esa-sceva/satcom-chunk-collection huggingface dataset:
# Integration of Multi-Branch GCNs Enhancing Aspect Sentiment Triplet Extraction
###### Abstract  
Aspect Sentiment Triplet Extraction (ASTE) is a complex and challenging task in Natural Language Processing (NLP). It aims to extract the triplet of aspect term, opinion term, and their associated sentiment polarity, which is a more fine-grained study in Aspect Based Sentiment Analysis. Furthermore, there have been a large number of approaches being proposed to handle this relevant task. However, existing methods for ASTE suffer from powerless interactions between different sources of textual features, and they usually exert an equal impact on each type of feature, which is quite unreasonable while building contextual representation. Therefore, in this paper, we propose a novel Multi-Branch GCN (MBGCN)-based ASTE model to solve this problem. Specifically, our model first generates the enhanced semantic features via the structure-biased BERT, which takes the position of tokens into the transformation of self-attention. Then, a biaffine attention module is utilized to further obtain the specific semantic feature maps. In addition, to enhance the dependency among words in the sentence, four types of linguistic relations are defined, namely part-of-speech combination, syntactic dependency type, tree-based distance, and relative position distance of each word pair, which are further embedded as adjacent matrices. Then, the widely used Graph Convolutional Network (GCN) module is utilized to complete the work of integrating the semantic feature and linguistic feature, which is operated on four types of dependency relations repeatedly. Additionally, an effective refining strategy is employed to detect whether word pairs match or not, which is conducted after the operation of each branch GCN. At last, a shallow interaction layer is designed to achieve the final textual representation by fusing the four branch features with different weights. To validate the effectiveness of MBGCNs, extensive experiments have been conducted on four public and available datasets. Furthermore, the results demonstrate the effectiveness and robustness of MBGCNs, which obviously outperform state-of-the-art approaches.  
ASTE; biaffine attention; structure-biased BERT; GCN; linguistic feature +
Footnote †: journal: _Applied Sciences_
# An Improved Differentiable Binarization Network for Natural Scene Street Sign Text Detection
## 4 Result and Discussion
### Experimental Datasets  
According to the literature survey, so far, no street sign dataset can be used in the natural scene of the experiment. Having a good dataset is the first step for street sign text detection experiments, so in this paper, we created our dataset, NSSS, to validate our proposed method. All of the datasets came from images taken in real environments, including street signs on both sides of daily streets, street signs in schools and shopping malls, street signs in scenic parks, and many other scenes, some data set images are shown in Figure 8. In order to simulate the noise that might be introduced by lighting and weather in natural scenes, some images were randomly adjusted the brightness and contrast. To obtain the true text areas (ground truth) of the images, all the images were manually annotated with text and 2000 images were selected from them for the experiment, of which 1500 were used for the training set and 500 were used for the test set.  
To further test the effectiveness of our proposed method, the publicly available dataset ICDAR2015 [31] and dataset ICDAR2017-MLT [32] were used for our experiments for evaluation. ICDAR2015 consisted of 1500 images taken in natural scenes and contained a large amount of irregular text. Of these images, 1000 were used as the training set, and 500 were used as the test set. ICDAR2017-MLT is a text dataset composed of multiple languages, including Chinese, English, Korean, Japanese, Italian, French, Indian, Arabic, and German. There are 9000 images in the data set, of which 7200 are used for the training set and 1800 are used for the test set.  
Figure 8: Partial images of the NSSS dataset.
## 3 Results
### Comparison of the Spatial Variability of Phenometrics  
The spatial distributions of mean \(D_{til}\), \(D_{head}\), and \(D_{mat}\) for Sites A and B during the period 2013-2017 are presented in Figure 6. Generally, all of these three phenometrics in Site A were earlier than those in Site B. For either the NDVI- or the phase-space-based phenometrics, the overall mean \(D_{til}\) and \(D_{head}\) in Site A were approximately 5-7 days earlier than those in Site B, while the overall mean \(D_{mat}\) in Site A was only 2-3 days earlier than that in Site B. The NDVI- and the phase-space-based phenometrics were close to each other. For both experimental sites, the overall mean values of NDVI-based \(D_{til}\) and \(D_{head}\) were 1-3 days earlier than those based on phase-space, while the overall mean values of NDVI-based \(D_{mat}\) were almost the same or a little later (approximately 1 day) compared to the phase-space-based ones.  
The spatial distributions of the mean \(L_{season}\), \(L_{veg}\), and \(L_{rep}\) in Sites A and B during the period 2013-2017 are presented in Figure 7. For either the NDVI- or the phase-space-based phenometrics, the overall mean \(L_{season}\) and \(L_{rep}\) in Site A were 2-3 days longer than those in Site B, while the overall mean \(L_{veg}\) in Site A was 1-2 days shorter than that in Site B. For both experimental sites, the overall mean values of the NDVI-based \(L_{season}\) and \(L_{rep}\) were 1-2 days longer than those derived from the phase-space, while the overall mean values of NDVI-based \(L_{veg}\) were the same or a little longer (1 day) compared to the phase-space-based ones. Noticeably, the spatial standard deviations of the 5-year mean \(L_{season}\) (2.8 to 3.1), \(L_{veg}\) (2.0 to 2.4), and \(L_{rep}\) (1.9 to 2.2) were distinctly smaller than those of the 5-year mean \(D_{til}\) (5.0 to 5.5), \(D_{head}\) (4.0 to 4.4), and \(D_{mat}\) (3.4 to 3.9).  
Figure 5: Comparison between satellite-derived rice phenometrics and plot-based rice growth stages for Site A and B in 2014. The satellite-derived phenometrics include the dates of active tillering (\(D_{til}\)), middle heading (\(D_{head}\)), and maturity (\(D_{mat}\)), from left to right, derived from NDVI and phase–space, respectively. The rice phenological stages from the 10 sample plots were acquired from the in-situ investigations. The end of the maturity stage represents the harvesting date.  
Figure 6: Spatial distributions of the mean dates of active tillering (\(D_{tilt}\)) (**a–d**), middle heading (\(D_{had}\)) (**e–h**), and maturity (\(D_{mat}\)) (**i–l**) for Sites A and B during the period 2013–2017. Each of the phenometrics was derived from NDVI (**a**,**c**,**e**,**g**,**i**,**k**) and phase–space (**b**,**d**,**f**,**h**,**j**,**l**).  
Figure 7: Spatial distributions of the mean lengths of growth season (\(L_{season}\)) (**a–d**), vegetative phase (\(L_{vcg}\)) (**e–h**), and reproductive phase (\(L_{rep}\)) (**i–l**) for Sites A and B during the period 2013–2017. Each of the phenometrics was derived from NDVI (**a**,**c**,**e**,**g**,**i**,**k**) and phase–space (**b**,**d**,**f**,**h**,**j**,**l**).
## 2 Methodology  
The methodology of this study is shown in Figure 1. It can be divided into two main steps. In the first step, statistical features were extracted from the AE signals acquired from the spherical tank under normal and cracked states. In the second step, these extracted statistical features were provided to SVM to identify whether each input instance belonged to the normal spherical tank or the cracked state.
## 3 Simulation and Results
### Procedure  
The following steps illustrate the procedure for our simulation.  
1. A confined urban area is selected as the simulation environment. Subsequently, the characteristics of the area, such as boundary coordinates, streets, and building locations and heights, are obtained from the OSM. For this research, two different urban areas, named area-A and area-B are selected.
2. Within the simulation environment, APs and SNs are randomly distributed in open areas according to a homogeneous PPP with specific densities. Coordinates are assigned to each AP and SN.
3. The Phased Array System Toolbox 2023 in MATLAB is used to implement the antenna arrays for APs and SNs.
4. MATLAB SiteViewer 2023 is used to perform various operations in 3D, as follows: 1. Import and visualize the simulation environment, APs transmitters, and SNs receivers in 3D. 2. Compute the distances between nodes, such as APs and SNs. 3. Perform ray tracing analysis by computing and plotting propagation paths between APs and SNs. The properties of each ray are computed, including the power loss, delay, azimuth angle of departure (AoD), azimuth angle of arrival (AoA), connection type (LoS or NLoS), and number of reflections in the case of NLoS. 4. Compute RSS and SINR at each SN.
5. The number of SNs attached to each cell is computed considering the maximum RSS cell association. The USE, CSE, and ASE are then calculated according to (6)-(8).
6. Using the output of the ray tracing analysis, customized CDL channel profiles are constructed for each channel.
7. For each channel profile, a model is developed for measuring the throughput of the 5G physical downlink shared channel (PDSCH) using MATLAB [48]. The model is customized to compute the maximum CQI and MCS that result in a block error rate (BLER) not exceeding 0.1. It is worth noting that the BLER may exceed 0.1 if a channel profile has poor radio conditions, such as low SINR, even if the lowest CQI and MCS are selected.
8. After identifying the channel profiles and their CQI indices, the results are used in a model developed in MATLAB [49] to compute the network performance in terms of DL throughput considering the round-robin scheduling scheme. In this step, all the results of the reference model are obtained.
9. The output of step 5 is used to identify the APs with the lowest SN association in order to replace them with UAVs, with the assumption that 50% of existing APs can be replaced by UAVs.  
10. Starting from an altitude of 15 m for the UAVs, steps 3, 4, and 5 are repeated to identify the optimum altitude that ensures LoS and results in the maximum ASE.
11. After identifying the optimal altitude, steps 6, 7, and 8 are executed to obtain the results for the UAV-assisted model.
# Modeling and Performance Analysis of mmWave and WiFi-Based Vehicle Communications
## 3 Performance Analysis
### Successful Transmission Probability (\(Stp\))
#### 3.1.1 WiFi Communications Scenario  
The \(STP\) expressions in LoS and NLoS WiFi communications scenarios are derived as follows:  
(a) LoS Scenario:  
Based on (3), the \(STP\) expression in an LoS WiFi communications scenario can be written as follows:  
\[STP_{{}_{WiFi}}^{{}^{LoS}}=\Pr\Big{\{}SNR_{{}_{WiFi}}>=\gamma_{{}_{th}}\Big{\}} =\Pr\Bigg{\{}H_{{}_{1}}\geq\frac{\gamma_{{}_{th}}\mathcal{P}_{{}_{N}}r^{\beta_ {1}}}{\mathcal{P}_{{}_{T}}A_{{}_{W}}}\Bigg{\}}, \tag{17}\]  
where the transmission distance \(r\) can be expressed as follows:  
\[r=\sqrt{d^{2}+[\chi-\eta]^{2}}, \tag{18}\]  
with \(\eta=\min(h_{{}_{Tx}},h_{{}_{Rx}})\), and \(\chi=\max(h_{{}_{Tx}},h_{{}_{Rx}})\).  
By using the PDF expressions of \(H_{{}_{1}}\), \(STP_{{}_{WiFi}}^{{}^{LoS}}\) is derived as follows:  
\[STP_{{}_{WiFi}}^{{}^{LoS}}=\int_{\frac{\gamma_{{}_{th}}\mathcal{P }_{{}_{N}}(d^{2}+[\chi-\eta]^{2})}{\gamma_{{}_{T}}A_{{}_{W}}}}^{+\infty}p_{{}_ {{}_{1}}}(x)\ dx\] \[\approx\frac{\mathcal{K}+1}{\mu_{1}}\sum_{n=0}^{N}\left(\frac{ \mathcal{K}[\mathcal{K}+1]}{\mu_{1}}\right)^{n}\exp\left(-\mathcal{K}\right) \int_{\frac{\gamma_{{}_{th}}\mathcal{P}_{{}_{N}}(d^{2}+[\chi-\eta]^{2})}{ \gamma_{{}_{th}}\mathcal{P}_{{}_{T}}A_{{}_{W}}}}^{+\infty}\exp\left(-\frac{[ \mathcal{K}+1]}{\mu_{1}}x\right)\frac{x^{n}}{n!\ \Gamma(n+1)}dx. \tag{19}\]  
By evaluating the integration in (15), with respect to \(x\), using (3.351.21) in [34], and after some simplifications, the expression of \(STP_{{}_{WiFi}}\) yields to  
\[STP_{{}_{WiFi}}^{{}^{LoS}}\approx\tau\exp\Big{(}-\chi_{{}_{1}}\tau-\mathcal{K }\Big{)}\sum_{n=0}^{N}\left\{\frac{\left(\mathcal{K}\tau\right)^{n}}{n!\ \Gamma(n+1)}\sum_{k=0}^{n}\left\{\frac{n!}{k!}\frac{\chi_{{}_{1}}^{k}}{\tau^{n -k+1}}\right\}\right\}, \tag{20}\]  
where \(\tau=\frac{\mathcal{K}+1}{\mu_{1}},\chi_{{}_{1}}=\frac{\gamma_{{}_{th}} \mathcal{P}_{{}_{N}}(d^{2}+[\chi-\eta]^{2})}{\mathcal{P}_{{}_{T}}A_{{}_{W}}}\).  
(b) NLoS Scenario:  
For an NLoS WiFi communications scenario, and based on (3), the \(STP\) expression can be written as follows:  
\[STP_{{}_{WiFi}}^{{}^{NLoS}}=\Pr\Bigg{\{}H_{{}_{2}}\geq\frac{\gamma_{{}_{th}} \mathcal{P}_{{}_{N}}r^{\beta_{2}}}{\mathcal{P}_{{}_{T}}A_{{}_{W}}}\Bigg{\}}, \tag{21}\]By using the PDF expression of \(H_{2}\), the expression of \(STP_{{}_{WPi}}^{{}^{NLoS}}\) can be written as follows  
\[STP_{{}_{WPi}}^{{}^{NLoS}} =\int_{\frac{\gamma_{th}P_{N}(d^{2}+[\chi-\eta]^{2})}{\mathcal{T}_{T }A_{W}}}^{+\infty}p_{{}_{H_{2}}}(x)\;dx\] \[=\int_{\frac{\gamma_{th}P_{N}(d^{2}+[\chi-\eta]^{2})}{\mathcal{T}_ {T}A_{W}}}^{+\infty}\frac{\gamma_{th}P_{N}(d^{2}+[\chi-\eta]^{2})}{\mathcal{T}_ {T}A_{W}}\frac{1}{\mu_{2}}\exp\left(-\frac{x}{\mu_{2}}\right)\;dx. \tag{22}\]  
By evaluating the integration in (22), with respect to \(x\), the final expression of \(STP_{{}_{WPi}}^{{}^{NLoS}}\) is given by  
\[STP_{{}_{WPi}}^{{}^{NLoS}}=\exp\left(-\frac{\gamma_{th}\mathcal{P}_{N}(d^{2}+ [\chi-\eta]^{2})^{\frac{\rho_{2}}{2}}}{\mathcal{P}_{T}A_{W}\mu_{2}}\right). \tag{23}\]
## 2 Definitions and Notations  
This section presents definitions and notations used in the paper [1; 2; 3; 4; 6; 7; 8; 10; 46].  
**Definition 1**.: _A Petri net \(N\) is a 4-tuple: \(N=(P,T,F,M_{0})\), where:_  
* \(P=\{p_{1},p_{2},\ldots,\ p_{n}\}\) _is a finite set of places,_
* \(T=\{t_{1},t_{2},\ldots,\ t_{m}\}\) _is a finite set of transitions,_
* \(F\subseteq(P\times T)\cup(T\times P)\) _is a finite set of arcs,_
* \(\mathrm{M}_{0}:P\rightarrow\ \mathbb{N}\) _is an initial marking (state)._  
**Definition 2**.: _A marking (state) \(M\) of a Petri net \(N=(P,T,F,M_{0})\) is defined as a subset of its places: \(M\subset P\). A place belonging to a marking is called a marked place. A marking can be changed by transition frings. Firing of a transition \(t\) changes marking by removing a token from each input places of \(t\), and adds a token to each output place of \(t\)._  
**Definition 3**.: _Incidence matrix of a Petri net \(N=(P,T,F,M_{0})\) with \(n\) places and \(m\) transitions is a matrix \(A_{|T|\times|P|}\) of integers, given by:_  
\[a_{ij}=\left\{\begin{array}{ll}-1,&\left(p_{j},t_{i}\right)\in F\\ 1,&\left(t_{i},p_{j}\right)\in F\\ 0,&\mathrm{otherwise}\end{array}\right.. \tag{1}\]  
_A cell \(a_{ij}\) of matrix \(A\) refers to transition \(t_{i}\) and place \(p_{j}\). The columns of the matrix correspond to places, while the rows refer to transitions of a Petri net._  
**Definition 4**.: _A place invariant (\(p\)-invariant) of a Petri net \(N=(P,T,F,M_{0})\) is a \(n\)-vector \(\overrightarrow{x}\) of integers such that \(A\overrightarrow{x}=0\). The set of places corresponding to nonzero entries in a \(p\)-invariant \(\overrightarrow{x}\geq 0\)is a support of the p-invariant. A net is covered by p-invariants if every place \(p\in P\) belongs to at least one support of a p-invariant._  
**Definition 5**.: _A state machine is a Petri net for which every transition has exactly one input place and exactly one output place, i.e., \(\forall t\in T:\left|\bullet t\right|=\left|t\bullet\right|=1\)._  
**Definition 6**.: _A state machine component (SMC) of a Petri net \(N=(P,T,F,M_{0})\) is a subnet \(S=(P^{\prime},T^{\prime},F^{\prime},M^{\prime}_{0})\) generated by \(P^{\prime}\subseteq P\) such that:_  
* \(S\) _is a state machine;_
* \(P^{\prime}\in P\)_,_ \(T^{\prime}\in T\)_,_ \(F^{\prime}\in F\cap(P^{\prime}\times T^{\prime})\cup(T^{\prime}\times P^{ \prime})\) _;_
* \(S\) _is strongly connected and has exactly one token in initial marking._  
**Definition 7**.: _A state machine decomposition (S-decomposition) of a Petri net \(N=(P,T,F,M_{0})\) is a set \(\mathcal{S}=\{S_{1},\ldots,S_{n}\}\) such that each component \(S_{j}\in\mathcal{S}\) is an SMC and each place \(p_{i}\in P\) belongs to exactly one component \(S_{j}\in\mathcal{S}\). If place \(p_{i}\in P\) exists in more than one \(S_{j}\in\mathcal{S}\), it is replaced in all remaining components (except one) by a non-operational place (NOP)._
# Cooperative UAV Scheme for Enhancing Video Transmission and Global Network Energy Efficiency
## 4 Evaluation
### Simulation Description and Metrics
On the other hand, \(UAV_{1}\) and \(UAV_{2}\) videos are captured from a UAV flying in a city and in a rural environment, but \(UAV_{2}\) has a higher motion level than \(UAV_{1}\) caused by UAV instability during the flight. We encoded those videos with an H.264 codec at 300 kbps, 30 fps, Group of Pictures (GoP) size of 20 frames, and common intermediate format (352 \(\times\) 288 pixels). The decoder applies a Frame-Copy method for error concealment to replace each lost frame with the last received one, reducing frame loss and maintaining the video quality. We consider the number of alive nodes and the remaining energy of alive UAVs to measure the effects of energy consumption on the UAV flying and packet transmissions. In terms of video quality evaluation, QoS schemes alone are not enough to assess the quality level of multimedia applications because they fail in capturing subjective aspects of video content related to human experience [6]. QoE metrics overcome those limitations, and thus we rely on a well-known objective QoE metrics, namely Structural Similarity (SSIM). Specifically, SSIM \(\in\) [0,1] is based on a frame-by-frame assessment of three video components, i.e., luminance, contrast, and structural similarity. Higher SSIM values mean better video quality. We used the MSU Video Quality Measurement Tool (VQMT) to measure the SSIM values for each transmitted video.
## 1 Introduction  
Terahertz (THz) technology has proven to be quite elusive; this is the electromagnetic spectrum of THz radiation situated between infrared light and microwave radiation. Therefore, there is no ionization hazard of THz radiation and penetration through dielectric materials is reasonably good [1]. The atmospheric scattering of THz radiation is relatively low [2]. Thus, devices based on this wavelength are becoming increasingly important in a diversity of human activity applications [3, 4], e.g., security, imaging, biological, communications, medicine, etc. This led to the development of THz detection systems such as bolometers, which are essentially heat detectors [5]. Although bolometers have large spectral rangeand less noise-equivalent power, they have a long response time [6]. Electro-optic sampling in nonlinear crystals, e.g., ZnTe, GaAs, Gap, Dast, etc, has been investigated. Electro-optic detection is a second-order nonlinear optical process in which an applied electric field induces a refractive index change in nonlinear crystals at THz frequencies that is proportional to the applied field [7]. Electro-optic detection requires dealing with phase matching issues. Microfabricated antennas on semiconductors measure the THz pulse but require specific expertise to make them using cleanroom facilities [8]. There is a high priority to using THz detectors based on the free-space electro-optic process in nonlinear crystals [9]. Due to the diffraction-limited spatial resolution, femtosecond temporal resolution, THz frequency bandwidth, mV/cm field sensitivity, and signal to noise ratio (SNR) are relatively high [10, 11]. Typically the ZnTe crystals are used for electro-optic sampling to detect ultrashort THz pulses using Ti: sapphire fs-lasers at a wavelength around 800 nm lasers [12]. In contrast to usual Ti: sapphire fs-lasers at a wavelength of 800 nm the fiber lasers at a wavelength of 1550 nm systems are compact, low-cost, and in the optical communication range. Recently, various researches have examined and offered methods for improving the electro-optic process detection using ZnTe crystals. The researches which have made a comparison between the ZnTe and GaAs crystals represent the free space detection methods without increasing or improving the output signal [13]. This paper presents an improvement of electro-optic detection of THz single pulse by GaAs crystal by using a fiber coupling system. Gallium Arsenide (GaAs) crystals have been employed in THz radiation detection [14]. The preference for GaAs crystal is because of its detection frequency range, which reaches 6 THz. The coupling into fiber provides an easy way to increase the optic output signal, which can increase the dynamic range of the detector. Experiment results show an improvement and increase of the output signal of the electro-optic detector. Further, the coupling of the optical output signal into fiber allows stretching the THz pulse, which can be done by using the electro-optic process to detect short THz pulses [15].
## 6 Phase Field Lattice Boltzmann Method
### Solving the Cahn-Hillard Equation for Interface Capturing with the WENO Scheme  
In the phase field model, a local order parameter, \(\varnothing\), is defined to differentiate between two phases, i.e., \(\varnothing_{1}\) and \(\varnothing_{g}\), referring to the liquid and gas phase, respectively. The local order parameter controls the kinetics and evolution of each phase throughout the simulation. In the CH equation, the motion of the interface is defined as:  
\[\frac{\partial\varnothing}{\partial t}+(\mathbf{u}.\
abla)\varnothing=\
abla. (\lambda\
abla\mu_{\varnothing}) \tag{22}\]  
This equation contains terms related to convection and diffusion of the interface. \(\lambda\) is a diffusion coefficient called mobility. \(\mu_{\phi}\) is the chemical potential, which is defined as the derivative of the free energy (\(\Psi_{\phi}^{\prime}\)) with respect to the order parameter as \(\mu_{\varnothing}=\Psi_{\phi}^{\prime}(\varnothing)-\kappa\
abla^{2}\varnothing\). If the free energy takes the double well form as \(\Psi_{\phi}=\beta\varnothing^{2}(\varnothing-1)^{2}\), then the chemical potential becomes \(4\beta\varnothing(\varnothing-0.5)(\varnothing-1)-\kappa\
abla^{2}\varnothing\), where both \(\beta\) and \(\kappa\) are related to the surface tension, \(\sigma\), and interface width, W, as \(\beta=\frac{12\sigma}{\text{w}\left(\rho_{1}-\rho_{g}\right)^{4}}\) and \(\kappa=1.5\frac{\text{W}\sigma}{\left(\rho_{1}-\rho_{g}\right)^{2}}\). For multiphase flow with large density ratios between phases, the order parameter distribution across the interface may show a high gradient, which causes instability in the simulation. To overcome this problem, it is preferred to use an upstream condition for the advection term. In this article, the upwind WENO scheme with third order Runge Kutta total variation diminishing (TVD) [57] is implemented to discretize the convection and temporal term, respectively. Equation (22) can then be rewritten as:  
\[\frac{\partial\phi}{\partial t}=\text{R}(\varnothing)\,\text{R}(\varnothing)= -\
abla.(\text{l})+\
abla.(\lambda\
abla\mu_{\varnothing}) \tag{23}\]  
The term, \(\upomega\), in Equation (22) is labeled as l for simplification. The discretization of the first term in the right-hand side of Equation (23) can be achieved in a variety of ways. In this paper, we consider the Lax-Friedrichs flux splitting, which uses three stencils, formed by five points [58]. In this method, the convection term is calculated by flux terms, which are determined by:  
\[\text{l}^{+}=\frac{1}{2}(\upomega+\text{m}\varnothing)\text{l}^{-}=\frac{1}{ 2}(\upomega-\text{m}\varnothing) \tag{24}\]  
where m is selected as \(\text{max}|(\upomega)^{\prime}|\) to make the scheme stable. Since the discretization in the any coordinate direction follows the same procedure, the discretization in the \(x\) direction is going to be explained.