TITLE:
Morphological Group Theory of Material Structure
AUTHORS:
Ziqiang Zhou
KEYWORDS:
Morphological Equation, Morphological Function, Algebraic Geometry Model, Complex Geometric Structures, Hodge Conjecture
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.6 No.1,
January
5,
2018
ABSTRACT:
The correct formulation and understanding of micro-images is one of the difficulties that occur to microstructures science today, which need to develop a new appropriate mathematics for micro-images of matter system. Here I study the image mathematics and physics description of micro-images of material system by topology, set theory, symbolic logic and show that there is a naturally morphological equation, that is a law of qualitative structure of matter system, the law of the unity of two kinds of morphological structure (Jordan and hidden structure), which can be used to describe not only the common feature of different correlated matter, but also to correct classify the micro-images into different classes, so that to study the morphology groups for materials science and Algebraic geometry. The morphology equation can be found a number of applications for the observation and analysis of micro-images of material system and other natural sciences, some important basic concepts of algebraic geometry can also be newly explained by the morphology equation, such as: 1) To construct the image-mathematical language and to construct the image mathematics model (IMM) for microstructures; 2) To construct complex geometric structures (Concave polygon) then analyze these complex shape structure by analytic geometry and algebraic geometry, to study complicated operators on complicated spaces; 3) A new explanation for the logical basis, concept definition and proof way of algebraic geometry and uses it to analyze morphological structure of the new and parent phase and the problem of Hodge’s theory and structure type, and points out that there may be a counterexamples for Hodge’s conjecture.