These notes are to help with the Homework. Not everything is as mathematically precise as one would find in a textbook. 1. Groups The information in this section is well know and can be found in most introductory books on group theory. See for example [4], [6], or [3]. 1.1. Basic Examples of Groups. Definition 1. A group is a nonempty set G with a binary operation ∗ which satisfies the followin...

Recently, the classical Freudenthal Magic Square has been extended over fields of characteristic 3 with two more rows and columns filled with (mostly simple) Lie superalgebras specific of this characteristic. This Supermagic Square will be reviewed and some of the simple Lie superalgebras that appear will be shown to be isomorphic to the Tits-Kantor-Koecher Lie superalgebras of some Jordan supe...

We propose and analyze the possibility of performing two-photon continuouswave Doppler-cooling of hydrogen atoms using the 1S − 2S transition. “Quenching” of the 2S level (by coupling with the 2P state) is used to increase the cycling frequency, and to control the equilibrium temperature. Theoretical and numerical studies of the heating effect due to Doppler-free two-photon transitions evidence...

The development of a theoretical model to describe the scattering mechanisms involved in the remote sensing of rice crops is essential, as it ensures correct application of remote sensing data for rice monitoring. The theoretical model used in this study is based on the radiative transfer theory applied on a layered dense discrete random medium. The dense medium phase and amplitude correction t...

Regions of positive slope in the pressure-discharge characteristics can result in a reduction or even lack of damping of system instabilities. They are therefore one of the major concerns in design and operation of centrifugal pumps as they are limiting the admissible operating range. The considered preliminary hydraulic design of an industrial double suction pump of medium specific speed ν = 0...

This paper analyzes a Monte Carlo algorithm for solving Smoluchowski’s coagulation equation. A finite number of particles approximates the initial mass distribution. Time is discretized and random numbers are used to move the particles according to the coagulation dynamics. Convergence is proved when quasi-random numbers are utilized and if the particles are relabeled according to mass in every...