The proof of the Dual State Lemma in Chapter 2.3 of Gilbert and Porter's Knots and Surfaces has some problems with the exposition and a critical typo. This document reworks the proof to make it more correct and comprehensible. (This material starts on page 33.)

Muons compose the penetrating component of Cosmic Rays. At sea level, they constitute the largest part of Secondary Cosmic Rays, giving an average flux of ≈ 100 m−2s−1sr−1. The aim of our experiment is to estimate, from muon decay, the mean lifetime and the mass of invisible products. Our experimental setup includes four detectors: three of them are plastic scintillators and compose the trigger system, while the last one is a liquid scintillator which measures the particles energy. All these scintillators are read by photomultipliers. Trigger and pulse thresholds are computed by logical and temporal modules in a VME crate. The Data Acquisition System has been verified to work properly. It is composed of two fADCs modules, one I/O Register, one Motorola computer and a Farm. The liquid scintillator has been calibrated in energy using both passing muons and 60CO gamma source. Thanks to the charge-energy conversion factor we estimated electron energy spectrum. In particular we selected a sample of decay events by estimating muon mean lifetime τμ = 2.19 ± 0.34 μs; then we finally extrapolated an upper limit for invisible products mass mν < 5.99 ± 0.73 MeV/c2.

For an electron moving in a circular path in a magnetic field, if we know the magnetic field strength, accelerating voltage, and radius of the electron's trajectory, then we can make an estimation of the electron's charge to mass ratio. We calculated an average charge to mass ratio of \(2.08 \times 10^{11} \pm 1.81 \times 10^8\) Coulombs per kilogram.

We will study some aspects of the damped driven pendulum. We will see that the trajectory of damped driven pendulum is unpredictable under certain ranges of parameters.
Under some conditions, the trajectory behaves like random. A perturbation of initial condition will lead to different result. Several mathematical tools will be introduced in this study.