esa-sceva/satcom-chunk-collection huggingface dataset:
## 4 SDOF Systems with Nonlinear Eddy Current Dampers
### Response to Seismic Excitations
#### 4.3.3. Energy Dissipation Analysis Under Seismic Excitations
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Seismic**} & \multicolumn{3}{c}{**SDOF-FVD**} & \multicolumn{3}{c}{**SDOF-EDD**} \\ \cline{2-6}  & \(\alpha\) & **Normalized Energy** & **ER** & **Normalized Energy** & **ER** \\ \hline \multirow{6}{*}{Electron} & 0.2 & 0.9974 & 383.51 & 0.9967 & 302.27 \\  & 0.4 & 0.9946 & 183.40 & 0.9946 & 185.37 \\  & 0.6 & 0.9892 & 91.83 & 0.9915 & 117.22 \\  & 0.8 & 0.9801 & 49.14 & 0.9870 & 75.77 \\  & 1 & 0.9648 & 27.38 & 0.9808 & 50.98 \\ \hline \multirow{6}{*}{Irpinia} & 0.2 & 0.9955 & 219.02 & 0.9950 & 199.28 \\  & 0.4 & 0.9914 & 114.72 & 0.9920 & 123.24 \\  & 0.6 & 0.9857 & 68.99 & 0.9881 & 83.02 \\  & 0.8 & 0.9771 & 42.62 & 0.9837 & 60.25 \\  & 1 & 0.9649 & 27.50 & 0.9783 & 45.18 \\ \hline \multirow{6}{*}{ChiChi} & 0.2 & 0.9976 & 424.25 & 0.9973 & 372.16 \\  & 0.4 & 0.9947 & 187.15 & 0.9950 & 200.18 \\  & 0.6 & 0.9894 & 92.95 & 0.9919 & 122.12 \\  & 0.8 & 0.9798 & 48.40 & 0.9877 & 80.03 \\  & 1 & 0.9649 & 27.50 & 0.9822 & 55.29 \\ \hline \multirow{6}{*}{Kobe} & 0.2 & 0.9919 & 122.42 & 0.9926 & 133.27 \\  & 0.4 & 0.9892 & 91.89 & 0.9898 & 97.12 \\  & 0.6 & 0.9839 & 60.99 & 0.9868 & 74.52 \\  & 0.8 & 0.9759 & 40.58 & 0.9833 & 58.78 \\  & 1 & 0.9649 & 27.50 & 0.9792 & 47.06 \\ \hline \multirow{6}{*}{Northridge} & 0.2 & 0.9974 & 383.35 & 0.9974 & 381.09 \\  & 0.4 & 0.9943 & 173.06 & 0.9946 & 185.42 \\  & 0.6 & 0.9889 & 89.46 & 0.9912 & 113.20 \\  & 0.8 & 0.9799 & 48.71 & 0.9871 & 76.30 \\  & 1 & 0.9649 & 27.50 & 0.9818 & 54.08 \\ \hline \multirow{6}{*}{Niggat} & 0.2 & 0.9968 & 315.39 & 0.9969 & 322.72 \\  & 0.4 & 0.9944 & 178.90 & 0.9947 & 189.16 \\  & 0.6 & 0.9889 & 89.25 & 0.9917 & 119.90 \\  & 0.8 & 0.9789 & 46.43 & 0.9874 & 78.53 \\  & 1 & 0.9649 & 27.50 & 0.9824 & 55.89 \\ \hline \multirow{6}{*}{Loma Prieta} & 0.2 & 0.9910 & 110.46 & 0.9930 & 140.99 \\  & 0.4 & 0.9895 & 94.66 & 0.9902 & 101.01 \\  & 0.6 & 0.9843 & 62.67 & 0.9871 & 76.57 \\  & 0.8 & 0.9763 & 41.12 & 0.9835 & 59.72 \\  & 1 & 0.9649 & 27.47 & 0.9794 & 47.57 \\ \hline \multirow{6}{*}{Iwate} & 0.2 & 0.9966 & 296.45 & 0.9963 & 267.73 \\  & 0.4 & 0.9942 & 171.17 & 0.9939 & 162.51 \\ \cline{1-1}  & 0.6 & 0.9882 & 83.77 & 0.9909 & 108.47 \\ \cline{1-1}  & 0.8 & 0.9785 & 45.54 & 0.9869 & 75.48 \\ \cline{1-1}  & 1 & 0.9649 & 27.50 & 0.9824 & 55.78 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Energy dissipation ratios (ERs) of SDOF-ECDs and SDOF-FVDs. Figure 12: Energy dissipation ratios under different earthquakes.
## 6 Test and Analysis of Results
The results show that with EBRP-TL, the network experiences the latest FND, occurring after 130 rounds. The number of cluster heads consistently remains around 5% of the total nodes, and the total network energy decreases uniformly with each round. This result is expected because, during clustering, we not only adopted the existing unequal clustering method to reduce the reclustering energy of cluster head nodes closer to the sink but also introduced a timing mechanism. This mechanism determines the order in which nodes become cluster heads by Formula (12). Additionally, the timing mechanism ensures that the nodes prioritized as cluster heads absorb all neighboring nodes into their cluster, preventing the creation of closely spaced clusters, which would result in an excessive number of redundant clusters. During the routing phase, we selected the optimal next-hop cluster head for cluster heads far from the sink based on Formula (18), considering the cluster head's energy, load, and location. This effectively protects low-energy, heavily loaded cluster heads and ensures the forward transmission of data. Figure 13: Total system energy.
## 2 YOLOv8 Reference Network  
YOLOv8, developed by the company Ultralytics, is the eighth-generation model in the YOLO series. It introduces several architectural improvements to enhance performance and flexibility in object detection tasks [24]. The standard YOLOv8 network consists of three main components: the backbone, the neck, and the head networks.  
In YOLOv8, the backbone network replaces the earlier C3 (Cross Stage Partial Network v3) module with the C2f (Cross Stage Partial Network v2 with Fusion) module for feature extraction. The C2f module integrates the bottleneck structure, convolutional layers, and skip connections, improving gradient flow while reducing model complexity. Additionally, the SPPF (Spatial Pyramid Pooling-Fast) module is incorporated for multi-scale feature extraction and fusion. The neck network employs the FPN (Feature Pyramid Network) and PAN (path aggregation network) architectures to enhance feature fusion for multi-scale object detection. The head network adopts a decoupled structure with an anchor-free design, where the classification task uses the BCE (Binary Cross Entropy) loss function and the regression task employs the Distribution Focal Loss and CIOU (complete intersection over union) loss functions.
## 4 The Missing Data Completion Algorithm Based on Residual Value Tensor Decomposition  
### Tensor Model for Traffic Flow Data  
Considering the multi-mode characteristics of traffic flow data, we model traffic flow data as a 3-order tensor \(\chi_{t,d,n}\in R^{T\times D\times N}\) (\(t=1,2,\ldots,T-1,T\), \(d=1,2,\ldots,D-1,D\), \(n=1,2,\ldots,N-1,N\)), where \(T\) is the total number of time slices, \(D\) is the total number of days, \(N\) is the total number of links. For example, if the sample data are collected in two minutes, 720 time slices can be collected in a day, so the value of \(T\) is 720. If the data are collected for 30 days, then the value of \(D\) is 30. If the road network has 132 links, then  
Figure 1: CP decomposition of a 3-order tensor.  
the value of \(N\) is 132. As shown in Figure 2, the potential data distribution contained in adjacent time slices can be obtained by the dimension of \"time slices\". The fluctuation in the data on different days can be observed by the dimension of \"days\", which also includes periodic variation, such as weeks and months. The correlation of data of adjacent links can be obtained by the dimension of \"links\".  
The advantages of tensor-based traffic flow data representation are as following.  
1. When the tensor has structure of multi-dimension, it is intuitively easy to represent the multi-mode characteristics of traffic flow data;
2. The tensor can effectively preserve the structural features of the original traffic flow data;
3. The tensor can effectively solve the problems of dimension disasters and matrix singularity.  
In the collection of traffic flow data, missing data is inevitable due to equipment failure, bad weather, and transmitting interference. There are missing data and non-missing data in the original traffic flow tensor. The missing data is denoted as \(\widetilde{\chi}_{t,d,n}\) while the non-missing data is denoted as \(\widetilde{\chi}_{t,d,n}\). Therefore, \(\chi_{t,d,n}=\widetilde{\chi}_{t,d,n}\cup\widetilde{\chi}_{t,d,n}\), \(\widetilde{\chi}_{t,d,n}\cap\widetilde{\chi}_{t,d,n}=\varnothing\). The problem of missing data restoration is how to recover the missing data by using the non-missing data.  
Set the non-negative weight tensor \(\omega\) has the same size with the original traffic flow tensor, that means \(\omega\in R^{I_{1}\times I_{2}\times I_{3}}\), \(I_{1}=T,I_{2}=D,I_{3}=N\). The corresponding element of tensor \(\omega\) is as follows:  
\[\omega_{i_{1}i_{2}i_{3}}=\left\{\begin{array}{l}0\text{ {if }}\chi_{i_{1}i_{2}i_{3}}\text{ is missing}\\ 1\text{ {if }}\chi_{i_{1}i_{2}i_{3}}\text{ is non-missing}\end{array}\right. \tag{8}\]  
where \(i_{1}=1,2,...,T-1,T,i_{2}=1,2,...,D-1,D,i_{3}=1,2,...,N-1,N\). Let \(\chi_{t,d,n}=\chi_{t,d,n}*\omega\), then, the missing data is set to 0, and the non-missing data remains unchanged.
## 2 Model and Methodology
### Grids Generation and Independence Analysis  
This research utilizes the algorithm of an unstructured tetrahedral grid generation method. For the boundary layer flow pattern, pentahedral prism grids are employed. The thickness of the first boundary layer grid near the communication port is set at 0.05 mm, approximately 0.17% of the diameter of the communication port, and it consists of four layers in total.  
Figure 4: Experimental and simulation result comparison [17].  
The obtained global and local mesh division results are presented in Figure 5. The total number of elements is 7,863,406, and the number of nodes is 2,318,100. The element quality exceeds 0.318, with an average value of 0.846, while the element skewness is less than 0.74865, with an average value of 0.20196.  
To ensure gird independence, a study is conducted with two sets of grids as an example. In the denser mesh, the grid size is reduced by 20%, and the first boundary layer grid near the communication port is reduced to 0.04 mm. This results in an increase in the total number of elements to 15,903,075, which is approximately twice the number in the normal mesh. Both the normal and dense meshes produce nearly identical solutions of pressure difference \(\Delta P\) across the communication port and the mass flow rate \(\dot{m}\) in it, as illustrated in Figure 6.  
The linear resistance obtained through simulation on the normal mesh is 0.008607 MPa\(\cdot\)s/kg, while on the dense mesh, it is 0.008636 MPa\(\cdot\)s/kg, with a deviation of less than 0.34%. Therefore, it can be concluded that this research meets the gird independence requirements.  
Figure 5: Mesh result for simulation research.  
Figure 6: Simulation results from two different grids.
# Dynamic Deflection Measurement on Stiff Bridges with High Piers by Preloaded Spring Method
###### Abstract  
The deflection dynamic load allowance (DLA) of stiff bridges with high piers requires sub-millimeter accuracy. New technologies such as the vision-based optical method and GNSS are not yet recognized for use in DLA measurements due to their smaller SNR. Presently, the scaffolding method is widely utilized for dynamic deflection measurements in dynamic load tests owing to the reliability of employing rigid contact. When scaffolding is not available, engineers have to resort to a suspension hammer system. However, the mass eccentricity of the hammer, stretched-wire length, and wind will decrease the measurement accuracy. To overcome these drawbacks of the suspension hammer method (SHM), a preloaded spring method (PSM) and the related stretched-wire-spring system (SWSS) are proposed in this paper. The dynamic deflection of the coupled vehicle-bridge-SWSS was obtained by vehicle-bridge interaction (VBI) analysis. The sensitivity parameters of the PSM were analyzed and optimized to minimize the measurement error. Indoor experiments and field dynamic load tests were conducted to validate the feasibility and accuracy of the PSM. Additionally, the differences in dynamic deflection measurements between the PSM and SHM in windy environments were compared. The results show that, in a windless environment, the DLAs of the PSM are affected by the spring stiffness, stretched-wire length, and stretched-wire section stiffness, independently of the preload force. When the wind speed is less than or equal to \(8\) m/s and the pier height is less than \(30\) m, the maximum deflection measurement error of the PSM is \(-2.53\%\), while that of the SHM is \(-15.87\%\). Due to its low cost and high accuracy, the proposed method has broad application prospects in the dynamic deflection measurement of stiff bridges with high piers.  
stiff bridge; dynamic deflection; preloaded spring method; contact measurement; suspension hammer systems; dynamic load allowance +
Footnote †: journal: Applied Sciences
## 5 Future Challenges and Scope
### Terahertz and Optical Frequencies  
Terahertz and optical frequency bands ranging from 100 GHz to 3 THz have potential for next generation wireless communication with a vast scope of applications due to the lack of use and unexplored spectrum. For instance, a circularly polarized graphene-based transmitarray for 6 THz is explored in [38]. Another sub-THz anisotropic fixed-configuration three-layer transmitarray is investigated in [114]. A fixed beam wideband high efficient transmitarray for D-band is studied in [115]. The implementation of reconfigurable THz and optical frequencies has many challenges ahead. The working principles of THz and above frequencies are similar to millimeter and microwave frequencies to design a transmitarray. Reflectarrays and transmitarrays provide the concept of low insertion losses and simple electronic beam control solutions. The most prominent challenge is the fabrication of higher frequency switchable and other devices. Mostly, conductors, dielectric materials, and polymers are lossy and economical to manufacture for higher frequencies.  
Beam control terahertz and optical frequency transmitarrays become a potential candidate for many applications of the terahertz spectrum, such as wireless cognition, sensing, imaging, etc. An effort ha been put into terahertz MEMS switches for a reconfigurable reflectarray at 120 GHz frequency [116]. However, MEMS switches are electrically large elements that can easily be loaded with the array. In particular, reconfigurable materials-based technologies should be considered. Some research articles have investigated liquid crystals, particularly graphene-based cells topology. Therefore, beam scanning reconfigurable transmitarrays realization at terahertz and optical frequencies represents a fascinating frontier in which technological challenges are likely to play a far more significant role than in lower frequency applications.
## 4 Results  
### Full-Factor Index Calculation Results  
#### 4.1.1 Characteristics of Input-Output Indicators for Each Station Domain  
The differences in the various indicators among the 17 stations are significant (Figure 4), reflecting the variations in land use, transportation infrastructure, commercial development, and population distribution. For the input indicators, X1 and X2 are fairly evenly distributed across stations, although a few stations show clear advantages. X3 and X7 are relatively low, indicating that some stations have less public space and lower transportation integration. X6 and X8 are higher at most stations, reflecting a strong commercial presence and population clustering.  
In terms of output indicators, Y1 and Y3 show considerable differences, with some stations offering higher ecological value. Y2 and Y4 reveal differences in transportation accessibility and coverage. Y5 is relatively high, indicating that parking resources are well-utilized at most stations. The differences in Y6 and Y7 further highlight the disparities in economic vitality and commercial value between stations. Overall, each station has distinct economic, ecological, and transportation characteristics, providing valuable data for urban planning and resource allocation.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{2}{c}{**Variable**} & \multirow{2}{*}{**Min**} & \multirow{2}{*}{**Max**} & \multirow{2}{*}{**Mean**} & \multirow{2}{*}{**SD**} \\ \hline \multirow{3}{*}{Input Variables} & **Primary Indicators** & **Secondary Indicators** & & & & \\ \cline{2-7}  & Land Resources & X1, total land area (m\({}^{2}\)) & 0.14 & 0.25 & 0.19 & 0.1 \\ \cline{2-7}  & Financial Resources & X6, number of commercial stores & 2198 & 11,656 & 6369.65 & 6,107,097 \\ \hline \multirow{3}{*}{Output Variables} & \multirow{3}{*}{Land Efficiency Output} & Y1, land diversity & 0.91 & 1.95 & 1.54 & 0.1 \\ \cline{2-7}  & & Y2, subway passenger flow & & & & \\ \cline{1-1} \cline{2-7}  & & (1000 people/day) & 1.65 & 49.87 & 22.23 & 174.99 \\ \cline{1-1} \cline{2-7}  & Spatial Efficiency Output & Y7, rent level (CNY/m\({}^{2}\)/month) & 32.19 & 50.16 & 40.62 & 21.85 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Reduced dimension variables and descriptive statistics.  
Figure 4: Histograms of input and output indicators.
# Bi-Directional Point Flow Estimation with Multi-Scale Attention for Deformable Lung CT Registration
## 4 Experiments
### Ablation Studies
#### 4.5.4 The Number of Attention Heads  
We conducted an ablation study on the number of attention heads \(N_{h}\) used in both the geometric keypoint and flow attention modules. As shown in Table 6, increasing the number of heads improved the alignment up to a point, but diminishing returns were observed beyond \(N_{h}=4\). We found this achieved the best balance between accuracy and stability, producing the lowest TRE and hthe highest DSC, and thus we used \(N_{h}=4\) in our final model.  
\begin{table}
\begin{tabular}{c c c c|c c c c|c} \hline \hline \multirow{2}{*}{**Ref.**} & \multirow{2}{*}{**\begin{tabular}{c} **\(\mathbf{K}\)**epoint** \\ **Attn.** \\ \end{tabular}} & \multirow{2}{*}{**\begin{tabular}{c} **Flow** \\ **Attn.** \\ \end{tabular}} & \multirow{2}{*}{**\begin{tabular}{c} **Bi-Dir.** \\ **Refine** \\ \end{tabular}} & \multirow{2}{*}{**\begin{tabular}{c} **TRE** \(\downarrow\) \\ \end{tabular}} & \multirow{2}{*}{**\begin{tabular}{c} **25\%** \\ \end{tabular}} & \multirow{2}{*}{**\
\begin{tabular}{c} **50\%** \\ \end{tabular}} & \multirow{2}{*}{**
\begin{tabular}{c} **75\%** \\ \end{tabular}} & \multirow{2}{*}{**DSC \(\uparrow\)**} \\ \hline (a) & & & & 2.85 & 1.52 & 2.33 & 3.54 & 89.9 \\ (b) & ✓ & & & 2.83 & 1.54 & 2.28 & 3.32 & 89.9 \\ (c) & & ✓ & & 2.67 & 1.38 & 2.27 & 3.40 & 90.1 \\ (d) & ✓ & ✓ & & 2.54 & 1.29 & 2.10 & 3.12 & 90.2 \\ Ours & ✓ & ✓ & ✓ & 2.24 & 1.16 & 1.79 & 2.77 & 91.4 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Ablation study on model components.  
\begin{table}
\begin{tabular}{c c c|c c} \hline \hline \multirow{2}{*}{**Ref.**} & \multirow{2}{*}{**\begin{tabular}{c} **\(\mathbf{K}\)**epoint** \\ **Attn.** \\ \end{tabular}} & \multirow{2}{*}{**\begin{tabular}{c} **Flow** \\ **Attn.** \\ \end{tabular}} & \multirow{2}{*}{**\begin{tabular}{c} **Bi-Dir.** \\ **Refine** \\ \end{tabular}} & \multirow{2}{*}{**\begin{tabular}{c} **TRE** \(\downarrow\) \\ \end{tabular}} & \multirow{2}{*}{**\begin{tabular}{c} **25\%** \\ \end{tabular}} & \multirow{2}{*}{**\
\begin{tabular}{c} **50\%** \\ \end{tabular}} & \multirow{2}{*}{**
\begin{tabular}{c} **75\%** \\ \end{tabular}} & \multirow{2}{*}{**DSC \(\uparrow\)**} \\ \hline (a) & & & & 2.85 & 1.52 & 2.33 & 3.54 & 89.9 \\ (b) & ✓ & & & 2.83 & 1.54 & 2.28 & 3.32 & 89.9 \\ (c) & & ✓ & & 2.67 & 1.38 & 2.27 & 3.40 & 90.1 \\ (d) & ✓ & ✓ & & 2.54 & 1.29 & 2.10 & 3.12 & 90.2 \\ Ours & ✓ & ✓ & ✓ & 2.24 & 1.16 & 1.79 & 2.77 & 91.4 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Ablation study on loss function.  
Overall, these ablation studies demonstrate that each proposed component--ranging from the attention modules and bi-directional training objectives to the key hyperparameter configurations--contributes meaningfully to the improved accuracy, stability, and robustness of our method.
# Estimation of Skid-Steered Wheeled Vehicle States Using STUKF with Adaptive Noise Adjustment
## 3 System Modeling
### Motion Analysis  
To model a skid-steered vehicle maneuvering on the ground, two coordinate frames are introduced, as shown in Figure 1. The XOY-coordinate is the geodetic coordinate system, and the XOY-coordinate is the vehicle reference coordinate whose origin is located on the geometric center. In Figure 1, ICR\({}_{\rm v}\) denotes the vehicle instantaneous rotation center, ICR\({}_{\rm i}\) and ICR\({}_{\rm r}\) denote the instantaneous rotation center of tire slip velocity on the left and right sides, respectively.  
In the vehicle reference coordinate, the velocity and yaw rate are expressed as:  
\[\left\{\begin{array}{l}\dot{x}=\frac{\mathbb{R}}{2}[\omega_{l}(1-i_{l})+ \omega_{r}(1-i_{r})]\\ \dot{y}=\frac{\mathbb{R}}{2}[\omega_{l}(1-i_{l})+\omega_{r}(1-i_{r})]\tan \beta=\dot{x}\tan\beta\\ \dot{\phi}=\frac{\mathbb{R}[-\omega_{l}(1-i_{l})+\omega_{l}(1-i_{r})]}{B}\\ \end{array}\right. \tag{1}\]  
where \(\dot{x}\) and \(\dot{y}\) denote longitudinal and lateral velocity on the vehicle coordinate, respectively; \(\omega_{l}\) and \(\omega_{r}\) denote right and left side rotating velocity, respectively; \(B\) denotes the width of the vehicle; R is the wheel radius; \(\beta\) is the sideslip angle of the vehicle body; \(i_{l}\) and \(i_{r}\) denote the slip rate of the wheels on each side, respectively, which is expressed as:  
\[i_{i}=\frac{Rw_{i}{-}v_{xi}}{Rw_{i}} \tag{2}\]  
On the basis of the assumption (4), the deformation of suspension is ignored. Thus, the same side wheels are at the same longitudinal velocity. The longitudinal velocity of each side is expressed as:  
\[u_{x,ij}=\left\{\begin{array}{l}\dot{x}-\frac{\mathbb{R}}{2}\dot{\phi}\quad j =l\\ \dot{x}+\frac{\mathbb{R}}{2}\dot{\phi}\quad j=r\\ \end{array}\right. \tag{3}\]  
Figure 1: The ICRs of the vehicle body and the wheels on each side when a skid-steered vehicle turns.  
Then, the longitudinal slip rate is expressed as:  
\[\left\{\begin{array}{l}i_{l}=\frac{R\omega_{l-}\left(\dot{x}-\frac{B}{2}\dot{ \phi}\right)}{R\omega_{l}}\\ i_{r}=\frac{R\omega_{r-}\left(\dot{x}+\frac{B}{2}\dot{\phi}\right)}{R\omega_{r} }\end{array}\right. \tag{4}\]  
It is easy to learn that the tires on the same axis are on the same lateral velocity as follows:  
\[u_{y,ij}=\left\{\begin{array}{ll}\dot{y}+a\dot{\phi}&i=f\\ \dot{y}+b\dot{\phi}&i=m\\ \dot{y}+c\dot{\phi}&i=r\end{array}\right. \tag{5}\]  
where \(a\), \(b\), and \(c\) denote the distance from each axis to the CM. Then, the tire sideslip angle on each wheel is expressed as:  
\[\alpha_{ij}=\frac{u_{y,i}}{u_{x,ij}}=\arctan\left(\frac{\dot{y}+\alpha_{1}\dot {\phi}}{\dot{x}+\alpha_{2}\frac{B}{2}\dot{\phi}}\right)=\arctan\left(\frac{ \tan\beta+\alpha_{1}\frac{\dot{\phi}}{x}}{1+\alpha_{2}\frac{B}{2}\frac{\dot{ \phi}}{x}}\right) \tag{6}\]  
where the coefficient \(\alpha_{1}\) and \(\alpha_{2}\) are defined as:  
\[\alpha_{1}=\left\{\begin{array}{ll}a&i=f\\ b&i=m\\ c&i=r\end{array}\right.,\;\alpha_{2}=\left\{\begin{array}{ll}-1&j=l\\ 1&j=r\end{array}\right. \tag{7}\]