esa-sceva/satcom-chunk-collection huggingface dataset:
## 3 Results and Discussion
### Chromatic Parameters and Phenolic Compounds  
According to colorimetric measurements, Marselan wines from HZB, LP, and XJ regions displayed lower \(L\) values and higher values of \(a\) and \(C_{\text{ab}}\) compared with the wines from JDP and BHB (Table 3). These differences may contribute to the relatively deeper color intensity and higher red-hue intensity in wines from HZB, LP, and XJ regions. No significant differences in \(b\) and \(H_{\text{ab}}\) values existed among different wines. The OPLS-DA model (\(R^{2}X\) = 0.591, \(R^{2}Y\) = 0.476, \(Q^{2}\) = 0.183) could not discriminate well between the Marselan wines from different regions based on the quantitative data of the phenolic compounds (Figure 2C,D). However, the Marselan wines from XJ were separated from the wines from other regions due to their higher concentrations of several flavonoids (e.g., myricetin-3-O-glucoside, quercetin-3-O-galactoside, isorhamentin-3-O-glucoside, syringetin-3-O-glucoside) (\(p\) < 0.05) (Table 4). This suggests that aromatic and phenolic compounds in wines from different regions were influenced by climatic conditions in a different manner. The average total flavon concentrations in XJ Marselan wines were approximately 2.98-4.31-fold higher than the wines from other regions. This is probably because the XJ region has the longest sunshine duration and highest PAR (Figure 1), which induces a more active synthesis of flavonoids to resist the UV radiation [56]. Our previous study also found that more flavonoids were accumulated in grapes from the GT region before the berry harvest than in those from the CL region [57]. \begin{table}
\begin{tabular}{c c c c c c} \hline
**CIELab** & **Jiaodong** & **Bohai Bay** & **Huaizhuo Basin** & **Loess Plateau** & **Xinjiang** \\  & **Peninsula** & & & & \\ \hline \(L\) & 66.32 \(\pm\) 13.31 a & 61.54 \(\pm\) 20.24 ab & 42.58 \(\pm\) 10.02 c & 48.90 \(\pm\) 8.62 bc & 39.11 \(\pm\) 12.07 c \\ \(a\) & 34.74 \(\pm\) 11.95 b & 34.49 \(\pm\) 14.90 b & 49.98 \(\pm\) 8.76 a & 45.64 \(\pm\) 9.33 ab & 52.52 \(\pm\) 12.07 a \\ \(b\) & 6.94 \(\pm\) 5.50 a & 7.10 \(\pm\) 1.58 a & 7.91 \(\pm\) 3.79 a & 9.22 \(\pm\) 4.43 a & 9.07 \(\pm\) 7.95 a \\ \(H_{\text{ab}}\) & 10.78 \(\pm\) 5.62 a & 13.70 \(\pm\) 8.28 a & 9.76 \(\pm\) 6.27 a & 11.46 \(\pm\) 5.12 a & 11.18 \(\pm\) 8.74 a \\ \(C_{\text{ab}}\) & 35.62 \(\pm\) 12.56 b & 35.39 \(\pm\) 14.41 b & 50.85 \(\pm\) 8.02 a & 46.72 \(\pm\) 9.43 ab & 53.93 \(\pm\) 11.42 a \\ \hline \end{tabular}  
* Averages and standard deviation followed by different letters are significant at \(p\) < 0.05, [PERSON]’s multiple range test.  
\end{table} 
Table 3: CIELab parameters (\(L\), \(a\), \(b\), \(H_{\text{ab}}\), and \(C_{\text{ab}}\)) in Marselan wines from different regions in China*.
# An Efficient Time-Variant Reliability Analysis Method with Mixed Uncertainties
## 4 A New Analysis Method of Structural Time-Variant Reliability with Random and Interval Variables
### Procedure  
Based on the equivalent analysis above, the original structural time-variant reliability model of mixed variables is transformed into the equivalence model of the structural time-variant reliability model of mixed variables. As shown in the Figure 3, the specific steps are as follows:  
1. The uncertainty in the structure is simply divided into two types according to the information given, including the probability random vector \(\mathbf{X}\) and the non-probability vector \(\mathbf{Y}\) (interval vector).
2. The interval variable \(\mathbf{Y}\) is treated as a uniform distribution, which is regarded as a random variable.
3. The limit state function of the time-variant reliability model of the mixed uncertainties only contains the random variables \(\mathbf{X}\) and \(\mathbf{Y}\). Then, the equivalent substitution model of the mixed reliability analysis is established.
4. The time parameters are discretized into m periods \(t_{i}\) by the static transformation method. Meanwhile, the generalized resistance random process \(R(t)\) and the variable load \(Q(t)\) can be discretized. Therefore, the time-variant problem is changed into a time-invariant problem.
5. FORM is adopted to compute and solve the transformed equivalent mode in Equation (19). Then, an optimum \((\mathbf{X}^{\prime},\mathbf{Y}^{\prime})\) can be obtained in each period.
6. The minimum reliability and the maximum failure probability of the model in each period is obtained.
## 3 Materials and Methods
### Voiceprint Embedding Extraction  
To obtain more robust voiceprint embeddings, Chrono-ECAPA-TDNN aggregates the output features of all Chrono modules. Previous studies have shown that shallow features  
Figure 2: Architecture of the Chrono module.  
play a crucial role in capturing key information in speech and can significantly improve the representational power of voioceprint embeddings. In ECAPA-TDNN, the output features of all SE-Res2 Net modules are aggregated before the Attentive Statistics Pooling (ASP) layer, which effectively enhances model performance. Similarly, in Chrono-ECAPA-TDNN, the output features of each Chrono module are aggregated before the LayerNorm layer (see Figure 2).  
First, the output features of all Chrono modules are integrated through feature concatenation to form the following, as shown in Equation (14):  
\[H^{\prime}=\text{Concat}(h_{m}^{2},h_{n}^{2}), \tag{14}\]  
where \(H^{\prime}\in R^{3\text{C}\times T}\), \(C\) is the convolution channels of the Chrono modules, and \(T\) is the number of time frames.  
Then, the concatenated features are normalized to obtain the standardized aggregated feature matrix \(H\), as shown in Equation (15).  
\[H=LayerNorm(H^{\prime}), \tag{15}\]  
The features are then passed into the Attentive Statistics Pooling layer to obtain the weight coefficients for each frame and extract global features. For the frame-level feature \(H_{t}\) at time frame \(t\), the weight coefficient \(\alpha_{t}\) is first computed as follows, as shown in Equations (16) and (17):  
\[\epsilon_{t}=\mathbf{v}^{T}f(WH_{t}+b)+k, \tag{16}\]  
\[\alpha_{t}=\frac{\exp(\epsilon_{t})}{\sum\limits_{t=1}^{T}\exp(\epsilon_{\tau })}, \tag{17}\]  
where \(t=1,2,\ldots,T,W\in R^{D\times D},b\in R^{D\times 1},D=3\text{C}\times T\), \(k\in R\) represents the learnable parameters of the ASP pooling layer, and \(f(*)\) represents Tan\(h\) Activation function.  
Using the weight coefficient \(\alpha_{t}\), calculate the weighted mean vector \(\upmu\) and weighted standard deviation \(\sigma\), as shown in Equations (18) and (19):  
\[\upmu=\sum\limits_{t=1}^{T}\alpha_{t}H_{t}, \tag{18}\]  
\[\sigma=\sqrt{\sum\limits_{t=1}^{T}\alpha_{t}(H_{t}\odot H_{t})-\upmu\odot \upmu}, \tag{19}\]  
where \(\odot\) represents the Hadamard product (element-wise multiplication).  
Finally, the output of the ASP pooling layer is obtained by concatenating \(\upmu\) and \(\sigma\), as shown in Equation (20):  
\[Z=\text{Concat}(\upmu,\sigma), \tag{20}\]  
The output is then passed through a BatchNorm layer and a Linear layer, further reducing the dimensionality to a fixed-length voioceprint embedding vector \(Z\in R^{N}\) of length \(N\), which is used for subsequent classification tasks.
# Bit Independence Criterion Extended to Stream Ciphers
## 3 BIC Algorithm in Stream Ciphers
### BIC Test Algorithm  
Given a set \(D=\left\{X_{1},\ldots,\ X_{l}\right\}\) of \(l\) randomly chosen \(n\) bits inputs to the function \(f\), constructs for each binary vector \(e_{i}\) (\(1\leq i\leq n\)) its associated SAC matrix \(H^{i}\) and for all for \(j,k\) with \(j\
eq k\), it is checked if \(HW(s^{i}_{jk})\) follow the \(B\left(l,\frac{1}{2}\right)\) distribution, see the proposed Algorithm 3.  
```
0:\(f\) function to evaluate, \(n\) size of the inputs of \(f\), \(m\) size of the outputs of \(f\), \(\alpha_{1}\) and \(\alpha_{2}\) levels of significance, \(D\) set of \(l\) inputs to the function \(f\).
0: If \(f\) satisfies the BIC
1:\(T=0\)
2:for\(i=1\to n\)do
3:for\(r=1\to l\)do\(\triangleright\) Matrix Construction \(H_{i}\)
4: Compute \(V^{i}_{r}=Y_{r}\oplus Y^{i}_{r}\)
5:endfor
6:for each \((j,k)\)do\(\triangleright\) Independence check between \(\sigma^{i}_{,j}\) and \(\sigma^{i}_{,k}\)
7:if\(\chi^{2_{s^{i}_{jk}}}>\chi^{2}_{s_{1},1}\)then
8:\(T=T+1\)\(\triangleright\) Independence is rejected between \(\sigma^{i}_{,j}\) and \(\sigma^{i}_{,k}\)
9:endif
10:endfor
11:endfor
12:if\(Z_{T}>Z_{1-\alpha_{2}}\)then\(f\) does not satisfy the BIC
13:else\(f\) satisfies the BIC
14:endif
```  
**Algorithm 3** BIC stream ciphers algorithm
# Fingerprint Classification Based on Multilayer Extreme Learning Machines
###### Abstract  
Fingerprint recognition is one of the most effective and widely adopted methods for person identification. However, the computational time required for the querying of large databases is excessive. To address this, preprocessing steps such as classification are necessary to speed up the response time to a query. Fingerprints are typically categorized into five classes, though this classification is unbalanced. While advanced classification algorithms, including support vector machines (SVMs), multilayer perceptrons (MLPs), and convolutional neural networks (CNNs), have demonstrated near-perfect accuracy (approaching 100%), their high training times limit their widespread applicability across institutions. In this study, we introduce, for the first time, the use of a multilayer extreme learning machine (M-ELM) for fingerprint classification, aiming to improve training efficiency. A comparative analysis is conducted with CNNs and unbalanced extreme learning machines (W-ELMs), as these represent the most influential methodologies in the literature. The tests utilize a database generated by SFINGE software, which simulates realistic fingerprint distributions, with datasets comprising hundreds of thousands of samples. To optimize and simplify the M-ELM, widely recognized descriptors in the field--Capelli02, Liu10, and Hong08--are used as input features. This effectively reduces dimensionality while preserving the representativeness of the fingerprint information. A brute-force heuristic optimization approach is applied to determine the hyperparameters that maximize classification accuracy across different M-ELM configurations while avoiding excessive training times. A comparison is made with the aforementioned approaches in terms of accuracy, penetration rate, and computational cost. The results demonstrate that a two-layer hidden ELM achieves superior classification of both majority and minority fingerprint classes with remarkable computational efficiency.  
Keywords:feature descriptors; fingerprint classification; identification systems; biometry; multilayer extreme learning machines +
Footnote †: journal: Applied Sciences
# Real-Time 3D PET Image with Pseudoinverse Reconstruction
### Theory
#### 1.2.2 Computation of the SRM  
SRMs can be obtained from Monte Carlo simulations [9; 20; 31] approximated by analytical expressions [35] ([PERSON] 2003), or even measured experimentally [36]. In this work, we used analytic SRMs. To model the problem more realistically for each LOR, we defined a tube of response (TORs) [37], where its width reflects the physical effects. In this work, TORs were assumed to have a Gaussian cross section [18; 38; 39], whose width (\(\sigma_{x,y},\sigma_{z}\)) was obtained empirically for a given scanner, being the same for all LORs of the same scanner.  
To derive the axial SRM, we considered an image space discretized in axial (z) and length coordinates (w) along the LOR (Figure 1b). The probability for each voxel to be connected to a given LOR was obtained as follows:  
\[A_{z}(z,w,z_{1},z_{2})=e^{(-\frac{1}{2}).\frac{d(z,w,z_{1},z_{2})^{2}}{\sigma_ {z}^{2}}}, \tag{2}\]  
where \(d(z,w,z_{1},z_{2})\) is the minimum distance between the voxel at \((z,w)\) and the LOR given by \((z_{1},z_{2})\) (Figure 1b).  
In a similar way, each element of the transaxial SRM can be modeled as (Figure 1c):  
\[A_{x,y}(x,\ y,\rho,\ \theta)=N\cdot e^{(-\frac{1}{2}).\frac{d(z,w,\rho)^{2}}{ \sigma_{xy}^{2}}}, \tag{3}\]  
where \(d(z,y,\rho,\theta)\) is the distance from the voxel \((x,y)\) to the LOR \((\rho,\theta)\)[39].  
In order to ease operations with these matrices, we combined some variables in the same bin index. We used \(A_{z}(zw,z_{1}z_{2})\) and \(A_{xy}(x,y,\rho\theta)\) instead of \(A_{z}(z,w,z_{1},z_{2})\) and \(A_{x,y}(x,y,\rho,\theta)\). In what follows, we will use this notation.
# Robust Image Inpainting Forensics by Using an Attention-Based Feature Pyramid Network
## 3 Methods
### Attention Mechanism  
We utilize a powerful attention mechanism called MSCA, which is illustrated in Figure 4.  
We utilize the CPFE module to obtain high-level features of multi-scale and multi-receiving fields. However, the high-level feature map in Figure 1e requires further appropriate optimization. The experimental results demonstrate that the MSCA effectively enhances the inpainting traces for high-level feature maps, see Figure 1g. Furthermore, the low-level feature map keeps the original state, as improper handling may result in the magnification of noisy features, including the contour and texture, as depicted in Figure 1d, negatively influencing the inpainting forensics process.  
Figure 4: Illustration of the multi-scale convolution attention (MSCA).  
Figure 3: Detailed structure of the context-aware pyramid feature extraction (CPFE).  
The MSCA module consists of three parts: the depth-wise convolution of aggregating local information, the multi-branch depth-wise strip convolution of capturing the multi-scale context, and the \(1\times 1\) convolution of simulating the relationship between different channels. The output of the \(1\times 1\) convolution is directly used as the attention weight to reweigh the input of the MSCA.  
The mathematical expression of the MSCA can be written as follows:  
\[Att=Conv_{(1\times 1)}(\sum_{i=0}^{3}Scale_{i}(DWConv(F))), \tag{1}\]  
\[Out=Att\otimes F. \tag{2}\]  
where \(F\) represents the optimized feature maps, and \(Att\) and \(Out\) represent the attention map and output, respectively. The \(\otimes\) operation involves matrix multiplication on a pixel-by-pixel basis. \(DWConv\) refers to the depth-wise convolution, and \(scale_{i}\), where \(i\in\{0,1,2,3\}\) represents the \(i\)th branch in Figure 4. \(scale_{0}\) represents the identity connection.  
In order to simulate the standard depth convolution with a large kernel, two depth-wise strip convolutions are employed in each of the three branches, with kernel sizes of 7, 11, and 21, respectively. The selection of depth-wise strip convolution is motivated by two primary factors. Firstly, strip convolution is characterized by its lightweight nature, requiring only a pair of \(7\times 1\) and \(1\times 7\) convolutions to simulate the standard 2D convolution with a kernel size of \(7\times 7\). Secondly, most of the inpainted areas contain some strip objects, such as people and slogans, increasing the difficulty of the inpainting forensics. As such, the implementation of strip convolution as a complementary technique to grid convolution can improve the detection of inpainting traces. In addition, compared with the attention used by AFPN in [26; 27; 28; 29; 30], MSCA is more portable, powerful, and suitable in image inpainting forensics.
# Integrated Sensing and Communication Using Random Padded OTFS with Reduced Interferences
## 1 Introduction
In [33] the authors propose a two-stage Bayesian learning-based approach for transceiver design and parameter estimation, which takes advantage of signal sparsity to improve ISAC system performance, and extend this to block sparse learning in mmWave ISAC systems. In some articles [15; 21], the authors argue that usage of the guard interval, consisting of inactive carriers in the DD domain, solves the problem of interference between data and pilot carriers, as in the case of ZP-OTFS or its ZP-BOX-OTFS modification. This is not completely true, especially in situations where the fractional reflection is close to the data carriers. An additional aggravating factor is that the power of the active pilot carriers in case of, for example, ZP-OTFS is concentrated in only one carrier, while in RP-OTFS, active pilot carriers are distributed over the whole pilot zone (in order to achieve the same power, the ZP-OTFS pilot carrier should be much stronger than in RP-OTFS). This means that the problems discussed in this paper are also applicable to other OTFS modifications. The main contribution of this paper is the proposal of a method to reduce interference between pilot symbols and data in the ISAC system based on the RP-OTFS signal. This new approach leads to a significant decrease in BER during data transmission while preserving the high probability of the detection of moving objects, performed at the same time. In consequence, after the proposed enhancement, the application of the ISAC system based on the modified RP-OTFS is more beneficial than that of the OFDM-based and ZP-OTFS-basedones, as shown in the paper by simulation. Furthermore, in this work, a method is proposed for the simultaneous use of the DD grid by multiple devices in the presence of fractional delays, which is achieved by redistribution of interference power in the DD domain. Last but not least, the paper also provides insight into the selection of DD grid parameters for ISAC systems. The structure of this article is as follows: In Section 2 the OTFS modulation and its different variants are presented and compared with OFDM. In Section 3, the random-padded OTFS modification is discussed in detail: its computational structure, the generation of random pilots, and the limitations of the RP-OTFS. Then, in Section 4, the system model for communication and radar subsystems is introduced. In turn, in Section 5 a new RP-OTFS version with reduced ICI, obtained by random pilot repetition, is proposed. Finally, in Section 6, some examples of the choice of DD grid parameters for the ICI-reduced RP-OTFS method are presented, and in Section 7, simulation results are shown that confirm the efficiency of the ICI-reduced RP-OTFS application compared with its predecessor, the regular RP-OTFS, as well as ZP-OTFS and OFDM. The paper ends with a discussion and conclusions.
# The Combined Effect of Indoor Air Quality and Socioeconomic Factors on Health in Northeast China
## 3 Results
### Model Result and Evaluation  
Figure 4 shows the structural equation modeling results inferred from field measurement data. The results indicate that the indoor air quality posed a direct impact on the occupants' health with the path coefficient of 0.105, whereas the effects from socioeconomic status were 0.413. The effect of socioeconomic status on indoor air quality was 0.381. As for the measurement model of the indoor air quality, the greatest weight came from PM\({}_{2.5}\) (0.813), followed by CO\({}_{2}\) (0.385), TVOC (0.218), and formaldehyde (0.142). Income level (0.595) was the major indicator contributing to socioeconomic status, followed by education level (0.551) and occupation prestige (0.508). With respect to the measurement model of health status, physical functioning (0.849) had the greatest absolute contribution, followed by role emotional (0.773), vitality (0.736), role physical (0.677), mental health (0.565), general health (0.564), bodily pain (0.545), and social functioning (0.491).  
Table 4 summarizes the results of the model evaluation. In the formative measurement models, the indicators' variance inflation factor (VIF) were uniformly below the threshold value of 5. Hence, the level of collinearity was very low. Moreover, all indicators in the formative models were significant (\(p<0.01\)). In the reflective models, the values of composite reliability for health (0.853) were larger than the threshold value of 0.7. The average variance extracted (AVE) were greater than the threshold value of 0.5 for health (0.637), confirming the convergent validity. The square roots of AVE, for health (0.798) were larger than its correlations with other variables, indicating a well-established discriminant validity. Finally, all VIF values were lower than the threshold value of 5, suggesting that the collinearity was also not an issue in the structure model. All relationships were significant in the structural model. The coefficients of determination for health (0.305) and indoor air quality (0.220) were above the threshold value of 0.2. The predictive relevance values for the indoor air quality (0.110) and health (0.283) were greater than the threshold value of 0, proving the predictive relevance for the model. In summary, the model has met all requirements, thus proving that it was reliable and valid.  
Figure 4: The model results inferred from the field measurement data (compared with Figure 3). SES- socioeconomic status; PF-physical functioning; RP-role physical; BP-bodily pain; GH-general health; VT-vitality; SF-social functioning; RE-role emotional; MH-mental health.
## 2 Capture Probabilities Under Different Fading Channels
### Nakagami-m Fading Channel  
The instantaneous received power in Nakagami-\(m\) fading channel is given by [33]  
\[f_{\gamma}(\gamma)=\frac{m^{m}\gamma^{m-1}}{\tilde{\gamma}^{m}\Gamma(m)}e^{- \frac{m\gamma}{\gamma}},\gamma\geq 0 \tag{23}\]  
where \(m\) is the shape parameter, which ranges from \(\dfrac{1}{2}\) to \(\infty\). By adopting the Laplace transform method again, we have  
\[f_{\gamma}(\gamma)\leftrightarrow F_{\gamma}(s)=\frac{m^{m}}{\tilde{\gamma} ^{m}}\frac{1}{\left(s+\frac{m}{\tilde{\gamma}}\right)^{m}} \tag{24}\]  
Then, by making use of the convolution property of Laplace transform again, we can easily obtain  
\[f_{\gamma}(\gamma)^{\otimes n-1}\leftrightarrow G_{\gamma_{n-1}}(s)=\frac{m^ {m(n-1)}}{\tilde{\gamma}^{m(n-1)}}\frac{1}{\left(s+\frac{m}{\tilde{\gamma}} \right)^{m(n-1)}} \tag{25}\]  
Then, we apply the inverse Laplace transformation again and have  
\[G_{\gamma_{n-1}}(s)\leftrightarrow g_{\gamma_{n-1}}(\gamma_{n-1})=\frac{m^{m (n-1)}}{\tilde{\gamma}^{m(n-1)}\Gamma(mn-m)}\gamma_{n-1}^{m(n-1)-1}e^{-\frac{m \gamma_{n-1}}{\tilde{\gamma}}} \tag{26}\]  
To simplify the writing, we let \(\frac{m}{\tilde{\gamma}}=a\), and then the capture probability in Nakagami-\(m\) fading channel can be written as  
\[\begin{split} p_{cap}^{Nak}&=n\int_{0}^{\infty}\frac {a^{m}\gamma_{t}^{m-1}}{\Gamma(m)}e^{-a\gamma_{t}}\left[\int_{0}^{\frac{\gamma _{t}}{z}}\frac{a^{m(n-1)}}{\Gamma(mn-m)}\gamma_{n-1}^{m(n-1)-1}e^{-a\gamma_{n -1}}d\gamma_{n-1}\right]d\gamma_{t}\\ &=\frac{na^{mn}}{\Gamma(m)\Gamma(mn-m)}\int_{0}^{\infty}\gamma_{ t}^{m-1}e^{-a\gamma_{t}}\left[\int_{0}^{\frac{\gamma_{t}}{z}}\gamma_{n-1}^{mn-m-1}e^{-a \gamma_{n-1}}d\gamma_{n-1}\right]d\gamma_{t}\end{split} \tag{27}\]  
According to [30] (p. 900), i.e.,  
\[\gamma(\alpha,x)=\sum_{k=0}^{\infty}\frac{(-1)^{k}x^{\alpha+k}}{k!(\alpha+k)} \tag{28}\]  
we can obtain  
\[\begin{split}\int_{0}^{\frac{\gamma_{t}}{z}}\gamma_{n-1}^{mn-m-1 }e^{-a\gamma_{n-1}}d\gamma_{n-1}&=\frac{1}{a^{mn-m}}\gamma\left( mn-m,\frac{a\gamma_{t}}{z}\right)\\ &=\frac{1}{a^{mn-m}}\sum_{k=0}^{\infty}\frac{(-1)^{k}\left(\frac {a\gamma_{t}}{z}\right)^{mn-m+k}}{k!(mn-m+k)}\end{split} \tag{29}\]  
Substituting (29) into (27), we can obtain \[p_{cap}^{Nak} =\frac{na^{m}}{\Gamma(m)\Gamma(mn-m)}\int_{0}^{\infty}\gamma_{t}^{m-1 }e^{-a\gamma_{t}}\sum_{k=0}^{\infty}\frac{(-1)^{k}\left(\frac{a\gamma_{t}}{z} \right)^{mn-m+k}}{k!(mn-m+k)}d\gamma_{t} \tag{30}\] \[=\frac{na^{m}}{\Gamma(m)\Gamma(mn-m)}\left[\frac{\Gamma(mn)}{(mn- m)a^{m}z^{mn-m}}-\frac{\Gamma(mn+1)}{(mn-m+1)a^{m}z^{mn-m+1}}\right.\] \[\left.+\frac{\Gamma(mn+2)}{2!(mn-m+2)a^{m}z^{mn-m+2}}-\cdots+ \frac{(-1)^{k}\Gamma(mn+k)}{k!(mn-m+k)a^{m}z^{mn-m+k}}-\cdots\right]\] \[=\frac{n}{\Gamma(m)\Gamma(mn-m)}\sum_{k=0}^{\infty}\frac{(-1)^{k} \Gamma(mn+k)}{k!(mn-m+k)z^{mn-m+k}}\]  
In fact, we can get the expression of capture probability in Rayleigh fading channel by Equation (30) with \(m=1\). However, this equation, introducing infinite series, is not as simple as Equation (11).