Understanding Differential Equations Using Mathematica and Interactive Demonstrations

The solution of differential equations using the software package Mathematica is discussed in this paper. We focus on two functions, DSolve and NDSolve, and give various examples of how one can obtain symbolic or numerical results using these functions. An overview of the Wolfram Demonstrations Project (http://demonstrations.wolfram.com) is given, along with various novel user-contributed examples in the field of differential equations. The use of these Demonstrations in a classroom setting is elaborated upon to emphasize their significance for education

Convergence Radii for Eigenvalues of Tri--diagonal Matrices

Consider a family of infinite tri--diagonal matrices of the form $L+ zB,$ where the matrix $L$ is diagonal with entries $L_{kk}= k^2,$ and the matrix $B$ is off--diagonal, with nonzero entries $B_{k,{k+1}}=B_{{k+1},k}= k^\alpha, 0 \leq \alpha < 2.$ The spectrum of $L+ zB$ is discrete. For small $|z|$ the $n$-th eigenvalue $E_n (z), E_n (0) = n^2,$ is a well--defined analytic function. Let $R_n$ be the convergence radius of its Taylor's series about $z= 0.$ It is proved that $$ R_n \leq C(\alpha) n^{2-\alpha} \quad \text{if} 0 \leq \alpha <11/6.$

An analysis of the Sonata for Trumpet and Piano by Peter Maxwell Davies, identifying the use of historical forms, and the implications for performance.

The Sonata for Trumpet and Piano by Peter Maxwell Davies is one of his earliest works, and a notoriously difficult work to perform. While using serialism and other twentieth-century compositional techniques, this work also uses older historical forms, including sonata-allegro and sonata-rondo forms. An analysis of the work is presented, identifying the older historical forms, and considerations for performers when making decisions on how to perform the work are provided