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author | bnewbold <bnewbold@eta.mit.edu> | 2009-02-27 13:39:56 -0500 |
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committer | bnewbold <bnewbold@eta.mit.edu> | 2009-02-27 13:39:56 -0500 |

commit | 4d81434f2c00dc73b1364a92edd05a62cd4be2a7 (patch) | |

tree | 261e332170865aec212b0648fe246a5107d1ee06 | |

parent | cc2eb6e18f516f4ee7bf70272841842fd4ca9e53 (diff) | |

download | 8thesis-4d81434f2c00dc73b1364a92edd05a62cd4be2a7.tar.gz 8thesis-4d81434f2c00dc73b1364a92edd05a62cd4be2a7.zip |

draft notes

-rw-r--r-- | draft/motivation.txt | 26 | ||||

-rw-r--r-- | draft/physical-motivations.txt | 35 | ||||

-rw-r--r-- | draft/types-operators.txt | 47 |

3 files changed, 108 insertions, 0 deletions

diff --git a/draft/motivation.txt b/draft/motivation.txt new file mode 100644 index 0000000..0336bd4 --- /dev/null +++ b/draft/motivation.txt @@ -0,0 +1,26 @@ +====================================== +Motivation +====================================== + +* reuse of tools, models, system, code both "up" and "down": can be built upon + and extended, or built "under" and applied as an extension to other existing + systems + +* seperation/abstraction of model, methods, and types from implementation, + allows for metaprogramming, compilation, symbolic analysis, potential for + proofs + +* conceptual simplicity/purity of a reference implementation allows it to serve + as a blueprint or recipe for similar systems or reimplementation + +* open source allows the entire heirarchy, tool chain, and implementation to be + explored, shared, extended, abused + +* unambiguous notation and syntax checking + +* educational exploration + +* same tools for theoretical development (symbolic, checking, searches) and + efficient numerical computation (compilation, parallelizing combinators, + evaluation flow control, optimization) + diff --git a/draft/physical-motivations.txt b/draft/physical-motivations.txt new file mode 100644 index 0000000..5d665f4 --- /dev/null +++ b/draft/physical-motivations.txt @@ -0,0 +1,35 @@ +========================================= +Physical Observations, Applications +========================================= + + +Michealson-Morley and Gravitational Redshift <=> Locally Lorentz +-------------------------------------------------------------------- +(motivational) + + +Precession Perhelion of Mercury <=> Schwartzchild? +------------------------------------------------------ +(explicit calculation) + + +Deflection of Light <=> Geodesics +----------------------------------- +From Carroll p292, +G M_sun /(c^2) = 1.48e5 cm, R_sun = 6.96e10 cm, deflection angle of 1.75 arcsec + + +Gravitational Waves <=> Weak Field Approximation, Perturbation Theory +--------------------------------------------------------------------------- +See Carroll p293 + + +GPS Corrections <=> Geodesics with E&M +------------------------------------------ +(order of magnitude?) + + +Gravity Probe B <=> Frame Dragging, geodetic +---------------------------------------------------------------------------- +(probably not...) + diff --git a/draft/types-operators.txt b/draft/types-operators.txt new file mode 100644 index 0000000..68bd058 --- /dev/null +++ b/draft/types-operators.txt @@ -0,0 +1,47 @@ +=============================== +Types and Operators +=============================== + +The careful definition of abstract types and the manipulation and handling of +instances by operators is the heart of any computational algebra system +(or indeed any data-driven symbolic system). + +The [R5R2] defines a simple "tower" of number subtypes (p.19): + + number + complex + real + rational + integer + +Each subtype is a subset of the subtype above it, so that each rational is +a complex but not every real is an integer. Additionally, every Scheme +number is either exact or inexact. The selection of the appropriate +implementation of operations like addition, exponentiation, or rounding for +arbitrary combinations of numbers and their return types is usually called +*operator dispatch*. For an operator of N arguments of M possible types, one +can imagine an M**N cell table defining the appropriate procedure (or +built-in hardware instruction) and return type for each combination of types. + +Such a table would be come unmanagably large for systems with even a modest +number of operators and types: for example each of the basic binary operators +(*, /, -, +, exp) would require it's own 2**(2*5)=1024 element table for the +basic types described above. Procedural operator dispatch and coersion allow +for more natural handling of type heirarchies. + +More rigorously defined algebras do not necessarily follow the simple strict +ordering of types like the Scheme number tower, or even a proper tree +structure: + +[figure: geometrical spaces from Merseve Geometry] +[figure: number types from Merseve Algebra] +[figure: math object hierarchy from Tegmark?] + +Types and Operators for Differential Geometry and Field Theories +------------------------------------------------------------------------ + +[topics: disjointness, + +Implementation +------------------------------------------------------------------------ + |