Ermakov System and Plane Curve Motions in Affine Geometries

It is shown that the Pinney equation, Ermakov systems, and their higher-order generalizations describeself-similar solutions of plane curve motions in centro-affine and affine geometries.

On the Origin of Charge-Asymmetric Matter. I. Geometry of the Dirac Field

This work presents new round of the author’s pursuit for consistent description of the finite sized objects in classical and quantum field theory. Current paper lays out an adequate mathematical background for this quest. A novel framework of the matter-induced physical affine geometry is developed. Within this framework, (1) an intrinsic nonlinearity of the Dirac equation becomes self-explanatory;(2) the spherical symmetry of an isolated localized object is of dynamic origin;(3) the auto-localization is a trivial consequence of nonlinearity and wave nature of the Dirac field;(4) localized objects are split into two major categories that are clearly associated with the positive and negative charges;(5) of these, only the former can be stable as isolated objects, which explains the global charge asymmetry of the matter observed in Nature. In the second paper, the nonlinear Dirac equation is written down explicitly. It is solved in one-body approximation (in absence of external fields). Its two analytic solutions unequivocally are positive (stable) and negative (unstable) isolated charges. From the author’s current perspective, the so for obtained results must be developed further and applied to various practical and fundamental problems in particle and nuclear physics, and also in cosmology.

PRIMARY RESEARCH ON FRACTAL GEOMETRY OF MERIDIAN THEORY

In meridian theory of traditional Chinese medicine (TCM), the geometrical descriptions can be traced back to the remote ancient times in China, mainly in The Yellow Emperor’s Internal Classic (The Internal Classic in short). Euclid’s geometry, topology and other classic mathematics are all at their wit’s end to explain the high complexity and non clinear phenomenon of the meridian. In recent over 2000 years, the meridian phenomenon has been being the challenge to fundamental mathematics. Fractral geometry, founded by Mandelbrot (1975), is a branch of learning for investigating irregular geometrical curves. It has successfully solved some qualitative and quantitative problems about the topographical structure of molecular Brown’s movement curve and other irregular complicated curves and geometrical characters. The characteristics of geometrical topographical structure of meridian and its phenomenon belong to the research category of Fractal Geometry. The author of this paper believes that Fractal Geometry may provide a useful mathematical tool and a possible way for revealing the enigma of acup moxibustion meridian theory. The human body is of basic characters of Fractal Geometry in structure, while meridian is the expression form of Fractal structure of the human body. The basic Fractal geometrical characters of meridian are: self similarity, self affinity, symmetry, minute structure and self avoidance, which has been applied for thousands of years in clinic, such as “taking the acupoints on the right side of the body in cases of disorders appearing on the left side and vice versa". The basic characters of meridians are 1) symmetry of the 12 regular meridians on the bilateral sides of the body (symmetry); 2) similarity in characters and actions of acupoints of the same one meridian (self similarity); 3) taking acupoints on the lower part of the body when disorders occurring on the upper part of the body; and taking acupoints on the upper part of the body if disorders appearing on the lower part (self affinity); 4) micro acupuncture system including hand acupuncture, foot acupuncture, scalp acupuncture, auricular acupuncture and eye acupuncture (minute structure); and 5) systematical running of needling sensation (self avoidance).

Some Theorems in Affine Differential Geometry

In this paper we prove that an affine hypersphere with scalar curvature zero in a unimodular affine space of dimensionn+1 must be contained either in an elliptic paraboloid or in an affine image of the hypersurfacex 1 x 2...x n+1=const. We prove also that an affine complete, affine maximal surface is an elliptic paraboloid if its affine normals omit 4 or more directions in general position.

Affine bracket algebra theory and algorithms and their applications in mechanical theorem proving

This paper discusses two problems:one is some important theories and algorithms of affine bracket algebra;the other is about their applications in mechanical theorem proving.First we give some efficient algorithms including the boundary expanding algorithm which is a key feature in application.We analyze the characteristics of the boundary operator and this is the base for the implementation of the system.We also give some new theories or methods about the exact division,the representations and structure of affine geometry and so on.In practice,we implement the mechanical auto-proving system in Maple 10 based on the above algorithms and theories.Also we test about more than 100 examples and compare the results with the methods before.

Blaschke Hypersurfaces with Symmetric Shape Operator

A Blaschke hypersurface admits S symmetry if and only if S（X, Y） = S（Y, X）. We prove that the shape operator has only one eigenvalue. And such Blaschke surfaces are classified as affine spheres or ruled surfaces.