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.
Highlights
Morphology is a branch in biology which dealing with the form and structure of organisms and their specific structural features
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 the common feature of different correlated matter, and to correct classify the micro-images into different classes, so that to study the morphology groups for materials science and Algebraic geometry
How to construct an images mathematics model (IMM), which can to convert from 2D images to 3D images, from local region images to whole region images, from micro images to macro images, these are the basic problems to be solved for correlated matter science including materials science and biology
Summary
Morphology is a branch in biology which dealing with the form and structure of organisms and their specific structural features. The different types of mathematicians has his emphasis to study the algebraic geometry problem, some focus on algebra, some focus on geometry, in which the advantage of algebraic methods can be far away from geometric intuition, the study of n-dimensional problem is not too difficult It brings up some problems, it is easy to predict structure types, but it is difficult to construct these complex types. 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 [9] [10] based Jordan curve theorem [11], through introducing a method of set theory and symbolic logic [12] [13] to construct the model of algebraic geometry problem (AGM), the middle step in the process consists of finding a image-mathematical solution to the image problem by algebraic geometry and the final to interpret for these complex and changing images of different matters for morphology physics by the equation as an interdisciplinary image mathematics model (IMM)
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