A p-adic model of DNA sequence and genetic code

Using basic properties of p-adic numbers, we consider a simple new approach to describe main aspects of DNA sequence and the genetic code. In our investigation central role plays an ultrametric p-adic information space whose basic elements are nucleotides, codons and genes. We show that a 5-adicmodel is appropriate for DNA sequence. This 5-adicmodel, combined with 2-adic distance, is also suitable for the genetic code and for amore advanced employment in genomics. We find that genetic code degeneracy is related to the p-adic distance between codons. The text was submitted by the authors in English. This paper is a slight modification of an article available in the electronic archive form arXiv:qbio. GN/0607018v1 (July 2006). Since that time some other papers on this subject have appeared, e.g. [1], [2].     

Zero-pole interpolation for meromophic matrix functions on an algebraic curve and transfer functions of 2D systems

We formulate and solve the problem of constructing a meromorphic bundle map over a compact Riemann surface X having a prescribed zero-pole structure (including directional information). The output bundle together with the zero-pole data is prespecified while the input bundle and the bundle map are to be determined. The Riemann surface X is assumed to be (birationally) embedded as an irreducible algebraic curve in ² and both input and output bundles are assumed to be equal to the kernels of determinantal representations for X. In this setting the solution can be found as the joint transfer function of a Livsic-Kravitsky two-operator commutative vessel (2D input-output dynamical system). Also developed is the basic theory of two-operator commutative vessels and the correct analogue of the transfer function for such a system (a meromorphic bundle map between input and output bundles defined over an algebraic curve associated with the vessel) together with a state space realization, a Mittag-Leffler type interpolation theorem and the state space similarity theorem for such bundle mappings. A more abstract version of the zero-pole interpolation problem is also presented.     

On the existence of generalized gibbs measures for the one-dimensional <Emphasis Type="Italic">p</Emphasis>-adic countable state Potts model

We consider the one-dimensional countable state p-adic Potts model. A construction of generalized p-adic Gibbs measures depending on weights λ is given, and an investigation of such measures is reduced to the examination of a p-adic dynamical system. This dynamical system has a form of series of rational functions. Studying such a dynamical system, under some condition concerning weights, we prove the existence of generalized p-adic Gibbs measures. Note that the condition found does not depend on the values of the prime p, and therefore an analogous fact is not true when the number of states is finite. It is also shown that under the condition there may occur a phase transition.     

On chaotic behaviour of the p-adic generalized Ising mapping and its application

In the present paper, by conducting research on the dynamics of the p-adic generalized Ising mapping corresponding to renormalization group associated with the p-adic Ising-Vannemenus model on a Cayley tree, we have determined the existence of the fixed points of a given function. Simultaneously, the attractors of the dynamical system have been found. We have come to a conclusion that the considered mapping is topologically conjugate to the symbolic shift which implies its chaoticity and as an application, we have established the existence of periodic p-adic Gibbs measures for the p-adic Ising-Vannemenus model.     

THE BINARY IMAGE OF THE DUAL OF QUATERNARY GOETHALS CODE

IIntroductlonA．Galois ringLet GR(4”)be the Gajois ring of characteristic 4 with 4”elements．In GR(4”)thereexists a nonzero element ’of order 2”一1．Let T＝{0,1,,…,矿－‘};then any elementc e GR(4”)can be writt。unlqllely sc＝a十 Zb,a。 E丁,whi山 is called 2－adic representation ofc．The elemem c can also be written unl叩ely sc＝ac ＋ al卜…＋am矿‘,a;E凤,(0＜6＜。一 1),whi血is called the additive representation of c．It is。11协。n that GR(”)/切。见。,where (2 is the pr…     

Mumford dendrograms and discrete p-adic symmetries

In this article, we present an effective encoding of dendrograms by embedding them into the Bruhat-Tits trees associated to p-adic number fields. As an application, we show how strings over a finite alphabet can be encoded in cyclotomic extensions of ℚ _p and discuss p-adic DNA encoding. The application leads to fast p-adic agglomerative hierarchic algorithms similar to the ones recently used e.g. by A. Khrennikov and others. From the viewpoint of p-adic geometry, to encode a dendrogram X in a p-adic field K means to fix a set S of K-rational punctures on the p-adic projective line ℙ₁. To ℙ₁ \ S is associated in a natural way a subtree inside the Bruhat-Tits tree which recovers X, a method first used by F. Kato in 1999 in the classification of discrete subgroups of PGL₂(K). Next, we show how the p-adic moduli space of ℙ¹ with n punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on ℙ¹. In this way, we can associate to certain classes of dynamical systems a Mumford curve, i.e. a p-adic algebraic curve with totally degenerate reduction modulo p. Finally, we indicate some of our results in the study of general discrete actions on ℙ¹, and their relation to p-adic Hurwitz spaces. The text was submitted by the author in English.     

On minimal decomposition of p-adic polynomial dynamical systems

A polynomial of degree ?2 with coefficients in the ring of p-adic numbers Zp is studied as a dynamical system on Zp. It is proved that the dynamical behavior of such a system is totally described by its minimal subsystems. For an arbitrary quadratic polynomial on Z₂, we exhibit all its minimal subsystems.     

K-groups associated with substitution minimal systems

Two ordered Bratteli diagrams can be constructed from an aperiodic substitution minimal dynamical system. One, the proper diagram, has a single maximal path and a single minimal path and the Vershik map on the path space can be extended homeomorphically to a map conjugate to the substitution system. The other, the improper diagram, encodes the substitution more naturally but often has many maximal and minimal paths and no continuous compact dynamics. This paper connects the two diagrams by considering theirK ₀-groups, obtaining the equation

Human memory as ap-adic dynamic system

We propose a mathematical model of the human memory-retrieval process based on dynamic systems over a metric space ofp-adic numbers. The elements of this space represent ideas. We assume that two ideas are close if they have a sufficiently long initial segment in common. We also assume that this dynamic system is located in the subconscious and is controlled by the conscious, which specifies the system parameters and provides the ideas that initiate the iteration of the dynamic system. We show that even simplep-adic dynamic systems describe essential features of the human memory-retrieval process. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 117, No. 3, pp. 385–396, December, 1998     

Global periodicity and complete integrability of discrete dynamical systems

Consider the discrete dynamical system generated by a map F. It is said that it is globally periodic if there exists a natural number p such that F ^p (x)=x for all x in the phase space. On the other hand, it is called completely integrable if it has as many functionally independent first integrals as the dimension of the phase space. In this paper, we relate both concepts. We also give a large list of globally periodic dynamical systems together with a complete set of their first integrals, emphasizing the ones coming from difference equations.     

Quadratic Newton Iteration for Systems with Multiplicity

Newton's iterator is one of the most popular components of polynomial equation system solvers, either from the numeric or symbolic point of view. This iterator usually handles smooth situations only (when the Jacobian matrix associated to the system is invertible). This is often a restrictive factor. Generalizing Newton's iterator is still an open problem: How to design an efficient iterator with a quadratic convergence even in degenerate cases? We propose an answer for an m -adic topology when the ideal m can be chosen generic enough: compared to a smooth case we prove quadratic convergence with a small overhead that grows with the square of the multiplicity of the root.     

On a class of rational p-adic dynamical systems

In this paper we investigate the behavior of trajectories of one class of rational p-adic dynamical systems in complex p-adic field Cp. We studied Siegel disks and attractors of such dynamical systems. We found the basin of the attractor of the system. It is proved that such dynamical systems are not ergodic on a unit sphere with respect to the Haar measure.     

On the chaotic behavior of a generalized logistic p-adic dynamical system

In the paper we describe basin of attraction p-adic dynamical system G(x)=(ax)²(x+1). Moreover, we also describe the Siegel discs of the system, since the structure of the orbits of the system is related to the geometry of the p-adic Siegel discs.     

Cognitive processes of the brain: An ultrametric model of information dynamics in unconsciousness

We discuss differences in mathematical representations of the physical and mental worlds. Following Aristotle, we present the mental space as discrete, hierarchic, and totally disconnected topological space. One of the basic models of such spaces is given by ultrametric spaces and more specially by m-adic trees. We use dynamical systems in such spaces to model flows of unconscious information at different level of mental representation hierarchy, for “mental points”, categories, and ideas. Our model can be interpreted as an unconventional computational model: non-algorithmic hierarchic “computations” (identified with the process of thinking at the unconscious level).     

Rigged Hilbert Space of Free Coherent States and p-Adic Numbers

We investigate the rigged Hilbert space of free coherent states. We prove that this rigged Hilbert space is isomorphic to the space of generalized functions over a p-adic disk. We discuss the relation of the described isomorphism of rigged Hilbert spaces and noncommutative geometry and show that the considered example realizes the isomorphism between the noncommutative line and the p-adic disk.     

Behaviour of Hensel perturbations of p-adic monomial dynamical systems

In the framework of non-Archimedean (p-adic) analysis we study cyclic behaviour of polynomial discrete dynamical systems (iterations of polynomial maps). One of the main tools of our investigation is Hensel's lemma (a p-adic analogue of Newton's method). Our considerations will lead to formulas for the number cycles of a specific length and for the total number of cycles. We will also study the distribution of cycles in the different p-adic fields.     

A methodology for the constructive approximation of nonlinear operators defined on noncompact sets

We consider the constructive approximation of a non-linear operator that is known on a bounded but not necessarily compact set. Our main result can be regarded as an extension of the classical Stone-Weierstrass Theorem and also shows that the approximation is stable to small

disturbances.

This problem arises in the modelling of real dynamical systems where an input-output mapping is known only on some bounded subset of the input space. In such cases it is desirable to construct a model of the real system with a complete input-output map that preserves, in some approximate sense, the known mapping. The model is normally constructed from an algebra of elementary continuous functions.

We will assume that the input space is a separable Hilbert space. To solve the problem we introduce a special weak topology and show that uniform continuity of the given operator in the weak topology provides an alternative compactness condition that is sufficient to justify the desired approximation.     

Rational map ax + 1/x on the projective line over Q_2 Dedicated to the Memory of Professor Lei Tan

The dynamical structure of the rational map ax+1/x on the projective line P~1(Q_2) over the field Q_2 of 2-adic numbers, is fully described.