esa-sceva/satcom-chunk-collection huggingface dataset:
## 3 Results and Discussion
### Results of Numerical Simulation
After 7.5 years, there was a clear saturation line in the whole dam body, and then the saturation line gradually lifted with the rainfall years, and the lifting speed was faster. After 50 years, the rise in the saturation line was not obvious, and then the saturation line gradually stabilized, approaching a saturation line similar to the steady state. Comparing Figure 6a-c, it can be seen that the evolution laws under different rainfall patterns are basically the same. As shown in Figures 7 and 8, when the annual rainfall is 800 mm or 1200 mm, the evolution law of the saturation line in the dry-stack tailings pond is similar to that when the annual rainfall is 400 mm. The difference is that when the rainfall is 800 mm or 1200 mm, the saturation line appears in the local area of the dam in the 2.5 th year of rainfall, and after 5 years, the whole dam appears with an obvious saturation line, and the saturation line of annual rainfall of 1200 mm is higher than 800 mm. Thereafter, the saturation line gradually lifted with the increase in rainfall years and finally approached a relatively stable state. 1.3 The Concept and Significance of the Relatively Stable Seepage Field of the Dry-Stack Tailings Pond  
In summary, from the perspective of the saturation line of the seepage field, under the action of long-period rainfall, the obvious saturation line is gradually formed at the bottom of the tailings pond. With the continuous rainfall, the saturation line in the tailings pond  
Figure 8: The spatial-temporal evolution law of the saturation line in the tailings pond (Guiyang, 1200 mm): (**a**) Guiyang 2018 (casell-g); (**b**) Guiyang 2019 (casell-h); (**c**) Guiyang 2020 (casell-i). gradually rises and gradually tends to be in a relatively stable state. From the perspective of seepage field pore pressure, under the action of long-term rainfall, the pore pressure in the tailings pond gradually increases and tends to be relatively stable, that is, the final pore pressure remains basically unchanged and fluctuates in a small range. The seepage field in this relatively stable state is called the'relatively stable seepage field (RSSF)' in this study. According to the above research, the characteristics of RSSF are as follows: (1) under the action of multi-year periodic rainfall, a RSSF will be formed inside the dry-stack tailings pond; (2) the distribution of pore pressure and the saturation line in the seepage field is basically stable; (3) the pore pressure and saturation line in the RSSF fluctuate with the periodic rainfall, which is related to seasonal effects. The RSSF is the final form of the seepage field in the dry-stack tailings pond under multi-year rainfall. Considering that the dry-stack tailings pond has a long design life and is a permanent structure, the RSSF can be used as the basis for the drainage design and stability analysis of dry-stack tailings ponds.
# Flavone Hybrids and Derivatives as Bioactive Agents
## 2 Flavone Hybrids with Different Pharmacophores
### Cyclic Saturated Amines in Flavone Derivatives  
Chrysin analogs (**37**) with antibacterial activities were synthesized using the maximum number of 6C long spacers (Figure 15).  
Morpholine, piperidine, and \(N\)-methylpiperazine were the coupling components and the key intermediate bronoalkyl chain was prepared from chrysin with dibromoalkanes [25]. The sulfonylpiperazine hybrids (**37d**) proved to be antioxidant and cytotoxic agents [26]. The halogen analogs (**37d**, R\({}^{1}\)=Cl, diCl, diF) showed anticancer effects against the SK-OVS (human ovarian cancer), HeLa (human cervical cancer), and HT-29 (human colon adenocarcinoma) cell lines; the methoxy and dimethoxy derivatives proved to be effective against the A-549 (human non-small cell lung carcinoma) and HT-29 cell lines (minimum value of \(IC_{50}\) = 4.67 \(\upmu\)g/mL). At the same time, it should be mentioned that no positive control was used in the cytotoxicity assays, only literature data for gefitinib were obtained.  
Several flavone hybrids substituted in position 8 with piperazine (**38**) and piperidine (**39**) were prepared (Figure 16). The key step in the synthesis was a typical Mannich reaction between flavone **40** and the piperazine derivative in the presence of paraformaldehyde resulting in the desired product **38**. Compound **38** (R=thiophen-2-ylmethyl) showed an excellent \(IC_{50}\) for PARP-1 (14.7 nM) and in PARP-2 (0.9 \(\upmu\)M) inhibition; thus, this hybrid could be considered a lead molecule in cancer treatment [27]. Moreover, the above-mentioned compound exerted a selective cytotoxic effect on breast cancer gene 1 (BRCA1)-deficient cells (SK-OV-3 cells) via inhibiting PARP-1. Poly(ADP-ribose) polymerase-1 (PARP-1) is a potential target for the discovery of anticancer drugs.  
Figure 16: Piperazine- and piperidine-substituted flavones.  
Figure 15: Flavone–cyclic amine hybrids in position 7.  
The nipecotic acid-flavone hybrids (**39**) were characterized as anti-Alzheimer agents [54] with a similar synthetic route.
## 5 Illustrative Example
### Identification of GIVs
Using symbols from Equation (8) this combination is represented as \(1_{1}\{j_{1}\}\)\(\{\)\(\)\(\}\), \(2_{1}\{j_{2}\}\)\(\{\)\(\}\), \(3_{1}\{j_{3}\}\)\(\{\)\(\}\), \(4_{1}\{j_{4}\}\)\(\{\)\(\}\), \(5_{1}\{j_{5}\}\)\(\{\)\(\}\) and using symbols from Figure 11 it looks like \(1_{1}\{p_{1}\}\), \(2_{1}\{p_{3}\}\), \(3_{1}\{p_{4}\}\), \(4_{1}\{p_{4}\}\), \(5_{1}\{p_{1}\}\). Thus, the position of particle #1 at iteration \(t\) can be expressed using the symbols in Equation (8) and Figure 11 as \(x_{1}\{t\}\) =[\(1_{1}\{p_{1}\}\), \(2_{1}\{p_{3}\}\), \(3_{1}\{p_{4}\}\), \(4_{1}\{p_{4}\}\), \(5_{1}\{p_{1}\}\)]\({}^{\rm T}\). It is assumed that at iteration \(t+1\), after applying Equation (9), particle #1 has the velocity \(v_{1}(t+1)\) = \(\{v_{1}\{p_{1}\}\), \(v_{1}\{p_{1}\}\), \(v_{1}\{p_{2}\}\), \(v_{1}\{p_{3}\}\), \(v_{1}\{p_{4}\}\}\)\({\}^{\rm T}\). Using the formalism from Figure 11, the sum \(x_{1}(t)\oplus v_{1}(t+1)\) leads to a new position of particle #1, corresponding to iteration \(t+1\), \(x_{1}(t+1)\) = \(\{1_{1}\{p_{1}\}\)\(\oplus\)\(p_{1}\}\), \(2_{1}\{p_{3}\}\)\(\oplus\)\(p_{1}\), \(3_{1}\{p_{4}\}\)\(\oplus\)\(p_{2}\), \(4_{1}\{p_{4}\}\)\(\oplus\)\(p_{3}\), \(5_{1}\{p_{1}\}\)\(\oplus\)\(p_{4}\}\)\(\}^{\rm T}\) or \(x_{1}(t+1)\) = \(\{1_{1}\{p_{2}\}\), \(2_{1}\{p_{4}\}\), \(3_{1}\{p_{2}\}\), \(4_{1}\{p_{3}\}\), \(5_{1}\{p_{1}\}\)\(\}^{\rm T}\). Translating this result in the space of GIVs, the position \(x_{1}(t+1)\) of particle #1 at iteration \(t+1\) is given by GIV #18 for the performance indicator #1, GIV #38 for the performance indicator #2, GIV #32 for the performance indicator #3, GIV #15 for the performance indicator #4, and GIV #1 for the performance indicator #5 (see notations from Table 2). Subtraction is done on the same logic, but matrices from Figure 11 associated with this operation are considered. For example, it is assumed that \(\underline{u}_{t}\) is the position of particle #2 at iteration \(t-5\). In addition, it is assumed that \(x_{2}(t\)\) - \(5\)) = \(\{1_{2}\{p_{2}\}\), \(2_{2}\{p_{2}\}\), \(3_{2}\{j_{3}\}\)\(\{\)\(\}\), \(4_{2}\{p_{2}\}\), \(5_{2}\{p_{1}\}\)\(\}^{\rm T}\). The subtraction \(\underline{u}_{t}\)\(\sim\)\(x_{1}(t)\) becomes \(x_{2}(t-5)\)\(\sim\)\(x_{1}(t)\) = \(\{1_{2}\{p_{2}\)\(-\)\(p_{1}\}\), \(2_{2}\{p_{2}\)\(\}\), \(2_{2}\{p_{2}\)\(\}\), \(3_{2}\{p_{3}\}\)\(\rightarrow\)\(x_{1}\), \(4_{2}\{p_{2}\)\(-\)\(p_{4}\}\), \(* Particle #2: indicator #1[GIV #10], indicator #2 {GIV #13}, indicator #3 {GIV #26}, indicator #4 {GIV #10}, indicator #5 {GIV #1},
* Particle #3: indicator #1[GIV #10], indicator #2 {GIV #28}, indicator #3 {GIV #26}, indicator #4 {GIV #10}, indicator #5 {GIV #1}. Symbolically, the initialization condition looks like: 11{p1}, 21{p1}, 31{p1}, 41{p1}, 51{p1}; 12{p1}, 22{p2}, 32{p1}, 42{p1}, 52{p1}; 13{p1}, 23{p3}, 33{p1}, 43{p1}, 53{p1}.
## 4 Methodology
### Digital Image Processing
#### 4.2.3 Band Ratio (BR)  
Band ratio (BR) is one of the commonly used image-enhancement techniques [72; 73]. It is a simple and helpful technique for reducing the effect of topography and enhancing the subtle differences in the spectral reflectance characteristics of rocks and soils [20; 74]. Several band ratios and false color composites of ETM+ in RGB have been suggested for enhancing lithology and alteration zones, for example, 5/7 for hydroxyl-bearing minerals and 3/1 for iron oxide [57; 61; 75; 76; 77; 78; 79]. Many previous studies have used band ratios to identify mantle periodtites. For example, [PERSON] used an ETM (5/7, 5/1, 5/4 \(\times\) 3/4) false-color image to delineate ultrabasic and basic rocks from silica-rich granitic rocks [24].
# Optimal Proactive Caching for Multi-View Streaming Mobile Augmented Reality
## 5 Conclusions  
Multi-view streaming applications with embedded augmented reality objects (AROs) are highly delay sensitive and confronted with a series of constraints in terms of storage and computing resources. In this paper, an optimization framework is proposed to realize optimal service decomposition in an edge-cloud supported 5G and beyond wireless network aiming to balance service latency and content popularity by taking into account the inherent user mobility. In addition, a nominal long short-term memory (LSTM) neural network is proposed that is trained using optimal solutions to explore real-time decision making for supporting edge proactive caching for multi-view streaming with popular augmented reality objects. It is important to note that the proposed framework is generic enough to be applied in a plethora of mobile-augmented reality applications that are composed by multiple view streams and popular augmented objects per stream. Numerical investigations reveal that the proposed schemes outperform a number of previously defined baseline schemes. More specifically, gains up to 38% are reported in terms of reducing the delay compared to previously proposed techniques that do not explicitly take into account user mobility and the inherent decomposition of the MAR services. in the proposed framework, during the online inference stage, the LSTM-based scheme is able to efficiently provide competitive predictions approaching the optimal solutions. On the other hand, the learning and training stage of the LSTM network require longer processing times, but those phases can be executed in an offline manner where high-quality solutions from the defined optimization framework could be explored. In that way, the proposed framework is able to provide high-quality decision making in an online manner by exploring the capabilities of the LSTM neural network and the powerful integer programming formulation that is able to provide optimal decision making that can be used for training the neural network.  
[PERSON]; writing--original draft, writing--review and editing, [PERSON]; writing--review and editing, supervision. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
**Institutional Review Board Statement:** Not applicable.  
**Informed Consent Statement:** Not applicable.  
Figure 12: Average Processing Time for Different Stages in the LSTM Scheme.  
Data Availability Statement: All data included in this study are available upon request by contact with the corresponding author.
## 9 Cramer-Rao Lower Bound  
In this section, the overview of the estimation error measure and the derivation of the appropriate CRLB is provided, which is used to assess the performance of considered estimation methods. The CRLB, which is given by the inverse of the Fisher information matrix (FIM), denoted as \(\text{F}(\mathbf{x})\), sets the lower limit for the variance (or covariance matrix) of any unbiased estimator [48]. The FIM \(\text{F}(\mathbf{x})\) represents a measure of the knowledge contained in the observed data about the unknown parameter vector, and it may be computed as follows  
\[\begin{split}&\text{F}(\mathbf{x})=\begin{bmatrix}F_{11}&F_{12}\\ F_{21}&F_{22}\end{bmatrix}=\text{E}\Bigg{[}\left(\frac{\partial\ln(f(\bar{ \mathbf{r}}|\mathbf{x}))}{\partial\mathbf{x}}\right)\left(\frac{\partial\ln (f(\bar{\mathbf{r}}|\mathbf{x}))}{\partial\mathbf{x}}\right)^{T}\Bigg{]}\\ &=-\,\text{E}\Bigg{[}\frac{\partial^{2}\ln(f(\bar{\mathbf{r}}| \mathbf{x}))}{\partial\mathbf{x}\partial\mathbf{x}^{T}}\Bigg{]}.\end{split} \tag{63}\]  
with the corresponding elements of the matrix determined according to the expressions  
\[F_{11}=\frac{1}{\sigma^{2}}\sum_{i=1}^{N}\Bigg{(}\frac{x-x_{r}}{\|\mathbf{x}- \mathbf{x}_{r}\|}+\frac{x-x_{i}^{t}}{\|\mathbf{x}-\mathbf{x}_{i}^{t}\|_{2}} \Bigg{)}^{2}, \tag{64}\]\[F_{12}=F_{21} =\frac{1}{\sigma^{2}}\sum_{i=1}^{N}\left(\frac{x-x_{r}}{\left\| \mathbf{x}-\mathbf{x}_{r}\right\|}+\frac{x-x_{i}^{t}}{\left\|\mathbf{x}- \mathbf{x}_{i}^{t}\right\|_{2}}\right) \tag{65}\] \[\times\left(\frac{y-y_{r}}{\left\|\mathbf{x}-\mathbf{x}_{r}\right\| }+\frac{y-y_{i}^{t}}{\left\|\mathbf{x}-\mathbf{x}_{i}^{t}\right\|_{2}}\right),\]  
\[F_{22}=\frac{1}{\sigma^{2}}\sum_{i=1}^{N}\left(\frac{y-y_{r}}{\left\|\mathbf{x }-\mathbf{x}_{r}\right\|}+\frac{y-y_{i}^{t}}{\left\|\mathbf{x}-\mathbf{x}_{i} ^{t}\right\|_{2}}\right)^{2}. \tag{66}\]  
The process of deriving the Equations (64)-(66) is described in detail in Appendix A. Therefore, the CRLB inequality bound of the unbiased estimator \(\mathbf{\hat{x}}\) of parameter \(\mathbf{x}\) is represented with the relationship between covariance \(\mathrm{cov}(\mathbf{\hat{x}})=E[\mathbf{\hat{x}}]\)and the inverse of the FIM, as follows  
\[E\Big{[}(\mathbf{\hat{x}}-\mathbf{x})\,(\mathbf{\hat{x}}-\mathbf{x})^{T} \Big{]}\geq\mathrm{Tr}\Big{\{}\mathbf{F}(\mathbf{x})^{-1}\Big{\}}=\mathrm{ CRLB}(\mathbf{x}), \tag{67}\]  
where \(\mathbf{\hat{x}}\) denotes the estimated value of \(\mathbf{x}\).
## 2 RLBM-LES and Boundary Condition  
### RLBM-LES  
RLBM is an improved model based on LBM. Therefore, RLBM can be derived from the introduction of LBM. LBM deduced from the kinetic theory consists of three parts: a discrete lattice velocity model, equilibrium distribution functions (EDFs), and a time evolution equation [21,46]. The time evolution equation of LBM can be written as:  
\[f_{i}(x+e_{ix}\delta_{t},y+e_{iy}\delta_{t},z+e_{iz}\delta_{t},t+\delta_{t})-f_ {i}(x,y,z,t)=-\frac{1}{\tau}[f_{i}(x,y,z,t)-f_{i}^{eq}(x,y,z,t)], \tag{1}\]  
where \(f_{i}(x,y,z,t)\) is the particle distribution function (PDF) at site \((x,y,z)\) and time \(t\), \(f_{i}^{eq}(x,y,z,t)\) is the equilibrium distribution function at site \((x,y,z)\) and time \(t\), \(\delta_{t}\) is the lattice time step, \(e_{i}\) is the discrete velocity, and \(\tau\) is the dimensionless relaxation time, which is determined by the fluid viscosity \(\upsilon\) of the fluid:  
\[\tau=\frac{\upsilon}{c_{s}^{2}}+\frac{\Delta t}{2}, \tag{2}\]  
where \(c_{s}\) is the lattice speed of sound, \(c_{s}=\sqrt{3}/3\).  
According to the characteristics of LBM, Equation (1) consists of the following parts: collision and streaming,  
\[f_{i}^{+}(x,y,z,t)=f_{i}(x,y,z,t)+(-\frac{1}{\tau}[f_{i}(x,y,z,t)-f_{i}^{eq}(x,y,z,t)]), \tag{3}\]  
\[f_{i}(x+e_{ix}\delta_{t},y+e_{iy}\delta_{t},z+e_{iz}\delta_{t},t+\delta_{t})=f_ {i}^{+}(x,y,z,t), \tag{4}\]  
where \(f_{i}^{+}(x,y,z,t)\) denotes the post-collision state of PDF.  
When using LBM to simulate three-dimensional fluif flow, the D3Q19 (see Figure 1) model is the most commonly used, and the corresponding discrete velocities \(e_{i}\) are:  
Figure 1: Velocity vectors in the D3Q19 model.  
In order to simulate the turbulent flow with LES, the Smagorinsky sub-grid scale (SGS) model is incorated into the RLBM framwork. According to [47], the relationship between the total viscosity \(\
u_{total}\) and the total relaxation time \(\tau_{total}\) can be expressed by the following formula,  
\[\
u_{total}=\
u+\
u_{t}=3(\tau_{total}-\frac{1}{2}), \tag{14}\]  
where \(\
u_{t}\) is an additional viscosity called the turbulent eddy viscosity.  
The turbulent eddy viscosity can be expressed as  
\[\
u_{t}=C_{s}\delta_{x}^{2}|S_{\alpha\beta}|, \tag{15}\]  
where \(C_{s}\) denotes the Smagorinsky constant, \(\delta_{x}\) represents the lattice space, and \(S_{\alpha\beta}\) (\(\alpha,\beta\in x,y,z\)) in Cartesian coordinates is the strain rate tensor, i.e., \(S_{\alpha\beta}=(\partial_{\alpha}u_{\beta}+\partial_{\beta}u_{\alpha})/2\) and \(|S_{\alpha\beta}|=\sqrt{2S_{\alpha\beta}S_{\alpha\beta}}\). In the RLBM, \(S_{\alpha\beta}\) can be computed directly from the non-equilibrium stress tensor \(\prod_{\alpha\beta}^{\text{neq}}\) and the calculation of \(S_{\alpha\beta}\) according to the following formula as in [PERSON] et al. [48]:  
\[|S_{\alpha\beta}|=\frac{1}{6C_{s}\delta_{x}^{2}}(\sqrt{\
u^{2}+18C_{s}^{2} \delta_{x}^{2}\sqrt{\prod_{\alpha\beta}\prod_{\alpha\beta}}}-\
u). \tag{16}\]  
Therefore, \(\tau_{total}\) can be obtained by \(\
u_{total}\), and \(\tau_{total}\) is used to replace the original relaxation time \(\tau\) in Equation (12).
## 3 Results
### Comparison of Sentinel-1 and Sentinel-2 Combinations
#### 3.2.2 Qualitative Evaluation
From Figure 15, it is evident that in clear conditions, the Sentinel-1 image and the combinations of VV, VH and NIR; VV, VH and SWIR; VV, VH and RGB; VV and VH; and VV, VH and NDVI all resulted in the model having much less confidence in its decision-making, as exemplified by the inconsistency and low magnitude of the heatmaps. Conversely, the models trained with Sentinel-2 or with combinations of Sentinel-1 and Sentinel-2, that is, VV, VH, RGB and SWIR; and VV, VH, RGB, NIR and SWIR, all demonstrated much more appropriate and consistent focus. Notably, in these examples the areas in the upper left which were falsely identified as flooded in the inundation maps had a very low magnitude of importance. Figure 15: Sentinel-1 image, Sentinel-2 image, ground truth, and heatmap explanations, with higher temperature (with blue being a low temperature and red a high temperature) indicating greater feature importance, from each input-data combination for an example image in clear conditions. The heatmaps shown in Figure 16 show that in cloud-covered conditions, Sentinel-1 was the only image to provide the model with a reasonable level of confidence in its prediction, although the magnitude here was still relatively inconsistent. Although the map generated by the combination of VV, VH and NDVI was less accurate than when the combinations of VV, VH and SWIR; or VV, VH and MNDWI were used, the heatmap showed that this combination was actually placing more consistent and higher magnitude of important on the more relevant areas.
## 2 Fundamental Principle and Main Formula
### SHAP Interpretability Method  
SHAP is a unified interpretable machine learning method proposed by [PERSON] and [PERSON] based on game theory. It interprets model predictions by calculating the contribution of each feature when added to the model, where the sum of all feature contributions equals the final output of the model. This approach effectively integrates both global and local interpretability [15]. In the SHAP method, the model output is based on the linear summation of the SHAP values of the input features, meaning that the effect of each input feature on the final prediction is quantified as a specific SHAP value, which can be expressed as follows:  
\[h(x)=\mathrm{g}\big{(}x^{\prime}\big{)}=\varphi_{0}+\sum_{i=1}^{N}\varphi_{i}x ^{\prime} \tag{4}\]  
where \(h(x)\) represents the original prediction model; \(\mathrm{g}(x^{\prime})\) represents the corresponding interpretation model; \(x\) denotes the input feature; \(x^{\prime}\) represents the simplified input feature that are mapped to \(x\); \(\varphi_{0}\) is the predicted mean of all samples, also known as the baseline;represents the total number of features; and \(\varphi_{i}\) is the SHAP value of the \(\hat{\mathbf{r}}\)'th feature, which can be calculated by Equation (5).  
\[\varphi_{i}(f,x)=\sum_{z^{\prime}\subseteq x^{\prime}}\frac{|z^{\prime}|!(N-|z^ {\prime}|-1)!}{N!}\bigg{[}f_{x}(z^{\prime})-f_{x}(\frac{z^{\prime}}{i})\bigg{]} \tag{5}\]  
where \(\mathbf{N}\) represents the feature complete set; \(\mathbf{z^{\prime}}\) represents the feature subset; and \(f_{x}(z^{\prime})=f_{x}(h_{x}(z^{\prime}))=E[g(x)|z_{s}]\) represents the expected value on the subset \(\mathbf{z^{\prime}}\).  
Figure 2 briefly illustrates how SHAP analyzes the model to obtain the predicted value. It can be seen that SHAP interprets the final prediction of the model for the sample as the sum of the influence \(\varphi_{i}\) of each feature that introduces the conditional expectation. In the figure, red indicates that the feature contribution is positive, and blue indicates that the feature contribution is negative. Therefore, SHAP can quantify the influence of each feature in a given sample and its direction of correlation (positive or negative). It not only gives the importance of input features, but also shows how each feature affects the final prediction.
# A Linear-Time Algorithm for the Isometric Reconciliation of Unrooted Trees
## 1 Introduction
This problem can be solved in linear time for two rooted trees [7; 8]. For unrooted gene trees, it is possible to find all optimal root locations in linear time [9]. Rooting an unrooted species tree using a set of unrooted gene trees was also considered [10]. Edge lengths were used in the context of probabilistic models of gene duplication and deletion [11; 12; 13] and in models allowing horizontal transfer of genes between species [14; 15]. Figure 1: An example of a reconciliation of a gene tree and a species tree. Leaves of the gene tree are labeled by the species of origin; for example, \(b_{1}\) and \(b_{2}\) are genes from species \(b\). In the history implied by the reconciliation, there was originally a single copy of the gene, which underwent speciation corresponding to nodes \(q\) and \(r\) and was passed without change to species \(c\). On the other lineage, the gene was first duplicated (node \(u\)), and then underwent speciation in node \(x\), followed by one loss in each of the new lineages. As a result, species \(a\) had only one copy of the gene. Finally, the gene was again duplicated on the edge leading to \(b\) (node \(v\)). Note that if \(u\) was mapped to node \(x\) by the reconciliation, there would be only a single duplication (node \(v\)) and no losses in the corresponding history. In the rest of this section, we introduce necessary notation and formally define the isometric reconciliation problem. In Section 2, we describe our algorithm and supporting proofs, including the characterization of the solution set. We provide further discussion of our results and several open problems in Section 3.
# Federated Reinforcement Learning Based AANs with LEO Satellites and UAVs
## 3 System Design
### Reinforcement Learning Algorithm  
In this subsection, we explain the RL algorithm used in the proposed system. The PPO algorithm is based on the actor-critic concept which uses two separate networks [27]. The actor network determines the optimal behavior of an agent, while the critic network evaluates the policy and trains the actor using rewards. PPO was inspired by TRPO, and PPO provides a more direct approach to implementing and coordinating tasks for learning in comparison with TRPO. Furthermore, PPO is known to provide simpler and superior performance than TRPO in many areas [33]. Thus, the PPO algorithm is suitable for various tasks performed by UAVs in the context of UAV system control because of the algorithm's short calculation time [34]. Actually, many researches on UAVs utilized the PPO as RL algorithms, and many results showed that the PPO outperforms other algorithms in various UAV operations, such as attitude control, landing, waypoint navigation, etc [34]. Thus, we selected the PPO as a learning algorithm for our system.