esa-sceva/satcom-chunk-collection huggingface dataset:
## 3 Dynamic Analysis
### Primary Frequency--Amplitude Relation
\right) \tag{25}\]  
The solution of Equation (13) is:  
\[\dot{u}=\dot{a}_{u}\cos(\Omega\tau-\gamma) \tag{26}\]  
As the amplitude and phase are unchanged, the steady state motion will occur, meaning that  
\[ \begin{array}{ll}\frac{1}{2}K\sin\gamma=\dot{\varsigma}\dot{a}_{u}\\ \frac{K}{2}\cos\gamma=G(\alpha_{1},\alpha_{2})\dot{a}_{u}^{3}+F(\alpha_{1}, \alpha_{2})\dot{a}_{u}^{5}-\dot{a}_{u}\sigma\end{array}  \tag{27}\]  
The frequency-response relationship of Equation (14) is obtained as follows:  
\[ \begin{array}{ll}\left(\dot{\varsigma}\dot{a}_{u}\right)^{2}+\left(G(\alpha _{1},\alpha_{2})\dot{a}_{u}^{3}+F(\alpha_{1},\alpha_{2})\dot{a}_{u}^{5}-\dot{a }_{u}\sigma\right)^{2}=\frac{1}{4}K^{2}\end{array}  \tag{28}\]  
From Equation (28), we have:  
\[ \begin{array}{ll}\sigma_{1,2}=\frac{G(\alpha_{1},\alpha_{2})\dot{a}_{u}^{3}+F (\alpha_{1},\alpha_{2})\dot{a}_{u}^{5}\pm\sqrt{\frac{1}{4}K^{2}-\left(\dot{ \varsigma}\dot{a}_{u}\right)^{2}}}{a_{u}}\end{array}  \tag{29}\]  
The peak amplitude (\(\dot{a}_{up}\)) and frequency (\(\sigma_{p}\)) are calculated as below:  
\[ \begin{array}{ll}\dot{a}_{up}=\frac{1}{2^{5}}K\\ \sigma_{p}=\frac{G(\alpha_{1},\alpha_{2})\dot{a}_{up}^{3}+F(\alpha_{1},\alpha_ {2})\dot{a}_{up}^{5}}{a_{up}}\end{array}  \tag{30}\]Considering \(\hat{a}_{uo}\&\gamma_{o}\) along with a set of steady state solutions and its neighborhood are determined by introducing small variations \(\hat{a}_{u1}\&\gamma_{u1}\) as follows:  
\[ \begin{array}{l}\hat{a}_{u}=\hat{a}_{uo}+\hat{a}_{u1}\\ \gamma=\gamma_{uo}+\gamma_{u1}\end{array}  \tag{31}\]  
To accomplish the stable analysis, Equation (24) is rewritten as  
\[ \begin{array}{lcl}\hat{a}^{\prime}_{uo}=&-\xi(\hat{a}_{uo}+\hat{a}_{u1})+ \frac{1}{2}K\sin(\gamma_{uo}+\gamma_{u1})\\ \gamma^{\prime}=&\sigma-G(\alpha_{1},\alpha_{2})(\hat{a}_{uo}+\hat{a}_{u1})^{2 }-F(\alpha_{1},\alpha_{2})(\hat{a}_{uo}+\hat{a}_{u1})^{4}\\ &+\frac{K}{2(\hat{a}_{uo}+\hat{a}_{u1})}\cos(\gamma_{uo}+\gamma_{u1})\end{array}  \tag{32}\]  
Based on Routh-Hurwitz criterion for nonlinear system, the stability of the steady state solution depending on the eigenvalues \(\lambda\) is determined as below:  
\[\left| \begin{array}{cc}-\xi-\lambda&G(\alpha_{1},\alpha_{2})\hat{a}_{uo}+F (\alpha_{1},\alpha_{2})\hat{a}_{uo}^{3}-\hat{a}_{u}\sigma\\ -2G(\alpha_{1},\alpha_{2})\hat{a}_{uo}-4F(\alpha_{1},\alpha_{2})\hat{a}_{uo}^{ 3}-\frac{K}{2\hat{a}_{uo}^{2}}\cos\gamma_{uo}&-\xi-\lambda\end{array} \right|=0 \tag{33}\]  
The unstable region of the steady state motion above is obtained as follows:  
\[\xi^{2}+\left( \begin{array}{c}2G(\alpha_{1},\alpha_{2})\hat{a}_{uo}+4F(\alpha _{1},\alpha_{2})\hat{a}_{uo}^{3}\\ +\frac{K}{2\hat{a}_{uo}^{2}}\cos\gamma_{uo}\end{array} \right)\left(G(\alpha_{1 },\alpha_{2})\hat{a}_{u}^{3}+F(\alpha_{1},\alpha_{2})\hat{a}_{u}^{5}-\hat{a}_{ u}\sigma\right)<0 \tag{34}\]
# Combined Channel Estimation with Interference Suppression in CPSS
## 4 Complexity Analysis and Performance Simulation
### Complexity Analysis  
We evaluate the computational complexity of DFT-KNN with others by multiply-accumulate (MAC) and multiply-add (MA) times. The computation of the algorithms modules are shown in Table 1, where core operator represents data width in parallel computing.  
Figure 2: The integrated process of channel estimation with interference suppression.  
According to the process shown in Figure 2, the amount of computation of DFT-KNN is 2641.92 MMAC (million MAC). There are two times of FFT transformation, one time LS estimation, one time noise cancellation by widowing in DFT-WF, and the amount of computation of DFT-WF is 2396.16 MMAC. There are two times of FFT transformation, one time LS estimation, one time noise cancellation by Mahalanobis distance in DFT-DA, and the amount of computation of DFT-WF is 2519.04 MMAC.  
The sum of computation of interference detection and interference suppression in DFT-KNN is less than 125 MMAC. DFT-KNN achieves interference suppression with amazing low computation load. The computational complexity of DFT-KNN than that of DFT-WF increased by 12.7%, and the computational complexity of DFT-KNN than that of DFT-DA increased by 3.7%.  
The current processors can provide 100 GMAC (giga MAC) computing power, such as ZYNQ, C66x, in which DFT-KNN can be easily implemented [28]. Therefore, the algorithm is useful for node equipment in CPSS.
## 6 Results and Discussion
### Analysis of Transmission Loss  
The AMCTD protocol's loss of transmission with different attacker nodes in various scenarios are explained in Figure 6. At the completion of the simulation in AMCTD, the results are as follows:  
* It has much lower transmission loss without attack.
* It has increased transmission loss with 4, 8, and 12 attacker nodes.  
This indicates the performance of the AMCTD protocol is reduced substantially in terms of transmission loss due to the active attack.  
Figure 7 displays the AMCTD protocol's transmission loss under various conditions with various attacker node variations. The data in Figure 7 compares the without attacker scenario to active attack scenarios with 4, 8, and 12 attacker nodes; the transmission loss in the AMCTD scenario is much lower in without attacker node scenario. Additionally, the transmission loss increases along with the number of attacker nodes in the scenario. Figure 7 makes clear that an active attack raises the AMCTD protocol's transmission loss by up to 7%.  
Figure 6: AMCTD: Transmission loss without attack and with 4, 8 and 12 Attacker Nodes.  
Figure 7: Transmission loss of AMCTD without Attack, AMCTD with 4, 8 and 12 Attacker Nodes.
## 4 Analysis of Energy Evolution Characteristics of Triaxial Unloading Confining Sandstone
### Energy Evolution Analysis  
According to the above calculation principles, the relationship between the energy evolution characteristics and the stress-strain curve was drawn, as shown in Figure 6.  
Figure 6: Energy evolution curve of sandstone under unloading confining pressure: (**a**) \(\sigma_{3}\) = 5 MPa, (**b**) \(\sigma_{3}\) = 10 MPa, (**c**) \(\sigma_{3}\) = 15 MPa, and (**d**) \(\sigma_{3}\) = 20 MPa.  
As can be seen from Figure 6, the evolution of the axial absorption energy \(U_{1}\) is similar under different confining pressures. With the increase in axial strain, \(U_{1}\) increases slowly before the crack closure stress \(\sigma_{\rm cc}\). Before reaching the peak stress \(\sigma_{\rm p}\), the growth rate increases. However, after \(\sigma_{\rm p}\), the growth rate gradually slows down. The overall growth rate is a slow-fast-slow process, which is related to the characteristics of the stress-strain curve. The maximum axial absorbed energy \(U_{1}\) increases with the increase in confining pressure.  
Different confining pressures have similar evolutions to \(U_{3}\) energy consumption. With the increase in axial strain, \(U_{3}\) shows a different growth trend before and after peak stress. The growth is slow before \(\sigma_{\rm p}\), especially in the crack closure stress \(\sigma_{\rm cc}\), while the growth is rapid after the peak stress. \(U_{3}\) is positively correlated to the confining pressure.  
The evolution law of elastic energy \(U^{e}\) is consistent with the stress-strain curve. Before the peak stress, \(U^{e}\) accounts for a large proportion of \(U_{1}\), while \(U_{3}\) and \(U^{d}\) are small. After reaching \(\sigma_{\rm p}\), \(U^{e}\) decreases rapidly, while \(U_{3}\) and \(U^{d}\) increase rapidly, and \(U^{d}\) rapidly becomes the main energy. As the radial strain increases, the difference between \(U^{e}\) and \(U_{1}\) becomes larger and larger, and the \(U^{e}\) in peak stress is about one-third of that of \(U_{1}\). \(U^{e}\) is positively correlated with confining pressure.  
The evolution trend of the dissipated energy \(U^{d}\) is similar to that of \(U_{3}\), with slow growth before \(\sigma_{\rm p}\) but rapid growth after \(\sigma_{\rm p}\). The difference is that \(U_{3}\) is a very small part of the total energy; thus, it is always small, while \(U^{d}\) grows almost exponentially after \(\sigma_{\rm p}\) and rapidly becomes the main part of the energy. The maximum of \(U^{d}\) is different and increases with confining pressures. After \(\sigma_{\rm p}\), the \(U^{d}\) growth rate increases slightly.
# Influence of Hyperparameters in Deep Learning Models for Coffee Rust Detection
## 4 Results
### Evaluation by the Optimizer  
Given the strong influence of the backbone used, the next step in the evaluation is to group the results by optimizer for each pre-trained network; it means grouped by 36 models in this study (see Figure 5). This is because an optimizer may work very well for one network but poorly for another, and the analysis must be performed separately.  
For all the backbones selected in this study, except for ResNet50, the optimizer that provides the lowest F1-score is SGD. For example, when using MobileNetv2 as the backbone and SGD as the optimizer, the F1-score result is 64.4%, while when the optimizer is changed to RMSprop, the value is 88.4%. In another example, when using InceptionResNetV2 and SGD as the optimizer, the F1-score is 52.8%, but when the optimizer is Nadam, the value  
Figure 4: Summary of evaluation results by pre-trained model used as a backbone, in terms of F1-score. Points marked with + represent outliers beyond the first and third quartiles.  
Figure 5: Summary of evaluation results by optimizer, in terms of F1-score.  
is 76.3%. The results obtained with Nadam and RMSprop are very similar to each other, and in general SGD is not a suitable optimizer for this type of classification problem.  
In summary, this hyperparameter has a moderate impact on model performance (with variations of up to 25% among them).
# IEC-61108-7 SBAS Standard for Shipborne Receivers: Preliminary Testing Validation Activities +
## 3 Sample Results
### TC-003: Dynamic Accuracy Test--A  
The test aims to verify the receiver accuracy under dynamic conditions; the key elements of the test are shown in Figure 4. From the upper left box, it can be noted that the receiver switches to SBAS/DGPS mode about 6 min from the start of the scenario. The number of satellites varies between five and nine in the initial part of the test (from 12:00 to 12:07), while it remains constant at nine for the remaining part of the test. The horizontal and vertical error time evolution are shown in the right-hand boxes together with the relative geometric parameters. In these dynamic conditions, the receiver under test was able to provide sub-meter errors--the 95% horizontal error is 0.66 m.
# Coexistence in Wireless Networks: Challenges and Opportunities
## 4 Spectrum Sharing and Coexistence
### Coexistence in Terrestrial Communication
INR measures interference relative to noise, while SINR evaluates communication quality. BER quantifies transmission errors, and throughput assesses data transfer rates. Latency measures delays  
Figure 8: Base station coverage and coexistence. from interference, spectral efficiency evaluates spectrum utilization, and PDR determines packet delivery reliability. Table 1 gives an overview of coexistence with the references for all coexistence scenarios.
## 4 Designing Sensors for Soft Robots
### Testing Procedures
In this way, we could ensure that the change in effective wire length genuinely mimics the curvature of each channel [38]. This new approach, using an optical fibre-based curvature sensor in conjunction with soft manipulators is demonstrably promising. The model has been tested and shows encouraging results as a tool with which to estimate the pose of a flexible cylindrical manipulator segment. Other advantages over those currently available include its potentially low-cost manufacture and ease of integration into a surgical system. Although techniques that use standard optic fibres, with mechanisms that displace a reflective surface in response to changes in the measurand, are suitable for remote light-based sensing in flexible structures, there are still challenges in creating and integrating such structures into soft bodies. Issues such as large size of the channel compared to the optical fibre, the friction between the moving fibre optic cables and the surrounding sheath and compact integration require further study. Since the STIFF-FLOP project, we have seen advances in many other approaches to soft-robot proprioception. Related to optical fibre-based sensing, an approach which involves the measurement of optical losses within a single optical fibre or lightguide to characterize the bending strain, or applied stimuli such as normal force, is emerging as a versatile technique for creating flexible and stretchable sensors that can be integrated into \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Bend Angle \(\theta\) (\(\beta\)=0)** & \(I_{1}\) **mm** & \(I_{2}\) **mm** & \(I_{3}\) **mm** \\ \hline \(-90^{\circ}\) & \(-6.5\) & \(15.0\) & \(5.1\) \\ \hline \(-60^{\circ}\) & \(-4.0\) & \(9.5\) & \(3.0\) \\ \hline \(-30^{\circ}\) & \(-2.0\) & \(6.0\) & \(2.0\) \\ \hline \(30^{\circ}\) & \(4.9\) & \(-2.3\) & \(1.1\) \\ \hline \(60^{\circ}\) & \(10.1\) & \(-4.0\) & \(1.7\) \\ \hline \(90^{\circ}\) & \(14.9\) & \(-5.3\) & \(5.0\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Measured wire length changes at shown angular positions [38]. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \(\theta_{m}\) (\(\beta\) = 0) & \(C_{x}\) & \(C_{x}\) & \(\theta_{s}\) & \(\beta_{s}\) & \(R\) \\ (Measured) & (Measured) & (Calculated) & (Sensor) & (Sensor) & (Sensor) \\ \hline \(-90^{\circ}\) & \(-26.5\) & \(-35.0\) & \(84.1\) & \(-177.3\) & \(37.4\) \\ \hline \(-60^{\circ}\) & \(-51.0\) & \(-52.5\) & \(54.6\) & \(-178.7\) & \(57.7\) \\ \hline \(-30^{\circ}\)  
soft robots. Although most of the studies in relation to deformable lightguides for sensing focus on proprioception in soft structures or soft robotic hands, these techniques would be equally applicable to soft surgical manipulators. Hence, we present a brief overview and discussion of sensing, using deformable lightguide-based sensors.
# Variational Autoencoders for Network Lifetime Enhancement in Wireless Sensors
## 2 Literature Review  
### Energy Efficiency Improvement Techniques in WSNs  
An improved grey wolf optimization technique [1] was developed to increase the energy efficiency of WSNs in industrial IoT applications. This improved model addressed the limitations of the traditional grey wolf method in terms of premature convergence and reduced population diversity. The simulation analysis indicated 333.51% better over the regular grey wolf optimization method on the network stability analysis [1]. A threshold-enabled scalable and energy-efficient scheme [2] was proposed to support the wireless networks that was implemented in a large-scale IoT scenario with around five or more different sensor nodes for regulating the data transmission process by avoiding unnecessary data transmission. It was achieved by implementing a hybrid threshold- based minimum cost cross layer transmission at every node to stabilize the energy saving mode. Hence, the model allowed data transmission only when it is necessary by understanding the static and dynamic factors of the connected sensor nodes at different stages. The simulation result indicated a betterment of 29% on energy efficiency and 68% of network lifetime enhancement over the traditional scalable and energy efficient scheme [2]. An alternative protocol approach [3] was designed to improve the energy efficiency by adjusting the time division and carrier sense multiple access in industrial wireless sensor networks. The work utilized carrier sense multiple access on light-weight data transmission and a modified selective activation technique for energy optimization and interference reduction between the redundant nodes. The work minimized the end-to-end delay and reduced the lower packet loss up to 25% on heavy traffic conditions when compared to the traditional industrial wireless network model implemented with IEEE 802.15.4 [3].  
An advanced distributed energy-efficient clustering technique [4] was structured to improve energy efficiency in the autonomous cellular networks. The work transfers the data information in a better way through heterogeneous contexts with a 19% improvement on throughput over the LEACH [4]. A hybrid particle swarm optimization technique was incorporated with improved low-energy adaptive clustering hierarchy [5] for improving the energy efficiency by determining a best cluster head and corresponding cluster nodes. The simulation work indicated an improvement of 28% in energy efficiency and 55% in network lifetime over the previous methods [5]. A collaborative energy-efficient routing protocol [6] was designed for making a sustainable network model in the 5G and 6G communication systems. The work utilized the reinforcement learning approach for making a reliable cluster network for providing a successful transmission with the available energy. The multi-objective improved seagull algorithm was also included in the work for optimizing the overall performances. The experimental work indicated an improved energy efficiency of 50% over the previous methods [6].
## 3 Methods
### Clustering and Overlap Formulation
Finally, the balanced dataset was constructed, collecting 5000 samples from each of the six cases and subsampling nine windows of \(5\times 5\) for each of them, reaching a combined number of 270,000 samples for the training. As it can be seen in Table 2, there is an RMSE value for each datum case. This gives an overview of the precision of the network as a function of the data complexity. Since case 1 stands for samples in which there are no clusters around, it is expected to have the lowest RMSE value. Once the samples start increasing in number of clusters involved, the RMSE value goes up accordingly. It is also expected to observe an increase in the error with the increment on data complexity. A good \"symptom\" is to see the case 6 samples, which have increased complexity regarding the energy difference between clusters, showing to have good performance. Even though the RMSE values are considerably low inside the full calorimeter range, it must be stated that the average energy value seen in the dataset used is of 1202.64 MeVs. With respect to that value, the maximum RMSE obtained from group 5 represents 20.3%. However, it must be taken into account that the RMSE metric is sensitive to out layers. A summary of the network parameters for this reconstruction step is provided in Table 3. Given the regressive nature of this network, results, in terms of training performance, are measured with the RMSE metric, comparing all the values predicted by the network to the results of applying the CA rules to the training input samples. Considering the reference of the mean energy value seen in the training samples, the RMSE of the network represents 14% of the average energy. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Case**} & **Number of** & **Overlap with** & **Samples on** & **RMSE** \\  & **Clusters (K)** & **Central Cell** & **2k Events** & \\ \hline
1 & 0 & No & 153,519 & 96.281 \\
2 & 1 & No & 121,316 & 135.616 \\
3 & 2 & Yes (1 cluster) & 45,066 & 147.501 \\
4 & 3 & Yes (2 clusters) & 9937 & 199.644 \\
5 & 4+ & Yes (3+ clusters) & 1367 & 244.312 \\
6 & \textgreater{}1 & Yes (energy difference of 1 & 6816 & 181.693 \\  & & order of magnitude) & & \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Case characteristics in the balanced dataset for the MLP training. Case 6 is a selection of samples with overlap with at least one cluster, but where the energy difference between clusters is bigger than an order of magnitude. The RMSE values were extracted, comparing the network predicted values and the samples generated with the application of the CA rule. Figure 4: Samples of the data used for the training of the clustering and overlap network. The three streams of data (**c–e**) are extracted from the analysis of a \(7\times 7\) window (**a**). Image (**b**) shows the output predicted by the network representing the reconstructed cluster located at the centre of the (**a**) sample.
## 3 Results
### Burned Area Prediction under Climate Change
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Zone** & **Metric** & **Training** & **Validation** & **Test** \\ \hline \multirow{4}{*}{Zone 1} & MAE (m\({}^{2}\)) & 40.882.33 & 22,644.32 & 39,377.21 \\  & RMSE (m\({}^{2}\)) & 72,770.19 & 28,197.83 & 79,424.98 \\  & MAPE (\%) & 32.54 & 34.10 & 25.88 \\  & Bias (m\({}^{2}\)) & 37,939.93 & 18,186.42 & –37,461.56 \\ \hline \multirow{4}{*}{Zone 2} & MAE (m\({}^{2}\)) & 5643.24 & 6060.23 & 28,262.64 \\  & RMSE (m\({}^{2}\)) & 7353.61 & 7087.41 & 34,996.43 \\  & MAPE (\%) & 11.70 & 17.21 & 37.38 \\  & Bias (m\({}^{2}\)) & –2535.66 & –5702.80 & –21,409.82 \\ \hline \multirow{4}{*}{Zone 3} & MAE (m\({}^{2}\)) & 5695.49 & 18,663.88 & 16,867.68 \\  & RMSE (m\({}^{2}\)) & 9186.30 & 26,384.25 & 26,047.79 \\  & MAPE (\%) & 11.83 & 28.20 & 18.83 \\  & Bias (m\({}^{2}\)) & 8.89 & 14,958.76 & –14,095.28 \\ \hline \multirow{4}{*}{Zone 4} & MAE (m\({}^{2}\)) & 47,778.19 & 14,211.44 & 20,903.01 \\  & RMSE (m\({}^{2}\)) & 65,997.94 & 17,194.42 & 25,914.53 \\ \cline{1-1}  & MAPE (\%) & 42.77 & 29.92 & 40.53 \\ \cline{1-1}  & Bias (m\({}^{2}\)) & 29,474.66 & –4890.40 & 18,471.57 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Goodness-of-fit tests of an ANN (60-20-20 correspond to training, validation, and test, respectively) in prediction of wildfires surface larger than 30,000 m\({}^{2}\) in the 2000–2014 period. Figure 6: Comparison of average percentage monthly wildfires in recorded period (2000–2014) (blue line) and RCP4.5 (red line) and RCP8.5 (green line) scenarios in long-term period (2070–2100).
## 4 Reliability and Reliability Sensitivity Analysis Method of the Igniter  
### Kriging Model of the Igniter  
The limit state functions \(I(\mathbf{x})\) are highly nonlinear and implicit. The Kriging model with a three-level Full Factorial method (TLFF) [39] is employed to predict the limit state functions due to its high computational accuracy and efficiency. An initial Kriging model is established based on the Design of Experiments (DoE) and the initial DoE is constructed by a TLFF method in this paper. According to the Kriging theory, the output function, \(\hat{I}(\mathbf{x})\), and random variable, \(\mathbf{x}\), are related as follows [40]:  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Parameter** & \(\mathbf{P_{m}}\)**/MPa** & \(\mathbf{P_{i}}\)**/MPa** & \(\mathbf{t_{s}}\)**/s** & \(\mathbf{t_{v}}\)**/s** \\ \hline Simulation model & 11.07 & 6.17 & 0.16 & 2.14 \\ Experiment 1 & 10.76 & 5.38 & 0.15 & 2.17 \\ Experiment 2 & 10.48 & 5.60 & 0.14 & 2.23 \\ Experiment 3 & 11.49 & 6.81 & 0.13 & 1.91 \\ Difference with respect to the average & & & & \\ value of exp. Results (\%) & 1.35 & 4.04 & 14.28 & 1.90 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Comparison between the Simulation model and the experiment characteristic parameters.  
Figure 5: The experiment setup.  
Figure 6: Comparison between the simulation model and the experiment results.  
According to the basic principle of MCS [42], the estimated probability of failure is:  
\[\hat{P}_{f}=\frac{1}{N}\sum_{i=1}^{N}I_{F}(\mathbf{x}_{i})=\frac{\mathbf{N}_{F}}{\mathbf{N}} \tag{30}\]  
where \(\mathbf{N}\) is the number of sample points extracted from the joint probability density function of random variables, \(\mathbf{N}_{F}\) is the number of sample points falling in the failure domain.  
Therefore, the ignition reliability of the igniter under low temperature combustion instability failure mode is:  
\[\hat{R}=1-\hat{P}_{f} \tag{31}\]