esa-sceva/satcom-chunk-collection huggingface dataset:
## 4 Closer Look into PPP
The historical definition of PPP requires the correlation to be positive because only in that case will the GLS estimate lie outside the range of the data for certain \(\Sigma\). Next rewrite Equation 8 as \(\hat{\mu}=a_{1}^{\prime}y_{1}+a_{2}^{\prime}y_{2}\) with \(a_{1}^{\prime}=\frac{a_{1}}{a_{1}+a_{2}}\) and \(a_{2}^{\prime}=\frac{a_{2}}{a_{1}+a_{2}}\). A condition equivalent to Equation 10 is obtained as follows. By setting the first derivative with respect to \(a_{1}^{\prime}\) to zero, one can solve for the value of \(a_{1}^{\prime}\) such that the variance of \(a_{1}^{\prime}y_{1}+a_{2}^{\prime}y_{2}\) is minimum subject to \(a_{1}^{\prime}+a_{2}^{\prime}=1\). The result is \(a_{1}^{\prime}=\frac{\sigma_{2}^{2}-\sigma_{12}^{2}}{\sigma_{1}^{2}+\sigma_{ 2}^{2}-2\sigma_{12}^{2}}\). Therefore, a condition equivalent to Equation 10 is: PPP occurs if \(a_{1}^{\prime}<0\) or \(a_{1}^{\prime}>1\). Another interesting fact is that if PPP occurs (so Equation 10 does not hold), then either \(a_{1}^{\prime}<0\) and \(a_{2}^{\prime}>0\) or \(a_{1}^{\prime}>0\) and \(a_{2}^{\prime}<0\), with \(a_{1}^{\prime}+a_{2}^{\prime}=1\) so that \(\hat{\mu}\) is unbiased. We can now state and prove Theorem 2. _Theorem 2_. Suppose \(a_{1}^{\prime}\) and \(a_{2}^{\prime}\) have opposite signs. Then either \(\hat{\mu}<\min(y_{1},y_{2})\) or \(\hat{\mu}>\max(y_{1},y_{2})\). That is, \(\hat{\mu}\) will _always_ fall outside the range of the \((y_{1},y_{2})\). Proof. First assume \(a_{1}^{\prime}>0\) and \(a_{2}^{\prime}<0\). If \(y_{1}<y_{2}\) then \(\hat{\mu}=a_{1}^{\prime}y_{1}+a_{2}^{\prime}y_{2}<a_{1}^{\prime}y_{1}+a_{2}^{ \prime}y_{1}=y_{1}\) because \(a_{2}^{\prime}<0\). If \(y_{1}>y_{2}\) then \(\hat{\mu}=a_{1}^{\prime}y_{1}+a_{2}^{\prime}y_{2}>a_{1}^{\prime}y_{1}+a_{2}^{ \prime}y_{1}=y_{1}\) because \(a_{2}^{\prime}<0\). The proof is completed by next assuming \(a_{1}^{\prime}<0\) and \(a_{2}^{\prime}>0\), and following similar steps.
# Luminescence Materials for Dye-Sensitized Solar Cells: Advances and Directions
## 4 Luminescent Materials in DSSCs
### How to Incorporate Luminescent Materials in DSSCs  
The general approach in integrating luminescent materials in DSSCs follows some careful sequential steps and involves photo-anode preparation, the sensitization of the photo-anode with dye molecules, and the assembly of the cell. The following activities depict the procedural approach of incorporating luminescent materials in DSSCs.  
* **Activity 1: Preparation of Photo-anode*
* Mix the TiO\({}_{2}\) nanomaterial with a suitable solvent to synthesize TiO\({}_{2}\) paste.
* Incorporate the synthesized luminescent material to the TiO\({}_{2}\) paste (either in varying concentrations or just the optimum concentration of the luminescent material) to study its effect on the photo-anode performance.
* Apply the TiO\({}_{2}\) mixed with the luminescent material onto fluorine-doped tin oxide (FTO) glass substrates using any appropriate material deposition technique to form a photo-anode [93; 94; 95].  
* Anneal the photo-anode at appropriate temperatures to remove the binder and enhance adhesion.
* **Activity 2: Sensitization with Dye*
* Prepare a dye solution, typically containing a ruthenium-based dye or organic dye, known for its high absorption in the visible range [96; 97].
* Immerse the annealed photo-anode in the dye solution for a few hours to allow thorough dye adsorption.
* Depending on the focus of the study, you could optimize dye loading conditions, such as concentration and immersion time, to maximize the light absorption capabilities.
* **Activity 3: Cell Assembly*
* Pair the sensitized photo-anode with a platinum-coated counter electrode to form a sandwich.
* Fill the inter-electrode space with a liquid electrolyte containing iodide/tri-iodide redox couple.
* Seal the cells to prevent electrolyte leakage and ensure stable operation.
* Finally, measure the photovoltaic current density--voltage performance of the device under standard solar illumination (Air Mass = 1.5 G) to determine the photovoltaic power conversion efficiency of the device.
# Geographical Distribution of Global Radiation and Sunshine Duration over the Island of Cyprus
## 4 Conclusions
### Nomenclature  
\begin{tabular}{l l} \(B\) & Hourly horizontal beam irradiance (W/m\({}^{2}\)) \\ \(B_{n}\) & Hourly normal beam irradiance (W/m\({}^{2}\)) \\ BSRN & Baseline Surface Radiation Network \\ CDF & Cumulative Density Function \\ CS & Campbell-Stokes sunshine recorder \\ CSD3 & Kipp \& Zonen Sunshine Duration sensor \\ \(d_{n}\) & Day number of the year (1.365) \\ \(D\) & Hourly diffuse horizontal irradiance (W/m\({}^{2}\)) \\ DCS & Clear skies \\ DIS & Intermediate skies \\ DOS & Overcast skies \\ ERL & Extremely Rare Limits \\ \(G\) & Hourly global solar irradiance (W/m\({}^{2}\)) \\ \(G_{cld}\) & Global irradiance under cloudy sky conditions \\ \(G_{pc}\) & Global irradiance under partly cloudy sky conditions \\ \(G_{clr}\) & Global irradiance under clear-sky conditions \\ \(G_{d}\) & Daily global solar irradiation (MJ/m\({}^{2}\)) \\ \(G_{0}\) & Hourly Extraterrestrial horizontal irradiance (W/m\({}^{2}\)) \\ \(G_{0n}\) & Normal irradiance at the top of the atmosphere (W/m\({}^{2}\)) \\ \(G_{0d}\) & Daily extraterrestrial irradiation (ETR) (MJ/m\({}^{2}\)) \\ \(G_{c}\) & Clear-sky global solar irradiance (W/m\({}^{2}\)) \\ \(G_{sc}\) & Solar constant (1367 Wm\({}^{2}\)) \\ \(G_{max}\) & Highest daily global solar irradiance (W/m\({}^{2}\)) \\ IQR & Interquartile Range \\ \(k_{b}\) & Fraction of Direct Horizontal to Global irradiance (\(B/G\)) \\ \(k_{d}\) & Fraction of Diffuse to Global irradiance (\(D/G\)) \\ \(k_{t}\) & Hourly clearness index (\(G/G_{0}\)) \\ \(K_{T}\) & Daily clearness index (\(G_{d}/G_{0d}\)) \\ \(LF\) & Conversion factor for PAR irradiance (4.57 \(\upmu\)mol s\({}^{-1}\) W\({}^{-1}\)) ([PERSON], 1972) \\ \(n\) & Number of observations \\ \(N\) & Number of days \\ PAR & Photosynthetic Active Radiation (W/m\({}^{2}\)) \\ PPFD & Photosynthetic Photon Flux Density (\(\upmu\)mol/s/m\({}^{2}\)) \\ PPL & Physically Possible Limits \\ Q1 & First Quartile \\ Q3 & Third Quartile \\ \(R^{2}\) & Coefficient of determination \\ \(S\) & Standard deviation of residuals \\ SD & Sunshine Duration \\ \(S_{d}\) & Daily sunshine duration (hours) \\ \(S_{0d}\) & Astronomical day length (hours) \\ StDev & Standard deviation (Std) \\ \(t\) & Local time (hour) \\ TMY & Typical Meteorological Year \\ \(z\) & Station's elevation (m) \\ Greek: \\ \(\alpha_{s}\) & Solar altitude angle (degrees) \\ \(\delta\) & Solar declination angle (degrees) \\ \(\varepsilon\) & Correction factor to mean solar distance \\ \(\theta_{\rm z}\) & Solar zenith angle (SZA) (degrees) \\ \(\lambda\) & Longitude of the station in degrees (East positive) \\ \(\sigma\) & Relative sunshine duration (\(S_{d}/S_{0d}\)) \\ \(\phi\) & Latitude of the station in degrees \\ \(\omega\) & Hour angle (degrees) \\ \(\omega_{s}\) & Sunset hour angle (degrees) \\ \end{tabular}
# Seismic Responses of a Cable-Stayed Bridge with Consideration of Uniform Temperature Load
## 5 Case Study and Discussion  
### Changes of Uniform Temperature Load  
The changes of temperature at the location of the bridge is hard to determine, but the intention of this paper is not investigating the complex changes of temperature. Thus, the temperature changes of \(\pm 20\)\({}^{\circ}\)C and \(\pm 35\)\({}^{\circ}\)C are adopted with consideration of the maximum temperature change of the bridge site. As a matter of fact, the temperature changes mentioned before may occur frequently during the lifetime of the bridge. The concurrent occurrence of the temperature and earthquake loadings should not be neglected. Otherwise, it may lead to non-conservative design. Figure 4a,b presents the time history of the bearing displacements at the abutment with temperature change of \(\pm 20\)\({}^{\circ}\)C and \(\pm 35\)\({}^{\circ}\)C, respectively. It is found that the seismic displacement of the bearing will deviate from the axis due to the initial displacement caused by temperature change. This effect will increase the damage of bearings of the bridge when subjected to a strong earthquake. Figure 4c,d shows the time history of the bounding force between the deck and abutment with different temperature changes. It is indicated that the initial gap between the deck and the abutment will decrease, attributed to the elongation of the deck with positive temperature change. Thus, the bounding at the expansion joints will frequently happen, and this increases the damage risk of the deck and abutments. As can be seen from the figure, the bending moment at the bottom of the pylon will also increase with the temperature changes. Some peak responses of the bridge are listed in Table 3. The results show a maximum increase of 16.5% for the compressive force of the deck and an increase of 11.7% for the tensile force of the deck. This increase is mainly attributed to the partially longitudinal restraints provided by the bearings between the deck and the pylons. The axial deformation of the deck due to temperature changes will be restrained by the bearings, and this will significantly affect the axial force of the deck. In Table 3, bearing displacement at the abutment will sharply increase with temperature changes and the same phenomenon will be seen about bounding force. These can be explained as follows: the gap between the deck and the abutments will be narrowed due to positive temperature loading, which will not only increase the displacement of the bearing, but also increase the probability of bounding.
## 2 Mixed Poisson Regression Model with Varying Dispersion
### Model Specification: The Poisson-Lognormal Regression Model with Varying Dispersion  
For expository purposes, we specify the lognormal distribution as the mixing distribution of \(z_{i}\) with the following pdf  
\[g(z_{i};\phi_{i})=\frac{1}{\sqrt{2\pi}\phi_{i}z_{i}}\exp\left[-\frac{1}{2\phi_ {i}^{2}}\left(\log(z_{i})+\frac{\phi_{i}^{2}}{2}\right)^{2}\right], \tag{7}\]  
where \(z_{i}>0\) and \(\phi_{i}>0\), with mean \(E(z_{i})=1\) and variance \(Var(z_{i})=\exp\left(\phi_{i}^{2}\right)-1\).  
Then, based on Equations (1) and (7), it is easy to see that the resulting distribution of \(k_{i}\) is the Poisson-lognormal (PLN) distribution with pmf  
\[P(k_{i})=\int\limits_{0}^{\infty}\frac{\exp(-\mu_{i}z_{i})(\mu_{i}z_{i})^{k_{i }}}{k_{i}!}\frac{\exp\left[-\frac{1}{2\phi_{i}^{2}}\left(\log(z_{i})+\frac{ \phi_{i}^{2}}{2}\right)^{2}\right]}{\sqrt{2\pi}\phi_{i}z_{i}}\mathrm{d}z_{i}, \tag{8}\]  
where \(\mu_{i}>0\) and \(\phi_{i}>0\) are given by Equations (5) and (6), respectively. Unfortunately, since \(g(z_{i};\phi_{i})\) is not conjugate to the Poisson, the integral in Equation (8) is mathematically intractable, but it can be easily calculated using numerical integration.  
Using the results in Equations (3) and (4), we calculate the mean and the variance of the PLN regression model with varying dispersion  
\[\mathrm{E}(k_{i})=\mathrm{E}_{z_{i}}[\mathrm{E}(k_{i}|z_{i})]=\mu_{i}\mathrm{ E}_{z_{i}}[z_{i}]=\mu_{i} \tag{9}\]and  
\[\begin{split}\text{Var}(k_{i})&=\text{E}_{Z_{i}}[\text{ Var}(k_{i}|Z_{i}=z_{i})]+\text{Var}_{Z_{i}}[\text{E}(k_{i}|Z_{i}=z_{i})]\\ &=\mu_{i}+\mu_{i}^{2}\Big{[}\text{exp}\left(\phi_{i}^{2}\right)- 1\Big{]}.\end{split} \tag{10}\]
## 5 Case Study  
The proposed method is also tested on the measurement data of a wind farm cluster in Northeast China. The proposed model is tested on Linux server Cluster (CPU: Intel Xeon (R) CPU E5-2650 v4 @ 2.10 GHz, GPU: NVIDIA Tesla P100) and deep learning framework Pytorch (1.4.0) with GPU acceleration to speed up the training process.
# Deep-Learning-Based Land Cover Mapping in Franciacorta Wine Growing Area
## 3 Experiment
### Backbones  
In our study, we performed two semantic segmentation approaches: with and without leveraging a backbone. Semantic segmentation without a backbone involves training models from scratch, without the use of pre-trained networks. This method relies solely on the model's architecture to extract features directly from input data. While this approach offers independence from pre-existing biases and data patterns embedded in pre-trained networks, it typically requires more extensive training and larger datasets to achieve performance comparable with pre-trained backbone models.  
Conversely, leveraging a pre-trained backbone is a widely adopted strategy aimed at enhancing model performance in semantic segmentation tasks. Backbones, such as ResNet, InceptionV2, DenseNet, and EfficientNet, are well-established architectures in CNN deep-learning. These models are pre-trained on large-scale datasets (such as ImageNet) and have learned to extract meaningful features from images by encompassing spatial relationships and contextual information crucial for accurate segmentation.  
By fine-tuning these pre-trained backbones for specific segmentation tasks, models can efficiently adapt to new datasets and extract relevant features tailored to the nuances in land cover classification. This process leverages transfer learning, where knowledge acquired from previous tasks (e.g., image classification) is transferred to improve performance on current segmentation tasks. This approach not only accelerates model convergence during training but also enhances segmentation accuracy by leveraging the rich feature representations learned by backbone networks.  
The choice of ResNet, InceptionV2, DenseNet, and EfficientNet as backbone architectures was motivated by their proven performance in various computer vision tasks and their distinct architectural strengths:  
Figure 5: Illustration of two randomly selected 128 \(\times\) 128-pixel image patches, extracted from the original images along with their corresponding ground truth masks. These patches are used to train and evaluate a deep learning model.
## 4 Major Components of Network Slicing System
### Intent-Based Networking (IBN) Tool
#### 4.2.3 Policy Store  
The policy store is a well-managed database repository that can store all the information related to network functions, RAN resources, orchestrator information, and network architecture information. To store the network resources information in a database repository is a unique idea where network operators can easily get the available network resources information and can create a service graph for a specific network slice. The policy store was designed by keeping the NFV dependencies with each other. It encompasses many database tables: Predefined xontracts, modules_architecture, modules_relations, architectures_supported, modules_versions, and snssais_supported tables etc. The information stored in each table is as follows; the xontracts table stores the information related to the contracts which are inserted from the GUI interface of the IBN system. The architectures_supported table stores the information of all registered architecture with name and unique ID; each architecture has its unique-ID. The module_architecture table can store the information of all modules (NFVs, eNBs) and their dependent nodes i.e., the network function SPGWC is a dependent node on MME and SPGWU network functions. This table stores the relation of each node to the other which will help to compute a service graph. The fourth important table named a modules_relations table that can store the information related to link specification between different modules. Lastly, the table snssais_supported stores the information related to each slice doamin with specific SNSSAI id. The design of the Policy store is shown in Figure 5 with a table definition and their attributes etc.  
Figure 5: Design of the policy store database.
# Security Scheme Based on Parameter Hiding Technic for Mobile Communication in a Secure Cyber World
## 3 Proposed Security Scheme
### Initial Attach with GUTI
#### 3.3.3. MME Changed and IMSI Needed  
The third case is shown as Figure 4. The third case is the one where MME connected with UE at the initial access is changed to new MME, and UE information connected at the time of initial access does not exist in MME. At the time of re-access, the initial attach process executes authentication using GUTI. The process that transmits GUTI is identical to the IMSI transmission process in Section 3.2.  
In case MME is changed, new MME requests UE information according to information such as GUTI, NAS-MAC, and NAS sequence number. At this moment, when relevant UE information does not exist in old MME, new MME requests IMSI to UE.  
In order to securely transmit IMSI in this process, UE generates a key by hashing KGUTI and \(\text{RN}_{\text{UE}\_2}\), and transmits to MME by encrypting IMSI using a generated key KIMSI. After IMSI is transmitted, processes of authentication, the NAS security setup, the location update, and the EPS session establishment are executed.  
Figure 2: Case 1: Mobile Management Entity (MME) unchanged.  
Figure 3: Case 2: MME changed.
## 3 Control Strategy Description
### Design of Backstepping Integral Sliding Mode Controller  
In order to solve the problem that the traditional backstepping sliding mode control algorithm is more sensitive to the disturbance of the unmatched channel, this chapter designs a backstepping integral sliding mode controller. The sliding mode surface is designed as follows:  
\[s=c_{1}e+x_{2}+c_{2}\int edt \tag{12}\]  
where \(c_{2}\) is the parameter of the sliding mode surface, and its value cannot be zero. The state equation of the sliding surface is shown in Equation (13).  
\[\dot{s}=c_{1}\dot{e}+\dot{x}_{2}+c_{2}e \tag{13}\]  
Choose the constant approximation law to design the controller, and the control output signal is shown in Equation (14).  
\[u=-g^{-1}[(ac_{1}+c+c_{2})e+(bc_{1}+d)x_{2}+(ac_{1}+c)x_{1}d+K\text{sgn}(s)] \tag{14}\]  
Substituting the designed control output signal expression (14) into Equation (6), and at the same time, we can obtain the following Equation:  
\[\ddot{e}+(bc_{1}-a)\dot{e}+bc_{2}e=a\dot{x}_{1d}+\dot{d}(t) \tag{15}\]  
In the PMSM speed control system, the desired value of the speed is set as a constant, so that Equation (15) can be simplified into Equation (16).  
\[\ddot{e}+(bc_{1}-a)\dot{e}+bc_{2}e=\dot{d}(t) \tag{16}\]  
Analyzing Equation (16) can prove that if the disturbance is a parameter that changes very slowly with time, then the equilibrium point of the system will converge to the origin. In other words, the speed of the PMSM will also converge to the sliding surface in a limited time, thereby realizing the tracking control of the speed. The backsteppingintegral sliding mode controller can effectively eliminate the steady-state bias caused by the system mismatched channel disturbance, and the entire PMSM speed control system has better robustness. However, in addition to the above-mentioned advantageous factors, the addition of the integral mode will greatly increase the adjustment time of the system and bring about a larger overshoot.
## 7 Case Analysis
### Spare Parts Support Capacity Evaluation  
By reference to the evaluation process described in Sections 5.2 and 6, the evaluation process for the spare parts support capacity is described as follows:  
Step 1: According to the decision matrix (Table 2) given by experts, the weight value of each attribute is calculated as per Formulas (11) and (10):  
\[w_{1}=0.2425,\ w_{2}=0.2489,\ w_{3}=0.2598,\ w_{4}=0.2488\]  
Step 2: The positive ideal solution and negative ideal solution are set as \(h^{+}=<(1,0)\), \((1,0)\), \((1,0)\), \((1,0)\) and \(h^{-}=<(0,1)\), \((0,1)\), \((0,1)\), \((0,1)\), respectively.  
Step 3: The weighted distances from the evaluation values for the four subsidiaries to the positive and negative ideal solutions are calculated as per Formulas (13) and (14), The results are shown in Table 3.  
\[\begin{array}{l}D_{1}^{+}=0.2635,\ D_{2}^{+}=0.2415,\ D_{3}^{+}=0.2632,\ D_{4}^{+}=0.1860\\ D_{1}^{-}=0.6632,\ D_{2}^{-}=0.6618,\ D_{3}^{-}=0.6584,\ D_{4}^{-}=0.7102\\ \end{array}\]  
Step 4: Assuming that the decision-maker holds a steady and firm attitude towards risk aversion, \(\alpha=0.5\) is set. The closeness of each subsidiary is calculated according to Formula (15):  
\[RC_{1}=0.4313,\ RC_{2}=0.4653,\ RC_{3}=0.4289,\ RC_{4}=0.5849\]  
Step 5: \(RC_{4}>RC_{2}>RC_{1}>RC_{3}\) can be obtained based on closeness, indicating that Subsidiary 4 has the more outstanding spare parts support capacity, so the spare parts support capacity solution of Subsidiary 4 is chosen for promotion.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Subsidiary**} & \multicolumn{4}{c}{**Evaluation Indicator**} \\ \cline{2-5}  & \(M_{1}\) & \(M_{2}\) & \(M_{3}\) & \(M_{4}\) \\ \hline \multirow{2}{*}{\(B_{1}\)} & \(<\)([0.7,0.8,0.9], \{0.3,0.4\}), & \(<\)([0.6,0.8,0.9], \{0.2,0.3\}), & \(<\)([0.8,0.9], \{0.3,0.4\}), & \(<\)([0.6,0.8], \{0.3,0.5\}), \\  & ([0.7,0.8], [0.4,0.5])\(>\) & ([0.6,0.8], [0.3,0.4])\(>\) & ([0.7,0.9], [0.2,0.3])\(>\) & ([0.6,0.7,0.8], [0.2,0.4])\(>\) \\ \hline \multirow{2}{*}{\(B_{2}\)} & \(<\)([0.7,0.9], \{0.3,0.4\}), & \(<\)([0.7,0.8], \{0.3,0.5\}), & \(<\)([0.6,0.7,0.8], \{0.4,0.5\}), & \(<\)([0.8,0.9], \{0.2,0.3\}), \\  & ([0.8,0.9], \{0.2,0.3\})\(>\) & ([0.7,0.8], \{0.2,0.3\})\(>\) & ([0.6
## 2 Materials and Methods
### System Reliability Analysis Combined with FORM  
In the paper, system reliability analysis is computed with the First Order Reliability Method (FORM). Using approximation methods (such as FORM) is essential to transform the limit state function from normal space to Gaussian normal space, which simplifies mathematical operation (because the mean value equals to 1 and standard deviation equals to zero):  
\[g_{i}(\mathbf{X}),\ i=1,\ 2,\ldots,\ m\ \ \ \ \ \rightarrow\ \ \ \ \ G_{i}(\mathbf{Z})\ i=1,\ 2,\ldots,\ m \tag{12}\]  
In the equation above, \(g_{i}\) and \(G_{i}\) are limit state functions for the \(i\)-th elements. \(\mathbf{X}\) and \(\mathbf{Z}\) are vectors of random variables in appropriate spaces: original and standardized, respectively.  
All the limit state functions divide the space into two parts: safe and failure areas. For few limit state functions, the total failure area is defined in different ways depending on the system type. In the case of a series system, the total failure area is the sum of individual failure areas (Figure 1a); in the case of a parallel system, this is the common part (Figure 1b).  
The failure area (\(\Omega_{\ell}\)) and the probability of failure (\(P_{f}\)) of the series system are defined as follows:  
\[\Omega_{f}=\bigcup_{i=1}^{n}\{G_{i}(\mathbf{Z})\leq 0\} \tag{13}\]  
\[P_{f}=P\Big{[}\bigcup_{i=1}^{n}\{G_{i}(\mathbf{Z})\leq 0\}\Big{]} \tag{14}\]  
where: \(n\)--number of elements of the series system  
For the parallel system, these values are defined as follows:  
\[\Omega_{f}=\bigcap_{i=1}^{m}\{G_{i}(\mathbf{Z})\leq 0\} \tag{15}\]  
Figure 1: Failure areas for: (**a**) series system, (**b**) parallel system.  
The probability of failure for a parallel system can be computed according to the following formula:  
\[P\left[\bigcap\limits_{i=1}^{m_{A}}(\beta_{i}^{*}\leq Y_{i}^{*})\right]=P\left[ \bigcap\limits_{i=1}^{m_{A}}(Y_{i}^{*}<-\beta_{i}^{*})\right]=\Phi_{m}(-\mathbf{B}^ {*},\mathbf{R}^{*}) \tag{24}\]  
where:  
\(\mathfrak{m}_{\Lambda}\)--number of active limit state functions,  
\(\mathbf{B}^{*}\)--vector of reliability indices for active functions  
\(\mathbf{R}^{*}\)--matrix of correlation coefficients \(r_{ij}\).(only for active functions)