Irakli Avalishvili

Morphological and functional graphs of the cell are constructed, expressing dynamic ontogenetic transformations between six different types of membranes of organelles and the interrelations between six different functions of vital significance for the cell. Morphisms (epimorphisms) from different biological graphs to a functional graph are constructed. It is shown that there exists an isomorphism between morphological and functional graphs which expresses the most essential forms of vital processes. It is further shown that the components with vital functions transform and transfer to each other. To classify and construct cell structures the apparatus for extensions of graphs developed by the author is used, which enables hierarchical biological systems to be described.

In this brief chapter, attempts are made to construct a model of the interstate relationships of human society as nation states based on fundamental sciences. States and state groups act on each other and change each other's political organization. Based on these interactions, a scheme of interstate relationships is constructed in the form of a graph. Whereas basic functions of each state or group of states form a functional scheme in the form of another graph, the interstate relationships graph and the functional graph are isomorphic to each other.

Arbitrary systems, will it be biological, physical, cybernetical, etc. may be described by a mathematical function, namely by a graph extension function, which we also call hierarchical function (and which mathematically shows hierarchical nature of science). This function can be also used to describe mathematical objects themselves, which in the paper is shown on the example of the action of the graph extension function on the set of integers. A new theory of graph extensions, similar to group extension theory, is outlined. A theorem about the equivalence of different extension functions is proved. There exists an isomorphism between the modified functional graph of the cell(functional block-scheme) and the morphological graph ofthe cell (the graph expressing topological membrane intertransformations of the cell) which expresses the most essential features of for the biology of the cell and captures one of the specific differences between living and non-living systems. It is shown that the construction of the graph of a complex organism from the primordial graph given by Rashevsky is nothing but an extension of the primordial graph by the graph extension function. It is described that there exist morphisms from biological graphs described by various authors to our functional graph.