esa-sceva/satcom-chunk-collection huggingface dataset:
# WiFi FTM, UWB and Cellular-Based Radio Fusion for Indoor Positioning
## 2 Overview of Location Technologies
### Ultra-Wide Band  
UWB technology provides a high ranging accuracy based on the RTT protocol, even in environments with harsh propagation characteristics [18]. This technology has multiple  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Technology**} & **Access** & **Positioning** & **Positioning** & \multirow{2}{*}{**Advantages**} & **Disadvantages** \\  & **Point** & **Accuracy** & **Method** & & \\ \hline \multirow{3}{*}{Cellular network} & \multirow{3}{*}{Cellular tower} & \multirow{3}{*}{\(>\)30 m} & & \multirow{3}{*}{Trilateration} & World-wide & \multirow{3}{*}{Low-precision} \\  & & & & coverage; & No extra & \textgreater{}100 m \\  & & & & infrastructure needed & \\ \hline \multirow{3}{*}{UWB} & \multirow{3}{*}{UWB anchor} & \multirow{3}{*}{cm-m} & \multirow{3}{*}{Trilateration} & Robust against & \multirow{3}{*}{High-cost} \\  & & & multipath; & & high-accuracy; & \\ \cline{1-1}  & & & & easy-deployment & \\ \hline WiFi-FTM & Router & cm-m & Trilateration & Low cost; & Not yet \\  & & & high-accuracy & widely deployed \\ \hline Bluetooth & Beacon & m & Trilateration; & Low cost; & \multirow{2}{*}{Low-stability} \\  & & & fingerprinting & easy-deployment & \\ \hline \multirow{3}{*}{INS} & \multirow{3}{*}{N/A} & \multirow{3}{*}{m} & \multirow{3}{*}{PDR} & \multirow{3}{*}{Self-sufficient} & Accumulative error; \\  & & & & & Smartphone-based \\ \cline{1-1}  & & & & & calculation \\ \hline \multirow{3}{*}{Geomagnetism} & \multirow{3}{*}{N/A} & \multirow{3}{*}{m} & \multirow{3}{*}{Fingerprinting} & No infrastructure; & Need data \\ \cline{1-1}  & & & low-cost; ubiquitous & & collection; \\ \cline{1-1}  & & & & low-cost; ubiquitous & affected by temporal \\ \cline{1-1}  & & & & electrical equipment; \\ \hline \hline \end{tabular}
\end{table}
Table 2: Overview of indoor positioning technology.  
advantages, such as centimetre-level ranging precision, good obstacle-penetration capabilities [4], and multipath mitigation in dense scenarios [2], making it indispensable for indoor positioning. UWB is also a wireless communication technology that supports a high throughput owing to the use of a very large spectrum. UWB uses very short time pulses of few nanoseconds that take a wide bandwidth. The Federal Communication Commission (FCC) authorised the unlicensed use of UWB in the range of 3.1 to 10.6 GHz [3]. UWB signals are centered at 3.5 GHz with a bandwidth higher than 500 MHz. The latest market trends show that UWB will soon become a de-facto standard for positioning and will eventually be addressed by 3 GPP standards [7]. Accordingly, some smartphones have integrated UWB chipsets in the recent years [19]. As a drawback, to achieve the short pulse width the UWB device has a high energy consumption.
## 4 Our DRL-JCBRTP Framework
### DRL-JCBRTP NN Architecture Design Based on SAC Algorithm
**Critic NN:** The critic network is updated by minimizing the mean square error between the expected target Q value \(r(s_{t},a_{t})+\gamma\,\mathbb{E}_{(s_{t+1})\sim p}[V_{\theta}(s_{t+1})]\) and the current Q-value \(Q_{\theta}(s_{t},a_{t})\) predicted by the critic network, i.e.,  
\[J_{Q}(\theta)=\mathbf{E}_{(s_{t},a_{t})\sim\mathcal{D}}\Bigg{[}\frac{1}{2} \bigg{(}Q_{\theta}(s_{t},a_{t})-\Big{(}r(s_{t},a_{t})+\gamma\mathbf{E}_{(s_{t+ 1})\sim p}[V_{\theta}(s_{t+1})]\Big{)}\bigg{)}^{2}\Bigg{]}, \tag{2}\]  
where the expectation is taken over a minibatch of state-action pairs (\(s_{t},a_{t}\)) sampled from the replay buffer \(\mathcal{D},r(s_{t},a_{t})\) is the immediate reward received after taking action \(a_{t}\) in state \(s_{t}\), \(\gamma\) is the discount factor, and the nested expectation is on the next states \(s_{t+1}\) sampled from the environment's dynamics \(p\). The critic NN in our DRL-JCBRTP employs the MLP to approximate the value function due to its capability to model complex, non-linear relationships between states and actions. The MLP enables the critic NN to accurately approximate the action-value function \(Q_{\theta}(s_{t},a_{t})\), which is essential for evaluating the expected return of different state-action pairs. By leveraging the MLP in the critic NN, the SAC algorithm can accurately assess the value of actions proposed by the actor NN, thereby facilitating the learning of robust policies that optimize long-term rewards in V2X communication scenarios.
## 3 Methodology of the Resampling Strategy and Variable Circular Correlation Time
### Performance Evaluation of Signal Acquisition  
For signal acquisition, time cost is usually a stochastic variable. Mean value of time cost is concerned as one of the most important measurements to assess the performance of acquisition algorithms. Computation complexity is another crucial evaluation index. Usually, convolution operation in the time domain is converted into multiplication in the frequency domain by FFT to improve the efficiency of signal acquisition algorithms. We choose amounts of multiplication and summation operations to evaluate the computational complexity of signal acquisition.  
Equation (25) presents computations of FFT (\(O_{FFT}\)), IFFT (\(O_{IFFT}\)), amplitude calculation (\(O_{AMP}\)), vector multiplication (\(O_{VM}\)) and comparison (\(O_{CMP}\)). \(N\) refers to the length of the FFT sequence. O\({}^{M}\) and O\({}^{A}\) denote the amounts of multiplication and summation operations, respectively.  
\[\left\{\begin{array}{c}O_{FFT}(N)=O_{IFFT}(N)=O^{M}\Big{(}\frac{N}{2}log_{2} \Big{(}\frac{N}{2}\Big{)}\Big{)}+O^{A}(Nlog_{2}N)\\ O_{AMP}(N)=O_{\text{ VM}}(N)=O^{M}(N)+O^{A}(N-1)\\ O_{CMP}(N)=O^{A}(N-1)\end{array}\right. \tag{25}\]  
Note the computations of local code replica generation and correlation ratio calculation as \(O_{lc}\) and \(O_{pk}\). And the computations of coarse and fine acquisition are denoted as \(O_{ca}\) and \(O_{fa}\), respectively. \(N_{SV}\) is the number of satellites in the search list, and \(N_{f}\) is the amount of carrier frequency bins.  
\[O_{lc}=O_{FFT}(L_{r})\]  
\[O_{ca}=2O_{VM}(L_{r})+O_{FFT}(L_{r})+O_{VM}(L_{r})+O_{IFFT}(L_{r})+O_{AMP}(L_{ r})\]  
\[O_{pk}=O_{CMP}\Big{(}N_{f}\times L_{r}\Big{)}+O_{CMP}\Big{(}N_{f}\times L_{x} \Big{)}+O^{M}\Big{(}N_{f}\times L_{r}\Big{)}\]  
\[O_{fa}=O_{VM}(L_{x})+O_{FFT}(L_{x})+O_{AMP}(L_{x})+O_{CMP}(L_{x})\]  
Therefore, the computational complexity of signal acquisition \(O_{total}\) is formulated as  
\[O_{total}=\Big{(}O_{lc}+N_{f}O_{ca}+O_{pk}+O_{fa}\Big{)}N_{SV}=O_{total}^{M}~ {}+~{}O_{total}^{A} \tag{26}\]  
in which,  
\[O_{total}^{M}=\bigg{[}\bigg{(}2N_{f}+\frac{3}{2}\bigg{)}L_{x}\log_{2}L_{x}+10N_ {f}L_{x}+\frac{3}{2}L_{x}\bigg{]}N_{SV} \tag{27}\]  
\[O_{total}^{A}=\Big{[}\Big{(}4N_{f}+3\Big{)}L_{x}\log_{2}L_{x}+15N_{f}L_{x}+5L_{ x}-4N_{f}-5\Big{]}N_{SV} \tag{28}\]
# Offline Payment of Central Bank Digital Currency Based on a Trusted Platform Module
## 5 Conclusions  
The potential of central bank digital currencies can be enhanced by increasing convenience while preserving the essential functions of cash. However, CBDCs, which share characteristics with electronic payment systems, are anticipated to face significant limitations in facilitating offline transactions. Specifically, these currencies require a constant internet connection to prevent double spending and to authenticate transactions, similarly to existing electronic payment methods and token-based blockchain systems. To address these challenges, this study has presented a framework for secure offline payments that does not rely on online ledger verification. This was achieved by storing individual token information within a TPM and employing a technique to ensure the security of each transaction. Nevertheless, the nature of peer-to-peer (P2P) payments allows for frequent fund transfers in offline environments, akin to cash transactions. Furthermore, independent terminal environments are perpetually vulnerable to security attacks, which represent a critical concern for digital currencies. A successful attack in an offline context could potentially result in the generation of unlimited funds, necessitating a cautious approach from monetary policy authorities. Additionally, the proposed model utilizes the internal counter provided by the TPM to mitigate security threats such as double spending; this counter is immutable and always monotonic. While TPMs do possess certain vulnerabilities, they are employed across various domains, including secure booting in Windows. If these vulnerabilities can be addressed, TPMs could prove to be highly beneficial for facilitating offline payments.  
This study focuses on the payment process associated with offline tokens, as opposed to the online-to-offline conversion of tokens. It has analyzed the security of offline transactions that utilize short-range wireless communication technologies, such as Bluetooth and NFC. Each payment message must be transmitted and received through a securely established cryptographic channel between offline wallets; however, the establishment of this cryptographic channel is not the primary focus of this research. Furthermore, the reliability of the wireless connection may be compromised due to low bandwidth and signal interference, indicating a need for further research aimed at enhancing the reliability of short-range wireless communication to facilitate the adoption of the CBDC.  
Conceptualization, J.Y.; methodology, Y.K.; formal analysis, J.Y.; supervision, Y.K.; Writing--Original Draft Preparation, J.Y.; Writing--Review & Editing, J.Y. and Y.K. All authors have read and agreed to the published version of the manuscript. This research received no external funding. The original data presented in the study openly available. The authors declare no conflicts of interest.
## 3 Kriging (Gaussian Process) Model
### Maximum Likelihood Estimation  
For the creation of a Kriging model, the hyperparameters denoted as \(\Theta\) are typically unknown and therefore necessitate estimation. This estimation process involves solving an optimization problem, the specifics of which depend on the chosen estimation method. In the case of the maximum likelihood estimation (MLE) method [31], the objective is to determine the set of Kriging parameters and appropriate variables in a manner that maximizes the likelihood of the observed data \(\boldsymbol{\mathcal{Y}}_{D}=\left\{\Gamma^{\text{\emph{trig}}}(\mathbf{x}^{ (1)}),\Gamma^{\text{\emph{trig}}}(\mathbf{x}^{(2)}),...,\Gamma^{\text{\emph{ trig}}}(\mathbf{x}^{(n)})\right\}^{T}\). In cases where no response noise exists, the likelihood function \(\mathcal{L}\) based on the multivariate normal PDF can be expressed as:  
\[\mathcal{L}(\mathbf{m},\sigma^{2},\Theta;\boldsymbol{\mathcal{Y}}_{D})=\frac{ det(\boldsymbol{\kappa})^{-0.5}}{(2\pi\sigma^{2})^{0.5n}}\cdot\exp\Bigl{(}-0.5( \boldsymbol{\mathcal{Y}}_{D}-\mathbf{Fm})^{T}(\sigma^{2}\boldsymbol{\kappa}) ^{-1}(\boldsymbol{\mathcal{Y}}_{D}-\mathbf{Fm})\Bigr{)} \tag{16}\]  
The hyperparameter \(\Theta\) is determined by solving the optimization problem expressed as:  
\[\max_{\Theta}\Bigl{(}\log\mathcal{L}(\mathbf{m},\sigma^{2},\Theta;\boldsymbol {\mathcal{Y}}_{D})\Bigr{)} \tag{17}\]
# Vibrational Analysis of Building Structures with Irregularities
## 3 Method of Solution
### Results from the Deterministic Model  
The deterministic model has demonstrated strong predictive capabilities for vibrations in the low-frequency range, specifically below 80 Hz, as shown in the previous studies [15; 27; 33]. A comparison between model predictions and experimental measurements of the root mean squared (RMS) velocity is displayed in Figure 2. This example features a 7 m \(\times\) 3.2 m structure with joists running along its length.  
Figure 2: Comparison of computed and measured surface root mean square velocity of the ceiling. Variations in design, from top to bottom, include additional layers on the upper surface. See [33] for a detailed description of these floor/ceiling designs, the experimental configuration, and measurement details.  
Several experimental floor/ceiling mock-ups were analyzed within the FWPRCD project [33], with three examples shown in Figure 2. The model successfully captured the variations across different configurations in the low-frequency range, affirming its adaptability to diverse structural designs. Design modifications tested included additional upper layers, variations in joint arrangements, changes in structural size, and extra joists on ceiling panels.  
However, above 80 Hz, structural vibration responses become increasingly sensitive to finer component details and design specifics, which are often difficult to measure accurately or subject to high variability. The next section explores how incorporating these structural irregularities and joint features can enhance model accuracy.
## 3 XNAV for Spacecraft in Formation
### Pulse Phase Delay Estimation  
In this section, we explain how the pulse phase delay can be abstained. Assume that spacecraft A and spacecraft B observe the same pulsar from \(t_{i}\) to \(t_{i+1}\), \(i=1\), \(2\),...\(I\) and receives \(\left\{\tau_{j}^{A}\right\}_{j=1}^{N_{A}}\) and \(\left\{\tau_{j}^{B}\right\}_{j=1}^{N_{B}}\), respectively. Applying (8) to \(\left\{\tau_{j}^{A}\right\}_{j=1}^{N_{A}}\) and \(\left\{\tau_{j}^{B}\right\}_{j=1}^{N_{B}}\), we have \(\left\{\phi\Big{(}\tau_{j,SSB}^{A}\Big{)}\right\}_{j=1}^{N}\) and \(\left\{\phi\Big{(}\tau_{j,SSB}^{B}\Big{)}\right\}_{j=1}^{N}\), where  
Figure 1: Diagram of XNAV for spacecraft in formation.  
\[\phi\Big{(}\tau_{j,SSB}^{A}\Big{)} =\phi_{i}^{A}+F_{i}^{orbit,A}\big{(}\tau_{j}-t_{i}\big{)}+\widetilde {\phi}\Big{(}\tau_{j,SSB}^{A}\Big{)}, \tag{30}\] \[\phi\Big{(}\tau_{j,SSB}^{B}\Big{)} =\phi_{i}^{B}+F_{i}^{orbit,B}\big{(}\tau_{j}-t_{i}\big{)}+\widetilde {\phi}\Big{(}\tau_{j,SSB}^{B}\Big{)}. \tag{31}\]  
As shown in Section 2.2, \(F_{i}^{orbit,A}\) and \(F_{i}^{orbit,B}\) can be estimated by (17). In Equation (19), \(\phi_{i}^{A}\) and \(\phi_{i}^{B}\) can be estimated by (18), and \(\Delta\phi_{i}^{A,B}\) can be calculated by (24). However, when estimating \(\phi_{i}^{A}\) and \(\phi_{i}^{B}\), respectively, we have to solve the optimization problem shown in (18) twice, and the template is indispensable. To improve computational efficiency and eliminate dependence on the template, we proposed a direct estimation method for \(\Delta\phi_{i}^{A,B}\).  
Similar to (19), we define the cost function for estimating \(\Delta\phi_{i}^{A,B}\) as follows:  
\[J\Big{(}\Delta\phi_{i}^{A,B}\Big{)}=\sum_{l=0}^{M_{\text{p}}}\Big{[}\lambda_{ EF}^{A}(\varphi_{l};\hat{F}_{l}^{orbit,A})-\lambda_{EF}^{B}(\varphi_{l};\hat{F}_{l}^ {orbit,B})\Big{]}^{2}. \tag{32}\]  
Therefore, \(\Delta\phi_{i}^{A,B}\) can be estimated by  
\[\Delta\hat{\phi}_{i}^{A,B}=\underset{\Delta\phi_{i}^{A,B}\in[0,1)}{\text{argmin }}J\Big{(}\Delta\phi_{i}^{A,B}\Big{)}. \tag{33}\]  
As shown in (32) and (33), \(\Delta\phi_{i}^{A,B}\) can be estimated by solving (33) once and is independent from the template of pulsars.
# Action Recognition Based on the Fusion of Graph Convolutional Networks with High Order Features
## 4 Experiments
### Comparison with the State-of-the-Art  
We compare the final model with the state-of-the-art skeleton-based action recognition methods on NTU-RGBD dataset and NTU-RGBD-120 dataset. The results of the comparison are shown in Tables 5 and 6. The methods used for comparison include the handcraft-feature-based methods [33], RNN-based methods [28; 29; 34; 35], CNN-based methods [36; 37], and GCN-based methods [6; 7; 8; 9; 10]. From Table 5, we can see that our proposed method achieves the best performances of 96.8% and 91.7% in terms of two criteria on the NTU-RGBD dataset.  
Since the NTU-RGBD-120 dataset was released in 2019, there are no related works on this dataset yet. Therefore, we only cite the result of relevant methods mentioned in the original paper of this dataset. As shown in the Table 6, our method is significantly better than the others.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Methods** & **Accuracy (\%) (No Augmentation)** \\ \hline Joint & 84.3 \\ Bone & 84.5 \\ Joint-Velocity & 81.4 \\ Bone-Velocity & 82.3 \\ Relative-Distance & 84.5 \\ \hline J+B+JV+BV+RD & **89.2** \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparisons of the validation accuracy with different input modalities on cross-setup benchmark of NTU-RGBD-120 dataset.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Methods** & **Cross-Subject (\%)** & **Cross-View (\%)** \\ \hline Lie Group(2014) [33] & 50.1 & 82.8 \\ \hline Trust Gate ST-LSTM(2016) [29] & 69.2 & 77.7 \\ Two-stream RNN(2017) [34] & 71.3 & 79.5 \\ STA-LSTM(2017) [28] & 73.4 & 81.2 \\ VA-LSTM(2017) [35] & 79.4 & 87.6 \\ \hline SR-TSL(2018) [37] & 84.8 & 92.4 \\ HCN(2018) [36] & 86.5 & 91.1 \\ \hline ST-GCN (2018) [6] & 81.5 & 88.3 \\ AS-GCN(2018) [9] & 86.8 & 94.2 \\ PB-GCN (2018) [8] & 87.5 & 93.2 \\
2s-AGCN(2019) [7] & 88.5 & 95.1 \\ AGC-LSTM(2019) [10] & 89.2 & 95.0 \\ \hline
**ours** & **91.7** & **96.8** \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparisons of the validation accuracy with state-of-the-art methods on the NTU-RGBD dataset.
## 5 Findings of the Study
### Intrinsic Evaluation
#### 5.2.2. Cosine Similarity  
As an alternative to human judgement, cosine similarity has been applied by [PERSON] et al. [58] to measure the similarity of topics modeled by LDA variants especially in the contexts of psychology. This type of metric-based validation enables the researchers to seek the ground truth and reliability of topics generated by the automatic models [58]. Therefore, we used cosine similarity as a metric to infer the intra-similarities of topics modeled by SLDA. As presented in Table 5, the cosine intra-similarity for _myPersonality_ was 0.832 and 0.827 for multiclass and one vs. all distribution, respectively, whereas for _Sentiment140_ it was 0.772 (multiclass) and 0.764 (one vs. all). The similarities indicated that topics in _myPersonality_ were more homogeneous compared to _Sentiment140_ where the topics or genres of discussion were naturally more diverse or overlapping.
# Urban Free-Space Optical Network Optimization
## 2 Problems Statement
### Network with Maximum Number of Terminal Points  
The problem of finding a network with the maximum number of endpoints serves as the baseline problem in this paper. It is motivated by the property of endpoints being less expensive in \"real-world\" FSO scenarios. This problem is known in the literature as the _max-leaf_ problem.  
Figure 1: Visibility graph simulation.  
**Definition 1**.: _For a graph \(G_{v}=(V,E)\), find a spanning tree with maximum leaves, where a leaf is a vertex with only one adjacent vertex. Figure 2 demonstrates the max-leaf problem on a small graph._  
The max-leaf problem is known to be NP-hard [30]. There is a polynomial-time algorithm for a graph with maximum degree 3 [31]. In our case, the visibility graph is often dense, and the maximum degree is not bounded by 3. Furthermore, the general max-leaf problem has no polynomial-time approximation scheme, meaning no algorithm can give an infinitely close solution for this problem, assuming \(P\
eq NP\). However, there exists a 2-factor approximation in polynomial time for this problem. In this scenario, a solution that contains up to two times the amount of nodes than necessary is not reliable. There is also a solution with reasonable run time for this problem in small graphs of up to 50 nodes [32]. Moreover, such an assumption is inapplicable in many urban visibility graphs.
# Harmonizing Multi-Source Remote Sensing Images for Summer Corn Growth Monitoring
## 2 Materials and Methods
### Data Sources and Preprocessing
These automated weather stations were installed for long-term climate observation and provided data on daily maximum, minimum, and average temperatures, as well as daily precipitation. As can be seen from Figure 1, meteorological datasets are single-point datasets, and all of the stations lie outside the study area, so the data need to be spatially processed. In this study, we used the Kriging method to produce a dataset with a spatial resolution of 30 m by 30 m from the agrometeorological station data. Then, the average value of the data in the study area was calculated, including daily maximum, minimum, average temperatures, and daily precipitation. The LAI measurements were carried out on 10 samples of summer corn in the study area in the DOY 205 and DOY 224, which corresponded to the flare opening stage and the tasseling and anthesis silking of summer corn, respectively.
# Resource Mapping Allocation Scheme in 6G Satellite Twin Network
## 3 System Model of 6G Satellite Twin Network
### Correlation between Domains within the Network Simulation Space  
According to the parameters of the space and domain, each domain in the simulation space is functionally independent of the other. The independence coefficients of the space D can be expressed as  
\[\begin{array}{rl}{Acvs(D)=\left\{\begin{matrix}{Ccvs(G^{COM},G^{S},G^{C}), \alpha(G^{COM})+\alpha(G^{S})+\alpha(G^{C})>1},\\ {SIM(G^{COM},G^{S},G^{C}),\textit{ others}}\end{matrix}\right.},\end{array} \tag{5}\]  
where the \(Ccvs(G^{COM},G^{S},G^{C})\) can be expressed separately as  
\[\begin{array}{rl}{Ccvs(G^{COM},G^{S},G^{C})}&=\sum\limits_{j=1}^{l}{\alpha( G^{COM})\times\log(|r_{j}^{s}|)/\log\biggl{\{}-\frac{\max\{R^{S}\}}{\sum_{p=1}^{p} \sum_{q=1}^{Q}|S_{pq}|}\biggr{\}}}\\ &=\sum\limits_{j=1}^{l}{0.5T|p_{d}v_{1}(R^{COM})|}\\ &\times\log(\sqrt{(\mathrm{cap}f_{j}^{S})^{2}+(\mathrm{mem}f_{j}^{S})^{2}+( \mathrm{bar}f_{j}^{S})^{2})}\\ &/\log\biggl{\{}-\frac{\max\{R^{S}\}}{\sum_{p=1}^{p}\sum_{q=1}^{Q}\sqrt{( \mathrm{cap}S_{pq})^{2}+(\mathrm{mem}S_{pq})^{2}+(\mathrm{bm}S_{pq})^{2}}} \biggr{\}},\end{array} \tag{6}\]  
The correlation coefficient between domains is related to the extension factor \(\alpha\) of the domain and the resource R of the domain. The correlation coefficient of the domain is obtained based on the CPU cap\(f_{j}\), memory mem\(f_{j}\), bandwidth ban\(f_{j}\), and the extension factor \(\alpha\). The SIM\((G^{COM},G^{S},G^{C})\) can be expressed as  
\[\begin{array}{rl}{\mathrm{SIM}(G^{COM},G^{S},G^{C})=\frac{V_{G^{COM}/G^{S}}} {V_{G^{S}/G^{C}}}=\frac{\int_{p_{min}^{max}\alpha(G^{COM})\cdot\alpha(G^{S}) \cdot v_{G^{COM}/G^{S}}(\mathrm{R})dp}}{\int_{p_{min}^{max}\alpha(G^{S})\cdot \alpha(G^{C})\cdot v_{G^{S}/G^{C}}(\mathrm{R})dp}},\end{array} \tag{7}\]  
where \(v_{G^{COM}/G^{S}}(\mathrm{R})=max\bigl{\{}R^{COM}-sum(R^{COM}/R^{S}),R^{S}- sum(R^{S}/R^{COM})\bigr{\}}\) and \(v_{G^{S}/G^{C}}(\mathrm{R})=max\bigl{\{}R^{S}-sum(R^{S}/R^{C}),R^{C}-sum(R^{C}/ R^{S})\bigr{\}}\). We calculate the maximum value of the residual of the ratio of resources between domains and the resources in the domain to obtain the relationship of domains in one space.