Some history of integration

This post is based mainly on a chapter in A Radical Approach to Lebesgue’s Theory of Integration by David Bressoud in which he explores the history of the Lebesgue integral. The story I will tell is closer to folklore than a historical account, but nevertheless enlightening.

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Linear representation of additive groups and the Fourier Transform: Part 1

In this article I will show that the cyclic group of order n, that is the set \{0,1,2,\ldots,n-1\} under addition modulo n motivates the discrete Fourier transform on a particular finite dimensional complex inner product space, and gives many of its properties. In a subsequent article I will extend this to the general Fourier transform and its relation to the group of integers and real numbers under addition.

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From polynomials to transcendental numbers

In a previous post I discussed finding the zeros of low degree polynomials; I want to extend that discussion to algorithmically finding the zeros of polynomials, more on solving the quintic and a brief discussion of transcendental numbers.

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Symmetry, Lie Algebras and Differential Equations Part 3

There is a deep relationship between the technique of separation of variables for solving partial differential equations and the symmetries of the underlying differential equations, as well as the special functions that often arise in this procedure.

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Symmetry, Lie Algebras and Quantum Differential Equations Part 2

In this article I will apply the ideas from part 1 to the theory of rotations in three dimensions. (The theory of rotations in an arbitrary number of dimensions is similar, but for reasons of familiarity and simplicity I will stick to 3 dimensions).

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Symmetry, Lie Algebras and Differential Equations Part 1

There is a deep relationship between being able to solve a differential equation and its symmetries. Much of the theory of second order linear differential equations is really the theory of infinite dimensional linear algebra. In particular Sturm-Liouville theory is the diagonalization of an infinite dimensional Hermitian operator. However there are deeper relationships, as Miller points out in “Lie theory and special functions”; the relationships between special functions such as Rodrigues’ formulae are related to the Lie algebra and symmetries of the system. Even better in some cases the solutions can be found almost entirely algebraically. Some examples from physics come from the Simple Harmonic Oscillator, the theory of Angular Momentum and the Kepler Problem (using the Laplace Runge Lenz vector). The rest of this article will be devoted to exploring a special case of these relations the Quantum Simple Harmonic Oscillator.

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Do you really mean S^1?

This is a follow up post to my previous post on \mathbb{R}^n. Mathematicians will often write S^1 without being clear of the context and structure associated with it.

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