A Recipe for Symbolic Geometric Computing: Long Geometric Product, BREEFS and Clifford Factorization

In symbolic computing, a major bottleneck is middle expression swell. Symbolic geometric computing based on invariant algebras can alleviate this difficulty. For example, the size of projective geometric computing based on bracket algebra can often be restrained to two terms, using final polynomials, area method, Cayley expansion, etc. This is the "binomial" feature of projective geometric computing in the language of bracket algebra. In this paper we report a stunning discovery in Euclidean geometric computing: the term preservation phenomenon. Input an expression in the language of Null Bracket Algebra (NBA), by the recipe we are to propose in this paper, the computing procedure can often be controlled to within the same number of terms as the input, through to the end. In particular, the conclusions of most Euclidean geometric theorems can be expressed by monomials in NBA, and the expression size in the proving procedure can often be controlled to within one term! Euclidean geometric computing can now be announced as having a "monomial" feature in the language of NBA. The recipe is composed of three parts: use long geometric product to represent and compute multiplicatively, use "BREEFS" to control the expression size locally, and use Clifford factorization for term reduction and transition from algebra to geometry. By the time this paper is being written, the recipe has been tested by 70+ examples from \cite{chou}, among which 30+ have monomial proofs. Among those outside the scope, the famous Miquel's five-circle theorem \cite{chou2}, whose analytic proof is straightforward but very difficult symbolic computing, is discovered to have a 3-termed elegant proof with the recipe

Applications of combinatorial groups to Hopf invariant and the exponent problem

Combinatorial groups together with the groups of natural coalgebra transformations of tensor algebras are linked to the groups of homotopy classes of maps from the James construction to a loop space. This connection gives rise to applications to homotopy theory. The Hopf invariants of the Whitehead products are studied and a rate of exponent growth for the strong version of the Barratt Conjecture is given.Comment: This is the version published by Algebraic & Geometric Topology on 29 November 200

Modular dynamics in diamonds

We investigate the relation between the actions of Tomita-Takesaki modular operators for local von Neumann algebras in the vacuum for free massive and massless bosons in four dimensional Minkowskian spacetime. In particular, we prove a long-standing conjecture that says that the generators of the mentioned actions differ by a pseudo-differential operator of order zero. To get that, one needs a careful analysis of the interplay of the theories in the bulk and at the boundary of double cones (a.k.a. diamonds). After introducing some technicalities, we prove the crucial result that the vacuum state for massive bosons in the bulk of a double cone restricts to a KMS state at its boundary, and that the restriction of the algebra at the boundary does not depend anymore on the mass. The origin of such result lies in a careful treatment of classical Cauchy and Goursat problems for the Klein-Gordon equation as well as the application of known general mathematical techniques, concerning the interplay of algebraic structures related with the bulk and algebraic structures related with the boundary of the double cone, arising from quantum field theories in curved spacetime. Our procedure gives explicit formulas for the modular group and its generator in terms of integral operators acting on symplectic space of solutions of massive Klein-Gordon Cauchy problem.Comment: 48 page