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ABSTRACT We prove that for any graph , the total chromatic number of is at most . This saves one color in comparison with the result of Hind from 1992. In particular, our result says that if , then has a total coloring using at most colors. When is regular and has a sufficient number of vertices, we can actually save an additional two colors. Specifically, we prove that for any , there exists such that: if is an ‐regular graph on vertices with , then . This confirms the Total Coloring Conjecture for such graphs . 

Recent studies suggest that deep neural networks (DNNs) have the potential to outperform neural networks (NNs) in approximating complex dynamics, which may enhance the tracking performance of a control system. However, unlike NN-based nonlinear control systems, designing update policies for the inner-layer weights of a DNN using Lyapunov-based stability methods is problematic since the inner-layer weights are nested within activation functions. Traditional DNN training methods (e.g., gradient descent) may improve the approximation capability of a DNN and thus could enhance a DNN-based controller’s tracking performance; however, traditional DNN-based control approaches lack stability guarantees and may be ineffective in training the DNN without large data sets, which could hinder the tracking performance. In this work, a DNN-based control structure is developed for a hybrid exoskeleton, which combines a rehabilitative robot with functional electrical stimulation (FES). The proposed control system updates the DNN weights in real-time and a rigorous Lyapunov-based stability analysis is performed to ensure semi-global asymptotic trajectory tracking, even without the presence of a data set. Specifically, a Lyapunov-based update law is developed for the output-layer DNN weights and Lyapunov-based constraints are established for the adaptation laws of the inner-layer DNN weights. Additionally, the DNN-based FES controller was designed to be saturated to increase the comfort and safety of the participant. 

Functional electrical stimulation (FES) is widely used for rehabilitating individuals with total or partial limb paralysis, but challenges like muscle fatigue and discomfort limit its effectiveness. Hybrid exoskeletons combine the rehabilitative benefits of exoskeletons and FES, while mitigating the drawbacks of each. However, despite hybrid exoskeletons being highly effective in rehabilitation, the dynamics associated with these systems are complex. Deep neural networks (DNNs) can approximate these complex hybrid exoskeleton dynamics; however, they traditionally lack stability guarantees and robustness, hindering their application in real-world systems. Moreover, traditional training methods (e.g., gradient descent) require an extensive dataset and offline training, further hindering a DNNs practical use. Therefore, this paper presents an innovative Lyapunov-based adaptation law, with a gradient descent-like structure, that is designed to update all layer weights of a DNN in real-time for a DNN-based hybrid exoskeleton control framework. To promote user comfort and safety, a saturation limit was implemented on the DNN-based FES controller and the excess control effort was redirected to the exoskeleton. A Lyapunov-based stability analysis was performed on the DNN-based hybrid exoskeleton control system to prove global asymptotic trajectory tracking. A numerical simulation of the designed controller was performed to validate the results. 

ABSTRACT This study utilizes linear elastic fracture mechanics to assess the fatigue criticality of volumetric defects in notched specimens with varying geometries. Contrasting to the existing literature, this study assesses the fatigue criticality of defects, prior to fracture, via a non‐destructive inspection technique, that is, X‐ray computed tomography (XCT). Treating volumetric defects as cracks, based on Murakami's definition, the approach calculates their Mode‐I stress intensity factor (SIF) with their local stresses obtained via linear elastic finite element analysis and utilizes the SIF to represent their criticality. For validation, cylindrical and flat specimens with notch root radii of 5 and 50 mm of AlSi10Mg and 17‐4 precipitation hardened stainless steel were fabricated, XCT scanned, and tested under fatigue loading. All crack initiating defects, observed from fractography, fell within the 99.3 percentile of the defects with the highest stress intensity factor in the respective specimens. 

Abstract Solar wind directional discontinuities, such as rotational discontinuities (RDs), significantly influence energy and transport processes in the Earth's magnetosphere. A recent observational study identified a long‐lasting double cusp precipitation event associated with RD in solar wind on 10 April 2015. To understand the magnetosphere‐ionosphere response to the solar wind RD, a global hybrid simulation of the magnetosphere was conducted, with solar wind conditions based on the observation event. The simulation results show significant variations in the magnetopause and cusp regions caused by the passing RD. After the RD propagates to the magnetopause, ion precipitation intensifies, and a double cusp structure at varying latitudes and longitudes forms near noon in the northern hemisphere, which is consistent with the satellite observations by Wing et al. (2023,https://doi.org/10.1029/2023gl103194). Regarding dayside magnetopause reconnection, the simulation reveals that the high‐latitude reconnection process persists during the RD passing, regardless of whether the interplanetary magnetic field (IMF) with a highB_y/B_zratio has a positive or negativeB_zcomponent, and low‐latitude reconnection occurs after the RD reaches the magnetopause at noon when the IMF turns southward. By examining the ion sources along the magnetic field lines, a connection is found between the single‐ or double‐cusp ion precipitation and the solar wind ions entering from both high‐latitude and low‐latitude reconnection sites. This result suggests that the double‐cusp structure can be triggered by magnetic reconnection occurring at both low latitudes and high latitudes in the opposite hemispheres, associated with a largeB_y/B_zratio of the IMF around the RD. 

People suffering from neurological conditions (NCs) can benefit from motorized functional electrical stimulation (FES)-based rehabilitation equipment, called hybrid exoskeletons. These hybrid exoskeletons incorporate muscle-motor interaction that requires both the control of human muscles (i.e., FES) and robot motors to obtain a desirable performance. Two types of controllers (deep neural networks (DNN)-based and Admittance-based) were developed in this paper for a hybrid exoskeleton to control both human muscles and the exoskeleton’s motors. The uncertain dynamics of the hybrid exoskeleton are approximated by DNN to enable efficient FES control. The approximated DNN weights and biases were implemented in a control law where they were updated in multiple timescales. Specifically, the inner-layer DNN weights were updated iteratively offline while the outer-layer weights were updated online in real-time. The update law for the output-layer DNN weights was augmented with a concurrent learning (CL) inspired term to improve the learning performance of the DNN and, consequently, the overall system performance. The admittance-based motor controller uses torque feedback and desired torque contribution from the participant to modify the motor’s desired trajectory without forcing the participant to follow along predetermined trajectories and to promote the overall safety of the system. A Lyapunov-based stability analysis was completed for both control systems to ensure overall system performance. 

A highly air- and water-stable Fe(ii) complex with a fluorinated ligand has a strong¹⁹F MRI signal but is a poorT₁-weighted¹H MRI contrast agent. Upon oxidation by H₂O₂, the¹⁹F MRI signal decays as the relaxivity for¹H MRI markedly improves.