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

Some aspects of the mm-adic analysis and its applications to mm-adic stochastic processes

In this paper we consider a generalization of analysis on $p$-adic numbers field to the $m$ case of $m$-adic numbers ring. The basic statements, theorems and formulas of $p$-adic analysis can be used for the case of $m$-adic analysis without changing. We discuss basic properties of $m$-adic numbers and consider some properties of $m$-adic integration and $m$-adic Fourier analysis. The class of infinitely divisible $m$-adic distributions and the class of $m$-adic stochastic Levi processes were introduced. The special class of $m$-adic CTRW process and fractional-time $m$-adic random walk as the diffusive limit of it is considered. We found the asymptotic behavior of the probability measure of initial distribution support for fractional-time $m$-adic random walk.Comment: 18 page

Prebiotic Homochirality as a Critical Phenomenon

The development of prebiotic homochirality on early-Earth or another planetary platform may be viewed as a critical phenomenon. It is shown, in the context of spatio-temporal polymerization reaction networks, that environmental effects -- be them temperature surges or other external disruptions -- may destroy any net chirality previously produced. In order to understand the emergence of prebiotic homochirality it is important to model the coupling of polymerization reaction networks to different planetary environments.Comment: 6 Pages, 1 Figure, In Press: Origins of Life and Evolution of Biosphere

Dissociation in a polymerization model of homochirality

A fully self-contained model of homochirality is presented that contains the effects of both polymerization and dissociation. The dissociation fragments are assumed to replenish the substrate from which new monomers can grow and undergo new polymerization. The mean length of isotactic polymers is found to grow slowly with the normalized total number of corresponding building blocks. Alternatively, if one assumes that the dissociation fragments themselves can polymerize further, then this corresponds to a strong source of short polymers, and an unrealistically short average length of only 3. By contrast, without dissociation, isotactic polymers becomes infinitely long.Comment: 16 pages, 6 figures, submitted to Orig. Life Evol. Biosp

Phase transitions for PP-adic Potts model on the Cayley tree of order three

In the present paper, we study a phase transition problem for the $q$-state $p$-adic Potts model over the Cayley tree of order three. We consider a more general notion of $p$-adic Gibbs measure which depends on parameter \rho\in\bq_p. Such a measure is called {\it generalized $p$-adic quasi Gibbs measure}. When $\rho$ equals to $p$-adic exponent, then it coincides with the $p$-adic Gibbs measure. When $\rho=p$, then it coincides with $p$-adic quasi Gibbs measure. Therefore, we investigate two regimes with respect to the value of $|\rho|_p$. Namely, in the first regime, one takes $\rho=\exp_p(J)$ for some J\in\bq_p, in the second one $|\rho|_p<1$. In each regime, we first find conditions for the existence of generalized $p$-adic quasi Gibbs measures. Furthermore, in the first regime, we establish the existence of the phase transition under some conditions. In the second regime, when $|\r|_p,|q|_p\leq p^{-2}$ we prove the existence of a quasi phase transition. It turns out that if $|\r|_p<|q-1|_p^2<1$ and \sqrt{-3}\in\bq_p, then one finds the existence of the strong phase transition.Comment: 27 page

Homochiral growth through enantiomeric cross-inhibition

The stability and conservation properties of a recently proposed polymerization model are studied. The achiral (racemic) solution is linearly unstable once the relevant control parameter (here the fidelity of the catalyst) exceeds a critical value. The growth rate is calculated for different fidelity parameters and cross-inhibition rates. A chirality parameter is defined and shown to be conserved by the nonlinear terms of the model. Finally, a truncated version of the model is used to derive a set of two ordinary differential equations and it is argued that these equations are more realistic than those used in earlier models of that form.Comment: 20 pages, 6 figures, Orig. Life Evol. Biosph. (accepted

Toward homochiral protocells in noncatalytic peptide systems

The activation-polymerization-epimerization-depolymerization (APED) model of Plasson et al. has recently been proposed as a mechanism for the evolution of homochirality on prebiotic Earth. The dynamics of the APED model in two-dimensional spatially-extended systems is investigated for various realistic reaction parameters. It is found that the APED system allows for the formation of isolated homochiral proto-domains surrounded by a racemate. A diffusive slowdown of the APED network such as induced through tidal motion or evaporating pools and lagoons leads to the stabilization of homochiral bounded structures as expected in the first self-assembled protocells.Comment: 10 pages, 5 figure

p-Adic Mathematical Physics

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

Глубокая очистка теллура для производства материалов электроники и фотоники

The regularities of impurity distribution between the distillate and the still as well as the spatial  distribution of impurities  along  the distillate length have been studied. We conclude that some impurities such as s−metals, Zn, Ni, V and rare metals distribute uniformly along the distillate length (20 cm). Contrarily, Se tends to concentrate in the distant (from the still) region  of distillate with more  than one order  of magnitude higher concentration compared to the nearest region.Для проведения процесса получе ния высокочистого теллура методом вакуумной  дистилляции предложена конструкция реактора из высокочистых кварцевого стекла и графита. В ходе процесса расплав теллура испаряется, пар переносится из горячей части системы в более холодную и конденсируется в виде твердой фазы (дистиллята) без образования жидкости. Изучены закономерности перераспределения примесей между дистиллятом и кубовым остатком, а также пространственное распределение примесей в дистилляте при проведении очистки металлического теллура. Установлено, что часть примесей, например щелочные металлы, Zn, Ni, V, редкоземельные металлы распределены равномерно по длине  дистиллята (20  см). В то же время концентрация Se в дальней (от перегонного куба) части дистиллята превышает концентрацию в ближней части на порядок

Network Geometry and Complexity

(28 pages, 11 figures)Higher order networks are able to characterize data as different as functional brain networks, protein interaction networks and social networks beyond the framework of pairwise interactions. Most notably higher order networks include simplicial complexes formed not only by nodes and links but also by triangles, tetrahedra, etc. More in general, higher-order networks can be cell-complexes formed by gluing convex polytopes along their faces. Interestingly, higher order networks have a natural geometric interpretation and therefore constitute a natural way to explore the discrete network geometry of complex networks. Here we investigate the rich interplay between emergent network geometry of higher order networks and their complexity in the framework of a non-equilibrium model called Network Geometry with Flavor. This model, originally proposed for capturing the evolution of simplicial complexes, is here extended to cell-complexes formed by subsequently gluing different copies of an arbitrary regular polytope. We reveal the interplay between complexity and geometry of the higher order networks generated by the model by studying the emergent community structure and the degree distribution as a function of the regular polytope forming its building blocks. Additionally, we discuss the underlying hyperbolic nature of the emergent geometry and we relate the spectral dimension of the higher-order network to the dimension and nature of its building blocks