esa-sceva/satcom-chunk-collection huggingface dataset:
# Workload Balancing on Identical Parallel Machines: Theoretical and Computational Analysis
## 3 Theoretical Analysis of Different Criteria
### Case with Two Machines  
**Proposition 3**: _In the case of two parallel machines, the problems \(P2||C_{max}\), \(P2||C_{\Delta}\) and \(P2||NSSWD\) are equivalent._  
**Proof of Proposition 3.** First, we establish that a \(C_{max}\)-optimal solution is also \(C_{\Delta}\)-optimal one. As a preliminary result, we assume the well-established relationship: \(\max\{a,b\}=\frac{1}{2}\times(a+b+|b-a|)\).  
Considering this relationship and the definition of the maximum completion time, we can write:\[C_{max}=\max\{C_{1},C_{2}\}=\frac{C_{1}+C_{2}}{2}+\frac{1}{2}\times(|C_{2}-C_{1}|)= \mu+\frac{1}{2}\times C_{\Delta} \tag{9}\]  
Since \(\mu=\frac{C_{1}+C_{2}}{2}\) is constant and \(C_{\Delta}=C_{2}-C_{1}\), it is clear that minimizing \(C_{max}\) implies the minimization of the difference \(C_{\Delta}\).  
Then, we establish that a \(C_{\Delta}\)-optimal solution is also (\(NSSWD\))-optimal. Based on the Normalized Sum of Square for Workload Deviations definition, we can write:  
\[\begin{array}{ll}NSSWD&=\frac{1}{\mu}\times\left[\sum\limits_{m=1}^{2}{(C_{ m}-\mu)^{2}}\right]^{\frac{1}{2}}=\frac{1}{\mu}\times\left[{(C_{1}-\mu)^{2}+(C_{2}- \mu)^{2}}\right]^{\frac{1}{2}}\\ &=\frac{1}{\mu}\times\left[{\left(C_{1}-\frac{C_{1}+C_{2}}{2}\right)^{2}+ \left(C_{2}-\frac{C_{1}+C_{2}}{2}\right)^{2}}\right]^{\frac{1}{2}}=\frac{1}{ \mu}\times\left[{\left(\frac{C_{1}-C_{2}}{2}\right)^{2}+\left(\frac{C_{2}-C_ {1}}{2}\right)^{2}}\right]^{\frac{1}{2}}\\ &=\frac{1}{\mu}\times\left[{\frac{1}{2}(C_{2}-C_{1})^{2}}\right]^{\frac{1}{2}}= \frac{1}{\sqrt{2}\times\mu}\times(C_{max}-C_{min})=\frac{1}{\sqrt{2}\times\mu }\times C_{\Delta}\end{array} \tag{10}\]  
Since \(\mu\) is constant, minimizing (\(C_{\Delta}\)) implies the minimization of \(NSSWD\). \(\square\)
# Enhancing Precision in Magnetic Map Interpolation for Regions with Sparse Data
## 2 Improved Variogram of Kriging Algorithm for Sparse Data
### Principles of Kriging Interpolation Algorithm
#### 2.1.3 Experimental Variogram  
The experimental variogram \(\gamma^{*}(l)\) is an estimate of the variogram \(\gamma(l)\) constructed from the measured data. With the second-order stationary hypothesis, the increment of \(Z(x)\) depends only on the lag \(l\) between the two points, and not on the position \(x\). The formula is as follows [26]:  
\[\gamma^{*}(l)=\frac{1}{2N(l)}\sum_{i=1}^{N(l)}[Z(x_{i})-Z(x_{i}+l)]^{2}, \tag{6}\]  
where \(N(l)\) represents the logarithm of the data pair divided by the vector \(l\). For different lags \(l\), the values of \(\gamma^{*}(l)\) can be calculated according to Equation (6), and points \([l_{i},\gamma(l_{i})]\) can be marked in the graph of the experimental variogram, which can directly reflect the relevant situation of measurement data [27].  
In order to predict the unknown position of the regionalized variable, it is necessary to analyze the experimental variogram and find the best-fitting variogram model. Selecting the variogram model and parameters under optimal conditions is the key to improving the accuracy of Kriging interpolation [28].
## 2 Materials and Methods
### SEM Analysis  
The microstructures of the films were evaluated by scanning electron microscopy (SEM) using an FEI Quanta 200 FEG (FEI, OR, USA), which is a high resolution field emission scanning electron microscope with a Schottky-type electron gun where the samples can be investigated under controlled pressure water steam atmosphere. The SEM was operated at 10 kV with magnification of 5000-10,000 times.
# Improving Structural Power Content Analysis Robustness for Satellite Navigation Applications +
###### Abstract  
Many critical applications exhibit a growing dependency on Global Navigation Satellite Systems (GNSS), which has led to GNSS jamming and spoofing becoming an increasing threat to society. The Structural Power Content Analysis (SPCA) algorithm is a pre-despreading, low-complexity, and effective method to detect spoofing events, but in practice it is seen that false alarms are being generated in certain jamming scenarios. To mitigate these effects, alternative filtering techniques are evaluated and tested on both simulated data and publicly available spoofing datasets. Effective false alarm reduction with only a minor degradation in spoofing detection sensitivity is demonstrated.  
GNSS; spoofing detection; SPCA; filter techniques; jamming +
Footnote †: Academic Editor: [PERSON] Published: 14 April 2025
## 2 System Architecture
### Analog Component  
The analog component of the proposed system is a transceiver composed of frequency converter modules, between IF and RF, and also by the antenna arrays that transmit and collect the involved signals. The operating frequency was selected in the Ka band, at 28 GHz, and the chosen IF was 3.84 GHz.  
Therefore, the RF upconverter module translates the modulated 3.84 GHz IF signal to the output operating frequency of 28 GHz. The block diagram of the upconverter is presented in Figure 5a, and its measurements are shown in Figure 6. The RF downconverter transforms the received 28 GHz signals down to 3.84 GHz IF, to be processed in the USRP. The block diagram of the downconverter can be seen in Figure 7a, and its measured results are provided in Figure 8.
# Flutter Reliability Analysis of Xiangshan Harbor Highway Cable-Stayed Bridges in Service
## 2 The Flutter Analysis Framework for Flexible Bridge
### Establishment of Bridge Finite Element Model  
The main bridge of the Xiangshan Harbor Highway cable-stayed bridge is a two-tower, two-cable plane, five-span continuous semi-floating system cable-stayed bridge, and the specific layout is shown in Figure 3. The main girder is a streamlined steel box girder, and the specific size is shown in Figure 4. The cable tower is diamond shaped tower, and the foundation of the main tower, auxiliary pier and transition pier is a drilling pole foundation. The whole bridge is simulated with a \"fish bone spur\" model.  
The finite element software ANSYS was used for modeling and analysis. Xiangshan Harbor Bridge in Ningbo was simplified into a single-backbone girder model, and the finite element model of Xiangshan Harbor Bridge was established by the APDL command flow method. BEAM188 element was used to simulate the main girder, tower and pier. The main girder and stay cables are connected by rigid arms, which were simulated by BEAM4 element. The cable was simulated by LINK10 element with axial tension only. MASS21 element was used to simulate ballasting weight and second-phase dead load. The connections of the main girder, bridge tower and pier were simulated by coupling, and the bridge tower and pier were consolidated at the bottom. The finite element model of Xiangshan Harbor Bridge is shown in Figure 5.  
The modal analysis of the bridge was carried out according to the established finite element model, and this allowed for the frequency and mode shape results of the bridge to be obtained. The calculation results of the first ten order frequencies are listed in Table 2, where they are compared with the measured frequencies of Xiangshan Harbor Bridge [40],  
Figure 4: Cross-sectional layout of steel box girder (mm).  
Figure 5: Finite element model of Xiangshan Harbor Bridge.  
Figure 3: Layout of main bridge (m).  
where the natural frequencies of this bridge were measured by dynamic-load tests, and the detailed contents of this experiment is from Ref. [41]. It can be seen from the table that the natural vibration frequency calculated by the finite element model was not much different from the measured value of the structure, which is within an acceptable range. Therefore, it is considered that the established finite element model can be used for subsequent analysis.
# Delamination Localization in Multilayered CFRP Panel Based on Reconstruction of Guided Wave Modes
## 2 Materials and Methods
### Guided Wave Frequency Tuning Curves  
In GW inspection, fundamental or higher-order modes can be selected for the inspection according to dispersion curves. The dispersion curves obtained give only velocity and frequency, but the sensitive mode cannot be selected. For this purpose, GW tuning curves are important to select sensitive mode and optimal frequency for the inspection.  
To obtain the frequency tuning curves, the simulation was carried out in COMSOL with a contact transducer having similar properties to those of the transducer used in the experiments. Figure 4 shows the schematic diagram of the inspection setup. GW excited with an excitation frequency range from 10 to 300 kHz on a 3.5 mm CFRP plate. The maximum displacements of both S\({}_{0}\) and A\({}_{0}\) were calculated for each 10 kHz frequency step. The maximum displacements of the GW modes are presented in Figure 5. Frequency tuning curves are used to select the optimal GW mode for the inspection.  
Figure 4: The schematic diagram of the inspection setup used to obtain frequency tuning curves.  
Figure 3: Guided wave dispersion curves in the CFRP panel: phase velocity versus frequency (**a**) and group velocity versus frequency (**b**).  
The A\({}_{0}\) mode is sensitive to delamination-type defects and is also sensitive to the length of the delamination. The displacements of the A\({}_{0}\) mode are higher than those of the S\({}_{0}\) mode at 150 kHz frequency and are dominant. Therefore, the A\({}_{0}\) mode is selected for inspection.
# Development of Navigation Network Models for Indoor Path Planning Using 3D Semantic Point Clouds
## 3 Experiments and Results  
To validate the effectiveness and robustness of both our pipeline and algorithm for constructing the navigation network model, we conducted a series of experiments using the S3 DIS dataset [32] and our self-collected dataset with HoloLens 2. First, we introduced the dataset used for the experiments in Section 3.1. Subsequently, in Section 3.2, we visualized and checked the distribution of nodes and edges in the extracted navigation network. Finally, in Section 3.3, we used [PERSON]'s algorithm [37] to verify the shortest path planning capabilities of the extracted navigation networks.
# Social)GCF: Social Recommendation with Jacobi Polynomial-Based Graph Collaborative Filtering
## 4 Experiments
### Comparisons of Graph Fusion Methods  
To demonstrate the proposed graph fusion method used in this paper, we compare the performance of SocialJGCF with the graph fusion method used in SocialLGN [40], which is a modification of GraphSage fusion [18]. The fusion method in GraphSage simply adopts concatenation as the fusion method,  
\[E_{GraphSage}=\sigma(W(E_{S}||E_{UI})),\]  
while SocialLGN applies different weight matrices before concatenation,  
\[E_{SocialLGN}=W_{3}[\sigma(W_{2}E_{S})||\sigma(W_{1}E_{UI})].\]  
Figure 2 shows the comparison results. The graph fusion method used in this paper achieves approximately 2\(\times\) recommendation performance for all three metrics and on both datasets.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Dataset** & **Metric** & **BPR** & **LightGCN** & **SocialLGN** & **SocialJGCF** & **Improvement** \\ \hline \multirow{8}{*}{LastFM} & Precision@10 & 0.0333 & 0.0417 & 0.0458 & 0.0583 & 27.2926\% \\ \cline{2-7}  & Precision@20 & 0.0188 & 0.0312 & 0.0333 & 0.0375 & 12.6126\% \\ \cline{2-7}  & Recall@10 & 0.1858 & 0.1727 & 0.1974 & 0.2297 & 16.3627\% \\ \cline{2-7}  & Recall@20 & 0.1910 & 0.2416 & 0.2663 & 0.2811 & 5.5576\% \\ \cline{2-7}  & NDCG@10 & 0.1212 & 0.1374 & 0.1419 & 0.1728 & 21.7759\% \\ \cline{2-7}  & NDCG@20 & 0.1240 & 0.1596 & 0.1643 & 0.1902 & 15.7638\% \\ \hline \multirow{8}{*}{Epinions} & Precision@10 & 0.0127 & 0.0139 & 0.0127 & 0.0133 & 4.7244\% \\ \cline{2-7}  & Precision@20 & 0.0097 & 0.0109 & 0.0102 & 0.0107 & 4.9020\% \\ \cline{2-7}  & Recall@10 & 0.0369 & 0.0403 & 0.0371 & 0.0384 & 3.5040\% \\ \cline{2-7}  & Recall@20 & 0.0571 & 0.0643 & 0.0603 & 0.0620 & 2.8192\% \\ \cline{2-7}  & NDCG@10 & 0.0273 & 0.0292 & 0.0266 & 0.0286 & 7.5188\% \\ \cline{2-7}  & NDCG@20 & 0.0345 & 0.0378 & 0.0348 & 0.0370 & 6.3218\% \\ \hline \hline \end{tabular}
\end{table}
Table 4: Comparison between SocialJGCF and other baselines on cold-start LastFM and epinions dataset.  
Figure 2: Comparison of different graph fusion methods. (**a**) LastFM. (**b**) epinions.
# Light-YOLOv5: A Lightweight Algorithm for Improved YOLOv5 in Complex Fire Scenarios
## 2 Related Work  
Due to the irregular shape of smoke, uneven spatial distribution, and short existence time, it is not easy to accurately detect smoke. Traditional methods usually detect smoke by some its features, such as color, texture, and shape. [PERSON] et al. [1] used dynamic texture features to detect smoke using two- and three-dimensional LBP histograms, which can exclude wind interference in static scenes. [PERSON] et al. [2] used the HSV model and adaptive median algorithm for pre-processing and a high-order linear dynamic system for the dynamic texture analysis of smoke, significantly improving the overall detection accuracy. [PERSON] et al. [3] proposed a flame-detection method that combines the dynamic and static features of a flame in a video and reduces the influence of the environment by combining flame-color and local features.  
In recent years, with the rapid development of machine learning techniques, the latest target-detection algorithms in deep learning have been applied to fire-detection practices. [PERSON] et al. [4] used an improved YOLOv4 network for real-time smoke and fire detection, dramatically reducing the number of parameters to improve the overall detection speed and successfully deploying this to UAVs, but with a lower accuracy than the original algorithm. [PERSON] et al. [5] proposed a T-YOLOX fire-detection algorithm using the VIT technique to improve the accuracy of detecting smoke, fire, and people, but did not discuss the number of parameters or the computational effort involved. [PERSON] et al. [6] proposed an improved Fire-YOLO algorithm for forest fires to enhance the detection of small targets in fires and reduce the model size. Nonetheless, they did not discuss the number of parameters or computational effort. [PERSON] et al. [7] improved the algorithm of YOLOv3-tiny to improve the fire-detection accuracy by multi-scale fusion and k-means clustering. However, the detection speed was not ideal, and the application scenario was singular. [PERSON] et al. [8] reduced the false-detection rate by increasing the resolution of the feature map and expanding the perceptual field, but the detection speed was not satisfactory. However, some of these articles mentioned above are unsatisfactory in terms of speed and accuracy, and some are too homogeneous in terms of the environment to balance these three issues. Therefore, this paper proposes a Light-YOLOv5s method for the purpose of fire detection to achieve a balance of speed and accuracy in complex environments.
# Optimized Scheduling of IoT Devices in Healthcare Facilities: Balancing Cost and Quality of Care
## Appendix A Class and Dependency Diagram  
The UML diagram in Figure 11 illustrates the main classes involved in an IoT-based scheduling system, highlighting their attributes, methods, and relationships. The Patient class represents individuals requiring medical device assignments, while the Device class models the available equipment. The Assignment class links patients to devices for a specific time period. The Schedule class manages assignments and computes relevant objectives, such as cost and quality of care. Finally, the IoTSchedulingProblem class orchestrates the scheduling process, providing methods for problem initialization, optimization, and result visualization. The relationships between classes illustrate dependencies, ensuring a structured and logical representation of the scheduling framework.  
The dependency relationships are reported in the dependency diagram in Figure 12. Patient and Device both contribute to creating Assignment. Schedule contains multiple Assignment instances. IoTSchedulingProblem references and manages all other classes, acting as the central coordination point. The diagram suggests a modsigned to optimize medical device scheduling using the NSGA-II advanced computational technique for multi-objective optimization.
## 2 Fluid-Structure Interaction Vibration Analysis of Single Pipeline
### Theoretical Model of Pipeline Vibration
Considering the fluid action on the pipeline, the overall structural dynamics equation is  
\[\mathrm{M}_{\mathrm{s}}\ddot{\delta}+\mathrm{C}_{\mathrm{s}}\dot{\delta}+ \mathrm{K}_{\mathrm{s}}\delta+\mathrm{f}_{\mathrm{o}}+\mathrm{F}_{\mathrm{d}}=0 \tag{5}\]  
where \(M_{\mathrm{s}}\) is the overall quality matrix for the pipeline; \(C_{\mathrm{s}}\) is the overall damping matrix; \(K_{\mathrm{s}}\) is global stiffness matrix; \(\delta\) is the displacement vector; \(f_{o}\) is the vector of fluid pipeline interaction at the fluid-solid interface; and \(F_{d}\) is the external excitation vector other than \(f_{o}\). Fluid structure coupling equation of pipeline:  
Figure 1: Single-pipe geometric modeling. The grid elements of the fluid model and solid model can be different, but they need to satisfy the conservation of variables such as force on the fluid-solid coupling surface (\(\tau\)), displacement (\(d\)), heat flow rate (\(q\)), and temperature (\(T\)), satisfying the following four equations:  
\[d_{f}=d_{s} \tag{6}\]  
\[\mathrm{n}\tau_{f}=\mathrm{n}\tau_{s} \tag{7}\]  
\[q_{f}=q_{s} \tag{8}\]  
\[T_{f}=T_{s} \tag{9}\]  
In the equation, \(d_{f}\) and \(d_{s}\) are the displacements of the fluid and solid on the fluid-solid coupling surface; \(\tau_{f}\),\(\tau_{s}\) are the shear stress of the fluid and solid on the fluid-solid coupling surface; \(q_{f}\), \(q_{s}\) is the heat flux of the fluid and solid on the fluid-solid coupling surface; and \(T_{f}\),\(T_{s}\) is the temperature of the fluid and solid on the fluid-solid coupling surface. When temperature changes are not considered, both the displacement balance equation and the stress balance equation should be satisfied; that is, by satisfying the conservation of these two variables, the transfer of analytical parameters between the fluid and the solid can be achieved. In Figure 1, U, V, W are coordinate axis. h is the thickness of the pipe wall, and a is the average radius of the pipeline. r is the direction vector of the pipeline cross-section, so \(\theta\) is the angle between the r vector and the vertical direction.
## 3 Solution Method for Orbital Pursuit-Evasion-Defense Game
### Hybrid Numerical Method  
So far, the orbital pursuit-evasion-defense problem has been transformed into a 12-dimensional TPBVP. Generally, this kind of problem cannot be solved analytically, and numerical algorithms must be employed [9]. Numerical algorithms for solving this kind of problems include collocation method [29] and multiple shooting method [30]. The collocation method suffers from poor accuracy and high computational burden, while the multiple shooting method has high accuracy but is very sensitive to the initial guess. As noted by [PERSON] [9], evolutionary methods constitute an effective statistical search technique for selecting the best parameters. Thus, we apply evolutionary methods to generate the initial guess for the multiple shooting method. A hybrid method combining the multi-objective genetic algorithm and the multiple shooting method is proposed to obtain the solution of the orbital pursuit-evasion-defense game. First, the initial guesses of unknown parameters are obtained by usingthe multi-objective genetic algorithm. Then the exact solution of the TPBVP is solved by using the multiple shooting method.  
For the sake of clarity, the state equations, the co-state equations, the initial conditions, and the terminal conditions are arranged.  
Combining Equations (10), (20), and (26), the state equations and the co-state equations can be concluded as follows:  
\[\left\{\begin{array}{l}\dot{\mathbf{Z}}_{\rm PE}=\mathbf{D}\mathbf{\Phi}T_{\rm P}\mathbf{U}_ {\rm P}^{\rm r}-\mathbf{D}\mathbf{\Phi}T_{\rm E}\mathbf{U}_{\rm E}^{\rm r}\\ \dot{\mathbf{Z}}_{\rm DP}=\mathbf{D}\mathbf{\Phi}T_{\rm D}\mathbf{U}_{\rm D}^{\rm r}-\mathbf{D}\mathbf{ \Phi}T_{\rm P}\mathbf{U}_{\rm P}^{\rm r}\\ \dot{\mathbf{A}}_{\rm E}=0\\ \dot{\mathbf{A}}_{\rm D}=0\end{array}\right. \tag{28}\]  
The initial conditions of Equation (28) are expressed as follows:  
\[\left\{\begin{array}{l}\mathbf{Z}_{\rm PE}(0)=\mathbf{D}\mathbf{\Phi}(t_{\rm r},0)\mathbf{X} _{\rm PE}(0)\\ \mathbf{Z}_{\rm DP}(0)=\mathbf{D}\mathbf{\Phi}(t_{\rm r},0)\mathbf{X}_{\rm DP}(0)\end{array}\right. \tag{29}\]  
where \(\mathbf{X}_{\rm PE}(0)=\)\(\mathbf{X}_{\rm P}(0)-\mathbf{X}_{\rm E}(0)\), \(\mathbf{X}_{\rm DP}(0)=\)\(\mathbf{X}_{\rm D}(0)-\mathbf{X}_{\rm P}(0)\).  
According to Equation (21), the terminal conditions are written as follows:  
\[\left\{\begin{array}{l}A_{\rm E}(t_{\rm f})=\frac{\partial_{\rm I_{E}}}{ \partial\mathbf{Z}_{\rm PE}(t_{\rm f})}=-\frac{\mathbf{Z}_{\rm PE}(t_{\rm f})}{\|\mathbf{ Z}_{\rm PE}(t_{\rm f})\|}\\ A_{\rm D}(t_{\rm f})=\frac{\partial_{\rm I_{D}}}{\partial\mathbf{Z}_{\rm DP}(t_{ \rm f})}=\frac{\mathbf{Z}_{\rm DP}(t_{\rm f})}{\|\mathbf{Z}_{\rm DP}(t_{\rm f})\|} \end{array}\right. \tag{30}\]
## 2 Materials and Methods
### Fuzzy Logic  
Fuzzy logic, first introduced by [PERSON] [7], provides a systematic approach to handle uncertainty, imprecision, and vagueness. In this framework, a fuzzy set \(\bar{\xi}\) on a universal set of real numbers \(\mathbb{X}\) is defined by its membership function \(\mu_{\bar{\xi}}\) which maps \(\mathbb{X}\) to \([0,1]\). This function assigns to each element \(x\in\mathbb{X}\) a real number \(\mu_{\bar{\xi}}(x)\) in the interval \([0,1]\). To enable the use of fuzzy rules, Petri nets with various extensions have been integrated with fuzzy logic. In [13; 15], the initial applications of SPNs with fuzzy parameters, the authors demonstrated how continuous stochastic Petri nets with fuzzy parameters can be applied in systems biology, and discrete stochastic Petri nets with fuzzy parameters can model a flexible manufacturing cell, respectively. In [16], it was shown how Petri nets can be applied to identify effective targets for \(\beta\)-globin disorders.  
There are different types of fuzzy numbers: triangular, trapezoidal, pentagonal, hexagonal, heptagonal, etc. In our work, we use the triangular fuzzy number (TFN) to represent reaction rates and concentrations. An TFN \(\bar{\xi}\) is described by three values \(a<b<c\), where \([a,c]\) forms the base of the triangle and \(b\) represents its peak.  
Here is an explanation of how we determine a TFN for the reaction rates, which is also illustrated in Figure 3. We always have qualitative knowledge on reaction rates at our disposal, such as translation being almost ten times slower than transcription. First, based on existing qualitative knowledge, we normalize the crisp value of a reaction rate \(b\) relative to the transcription rate, which is set to 1. During the validation stage of the model, we fine-tune the reaction rates by adjusting those of the corresponding transcriptions. Once these two stages are completed, the reaction rate \(x=b\) is considered acceptable yieldings \(\mu(x)=1\); otherwise, \(\mu(x)=0\). Next, we define the TFN, \((a,b,c)\), of \"acceptable rates\" for each reaction by assigning a degree of membership to each rate value \(x\), where \(\mu(x)\in[0,1]\). Notably, \(\mu(x)\) increases with \(x\) in the interval \((a,b)\) and decreases in the interval \((b,c)\). In this research \(a=b-\frac{b}{2}\) and \(c=b+\frac{b}{2}\).  
Figure 3 illustrates the application of the above principle for the phenomenon of transcription. The TFN is implemented with two \(\alpha\) cuts: \(\alpha=0.5\) and \(\alpha=1.0\), yielding the crisp subsets \([0.75,1.25]\) and \(\{1.0\}\), respectively. Finally, the set of fuzzy rates for the transcription phenomenon is defined using these \(\alpha\) cuts as \(\theta=\{r_{1}=0.0,r_{2}=0.5,r_{3}=1.0,\)\(r_{4}=1.25,r_{5}=1.5\}\).