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As a kind of special terrain, the landslide disaster generated by liquefiable layer slopes under earthquake has become a major engineering challenge due to its large scale and long slip distance. In order to study the seismic response and damage mode of liquefiable layer slope, this paper follows the research line of “geological generalization, physical modeling, and result analysis”, and takes the liquefiable layer slope in the upper reaches of the Yellow River Class secondary terrace as the object of study, generalizes the physical model of the slope, and carries out shaking table test. Based on the PGA amplification coefficient, Fourier analysis and HHT time–frequency characterization of the model slope, it was found that the PGA amplification coefficient increases gradually along the slope height, reaches the maximum value at the top of the liquefiable layer and then decreases gradually, which indicates that the liquefiable sand layer has an obvious energy dissipation effect, and on the horizontal direction of model slope is a tendency to the surface effect; the seismic waves at the discontinuous interface change drastically, and the Hilbert time–frequency spectrum transforms from multiple peaks to a single peak; with the increase of the intensity, the intrinsic frequency of the overall model decreases, and the high-frequency component within the liquefiable sand layer decreases from 5–15Hz to 0–5Hz, indicating that the liquefiable layer has a filtering effect; the damage process of the liquefiable layer slope is the tensile crack at the top of the slope — seismic subsidence at the top of the slope — the shear yielding at the angle of the slope — shear surface penetration at the slope face — overall slope instability and flow-slip damage. The research results will provide a reference for the study of the disaster mechanism of the liquefiable layer slope.

In this paper, we propose a novel slope cellular automaton (CA) model to depict some physical properties of traffic flow with slopes. In our model, we present the effect of slopes on the acceleration/deceleration capabilities and safety distance of the vehicles in highways as in real traffic situations. By numerical simulations, we investigate the dependence of the vehicle capacities in highways on the length and grade of slopes. It is shown that the larger the slope grade, the more significant the effect of slopes on the traffic flow is. Especially when the slope grade is beyond a certain value (i.e. |σ| > 3%), the effect of slopes on traffic flow becomes quite markedly.

In this paper, we present an extended car-following model with consideration of the gravitational force. A new macroscopic model taking into account the slope effects is developed using the relationship between the microscopic and macroscopic variables. The proposed model is applied to reflect the effect of the slope on uniform flow, traffic waves and small perturbation. The simulation results demonstrate that both the angle and the length of the slope have important impacts on traffic flow. The effect of the slope becomes more significant with the increase of the slope angle.

To depict the effect of low-visibility foggy weather upon traffic flow on a highway with slopes, this paper proposes an extended car-following model taking into consideration the drivers’ misjudgment of the following distance and their active reduction of the velocity. By linear stability analysis, the neutral stability curves are obtained. It is shown that under all the three road conditions: uphill, flat road and downhill, drivers’ misjudgment of the following distance will change the stable regions, while having little effect on the sizes of the stable regions. Correspondingly, drivers’ active reduction of the velocity will increase the stability. The numerical simulations agree well with the analytical results. It indicates that drivers’ misjudgment contributes to a higher velocity. Meanwhile, their active reduction of the velocity helps mitigate the influences of small perturbation. Furthermore, drivers’ misjudgment of the following distance has the greatest effect on downhill and the smallest effect on uphill, so does drivers’ active reduction of the velocity.

An extended two-lane lattice model of traffic flow with consideration of the slope effect is proposed. The slope effect is reflected in both the maximal velocity and the relative current. The stability condition of the model is derived by applying the linear stability method. By using the nonlinear analysis method, we obtain the Korteweg–de Vries (KdV) equation near the neutral stability line and the modified Korteweg–de Vries (mKdV) equation near the critical point. The analytical and numerical results demonstrate that the stability of traffic flow is enhanced on the uphill but is weakened on the downhill when the slope angle increases.

In this paper, we deduced a macroscopic traffic model on the uphill and downhill slopes by employing the transformation relation from microscopic variables to macroscopic ones based on a microscopic car-following model considering the velocity difference between adjacent vehicles. The angle $𝜃$ of the uphill and downhill and the gravitational force have a great impact upon the stability of traffic flow. The linear stability analysis for macroscopic traffic model yielded the stability condition. The Korteweg–de Vries (KdV) equation is derived by nonlinear analysis and the corresponding solution to the density wave near the neutral stability line is obtained. By using the upwind finite difference scheme for simulation, the spatiotemporal evolution patterns of traffic flow on the uphill and downhill are attained. The unstable region is shrunken with slope of the gradient increasing and backward-traveling density waves gradually decrease and even disappear on uphill. Conversely, the unstable region on downhill is extended and density waves propagate quickly backward to the whole road with slope of the gradient increasing.

In this paper, for some typical local slump problems, through the site survey of engineering conditions, we observe the local slumping parts, collect a large number of production data, and adopt the Chiping polar projection method to evaluate the initial stability of the problem area. The unstable structure is combined with the solid scale projection to analyze the sliding direction, the sliding surface, the sliding amount, and the like. The study has a clearer understanding of the local slump problem, and based on the stability judgment results, combined with the site conditions, ultimately serves the engineering examples.

It is shown that by knowing the slope of a bounded from below lower semicontinuous convex function at each point of a Banach space, we know the function up to an additive constant.

Structure, equipment, statue or storage cask may experience free-standing rocking on slopes because of the nonuniform settlement of the base, construction error or foundation failure. The free-standing rigid block on a slope subjected to one-sine pulse excitation and earthquake motions is examined in this paper. First, considering the free-standing rigid block on a slope, the overturning acceleration spectrum under one-sine pulse is built. The spectrum covers two overturning modes, i.e. overturning with one impact either before or after the excitation and overturning without impact. Then the influences of the slope on the overturning acceleration spectrum are discussed in depth. The analytical solution is compared with the numerical solution based on both linear and nonlinear formulations. It reveals that the safe region of the minimum overturning acceleration spectrum between mode 1 and mode 2 depends on the angle of the slope. Further, the block on a slope is subjected to the Tianjin, Kobe and Northbridge earthquake motions. By simplifying the main pulse of the motions to one-sine pulse, although the overturning acceleration spectrum fails to predict the overturning acceleration of the block subjected to real earthquake motions, it can provide some guidelines to predict the practical performance of the rocking structures.

We aimed to investigate the effect of basic fitness function according to whole-body vibration (WBV) stimulus with slope during deadlift in adults. A total of 15 subjects performed deadlift exercise with a sound-wave vibrator. The subjects consisted of three groups: no slope and WBV group (control), WBV only group (group 1), and slope and WBV group (group 2). Slope was set at $5^{\circ }$, and the frequency and amplitude of WBV were 10Hz and 5mm, respectively. The participants performed Romanian deadlift 2 days a week for 4 weeks, including 10 trials per set and five sets a day. We measured basic fitness function factor including the isokinetic muscle contraction test using biodex system3. All groups showed an increase in strength of approximately $>20$%. Group 2 showed the highest increase. Moreover, maximal peak torque of the lumbar joint showed an increase trend similar to that of back muscle strength. An increase of 15.72%, 24.86%, and 51.44% was noted in the control, group 1, and group 2, respectively. The findings indicate that WBV exercise with slope is the most efficient exercise protocol for improving muscle function of the trunk. WBV with slope could help stimulate trunk muscles more and efficiently, could result in a more positive effect on muscle function compared with WBV only, and could be included in an exercise program for efficient patient rehabilitation.

Gait pattern performance, for its crucial significance in humanoid robot stabilization, has attracted the attention of researchers worldwide. Although simplified models highlight major features, bipedal walking has bewildered the researchers. Therefore, for a precise understanding of the bipedal model, a state-of-the-art, simplified model has been proposed in this paper which comprises a 3D-multilinked dual spring-loaded inverted pendulum (3D-MDSLIP) while acknowledging the vertical fluctuations of the center of mass (CoM). In addition, the model considers upper body movement and its effects on the stabilization of the humanoid robot. The mathematical modeling of a humanoid walking over the obstacle and slope is demonstrated to precisely understand the problem. The tuning process of the parameters and postures in a humanoid robot is complex and time-consuming. For proper walking of a robot over uneven terrains and slopes, tuning of the PID controller is achieved using converged teaching-learning based optimization (CTLBO) technique for a central pattern generator (CPG) gait, as introduced in the paper. The optimal gait angles are applied to the experimental and simulated NAO to successfully navigate the provided terrain. Thus, the experimental and simulation results jointly show that the proposed CPG-CTLBO gait learning technique is feasible for finding an optimal gait pattern for the humanoid robot within a deviation of 5%. The energy efficiency of the proposed controller is compared with the default controller of NAO based on the average electronic current in sagittal and lateral movement. Further, it is examined for the energy consumption for several slopes, and the results obtained are acceptable, showing the controller is efficient. Additionally, it has been compared with an existing technique for walking a humanoid robot on uneven terrains. The graph obtained using the proposed technique demonstrates the superiority of the proposed technique.

The present study aims to simulate the various features of wave-mud interaction on fine-grained shore profiles including wave height attenuation, wave-induced mud mass transport, gravity-driven flow of fluid mud and the reconfiguration of profile shape. A two-dimensional mud beach deformation model is presented considering the transport of fluid mud under continued wave action and downward gravity force. The wave height transformation is computed from the energy flux conservation law combining the effects of mud bed, shoaling and wave breaking. The rheological constitutive equations of visco-elastic-plastic model (Shibayama et al., 1990) are selected for numerical simulation. Wave flume experiments are carried out and the results are utilized for the verification of numerical model. The results of the numerical model are also compared with the laboratory data of Nakano (1994). It is concluded that the model is capable to predict the observed phenomena.

This study investigated the applicability of a nonlinear analysis method that considers progressive failure to evaluating the stability of rock slopes, including post-earthquake residual displacement. The vibration step of the nonlinear analysis, which begins after residual displacement occurs, was found to match that of a dynamic centrifugal model test. The nonlinear analysis was concluded to be relatively conservative because it calculated a slightly larger residual displacement than that observed during the model test. A real-scale soft rock slope was analyzed, and the results showed that the proposed analysis method is useful for evaluating slope seismic stability, including post-earthquake residual displacements.

A series of shaking table tests were conducted on reinforced slopes to study the slope dynamic characteristics. The influence of concrete-canvas tilt degrees on the seismic response was studied. By considering the effects of different concrete-canvas tilt degrees, the seismic responses of the reinforced slopes were analyzed, along with the accelerations, crest settlements, and horizontal displacements. The failure patterns of different model slopes were compared using white coral sand marks placed at designated elevations to monitor the internal slide of the reinforced slopes. Several round markers were placed on the slope surface to compare the deformation before and after shaking with different amplitudes. The results indicated that with the increase in concrete-canvas tilt degrees, a better reinforcing effect was obtained, and 30^° reinforcement reached a threshold level, the slide-out point shifts from the crest of the slope to the middle of the reinforced model. The bottom 2/7th zone of the slope was relatively stable during the earthquake and the reinforcement was ineffective at the bottom of the slope. When both considered the influence of reinforcing effect and construction difficulty, 20^° is the suitable tilt degree in concrete-canvas reinforced slopes. The characteristics of increasing strength of the concrete canvas make it suitable for the application in slope protection.

An experimental study was carried out to investigate the tsunami surge impacts on small balls climbing on different slopes. Dam-break flows were generated in a flume to simulate tsunami surge. The water surface profiles at the sluice gate were observed, the tsunami inundation height and the surge velocity in the flume were measured, and the maximum climbing heights of small balls on different slopes were recorded. Results show that the dam-break speed and the tsunami surge strength increase with increasing reservoir water level. The increasing tsunami inundation height, the decreasing ball density, and the decreasing ball diameter have positive effects on the maximum ball climbing height. Based on the normalized experimental data, equations for estimating the maximum ball climbing heights on different slopes were proposed as functions of the inundation height, the ball diameter, and the ball density. The calculated values from the equation are generally within $\pm 20$% of the measured values in the experimental ranges.

For any $g\ge 3$, we construct genus-$g$ Lefschetz fibrations over the two-sphere whose slopes are arbitrarily close to $2$. The total spaces of the Lefschetz fibrations can be chosen to be minimal and simply connected. It is also shown that the infimum and the supremum of slopes of all Lefschetz fibrations over the two-sphere are not realized as slopes.

The rigid pavements and rigid base supporting rails can be considered as a strip footing resting over an embankment. In addition, there can also be single or double-strip footings resting over an embankment. These strip footings are influenced by the properties of the soil, slope angle of an embankment, setback distance, and spacing between the footings and the height of embankment. In this study, the bearing capacity factors for the single- and double-strip footings resting over an embankment are evaluated using the finite element method (FEM). The bearing capacity factors $N_c^{\prime }$ and $N_{\gamma }^{\prime }$ are evaluated for different soil friction angles $(\varphi )$, slope angles of embankment $(\beta )$, setback distances $(S)$, embankment heights $(D)$ and spacing between the footings $(t)$. Considering the practical aspect, i.e., the surcharge is not present for footing on an embankment, the $N_q^{\prime }$ is not evaluated in the study. Based on the analysis, it is observed that the bearing capacity factors $N_c^{\prime }$ and $N_{\gamma }^{\prime }$ increase with increase in friction angle of the soil and setback distance and reduce with increase in the slope angle of the embankment. The effect of depth of embankment on the $N_c^{\prime }$ and $N_{\gamma }^{\prime }$ is negligible for smaller friction angles $(\varphi \le 20^{\circ })$, whereas it increases marginally for higher friction angles $(\varphi >20^{\circ })$. Furthermore, the $N_c^{\prime }$ and $N_{\gamma }^{\prime }$ are influenced by the spacing between the footings. This study’s results are compared with the results available in the literature. This study’s results are presented as design charts, and these could be adopted in the practice in routine designs of shallow foundations.

Prolonged standing is related to various health problems such as lower back pain and lower extremity discomfort. This study was to investigate the effects of prolonged standing on posture control and whether the sloped surface is beneficial to adults who are required to stand for a long period of time. Twenty young healthy adults (age: $20.5\pm 0.8$ years, height: $165.2\pm 8.5$ cm, weight: $56.6\pm 9.6$ kg, 6 males and 14 females) participated in this study. They were asked to perform a sixty-second quiet-standing evaluation first (i.e. the pre-test condition), then the thirty-minute standing test, and followed by the sixty-second standing test again (i.e. the post-test condition). They stood barefoot quietly on a force plate watching a video on television located 2 m ahead. Three sloped conditions, i.e. the level ground, inclined (with the ankle dorsiflexed), and declined (with the ankle plantarflexed), were randomly examined on separate days. The trajectory, maximal anteroposterior/mediolateral displacement, sway area, and complexity index (CI) of the center of pressure (CoP) during the standing tests were analyzed. Ten-point visual analogue scale (VAS) for perceived fatigue was also recorded. One-way ANOVA and paired t-test were used to analyze postural changes among sloped conditions before and after the prolonged standing.

Signs of fatigue (VAS were $4.1\pm 1.9$, $4.6\pm 1.3$ and $3.5\pm 1.5$ for the level, inclined, and declined conditions, respectively) and significant increases in all CoP measures for the three slope conditions after thirty minutes of standing (all $p<0.05$) were noted. Trajectory was greatest under inclined, followed by the declined and level conditions ($p<0.05$). The CI was generally greater under the declined surface than the level and inclined surfaces along with the thirty-minute standing.

These findings indicated that prolonged standing resulted in fatigue and increased postural changes, particularly on the inclined surface. A greater complexity on the declined surface implied that participants had better adaptability while standing on a declined surface than a flat or inclined surface. Current findings suggested that a declined surface could be a suitable choice for a prolonged standing and further studies are warranted to evaluate its efficacy on different career workers.

The following sections are included:

• INTRODUCTION

• POPULATION PARAMETERS AND THE REGRESSION MODELS
  • Data Description
  • Building the Population Regression Model
  • Sample Versus Population Regression Model

• THE LEAST-SQUARES ESTIMATION OF α AND β
  • Scatter Diagram
  • The Method of Least Squares
  • Estimation of Intercept and Slope

• STANDARD ASSUMPTIONS FOR LINEAR REGRESSION

• THE STANDARD ERROR OF ESTIMATE AND THE COEFFICIENT OF DETERMINATION
  • Variance Decomposition
  • Standard Error of Residuals (Estimate)
  • The Coefficient of Determination

• THE BIVARIATE NORMAL DISTRIBUTION AND CORRELATION ANALYSIS
  • The Sample Correlation Coefficient
  • The Relationship Between r and b
  • The Relationship Between r and R²

• Summary

• Appendix 13A Derivation of Normal Equations and Optimal Portfolio Weights

• Appendix 13B The Derivation of Equation 13.16

• Appendix 13C The Bivariate Normal Density Function
  • Using a Mathematics Aptitude Test to Predict Grade in Statistics

• Appendix 13D American Call Option and the Bivariate Normal CDF
  • Valuating American Option

• Questions and Problems

The quantitative estimation of forest biomass provides important reference information for global carbon storage and carbon cycle research. This study based on the four periods TM remote sensing data of 1970s, 1980s, 1990s and early 21st century and forest resources inventory sample data during the same period, with ArcGIS analyzed the forest biomass spatial distribution characteristics of Lesser Khingan Range, and acquired the forest biomass distribution law with elevation and slope change, the study results are important to maintain the ecological balance of the three northeast provinces of China. Research results shows that: in this study area with the increasing of elevation the forest biomass and distribution have a lesser proportion, the sequence of forest biomass distribution with slope change of each period from big to small is: slope class 1 > slope class 2 > slope class 3 > slope class 4 > slope class 5 = slope class 6.