Abstract:
We build a model of storage of well-defined positional information in probabilistic sequence patterns. Once a pattern is defined, it is possible to judge the effect of any mutation in it. We show that the frequency of beneficial mutations can be high in general and the same mutation can be either advantageous or deleterious depending on the pattern’s context. The model allows to treat positional information as a physical quantity, formulate its conservation law and to model its continuous evolution in a whole genome, with meaningful applications of basic physical principles such as optimal efficiency and channel capacity. A plausible example of optimal solution analytically describes phase transitions-like behavior. The model shows that, in principle, it is possible to store error-free information on sequences with arbitrary low conservation.The described theoretical framework allows one to approach from novel general perspectives such long-standing paradoxes as excessive junk DNA in large genomes or the corresponding G- and C-values paradoxes. We also expect it to have an effect on a number of fundamental concepts in population genetics including the neutral theory, cost-of-selection dilemma, error catastrophe and others.

Abstract:
In this note we present a universal formula in terms of theta functions for the Log- capacity of several segments on a line. The case of two segments was studied by N.I.Akhiezer (1930); three segments were considered by A.Sebbar and T.Falliero (2001).

Abstract:
Two recent scenarios of preheating-related baryogenesis are compared within the framework of Abelian Higgs model in (1+1) dimensions. It is shown that they shift baryon number in opposite directions. Once both scenarios can realize simultaneously as overlapped stages of the same physical process, even the sign of net generated asymmetry becomes dependent on initial parameters.

Abstract:
We discuss topological transitions during parametric resonance in the gauge sector of electroweak theory. It is shown that the resonance leads to separation of topological indices of the gauge and Higgs fields, resulting in topological transitions of non-sphaleron nature.

Abstract:
The recently proposed first-order parent formalism at the level of equations of motion is specialized to the case of Lagrangian systems. It is shown that for diffeomorphism-invariant theories the parent formulation takes the form of an AKSZ-type sigma model. The proposed formulation can be also seen as a Lagrangian version of the BV-BRST extension of the Vasiliev unfolded approach. We also discuss its possible interpretation as a multidimensional generalization of the Hamiltonian BFV--BRST formalism. The general construction is illustrated by examples of (parametrized) mechanics, relativistic particle, Yang--Mills theory, and gravity.

Abstract:
The de Haas - van Alphen effect in quasi-two-dimensional metals is studied at arbitrary parameters. The oscillations of the chemical potential may substantially change the temperature dependence of harmonic amplitudes that is usually used to determine the effective electron mass. Hence, the processing of the experimental data using the standard Lifshitz-Kosevich formula (that assumes the chemical potential to be constant) may lead to substantial errors even in the limit of strong harmonic damping. This fact may explain the difference between the effective electron masses, determined from the de Haas - van Alphen effect and the cyclotron resonance measurements. The oscillations of the chemical potential and the deviations from the Lifshitz-Kosevich formula depend on the reservoir density of states, that exists in organic metals due to open sheets of Fermi surface. This dependence can be used to determine the density of electron states on open sheets of Fermi surface. We present the analytical results of the calculations of harmonic amplitudes in some limiting cases that show the importance of the oscillations of the chemical potential. The algorithm of the simple numerical calculation of the harmonic amplitudes at arbitrary reservoir density of states, arbitrary warping, spin-splitting, temperature and Dingle temperature is also described.

Abstract:
Substituting Skyrmion for nucleon, one can potentially see -- in real time -- how the monopole is catalysing the proton (or neutron) decay, and even obtain a plausible estimate for catalysis cross-section. Here we discuss the key aspects of a practical implementation of such approach and demonstrate how one can overcome the main technical problems: Gauss constraint violation and reflections at the boundaries.

Abstract:
Since a tropical Nullstellensatz fails even for tropical univariate polynomials we study a conjecture on a tropical {\it dual} Nullstellensatz for tropical polynomial systems in terms of solvability of a tropical linear system with the Cayley matrix associated to the tropical polynomial system. The conjecture on a tropical effective dual Nullstellensatz is proved for tropical univariate polynomials.

Abstract:
We introduce a concept of a fractional-derivatives series and prove that any linear partial differential equation in two independent variables has a fractional-derivatives series solution with coefficients from a differentially closed field of zero characteristic. The obtained results are extended from a single equation to $D$-modules having infinite-dimensional space of solutions (i. e. non-holonomic $D$-modules). As applications we design algorithms for treating first-order factors of a linear partial differential operator, in particular for finding all (right or left) first-order factors.

Abstract:
In the paper I study properties of random polynomials with respect to a general system of functions. Some lower bounds for the mathematical expectation of the uniform and recently introduced integral-uniform norms of random polynomials are established. {\sc Key words and phrases:} Random polynomial, estimates for maximum of random process, integral-uniform norm.