krimson's New Writeups

Freedom of the press and repression of the photocopier (essay)

2008-09-13T20:52:46Z

A tale of Swedish jurisprudence

This is a story of two Swedish free-speech cases which are similar, but with one crucial difference. Let's begin with the background.

The first case was against environmental activist Linus Brohult. In issue 1/1996 of the paper Ekologisten he wrote an article, headlined "Sabotage more...!", about sabotage as a tool for defending the environment, and in particular some techniques for sabotaging machines for road-building. At the time there was a campaign against the building of motorways near Stockholm which involved such sabotage; Linus had once been arrested on suspicion of involvement, but released. In 1999, Linus was charged with serious inveiglement (in the sense of "persuading to commit a crime", maximum sentence 4 years prison) for writing the article in Ekologisten.

The other case was against an editor of the anarchist paper Brand. In issue 1/2000 they published an article, whose title roughly…

algebraic topology (idea)

2008-09-06T13:49:33Z

Introductory remark: This write-up is intentionally light on definitions; follow the hard-links for details.

A topological space is a space with a basic notion of shape, sufficient to define the notion of a continuous function. Two topological spaces can be considered equivalent if there is a homeomorphism between them, i.e. an invertible function which is continuous in both directions. A basic problem in topology is to determine whether two spaces are homeomorphic. If they are, then that can be proved by finding a homeomorphism. It is less obvious how to show that a pair of spaces is not homeomorphic. The easiest way to do it is usually to find some invariant that distinguishes between the spaces.

One simple property of topological spaces that is invariant under homeomorphisms is connectedness. Thus, if a pair of spaces has a different number of connected components then they cannot possibly be homeomorphic. But obviously this invariant is rather limited.…

Philip J. Fry's DNA (essay)

2008-07-01T09:42:28Z

In collaboration with sam512, BaronWR, StrawberryFrog and Andrew Aguecheek

Huser's write-up above is completely wrong, for the reason that sam512 explains: the genes are discrete, and cannot be mixed. However, the analysis in the original write-up is not entirely correct either, since it uses an insufficiently accurate model for genetic inheritance.

Genetic model

The human DNA blueprint consists of a large number of genes. Each gene can occur in one or more versions called alleles. One can think of each gene as encoding some characteristic, e.g. eye colour, and of the alleles as the possible values of this characteristic, e.g. "blue eyes" or "brown eyes". (Actually, eye colour is not determined by a single gene, but that's beside the point.)

Each gene is located on a chromosome. Humans have 23 pairs of chromosomes, and both chromosomes…

winding number (idea)

2008-06-29T15:46:34Z

The winding number of a closed curve C in the plane around a point is intuitively the number of times the curve goes around the point. It is possible to give a formula for winding numbers in terms of line integrals of complex functions. This turns out to be important in complex analysis, since allows you to interpret some integrals geomtrically, as in e.g. the residue theorem. Let us first give a naive definition:

Definition 1: Let C : [0, 1] → C be a closed curve that does not pass through the origin (C is the complex plane. Working here rather than the real plane gives neater notation, and anyway one of the main points is to get to the complex integral formula). Let A be a continuous choice of argument for C, i.e. a continuous function A : [0, 1] → R such that A(t) is a choice of argument for C(t) for all t. The winding number of C about the origin is n(C) = (A(1)-A(0))/2π. (The winding number around any other point is defined…

Taylor's theorem for complex functions (idea)

2003-05-23T23:15:54Z

There is an analogue for complex functions of the well-known Taylor theorem for real functions. It roughly states that any analytic (i.e. complex differentiable) function is locally equal to a power series. Taylor's theorem is nice because power series are (in particular the convergence of the power series is uniform).
As usual the complex result is much nicer than the corresponding real one. Contrasting them we note that:

1) For real functions we only get an approximation (with error bounds) while for complex functions we actually have that the power series is equal to the original function.
2) We only require a complex function to be once complex differentiable, while a real function has to be several times differentiable to apply the theorem.

Indeed the complex Taylor theorem allows us to deduce that any analytic function is in fact infinitely complex differentiable, while on the other hand even if a real function…

Taylor's Theorem (idea)

2003-05-23T23:10:06Z

Taylor's theorem roughly states that a real function that is sufficiently smooth can be locally well approximated by a polynomial: if f(x) is n times continuously differentiable then

f(x) = a₀x + a₁x + ... + a_n-1x^n-1 + o(xⁿ)

where the coefficients are a_k = f^(k)(0)/k! (for notation see little o notation and factorial; ^(k) denotes the kth derivative). Several formulations of this idea are given and proved below (unfortunately it is quite difficult to read when typeset in HTML).
The theorem is proved as a result in analysis, but its importance is mainly as a tool for calculations in applied mathematics. There the option of replacing an arbitrary function by some polynomial (adding some caveat about "for x sufficiently small") is often invaluable.
It is important to note that the theorem does not guarantee the…