Renormalizability of center-vortex sectors in continuum Yang-Mills theory

Recently, a novel approach to quantize SU(N) Yang-Mills theory was proposed, where the configuration space {Aμ} is split into sectors labeled by topological defects, and then the gauge is fixed by a sector dependent condition. As the procedure is local in {Aμ}, it could be free from Gribov copies. In this work, we review the renormalizability of sectors labeled by an arbitrary number of elementary center vortices.

Optimization and software-numerical implementation of miller's algorithm on a four-frequency model of solid vibration movement

On the basis of a programmed-numerical approach, new values of the coefficients in the Miller orientation algorithm are obtained. For this, an analytical reference model of the angular motion of a rigid body was applied in the form of a four-frequency representation of the orientation quaternion.The numerical implementation of the reference model for a given set of frequencies is presented in the form of constructed trajectories in the configuration space of orientation parameters. A software-numerical implementation of Miller's algorithm is carried out for different values of the coefficients and the values of the coefficients are obtained, which optimize the error of the accumulated drift. It is shown that for the presented reference model of angular motion, Miller's algorithm with a new set of coefficients provides a lower computational drift error compared to with the classic Miller algorithm and the Ignagni modification, which are optimized for conical motion.

Rate of Entropy Production in Stochastic Mechanical Systems

Entropy production in stochastic mechanical systems is examined here with strict bounds on its rate. Stochastic mechanical systems include pure diffusions in Euclidean space or on Lie groups, as well as systems evolving on phase space for which the fluctuation-dissipation theorem applies, i.e., return-to-equilibrium processes. Two separate ways for ensembles of such mechanical systems forced by noise to reach equilibrium are examined here. First, a restorative potential and damping can be applied, leading to a classical return-to-equilibrium process wherein energy taken out by damping can balance the energy going in from the noise. Second, the process evolves on a compact configuration space (such as random walks on spheres, torsion angles in chain molecules, and rotational Brownian motion) lead to long-time solutions that are constant over the configuration space, regardless of whether or not damping and random forcing balance. This is a kind of potential-free equilibrium distribution resulting from topological constraints. Inertial and noninertial (kinematic) systems are considered. These systems can consist of unconstrained particles or more complex systems with constraints, such as rigid-bodies or linkages. These more complicated systems evolve on Lie groups and model phenomena such as rotational Brownian motion and nonholonomic robotic systems. In all cases, it is shown that the rate of entropy production is closely related to the appropriate concept of Fisher information matrix of the probability density defined by the Fokker–Planck equation. Classical results from information theory are then repurposed to provide computable bounds on the rate of entropy production in stochastic mechanical systems.

Overcoming Kinematic Singularities for Motion Control in a Caster Wheeled Omnidirectional Robot

Omnidirectional planar robots are common these days due to their high mobility, for example in human–robot interactions. The motion of such mechanisms is based on specially designed wheels, which may vary when different terrains are considered. The usage of actuated caster wheels (ACW) may enable the usage of regular wheels. Yet, it is known that an ACW robot with three actuated wheels needs to overcome kinematic singularities. This paper introduces the kinematic model for an ACW omni robot. We present a novel method to overcome the kinematic singularities of the mechanism’s Jacobian matrix by performing the time propagation in the mechanism’s configuration space. We show how the implementation of this method enables the estimation of caster wheels’ swivel angles by tracking the plate’s velocity. We present the mechanism’s kinematics and trajectory tracking in real-world experimentation using a novel robot design.

ON THE STRUCTURE OF THE CONFIGURATION SPACE OF CHARGED BLACK HOLES

Some properties of the configuration space (CS) of charged black holes (BH) we are considered. A reduced action for the spherically symmetric configuration of the gravitational and electromagnetic fields is constructed. We restrict ourselves to considering of T-region, where the studied fields have a dynamic meaning. Using the Hamiltonian constraint, we exclude the nondynamic degree of freedom. This leads to the action of the system in the CS with the corresponding supermetric. It turns out that the CS is flat, and its metric admits a twoparametric group of motions. This group generates conservation laws for the geodesic equations. The first law is the charge conservation law, and second is the mass conservation law (the mass function). Using the Hamiltonian constraint, they allow one to find momenta as a function of the field variables andcalculate the action as a function of the conserved quantities and field variables in CS. We emphasize that to find this action, we use only the integrability condition for a differential form. The quantization of the system is reduced to the uantization of a free particle in a three-dimensional pseudo-Euclidean space. The natural measure corresponding to the CS metric is used to construct the Hermitian DeWitt and mass operators. Based on the self-consistent solution of quantum DeWitt equations and equations for the eigenvalues of the mass and charge operators, the wave function for the spherically symmetric configuration of the gravitational and electromagnetic fields in the T- region is constructed. As a result, we get a model of charged BH with continuous mass and charge spectra.

A consistent two-skyrmion configuration space from instantons

To study a nuclear system in the Skyrme model one must first construct a space of low energy Skyrme configurations. However, there is no mathematical definition of this configuration space and there is not even consensus on its fundamental properties, such as its dimension. Here, we propose that the full instanton moduli space can be used to construct a consistent skyrmion configuration space, provided that the Skyrme model is coupled to a vector meson which we identify with the ρ-meson. Each instanton generates a unique skyrmion and we reinterpret the 8N instanton moduli as physical degrees of freedom in the Skyrme model. In this picture a single skyrmion has six zero modes and two non-zero modes: one controls the overall scale of the solution and one the energy of the ρ-meson field. We study the N = 1 and N = 2 systems in detail. Two interacting skyrmions can excite the ρ through scattering, suggesting that the ρ and Skyrme fields are intrinsically linked. Our proposal is the first consistent manifold description of the two-skyrmion configuration space. The method can also be generalised to higher N and thus provides a general framework to study any skyrmion configuration space.

The Topology of the Configuration Space of a Mathematical Model for Cycloalkenes

As a mathematical model for cycloalkenes, we consider equilateral polygons whose interior angles are the same except for those of the both ends of the specified edge. We study the configuration space of such polygons. It is known that for some case, the space is homeomorphic to a sphere. The purpose of this chapter is threefold: First, using the h-cobordism theorem, we prove that the above homeomorphism is in fact a diffeomorphism. Second, we study the best possible condition for the space to be a sphere. At present, only a sphere appears as a topological type of the space. Then our third purpose is to show the case when a closed surface of positive genus appears as a topological type.