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<head><title>bnewbold thesis</title></head>
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<h1 style="border-bottom: 2px solid;">
Journal: Feb 26, 2009</h1>
<i>Bryan Newbold, <a href="mailto:bnewbold@mit.edu">bnewbold@mit.edu</a></i><br />
<i><a href="http://web.mit.edu/bnewbold/thesis/">
http://web.mit.edu/bnewbold/thesis/</a></i>
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I've started writing some things up, look in the <a href="../draft/">draft</a>
folder. Very rough, just putting down ideas to see what structure emerges.
<h3>Type Hierarchies</h3>
The <b>sage</b> math rough documentation on coercion and arithmetic is at
<a href="http://www.sagemath.org/doc/prog/node22.html">
http://www.sagemath.org/doc/prog/node22.html</a>. It mostly uses the python
object system with a variety of Fields to handle arbitrary precision numerical
data (RealField and ComplexField can be specified by precision).
<p />
<b>AXIOM</b> has great graphs of their huge algebraic hierarchy (warning: large
slow svg files!): <a href="http://axiom-developer.org/axiom-website/dotabb.html">
abreviated category and domain</a>,
<a href="http://axiom-developer.org/axiom-website/hp.svg">everything?</a>.
<br/>
Axiom uses strict type checking (everything gets compiled down?); every type is
a category which allows for checking of properties and mathematical
correctness. The interpreter guesses what type/category user input should be.
There is a good overview <a
href="http://axiom-developer.org/axiom-website/hyperdoc/axbook/section-0.2.xhtml">here</a>.
To paraphrase: every object is of a single type called it's domain ("domain of
computation", eg "String", "Float". Domains themselves have a type called a
Category (disclaimer: I don't know anything about category theory). The
Categories for hierarchies ("directed acyclic graphs"), eg:
<pre>
SetCategory +---- Ring ---- IntegralDomain ---- Field
|
+---- Finite ---+
| \
+---- OrderedSet -----+ OrderedFinite
</pre>
Packages are special domains which just have associated polymorphic operators
(so they are generic on inputs?).
<br />From the documentation:
"Roman numerals are also available for those special occasions."
<br />Interesting stuff! The main book explains a lot about how types are
coerced together for operations etc.
<p />
<b>Singular</b> has a list of types
<a href="http://www.singular.uni-kl.de/Manual/latest/sing_67.htm#SEC107">here
</a>; it includes:
bigint def ideal int intmat intvec link
list map matrix module number package poly
proc qring resolution ring string vector.
<p />
<b>Cadabra</b> is an interactive C++ system designed specifically for tensor
manipulation in physics (usually HEP?). It describes some of it's important
features as:
<ul>
<li />Built-in understanding of dummy indices and dummy symbols, including
their automatic relabelling when necessary.
<li />Powerful algorithms for canonicalisation of objects with index
symmetries, both mono-term and multi-term.
<li />A new way to deal with products of non-commuting objects, enabling a
notation which is identical to standard physicist's notation (i.e. no need for
special non- commuting product operators).
<li />A flexible way to associate meaning ("type information") to tensors by
attaching them to "properties".
</ul>
<p />
With respect to some sort of absolute mathematical type hierarchy, one of my
favorite papers by Max Tegmark ("The Mathematical Universe"):
<br />
<img src="26feb2009-tegmark-toe.gif" />
<h3>Tools Needed</h3>
Basic multivariable calculus tools to be extended for differential geometry:
<dl>
<dt />integral
<dt />derivative
<dt />partial derivative
<dt />curl
<dt />divergence
<dt />taylor expansion
</dl>
Eg, <code>(taylor function variable point number)</code> would give a list
of the first *number* terms of the taylor expansion of *function* by
*variable* around *point*.
<br /><br />
<a href="19feb2009.html"><i>(previous entry)</i></a>
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