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Problems

$\begin{axiom} For the simple harmonic oscillator, find a bound on $du/dt$\ in te... ...ve constants $a_{1}$\ and $a_{2}$\ in terms of initial conditions. \end{axiom}$

$\begin{axiom} For the scheme \eqref{shofd2}, find a bound on the solution size i... ...rough the expressions for the energy given in \eqref{shoegyother}. \end{axiom}$

$\begin{axiom} Prove that the differential equation given in \eqref{hbardef} holds in the case of the linear oscillator with loss \eqref{sholoss}. \end{axiom}$

$\begin{axiom} Find a general real-valued solution to \eqref{sholoss} under the c... ...ermine the values of any constants in terms of initial conditions. \end{axiom}$

$\begin{axiom} The deviation in cents\index{cent} of a frequency $f$\ (in Hertz) ... ...e of $\alpha = 0.82$, what is the minimum value of this deviation? \end{axiom}$

$\begin{axiom} Consider the expression \eqref{sholossroots} for the roots of th... ...f solution will be produced, and will it be physically reasonable? \end{axiom}$

$\begin{axiom} Considering scheme \eqref{shofdloss2} for the simple harmonic os... ...\delta_{tt}u = -\omega_{0}^2 u-2\sigma\delta_{t+}u \end{equation} \end{axiom}$

$\begin{axiom} Consider the following approximation to the lossy SHO \eqref{sholo... ...ss conditions, and show that it will be dependent on $\omega_{0}$. \end{axiom}$

$\begin{axiom} Considering the exact two-step recursion \eqref{sholossexact} f... ...amma$\ and $\epsilon$\ in terms of $\sigma$\ and $\omega_{0}$. \par \end{axiom}$

$\begin{axiom} Consider the finite difference scheme for the oscillator given by... ...characteristic polynomial for this scheme, and find the roots, \par \end{axiom}$

$\begin{axiom} For scheme \eqref{shofd}, under the condition \eqref{shocfl}, prov... ...in analysis and energetic techniques, respectively, are identical. \end{axiom}$

Stefan Bilbao 2006-11-15