esa-sceva/satcom-chunk-collection huggingface dataset:
# Dense Small Object Detection Based on an Improved YOLOv7 Model
## 4 Experiments and Analysis of Results
### Experimental Analysis of Different Improvement Points
#### 4.4.2 C3R2N Multi-Scale Feature Extraction Module  
In order to verify the effect of introducing the designed C3R2N module into the model, the following experiments were conducted on the basis of adding the P2 detection head and removing the P5 detection layer: in Experiment 1, the third ELAN_1 of the backbone network is replaced with the C3R2N module; in Experiment 2, the second and third ELAN_1 modules of the backbone network are replaced with C3R2N modules; in Experiment 3, all ELAN_1 modules are replaced with C3R2N modules; in Experiment 4, the third ELAN_2 in the neck network is replaced by the C3R2N module; in Experiment 5, the third ELAN_2 in neck network and the third ELAN_1 in backbone network are replaced with C3R2N modules; in Experiment 6, on the basis of Experiment 5, the second ELAN_2 of neck network is replaced with C3R2N module; in Experiment 7, on the basis of Experiment 6, the first ELAN_2 in neck network is replaced with C3R2N module; the results are shown in Table 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Experiment** & **mAP(s)\%** & **mAP(m)\%** & **mAP(l)\%** & **[EMAIL_ADDRESS]\%** & **Parameters/M** \\ \hline
1 & 18.6 & 38.7 & 48.8 & 48.8 & 36.53 \\
2 & 21.0 & 40.7 & 47.8 & 50.5 & 37.10 \\
3 & 21.3 & 40.8 & 47.2 & 50.9 & 15.49 \\
4 & 21.6 & 40.7 & 49.0 & 51.4 & 11.31 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Experiment of the detection head.  
Figure 11: Removal of the entire P5 layer.  
As can be seen from the results of the experiment on the C3R2N module results in Table 3, replacing various ELAN_1 and ELAN_2 modules with the C3R2N module generally resulted in improvements in detection performance compared to the baseline model. The best performance in terms of [EMAIL_ADDRESS]% was achieved in Experiment 1, where the third ELAN_1 of the backbone network was replaced, resulting in an [EMAIL_ADDRESS]% of 52.1%, which is higher than the baseline of 51.4%. The model parameters were reduced significantly with the introduction of C3R2N modules. The baseline model had 11.31 M parameters, whereas Experiment 7, which replaced the most ELAN modules, resulted in the fewest parameters (9.01 M). This suggests that the C3R2N module can reduce model complexity while maintaining competitive performance. While Experiment 1 achieved the highest [EMAIL_ADDRESS]%, Experiment 5 offered a good balance between high detection accuracy (51.9% [EMAIL_ADDRESS]%) and lower model parameters (9.30 M), making it a potentially more efficient choice. Therefore, we chose to replace the third ELAN_2 on the neck network and the third ELAN_1 on the backbone network with C3R2N.
## 2 Materials and Methods
### Multi-Temporal SPOT5 Pan-Sharpened Surface Reflectance
#### 2.1.4 Mask Creation  
Masks were created for each image at 10 m resolution. Although most images contained no or very little cloud, pixels contaminated by cloud and cloud-shadow were masked with a semi-automated object-based method, which included manual checking and editing [46]. Surface water was identified and masked using a multi-dimensional water index method [47]. The cast-shadow mask identified those pixels that lie in shadows cast from the surrounding topography, using a ray-casting technique [48], which was simplified through assuming parallel rays of light. The extreme-angles mask identified those pixels with incidence (angle between surface normal and vector to sun) and exitance (angle between surface normal and vector to satellite) angles greater than 80\({}^{\circ}\), which represent pixels with low signal to noise ratio due to little incidence or reflected light. The per-pixel surface normal, used in the calculation of these angles, was derived from the SRTM DEM.
## 3 Literature Review and Related Work
### Positioning with Discretization of the Positioning Areas and Classification-Based Models  
The following research works are related to our work in terms of the discretization of the positioning area and the use of a classification-based model to estimate the positions.  
In [73], the positioning area is discretized into different classes. The position is estimated through classification, through a weighted average of the probability for each class and the associated position for each class.  
In [207], CSI features are proposed as fingerprints for the uplink case. The positioning area is discretized into so-called grid points. An NN with a softmax output layer estimates the likelihood probability for the users for each grid point, arranged into a so-called probability map. The probability maps are fused across multiple access points to estimate the position of the users.  
In [208], the authors discretize the positioning area in an indirect way by assigning a class to a number of reference points distributed across the positioning area. An FNN with a softmax output layer is trained to map an input vector of fingerprints to a one-hot-encoded vector associated with the class of the reference point where the measurement of the fingerprints is performed. In the testing phase, fingerprints are collected at an arbitrary position. Then, the softmax layer outputs the likelihood probability for each reference point. By relying on the tabulated relationship of each reference point to its coordinates, the position is estimated using the weighted average of the likelihood probability of the class multiplied by the reference position. In addition, the past position estimates are aggregated in the position domain with the current position estimate through a weighted average, influenced by a component from the fingerprint domain given by the similarity of the current vector of fingerprints to past ones.  
Other works propose dividing the positioning area into grids and implementing a classification-based model to estimate the positions associated with these grids [120,150,193].
# Methodology for Improving Scanning Performance Loading an Array Element with a 3D All-Metal WAIM
## 2 Proposed Unit-Cell and Equivalent Circuit
### EC Validation  
As a validation example, a unit-cell is designed at \(f_{0}=18\) GHz. It has the following geometrical parameters: \(p=4.5\) mm, \(t=0.42\) mm (metal thickness) \(w_{y}=0.15p\) and \(l=0.352\lambda_{0}\) where \(\lambda_{0}\) is \(c/f_{0}\). The validation of the EC is performed comparing the unit-cell full-wave _S-Parameters_ with the EC _S-Parameters_. The considered frequency band is [13; 14; 15; 16; 17; 18] GHz. This validation is realized for five \(\theta\) angles and both _E-_ and _H-planes_.  
Firstly, the validation of the EC for the unit-cell without any load is shown in Figure 3.  
These results show a very good agreement for all \(\theta\) values and for both scanning planes.  
The second example is the cell loaded by two vertical columns in the loading plane of the cell as in Figure 4, with the geometrical parameters: \(A_{x}=0.775\) mm, \(R=0.32\) mm and \(l_{1}=l_{2}=\frac{l}{2}\).  
The results are shown in Figure 5. These results also show a quite good agreement for the five \(\theta\) cases and for both incident planes, although the discrepancy with the full-wave simulations is a bit larger. The mutual coupling between the load and the input/output discontinuities may affect the accuracy of the EC, especially because the cell length is quite small. Moreover, the modeling does not account for the real dimensions of the columns  
Figure 3: Reflection coefficient validation unloaded unit-cell: **(a,b)**_H-plane_ and **(c,d)**_E-plane_, magnitude and phase, respectively.  
(especially their radius \(R\)), as it assumes the loading is concentrated in a single vertical plane along the cell.  
The load used in this example mostly produces an inductive effect. As \(\theta\) varies from \(0^{\circ}\) to \(55^{\circ}\), the extracted \(Z_{x}(\theta)\) changes from 60j to 54j in the _H-plane_, while in the _E-plane_, it does not depend on the angle. Therefore, for the sake of simplicity, in the next section, it is considered that \(Z_{x}\) does not depend on \(\theta\).
## Materials and Methods
### Search Methods for Identification of Studies  
The literature search was performed following PRISMA guidelines [19]. The databases were searched up to October 2021. MEDLINE was searched with the PubMed search engine, www.ncbi.nlm.nih.gov (accessed on 5 October 2021). The other databases were Scopus, (www.scopus.com, accessed on 5 October 2021), Embase (www.embase.com, accessed on 5 October 2021), and Cochrane Central (www.cochranelibrary.com, accessed on 5 October 2021). The population, intervention, comparison, and outcome (PICO) strategy was 1. Population: Extracted mature permanent human teeth undergoing root canal preparation using needle irrigation. 2. Intervention: NaOCI or CHX irrigation solutions used after the removal of the smear layer by EDTA. 3. Comparisons: Extracted mature permanent human teeth with complete root formation undergoing root canal preparation either without irrigation or with irrigation with saline. 4. Outcome: Depth of bacterial penetration into the dentinal tubules. The search terms were: dentin OR dentinal tubule OR dentinal tubule penetration OR root canal biofilm OR _Enterococcus faecalis_ OR _E. faecalis_ OR bacterial penetration OR bacteria penetration AND irrigant OR needle irrigation OR saline OR NaOCI OR sodium hypochlorite OR saline AND confocal OR SEM OR scanning electron microscopy OR dentinal shavings OR CFU. The search was limited to the English language. The bibliographies of included articles were searched manually in order to identify extra studies not detected by the electronic search. MeSH received the following: ((\"dentin\"[MeSH Terms] OR \"dentin\"[All Fields] OR \"dentine\"[All Fields] OR \"dentine\"[All Fields] OR \"dentin s\"[All Fields] OR \"dentinal\"[All Fields] OR \"dentine s\"[All Fields] OR (\"dentin\"[MeSH Terms] OR \"dentin\"[All Fields] OR \"dentine\"[All Fields] OR \"dentines\"[All Fields] OR \"dentines\"[All Fields] OR \"dentines\"[All Fields] OR \"dentins\"[All Fields] OR \"dentins\"[All Fields] OR \"dentins\"[All Fields])) AND (\"tubule\"[All Fields] OR \"dentine s\"[All Fields]) AND (\"tubule\"[All Fields] OR \"tubule s\"[All Fields] OR \"tubular\"[All Fields] OR \"tubules\"[All Fields])) OR ((\"dentin\"[MeSH Terms] OR \"dentin\"[All Fields] OR \"dentine\"[All Fields] OR \"dentines\"[All Fields] OR \"dentin\"[All Fields] OR \"dentine\"[All Fields]) OR \"dentine\"[All Fields] OR \"dentine\"[All Fields]) AND (\"tubule\"[All Fields] OR \"dentine\"[All Fields] OR \"tubular\"[All Fields]) AND (\"penetrability\"[All Fields] OR \"penetrable\"[All Fields] OR \"penetrated\"[All Fields] OR \"penetrates\"[All Fields] OR \"penetrating\"[All Fields] OR \"penetrations\"[All Fields] OR \"penetrations\"[All Fields]) OR \"root canal\"[All Fields]) AND (\"biofilm s\"[All Fields] OR \"biofilmed\"[All Fields] OR \"biofilms\"[MeSH Terms] OR \"biofilms\"[All Fields] OR \"biofilm\"[All Fields] OR \"_Enterococcus faecalis_\"[All Fields] OR \"_E.
## 2 Materials and Methods
### Blast Experimentation  
1. The four OB benches chosen for experimentation were 1 de-coaled, 3A coal seam, and 3A de-coaled with bench heights of 12 m, 10.5 m, 11 m, and 9.5 m, respectively. Because the mine was previously worked underground, there was a high risk of disturbances in strata and induction of cracks. To address this issue, benches were initially cleaned to a depth of 0.3 m for improved visibility and identification of cracks and joint planes, as shown in Figure 9a,b.
2. The joint planes of the benches were identified by STRAYOS software and marked with white powder on the bench top surface to avoid the seizing-up of drilling bits in joints and to have a reference in deciding blast pattern and connections, as shown in Figure 10. Table 1 shows the design burden and spacing values in the O-PITBLAST that yielded good predicted results. The drill bit diameter was 150 mm, which was adequate for the existing bench height, burden, and spacing, and the drilled holes are shown in Figure 11.  
Figure 8: (a,b) Blast Verification and Wire Mesh model with charging.  
Figure 7: Blast design in O-PITBLAST.  
Figure 9: (a,b) Bench leveling and preparation for drilling.  
A total of 152 blasts were performed to determine the effect of Se/Be ratio, total explosive quantity, and firing pattern on rock fragmentation and ground vibration. In all these blasts, site mixed emulsion (SME) explosive was used with a booster-nonel combination, and Down hole delays of 425 and 450 ms were chosen, with hole-to-hole delays of 17 milliseconds and row-to-row delays of 25 and 42 ms, adopted, respectively, according to rock conditions. The average explosive quantity was between 45-55 kg and 2 boosters per 1 m of the hole were used.  
152 blasts were performed in four phases in 27 benches. The phases are A, B, and C, as shown below.  
**Phase A:** All blast design parameters were maintained the same, but the firing pattern was altered with respect to joint angle.  
**Phase B:** All blast design parameters were maintained the same, but the Spacing Burden Ratio was altered with respect to joint angle.  
**Phase C:** All blast design parameters were maintained the same, but the Explosive Quantity was altered with respect to joint angle.
## 4 Simulation
### Impact of Buffer Size  
To evaluate the impact of buffer space on the performance, we vary the buffer size from 50 to 250 messages and get the delivery ratios and average delay of four protocols (see Figure 4). From Figure 4(a), we find that the data delivery ratio of all evaluated protocols increases as the buffer size increases, which shows the node buffer size has a remarkable influence on the performance of the multiple copy delivery protocols. Epidemic protocol is more sensitive to the buffer size compared to the others because it generates many redundant messages so that needs much more buffer space. Due to the fact that DRADG generates much less message replicas than the epidemic, FAD and Spray-and-Wait protocols, it needs less amount of buffer space. Therefore, DRADG outperforms the other three protocols most of the time. In Figure 4(b), it can be seen that the average delivery delays of all four protocols show an incremental trend when the node buffer size increases. The reason is that larger node buffer size leads to more data message exchange between nodes, so the chance of network congestion also increases.
## 3 Results and Discussion  
### Structure Characterization  
Figure 2a,b shows the crystal structure of the perovskites (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbBr\({}_{4}\) and (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbI\({}_{4}\). The Pb atoms are six-coordinated by the halogen to form [PbX\({}_{6}\)] octahedrons, which are corner-sharing to constitute three-dimensional frameworks. It can be observed that the perovskites (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbBr\({}_{4}\) and (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbI\({}_{4}\) exhibit the structure with alternating organic and inorganic layers. The growth directions of the layers are along the a axis and b axis, respectively. The lattice parameters of the perovskite (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbBr\({}_{4}\) and (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbI\({}_{4}\) are a = 3.3394 nm, b = 0.8153 nm, c = 0.8131 nm and a = 0.8689 nm, b = 2.878 nm, c = 0.9162 nm, which are consistent with reports in the literature [23].  
Figure 1: The experiment optical path of photoluminescence test system. PZT, piezoelectric ceramic transducer; O, objective; A, half-wave plate; B, polarizer; C, low pass filter; M, reflector; CCD, charge couple device; S, spectrometer; PMT, photomultiplier tube.  
Figure 3a,b shows the XRD patterns of the perovskite powders (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbBr\({}_{4}\) and (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbI\({}_{4}\). The diffraction peaks of the perovskite (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbBr\({}_{4}\) are located at 6.41\({}^{\circ}\), 10.63\({}^{\circ}\), 15.96\({}^{\circ}\), 21.32\({}^{\circ}\), 26.72\({}^{\circ}\), 32.18\({}^{\circ}\) and 37.73\({}^{\circ}\), which correspond to the lattice plane (X00, \(X\) = 2, 4, 6, 8,... ). The diffraction peaks of the perovskite (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbI\({}_{4}\) are located at 6.14\({}^{\circ}\), 12.29\({}^{\circ}\), 18.48\({}^{\circ}\), 24.73\({}^{\circ}\), 31.05\({}^{\circ}\) and 37.47\({}^{\circ}\), which are correspond to the lattice plane (0X0, \(X\) = 2, 4, 6, 8,... ). The calculated interlayer spacing of the perovskites (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbBr\({}_{4}\) and (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbI\({}_{4}\) are 1.44 nm and 1.68 nm, respectively, using the Bragg formula: 2 dsin\(\theta\) = n\(\lambda\). The experimental XRD patterns of the perovskites (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbBr\({}_{4}\) and (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbI\({}_{4}\) are consistent with calculated values, confirming there is no tri-halide perovskite or PbX\({}_{2}\) phase [23]. All the patterns confirmed that the perovskites (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbBr\({}_{4}\) and (C\({}_{6}\)H\({}_{5}\)CH\({}_{2}\)NH\({}_{3}\))\({}_{2}\)PbI\({}_{4}\) are well crystallized and oriented.
## 2 Review of Design Provisions and Concepts
### Australia Masonry Standards  
The guidelines for designing URM walls under axial compression in Australia, as outlined in AS 3700, 2018 [26], ensure the safe and efficient construction of masonry walls that do not contain any reinforcing elements such as steel bars or mesh. In the previous edition of the Australian masonry design standards, AS 3700, 2011 [39], the maximum compressive strength for grouted masonry was set at 1.3 times the compressive strength of the masonry units. The lower limit for grout strength was 12 MPa. However, the approach to incorporating grout strength has been revised in the updated version. The new requirement stipulates that the grout strength must be equal to or greater than the characteristic compressive strength of the masonry, with an upper limit of 50 MPa. Consequently, the design expression for URM walls subjected to axial compressive loads has been revised based on the expression provided in Equations (5) and (6).  
\[\mathrm{P}=\Phi f^{\prime}_{m}\mathrm{A_{b}}\qquad\qquad\mathrm{ungrouted\ URM} \tag{5}\]  
\[\mathrm{P}=\Phi\Bigg{\{}f^{\prime}_{m}\mathrm{A_{b}}+\ f_{\mathrm{g}}\frac{f^{ \prime}_{\mathrm{cg}}}{1.3}\ \left(0.55+0.005f_{\mathrm{cg}}\right)\ \mathrm{A_{g}}\Bigg{\}}\quad\mathrm{ grouted\ URM} \tag{6}\]  
where \(\Phi\) is the capacity reduction factor, which takes different values depending on the type of masonry: 0.5 for unreinforced hollow masonry, 0.6 for unreinforced grouted masonry, and 0.75 for unreinforced solid masonry. \(\mathrm{A_{b}}\) refers to the bedded area of the masonry unit, while \(f_{\mathrm{g}}\) denotes the compressive strength of grout. \(F_{m}\) is the characteristic compressive strength of the masonry, while \(f_{cg}\) is the design characteristic compressive strength of the grout. \(\mathrm{A_{g}}\) denotes the design cross-sectional area of the grout.  
AS3700 permits a simple assumption that when a roof or floor connects to a wall, its load can be regarded as acting with an eccentricity equal to one-third of the bearing thickness.  
Each side of the wall is considered to be supported by half of the bearing thickness if the floor or roof spans across the wall. In this approach, the vertical load coming from the level above is assumed to be centred at the level under consideration. Figure 2 depicts these simplified assumptions for evaluating eccentricity. In the case of a slab connected to either a solid wall or one leaf of a cavity wall (as shown in Figure 2a), the resulting eccentricity is computed using Equation (7). For a slab that spans continuously over a solid wall (as shown in Figure 2b), the resulting eccentricity is calculated based on Equation (8)  
\[\mathrm{e}=\frac{\mathrm{W}_{2}\times\frac{\mathrm{t}_{\mathrm{w}}}{6}}{ \mathrm{W}_{1}+\mathrm{W}_{2}} \tag{7}\]  
\[\mathrm{e}=\frac{(\mathrm{W}_{2}-\mathrm{W}_{1})\times\frac{\mathrm{t}_{ \mathrm{w}}}{6}}{\mathrm{W}_{1}+\mathrm{W}_{2}+\mathrm{W}_{3}} \tag{8}\]
# A Randomized Algorithm for Optimal PID Controllers
## 3 Statement of the Problem
Therefore \(\frac{J\left(\hat{x}_{0},\hat{K}\right)}{\left\|\hat{x}_{0}\right\|^{2}}\leq \sigma_{max}\left(\hat{K}\right)\), with equality in the worst case. Let \(\Omega\) denote a subset of \(\mathbb{C}_{-}\), which is symmetric with respect to the \(x\)-axis, containing a positive length segment of the real axis, with some neighborhood of it, and such that \(\Omega=\overline{int\left(\Omega\right)}\) (i.e., \(\Omega\) is the closure of its interior). Let \(\mathcal{S}^{q\times r}\) denote the set of all matrices \(\hat{K}\in\mathbb{R}^{q\times r}\), such that \(\hat{A}_{\varepsilon\ell}\left(\hat{K}\right)\) is stable, i.e., \(\sigma\left(\hat{A}_{\varepsilon\ell}\left(\hat{K}\right)\right)\subset \mathbb{C}_{-}\). By \(\mathcal{S}^{q\times r}_{\Omega}\), we denote the set of all matrices \(\hat{K}\in\mathbb{R}^{q\times r}\), such that \(\sigma\left(\hat{A}_{\varepsilon\ell}\left(\hat{K}\right)\right)\subset \Omega\). In this case, we say that \(\hat{A}_{\varepsilon\ell}\left(\hat{K}\right)\) is \(\Omega\)-stable (or, the augmented system (1) is \(\Omega\)-stable). In the sequel, we will occasionally write \(\mathcal{S}_{\Omega}\) instead of \(\mathcal{S}^{q\times r}_{\Omega}\), when it is clear what the size of the related matrices is. Please note that \(\mathcal{S}_{\Omega}\) is a closed set, since \(\Omega\) is closed and that \(\mathcal{S}_{\Omega}\) has nonempty interior (if it is nonempty) since \(\Omega\) has nonempty interior. Under the assumptions that \(D_{2,1}=0\) and that \(\left(\hat{A},\hat{B}_{2}\right),\left(\hat{A}^{T},\hat{C}_{2}^{T}\right)\) are controllable (and the additional assumption (5) when dealing with the \(H_{2}\)-norm minimization), for the PID controller problem with minimal \(H_{\infty}\)-norm, we need to solve:  
\[\hat{D}_{1,1}=D_{1,1}-D_{1,2}M^{-1}K_{D}C_{2}B_{1}=D_{1,1}-D_{1,2}\hat{K}_{3}C _{2}B_{1}=0, \tag{4}\]2 (12) such that: \[\left\{ \begin{aligned} &\hat{K}\in\mathcal{S}_{\Omega}\\ & I-\hat{K}_{3}C_{2}B_{2}\text{ is invertible}\end{aligned} \right.\]  
where \(T_{\hat{x},\hat{z}}\) is given by (3). For the PID controller problem with minimal \(H_{2}\)-norm, we need to solve:  
\[\min\;f_{H_{2}}\left(\hat{K}\right):=\left\|T_{\hat{x},\hat{z}} \right\|_{H_{2}}\] (13) such that: \[\left\{ \begin{aligned} &\hat{K}\in\mathcal{S}_{\Omega}\\ &\hat{K}_{3}\text{ has the structure (\ref{eq:P})}\\ & I-\hat{K}_{3}C_{2}B_{2}\text{ is invertible}\end{aligned} \right.\]  
where \(T_{\hat{x},\hat{z}}\) is given by (3) and, for the PID controller problem with minimal LQR functional-value, we need to solve:  
\[\min\;f_{LQR}\left(\hat{K}\right):=\sigma_{max}\left(\hat{K}\right)\] (14) such that: \[\left\{ \begin{aligned} &\hat{K}\in\mathcal{S}_{\Omega}\\ & I-\hat{K}_{3}C_{2}B_{2}\text{ is invertible}\end{aligned} \right.\]where \(\sigma_{max}\left(\hat{K}\right)\) is given by (11).