Harmonic sections of vector bundles with spherically symmetric metrics

We equip an arbitrary vector bundle over a Riemannian manifold, endowed with a fiber metric and a compatible connection, with a spherically symmetric metric (cf. [4]), and westudy harmonicity of its sections firstly as smooth maps and then as critical points of the energy functional with variations through smooth sections.We also characterize vertically harmonic sections. Finally, we give some examples of special vector bundles, recovering in some situations some classical harmonicity results.

Generation and characterization of complex vector modes with digital micromirror devices: a tutorial

Complex vector light modes with a spatial variant polarization distribution have become topical of late, enabling the development of novel applications in numerous research fields. Key to this is the remarkable similarities they hold with quantum entangled states, which arises from the non-separability between the spatial and polarisation degrees of freedom (DoF). As such, the demand for diversification of generation methods and characterization techniques have increased dramatically. Here we put forward a comprehensive tutorial about the use of DMDs in the generation and characterization of vector modes, providing details on the implementation of techniques that fully exploits the unsurpassed advantage of Digital Micromirrors Devices (DMDs), such as their high refresh rates and polarisation independence. We start by briefly describing the operating principles of DMD and follow with a thorough explanation of some of the methods to shape arbitrary vector modes. Finally, we describe some techniques aiming at the real-time characterization of vector beams. This tutorial highlights the value of DMDs as an alternative tool for the generation and characterization of complex vector light fields, of great relevance in a wide variety of applications.

Simulation Evaluation for Methods Used to Determine Muscular Internal Force Based on Joint Stiffness Using Muscular Internal Force Feedforward Controller for Musculoskeletal System

This study proposes two novel methods for determining the muscular internal force (MIF) based on joint stiffness, using an MIF feedforward controller for the musculoskeletal system. The controller was developed in a previous study, where we found that it could be applied to achieve any desired end-point position without the use of sensors, by providing the MIF as a feedforward input to individual muscles. However, achieving motion with good response and low stiffness using the system, posed a challenge. Furthermore, the controller was subject to an ill-posed problem, where the input could not be uniquely determined. We propose two methods to improve the control performance of this controller. The first method involves determining a MIF that can independently control the response and stiffness at a desired position, and the second method involves the definition of an arbitrary vector that describes the stiffnesses at the initial and desired positions to uniquely determine the MIF balance at each position. The numerical simulation results reported in this study demonstrate the effectiveness of both proposed methods.

CALCULATION OF FLOWS IN AN ACCELERATOR OF ELEMENTARY PARTICLES

The Schrödinger and Klein-Gordon equations have a finite velocity solution using the Navier-Stokes equation. But the Dirac equation did not lend itself to solving with the help of a finite formula for an arbitrary vector and scalar potential. Finally, the transition from the derivative of the function to the derivative of the logarithm of the function worked. Then we managed to solve a linear equation with respect to the derivative of the logarithm of the function, which can be integrated. Moreover, it turned out that it is possible to describe many particles. At accelerators, the trajectories of particles with an error are described, i.e. complex trajectories. In this article, the task is to calculate the accelerator in the complex plane, where the imaginary part is the error of the mean — the real part.

Off-Shell Noether Currents and Potentials for First-Order General Relativity

We report off-shell Noether currents obtained from off-shell Noether potentials for first-order general relativity described by n-dimensional Palatini and Holst Lagrangians including the cosmological constant. These off-shell currents and potentials are achieved by using the corresponding Lagrangian and the off-shell Noether identities satisfied by diffeomorphisms generated by arbitrary vector fields, local SO(n) or SO(n−1,1) transformations, ‘improved diffeomorphisms’, and the ‘generalization of local translations’ of the orthonormal frame and the connection. A remarkable aspect of our approach is that we do not use Noether’s theorem in its direct form. By construction, the currents are off-shell conserved and lead naturally to the definition of off-shell Noether charges. We also study what we call the ‘half off-shell’ case for both Palatini and Holst Lagrangians. In particular, we find that the resulting diffeomorphism and local SO(3,1) or SO(4) off-shell Noether currents and potentials for the Holst Lagrangian generically depend on the Immirzi parameter, which holds even in the ‘half off-shell’ and on-shell cases. We also study Killing vector fields in the ‘half off-shell’ and on-shell cases. The current theoretical framework is illustrated for the ‘half off-shell’ case in static spherically symmetric and Friedmann–Lemaitre–Robertson–Walker spacetimes in four dimensions.

Author Correction: Time reversed optical waves by arbitrary vector spatiotemporal field generation

A Correction to this paper has been published: https://doi.org/10.1038/s41467-021-20944-8.