1,342 research outputs found
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
Tarski's influence on computer science
The influence of Alfred Tarski on computer science was indirect but
significant in a number of directions and was in certain respects fundamental.
Here surveyed is the work of Tarski on the decision procedure for algebra and
geometry, the method of elimination of quantifiers, the semantics of formal
languages, modeltheoretic preservation theorems, and algebraic logic; various
connections of each with computer science are taken up
Spectral Triples on Thermodynamic Formalism and Dixmier Trace Representations of Gibbs: theory and examples
In this paper we construct spectral triples on the symbolic space
when the alphabet is finite. We describe some new results for the associated
Dixmier trace representations for Gibbs probabilities (for potentials with less
regularity than H\"older) and for a certain class of functions. The Dixmier
trace representation can be expressed as the limit of a certain zeta function
obtained from high order iterations of the Ruelle operator. Among other things
we consider a class of examples where we can exhibit the explicit expression
for the zeta function. We are also able to apply our reasoning for some
parameters of the Dyson model (a potential on the symbolic space
) and for a certain class of observables. Nice results by
R. Sharp, M.~Kesseb\"ohmer and T.~Samuel for Dixmier trace representations of
Gibbs probabilities considered the case where the potential is of H\"older
class. We also analyze a particular case of a pathological continuous potential
where the Dixmier trace representation - via the associated zeta function - is
not true.Comment: the tile was modified and there are two more author
Formalized proof, computation, and the construction problem in algebraic geometry
An informal discussion of how the construction problem in algebraic geometry
motivates the search for formal proof methods. Also includes a brief discussion
of my own progress up to now, which concerns the formalization of category
theory within a ZFC-like environment
- β¦