Preprints & selected publications:
2012 2011 2010 2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1993 1992 1991 1980-1990
Nadya Morozova, Andrei
Zinovyev, Nora
Nonne, Linda-Louise
Pritchard, Alexander
N. Gorban, and Annick
Harel-Bellan,
Kinetic signatures of microRNA modes of action, RNA,
Vol. 18, No. 9 (2012). Published
in Advance July
31, 2012, doi:10.1261/rna.032284.112
MicroRNAs (miRNAs) are key
regulators of all important biological processes, including development,
differentiation, and cancer. Although remarkable progress has been made in
deciphering the mechanisms used by miRNAs to regulate translation, many
contradictory findings have been published that stimulate active debate in this
field. Here we contribute to this discussion in three ways. First, based on a
comprehensive analysis of the existing literature, we hypothesize a model in
which all proposed mechanisms of microRNA action coexist, and where the
apparent mechanism that is detected in a given experiment is determined by the
relative values of the intrinsic characteristics of the target mRNAs and
associated biological processes. Among several coexisting miRNA mechanisms, the
one that will effectively be measurable is that which acts on or changes the
sensitive parameters of the translation process. Second, we have created a
mathematical model that combines nine known mechanisms of miRNA action and
estimated the model parameters from the literature. Third, based on the mathematical
modeling, we have developed a computational tool for discriminating among
different possible individual mechanisms of miRNA action based on translation
kinetics data that can be experimentally measured (kinetic signatures). To
confirm the discriminatory power of these kinetic signatures and to test our
hypothesis, we have performed several computational experiments with the model
in which we simulated the coexistence of several miRNA action mechanisms in the
context of variable parameter values of the translation.
Ovidiu Radulescu, Alexander N. Gorban, Andrei Zinovyev, Vincent Noel
Reduction of dynamical biochemical reaction networks in computational
biology, Frontiers in Genetics (Bioinformatics and Computational
Biology). July2012, Volume3, Article 131. (e-print arXiv:1205.2851 [q-bio.MN])
Biochemical networks are used in
computational biology, to model mechanistic details of systems involved in cell
signaling, metabolism, and regulation of gene expression. Parametric and
structural uncertainty, as well as combinatorial explosion are strong obstacles
against analyzing the dynamics of large models of this type. Multiscaleness, an
important property of these networks, can be used to get past some of these
obstacles. Networks with many well separated time scales, can be reduced to
simpler models, in a way that depends only on the orders of magnitude and not
on the exact values of the kinetic parameters. The main idea used for such
robust simplifications of networks is the concept of dominance among model
elements, allowing hierarchical organization of these elements according to
their effects on the network dynamics. This concept finds a natural formulation
in tropical geometry. We revisit, in the light of these new ideas, the main
approaches to model reduction of reaction networks, such as quasi-steady state
(QSS) and quasi-equilibrium approximations (QE), and provide practical recipes
for model reduction of linear and non-linear networks. We also discuss the
application of model reduction to the problem of parameter identification, via
backward pruning machine learning techniques.
Local Equivalence of Reversible and General Markov Kinetics, arXiv:1205.2052 [physics.chem-ph]
We consider continuous--time Markov
kinetics with a finite number of states and a given positive equilibrium
distribution P*. For an arbitrary
probability distribution P we study
the possible right hand sides, dP/dt, of the Kolmogorov (master) equations.
We describe the cone of possible values of the velocity, dP/dt, as a function of P and P*. We prove that, surprisingly, these cones coincide for the class
of all Markov processes with equilibrium P*
and for the reversible Markov processes with detailed balance at this
equilibrium. Therefore, for an arbitrary probability distribution P and a general system there exists a
system with detailed balance and the same equilibrium that has the same
velocity dP/dt at point P. The set of
Lyapunov functions for the reversible Markov processes coincides with the set
of Lyapunov functions for general Markov kinetics. The results are extended to
nonlinear systems with the generalized mass action law.
Alexander N. Gorban, Andrei Zinovyev, Nadya Morozova, Annick Harel-Bellan
Modeling coupled transcription, translation and degradation and miRNA-based
regulation of this process, arXiv:1204.5941 [q-bio.MN]
The translation-transcription process with the description of the most basic
"elementary" processes consists in: 1) production of mRNA molecules,
2) initiation of these molecules by circularization with help of initiation
factors, 3) initiation of translation, recruiting the small ribosomal subunit,
4) assembly of full ribosomes, 5) elongation, i.e. movement of ribosomes along
mRNA with production of protein, 6) termination of translation, 7) degradation
of mRNA molecules. A certain complexity in the mathematical formulation of this
process arises when one tries to take into account the phenomenon of polysome
first, when several ribosomes are producing peptides on a single mRNA at the
same time. This leads to multiplicity of possible states of mRNA with various
numbers of ribosomes with potentially different dynamics, interaction between
ribosomes and other difficulties. In this preprint we provide 1) detailed
mechanistic description of the translation process with explicit representation
of every state of translating mRNA, followed by 2) deriving the simplest and
basic ODE model of coupled transcription, translation and degradation, and 3)
developing a model suitable for describing all known mechanisms of miRNA action
on translation. The basic model is constructed by correct lumping of the
detailed model states and by separating the description of ribosomal turnover.
It remains linear under assumption of that the translation is not limited by
availability of ribosomal subunits or initiation factors. The only serious
limitation of this type of translation modeling is in that it does not take
into account possible interactions between ribosomes. The latter might lead to
more complex phenomena which can be taken into account in simulatory models of
the detailed representation of translation at the cost of more difficult
analytical analysis of the model.
A. Zinovyev, N. Morozova, A. N. Gorban, A. Harel-Belan
Mathematical modeling of microRNA-mediated mechanisms of translation
repression, arXiv:1202.1243 [q-bio.MN]
MicroRNAs can affect the protein translation using nine mechanistically
different mechanisms, including repression of initiation and degradation of the
transcript. There is a hot debate in the current literature about which
mechanism and in which situations has a dominant role in living cells. The
worst, same experimental systems dealing with the same pairs of mRNA and miRNA
can provide ambiguous evidences about which is the actual mechanism of
translation repression observed in the experiment. We start with reviewing the
current knowledge of various mechanisms of miRNA action and suggest that
mathematical modeling can help resolving some of the controversial
interpretations. We describe three simple mathematical models of miRNA
translation that can be used as tools in interpreting the experimental data on
the dynamics of protein synthesis. The most complex model developed by us
includes all known mechanisms of miRNA action. It allowed us to study possible
dynamical patterns corresponding to different miRNA-mediated mechanisms of
translation repression and to suggest concrete recipes on determining the
dominant mechanism of miRNA action in the form of kinetic signatures. Using
computational experiments and systematizing existing evidences from the
literature, we justify a hypothesis about co-existence of distinct
miRNA-mediated mechanisms of translation repression. The actually observed
mechanism will be that acting on or changing the limiting "place" of
the translation process. The limiting place can vary from one experimental
setting to another. This model explains the majority of existing controversies
reported.
Thermodynamic Tree: The Space of Admissible Paths, arXiv:1201.6315
[cond-mat.stat-mech]
Is a spontaneous
transition from a state x to a state y allowed by thermodynamics? Such a
question arises often in chemical thermodynamics and kinetics. We ask the more
formal question: is there a continuous path between these states, along which
the conservation laws hold, the concentrations remain non-negative and the
relevant thermodynamic potential G
(Gibbs energy, for example) monotonically decreases? The obvious necessary
condition, G(x)≥G(y), is not sufficient, and we construct
the necessary and sufficient conditions. For example, it is impossible to
overstep the equilibrium in 1-dimensional (1D) systems (with n components and
n-1 conservation laws). The system cannot come from a state x to a state y if
they are on the opposite sides of the equilibrium even if G(x) >G(y).
We find the general multidimensional analogue of this 1D rule and
constructively solve the problem of the thermodynamically admissible
transitions.
We study dynamical systems, which are given in a positively invariant convex
polyhedron and have a convex Lyapunov function G. An admissible path is a continuous curve along which G does not increase. For x,y from D, x>y (x precedes y) if there exists an admissible path
from x to y and x~y if x>y and y>x. The tree of G in D is a quotient space D/~. We provide an algorithm for the
construction of this tree. In this algorithm, the restriction of G onto the 1-skeleton of D (the union of edges) is used. The
problem of existence of admissible paths between states is solved
constructively. The regions attainable by the admissible paths are described.
2011
A.N. Gorban, G.S.Yablonsky
Extended detailed balance for systems with irreversible reactions, Chemical Engineering Science 66 (2011) 5388–5399.
The principle of detailed balance states that in
equilibrium each elementary process is equilibrated by its reverse process. For
many real physico-chemical complex systems (e.g. homogeneous combustion,
heterogeneous catalytic oxidation, most enzyme reactions etc), detailed
mechanisms include both reversible and irreversible reactions. In this case,
the principle of detailed balance cannot be applied directly. We represent
irreversible reactions as limits of reversible steps and obtain the principle
of detailed balance for complex mechanisms with some irreversible elementary
processes. We proved two consequences of the detailed balance for these
mechanisms: the structural condition and the algebraic condition that form
together the extended form of detailed balance. The algebraic condition is the
principle of detailed balance for the reversible part. The structural condition
is: the convex hull of the stoichiometric vectors of the irreversible reactions
has empty intersection with the linear span of the stoichiometric vectors of
the reversible reaction. Physically, this means that the irreversible reactions
cannot be included in oriented pathways.
The systems with the extended form of
detailed balance are also the limits of the reversible systems with detailed
balance when some of the equilibrium concentrations (or activities) tend to
zero. Surprisingly, the structure of the limit reaction mechanism crucially
depends on the relative speeds of this tendency to zero.
A. N.
Gorban, D.
Packwood
Possibility and Impossibility of the Entropy
Balance in Lattice Boltzmann Collisions, arXiv:1111.5994
[physics.comp-ph]
We demonstrate that in the space of distributions operated on by lattice
Boltzmann methods that there exists a vicinity of the equilibrium where
collisions with entropy balance are possible and, at the same time, there exist
an area of nonequilibrium distributions where such collisions are impossible.
We calculate and graphically represent these areas for some simple entropic
equilibria using single relaxation time models. Therefore it is shown that the
definition of an entropic LBM is incomplete without a strategy to deal with
certain highly nonequilibrium states. Such strategies should be explicitly
stated as they may result in the production of additional entropy.
R.A.
Brownlee, J.
Levesley, D.
Packwood, A.N.
Gorban
Add-ons for Lattice Boltzmann Methods: Regularization, Filtering and
Limiters, arXiv:1110.0270 [physics.comp-ph]
We describe how regularization of lattice
Boltzmann methods can be achieved by modifying dissipation. Classes of
techniques used to try to improve regularization of LBMs include flux limiters,
enforcing the exact correct production of entropy and manipulating
non-hydrodynamic modes of the system in relaxation. Each of these techniques
corresponds to an additional modification of dissipation compared with the
standard LBGK model. Using some standard 1D and 2D benchmarks including the shock
tube and lid driven cavity, we explore the effectiveness of these classes of
methods.
F.
Spahn, E. V.
Neto, A. H.
F. Guimaraes, A. N.
Gorban, N. V.
Brilliantov
A Statistical Model of Aggregates Fragmentation, arXiv:1106.2721
[cond-mat.stat-mech]
A statistical model of fragmentation of aggregates is proposed, based on
the stochastic propagation of cracks through the body. The propagation rules
are formulated on a lattice and mimic two important features of the process --
a crack moves against the stress gradient and its energy depletes as it grows.
We perform numerical simulations of the model for two-dimensional lattice and
reveal that the mass distribution for small and intermediate-size fragments
obeys a power-law, F(m)\propto m^(-3/2), in agreement with experimental
observations. We develop an analytical theory which explains the detected
power-law and demonstrate that the overall fragment mass distribution in our
model agrees qualitatively with that, observed in experiments.
A.N.
Gorban, H.P. Sargsyan and H.A. Wahab
Quasichemical Models of
Multicomponent Nonlinear Diffusion, Mathematical Modelling of Natural Phenomena,
Volume 6 /
Issue 05,
(2011), 184−262.
Diffusion preserves the positivity of concentrations, therefore, multicomponent
diffusion should be nonlinear if there exist non-diagonal terms. The vast
variety of nonlinear multicomponent diffusion equations should be ordered and
special tools are needed to provide the systematic construction of the
nonlinear diffusion equations for multicomponent mixtures with significant
interaction between components. We develop an approach to nonlinear
multicomponent diffusion based on the idea of the reaction mechanism borrowed
from chemical kinetics.
Chemical kinetics gave rise to very seminal tools for the modeling of processes. This is the stoichiometric algebra supplemented by the simple kinetic law. The results of this invention are now applied in many areas of science, from particle physics to sociology. In our work we extend the area of applications onto nonlinear multicomponent diffusion.
We demonstrate, how the mechanism based approach to multicomponent diffusion can be included into the general thermodynamic framework, and prove the corresponding dissipation inequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementary process cannot have an arbitrary form. For the general kinetic law (the generalized Mass Action Law), additional conditions are proved. The cell–jump formalism gives an intuitively clear representation of the elementary transport processes and, at the same time, produces kinetic finite elements, a tool for numerical simulation
A. Gorban and S. Petrovskii
Collective dynamics: when one plus one does not make two, Mathematical Medicine and Biology (2011) 28, 85−88.
A brief introduction into the interdisciplinary field of collective dynamics is given, followed by an overview of ‘Mathematical Models of Collective Dynamics in Biology and Evolution’ (University of Leicester, 11–13 May 2009). Collective dynamics—understood as the dynamics arising from the interplay between the constituting elementary argents or parts of a more complex system—has been one of the main paradigms of the natural sciences over the last several decades.
A.N. Gorban and M. Shahzad
The Michaelis-Menten-Stueckelberg Theorem. Entropy 2011, 13, 966-1019.
We study chemical reactions with complex mechanisms under two assumptions: (i) intermediates are present in small amounts (this is the quasi-steady-state hypothesis or QSS) and (ii) they are in equilibrium relations with substrates (this is the quasiequilibrium hypothesis or QE). Under these assumptions, we prove the generalized mass action law together with the basic relations between kinetic factors, which are sufficient for the positivity of the entropy production but hold even without microreversibility, when the detailed balance is not applicable. Even though QE and QSS produce useful approximations by themselves, only the combination of these assumptions can render the possibility beyond the “rarefied gas” limit or the “molecular chaos” hypotheses. We do not use any a priori form of the kinetic law for the chemical reactions and describe their equilibria by thermodynamic relations. The transformations of the intermediate compounds can be described by the Markov kinetics because of their low density (low density of elementary events). This combination of assumptions was introduced by Michaelis and Menten in 1913. In 1952, Stueckelberg used the same assumptions for the gas kinetics and produced the remarkable semi-detailed balance relations between collision rates in the Boltzmann equation that are weaker than the detailed balance conditions but are still sufficient for the Boltzmann H-theorem to be valid. Our results are obtained within the Michaelis-Menten-Stueckelbeg conceptual framework.
G. S. Yablonsky, A. N. Gorban, D. Constales, V.
V. Galvita and G. B. Marin
Reciprocal
relations between kinetic curves, EPL, 93 (2011) 20004.
We study coupled irreversible processes. For
linear or linearized kinetics with microreversibility, 
,
the kinetic operator K is symmetric in the entropic inner product. This
form of Onsager's reciprocal relations implies that the shift in time, exp(Kt),
is also a symmetric operator. This generates the reciprocity relations between
the kinetic curves. For example, for the Master equation, if we start the
process from the i-th pure state and measure the probability pj(t)
of the j-th state (j≠i), and, similarly, measure pi(t)
for the process, which starts at the j-th pure state, then the ratio of these
two probabilities pj(t)/pi(t)
is constant in time and coincides with the ratio of the equilibrium
probabilities. We study similar and more general reciprocal relations between
the kinetic curves. The experimental evidence provided as an example is from
the reversible water gas shift reaction over iron oxide catalyst. The
experimental data are obtained using Temporal Analysis of Products (TAP)
pulse-response studies. These offer excellent confirmation within the
experimental error.
A.N. Gorban, D. Roose
Preface, In: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.), Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. V-VI.
A mathematical model is an intellectual device that works. …
A.N. Gorban
Self-simplification in Darwin’s Systems, In: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.), Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. 311-344
We prove that a non-linear kinetic system with conservation of supports for distributions has generically limit distributions with final support only. The conservation of support has a biological interpretation: inheritance. We call systems with inheritance “Darwin’s systems”. Such systems are apparent in many areas of biology, physics (the theory of parametric wave interaction), chemistry and economics. The finite dimension of limit distributions demonstrates effects of natural selection. Estimations of the asymptotic dimension are presented. After some initial time, solution of a kinetic equation with conservation of support becomes a finite set of narrow peaks that become increasingly narrow over time and move increasingly slowly. It is possible that these peaks do not tend to fixed positions, and the path covered tends to infinity as t → ∞. The drift equations for peak motion are obtained. They describe the asymptotic layer near the omega-limit distributions with finite support .
D.J. Packwood, J. Levesley, and A.N. Gorban
Time step expansions and the invariant manifold approach to lattice Boltzmann models, In: Coping with Complexity: Model Reduction and Data Analysis, A.N. Gorban and D. Roose (eds.), Lecture Notes in Computational Science and Engineering, 75, Springer: Heidelberg – Dordrecht - London -New York, 2011, pp. 169-206.
The classical method for deriving the macroscopic dynamics of a lattice Boltzmann system is to use a combination of different approximations and expansions. Usually a Chapman-Enskog analysis is performed, either on the continuous Boltzmann system, or its discrete velocity counterpart. Separately a discrete time approximation is introduced to the discrete velocity Boltzmann system, to achieve a practically useful approximation to the continuous system, for use in computation. Thereafter, with some additional arguments, the dynamics of the Chapman-Enskog expansion are linked to the discrete time system to produce the dynamics of the completely discrete scheme. In this paper we put forward a different route to the macroscopic dynamics. We begin with the system discrete in both velocity space and time. We hypothesize that the alternating steps of advection and relaxation, common to all lattice Boltzmann schemes, give rise to a slow invariant manifold. We perform a time step expansion of the discrete time dynamics using the invariance of the manifold. Finally we calculate the dynamics arising from this system. By choosing the fully discrete scheme as a starting point we avoid mixing approximations and arrive at a general form of the microscopic dynamics up to the second order in the time step. We calculate the macroscopic dynamics of two commonly used lattice schemes up to the first order, and hence find the precise form of the deviation from the Navier-Stokes equations in the dissipative term, arising from the discretization of velocity space.
Finally we perform a short wave perturbation on the dynamics of these example systems, to find the necessary conditions for their stability.
A.N. Gorban
Kinetic path summation, multi-sheeted extension of master equation, and evaluation of ergodicity coefficient, Physica A 390 (2011) 1009–1025.
We study the master equation with time-dependent coefficients, a linear kinetic equation for the Markov chains or for the monomolecular chemical kinetics. For the solution of this equation a path summation formula is proved. This formula represents the solution as a sum of solutions for simple kinetic schemes (kinetic paths), which are available in explicit analytical form. The relaxation rate is studied and a family of estimates for the relaxation time and the ergodicity coefficient is developed. To calculate the estimates we introduce the multi-sheeted extensions of the initial kinetics. This approach allows us to exploit the internal (‘‘micro’’) structure of the extended kinetics without perturbation of the base kinetics.
A.N. Gorban, L.I. Pokidysheva,·E,V. Smirnova,
T.A. Tyukina.
Law of the Minimum Paradoxes, Bull Math Biol 73(9) (2011), 2013-2044; Online first 19.11.2010,
The “Law of the
Minimum” states that growth is controlled by the scarcest resource (limiting
factor). This concept was originally applied to plant or crop growth (Justus
von Liebig, 1840) and quantitatively supported by many experiments. Some
generalizations based on more complicated “dose-response” curves were proposed.
Violations of this law in natural and experimental ecosystems were also reported.
We study models of adaptation in ensembles of similar organisms under load of
environmental factors and prove that violation of Liebig’s law follows from
adaptation effects. If the fitness of an organism in a fixed environment
satisfies the Law of the Minimum then adaptation equalizes the pressure of
essential factors and, therefore, acts against the Liebig’s law. This is the the
Law of the Minimum paradox: if for a randomly chosen pair “organism–environment”
the Law of the Minimum typically holds, then in a well-adapted system, we have
to expect violations of this law.
For the opposite
interaction of factors (a synergistic system of factors which amplify each
other), adaptation leads from factor equivalence to limitations by a smaller
number of factors.
For analysis of
adaptation, we develop a system of models based on Selye’s idea of the
universal adaptation resource (adaptation energy). These models predict that
under the load of an environmental factor a population separates into two
groups (phases): a less correlated, well adapted group and a highly correlated
group with a larger variance of attributes, which experiences problems with
adaptation. Some empirical data are presented and evidences of
interdisciplinary applications to econometrics are discussed.
A.N. Gorban,
E.V. Smirnova, T.A. Tyukina,
Correlations,
risk and crisis: From physiology to finance,
Physica A, Vol. 389, Issue 16,
2010, 3193-3217. Number 9 in
the Top Hottest Articles in the Journal, April to June 2010
We study the dynamics of correlation and variance in
systems under the load of environmental factors. A universal effect in ensembles
of similar systems under the load of similar factors is described: in crisis,
typically, even before obvious symptoms of crisis appear, correlation
increases, and, at the same time, variance (and volatility) increases too. This
effect is supported by many experiments and observations of groups of humans,
mice, trees, grassy plants, and on financial time series.
A general approach to the explanation of the effect through
dynamics of individual adaptation of similar non-interactive individuals to a
similar system of external factors is developed. Qualitatively, this approach
follows Selye’s idea about adaptation energy.
A.N. Gorban
We study the Master equation with
time--dependent coefficients, a linear kinetic equation for the Markov chains
or for the monomolecular chemical kinetics. For the solution of this equation a
paths summation formula is proved. This formula represents the solution as a
sum of solutions for simple kinetic schemes (kinetic paths), which are
available in explicit analytical form. The relaxation rate is studied and a
family of estimates for the relaxation time and the ergodicity coefficient is
developed. To calculate the estimates we introduce the multi--sheeted
extensions} of the initial kinetics. This approach allows us to exploit the
internal ("micro")structure of the extended kinetics without
perturbation of the base kinetics.
A. N. Gorban, A. Zinovyev.
Principal manifolds and graphs in practice: from molecular biology to dynamical systems, International Journal of Neural Systems, Vol. 20, No. 3 (2010) 219–232.
We present several applications of non-linear data modeling, using
principal manifolds and principal graphs constructed using the metaphor of
elasticity (elastic principal graph approach). These approaches are
generalizations of the Kohonen’s self-organizing maps, a class of artificial
neural networks. On several examples we show advantages of using non-linear
objects for data approximation in comparison to the linear ones. We propose
four numerical criteria for comparing linear and non-linear mappings of
datasets into the spaces of lower dimension. The examples are taken from
comparative political science, from
analysis of high-throughput data in molecular biology, from analysis of
dynamical systems.
E. Chiavazzo, I.V.
Karlin, A.N. Gorban, K. Boulouchos,
Coupling of the model reduction technique
with the lattice Boltzmann method, Combustion and Flame 157 (2010)
1833–1849 
doi:10.1016/j.combustflame.2010.06.009
A new framework of simulation of reactive flows is proposed
based on a coupling between accurate reduced reaction mechanism and the lattice
Boltzmann representation of the flow phenomena. The model reduction is developed
in the setting of slow invariant manifold construction, and the simplest
lattice Boltzmann equation is used in order to work out the procedure of
coupling of the reduced model with the flow solver. Practical details of
constructing slow invariant manifolds of a reaction system under various
thermodynamic conditions are reported. The proposed method is validated with
the two-dimensional simulation of a premixed counterflow flame in the
hydrogen-air mixture.
Gorban
A.N., Gorban P.A., Judge G.
Entropy: The Markov Ordering Approach. Entropy.
2010; 12(5):1145-1193. GorbanGorbanJudgeEntropy2010.pdf
The
focus of this article is on entropy and Markov processes. We study the
properties of functionals which are invariant with respect to monotonic
transformations and analyze two invariant “additivity” properties: (i)
existence of a monotonic transformation which makes the functional additive with
respect to the joining of independent systems and (ii) existence of a monotonic
transformation which makes the functional additive with respect to the
partitioning of the space of states. All Lyapunov functionals for Markov chains
which have properties (i) and (ii) are derived. We describe the most general
ordering of the distribution space, with respect to which all continuous-time
Markov processes are monotonic (the Markov order). The
solution differs significantly from the ordering given by the inequality of
entropy growth. For inference, this approach results in a convex compact set of
conditionally “most random” distributions.
A. N. Gorban and V. M. Cheresiz,
Slow Relaxations and Bifurcations of the Limit
Sets of Dynamical Systems. I. Bifurcations of Limit Sets,
Journal of Applied and Industrial Mathematics,
2010, Vol. 4, No. 1, pp. 54–64.
We consider one-parameter semigroups of homeomorphisms depending continuously on the parameters. We study the phenomenon of slow relaxation that consists in anomalously slow motion to the limit sets. We investigate the connection between slow relaxations and bifurcations of limit sets and other singularities of the dynamics. The statements of some of the problems stem from mathematical chemistry.
A. N.
Gorban and V. M. Cheresiz,
Slow Relaxations and Bifurcations of the Limit
Sets of Dynamical Systems. II. Slow Relaxations of a Family of Semiflows,
Journal of Applied and Industrial Mathematics,
2010, Vol. 4, No. 2, pp. 182–190.
We propose a number of approaches to the notion of the relaxation time of a dynamical system which are motivated by the problems of chemical kinetics, give exact mathematical definitions of slow relaxations, study their possible reasons, among which an important role is played by bifurcations of limit sets.
E. Chiavazzo, I.V. Karlin, and A.N. Gorban,
The Role of Thermodynamics in Model Reduction when Using Invariant Grids, Commun. Comput. Phys., Vol. 8, No. 4 (2010), pp. 701-734.
In the present work, we develop in detail the process leading to reduction of models in chemical kinetics when using the Method of Invariant Grids (MIG). To this end, reduced models (invariant grids) are obtained by refining initial approximations of slow invariant manifolds, and used for integrating smaller and less stiff systems of equations capable to recover the detailed description with high accuracy. Moreover, we clarify the role played by thermodynamics in model reduction, and carry out a comparison between detailed and reduced solutions for a model hydrogen oxidation reaction.
Andrei Zinovyev, Nadya Morozova, Nora Nonne,
Emmanuel Barillot, Annick Harel-Bellan, Alexander N Gorban
Dynamical modeling of microRNA
action on the protein translation process, BMC Systems Biology 2010, 4:13 (24 February 2010)
Background
Protein translation is a multistep process which can be represented as a cascade of biochemical reactions (initiation, ribosome assembly, elongation, etc.), the rate of which can be regulated by small non-coding microRNAs through multiple mechanisms. It remains unclear what mechanisms of microRNA action are the most dominant: moreover, many experimental reports deliver controversal messages on what is the concrete mechanism actually observed in the experiment. Nissan and Parker have recently demonstrated that it might be impossible to distinguish alternative biological hypotheses using the steady state data on the rate of protein synthesis. For their analysis they used two simple kinetic models of protein translation.
Results
In contrary to the study by Nissan and Parker, we show that dynamical data allow to discriminate some of the mechanisms of microRNA action. We demonstrate this using the same models as developed by Nissan and Parker for the sake of comparison but the methods developed (asymptotology of biochemical networks) can be used for other models. We formulate a hypothesis that the effect of microRNA action is measurable and observable only if it affects the dominant system (generalization of the limiting step notion for complex networks) of the protein translation machinery. The dominant system can vary in different experimental conditions that can partially explain the existing controversy of some of the experimental data.
Conclusions
Our analysis of the transient protein translation dynamics shows that it gives enough information to verify or reject a hypothesis about a particular molecular mechanism of microRNA action on protein translation. For multiscale systems only that action of microRNA is distinguishable which affects the parameters of dominant system (critical parameters), or changes the dominant system itself. Dominant systems generalize and further develop the old and very popular idea of limiting step. Algorithms for identifying dominant systems in multiscale kinetic models are straightforward but not trivial and depend only on the ordering of the model parameters but not on their concrete values. Asymptotic approach to kinetic models allows to put in order diverse experimental observations in complex situations when many alternative hypotheses co-exist.
A. N.
Gorban, O.
Radulescu, A. Y.
Zinovyev,
Asymptotology
of chemical reaction networks, Chemical Engineering Science 65 (2010) 2310–2324
GorbRadZinCES2010Rev.pdf
The concept of the limiting step is extended to the asymptotology of multiscale reaction networks. Complete theory for linear networks with well separated reaction rate constants is developed. We present algorithms for explicit approximations of eigenvalues and eigenvectors of kinetic matrix. Accuracy of estimates is proven. Performance of the algorithms is demonstrated on simple examples. Application of algorithms to nonlinear systems is discussed.
A.N.
Gorban, E.V.
Smirnova, T.A.
Tyukina
General Laws of Adaptation to Environmental
Factors: from Ecological Stress to Financial Crisis. Math.
Model. Nat. Phenom. Vol. 4, No. 6, 2009, pp. 1-53
We study ensembles of similar systems under load of environmental factors. The phenomenon of adaptation has similar properties for systems of different nature. Typically, when the load increases above some threshold, then the adapting systems become more different (variance increases), but the correlation increases too. If the stress continues to increase then the second threshold appears: the correlation achieves maximal value, and start to decrease, but the variance continue to increase. In many applications this second threshold is a signal of approaching of fatal outcome. This effect is supported by many experiments and observation of groups of humans, mice, trees, grassy plants, and on financial time series.
A general approach to explanation of the effect through dynamics of adaptation is developed. H. Selye introduced “adaptation energy” for explanation of adaptation phenomena. We formalize this approach in factors – resource models and develop hierarchy of models of adaptation. Different organization of interaction between factors (Liebig’s versus synergistic systems) lead to different adaptation dynamics. This gives an explanation to qualitatively different dynamics of correlation under different types of load and to some deviation from the typical reaction to stress. In addition to the “quasistatic” optimization factor – resource models, dynamical models of adaptation are developed, and a simple model (three variables) for adaptation to one factor load is formulated explicitly.
A. N.
Gorban, A. Y.
Zinovyev
Principal Graphs and Manifolds,
Chapter 2 in: Handbook of Research on Machine Learning Applications and Trends:
Algorithms, Methods, and Techniques, Emilio Soria Olivas et al. (eds), IGI
Global, Hershey, PA, USA, 2009, pp. 28-59.
In
many physical, statistical, biological and other investigations it is desirable
to approximate a system of points by objects of lower dimension and/or
complexity. For this purpose, Karl Pearson invented principal component
analysis in 1901 and found ‘lines and planes of closest fit to system of
points’. The famous k-means algorithm solves the approximation problem too, but
by finite sets instead of lines and planes. This chapter gives a brief
practical introduction into the methods of construction of general principal
objects (i.e., objects embedded in the ‘middle’ of the multidimensional data
set). As a basis, the unifying framework of mean squared distance approximation
of finite datasets is selected. Principal graphs and manifolds are constructed
as generalisations of principal components and k-means principal points. For
this purpose, the family of expectation/maximisation algorithms with nearest
generalisations is presented. Construction of principal graphs with controlled
complexity is based on the graph grammar approach.
A.N.
Gorban, L.I.
Pokidysheva, E.V.
Smirnova, T.A.
Tyukina
Law of the Minimum Paradoxes, e-print http://arxiv.org/abs/0907.1965
The "law of the minimum" states that growth is controlled by the
scarcest resource (limiting factor) (Justus von Liebig (1840)). This concept
was originally applied to plant or crop growth and quantitatively supported by
many experiments. Some generalizations based on more complicated
"dose-response" curves were proposed. Violations of this law in
natural and experimental ecosystems were also reported. We study models of
adaptation in ensembles of similar organisms under load of environmental
factors and prove that violation of the Liebig law follows from adaptation
effects. If the fitness of an organism in fixed environment satisfies the law
of the minimum then adaptation equalizes the pressure of essential factors and
therefore acts against the law. This is the the law of the minimum paradox: if
for a randomly chosen pair "organism--environment" the law of the
minimum typically holds, then, in a well-adapted system, we have to expect
violations of this law. For the opposite interaction of factors (a synergistic
system of factors which amplify each other) adaptation leads from factor
equivalence to limitations by a smaller number of factors. For analysis of adaptation
we develop a system of models based on Selye's idea of the universal adaptation
resource (adaptation energy). These models predict that under the load of an
environmental factor a population separates into two groups (phases): a less
correlated, well adapted group and a highly correlated group with a larger
variance of attributes, which experiences problems with adaptation. Some
empirical data are presented and some evidences of interdisciplinary
applications to econometrics are discussed.
E. Chiavazzo, I. V. Karlin, A. N. Gorban and K Boulouchos,
Combustion simulation via lattice Boltzmann and reduced chemical kinetics, J. Stat. Mech. (2009) P06013, MIG-LB_StatMech_2009.pdf
We present and validate a methodology for coupling reduced models of detailed combustion mechanisms within the lattice Boltzmann framework. A detailed mechanism (9 species, 21 elementary reactions) for modeling reacting mixtures of air and hydrogen is considered and reduced using the method of invariant grids (MIG). In particular, a 2D quasi-equilibrium grid is constructed, further refined via the MIG method, stored in the form of tables and used to simulate a 1D flame propagating freely through a homogeneous premixed mixture. Comparisons between the detailed and reduced models show that the technique presented enables one to achieve a remarkable speedup in the computations with excellent accuracy.
A. N.
Gorban, E. V.
Smirnova, T. A.
Tyukina,
Correlations, Risk
and Crisis: from Physiology to Finance, e-print: http://arxiv.org/abs/0905.0129. Available at
SSRN: http://ssrn.com/abstract=1397677.
We study the dynamics of correlation and variance
in systems under the load of environmental factors. A universal effect in
ensembles of similar systems under load of similar factors is described: in
crisis, typically, even before obvious symptoms of crisis appear, correlation
increases, and, at the same time, variance (and volatility) increases too.
After the crisis achieves its bottom, it can develop into two directions: recovering
(both correlations and variance decrease) or fatal catastrophe (correlations
decrease, but variance not). This effect is supported by many experiments and
observation of groups of humans, mice, trees, grassy plants, and on financial
time series. A general approach to explanation of the effect through dynamics
of adaptation is developed. Different organization of interaction between
factors (Liebig's versus synergistic systems) lead to different adaptation
dynamics. This gives an explanation to qualitatively different dynamics of
correlation under different types of load.
A. N.
Gorban, O.
Radulescu, A. Y.
Zinovyev,
Limitation and Asymptotology of Chemical
Reaction Networks,
e-print: http://arxiv.org/abs/0903.5072
The concept of the limiting step is extended to
the asymptotology of multiscale reaction networks. Complete theory for linear
networks with well separated reaction rate constants is developed. We present
algorithms for explicit approximations of eigenvalues and eigenvectors of
kinetic matrix. Accuracy of estimates is proven. Performance of the algorithms
is demonstrated on simple examples. Application of algorithms to nonlinear
systems is discussed.
A.
Gorban, I.
Tyukin, E.
Steur, H.
Nijmeijer
Positive Invariance Lemmas for Control
Problems with Convergence to Lyapunov-unstable Sets, e-print http://arxiv.org/abs/0901.3577
We provide Lyapunov-like characterizations of
positive invariance, boundedness and convergence of non-trivial solutions for a
class of systems with unstable invariant sets. The systems of this class
comprise of a stable part coupled with a one-dimensional unstable or critically
stable subsystem. Examples of these systems appear in the problems of nonlinear
output regulation, parameter estimation and adaptive control. We demonstrate
that, for a large class of systems with unstable equilibria and solutions that
might escape to infinity in finite time, it is always possible to determine
simple criteria for positive invariance and boundedness of the system's
nontrivial solutions. Conversely, it is possible to characterize domains of
initial conditions that lead to solutions escaping from the origin. In contrast
to other works addressing convergence issues in unstable systems, our results
do not rely on the availability of input-output gains or contraction rates that
are usually required for the stable compartment.
Principal Graphs and Manifolds, e-print: http://arxiv.org/abs/0809.0490
In many physical statistical, biological and other
investigations it is desirable to approximate a system of points by objects of
lower dimension and/or complexity. For this purpose, Karl Pearson invented
principal component analysis in 1901 and found "lines and planes of
closest fit to system of points". The famous k-means algorithm solves the
approximation problem too, but by finite sets instead of lines and planes. This
chapter gives a brief practical introduction into the methods of construction
of general principal objects, i.e. objects embedded in the "middle"
of the multidimensional data set. As a basis, the unifying framework of mean
squared distance approximation of finite datasets is selected. Principal graphs
and manifolds are constructed as generalisations of principal components and
k-means principal points. For this purpose, the family of
expectation/maximisation algorithms with nearest generalisations is presented.
Construction of principal graphs with controlled complexity is based on the
graph grammar approach.
Ovidiu Radulescu, Alexander N Gorban,
Andrei
Zinovyev, and Alain Lilienbaum
Robust simplifications of multiscale biochemical networks, BMC Systems Biology 2008, 2:86 doi:10.1186/1752-0509-2-86
The most accessed paper in BMC Systems
Biology in November 2008
Background
Cellular processes such as metabolism, decision making in development and differentiation, signalling, etc., can be modeled as large networks of biochemical reactions. In order to understand the functioning of these systems, there is a strong need for general model reduction techniques allowing to simplify models without loosing their main properties. In systems biology we also need to compare models or to couple them as parts of larger models. In these situations reduction to a common level of complexity is needed.
Results
We propose a systematic treatment of model reduction of multiscale biochemical networks. First, we consider linear kinetic models, which appear as "pseudo-monomolecular" subsystems of multiscale nonlinear reaction networks. For such linear models, we propose a reduction algorithm which is based on a generalized theory of the limiting step that we have developed in (Gorban and Radulescu 2008). Second, for non-linear systems we develop an algorithm based on dominant solutions of quasi-stationarity equations. For oscillating systems, quasi-stationarity and averaging are combined to eliminate time scales much faster and much slower than the period of the oscillations. In all cases, we obtain robust simplifications and also identify the critical parameters of the model. The methods are demonstrated for simple examples and for a more complex model of NFkB pathway.
Conclusions
Our approach allows critical parameter identification and produces hierarchies of models. Hierarchical modeling is important in "middle-out" approaches when there is need to zoom in and out several levels of complexity. Critical parameter identification is an important issue in systems biology with potential applications to biological control and therapeutics. Our approach also deals naturally with the presence of multiple time scales, which is a general property of systems biology models.
A.N. Gorban and O. Radulescu,
Dynamic
and Static Limitation in Multiscale Reaction Networks, Revisited, Advances
in Chemical Engineering 34,
103-173. GorbanRadulescuAdvChemEng2008.pdf
The concept of the limiting step gives the limit simplification: the whole network behaves as a single step. This is the most popular approach for model simplification in chemical kinetics. However, in its elementary form this idea is applicable only to the simplest linear cycles in steady states. For simple cycles the nonstationary behavior is also limited by a single step, but not the same step that limits the stationary rate. In this chapter, we develop a general theory of static and dynamic limitation for all linear multiscale networks. Our main mathematical tools are auxiliary discrete dynamical systems on finite sets and specially developed algorithms of ‘‘cycles surgery’’ for reaction graphs. New estimates of eigenvectors for diagonally dominant matrices are used.
Multiscale ensembles of reaction networks with well-separated constants are introduced and typical properties of such systems are studied. For any given ordering of reaction rate constants the explicit approximation of steady state, relaxation spectrum and related eigenvectors (‘‘modes’’) is presented. In particular, we prove that for systems with well-separated constants eigenvalues are real (damped oscillations are improbable). For systems with modular structure, we propose the selection of such modules
that it is possible to solve the kinetic equation for every module in the explicit form. All such ‘‘solvable’’ networks are described. The obtained multiscale approximations, that we call ‘‘dominant systems’’ are computationally cheap and robust. These dominant systems can be used for direct computation of steady states and relaxation dynamics, especially when kinetic information is incomplete, for design of experiments and mining of experimental data, and could serve as a robust first approximation in perturbation theory or for preconditioning.
R. A. Brownlee, A. N.
Gorban, and J. Levesley,
Nonequilibrium entropy limiters in lattice Boltzmann methods, Physica A: Statistical Mechanics and its
Applications
Volume 387, Issues 2-3, 15 January 2008, Pages 385-406 BrownGorbLevPhysA2007FinFin.pdf
We construct a system of nonequilibrium entropy limiters for the lattice
Boltzmann methods (LBM). These limiters erase spurious oscillations without
blurring of shocks, and do not affect smooth solutions. In general, they do the
same work for LBM as flux limiters do for finite differences, finite volumes
and finite elements methods, but for LBM the main idea behind the construction
of nonequilibrium entropy limiter schemes is to transform a field of a scalar
quantity — nonequilibrium entropy. There are two families of limiters: (i)
based on restriction of nonequilibrium entropy (entropy “trimming”) and (ii)
based on filtering of nonequilibrium entropy (entropy filtering). The physical
properties of LBM provide some additional benefits: the control of entropy
production and accurate estimation of introduced artificial dissipation are
possible. The constructed limiters are tested on classical numerical examples:
1D athermal shock tubes with an initial density ratio 1:2 and the 2D lid-driven
cavity for Reynolds numbers between 2000 and 7500 on a coarse
100×100 grid. All limiter constructions are applicable both for entropic and
for non-entropic equilibria.
2007
A.
Gorban, B. Kegl, D. Wunsch, A. Zinovyev (Eds.),
Principal Manifolds for Data Visualisation and Dimension
Reduction, Lecture Notes in Computational Science and Engineering,
Vol. 58, Springer, Berlin –
Heidelberg – New York, 2007. (ISBN 978-3-540-73749-0)
In 1901, Karl Pearson invented Principal
Component Analysis (PCA). Since then, PCA serves as a prototype for many other
tools of data analysis, visualization and dimension reduction: Independent
Component Analysis (ICA), Multidimensional Scaling (MDS), Nonlinear PCA
(NLPCA), Self Organizing Maps (SOM), etc. The book starts with the quote of the
classical Pearson definition of PCA and includes reviews of various methods:
NLPCA, ICA, MDS, embedding and clustering algorithms, principal manifolds and
SOM. New approaches to NLPCA, principal manifolds, branching principal
components and topology preserving mappings are described as well. Presentation
of algorithms is supplemented by case studies, from engineering to astronomy,
but mostly of biological data: analysis of microarray and metabolite data. The
volume ends with a tutorial "PCA and K-means decipher genome". The
book is meant to be useful for practitioners in applied data analysis in life
sciences, engineering, physics and chemistry; it will also be valuable to PhD
students and researchers in computer sciences, applied mathematics and
statistics.
A. N. Gorban,
Selection Theorem for Systems with
Inheritance, Math.
Model. Nat. Phenom., Vol. 2, No. 4, 2007, pp. 1-45. GOtborMMNP2(4)2007.pdf The original publication is available
at www.edpsciences.org
The
problem of finite-dimensional asymptotics of infinite-dimensional dynamic
systems is studied. A non-linear kinetic system with conservation of
supports for distributions has generically finite-dimensional asymptotics.
Such systems are apparent in many areas of biology, physics (the theory of
parametric wave interaction), chemistry and economics. This conservation of
support has a biological interpretation: inheritance. The
finite-dimensional asymptotics demonstrates effects of natural selection.
Estimations of the asymptotic dimension are presented. After some initial time,
solution of a kinetic equation with conservation of support becomes a finite
set of narrow peaks that become increasingly narrow over time and move
increasingly slowly. It is possible that these peaks do not tend to fixed
positions, and the path covered tends to infinity as t→∞.
The drift equations for peak motion are obtained. Various types of
distribution stability are studied: internal stability (stability with respect
to perturbations that do not extend the support), external stability or
uninvadability (stability with respect to strongly small perturbations that
extend the support), and stable realizability (stability with respect to small
shifts and extensions of the density peaks). Models of self-synchronization of
cell division are studied, as an example of selection in systems with
additional symmetry. Appropriate construction of the notion of typicalness in
infinite-dimensional space is discussed, and the notion of “completely thin”
sets is introduced.
A.N.
Gorban and O. Radulescu
Dynamical robustness of biological
networks with hierarchical distribution of time scales, IET
Syst. Biol.,
2007, 1, (4), pp. 238–246 Gorban2007IEESystemsBiology.pdf
Concepts
of distributed robustness and r-robustness proposed by biologists to explain a
variety of stability phenomena in molecular biology are analysed. Then, the
robustness of the relaxation time using a chemical reaction description of genetic
and signalling networks is discussed. First, the following result for linear
networks is obtained: for large multiscale systems with hierarchical
distribution of time scales, the variance of the inverse relaxation time (as
well as the variance of the stationary rate) is much lower than the variance of
the separate constants. Moreover, it can tend to 0 faster than 1/n, where n is
the number of reactions. Similar phenomena are valid in the nonlinear case as
well. As a numerical illustration, a model of signalling network is used for
the important transcription factor NFkB.
A.N. Gorban and A.Y. Zinovyev
The Mystery of Two Straight Lines in
Bacterial Genome Statistics, Bulletin of Mathematical Biology (2007) DOI 10.1007/s11538-007-9229-6 (Online
First) GorbanZinovyev2007BMB1.pdf
In special coordinates (codon position-specific nucleotide frequencies),
bacterial genomes form two straight lines in 9-dimensional space: one line for
eubacterial genomes, another for archaeal genomes. All the 348 distinct
bacterial genomes available in Genbank in April 2007, belong to these lines
with high accuracy. The main challenge now is to explain the observed high
accuracy. The new phenomenon of complementary symmetry for codon
position-specific nucleotide frequencies is observed. The results of analysis
of several codon usage models are presented.We demonstrate that the mean-field
approximation, which is also known as context-free, or complete independence
model, or Segre variety, can serve as a reasonable approximation to the real
codon usage. The first two principal components of codon usage correlate
strongly with genomic G+C content and the optimal growth temperature,
respectively. The variation of codon usage along the third component is related
to the curvature of the mean-field approximation. First three eigenvalues in
codon usage PCA explain 59.1%, 7.8% and 4.7% of variation. The eubacterial and
archaeal genomes codon usage is clearly distributed along two third order
curves with genomic G+C content as a parameter.
A.N.
Gorban, O.
Radulescu
Dynamic and static limitation in reaction
networks, revisited, http://arxiv.org/abs/physics/0703278 [physics.chem-ph] GorRadLimarXiv0703278v2.pdf
The concept of limiting step gives the limit simplification: the whole network
behaves as a single step. This is the most popular approach for model
simplification in chemical kinetics. However, in its simplest form this idea is
applicable only to the simplest linear cycles in steady states. For such the
simplest cycles the nonstationary behaviour is also limited by a single step,
but not the same step that limits the stationary rate. In this paper, we
develop a general theory of static and dynamic limitation for all linear multiscale
networks, not only for simple cycles. Our main mathematical tools are auxiliary
discrete dynamical systems on finite sets and specially developed algorithms of
``cycles surgery" for reaction graphs. New estimates of eigenvectors for
diagonally dominant matrices are used.
Multiscale
ensembles of reaction networks with well separated constants are introduced and
typical properties of such systems are studied. For any given ordering of
reaction rate constants the explicit approximation of steady state, relaxation
spectrum and related eigenvectors (``modes") is presented. In particular,
we proved that for systems with well separated constants eigenvalues are real
(damped oscillations are improbable). For systems with modular structure, we
propose to select such modules that it is possible to solve the kinetic
equation for every module in the explicit form. All such ``solvable"
networks are described. The obtained multiscale approximations that we call
``dominant systems" are computationally cheap and robust. These dominant
systems can be used for direct computation of steady states and relaxation
dynamics, especially when kinetic information is incomplete, for design of
experiments and mining of experimental data, and could serve as a robust first
approximation in perturbation theory or for preconditioning.
R.A.
Brownlee, A.N.
Gorban, J.
Levesley,
Nonequilibrium entropy limiters in
lattice Boltzmann methods, arXiv:0704.0043v1
[cond-mat.stat-mech] BrowGorLevLimitersArXiv.pdf
We construct a system of nonequilibrium entropy
limiters for the lattice Boltzmann methods (LBM). These limiters erase spurious
oscillations without blurring of shocks, and do not affect smooth solutions. In
general, they do the same work for LBM as flux limiters do for finite
differences, finite volumes and finite elements methods, but for LBM the main
idea behind the construction of nonequilibrium entropy limiter schemes is to
transform a field of a scalar quantity - nonequilibrium entropy. There are two
families of limiters: (i) based on restriction of nonequilibrium entropy
(entropy "trimming") and (ii) based on filtering of nonequilibrium
entropy (entropy filtering). The physical properties of LBM provide some
additional benefits: the control of entropy production and accurate estimate of
introduced artificial dissipation are possible. The constructed limiters are
tested on classical numerical examples: 1D athermal shock tubes with an initial
density ratio 1:2 and the 2D lid-driven cavity for Reynolds numbers Re between
2000 and 7500 on a coarse 100*100 grid. All limiter constructions are
applicable for both entropic and non-entropic quasiequilibria.
R. A. Brownlee, A. N.
Gorban, and
Stability and stabilization of the lattice
Boltzmann method, Phys. Rev. E 75, 036711 (2007) (17
pages) BGJPhyRev2007.pdf
We revisit the classical stability versus accuracy dilemma for the
lattice Boltzmann methods (LBM). Our goal is a stable method of
second-order accuracy for fluid dynamics based on the lattice
Bhatnager-Gross-Krook method (LBGK). The LBGK scheme can be
recognized as a discrete dynamical system generated by free flight
and entropic involution. In this framework the stability and
accuracy analysis are more natural. We find the necessary and
sufficient conditions for second-order accurate fluid dynamics
modeling. In particular, it is proven that in order to guarantee
second-order accuracy the distribution should belong to a
distinguished surface—the invariant film (up to second order in the
time step). This surface is the trajectory of the (quasi)equilibrium
distribution surface under free flight. The main instability mechanisms
are identified. The simplest recipes for stabilization add no artificial
dissipation (up to second order) and provide second-order accuracy of
the method. Two other prescriptions add some artificial dissipation locally
and prevent the system from loss of positivity and local blowup.
Demonstration of the proposed stable LBGK schemes are provided by
the numerical simulation of a one-dimensional (1D) shock tube and
the unsteady 2D flow around a square cylinder up to Reynolds number
Re~20,000.
E.
Chiavazzo, A.N. Gorban, and
I.V. Karlin,
Comparison
of Invariant Manifolds for Model Reduction in Chemical Kinetics, Commun.
Comput. Phys. Vol. 2, No. 5 (2007), pp. 964-992 CiCP2007vol2_n5_p964.pdf
A
modern approach to model reduction in chemical kinetics is often based on the
notion of slow invariant manifold. The goal of this paper is to give a
comparison of various methods of construction of slow invariant manifolds using
a simple Michaelis-Menten catalytic reaction. We explore a recently introduced
Method of Invariant Grids (MIG) for iteratively solving the invariance
equation. Various initial approximations for the grid are considered such as
Quasi Equilibrium Manifold, Spectral Quasi Equilibrium Manifold, Intrinsic Low
Dimensional Manifold and Symmetric Entropic Intrinsic Low Dimensional Manifold.
Slow invariant manifold was also computed using the Computational Singular
Perturbation (CSP) method. A comparison between MIG and CSP is also reported.
A.N. Gorban,
N.R. Sumner, and A.Y. Zinovyev,
Topological
grammars for data approximation, Applied
Mathematics Letters Volume 20, Issue 4 (2007),
382-386 GorSummnZinAML2006.pdf
A method of topological grammars is proposed for
multidimensional data approximation. For data with complex topology we define a
principal cubic complex of low dimension and given complexity that gives
the best approximation for the dataset. This complex is a generalization of
linear and non-linear principal manifolds and includes them as particular
cases. The problem of optimal principal complex construction is transformed
into a series of minimization problems for quadratic functionals. These
quadratic functionals have a physically transparent interpretation in terms of
elastic energy. For the energy computation, the whole complex is represented as
a system of nodes and springs. Topologically, the principal complex is a
product of one-dimensional continuums (represented by graphs), and the grammars
describe how these continuums transform during the process of optimal complex construction.
This factorization of the whole process onto one-dimensional transformations
using minimization of quadratic energy functionals allows us to construct
efficient algorithms.
A.N. Gorban,
Order–disorder separation: Geometric
revision, Physica A
Volume 374, Issue 1 , 15 January 2007,
Pages 85-102 GorPhysA2006Order.pdf
After Boltzmann and Gibbs, the notion of disorder in statistical physics
relates to ensembles, not to individual states. This disorder is measured by the
logarithm of ensemble volume, the entropy. But recent results about measure
concentration effects in analysis and geometry allow us to return from the
ensemble-based point of view to a state-based one, at least, partially. In this
paper, the order–disorder problem is represented as a problem of relation
between distance and measure. The effect of strong order–disorder separation
for multiparticle systems is described: the phase space could be divided into
two subsets, one of them (set of disordered states) has almost zero diameter,
the second one has almost zero measure. The symmetry with respect to
permutations of particles is responsible for this type of concentration.
Dynamics of systems with strong order–disorder separation has high average
acceleration squared, which can be interpreted as evolution through a series of
collisions (acceleration-dominated dynamics). The time arrow direction from
order to disorder follows from the strong order–disorder separation. But,
inverse, for systems in space of symmetric configurations with “sticky
boundaries” the way back from disorder to order is typical (Natural selection).
Recommendations for mining of molecular dynamics results are also presented.
2006
Ovidiu Radulescu, Alexander N. Gorban,
Sergei Vakulenko, Andrei Zinovyev
Hierarchies
and modules in complex biological systems, In: Proceedings of European
Conference on Complex Systems (paper ECCS06-114), Oxford, UK, September 2006 OxfordHiModP114.pdf
We review several
mathematical methods allowing to identify modules and hierarchies with several
levels of complexity in biological systems. These methods are based either on
the properties of the input-output characteristic of the modules or on global
properties of the dynamics such as the distribution of timescales or the
stratification of attractors with variable dimension. We also discuss the
consequences of the hierarchical structure on the robustness of biological
processes. Stratified attractors lead to Waddington's type canalization
effects. Successive application of the many to one mapping relating parameters
of different levels in an hierarchy of models (analogue to the renormalization
operation from statistical mechanics) leads to concentration and robustness of
those properties that are common to many levels of complexity. Examples such as
the response of the transcription factor NF·B to signalling, and the
segmentation patterns in the development of Drosophila are used as
illustrations of the theoretical ideas.
R. A. Brownlee, A. N. Gorban, and
Stabilization
of the lattice Boltzmann method using the Ehrenfests' coarse-graining idea,
Phys. Rev. E 74, 037703 (2006) RobBrowGorbLeveslPRE2006.pdf
The lattice Boltzmann method (LBM) and its variants have emerged as
promising, computationally efficient and increasingly popular numerical
methods for modeling complex fluid flow. However, it is acknowledged
that the method can demonstrate numerical instabilities, e.g., in
the vicinity of shocks. We propose a simple technique to stabilize
the LBM by monitoring the difference between microscopic and
macroscopic entropy. Populations are returned to their equilibrium
states if a threshold value is exceeded. We coin the name Ehrenfests'
steps for this procedure in homage to the vehicle that we use to
introduce the procedure, namely, the Ehrenfests' coarse-graining
idea.
A.N. Gorban, B.M. Kaganovich, S.P. Filippov, A.V. Keiko, V.A.
Shamansky, I.A. Shirkalin,
Thermodynamic
Equilibria and Extrema: Analysis of Attainability Regions and Partial
Equilibria, Springer, Berlin-Heidelberg-New York, 2006.
Model
Reduction and Coarse--Graining Approaches for Multiscale Phenomena,
Ed. by Alexander N. Gorban, Nikolaos
Kazantzis, Ioannis G. Kevrekidis, Hans Christian Öttinger, Constantinos
Theodoropoulos , Springer,
Invariant Grids: Method of Complexity
Reduction in Reaction Networks, Complexus, V. 2, 110–127. ComPlexUs2006.pdf
Complexity in the description of big chemical reaction
networks has both structural (number of species and reactions) and temporal
(very different reaction rates) aspects. A consistent way to make model
reduction is to construct the invariant manifold which describes the asymptotic
system behaviour. In this paper we present a discrete analogue of this object:
an invariant grid. The invariant grid is introduced independently from the
invariant manifold notion and can serve to represent the dynamic system
behaviour as well as to approximate the invariant manifold after refinement.
The method is designed for pure dissipative systems and widely uses their
thermodynamic properties but allows also generalizations for some classes of
open systems. The method is illustrated by two examples: the simplest catalytic
reaction (Michaelis-Menten mechanism) and the hydrogen oxidation.
A.N. Gorban,
Basic Types of Coarse-Graining, e-print http://arxiv.org/abs/cond-mat/0602024
(local copy CoaGrWorkSpri7.pdf).
42 pgs, 11 figs. A talk given at the
research workshop: "Model Reduction and Coarse-Graining
Approaches for Multiscale Phenomena,"
We consider two
basic types of coarse-graining: the Ehrenfest's coarse-graining and its
extension to a general principle of non-equilibrium thermodynamics, and the
coarse-graining based on uncertainty of dynamical models and $\epsilon$-motions
(orbits). Non-technical discussion of basic notions and main coarse-graining
theorems are presented: the theorem about entropy overproduction for the
Ehrenfest's coarse-graining and its generalizations, both for conservative and
for dissipative systems, and the theorems about stable properties and the Smale
order for $\epsilon$-motions of general dynamical systems including
structurally unstable systems. A brief discussion of two other types,
coarse-graining by rounding and by small noise, is also presented.
Computational kinetic models of macroscopic dynamics are considered. We
construct a theoretical basis for these kinetic models using generalizations of
the Ehrenfest's coarse-graining.
A.N. Gorban, I.V. Karlin,
Quasi-Equilibrium Closure Hierarchies for the
Boltzmann Equation, Physica A 360 (2006) 325–364 GKQEBoltzPhysA2006.pdf
In this paper, explicit method of
constructing approximations (the
Triangle Entropy Method) is developed for nonequilibrium problems. This method enables one to treat any
complicated nonlinear functionals that fit best the physics of a problem (such as, for
example, rates of processes) as new independent variables.
The work of the method was demonstrated on the Boltzmann's - type kinetics.
New macroscopic variables are
introduced (moments of the Boltzmann
collision integral, or scattering rates). They are treated as
independent variables rather than as infinite moment series. This approach gives the complete account
of rates of scattering
processes. Transport equations for scattering rates are obtained
(the second hydrodynamic chain), similar
to the usual moment chain (the first
hydrodynamic chain). Various examples of the closure of the first, of the
second, and of the mixed hydrodynamic chains
are considered for the hard spheres model. It is shown, in particular, that the complete
account of scattering processes leads to a renormalization of transport
coefficients.
The method gives the explicit solution for the closure problem, provides
thermodynamic properties of reduced models, and can be applied to any kinetic
equation with a thermodynamic Lyapunov function
Elastic
Principal Graphs and Manifolds and their Practical Applications,
Computing 75, 359–379 (2005), (DOI) 10.1007/s00607-005-0122-6
, GorbZin2005Computing.pdf
Principal manifolds serve as useful tool for many practical applications. These manifolds are defined as lines or surfaces passing through “the middle” of data distribution. We propose an algorithm for fast construction of grid approximations of principal manifolds with given topology. It is based on analogy of principal manifold and elastic membrane. First advantage of this method is a form of the functional to be minimized which becomes quadratic at the step of the vertices position refinement. This makes the algorithm very effective, especially for parallel implementations. Another advantage is that the same algorithmic kernel is applied to construct principal manifolds of different dimensions and topologies. We demonstrate how flexibility of the approach allows numerous adaptive strategies like principal graph constructing, etc. The algorithm is implemented as a C++ package elmap and as a part of stand-alone data visualization tool VidaExpert, available on the web. We describe the approach and provide several examples of its application with speed performance characteristics.
A.N. Gorban, I.V. Karlin,
Invariance correction to Grad's equations: Where to go beyond
approximations? Continuum Mechanics and Thermodynamics, 17(4) (2005), 311–335, GorKarCMT_05.pdf, http://arxiv.org/abs/cond-mat/0504221
We review some recent developments of Grad's approach to solving the Boltzmann
equation and creating reduced description. The method of invariant manifold is
put forward as a unified principle to establish corrections to Grad's
equations. A consistent derivation of regularized Grad's equations in the
framework the method of invariant manifold is given. A new class of kinetic
models to lift the finite-moment description to a kinetic theory in the whole
space is established. Relations of Grad's approach to modern mesoscopic
integrators such as the entropic lattice Boltzmann method are also discussed.
A.N. Gorban, T.G.Popova, A.Yu. Zinovyev,
Codon usage trajectories and 7-cluster
structure of 143 complete bacterial genomic sequences Physica A
353C
(2005), 365-387. CodonPhysA2005.pdf (Number 11
in TOP25
articles within the journal: Physica A: Statistical Mechanics and its Applications, APR - JUN 2005 Top25.pdf)
Three results are presented. First, we prove the existence of a universal
7-cluster structure in all 143 completely sequenced bacterial genomes available
in Genbank in August 2004, and explained its properties. The 7-cluster
structure is responsible for the main part of sequence heterogeneity in
bacterial genomes. In this sense, our 7 clusters is the basic model of
bacterial genome sequence. We demonstrated that there are four basic
``pure" types of this model, observed in nature: ``parallel
triangles", ``perpendicular triangles", degenerated case and the
flower-like type.
Second, we answered the question: how big are the position-specific information
and the contribution connected with correlations between nucleotide. The
accuracy of the mean-field (context-free) approximation is estimated for
bacterial genomes.
We show that codon usage of bacterial genomes is a multi-linear function of
their genomic G+C-content with high accuracy (more precisely, by two similar
functions, one for eubacterial genomes and the other one for archaea).
Description of these two codon-usage trajectories is the third result.
All 143 cluster animated 3D-scatters are collected in a database and is made
available on our web-site: http://www.ihes.fr/~zinovyev/7clusters
.
A.N. Gorban, T.G.Popova, A.Yu. Zinovyev,
Four basic symmetry types in the universal 7-cluster structure of microbial
genomic sequences, In
Silico Biology, 5 (2005), 0039. Internet
site CLUSTER
STRUCTURE IN GENOME with analysis of all bacterial genomes.
Coding information is the main source of heterogeneity (non-randomness) in the
sequences of microbial genomes. The heterogeneity corresponds to a cluster
structure in triplet distributions of relatively short genomic fragments
(200-400bp). We found a universal 7-cluster structure in microbial genomic
sequences and explained its properties. We show that codon usage of bacterial
genomes is a multi-linear function of their genomic G+C-content with high
accuracy. Based on the analysis of 143 completely sequenced bacterial genomes
available in Genbank in August 2004, we show that there are four "pure"
types of the 7-cluster structure observed. All 143 cluster animated 3D-scatters
are collected in a database which is made available on our web-site (http://www.ihes.fr/~zinovyev/7clusters).
The findings can be readily introduced into software for gene prediction,
sequence alignment or microbial genomes classification.
A.N. Gorban, I.V. Karlin,
Invariant Manifolds for Physical and
Chemical Kinetics, Lect.
Notes Phys. 660, Springer,
The concept of the slow invariant manifold is recognized as the central idea
underpinning a transition from micro to macro and model reduction in kinetic
theories. We present the constructive methods of invariant manifolds for model
reduction in physical and chemical kinetics, developed during last two decades.
The physical problem of reduced description is studied in the most general form
as a problem of constructing the slow invariant manifold. The invariance
conditions are formulated as the differential equation for a manifold immersed
in the phase space (the invariance equation). The equation of motion for immersed
manifolds is obtained (the film extension of the dynamics).
Invariant manifolds are fixed points for this equation, and slow invariant
manifolds are Lyapunov stable fixed points, thus slowness is presented as
stability.
A collection of methods to derive analytically and to compute numerically the slow invariant manifolds is presented. Among them, iteration methods based on incomplete linearization, relaxation method and the method of invariant grids are developed. The systematic use of thermodynamic structures and of the quasi-chemical representation allows us to construct approximations which are in concordance with physical restrictions.
The following examples of applications are presented: Nonperturbative derivation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the moment equations for nonequilibrium media and their dynamical correction (instead of extension of the list of variables) in order to gain more accuracy in description of highly nonequilibrium flows; kinetic theory of phonons; model reduction in chemical kinetics; derivation and numerical implementation of constitutive equations for polymeric fluids; the limits of macroscopic description for polymer molecules, cell division kinetics.
Keywords: Model Reduction; Invariant Manifold; Entropy; Kinetics; Boltzmann Equation; Fokker--Planck Equation; Navier-Stokes Equation; Burnett Equation; Quasi-chemical Approximation; Oldroyd Equation; Polymer Dynamics; Molecular Individualism; Accuracy Estimation; Post-processing.
PACS codes: 05.20.Dd Kinetic theory, 02.30.Mv Approximations and expansions, 02.70.Dh Finite-element and Galerkin methods, 05.70.Ln Nonequilibrium and irreversible thermodynamics.
A.N. Gorban
Order--disorder separation: Geometric revision, E-print: http://arxiv.org/abs/cond-mat/0507644
After Boltzmann and Gibbs, the notion of disorder in statistical physics
relates to ensembles, not to individual states. This disorder is measured by
the logarithm of ensemble volume, the entropy. But recent results about measure
concentration effects in analysis and geometry allow us to return from the
ensemble--based point of view to a state--based one, at least, partially. In
this paper, the order--disorder problem is represented as a problem of relation
between distance and measure. The effect of strong order--disorder separation
for multiparticle systems is described: the phase space could be divided into
two subsets, one of them (set of disordered states) has almost zero diameter,
the second one has almost zero measure. The symmetry with respect to
permutations of particles is responsible for this type of concentration.
Dynamics of systems with strong order--disorder separation has high average
acceleration squared, which can be interpreted as evolution through a series of
collisions (acceleration--dominated dynamics). The time arrow direction from
order to disorder follows from the strong order--disorder separation. But,
inverse, for systems in space of symmetric configurations with ``sticky
boundaries" the way back from disorder to order is typical (Natural
selection). Recommendations for mining of molecular dynamics results are
presented also.
S. Ansumali, S. Archidiacono, S. Chikatamarla, A.N. Gorban,
I.V. Karlin,
Regularized Kinetic Theory, E-print: http://arxiv.org/abs/cond-mat/0507601
A new approach to model hydrodynamics at the level of one-particle distribution
function is presented. The construction is based on the choice of
quasi-equilibria pertinent to the physical context of the problem. Kinetic
equations for a single component fluid with a given Prandtl number and models
of mixtures with a given Schmidt number are derived. A novel realization of these
models via an auxiliary kinetic equation is suggested.
A.N.
Gorban, G.S. Yablonsky
Thermodynamic theory of kinetic overshoots, IMACS2005 extended abstract,
E-print: http://arxiv.org/abs/physics/0505135
Everything that is not prohibited is permissible. So, what is prohibited in the
course of chemical reactions, heat transfer and other dissipative processes? Is
it possible to "overshoot" the equilibrium, and if yes, then how far?
Thermodynamically allowed and prohibited trajectories of processes are
discussed by the example of effects of equilibrium encircling. The complete
theory of thermodynamically accessible states is presented. The space of all
thermodynamically admissible paths is presented by projection on the
"thermodynamic tree", that is the tree of the related thermodynamic
potential (entropy, free energy, free enthalpy) in the balance polyhedron. The
stationary states and limit points for open systems are localized too.
A.N.
Gorban, M. Kudryashev, T. Popova,
On the Way to Protein Alphabet: Informational
Classification of Amino Acids in Comparison to Other Classifications, E-print: http://arxiv.org/abs/q-bio.BM/0501019
What proteins are made from, as the working parts of the living cells protein
machines? To answer this question, we need a technology to disassemble proteins
onto elementary functional details and to prepare lumped description of such
details. This lumped description might have a multiple material realization (in
amino acids). Our hypothesis is that informational approach to this problem is
possible. We propose a way of hierarchical classification that makes the
primary structure of protein maximally non-random and compare them with other
classifications. The first step of the suggested research program is realized:
the analysis of protein binary alphabet in comparison with other amino acid
classifications.
A.N. Gorban,
A. Yu. Zinovyev
PCA deciphers genome, E-print: http://arxiv.org/abs/q-bio.QM/0504013 PCAdecGen.pdf
In this paper, we give a tutorial for undergraduate students studying statistical
methods and/or bioinformatics. The students learn how data visualization can
help in genomic sequences analysis. Students start with a fragment of genetic
text of a bacterial genome and analyze its structure. By means of principal
component analysis they ``discover'' that the information in genome is encoded
by non-overlapping triplets. Next, they learn to find gene positions. This
exercise on principal component analysis and K-Means clustering gives a
possibility for active study of the basic bioinformatics notions. In Appendix
the program listings for MatLab are published.
2004
A.N. Gorban,
D.A. Rossiyev, M.G. Dorrer
MultiNeuron - Neural Networks Simulator For
Medical, Physiological, and Psychological Applications, The talk for the 1995 World Congress on
Neural Networks, E-print: http://arxiv.org/abs/q-bio.QM/0411034
This work describes neural software applied in medicine and physiology to:
investigate and diagnose immune deficiencies; diagnose and study allergic and
pseudoallergic reactions; forecast emergence or aggravation of stagnant cardiac
insufficiency in patients with cardiac rhythm disorders; forecast development
of cardiac arrhythmia after myocardial infarction; reveal relationships between
the accumulated radiation dose and a set of immunological, hormonal, and
bio-chemical parameters of human blood and find a method to be able to judge by
these parameters the dose value; propose a technique for early diagnosis of
chor-oid melanomas; Neural networks also help to predict human relations within
a group.
A.N.
Gorban, A.Yu. Zinovyev,
The Mystery of Two Straight Lines in Bacterial
Genome Statistics,
E-print: http://arxiv.org/abs/q-bio.GN/0412015
In special coordinates (codon position--specific nucleotide frequencies)
bacterial genomes form two straight lines in 9-dimensional space: one line for
eubacterial genomes, another for archaeal genomes. All the 175 known bacterial
genomes (Genbank, March 2005) belong these lines with high accuracy, and these
two lines are certainly different. The results of PCA analysis of codon usage
and accuracy of mean--field (context--free) approximation are presented. The
first two principal components correlate strongly with genomic G+C-content and
the optimal growth temperature respectively. The variation of codon usage along
the third component is related to the curvature of the mean-field
approximation. The eubacterial and archaeal genomes codon usage are clearly
distributed along two third order curves with genomic G+C-content as a
parameter.
A.N. Gorban, T.G. Popova, A.Yu. Zinovyev,
Four basic symmetry types in the universal 7-cluster structure of 143
complete bacterial genomic sequences E-print: http://arxiv.org/abs/q-bio/0410033
The coding information is the main source of heterogeneity (non-randomness) in
the sequences of bacterial genomes. This information can be naturally modeled
by analysing cluster structures in the "in-phase" triplet
distributions of relatively short genomic fragments (200-400bp). We found a
universal 7-cluster structure in bacterial genomic sequences and explained its
properties. We show that codon usage of bacterial genomes is a multi-linear
function of their genomic G+C-content with high accuracy. Based on the analysis
of 143 completely sequenced bacterial genomes available in Genbank in August
2004, we show that there are four "pure" types of the 7-cluster
structure observed. All 143 cluster animated 3D-scatters are collected in a
database and is made available on our web-site: http://www.ihes.fr/~zinovyev/7clusters.
The finding can be readily introduced into any software for gene prediction,
sequence alignment or bacterial genomes classification.
Gorban A.N., Popova T.G., Zinovyev A.Yu.,
Seven clusters and unsupervised gene prediction, Proceedings of the Fourth International Conference on Bioinformatics of Genome Regulation and Structure, BGRS’ 2004, Novosibirsk, Russia, July 25 - 30, 2004, IC&G, Novosibirsk, 2004, pp. 277-280.
Motivation: The effectiveness of most unsupervised
gene-detection algorithms follows from a cluster structure in oligomer
distributions. Existence of this structure is implicitly known but it was never
visualized and studied in terms of data exploration strategies. Visual
representation of the structure allows deeper understanding of its properties
and can serve to display and analyze characteristics of existing gene-finders.
Results: The cluster structure of genome fragments
distribution in the space of their triplet frequencies was revealed by pure
data exploration strategy. Several complete genomic sequences were analyzed,
using visualization of distribution of 64-dimensional vectors of triplet
frequencies in a sliding window. The structure of distribution was found to
consist of seven clusters, corresponding to proteincoding genome fragments in
three possible phases in each of the two complementary strands and to the
non-coding regions with high accuracy. The self-training technique for
automated gene recognition both in entire genomes and in unassembled ones is
proposed.
Gorban, A.N., Zinovyev, A.Y.
Elastic principal manifolds and their practical applications E-print http://arxiv.org/abs/cond-mat/0405648
Principal manifolds defined as lines or surfaces passing through "the
middle" of the data distribution serve as useful objects for many
practical applications. We propose a new algorithm for fast construction of grid
approximations of principal manifolds with given topology. One advantage of the
method is a new form of the functional to be minimized, which becomes quadratic
at the step of the vertexes positions refinement. This makes the algorithm very
effective, especially for parallel implementations. Another advantage is that
the same algorithmic kernel is applied to construct principal manifolds of
different dimensions and topologies. We demonstrate how flexibility of the
approach allows easily numerous adaptive strategies like principal graph
constructing, etc. The algorithm is implemented as a C++ package elmap and as a
part of stand-alone data visualization tool VidaExpert, available on the web.
We describe the approach and provide several examples of its applications with
speed performance characteristics.
Gorban, A.N.
Systems with inheritance: dynamics of
distributions with conservation of support, natural selection and
finite-dimensional asymptotics E-print: http://arxiv.org/abs/cond-mat/0405451
If we find a representation of an infinite-dimensional dynamical system as a
nonlinear kinetic system with {\it conservation of supports} of distributions,
then (after some additional technical steps) we can state that the asymptotics
is finite-dimensional. This conservation of support has a {\it quasi-biological
interpretation, inheritance} (if a gene was not presented initially in a
isolated population without mutations, then it cannot appear at later time).
These quasi-biological models can describe various physical, chemical, and, of
course, biological systems. The finite-dimensional asymptotic demonstrates
effects of {\it "natural" selection}. The estimations of asymptotic
dimension are presented. The support of an individual limit distribution is
almost always small. But the union of such supports can be the whole space even
for one solution. Possible are such situations: a solution is a finite set of
narrow peaks getting in time more and more narrow, moving slower and slower. It
is possible that these peaks do not tend to fixed positions, rather they
continue moving, and the path covered tends to infinity at $t \to \infty$. The
{\it drift equations} for peaks motion are obtained. Various types of stability
are studied. In example, models of cell division self-synchronization are
studied. The appropriate construction of notion of typicalness in
infinite-dimensional spaces is discussed, and the "completely thin" sets
are introduced
Gorban, A.N.
Singularities of transition processes in
dynamical systems: Qualitative theory of critical delays Electron.
J. Diff. Eqns. Monograph 5, 2004, 55 p.Slorelax2004EJDE.pdf
Online: http://ejde.math.txstate.edu/Monographs/05/abstr.html
This monograph presents a systematic analysis of the singularities in the
transition processes for dynamical systems. We study general dynamical systems,
with dependence on a parameter, and construct relaxation times that depend on
three variables: Initial conditions x, parameters k of the system, and accuracy
e of the relaxation. We study the singularities of relaxation times as
functions of (x,k) under fixed e, and then classify the bifurcations
(explosions) of limit sets. We study the relationship between singularities of
relaxation times and bifurcations of limit sets. An analogue of the Smale order
for general dynamical systems under perturbations is constructed. It is shown
that the perturbations simplify the situation: the interrelations between the
singularities of relaxation times and other peculiarities of dynamics for
general dynamical system under small perturbations are the same as for the
Morse-Smale systems
Gorban, A.N.;Gorban, P.A.;Karlin, I.V.
Legendre integrators, post-processing and
quasiequilibrium J. Non-Newtonian
Fluid Mech. 120 (2004) 149-167 GoGoKar2004.pdf
Online: http://arxiv.org/abs/cond-mat/0308488
A toolbox for the development and reduction of the dynamical models of
nonequilibrium systems is presented. The main components of this toolbox are:
Legendre integrators, dynamical post-processing, and the thermodynamic
projector. The thermodynamic projector is the tool to transform almost any
anzatz to a thermodynamically consistent model. The post-processing is the
cheapestway to improve the solution obtained by the Legendre integrators.
Legendre integrators give the opportunity to solve linear equations instead of
nonlinear ones for quasiequilibrium (maximum entropy, MaxEnt) approximations.
The essentially new element of this toolbox, the method of thermodynamic
projector, is demonstrated on application to the FENE-P model of polymer
kinetic theory. The multi-peak model of polymer dynamics is developed.
Gorban, A.N.;Karlin, I.V.
Uniqueness of thermodynamic projector and
kinetic basis of molecular individualism Physica A, 336, 2004, 391-432 UniMolIndRepr.pdf
Online: http://arxiv.org/abs/cond-mat/0309638
Three results are presented: First, we solve the problem of persistence of
dissipation for reduction of kinetic models. Kinetic equations with
thermodynamic Lyapunov functions are studied. Uniqueness of the thermodynamic
projector is proven: There exists only one projector which transforms any
vector field equipped with the given Lyapunov function into a vector field with
the same Lyapunov function for a given anzatz manifold which is not tangent to
the Lyapunov function levels. Second, we use the thermodynamic projector for
developing the short memory approximation and coarse-graining for general
nonlinear dynamic systems. We prove that in this approximation the entropy
production increases. (The theorem about entropy overproduction.) In example,
we apply the thermodynamic projector to derive the equations of reduced
kinetics for the Fokker-Planck equation. A new class of closures is developed,
the kinetic multipeak polyhedra. Distributions of this type are expected in
kinetic models with multidimensional instability as universally as the Gaussian
distribution appears for stable systems. The number of possible relatively
stable states of a nonequilibrium system grows as 2^m, and the number of
macroscopic parameters is in order mn, where n is the dimension of configuration
space, and m is the number of independent unstable directions in this space.
The elaborated class of closures and equations pretends to describe the effects
of molecular individualism. This is the third result.
Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.
Constructive methods of invariant manifolds for
kinetic problems Phys. Rep.,
396, 2004, 197-403 PhysRepCorr.pdf Online: http://arxiv.org/abs/cond-mat/0311017
The concept of the slow invariant manifold is recognized as the central idea
underpinning a transition from micro to macro and model reduction in kinetic
theories. We present the Constructive Methods of Invariant Manifolds for model
reduction in physical and chemical kinetics, developed during last two decades.
The physical problem of reduced description is studied in the most general form
as a problem of constructing the slow invariant manifold. The invariance conditions
are formulated as the differential equation for a manifold immersed in the
phase space (the invariance equation). The equation of motion for immersed
manifolds is obtained (the film extension of the dynamics). Invariant manifolds
are fixed points for this equation, and slow invariant manifolds are Lyapunov
stable fixed points, thus slowness is presented as stability.
A collection of methods to derive analytically and to compute numerically the
slow invariant manifolds is presented. Among them, iteration methods based on
incomplete linearization, relaxation method and the method of invariant grids
are developed. The systematic use of thermodynamics structures and of the
quasi-chemical representation allow to construct approximations which are in
concordance with physical restrictions.
The following examples of applications are presented: nonperturbative
derivation of physically consistent hydrodynamics from the Boltzmann equation
and from the reversible dynamics, for Knudsen numbers Kn~1; construction of the
moment equations for nonequilibrium media and their dynamical correction
(instead of extension of list of variables) to gain more accuracy in
description of highly nonequilibrium flows; determination of molecules
dimension (as diameters of equivalent hard spheres) from experimental viscosity
data ; model reduction in chemical kinetics; derivation and numerical
implementation of constitutive equations for polymeric fluids; the limits of
macroscopic description for polymer molecules, etc.
Gorban, A.N.;Karlin, I.V.;Zinovyev, A.Y.
Invariant grids for reaction kinetics Physica A, 333, 2004 106-154 ChemGrPhA2004.pdf Online: http://arxiv.org/abs/cond-mat/0307076
In this paper, we review the construction of low-dimensional manifolds of
reduced description for equations of chemical kinetics from the standpoint of
the method of invariant manifold (MIM). MIM is based on a formulation of the
condition of invariance as an equation, and its solution by
A. Yu. Zinovyev, A. N. Gorban, T. G. Popova
Self-Organizing Approach for Automated Gene
Identification Open Sys. &
Information Dyn., 10, 2003, 321-333 GoZiPo2003final.pdf
Self-training technique for automated gene recognition both in entire genomes
and in unassembled ones is proposed. It is based on a simple measure (namely,
the vector of frequencies of non-overlapping triplets in sliding window), and
needs neither predetermined information, nor preliminary learning. The sliding
window length is the only one tuning parameter. It should be chosen close to
the average exon length typical to the DNA text under investigation. An essential
feature of the technique proposed is preliminary visualization of the set of
vectors in the subspace of the first three principal components. It was shown,
the distribution of DNA sites has the bullet-like structure with one central
cluster (corresponding to non-coding sites) and three or six ank ones
(corresponding to protein-coding sites). The bullet-like structure itself
revealed in the distribution seems to be very interesting illustration of
triplet usage in DNA sequence. The method was examined on several genomes
(mitochondrion of P.wickerhamii, bacteria C.crescentus and primitive eukaryot
S.cerevisiae). The percentage of truly predicted nucleotides exceeds 90%.
In October 2004 this paper was mentioned as one of the five most viewed
paper published in the Journal since 1997 http://www.kluweronline.com/issn/1230-1612
.
A. N. Gorban, A. Yu. Zinovyev, T. G. Popova
Seven clusters in genomic triplet distributions In Silico Biology, 3, 2003, 471-482 (0039), Online: http://arXiv.org/abs/cond-mat/0305681
29 May 2003 Seven03.pdf
Motivation: In several recent papers new algorithms were proposed for detecting
coding regions without requiring learning dataset of already known genes. In
this paper we studied cluster structure of several genomes in the space of
codon usage. This allowed to interpret some of the results obtained in other
studies and propose a simpler method, which is, nevertheless, fully functional.
Results: Several complete genomic sequences were analyzed, using visualization
of tables of triplet counts in a sliding window. The distribution of 64-dimensional
vectors of triplet frequencies displays a well-detectable cluster structure.
The structure was found to consist of seven clusters, corresponding to
protein-coding information in three possible phases in one of the two
complementary strands and in the non-coding regions. Awareness of the existence
of this structure allows development of methods for the segmentation of
sequences into regions with the same coding phase and non-coding regions. This
method may be completely unsupervised or use some external information. Since
the method does not need extraction of ORFs, it can be applied even for
unassembled genomes. Accuracy calculated on the base-pair level (both
sensitivity and specificity) exceeds 90%. This is not worse as compared to such
methods as HMM, however, has the advantage to be much simpler and clear.
Availability: The software and datasets are available at http://www.ihes.fr/~zinovyev/bullet
Gorban, A.N.;Karlin, I.V.,
Method of invariant manifold for chemical kinetics,
Chem. Eng. Sci. 58, 2003,
4751-4768 ChemEngSci2003.pdf
NEW:
Elsevier Most Cited Paper Award for this paper DIPLOMA (jpg)
In this paper, we review the construction of low-dimensional manifolds of
reduced description for equations of chemical kinetics from the standpoint of
the method of invariant manifold (MIM). The MIM is based on a formulation of
the condition of invariance as an equation, and its solution by
A. N. Gorban, A. Y. Zinovyev, D.C. Wunsch
Application of
The Method of Elastic Maps In Analysis of Genetic Texts, Proceedings
of IJCNN2003 GZW2003.pdf
Method of elastic maps allows to construct efficiently 1D, 2D and 3D non-linear
approximations to the principal manifolds with different topology (piece of
plane, sphere, torus etc.) and to project data onto it. We describe the idea of
the method and demonstrate its applications in analysis of genetic
sequences.
Gorban A. N.,
Quasi-Equilibrium
Closure Hierarchies for The Boltzmann Equation E-print, http://arXiv.org/abs/cond-mat/0305599
v1 26 May 2003 Triangl2003.pdf
Explicit method of constructing of approximations (Triangle Entropy Method) is
developed for strongly nonequilibrium problems of Boltzmann's--type kinetics,
i.e. when standard moment variables are insufficient. This method enables one
to treat any complicated nonlinear functionals that fit the physics of a
problem (such as, for example, rates of processes) as new independent
variables. The method is applied to the problem of derivation of hydrodynamics
from the Boltzmann equation. New macroscopic variables are introduced (moments
of the Boltzmann collision integral, or collision moments). They are treated as
independent variables rather than as infinite moment series. This approach
gives the complete account of rates of scattering processes. Transport
equations for scattering rates are obtained (the second hydrodynamic chain),
similar to the usual moment chain (the first hydrodynamic chain). Using the
triangle entropy method, three different types of the macroscopic description
are considered. The first type involves only moments of distribution functions,
and results coincide with those of the Grad method in the Maximum Entropy
version. The second type of description involves only collision moments.
Finally, the third type involves both the moments and the collision moments
(the mixed description). The second and the mixed hydrodynamics are sensitive
to the choice of the collision model. The second hydrodynamics is equivalent to
the first hydrodynamics only for Maxwell molecules, and the mixed hydrodynamics
exists for all types of collision models excluding Maxwell molecules. Various
examples of the closure of the first, of the second, and of the mixed
hydrodynamic chains are considered for the hard spheres model. It is shown, in
particular, that the complete account of scattering processes leads to a
renormalization of transport coefficients.
The paper gives English translation of the first part of the paper: Gorban, A.
N., Karlin, I. V., Quasi-equilibrium approximation and non-standard expansions
in the theory of the Boltzmann kinetic equation, in: "Mathematical
Modelling in Biology and Chemistry. New Approaches", ed. R. G. Khlebopros,
Nauka,
Gorban A. N.
Neuroinformatics: What are us, where are we going, how to
measure our way? The lecture was given at the USA-NIS
Neurocomputing opportunities workshop,
What is neuroinformatics? We can define it as a direction of science and
information technology, dealing with development and study of the methods for
solution of problems by means of neural networks. A field of science cannot be
determined only by fixing what it is "dealing with". The main
component, actually constituting a scientific direction, is "THE GREAT
PROBLEM", around which the efforts are concentrated. One may state even
categorically: if there is no a great problem, there is no a field of science,
but only more or less skilful imitation. What is "THE GREAT PROBLEM"
for neuroinformatics? The problem of effective parallelism, the study of brain
(solution of mysteries of thinking), etc are discussed. The neuroinformatics
was considered not only as a science, but as a services sector too. The main
ideas of generalized technology of extraction of explicit knowledge from data
are presented. The mathematical achievements generated by neuroinformatics, the
problem of provability of neurocomputations, and benefits of neural network
realization of solution of a problem are discussed.
Gorban A. N., Karlin I. V.
Geometry of irreversibility: The film of
nonequilibrium states E-print: http://arxiv.org/abs/cond-mat/0308331
A general geometrical framework of nonequilibrium thermodynamics is developed.
The notion of macroscopically definable ensembles is developed. The thesis
about macroscopically definable ensembles is suggested. This thesis should play
the same role in the nonequilibrium thermodynamics, as the Church-Turing thesis
in the theory of computability. The primitive macroscopically definable ensembles
are described. These are ensembles with macroscopically prepared initial
states. The method for computing trajectories of primitive macroscopically
definable nonequilibrium ensembles is elaborated. These trajectories are
represented as sequences of deformed equilibrium ensembles and simple quadratic
models between them. The primitive macroscopically definable ensembles form the
manifold in the space of ensembles. We call this manifold the film of
nonequilibrium states. The equation for the film and the equation for the
ensemble motion on the film are written down. The notion of the invariant film
of non-equilibrium states, and the method of its approximate construction
transform the the problem of nonequilibrium kinetics into a series of problems
of equilibrium statistical physics. The developed methods allow us to solve the
problem of macro-kinetics even when there are no autonomous equations of
macro-kinetics
Iliya V. Karlin, Larisa L. Tatarinova, Alexander N. Gorban, Hans Christian
Ottinger
Irreversibility in the short memory
approximation Physica A, 327, 2003, 399-424
Online: http://arXiv.org/abs/cond-mat/0305419
v1 18 May 2003 KTGOe2003LANL.pdf
A recently introduced systematic approach to derivations of the macroscopic
dynamics from the underlying microscopic equations of motions in the
short-memory approximation [Gorban et al, Phys. Rev. E 63 , 066124 (2001)] is
presented in detail. The essence of this method is a consistent implementation
of Ehrenfest's idea of coarse-graining, realized via a matched expansion of
both the microscopic and the macroscopic motions. Applications of this method
to a derivation of the nonlinear Vlasov-Fokker-Planck equation, diffusion
equation and hydrodynamic equations of the uid with a long-range mean field
interaction are presented in full detail. The advantage of the method is
illustrated by the computation of the post-Navier-Stokes approximation of the
hydrodynamics which is shown to be stable unlike the Burnett hydrodynamics.
Alexander N. Gorban, Iliya V. Karlin
Family
of additive entropy functions out of thermodynamic limit,
Physical Review E 67, 016104, 2003. Online: http://arXiv.org/abs/cond-mat/0205511
24 May 2002. PRE162003.pdf
We derive a one-parametric family of entropy functions that respect the
additivity condition, and which describe effects of finiteness of statistical
systems, in particular, distribution functions with long tails. This
one-parametric family is different from the Tsallis entropies, and is a convex
combination of the Boltzmann- Gibbs-Shannon entropy and the entropy function
proposed by Burg. An example of how longer tails are described within the
present approach is worked out for the canonical ensemble. We also discuss a
possible origin of a hidden statistical dependence, and give explicit recipes
on how to construct corresponding generalizations of the
master equation.
Gorban A. N., Karlin I. V.,
Reconstruction
Lemma and Fluctuation-Dissipation Theorem, Revista Mexicana De F´isica 48 Suplemento 1, Septiembre 2002, 238 –
242. Mexico_48_1_238.pdf
We discuss a new approach to nonequilibrium statistical thermodynamics
based on mappings of the microscopic dynamics into the macroscopic dynamics.
Near stationary solutions, this mapping results in a compact formula for the
macroscopic vector field without a hypothesis of a separation of time scales.
Relations of this formula to short-memory approximation, the Green-Kubo
formula, and expressions of transport coefficients in terms of Lyapunov
exponents are discussed.
Keywords: Nonequilibrium statical mechanics, coarse-graining, exact
fluctuation-dissipation relation
Gorban A. N., Karlin I. V.
Geometry
of irreversibility, in: Recent Developments in Mathematical and
Experimental Physics, Volume C: Hydrodynamics and Dynamical Systems, Ed. F.
Uribe (Kluwer,
A general geometrical setting of nonequilibrium thermodynamics is developed.
The approach is based on the notion of the natural projection which generalizes
Ehrenfests' coarse-graining. It is demonstrated how derivations of irreversible
macroscopic dynamics from the microscopic theories can be addressed through a
study of stability of quasiequilibrium manifolds.
Recovering
data gaps through neural network methods, International
Journal of Geomagnetism and Aeronomy vol. 3, no. 2, pages 191-197, December
2002 geomag02.pdf
A new method is presented to recover the lost data in geophysical time series.
It is clear that gaps in data are a substantial problem in obtaining correct
outcomes about phenomenon in time series processing. Moreover, using the data
with irregular coarse steps results in the loss of prime information during
analysis. We suggest an approach to solving these problems, that is based on
the idea of modeling the data with the help of small-dimension manifolds, and
it is implemented with the help of a neural network. We use this approach on
real data and show its proper use for analyzing time series of cosmogenic
isotopes. In addition, multifractal analysis was applied to the recovered 14C
concentration in the Earth's atmosphere.
Gorban A.N., Karlin I.V.
Methods of nonlinear kinetics, Contribution to the "Encyclopedia of
Life Support Systems" (EOLSS Publishers,
Nonlinear kinetic equations are reviewed for a wide audience of specialists and
postgraduate students in physics, mathematical physics, material science,
chemical engineering and interdisciplinary research.
Contents:
1. The Boltzmann equation
2. Phenomenology of the Boltzmann equation
3. Kinetic models
4. Methods of reduced description
4.1. The Hilbert method
4.2. The Chapman-Enskog method
4.3. The Grad moment method
4.4. Special approximations
4.5. The method of invariant manifold
4.6. Quasi-equilibrium approximations
5. Discrete velocity models
6. Direct simulation
7. Lattice Gas and Lattice Boltzmann models
8. Other kinetic equations
8.1. The Enskog equation for hard spheres
8.2. The Vlasov equation
8.3. The Smoluchowski equation
Gorban A.N., Karlin I.V.
Method
of invariant manifold for chemical kinetics Online:
http://arXiv.org/abs/cond-mat/0207231
v1 9 Jul 2002 InvManLANL2002.pdf
In this paper, we review the construction of low-dimensional manifolds of reduced
description for equations of chemical kinetics from the standpoint of the
method of invariant manifold (MIM). MIM is based on a formulation of the
condition of invariance as an equation, and its solution by
Karlin I.V., Gorban A.N.
Hydrodynamics
from Grad's equations: What can we learn from exact solutions?
Annalen der Physics, 2002. Online: http://arXiv.org/abs/cond-mat/0209560 v1 24 Sep 2002. annphys02.pdf
A detailed treatment of the classical Chapman-Enskog derivation of hydrodynamics
is given in the framework of Grad's moment equations. Grad's systems are
considered as the minimal kinetic models where the Chapman-Enskog method can be
studied exactly, thereby providing the basis to compare various approximations
in extending the hydrodynamic description beyond the Navier-Stokes
approximation. Various techniques, such as the method of partial summation,
Pad_e approximants, and invariance principle are compared both in linear and
nonlinear situations.
Karlin I.V., Grmela M., Gorban A.N.
Duality
in nonextensive statistical mechanics. Physical Review E, 2002, Volume 65,
036128. P.1-4. PRE362002.pdf
We revisit recent derivations of kinetic equations based on Tsallis’ entropy
concept. The method of kinetic functions is introduced as a standard tool for
extensions of classical kinetic equations in the framework of Tsallis’
statistical mechanics. Our analysis of the Boltzmann equation demonstrates a remarkable
relation between thermodynamics and kinetics caused by the deformation of
macroscopic observables.
Gorban A.N., Karlin I.V., Ottinger H.C.
The additive generalization of the Boltzmann
entropy, Physical Review E, 2003,
Volume 67, 067104,. Online: http://arXiv.org/abs/cond-mat/0209319 v1 13 Sep
2002 ProofMS2003.pdf
There exists only one generalization of the classical Boltzmann-Gibbs-Shannon
entropy functional to a one-parametric family of additive entropy functionals.
We find analytical solution to the corresponding extension of the classical
ensembles, and discuss in some detail the example of the deformation of the
uncorrelated state.
Gorban A.N., Karlin I.V.
Macroscopic
dynamics through coarse-graining: A solvable example,
Physical Review E, 2002, Volume 65, 026116, P.1-5. PREEhr02.pdf
The recently derived fluctuation-dissipation formula (A. N. Gorban et al.,
Phys. Rev. E 63, 066124. 2001) is illustrated by the explicit computation for
McKean’s kinetic model (H. P. McKean, J. Math. Phys. 8, 547. 1967). It is
demonstrated that the result is identical, on the one hand, to the sum of the Chapman-Enskog
expansion, and, on the other hand, to the exact solution of the invariance
equation. The equality between all three results holds up to the crossover from
the hydrodynamic to the kinetic domain.
Gorban' A., Braverman M., and Silantyev V.
Modified Kirchhoff flow with a partially penetrable
obstacle and its application to the efficiency of free flow turbines, Mathematical and Computer Modelling, Volume 35, Issue
13, June 2002, P. 1371-1375. MCM2002-2.pdf
An explicitly solvable analog of the Kirchhoff flow for the case of a
semipenetrable obstacle is considered. Its application to estimating the
efficiency of free flow turbines is discussed.
Gorban' A., Silantyev V.
Riabouchinsky flow with partially penetrable
obstacle, Mathematical and Computer Modelling, Volume
35, Issue 13, June 2002, P. 1365-1370. MCM2002-1.pdf
An explicitly solvable Riabouchinsky model with a partially penetrable obstacle
is introduced. This model applied to the estimation of the efficiency of free
flow turbines allows us to take into account the pressure drop past the lamina.
Gorban' A.N., Gorlov A.N., Silantyev
V.M.
Limits
of the Turbine Efficiency for Free Fluid Flow,
Journal of Energy Resources Technology - December 2001 - Volume 123, Issue 4,
pp. 311-317. Gorlov2001.pdf
An accurate estimate of the theoretical power limit of turbines in free fluid
flows is important because of growing interest in the development of wind power
and zero-head water power resources. The latter includes the huge kinetic
energy of ocean currents, tidal streams, and rivers without dams. Knowledge of
turbine efficiency limits helps to optimize design of hydro and wind power
farms. An explicitly solvable new mathematical model for estimating the maximum
efficiency of turbines in a free (nonducted) fluid is presented. This result
can be used for hydropower turbines where construction of dams is impossible
(in oceans) or undesirable (in rivers), as well as for wind power farms. The
model deals with a finite two-dimensional, partially penetrable plate in an
incompressible fluid. It is nearly ideal for two-dimensional propellers and
less suitable for three-dimensional cross-flow Darrieus and helical turbines.
The most interesting finding of our analysis is that the maximum efficiency of
the plane propeller is about 30 percent for free fluids. This is in a sharp
contrast to the 60 percent given by the Betz limit, commonly used now for
decades. It is shown that the Betz overestimate results from neglecting the
curvature of the fluid streams. We also show that the three-dimensional helical
turbine is more efficient than the two-dimensional propeller, at least in water
applications. Moreover, well-documented tests have shown that the helical
turbine has an efficiency of 35 percent, making it preferable for use in free
water currents.
Gorban A.N., Zinovyev A.Yu.
Visualization of
Data by Method of Elastic Maps and its Applications in Genomics, Economics and
Sociology, Institut des Hautes Etudes Scientifiques
Preprint. IHES M/01/36. Online: http://www.ihes.fr/PREPRINTS/M01/Resu/resu-M01-36.html
elmap.pdf
Technology of data visualization and data modeling is suggested. The basic of
the technology is original idea of elastic net and methods of its construction
and application. A short review of relevant methods has been made. The methods
proposed are illustrated by applying them to the real biological, economical,
sociological datasets and to some model data distributions.
Gorban A.N., Karlin I.V., Ilg P.,
Ottinger H.C.
Corrections
and enhancements of quasi-equilibrium states, J.
Non-Newtonian Fluid Mech. 2001, 96, P. 203-219. NonNew01.pdf
We give a compact non-technical presentation of two basic principles for
reducing the description of nonequilibrium systems based on the
quasi-equilibrium approximation. These two principles are: construction of
invariant manifolds for the dissipative microscopic dynamics, and coarse-graining
for the entropy-conserving microscopic dynamics. Two new results are presented:
first, an application of the invariance principle to hybridization of
micro-macro integration schemes is introduced, and is illustrated with
non-linear dumbbell models; second, Ehrenfest’s coarse-graining is extended to
general quasi-equilibrium approximations, which gives the simplest way to
derive dissipative equations from the Liouville equation in the short memory
approximation.
Gorban A.N., Karlin I.V., Ottinger H.C.,
Tatarinova L.L.
Ehrenfest’
argument extended to a formalism of nonequilibrium thermodynamics,
Physical Review E, 2001. Volume 63, 066124, P.1-6. PREEhr01.pdf
A general method of constructing dissipative equations is developed, following
Ehrenfest’sidea of coarse graining. The approach resolves the major issue of
discrete time coarse graining versus continuous time macroscopic equations.
Proof of the H theorem for macroscopic equations is given, several examples
supporting the construction are presented, and generalizations are suggested.
Gorban A.N., Zinovyev A.Yu., Popova
T.G.
Self-organizing
approach for automated gene identification in whole genomes,
Institut des Hautes Etudes Scientifiques Preprint. IHES. December 12, 2001,
Online: http://arXiv.org/abs/physics/0108016
v1 10 Aug 2001 lanlgpz01.pdf
An approach based on using the idea of distinguished coding phase in explicit
form for identi cation of protein-coding regions in whole genome has been
proposed. For several genomes an optimal window length for averaging GC-content
function and calculating codon frequencies has been found. Self-training
procedure based on clustering in multidimensional space of triplet frequencies
is proposed.
Gorban A.N., Zinovyev A.Yu., Popova T.G.
Statistical approaches to automated gene identification
without teacher. Institut des Hautes Etudes Scientifiques Preprint.
IHES M/01/34. Online: http://www.ihes.fr/PREPRINTS/M01/Resu/resu-M01-34.html
geneid.pdf
Overview of statistical methods of gene identification is made. Particular
attention is given to the methods which need not a training set of already
known genes. After analysis several statistical approaches are proposed for
computational exon identification in whole genomes. For several genomes an
optimal window length for averaging GC-content function and calculating codon
frequencies has been found. Self-training procedure based on clustering in
multidimensional codon frequencies space is proposed.
A. N. Gorban, K. O. Gorbunova, D. C.
Wunsch II
Liquid
Brain: Kinetic Model of Structureless Parallelism,
liquidbrain.pdf
A new formal model of parallel computations, the Kirdin kinetic machine, is
suggested. It is expected that this model will play the role for parallel
computations similar to Markov normal algorithms, Kolmogorov and Turing machine
or Post schemes for sequential computations. The basic ways in which
computations are realized are described; correctness of the elementary programs
for the Kirdin kinetic machine is investigated. It is proved that the
determined Kirdin kinetic machine is an effective calculator. A simple
application of the Kirdin kinetic machine, heap encoding, is suggested.
Subprograms similar to usual programming enlarge the Kirdin kinetic machine.
Gorban A.N., Karlin I.V., Zmievskii
V.B., Dymova S.V.
Reduced
description in the reaction kinetics, Physica A, 2000, 275,
P.361-379. GKZD2000.pdf
Models of complex reactions in thermodynamically isolated systems often
demonstrate evolution towards low-dimensional manifolds in the phase space. For
this class of models, we suggest a direct method to construct such manifolds, and
thereby to reduce the effective dimension of the problem. The approach realizes
the invariance principle of the reduced description, it is based on iterations
rather than on a small parameter expansion, it leads to tractable linear
problems, and is consistent with thermodynamic requirements. The approach is
tested with a model of catalytic reaction.
Gorban A.N., Popova T.G., Sadovsky M.G.
Classification
Of Symbol Sequences Over Thier Frequency Dictionaries: Towards The Connection
Between Structure And Natural Taxonomy, Open Sys.
& Information Dyn. 7: 1-17, 2000. opsygps00.pdf
The classifications of bacterial 16S RNA sequences developed over the real and
transformed frequency dictionaries have been studied. Two sequences considered
to be close each other, when their frequency dictionaries were close in
Euclidean metrics. A procedure to transform a dictionary is proposed that makes
clear some features of the information pattern of a symbol sequence. A
comparative study of classifications developed over the real frequency
dictionaries vs. the transformed ones has been carried out. A correlation
between an information pattern of nucleotide sequences and taxonomy of the
bearer of the sequence was found. The sites with high information value are
found, that were the main factors of the difference between the classes in a
classification. The classification of nucleotide sequences developed over the
real frequency dictionaries of the thickness 3 reveals the best correlation to
a gender of bacteria. A set of sequences of the same gender is included
entirely into one class, as a rule, and the exclusions occur rarely. A
hierarchical classification yields one or two taxonomy groups on each level of
the classification. An unexpectedly often (in comparison to the expected), or
unexpectedly rare occurrence of some sites within a sequence makes a basic
difference between the structure patterns of the classes yielded; a number of
those sites is not too great. Further investigations are necessary in order to
compare the sites revealed with those determined due to other methodology.
A. N.
Gorban, I.V. Karlin, and V.B. Zmievskii
Two-Step Approximation of
Space-Independent Relaxation, TRANSPORT THEORY AND STATISTICAL
PHYSICS, 28(3) (1999), 271-296. GorKarZmiTTSP99.pdf
In this
paper we introduce a new method of constructing approximate trajectories for
space independent kinetic equations confirming to the second law of
thermodynamics. Classical examples are the space independent Boltzmann equation
and chemical kinetics equations for closed
homogeneous systems. This family of kinetic equations is characterized by the
following general properties:
(1).
There exists a set of functions which remain constant on a solution (these are
density, momentum and energy in context of the Boltzmann equation).
(ii).
There exists a convex function which monotonically decreases along any solution
from its value in the initial state to an absolute minima in the final
equilibrium state (this is the H-theorem for the Boltzmann equation) .
Usually
we do know only the initial and the final (equilibrium) states, and the kinetic
equation neither can be solved exactly, nor contains small parameters to
develop a reliable perturbation theory. Still, we would like to get (perhaps a
rather rough but a simple) approximation of the relaxation trajectory.
An express method to
approximate trajectories of space independent kinetic equations is developed.
It involves a two-step treatment of relaxation through a quasiequilibria
located on a line emerging from the initial state in the direction prescribed
by the kinetic equation. A test for the Boltzmann equation shows the
validity of the method.
A.N. Gorban, A.A. Rossiev, D. C. Wunsch II
Neural Network
Modeling of Data with Gaps: Method of Principal Curves, Carleman's Formula, and
Other, The talk was given at the USA-NIS Neurocomputing
opportunities workshop,
Online: http://arXiv.org/abs/cond-mat/0305508
21 May 2003 gaps.pdf
A method of modeling data with gaps by a sequence of curves has been developed.
The new method is a generalization of iterative construction of singular
expansion of matrices with gaps. Under discussion are three versions of the
method featuring clear physical interpretation:
1) linear: modeling the data by a sequence of linear manifolds of small
dimension;
2) quasilinear: constructing "principal curves": (or "principal
surfaces"), univalently projected on the linear principal components;
3) essentially non-linear, based on constructing "principal curves":
(principal strings and beams) employing the variation principle; the iteration
implementation of this method is close to Kohonen self-organizing maps.
The derived dependencies are extrapolated by Carleman’ formulas. The method is
interpreted as a construction of neural network conveyor designed to solve the
following problems:
1) to fill gaps in data;
2) to repair data, to correct initial data values in such a way as to make the
constructed models work best;
3) to construct a calculator to fill gaps in the data line fed to the input.
Gorban A. N.
Neuroinformatics:
What are us, where are we going, how to measure our way?
The lecture was given at the USA-NIS Neurocomputing opportunities workshop,
What is neuroinformatics? For me here and now neuroinformatics is a direction
of science and information technology, dealing with development and study of
the methods for solution of problems by means of neural networks. A base
example of artificial neural network, which will be referred to below, is a
feed-forward network from standard neurons.
Alexander N. Gorban, Eugeniy M. Mirkes and Victor
G. Tsaregorodtsev
Generation of
Explicit Knowledge from Empirical Data through Pruning of Trainable Neural
Networks, International Joint Conference on Neural
Networks, Washington, DC July 10-16, 1999. know.pdf
E-print: http://arxiv.org/abs/cond-mat/0307083
This paper presents a generalized technology of extraction of explicit
knowledge from data. The main ideas are:
1) maximal reduction of network complexity (not only removal of neurons or
synapses, but removal all the unnecessary elements and signals and reduction of
the complexity of elements),
2) using of adjustable and flexible pruning process (the pruning sequence
shouldn't be predetermined - the user should have a possibility to prune
network on his own way in order to achieve a desired network structure for the
purpose of extraction of rules of desired type and form),
3) extraction of rules not in predetermined but any desired form.
Some considerations and notes about network architecture and training process
and applicability of currently developed pruning techniques and rule extraction
algorithms are discussed. This technology, being developed by us for more than
10 years, allowed us to create dozens of knowledge-based expert systems.
A.
N. Gorban, I. V. Karlin
Schrodinger operator in an overfull set,
Europhys. Lett., 42 (2) (1998), 113-117. GK98Shro.pdf
Operational simplicity of an expansion of a wave function over a basis in the
Hilbert space is an undisputable advantage for many non-relativistic
quantum-mechanical computations. However, in certain cases, there are several
\natural" bases at one's disposal, and it is not easy to decide which is
preferable. Hence, it sounds attractive to use several bases simultaneously,
and to represent states as expansions over an overfull set obtained by a
junction of their elements. Unfortunately, as is well known, such a
representation is not unique, and lacks many convenient properties of full sets
(e.g., explicit formulae to compute coeffcients of expansions). Because of this
objection, overfull sets are used less frequently than they, perhaps, deserve.
We introduce a variational principle which eliminates this ambiguity, and
results in an expansion which provides “the best" representation to a
given Schrodinger operator.
N. N. Bugaenko, A. N. Gorban and M. G.
Sadovsky,
Maximum Entropy Method in Analysis of Genetic Text
and Measurement of its Information Content, Open Systems
& Information Dynamics, 1998, Volume 5, Number 3, Pages 265-278.
The information capacity in frequency dictionaries of nucleotide sequences is
estimated through the efficiency of reconstruction of a longer frequency
dictionary from a short one. This reconstruction is performed by the maximum
entropy method. Real nucleotide sequences are compared to random ones (with the
same nucleotide composition). Phages genes from NCBl bank were analyzed. THe
significant difference of real genetic text from random sequences is observed
for the dictionary length q=2,5 and
6.
Karlin
I.V., Gorban A.N., Dukek G., Nonnenmacher T. F.
Dynamic
correction to moment approximations. Physical Review E, February 1998 Volume 57, Number
2, P.1668-1672. KGDN98.pdf
Considering the Grad moment ansatz as a suitable first approximation to a closed
finite-moment dynamics, the correction is derived from the Boltzmann equation.
The correction consists of two parts, local and nonlocal. Locally corrected
thirteen-moment equations are demonstrated to contain exact transport
coefficients. Equations resulting from the nonlocal correction give a
microscopic justification to some phenomenological theories of extended
hydrodynamics.
Gorban
A. N.
Approximation
of Continuos Functions of Several Variables by an Arbitrary Nonlinear
Continuous Function of One Variable, Linear Functions, and Their
Superpositions, Appl. Math. Lett., Vol. 11, No. 3, pp 45-49,
1998 approx98.pdf
Linear spaces
of continuous functions of real variables closed under the superposition
operation are considered. It has been proved that when such a space contains
constants, linear functions, and at least one nonlinear function, it is dense
in the space of all continuous functions in the topology of uniform convergence
on compact sets. So, the approximation of continuous functions of several
variables by an arbitrary nonlinear continuous function of one variable, linear
functions, and their superpositions is possible.
Karlin I.V., Gorban A.N., Succi S., Boffi V.
Maximum
Entropy Principle for Lattice Kinetic Equations.
Physical Review Letters Volume 81, Number 1, 6 July 1998, P.6-9. p6_11998.pdf
The entropy maximum approach to constructing equilibria in lattice kinetic
equations is revisited. For a suitable entropy function, we derive explicitly
the hydrodynamic local equilibrium, prove the H theorem for lattice
Bhatnagar-Gross-Krook models, and develop a systematic method to account for
additional constraints.
Gorban A.N., Shokin Yu.I., Verbitskii
V.I.
Simultaneously
dissipative operators and the infinitesimal wrapping effect in interval spaces,
Computational Technologies, 2 (4) (1997), 16-48. Online: http://arXiv.org/abs/physics/9702021
, 1997. GorbanShokVerVychTechnol.pdf
We study simultaneously dissipative linear operators. The family of linear
operators is simultaneously dissipative, if there exists a norm relative to
which all the operators are dissipative. We construct various sufficient
conditions for existence of such a norm. We consider two examples of
applications for this theory: stability of chemical kinetics and phenomenon of
interval expansion.
In solving a system of ordinary differential equations by an interval method
the approximate solution at any considered moment of time t represents a set
(called interval) containing the exact solution at the moment t. The intervals
determining the solution of a system are often expanded in the course of time
irrespective of the method and step used.
The phenomenon of interval expansion, called the wrapping or
M.Yu. Senashova, A.N. Gorban, D. C.
Wunsch II
Back-propagation
of accuracy, The talk given on ICNN97 (The 1997 IEEE
International Conference on Neural Networks, Houston, USA), Online: http://arXiv.org/abs/cond-mat/0305527
gorsenwu.pdf
In this paper we solve the problem: how to determine maximal allowable errors,
possible for signals and parameters of each element of a network proceeding
from the condition that the vector of output signals of the network should be
calculated with given accuracy? "Back-propagation of accuracy" is
developed to solve this problem.
A. N: Gorban, Ye. M. Mirkes, D.C. Wunsch
II
High order
ortogonal tensor networks: information capacity and reliability.
The talk given on ICNN97 (The 1997 IEEE International Conference on
Neural Networks, Houston, USA), gomirwu1.pdf
Neural networks based on construction of ortogonal projectors in the tensor
power of space of signals are described. A sharp estimate of their ultimate
information capacity is obtained. The numbers of stored prototype patterns
(prototypes) can many times exceed the number of neurons. A comparison with the
error control codes is made.
Gorban A.N., Karlin I.V.
Short-Wave
Limit of Hydrodynamics: A Soluble Example. Physical
Review Letters, Volume 77, Number 2, 8 July 1996. P. 282-285. p282_11996.pdf
The Chapman-Enskog series for shear stress is summed up in a closed form for a
simple model of Grad moment equations. The resulting linear hydrodynamics is
demonstrated to be stable for all wavelengths, and the exact asymptotic of the
acoustic spectrum in the short-wave domain is obtained.
Gorban A.N., Karlin I.V. Nonnenmacher
T. F., Zmievskii V.B.
Relaxation
Trajectories: Global approximation. Physica A, 1996, 231, P.648-672. GKZNPhA96.pdf
Gorban A. N., Karlin I. V.
Scattering
rates versus moments: Alternative Grad equations,
Physical Review E October 1996 Volume 54, Number 4, P. 3109-3112. pR3109_11996.pdf
Scattering rates (moments of collision integral) are treated as independent
variables, and as an alternative to moments of the distribution function, to
describe the rarefied gas near local equilibrium. A version of the entropy
maximum principle is used to derive the Grad-like description in terms of a
finite number of scattering rates. The equations are compared to the Grad
moment system in the heat nonconductive case. Estimations for hard spheres
demonstrate, in particular, some 10% excess of the viscosity coefficient
resulting from the scattering rate description, as compared to the Grad moment
estimation.
Gorban A. N., Karlin I. V.
On “Solid
Liquid” limit of Hydrodynamic Equations, Transport Theory and
Statistical Physics 24 (9) (1995), 1419-1421. GKSolJet95s.pdf
An “infinitely viscid threshold” for compressible liquid is described. A rapid
compression of a flux amounts to a strong deceleration of particles (particles
loose velocity comparable to heat velocity on a distance compatible to the main
free path). Such a strong deceleration is described in the frames of
hydrodynamic equations by a divergency of viscosity. A fluid becomes “solid”.
A.N. Gorban, C. Waxman,
Neural Networks for Political Forecast. Proceedings of the 1995 World Congress On Neural Networks, A Volume in the INNS Series of Texts, Monographs, and Proceedings, Vol. 1, 1995. (A preliminary 1992 publication of Cory Waxman, the student of A.Gorban, is available in electronic form – see below)
Cory Waxman,
The History of US Presidential Elections from Siberian NC Point of View, In: Neuroinformatics and Neurocomputers, 7-10 Oct 1992, Rostov-on-Don, Russia, Proc. RNNS/IEEE Symposium, vol.2, pp. 1000 – 1010, IEEE press, 1992. Cory.pdf http://ieeexplore.ieee.org/stamp/stamp.jsp?arnumber=00268530
Tests were performed
with the program "US Presidential Elections" and the future
relationship between neurocomputers and
the human sciences was discussed.
This paper will discuss the type of neurocomputer being developed in Krasnoyarsk (by S. E. Gilev, A. N. Gorban, E. M. Mirkes), describe the results of some experiments, and conclude with a discussion on possible future applications of neurocomputers in the human sciences.
Perhaps the most revolutionary aspect of neurocomputers is that they can be applied to problems of which we have very little understanding. This is quite different than the standard use of computers in science. Often scientists apply computers to algorithmic problems (in which the problem can be solved by a predefined series of steps). For such problems traditional computers are of tremendous value, and can work thousands of times faster than humans. But there is another area of science where the exact nature or form of the problem is rarely well understood – the human sciences. In History, Political Science, Psychology, and Education sciences there are many possible applications of NC’s. We have already discussed some direct applications in history and political science. We also saw how new questions might be formed in the course of these applications. T his ability to find new questions should not be overlooked as it has been said that sometimes the question is much more important than the answer.
Dorrer, M.G., Gorban, A.N., Zenkin, V.I.
Neural networks in psychology: classical explicit diagnoses, In: Neuroinformatics and Neurocomputers, 1995, Second International Symposium, 20-23 Sep 1995, Rostov on Don, Russia, pp. 281-284, DOI: 10.1109/ISNINC.1995.480869
The purpose of this work is to employ trainable neural networks to start solving the problem facing the designers and users of computer psychological tests: cultural, national and social adaptation of tests. Mathematical construction of up-to-date objective diagnostic tests is based on a comparison of the revealed condition with the norm standard. It is understandable that the norms worked out for one socio-cultural group are not necessarily the same for the other. By way of example it is possible to cite the difficulties to be reckoned with in adapting foreign techniques. Neural networks have been successfully used for classical explicit diagnoses. A typical experiment is described
Alexander N. Gorban, Iliya V. Karlin
Method of invariant manifolds and regularization of acoustic spectra, Transport Theory and Statistical
Physics 23 (5) (1994), 559-632. GorbanKarlinTTSP94.pdf
A new approach to the problem of reduced description for Boltzmann-type systems
is developed. It involves a direct solution of two main problems:
thermodynamicity and dynamic invariance of reduced description. A universal
construction is introduced, which gives a thermodynamic parameterization of an
almost arbitrary approximation. Newton-type procedures of successive
approximations are developed which correct dynamic noninvariance. The method is
applied to obtain corrections to the local Maxwell manifold using parametrics
expansion instead of
Alexander N. Gorban', Iliya V. Karlin
General
approach to constructing models of the Boltzmann equation,
Physica A, 1994, 206, P.401-420. GKPhA94.pdf
The problem of thermodynamic parameterization of an arbitrary approximation of
reduced description is solved. On the base of this solution a new class of
model kinetic equations is constructed that gives a model extension of the
chosen approximation to a kinetic model. Model equations describe two
processes: rapid relaxation to the chosen approximation along the planes of
rapid motions, and the slow motion caused by the chosen approximation. The
H-theorem is proved for these models. It is shown, that the rapid process
always leads to entropy growth, and also a neighborhood of the approximation is
determined inside which the slow process satisfies the H-theorem. Kinetic
models for Grad moment approximations and for the Tamm-Mott-Smith approximation
are constructed explicitly. In particular, the problem of concordance of the
ES-model with the H-theorem is solved.
A.N. Gorban, I. V. Karlin,
Nonarbitrary regularization of acoustic spectra, Transport Theory and Statistical Physics, 22(1), 121-124.
We
suggest a method of constructing dynamic invariant manifolds for the Boltzmann
equation. It aims to improve the Chapman-Enskog expansion (CE) free of ad hoc
assumptions. The problems of the CE method are well known, for example, a
short-wave instability of the Burnett approximation. Many attempts were
made to improve the CE expansion. In particular, in our previous work we used
the idea of partial summing. However, all these attempts have an ad hoc
character. The famous KAM theory serves us as a prototype. In KAM, the rapidly
converging Newton method is used instead of diverging Taylor expansion, and one
searches for an invariant manifold rather than for a solution. Following ides
of KAM, we use the Newton method. Each iteration is concordant with the H-theorem.
Our method consists of two main parts:
1. Construction of a special thermodynamic parameterization for an arbitrary
manifold which gives dynamic
equations
on this manifold (this part has no analogue in KAM and it is caused by the
necessity to satisfy the H-theorem at every step).
2. Correction of the dynamic
noninvariance of a manifold by the Newton method.
We describe the method for a general dynamic system with a global convex
H-function.
Alexander N. Gorban' , Iliya V. Karlin
Thermodynamic
parameterization, Physica A, 1992, 190, P.393-404 GKPhA92.pdf
A new method of successive construction of a solution is developed for problems
of strongly nonequilibrium Boltzmann kinetics beyond normal solutions. Firstly,
the method provides dynamic equations for any manifold of distributions where
one looks for an approximate solution. Secondly, it gives a successive
procedure of obtaining corrections to these approximations. The method requires
neither small parameters, nor strong restrictions upon the initial
approximation; it involves solutions of linear problems. It is concordant with
the H-theorem at every step. In particular, for the Tamm-Mott-Smith approximation,
dynamic equations are obtained, an expansion for the strong shock is
introduced, and a linear equation for the first correction is found.
Alexander
N. Gorban', Iliya V. Karlin
Structure
and approximations of the
Chapman-Enskog expansion for the linearized Grad Equations, Transport Theory and Statistical Physics,
21(1&2), 101-117 (1992).
A detailed structure of the Chapman-Enskog expansion for the linearized
Grad moment equations is determined. A method of partial summing of the
Chapman-Enskog series is introduced, and is used to remove short-wave
instability of the Burnett approximations.
Gilev, S.Y., Gorban, A.N., Mirkes, Y.M.,
Internal conflicts in neural networks, In: Neuroinformatics and Neurocomputers, 1992., RNNS/IEEE Symposium, Vol. 1, pp. 219-225. DOI: 10.1109/RNNS.1992.268591
Hierarchical neural networks consisting of small expert-networks are considered. Algorithms of fast parallel learning are proposed. The approach proposed greatly enlarges the information capacity of the network and accelerates learning.
V. I. Verbitskii and A. N. Gorban'
Jointly
dissipative operators and their applications, Siberian Mathematical
Journal, Volume 33, Number 1 (1992), 19-23, DOI: 10.1007/BF00972932
The jointly dissipative operators were introduced by Verbitskii and Gorban'
(1989). Let E be an n-dimensional real or complex linear
space, and let L(E) be the space of linear operators in E. Let us introduce a norm ||…|| on E and the corresponding norm in L(E). An operator A from L(E) is said to be dissipative if ||exp(tA)||≤1 for all t≥0. It is stable dissipative (in the paper due to the interpreter mistake a
term “roughly dissipative” is used)
if there is ε > 0 such that ||exp(tA)||≤exp(-εt) for all t≥0. For the existence of a norm with respect to which the
operator A is be roughly dissipative it is necessary and sufficient that the
system (i) be asymptotically stable, i.e., that the matrix of A be stable
(i.e., that the spectrum of A lie in the open left halfplane). A family of
operators is said to be jointly dissipative (resp. jointly
roughly dissipative) if there exists a norm with respect to which all operators
from this family are dissipative (resp., roughly dissipative). The jointly dissipative operators find
application in the analysis of dynamical properties of nonlinear systems of
ordinary differential equations and in some applications (chemical kinetics, numerical
analysis). In the present paper we discuss the properties of jointly
dissipative operators and some of their applications. For example, the
following theorems are proved: (Theorem 1) Suppose the family {A} is compact, generates a solvable
Lie algebra, and all matrices in {A}
are stable. Then {A} is jointly
roughly dissipative. (Theorem 2) Suppose the family {A} is finite, generates a nilpotent Lie algebra, and for each
operator from {A} there exists a
norm with respect to which it is dissipative. Then the family {A} is jointly dissipative.
1991
N. N. Bugaenko,
A. N. Gorban', and I. V. Karlin
Universal
expansion of three-particle distribution function, Theoretical
and Mathematical Physics, Vol. 88,
No. 3, 1991. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol.
88, No. 3, pp. 430-441, September, 1991.TMF1990.pdf
A universal, i.e., not dependent on the Hamiltonian of the two-particle
interaction, expansion of the equilibrium three-particle distribution function
with respect to the two-particle correlation functions is constructed. A
diagram technique that permits systematic calculation of the coefficients of
this expansion is proposed. In particular, it is established that allowance for
the first four orders in the absence of long-range correlations gives the
G.S.Yablonskii, V.I.Bykov, A.N. Gorban, and
V.I.Elokhin
Kinetic Models of Catalytic Reactions (Comprehensive
Chemical Kinetics, V.32, ed. by R.G. Compton), Elsevier, Amsterdam,
1991, 396p.
Synopsis
This book has been written by a group of mathematicians and chemists whose
common interest is in the complex dynamics of catalytic reactions. Based on
developments in mathematical chemistry, a general theory is described that
allows the investigation of the relationships between the kinetic
characteristics of complex reactions and their detailed reaction mechanism. Furthermore,
a comprehensive analysis is made of some typical mechanism of catalytic
reactions, in particular for the oxidation of carbon monoxide on platinum
metals. In fact, the book presents "three kinetics": (a) detailed,
oriented to the elucidation of a detailed reaction mechanism according to its
kinetic laws; (b) applied, with the aim of obtaining kinetic relationships for
the further design of chemical reactors; and (c) mathematical kinetics whose
purpose is the analysis of mathematical models for heterogeneous catalytic
reactions taking place under steady- or unsteady-state conditions.
Contents
1. Minimum minimorum. 2. The development of basic concepts of chemical kinetics
in heterogeneous catalysis. 3. Formalism of chemical kinetics. 4. Graphs in chemical
kinetics. 5. Simplest non-linear mechanisms of catalytic reactions producing
critical phenomena. 6. Studies of kinetic models for oxidation reactions over
metals (exemplified by CO oxidation). 7. Critical retardation effects and slow
relaxations. 8. Conclusions. Index.
(Review on this book: Journal of American Chemical Society
(JAChS), V.114, n 13, 1992; sections “Reviews on the book”, W. Henry
Weinberg, review on the book "Comprehensive Chemical Kinetics",
Volume 32, Kinetic Models of Catalytic Reactions, Elsevier, 1991).
A.
N. Gorban', E. M. Mirkes, A. N. Bocharov, and V. I. Bykov,
Thermodynamic consistency of kinetic data, Combustion,
Explosion, and Shock Waves, Volume 25, Number 5 / September, 1989, 593-600, DOI: 10.1007/BF00772975 Consistency1989.pdf
It
is well known that the rate constants of different elementary reactions are
often interdependent. Relationships determined by the principle of detailed
balance exist between them when microreversibility is valid and by the
generalizations of that principle when it is not (for example, in magnetic
fields, during macroscopic rotations, etc.). Nevertheless, in practice the
verification of consistency in the kinetic constants for complicated
transformation schemes involves a certain amount of technical difficulty. The
problem of consistency in the kinetic constants arises especially sharply in
connection with the creation of kinetic data banks intended for widespread use.
Here it is impossible to avoid solving that problem or examining each
multistage reaction separately, without leaving the user with the burden of
finding a way to carry out this analysis. Thus, the methods for establishing
the consistency of these constants, along with the conditions under which this
consistency may fail, must be analyzed and suitable algorithms and programs
have to be developed. We proposed such methods, developed algorithms,
implemented and tested them.
Gorban A.N., Bykov V.I.
A model of
autooscillations in association reactions, Chemical
Engineering Science. 1987, Vol. 42, No. 5. P. 1249-1251. BG1987.pdf
The aim of this paper is to show that association reactions can result in the
appearance of autooscillations in nonlinear systems.
Gorban A.N., Bykov V.I., Yablonskii G.S.
Thermodynamic
function analogue for reactions proceeding without interaction of various
substances, Chemical Engineering Science, 1986. Vol. 41,
No. 11. P. 2739-2745. BGYa1986.pdf
Function similar to Lyapunov’s function has been constructed for reactions with
$a_i A_i \to b_j A_j$ stages. This provides for the quasi-thermodynamics of the
appropriate kinetic model, which implies steady-state uniqueness and global
stability in the reaction polyhedron. The kinetic law generalizing the
Marcelin-de Donder kinetics has been written for a separate stage. Explicit
Lyapunov thermodynamic functions have been written for various conditions of
the reaction proceeding in closed systems. The matrix of linear approximation
close to equilibrium is expressed by means of the introduced scalar product.
Particularly, the absence of damped oscillations as equilibrium is approached
as shown.
V. I. Bykov, A. N. Gorban and G. S. Yablonskii,
Description
of nonisothermal reactions in terms of Marcelin-De-Donder kinetics and its
generalizations, React. Kinet. Catal. Lett., Vol. 20, Nos. 3-4 (1982).
A general form for the description of nonisothermal reactions in closed
chemical systems in terms of the Marcelin-de-Donder kinetics and explicit forms
of the Lyapunov functions for the systems treated under various conditions are
suggested.
V. I. Bykov, A. N. Gorban', and T. P.
Pushkareva
Singularities
in the relaxation periods in the oxidation of CO on Platinum, Teoreticheskaya i
Eksperimental'naya Khimiya, Vol. 18, No, 4, pp 431-439, July-August, 1982.
Original article submitted July 13, 1981. SloRelCO1982.pdf
(Translated from Russian by Plenum, in the file some of the Plenum translation
mistakes are corrected).
When studying the process dynamics of chemical reactions the first problem is
generally considered to be its limiting (for t → ∞) conditions. But besides a reply to the question "what will happen at t →
∞ ?" it is also important to know how rapidly the limiting behavior
is established. The slow establishment of chemical equilibrium, associated with
delays in the reaction far from equilibrium (the induction periods) has been
studied in chemistry since the time of van't Hoff. At present, interest in slow
relaxations arises from experiments in which it was found that for certain
chemical (including heterogeneous catalytic) reactions the reactant
concentrations may slowly approach their limiting (steady-state) values,
although the observed rate of reaction may remain fairly high. Where are the
reasons of such a situation in "intrinsic" relaxation processes which
are determined directly by the reaction mechanism, or in "extrinsic"
relaxation processes arising from reasons of a non-kinetic nature (the
diffusion of the substances within the catalyst, a slow variation in its
structure, etc.). Slow relaxations of a purely kinetic (intrinsic) nature are
possible. This possibility has been demonstrated for the oxidation of CO on Pt.
The surface of the singularities in the relaxation time has been constructed
for this specific catalytic oxidation reaction.
Gorban A.N., Bykov V.I.
Macroscopic
clusters induced by diffusion in a catalytic oxidation reactions,
ChemicaI Engineering Science, 1980. Vol. 35, P. 2351-2352 BG1980.pdf
V. I. Elokhin, G. S. Yablonskii, A. N. Gorban and V. M. Cheresiz,
Dynamics of chemical reactions and nonphysical steady states, React. Kinet. Catal. Lett., Vol. 15, No. 2 (1980), 245-250 RKCL_80_EYaGCh.pdf
Data on the position of nonphysical (lying beyond the region of determination) steady states are shown to be of use for understanding the dynamic behavior of chemical reactions, in particular, the reasons for slow relaxations. As a rule, the kinetic equations are nonlinear and should have several steady-state solutions, but not all of them are physically meaningful (negative and complex steady-state solutions are possible). But as has been shown, slow transient regimes can also be observed when the physically meaningless steady-state solutions are positioned near the reaction polyhedron.
Gorban A.N.
Singularities of Transition Processes In
Dynamical Systems. http://arXiv.org/abs/chao-dyn/9703010
v1 18 Mar 1997, Translation of Candidate (Ph.D) Thesis, 1980 PhDslowrelax.pdf
The paper gives the systematic analysis of singularities of transition
processes in general dynamical systems. Dynamical systems depending on
parameter are studied. A system of relaxation times is constructed. Each
relaxation time depends on three variables: initial conditions, parameters k of
the system and accuracy \epsilon of relaxation. This system of times contains:
the time before the first entering of the motion into \epsilon -neighbourhood
of the limit set, the time of final entering in this neighbourhood and the time
of stay of the motion outside the \epsilon -neighbourhood of the limit set. The
singularities of relaxation times as functions of (x_0; k) under fixed \epsilon
are studied. A classification of different bifurcations (explosions) of limit
sets is performed. The bifurcations fall into those with appearance of new
limit points and bifurcations with appearance of new limit sets at finite distance
from the existing ones. The relations between the singularities of relaxation
times and bifurcations of limit sets are studied. The peculiarities of dynamics
which entail singularities of transition processes without bifurcations are
described as well. The peculiarities of transition processes under
perturbations are studied. It is shown that the perturbations simplify the
situation: the interrelations between the singularities of relaxation times and
other peculiarities of dynamics for general dynamical system under small
perturbations are the same as for smooth two-dimensional structural stable
systems.
Gorban
A.N.
Invariant sets for kinetic equations, React. Kinet. Catal. Lett., Vol. 10, No. 2 (1979), 187-190. RKCL1978.pdf
Some sets in the space of compositions possessing an invariance property are
considered for a closed system, where a complex chemical reaction of a known
mechanism proceeds. If the vector of concentrations belongs to such a set at a
certain moment of time, it will remain within it at any succeeding moment. Some
possible applications are discussed. The most important circumstance of the
above analysis is the fact that these positively invariant sets are strongly
dependent on the detailed reaction mechanism. This may be used for
discrimination of various mechanisms under consideration.
A. N. Gorban' and V. B. Melamed
Certain
properties of Fredholm analytic sets in Banach spaces, Sibirskii Matematicheskii
Zhurnal, Vol. 17, No. 3, pp. 682-685, May-June, 1976. Original article
submitted December 9, 1974. SMZh1976.pdf
With the aid of the Lyapunov-Schmidt method of transition to a finite-dimensional
equation, we prove in this paper certain assertions about analytic sets in
complex Banach spaces. The principal result is a counterpart of the
finite-dimensional Remmert~Stein theorem, stating that an analytic set in an
open set U is either discrete , or it contains points that are as close as
desired to the boundary of U. As an application we shall prove the
nonnegativeness of the rotation of the vector field x~Ax with an analytic and
completely continuous operator A; we also consider the finiteness of the number
of solutions of an equation that depends on a parameter.