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S. V. Kozyrev

Publications and source records attributed to S. V. Kozyrev.

At least 37 records · Page 2Linked to original sources
arXiv2012

Lattice model of protein conformations

We introduce a lattice model of protein conformations which is able to reproduce second structures of proteins (alpha--helices and beta--sheets). This model is based on the following two main ideas. First, we model backbone parts of amino acid residues in a peptide chain by edges in the cubic lattice which are not parallel to the coordinate axes. Second, we describe possible contacts of amino acid residues using a discrete model of the Ramachandran plot. This model allows to describe hydrogen bonds between the residues in the backbone of the peptide chain. In particular the lattice secondary structures have the correct structure of hydrogen bonds. We also take into account the side chains of amino acid residues and their interaction. The expression for the energy of conformation of a lattice protein which contains contributions from hydrogen bonds in the backbone of the peptide chain and from interaction of the side chains is proposed. The lattice secondary structures are local minima of the introduced energy.

cond-mat.soft↗
arXiv2012

Clustering by hypergraphs and dimensionality of cluster systems

In the present paper we discuss the clustering procedure in the case where instead of a single metric we have a family of metrics. In this case we can obtain a partially ordered graph of clusters which is not necessarily a tree. We discuss a structure of a hypergraph above this graph. We propose two definitions of dimension for hyperedges of this hypergraph and show that for the multidimensional p-adic case both dimensions are reduced to the number of p-adic parameters. We discuss the application of the hypergraph clustering procedure to the construction of phylogenetic graphs in biology. In this case the dimension of a hyperedge will describe the number of sources of genetic diversity.

cs.DS↗
arXiv2012

Classification by Ensembles of Neural Networks

We introduce a new procedure for training of artificial neural networks by using the approximation of an objective function by arithmetic mean of an ensemble of selected randomly generated neural networks, and apply this procedure to the classification (or pattern recognition) problem. This approach differs from the standard one based on the optimization theory. In particular, any neural network from the mentioned ensemble may not be an approximation of the objective function.

cs.NE↗
arXiv2011

Replica procedure for probabilistic algorithms as a model of gene duplication

In the present paper we propose to describe gene networks in biological systems using probabilistic algorithms. We describe gene duplication in the process of biological evolution using introduction of the replica procedure for probabilistic algorithms. We construct the examples of such a replica procedure for hidden Markov models. We introduce the family of hidden Markov models where the set of hidden states is a finite additive group with a p-adic metric and build the replica procedure for this family of markovian models.

q-bio.MN↗
arXiv2011

Pseudodifferential p-adic vector fields and pseudodifferentiation of a composite p-adic function

We discuss transformation of p-adic pseudodifferential operators (in the one-dimensional and multidimensional cases) with respect to p-adic maps which correspond to automorphisms of the tree of balls in the corresponding p-adic spaces. In the dimension one we find a rule of transformation for pseudodifferential operators. In particular we find the formula of pseudodifferentiation of a composite function with respect to the Vladimirov p-adic fractional differentiation operator. We describe the frame of wavelets for the group of parabolic automorphisms of the tree of balls in the p-adic field. In many dimensions we introduce the group of mod p-affine transformations, the family of pseudodifferential operators corresponding to pseudodifferentiation along vector fields on the tree of balls in p-adic miltidimensional space and obtain a rule of transformation of the introduced pseudodifferential operators with respect to mod p-affine transformations.

math.MG↗
arXiv2011

Multidimensional p-adic wavelets for the deformed metric

The approach to p-adic wavelet theory from the point of view of representation theory is discussed. p-Adic wavelet frames can be constructed as orbits of some p-adic groups of transformations. These groups are automorphisms of the tree of balls in the p-adic space. In the present paper we consider deformations of the standard p-adic metric in many dimensions and construct some corresponding groups of transformations. We build several examples of p-adic wavelet bases. We show that the constructed wavelets are eigenvectors of some pseudodifferential operators.

math.FA↗
arXiv2010

The Arrhenius formula in kinetic theory and Witten's spectral asymptotics

A new approach to the proof of the Arrhenius formula of kinetic theory is proposed. We prove this formula starting from the equation of diffusion in a potential. We put this diffusion equation in the form of evolutionary equation generated by some Schroedinger operator. We show that the Arrhenius formula for the rate of over the barrier transitions follows from the formula for the rate of quantum tunnel transitions for the considered Schroedinger operator. Relation of the proposed approach and the Witten method of the proof of the Morse inequalities is discussed. In our approach the Witten spectral asymptotics takes the form of the low temperature limit and the Arrhenius formula is a correction to the Witten asymptotics.

math-ph↗
arXiv2009

p-Adic numbers in bioinformatics: from genetic code to PAM-matrix

In this paper we denonstrate that the use of the system of 2-adic numbers provides a new insight to some problems of genetics, in particular, generacy of the genetic code and the structure of the PAM matrix in bioinformatics. The 2-adic distance is an ultrametric and applications of ultrametrics in bioinformatics are not surprising. However, by using the 2-adic numbers we match ultrametric with a number theoretic structure. In this way we find new applications of an ultrametric which differ from known up to now in bioinformatics. We obtain the following results. We show that the PAM matrix A allows the expansion into the sum of the two matrices A=A^{(2)}+A^{(\infty)}, where the matrix A^{(2)} is 2-adically regular (i.e. matrix elements of this matrix are close to locally constant with respect to the discussed earlier by the authors 2-adic parametrization of the genetic code), and the matrix A^{(\infty)} is sparse. We discuss the structure of the matrix A^{(\infty)} in relation to the side chain properties of the corresponding amino acids.

q-bio.GN↗
arXiv2009

Multidimensional basis of p-adic wavelets and representation theory

A multidimensional basis of p-adic wavelets is constructed. The relation of the constructed basis to a system of coherent states (i.e. orbit of action) for some $p$-adic group of linear transformations is discussed. We show that the set of products of the vectors from the constructed basis and p-roots of one is the orbit of the corresponding p-adic group of linear transformations.

math-ph↗
arXiv2008

Frames of p-adic wavelets and orbits of the affine group

The general construction of frames of p-adic wavelets is described. We consider the orbit of a mean zero generic locally constant function with compact support (mean zero test function) with respect to the action of the p-adic affine group and show that this orbit is a uniform tight frame. We discuss relation of this result to the multiresolution wavelet analysis.

math-ph↗
arXiv2008

Towards ultrametric theory of turbulence

Relation of ultrametric analysis, wavelet theory and cascade models of turbulence is discussed. We construct the explicit solutions for the nonlinear ultrametric integral equation with quadratic nonlinearity. These solutions are built by means of the recurrent hierarchical procedure which is analogous to the procedure used for the cascade models of turbulence.

math-ph↗
arXiv2007

Landscape dynamics, interbasin kinetics and ultrametric diffusion

We discuss the interbasin kinetics approximation for random walk on a complex landscape. We show that for a generic landscape the corresponding model of interbasin kinetics is equivalent to an ultrametric diffusion, generated by an ultrametric pseudodifferential operator on the ultrametric space related to the tree of basins. The simplest example of ultrametric diffusion of this kind is described by the p-adic heat equation.

cond-mat.dis-nn↗
arXiv2007

Genetic code on the dyadic plane

We introduce the simple parametrization for the space of codons (triples of nucleotides) by 8\times 8 table. This table (which we call the dyadic plane) possesses the natural 2-adic ultrametric. We show that after this parametrization the genetic code will be a locally constant map of the simple form. The local constancy of this map will describe degeneracy of the genetic code. The map of the genetic code defines 2-adic ultrametric on the space of amino acids. We show that hydrophobic amino acids will be clustered in two balls with respect to this ultrametric. Therefore the introduced parametrization of space of codons exhibits the hidden regularity of the genetic code.

q-bio.QM↗
arXiv2007

Multidimensional ultrametric pseudodifferential equations

We develop an analysis of wavelets and pseudodifferential operators on multidimensional ultrametric spaces which are defined as products of locally compact ultrametric spaces. We introduce bases of wavelets, spaces of generalized functions and Lizorkin generalized functions on multidimensional ultrametric spaces. We also consider some family of pseudodifferential operators on multidimensional ultrametric spaces. The notions of Cauchy problem for ultrametric pseudodifferential equations and of ultrametric characteristics are introduced. A theorem about existence and uniqueness of the solution for the Cauchy problem (the analogue of the Kovalevskaya theorem) is proven.

math-ph↗
arXiv2007

Coincidence of the continuous and discrete p-adic wavelet transforms

We show that translations and dilations of a p-adic wavelet coincides (up to the multiplication by some root of one) with a vector from the known basis of discrete p-adic wavelets. In this sense the continuous p-adic wavelet transform coincides with the discrete p-adic wavelet transform. The p-adic multiresolution approximation is introduced and relation with the real multiresolution approximation is described. Relation of application of p-adic wavelets to spectral theory of p-adic pseudodifferential operators and the known results about sparsity of matrices of some real integral operators in the bases of multiresolution wavelets is discussed.

math-ph↗