krimson's New Writeupshttp://everything2.com/?node=New%20Writeups%20Atom%20Feed&foruser=krimson2008-09-13T20:52:46ZFreedom of the press and repression of the photocopier (essay)http://m.everything2.com/user/krimson/writeups/Freedom+of+the+press+and+repression+of+the+photocopierkrimsonhttp://m.everything2.com/user/krimson2008-09-13T20:52:46Z2008-09-13T20:52:46Z<em>A tale of Swedish jurisprudence</em>
<p>This is a story of two Swedish <a href="/title/free+speech">free-speech</a> cases which are similar, but with one crucial difference. Let's begin with the background.</p>
<p>The first case was against <a href="/title/environmental+activist">environmental activist</a> <a href="/title/Linus+Brohult">Linus Brohult</a>. In issue 1/1996 of the paper <a href="/title/Ekologisten">Ekologisten</a> he wrote an article, headlined "Sabotage more...!", about <a href="/title/sabotage">sabotage</a> 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 <a href="/title/motorway">motorway</a>s near <a href="/title/Stockholm">Stockholm</a> which involved such sabotage; Linus had once been <a href="/title/arrest">arrest</a>ed on <a href="/title/suspicion">suspicion</a> of involvement, but released. In 1999, Linus was charged with serious <a href="/title/inveiglement">inveiglement</a> (in the sense of "persuading to commit a crime", maximum <a href="/title/sentence">sentence</a> 4 years prison) for writing the article in Ekologisten.</p>
<p>The other case was against an <a href="/title/editor">editor</a> of the <a href="/title/anarchist">anarchist</a> paper <a href="/title/Brand">Brand</a>. In issue 1/2000 they published an article, whose title roughly<!-- close unclosed tag --></p>…algebraic topology (idea)http://m.everything2.com/user/krimson/writeups/algebraic+topologykrimsonhttp://m.everything2.com/user/krimson2008-09-06T13:49:33Z2008-09-06T13:49:33Z<p><i>Introductory remark</i>: This write-up is intentionally light on <a href="/title/definition">definition</a>s; follow the <a href="/title/hard-link">hard-link</a>s for details.</p>
<p>A <a href="/title/topological+space">topological space</a> is a space with a basic notion of <a href="/title/shape">shape</a>, sufficient to define the notion of a <a href="/title/continuous+function">continuous function</a>. Two topological spaces can be considered <a href="/title/equivalent">equivalent</a> if there is a <a href="/title/homeomorphism">homeomorphism</a> between them, i.e. an <a href="/title/invertible">invertible</a> function which is continuous in both directions. A basic problem in <a href="/title/topology">topology</a> is to determine whether two spaces are <a href="/title/homeomorphic">homeomorphic</a>. 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 <a href="/title/invariant">invariant</a> that distinguishes between the spaces.</p>
<p>One simple property of topological spaces that is invariant under homeomorphisms is <a href="/title/connected">connected</a>ness. 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.<!-- close unclosed tag --></p>…Philip J. Fry's DNA (essay)http://m.everything2.com/user/krimson/writeups/Philip+J.+Fry%2527s+DNAkrimsonhttp://m.everything2.com/user/krimson2008-07-01T09:42:28Z2008-07-01T09:42:28Z<p><i>In <a href="/title/Running+the+Reality+Checkpoint%253A+a+Sun-dappled+Nodermeet+in+Cambridge">collaboration</a>
with <a href="/title/sam512">sam512</a>, <a href="/title/BaronWR">BaronWR</a>, <a href="/title/StrawberryFrog">StrawberryFrog</a> and
<a href="/title/Andrew+Aguecheek">Andrew Aguecheek</a></i></p>
<p>Huser's write-up above is completely wrong,
for the reason that <a href="/title/sam512">sam512</a> explains: the <a href="/title/gene">gene</a>s are
<a href="/title/discrete">discrete</a>, and cannot be mixed. However, the <a href="/title/analysis">analysis</a>
in the original write-up is not entirely correct either,
since it uses an insufficiently accurate <a href="/title/model">model</a> for
genetic <a href="/title/inheritance">inheritance</a>.</p>
<h3>Genetic model</h3>
<p>
The <a href="/title/human">human</a> <a href="/title/DNA">DNA</a> <a href="/title/blueprint">blueprint</a> consists of a large number
of <a href="/title/gene">gene</a>s. Each gene can occur in one or more versions
called <a href="/title/allele">allele</a>s. 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.)</p>
<p>
Each gene is located on a <a href="/title/chromosome">chromosome</a>. Humans have 23
pairs of chromosomes, and both chromosomes<!-- close unclosed tag --></p>…winding number (idea)http://m.everything2.com/user/krimson/writeups/winding+numberkrimsonhttp://m.everything2.com/user/krimson2008-06-29T15:46:34Z2008-06-29T15:46:34ZThe 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 <a href="/title/line+integral">line integral</a>s of complex functions. This turns out to be important in <a href="/title/complex+analysis">complex analysis</a>, since allows you to interpret some integrals geomtrically, as in e.g. the <a href="/title/residue+theorem">residue theorem</a>. Let us first give a naive definition:<p>
<b>Definition 1</b>: Let C : [0, 1] → <b>C</b> be a closed curve that does not pass through the origin (<b>C</b> 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] → <b>R</b> such that A(t) is a choice of <a href="/title/argument">argument</a> 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<!-- close unclosed tag --></p>…Taylor's theorem for complex functions (idea)http://m.everything2.com/user/krimson/writeups/Taylor%2527s+theorem+for+complex+functionskrimsonhttp://m.everything2.com/user/krimson2003-05-23T23:15:54Z2003-05-23T23:15:54ZThere is an analogue for <a href="/title/complex+number">complex</a> <a href="/title/function">function</a>s of the well-known <a href="/title/Taylor%2527s+theorem">Taylor theorem for real functions</a>. It roughly states that any <a href="/title/analytic">analytic</a> (i.e. <a href="/title/complex-differentiable">complex differentiable</a>) <a href="/title/function">function</a> is locally equal to a <a href="/title/power+series">power series</a>. Taylor's theorem is nice because <a href="/title/power+series">power series</a> are (in particular the convergence of the power series is <a href="/title/uniform+convergence">uniform</a>).<br>
As usual the complex result is much nicer than the corresponding real one. Contrasting them we note that:<p>
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.<br>
2) We only require a complex function to be once complex differentiable, while a real function has to be several times <a href="/title/differentiable">differentiable</a> to apply the theorem.<p>
Indeed the complex Taylor theorem allows us to deduce that any analytic function is in fact infinitely complex differentiable, while on the other hand<!-- close unclosed tag --></p><!-- close unclosed tag --></p>…Taylor's Theorem (idea)http://m.everything2.com/user/krimson/writeups/Taylor%2527s+Theoremkrimsonhttp://m.everything2.com/user/krimson2003-05-23T23:10:06Z2003-05-23T23:10:06ZTaylor's <a href="/title/theorem">theorem</a> roughly states that a <a href="/title/real+function">real function</a> that is sufficiently <a href="/title/smooth">smooth</a> can be locally well <a href="/title/approximate">approximate</a>d by a <a href="/title/polynomial">polynomial</a>: if f(x) is n times <a href="/title/continuous">continuous</a>ly <a href="/title/differentiable">differentiable</a> then<br><br>
f(x) = a<sub>0</sub>x + a<sub>1</sub>x + ... + a<sub>n-1</sub>x<sup>n-1</sup> + <a href="/title/little+o+notation">o</a>(x<sup>n</sup>)<br><br>
where the <a href="/title/coefficient">coefficient</a>s are a<sub>k</sub> = f<sup><a href="/title/derivative">(k)</a></sup>(0)/k<a href="/title/factorial">!</a> (for notation see <a href="/title/little+o+notation">little o notation</a> and <a href="/title/factorial">factorial</a>; <sup>(k)</sup> 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).<br>
The theorem is proved as a result in <a href="/title/analysis">analysis</a>, but its importance is mainly as a tool for calculations in <a href="/title/applied+mathematics">applied mathematics</a>. There the option of replacing an arbitrary function by some polynomial (adding some <a href="/title/caveat">caveat</a> about "for x sufficiently small") is often invaluable.<br>
It is important to note that the theorem does <i>not</i> guarantee the…