eien_meru's New Writeups

Re-expansion Theorem (idea)

2009-11-27T08:28:40Z

Def'n 2.1: The series Σa_n is said to converge to a complex number a if the limit of the sequence of partial sums exists. That's to say,

Σ_n=1^∞a_n = lim_n→∞ Σ_k=1ⁿa_k

whenever the right-hand side exists. A series is said to converge absolutely if the sum of the moduli of the terms converges. Absolutely convergent series are nice because it is possible to rearrange terms freely.

Def'n 2.2: The series of complex-valued functions Σf_n(z) is said to converge uniformly on a compact set K if there exists some function f(z) such that

∀ε>0 ∃N ∀n≥N ∀z∈K |(Σ_k=1ⁿf_k(z)) - f(z)| < ε

…

Complex-differentiable (idea)

2009-11-27T07:48:30Z

Course in Complex Analysis, Section 1

Def'n 1.1: The complex numbers are the algebraic closure of the real numbers. Every complex number can be written in the form a + bi, where a and b are real and i^2 = -1. The complex numbers are a field under the addition and multiplication induced by the reals. The multiplication of two complex numbers is defined as (a + bi)(c + di) = (ac - bd) + (ad + bc)i. The complex numbers as a set are labeled C.

In a natural way the complex numbers look a lot like the real Euclidean plane — viewing them this way is called the Argand plane, after an electrical engineer who used the idea to think about alternating current.

In addition to the field structure, there are three other common operations defined on complex numbers. First, there is complex conjugation, usually written as an overline or a star. If z = x + yi, z* = x - yi. The second is the modulus, which represents the length of a complex number…

commutative ring (idea)

2009-11-22T08:05:51Z

(Algebra:)

A commutative ring is a ring, meaning that it has two binary operations, + and *, which act the way + and * act for the integers. Most of the theory surrounding commutative rings, which is now known as commutative algebra, was created in order to understand the integers better. Unlike usual rings, commutative rings have the additional property that multiplication commutes, i.e., ab = ba for all of the elements in a ring. Addition is always commutative in a ring, so a + b = b + a in commutative rings as well.

The extra niceness of a commutative ring makes it easier to work with, and there are fewer pathological examples to worry about. For example, the definition of an ideal in a not-necessarily-commutative ring breaks down into left-sided and right-sided variants, complicating the statement of theorems. In a commutative ring, all ideals are by nature two-sided.

Examples of commutative rings include the integers, the Gaussian integers, the…

Solids of Revolution (how-to)

2009-11-14T05:12:13Z

(Calculus:)

A solid of revolution is a solid figure made by rotating a curve about an axis, in a way reminiscent of the way a lathe works. The general method for determining the volume of these things was worked out several centuries ago by Riemann, who is generally credited with the development of the subfield of mathematics that this sort of thing falls under: differential geometry.

The simplest solid of revolution is a cylinder. In this example, the curve that generates the volume is a line parallel to the axis of revolution, which, while sounding the United States' next enemy combatant or a really cool rock band, is usually something boring like the x-axis or y-axis on the Cartesian plane. No calculus is needed to find the volume of a cylinder. Even in ancient times it was known that the volume of a prism is equal to the height times the area of the base, and in this case the base is a circle with area πr².

The next simplest solid…

running up the mountain (essay)

2009-11-02T03:44:55Z

introduction
section 1, part 2

"...who were expelled from the academies for crazy &
publishing obscene odes on the windows of the skull,
who cowered in unshaven rooms in underwear,
burning their money in wastebaskets and listening
to the Terror through the wall,"

"Howl", Allen Ginsburg

The eien_meru of four years ago was a budding logician at the height of his undergraduate career. As the only pure math major in his class, the university was his oyster, and he practically had two whole departments full of professors at his disposal. I don't really resemble anything like that, anymore, but the reason I changed has a lot to do with the peculiar tone of urgency of the first two parts.

After all, the conclusion of cognitive limits seems rather outlandish, like something I must have thought up after reading too much Frank Herbert. Is…

cognitive limits (essay)

2009-11-01T08:16:42Z

introduction

Section 1: "I have not the time"

"Do not go gentle into that good night,
Old age should burn and rave at close of day;
Rage, rage against the dying of the light."

"Do not go gentle into that good night," Dylan Thomas .

In general there are two biological limits that restrict the productivity of the mathematician. First, it takes a substantial period of time to develop the skills and knowledge base necessary for research. We look on fondly at the history of mathematicians past; people like Evariste Galois were able to revolutionize the field of their time before turning twenty-five (though in the modern era, Australian mathematician Terence Tao certainly tried). Aside from these outliers, I would hazard to guess that the average mathematician requires a solid twenty-five to thirty years to reach both a working level of skill in…