arXiv subjects

A. N. Gorban

Publications and source records attributed to A. N. Gorban.

87 records · Page 5Linked to original sources

arXiv2005

Invariance correction to Grad's equations: Where to go beyond approximations?

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.

cond-mat.stat-mech↗

arXiv2004

Seven clusters in genomic triplet distributions

In several recent papers new gene-detection algorithms were proposed for detecting protein-coding regions without requiring learning dataset of already known genes. The fact that unsupervised gene-detection is possible closely connected to existence of a cluster structure in oligomer frequency distributions. In this paper we study cluster structure of several genomes in the space of their triplet frequencies, using pure data exploration strategy. Several complete genomic sequences were analyzed, using visualization of tables of triplet frequencies 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 with high accuracy (higher than 90% on the nucleotide level). Visualizing and understanding the structure allows to analyze effectively performance of different gene-prediction tools. Since the method does not require extraction of ORFs, it can be applied even for unassembled genomes. The information content of the triplet distributions and the validity of the mean-field models are analysed.

cond-mat.dis-nn↗

arXiv2004

MultiNeuron - Neural Networks Simulator For Medical, Physiological, and Psychological Applications

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.

q-bio.QM↗

arXiv2004

Back-propagation of accuracy

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. The calculation of allowable errors for each element of network by back-propagation of accuracy is surprisingly similar to a back-propagation of error, because it is the backward signals motion, but at the same time it is very different because the new rules of signals transformation in the passing back through the elements are different. The method allows us to formulate the requirements to the accuracy of calculations and to the realization of technical devices, if the requirements to the accuracy of output signals of the network are known.

cond-mat.dis-nn↗

arXiv2004

Four basic symmetry types in the universal 7-cluster structure of 143 complete bacterial genomic sequences

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.

q-bio.GN↗

arXiv2004

Singularities of Transition Processes in Dynamical Systems: Qualitative Theory of Critical Delays

The paper gives a systematic analysis of singularities of transition processes in dynamical systems. General dynamical systems with dependence 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 $ε$ of the relaxation. The singularities of relaxation times as functions of $(x_0,k)$ under fixed $ε$ are studied. The classification of different bifurcations (explosions) of limit sets is performed. 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 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.

chao-dyn↗

arXiv2004

Methods of nonlinear kinetics

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: The Boltzmann equation, Phenomenology and Quasi-chemical representation of the Boltzmann equation, Kinetic models, Discrete velocity models, Direct simulation, Lattice Gas and Lattice Boltzmann models, Minimal Boltzmann models for flows at low Knudsen number, Other kinetic equations: The Enskog equation for hard spheres, The Vlasov equation, The Fokker-Planck equation, Equations of chemical kinetics and their reduction.

cond-mat.stat-mech↗

arXiv2004

Constructive Methods of Invariant Manifolds for Kinetic Problems

We present the Constructive Methods of Invariant Manifolds for model reduction in physical and chemical kinetics, developed during last two decades. The problem of reduced description is studied 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 for construction of slow invariant manifolds is presented, in particular the analogue of KAM methods for dissipative systems. We systematically consider a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, invariant grids. The following examples of applications are presented: Nonperturbative deviation of physically consistent hydrodynamics from the Boltzmann equation and from the reversible dynamics, for Knudsen numbers ~ 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; invariant grids for chemical reactions; universal continuous media description of dilute polymeric solution, etc.

cond-mat↗

arXiv2004

Uniqueness of thermodynamic projector and kinetic basis of molecular individualism

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 the arbitrary 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. Moreover, from the requirement of persistence of the {\it sign} of dissipation follows that the {\it value} of dissipation (the entropy production) persists too. The explicit construction of this {\it thermodynamic projector} is described. Second, we use this projector for developing the short memory approximation and coarse-graining for general nonlinear dynamic systems. We prove that in this approximation the entropy production grows. ({\it The theorem about entropy overproduction.}) In example we apply the thermodynamic projector to derivation the equations of reduced kinetics for the Fokker-Planck equation. The new class of closures is developed: The kinetic multipeak polyhedrons. Distributions of this type are expected to appear in each kinetic model with multidimensional instability as universally, as Gaussian distribution appears for stable systems. The number of possible relatively stable states of 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.

cond-mat↗

arXiv2003

Legendre Integrators, Post-Processing and Quasiequilibrium

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 postprocessing, and thermodynamic projector. Thermodynamic projector is the tool to transform almost arbitrary anzatz to a thermodynamically consistent model, the postprocessing is the cheapest way to improve the solution, obtained by the Legendre integrators. Legendre Integrators give the opportunity to solve linear equations instead of nonlinear ones for quasiequilibrium (MaxEnt) approximations. The essentially new element of this toolbox, the method of thermodynamic projector, is demonstrated on application to FENE-P model of polymer kinetic theory. The multy-peak model of polymer dynamics is developed. The simplest example, discussed in details, is the two peaks model for Gaussian manifold instability in polymer dynamics. This type of models opens a way to create the computational models for the "molecular individualism".

cond-mat↗

arXiv2003

Geometry of irreversibility: The film of nonequilibrium states

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.

cond-mat.stat-mech↗

arXiv2003

Neuroinformatics: What are us, where are we going, how to measure our way?

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.

cond-mat.dis-nn↗

arXiv2003

Generation of Explicit Knowledge from Empirical Data through Pruning of Trainable Neural Networks

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), and 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. In this paper we present a generalized three-step technology of extraction of explicit knowledge from empirical data.

cond-mat↗

arXiv2003

Invariant grids for reaction kinetics

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 Newton iterations. A grid-based version of MIM is developed. Generalizations to open systems are suggested. The set of methods covered makes it possible to effectively reduce description in chemical kinetics. The most essential new element of this paper is the systematic consideration of a discrete analogue of the slow (stable) positively invariant manifolds for dissipative systems, {\it invariant grids}. We describe the Newton method and the relaxation method for the invariant grids construction. The problem of the grid correction is fully decomposed into the problems of the grid's nodes correction. The edges between the nodes appears only in the calculation of the tangent spaces. This fact determines high computational efficiency of the invariant grids method.

cond-mat.mtrl-sci↗

arXiv2003

Neural network modeling of data with gaps: method of principal curves, Carleman's formula, and other

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: linear - modeling the data by a sequence of linear manifolds of small dimension; quasilinear - constructing "principal curves: (or "principal surfaces"), univalently projected on the linear principal components; 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's formulas. The method is interpreted as a construction of neural network conveyor designed to solve the following problems: to fill gaps in data; to repair data - to correct initial data values in such a way as to make the constructed models work best; to construct a calculator to fill gaps in the data line fed to the input.

cond-mat.dis-nn↗