esa-sceva/satcom-chunk-collection huggingface dataset:
## 3 Results
### Comparison with Different Univariate Models
\begin{table}
\begin{tabular}{l l l l l l l l l l l l l l} \hline  & & **S1** & **S2** & **S3** & **S4** & **S5** & **S6** & **S7** & **S8** & **S9** & **S10** & **S11** & **S12** & **Mean** \\ \hline  & BLSTM & 25.7021 & 23.8906 & 21.1004 & 28.3287 & 26.4759 & 28.4701 & 22.4960 & 27.7977 & 27.5000 & 25.0629 & 25.2665 & 28.4127 & 25.8753 \\ RMSE & BLSTM-attention & 25.7708 & 24.0441 & 21.4432 & 28.3129 & 26.4545 & 28.3070 & 22.5573 & 27.8118 & 27.6883 & 25.0518 & 25.2951 & 28.6464 & 25.9539 \\  & TF-BLSTM & 19.1547 & 19.7329 & 21.8664 & 20.4544 & 21.4374 & 18.8939 & 22.0707 & 20.6508 & 18.9912 & 21.8951 & 28.8156 & 24.0011 \\  & TF-BLSTM-attention & 19.0154 & 19.9044 & 17.8012 & 21.6904 & 20.4426 & 21.4567 & 18.7613 & 21.9312 & 20.4372 & 18.9155 & 19.8894 & 22.37143 & 20.3330 \\ \hline  & BLSTM & 14.9132 & 13.5571 & 11.7857 & 16.2541 & 15.4316 & 15.8921 & 11.7416 & 15.8227 & 15.4688 & 14.7648 & 14.2116 & 15.9601 & 14.6474 \\ MAE & 14.9621 & 13.5576 & 11.9183 & 16.3262 & 15.4384 & 15.8599 & 11.8414 & 15.9867 & 15.7161 & 14.8636 & 14.2167 & 116.1878 & 14.7367 \\  & TF-BLSTM & 10.3375 & 10.3115 & 9.35811 & 11.1299 & 11.0372 & 13.8072 & 11.105 & 10.5434 & 10.6783 & 10.0862 & 11.7263 & 10.4787 \\  & TF-BLSTM-attention & 10.2423 & 10.3312 & 9.3309 & 10.9785 & 11.0520 & 10.7917 & 6.6934 & 10.9732 & 10.4370 & 10.6395 & 10.1646 & 11.6934 & **10.4440** \\ \hline  & BLSTM & 0.9052 & 0.8916 & 0.8996 & 0.9035 & 0.9075 & 0.8988 & 0.89300 & 0.9004 & 0.8869 & 0.9115 & 0.9041 & 0.9048 & 0.9003 \\ R2 & BLSTM-attention & 0.9047 & 0.9802 & 0.89033 & 0.9040 & 0.9077 & 0.8998 & 0.9082 & 0.9003 & 0.8854 & 0.9116 & 0.9003 & 0.9032 & 0.9096 \\  & TF-BLSTM & 0.9473 & 0.9261 & 0.9258 & 0.9425 & 0.9448 & 0.9426 & 0.9249 & 0.9372 & 0.9362 & 0.9492 & 0.9405 & 0.9331 & 0.9375 \\  & TF-BLSTM-attention & 0.9481 & 0.9245 & 0.9264 & 0.9344 & 0.9449 & 0.9425 & 0.9256 & 0.9380 & 0.9376 & 0.9496 & 0.9346 & 0.9337 & **0.9379** \\ \hline  & BLSTM & 0.4400 & 0.4512 & 0.5211 & 0.4775 & 0.49933 & 0.4192 & 0.4630 & 0.4646 & 0.4818 & 0.4321 & 0.4454 & 0.4549 & 0.4575 \\ MAPE & BLSTM-attention & 0.4463 & 0.4429 & 0.5183 & 0.4830 & 0.4425 & 0.4209 & 0.4659 & 0.4795 & 0.5054 & 0.4481 & 0.4375 & 0.4688 & 0.4632 \\  & TF-BLSTM & 0.2899 & 0.3227 & 0.4243 & 0.3270 & 0.2868 & 0.2482 & 0.3479 & 0.3409 & 0.3057 & 0.3124 & 0.2907 & 0.3283 & **0.3158** \\  & TF-BLSTM-attention & 0.2850 & 0.3289 & 0.4278 & 0.3240 & 0.2876 & 0.2510 & 0.3465 & 0.2920 & 0.2985 & 0.3204 & 0.2963 & 0.3397 & 0.3165 \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of different models using PM2.5 as the input in terms of RMSE, MAE, R\({}^{2}\), and MAPE.
# Attosecond Pulse Shaping by Multilayer Mirrors
## 5 Outlook
### The Way Toward Ever-Shorter Pulses: Approaching the Atomic Unit of Time  
The cornerstone of attosecond physics was the key for ever-shorter pulses, since the ultrafast femtosecond pulses were superseded after several decades by attosecond pulses. The first experiment  
Figure 11: Photoelectron spectroscopy utilizing attosecond pulse trains. (**a**) Measured EUV spectrum including a 300 nm aluminum filter (black line) and reflectivity design curve of the employed multilayer mirror (red area). (**b**) Photoelectrons (black) from a tungsten target and EUV spectrum (red) after spectral filtering with the EUV multilayer mirror. The latter was obtained by multiplying the reflectivity curve and the measured EUV spectrum. The bottom axis (black) represents the photoelectron kinetic energy and the top axis (red) the EUV photon energy in eV. The two axes are shifted with respect to each other by the tungsten (110) work function of 5.3 eV [86]. This illustrates the good agreement between the two curves.  
for the generation of isolated pulses proved a temporal duration of 650 as [1], and lots of subsequent experiments have realized even shorter pulses, resulting in the nowadays shortest flashes of light below 100 as. For instance, isolated attosecond pulses with a temporal duration of 80 as [55] and 67 as [87] have been measured.  
With the investigation of ever more intense laser systems (HHG cut-off energy: \(\sim\)intensity) and, in parallel, moving toward laser systems with longer wavelengths (HHG cut-off energy: \(\sim\)\(\lambda^{2}\)), an ever higher cut-off energy, up to the keV, was achieved [88] and, as a consequence, broader spectra are accessible. Therefore, shorter pulses can basically be achieved. A timescale often targeted and provided is the time it takes an electron to complete one radian of its orbit in the ground state of hydrogen [89]: \(\tau_{\rm atomic}=\hbar/2U_{\rm H}\approx 24\) as (often called the atomic unit of time), where \(U_{\rm H}\) is the binding energy of the hydrogen atom (13.6 eV). The key is the multicolor high-peak-power excitation in the HHG process [90]. So, the HHG field is aiming toward high-energy few-cycle femtosecond mid-infrared lasers as prospective driver sources for strong-field applications, offering a high photon flux in a broad energy window and the opportunity for ever shorter time resolutions, approaching the atomic unit of time. Very recently, the first accuracy below the attosecond timescale was achieved. Scientists succeeded in determining the photoemission time in helium with an accuracy of 0.85 as [91].
## 6 MATLAB Simulation
### Scenario 1
#### 6.3.2 Comparison with Other State-of-the-Art Guidance Laws (Scenario 1)  
To the best of our knowledge, this article is novel in developing a ship defense approach using clustered multiple drones. For comparison, we consider the case where only one drone is used, and the virtual agent applies the 3D PNG law in Section 6.2 to capture the target. We also consider the case where only one drone is used, and the drone applies the 3D Motion Camouflage Guidance (MCG) law in [18] to chase the target. We further consider the case where only one drone is used, and the drone applies the 3D Command to Line-Of-Sight (CLOS) guidance law in [53].  
Table 2 shows the MC simulation comparison results of Scenario 1. We run \(mC\) MC simulations per each control law. Let \(PRO_{q}\) indicate the proposed guidance law using \(G=q\) in Equation (4). Note that only one drone is used in \(PRO_{1}\). Moreover, nine drones are used in \(PRO_{3}\).  
In Table 2, \(MCG\) presents the case where the 3D MCG law is used. \(PNG\) presents the case where the 3D PNG law is used. \(CLOS\) presents the case where the 3D CLOS law is used. See that the proposed control outperforms all other state-of-the-art controls.  
Note that Equation (31) can be satisfied even when \(v_{max}<v_{k}^{t}\). This implies that even slow drones can capture a fast target when the drones stay close to the ship while staying in the lineState at all times. However, other state-of-the-art guidance laws (\(MCG\), \(PNG\), and \(CLOS\)) make a drone continue to chase the target. This maneuver makes the drone move away from the ship, which is not desirable for capturing a fast target using a slow interceptor.  
Other state-of-the-art guidance laws (\(MCG\), \(PNG\), and \(CLOS\)) make a single drone chase the target. In our paper, we let multiple drones form a planar grid formation, which can be considered a \"net\" structure to capture the incoming target. The target may perform elusive maneuvers, and there may be measurement noise in measuring the target position. By maximizing the grid formation size, the capture rate is 100, even when there exists error in the prediction of the target's position. Since the target is guided to hit its goal (ship), the drones can effectively block the target using this grid formation.
# Phased Array Antenna Analysis Workflow Applied to Gateways for LEO Satellite Communications
## 5 Phased Array Modelling
### Reception Phased Array Modelling with SystemVue
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Satellite** & **Path Distance** & **FSPL** & **Atm Losses** & **Sensitivity** \\  & **(km)** & **(dB) (18)** & **(dB)** & **(dB)** & **(dBm) (17)** \\ \hline SpaceX & 2439 & \(-185.8\) & 2 & \(-113\) \\ \hline Telesat & 1684 & \(-182.5\) & 2 & \(-102\) \\ \hline OneWeb & 1504 & \(-181.6\) & 2 & \(-114\) \\ \hline Amazon & 590 & \(-173.4\) & 2 & \(-106\) \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Sensitivity calculations for each constellation using Equation (17). Figure 18: Antenna gain vs. number of radiating elements for reception obtained with Systemvue. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**LEO Constellation** & **Aphotem (m)** & **Side (m)** & **Area (m\({}^{2}\))** \\ \hline SpaceX & 0.95 & 0.85 & 2.44 \\ \hline Telesat & 0.91 & 0.82 & 2.22 \\ \hline OneWeb & 1.6 & 1.5 & 7.6 \\ \hline Amazon & 1.4 & 1.3 & 5.6 \\ \hline \hline \end{tabular}
\end{table} 
Table 10: Area used by phased array antennas in reception. Figure 19: Radiation pattern of the reception antenna for hexagonal shape configuration, 625 elements and different elevations. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Constellation** & **SpaceX** & **Telesat** & **OneWeb** & **Amazon** \\ \hline Operation & & & \\ Frequency & 18.55 & 19 & 18.55 & 19 \\ (GHz) & & & \\ \hline Required Gain & 46.9 & 46.5 & 51.5 & 50.2 \\ (dB) & & & \\ \hline Theoretical & & & \\ Radiant & 11,749 & 8512 & 33,885 & 25,119 \\ Elements (16) & & & \\ \hline SystemVue & & & \\ Radiant & 8910 & 7956 & 26,791 & 19,927 \\ Elements (20) & & & \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Reception antenna size estimation for hexagonal configuration. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**LEO Constellation** & **Diameter** & **Area (m\({}^{2}\))** & **Above Ground Level (m)** \\ \hline SpaceX & 1.5 & 1.8 & 1.95 \\ \hline Telesat & 1.5 & 1.8 & 3.2 \\ \hline OneWeb & 3.5 & 9.6 & 4.55 \\ \hline Amazon & 2.4 & 4.5 & - \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Area used by parabolic antennas in reception.
# High Performance Receiver Design for RX Carrier Aggregation
## 3 LNTA Topologies and Simulation Results
### Cascade Shut-off LNTA  
Figure 9 shows the cascode shut-off LNTA. \(g_{m}\) is separated between different CA paths. Noise contribution due to the cascode device is the same as a conventional LNTA without sharing \(g_{m}\) as in the split-cascode LNTA. In non-CA operation, gate bias of the second gm transistor is properly ON. At the same time, the cascode bias of CA2 is shut-off to avoid throwing additional current. With this bias scheme, the second \(g_{m}\) transistor operates in saturation and triode regions in CA and non-CA, respectively. Impedance matching (\(S_{11}\)) is slightly disturbed due to the change in \(C_{gs}\) and \(g_{m}\) for different CA modes. Input impedance (\(Z_{in}\)) of the cascode shut-off LNTA is expressed in Equations (7) and (8) for non-CA and CA mode, respectively.  
\[Z_{\text{in,non-CA}}=Z_{CA1}\parallel Z_{CA2} \tag{7}\]  
\[Z_{\text{in,non-CA}}=Z_{CA1}\parallel\left(j\omega L_{deg}+\frac{1}{j\omega C_ {gs}}\right) \tag{8}\]  
where \(Z_{CA1}\) (\(=Z_{CA2}\)) is the input impedance of CA1 path (= CA2 path), and it can be expressed as \(\omega_{t}\cdot L_{deg}+j\omega L_{deg}+\frac{1}{j\omega C_{gs}}\). Two inductors for source degeneration purposes can be merged into the shared degeneration without losing the property of the cascode shut-off LNTA. Since two inductors are merged in parallel, the size of the inductor decreases, and thus the size of the inductor is minimized accordingly. Table 3 summarizes the simulation results with cascode shut-off LNTA approach.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Gain (dB)** & **NF (dB)** & \(\mathbf{II_{3}}\) (dBm) & \(\mathbf{I_{d}}\) (mA)** \\ \hline Reference (SISO) & 56.6 & 1.33 & \(-\)6.2 & 7.4 \\ Split-Cascode, non-CA & 56.7 & 1.46 & \(-\)6.7 & 7.4 \\ Split-Cascode, CA (1 \(\times\)\(I_{d}\)) & 51.9 & 2.55 & \(-\)4.8 & 7.8 \\ Split-Cascode, CA (3 \(\times\)\(I_{d}\)) & 54.2 & 2.09 & \(-\)4.2 & 19.8 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Simulation results with split-cascode LNTA.  
Figure 8: Split-cascode LNTA.
# Hierarchical Self-Supervised Learning for Knowledge-Aware Recommendation
## 4 Methodology
### Parallel Cross-View Contrastive Learning
#### 4.2.2 Contrastive Learning  
In cases where augmented KG views and UIG views are generated, we employ the message propagation strategy of LightGCN [41] on UIG views due to its effectiveness and lightweight architecture for learning user-item representations, which is updated as follows:  
\[\mathbf{e}_{it}^{(l+1)}=\sum_{i\in\mathcal{N}_{u}}\frac{\mathbf{e}_{i}^{(l)}}{ \sqrt{|\mathcal{N}_{u}||\mathcal{N}_{i}|}} \tag{14}\]  
\[\mathbf{e}_{i}^{(l+1)}=\sum_{u\in\mathcal{N}_{i}}\frac{\mathbf{e}_{u}^{(l)}}{ \sqrt{|\mathcal{N}_{i}||\mathcal{N}_{u}|}} \tag{15}\]  
where \(\mathbf{e}_{u}^{(l+1)}\) and \(\mathbf{e}_{i}^{(l+1)}\) denote the \((l+1)\)th layer's representations of user and item. \(\mathcal{N}_{u}\) and \(\mathcal{N}_{i}\) denote the interaction items of user \(u\) and the set of connected users of item \(i\), respectively. To create a unified representation space for subsequent contrastive learning, facilitating the comparison of embeddings of cross-views, we take the representations as inputs of an MLP to map them to a shared space where the contrastive loss is calculated:  
\[\mathbf{z}_{u,i}^{*}=\sigma((\mathbf{e}_{i}^{G_{u}^{*}})^{\top}\mathbf{W}_{1} ^{G_{u}^{*}}+\mathbf{b}_{1}^{G_{u}^{*}})^{\top}\mathbf{W}_{2}^{G_{u}^{*}}+ \mathbf{b}_{2}^{G_{u}^{*}} \tag{16}\]\[\mathbf{z}_{k,i}^{*}=\sigma((\mathbf{e}_{i}^{\mathcal{G}_{k}^{*}})^{\top}\mathbf{W}_{1}^{ \mathcal{G}_{k}^{*}}+\mathbf{b}_{1}^{\mathcal{G}_{k}^{*}})^{\top}\mathbf{W}_{2}^{ \mathcal{G}_{k}^{*}}+\mathbf{b}_{2}^{\mathcal{G}_{k}^{*}} \tag{17}\]  
where \(*\in(1,2)\) denote the superscripts of structural views constructed by edge-level dropout and neighbor-level dropout, respectively, and \(\mathbf{W}^{(\cdot)},\mathbf{b}^{(\cdot)}\) are trainable parameters.  
Positive pairs (i.e., item embedding \(z_{u,i}^{1}\) in UIG view \(\mathcal{G}_{u}^{1}\) and its corresponding entity embedding \(z_{k,i}^{1}\) in KG view\(\mathcal{G}_{k}^{1}\)) promote consistency among different perspectives of the same node. On the other hand, negative pairs (i.e., any two node representations in cross-views without relatedness) emphasize distinctions between different nodes. In what follows, we design the contrastive learning loss based on InfoNCE loss [42] to minimize the distance of positive pairs and maximize that of negative pairs:  
\[\mathcal{L}_{C}=\sum_{*\in(1,2)}\sum_{i\in\mathcal{I}}-\log\frac{\mathbf{\mathrm{ e}}^{(\mathbf{z}_{u,i}^{*}\mathbf{z}_{k,i}^{*})/\tau}}{\mathbf{\mathrm{e}}^{(\mathbf{z}_{u,i}^{*} \mathbf{z}_{k,i}^{*})/\tau}+\sum\limits_{j\in\{i\in\mathcal{I},i^{\prime},i^{ \prime}\}}\mathbf{\mathrm{e}}^{(\mathbf{z}_{u,j}^{*}\mathbf{z}_{k,i}^{*})/\tau)}} \tag{18}\]  
where \(s(\centerdot)\) denotes cosine similarity function and \(i^{\prime}\), \(i^{\prime\prime}\) denote the randomly sampled negative candidates for item \(i\). Multiple cross-view contrastive learning creates augmented data from two perspectives, enhancing the robustness of our HKRec.
## 2 Results
### Impact of SLC10A1 Missense Variants on SLC10A1/NTCP Function
(**C**) Immunolocalization of NTCP (green) in vector-transfected control HEK cells and HEK cells stably expressing NTCP reference sequence or the respective missense variant using affinity-purified NTCP-specific GNG antibody [58] and confocal laser scanning microscopy. Red: nuclei. Bar, 20 \(\upmu\)m. (**D**) Time-dependent uptake of 25 nM taurocholate (prototypic substrate) into HEK cells stably expressing NTCP reference in the presence (filled circle) or absence (open circle) of sodium. Results are means \(\pm\) SD of 3 wells. (**E**) Concentration-dependent uptake of taurocholate into HEK cells stably expressing NTCP reference sequence or the respective missense variant in the presence (filled circle) or absence (open circle) of sodium determined at an incubation time of 30 s. Results are means \(\pm\) SD of 12 wells. Kinetic parameters were calculated by subtracting taurocholate uptake in the absence of sodium from the uptake in the presence of sodium. (**F**) Inhibition of the uptake of prototypic substrate taurocholate by remdesivir into HEK cells expressing NTCP reference sequence or missense variant R185C or S213R. Cells were incubated with 15 \(\upmu\)M taurocholate in the presence of different remdesivir concentrations and cellular accumulation was determined after 30 sec (filled circles). Results are presented as a percentage of uninhibited taurocholate uptake in the absence of remdesivir (100%, open circle). The filled square indicates taurocholate accumulation in the absence of sodium. Results are means \(\pm\) SD from 9 wells. Recently, the small molecule nucleoside analog remdesivir has been approved for the treatment of COVID-19 [59]. Because remdesivir was proposed as a novel inhibitor of NTCP [60] and transporter inhibition may be genotype-dependent [61], we analyzed whether both missense variants are also inhibited by remdesivir. The IC\({}_{50}\) values were 90.3 \(\upmu\)M, 65 \(\upmu\)M, and 63.1 \(\upmu\)M for NTCP reference, p.R185C and p.S213R, respectively (Figure 3F), and the R values calculated according to FDA guidelines [38] were 1.01 for NTCP reference and 1.02 for both variants. We then analyzed whether remdesivir is also a substrate of NTCP. We could identify a time- and sodium-dependent uptake of remdesivir into the NTCP transfectants (Supplementary Figure S5A). When analyzing the concentration-dependent uptake of remdesivir, we observed strong sodium-independent uptake into the transfectants (Supplementary Figure S5B). The sodium-dependent uptake was saturable with K\({}_{\text{m}}\) values of 71.0 \(\upmu\)M, 43.2 \(\upmu\)M, and 219 \(\upmu\)M for NTCP reference, p.R185C, and p.S213R, respectively. This data of high remdesivir uptake even in the absence of sodium indicates that NTCP or the two missense variants are not important determinants of hepatocellular remdesivir uptake.
## 2 Materials and Methods
### Convergence Analysis
If case (i) were not true, then (5) and Lemma 3.1(b) would give \(\|\mathbf{x}_{k}-\hat{\mathbf{x}}_{k(\ell)}\|^{2}\geq\|\mathbf{x}_{k-1}-\hat{ \mathbf{x}}_{k(\ell)}\|^{2}+\left(\frac{\alpha\Delta}{\mathrm{L}}\right)^{2}\), for all \(k\in K^{\ell}\) greater than the iteration \(\overline{k}\) of the last coordination iterate. Applying this inequality recursively up to \(\overline{k}\), we obtain \(\mathit{Diam}(X)^{2}\geq\|\mathbf{x}_{k}-\hat{\mathbf{x}}_{k(\ell)}\|^{2}\geq (k-\overline{k}-1)\big{(}\frac{\alpha\Delta}{\mathrm{L}}\big{)}^{2}\). However, this inequality, together with H1 and \(L<\infty\)(due to H3) contradicts the fact that \(\mathcal{X}\) is bounded. Therefore, item (i) holds. We now turn our attention to the item (ii): Let \(\mathsf{s},\mathsf{s}^{\prime}\in K^{\ell}\) such that \(\mathsf{s}<\mathsf{s}^{\prime}\) be the iteration indices of any two coordination steps. At the moment in which \(\overline{\mathbf{x}}_{\mathsf{s}^{\prime}}\) is computed, the information \((f(\overline{\mathbf{x}}_{\mathsf{s}}),g_{\mathsf{s}}^{\mathsf{j}})\) is available at the MP for all \(\mathsf{j}=1,\ldots,\mathsf{w}\). As a result of the MP definition, the following constraints are satisfied by \(\overline{\mathbf{x}}_{\mathsf{s}}\):  
\[f^{\mathsf{j}}(\overline{\mathbf{x}}_{\mathsf{s}})+\langle g_{\mathsf{s}^{ \prime}}^{\mathsf{j}}\overline{\mathbf{x}}_{\mathsf{s}^{\prime}}-\overline{ \mathbf{x}}_{\mathsf{s}}\rangle\leq r^{\mathsf{j}}\text{ and }\mathsf{c}^{\mathsf{T}}\overline{\mathbf{x}}_{\mathsf{s}^{\prime}}+\sum \limits_{\mathsf{j}=1}^{\mathsf{w}}r^{\mathsf{j}}\leq f_{\mathsf{s}^{\prime}- 1}^{\mathsf{dev}}. \tag{7}\]  
By assuming these inequalities and rearranging terms, we get \(f(\overline{\mathbf{x}}_{\mathsf{s}})-f_{\mathsf{s}^{\prime}-1}^{\mathsf{dev} }\leq\langle\mathsf{c}+\sum\limits_{\mathsf{j}=1}^{\mathsf{w}}g_{\mathsf{s}^{ \prime}}^{\mathsf{j}}\overline{\mathbf{x}}_{\mathsf{s}}-\overline{\mathbf{x}} _{\mathsf{s}^{\prime}}\rangle\leq\Gamma\|\overline{\mathbf{x}}_{\mathsf{s}}- \overline{\mathbf{x}}_{\mathsf{s}^{\prime}}\|\), where the constant \(\infty>\Gamma\geq\Gamma\geq\|\mathsf{c}+\sum\limits_{\mathsf{j}=1}^{\mathsf{ w}}g_{\mathsf{s}}^{\mathsf{j}}\|\) is ensured by H3. The definition of \(f_{\mathsf{s}^{\prime}}^{\mathsf{dev}}=f_{\mathsf{s}^{\prime}}^{\mathsf{up}}- \alpha\Delta_{\mathsf{s}^{\prime}}\) and inequality \(f(\overline{\mathbf{x}}_{\mathsf{s}})\geq f_{\mathsf{s}^{\prime}}^{\mathsf{ up}}\) gives \(\|\overline{\mathbf{x}}_{\mathsf{s}}-\overline{\mathbf{x}}_{\mathsf{s}^{\prime}}\|\geq \alpha\frac{\Delta_{\mathsf{s}^{\prime}}}{\Gamma}\geq\alpha\frac{\Delta_{ \mathsf{s}^{\prime}}}{\Gamma}>0\). If there was an infinite number of coordination steps inside \(K^{\ell}\), the compactness of \(\mathcal{X}\) would allow us to extract a converging subsequence, and this would contradict the above inequality. The number of coordination steps inside \(K^{\ell}\) is thus finite. As a conclusion of (i) and (ii), the index-set \(K^{\ell}\) is hence finite, and the chain (6) concludes the proof.
## 3 Results and Discussion
### Consumption of Type B Nodes  
The same measurements and calculations made for node A were carried out for type B node. These calculations are shown in Figures 8-10, Tables 6-8, and Equations (5)-(7). In this case, node B had a standby current of 1.5 mA.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**State** & \(1\) & 6 & 5 & 3, 4 \\ \hline \multirow{2}{*}{**Description**} & Time & Sensors & Access to & Access to \\  & synchronization & preheating & microSD & microSD and \\ \hline \multirow{2}{*}{\(\mathbf{I_{average}\) (mA)}} & 61 & 4.6 & 21 & 74 \\ \hline \multirow{2}{*}{\(\mathbf{Duration\) (s)}} & 13 & 6 & 0.5 & 10 \\ \hline \hline \end{tabular}
\end{table}
Table 6: Description of functional states of node B—mode (i).  
Figure 8: Node B—mode (i): NTP sync, sensor reading and data sending (once every 24 h).  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**State** & 2 & 5 & 2 & 3 \\ \hline \multirow{2}{*}{**Description**} & \multirow{2}{*}{Task signaling} & \multirow{2}{*}{Sensors reading} & \multirow{2}{*}{Task signaling} & Access to \\  & & & microSD \\ \hline \multirow{2}{*}{\(\mathbf{I_{average}\) (mA)}} & 4 & 208 & 7 & 16 \\ \hline \multirow{2}{*}{\(\mathbf{Duration\) (s)}} & 2 & 16 & 3.5 & 1.5 \\ \hline \hline \end{tabular}
\end{table}
Table 5: Description of functional states of node A—mode (iii).  
\[\overline{T}_{B\_total\_i}=\frac{\sum_{B\_state=1}^{B\_state=3,4}\left(\overline{T}_ {B\_state}-\overline{T}_{B\_standby}\right)\cdot t_{B\_state}}{24\text{-}60 \text{-}60}=0.111\text{ mA} \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline
**State** & 6 & 5 & 3 & 4 \\ \hline
**Description** & \
\begin{tabular}{c} Sensors \\ preheating \\ \end{tabular} & Sensors reading &
\begin{tabular}{c} Access to \\ microSD \\ \end{tabular} & Data sending \\ \hline
**I\({}_{\text{average}}\) (mA)** & 46 & 170 & 150 & 100 \\ \hline
**Duration (s)** & 14 & 17 & 1 & 10 \\ \hline \hline \end{tabular} \[\overline{T}_{B\_total\_ii}=\frac{\sum_{B\_state=4}^{B\_state=4}\left( \overline{T}_{B\_state}-\overline{T}_{B\_standby}\right)\cdot t_{B\_state}}{30 \text{-}60}=2.57\text{ mA}\] (6)  
\end{table}
Table 7: Description of functional states of node B—mode (ii).  
Figure 10: Node B—mode (iii): Sensor reading without data sending mode (every 30 min).  
Figure 9: Node B—mode (ii): Sensor reading and data sending (every 30 min).  
\[T_{B\_total\_iii}=\frac{\sum_{B\_state=3}^{B\_state=3}\left(\overline{I}_{B\_state }-\overline{I}_{B\_standby}\right)\cdot t_{B\_state}}{30\cdot 60}=2.02\text{ mA} \tag{7}\]  
The autonomy of node B, provided that the battery is not charged by any means, is depicted in Equation (8).  
\[Number\ of\ days=\frac{3000\text{ mA}\cdot\text{h}}{\left(\overline{I}_{B\_total\_i}+ \overline{I}_{B\_total\_ii}+\overline{I}_{B\_total\_iii}+\overline{I}_{B\_ standby}\right)\cdot 24\text{ h}}=20.15\text{ days} \tag{8}\]