arXiv subjects

Robert P. Schneider

Publications and source records attributed to Robert P. Schneider.

7 recordsLinked to original sources

arXiv2019

A Golden Pair of Identities in the Theory of Numbers

We find an interesting relationship between the golden ratio, the Moebius function, the Euler totient function and the natural logarithm - central players in the theory of numbers. A number of identities involving the golden ratio and its reciprocal are proved, including an expression for the base of the natural logarithm. The theorem and corollaries highlight a connection between the golden ratio and the factorization of integers that is not obvious; and display a sort of inverse relationship between the Moebius function and Euler totient function.

math.HO↗

arXiv2014

A Golden Product Identity for $e$

We prove an infinite product representation for the constant $e$ involving the golden ratio, the Möbius function and the Euler phi function - prominent players in number theory.

math.NT↗

arXiv2013

A Non-Pythagorean Musical Scale Based on Logarithms

A new musical scale devised by the author, based on natural logarithms, is described. Most of the logarithmic pitches bear no correspondence to the twelve tones of the ancient tuning system attributed to Pythagoras, based on ratios of whole numbers, nor to the chromatic tones of scales in equal temperament used widely in the modern era. Logarithms obey a special arithmetic compared to whole and rational numbers, which can be heard in beat frequencies between tones of the scale, suggesting extended harmonic possibilities by incorporating the beat frequencies into compositions. The author uses the broad term "non-Pythagorean" to describe the logarithmic musical scale, as the ratios of pitches are usually irrational numbers.

math.HO↗

arXiv2013

Al-Jabar: A Mathematical Game of Strategy

We detail the rules and mathematical structure of Al-Jabar, a game invented by the authors based on intuitive concepts of color-mixing and ideas from abstract algebra. Game-play consists of manipulating colored game pieces; we discuss how these colored pieces form a group structure and how this structure, along with an operation used to combine the pieces, is used to create a game of strategy. We also consider extensions of the game rules to other group structures. Note: While this is an article for general readership originally published online by Gathering for Gardner in honor of Martin Gardner's birthday (Oct. 2011), Al-Jabar has been played in university abstract algebra courses as a teaching tool, as well as by game enthusiasts, since its release. Moreover, the algebraic game structure described has sparked further work by other mathematicians and game designers. Thus, we submit this article to the ArXiV as a resource for educators as well as those interested in mathematical games.

math.HO↗

arXiv2013

A "Strange" Vector-Valued Quantum Modular Form

Since their definition in 2010 by Zagier, quantum modular forms have been connected to numerous different topics such as strongly unimodal sequences, ranks, cranks, and asymptotics for mock theta functions near roots of unity. These are functions that are not necessarily defined on the upper half plane but a priori are defined only on a subset of $\Q$, and whose obstruction to modularity is some analytically "nice" function. Motivated by Zagier's example of the quantum modularity of Kontsevich's "strange" function $F(q)$, we revisit work of Andrews, Jimenez-Urroz, and Ono to construct a natural vector-valued quantum modular form whose components are similarly "strange".

math.NT↗

arXiv2013

Combinatorial Applications of Möbius Inversion

In important work on the parity of the partition function, Ono related values of the partition function to coefficients of a certain mock theta function modulo 2. In this paper, we use Möbius inversion to give analogous results which relate several combinatorial functions via identities rather than congruences.

math.NT↗

arXiv2012

Uncovering Ramanujan's "Lost" Notebook: An Oral History

Here we weave together interviews conducted by the author with three prominent figures in the world of Ramanujan's mathematics, George Andrews, Bruce Berndt and Ken Ono. The article describes Andrews's discovery of the "lost" notebook, Andrews and Berndt's effort of proving and editing Ramanujan's notes, and recent breakthroughs by Ono and others carrying certain important aspects of the Indian mathematician's work into the future. Also presented are historical details related to Ramanujan and his mathematics, perspectives on the impact of his work in contemporary mathematics, and a number of interesting personal anecdotes from Andrews, Berndt and Ono.

math.HO↗