esa-sceva/satcom-chunk-collection huggingface dataset:
# A Study on the Feasibility of Natural Frequency-Based Crack Detection
## 5 Numerical Examples and Discussion
### Cracked Plates Made of Brittle Metals
This is apparently unfavourable to natural frequency-based crack detection. For all the scenarios listed in Table 6, the magnitude of frequency drop is under 0.1 Hz. It is very likely that brittle fracture occurs before the frequency drop is measurable. It indicates that the natural frequency-based early crack detection may be inapplicable to a high-strength structural component made of brittle metals as this type of material is not tolerant enough toward cracks. When the crack is severe enough to cause a measurable frequency drop, the component may already face the risk of brittle fracture. As a comparison, the results of cracked plates made of a 'less brittle' metal, 2014-T651 aluminium alloy, are listed in Table 7. Compared with the former material, 2014-T651 aluminium alloy is more capable of accommodating cracks. When \(\chi=10\), the maximum critical crack length ratio is 8.887% (with \(c_{s}\) approximately 71.1 mm). This is nearly 59 times the maximum \(c_{s}\) for AISI 52100 alloy steel when \(\chi=10\). It implies that as the metal becomes less brittle, a crack of medium size can be contained when \(\chi\) is large. However, when \(\chi\) is small, the maximum critical crack length ratio is still below 2%. As \(\chi\) increases, the magnitude of the natural frequency drop \(\Delta\omega\) increases. For the first mode, \(\Delta\omega\) is below 0.1 Hz even when \(\chi=10\); for the second mode, \(\Delta\omega\) becomes measurable when \(\chi=8\); while for the sixth mode, \(\Delta\omega\) becomes measurable when \(\chi=4.5\). For a given \(\chi\) (or \(c_{s}\)), the \(\Delta\omega\) of high modes is larger than that of low modes. As \(\chi\) (or \(c_{s}\)) increases, the \(\Delta\omega\) of high modes becomes measurable sooner. It can be inferred that natural frequency-based crack detection may be applicable to cracked structural components made of metals that are not too brittle, with the condition that the measurement for high modes should be guaranteed. Although theoretically achievable, measuring the natural frequency of high modes can be challenging. The difficulty lies in the balance between frequency resolution and measurement duration. For example, for a common signal analyser, a large frequency bandwidth should be used when measuring high natural frequencies. To ensure a small \(\Delta\omega\) can be captured, the frequency resolution should be small enough. A large frequency bandwidth together with a small frequency resolution leads to a long measurement duration, which may be inconvenient in practical applications. It should be noted that in Tables 6 and 7 (also in Table 8 in the next subsection), as \(\chi\) increases \(\Delta\omega\) may become negative, which indicates the natural frequency increases after
# Xanthone Derivatives in the Fight against Glioblastoma and Other Cancers
## 2 Antineoplastic Activity of Xanthone Derivatives
### Garcinoxanthocin A and B  
Garcinoxanthocin A (7, Figure 5A) and Garcinoxanthocin B (8, Figure 5B) are two examples of prenylated xanthone derivatives isolated from the fruits of _Garcinia xanthochymus_. They show an inhibitory effect against **U251 MG glioblastoma** cells (IC\({}_{50}\) values 3.5 and 1.8 \(\upmu\)M, respectively). The postulated mechanism of action involves the STAT3 transcription factor (signal transducer and activator of transcription 3), the abnormal activity of which characterizes the U251 MG cells. However, it should be noted that mechanism studies were performed for other derivatives in the series, containing structural modifications of the xanthone moiety [59].  
Figure 4: Chemical structure of Gartanin (**6**).
## 3 Results
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Question** & **Answer** & **Frequency** & **Percentage** \\ \hline \multirow{2}{*}{Q1. Gender} & Male & 42 & 50.0 \\  & Female & 42 & 50.0 \\ \hline \multirow{4}{*}{Q2. Civil Status} & Single & 20 & 23.8 \\  & Married & 55 & 65.5 \\  & Divorced & 7 & 8.3 \\  & Widower & 2 & 2.4 \\ \hline \multirow{4}{*}{Q3. Profession} & Employed & 55 & 65.5 \\  & Unemployed & 11 & 13.1 \\  & Student & 3 & 3.6 \\  & Retired & 15 & 17.9 \\ \hline \multirow{4}{*}{Q4. City of Residence} & Rome & 41 & 48.8 \\  & Other City & 43 & 51.2 \\ \hline \multirow{4}{*}{Q5. Age} & 21–30 & 4 & 4.7 \\  & 31–40 & 11 & 13.1 \\  & 41–50 & 12 & 14.3 \\  & 51–60 & 32 & 38.1 \\  & 61–75 & 25 & 29.8 \\ \hline \multirow{4}{*}{Q6. Mean annual days accessed to the ambulatory} & 1–3 times & 49 & 58.3 \\  & 4–8 times & 23 & 24.4 \\  & \(>\)9 times & 12 & 14.3 \\ \hline \multirow{4}{*}{Q7. Time taken to get to the hospital from home} & 00:05 h-01:00 h & 58 & 69.0 \\  & 01:15 h-02:00 h & 22 & 26.2 \\  & \(>\)02:15 h & 4 & 4.8 \\ \hline \multirow{4}{*}{Q8. Personal impact of home-hospital distance} & Null & 39 & 46.4 \\  & Low & 20 & 23.8 \\  & Medium & 18 & 21.4 \\  & High & 6 & 7.2 \\  & Very high & 1 & 1.2 \\ \hline \multirow{4}{*}{Q9. Way of reaching the hospital} & Alone & 55 & 65.5 \\  & Accompanied & 29 & 34.5 \\ \hline \multirow{4}{*}{Q10. Average waiting-time before the visit} & 00:05 h-00:30 h & 26 & 31.0 \\  & 00:30 h-01:00 h & 17 & 20.2 \\  & 01:15 h-02:00 h & 27 & 32.1 \\  & \(>\)02:15 h & 14 & 16.7 \\ \hline \multirow{4}{*}{Q11. Personal impact of waiting-time before the visit} & Null & 32 & 38.1 \\  & Low & 19 & 22.6 \\ \cline{1-1}  & Medium & 20 & 23.8 \\ \cline{1-1}  & High & 10 & 11.9 \\ \cline{1-1}  & Very high & 3 & 3.6 \\ \hline \multirow{4}{*}{Q12. Payment exemption of medical service} & Yes & 42 & 50.0 \\ \cline{1-1}  & No & 42 & 50.0 \\ \hline \multirow{4}{*}{Q13. Access to an IT device} & Yes & 80 & 95.2 \\ \cline{1-1}  & No & 4 & 4.8 \\ \hline \multirow{4}{*}{Q14. Ability to use IT devices} & Null & 17 & 20.2 \\ \cline{1-1}  & Low & 21 & 25.0 \\ \cline{1-1}  & Medium & 37 & 44.0 \\ \cline{1-1}  & High & 9 & 10.7 \\ \cline{1-1}  & Very high & 0 & 00.0 \\ \hline \multirow{4}{*}{Q15. Ability to use a smartphone camera} & Null & 18 & 21.4 \\ \cline{1-1}  & Low & 17 & 20.2 \\ \cline{1-1}  & Medium & 35 & 41.7 \\ \cline{1-1}  & High & 14 & 16.7 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Frequencies and percentages of answers to our survey.
# Shape Effect and Accuracy Analysis of Rock Tensile Strength Test
###### Abstract
However, current research on splitting tests mainly focuses on the relationship between the height-to-diameter ratio of disc specimens and their tensile strength, with little attention given to the size effect of the disc diameter [17]. Given the general lack of systematic studies on the effects of specimen shape and size in existing research, especially the absence of an in-depth comparison of the correlation between tensile strength results obtained by direct tension and platform splitting tests, this study combines laboratory experiments with numerical simulations to comprehensively analyze the shape effects of various testing methods. Additionally, it focuses on comparing the accuracy and correlation of tensile strength results from two different shaped specimens under the direct tension method and disc specimens in platform splitting tests. This research not only provides new insights into the mechanical mechanisms reflected by specimens of different shapes and sizes in determining rock tensile strength but also offers valuable guidance for selecting appropriate specimen shapes and sizes in engineering practices to obtain more reliable experimental results.
## 5 Evaluation and Discussion
### Query Comparison Analysis  
For the evaluation of the workflow-based query system, ZTF DR1 data were used and analyzed. This data set contains images, metadata with image header information, and relations. As stated in Section 3.3, the metadata and relations are saved in the local Postgres database. We downloaded data, made a schema and 20 queries were constructed for performance evaluation, as presented below.  
1. Find all the images where Fields ID = 841;
2. Find all the images where Exposures ID = 44316126;
3. Find all images from fields where OBSJD = 2458197.6612616;
4. Find all the images where Night ID = 443;
5. Find all the images where Host galaxies where HOSTTAG = m81;
6. Find all images by observation date between 2018-04-01 and 2018-04-30;
7. Find all the images where Filters = 2;
8. Find all the images with R-band filters = zr;
9. Find all the images with CCD ID = 16;
10. Find all the images with Night ID = 443 and Fields ID = 809;
11. Find all the images with Night ID = 443 and CCD ID = 5;
12. Find all the images with Night ID = 443 and filtercode = zg;
13. Find all the images with Field ID = 841 and Exposure ID = 44316126;
14. Find all the images with Field ID = 809 and filtercode zg;
15. Find all the images where Exposures ID = 44316126 and Filters ID = 2;
16. Find all the images from date 2018-04-01 and 2018-04-30 and Field ID = 841 and CCD ID = 5 with R-band filters;
17. Find all the images with Night ID = 443 and Fields ID = 809 and CCD ID = 12 with g-band filter;
18. Find all the images from the Fields table or exposures tables;
19. Find all the Science Exposures images where Host galaxies name = m81;
20. Find all the References Images and Science Images where Host Galaxies name = ngc 13.  
The proposed system and the source of ZTF data (IRSA web system) are used to validate the queries. The queries may involve the retrieval of data from a single or numerous objects. We have used all the possible queries by example, which is already predefined by the proposed system for those users who have a lack of query language knowledge (for novice users). The outputs of the queries were compared and depicted in Figure 9.
# An L-Slot Frequency Reconfigurable Antenna Based on MEMS Technology
## 3 Simulation Result
It can be predicted that after adding the MEMS switch on the L-slot, the resonant frequency of the antenna will float between the resonant frequencies of the antennas in Step 1 and Step 2. Step 3: Build the antenna model with slots and switches. Based on the model in Step 2, the MEMS switch is added at the corner of the L-slot. The activated/deactivated state of the switch is controlled by applying and removing the voltage to the drive electrode. The S11 curve of the antenna when the state of the switch is tuned is shown in Figure 6. The left switch is on as state 1, the right switch is on as state 2, both left and right switches are on as state 3, and both left and right switches are off as state 4. The \(-\)10 dB impedance bandwidths for the four states are 1.43 GHz (56.01-57.44 GHz), 1.84 GHz (46.76-48.60 GHz), 1.48 GHz (41.63-43.11 GHz), and 2.63 GHz (51.88-54.51 GHz). The impedance bandwidthof the antenna in all four states is above 1 GHz, with a minimum S11 value of 14.7 dB and a maximum of 28.9 dB. Thus far, the L-slot frequency reconfigurable antenna model based on RF MEMS switches has been obtained after three optimization steps. Figure 7 shows the far-field radiation direction maps of the antennas in the E-plane and the H-plane at the four operating frequency points. As can be seen in Figure, the antenna orientation is fundamentally the same across the four states, exhibiting stable directional radiation properties. Figure 7a shows the radiation direction diagram of the antenna working at 56.72 GHz when the  
Figure 5: S11 results during antenna model optimization. (**a**) Theoretical computational model; (**b**) Patch width optimization process; (**c**) Optimized model with slots. Figure 6: S11 results for the antenna model with slots and switches after Step 3: (**a**) state 1; (**b**) state 2; (**c**) state 3; (**d**) state 4.  
left switch is on. Although its H-plane curve is slightly distorted compared to the other frequency points, the main radiation direction range is almost the same, within acceptable limits. In this paper, the current distribution of the antenna in the four modes is shown in Figure 8. It can be found that when the switch is on, the current is mainly concentrated near the switch. When the switch is switched off, the current rushes along the L-slot to the end. The difference in the current distribution of the antenna in the four states is significant enough to conclude that RF MEMS switches are the key factor affecting the frequency variation. There is inevitable current convergence due to the presence of the driving electrodes. The gain of the antenna in different operating modes is shown in Figure 4:0 The maximum gain of the antennas in the four operating states is 7.88 dB, 7.60 dB, 7.90 dB, and 6.20 dB. Overall, the gain of the antenna is considerable.
# Fundus Image Registration Technique Based on Local Feature of Retinal Vessels
###### Abstract  
Feature-based retinal fundus image registration (RIR) technique aligns fundus images according to geometrical transformations estimated between feature point correspondences. To ensure accurate registration, the feature points extracted must be from the retinal vessels and throughout the image. However, noises in the fundus image may resemble retinal vessels in local patches. Therefore, this paper introduces a feature extraction method based on a local feature of retinal vessels (CURVE) that incorporates retinal vessels and noises characteristics to accurately extract feature points on retinal vessels and throughout the fundus image. The CURVE performance is tested on CHASE, DRIVE, HRF and STARE datasets and compared with six feature extraction methods used in the existing feature-based RIR techniques. From the experiment, the feature extraction accuracy of CURVE (86.021%) significantly outperformed the existing feature extraction methods (\(p\leq 0.001\)*). Then, CURVE is paired with a scale-invariant feature transform (SIFT) descriptor to test its registration capability on the fundus image registration (FIRE) dataset. Overall, CURVE-SIFT successfully registered 44.030% of the image pairs while the existing feature-based RIR techniques (GDB-ICP, Harris-PIIFD, Ghassabi's-SIFT, H-M 16, H-M 17 and D-Saddle-HOG) only registered less than 27.612% of the image pairs. The one-way ANOVA analysis showed that CURVE-SIFT significantly outperformed GDB-ICP (\(p\) = 0.007*), Harris-PIIFD, Ghassabi's-SIFT, H-M 16, H-M 17 and D-Saddle-HOG (\(p\leq 0.001\)*).  
image registration; fundus image; feature extraction Article
# Sparse Parabolic Radon Transform with Nonconvex Mixed Regularization for Multiple Attenuation
## 4 Real Data Application
### Prestack Field Data  
The SPRTL\({}_{q1}-L_{q2}\) method was further tested with multi-shot prestack field data. In order to comprehensively evaluate the multiple suppression effect, we used a 3D method to display the multi-shot data volume, then analyzed the effectiveness of the method from the time slices, the common middle point gathers and the common offset gathers.  
Figure 10a is original data, and multiples can be clearly seen at the arrow position in the figure. Figure 10b,c are the multiple attenuation results obtained by the LSPRT and the SPRTL\({}_{1}\) methods; Figure 10d is the result obtained by the SPRTL\({}_{q1}-L_{q2}\). Overall, all three methods suppress multiples, but the effects are different. In Figure 10b,c, it can be obviously observed from the time slice that a certain amount of the primary energy is lost when multiples are suppressed. In the CMP gathers the multiples of far offset are sufficiently suppressed by SPRTL\({}_{q1}-L_{q2}\), as indicated by the arrow. In the common offset gathers, the effectiveness of the SPRTL\({}_{q1}-L_{q2}\) method is demonstrated by the continuity of the seismic events. This method has a sparser representation of the seismic data and fully retains the characteristics and information of the primaries, so as to improve the continuity of the seismic events.  
Figure 8: Multiple attenuation result: **(a)** CMP gather of marine real data; **(b)** by LSPRT method; **(c)** by SPRTL\({}_{1}\) method; **(d)** by SPRTL\({}_{q1}-L_{q2}\) method. The black arrows indicate the continuity of the seismic events are improved in the Figure 8d.  
Figure 9: Comparison of single trace amplitude after multiple attenuation by three algorithms. The red rectangle indicate that the amplitude of multiple decays more significantly in the proposed method.  
Figure 11 shows the stack section after multiple attenuation by three methods. The residual of multiples leads to the discontinuity of the events of the primaries (black rectangle), which increases the difficulty of subsequent interpretation. The continuity of the seismic events is obviously enhanced in the stack sections using the method presented in this paper, and the effect of multiple attenuation between 0.5 s and 1 s has a remarkable effect. In addition, as can be seen by the black arrow region of Figure 11d, there is almost no multiple energy compared to Figure 11b,c.  
Figure 10: Multiple attenuation result: (**a**) pre-stack real data; (**b**) LSPRT method; (**c**) SPRTL\({}_{1}\) method; (**d**) SPRTL\({}_{q1}-L_{q2}\) method. The red arrows indicate the cleaner multiple attenuation in the Figure 10d.  
Therefore, the SPRTL\({}_{q1}-L_{q2}\) method has some advantages in suppressing multiple reflections and retaining effective reflections.
## 2 Methods
### Data Processing  
Data processing included two main parts: the reflectance data process for calculating the VI and the photosynthetic capacity calculation processes. At the leaf scale, we analyzed the relationship between SGLI band reflectance and chlorophyll content. At the canopy scale, we analyzed the relationship between SGLI band reflectance and Pmax_capacity. At the satellite scale, we analyzed the relationship between MOD09A1 and Pmax_capacity. Pmax_capacity was calculated from EC flux data by fitting the light-response curve.
## 5 Performance Evaluation  
In this section, we evaluate the performance of the proposed method using simulation studies. The default parameter values are as follows, if not specified otherwise. A MES is equipped with a \(f_{mes}=1.0\) GHz CPU core, whose effective switching capacity is \(\kappa=10^{-27}\). The length of a time slot is \(\tau=1\) s and \(V=10^{15}\). A MES supports \(|N|=3\) classes, whose delay requirement is configured as \(\delta_{i}=10i\) ms for each \(i\in N\). To emulate the task arrival process for each class, we used the traffic traces from a cellular network containing the time that a device generates a call. We used the dataset that results from a computation over the call detail records (CDRs) generated by the Telecom Italia cellular network for nine days from the city [40]. Assuming that the call generation pattern of a device is similar to the task offloading pattern of the device, we divided the real traffic traces into time slots and calculated the number of tasks generated during each time slot. The workload imposed by a task is influenced by the applications to which a task belongs [41]. In this study, for each task class \(i\), the workload imposed by a task is randomly selected in \([w_{i}^{min},w_{i}^{max}]\) according to the uniform distribution. For each class \(i\in N\), we set \([w_{i}^{min},w_{i}^{max}]=[i\times 10.0,i\times 31.6]\) kilocycles. To focus on the service-time allocation problem at a MES, we considered the situation where a MES is not overloaded such that \(\lim_{T\to\infty}1/T\sum_{t=1}^{T}\sum_{i\in N}w_{i}(t)<\tau f_{mes}\).
## 2 Materials and Methods
### Polynomial Response Surface  
[PERSON] and [PERSON] first proposed [36] the PRS model, which is one of the most commonly used surrogates. The most widely utilized is the second-order polynomial function, because overfitting occurs easily when the order exceeds 3. The second-order PRS can be represented as follows:  
\[\hat{y}(\mathbf{x})=\beta_{0}+\sum_{i=1}^{m}\beta_{i}x_{i}+\sum_{i=1}^{m}\beta_{ ii}x_{i}^{2}+\sum_{i=1}^{m-1}\sum_{j=i+1}^{m}\beta_{ij}x_{i}x_{j} \tag{1}\]  
where \(\hat{y}(x)\) is the predicted value of the surrogate, \(m\) denotes the number of input variables, and \(\beta=\left[\beta_{0},\ \beta_{i},\ \beta_{ii},\ \beta_{ij}\right]\) represents the coefficient that needs to be determined.  
PRS has the advantages of strong interpretability, simple computation, high efficiency, and good noise filtering, which make it effective for low-dimensional nonlinear problems. However, with the increase in problem dimensions, the accuracy of PRS is relatively low.  
\begin{table}
\begin{tabular}{c c c c} \hline
**Algorithm** & **Author, References** & **Surrogate Models** & **Evolutionary Algorithms** & **Year** \\ \hline MFA & [PERSON] et al. [34] & \(\times\) & MFAE & 2016 \\ G-MFEA & [PERSON] et al. [35] & \(\times\) & GMFEA & 2017 \\ FSAPSO & [PERSON] et al. [15] & RBF & PSO & 2020 \\ BiS-SAHA & [PERSON] et al. [25] & RBF & SL-PSO, DE & 2021 \\ GLS-SADE & [PERSON] et al. [21] & RBF, Kriging & DE & 2023 \\ LSADE & [PERSON] et al. [22] & RBF, Lipschitz & DE & 2023 \\ DSP-SAEA & [PERSON] et al. [27] & RBF, Kriging, and polynomial regression (PR) & PSO & 2023 \\ SAMTO & [PERSON] et al. [31] & Kriging, RBF & MFAE & 2023 \\ HSAOA & [PERSON] et al. [28] & RBF, PRS, and SVR & TLBO, DE & 2024 \\ \hline \end{tabular}
\end{table}
Table 1: Currently important algorithms, including EAs and SAEAs.
## 2 Theoretical Model
### Load Transfers between Pile and Soil
#### 2.2.2 The Negative Friction Zone  
As shown in Figure 4, since the soil moves down relative to the pile in the negative friction zone, the pile is subjected to a downward friction. The Randolph method [33] is adopted to calculate the friction, and it can be shown as follows:  
\[\tau=\left\{\begin{array}{ll}-\mu_{1}k_{as}p_{s}(z^{**})&L_{0}<z^{**}<L_{1} \\ \mu_{2}k_{as}p_{s}(z^{**})&L_{1}<z^{**}<L_{p}\end{array}\right. \tag{21}\]  
where \(k_{as}\) is the active earth pressure coefficient and \(k_{as}=\tan^{2}(45-\rho_{s}/2)\). \(\mu_{1}\) and \(\mu_{2}\) respectively stand for the negative and positive friction coefficients; [PERSON] [34] suggested the two values should be 0.3.  
Similarly, the analysis is carried out using a soil unit with a thickness of d\(z^{**}\). The equilibrium of the vertical force can be expressed as  
\[p_{s}(z^{**})A_{s}=p_{s}(z^{**})A_{s}+\mathrm{d}p_{s}(z^{**})A_{s}+\tau A_{b1} \mathrm{d}z^{**} \tag{22}\]  
where the soil area around the pile is referred as \(A_{s}=\pi(r_{e}\)2\(-\)\(r_{2}\)2), and \(A_{b1}\) is the perimeter of the pile.  
Substituting Equation (21) into Equation (22) and integrating Equation (22), while introducing the stress continuity condition at \(z^{**}=L_{0}\), the following formula is obtained:  
\[p_{s}(z^{**})=e^{(\frac{2\mu_{1}k_{as}\eta_{s}(z^{**}-l_{0})}{r_{2}^{*}-r_{e} ^{2}})}[(r_{1}-r_{2})(r_{1}+r_{2})]^{\frac{D\!p}{C\!s_{1}}}[(r_{1}-k_{1}L_{0}) ^{2}-r_{e}^{2}]^{\frac{D\!p}{C\!s_{1}}}\sigma_{o}(h) \tag{23}\]Similarly, the sum of the vertical stress on the pile and soil is equal to the gravity.  
\[m_{2}p_{p}(z^{**})+(1-m_{2})p_{s}(z^{**})=\gamma_{c}h+\gamma_{s}z^{**} \tag{24}\]  
where \(m_{2}\) is the area replacement rate at the lower side of the conical head, and \(m_{2}={r_{2}}^{2}/{r_{e}}^{2}\).  
Equation (25) could be deduced by substituting Equation (23) into Equation (24):  
\[\begin{array}{l}p_{p}(z^{**})=\frac{r_{e}^{2}}{r_{2}^{2}}[(\gamma_{c}h+\gamma _{s}z^{**})\\ -e^{\frac{2\mu_{1}k_{1}\gamma_{2}(z^{**}-l_{0})}{r_{2}^{2}-r_{e}^{2}}}(1-\frac{ r_{e}^{2}}{r_{e}^{2}})((r_{1}-r_{e})(r_{1}+r_{e}))^{\frac{-\Omega_{\rm B}}{ \Omega_{1}}}((r_{1}-k_{1}L_{0})^{2}-r_{e}^{2})^{\frac{\Omega_{\rm B}}{\Omega_{ 1}}}\sigma_{o}(h)]\end{array} \tag{25}\]  
In the negative friction zone, the vertical stress of piles and soft soil can be obtained by Equation (25) and Equation (23), respectively. Analogically, the deformation of the soil and pile can be obtained.  
\[S_{s2}=\int\limits_{L_{0}}^{L_{1}}\frac{p_{s}(z^{**})}{E_{s}}\mathrm{d}z^{**} \tag{26}\]  
\[S_{p2}=\int\limits_{L_{0}}^{L_{1}}\frac{p_{p}(z^{**})}{E_{p}}\mathrm{d}z^{**} \tag{27}\]