[DRAFT]
Arithmetic Geometry Language
4 (Theorem Gillet-Soule)
n-dimensional real vector space and its inner product
(V,
h)
x, y (element) Lambda
h(x,
y) (element) Z
Free base of Lambda
{w1, ...,
wn}
(h(wi, wj)) (element)
GL (n, Z)
Boundary symmetry vertex set
K
6(upper)-n <= #(K (cap)
Lambda) / #(K*(cap) Lambda)volh(K) <=(3/2)(upper)n(n!)(upper)2
5 (Lemma)
Boundary symmetry vertex set with inner point
K(cap)Lambda generates V.
#(K(cap)Lambda)volh(K*)volh(V/A)<=6(upper)n
(Notice)
5(Lemma) is useful as generation theorem for
language.
6 (Lemma)
(V, h, Lambda) is uni-modular.
Saturated sub Z module of Lambda
T
T(orthogonal) = {x(element)Lambda
| for all y(element)T, h(x, y)
= 0}
V' s sub-vector space generated by T W
W's orthogonal complement W(orthogonal)
T(orthogonal) is
lattice point set of W(orthogonal).
7 (Notice)
6(Lemma) ideally satisfies language that is constructed by lattice.
It is called <lattice language>.
[Reference]
Nested Torus Theory / Plane, Pillar, Torus / Tokyo May
27, 2006
Tokyo August
11, 2009
Sekinan Research Field of Language
sekinan.org