From cc2eb6e18f516f4ee7bf70272841842fd4ca9e53 Mon Sep 17 00:00:00 2001 From: bnewbold Date: Fri, 27 Feb 2009 13:39:43 -0500 Subject: journal entry --- journal/26feb2009.html | 106 +++++++++++++++++++++++++++++++++++++++++++++++++ 1 file changed, 106 insertions(+) create mode 100644 journal/26feb2009.html (limited to 'journal/26feb2009.html') diff --git a/journal/26feb2009.html b/journal/26feb2009.html new file mode 100644 index 0000000..052534d --- /dev/null +++ b/journal/26feb2009.html @@ -0,0 +1,106 @@ + + +bnewbold thesis + +

+Journal: Feb 26, 2009

+Bryan Newbold, bnewbold@mit.edu
+ +http://web.mit.edu/bnewbold/thesis/ +

+ + + +I've started writing some things up, look in the draft +folder. Very rough, just putting down ideas to see what structure emerges. + +

Type Hierarchies

+The sage math rough documentation on coercion and arithmetic is at + +http://www.sagemath.org/doc/prog/node22.html. 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). +

+AXIOM has great graphs of their huge algebraic hierarchy (warning: large +slow svg files!): +abreviated category and domain, +everything?. +
+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 here. +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?). +
From the documentation: +"Roman numerals are also available for those special occasions." +
Interesting stuff! The main book explains a lot about how types are +coerced together for operations etc. + +

+Singular has a list of types +here +; it includes: +bigint def ideal int intmat intvec link +list map matrix module number package poly +proc qring resolution ring string vector. + +

+Cadabra is an interactive C++ system designed specifically for tensor +manipulation in physics (usually HEP?). It describes some of it's important +features as: +

Built-in understanding of dummy indices and dummy symbols, including +their automatic relabelling when necessary. +
Powerful algorithms for canonicalisation of objects with index +symmetries, both mono-term and multi-term. +
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). +
A flexible way to associate meaning ("type information") to tensors by +attaching them to "properties". +

+ +

+With respect to some sort of absolute mathematical type hierarchy, one of my +favorite papers by Max Tegmark ("The Mathematical Universe"): +
+ + +

Tools Needed

+Basic multivariable calculus tools to be extended for differential geometry: +

integral +
derivative +
partial derivative +
curl +
divergence +
taylor expansion +

+ +Eg, (taylor function variable point number) would give a list +of the first *number* terms of the taylor expansion of *function* by +*variable* around *point*. + +

+(previous entry) + + + + + -- cgit v1.2.3