esa-sceva/satcom-chunk-collection huggingface dataset:
# Secure Control of Linear Controllers Using Fully Homomorphic Encryption
## 3 Preliminaries
### Encryppted Controller  
The discrete-time linear controller case that is under consideration in this work is summarized in the following form:  
\[f:\begin{cases}\mathbf{x}(t+1)=\mathbf{A}\mathbf{x}(t)+\mathbf{B}\mathbf{y}(t)\\ \mathbf{u}(t)=\mathbf{C}\mathbf{x}(t)+\mathbf{D}\mathbf{y}(t),\end{cases} \tag{1}\]  
where \(\mathbf{y}(t)\in\mathbb{R}^{m}\) is a controller input (or a plant output), \(\mathbf{u}(t)\in\mathbb{R}^{l}\) is a controller output (or a plant input), and \(t\) is a step. \(\mathbf{A},\mathbf{B},\mathbf{C}\), and \(\mathbf{D}\) are controller parameter values. The following is an equivalent rewriting of Equation (1):  
\[\begin{bmatrix}\mathbf{x}(t+1)\\ \mathbf{u}(t)\end{bmatrix}=f(\mathbf{\Phi},\mathbf{\xi}(t))=\mathbf{\Phi} \mathbf{\xi}(t), \tag{2}\]  
where the parameter \(\mathbf{\Phi}\) and the input \(\mathbf{\xi}\) are represented in the following form:  
\[\mathbf{\Phi}:=\begin{bmatrix}\mathbf{A}&\mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{bmatrix}\in\mathbb{R}^{\alpha\times\beta},\qquad \mathbf{\xi}:=\begin{bmatrix}\mathbf{x}\\ \mathbf{y}\end{bmatrix}\in\mathbb{R}^{\beta}.\]  
with \(\alpha:=n+l\) and \(\beta:=n+m\).  
**Definition 4** ([27]).: _For an NCS, we assume that given a linear controller \(f\) in (1) for an NCS, the controller's input \(y\) and output \(u\) are encrypted using the encryption algorithm \(E=(\mathbf{Gen},\mathbf{Enc},\mathbf{Dec})\). If a map \(f_{E}\) exists such that the equation_  
\[f_{E}(\mathbf{Enc}(k_{p},\overline{\mathbf{\Phi}}),\mathbf{Enc}(k_{p}, \overline{\mathbf{\xi}}))=\mathbf{Enc}(k_{p},\overline{f}(\mathbf{\Phi}, \mathbf{\xi})) \tag{3}\]  
_holds, then we call \(f_{E}\) the encrypted controller of \(f\). Here, \(\overline{\mathbf{\Phi}}\in\mathcal{M}^{\alpha\times\beta}\), \(\overline{\mathbf{\xi}}\in\mathcal{M}^{\beta}\), and \(\overline{f}(\cdot)\in\mathcal{M}^{\alpha}\) are the plaintexts, rounded to ensure that each component can be represented as an element of the information space._
# Research on Hard, Transparent and Hydrophobic Coating on PMMA Sheet
## 3 Results and Discussions
### Scouring Resistance of Coatings  
Relating to the wettability, hardness, adhesive strength and transparency of the coatings comprehensively, the S05 coating was chosen as the optimal coating and underwent the water scouring test. As shown in Figure 7, the wettability of the coating slightly increased, where the water contact angles were 110.6\({}^{\circ}\) after 20 h and 108.3\({}^{\circ}\) after 40 h. The visible light transmission spectra of the S05 coating after the scouring test are shown in Figure 8, implying that the spectral transmittance was decreased in the whole observed wave band. The total visible light transmittance of the coating was 85.6% after 20 h and 83.9% after  
Figure 6: Visible light transmission spectra of PMMA and nano SiO\({}_{2}\) particles modified coatings.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Samples** & **PMMA** & **S0** & **S05** & **S1** & **S2** \\ \hline \(\tau_{v}\) (\%) & 90.1 & 89.5 & 88.6 & 86.9 & 83.5 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Visible light transmittance of PMMA and coatings.  
40 h, correspondingly. The roughness of the coating surface increased with increasing the scouring time, which reduced the transparency of the coating. Meanwhile, the coating surface area increased, which promoted the wetting behavior between the water drop and the coating.
# Transcription of _Listeria monocytogenes_ Key Virulence Genes on Tomato, Cucumber and Carrot
## 3 Results
Presence of asterisks indicates that the relative transcription was above 1 or below \(-1\) (the values that were used as threshold) at \(p\) value indicated by the number of asterisks (*, \(p\) \(<\) 0.05; **, \(p\) \(<\) 0.01; ***, \(p\) \(<\) 0.001). As far as the effect of temperature on the relative transcription of the genes under study was concerned, it was not significant in the majority (67.7%) of the cases. More accurately, transcription of _sigB_, _inlB_ and _inlC_ seemed to be unaffected by temperature. Transcription  
\begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{4}{c}{**Effect of Substrate**} & \multicolumn{4}{c}{**Effect of Temperature**} \\ \hline log\({}_{2}\)(FC) & total & cucumber & tomato & carrot & total & BHI broth & cucumber & tomato & carrot \\ \hline _sigB_ & & & & & & & & & \\ \hline \(\sim\)\(-1\) & 6 (16.7) & 1 (8.3) & 4 (33.3) & 1 (8.3) & 2 (4.3)  
of _prfA_, _hly_ and _inLA_ on tomato exhibited a mixed response, i.e., up- and down- regulation at different sampling times. The same response was observed for _plcB_ transcription in BHI broth and on cucumber. Transcription on _plcA_ was mostly affected on carrot; in the majority of the cases temperature increase resulted in upregulation. Transcription of _hly_ in BHI broth and _inlJ_ on cucumber presented a similar trend. In the first case, the relative transcription at 10 and 30 \({}^{\circ}\)C compared to the one at 4 \({}^{\circ}\)C indicated upregulation but the relative transcription at 30 \({}^{\circ}\)C compared to the one at 10 \({}^{\circ}\)C indicated downregulation. In the case of _inlJ_, the opposite was evident; the relative transcription at 10 and 30 \({}^{\circ}\)C compared to the one at 4 \({}^{\circ}\)C indicated downregulation but the relative transcription at 30 \({}^{\circ}\)C compared to the one at 10 \({}^{\circ}\)C indicated upregulation. In only one case, the same transcriptomic response was observed in all four sampling times, namely _hly_ upregulation in BHI broth at 10 \({}^{\circ}\)C compared to 4 \({}^{\circ}\)C.  
The correlation between the transcriptomic responses of the genes under study were collectively assessed by the Pearson correlation coefficient (Table 3). A total of 36 assessments were performed; no statistically significant correlation was observed in seven cases and positive correlation was observed in the remaining 29 cases. Weak correlation (0.00 \(<\) r \(<\) 0.29) was indicated in 5 cases, moderate correlation (0.30 \(<\) r \(<\) 0.49) was indicated in 14 cases and strong correlation (0.50 \(<\) r \(<\) 0.99) in 10 cases. The strongest correlation (r \(>\) 0.70) was indicated between _prfA_ and _inlA_, _inlB_, _inlC_, between _inlA_ and _inlB_, as well as between _inlB_ and _inlC_.
# Analytical Model for the Structural Behavior of Pipelines During Lowering-In
## Appendix A
The equations and lengths for each element are summarized in Table A3. The integral constants of the equations for each element in Table A3 can be obtained by using the following boundary conditions. \[ \begin{split}\text{TDP}\bigg{\{}\begin{array}{l}w_{1}(0)=2.0\\ M_{1}(0)=0\end{array}\text{, LP1}\bigg{\{}\begin{array}{l}w_{1}(51.752)=1.5\\ w_{2}(0)=1.5\\ \theta_{1}(51.752)=\theta_{2}(0)\\ M_{1}(51.752)=M_{2}(0)\end{array}\text{, LP2}\bigg{\}\begin{array}{l}w_{2}(20)=1.0\\ w_{3}(0)=1.0\\ \theta_{2}(20)=\theta_{3}(0)\\ M_{2}(20)=M_{3}(0)\end{array}\text{, LP3}\bigg{\}}\begin{array}{l}w_{3}(15)=0.5\\ w_{4}(0)=0.5\\ \theta_{3}(15)=\theta_{4}(0)\\ M_{3}(15)=M_{4}(0)\end{array}\text{,}\\ \end{split}  \tag{10}\]  
By substituting the integral constants obtained through the above procedure into the equations for each element, the behavior of the pipeline in the lateral plane can be analyzed. The sectional force of the segmental pipeline model can be evaluated using the elastic theory with \(M(x)=Elv^{\prime\prime}(x)\) and \(V(x)=Elv^{\prime\prime\prime}(x)-Tv^{\prime}(x)\). Figures A3 and A4 show the analysis result for the segmental pipeline model in the vertical plane and the lateral plane. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & **Equations for Each Element** & **Elemental Length** \\ \hline TDP to LP1 & PSP 1 & \(w_{1}(x_{1})=c_{1}+c_{2}x_{1}+c_{3}\text{sinh}(0.0102x_{1})+c_{4}\text{cosh} (0.0102x_{1})\) & 51.752 m \\ LP1 to LP2 & PSP 2 & \(w_{2}(x_{2})=c_{5}+c_{6}x_{2}+c\text{sinh}(0.0102x_{2})+c\text{cosh} (0.0102x_{2})\) & 20 m \\ LP2 to LP3 & PSP 3 & \(w_{3}(x_{3})=c_{9}+c_{10}x_{3}+c_{11}\text{sinh}(0.0102x_{3})+c_{12}\text{ cosh}(0.0102x_{3})\) & 15 m \\ LP3 to SSP & PSP 4 & \(w_{4}(x_{4})=c_{13}+c_{14}x_{4}+c_{15}\text{sinh}(0.0102x_{4})+c_{16}\text{ cosh}(0.0102x_{4})\) & 30.078 m \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The equations for each element of the segmental pipeline model in the lateral plane. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multicolumn{2}{c}{**Front Side of LP2**} & \multicolumn{2}{c}{**Back Side of LP2**} \\ \hline elements & Length of elements & elements & Length of elements \\ PCS (\(l_{z}\)) & 128.249 m & PSP 4 (\(s_{2}\)) & 15 m \\ PSP 1 (\(l_{1}\)) & 39.606 m & PSP 5 (\(l_{3}\)) & 23.223 m \\ PSP 2 (\(l_{2}\)) & 12.145 m & PSP 6 (\(l_{4}\)) & 6.855 m \\ PSP 3 (\(s_{1}\)) & 20 m & PSS (\(l_{s}\)) & 154.922 m \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The unknown lengths of each element. Figure A2: Segmental pipeline model in the lateral plane with 3 pipelayers. Using the segmental pipeline model, the assessment for the 3-D behavior of a pipeline is performed on the sectional force and Von-Mises stress by root means square (RMS). The 3-D deflections, sectional forces (bending moments and shear forces) obtained by using RMS and Von-Mises stress of segmental pipeline model are represented as shown in Figure A5.
# Current Trends on Green Wireless Sensor Networks
## 2 Energy Conservation
### Sleep/Wake-Up Schemes
#### 2.3.3 Wake-Up Radios
Though sensor nodes are settled at one-hop distance from the BS (from the point of view of the main radio transmissions) they succeed to propagate WuR signals using a forwarding decision table built in an early topology discovery phase executed by the sink. This way, the covered area can be much larger. They also use sensor nodes' state of charge information so that the BTS, knowledgeable about the application requirements, can properly poll deployed devices. 2. In RI-LD-WuR (RI Low Delay MAC), [PERSON] and [PERSON] [113] propose partitioning sensor nodes in k (almost) equal sized groups to reduce packet collisions in a broadcast-based RI WuR enabled WSN: the sink commands its cluster broadcasting a request in slot0--dynamic TDMA with contention--firing the data collection phase. N transmissions fit in each slot and only the sensor nodes of the respective group are able to contend for sending. They also contribute with a distributed algorithm for the initial composition of the groups. 3. In RI-CPT-WuR (RI Consecutive Packet Transmission MAC) [114], to cancel out the time wasted in collisions, the protocol enables transmitting multiple packets through a single competition. 4. In addition, precisely to avoid collisions, refs. [114; 115] present different strategies to face such issue in typical asynchronous TI WuR implementations: The proposal of [PERSON] et al. [114] uses an early backoff at time to send the WuR signal, arguing that in the case of success the transmitter will not need any additional backoff to use the main radio. [PERSON] et al. [115], on the contrary, advocate to incorporate to the WuC (wake-up channel) a more complete MAC protocol using CCA (CCA-WuR), CSMA (CSMA-WuR) or both adaptively (ADP-WuR). In the last, a threshold determines switching between CCA and CSMA modes if CCA fails such number of consecutive attempts. 5. Finally, in SNW-MAC (Star Network WuRx MAC) [116], an Energy Management (EM) scheme is incorporated which uses the node's residual energy to optimize the energy usage in truly EH-WSN scenarios. The protocol assumes that the time is divided into time slots of equal duration T, and the EM is executed at the end of each slot to set the throughput of the node for the next interval. First, the Energy Budget Computation module evaluates the energy that the node can consume in the next time slot k to remain sustainable; and then Throughput Computation module calculates the wake-up interval TWI[k] according to the energy budget eB[k], so determining the frequency at which the node performs sensing and sends the so-obtained data.
## 3 Artificial Intelligence Approach
### Deep Learning  
Deep learning is a higher level of machine learning, which involves multi-layer artificial neural networks to interpret large volumes of data for specific purposes, even complex ones. Deep learning addresses training neural networks themselves to extract features from the raw data. Multiple layers interconnected in deep neural networks allow for representing complex relationships between inputs and outputs [145]. Deep learning will further enhance HSI capabilities beyond those provided by conventional methods by modeling complex nonlinear relationships within the spectral data. One such application of deep learning has been with the integration of NIR-HSI to estimate nutrition in foods. The introduction of OptmWave was another approach involving deep learning that coupled the selection of wavelength characteristics with modeling in order to optimize the analysis. Applied to a dataset of scrambled eggs with tomatoes, the OptmWave method returned a high determination coefficient of 0.9913 and a low RMSE of 0.3548 [146]. Applications that can show high potentials for deep learning include microbial detection and food safety. In this regard, deep learning frameworks have been applied to identify and quantify Clostridium sporogeness spores within food matrices. Their comparison of the 1D convolutional neural network against random forest models showed that CNN significantly outperformed the RF model by improving the accuracy in spore quantification up to 8% [147]. While in difficult conditions, such as snapshot HSI systems, the robustness of deep learning forms its essential advantage, where traditional methods often failed. In the meat industry, the use of deep learning was applied by researchers, classifying red meat using hyperspectral images from the 3D convolutional neural network. It introduced novel graph-based post-processing to improve the accuracy. This achieved an over 96% classification accuracy in the meat classification problem with the use of both NIR and visible HSI data [148]. Despite certain challenges, deep learning offers significant advantages over traditional machine learning approaches, including the capability of handling large and complex datasets, making it particularly well-suited for detecting adulteration and ensuring food safety [34] (Table 3). Diverse applications of deep learning can make it useful in different processing methods and different food products like meat, diary, fruits, and vegetables. Overall, as technology continues to advance, deep learning is expected to play an increasingly critical role in the future of food analysis and quality control.
## 3 Results and Discussion
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Properties** & **Units** & **Values** & **Test Reference** \\ \hline
**Dimensions** & Diameter & mm & 6–10 & EN16127 \\  & Lenght & mm & 3.15–35 & EN16127 \\  & Moisture & \% & 3–8 & EN14774-2 \\
**Thermogravimetric** & Volatiles & \% & 60–68 & EN15148 \\
**Analysis (db)** & Fixed Carbon & \% & 25–30 & — \\  & Ashes & \% & \textless{}3 & EN14775 \\  & C & \% & 52–58 & EN15104 \\  & H & \% & 4–6 & EN15104 \\
**Elemental Analysis (db)** & O & \% & 37–39 & EN15104 \\  & N & \% & \textless{}0.500 & EN15104 \\  & S & \% & \textless{}0.03 & EN15289 \\  & Cl & \% & \textless{}0.02 & EN15289 \\  & HHV (ar) & MJ/kg & 22–23 & EN14918 \\
**Heating Value** & LHV (ar) & MJ/kg & 19.5–22 & EN14918 \\
**Bulk Density** & kg/m\({}^{3}\) & & 650–750 & EN15103 \\
**Mechanical Durability** & Durability & \% & 97.5–99 & EN15210-1 \\  & Fines & \% & \textless{}3 & EN15210-1 \\  & Al & mg/kg & 3500 & EN15290/EN15297 \\  & As & mg/kg & \textless{}1 & EN15290/EN15297 \\  & Cd & mg/kg & \textless{}0.5 & EN15290/EN15297 \\  & Ca & mg/kg & 1500 & EN15290/EN15297 \\  & Cr & mg/kg & \textless{}10 & EN15290/EN15297 \\  & Cu & mg/kg & \textless{}10 & EN15290/EN15297 \\  & Fe & mg/kg & 2800 & EN15290/EN15297 \\  & Pb & mg/kg & \textless{}10 & EN15290/EN15297 \\
**Minor Elements** & Mg & mg/kg & 2000 & EN15290/EN15297 \\
**Analysis—ICP (db)** & P & mg/kg & 200 & EN15290/EN15297 \\  & K & mg/kg & 2000 & EN15290/EN15297 \\  & Na & mg/kg & 200 & EN15290/EN15297 \\  & Si & mg/kg & 13,000 & EN15290/EN15297 \\  & Ti & mg/kg & 250 & EN15290/EN15297 \\  & Mn & mg/kg & 180 & EN15290/EN15297 \\  & Hg & mg/kg & \textless{}0,1 & EN15290/EN15297 \\  & Ni & mg/kg & \textless{}10 & EN15290/EN15297 \\  & Zn & mg/kg & \textless{}100 & EN15290/EN15297 \\
**Ashes** & Shrinking T. & \({}^{\circ}\)C & 1100 & CEN/TS15370-1 \\
**Fusibility—Oxidizing** & Softening T. & \({}^{\circ}\)C & 1250 & CEN/TS15370-1 \\
**Atmosphere** & Hemispherical T. & \({}^{\circ}\)C & 1250 & CEN/TS15370-1 \\  & Fluid T. & \({}^{\circ}\)C & 1250 & CEN/TS15370-1 \\
**Ashes** & Shrinking T. & \({}^{\circ}\)C & 800 & CEN/TS15370-1 \\
**Fusibility—Reducing** & Softening T. & \({}^{\circ}\)C & 1250 & CEN/TS15370-1 \\
**Atmosphere** & Hemispherical T. & \({}^{\circ}\)C & 1250 & CEN/TS15370-1 \\  & Fluid T. & \({}^{\circ}\)C & 1250 & CEN/TS15370-1 \\ \hline \hline \end{tabular} Notes: HHV—High Heating Value; LHV—Low Heating Value; ar—as received; db—dry basis.  
\end{table} 
Table 1: Summary of the results obtained during the different phases of tests carried out between 2013 and 2016.
## 3 Wind Speed Data Characteristics and Analysis  
Usually, a 30-year period is considered necessary to estimate the long-term wind conditions at a certain location, but a period of at least 10 years may be sufficient to obtain an acceptable estimate [55]. The data used in this study were collected over a 17-year period (2005-2021) and counted a total of 148,001 wind speed measured data. Throughout the quality check, errors and missing data were removed from the analysis. After the check, the percentage of data removed was very low--0.68%; this indicates a high reliability of the monitoring system. The data were analysed and modelled as a whole. Further, they were split into groups by months and seasons--these datasets were analysed and modelled, too. Tables 1-3 present the descriptive statistics of the monthly, seasonal, and total wind speed, including  
Figure 2: The topography of the Poprad basin and its surroundings. Source [54]. the mean, minimum, maximum, standard deviation, upper and lower quartile, median, coefficient of variation, skewness, and kurtosis. **Monthly wind speed analysis** (Table 1): It was observed that during the studied period, the lowest monthly mean wind speed with value of 2.75 m/s was in August, while in March there was the highest mean wind speed with value of 3.85 m/s. The standard deviation was used to assess the variability in the wind speed. Here, the standard deviation varied from 1.86 m/s in August to 2.87 m/s in January. In general, the winter and spring months have a higher mean wind speed and variability in the wind speed than the summer and autumn months. The coefficient of variation (CV) is useful for identifying months with a higher variability in the wind speed. According to [56], the value of the CV \(>\) 40% is classified as a very high variability and CV \(>\) 70% indicates an extremely high variability in the wind speed. The coefficient of variation ranged from 66.10% in June to 86.49% in January. Based on this, the results imply that the wind speed in all months could be classified as having very high variability. During the months of September-March, there was an observed extremely high variability in the wind speed in this location. Skewness and kurtosis measure the asymmetry and the peakness of the wind speed distribution, respectively. The coefficients of skewness ranged from 0.95 in March to 1.31 in November, indicating that all distributions were right skewed. Because skewness for all months was greater than 1, the wind speed data could be regarded as highly right skewed, except for March, when the skewness of 0.95 corresponded to a moderately right skewed distribution. The coefficient of kurtosis ranged from 3.31 in March to 4.57 in August. This indicated a highly leptokurtic distribution when compared to the normal distribution. **Seasonal wind speed analysis** (Table 2): The results show that the highest seasonal mean wind speed was observed in spring with value of 3.56 m/s, whereas in autumn, there was observed the lowest mean wind speed with value of 2.97 m/s. CV ranged from
## 2 Materials and Methods
### Optimization of the Mispe Condition  
An empty plastic solid phase extraction cartridge was used for this study. Cartridges with frits at both ends were filled with 200 mg of dry polymer. These are called MISPE and NISPE. Optimization conditions were used to determine the conditioning, loading, washing, and elution solvent that yielded the highest SLX recovery.  
\[\text{Recovery}\ =\ \frac{\text{The area of eluate}}{\text{The area of standard}}\times 100\%.\]  
Application of Molecularly Imprinted Solid Phase Extraction (MISPE) and Non-Imprinted Solid Phase Extraction (NISPE) on Spiked Blood Serum  
The application of MIP and NIP was carried out on spiked blood serum with SLX standard solution alone and with SLX solution with analogue compounds, namely salbutamol (SAL) and terbutaline (TER). Three-milliliter empty SPE cartridges were filled with 200 mg of MIP and NIP. The SPE process was carried out using a 12-port SPE vacuum manifold from Phenomenex. SPE optimization was carried out with various solvents to obtain optimal conditions during the conditioning, washing, and eluting stages. The optimal conditions were determined from the highest percent recovery obtained through analysis with HPLC. This condition was used further to extract 2 mg/L SLX-spiked blood serum only and with a mixture of SLX with analogue substances. To ensure efficient treatment usingMIP, blank serum treated with protein deposition using acetonitrile 3 times the volume of serum and centrifuged for 15 min at 5,000 rpm was injected. All MISPE and NISPE analyses were performed using validated HPLC conditions, with a correlation coefficient for linearity (r) of 0.9995, accuracy 95.81 \(\pm\) 2.21%, and %RSD for intermediate precision of 1.21%, using acetonitrile:water in gradient condition with 0.8 mL/min flow rate [7].  
Physical Characterization of Sorbent with Fourier Transform Infrared (FTIR), Brunauer-Emmett-Teller (BET), and Scanning Electron Microscope (SEM)  
The chemical structure of MIP before and after extraction and NIP was characterized by FTIR. Two hundred micrograms of KBr were mixed with 2 mg of sample and then flattened into pellets. Transmission is measured at 400-4000 cm\({}^{-1}\). The morphology of MIP and NIP sorbents was observed using SEM. The specific surface area of the MIP and NIP was determined using a BET surface analyzer. In the BET method, the specific surface area of the beads is correlated to the amount of N\({}_{2}\) gas absorbed on the surface of the beads.
# Semi-Empirical Satellite-to-Ground Quantum Key Distribution Model for Realistic Receivers
## Appendix A Derivation of Equation (2)  
In this section, we derive the elevation angle for the satellite passage to describe the quantum communication link geometry, as illustrated in Figure 1. The following equations hold for a triangle with vertices at the satellite (S), the ground station (G), and the centre of the earth (O), where all the quantities are defined in the Section 2:  
\[\begin{cases}d^{2}=R_{E}{}^{2}+R^{2}-2R_{E}R\cos\alpha\\ \frac{\sin\alpha}{d}=\frac{\sin\left(90+\theta_{E}\right)}{R}\end{cases} \tag{10}\]  
By removing variable of the distance \(d\) between the ground station and the satellite from the equation system, we get:  
\[R^{2}\sin^{2}\alpha=\left[R_{E}{}^{2}+R^{2}-2R_{E}R\cos\alpha\right]\cos^{2} \theta_{EI} \tag{11}\]  
Then, we obtain the required Equation (2), which is given as:  
\[\cos\theta_{EI}=\sqrt{\frac{\sin^{2}\alpha}{1+\left(\frac{R_{E}}{R}\right)^{2} -2\left(\frac{R_{E}}{R}\right)\cos\alpha}} \tag{12}\]