Affine Geometry

Formal Definition Affine theory deals with 2 sets: One which elements are called locations and another, which elements are called directions. The directions form an additive group. Every additive group carries a derived composition between its elements and scalars. It is defined recursively as (composition symbol implied). Greek letters will be reserved for scalar names… Read More »

Numbers

Formal Definition A Field is a set which elements are called Scalars. 2 scalars can be combined in 2 ways, One is called their sum and is denoted by . Another is called their product and is denoted  by (with composition symbol implied). Both compositions are associative and commutative and both have neutral elements called… Read More »

Language of Mathematics

Introduction Mathematics is a language. Like any other language it is written by combining characters into words and words into grammatical sentences. In this post we use plain English to describes this grammar and situations where the language can be useful. Mathematics make statements about elements of a Set. It does not matter what Sets… Read More »

Anatomy of Software Project

Scope of Discussion Here we talk primarily about JavaScript projects but include Java as well. The major tools at our disposal here are Git for source control, Maven and Npm for dependency/task management of Java and JavaScript respectively. Java is compiled and packaged by JDK while in JavaScript world these tasks are performed by WebPack… Read More »

Setup of New React Project

References Code Redux tutorial Webpack intro Dual-Deploy React Project A dual-deploy project can be served in a NODE HTTP server as well as a SERVLET container such as Tomcat. The only NODE-specific part of the project is the landing page (index.html). The SERVLET container will supply its own landing page. Sample Project Directory Structure These… Read More »

Exterior Algebra

Interior Antiderivations If is a vector space then any form extends to a unique anti-derivation of the exterior algebra , i.e., for homogenous If is a basis of V, the dual basis and if denotes we have the following action on monomials (The sign is negative if the number of terms preceding the matching one… Read More »

Orthogonal Overview

Overview Orthogonal Group is a symmetry group of a vector space equipped with a (non-degenerate) scalar product that is a group of linear transformations such that . If we present as a matrix with respect to some vector basis then is orthogonal when its coefficients satisfy certain algebraic equations which depend on how the basic… Read More »

References

Reps of Lie Algebras & PDEs by Xu V. Kac Lecture Notes Samelson Lecture Notes Baez Octonions Zuber Home Stienstra Lecture Notes on Hypergeometric Structures

References Intro to Lie Algebras by V. Kac