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"ALGHCA.cdf is a package of procedures to create, manipulate, identify, and \
use small Algebraic Loops, Groups, \[OpenCurlyDoubleQuote]Hoops\
\[CloseCurlyDoubleQuote] and Clifford algebras. Hoops are relevant to physics \
because their Moufang vector-division and Frobenius conservation properties \
provide the conserved symmetries that define forces and particles. A database \
includes over 80 hoops and most groups with up to 96 elements. \
ALGHACAtest.cdf validates the database, ALGHCAGloss.cdf explains the terms \
employed, Hoops.cdf explains the procedures and provides example \
calculations. \nThe notebooks are designed to be read by non-Mathematica \
users who have downloaded CDFPlayer; Mathematica users (Version 7 or later) \
can test them with their own data. They provide background material on \
algebraic loops and groups and then demonstrate \
\[OpenCurlyDoubleQuote]Symmetry-conserving Generalised-sign Vector-division \
Algebras\[CloseCurlyDoubleQuote] or \[OpenCurlyDoubleQuote]Hoops\
\[CloseCurlyDoubleQuote]. Hoops unify most standard algebras - complex, \
Cayley-Dickson, Clifford, Davenport, Dirac, non-commutative, octonion, \
Olariu, Pauli-sigma, quaternion, real, spinor, wedge, etc. algebras. A \
database of over 900 loops is provided. Many \
\[OpenCurlyDoubleQuote]mathematical truths\[CloseCurlyDoubleQuote] are only \
valid in the context of real and complex numbers because they are degenerate \
cases of more general \[OpenCurlyDoubleQuote]Hoop\[CloseCurlyDoubleQuote] \
relationships.\nThe key features of Hoop algebras are: \n1) Algebraic Loops \
define multiplication of \[OpenCurlyDoubleQuote]vecs\[CloseCurlyDoubleQuote] \
(sets of unsigned \[OpenCurlyDoubleQuote]Primal\[CloseCurlyDoubleQuote] \
numbers). Loops having C",
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" elements) as a central subgroup can be \[OpenCurlyDoubleQuote]folded\
\[CloseCurlyDoubleQuote] to introduce ",
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"\[CloseCurlyQuote]th roots of unity as \[OpenCurlyDoubleQuote]generalised \
signs\[CloseCurlyDoubleQuote].\n2) \[OpenCurlyDoubleQuote]Folding\
\[CloseCurlyDoubleQuote] converts loops (with one operation, multiply) into \
algebras with generalised addition/subtraction as a second operation; vecs \
fold to vectors. Real and complex numbers are folded from pairs (r=2) or \
quads (r=4) of primal numbers.\n3) Hoops are algebras possessing Frobenius\
\[CloseCurlyQuote]s determinant-conservation property. This provides \
symmetries (or \[OpenCurlyDoubleQuote]sizes\[CloseCurlyDoubleQuote], factors \
of the symbolic multiplication table) that are conserved on multiplication. \
Sizes are also denominators for the partial-fraction formulation of \
multiplicative inverses.\n4) Multiplication and division are implemented by \
the \[OpenCurlyDoubleQuote]hoopTimes\[CloseCurlyDoubleQuote] procedure; \
division is pre-multiplication by the inverse, given by \
\[OpenCurlyDoubleQuote]hoopInverse\[CloseCurlyDoubleQuote].\n5) Sizes can be \
zero; operations are then \[OpenCurlyDoubleQuote]projected\
\[CloseCurlyDoubleQuote] onto a sub-algebra (of reduced symmetry) with the \
same sizes zeroed. To maintain size conservation, hoopTimes \
\[OpenCurlyDoubleQuote]projects\[CloseCurlyDoubleQuote] results into the \
sub-algebra and \[OpenCurlyDoubleQuote]ejects\[CloseCurlyDoubleQuote] \
remainders. This resembles particle interactions and decays, with remainders \
corresponding to ejected particles with different symmetries.\n6) Some hoops \
have Polar-duals with additive angles, generalizing the x,y and r,theta \
complex-plane duality. This allows the calculation of vector powers and \
roots, via \[OpenCurlyDoubleQuote]hoopPower\[CloseCurlyDoubleQuote].\n7) Some \
unital Polar-duals have (deBroglie-like) multi-phase orbits and intrinsic \
(Planck-like) radii. Their elements include half-spin quantum operators that \
are neither real nor complex. Hoop algebras give tantalizing suggestions \
about the re-interpretation of mathematical physics - they may describe \
bosons and fermions. Ternary (quark) symmetries occur without invoking \
octonion triality. Kaluza-Klein orbital velocities could provide mass. Unital \
orbits may explain the \[OpenCurlyDoubleQuote]law of large numbers\
\[CloseCurlyDoubleQuote]. Hoop maths may describe non-point-like particles \
with an intrinsic sub-Planck scale.\nComplex numbers can express much of \
mathematics, but are a Procrustean bed; some parts relevant to physics do not \
fit on it. I have scratched the surface of a more fundamental mathematics, \
and found many features that may be relevant to mathematical physics. 8 May \
2012."
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