S. V. Kozyrev

On p-Adic Mathematical Physics

A brief review of some selected topics in p-adic mathematical physics is presented.

Pseudodifferential operators on ultrametric spaces and ultrametric wavelets

A family of orthonormal bases, the ultrametric wavelet bases, is introduced in quadratically integrable complex valued functions spaces for a wide family of ultrametric spaces. A general family of pseudodifferential operators, acting on complex valued functions on these ultrametric spaces is introduced. We show that these operators are diagonal in the introduced ultrametric wavelet bases, and compute the corresponding eigenvalues. We introduce the ultrametric change of variable, which maps the ultrametric spaces under consideration onto positive half-line, and use this map to construct non-homogeneous generalizations of wavelet bases.

Genetic code on the diadic plane

We introduce the simple parametrization for the space of codons (triples of nucleotides) by 8×8 table. This table (which we call the diadic 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.

Replica symmetry breaking related to a general ultrametric space I: replica matrices and functionals

Family of replica matrices, related to general ultrametric spaces, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed.

ULTRAMETRIC RANDOM FIELD

Gaussian random field on general ultrametric space is introduced as a solution of pseudodifferential stochastic equation. Covariation of the introduced random field is computed with the help of wavelet analysis on ultrametric spaces. Notion of ultrametric Markovianity, which describes independence of contributions to random field from different ultrametric balls is introduced. We show that the random field under investigation satisfies this property.

Replica symmetry breaking related to a general ultrametric space—II: RSB solutions and the n→0 limit

Replica symmetry breaking solutions for the new replica anzats, related to general ultrametric spaces, are investigated. A variant of analysis on trees is developed and applied to the computation of the nto0 limit in the new replica anzats.

Replica symmetry breaking related to a general ultrametric space III: The case of general measure

Family of replica matrices, related to general ultrametric spaces with general measures, is introduced. These matrices generalize the known Parisi matrices. Some functionals of replica approach are computed. Replica symmetry breaking solution is found.

2-Adic clustering of the PAM matrix

In this paper we demonstrate that the use of the system of 2-adic numbers provides a new insight to some problems of genetics, in particular, degeneracy of the genetic code and the structure of the PAM matrix in bioinformatics. The 2-adic distance is an ultrametric and applications of ultrametric 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((infinity)), 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((infinity)) is sparse. We discuss the structure of the matrix A((infinity)) in relation to the side chain properties of the corresponding amino acids. We introduce the family of substitution matrices A(alpha,beta)=alpha A((2))+beta A((infinity)), alpha,beta>or=0 which should allow to vary the alignment procedure in order to take into account the different chemical and geometric properties of the amino acids.

Dynamics on rugged landscapes of energy and ultrametric diffusion

We discuss the interbasin kinetics approximation for random walk on a complex (rugged) landscape of energy. In this approximation the random walk is described by the system of kinetic equations corresponding to transitions between the local minima of energy. If we approximate the transition rates between the local minima by the Arrhenius formula then the system of kinetic equations will be hierarchical. We discuss for a generic landscape of energy the anzats of interbasin kinetics which is equivalent to the ultrametric diffusion generated by an ultrametric pseudodifferential operator.

APPLICATION OF p-ADIC ANALYSIS TO TIME SERIES

Time series defined by a p-adic pseudo-differential equation is investigated using the expansion of the time series over p-adic wavelets. Quadratic correlation function is computed. This correlation function shows a degree-like behavior and is locally constant for some time periods. It is natural to apply this kind of models for the investigation of avalanche processes and punctuated equilibrium as well as fractal-like analysis of time series generated by measurement of pressure in oil wells.