Vladimir V. Bochkarev

Gradient models of the axion-photon coupling

We establish an extended version of the Einstein–Maxwell-axion model by introducing into the Lagrangian cross-terms, which contain the gradient four-vector of the pseudoscalar (axion) field in convolution with the Maxwell tensor. The gradient model of the axion–photon coupling is applied to cosmology: we analyze the Bianchi-I type Universe with an initial magnetic field, electric field induced by the axion–photon interaction, cosmological constant and dark matter, which is described in terms of the pseudoscalar (axion) field. Analytical, qualitative and numerical results are presented in detail for two distinguished epochs: first, for the early Universe with magnetic field domination; second, for the stage of late-time accelerated expansion.

ARCHIMEDEAN-TYPE FORCE IN A COSMIC DARK FLUID: I. EXACT SOLUTIONS FOR THE LATE-TIME ACCELERATED EXPANSION

We establish a new self-consistent model in order to explain from a unified viewpoint two key features of the cosmological evolution: the inflation in the early Universe and the late-time accelerated expansion. The key element of this new model is the Archimedean-type coupling of the dark matter with dark energy, which form the so-called cosmic dark fluid. We suppose that dark matter particles immersed into the dark energy reservoir are affected by the force proportional to the four-gradient of the dark energy pressure. The Archimedean-type coupling is shown to play a role of effective energy-momentum redistributor between the dark matter and the dark energy components of the dark fluid, thus providing the Universe evolution to be a quasiperiodic and/or multistage process. In the first part of the work we discuss a theoretical base and new exact solutions of the model master equations. Special attention is focused on the exact solutions, for which the scale factor is presented by the anti-Gaussian function: these solutions describe the late-time acceleration and are characterized by a nonsingular behavior in the early Universe. The second part contains qualitative and numerical analysis of the master equations; we focus there on the solutions describing a multi-inflationary Universe.

ARCHIMEDEAN-TYPE FORCE IN A COSMIC DARK FLUID: III. BIG RIP, LITTLE RIP AND CYCLIC SOLUTIONS

We analyze late-time evolution of the Universe in the framework of the self-consistent model, in which the dark matter is influenced by the Archimedean-type force proportional to the four-gradient of the dark energy pressure. The dark energy is considered as a fluid with the equation of state of the relaxation type, which takes into account a retardation of the dark energy response to the Universe accelerated expansion. The dark matter is guided by the Archimedean-type force, which redistributes the total energy of the dark fluid between two its constituents, dark energy and dark matter, in the course of the Universe accelerated expansion. We focus on the constraints for the dark energy relaxation time parameter, for the dark energy equation of state parameter, and for the Archimedean-type coupling constants, which guarantee the Big Rip avoidance. In particular, we show that the Archimedean-type coupling protects the Universe from the Big Rip scenario with asymptotically infinite negative dark energy pressure, and that the Little Rip is the fate of the Universe with the Archimedean-type interaction inside the dark fluid.

The Zipf Law for Random Texts with Unequal Letter Probabilities and the Pascal Pyramid

We study a model of generating words with independent unequal probabilities of letters. We prove that the probability p(r) of words of rank r has a power asymptotics. As distinct from a paper published earlier by B. Conrad andM.Mitzenmacher, we give a short proof, using elementary methods, and obtain an explicit formula for the degree of the power law.

Strong power and subexponential laws for an ordered list of trajectories of a Markov chain

Consider a homogeneous Markov chain with discrete time and with a finite set of states E0, . . . , En such that the state E0 is absorbing and states E1, . . . , En are nonrecurrent. The frequencies of trajectories in this chain are studied in this paper, i.e., “words” composed of symbols E1, . . . , En ending with the “space” E0. Order the words according to their probabilities; denote by p(t) the probability of the tth word in this list. As was proved recently, in the case of an infinite list of words, in the dependence of the topology of the graph of the Markov chain, there exists either the limit ln p(t)/ ln t as t → ∞ or that of ln p(t)/t1/D , where D ∈ N (weak power and subexponential laws). As appeared, in the latter case the decreasing order of the function p(t) is always subexponential (the strong subexponential law). In the first case, this paper describes necessary and sufficient conditions of the power order (the strong power law). These conditions are fulfilled, in particular, if the graph of the Markov chain that corresponds to states E1, . . . , En is strongly connected.

Light propagation with non-minimal couplings in a two-component cosmic dark fluid with an Archimedean-type force and unlighted cosmological epochs

During the evolution of the universe there are at least two epochs during which electromagnetic waves cannot scan the universes internal structure neither bring information to outside observers. The first epoch is when photons are in local thermodynamic equilibrium with other particles, and the second is when photon scattering by charged particles is strong. One can call these two periods of cosmological time as standard unlighted epochs. After the last scattering surface, photons become relic photons and turn into a source of information about the universe. Unlighted cosmic epochs can also appear when one considers non-minimal theories, i.e., theories in which the electromagnetic field is coupled in an intricate way with the cosmological gravitational field. By considering a cosmological model where the dark sector, i.e., the dark energy and dark matter, self-interacts via an Archimedean-type force, and taking into account a non-minimal coupling theory for the electromagnetic field, we discuss the appearance of unlighted epochs. In the framework of our non-minimal theory, a three-parameter non-minimal Einstein-Maxwell model, the curvature coupling can be formulated in terms of an effective refraction index n(t). Then, taking advantage of a well-known classical analogy, namely, in a medium with n^2<0 electromagnetic waves do not propagate and their group velocity, i.e., energy transfer velocity, has zero value at the boundary of the corresponding zone, one can search for the unlighted epochs arising in the interacting dark fluid cosmological model. We study here, both analytically and numerically, cosmological models admitting unlighted epochs.

Zipf exponent of trajectory distribution in the hidden Markov model

This paper is the first step of generalization of the previously obtained full classification of the asymptotic behavior of the probability for Markov chain trajectories for the case of hidden Markov models. The main goal is to study the power (Zipf) and nonpower asymptotics of the frequency list of trajectories of hidden Markov frequencys and to obtain explicit formulae for the exponent of the power asymptotics. We consider several simple classes of hidden Markov models. We prove that the asymptotics for a hidden Markov model and for the corresponding Markov chain can be essentially different.

Deviations in the Zipf and Heaps laws in natural languages

This paper is devoted to verifying of the empirical Zipf and Hips laws in natural languages using Google Books Ngram corpus data. The connection between the Zipf and Heaps law which predicts the power dependence of the vocabulary size on the text size is discussed. In fact, the Heaps exponent in this dependence varies with the increasing of the text corpus. To explain it, the obtained results are compared with the probability model of text generation. Quasi-periodic variations with characteristic time periods of 60-100 years were also found.

Calculation of Precise Constants in a Probability Model of Zipf’s Law Generation and Asymptotics of Sums of Multinomial Coefficients

Let be a full set of outcomes (symbols) and let positive , , be their probabilities . Let us treat as a stop symbol; it can occur in sequences of symbols (we call them words) only once, at the very end. The probability of a word is defined as the product of probabilities of its symbols. We consider the list of all possible words sorted in the nonincreasing order of their probabilities. Let be the probability of the th word in this list. We prove that if at least one of the ratios , , is irrational, then the limit exists and differs from zero; here is the root of the equation . The limit constant can be expressed (rather easily) in terms of the entropy of the distribution .

Variations of Social Psychology of Russian Society in Last 100 Years

The study compares the dynamics of a number of important socio-demographic parameters with the frequency of the use of keywords for Russian society. We have shown that the frequency of words typical for the patriarchal way of life of Russian society in the 19th century decreases with the reduction of rural population. The paper discusses the interaction of moral values of society, its state ideology and environmental changes. Similar studies were carried out earlier for the English and Chinese languages. Our results confirm the findings of these studies partially. We also have considered the connection of these processes with the frequency of the use of emotive language. Google Books Ngram was used as a source of lexical data.