From cc2eb6e18f516f4ee7bf70272841842fd4ca9e53 Mon Sep 17 00:00:00 2001 From: bnewbold Date: Fri, 27 Feb 2009 13:39:43 -0500 Subject: journal entry --- journal/19feb2009.html | 3 +- journal/26feb2009-tegmark-toe.gif | Bin 0 -> 20661 bytes journal/26feb2009.html | 106 ++++++++++++++++++++++++++++++++++++++ 3 files changed, 108 insertions(+), 1 deletion(-) create mode 100644 journal/26feb2009-tegmark-toe.gif create mode 100644 journal/26feb2009.html diff --git a/journal/19feb2009.html b/journal/19feb2009.html index 54183f6..5e60292 100644 --- a/journal/19feb2009.html +++ b/journal/19feb2009.html @@ -45,7 +45,8 @@ still looks fun! Grafted from readscheme.org

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+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: +

+ +

+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