Research programme and research problems of the laboratory-chair (research, post-graduate, graduate and student works)

- The formation construction
- Metabolic scales and clocks
- Construction of a picture of the World
- Substitutional motion
- Category and functor modelling of systems
- Extremal principle of substitutional motion
- Variational derivation of the equations of motion
- Systems' entropy time

**The formation construction.**- Generating flows as the World objects' variability sources. Natural references of the "course" of time, or formation. Formation as a change in the amount of a substance in objects.
- Confrontation of relational and substantial approaches to time modelling (see the materials of laboratories headed by V.V. Aristov, Yu.S. Vladimirov, I.M. Dmitrievskii, S.M. Korotaev, M.Kh. Shul'man).
- Confrontation of the phenomenal and noumenal treatments of time in the framework of the substitutional approach to the origin of the World's variability.
- Analysis of the origin of the linear ordering principle for element substitutions in natural systems as perceived by an observer. Formulation of the ordering principle in the framework of the substitutional approach.
- Development of methods for the identification of generating flows from deep levels of the structure of matter, on the basis of presenting observational consequences of the existence of generating flows and calculations of the expected values of the possible effects. For instance, when modelling time by a discrete substantial flow, one can study the possible effect of the flow on non-equilibrium processes, such as chemical and biological reactions, cellular electrophoresis, gaseous discharge, radioactive decay (S.E. Shnol' effect: S.E. Shnol', E.V. Pozharskii, V.A. Kolombet, I.M. Zvereva, T.A. Zenchenko, A.A. Konradov, "On the discreteness of time course measurement results for processes of different nature, created by space-physical factors". Rossiyskiy Khimicheskiy Zhurnal, 1997, No. 3, p. 30; S.E. Shnol', V.A. Kolombet, E.V. Pozharskii, T.A. Zenchenko, I.M. Zvereva, A.A. Konradov, "On the realization of discrete states in the course of fluctuations in macroscopic processes", Uspekhi Fizicheskikh Nauk, v. 168, No. 10, 1998, pp. 1129-1140.) Calculate the magnitude of the effect on the shape of the process histograms. Comparing the model calculations with S.E. Shnol's experimental results, try to describe the properties of the substantial flows necessary for their agreement. (See the materials of S.M. Korotaev's laboratory-chair).
- Confrontation of the properties of generating flows and N.A. Kozyrev's "time flow". (See the materials of S.M. Korotaev's laboratory-chair and the publications: N.A. Kozyrev, Selected Works, Leningrad University Press, Leningrad, 1991; A.P. Levich, A substantional interpretation of N.A.Kozyrev's conception of time // On the Way to Understanding the Time Phenomenon: the Constructions of Time in Natural Science. Part 2. The "active" properties of time according to N.A.Kozyrev. World Scientific. 1996. Pp.1-42). See also electronic publications by L.S. Shikhobalov, S.M. Korotaev and A.G. Parkhomov on the Institute's site.

**Metabolic scales and clocks.**- Measurement of the object variability, created by the generating flows, by the number of substituted elements (introduction of substitutional, or metabolic clocks).
- Formal description of the time "flow" non-uniformity by introducing congruencies (equivalences on ordered sets) for time scales at different hierarchic levels of the system. Possible application of the Radon-Nicodim theorem on measure differentiability.
- The system-specific nature of substitutional time.
- Explanation of the logarithmic relation between the substitutional time scales from different hierarchic levels of the system both for organisms (G. Backman, "Wachtum und Organische Zeit", Leipzig, 1943) and for the "atomic" and "gravitational" times (E.A. Milne, "Kinematic Relativity", Oxford, 1948).
- Make an attempt to represent the transition from the equations of mechanics in Euclidean space with the Newtonian gravitational potential to the Einstein equations for geodesic motion without a force potential as a result of a nonlinear transformation of the time scale only. Find the form of this transformation. Compare it with E. Milne's time scale transformation.

**Construction of a picture of the World.**- Construction of a description of an open World taking into
account the existence of generating flows. Rigorous
constructions:
- Charges as sources and drains of generating flows.
- Lesage's interaction mechanism. (The modern proof of Newton's theorem rests on hydrodynamical considerations, dating back to Laplace. The point is that the only spherically symmetric flow of an incompressible fluid is a radial flow whose velocity is inversely proportional to squared distance from the centre... Thus the force field of attraction to a point mass mathematically coincides with the velocity field of an incompressible fluid flow." - V.I. Arnold, "300 Years of Mathematical Natural Science and Celestial Mechanics", Priroda, 1987, No. 8, pp. 5-15.)
- The space of a system as a union of its generating flow substances. Creation of a topology of neighbourhoods and a metric in the system space. (See also the construction of inductor spaces and its development at A.V. Koganov's laboratory-chair). Substitutional origin of space dimensionality.

- Properties of charges and interactions as dynamical characteristics of generating flows. Investigation of the space dependence of objects' interaction in the system space. Calculations of "action radii" of the interactions created by generating flows of different system structure levels. Taking into account the stratification theorem (see. the section on variational modelling of systems).
- Organisms as charges. The "flask" model of their development and aging. Measurement of life duration by molecular flows through the organism. Calculation of Rubner's constants. Flow interpretation of the organism age characteristics, of the nutrition restriction effect, the dependence of life duration on the body mass and on cephalization characteristics. (A.I. Zotin and A.A. Zotin, "Direction, Rate and Mechanisms of Progressive Evolution". Nauka, Moscow, 1999.
- Basic problems of "Theoretical Biology" by E. Bauer: a search for a theory of generalized motion and sources of the non-equilibrium nature of living matter. // In: Erwin Bauer and Theoretical Biology, Pushchino, 1993, pp.91-101, in Russian.

- Construction of a description of an open World taking into
account the existence of generating flows. Rigorous
constructions:
**Substitutional motion.**- Description of systems' substitutional motion in terms of generating flows. Construction of kinematic and dynamic characteristics of motion (translation, velocity, inertial mass, momentum, energy, Lagrange function, force,..,)
- Derivation of the equations of motion. Description of the properties of substitutional motion: nonlocality, the existence of a maximum velocity, invariance with respect to motion reversion and non-invariance with respect to time reversion, absence of "ether" friction, etc.
- Simultaneity construction. Derivation of a substitutional velocities adding formula.
- Investigation of the distance dependence of the substitutional flow density on in the substitutional space.
- Establishing a relation between the objects' inertial mass and charges.
- Studying the properties of rotational substitutional motion.
- Quantitative analysis of the hypothesis on the expansion of the Universe as a spatial analogue of the "time flow" due to the existence of generating flows and the lack of drain for the "gravitational" generating flow.

**Category and functor modelling of systems.**- Functor comparison of system structures.
- Foundation of the generalized entropy construction (generalization of the Lagrange theorem to semigroups of categories, construction of proper invariants, application of non-standard functors).
- Category description of systems which do not employ a priori axiomatics of mathematical structures.
- Calculations of invariants for mathematical structures (systems with two, three and more hierarchic structure levels).
- Using complex-valued variables for the description of hierarchic objects.
- Discovery of equivalence or non-equivalence of symmetry descriptions with the aid of group theory and with the aid of a restriction of the set of morphisms in a category-theoretical structure formalization. Investigation of the analogy between the derivation of the Lagrange function in theoretical physics from symmetry requirements (the existence of transformation groups in the basic space) and the choice of admissible transformations in the category and functor derivation of functionals. Making clear the connection with the Noether theorem.
- Information as system structure.
- References:

A.P. Levich and A.V. Solovyov, "Category and functor modelling of natural systems". In: "System Analysis on the Eve of the XXI Century". Intellekt, Moscow, 1997, pp. 66-78.

A.P. Levich, "Entropy as a generalization of the concept of number of elements for finite sets" // Filosofskie Issledovaniya, 2001, No. 1, pp. 59-72, (in Russian).

**Extremal principle of substitutional motion.**- Choice of functionals and formulations for the global extremal principle.
- Interpretations of the entropy extremal principle and its relationships with the extremal principles of natural science (P. Fermat's least time principle, P. Maupertius' least action principle, L. Boltzmann and J. Gibbs' maximum entropy principle, the L. Onsager - I. Gyarmati principle of minimum dissipation energy, P. Glennsdorf and I. Prigogine's minimum entropy production principle, the maximum living matter expansion principle, the maximum diversity principle for biological communities, the principle of full consumption of limiting resources, the principle of realization of the most complex states and others). Interpretation of the extremal principle as a consequence of motion stability and interpretation of the extremal functional as the Lyapunov function. Interpretation of the extremal principle as a consequence of the path integral formalism. Information treatments of the extremal principle.
- Proof of an analogue of Gibbs' theorem (on the equivalence between the entropy extremal principle and the minimal substitutional time principle) for multilevel systems. Applications of the theorem for obtaining analogues of the minimum action principle.
- Formulation of a local (infinitesimal) entropy principle.
- References:

A.P. Levich and A.B. Lebed', "Biological species' demands for nutrition components and the environmental factors consumption by an ecological community". In: "Problems of Ecological Monitoring and Ecosystem Modelling", 1987, v. 10, pp. 268-283, (in Russian).

A.P. Levich and V.L. Alexeev, "Entropy extremal principle in the ecology of communities: results and discussion". Biofizika, 1997, v. 42, 2nd issue, pp. 534-541, (in Russian).

A.P. Levich, Variational modelling theorems and algocoenoses functioning principles // Ecological Modelling. 2000. V.131. Pp.207-227.

S.D. Khaitun, "Mechanics and Irreversibility". Yanus, Moscow, 1996, (in Russian).

A.M. Khazen, "Introduction of the Information Measure to the Axiomatic Basis of Mechanics". Moscow, 1996, (in Russian).

**Variational derivation of the equations of motion.**- Two-level functionals:
- Statement of the variational problem and obtaining the Euler-Lagrange equations. Existence, uniqueness, stratification and optimization theorems for variational problems with different functionals.
- Comparison between the properties of generalized entropy and solutions to the variational problem obtained without statistical considerations and their analogues in statistical physics (logarithm of the number of states in a system and the Gibbs distribution).
- Physical interpretation of solutions to the variational problem when modelling natural objects by sets with partition structure.
- An attempt to take into account the "astronomical" time dependence of the solutions to the variational problem by invoking restrictions upon the generating flows - hypothetical references of physical time.

- Calculations with three-level and multilevel functionals. Inclusion of substitutional motion velocities into the functionals.
- Entropy growth along trajectories and irreversibility of the Euler-Lagrange equations for problems with a local extremal principle.
- Application of the category-theoretical description to modelling
the substitutional motion.
- Choice of a mathematical structure for the description of "free" and interacting hierarchic "metabolic objects" (non-standard analysis, Boolean-valued sets, fibre bundles, multilevel hierarchies of partitioned sets,...). Choice of motion morphisms.
- Calculation of invariants for the corresponding structures.
- Description of restrictions connected with the parameters of generating flows through a metabolic object.
- Statement of a variational problem, obtaining the equations of motion as Euler-Lagrange equations.
- Comparison with the equations of substitutional motion.
- Variational description of interaction between metabolic objects.

- Application of stratification and optimization theorems to the dynamics of physical, chemical and biological systems.
- Investigation of the validity of the correspondence principle between the equations of substitutional motion and the equations of classical, relativistic and quantum mechanics and electrodynamics.
- References:

A.P. Levich, "Possible ways of finding dynamical equations in communities ecology". Zhurnal Obshchey Biologii, 1988, v. 49, No. 2, pp. 245-254, (in Russian).

A.P. Levich, V.L. Alexeev and V.A. Nikulin, "Mathematical aspects of variational modelling in communities ecology". Matematicheskoe Modelirovanie, 1994, v.6, No. 5, pp. 55-71, (in Russian).

A.P. Levich, V.L. Alexeyev, S. Yu. Rybakova, "Optimization of the structure of ecological communities: model analysis".

V.L.Alexeyev, A.P. Levich, A search for maximum species abundences in ecological communities under conditional diversity optimization // Bull. of Mathemat. Biology. 1997. V.59. №4. Pp.649-677.

A.P. Levich, V.N. Maximov and N.G. Bulgakov, "Theoretical and experimental ecology of phytoplankton". Nauka, Iskusstvo, Literatura. Moscow, 1997, (in Russian).

A.P. Levich, Variational modelling theorems and algocoenoses functioning principles // Ecological Modelling. 2000. V.131. Pp.207-227.

- Two-level functionals:
**Systems' entropy time**- The entropy parametrization of time (the "entropy" clock).
- Returning the universal status to time (entropy time as an "averager" of metabolic times).
- Substitutional, entropy and category time. The "Boltzmann theorem" on the monotonicity of systems' metabolic and entropy times.
- References:

A.P. Levich, "Entropy as a measure of structuredness of complex systems". // Filosofskie Issledovaniya, 2001, No. 1, 59-72. Proceedings of seminar " Time, chaos and mathematical problems ". V. 2. 2000. M.: Institute of mathematical researches of complex systems. Pp. 163-176. (in Russian).

A.P. Levich, "Sets Theory, the Language of Category Theory and Their Application in Theoretical Biology''. Moscow University Press, Moscow, 1982, 180 pp., (in Russian).