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 genetic code. Central role in our investigation plays an ultrametric p-adic information space which basic elements are nucleotides, codons and genes. We show that a 5-adic model is appropriate for DNA sequence. This 5-adic model, combined with 2-adic distance, is also suitable for genetic code and for a more advanced employment in genomics. We find that genetic code degeneracy is related to the p-adic distance between codons.Comment: 13 pages, 2 table

p-Adic Mathematical Physics

A brief review of some selected topics in p-adic mathematical physics is presented.Comment: 36 page

Alien introgressions and chromosomal rearrangements do not affect the activity of gliadin-coding genes in hybrid lines of Triticum aestivum L. × Aegilops columnaris Zhuk

Using chromosome C-banding and electrophoresis of grain storage proteins, gliadins, 17 Triticum aestivumAegilops columnaris lines with substitutions of chromosomes of homoeologous groups 1 and 6 were examined. Based on their high polymorphism, gliadins were used to identify alien genetic material. For all of the lines examined, electrophoretic analysis of gliadin spectra confirmed substitution of wheat chromosomes 6A, 6D or 1D for the homoeologous Aegilops chromosomes of genomes Uс or Xс. The substitution manifested in the disappearance of the products of gliadin-coding genes on chromosomes 6A, 6D or 1D with the simultaneous appearance of the products of genes localized on alien chromosomes of genomes Uс or Xс. Thus, Aegilops chromosomes were shown to be functionally active in the alien wheat genome. The absence of alien genes expression in the lines carrying a long arm deletion in chromosome 6Xc suggested that the gliadin-coding locus moved from the short chromosome arm (its characteristic position in all known wheat species) to the long one. This is probably associated with a large species-specific pericentric inversion. In spite of losing a part of its long arm and combination with a non-homologous chromosome of a different genome (4BL), chromosome 1D was fully functioning. For Aegilops, the block type of gliadin components inheritance was shown, indicating similarity in the structural organization of gliadin-coding loci in these genera. Based on determining genetic control of various polypeptides in the electrophoretic aegilops spectrum, markers to identify Ae. columnaris chromosomes 1Xс, 6Xс and 6Uс were constructed

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 $\mathbb{Q}_p$ 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 $\mathbb{P}^1$. To $\mathbb{P}^1\setminus S$ 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 $\textrm{PGL}_2(K)$. Next, we show how the $p$-adic moduli space $\mathfrak{M}_{0,n}$ of $\mathbb{P}^1$ with $n$ punctures can be applied to the study of time series of dendrograms and those symmetries arising from hyperbolic actions on $\mathbb{P}^1$. 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 $\mathbb{P}^1$, and their relation to $p$-adic Hurwitz spaces.Comment: 14 pages, 6 figure

Functional Classical Mechanics and Rational Numbers

The notion of microscopic state of the system at a given moment of time as a point in the phase space as well as a notion of trajectory is widely used in classical mechanics. However, it does not have an immediate physical meaning, since arbitrary real numbers are unobservable. This notion leads to the known paradoxes, such as the irreversibility problem. A "functional" formulation of classical mechanics is suggested. The physical meaning is attached in this formulation not to an individual trajectory but only to a "beam" of trajectories, or the distribution function on phase space. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values and there are corrections to the Newton trajectories. We give a construction of probability density function starting from the directly observable quantities, i.e., the results of measurements, which are rational numbers.Comment: 8 page

Development of the genetic classification of Aegilops columnaris Zhuk. chromosomes based on the analysis of introgression lines Triticum aestivum×Ae. columnaris

Aegilops columnaris Zhuk. is a potential source of new genes for wheat improvement. However, this species has not yet been used in practical breeding. In the present work we have for the first time reported the development and molecular-cytogenetic characterization of T. aestivum×Ae. columnaris introgression lines. Analysis has not revealed alien genetic material in five of the 20 lines we have studied, while the remaining lines carried from 1 to 3 pairs of Aegilops chromosomes as addition(s) or substitution(s) to wheat chromosomes. Altogether, five different chromosomes of Aegilops columnaris have been detected in the karyotypes of 15 lines by C-banding and fluorescent in-situ hybridization (FISH). Based on substitution spectra, these chromosomes were identified as 3Ае1, 3Ае2, 5Ае2, 6Ае1 and 6Ае2. In addition, another Aegilops chromosome has been found in the line 2305/1 as a monosomic addition; due to the lack of group-specific markers we were unable to assign this chromosome to a particular genome or a genetic group and therefore it was designated Ае-а. In several lines acrocentric and telocentric chromosomes have been revealed (Ae-b and Ae-c). It is most likely that these chromosomes were derived from unknown Aegilops chromosomes due to a large deletion. A comparison of electrophoretic spectra of gliadins in introgression lines L-2310/1 and L-2304/1 with substitutions of chromosome 6D with two different chromosomes of Ae. columnaris (these lines were assigned to the 6th homoeologous group based on C-banding data) has shown that they carry different alleles of the gliadin loci. This observation confirmed that lines L-2310/1 and L-2304/1 contained non-identical 6Ae chromosomes. Taking into consideration our previous results of FISH analyses, three other Ae. columnaris chromosomes can be assigned to homoeologous groups 1, 5 and 7 of the U-genome based on the location of 5S and 45S rDNA loci (1U and 5U) or pSc119.2 probe distribution (7U). Thus, based on our current data as well as on the results of earlier work, we can identify eight out of the 14 chromosomes of Aegilops columnaris

Randomness in Classical Mechanics and Quantum Mechanics

The Copenhagen interpretation of quantum mechanics assumes the existence of the classical deterministic Newtonian world. We argue that in fact the Newton determinism in classical world does not hold and in classical mechanics there is fundamental and irreducible randomness. The classical Newtonian trajectory does not have a direct physical meaning since arbitrary real numbers are not observable. There are classical uncertainty relations, i.e. the uncertainty (errors of observation) in the determination of coordinate and momentum is always positive (non zero). A "functional" formulation of classical mechanics was suggested. The fundamental equation of the microscopic dynamics in the functional approach is not the Newton equation but the Liouville equation for the distribution function of the single particle. Solutions of the Liouville equation have the property of delocalization which accounts for irreversibility. The Newton equation in this approach appears as an approximate equation describing the dynamics of the average values of the position and momenta for not too long time intervals. Corrections to the Newton trajectories are computed. An interpretation of quantum mechanics is attempted in which both classical and quantum mechanics contain fundamental randomness. Instead of an ensemble of events one introduces an ensemble of observers.Comment: 12 pages, Late

Linearization in ultrametric dynamics in fields of characteristic zero - equal characteristic case

Let $K$ be a complete ultrametric field of charactersitic zero whose corresponding residue field $\Bbbk$ is also of charactersitic zero. We give lower and upper bounds for the size of linearization disks for power series over $K$ near an indifferent fixed point. These estimates are maximal in the sense that there exist exemples where these estimates give the exact size of the corresponding linearization disc. Similar estimates in the remaning cases, i.e. the cases in which $K$ is either a $p$-adic field or a field of prime characteristic, were obtained in various papers on the $p$-adic case (Ben-Menahem:1988,Thiran/EtAL:1989,Pettigrew/Roberts/Vivaldi:2001,Khrennikov:2001) later generalized in (Lindahl:2009 arXiv:0910.3312), and in (Lindahl:2004 http://iopscience.iop.org/0951-7715/17/3/001/,Lindahl:2010Contemp. Math) concerning the prime characteristic case

On hyperbolic fixed points in ultrametric dynamics

Let K be a complete ultrametric field. We give lower and upper bounds for the size of linearization discs for power series over K near hyperbolic fixed points. These estimates are maximal in the sense that there exist examples where these estimates give the exact size of the corresponding linearization disc. In particular, at repelling fixed points, the linearization disc is equal to the maximal disc on which the power series is injective.Comment: http://www.springerlink.com/content/?k=doi%3a%2810.1134%2fS2070046610030052%2