tag:blogger.com,1999:blog-37766505901765525302017-11-29T19:47:37.067-08:00TeX-nical StuffRandom scratchwork on Mathematical Physics and sometimes Programming. Diagrams done with <a href="http://www.paultaylor.eu/diagrams/"><code>commutative</code></a> and <a href="http://www.nought.de/tex2im.php"><code>tex2im</code></a>. <br><br>"<em>Dimidium facti qui coepit habet: sapere aude!</em>" (He who has begun is half done: dare to know!) - Horace.pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.comBlogger69125tag:blogger.com,1999:blog-3776650590176552530.post-64262497676847862312012-12-16T09:33:00.000-08:002012-12-16T09:33:37.747-08:00TeX macro for normal operator ordering<p>I've always been bothered with normal operator ordering, writing <code>$:O(a)O(b):$</code> always produces bad results. <br />
</p><p>The quick fix I've been using is the following:<br />
</p><pre>\def\normOrd#1{\mathop{:}\nolimits\!#1\!\mathop{:}\nolimits}
%%
% example:
% \begin{equation}
% \normOrd{a(z)b(\omega)} = a(z)_{+}b(\omega)+(-1)^{\alpha\beta}b(\omega)a(z)_{-}
% \end{equation}
%%
</pre>Which in practice looks like:<br />
<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/-7igR_t95KTI/UM4Dlaje3JI/AAAAAAAAA0s/wmlZ8Sb0vRQ/s1600/foo.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="25" width="400" src="http://4.bp.blogspot.com/-7igR_t95KTI/UM4Dlaje3JI/AAAAAAAAA0s/wmlZ8Sb0vRQ/s400/foo.png" /></a></div><h2>How I got this solution</h2>I determined this solution iteratively after many different attempts, which I shall enumerate along with the problems they each had.<br />
<br />
However, using mere colons <code>:a(z)b(\omega): = ...</code> produces the following:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://3.bp.blogspot.com/-rXEbDTPo9eg/UM4D4SDdVlI/AAAAAAAAA08/MdGSaorSWaI/s1600/bad1.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="24" width="400" src="http://3.bp.blogspot.com/-rXEbDTPo9eg/UM4D4SDdVlI/AAAAAAAAA08/MdGSaorSWaI/s400/bad1.png" /></a></div>Being clever, I asked myself "Hey, why not write <code>:x\colon</code> for the normal ordering?" This was clever, but wrong. Consider the following example:<br />
<pre>g = :x\colon
</pre>Producing:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://1.bp.blogspot.com/-9MbaD1ItxLU/UM4EaoPSwvI/AAAAAAAAA1E/aCX9JUAVaTM/s1600/bad2.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="30" width="166" src="http://1.bp.blogspot.com/-9MbaD1ItxLU/UM4EaoPSwvI/AAAAAAAAA1E/aCX9JUAVaTM/s400/bad2.png" /></a></div>Not one to give up easily, I found a <code>\cocolon</code> definition on <a href="http://tex.stackexchange.com/q/2300">tex.stackexchange</a>. Trying that instead:<br />
<pre>g = \cocolon x\colon = y
</pre>Produces strange extra whitespace on the right:<br />
<div class="separator" style="clear: both; text-align: center;"><a href="http://4.bp.blogspot.com/--Lr6lUbwc-o/UM4FAOmfftI/AAAAAAAAA1U/DO3oqTrM88k/s1600/bad3.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="30" width="264" src="http://4.bp.blogspot.com/--Lr6lUbwc-o/UM4FAOmfftI/AAAAAAAAA1U/DO3oqTrM88k/s400/bad3.png" /></a></div>After examining the co-colon code, I just determined that something along the lines of<br />
<pre>% rough draft definition #1
\def\normOrd#1{\mathrel{:}\!#1\!\mathrel{:}}
</pre>would work. This didn't quite work, the whitespacing was strange. So instead I just use <code>\mathop{:}\nolimits...</code>, which produces the desired result.pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com3tag:blogger.com,1999:blog-3776650590176552530.post-81940705587074861242012-08-12T09:00:00.000-07:002012-08-12T09:00:03.035-07:00Revising my Notes on General Relativity<p>So I've been revising my notes on general relativity, and I've found several things worth mentioning.<br />
</p><p><b>1. Equivalence Principle.</b> The equivalence principle gives us geometry. This is often poorly described (I too committed this error in my drafts). <br />
</p><p>The equivalence principle tells us neither the composition of a body nor its mass determines its trajectory in a gravitational field. So gravity determines <em>paths</em>, and this gives us geometry.<br />
</p><p>Moreover, there are <em>different</em> equivalence principles which should be mentioned. I yielded to this, and became incoherent (alas!). The trick is to stick this into a box, for the interested reader to find out more about it, but not obstruct the writing.<br />
</p><p><b>2. Coordinates for Black Hole.</b> Different coordinates for the Schwarzschild solution are described beautifully in Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, and Eduard Herlt's <em>Exact Solutions of Einstein’s Field Equations</em> (Cambridge University Press, 2d edition, 2009).<br />
</p><p><b>3. Manifolds, Mathematics.</b> I think I ought to examine Christopher Isham's <em>Modern Differential Geometry for Physicists</em> for a Physicist's differential geometry.<br />
</p><p>I should like to discuss the exponential map, which relates paths to geometry (as alluded in the equivalence principle discussion). <br />
</p><p>Most readers probably will agree that "Part II" of my notes (which specifically discuss differential geometry) are the toughest part of the notes. <br />
</p><p>Probably, I should mention a few examples of manifolds and explicitly study their coordinates in lecture 5.<br />
</p><p><b>3.1. Functions.</b> I never discussed what it means for a function on a manifold (a) to exist, (b) to be smooth.<br />
</p><p>Really, this let us discuss curves too. Why? A curve is just a function γ:<em>I</em>→<em>M</em> where <em>I</em> is just a closed interval, and <em>M</em> is the manifold.<br />
</p><p><b>3.2. Diffeomorphisms.</b> This word is thrown around a lot, but never defined rigorously (or at all!). So I should re-investigate this a bit.<br />
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-57696427649104426632012-05-25T17:34:00.001-07:002012-05-28T19:45:55.011-07:00Vertex algebras<p>Think about your favorite Lie algebra Lie(G). We have a mapping on it, namely, the <a href="http://en.wikipedia.org/wiki/Adjoint_representation_of_a_Lie_algebra">adjoint representation</a>:<br />
</p><p style="padding-left:1cm;">ad:Lie(G) → End[Lie(G)]<br />
</p><p>where "End[Lie(G)]" are the endomorphisms of the Lie algebra Lie(G).<br />
</p><p>Normally this is of the form "ad(u)v∈Lie(G)" and is shorthand for "ad(u)=[u,-]".<br />
</p><p>The Jacobi identity looks like:<br />
</p><p style="padding-left:1cm;">ad(u)ad(v)-ad(v)ad(u)=ad(ad(u)v).<br />
</p><p>This is the most important identity. Vertex operator algebras are an algebra with a similar property.<br />
</p><p>A vertex operator algebra consists of a vector space V equipped with a mapping usually denoted<br />
</p><p style="padding-left:1cm;">Y:V→(End V)[[x,x<sup>-1</sup>]].<br />
</p><p>In this form, it looks like left-multiplication operator...or that's the intuition anyways. So if "v∈V", we should think Y(v,x) belongs to "(End V)[[x,x<sup>-1</sup>]]" and acts on the left.<br />
</p><p>Really through <a href="http://en.wikipedia.org/wiki/Currying">currying</a> this should be thought of as "V⊗V→V[[x,x<sup>-1</sup>]]", i.e., a sort of multiplication operator with a parameter "x". (This is related to the "state-operator correspondence" physicists speak of with conformal field theories.)<br />
</p><p>Just like a Lie algebra, the Vertex Operator algebra satisfies a Jacobi identity and it is the most important defining property for the VOA.<br />
</p><p>Lets stop and look at this structure again:<br />
</p></p><p style="padding-left:1cm;">Y:V→(End V)[[x,x<sup>-1</sup>]].<br />
</p><p>What's the codomain exactly? Well, it's a formal <em>distribution</em> (<b>not</b> a mere formal power series!).<br />
</p><p>So what does one look like? Consider δ(z-1) = Σ z<sup>n</sup> where the summation ranges over n∈ℤ. This series representation is a formal distribution, and behaves in the obvious way. Lets prove this!<br />
</p><p><b>Desired Property:</b> δ(z-1) vanishes almost everywhere. <br />
</p><p>Consider the geometric series f(z) = Σz<sup>n</sup> where n is any non-negative integer (n=0,1,...).<br />
</p><p>Observe that δ(z-1) = f(z) + z<sup>-1</sup>f(z<sup>-1</sup>). Lets now substitute in the resulting geometric series:<br />
</p><p style="padding-left:1cm;">δ(z-1) = [1/(1-z)] + z<sup>-1</sup>[1/(1-z<sup>-1</sup>)]<br />
</p><p>and after some simple arithmetic we see for z≠1 we have δ(z-1)=0. <br />
</p><p><b>Desired Property:</b> for any Laurent polynomial f(z) we have δ(z-1)f(z)=δ(z-1)f(1).<br />
</p><p>This turns out to be true, thanks to the magic of infinite series; but due to html formatting, I omit the proof. The proof is left as an exercise to the reader (the basic sketch is consider δ(z-1)z<sup>n</sup>, then prove linearity, and you're done).<br />
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-65335284522256622422012-05-18T12:36:00.001-07:002012-05-18T12:36:54.428-07:00Finite Field with Four Elements<p>Small note to myself on notational problems when facing finite groups.<br />
</p><p>Recall the finite field with four elements is ℤ<sub>2</sub>[x]/(1+x+x<sup>2</sup>). <br />
</p><p>People often write ω = 1+x and <span style="text-decoration:overline;">ω</span>=x. Observe then that <span style="text-decoration:overline;">ω</span><sup>2</sup> = ω, and ω<sup>2</sup> = <span style="text-decoration:overline;">ω</span>. Moreover ω<span style="text-decoration:overline;">ω</span>=1 and 1+ω+<span style="text-decoration:overline;">ω</span>=0.<br />
</p><p>I have only seen this ω notation specified in Pless' <em>Error Correcting Codes</em>, Third ed., page 102 et seq.<br />
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-75632390247699909652012-03-16T12:03:00.000-07:002012-03-16T12:03:22.491-07:00End notes and Foot notes in LaTeX<p>
So I was writing up notes for a reading group on Afghanistan, and it has apparently become fashionable to use endnotes in the humanities. Being fond of Edward Gibbon, I use footnotes excessively. Irritating everyone, I use both while making it indistinguishable whether I refer to a footnote or endnote by a superscripted number.
</p>
<p>
How to do this in LaTeX? Quite simple:
</p>
<pre>
\documentclass{article}
\usepackage{endnotes}
\makeatletter
\newcommand*{\dupcntr}[2]{%
\expandafter\let\csname c@#1\expandafter\endcsname\csname c@#2\endcsname
}
\dupcntr{endnote}{footnote}
\renewcommand\theendnote{\thefootnote}
\makeatother
\begin{document}
Blah blah blah.
\end{document}
</pre>
<p>
It turns out to work quite well.
</p>
<p>
I modified some macros from a <a href="http://tex.stackexchange.com/questions/33898/slave-duplicate-counter">TeX Stackexchange</a> discussion on "slave counters"...so I get only partial credit.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-73850860216013854982012-02-21T10:54:00.000-08:002012-02-21T10:55:29.284-08:00Metapost and Labels<p>
This is just a quick note to myself. When I want to write a label with a smaller font, I should use <code>\scriptstyle</code>...but it is tricky since it requires <b>math mode!</b>
</p>
<p>
So an example, consider this diagram describing an experiment for gravitational redshift:
</p>
<pre>
numeric u;
u = 1pc;
beginfig(0)
path earth;
pair clock[];
earth = fullcircle scaled u;
clock[0] = (0,2u);
clock[1] = (0,4u);
draw (0,0)--(0,5u) dashed evenly;
for i=0 upto 1:
label(btex $\bullet$ etex, clock[i]);
endfor;
fill earth withcolor 0.75white;
draw earth;
label.rt(btex ${\scriptstyle\rm Earth}$ etex, (.5u,0));
label.rt(btex ${\scriptstyle\rm Satellite\ 1}$ etex, clock[0]);
label.rt(btex ${\scriptstyle\rm Satellite\ 2}$ etex, clock[1]);
endfig;
end;
</pre>
<p>
Just remember to use "<code>\ </code>" for spaces. Otherwise it will all run together horribly!
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-59300782745271084822012-02-14T13:21:00.000-08:002012-02-14T13:26:55.862-08:00Feynman Diagrams and Motives<p>
I have been re-reading the following book:
</p>
<p style="margin-left:3em;">
Alain Connes and Matilde Marcolli, <br />
<i>Noncommutative Geometry, Quantum Fields, and Motives</i>, <br />
Colloquium Publications, Vol.55, American Mathematical Society, 2008.
</p>
<p>
It turns out that Dr Marcolli has taught a <a href="http://www.its.caltech.edu/~matilde/course2008.html">course</a> on related material back in 2008! It is mostly dealing with the first chapter of the book.
</p>
<h2>Hopf Algebras and Feynman Calculations</h2>
<p>
There is a nice review of Hopf algebras used in Feynman diagram calculations:
</p>
<p style="margin-left:3em;">
Kurusch Ebrahimi-Fard, Dirk Kreimer, <br />
"Hopf algebra approach to Feynman diagram calculations". <br />
Eprint <a href="http://arxiv.org/abs/hep-th/0510202v2">arXiv:hep-th/0510202v2</a>, 30 pages.
</p>
<p>
For another specifically reviewing the noncommutative approach discussed in Connes and Matilde's book, see:
</p>
<p style="margin-left:3em;">
Herintsitohaina Ratsimbarison, <br />
"Feynman diagrams, Hopf algebras and renormalization."<br />
Eprint <a href="http://arxiv.org/abs/math-ph/0512012v2">arXiv:math-ph/0512012v2</a>, 12 pages.
</p>
<p>
What is a "Hopf algebra", anyways?
</p>
<p style="margin-left:3em;">
Pierre Cartier, <br />
"A primer of Hopf algebras." <br />
<a href="http://www.math.osu.edu/~kerler.2/VIGRE/InvResPres-Sp07/Cartier-IHES.pdf">Eprint <tt>[math.osu.edu]</tt></a>, 81 pages.
</p>
<h3>Hopf Algebras</h3>
<p>
What the deuce is a "Hopf algebra"? That's a very good question, and I'm very glad you asked it.
Wikipedia has its <a href="">definition</a>, which may or may not be enlightening.
</p>
<p>
Lets consider a concrete example. Consider a finite group G, and the field of complex number ℂ. We assert the collection Hom(G,ℂ) is a Hopf algebra.
</p>
<p>
Recall we have multiplication of group elements. This is a mapping G×G→G.
</p>
<p>
Now, observe we have functoriality to give us a mapping Hom(G×G→G,ℂ) = ℂ<sup>G</sup>→ℂ<sup>G</sup>×ℂ<sup>G</sup>. Lets call this thing Δ
</p>
<p>
Great, but what does it do? Good question!
</p>
<p>
Take some f∈Hom(G,ℂ) then what is Δ(f)?
</p>
<p>
It is a function of two variables, [Δ(f)](x,y). Functoriality demands, if we fix one of the arguments to be the identity element e∈G of the group, then [Δ(f)](e,y)=f(y) and [Δ(f)](x,e)=f(x).
</p>
<p>
It follows logically that [Δ(f)](x,y)=f(xy).
</p>
<p>
We also need to consider the antipode map S:ℂ<sup>G</sup>→ℂ<sup>G</sup>. We have [S(f)](x) be determined by the Hopf property, and a long story short [S(f)](x)=f(x<sup>-1</sup>).
</p>
<p>
Note that the antipode map is a <b>generalization</b> of the "group inverse" notion.
</p>
<p>
The other algebraic structure is a triviality, lets consider other interesting applications!
</p>
<h3>Feynman Diagrams</h3>
<p>
Now, I have written <a href="http://notebk.googlecode.com/files/feynman.pdf">some notes <tt>[pdf]</tt></a> on the basic algorithm evaluating Feynman diagrams and producing a number (the "probability amplitude").
</p>
<p>
As I understand it (and I don't!!) Ebrahimi-Fard and Kreimer suggest considering the Hopf algebra of "Feynman graphs" (which are just considered as colored graphs representing physical processes).
</p>
<p>
The basic algorithm to evaluating Feynman diagrams are based on the "Feynman rules" (what we assign to each edge, vertex, etc.). So Feynman rules are linear and multiplicative maps, associating to each Feynman graph (again, seen as a collection of vertices and edges) its corresponding <b>Feynman integral</b>.
</p>
<p>
So these maps are the important things, which enable us to algorithmically do stuff.
</p>
<p>
Lets stop! I said "Feynman integrals" are assigned to each graph...am I drunk, or is that correct?
</p>
<p>
Yes yes, the answer is "yes" ;)
</p>
<p>
What a horrible joke...but what I mean is: the scattering process of electrons, for example, is the infinite sum taking into account all the virtual processes.
</p>
<p>
Usually we only care up to a few orders.
</p>
<p>
Of course, this is my understanding of the Hopf algebra treatment of Feynman diagrams...and I openly admit: <b>I could be completely wrong!</b>
</p>
<p>
So to figure it out, I'll stop rambling, and continue reading.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-39626704031083745202012-02-06T10:13:00.000-08:002012-02-06T10:13:26.704-08:00(Ramblings on) Writing Notes on Quantum Field Theory<p>
So, in the long run, my aim is to write great notes on quantum field theory and quantum gravity. Since quantum gravity <em>depends</em> on quantum field theory, it makes sense to begin there!
</p>
<p>
For those uninterested in my rambling thought process, here's the punch line: just as integrals and derivatives are first covered symbolically in calculus, then rigorously in analysis...we likewise believe that a naive and symbolic approach first ought to be covered, then a rigorous and axiomatic approach second.
</p>
<p>
Greiner and Reinhardt's <em>Field Quantization</em> provides a good level for detail, at least for the naive/symbolic treatment of field theory.
</p>
<h2>What Other People Do</h2>
<p>
The basic approach other books take is "Well, here's Feynman diagrams. Quantum Field Theory just plays with these...here's how you get Feynman rules...and here's renormalization, the end."
</p>
<p>
This is not terrible. But it is lacking a certain <em>je ne sais quoi</em>.
</p>
<p>
So instead, perhaps I should look at it from the mathematical perspective. This has its own problems.
</p>
<h2>Depends on...</h2>
<p>
The problem I have is with <b>dependencies!</b> It doesn't make sense to write about quantum field theory without first writing about classical field theory, quantum mechanics, and a bit about functional analysis.
</p>
<p>
I have written a note about <a href="http://notebk.googlecode.com/files/rqm.pdf">Relativistic Quantum Mechanics <tt>[pdf]</tt></a> (which may make more sense after reading my notes on <a href="http://notebk.googlecode.com/files/lieGroups.lieAlgebras.Representations.gamma.pdf">Lie Groups, Lie Algebras, and their Representations <tt>[pdf]</tt></a>).
</p>
<p>
However, there is still more to do with quantum mechanics. Particularly, the subject of <b>scattering theory</b> is lacking. (Erik Koelink has some great <a href="http://fa.its.tudelft.nl/~koelink/dictaat-scattering.pdf">Lecture Notes <tt>[tudelft.nl]</tt></a> too)
</p>
<p>
Despite my notes on <a href="http://notebk.googlecode.com/files/functionalQFT.pdf">Functional Techniques in Path Integral Quantization <tt>[pdf]</tt></a>, I still feel lacking in the "path integral" department.
</p>
<p>
Perhaps I should write notes on measure theory, functional analysis, then tackle Glimm and Jaffe's <em>Quantum physics: a functional integral point of view</em>?
</p>
<p>
With classical field theory, the subject quickly becomes a can of worms (sadly enough).
</p>
<p>
Gauge theory, as Derek Wise notes in his blog post <a href="http://dkwise.wordpress.com/2012/01/30/the-geometric-role-of-symmetry-breaking-in-gravity/">"The geometric role of symmetry breaking in gravity"</a>, is intimately connected to Cartan Geometry.
</p>
<p>
There are dozens of exercises/examples to consider in gauge theory: Yang-Mills Theory, Born-Infield Action, Non-linear Sigma-Model, Non-linear Electrodynamics, Chern-Simons Theory, etc.
</p>
<h2>What Outline</h2>
<p>
So far, I've been considering my obstacles...but what about an outline?
</p>
<p>
The model I am following is the treatment of integration and differentiation in mathematics. First we have the <b>naive</b> symbolic manipulations (as done in calculus), then later we have the <b>formal and rigorous</b> proof based approach (as done in analysis).
</p>
<p>
Perhaps we should begin with <b>naive field theory</b>, where we obtain classical field theory "naively" from a "many body" problem.
</p>
<p>
This has merit from modelling fields as densities on the intuitive level.
</p>
<p>
Canonical quantization of this scheme becomes a triviality.
</p>
<p>
The problem with this approach is: what about the treatment of gauge theories, and their quantization?
</p>
<p>
After a few miracles, I expect to end up working with path integral quantization and formal calculus.
</p>
<p>
Naive treatment on quantizing gauge systems ought to be considered a bit more closely...
</p>
<p>
So that concludes the "naive" approach, and we begin the <b>Axiomatic Approach</b>. We should clarify the term "Axiom" means <b>specification</b> (not "God given truth", as dictionaries insist!).
</p>
<p>
The axiomatic approach would be done in a "guess-and-check" manner, modifying the axioms as necessary.
</p>
<p>
We naturally begin with Wightman axioms for the canonical approach, and the Osterwalder-Schrader axioms for the path integral approach. (Quickly, we ought to prove these two are equivalent!)
</p>
<p>
Kac's <em>Vertex Algebras for Beginners</em> takes the Wightman axioms, then extends them to conformal field theory through some magic. Perhaps this would be a worth-while example to consider?
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-14218549286783092812012-01-20T08:55:00.000-08:002012-01-20T09:02:50.304-08:00Studying Tactics<p>
My friend asked me for help on military tactics. "Where'd you get the idea I know anything about that?" I asked.
</p>
<p>
"I thought game theory was known to all mathematicians," my friend sheepishly replied.
</p>
<p>
Well, he's wrong about that. But I later found Clausewitz's suggestion <a href="http://www.gutenberg.org/files/1946/1946-h/1946-h.htm#2HCH0014">on examples</a>:
</p>
<blockquote>
<p>
EXAMPLES from history make everything clear, and furnish the best description of proof in the empirical sciences. This applies with more force to the Art of War than to any other. General Scharnhorst, whose handbook is the best ever written on actual War, pronounces historical examples to be of the first importance, and makes an admirable use of them himself. Had he survived the War in which he fell, the fourth part of his revised treatise on artillery would have given a still greater proof of the observing and enlightened spirit in which he sifted matters of experience.
</p>
<p>
But such use of historical examples is rarely made by theoretical writers; the way in which they more commonly make use of them is rather calculated to leave the mind unsatisfied, as well as to offend the understanding. We therefore think it important to bring specially into view the use and abuse of historical examples.
</p>
<p>
[...]
</p>
<p>
Now, if we consider closely the use of historical proofs, four points of view readily present themselves for the purpose.
</p>
<p>
First, they may be used merely as an EXPLANATION of an idea. In every abstract consideration it is very easy to be misunderstood, or not to be intelligible at all: when an author is afraid of this, an exemplification from history serves to throw the light which is wanted on his idea, and to ensure his being intelligible to his reader.
</p>
<p>
Secondly, it may serve as an APPLICATION of an idea, because by means of an example there is an opportunity of showing the action of those minor circumstances which cannot all be comprehended and explained in any general expression of an idea; for in that consists, indeed, the difference between theory and experience. Both these cases belong to examples properly speaking, the two following belong to historical proofs.
</p>
<p>
Thirdly, a historical fact may be referred to particularly, in order to support what one has advanced. This is in all cases sufficient, if we have ONLY to prove the POSSIBILITY of a fact or effect.
</p>
<p>
Lastly, in the fourth place, from the circumstantial detail of a historical event, and by collecting together several of them, we may deduce some theory, which therefore has its true PROOF in this testimony itself.
</p>
<p>
For the first of these purposes all that is generally required is a cursory notice of the case, as it is only used partially. Historical correctness is a secondary consideration; a case invented might also serve the purpose as well, only historical ones are always to be preferred, because they bring the idea which they illustrate nearer to practical life.
</p>
<p>
The second use supposes a more circumstantial relation of events, but historical authenticity is again of secondary importance, and in respect to this point the same is to be said as in the first case.
</p>
<p>
For the third purpose the mere quotation of an undoubted fact is generally sufficient. If it is asserted that fortified positions may fulfil their object under certain conditions, it is only necessary to mention the position of Bunzelwitz [Frederick the Great's celebrated entrenched camp in 1761] in support of the assertion.
</p>
<p>
But if, through the narrative of a case in history, an abstract truth is to be demonstrated, then everything in the case bearing on the demonstration must be analysed in the most searching and complete manner; it must, to a certain extent, develop itself carefully before the eyes of the reader. The less effectually this is done the weaker will be the proof, and the more necessary it will be to supply the demonstrative proof which is wanting in the single case by a number of cases, because we have a right to suppose that the more minute details which we are unable to give neutralise each other in their effects in a certain number of cases.
</p>
<p>
If we want to show by example derived from experience that cavalry are better placed behind than in a line with infantry; that it is very hazardous without a decided preponderance of numbers to attempt an enveloping movement, with widely separated columns, either on a field of battle or in the theatre of war—that is, either tactically or strategically—then in the first of these cases it would not be sufficient to specify some lost battles in which the cavalry was on the flanks and some gained in which the cavalry was in rear of the infantry; and in the tatter of these cases it is not sufficient to refer to the battles of Rivoli and Wagram, to the attack of the Austrians on the theatre of war in Italy, in 1796, or of the French upon the German theatre of war in the same year. The way in which these orders of battle or plans of attack essentially contributed to disastrous issues in those particular cases must be shown by closely tracing out circumstances and occurrences. Then it will appear how far such forms or measures are to be condemned, a point which it is very necessary to show, for a total condemnation would be inconsistent with truth.
</p>
<p>
It has been already said that when a circumstantial detail of facts is impossible, the demonstrative power which is deficient may to a certain extent be supplied by the number of cases quoted; but this is a very dangerous method of getting out of the difficulty, and one which has been much abused. Instead of one well-explained example, three or four are just touched upon, and thus a show is made of strong evidence. But there are matters where a whole dozen of cases brought forward would prove nothing, if, for instance, they are facts of frequent occurrence, and therefore a dozen other cases with an opposite result might just as easily be brought forward. If any one will instance a dozen lost battles in which the side beaten attacked in separate converging columns, we can instance a dozen that have been gained in which the same order was adopted. It is evident that in this way no result is to be obtained.
</p>
<p>
Upon carefully considering these different points, it will be seen how easily examples may be misapplied.
</p>
<p>
An occurrence which, instead of being carefully analysed in all its parts, is superficially noticed, is like an object seen at a great distance, presenting the same appearance on each side, and in which the details of its parts cannot be distinguished. Such examples have, in reality, served to support the most contradictory opinions. To some Daun's campaigns are models of prudence and skill. To others, they are nothing but examples of timidity and want of resolution. Buonaparte's passage across the Noric Alps in 1797 may be made to appear the noblest resolution, but also as an act of sheer temerity. His strategic defeat in 1812 may be represented as the consequence either of an excess, or of a deficiency, of energy. All these opinions have been broached, and it is easy to see that they might very well arise, because each person takes a different view of the connection of events. At the same time these antagonistic opinions cannot be reconciled with each other, and therefore one of the two must be wrong.
</p>
<p>
Much as we are obliged to the worthy Feuquieres for the numerous examples introduced in his memoirs—partly because a number of historical incidents have thus been preserved which might otherwise have been lost, and partly because he was one of the first to bring theoretical, that is, abstract, ideas into connection with the practical in war, in so far that the cases brought forward may be regarded as intended to exemplify and confirm what is theoretically asserted—yet, in the opinion of an impartial reader, he will hardly be allowed to have attained the object he proposed to himself, that of proving theoretical principles by historical examples. For although he sometimes relates occurrences with great minuteness, still he falls short very often of showing that the deductions drawn necessarily proceed from the inner relations of these events.
</p>
<p>
Another evil which comes from the superficial notice of historical events, is that some readers are either wholly ignorant of the events, or cannot call them to remembrance sufficiently to be able to grasp the author's meaning, so that there is no alternative between either accepting blindly what is said, or remaining unconvinced.
</p>
<p>
It is extremely difficult to put together or unfold historical events before the eyes of a reader in such a way as is necessary, in order to be able to use them as proofs; for the writer very often wants the means, and can neither afford the time nor the requisite space; but we maintain that, when the object is to establish a new or doubtful opinion, one single example, thoroughly analysed, is far more instructive than ten which are superficially treated. The great mischief of these superficial representations is not that the writer puts his story forward as a proof when it has only a false title, but that he has not made himself properly acquainted with the subject, and that from this sort of slovenly, shallow treatment of history, a hundred false views and attempts at the construction of theories arise, which would never have made their appearance if the writer had looked upon it as his duty to deduce from the strict connection of events everything new which he brought to market, and sought to prove from history.
</p>
<p>
When we are convinced of these difficulties in the use of historical examples, and at the same time of the necessity (of making use of such examples), then we shall also come to the conclusion that the latest military history is naturally the best field from which to draw them, inasmuch as it alone is sufficiently authentic and detailed.
</p>
<p>
In ancient times, circumstances connected with War, as well as the method of carrying it on, were different; therefore its events are of less use to us either theoretically or practically; in addition to which, military history, like every other, naturally loses in the course of time a number of small traits and lineaments originally to be seen, loses in colour and life, like a worn-out or darkened picture; so that perhaps at last only the large masses and leading features remain, which thus acquire undue proportions.
</p>
<p>
If we look at the present state of warfare, we should say that the Wars since that of the Austrian succession are almost the only ones which, at least as far as armament, have still a considerable similarity to the present, and which, notwithstanding the many important changes which have taken place both great and small, are still capable of affording much instruction. It is quite otherwise with the War of the Spanish succession, as the use of fire-arms had not then so far advanced towards perfection, and cavalry still continued the most important arm. The farther we go back, the less useful becomes military history, as it gets so much the more meagre and barren of detail. The most useless of all is that of the old world.
</p>
<p>
But this uselessness is not altogether absolute, it relates only to those subjects which depend on a knowledge of minute details, or on those things in which the method of conducting war has changed. Although we know very little about the tactics in the battles between the Swiss and the Austrians, the Burgundians and French, still we find in them unmistakable evidence that they were the first in which the superiority of a good infantry over the best cavalry was, displayed. A general glance at the time of the Condottieri teaches us how the whole method of conducting War is dependent on the instrument used; for at no period have the forces used in War had so much the characteristics of a special instrument, and been a class so totally distinct from the rest of the national community. The memorable way in which the Romans in the second Punic War attacked the Carthaginan possessions in Spain and Africa, while Hannibal still maintained himself in Italy, is a most instructive subject to study, as the general relations of the States and Armies concerned in this indirect act of defence are sufficiently well known.
</p>
<p>
But the more things descend into particulars and deviate in character from the most general relations, the less we can look for examples and lessons of experience from very remote periods, for we have neither the means of judging properly of corresponding events, nor can we apply them to our completely different method of War.
</p>
<p>
Unfortunately, however, it has always been the fashion with historical writers to talk about ancient times. We shall not say how far vanity and charlatanism may have had a share in this, but in general we fail to discover any honest intention and earnest endeavour to instruct and convince, and we can therefore only look upon such quotations and references as embellishments to fill up gaps and hide defects.
</p>
<p>
It would be an immense service to teach the Art of War entirely by historical examples, as Feuquieres proposed to do; but it would be full work for the whole life of a man, if we reflect that he who undertakes it must first qualify himself for the task by a long personal experience in actual War.
</p>
<p>
Whoever, stirred by ambition, undertakes such a task, let him prepare himself for his pious undertaking as for a long pilgrimage; let him give up his time, spare no sacrifice, fear no temporal rank or power, and rise above all feelings of personal vanity, of false shame, in order, according to the French code, to speak THE TRUTH, THE WHOLE TRUTH, AND NOTHING BUT THE TRUTH.
</p>
</blockquote>
<p>
<b>Addendum:</b> one may be interested in <a href="http://www.napoleonic-literature.com/Book_8/MaximsOfNapoleon.html">Napoleon's Maxims</a>, <a href="http://regimentalrogue.com/library/library-patton.htm">Patton's Reading List</a>, and Roosevelt's <a href="http://trevorhuxham.wordpress.com/2009/10/24/teddy-roosevelt%E2%80%99s-pigskin-library/">Pigskin Library</a>.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-35805855691167651822012-01-11T13:28:00.000-08:002012-01-11T13:28:54.385-08:00Puzzles<p>
Recently I've been more interested in puzzles.
</p>
<p>
<a href="http://projecteuler.net/">Project Euler</a> is the classic example of puzzles which require either higher math or computational skill (or both!).
</p>
<p>
Facebook has a <a href="http://www.facebook.com/careers/puzzles.php">collection of puzzles</a> too, motivated from the engineering perspective.
</p>
<p>
But note: Facebook uses these puzzles for hiring people. Plus, the puzzles are not always mathematically oriented.
</p>
<p>
I suppose a good mathematician should always set up puzzles for themselves. As Socrates remarked:
</p>
<blockquote>
<p>
SOCRATES: Indeed, Lysimachus, I should be very wrong in refusing to aid
in the improvement of anybody. And if I had shown in this conversation
that I had a knowledge which Nicias and Laches have not, then I admit
that you would be right in inviting me to perform this duty; but as we
are all in the same perplexity, why should one of us be preferred
to another? I certainly think that no one should; and under these
circumstances, let me offer you a piece of advice (and this need not go
further than ourselves). <b>I maintain, my friends, that every one of us
should seek out the best teacher whom he can find, first for ourselves,
who are greatly in need of one, and then for the youth, regardless of
expense or anything. But I cannot advise that we remain as we are. And
if any one laughs at us for going to school at our age, I would quote to
them the authority of Homer, who says, that
</p>
<p>
'Modesty is not good for a needy man.'
</p>
<p>
Let us then, regardless of what may be said of us, make the education of
the youths our own education.</b> (Emphasis added, from Plato's <a href="http://www.gutenberg.org/dirs/1/5/8/1584/1584.txt">Laches</a>)
</p>
</blockquote>
<p>
For example, I know a little bit about representations of Lie groups and Lie algebras (one can always learn more!)...but what about the representation of the quaternion group induced from the irreducible representations of SU(2)? How does it decompose into irreps? Etc.
</p>
<p>
Knuth remarked somewhere what helped him understand the representation theory for the symmetric group was writing a program which generated the permutation matrix representations.
</p>
<p>
I suspect writing a program which does these sorts of computations is a great puzzle for any mathematician that's savvy with programming.
</p>
<h2>Reading Material</h2>
<p>
And now, for something completely different.
</p>
<p>
A few papers I want to read: <br />
<a href="http://arxiv.org/abs/1201.1975">When physics helps mathematics: calculation of the sophisticated multiple integral</a>, 13 pages; <br />
<a href="http://arxiv.org/abs/1201.1992">Some algebraic properties of differential operators</a>, 15 pages; <br />
<a href="http://arxiv.org/abs/1112.3502">Introduction to supergravity</a>, 152 pages; <br />
<a href="http://arxiv.org/abs/1112.4669">Fermionic impurities in Chern-Simons-matter theories</a>, 31 pages; <br />
<a href="http://arxiv.org/abs/1201.2120">Spinors and Twistors in Loop Gravity and Spin Foams</a>, 16 pages; <br />
<a href="http://arxiv.org/abs/0711.2699">Quaternionic Analysis, Representation Theory and Physics</a>, 60 pages.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com1tag:blogger.com,1999:blog-3776650590176552530.post-18405458805435117392012-01-06T08:55:00.000-08:002012-01-06T08:55:58.569-08:00Problem Notebooks<p>
I've been looking for the infamous Kourovka notebook, but there are apparently others.
</p>
<p>
Recall the Kourovka notebook is a collection of open problems in group theory. There are other notebooks with open problems in other fields.
</p>
<p>
For example, the <a href="http://math.usask.ca/~bremner/research/translations/dniester.pdf">Dniester Notebook <tt>[usask.ca]</tt></a> (pdf, 56 pages) has problems in ring theory.
</p>
<p>
The notebook stopped being published years ago. Now it's been open sourced.
</p>
<p>
The Sverdlovsk notebook stopped publishing back in 1989; it was a collection of open problems for semigroups, but it cannot be found.
</p>
<p>
Other problem notebooks might exist too, but I am unaware of them...
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-67397326066397165402012-01-06T08:37:00.000-08:002012-01-06T08:38:37.172-08:00People don't read anymore :(<p>
I have been wandering around book stores recently, and stumbled upon the <em>Landmark Herodotus</em>. If you recall, I wrote a <a href="http://texnicalstuff.blogspot.com/search/label/Herodotus">few posts</a> on Herodotus.
</p>
<p>
People ought to be reading the text, and produce their own notes, producing a similar result (as <em>Landmark</em>) which is personalized.
</p>
<p>
Don't mistake me: I think <em>Landmark</em> is a wonderful resource! It has helped with maps, and so on...but people should be doing this on their own.
</p>
<p>
The problem <em>Landmark</em> posed (to me) parallels <em>Cliff Notes</em>. The books are produced as an example of how to take notes while reading books<b>...not a replacement for reading!</b>
</p>
<p>
Extemporaneously producing maps is an invaluable skill. The map doesn't have to be precise, e.g. drawing Asia minor as a rectangle, or Greece as three rectangles.
</p>
<p>
There are some elements to, e.g., the <em>Landmark Herodotus</em> that are pleasant, and makes it a great resource <b>to borrow from.</b> For example: what did Herodotus get wrong?
</p>
<p>
It appears that he got a lot of Egyptian history wrong, for example.
</p>
<p>
But my point is: <b>reading is studying!</b> You have to make a book <em>your own</em> through notes, maps, references to other books, etc.
</p>
<p>
<b>How</b> you study is entirely up to you; <b>how</b> you take notes, well, that is entirely up to you. But reading without taking notes is like dancing: nobody does it unless drunk or insane.
</p>
<p>
People just confuse <em>the mechanics of reading</em> (parsing letters into words) with <em>the procedure called "reading"</em> (obtaining information from books, and evaluating it). *sigh*
</p>
<p>
P.S. Happy New Year!
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-86240143384533027982011-12-29T12:34:00.000-08:002011-12-29T14:00:32.985-08:00Writing a History Book?<p>
So I have been thinking about writing a book on history, but a
different sort of book. The style is slightly inspired from
mathematical writing and Herodotus, where each chapter is divided
into "chunks".
</p>
<p>
A "chunk" discusses one subject, which is
summarized in a single sentence --- the sentence is highlighted
in bold, at the beginning of the chunk.
</p>
<p>
The idea is that there
would be an overview, summarizing the chapter, which consists of
just the "summary sentences". This way, the reader can get a
cursory understanding of history, while looking for details in
the appropriate section.
</p>
<p>
It would be nice to make it a website in the form of <a href="http://www.realworldhaskell.org/blog/2008/02/10/about-our-comment-system">Real World
Haskell</a>, <a href="http://www.djangobook.com/about/comments/">The Django Book</a>, or <a href="http://survivethedeepend.com/index/comments">Zend's Comment System</a>.
I think the Zend system is freely available via GIT.
</p>
<p>
Each paragraph ends with its own comments. However, with history, the author must be well prepared to deal with editing obscene comments. History is quite a sensitive subject.
</p>
<p>
After carefully examining the tools available, <a href="http://ucomment.org/contents/">ucomment</a> is the best choice.
</p>
<p>
It allows the author to dynamically update the document, while keeping the comments. Additionally, it uses the Sphinx markup language (which allows output to LaTeX).
</p>
<!--
<p>
I wonder if it would be possible to write a small python script that could convert the html to a <code>TeX</code> document, compile, and produce a pdf for publishing?
</p>
<p>
Also the <a href="http://ucomment.org/contents/">ucomment</a> system implements a Django-book-like comment system for free. <strike>Or <a href="http://www.remarkbox.com/">Remark Box</a> as an alternative.</strike> It may be worth looking in to...
</p>
-->pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-2234824537471533702011-12-24T10:38:00.000-08:002011-12-24T10:38:17.748-08:00My Differential Geometry Images<p>
So I am writing some notes on differential geometry, and using metapost for tricky diagrams. Here are a few of the tricky diagrams I have.
</p>
<p>
The first three images are:
</p>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://4.bp.blogspot.com/-3BGRGGOZ2sw/TvYYxBbDl7I/AAAAAAAAAv0/aTgG7B3sfow/s1600/img0.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="50" width="50" src="http://4.bp.blogspot.com/-3BGRGGOZ2sw/TvYYxBbDl7I/AAAAAAAAAv0/aTgG7B3sfow/s400/img0.png" /></a>
<a href="http://3.bp.blogspot.com/-zBdFEBGpqp8/TvYYxPl-9rI/AAAAAAAAAv8/Ux1WPjcY96c/s1600/img1.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="68" width="62" src="http://3.bp.blogspot.com/-zBdFEBGpqp8/TvYYxPl-9rI/AAAAAAAAAv8/Ux1WPjcY96c/s400/img1.png" /></a>
<a href="http://2.bp.blogspot.com/-770Fr3xShmM/TvYYxHP1BdI/AAAAAAAAAwM/fbj-HiT5gTA/s1600/img2.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" height="74" width="67" src="http://2.bp.blogspot.com/-770Fr3xShmM/TvYYxHP1BdI/AAAAAAAAAwM/fbj-HiT5gTA/s400/img2.png" /></a></div>
<p>
The third image is a tad bit bigger:
</p>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://3.bp.blogspot.com/-V2t8l_hsZdE/TvYaL8nHLYI/AAAAAAAAAwY/zABvNkPuyuA/s1600/img3.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" src="http://3.bp.blogspot.com/-V2t8l_hsZdE/TvYaL8nHLYI/AAAAAAAAAwY/zABvNkPuyuA/s400/img3.png" /></a></div>
<p>
Note the fonts in this picture are messed up, since I had to use postscript fonts. It looks far more beautiful in TeX using metapost, trust me!
</p>
<pre>
numeric u;
color yellow;
u := 1pc;
yellow = red+green;
verbatimtex \input amssym.tex etex;
% sphere
beginfig(0)
draw fullcircle scaled 4u;
draw (-2u,0)..(0,-.7u)..(2u,0);
draw (-2u,0)..(0,.7u)..(2u,0) dashed evenly;
endfig;
% plane + cylinder
beginfig(1)
draw (0,-u)--(0,3u)--(u,4u)--(u,0)--cycle;
draw (3u,0)..(4u,-.5u)..(5u,0);
draw (3u,0)..(4u,.5u)..(5u,0) dashed evenly;
draw (3u,4u)..(4u,4.5u)..(5u,4u);
draw (5u,4u)..(4u,3.5u)..(3u,4u);
draw (3u,0)--(3u,4u);
draw (5u,0)--(5u,4u);
endfig;
% saddle
beginfig(2)
path p[];
z[0] = (2u,4u);
z[1] = (2.5u,3u);
z[2] = (1.5u,2.5u);
z[3] = (4u,0u);
z[4] = (6.5u,u);
z[5] = (7.u,1.75u);
z[6] = (6.75u,0.5u);
p[0] = z[0]..z[1]..z[2];
p[1] = z[1]..z[3]..z[4];
p[2] = z[5]..z[4]..z[6];
p[3] = z[0]--(point 0.75*length(p[1]) of p[0]);
p[4] = z[2]--(1.75u,-2u);
p[5] = z[6]--(6.5u,-2u);
p[6] = z[5]--(point 0.9*length(p[1]) of p[2]);
for i=0 upto 6:
draw p[i];
endfor;
draw (4u,-u){up}..(point 0.6*length(p1) of p1);
draw (point 0.6*length(p1) of p1)..{down}(5.125u,-.75u) dashed evenly;
draw (1.75u,-2u)..(4u,-u)..(6.5u,-2u);
p7 = ((0,0)--(2u*unitvector(direction 0 of p[6]))) shifted point (length p6) of p[6];
draw p[7] dashed evenly;
p8 = ((0,0)--(3.5u*unitvector(direction 0 of p3))) shifted point (length p3) of p3;
draw p8 dashed evenly;
p9 = (point (length p8) of p8)..(5.125u,-.75u)..(point (length p7) of p7);
draw p9 dashed evenly;
endfig;
beginfig(3)
picture Rn;
picture sphere;
picture torus;
% R^n
Rn = image(
for i=1 upto 3:
draw (-.5u,i*u)--(3.5u,i*u) withcolor 0.75white;
draw (i*u,-.5u)--(i*u,3.5u) withcolor 0.75white;
endfor;
drawdblarrow (-.5u,0)--(3.5u,0);
drawdblarrow (0,-.5u)--(0,3.5u);
);
% sphere
sphere = image(
draw fullcircle scaled 4u;
draw ((-2u,0)..(0,-.7u)..(2u,0));
draw ((-2u,0)..(0,.7u)..(2u,0)) dashed evenly;
);
% torus
torus = image(
path hole;
path uhole;
draw
(0,2u)..(-2u,0)..(0,-2u)..(2u,-u)..(4u,-2u)..(6u,0)..(4u,2u)..(2u,u)..cycle;
hole = halfcircle rotated 180 scaled 2u shifted (4.5u,0);
draw hole;
uhole = (point 0.1*length(hole) of hole)
..(0.5[(point 0.1*length(hole) of hole),(point 0.9*length(hole) of hole)]+(0,.5u))
..(point 0.9*length(hole) of hole);
draw uhole;
);
draw Rn;
draw Rn shifted (20u,-5u);
draw torus shifted (8u,4u);
path localPatch;
path imageOfF;
path preimageOfG;
path imageOfG;
path imageOfPsi;
path intersection;
localPatch = (fullcircle scaled 2u shifted (1.5u,1.5u));
fill localPatch withcolor 0.75[blue,white];
draw localPatch dashed evenly;
z[0] = (((u,0)--(u,2u)) intersectionpoint localPatch);
draw (((0,u)--(u,u)) intersectionpoint localPatch)
--(((2u,u)--(4u,u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
draw (((0,2u)--(2u,2u)) intersectionpoint localPatch)
--(((2u,2u)--(4u,2u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
draw (((u,0)--(u,2u)) intersectionpoint localPatch)
--(((u,2u)--(u,4u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
draw (((2u,0)--(2u,2u)) intersectionpoint localPatch)
--(((2u,2u)--(2u,4u)) intersectionpoint localPatch) withcolor 0.75[black,blue];
imageOfF = fullcircle xscaled 3u yscaled 1.5u shifted (8u,4u);
fill imageOfF withcolor 0.75[blue,white];
draw imageOfF dashed evenly;
picture preG;
preG = image(
preimageOfG = unitsquare scaled 2u rotated 45 shifted (22u,-5u);
fill preimageOfG withcolor 0.5[yellow,white];
fill fullcircle xscaled u yscaled 5u rotated -23 shifted (22u,-3u)
withcolor 0.5[green,white];
draw (((21u,-4u)--(21u,-5u)) intersectionpoint preimageOfG)
--(((21u,-5u)--(21u,-3u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw (((22u,-2u)--(22u,-3u)) intersectionpoint preimageOfG)
--(((22u,-4u)--(22u,-8u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw (((23u,-3u)--(23u,-3.5u)) intersectionpoint preimageOfG)
--(((23u,-3.5u)--(23u,-8u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw (((21u,-4u)--(22u,-4u)) intersectionpoint preimageOfG)
--(((22u,-4u)--(25u,-4u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw (((21u,-3u)--(22u,-3u)) intersectionpoint preimageOfG)
--(((22u,-3u)--(25u,-3u)) intersectionpoint preimageOfG)
withcolor 0.5[black,yellow];
draw preimageOfG dashed evenly;
clip currentpicture to preimageOfG;
);
draw preG;
imageOfG = (19u,2.5u)--(20u,3u)--(21.5u,3u)--(21u,2.5u)--cycle;
fill imageOfG withcolor 0.5[white,yellow];
imageOfPsi = fullcircle xscaled 2u yscaled u rotated -45
shifted (20u,3u);
fill imageOfPsi withcolor 0.75[blue,white];
numeric t[];
(t[0],whatever) = imageOfPsi intersectiontimes ((19u,2.5u)--(20u,3u));
(t[1],whatever) = imageOfPsi intersectiontimes ((21u,2.5u)--(19u,2.5u));
(t[2],whatever) = imageOfPsi intersectiontimes ((20u,3u)--(23u,3u));
(t[3],whatever) = imageOfPsi intersectiontimes ((19u,2.5u)--(20u,2.5u));
intersection = (subpath(t0,t3) of imageOfPsi)--%(20u,3u)--
(subpath(t1,t2) of imageOfPsi)--(20u,3u)--cycle;
fill intersection withcolor 0.5[green,white];
draw imageOfPsi dashed evenly;
draw imageOfG dashed evenly;
draw sphere shifted (20u,4u);
p[0] := (point 0.25*length(localPatch) of localPatch)..(4u,4u)
..(point 0.5*length(imageOfF) of imageOfF);
p[1] := (point 0 of imageOfF)..(12u,5u)..(19.5u,3.5u);
p[2] := (21.5u,3u)..(23u,0)..(23u,-3u);
p[3] := (point .75*length(localPatch) of localPatch)..(1.5u,-u)
..(10u,-4u)..(21.5u,-4.5u);
drawarrow p[0];
drawarrow p1;
drawarrow p2;
drawarrow p3 dashed evenly;
label.top(btex $x$ etex, point 0.5*length(p0) of p0);
label.top(btex $\varphi$ etex, point 0.75*length(p1) of p1);
label.rt(btex $y^{-1}$ etex, point 0.5*length(p2) of p2);
label.bot(btex $y^{-1}\circ\varphi\circ x$ etex, point 0.6*length(p3) of p3);
label(btex $U$ etex, (1.5u,1.5u)) withcolor 0.5[black,blue];
draw (8u,4u) withpen pencircle scaled 3;
label.lft(btex $p$ etex, (8u,4u)) withcolor 0.75[blue,black];
label.bot(btex $M$ etex, (12u,2u));
label.rt(btex $N$ etex, (22u,4u));
label.lft(btex $\Bbb{R}^{n}$ etex, (20u,-2u));
label.lft(btex $\Bbb{R}^{m}$ etex, (0,3u));
endfig;
end;
</pre>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-2915043986251800712011-12-22T09:43:00.000-08:002011-12-22T09:47:21.459-08:00Outline of Commutative Geometry<p>
So, preparing for my discussion of noncommutative geometry, I need to discuss "commutative geometry".
</p>
<p>
What's going on here? Well, lets begin with the simplest notion of a space: a topological space.
</p>
<p>
I have discussed in my <a href="http://notebk.googlecode.com/files/rapidIntroTopology.pdf">notebk <tt>[googlecode.com]</tt></a> the notion of a topological space and continuous functions.
</p>
<p>
However, the algebra of continuous real-valued (or, more generally, complex-valued) functions encode the topology.
</p>
<p>
So I need to write notes reconstructing the topological data for $X$ from the ring structure and properties which $C(X)$ satisfy.
</p>
<h2>Vector Bundles</h2>
<p>
The next sort of space we can work with is a vector bundle. What's this guy?
</p>
<p>
Well, it's really a fibre bundle whose fibre forms a vector space. What's a fibre bundle?
</p>
<p>
It's a generalization of the product space where we fix one of the spaces.
</p>
<p>
Where does this occur? In vector calculus!
</p>
<p>
We are working with $\mathbb{R}^{3}$. A vector field assigns to each point in $\mathbb{R}^{3}$ a vector. But vectors live in "linear spaces" (or <em>vector spaces</em>).
</p>
<p>
So secretly we have $X=\mathbb{R}^{3}$ be the underlying space, and the total space be $E=F\times X$ consisting of "tangent vectors" (an ordered pair consisting of the vector assigned by the vector field, and its base point).
</p>
<p>
The fibre here is a vector space. Moreover, it is $F=\mathbb{R}^{3}$ as a vector space.
</p>
<p>
This is the simplest example of a vector bundle. So what?
</p>
<p>
Well, vector fields can be represented through ordered triples. That is, three smooth functions represent each component of the vector field (the x part, y part, and z part).
</p>
<p>
So algebraically we have $C^{\infty}(\mathbb{R}^{3})\times C^{\infty}(\mathbb{R}^{3})\times C^{\infty}(\mathbb{R}^{3})$ represent all possible vector fields on our space.
</p>
<p>
This is a free module over $C^{\infty}(\mathbb{R}^{3})$. So are vector bundles represented by free modules?
</p>
<p>
Not really, we use <em>projective modules</em> (which is more general).
</p>
<p>
There are a few other things to discuss on this matter, e.g., global sections, and so forth.
</p>
<h2>Spinor Bundles</h2>
<p>
This should be discussed in some detail, as there are few good references on the subject.
</p>
<p>
Even <a href="http://ncatlab.org/nlab/show/spinor+bundle">nLab's entry</a> on Spinor bundles is lacking, alas!
</p>
<p>
However, we need to encode this data in a spectral triple. See the <a href="http://ncatlab.org/nlab/show/spectral+triple">nLab's entry</a>, it is quite good.
</p>
<p>
See also Alain Connes' "On the spectral characterization of manifolds" (<a href="http://arxiv.org/abs/0810.2088">arXiv:0810.2088</a>) for details.
</p>
<p>
This would take some time to write up.
</p>
<h2>Noncommutative Rejoinder</h2>
<p>
I suspect that by taking an arbitrary ring, instead of the ring of smooth functions (or continuous functions, or...), we begin working with noncommutative geometry.
</p>
<p>
Is this algebraic geometry? No, not really. Algebraic geometers use polynomials to encode their geometric objects.
</p>
<p>
On the other hand, what we are doing here is considering the structure of rings and modules over our rings to encode geometric properties and data.
</p>
<p>
Of course, I may be misinformed on what algebraic geometers do...I frankly never understood it well enough to satisfy myself.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-34119319384104374152011-12-20T11:09:00.000-08:002012-01-06T07:46:26.534-08:00Ancient Geography<p>
Reading Herodotus requires a greater knowledge of geography than I <a href="http://texnicalstuff.blogspot.com/2011/08/reading-herodotus.html">previously implied.</a>
</p>
<p>
Fortunately, all the geography you need to know is encoded in a free map. Ironically it comes from a video game titled "Rome: Total Realism".
</p>
<p>
So, without further ado, here is the map:
</p>
<div class="separator" style="clear: both; text-align: center;">
<a href="http://upload.wikimedia.org/wikipedia/en/8/87/Rtw_political_trm60.png" imageanchor="1" style="margin-left:1em; margin-right:1em"><img border="0" width="90%" src="http://upload.wikimedia.org/wikipedia/en/8/87/Rtw_political_trm60.png" /></a></div>
<p>
Does it seem small? Well, it's a link, so click on it for the huge map.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-81314975304968703252011-12-18T13:35:00.000-08:002011-12-18T13:35:46.951-08:00Experiments as "registers" in notes<p>
Mathematics is about theorems, science is about experiments. But theorems are expressed rigorously in formal logic...what about experiments?
</p>
<p>
The "old fashioned" way would be to write up a lab report and <em>carry out the experiment!</em> However, nowadays, there are some programs to simulate experiments.
</p>
<p>
For example, the <a href="http://edu.kde.org/applications/science">KDE Education Project</a> has a few programs for physics, astronomy, and chemistry. Sadly, no biology (although I wouldn't know what that would look like...).
</p>
<p>
Additionally, if one is writing notes on history, there is a beautiful globe program one could use...
</p>
<p>
At any rate, for <em>classical</em> physics, this is perhaps the ideal set up. It allows the reader to play with the experiment, which is the entire idea of an experiment(!), and if so possessed one could set it up in real life.
</p>
<p>
Plus this is free, a small perk.
</p>
<h2>Physics</h2>
<p>
There is the <a href="http://edu.kde.org/step/">Step</a> program which is useful for classical mechanics.
</p>
<p>
Better, the physical systems being studied can be represented as a program.
</p>
<p>
And when you get to numerical analysis, you can perform a "How accurate was this?" series of exercises ;)
</p>
<p>
It is multiplatform, so anyone on Windows, Mac OS X, or *nix can use the KDE Education suite. It may be worth while to consider :)
</p>
<p>
However, I wonder if there is not a better program out there? The qt framework is always a wee bit heavy duty.
</p>
<p>
It would be nice if someone invented a scripting language that produced flash videos, or something along those lines...since qt runs slow on my 733MHz Pentium 3 processor :'(
</p>
<h2>Chemistry</h2>
<p>
I have not thought about writing up chemistry notes, although I feel if I write biology notes then it would become necessary to study chemistry.
</p>
<p>
Personally, chemistry seems to be a sibling to physics, it just is concerned with different scales. So experiment is very important in chemistry, and it is just as repeatable as in physics.
</p>
<p>
The <a href="http://docs.kde.org/stable/en/kdeedu/kalzium/">Kalzium suite</a> simulates chemistry <em>to a degree</em>. Everything is always "to a degree"!
</p>
<h2>Biology?</h2>
<p>
I am unaware of any similar such program for biology...but there are few experiments which are reproducible in biology!
</p>
<p>
Perhaps something like "virtual fruit flies" or "virtual peas" would be nice to study genetics, but I think it would be best to do it the "old fashioned way" (i.e., actually carry out the experiment!).
</p>
<h2>Take Home Message</h2>
<p>
If you are a mathematician writing notes on science, and are unable to perform the experiments: write a program.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-3599242935298052572011-12-16T09:35:00.000-08:002011-12-17T10:56:05.224-08:00Math to think about<p>
There are several interesting directions I'd like to investigate. So interesting, I have decided to let <b><i>you</i></b> in on it too!
</p>
<h2>Moonshine</h2>
<p>
No living man cannot deny interest in moonshine. Terry Gannon's "Monstrous moonshine and the classification of CFT" (<a href="http://arxiv.org/abs/math/9906167">arXiv:math/9906167</a>) provides a great review.
</p>
<p>
The basic idea is that we have a way to associate to "algebraic stuff" (e.g., groups) some "modular stuff".
</p>
<p>
What's great is, the word "stuff" is used in the <a href="http://texnicalstuff.blogspot.com/2009/08/object-oriented-math-category-theory-as.html">technical sense</a> of the word.
</p>
<p>
This would require reviewing group theory, finite groups, as well as some ring theory (for the "algebraic stuff"). To discuss "modular stuff", we'd need to review complex analysis, functional analysis. The connection here would require studying conformal field theory, to some degree.
</p>
<p>
Robert Wilson's <em>Finite Simple Groups</em> is a wonderful reference for finite simple groups; and as always <em>SPLAG</em> is a good reference too.
</p>
<h2>Noncommutative Geometry</h2>
<p>
People mean many things by "Noncommutative Geometry", here I mean Connes' approach.
</p>
<p>
I suppose this first requires us to consider what "commutative geometry" is!
</p>
<p>
This requires knowledge of commutative algebra and differential Geometry. The idea is to model "differential calculus over a commutative ring" (as Wikipedia calls it), i.e., consider the algebraic "grammar" underlying differential geometry.
</p>
<p>
Commutative algebra describes this algebraic "grammar". Studying this model in commutative algebra is precisely what I mean by "commutative geometry" (where projective modules correspond to vector bundles, and so on).
</p>
<p>
Noncommutative geometry, on the other hand, generalizes this model to the <i>noncommutative</i> setting!
</p>
<p>
I still need to write up my notes on differential geometry, but there are no good references for "commutative geometry"! There are a few books on commutative algebra, though...
</p>
<p>
I should type up my notes on algebraic topology too, since spin bundles are a "principal Spin bundle". Although I have some notes written on the Spin group (see my <a href="http://notebk.googlecode.com/files/lieGroups.lieAlgebras.Representations.gamma.pdf">Lie groups notes</a>), I should review it some more. Michelson and Lawson's <em>Spin Geometry</em>
is a wonderful book to consider...
</p>
<p>
Operator algebras need to be reviewed for considering spectral triples. The algebra we typically work with are <b>von Neumann algebras</b> which are related to <b>C* algebras</b>.
</p>
<p>
Some references for operator algebras:
<ol>
<li>Kadison and Ringrose, <em>Fundamentals of the theory of operator algebras</em> vol. I and II</li>
<li>Blackadar. <em>Operator Algebras: Theory of C*-Algebras and von Neumann Algebras</em>. Encyclopaedia of Mathematical Sciences. Springer-Verlag, 2005.</li>
<li>Yasuyuki Kawahigashi, "Conformal Field Theory and Operator Algebras" <a href="http://arxiv.org/abs/0704.0097">arXiv:0704.0097</a> (18 pages)</li>
<li>Meghna Mittal, Vern Paulsen, "Operator Algebras of Functions." <a href="http://arxiv.org/abs/0907.5184">arXiv:0907.5184</a></li>
<li>John M. Erdman <a href="http://www.mth.pdx.edu/~erdman/614/operator_algebras_pdf.pdf">Lecture Notes on Operator Algebras</a> (129 pages)</li>
<li>J. A. Erdos, <a href="http://www.mth.kcl.ac.uk/~jerdos/CS/CS.pdf">C*-Algebras</a> (51 pages).</li>
<li>N.P. Landsman, "Lecture notes on C*-algebras, Hilbert C*-modules, and quantum mechanics" <a href="http://arxiv.org/abs/math-ph/9807030">arXiv:math-ph/9807030</a> (89 pages).</li>
<li>Jacob Lurie's <a href="http://www.math.harvard.edu/~lurie/261y.html">Course Notes on Von Neumann Algebras</a>, quite comprehensive!</li>
<li>Wassermann, <a href="http://iml.univ-mrs.fr/~wasserm/OHS.ps">Operators on Hilbert space <tt>[ps]</tt></a> (70 pages)</li>
<li>VFR Jones, <a href="http://www.math.berkeley.edu/~vfr/MATH20909/VonNeumann2009.pdf">von Neumann algebras <tt>[pdf]</tt></a> (150 pages)</li>
<li>NP Landsman's <a href="http://www.math.ru.nl/~landsman/oa2011.pdf">Lecture Notes on Operator Algebras <tt>[pdf]</tt></a> (64 pages).</li>
<li>John Hunter and Bruno Nachtergaele, <a href="http://www.math.ucdavis.edu/~hunter/book/pdfbook.html">Applied Analysis</a> (free, legal ebook!)</li>
</ol>
</p>
<p>
And, of course, there is Connes' <a href="http://www.alainconnes.org/docs/book94bigpdf.pdf">Noncommutative Geometry <tt>[pdf]</tt></a>, as well as Connes and Marcolli's <a href="http://www.alainconnes.org/docs/bookwebfinal.pdf">Noncommutative Geometry, Quantum Fields and Motives <tt>[pdf]</tt></a>
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-46830628265789720822011-12-15T11:31:00.000-08:002011-12-15T11:31:15.333-08:00Basic Physics Macros<p>
Continuing from my post <a href="http://texnicalstuff.blogspot.com/2011/12/latex-macros-for-personal-notes.html">LaTeX Macros for Personal Notes</a>, I'd like to discuss some macros for physics.
</p>
<p>
I am using the "ISEE" approach to tackling examples, where we have four major steps:
</p>
<ol>
<li>"Identify" what do we have and what are we looking for?</li>
<li>"Set Up" what are the relevant concepts and equations? Set up the equations.</li>
<li>"Execute" Carry out the scratch work</li>
<li>"Evaluate" Look back, reflect, what were the key points and key ideas?</li>
</ol>
<p>
We are working with a lot of examples, and the examples are long (compared to math!). So we need to indicate when the examples are done.
</p>
<p>
Following Euclid, we introduce a <code>\qefsymbol</code> which is used at the end of examples and constructions. This is done just as QED is used at the end of proofs.
</p>
<p>
I will use the <code>amsthm</code> package.
</p>
<pre>
\usepackage{amsthm}
\theoremstyle{plain}
\newtheorem{thm}{Theorem}[chapter]
\newtheorem{prop}[thm]{Proposition}
\theoremstyle{definition}
\newtheorem{defn}[thm]{Definition}
\newtheorem{ex}[thm]{Example}
\newtheorem{fact}{Experimental Fact}
\newtheorem{prob}[thm]{Problem}
\newtheorem{construction}[thm]{Construction}
\newtheorem{con}[thm]{Conjecture}
\newtheorem*{notation}{Notation}
\newtheorem*{assume}{Assumption}
\newtheorem*{quest}{Question}
\theoremstyle{remark}
\newtheorem{rmk}[thm]{Remark}
\newtheorem{sch}[thm]{Scholium}
\newcommand\qefsymbol{\ensuremath\blacksquare}
%{\ensuremath\triangle} % perhaps \ensuremath\triangle if one prefers...
\makeatletter
\newenvironment{example}{\begin{ex} %
\let\qedsymbol\qefsymbol % this is a temporary "let"
\pushQED{\qed}}%
{\popQED\@endpefalse\end{ex}}
\newenvironment{construct}%
{\begin{construction}\pushQED{\qed}}%
{\popQED\end{construction}}
\makeatother
</pre>
<p>
Now, to keep track of which step of ISEE we are at, I'd like to introduce the following code:
</p>
<pre>
\font\manual=manfnt
\newcommand\identify{\noindent\llap{\manual\char'170\rm\kern.5em}\textbf{Identify:}}
\newcommand\setup{\noindent\llap{\manual\char'170\rm\kern.5em}\textbf{Set up:}}
\newcommand\execute{\noindent\llap{\manual\char'170\rm\kern.5em}\textbf{Execute:}}
\newcommand\evaluate{\noindent\llap{\manual\char'170\rm\kern.5em}\textbf{Evaluate:}}
</pre>
<p>
There are other matters to discuss, like units and so forth, which I'll tackle next time...
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-56443126885477705462011-12-15T10:48:00.000-08:002011-12-15T10:48:43.125-08:00MetaPost, Plotting, and numerical precision<p>
So, to write up diagrams in <code>LaTeX</code>, you need to use <code>Metapost</code>. But <code>Metapost</code> doesn't use floating point arithmetic.
</p>
<p>
As Claudio Beccari's "Floating point numbers and METAFONT,
METAPOST, TEX, and PostScript Type 1 fonts" (<a href="http://www.tug.org/TUGboat/tb23-3-4/tb75beccreal.pdf">TUGboat <tt>[pdf]</tt></a>) notes, 32 bit integers are used to represent real numbers. The first 16 bits form the fractional part, 14 bits the integer part, 1 bit for the sign, and 1 bit for special purposes.
</p>
<p>
So, that means we have 16 log(2)/log(10) digits of precision, or about 4 digits. Now lets remember:
</p>
<p>
1 PS point = 1.00375 points <br />
1 pica = 12 PS points <br />
1 inch = 72 PS points = 72.27 points = 6 pica<br />
1 cm = 28.3464567 PS points = 2.36220472 pica<br />
</p>
<p>
We have precision of 2<sup>-16</sup> points, or about 0.000868055556 inches, or 0.00220486111 centimeters.
</p>
<p>
That's decent for <em>output</em> but not for <em>intermediate computations</em>. For example, if we were to plot <i>x<sup>x</sup></i>, we may lose a lot of precision.
</p>
<h2>Plots in Metapost</h2>
<p>
Lets consider a simple plot of $f(x)=x^{2}$.
</p>
<pre>
numeric u;
u := 1pc; % units
vardef f(expr x) = x*x enddef;
beginfig(0)
% draw the axes
drawdblarrow (-3u-ahlength,0)--(3u+ahlength,0);
drawdblarrow (0,0-2ahlength)--(0,9u+ahlength);
% plot the function
draw (-3u,f(-3)*u)
for i=-3+0.05 step 0.05 until 3:
..(i*u,f(i)*u)
endfor;
endfig;
end;
</pre>
<p>
Remember that <code>ahlength</code> is the length of the arrow head.
</p>
<p>
This basic scheme can be generalized if we add numerics <code>x0</code> and <code>x1</code> which control where the plot begins and ends (respectively), as well as the step size <code>dx</code> which is taken to be "small enough".
</p>
<p>
Revising our code:
</p>
<pre>
numeric u;
numeric dx;
u := 1pc; % units
dx := 0.05;
vardef f(expr x) = x*x enddef;
beginfig(0)
numeric x[];
x0 := -3; % start plotting at x=-3
x1 := 3; % stop plotting at x=+3
% draw the axes
drawdblarrow (x0*u-ahlength,0)--(x1*u+ahlength,0); % x-axis
drawdblarrow (0,0-2ahlength)--(0,f(x1)*u+ahlength); % y-axis
% plot the function
draw (x0*u,f(x0)*u)
for i=x0+dx step dx until 3:
..(i*u,f(i)*u)
endfor;
endfig;
end;
</pre>
<p>
This makes things a little complicated. What we are doing is computing the pairs (x,y) and then scaling them, then plotting.
</p>
<p>
The <code>dx</code> is the change in <code>x</code> <em>before</em> scaling. The points plotted have a change in <code>x</code> that amounts to <code>dx*u=0.6pt</code> approximately.
</p>
<p>
But we can do <em>more!</em> If we specify how big we want this plot to be, i.e. it has to fit within <em>X</em> inches, then we can determine the scale <code>u</code> by this.
</p>
<p>
Specifically, <code>u := X/(x[1]-x[0])</code> is the scale factor definition.
</p>
<p>
If we demand that <code>dx*u=0.6pt</code> hold, which is "sufficiently good" for practical purposes, then we also define <code>dx := (3pt)/(5*u)</code>.
</p>
<p>
The interested reader may want to read <a href="http://staff.science.uva.nl/~heck/Courses/mptut.pdf">Learning Metapost by Doing</a>.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-29912391143613762812011-12-15T09:03:00.000-08:002012-11-22T19:19:42.669-08:00LaTeX Macros for Personal Notes<p>So <a href="http://texnicalstuff.blogspot.com/2011/12/personal-v-expository-notes.html">last time</a>, I discussed the notion of personal mathematical notes (as opposed to <em>expository</em> mathematical notes) and would like to discuss some LaTeX macros which enable writing personal notes. </p><p>The basic scheme is to write in "chunks" (to borrow a term from <a href="http://en.wikipedia.org/wiki/Literate_programming">literate programming</a>). We've all seen examples of this, Bagchi and Wells refer to it as "labeled style" in their paper <a href="http://www.cwru.edu/artsci/math/wells/pub/pdf/mathrite.pdf">Varieties of Mathematical Prose</a> </p><p>But each "chunk" is a self-contained concept, example, discussion, etc. </p><p>For a good example of this writing style, see <a href="http://at.yorku.ca/i/a/a/z/20.htm">On Euler's Footsteps</a>. </p><h2>LaTeX Code</h2><p>I am taking <code>CWEB</code>'s style. So, the code listing I have is as follows: </p><pre>% chunk.sty
\ProvidesPackage{chunk}[2011/12/15 Cunking commands for personal notes]
\makeatletter
\@ifundefined{@addpunct}{
\def\@addpunct#1{\ifnum\spacefactor>\@m \else#1\fi}
}{}
\newcounter{chunk@ctr}
\newcommand\M{\medbreak\noindent%
\refstepcounter{chunk@ctr}%
\textbf{\thechunk@ctr\@addpunct{.}}\quad\ignorespaces}
% deprecated macro:
% \newcommand\N[1]{\M\textbf{#1\@addpunct{.}}\quad\ignorespaces}
% superior implementation:
\def\N{\@ifstar
\NStar%
\NNoStar%
}
\def\NStar#1{\medbreak\noindent\textbf{#1\@addpunct{.}\quad}\ignorespaces}
\def\NNoStar#1{\M\textbf{#1\@addpunct{.}\quad}\ignorespaces}
% permits writing \N*{Un-numbered chunk} for a chunk
% without a numeric label!
\makeatother
% end of cunk.sty
</pre><p>Note that it is completely self-contained code, and you do not need <code>amsgen</code> package. If you already loaded it, then no worries! </p><p>Each chunk is numbered. We use <code>\M</code> for unlabeled chunks, and <code>\N{My favorite chunk!}</code> for labeled chunks (which is labeled "My favorite chunk!"). </p><p>So lets write up some example usage: </p><pre>\documentclass{article}
\usepackage{chunk} % make it in the same directory
% or put it in ~/texmf/tex/latex/ and run "<code>sudo texhash</code>"
\title{Example Notes}
\author{Alex Nelson}
\date{\today}
\begin{document}
\maketitle
\N{Introduction}
Today we will solve all the problems in the universe.
\M Lorem ipsum dolor sit amet, consectetur adipisicing elit,
sed do eiusmod tempor incididunt ut labore et dolore magna
aliqua. Ut enim ad minim veniam, quis nostrud exercitation
ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis
aute irure dolor in reprehenderit in voluptate velit esse
cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat
cupidatat non proident, sunt in culpa qui officia deserunt
mollit anim id est laborum.
\N{Conclusion?} Nobel Prize please!
\end{document}
</pre><p>As far as bugs, I don't think there are any...it's too minimalistic! </p><h2>To Do</h2><p>The chunk counter is rather minimalistic, and doesn't count within any section. This has to be changed by hand if the user wants to use these macros and number chunks within each chapter... </p><p>In the time honored tradition of mathematicians, this exercise is left for the reader! </p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-55744670647167066332011-12-13T18:27:00.000-08:002011-12-13T18:27:31.164-08:00Personal v. Expository notes<h2>Personal Notes Are For One's Self</h2>
<p>
I recently stumbled across <em>Thinking Mathematically</em> by J. Mason, L. Burton, K. Stacey (<a href="http://www.amazon.com/Thinking-Mathematically-J-Mason/dp/0201102382">Amazon</a>) which has an interesting approach to writing personal mathematical notes.
</p>
<p>
I say "personal" as opposed to "expository" because they are really personal scratchwork rather than explanations.
</p>
<p>
What's really cute is Mason, et al., espouse a sort of "markup language" approach (they call it a "rubric"). Let me review this a little.
</p>
<p>
When "entering" a mathematics problem, there
are three things to ask ourselves, which constitute the entry phase:
<ol>
<li> What am I know? (What is given?)</li>
<li> What do I want?</li>
<li> What can I introduce?</li>
</ol>
Know, want, introduce. These three things we should always, always, always ask ourselves! Even in long proofs, we should ask ourselves these questions! We should ask ourselves when we get stuck!
</p>
<p>
With introducing stuff. . . we can do several types of introduction.
<dl>
<dt>Notation:</dt> <dd>Assigning values and meanings to variables.</dd>
<dt>Organization:</dt> <dd>Recording and arranging what you know.</dd>
<dt>Representation:</dt> <dd>Choose particular representatives which are easier to manipulate.</dd>
</dl>
Bear in mind that rephrasing the question is particularly useful.
</p><p>
Now, reviewing your work is also critical. There are several different ways of doing this:
<dl>
<dt>Check:</dt> <dd>the resolution;</dd>
<dt>Reflect:</dt> <dd>on the key ideas and key moments;</dd>
<dt>Extend:</dt> <dd>to a wider context.</dd>
</dl>
The best way to get the most out of reviewing is to write up your resolution for someone else, in such a way that one can follow what you have done and why.
</p>
<p>
We can check several things:
<ol>
<li>Check calculations;</li>
<li>Check arguments to ensure computations are appropriate;</li>
<li>Consequences of conclusions to see if they are reasonable;</li>
<li>Check that the resolution fits the question.</li>
</ol>
This is sort of subconsciously done.
</p>
<p>
We also reflect in finitely many ways:
<ol>
<li>What are the key ideas and key moments?</li>
<li>What are the implications of conjectures and arguments?</li>
<li>Can the resolution be made clearer?</li>
</ol>
It helps one see things that otherwise would have been missed.
</p>
<p>
As far as <b>extending</b>, one should really be <em>generalizing</em> the problem. For example: how many squares are on a 3 × 3 chess board? There are 9 instances of 1 × 1 squares, 4 instances of 2 × 2 squares, and a single 3 × 3 square. Thus there are 14 squares altogether. Now, to extend:
<ol>
<li>How many squares are on an n × n board?</li>
<li>How many rectangles are on a 3 × 3 board? Extend this to n × n boards.</li>
<li>What if we start with an m × n board? How many squares are there in it?</li>
<li>Why work only in two dimensions?</li>
<li>Why count squares with edges parallel to the original?</li>
</ol>
</p>
<p>
When <b>stuck</b>, try re-entering the entry phase. This can be done through:
<ol>
<li>Summarize everything known and wanted;</li>
<li>Rephrase the question in a more appealing way;</li>
<li>Re-read or re-digest the problem.</li>
</ol>
This is useful sometimes.
</p>
<p>
Conjecturing is a cyclic four-step procedure:
<ol>
<li>Articulate a conjecture (and while making it, believe it);</li>
<li>Check the conjecture covers all known cases and examples;</li>
<li>Distrust the conjecture. Try to refute it by finding a nasty case or example; use it to make predictions
which can be checked;</li>
<li>Get a sense of why the conjecture is right, or how to modify it, on new examples (go back to step 1).</li>
</ol>
Note you can start anywhere in this procedure.
</p>
<p>
It's not too long until one gets to a state where one says "I don't believe it's possible" which leads to the questions
<ol>
<li>Why can it not be done?</li>
<li>All right, what can be done?</li>
</ol>
Asking "What can be done" is a critical step in conjecturing.
</p>
<p>
Now, critical mathematical thinking should be nurtured by thinking three things while doing or reading a proof:
<ol>
<li>Every statement made should be treated as a conjecture.</li>
<li>Try to defeat and prove conjectures simultaneously.</li>
<li>Look critically at other people's proofs.</li>
</ol>
There are some mathematical registers that some authors suggest using while writing notes. The collection is sometimes called a rubric:
<dl>
<dt>I Know:</dt> <dd>What is given? What is known?</dd>
<dt>I Want:</dt> <dd>What do we want to prove?</dd>
<dt>Introduce:</dt> <dd>Try contributing some:
<dl>
<dt>Notation:</dt> <dd>Assigning values and meanings to variables.</dd>
<dt>Organization:</dt> <dd>Recording and arranging what you know.</dd>
<dt>Representation:</dt> <dd>Choose particular representatives which are easier to manipulate.</dd>
</dl></dd>
<dt>Stuck!:</dt> <dd>"I do not understand...", "I do not know what to do about...", "I cannot see how to...",
"I cannot see why..." — try going back to the entry portion "want/know/introduce", or make a
conjecture.</dd>
<dt>AHA!:</dt> <dd>Whenever you have a good idea, write it down. Usually, they are of the form:
"AHA! Try...",
"AHA! Maybe...", or
"AHA! But why...".</dd>
<dt>Check:</dt> <dd>the mathematics. This means:
<ol>
<li>Check calculations;</li>
<li>Check arguments to ensure computations are appropriate;</li>
<li>Consequences of conclusions to see if they are reasonable;</li>
<li>Check that the resolution fits the question, i.e., our answer is the answer to the question asked.</li>
</ol></dd>
<dt>Reflect:</dt> <dd>meditate on:
<ol>
<li>What are the key ideas and key moments?</li>
<li>What are the implications of conjectures and arguments?</li>
<li>Can the resolution be made clearer?</li>
</ol></dd>
<dt>Extend:</dt> <dd>generalize to other settings.</dd>
</dl>
The real trick is to change "I'm stuck, panic!" to "I'm stuck, okay, so what can be done about it?"
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-44224156956433091162011-12-12T11:02:00.000-08:002011-12-12T11:02:21.356-08:00MathJax, Updates<h2>Updates</h2>
<p>
I've just updated the design of the blog. I think it looks negligibly better.
</p>
<p>
I'll also try to write more here rather than on my <tt>notebk</tt>'s <a href="http://code.google.com/p/notebk/w/list">wiki</a>.
</p>
<p>
I posted some <a href="http://notebk.googlecode.com/files/exercises_for_sean.pdf">exercises for Sean <tt>[pdf]</tt></a> which are fun calculus problems.
</p>
<p>
I suspect what I'll do next is write up my notes on differential geometry from Osserman's course I audited a couple years ago, as well as my notes on algebraic topology (from Dr Schwarz's courses). Then I'll work on spin geometry, "commutative geometry", analysis, and so on.
</p>
<p>
By "commutative geometry", I really mean spectral triples using commutative rings (Here I am sloppy, but meh I am a sloppy person!). It is also called <a href="http://en.wikipedia.org/wiki/Differential_calculus_over_commutative_algebras">Differential Calculus over Commutative Algebras</a>, although there are no real texts on the subject...
</p>
<p>
I'll have to review the prime spectrum of the commutative ring $C(M)$ of continuous functions on a topological space $M$ and how it relates to the topology of $M$.
</p>
<p>
If we let $M$ be a smooth manifold, then we work with $C^{\infty}(M)$ — I am told there is a theorem due to Shields which says if $M$ and $N$ are smooth manifolds and $C^{\infty}(M)$ is isomorphic to $C^{\infty}(N)$ then $M$ and $N$ are diffeomorphic. How interesting! But I cannot find this theorem...
</p>
<p>
At any rate, vector bundles over $M$ may be considered by looking at the projective modules over $C^{\infty}(M)$.
</p>
<p>
We consider algebraic analogs for sections, vector fields, covector fields, and so on. It is really quite cute.
</p>
<p>
Noncommutative geometry is similar in setting up a dictionary between "algebraic stuff" and "geometric stuff", at least how Connes approaches it. It's just that the "geometric stuff" we work with is a smooth Riemannian manifold $M$ equipped with a spin structure, we consider spin bundles over it, and so on.
</p>
<h2>MathJax</h2>
<p>
I am experimenting with MathJax on blogger, so bear with me people.
</p>
<p>
My reference for this subject is the <a href="http://webapps.stackexchange.com/questions/13053/mathjax-on-blogger">thread</a> at stackexchange on it.
</p>
<p>
Consider the "Harmonic Series"
\[
\sum^{\infty}_{n=1}\frac{1}{n}=1+\frac{1}{2}+\frac{1}{3}+\cdots
\]
which diverges famously.
</p>
<p>
MathJax uses the <code>\(...\)</code> or <code>$...$</code> for "inline mathematics" and <code>\[...\]</code> or <code>$$...$$</code> for "display math", e.g., the mathematics produced above.
</p>
<p>
I don't know whether to keep it or not, because MathJax is sluggish on some computers. But it is the "way of the future", like blimps and autogyros.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-55467987310514545842011-11-12T11:56:00.000-08:002011-12-15T09:43:00.291-08:00Some more References<p>
I found a number of free references on various subjects in mathematical physics, available <em>free</em>, <em>legally</em>, online.
</p>
<h2>Mathematics</h2>
<h3>Partial Differential Equations</h3>
<ol>
<li>
Alexander Komech, and Andrew Komech, <br />
"Book of Practical PDEs."<br />
Eprint <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-3307.html">Lecture note 33/2007</a>, 125 pages.
</li>
<li>
Alexander Komech, <br />
"Lectures on elliptic partial differential equations (Pseudodifferential operator approach)". <br />
Eprint <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-3207.html">Lecture note 32/2007</a>, 44 pages.
</li>
<li>
Norbert Ortner, and Peter Wagner, <br />
"Distribution-Valued Analytic Functions - Theory and Applications". <br />
Eprint <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-3708.html">Lecture note 37/2008</a>, 133 pages.
</li>
<li>
A.D.R. Choudary, Saima Parveen, Constantin Varsan, <br />
"Partial Differential Equations An Introduction". <br />
Eprint <tt><a href="http://arxiv.org/abs/1004.2134">arXiv:1004.2134v1</a> [math.AP]</tt>, 204 pages.
</li>
<li>
Robert Geroch, <br />
"Partial Differential Equations of Physics". <br />
Eprint <tt><a href="http://arxiv.org/abs/gr-qc/9602055">arXiv:gr-qc/9602055v1</a></tt>, 57 pages.
</li>
</ol>
<h3>Differential Geometry</h3>
<ol>
<li>
Jürgen Jost, <br />
"The principles and concepts of geometric analysis". <br />
Eprint <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-1201.html">Lecture note 12/2001</a>, 15 pages.<br />
Note: "geometric analysis" studies things like Morse functions, etc.
</li>
</ol>
<h3>Homological Algebra</h3>
<!--
More lecture notes on homological algebra:
1. http://www.math.umass.edu/~mirkovic/A.COURSE.notes/3.HomologicalAlgebra/HA/2.Spring06/A.pdf
2. http://www.maths.gla.ac.uk/~phk/kap1.pdf
3. http://www.cis.upenn.edu/~cis610/alg5.pdf
4. http://people.math.jussieu.fr/~schapira/lectnotes/HomAl.pdf
5. http://www.staff.science.uu.nl/~lukac101/homalg2007.pdf
6. http://www.math.uwo.ca/~jardine/papers/HomAlg/index.shtml
7. http://arxiv.org/abs/0903.2563
-->
<ol>
<li>
Volker Runde, <br />
"Abstract harmonic analysis, homological algebra, and operator spaces".<br />
Eprint <tt><a href="http://arxiv.org/abs/math/0206041">arXiv:math/0206041v6</a> [math.FA]</tt>, 12 pages.
</li>
<li>
Mohamed Barakat, Markus Lange-Hegermann, <br />
"An Axiomatic Setup for Algorithmic Homological Algebra and an Alternative Approach to Localization". <br />
Eprint <tt><a href="http://arxiv.org/abs/1003.1943v3">arXiv:1003.1943v3</a> [math.AC]</tt>, 30 pages.
</li>
<li>
Joseph Krasil'shchik, Alexander Verbovetsky, <br />
"Homological Methods in Equations of Mathematical Physics". <br />
Eprint <tt><a href="http://arxiv.org/abs/math/9808130"> arXiv:math/9808130v2</a> [math.DG]</tt>, 150 pages.
</li>
<li>
R. P. Thomas, <br />
"Derived categories for the working mathematician". <br />
Eprint <tt><a href="http://arxiv.org/abs/math/0001045"> arXiv:math/0001045v2</a> [math.AG]</tt>, 13 pages.
</li>
<li>
Semen Podkorytov, <br />
"On homology of map spaces". <br />
Eprint <tt><a href="http://arxiv.org/abs/1102.1645">arXiv:1102.1645v1</a> [math.AT]</tt>, 9 pages.
</li>
</ol>
<h3>K-Theory</h3>
<ol>
<li>
Max Karoubi, <br />
"K-theory. An elementary introduction". <br />
Eprint <tt><a href="http://arxiv.org/abs/math/0602082">arXiv:math/0602082v1</a> [math.KT]</tt>, 22 pages
</li>
<li>
Ioannis P. Zois, <br />
"18 Lectures on K-Theory". <br />
Eprint <tt><a href="http://arxiv.org/abs/1008.1346">arXiv:1008.1346v1</a> [math.KT]</tt>, 137 pages.
</li>
</ol>
<h3>Mathematical Physics</h3>
This is kind of physics stuff, kind of mathematics stuff, and doesn't really fit cleanly into either category.
<ol>
<li>
Andreas Knauf, <br />
"Number theory, dynamical systems and statistical mechanics". <br />
Eprint: <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-0398.html">Lecture note 3/1998</a>, 41 pages.
</li>
<li>
Bruce Hunt, <br />
"Geometry of super Yang-Mills and supergravity". <br />
Eprint: <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-0499.html">Lecture note 4/1999</a>, 59 pages.
</li>
<li>
Bruce Hunt, <br />
"Conformal and gauge symmetry in D = 2 QFT". <br />
Eprint: <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-0599.html">Lecture note 5/1999</a>, 115 pages.
</li>
<li>
Friedemann Brandt, <br />
"Lectures on Supergravity". <br />
Eprint: <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-1302.html">Lecture note 13/2002</a>, 50 pages.
</li>
<li>
Alexander Komech, <br />
"Lectures on Quantum Mechanics (nonlinear PDEs point of view)". <br />
Eprint: <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-2505.html">Lecture note 25/2005</a>, 231 pages.
</li>
</ol>
<h3>Physics</h3>
<ol>
<li>
Christopher J. Fewster, <br />
"Lectures on quantum field theory in curved spacetime". <br />
Eprint <a href="http://www.mis.mpg.de/publications/other-series/ln/lecturenote-3908.html">Lecture note 39/2008</a>, 62 pages.
</li>
</ol>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0tag:blogger.com,1999:blog-3776650590176552530.post-14220213428558107382011-11-01T09:11:00.000-07:002011-11-01T09:11:45.269-07:00Quantifiers<p>
I have been thinking about quantifiers, and the notation around them...since most books use differing notation that is not easy to read.
</p>
<p>
Since I am trying to be consistent throughout all of mathematics, it seems natural to suggest that the colon ":" should be read as "such that".
</p>
<p>
In this case, one should write "∃x : P(x)" since one usually writes in natural language "There is some <i>x</i> such that <i>P(x)</i>".
</p>
<p>
Likewise we often find expressions "If <i>x</i>∈<i>X</i>, then <i>P(x)</i>"...so it would be natural to use the notation "∀x∈X, P(x)" or "∀ x, x∈X and P(x)". After all, "x∈ X" is really a predicate "<code>isIn</code>(x,X)"...
</p>
<p>
I'm currently revising my notes on logic in Fascicles 0 of my <em>Elements of Mathematics</em>, which is why I bring this notational problem up! Hopefully, I will be done with revising and improving my logic chapter soon.
</p>pqnelsonhttp://www.blogger.com/profile/12779680952736168655noreply@blogger.com0