On the Schrödinger map for regular helical polygons in the hyperbolic space

The main purpose of this article is to understand the evolution of X t = X s ∧− X ss , with X(s, 0) a regular polygonal curve with a nonzero torsion in the three-dimensional Minkowski space. Unlike in the case of the Euclidean space, a nonzero torsion now implies two different helical curves. This generalizes recent works by the author with de la Hoz and Vega on helical polygons in the Euclidean space as well as planar polygons in the Minkowski space. Numerical experiments in this article show that the trajectory of the point X(0, t) exhibits new variants of Riemann’s non-differentiable function whose structure depends on the initial torsion in the problem. As a result, we observe that the smooth solutions (helices, straight line) in the Minkowski space show the same instability as displayed by their Euclidean counterparts and curves with zero-torsion. These numerical observations are in agreement with some recent theoretical results obtained by Banica and Vega.

Friedmann-like universes with weak torsion: a dynamical system approach

We consider Friedmann-like universes with torsion and take a step towards studying their stability. In so doing, we apply dynamical-system techniques to an autonomous system of differential equations, which monitors the evolution of these models via the associated density parameters. Assuming relatively weak torsion, we identify the system’s equilibrium points. These are found to represent homogeneous and isotropic spacetimes with nonzero torsion that undergo accelerated expansion. We then examine the linear stability of the aforementioned fixed points. Our results indicate that Friedmann-like cosmologies with weak torsion are generally stable attractors, either asymptotically or in the Lyapunov sense. In addition, depending on the equation of state of the matter, the equilibrium states can also act as intermediate saddle points, marking a transition from a torsional to a torsion-free universe.

Roll maneuvers are essential for active reorientation of Caenorhabditis elegans in 3D media

Locomotion of the nematode Caenorhabditis elegans is a key observable used in investigations ranging from behavior to neuroscience to aging. However, while the natural environment of this model organism is 3D, quantitative investigations of its locomotion have been mostly limited to 2D motion. Here, we present a quantitative analysis of how the nematode reorients itself in 3D media. We identify a unique behavioral state of C. elegans—a roll maneuver—which is an essential component of 3D locomotion in burrowing and swimming. The rolls, associated with nonzero torsion of the nematode body, result in rotation of the plane of dorsoventral body undulations about the symmetry axis of the trajectory. When combined with planar turns in a new undulation plane, the rolls allow the nematode to reorient its body in any direction, thus enabling complete exploration of 3D space. The rolls observed in swimming are much faster than the ones in burrowing; we show that this difference stems from a purely hydrodynamic enhancement mechanism and not from a gait change or an increase in the body torsion. This result demonstrates that hydrodynamic viscous forces can enhance 3D reorientation in undulatory locomotion, in contrast to known hydrodynamic hindrance of both forward motion and planar turns.

Nonholonomic jet deformations, exact solutions for modified Ricci soliton and Einstein equations

Let [Formula: see text] be a pseudo-Riemannian metric of arbitrary signature on a manifold [Formula: see text] with conventional [Formula: see text]-dimensional splitting, [Formula: see text] determined by a nonholonomic (nonintegrable) distribution [Formula: see text] defining a generalized (nonlinear) connection and associated nonholonomic frame structures. We work with an adapted linear metric compatible connection [Formula: see text] and its nonzero torsion [Formula: see text], both completely determined by [Formula: see text]. Our first goal is to prove that there are certain generalized frame and/or jet transforms and prolongations with [Formula: see text] into explicit classes of solutions of some generalized Einstein equations [Formula: see text], [Formula: see text], encoding various types of (nonholonomic) Ricci soliton configurations and/or jet variables and symmetries. The second goal is to solve additional constraint equations for zero torsion, [Formula: see text], on generalized solutions constructed in explicit forms with jet variables and extract Levi-Civita configurations. This allows us to find generic off-diagonal exact solutions depending on all space time coordinates on [Formula: see text] via generating and integration functions and various classes of constant jet parameters and associated symmetries. Our third goal is to study how such generalized metrics and connections can be related by the so-called “half-conformal” and/or jet deformations of certain subclasses of solutions with one, or two, Killing symmetries. Finally, we present some examples of exact solutions constructed as nonholonomic jet prolongations of the Kerr metrics, with possible Ricci soliton deformations, and characterized by nonholonomic jet structures and generalized connections.

Torsion in tensor powers of modules

Tensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modulesNwith the property thatM ⊗RNhas nonzero torsion unlessMis very special. An important example of such a moduleNis the Frobenius powerpeRover a complete intersection domainRof characteristicp> 0.

Torsion in tensor powers of modules

Tensor products usually have nonzero torsion. This is a central theme of Auslander's 1961 paper; the theme continues in the work of Huneke and Wiegand in the 1990s. The main focus in this article is on tensor powers of a finitely generated module over a local ring. Also, we study torsion-free modules N with the property that M ⊗R N has nonzero torsion unless M is very special. An important example of such a module N is the Frobenius power peR over a complete intersection domain R of characteristic p > 0.

Diameter and girth of Torsion Graph

Let R be a commutative ring with identity. Let M be an R-module and T (M)* be the set of nonzero torsion elements. The set T(M)* makes up the vertices of the corresponding torsion graph, ΓR(M), with two distinct vertices x, y ∈ T(M)* forming an edge if Ann(x) ∩ Ann(y) ≠ 0. In this paper we study the case where the graph ΓR(M) is connected with diam(ΓR(M)) ≤ 3 and we investigate the relationship between the diameters of ΓR(M) and ΓR(R). Also we study girth of ΓR(M), it is shown that if ΓR(M) contains a cycle, then gr(ΓR(M)) = 3.

THE RELATION BETWEEN THE MODEL OF A CRYSTAL WITH DEFECTS AND PLEBANSKI'S THEORY OF GRAVITY

In the present investigation, we show that there exists a close analogy of geometry of space–time in general relativity (GR) with a structure of defects in a crystal. We present the relation between the Kleinert's model of a crystal with defects and Plebanski's theory of gravity. We have considered the translational defects — dislocations and the rotational defects — disclinations — in the three- and four-dimensional crystals. The four-dimensional crystalline defects present the Riemann–Cartan space–time which has an additional geometric property — "torsion" — connected with dislocations. The world crystal is a model for the gravitation which has a new type of gauge symmetry: the Einstein's gravitation has a zero torsion as a special gauge, while a zero connection is another equivalent gauge with nonzero torsion which corresponds to the Einstein's theory of "teleparallelism". Any intermediate choice of the gauge with nonzero connection [Formula: see text] is also allowed. In the present investigation, we show that in the Plebanski formulation the phase of gravity with torsion is equivalent to the ordinary or topological gravity, and we can exclude a torsion as a separate dynamical variable.

CONFORMALLY INVARIANT LAGRANGIANS IN METRIC-AFFINE AND RIEMANN-CARTAN SPACES

Various types of generalized conformal transformations in non-Riemannian manifolds and corresponding transformation properties of the geometrical objects are reviewed. Possible structures of conformally invariant gravity and matter Lagrangians in arbitrary space dimensions are discussed. New electrovac exact solutions of R+R2+Q2 gravity with conformally flat Bertotti-Robinson metric and nonzero torsion have been found.