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An important research area in physiological and sport sciences is the analysis of the variations of the muscle reaction due to changes in walking speed. In this paper, we investigated the effect of walking speed variations on leg muscle reaction by the analysis of Electromyogram (EMG) signals at different walking inclines. For this purpose, we benefited from fractal theory and sample entropy to analyze how the complexity of EMG signals changes at different walking speeds. According to the results, although fractal theory could not show a clear trend between the variations of the complexity of EMG signals and the variations of the walking speed, however, based on the results, increasing the speed of walking in the case of different inclines is mapped on to the decrement of the sample entropy of EMG signals. Therefore, sample entropy could decode the effect of walking speed on the reaction of leg muscle. This analysis method could be applied to analyze the variations of other physiological signals of humans durin walking.

Some properties of (left) k-ideals and r-ideals of a semiring are considered by the help of the congruence class semiring. It is proved that a proper k-ideal of a semiring with an identity is prime if it is a maximal left k-ideal. An equivalent condition for a proper r-ideal of a semiring being a maximal (left) r-ideal is established. It is shown that (left) r-ideals and (left) k-ideals coincide for an additively idempotent semiring, though the former is a special kind of the latter in general. It is proved that a proper k-ideal of an incline with an identity is a maximal k-ideal if and only if the corresponding congruence class semiring is the Boolean semiring.

Recently, there are more people jogging with a treadmill at the gym or the home setting. The main available selected modes for treadmill jogging are speed and slope of incline. Increased speeds and incline slopes will not only increase the cardiopulmonary loading but may also alter the lower extremity (LE) movement patterns. There are few systematic investigations of the effect of the speed and incline on LE kinematics. Most studies have used 2D methods which focused on movements in sagittal plane only and this has limitations in the acquired data since lower extremity movements also include frontal and transverse planes. The current study aimed to investigate LE movement during jogging at different speeds and incline slopes using a high speed three-dimensional (3D) motion analysis system.

Eighteen young healthy males were recruited. The video-based motion capture system with six CCD cameras, HIRES Expert Vision System (Motion Analysis Corporation, CA, USA), was used to collect kinematic data at a sampling frequency of 120Hz. Nineteen passive reflective markers were attached to bilateral lower extremities of the subject. The joint angle is calculated by Euler angle using the rotation sequence: 2-1-3 (y-x′-z″). Four speeds were selected: 2 m/s, 2.5 m/s, 3 m/s, 3.5 m/s with the slope at 0, and four slopes were selected: 0%, 5%,10%,15% at a speed of 3 m/s. Repeated-measures ANOVA was used to test hypotheses regarding changes in jogging condition on LE kinematic variables. The significance level was set at 0.05.

As the jogging slope increased, the hip, knee and ankle demonstrated a significantly greater maximum flexion in swing phase (p<0.001), but the maximum extension angles in stance phase were relatively unchanged. Increased LE flexion during swing phase is important to ensure foot clearance with increased slope. For increased speed, the hip and ankle joints had significantly greater maximum joint extension angles during stance phase and the hip and knee joint had significantly larger maximum flexion angles in swing phase (p<0.001). Increased motion during swing phase account for a larger step length and increased motion during stance phase may facilitate the generation of power during forward propulsion as the jogging speed increased. As the slope and speed increased, LE movement patterns were changed in the transverse plane: the significantly increased (p<0.01) internal hip rotation at terminal stance, the increased toe-in of foot (p<0.001) during terminal stance phase and decreased (p<0.05) toe-out during swing phase. Increased hip motion in transverse plane could lengthen the stride distance and increase foot toe-in for providing a stable lever for push off to increase propulsion force as speed or slope is increased. By way of systematic 3D kinematic investigation of the LE in jogging, the results further elucidate the effect of changing speed and incline on LE joints movements. This information could provide guidelines for rehabilitation clinicians or coaches to select an appropriate training mode for jogging.