Simulation Systems Department.
Mathematical model decomposition problem investigations have been carried out in the department during approximately twenty years. These investigations are methodological support of basic direction of activity in the department: development of means, which provide spreading of mathematical modeling in new complicated line of investigations and practice. It was necessary to understand conditions complicated mathematical model may be presented by means of more simpler models family. It was necessary also conceptual instrument for analysis the problem of frontier inside which mathematical modeling is valid and the problem of interaction of mathematical and humanitarian methods of analysis and prognosis of complicated processes.
An approach used for study mathematical model decomposition problem was called 'geometric decomposition theory'. In the framework of this theory mathematical model is loaded into class of allied mathematical objects in which notion of isomorphism are defined. Among isomorphic objects are looking up such which is 'representation' of the original object by means of family of 'more simpler' objects. Consequence of such approach is mathematical models properties to have decomposition are preserved as the model is transformed by the isomorphism. Therefore mathematical models decomposition properties are formulated in terms which also preserved as isomorphic transformation act. Thus decomposition problem is allied to the problem canonic form finding respectively isomorphism.
Bourbakis formalism was chosen as the basic instrument for development of geometric decomposition theory. Mathematical models are treated as set supplied by bourbakis structure. The mathematical model is loaded thereby into class of allied mathematical objects in which notion of isomorphism are defined. The (bourbakis) sub-objects and factor-object are defined respectively some (bourbakis) morphism. (Bourbakis) sub-objects and factor-object are differ from their analog in category theory.) Draw attention: sub-object (factor-object) is transformed by isomorphism again in sub-object (factor-object), i.e. sub-object (factor-object) is preserved as isomorphic transformation act. Only two dual notion are in foundation of the language surroundings which is geometric decomposition theory: the notion about P-decomposition and the notion about F-decomposition of mathematical objects (model). If we simplify and plebeianise the situation one might that mathematical objects P-decomposition (F-decomposition) is the family of sub-objects (factor-objects) such that exist not more then single object which have this sub-objects (factor-objects) family. Mathematical model decomposition structure is system of its decomposition (P- and F-) sets properties and relationship between various decompositions. Mathematical model decomposition structure may be treated also as bourbakis structures is generated on the set of decomposition (P- and F-) by original bourbakis structure of the object (model).
The simplest examples of relationship between various objects decompositions is relation 'more simpler' on the set of P-decomposition of the objects: if from P-decomposition may be move off some of P-objects and residuary family of P-objects will be represent as before P-decomposition of the objects the last family is called 'more simpler' then the original family. This relation is connected tight with properties of compactness of topological spaces (see lower). The relation 'more simpler' is partial order relation. For the most part minimal P-decomposition (F-decomposition) in the sense of this partial order is of interest. Most essential among minimal P-decompositions respectively 'more simpler' relation are such P-decomposition P-objects of which are defined on a equivalences relation classes. If such P-decomposition exists then the original object fall to pieces of 'independent' P-objects. Such P-decomposition was called decomposition on disjunctive sum or 'CC-decomposition'. The set of CC-decomposition of object is supplied naturally by partial order relation. For instance the family of transitive group of transformation is generated by original group of transformation on the classes of equivalences relation associated with the original group is maximal CC-decomposition of the original group on family of CC-simple (see lower) sub-objects. Dual to CC-decomposition of a object is decomposition on Cartesian product of factor-object family of the object or DP-decomposition. Jordan representation of linear operator in finite-dimensional linear vector space over complex number field is maximal DP-decomposition of the operator on family of DP-simple (see lower) factor-object of the operator.
In the geometric decomposition theory are defined four type of simplicity. P-simple object is object, which have no nontrivial P-object. F-simple object is object, which have no nontrivial F-object. CC-simple object is object, which have no nontrivial CC-decomposition. DP-simple object is object, which have no nontrivial DP-decomposition. CC-simplicity and DP-simplicity is examples of decomposition structure properties of object. Many examples suggest that major mathematical objects properties are properties of the objects decomposition structure. For instance transitive property of transformation group is properties of its CC-simplicity and its primitive property is properties of DP-simplicity. Simple abstract group is F-simple object (every abstract group is CC-simple object). Simple field is P-simple object (every field is CC-simple and F-simple object). Connected topological space and connected graph are CC-simple objects. Compactness of topological spaces is follow properties decomposition structure of the space: for every P-decomposition of the space which consist of open subsets exist more simpler finite P-decomposition (open overlapping of topological spaces is its P-decomposition). Nontrivial stratified manifold of is DP-simple object which have P-decomposition every P-object of which have nontrivial DP-decomposition.
The key for solution of many problem in applied sciences is study of corresponding mathematical model decomposition structure. For instance observability of control dynamic system respectively observation system which measure some functions of control system phase variable is properties of the control system have no F-objects, among phase variable of which are measurable functions. In the contrary the properties of control dynamic system to have invariant functions is its properties to have decomposition of fixed character (V.I. Elkin.). The problem of building images of scenes is studied now in the department by means of geometric decomposition theory. The idea is: images of scenes is 'something preserving' as the scene is transform by a 'set of transformation'. Let bourbakis mathematical object is associated with scene. Then set of transformation is the set of isomorphism and 'something preserving' is decomposition structure of the mathematical object.
The language means of geometric decomposition theory permit to give some methodology statements, which are humanitarian superstructure on rigorous mathematical theory. Some example are given lower.
If structure is observed in some system (process, phenomena), then mechanism build into the system, which preserve the structure. Understanding of the mechanism is necessary condition of understanding of nature of the system. If adequate mathematical modeling of the system is possible, then the structure is decomposition structure of corresponding mathematical model. 'Chaos' is absence of decomposition structure in observed system (process, phenomena). Synergetic is field of research, which study beginnings of (dissipative) structure in physical and physical-chemical processes.
Decomposition of complicated control systems is realized by assignment part of control by function of measured systems characteristics (feedback) and by control restriction. Decomposition in such system is mean to realize hierarchy of value, which are achieved the system, to bring in correspondence with complexity of control tasks with resource for its solution and information technology inbuilt in the system. In very complicated systems (biological, ecological, social) on the lower level of the hierarchy is aim to retain some systems characteristics in very narrow restrictions. The sert of such characteristics are called 'internal media' of the system. Such systems exists until its are capable to preserved their 'internal media' and decomposition structure by feedback. Assembly of the feedback are called homeostasis. Homeostasis element are in social-economic systems. This elements have form of supplying stability in the social-economic systems. In particular moral and ethical norms in human society are mechanisms of homeostasis, which appear on definite stage of in human society evolution and which guarantee human society further development.
Some result of mathematical model decomposition problem investigations published in:
Elkin V.I., Pavlovsky J.N. Decomposition of models of control processes. Journal of Mathematical science. Vol. 88., No. 5, pp. 723-761, 1998.,
Pavlovsky J.N., Smirnova T.G. The Problem of Decomposition in Mathematical Modeling.Moscow. Phasis. 1998. 272 pp. (in Russian)
Elkin V.I. Methods of Algebra and Geometry in Control Theory. Affine Distribution and Affine systems. Moscow. MFTI. 1996. 112 pp. (In Russian).
Elkin V.I. Reduction of nonlinear control sistems. Kluver Academic Publishes. Dordrecht-Boston-London. 1999. 248 pp.
Danilov N.Yu., Pavlovsky Yu.N., Socolov V.I., Yakoveno G.N. Geometric Methods in Control Theory. Moscow. MFTI. 1999. 156 pp. (In Russian).
Ivashko D.G. Three-dimensional Control Systems.Moscow. CCAS. 2000. 130 pp. In Russian.
Elkin V.I. Reduction of Nonlinear Control Sistems. Decomposition disturbance invariance. Moscow. Phasis. 2004. 207 pp. In English.
Pavlovsky Yu.N., Smirnova T.G. Scales of Structure Kinds, Terms and Relationship Preserving by Isomorphism. Moscow. 2003. CCAS. 93 pp. (In Russian).
Pavlovsky Yu.N. Decomposition of Sets Supplied by Structure on Disjunctive Sum and Direct Product. Doklady RAN. 1995. V. 340. No. 3. pp. 314-316.
Pavlovsky Yu.N. About P- and F- Decomposition S-objects. Doklady RAN. 1996. V. 351. No. 5. pp. 603-605.
Pavlovsky Yu.N. Decomposition of Sets Supplied by Structure over Subordinate Structure.Doklady RAN. 1997. V.357. No. 5. pp. 314-316.
Pavlovsky Yu.N. About Structure Kind Scale. Doklady RAN. 1998. V. 363. No. 2. pp. 163-165.
Pavlovsky Yu.N. About HPF- and PF- morphism. .Doklady RAN. 1999. V. 369. No. 6. pp. 745-746.
Pavlovsky Yu.N. About Decomposition Method of Building of Images of Sub-Sets Supplied by Structure. Doklady RAN. 2000. V. 374. No. 4. pp. 450-452. ,
Pavlovsky Yu.N. About Decomposition Approach to Building of Images of Scene.Doklady RAN. 2003. V. 392. No. 6. pp. 336-338.
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