to nibble at it, by doing "finite" analogs. This is already interesting for its own sake, but we believe that it has the potential to lead to extremely interesting sharpening of the currently rather weak bounds. Let's hope! Along with introducing the theory behind the finite analogs of Szemerédi's Theorem, full Maple, Mathematica, and Java code is provided. See paper for further details.

math.CO↗

arXiv2008

Spanning Trees in Grid Graphs

Building on work by Desjarlais, Molina, Faase, and others, a general method is obtained for counting the number of spanning trees of graphs that are a product of an arbitrary graph and either a path or a cycle, of which grid graphs are a subclass. Results are obtained pertaining to recurrences obtained in this manner, and numerous new integer sequences are found.

math.CO↗

arXiv2006

A formally verified proof of the prime number theorem

The prime number theorem, established by Hadamard and de la Vall'ee Poussin independently in 1896, asserts that the density of primes in the positive integers is asymptotic to 1 / ln x. Whereas their proofs made serious use of the methods of complex analysis, elementary proofs were provided by Selberg and Erd"os in 1948. We describe a formally verified version of Selberg's proof, obtained using the Isabelle proof assistant.

cs.AI↗