esa-sceva/satcom-chunk-collection huggingface dataset:
# An Algorithm to Assist the Robust Filter for Tightly Coupled RTK/INS Navigation System
## 4 Results
### The Performance of the TC RTK/INS Integration with the Outlier Resistance
However, Figure 10 shows that the positioning errors provided by the IGG-based integration can be larger than those given by the integration without the outlier resistance at some epochs. This phenomenon is particularly evident in the Shinjuku dataset, which means that excessive robustness happens. Because the dots denoting positioning results in Figure 10 are too dense, we use two box plots to visualize the distribution of the positioning error for two datasets, as shown in Figure 11. A box plot can visualize the distribution of numerical data and skewness by showing the five-number summary of a set of data, including the lower adjacent, first quartile, median, third quartile, and upper adjacent values [84]. We divide the error results into ten groups by epochs and use a box with two whiskers to show the distribution of errors for each group. In Figure 11a, each group contains 120 results. The boxes of Group 3, Group 4, Group 5, and Group 8 clearly show the error results given by the integration with the outlier resistance have a smaller median, third quartile, and upper adjacent values when compared to those given by the integration without the outlier resistance, which means RKF is superior to EKF for these groups. Let us turn to the results at epochs corresponding to these groups in Figure 10a. We can find that the positioning errors provided by RKF are generally smaller than those provided by EKF. Figures 10a and 11a show that RKF either outperforms or performs comparably to EKF for all the groups. In Figure 11b, each group contains 200 results. The boxes of Group 4 and Group 5 also show the benefits of the IGG-based outlier resistance. The boxes of Group 8 and Group 9 are related to the positioning error results from the 1400 th epoch to the 1800 th epoch in Figure 10b. Figure 10b shows that excessive robustness happens during the period from time to time. The boxes of Group 8 and Group 9 also show that RKF degrades during this period. The box of Group 9 shows the error results given by RKF have larger third quartile values, and RKF provides a more diffuse error distribution. Our proposed novel algorithm can assist RKF in improving positioning accuracy. Figure 11: The box plots show comparison between the integration with and without the outlier resistance. Figure 10: The positioning accuracy comparison between the integration with and without the outlier resistance. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Techniques** & **RMSE (\(m\))** & **Continuity** & **The Fixed Rate** \\ \hline RTK & 7.99 & 86\% & 20\% \\ \hline TC RTK/INS (EKF) & 7.95 & 100\% & 26\% \\ \hline TC RTK/INS (RKF) & 1.75 & 100\% & 27\% \\ \hline \hline \end{tabular}
\end{table} 
Table 4: The positioning performance comparison (Odaiba).
## 2 Proposed System Design  
The proposed system consists of an MG, HSAPF, DC-DC boost converter (DBC). The MG is the combination of PV and WT. The MG system is connected to the BES and the non-linear loads via a bi-directional converter (BDC). The schematic structure is shown in Figure 1. The proposed system is designed with the MG interfaced power system through interfacing inverter, HSAPF at POI. The MG (consists of a PV, WT, and BESS) is connected to the DC link of VSI of HSAPF. The modeling of the PV and WT is presented in Appendices A.2 and A.2. BESS is employed for drawing the power intermittency of wind speed and solar-irradiance variations and load perturbations. In BESS, the process of charging/discharging is controlled by using the BDC and the modeling is presented in Appendix A.3. In BESS, the existence of the BDC is to sustain the desired bus voltage even at the maximum load conditions by supplying the surplus source from BESS. The PV is connected to the dc bus through a DBC. The solar modules are connected in series to form strings and are attached in shunt an array. To achieve the MPPT, the PV output voltage is appropriately attuned and can be done using the DBC and its modeling is provided in Appendices A.4 and A.5. Further, in WT, a permanent magnet synchronous generator (PMSG) is used for the production of power. With WT output, a DBC is connected, which regulates the DC bus voltage to follow-up the point of maximum power output. The WT can be operated in on MPPT, which is measured based on the power balance between the system and the state of battery charge (SOC) of the battery. The DC bus, inherently controlled by the DBC, provides the input to the three-phase grid side HSAPF. The VSI of HSAPF is used to control the alternating current (AC) bus voltage and frequency at load terminals and improving the PQ at POI.  
Figure 1: Proposed microgrid integrated power system network, DBC—DC-DC boost converter, BDC—bi-directional converter, PCC—point od coupling contact, MPPT—maximum power extraction technique.  
The proposed system is based on a two-step power conversion process. Initially, with the help of DBC, the maximum power extraction is determined. DBC also regulates the DC link voltage for MPPT. The LINC technique [22] is used for the peak power extraction of a PV and WT. The duty cycle is determined by using PV reference voltage and battery voltage. In the next stage, the VSI is employed for converting DC to AC power.  
The VSI is controlled by AMDWPT based SRF technique. The DC link of VSI is connected to a battery through a BDC. The charging and discharging of the battery depends on the variation of load and improves the system reliability. The MG is capable of operating either in grid-connected or islanded conditions just by changing the transfer switch at the POI.
# Covert Communication for Integrated Satellite-Terrestrial Relay Networks with Cooperative Jamming
## 3 Performance Analysis
### Effective Covert Rate  
When \(S\) forwards its source information, the signal received by \(B\) from the AF relay should be expressed as  
\[y_{B}(k)=\sqrt{G}h_{SR}\left[\sqrt{P_{S}}h_{SR}x_{S}(k)+r(k)+n_{R}(k)\right]+n _{B}(k). \tag{27}\]Through a series of derivations, the equivalent instantaneous end-to-end SNR of \(B\) is expressed as  
\[\gamma_{B}=\frac{\gamma_{SR}\gamma_{RB}}{\gamma_{SR}+\gamma_{RB}+1}, \tag{28}\]  
where \(\gamma_{SR}=P_{5}|h_{SR}|^{2}/\big{(}\sigma^{2}(1+\mu P_{R})\big{)}\), and further, \(\gamma_{RB}=P_{R}|h_{RB}|^{2}/\sigma^{2}\). In the noise-limited system, the outage probability denotes the probability that \(C_{SB}=\log_{2}(1+\gamma_{B})\), which represents the actual system capacity, falls below the required transmission rate \(R_{SB}\). Consequently, the OP should be written as  
\[P_{out}=\Pr\big{[}\gamma_{B}<2^{R_{SB}}-1\big{]}. \tag{29}\]  
To facilitate the derivations of the (29), \(\gamma_{B}\) can be approximately equivalent to by its upper boundary as  
\[\gamma_{B}\leq\min(\gamma_{SR},\gamma_{RB})\,. \tag{30}\]  
Based on this, a closed-form lower bound of \(P_{out}\) can be rewritten as  
\[P_{out}=C_{1}(\bar{\gamma}_{SB})+C_{2}(\bar{\gamma}_{SB})-C_{1}(\bar{\gamma}_ {SB})C_{2}(\bar{\gamma}_{SB}), \tag{31}\]  
where \(C_{1}(x)\stackrel{{\Delta}}{{=}}F_{\gamma_{SB}}(x)\), \(C_{2}(x)\stackrel{{\Delta}}{{=}}F_{\gamma_{RB}}(x)\), and \(\bar{\gamma}_{SB}=2^{R_{SB}}-1\).  
By combining Equations (4) and (31), a closed form of effective covert rate \(R_{c}\) is derived as  
\[\begin{array}{l}R_{c}=\alpha_{SR}\sum\limits_{i=0}^{m_{SB}-1}\sum\limits_{t= 0}^{i}\frac{(1-m_{SB})(-\delta_{SR})^{i}}{\mu(\gamma_{SR})^{i+1}t(\alpha_{SR})^ {i-t+1}}\.\\ \times(\bar{\gamma}_{SB})^{t}e^{-\big{(}\Delta_{SR}+\frac{1}{\bar{\gamma}_{RB }}\big{)}\gamma_{SB}}R_{SB}\end{array}. \tag{32}\]  
From derivation and discussion, we note that the \(R_{c}\) monotonically increases with the rising \(P_{S}\) and \(P_{R}\), where their sum is fixed. This indicates that we can employ \(\eta_{opt}\), the optimal factor for power allocation, to obtain the maximum \(R_{c}\). However, based on the optimization function (5), even if acquiring the absolute \(\eta_{opt}\) by (32), we should further discuss whether it is in the feasible region derived from covertness constraints--namely, \(P_{e}\geq 1-\varepsilon\) and (25). After that, the global optimum power allocation solution will be obtained to maximize the effective covert rate \(R_{c}\), which was solved and is shown subsequently.
## 4 Discussion
These studies demonstrate the effectiveness of IoMT systems in the early detection of health problems and patient monitoring in healthcare settings. However, it is essential to note that each IoMT system is unique and must be tailored to meet the specific needs of each healthcare environment. The main contribution of this work is the successful implementation of an IoMT system in a private medical clinic, which has been shown to improve the quality of medical care by allowing the real-time monitoring of patients' vital signs, even when they are not present in the clinic. Furthermore, the results obtained in the study show that the implemented IoMT system meets the security and sustainability requirements and has shown positive results in terms of response time, sensitivity, specificity, precision, accuracy, F-Score, positive predictive value, value predictive negative, false positive rate, false negative rate, and error rate. In addition, it is highlighted that the IoMT system has achieved a high success rate in transmitting data from the devices to the cloud system and a high precision in the vital signs data collected by the monitoring devices. These results support the reliability and accuracy of the system in data collection and transmission. Patients' frequency of use of the system has also been noted as high, indicating positive acceptance and improvement in healthcare by allowing patients to monitor their health status in real time, even outside the clinic. Additionally, the contribution of this work is mentioned by analyzing and summarizing similar studies that evaluate the effectiveness of IoMT systems in healthcare settings. These studies demonstrate that IoMT systems detect health problems and patient follow-up in various medical contexts.
# 3D In Situ Stress Estimation by Inverse Analysis of Tectonic Strains
## 3 Application in the In Situ Stress Determination of Three Gorges Dam
### Inverse Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline \multirow{2}{*}{**Test Points**} & \(\sigma_{x}\) & \(\sigma_{y}\) & \(\sigma_{z}\) & \(\sigma_{xy}\) & \(\sigma_{yz}\) & \(\sigma_{xz}\) \\  & (MPa) & (MPa) & (MPa) & (MPa) & (MPa) & (MPa) & (MPa) \\ \hline \multirow{4}{*}{D2} & Measurement & \(-4.73\) & \(-9.06\) & \(-3.95\) & \(1.93\) & \(0.09\) & \(-0.05\) \\ \cline{2-7}  & Stress based & \(-4.44\) & \(-9.28\) & \(-1.99\) & \(0.17\) & \(0.407\) & \(-0.058\) \\ \cline{2-7}  & Strain based & \(-4.55\) & \(-8.98\) & \(-3.79\) & \(1.19\) & \(0.30\) & \(-0.65\) \\ \hline \multirow{4}{*}{D3} & Measurement & \(-4.27\) & \(-9.47\) & \(-3.86\) & \(1.28\) & \(0.38\) & \(-0.16\) \\ \cline{2-7}  & Stress based & \(-4.92\) & \(-8.95\) & \(-1.92\) & \(0.28\) & \(1.18\) & \(-0.165\) \\ \cline{2-7}  & Strain based & \(-4.88\) & \(-9.52\) & \(-3.67\) & \(1.35\) & \(0.29\) & \(-0.34\) \\ \hline \multirow{4}{*}{D4} & Measurement & \(-4.17\) & \(-9.13\) & \(-4.10\) & \(1.00\) & \(-0.00\) & \(-0.13\) \\ \cline{2-7}  & Stress based & \(-4.87\) & \(-8.86\) & \(-1.83\) & \(0.24\) & \(1.11\) & \(-0.185\) \\ \cline{2-7}  & Strain based & \(-4.59\) & \(-7.82\) & \(-2.69\) & \(1.12\) & \(0.24\) & \(-0.12\) \\ \hline \multirow{4}{*}{D5} & Measurement & \(-5.46\) & \(-9.38\) & \(-5.36\) & \(1.48\) & \(0.14\) & \(-0.59\) \\ \cline{2-7}  & Stress based & \(-4.75\) & \(-8.03\) & \(-2.23\) & \(0.27\) & \(0.65\) & \(-0.043\) \\ \cline{2-7}  & Strain based & \(-4.86\) & \(-8.92\) & \(-3.14\) & \(1.24\) & \(-0.21\) & \(0.46\) \\ \hline \multirow{4}{*}{D6} & Measurement & \(-5.52\) & \(-10.24\) & \(-5.44\) & \(1.73\) & \(-0.08\) & \(-0.31\) \\ \cline{2-7}  & Stress based & \(-4.43\) & \(-8.54\) & \(-3.07\) & \(0.39\) & \(0.58\) & \(0.39\) \\ \cline{2-7}  & Strain based & \(-4.71\) & \(-9.24\) & \(-2.52\) & \(1.46\) & \(-0.62\) & \(-0.10\) \\ \hline \multirow{4}{*}{D7} & Measurement & \(-4.73\) & \(-8.13\) & \(-4.64\) & \(-0.23\) & \(0.01\) & \(-0.50\) \\ \cline{2-7}  & Stress based & \(-4.44\) & \(-8.63\) & \(-3.07\) & \(0.33\) & \(0.45\) & \(0.30\) \\ \cline{2-7}  & Strain based & \(-4.63\) & \(-8.26\) & \(-3.40\) & \(1.29\) & \(-0.20\) & \(-0.14\) \\ \hline \multirow{4}{*}{D8} & Measurement & \(-4.10\) & \(-7.08\) & \(-4.42\) & \(0.53\) & \(0.20\) & \(-0.51\) \\ \cline{2-7}  & Stress based & \(-4.33\) & \(-8.47\) & \(-2.97\) & \(0.042\) & \(1.09\) & \(0.036\) \\ \cline{2-7}  & Strain based & \(-4.81\) & \(-9.51\) & \(-4.34\) & \(0.94\) & \(0.45\) & \(-0.55\) \\ \hline \end{tabular} Note: In this table, the positive sign means compression, the negative sign means tension. Stress based stands for the stress-based approach. Displacement based is on behalf of the displacement based method. Strain based means the proposed approach.  
\end{table} 
Table 2: Contrast data of stress component under calculating coordinate system.
## 2 Background
### Basics of SSL/TLS
Note that the HostName field in the SNI extension includes the fully qualified domain name system (DNS) hostname of the server (i.e., another endpoint), which can primarily be used for hosting multiple virtual servers, also known as virtual hosts, in an end host (_for example_, to enable request direction to an appropriate virtual server without decryption). Next, the server responds with a ServerHello message containing the server's chosen ciphersuite among the client-offered ciphersuites, the server's chosen extensions among the client-requested extensions, a Certificate message containing a sequence (chain) of certificates for proving the identity of the server, and finally a ServerHelloDone message to indicate the end of the response. The client then verifies the authenticity of the given certificate chain. Once the ClientKeyExchange message is sent from the client, and ChangeCipherSpec and Finished messages are sent from both endpoints, the endpoints can exchange encrypted application data.
## 2 Materials and Methods  
### Generation of Electrolyzed Water  
The electrolyzed water was produced using a LAMI-50 electrolyzed water generator, supplied by Aquactiva Solutions (Valencia, Spain). This generator is equipped with a membrane electrolysis cell. A solution of saturated NaCl and deionized water was employed, eliminating the need for pre-treatment of the inlet water.  
For the characterization of the electrolyzed water, parameters such as pH, oxidation-reduction potential (ORP), and conductivity were measured using the pH/Ion/DO Multicenter SG68 provided by Mettler Toledo (Columbus, OH, USA). The FAC concentration was determined using the Handheld Colorimeter Chlorine UHR, supplied by Hanna Instruments (RI Woonsocket, Providence County, RI, USA). Each sample was prepared at the desired FAC level by adjusting the intensity value applied to the cell and by dilution. The characterization values are given in Table 1. If the desired FAC level was not obtained, a dilution was carried out to obtain the exact concentration. The %NaCl was adjusted by adding salt and measuring the conductivity of the sample until the value corresponding to this percentage in deionized water was obtained.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Sample** & **FAC (ppm)** & **ORP (mV)** & \begin{tabular}{c} **\%NaCl** \\ **(_ml_v_) \\ \end{tabular} & \begin{tabular}{c} **Conductivity** \\ **(mS/cm)** \\ \end{tabular} & **pH** \\ \hline SAEW5 & 5 & \begin{tabular}{c} 828 \\ 900 \\ \end{tabular} & \begin{tabular}{c} 1.5 \\ 3.0 \\ \end{tabular} & \begin{tabular}{c} 26.2 \\ 47.5 \\ \end{tabular} & \\ \hline SAEW25 & 25 & \begin{tabular}{c} 889 \\ 937 \\ \end{tabular} & \begin{tabular}{c} 1.5 \\ 3.0 \\ \end{tabular} & \begin{tabular}{c} 25.7 \\ 48.1 \\ \end{tabular} & 5.0 \\ \hline SAEW125 & 125 & \begin{tabular}{c} 943 \\ 960 \\ \end{tabular} & \
\begin{tabular}{c} 1.5 \\ 3.0 \\ \end{tabular} &
\begin{tabular}{c} 25.9 \\ 49.3 \\ \end{tabular} & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Physico-chemical parameters of the electrolyzed water samples used.
## 3 Results and Discussion
### Comparison of Unground and Ground Samples  
Since the hens' feed mixture consists of several components with different particle sizes, the influence of particle size on lutein extraction was investigated. The hens' feed sample was ground until no residue remained on a sieve mesh with a 300 \(\upmu\)m pore size. A comparison of unground (a) and ground and sieved samples (b) is shown in Figure 2. Extraction of lutein was carried out using the SM, and the resulting chromatogram is presented in Figure 3. Analysis of the ground samples yielded a 25% higher lutein concentration (in unground samples, the lutein concentration was 10.145 mg of lutein/kg of feed mixture vs. 12.753 mg of lutein/kg of feed mixture in ground samples). [PERSON] et al. investigated the influence of particle size on the efficiency of lutein extraction from marigold flower petals. They concluded that particle size does not significantly affect the extraction efficiency of lutein, but does reduce the extraction time [22]. In our study, better results were obtained with ground samples; therefore, all further analyses were performed on ground and sieved samples.  
Figure 1: Calibration curve for lutein (lutein concentrations \(\gamma=0.25\), 0.50, 1.25, 2.50, and 5.00 mg/L).
# An Approach to Multi-Objective Path Planning Optimization for Underwater Gliders
## 3 Experimental Setup
### Simulation Scenarios
With respect to the variability factor, the scenarios 2, 3, 6, and 7 exhibit lower stability in the currents temporal evolution, compared to scenarios 1, 4, 5, and 8, that are more stable.
# Disaster-Resilient Optical Network Survivability: A Comprehensive Survey
## 5 Congestion-Aware Routing and Wavelength Assignment
It is essential for service providers to employ optimized path computation techniques for efficient routing. OSPF and Intermediate System to Intermediate System (ISIS) routing protocols use Dijkstra algorithm to find the shortest path for every source-destination pair. As a result, short weighted links easily congested while other links remained underutilization. [PERSON] et al. in Reference [122] introduced a novel centralized Path Computation Element (PCE) that employs refined path computation algorithm with dynamic link cost metrics to enable network-embedded routing protocols in discovering paths that use available link capacity more efficiently. Further, PCE applies Software-Defined Optical Networking (SDON) [124] paradigms to separate path computing and path signaling functions, which gives operators more control over their network. [PERSON] et al. proposed that centralized PCE with Self-Tuned Adaptive Routing (STAR) algorithm contributes to the efficient consumption of link capacity for load balancing and avoids overloading the links (i.e., to avoid congestion and deadlocks) as well. Overall, single PCE or multiple PCEs in a network results in terms of more revenue-bearing traffic. The incident of Boston Bombing in 2013 introduced a new problem as capacity exhaustion during a disaster and might affect the ICT services. Unlike considerations used in previous studies to design resilient networks, this issue is relatively different. Human instinct drives him to get more and more information about what is going on during a disaster. This practice has been progressively increased since the introduction of social media, mobile broadband, and several other like services. According to [PERSON] et al. [125], two out of three persons use social media in the event of disasters. It is estimated that about 95% people make phone calls (caused semi-shutdown of mobile networks) and 76% use social media to post information regarding the incident [126; 127]. Live broadcasts and video streaming (i.e., TV breaking news, user-generated videos etc.) also got attraction [128]. However, ICT's physical infrastructure is not damaged but heavy traffic and a huge number of connection requests overload the access network capacity referred to as capacity exhaustion problem. The core optical network may have the capability to handle such traffic flood as a consequence of natural disasters and human threats so that violation of the SLA is avoided. Post-disaster traffic has a higher influence on the QoT of light paths in all-optical networks and failure of few seconds may cause massive losses both in terms of data volumes and decline of revenue. [PERSON] et al. in Reference [123] proposed a MILP to handle capacity exhaustion problem due to post-disaster traffic floods. Authors evaluated the network blocking for single node flooding with different traffic flood sizes. The proposed model has investigated four techniques to maximally serve the traffic floods using rerouting, excess capacity, traffic filtering and differentiated services.
# Source/Load-Pull Noise Measurements at K\({}_{\text{a}}\) [PERSON]
## 2 Materials and Methods
### De-Embedding and Processing  
Provided that the isolator in front of the receiver is sufficiently unidirectional, the nominal parameters of the receiver (REC) are common to the three noise power readings taken in the calibration and measurement steps:  
\[N_{c,cal} = k_{B}\big{(}T_{c,cal}+T_{eq,REC}\big{)}BG_{av,REC} \tag{16}\] \[N_{h,cal} = k_{B}\big{(}T_{h,cal}+T_{eq,REC}\big{)}BG_{av,REC}\] (17) \[N_{c,msm} = k_{B}\big{(}T_{c,msm}+T_{eq,REC}\big{)}BG_{av,REC}\;, \tag{18}\]  
where \(T_{c,cal}\), \(T_{h,cal}\) and \(T_{c,msm}\) are all referred to section _s-REC_:  
\[T_{c,cal} = T_{c,NS}G_{av,BLK0} \tag{19}\] \[T_{h,cal} = T_{h,NS}G_{av,BLK0}\] (20) \[T_{c,msm} = T_{out}G_{av,BLK2}\;. \tag{21}\]By linear interpolation/extrapolation, it is easily found that:  
\[T_{c,msm}=T_{c,cal}+(N_{c,msm}-N_{c,cal})\cdot\frac{T_{h,cal}-T_{c,cal}}{N_{h, cal}-N_{c,cal}}\, \tag{22}\]  
from which BLK2 is straightforwardly de-embedded by inverting (21). Then, (10) and (11) or, equivalently, (12)-(15) can be written down and solved by pseudo-inversion, as discussed in Section 2.1.  
Adopting both forward and reverse noise measurements allows to build up an over-determined system of equations, which improves the quality of the extractions. However, only a total of \(N=8\) equations is reached in this manner, which is still quite scarce as compared to a typical value of a few tens. Also, there is in practice no control on the actual spread of the terminations resulting at each measurement frequency. To the contrary, well spread termination patterns will result at some frequencies and not at others, easily leading to a rapid succession of good and bad extractions.  
To mitigate this issue, the Authors suggest extending the input data for each nominal frequency to the set of all measurements in its neighborhood. The idea is similar to what proposed in [16; 17], where a long transmission line is exploited at the input of the transistor under test. The present problem, however, has to tackle very long electrical paths which are not necessarily good approximations of transmission lines.  
Under these conditions, it can be assumed that, whereas the DUT's source reflection coefficients vary significantly from one frequency to another (with a frequency step of 100 MHz), the elements of the DUT's correlation matrix can be safely approximated to the first-order at frequencies \(f\) in a neighborhood \([f_{1},f_{2}]\) of a nominal frequency \(f_{0}\). This corresponds to modifying \(\mathbf{A}\) and \(\mathbf{x}\) of (14) and (15) into their 'expanded' counterparts:  
\[\mathbf{A}_{e} = \left[\begin{array}{c}\vdots\ \ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \\ a_{n,1}\ \ \ a_{n,1}\Delta f\ \ \ a_{n,2}\ \ a_{n,2}\Delta f\ \ \ a_{n,3}\ \ a_{n,3}\Delta f\ \ \ a_{n,4}\ \ a_{n,4}\Delta f\\ \vdots\ \ \ \ \vdots\ \ \ \ \vdots\ \ \ \vdots\ \ \ \vdots\ \ \ \vdots\ \ \ \vdots\ \ \ \vdots\ \ \ \ \vdots\ \ \ \ \ \vdots\ \ \ \ \ \ \vdots\