10998521's New Writeupshttp://everything2.com/?node=New%20Writeups%20Atom%20Feed&foruser=109985212003-02-12T03:43:46Zreal analysis (idea)http://m.everything2.com/user/10998521/writeups/real+analysis10998521http://m.everything2.com/user/109985212003-02-12T03:43:46Z2003-02-12T03:43:46Z<p><h1>Keepin' it real: Real analysis for the mathematically lacking</h1></p>
<p>So, you want to know what real analysis is all about? First of all, a warning: This is one of the two "hard bits" in the switchover to university level <a href="/title/mathematics">mathematics</a><sup>1</sup>. This is the first introduction to truly <a href="/title/rigorous+proof">rigorous proof</a>, and all the headaches that come with it. I'm going to try and keep the level of tediously anal argument to a minimum, but it really is necessary, at least until a certain point (generally about midway through your second year of study, unfortunately). With that out of the way, let us proceed to the main question.</p>
<p><h1>What is real analysis and why do I care?</h1></p>
<p>A very good question, Mr/Madam reader sir/ma'am. Real analysis is the study of <a href="/title/function">functions</a> on the <a href="/title/real+line">real line</a>. It's about what makes the <a href="/title/real+number">real numbers</a> special, as opposed to the <a href="/title/rational+number">rationals</a> or the <a href="/title/complex+number">complexes</a> (complices? I'm not sure). As for why you should care,…Maths for the masses (idea)http://m.everything2.com/user/10998521/writeups/Maths+for+the+masses10998521http://m.everything2.com/user/109985212003-01-07T01:18:16Z2003-01-07T01:18:16Z<p><h1>Maths for the masses</h1></p>
<p><i>Or "Everything you never wanted to know about mathematics but would really like if you two got together, had a few beers, and got to know each other a little better"</i></p>
<hr>
<p>This is a little pet project of mine which used to be housed on my <a href="/title/10998521">homenode</a> until <a href="/title/dodo37">some</a> <a href="/title/wertperch">people</a> shouted at me to node it properly. Here it is in all its perpetually incomplete glory.</p>
<p>Basically, here on <a href="/title/Everything2">Everything2</a> I'm always hearing complaints that mathematics nodes are completely incomprehensible to a non-mathematician. In a sense, this is fine. The concepts involved can be hard to understand even for experts. However, we are missing whole chunks of the stuff which people need to know in order to begin to understand these nodes. That's what I'm trying to collect. If you know of any nodes which are well-written and <b>explain</b> mathematical concepts in a way that the average person can understand, /msg me and I'll put them here. I've…Lipschitz equivalent (idea)http://m.everything2.com/user/10998521/writeups/Lipschitz+equivalent10998521http://m.everything2.com/user/109985212002-11-09T01:49:40Z2002-11-09T01:49:40Z<p>Given a <a href="/title/set">set</a> <i>A</i> and two <a href="/title/metric">metrics</a>, <i>d</i> and <i>d'</i>, we say that <i>d</i> and <i>d'</i> are Lipschitz equivalent on <i>A</i> if there exist positive constants <i>h</i> and <i>k </i> such that</p>
<p><i>hd'(x,y)</i> ≤ <i>d(x,y)</i> ≤ <i>kd'(x,y)</i></p>
<p>for any <i>x, y</i> in <i>A</i>.</p>
<p>Why is this a useful definition, you ask? Well, the most important thing about this definition is this:</p>
<p><b>Theorem:</b> <i>Any two Lipschitz equivalent metrics are <a href="/title/topologically+equivalent">topologically equivalent</a>.</i></p>
<p><b>Proof:</b> Let <i>B<sub>r</sub>(x;d)</i> denote the <a href="/title/open+ball">open ball</a> with radius <i>r</i>, centre <i>x</i> and metric <i>d</i>, i.e. <i>B<sub>r</sub>(x;d)</i> = {<i>y</i> ∈ <i>A</i> | <i>d(x,y) < r</i>}. Now observe that, for any positive <i>ε</i>,</p>
<p><i>B<sub>hε</sub>(x;d)</i> ⊆ <i>B<sub>ε</sub>(x;d')</i> and <i>B<sub>ε/k</sub>(x;d')</i> ⊆ <i>B<sub>ε</sub>(x;d)</i></p>
<p>From here we can easily see…eigenvalue (thing)http://m.everything2.com/user/10998521/writeups/eigenvalue10998521http://m.everything2.com/user/109985212002-11-07T14:27:06Z2002-11-07T14:27:06Z<p>In a more general setting, let <b>O</b> be an <a href="/title/operator">operator</a>, acting on a space <i>T</i> (usually a <a href="/title/topological+vector+space">topological vector space</a>, but anything with a <a href="/title/scalar+product">scalar product</a> will do). We say a <a href="/title/scalar">scalar</a> λ is an eigenvalue for <b>O</b> if there exists ψ ∈ <i>T</i>, such that <b>O</b>ψ = λψ. Here ψ is an eigen(<a href="/title/eigenvector">vector</a>/<a href="/title/eigenfunction">function</a>/whatever you're calling elements of <i>T</i>) for <b>O</b>.</p>
<p>Where a <a href="/title/determinant">determinant</a> and <a href="/title/trace">trace</a> are well-defined, it can be shown that det <b>O</b> is the product of the eigenvalues and tr <b>O</b> is the sum. Even more usefully, the characteristic function, c<sub><b>O</b></sub>(λ) = det <b>O</b> - λ<b>I</b>, where <b>I</b> is the identity operator, has roots (only) at the eigenvalues.</p>
<p>Over function spaces, eigenvalues become very important in <a href="/title/quantum+mechanics">quantum mechanics</a> where they represent values of <a href="/title/observable">observables</a>. For example, the (time-independent) <a href="/title/Schr%2526ouml%253Bdinger+equation">Schrödinger equation</a> can be represented as …modular arithmetic (idea)http://m.everything2.com/user/10998521/writeups/modular+arithmetic10998521http://m.everything2.com/user/109985212002-07-03T01:08:44Z2002-07-03T01:08:44Z<p><h1>What, like arithmetic in modules?</h1></p>
<p>Not exactly. The "modular" part comes from the word <i><a href="/title/modulo">modulo</a></i> (from the Latin meaning "with the measure"). That doesn't explain much, does it? OK, the idea is that we do normal arithmetic with the integers (..., -1, 0, 1, 2, ...) but instead of having a "number line" that stretches to infinity in both directions, we let it wrap around in a circle. It's because of this that modular arithmetic is sometimes called <a href="/title/clock+arithmetic">clock arithmetic</a>. So 6 + 7 isn't 13 on a clock, it's 1. Similarly, modulo 5, 4 + 3 = 2.</p>
<p><h1>Alright, so what's it good for?</h1></p>
<p>Well, part of the joy is, most things you can do to a normal equation, you can do to a modular equation. You can add, subtract, multiply and raise to powers. You can even divide, as long as the modulus is prime (see <a href="/title/Noether">Noether</a>'s writeup <a href="/title/integers+modulo+n">here</a> for a proof, etc.). However, many quantities simplify greatly in modular form, and this lets us prove many rather nifty things. For…Rolle's theorem (idea)http://m.everything2.com/user/10998521/writeups/Rolle%2527s+theorem10998521http://m.everything2.com/user/109985212002-06-03T22:57:02Z2002-06-03T22:57:02Z<p><b>Theorem (<a href="/title/Rolle">Rolle</a>):</b> <i>Given</i> f: [a,b] -> R <i><a href="/title/continuous">continuous</a> on</i> [a,b] <i>with </i>f(a) = f(b)<i> and <a href="/title/differentiable">differentiable</a> on</i> (a,b), <i>there exists </i>c<i> in</i> (a,b) <i>with</i> f'(c) = 0</p>
<p><h1>Help! He's talking in tongues!</h1></p>
<p>This is one of those very important pieces of mathematics that seems <a href="/title/blindingly+obvious">blindingly obvious</a> until you think about why it's true. The you have to think for a very long time before it becomes obvious again. What it says is, if you have a <a href="/title/smooth+curve">smooth curve</a> with endpoints at the same height, at some point it has to be flat and level. If you spend a few minutes drawing squiggly lines on a bit of paper, you'll see this is obviously true. However, <a href="/title/pure+mathematics">pure mathematicians</a> are <a href="/title/pedant">pedantic little gits</a> and so we have to prove things like this.</p>
<p><h1>OK, Mr Smartypants, how do we do that?</h1></p>
<p>Well, first we have to look at what we mean by a "smooth curve". The mathematical term is a <a href="/title/differentiable+function">differentiable function</a>.…