esa-sceva/satcom-chunk-collection huggingface dataset:
# An Efficient Carrier Synchronization Scheme for Demodulation Systems
## 3 Proposed Carrier Synchronization Algorithm
### Carrier Phase Recovery Algorithm  
After the CFO compensation, residual frequency offset and residual phase offset still exist in the signal, so residual phase estimation and phase compensation are also needed. In this paper, the PA phase linear interpolator algorithm is used for phase estimation and compensation, and the phase error detector uses the pilot block to estimate the phase error. The expression is:  
\[\theta_{\text{k}}=\text{arg}\left[\sum\limits_{n=1}^{36}x(n)\cdot p^{*}(n)\right] \tag{14}\]  
where \(x(n)\) is the input sequence of the CPR, \(p(n)\) is the known pilot sequence, and \(p^{*}(\text{n})\) is the conjugate of \(p(\text{n})\). The PA phase linear interpolator algorithm implements linear  
Figure 6: Block diagram of carrier frequency offset estimation regarding RLR algorithm.  
interpolation between the preamble pilot block and the current pilot block, which is shown in Figure 7. The phase linear interpolation expression is:  
\[\theta_{\mathrm{k_{i}}}=\theta_{\mathrm{k-1}}+\frac{(\theta_{\mathrm{k}}-\theta _{\mathrm{k-1}})\times i}{L_{k}} \tag{15}\]  
where \(\theta_{\mathrm{k}}\) represents the estimated phase of the \(i\)-th data symbol between the \((k-1)\)-th pilot block and the \(k\)-th pilot block, \(\theta_{\mathrm{k}}\) is the phase offset evaluation value of the \(k\)-th pilot block, and \(L_{k}\) denotes the number of symbols between the \((k-1)\)-th block and the \(k\)-th pilot block. The estimated range of the phase error detector should be between \([-\pi,\pi]\).  
As shown in Figure 8, the implementation structure of the CPR algorithm is composed of Unito and Unit7. The de-modulated pilot block is selected to accumulate. Moreover, the current pilot block phase \(\theta_{\mathrm{k}}\) is calculated in Unit6. The phase difference value \(\Delta\theta_{\mathrm{k}}\) between the current pilot block phase \(\theta_{\mathrm{k}}\) and the preamble block \(\theta_{\mathrm{k-1}}\) is calculated. Moreover, the phase difference \(\Delta\theta_{\mathrm{k}}\) is limited between \([-\pi,\pi)\) in Unit7. The index value \(index\) and the distance \(len\) between the preamble block and the current pilot block are used to calculate a linear interpolation \(\theta_{\mathrm{k}}\), which is limited to between \([-\pi,\pi)\), and the linear interpolation is input to the DDS for feedforward compensation.
## 4 Motion Planning Algorithm Based on Co-Optimization of Communication and Sensing
### Communication Topology Optimization Based on a Minimum Weighted Spanning Tree  
In this paper, the packet routing algorithm always chooses the route with the highest successful transmission probability. To find such a route, we define the weight of each link as:  
\[W^{m,n}=-\ln(P_{r}^{m,n}) \tag{42}\]  
Figure 3: Sketch of the distributed decision-making model for multiple UAV cooperative target tracking.  
where smaller weight, indicates a higher the probability of successful transmission. For UAV \(A_{i}\), the problem of finding the route with the highest successful transmission probability can be represented via solving:  
\[\text{route}^{*}_{i,k}=\arg\max_{\text{route}_{i,k}}\beta_{i,k}=\arg\min_{ \text{route}_{i,k}}\left[\sum_{(m,n)\in\text{route}_{i,k}}W^{m,n}\right] \tag{43}\]  
which can be solved by Dijkstra algorithm.  
For the entire multiple UAV system, to ensure that all the UAVs and the base station relate to the highest successful transmission probabilities, the concept of the minimum weighted spanning tree (MST) [26] from graph theory is used to determine the topology of the communication network. Figure 4 shows an example communication topology and its minimum weighted spanning tree. The base station is the root node that directly or indirectly connects to all the other nodes (the UAVs). The MST must connect all nodes with the condition that the sum of the weights is minimized. Thus, the overall probability of a message \"floating\" in the MST network is maximized.  
At each time \(k\), the MST can be constructed according to the positions of UAVs and the base station. Based on such MST communication topology, Dijkstra algorithm is used to find the route with the highest successful transmission probability between each UAV \(A_{i}\) and the base station. The MST communication topology determines the planning hierarchy, which is the order of the optimization calculation in the distributed frame. All UAVs are sorted in the ascending order of their total hops numbers of the optimal transmission route to the base station. If the total hops are the same, they are arranged in ascending order of their identity numbers. This sorted result is the planning hierarchy, in which each UAV plans its trajectory based on the planned results of the aircrafts which are higher in the hierarchy and sent the planned results to the aircrafts which are lower in the hierarchy. For example, in Figure 4, \(A_{1}\) acts a relay for \(A_{2}\), thus the successful transmission probability between \(A_{2}\) and the base station can be represented as \(\beta_{2,k+l}=\beta_{2\to 1\downarrow k\downarrow}\cdot\beta_{1,k\downarrow l}\). Where \(\beta_{1,k+l}\) denotes the transmission probability between \(A_{1}\) and the base station, and \(\beta_{2\to 1\downarrow k+l}\) denotes the transmission probability between \(A_{2}\) and \(A_{1}\). It can be seen that \(A_{1}\) computes its own trajectory by itself, while \(A_{2}\) computes the trajectory based on the planned results of \(A_{1}\).
## 5 Applicability of Static Testing to Dynamic Crash Performance  
While static testing does not replicate the strain rate effects and inertia forces present during actual crash events, it remains a widely accepted method for evaluating the ultimate capacity and failure mechanisms of structural components. In fact, static load testing has been used in prior qualification studies of GFRP-reinforced barriers (e.g., [26; 28; 29; 30; 32]) to complement full-scale crash testing and to validate design assumptions under controlled conditions.  
The Canadian Highway Bridge Design Code, CHBDC [62], specifies that barrier design is to be based on the use of relevant and existing full-scale crash test data as well as a commonly used method for ensuring that the barrier design satisfies the crash test requirements. CHBDC specifies that changes to the details of a traffic barrier or traffic barrier transition that meet the crash test requirements may be made, provided that any changes affecting the geometry, strength, or behavior of the traffic barrier can be demonstrated not to adversely affect vehicle-barrier interaction. The CHBDC clause for barrier anchorage to the deck states that the suitability of a traffic barrier anchorage shall be based on its performance during crash testing of the traffic barrier. For an anchorage to be considered acceptable, significant damage shall not occur in the anchorage or deck during crash testing. If crash testing results for the anchorage are not available, the anchorage and deck shall be designed to resist the maximum bending, shear, and punching loads that can be transmitted to them by the traffic barrier, except that these loads need not be taken as greater than those resulting from the CHBDC traffic loads (i.e., equivalent transverse vehicle impact loading). This can be achieved by conducting static tests to collapse on the anchorage and/or performing analysis using an approved numerical method such as finite-element computer modeling.  
The commentaries of CHBDC suggest the following examples where changes may potentially be acceptable: (a) changes to an anchorage system, where failure of the anchorage is not exhibited during crash testing and the strength of the replacement anchorage is determined to be equivalent to that of the original, and (b) substituting materials with properties identical or superior to the original material as far as behavior during crash testing is concerned.  
In conclusion, although dynamic effects such as inertia and strain-rate sensitivity were not directly simulated, the static testing methodology employed in this research is consistent with accepted engineering practices for validating ultimate capacity and failure mechanisms. The GFRP-reinforced systems demonstrated performance exceeding factored design loads, and their structural adequacy was further corroborated by previous full-scale crash tests cited in the study. Therefore, the findings remain applicable to real-world crash scenarios, especially in conjunction with the design provisions and interpretive clauses of the CHBDC that allow for such validation methods.
## 5 Simulation Verification  
### Impact of DC Cable Length  
Figure 14 shows the simulation results of the VSC-HVDC system with virtual impedance control strategy under different DC cable lengths. The voltage control parameters are \(k_{p\alpha c}=15\), \(k_{i\alpha c}=100\), \(k_{i}=10\), \(R_{eq}=0.5\)\(\Omega\), and \(L_{eq}=0.002\)\(\Omega\). At 2 s, the steady-state power command value is set from 500 MW to \(-500\) MW. As shown in Figure 14, the blue line indicates the simulation results when the DC cable length is 50 km, and the red line indicates 100 km. The DC voltage in the blue line drops by approximately 5% and begins to oscillate, and the system loses stability. The DC voltage in the red line drops by approximately 5%, smoothly restoring the instruction value, and the system remains stable. The simulation results are consistent with the theoretical analysis in Section 4.4.  
Figure 14: DC-link voltage and active power of the VSC–HVDC system under different DC cable lengths.  
Figure 13: Nichols plots of \(T_{m}\) under a different grid impedance.
## 3 Results and Discussion
### TPC, TFC, and Antioxidant Activities
#### 3.1.2 Molecular Docking Interaction  
The molecular level interaction of _BCL-2_, _Bax_, _p53_, _Casp 3_, and _Casp 9_ protein and the 15 compounds from _Aquilaria gallocha_ Roxb. and _Aquilaria malaccensis_ Lam. was analyzed using the molecular docking method. This work used the AutoDock Vina program to run a docking simulation using the crystal structures of the proteins _BCL-2_, _Bax_, _p53_, _Casp 3_, and _Casp 9_. The known inhibitor BDA-366 was utilized as a control, and the crucial active site residues were kept flexible. The creation of hydrogen bonds and the binding energy to the important active residues and the chosen compounds served as confirmation of the interaction results. According to the analysis of docking results, Grewin interacts with _BCL-2_ via two hydrogen bonds (ASP108 and ARG143) with binding affinities of \(-\)7.5 kcal/mol each. The control drug BDA-366 and Grewin establish two hydrogen bonds, one of which is similar to that of the control drug. The detailed docking results of the best compound with _BCL-2_ and the control ligand are shown in Table 3. An illustration showing the interactions of Grewin and BDA-366 with the residues in the active site of _BCL-2_ Figure 2A,B. The 2D interaction diagram for the top five ligands with each of the proteins employed in this investigation is also included in Supplementary Materials Tables S1-S11. Surprisingly, among all the 15 compounds from this study, Grewin ranked best among all the proteins used. These results showed that Grewin is a promising compound in treating cancer and inflammation. It also further reveals that the presence of Grewin and the fifteen other compounds in the extracts of _Aquilaria gallocha_ Roxb. and _Aquilaria malaccensis_ [PERSON].  
Figure 1: ADMET properties of (**A**) Grewin; (**B**) BDA-366. Abbreviations: MW: Molecular weight; nRig: number of rigid bonds; fChar: formal charge; nHet: number of heteroatoms; MaxRing: number of atoms in the biggest ring; nRing number of rings; nRot: number of rotatable bonds; TPSA: topological polar surface area; nHD: number of hydrogen bond donors; nHA: number of hydrogen bond acceptor; LogD: logP at physiological pH 7.4; logS: log of the aqueous solubility; and LogP: log of the octanol/water partition coefficient.  
might be the reason for its anti-cancer and anti-inflammatory activity, which needs further evaluation. The docked ligands Grewin and BDA-366 with _BCL-2_ were further subjected to MD simulation.
## 3 Results: Publication Analysis
### Relationships between Sense of Presence and the Study Variables
#### 3.4.1 Relationships between Sense of Presence and Environmental Stimuli  
The analyzed studies revealed that different environmental stimuli had significant effects on the intensity of presence experienced. The results are shown in Tables 10 and 11. Regarding biological stimuli, seven studies established that relationships existed between environmental stimuli and the experience of presence. One study [21] found greater presence at warm temperatures than at cold. Studies [22; 26] showed increases in presence when smells congruent with the virtual scenario were incorporated. Studies [23; 32] agreed on the positive relationship between smells and sense of presence but differed in terms of tactile stimulation. Study [32] found no differences in presence when wind-based tactile stimuli were added, while [23] established that a relationship existed between tactile stimuli and presence. The authors of study [24] proposed that a closer relationship existed between tactile (using textiles) and presence than existed between smell and presence, although it should be noted that they used a less advanced smell stimulation medium (aromatic oil) than [23; 32]. Regarding auditory stimuli, [23; 59] found increases in presence when sounds were presented, with [23] making a presence/absence examination of audition; [59] examined different audio configurations, finding that monoaural and artificial sounds should be avoided. \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Ref.**} & \multirow{2}{*}{**Categories**} & \multirow{2}{*}{**Human Responses (HR)**} & \multirow{2}{*}{**HR Behavior**} \\ \cline{5-6}  & & & & **Presence** & & **HR is (\textless{} / \textgreater{}) in (A) than in (B)** \\ \hline \multirow{2}{*}{[21]} & \multirow{2}{*}{Stimulus} & \multirow{2}{*}{Biological,} & Tactile (Temperature) & \multirow{2}{*}{Presence} & \multirow{2}{*}{(Heat)} & \multirow{2}{*}{\(>\)} & (Cold) \\ \cline{5-6}  & & & & & & \\ \hline \multirow{2}{*}{[22]} & \multirow{2}{*}{Stimulus} & \multirow{2}{*}{Biological,} & \multirow{2}{*}{Smell} & \multirow{2}{*}{Presence} & (Dissonant & \multirow{2}{*}{\(\equiv\)} & (Control) \\  & & & & & & (Dissonant \\  & & & & & & (Consonant \\  & & & & & & Smell) \\ \hline \multirow{2}{*}{[26]} & \multirow{2}{*}{Stimulus} & \multirow{2}{*}{Biological,} & \multirow{2}{*}{Smell} & \multirow{2}{*}{Spatial Presence} & (Smell) & \multirow{2}{*}{\(>\)} & (Control) \\ \cline{5-6}  & & & & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Effects of environmental stimuli on the sense of presence.
# Temporal Logic Planning and Receding Horizon Control for Signal Source Localization
## 6 Experimental Results  
In what follows, the effectiveness of Algorithm 1 is validated for signal source localization. The real testing environment is shown in Figure 6.  
The number of signal sources is 3, and their positions are (0.5, 3), (2.5, 3.5) and (3, 0.5). Table 4 illustrates the average results and standard errors on localization time and path length. Figures 7-9 show the movement trajectories of the mobile robot and the predicted position of the signal sources, where the green star denotes the real source positions, and the red dot refers to the predicted source positions. The red circle is the starting position, and the black blocks are the obstacles. Hence, RHC-TLP can control the mobile robot to find all of the signal sources. From Table 3, one can see that the localization time and path length increases significantly with the number of obstacles from 0 to 2. The main reason is that the robot avoids obstacles in the movement process. Figure 10 shows the average localization errors. One can see that as the number of obstacles increases, the average localization errors also increase.  
Figure 5: The average localization errors.  
Figure 6: The experimental environment.  
Figure 8: The robot movement trajectory in the real environment with 3 signal sources and 1 obstacle.  
\begin{table}
\begin{tabular}{l l l} \hline
**Obstacles** & **Localization Time** & **Path Length** \\ \hline
0 & 77.6 (3.9) & 10.23 (0.13) \\
1 & 91.6 (4.5) & 10.83 (0.21) \\
2 & 96.4 (5.2) & 11.05 (0.29) \\ \hline \end{tabular}
\end{table}
Table 4: Average results (Standard errors) on localization time (s) and path length (m) in light of 10 runs.  
Figure 7: The robot movement trajectory in the real environment with 3 signal sources and 0 obstacles.
# A Novel Energy Optimization Control Strategy for Electric Drive System Based on Current Angle
## 4 Steady State and Transient State Simulation Validation
### Transient State Simulation Validation
In Figure 15c, it can be seen that all kinds of losses of EDS have been greatly reduced, which showed that under the conditions of 4000 rpm/60 Nm, 4000 rpm/40 Nm, 5000 rpm/40 Nm, and 6000 rpm/40 Nm, the efficiency of EDS had increased from 88.5%, 87.7%, 86.5%, 86.2% to 90.7%, 91.3%, 92.4%, 92.7%, respectively. From the whole transient process, when the EDS speed did not exceed the base speed between 0-0.2 s, the efficiency of EDS under the control strategy in this paper was slightly higher than that of MTPA. After 0.2 s, the EDS speed exceeded the base speed, and the optimization value of EDS efficiency of this strategy gradually increased. When the motor was stable to 6000 rpm/40 Nm, it reached the maximum value. The highest value was 92.7%, which is 6.5% higher than 86.2% of the traditional control. Specific efficiency comparison is illustrated in Table 3.  
Figure 16 shows the \(\beta\) and \(i_{s}\) got by the proposed control in this paper during this transient process. It can be found that in region before the base speed, \(\beta\) and \(i_{s}\) constantly changed to meet the optimal EDS efficiency. When the motor speed exceeded the base speed, \(\beta\) and \(i_{s}\) obtained according to the current speed and torque demand of the EDS were gradually reduced, then they remained unchanged when the motor worked stably. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Time (s)**} & **Traditional Control** & **Proposed Control** & **Efficiency** \\  & **Efficiency (\%)** & **Efficiency (\%)** & **Improvement (\%)** \\ \hline
0.05 & 86.2 & 86.6 & 0.3 \\
0.105 & 80.3 & 80.3 & 0 \\
0.15 & 82.0 & 82.4 & 0.4 \\
0.21 & 86.6 & 85.4 & 1.2 \\
0.25 & 88.7 & 91.0 & 2.3 \\
0.31 & 87.5 & 89.1 & 1.6 \\
0.35 & 86.5 & 92.4 & 5.9 \\
0.41 & 88.0 & 90.5 & 2.5 \\
0.45 & 86.2 & 92.7 & 6.5 \\
0.5 & 86.2 & 92.7 & 6.5 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Efficiency comparison chart of transient process at different time. Figure 16: Change chart of \(\beta\) and \(i_{s}\) in proposed control strategy at transient process.
## 3 Unconfined Compression Test  
### Sample Preparation and Test Procedure  
An unconfined compression test was used to investigate the effect of the mixture ratio of the solidifying agent on the strength of the solidified Caofeidian dredger fill. The mix proportion design for the unconfined compression test is presented in Table 4. The 10% slag was used as the main solidifying agent, and three additives (sodium silicate, quicklime, and gypsum powder) at different content levels were applied to two approaches for sample preparation: single-additive and composite-additive. The orthogonal design is applied in the composite additive method.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Number** & **Test Type** & **Slag** & **Quicklime** & **Sodium** & **Gypsum** \\  & & **(\%)** & **(\%)** & **Silicate (\%)** & **Powder (\%)** \\ \hline A0 & & 0 & 0 & 0 & 0 \\ A1 & & 10 & 0.5 (Level-1) & 0 & 0 \\ A2 & & 10 & 1.0 (L-2) & 0 & 0 \\ A3 & & 10 & 1.5 (L-3) & 0 & 0 \\ A4 & & 10 & 0 & 0.6 (L-1) & 0 \\ A5 & & 10 & 0 & 0.8 (L-2) & 0 \\ A6 & & 10 & 0 & 1.0 (L-3) & 0 \\ A7 & & 10 & 0 & 0 & 0.5 (L-1) \\ A8 & & 10 & 0 & 0 & 1.0 (L-2) \\ A9 & & 10 & 0 & 0 & 1.5 (L-3) \\ \hline A10 & & 10 & 0.5 (L-1) & 0.6 (L-1) & 0.5 (L-1) \\ A11 & & 10 & 0.5 & 0.8 (L-2) & 1.0 (L-2) \\ A12 & & 10 & 0.5 & 1.0 (L-3) & 1.5 (L-3) \\ A13 & & 10 & 1.0 (L-2) & 0.6 & 1.0 \\ A14 & Composite-additive & 10 & 1.0 & 0.8 & 1.5 \\ A15 & & 10 & 1.0 & 1.0 & 0.5 \\ A16 & & 10 & 1.5 (L-3) & 0.6 & 1.5 \\ A17 & & 10 & 1.5 & 0.8 & 0.5 \\ A18 & & 10 & 1.5 & 1.0 & 1.0 \\ \hline \hline \end{tabular}
\end{table}
Table 4: Proportion design for unconfined compression test.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Chemical Composition** & **SiO\({}_{2}\)** & **Al\({}_{2}\)O\({}_{3}\)** & **CaO** & **MgO** & **Others** \\ \hline Mass percentage (\%) & 35–50 & 15–20 & 20–30 & 5–15 & Negligible \\ \hline \hline \end{tabular}
\end{table}
Table 3: Proportion of chemical composition of slag.  
The Caofeidian dredger fill was crushed and air-dried, and the impurities were removed from the soil before the tests. In addition, grains larger than 2 mm were removed. The soil was prepared with a water content of 16.4%. Next, the different types and proportions of solidifying agents were mixed well with wet soil. Finally, the soil samples were compacted in a cylindrical mould with a dry density of 1.63 g/cm\({}^{3}\) (99% of maximum dry density).  
Subsequently, the sample was demoulded and cured in a curing chamber at a temperature of 20 \(\pm\) 2 \({}^{\circ}\)C and humidity of 90%. After 7 and 28 days of curing, unconfined compression tests were carried out to obtain the 7 day and 28 day compressive strength, respectively, of the solidified Caofeidian dredger fill. The instrument used in the test was an electric limestone unconfined pressure tester (Tianjian, China). The test was controlled to a speed of 1 mm/min and terminated when the sample failed.
# Parameter Estimation for Some Probability Distributions Used in Hydrology
## 3 Methods
### Methods of Parameter Estimation
#### 3.2.11 Pearson V Distribution (PV3)  
For estimation with MOM, parameter \(\alpha\) can be evaluated numerically with the following approximation, depending on \(\mathrm{C_{s}}\geq 0.2\):  
\[\alpha=\exp\left(\begin{array}{l}3.04183273-1.532335271\cdot\ln(\mathrm{C_{s }})+0.350524466\cdot\ln(\mathrm{C_{s}})^{2}+\\ 0.114266418\cdot\ln(\mathrm{C_{s}})^{3}-0.018372078\cdot\ln(\mathrm{C_{s}})^{4 }-0.026493586\cdot\ln(\mathrm{C_{s}})^{5}-\\ 0.000919818\cdot\ln(\mathrm{C_{s}})^{6}+0.004375959\cdot\ln(\mathrm{C_{s}})^{7 }+\\ 0.000310213\cdot\ln(\mathrm{C_{s}})^{8}-0.000357638\cdot\ln(\mathrm{C_{s}})^{9 }\end{array}\right) \tag{119}\]  
\[\beta=\sigma\cdot(\alpha-2)\cdot\sqrt{\alpha-3} \tag{120}\]  
\[\gamma=\mu-\frac{\beta}{\alpha-2} \tag{121}\]  
For estimation with the L-moments, parameter \(\alpha\) has the following approximate forms depending on L-skewness:  
\[\alpha=\exp\left(\begin{array}{l}0.692740135-0.466055586\cdot\ln(|\tau_{3}| )+0.460508385\cdot\ln(|\tau_{3}|)^{2}+\\ 0.470172127\cdot\ln(|\tau_{3}|)^{3}+0.938027763\cdot\ln(|\tau_{3}|)^{4}+\\ 0.805363294\cdot\ln(|\tau_{3}|)^{5}+0.359846034\cdot\ln(|\tau_{3}|)^{6}+\\ 0.08945247\cdot\ln(|\tau_{3}|)^{7}+0.011775528\cdot\ln(|\tau_{3}|)^{8}+0.000641 664\cdot\ln(|\tau_{3}|)^{9}\end{array}\right) \tag{122}\]  
\[\beta=\frac{\mathrm{L}_{2}}{\frac{1}{\alpha-2}-2\cdot\mathrm{z}} \tag{123}\]\[\gamma=\mathrm{L}_{1}-\frac{\beta}{\alpha-2} \tag{124}\]  
where z has the following expression:  
\[\mathrm{z}=\exp\left(\begin{array}{l}4.083852739-11.442833473\cdot\ln(\alpha)+ 10.701793287\cdot\ln(\alpha)^{2}-\\ 6.513639901\cdot\ln(\alpha)^{3}+2.527142191\cdot\ln(\alpha)^{4}-\\ 0.640389591\cdot\ln(\alpha)^{5}+0.105546499\cdot\ln(\alpha)^{6}-\\ 0.010903032\cdot\ln(\alpha)^{7}+0.000641094\cdot\ln(\alpha)^{8}-0.000016371\cdot \ln(\alpha)^{9}\end{array}\right) \tag{125}\]