Search

The goal of this paper is to promote the use of R and Python as high-level, free, open-source programming environments that can be used as a computational and visualization tool for teaching differential equations. Both R and Python also allow for creating reproducible dynamic documents using Markdown, which combines live code, plain text, and expressions to generate different formats (pdf, html, Word), including the numerical and graphical output from the code, along with the code itself, properly formatted with minimum effort. This technology can be used for creating interactive teaching documents, randomized exams, homework and project reports, and even publication quality papers. We illustrate this technology with several examples implemented in R and Python. This approach has been classroom-tested, with some promising results. This article has online supplemental material posted on the PRIMUS website. [ABSTRACT FROM AUTHOR]

The article discusses the arrangement of stars on the American flag representing states in the U.S. The 50 star jack arrangement was designed by Robert G. Heft, designer of America's current national flag. A "Nice arrangement" of stars uses one of the six patterns of the Union Jack arrangement and exhibits a lot of symmetry which rarely changes with each adding star to the flag as number of states in the union increases.

For positive integers u and v , let L u = [ 1 0 u 1 ] and R v = [ 1 v 0 1 ]. Let G u , v be the group generated by L u and R v. In a previous paper, the authors determined a characterization of matrices M = [ a b c d ] in G u , v when u , v ≥ 3 in terms of the short continued fraction representation of b / d. We extend this result to the case where u v ≥ 5. Additionally, we compute [ G u , v : G u , v ] where G u , v = { [ 1 + u v n 1 v n 2 u n 3 1 + u v n 4 ] ∈ SL 2 (Z) : (n 1 , n 2 , n 3 , n 4) ∈ Z 4 } for u , v ≥ 1 , extending a result of Chorna, Geller, and Shpilrain and solving the membership problem for G u , v with u , v ≥ 1. [ABSTRACT FROM AUTHOR]

For positive integers u and v, let Lu = [1 & 0 \\ u & 1] and Rv = [1 & v \\ 0 & 1 ]. Let Su,v be the monoid generated by Lu and Rv, and let Gu,v be the group generated by Lu and Rv. In this paper we expand on a characterization of matrices M = [a & b \\ c & d] in Sk,k and Gk,k when k ≥ 2 given by Esbelin and Gutan to Su,v when u,v ≥ 2 and Gu,v when u,v ≥ 3. We give a simple algorithmic way of determining if M is in Gu,v using a recursive function and the short continued fraction representation of b/d. [ABSTRACT FROM AUTHOR]

In this paper, we consider a refinement, due to Nathanson, of the Calkin-Wilf tree. In particular, we study the properties of such trees associated with the matrices and , where and are non-negative integers. We extend several known results of the original Calkin-Wilf tree, including the symmetry, numerator-denominator, and successor formulas, to this new setting. Additionally, we study the ancestry of a rational number appearing in a generalized Calkin-Wilf tree. [ABSTRACT FROM AUTHOR]