### Morphological Group Theory of Material Structure

Ziqiang Zhou
2018 Journal of Applied Mathematics and Physics
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
more » ... 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. spaces." The time has arrived to study complicated operators on complicated spaces  . The algebraic geometry model required for morphological analysis is just such a kind of difficult question. Here I want to discuss the construction about IMM for descriptions of the different complex structures for materials science and find some basic internal relations between the three branches of mathematics, namely, analysis, topology, and algebraic geometry. I begin with a morphological equation   based Jordan curve theorem  , then through introducing a method of set theory and Definition Since the set A of points inside a simple closed curve C must be a connected region, I call the connected region as a morphological element (ME)  , but the set B of points outside the simple closed curve is also a connected region, so it is Z. Q. Zhou also a ME. To distinguish this two kinds of ME, we can call the former connected region as Jordan ME (or   J ), which can be used to describe the morphology of new phase α , and the latter one as   ME H , which can be used to describe the morphology of the parent phase austenite γ  . So the morphology M is consisted of two ME of In the morphological Equation (2.2),