Notes on water analysis

Some notes on He and Water NMR

I will derive the NMR-AFP lineshape, specify the parameters that define the signal size, set up the equations to calculate the flux, and derive the water signal lineshape.

Notes on Stability Analysis

1.1. Norms on Vector Spaces. We want to measure the “size” or “length” of a vector. The mathematical device for doing this in called a norm. Here is the definition. Definition 1.1. Let V be a vector space over K. A norm on V is a function ‖·‖ : V → R : v 7→ ‖v‖ that has the following properties. (1) ‖v‖ ≥ 0 for all v ∈ V , and ‖v‖ = 0 if and only if v = 0. (2) If λ ∈ K and v ∈ V , ‖λv‖ = |λ| ‖v...

Notes on Circuit Analysis

These notes are o ered to refresh your memory on some fundamental circuit analysis techniques. In particular, a review of reactive components and transformers is o ered, followed by a brief review of time domain circuit analysis. The notes conclude with a discussion of some basic power concepts. The orientation of these notes is from the perspective of the power electronics engineer. That is, w...

Notes on Numerical Analysis

This set of notes covers topics that most commonly show up on the Numerical Analysis qualifying exam in the Mathematics department at UCLA. Each section covers a specific area of numerical analysis. At the end of each section, we provide a list of key theorems. If the proof of the theorem is directly relevant to the qualifying exam, it will be included in this document. Otherwise, we simply inc...

Notes on Multivariate Analysis