Preprints & selected publications:
2009 2008 2007 2006 2005 2004 2003 2002 2001 2000 1999 1998 1997 1996 1995 1994 1992 1991 1980-1990
A.N. Gorban, 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.
2008
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.
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.
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.
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 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.
Ovidiu Radulescu, Alexander N. Gorban,
Sergei Vakulenko,
Hierarchies and modules in complex
biological systems, In: Proceedings of European Conference on Complex Systems (paper
ECCS06-114), Oxford, UK, September 2006. http://complexsystems.lri.fr/FinalReview/FILES/PDF/p114.pdf
or 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: Statistical and Theoretical Physics, 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.;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) 149-167GoGoKar2004.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.
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
Gorban A.N., Karlin I.V.
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.
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
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.
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.
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.
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 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.