Geometric sigma model of the Universe

The purpose of this work is to demonstrate how an arbitrarily chosen background of the Universe can be made a solution of a simple geometric sigma model. Geometric sigma models are purely geometric theories in which spacetime coordinates are seen as scalar fields coupled to gravity. Although they look like ordinary sigma models, they have the peculiarity that their complete matter content can be gauged away. The remaining geometric theory possesses a background solution that is predefined in the process of constructing the theory. The fact that background configuration is specified in advance is another peculiarity of geometric sigma models. In this paper, I construct geometric sigma models based on different background geometries of the Universe. Whatever background geometry is chosen, the dynamics of its small perturbations is shown to posses a generic classical stability. This way, any freely chosen background metric is made a stable solution of a simple model. Three particular models of the Universe are considered as examples of how this is done in practice.Comment: 31 pages, 9 figure

Pseudo-Riemannian Universe from Euclidean bulk

I develop the idea that our world is a brane-like object embedded in Euclidean bulk. In its ground state, the brane constituent matter is assumed to be homogeneous and isotropic, and of negligible influence on the bulk geometry. The analysis of this paper is model independent, in the sense that action functional of bulk fields is not specified. Instead, the behavior of the brane is derived from the universally valid conservation equation of the bulk stress tensor. The present work studies the behavior of a $3$-sphere in the $5$-dimensional Euclidean bulk. The sphere is made of bulk matter characterized by the equation of state $p=\alpha\rho$. It is shown that stability of brane vibrations requires $\alpha < 0$. Then, the stable brane perturbations obey Klein-Gordon-like equation with an effective metric of Minkowski signature. The argument is given that it is this effective metric that is detected in physical measurements. The corresponding effective Universe is analyzed for all the values of $\alpha<0$. In particular, the effective metric is shown to be a solution of Einstein's equations coupled to an effective perfect fluid. The effective energy density and pressure at the present epoch are calculated. So are the age of the Universe, and the effective cosmological constant. All the results are presented in two tables. As an illustration, one simple choice of the brane constituent matter is studied in detail.Comment: 21 pages, 2 figure

A class of regular bouncing cosmologies

In this paper, I construct a class of everywhere regular geometric sigma models that possess bouncing solutions. Precisely, I show that every bouncing metric can be made a solution of such a model. My previous attempt to do so by employing one scalar field has failed due to the appearance of harmful singularities near the bounce. In this work, I use four scalar fields to construct a class of geometric sigma models which are free of singularities. The models within the class are parametrized by their background geometries. I prove that, whatever background is chosen, the dynamics of its small perturbations is classically stable on the whole time axes. Contrary to what one expects from the structure of the initial Lagrangian, the physics of background fluctuations is found to carry 2 tensor, 2 vector and 2 scalar degrees of freedom. The graviton mass, that naturally appears in these models, is shown to be several orders of magnitude smaller than its experimental bound. I provide three simple examples to demonstrate how this is done in practice. In particular, I show that graviton mass can be made arbitrarily small.Comment: 34 pages, 7 figure

String background fields and Riemann-Cartan geometry

We study classical dynamics of cylindrical membranes wrapped around the extra compact dimension of a $(D+1)$-dimensional Riemann-Cartan spacetime. The world-sheet equations and boundary conditions are obtained from the universally valid conservation equations of the stress-energy and spin tensors. Specifically, we consider membranes made of macroscopic matter with maximally symmetric distribution of spin. In the narrow membrane limit, the dimensionally reduced theory is obtained. It describes how effective strings couple to the effective $D$-dimensional geometry. The striking coincidence with the string theory $\sigma$-model is observed. In this correspondence, the string background fields $G_{\mu\nu}$, $B_{\mu\nu}$, $A_{\mu}$ and $\Phi$ are related to the metric and torsion of the Riemann-Cartan spacetime.Comment: 15 pages, JHEP styl

Numerical Search for Periodic Solutions in the Vicinity of the Figure-Eight Orbit: Slaloming around Singularities on the Shape Sphere

We present the results of a numerical search for periodic orbits with zero angular momentum in the Newtonian planar three-body problem with equal masses focused on a narrow search window bracketing the figure-eight initial conditions. We found eleven solutions that can be described as some power of the "figure-eight" solution in the sense of the topological classification method. One of these solutions, with the seventh power of the "figure-eight", is a choreography. We show numerical evidence of its stability.Comment: 14 pages, 6 figures, submitted to Celestial Mechanics and Dynamical Astronom

Dirac particle in gravitational field

Classical dynamics of spinning zero-size objects in an external gravitational field is derived from the conservation law of the stress-energy and spin tensors. The resulting world line equations differ from those in the existing literature. In particular, the spin of the Dirac particle does not couple to the background curvature. As a check of consistency, the wave packet solution of the free Dirac equation is considered. The resulting equations are shown to include a constraint that relates the wave packet spin to its orbital angular momentum. In the zero-size limit, both contributions to the total angular momentum disappear simultaneously.Comment: 4 page

Topological Dependence of Kepler's Third Law for Collisionless Periodic Three-Body Orbits with Vanishing Angular Momentum and Equal Masses

We present results of numerical calculations showing a three-body orbit's period's $T$ dependence on its topology. This dependence is a simple linear one, when expressed in terms of appropriate variables, suggesting an exact mathematical law. This is the first known relation between topological and kinematical properties of three-body systems. We have used these results to predict the periods of several sets of as yet undiscovered orbits, but the relation also indicates that the number of periodic three-body orbits with periods shorter than any finite number is countable.Comment: 9 pages, 4 figure

Counterexamples to conjectures on graph distance measures based on topological indexes

In this paper we disprove three conjectures from [M. Dehmer, F. Emmert-Streib, Y. Shi, Interrelations of graph distance measures based on topological indices, PLoS ONE 9 (2014) e94985] on graph distance measures based on topological indices by providing explicit classes of trees that do not satisfy proposed inequalities. The constructions are based on the families of trees that have the same Wiener index, graph energy or Randic index - but different degree sequences.Comment: 9 pages, 2 figure

Can Human-Like Bots Control Collective Mood: Agent-Based Simulations of Online Chats

Using agent-based modeling approach, in this paper, we study self-organized dynamics of interacting agents in the presence of chat Bots. Different Bots with tunable ``human-like'' attributes, which exchange emotional messages with agents, are considered, and collective emotional behavior of agents is quantitatively analysed. In particular, using detrended fractal analysis we determine persistent fluctuations and temporal correlations in time series of agent's activity and statistics of avalanches carrying emotional messages of agents when Bots favoring positive/negative affects are active. We determine the impact of Bots and identify parameters that can modulate it. Our analysis suggests that, by these measures, the emotional Bots induce collective emotion among interacting agents by suitably altering the fractal characteristics of the underlying stochastic process.Positive-emotion Bots are slightly more effective than the negative ones. Moreover, the Bots which are periodically alternating between positive and negative emotion, can enhance fluctuations in the system leading to the avalanches of agent's messages that are reminiscent of self-organized critical states.Comment: 21 pages, 9 figures (multiple) colo

Generalizations of Wiener polarity index and terminal Wiener index

In theoretical chemistry, distance-based molecular structure descriptors are used for modeling physical, pharmacologic, biological and other properties of chemical compounds. We introduce a generalized Wiener polarity index $W_k (G)$ as the number of unordered pairs of vertices ${u, v}$ of $G$ such that the shortest distance $d (u, v)$ between $u$ and $v$ is $k$ (this is actually the $k$-th coefficient in the Wiener polynomial). For $k = 3$, we get standard Wiener polarity index. Furthermore, we generalize the terminal Wiener index $TW_k (G)$ as the sum of distances between all pairs of vertices of degree $k$. For $k = 1$, we get standard terminal Wiener index. In this paper we describe a linear time algorithm for computing these indices for trees and partial cubes, and characterize extremal trees maximizing the generalized Wiener polarity index and generalized terminal Wiener index among all trees of given order $n$.Comment: 3pages, 4 figure