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authorbnewbold <bnewbold@eta.mit.edu>2009-02-27 13:39:43 -0500
committerbnewbold <bnewbold@eta.mit.edu>2009-02-27 13:39:43 -0500
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-<a href="16feb2009.html"><i>(previous entry)</i></a>
+<a href="16feb2009.html"><i>(previous entry)</i></a> -
+<a href="26feb2009.html"><i>(next entry)</i></a>
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+<!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Transitional//EN"
+<html xmlns="http://www.w3.org/1999/xhtml">
+<head><title>bnewbold thesis</title></head>
+<body style="margin: 25px; font-family: helvetica;">
+<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/">
+<br /><p />
+<!-- ================================================================ -->
+<!-- ================================================================ -->
+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>.
+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
+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:
+SetCategory +---- Ring ---- IntegralDomain ---- Field
+ |
+ +---- Finite ---+
+ | \
+ +---- OrderedSet -----+ OrderedFinite
+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:
+<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".
+<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:
+ <dt />integral
+ <dt />derivative
+ <dt />partial derivative
+ <dt />curl
+ <dt />divergence
+ <dt />taylor expansion
+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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