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

Research on motion-emotion metaphor and its social cognitive mechanism—A case study of Chinese Mandarin, Yi language and English

Chinese motion-emotion metaphor and its social cognitive mechanism are explored, for the first time, with a comparison between Mandarin Chinese, the Yi language and English. The interaction between motions and emotions is the key to do the research from the perspective of cognitive functionalism. Cognitive functionalism argues that language reflects people’s consciousness, and the cognitive aspect of language interacts with the communicative function of language very well. According to this argument, motion-emotion metaphor, as a popular language phenomenon, can testify to such interactions. The comparative analysis of motion-emotion metaphors, from the perspective of cognitive functionalism, in this paper has proved to take the following aspects into consideration: the subjects’ experiences of physical motions and their effects on objects; the universality and the specificity of such experience; the emotions’ observable traits and their related motions; the common knowledge and normal beliefs among the motions’ subjects and their surrounding contexts