esa-sceva/satcom-chunk-collection huggingface dataset:
# A New Centralized Clustering Algorithm for Wireless Sensor Networks
## 1 Introduction
Then, the skip is set by an experimental evaluation of the network performance in terms of reliability and durability. The implementation of the technique demonstrates the extension of the network lifetime when compared with other clustering algorithms. Therefore, this paper presents the following contributions:  
* The demonstration of the benefits obtained when a skip is set. As described in hypothesis H1, a carefully selected skip value significantly improves the lifetime of a WSN.The paper also verifies that nodes elected as CHs cannot keep this role for an undetermined period because they will deplete their batteries prematurely and degrade network reliability. This last behaviour is compliant with hypothesis H2. * The definition and validation of a Type-2 fuzzy system to manage the uncertainty of the network information when deciding the CHs with reduced control messages. * The proposed algorithm was tested in a wide set of scenarios and compared with a significant number of clustering algorithms. The rest of the paper is structured as follows. The next section presents a review and a classification on the algorithms used to select the CHs. Section 3 describes the proposed fuzzy system, giving a detailed explanation about how this system was designed. Section 4 describes the simulation results obtained to evaluate the proposed algorithm. Finally, Section 5 includes the conclusions and the future research guidelines.
# Analytical Solution and Experimental Validation of the Electromagnetic Field in an IPT System
## 4 Electric and Magnetic Field in the IPT System
### Analytical Solution of the Electric and Magnetic Field  
As shown in Equation (7), there is a one-to-one correspondence between the electric field \(\mathbf{E}\) and the magnetic flux density \(\mathbf{B}\). When the analytical solution of \(\mathbf{E}\) is obtained, the analytical solution of \(\mathbf{B}\) can also be uniquely determined. Thus, in this section, the electric field is calculated only.  
An electric field illustration of the sampling point in the double-coil system is given in Figure 5. The radius of each coil is \(a\). The current in the primary and secondary coil is defined as \(I_{P}\) and \(I_{S}\), respectively. \(N_{P}\) and \(N_{S}\) are the numbers of turns in the two coils, respectively. The distances from \(Coil_{P}\) and \(Coil_{S}\) to the point Q are denoted as \(l_{2}\) and \(h_{1}\), respectively.  
The electric field only exists along the \(\phi\) direction. Thus, by combining Equations (2) and (5), when the system is operated at its resonant frequency, the following equation can be obtained:  
\[I_{S}=\frac{-\oint\limits_{Coil_{S}}\mathbf{E}_{P_{0}}dl_{S}}{R_{L}+R_{S}+\left(j \omega L_{S}+\frac{1}{j\omega C_{S}}\right)}=\frac{-\oint\limits_{Coil_{S}}\bm {E}_{P_{0}}dl_{S}}{R_{L}+R_{S}}, \tag{44}\]  
where \(E_{P_{0}}\) is the electric field at \(Coil_{S}\) produced by the current in \(Coil_{P}\):  
\[\mathbf{E}_{P_{0}}=-\frac{j\omega uNa_{P}I_{P}}{2}\int\limits_{0}^{\infty}\frac{ \lambda}{u}I_{1}(\lambda a)I_{1}(\lambda\rho)e^{-u(h_{1}+h_{2})}d\lambda\mathbf{e }_{\phi}, \tag{45}\]  
The total electric field at point Q is \(\mathbf{E}=\mathbf{E}_{P}+\mathbf{E}_{S}\), and  
\[\mathbf{E}_{P}=-\frac{\omega uNa_{P}I_{P}}{2}\int\limits_{0}^{\infty}\frac{ \lambda}{u}I_{1}(\lambda a)J_{1}(\lambda\rho)e^{-uh_{1}}d\lambda\mathbf{e}_{\phi}, \tag{46}\]  
\[\mathbf{E}_{S}=-\frac{j\omega\mu Na_{S}I_{S}}{2}\int\limits_{0}^{\infty}\frac{ \lambda}{u}I_{1}(\lambda a)J_{1}(\lambda\rho)e^{-uh_{2}}d\lambda\mathbf{e}_{\phi}, \tag{47}\]  
where \(\mathbf{E}_{P}\) and \(\mathbf{E}_{S}\) are the electric fields produced by the current in the primary and secondary coils, respectively. Equations (46) and (47) laid the base for the next step numerical calculations to find the distribution of the E and H field.
# Short-Term and Long-Term Travel Time Prediction Using Transformer-Based Techniques
## Appendix B Statistical Analysis of Predictive Performance  
To validate our findings, we conducted a statistical analysis comparing the MAEs and RMSEs of the presented models for statistical significance. We selected the Mann-Whitney U test due to its suitability for its robustness in handling non-normally distributed data [31]. Our analysis utilized 7 days of data, as presented in Table 5, to assess the stability and consistency of the SLIT model under varying conditions. Compared with the best competitive method, PASS2S, in MAE, we employed a one-tailed test based on our directional hypothesis, anticipating superior performance from SLIT.  
* Null Hypothesis (\(H_{0}\)): \(MAE_{SLIT}\geq MAE_{PASS2S}\)
* Alternative Hypothesis (\(H_{1}\)): \(MAE_{SLIT}<MAE_{PASS2S}\)  
Our findings (U statistic: 9.0, \(p\)-value: 0.0265) indicate that SLIT statistically outperforms PASS2S. This result is statistically significant as the \(p\)-value is less than 0.05, suggesting a significant advantage in decreasing errors across the prediction horizon.  
Conversely, for RMSE, initial tests were two-tailed to assess any performance differences without a predefined direction of superiority, given the competitive nature of SLIT and PASS2S in managing larger errors.  
* Null Hypothesis (\(H_{0}\)): \(RMSE_{SLIT}=RMSE_{PASS2S}\)
* Alternative Hypothesis (\(H_{1}\)): \(RMSE_{SLIT}\
eq RMSE_{PASS2S}\)  
The RMSE results (U statistic: 38.0, \(p\)-value: 0.9636) showed no significant difference, indicating that both models manage larger errors similarly. This finding suggests that there is no statistically significant difference in how each model handles larger errors under the tested conditions.  
Given these outcomes, we extended our analysis to include one-tailed tests against other competitive models, hypothesizing that SLIT's capabilities are likely to result in lower RMSE values.  
* Null Hypothesis (\(H_{0}\)): \(RMSE_{SLIT}\geq RMSE_{OtherModels}\)
* Alternative Hypothesis (\(H_{1}\)): \(RMSE_{SLIT}<RMSE_{OtherModels}\)  
Results from these one-tailed tests, detailed in Table 5, confirm that SLIT significantly outperforms DE-SLSTM, MTSMFF, DHM, and TFT in RMSE reduction. This evidence, indicated by \(p\)-values well below 0.05, underscores that SLIT's RMSE performance is not only statistically significant but also superior to these competitors under long-term prediction conditions.
# Signal Preprocessing for Enhanced IoT Device Identification Using Support Vector Machine
## 3 Results
### SVM-Based Classifier with Normalized Raw Signals  
The confusion matrix in Table 3 shows a marked improvement in classifier performance once the raw Bluetooth signals are normalized. Overall diagnostical accuracy rises to 51.06 %, and effective accuracy climbs to 75 %, confirming that normalization substantially enhances the separability of most classes.  
Normalization produces clear diagonal dominance for several devices. Notably, Class 7 (a Rusins wearable) attains a diagonal value of 0.93, and Class 8 (an Uzundurukan smartphone) reaches 0.97, indicating that these two devices become almost perfectly distinguishable after normalization. Within the Rusins set, Class 18 also shows very strong performance with 0.87 on the diagonal. However, Classes 1-3 still exhibit considerable mutual confusion: Class 1's diagonal of 0.65 is accompanied by off-diagonal peaks of 0.57 (Class 2), 0.52 (Class 3) and even 0.53 (Class 18), while Class 3, despite peaking at 0.40, remains frequently misassigned to Classes 1 and 2. These results suggest that although normalization reduces gross amplitude differences, the intrinsic RF similarities among these wearables persist.  
Among the Uzundurukan smartphone twins, normalization likewise brings significant gains. Classes 15 and 16 both achieve diagonals of 0.87 and 0.97, respectively, and Class 8 with 0.97 confirms its robust separability. Residual overlap remains in a few cases: Class 4, despite improving to 0.60 on its diagonal, still scatters into other rows; Class 9 climbs to 0.77 but maintains non-negligible entries in Classes 13 and 17; and Class 10 peaks at only 0.47 while still misclassifying many samples as Class 2, indicating cross-dataset confusion.
# Fusion of Telecommunications and IT Services Boosted by Application Programming Interfaces
## Abbreviations  
* Application Programming Interface
* Application Server
* Asynchronous Transfer Mode
* Business Logic
* Base Transceiver Station
* Customized Applications for Mobile network Enhanced Logic
* CAMEL Application Part  
Figure 9: Convergence of telco services.  
\begin{tabular}{l l} CNF & Cloud-native Network Function \\ CS & Circuit Switched \\ CSCF & Call Session Control Function \\ CSP & Communication Service Provider \\ CP & Control Plane \\ DC & Data Channel \\ EDGE & Enhanced Data rates for GSM Evolution \\ EPC & Evolved Packet Core \\ ETSI & European Telecommunications Standards Institute \\ FMC & Fixed-Mobile Convergence \\ GSM & Global System for Mobile Communications \\ HLR & Home Location Register \\ HSS & Home Subscriber Server \\ HW & hardware \\ IaaS & Infrastructure as a Service \\ IMS & IP Multimedia Subsystem \\ IN & Intelligent Network \\ INAP & Intelligent Network Application Protocol \\ IM-SSF & IP Multimedia Service Switching Function \\ IP & Internet Protocol \\ IT & Information Technology \\ IoT & Internet of Things \\ LTE & Long-Term Evolution \\ LTE-A & Advanced LTE \\ MEC & Multi-Access Edge Computing \\ MNO & Mobile Network Operator \\ MMS & Multimedia Messaging Service \\ MMTEL & Multimedia Telephony \\ MSC & Mobile Switching Center \\ MSS & MSC Server \\ M2M & Machine-to-Machine \\ NaaS & Networking as a Service \\ NE & Network Element \\ NR & New Radio \\ NSS & Network Switching Subsystem \\ NW & network \\ OEM & Original Equipment Manufacturer \\ OMA & Open Mobile Alliance \\ OpEx & Operational Expenditure \\ OSA & Open Services Architecture \\ OT & Operation Technology \\ OTT & Over-the-Top \\ PaaS & Platform as a Service \\ PDH & Plesiochronous Digital Hierarchy \\ POTS & Plain Old Telephone Service \\ PNF & Physical Network Function \\ PS & Packet Switched \\ PSTN & Public Switched Telephone Networks \\ REST & Representational State Transfer \\ RAN & Radio Access Network \\ RAT & Radio Access Technology \\ SaaS & Software as a Service \\ SBA & Service-Based Architecture \\ SBC & Session Border Controller \\ SCC AS & Service Centralization and Continuity Application Server \\ SCF & Service Control Function \\ SCP & Service Control Point \\ \end{tabular}
# Interactions in Augmented and Mixed Reality: An Overview
## 3 Research Results
### Audio-Based Modality
#### 3.2.1 Sound-Based  
_Sound source recognition_ and _sound visualization_ are two methods [PERSON] et al. used in [117] to augment human vision in identifying sound sources for people with hearing disabilities. The system can successfully perform identification based on recognition algorithms that exploit the microphone's capabilities and AR tags placed on variant objects to position a virtual object that indicates the sound source. [PERSON] et al. [118] reviewed research papers to determine challenges and opportunities in spatial soundscapes. Spatial soundscapes exploit the _spatial sound perception_ method to enhance situational awareness. _Audio superimposition_ is another method that is commonly used in audio augmented reality. In [119], the authors [PERSON] et al. combined virtual sounds with real sounds captured by a microphone and reproduced them with stereo earphones. However, the users that utilized this method expressed difficulties in separating the virtual from the real sounds.  
_Sound synthesis_, often used in real time for immersive realities, includes the techniques and algorithms that estimate the ground reaction force based on physical models exploited by a sound synthesis engine. [PERSON] et al. [120] proposed a system that affords real-time sound synthesis of footsteps on different materials. To feed the sound synthesis engine, they used inputs regarding the surface material. For solid surfaces (metal and wood), they exploited the impact and friction model to simulate the act of walking and the sound of creaking wood accordingly. For aggregate surfaces (gravel, sand, snow, etc.), they further enhanced the previous models by using some reverberation by convolving in real time the footstep sounds with the impulse response recorded in different indoor environments. In addition, _sound synthesis_ combined with _sound spatialization_ can create realistic environmental sounds. [PERSON] et al. [121] presented a 3D immersive synthesizer dedicated to environmental sounds. Based on [PERSON]'s work [122], three basic categories concerning environmental sounds stood out: liquid sounds, vibrating objects, and aerodynamic sounds. The proposed system reduces the computational cost per sound source compared with implementations and constitutes a reliable tool for creating multiple sound sources in a 3D space.
# Multi-Agent Modeling and Jamming-Aware Routing Protocols for Movable-Jammer-Affected WSNs
## 3 Methodology
### Jamming-Aware Routing Protocol (JRP)
#### 3.2.2 Path Discovery  
In the initialization phase, an initial value is assigned to the depth and jamming of each sensor node. The depth of the base station is set to 0, and a depth broadcast is initiated by the base station to assign depth values to each sensor node. The sensor nodes update their depth information and relay the depth broadcast based on their comparison of the received depth value. Similarly, the jamming value of each sensor node is set to 1 initially, and the jamming influence of the jammer is applied to update the jamming value of the sensor nodes. In order to determine the target value of each sensor node, the reciprocal of its depth and jamming values are summed and divided by 2. This target value is then broadcast to surrounding nodes to guide their selection of a successor node. The formula used to calculate the target value is given by  
\[T_{i}=\frac{\frac{1}{D_{i}}+J_{i}}{2} \tag{1}\]  
Among them, \(T_{i}\) is the target value of sensor node \(i\), which is used to guide the sensor node to select a successor node, \(D_{i}\) is the depth value of sensor node \(i\), and \(J_{i}\) is the jamming value of sensor node \(i\).  
During the path discovery stage, each sensor node selects the node with the highest target value as its successor node. The node then repeats this process until it reaches the base station. This process results in the formation of multiple paths from the sensor nodes to the base station, which ensures the reliable transmission of data. For example, Figure 3 illustrates the base station using a hexagon, while the sensor node is represented by a circle. The arrow denotes the actual direction of information transmission, whereas the dashed line signifies the potential communication connection. Node 6 selects either node 4 or node 5, depending on which has the higher target value, as its successor node to transmit its sensing data to the base station. This process is repeated until the base station is reached, resulting in two paths 6\(\rightarrow\)4\(\rightarrow\)3\(\rightarrow\)1\(\rightarrow\)base station and 5\(\rightarrow\)2\(\rightarrow\)base station that cover all the current sensor nodes.  
In summary, during the initialization phase, the depth and jamming values of each sensor node are assigned, and the target value is calculated and broadcasted to surrounding nodes. During the path discovery phase, each sensor node selects the node with the highest target value as its successor node to transmit its data to the base station.
# I/Q Imbalance and Imperfect SIC on Two-Way Relay NOMA Systems
## 3 Performance Analysis
### Diversity Orders  
In this subsection, the diversity order is analysed, which is defined as [16]:  
\[d=-\lim\limits_{\rho\rightarrow\infty}\frac{\log\left(\mathrm{P}_{D_{n}}^{\infty} \left(\rho\right)\right)}{\log\rho}, \tag{26}\]  
where \(\rho=\rho_{u}=\varsigma\rho_{r}\) and \(\mathrm{P}_{D_{n}}^{\infty}\) denotes the asymptotic outage probability of \(D_{n}\), \(n=1,2\).  
By using the definition in (26), the diversity orders for the IQI and ipSIC conditions of both \(D_{1}\) and \(D_{2}\) are respectively derived as follows:  
\[d_{D_{1}}^{IQI,ipSIC}=-\lim_{\rho\rightarrow\infty}\frac{\log\left(\mathrm{P}_{ out,D_{1}}^{\infty,IQI,ipSIC}\left(\rho\right)\right)}{\log\rho}=0, \tag{27}\]  
\[d_{D_{2}}^{IQI,ipSIC}=-\lim_{\rho\rightarrow\infty}\frac{\log\left(\mathrm{P}_ {out,D_{2}}^{\infty,IQI,ipSIC}\left(\rho\right)\right)}{\log\rho}=0. \tag{28}\]  
The diversity orders of both \(D_{1}\) and \(D_{2}\) for the considered system with the ideal condition (I/Q balance and pSIC) are derived as:  
\[d_{D_{1}}^{ideal}=-\lim_{\rho\rightarrow\infty}\frac{\log\left(\mathrm{P}_{ out,D_{1}}^{\infty,ideal}\left(\rho\right)\right)}{\log\rho}=0, \tag{29}\]  
\[d_{D_{2}}^{ideal}=-\lim_{\rho\rightarrow\infty}\frac{\log\left(\mathrm{P}_{ out,D_{2}}^{\infty,ideal}\left(\rho\right)\right)}{\log\rho}=0. \tag{30}\]  
**Remark 1**.: _An important conclusion from the analysis above is that due to the impact of IQI and residual interference with the use of ipSIC, the diversity orders of \(D_{1}\) and \(D_{2}\) are both zero due to the fixed outage probabilities at the high SNR region. As can be observed, there are error floors for \(D_{1}\) and \(D_{2}\) with IQI and ipSIC._
# A Low-Complexity Volterra Filtered-Error LMS Algorithm with a Kronecker Product Decomposition
## 4 Computational Complexity
With the growth of memory length, the computational complexity of the VFELMS algorithm can be intolerable for ANC systems. The computational complexity of the GFLNN and EMFNL structure is smaller than the VFELMS algorithm, while it is still large for a real-time system. However, by utilizing a decomposable Volterra structure and filtered-error structure, the proposed DVMFELMS algorithm can significantly reduce the computational complexity of the algorithm, which is promising in NANC systems, especially for the NSP case.
## 5 Simulation Examples
### Examples with No Consideration of Uncertainties
#### 5.1.1 Example 1--Coverage of the Required Load and Its Changes
For better clarity the BESS fuel price will remain unchanged throughout the whole simulation. However, if the designer would also like to change the price for BESS, the presented algorithm makes this possible. The initial value \(\mu\) and all changes are described as:  
\[\mu_{0} =[0.8,0,7,1,1,1],\quad\mu_{10,000}=[1.2,1.2,1.15,1.15,1],\quad\mu_{20,000 }=[1,1,1,1,1],\] \[\mu_{30,000} =[0.75,0.75,0.8,0.8,1],\quad\mu_{50,000}=[1,1,1,1,1],\quad\mu_{60,0 00}=[1.3,1.3,1.2,1.15,1],\] \[\mu_{750,000} =[1,1,1,1,1].\]  
The BESS control logic Y() operates as follows. When fuel prices fall to 80% or less of the original price, the BESS begins charging; conversely, if prices exceed 120% of the original, the BESS discharges; for prices in between, the BESS remains inactive. BESS discharge is assumed as a ramp for simulation clarity, but it can also be represented as a rectangle while ensuring total energy transfer consistency. You can see the simulation results for \(\lambda_{i}(t)\), \(P_{i}(t)\), BESS stored energy \(E_{b_{i}}(t)\), and fuel cost \(\mu(t)\) in Figures 8 and 9. For better clarity, the results were divided into two figures. Figures 8 and 9 illustrate how the algorithm responds to changes in fuel prices. The BESS was charged during 2 intervals: from \(t=0\) s to \(t=10\),000 s and from \(t=30\),000 s to \(t=50\),000 s. Conversely, energy is supplied to the grid when fuel prices rise above the threshold. Discharge occurred during the intervals from \(t=10\),000 s to \(t=20\),000 s and from \(t=50\),000 s to \(t=75\),000 s. No charging or discharging of the BESS occurred between \(t=20\),000 s and \(t=30\),000 s, as well as from \(t=75\),000 s until the end of the simulation. In Figure 8, the upper part depicts the evolution of \(\lambda_{i}(t)\) where the influence of the fuel price can be noticed on its value. Figure 8: Example 2 results for \(\lambda_{i}\) and \(P_{i}\). **Remark 6**. : _The control function BESS \(\mathrm{Y}()\) was considered very simply here. Its aim was only to show the possible integration of BESS into the network. A much more complex implementation of the control function will be needed in the real world. The potential employment of more complex approaches will be described in Section 6, Future Works._
## 3 Convergence Analysis  
To prove the theoretical convergence, the usual [PERSON]'s expansion (\(n\)-dimensional) has been applied. Hence, we recall some important results from [3]:  
Let \(\Phi:D\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) be sufficiently Frechet differentiable in \(D\). Assume that \(i\)th derivative of \(\Phi\) at \(u\in\mathbb{R}^{n}\), \(i\geq 1\), is the \(i\)-linear function \(\Phi^{(i)}(u):\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n}\longrightarrow \mathbb{R}^{n}\) such that \(\Phi^{(i)}(u)(v_{1},\ldots,v_{i})\in R^{n}\). Given \(\alpha+h\in\mathbb{R}^{n}\), which lies near the solution \(\alpha\) of the system of nonlinear equations \(\Phi(x)=0\), one may apply [PERSON]'s expansion (whenever Jacobian matrix \(\Phi^{\prime}(\alpha)\) is nonsingular) to get  
\[\Phi(\alpha+h)=\Phi^{\prime}(\alpha)\left[h+\sum_{i=2}^{p-1}C_{i}h^{i}\right]+ O(h^{p}), \tag{9}\]where \(C_{i}=(1/i! )[\Phi^{\prime}(\alpha)]^{-1}\Phi^{(i)}(\alpha)\), \(i\geq 2\). It is noted that \(C_{i}h^{i}\in\mathbb{R}^{n}\) since \(\Phi^{(i)}(\alpha)\in\mathcal{L}(\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n},\mathbb{R}^{n})\) and \([\Phi^{\prime}(\alpha)]^{-1}\in\mathcal{L}(\mathbb{R}^{n})\). Expanding \(\Phi^{\prime}(\alpha+h)\) using [PERSON]'s series, we get  
\[\Phi^{\prime}(\alpha+h)=\Phi^{\prime}(\alpha)\left[I+\sum_{i=2}^{p-1}iC_{i}h^{ i-1}\right]+O(h^{p}), \tag{10}\]  
where \(I\) denotes the identity matrix. We remark that \(iC_{i}h^{i-1}\in\mathcal{L}(\mathbb{R}^{n})\). The error is denoted as \(E^{(r)}=x^{(r)}-\alpha\) for the \(r\)th iteration. The equation \(E^{(r+1)}={LE^{(r)}}^{p}+O({E^{(r)}}^{p+1})\) is called the _error equation_, where \(L\) is a \(p\)-linear function \(L\in\mathcal{L}(\mathbb{R}^{n}\times\cdots\times\mathbb{R}^{n},\mathbb{R}^{n})\) and \(p\) denotes _order of convergence_. Also, \({E^{(r)}}^{p}=(E_{1}^{(r)},E_{2}^{(r)},\cdots,E_{n}^{(r)})^{p}\). By using the above concepts, the following theorems are established. **Theorem 1**. : _Let \(\Phi:D\subseteq\mathbb{R}^{n}\longrightarrow\mathbb{R}^{n}\) be sufficiently Frechet differentiable at each point of an open convex neighborhood \(D\) of \(\alpha\in\mathbb{R}^{n}\). Choose \(x^{(0)}\) close enough to \(\alpha\), where \(\alpha\) is a solution of the system \(\Phi(x)=0\) and \(\Phi^{\prime}(x)\) is continuous and nonsingular at \(\alpha\). The iterative formula (7) producing the sequence \(\{x^{(r)}\}_{r\geq 0}\) converges to \(\alpha\) locally with order five. The error expression is found to be_  
\[E^{(r+1)}=G_{\mathbb{S}^{\text{th}}P}\big{(}x^{(r)})-\alpha=L_{1}{E^{(r)}}^{5} +O({E^{(r)}}^{6}),\,L_{1}=4C_{2}^{4}-C_{2}C_{3}C_{2}.\]  
Proof. : Applying [PERSON]'s formula around \(\alpha\) we get  
\[\Phi(x^{(r)})=\Phi^{\prime}(\alpha)\left[E^{(r)}+C_{2}{E^{(r)}}^{2}+C_{3}{E^{ (r)}}^{3}+C_{4}{E^{(r)}}^{4}+C_{5}{E^{(r)}}^{5}+C_{6}{E^{(r)}}^{6}\right]+O({E ^{(r)}}^{7})\]  
and we express the differential of first order of \(\Phi(x^{(r)})\) as  
\[\Phi^{\prime}(x^{(r)})=\Phi^{\prime}(\alpha)\left[I+2C_{2}{E^{(r)}}+3C_{3}{E^{ (r)}}^{2}+4C_{4}{E^{(r)}}^{3}+5C_{5}{E^{(r)}}^{4}+6C_{6}{E^{(r)}}^{5}\right]+ O({E^{(r)}}^{6}),\]  
where \(C_{i}=(1/i!