esa-sceva/satcom-chunk-collection huggingface dataset:
## 2 Materials and Methods
### A Hybrid Approach for Diagnosis Refinement
```
0: Symptoms description word vector \(T\leftarrow(w_{1},\dots,w_{t})\)
0: Predicted condition \(D_{p}\)
1:procedureQMRBert:
2: Initialize BERT model with weights \(\theta\)
3: Let \(D\leftarrow\{d_{1},\dots,d_{m}\}\) be the set of diseases
4: Let \(S\leftarrow\{s_{1},\dots,s_{n}\}\) be the set of symptoms
5: Let \(G\) be the disease-symptoms network
6: Compute \(p(D|T)\) using BERT: \(f_{\theta}(T)\leftarrow\text{softmax}(W^{\top}\text{BERT}_{\text{enc}}(T))\)
7: Select top \(k\) diseases: \(D_{k}\leftarrow\text{top}_{k}(P(D|T))\)
8: Construct subnetwork \(G_{k}\) from \(G\) using \(D_{k}\)
9: Initialize evidence as \(E\leftarrow\{\}\)
10:for iteration \(i\in\{1\dots t\}\)do
11: Compute symptoms probabilities as \(P(s_{i}|T)\leftarrow\sum_{d}P(s_{i}|d)P(d|T)\)
12: Select \(t\) symptoms using \(S_{q}\leftarrow\text{top}_{q}(P(S|T))\)
13: Ask user to provide evidence \(e\) based on \(S_{q}\)
14: Update evidence vector \(E\gets E+e\)
15: Update \(P(d_{j}|S,E)\) for each disease \(d_{j}\) in \(G_{k}\)
16:\(P(d_{j}|S)\leftarrow\frac{1}{N}\sum_{d\in D_{k}\setminus\{d_{j}\}}P(S|d)P(d)\)
17:endfor
18: Final prediction: \(D_{p}\leftarrow\text{arg max}(P(D|T,E))\)
```  
**Algorithm 1** Disease diagnosis using BERT and QMR-inspired Bayesian network
## 2 Materials and Methods
### Segmentation and Object-Based Image Classification  
This study employed an object-based image analysis (OBIA) approach for LULC classification using Google Earth Engine (GEE). OBIA was chosen for its ability to reduce the \"salt-and-pepper\" effect common in pixel-based classifications and to incorporate spatial context into the classification process [18; 32]. The OBIA method consists of two primary steps: image segmentation and object classification.  
The segmentation process divides the entire image into multiple areas of varying sizes based on distinct spectral and textural properties. These segments are then merged to form larger objects corresponding to land cover classes. During this process, the geometric properties, topology, and adjacency relationships of the objects are exploited, with the anticipation that these objects will correspond more easily to land cover types than individual pixels would.  
For the segmentation step, this study utilized the Simple Non-Iterative Clustering (SNIC) algorithm, which efficiently groups similar pixels and identifies potential individual objects [33]. SNIC is initiated using a uniform grid of seeds, generated by the \"Image.Segmentation.seedGrid\" function, which requires specifying a superpixel seed location spacing in pixels to influence the size of the resulting clusters. The optimal spacing value can be determined through experimentation. The algorithm then identifies objects (clusters) based on input parameters and produces a multi-band raster output, which includes the clusters themselves and additional layers containing average values of the input features [34]. SNIC requires several key parameters for setting:  
* Compactness factor: Influences cluster shape, with larger values producing more compact clusters.
* Connectivity: Can be set to 4 or 8, determining whether to use Rook's or Queen's contiguity for merging adjacent clusters.
* Neighborhood size: Helps avoid tile boundary artifacts.  
After visual inspection of different combinations of parameter values and considering the characteristics of the LULC classes in the study areas, the following set of parameters was used in this study: compactness = 0, connectivity = 8, and neighborhood size = 256. Following segmentation, the OBIA approach combines spectral and spatial information with texture and contextual information from the image to perform the final object classification [35].
## Simulation Studies  
### Simulation Setup  
As mentioned previously, we consider heat pump and solar water heater as thermal appliances for current simulation studies. Table 1 contains the values of the parameters used in modeling of thermal appliances as described in Reference [49; 51]. To put a constraint on the indoor air temperature, (18 \({}^{\circ}\)C, 22 \({}^{\circ}\)C) is specified as the desired comfort range. These constraints can be chosen to be time varying also. The heat pump is considered to consume maximum of 1 kW power at each time step. Similarly, the water temperature in. the solar water heater tank is kept bounded between (50 \({}^{\circ}\)C, 70 \({}^{\circ}\)C) and the maximum power consumption of the water heater is considered to be 2 kW. Everyday water withdrawal demand profile is obtained from the DHWcalc toolbox [52], which generates realistic domestic hot water demand depending upon various characteristics like number of households, total mean daily draw off volume, etc.  
Without any loss of generality, a washing machine and dishwasher are considered as the non-thermal household appliances. Table 2 details the parameter values for the washing machine and the dishwasher. Every day, non-thermal appliance requests are activated at random time steps with a random deadline which is greater than the total running time of the appliance.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Parameter** & **Value** & **Unit** & **Parameter** & **Value** & **Unit** \\ \hline \(C_{p,f}\) & 3315 & kJ/\({}^{\circ}\)C & \((UA)_{fr}\) & 624 & kJ/\({}^{\circ}\)Ch \\ \(C_{p,r}\) & 810 & kJ/\({}^{\circ}\)C & \((UA)_{wf}\) & 28 & kJ/\({}^{\circ}\)Ch \\ \(C_{p,w}\) & 836 & kJ/\({}^{\circ}\)C & \((UA)_{wf}\) & 28 & kJ/\({}^{\circ}\)Ch \\ \(C_{t}\) & 3881.3 & kJ/\({}^{\circ}\)C & \((UA)_{t}\) & 29.84 & kJ/\({}^{\circ}\)Ch \\ \(\eta\) & 3 & - & \(p\) & 0.2 & - \\ \(\eta_{h}\) & 1 & - & & & \\ \hline \hline \end{tabular}
\end{table}
Table 1: Thermal appliance parameters.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Appliance** & **Running Time (f)** & **Rated-Power (P)** \\ \hline Washing machine & 2 h & 3 kW \\ Dishwasher & 2.5 h & 4 kW \\ \hline \hline \end{tabular}
\end{table}
Table 2: Non-thermal appliance parameters.  
The entire simulation setup is implemented in MATLAB. For simulation purpose, ambient temperature \(T_{a}\) and solar radiation \(\phi_{\mathrm{s}}\) are obtained from ASHRAE IWEC (International Weather for Energy Calculations) weather data files for Dublin, Ireland [53]. The electricity prices are taken from wholesale Danish Energy Market data [54] and employed for real time implementation of the proposed algorithm. A maximum capacity constraint of 4 kW is imposed on the total power consumption of all the appliances. In reality, the capacity constraints can be time varying. The setup is simulated over a 7-day period with a prediction horizon of 24 h (Mode 1). To solve the MILP, Gurobi solver is used in integration with YALMIP toolbox [55] and MATLAB. To summarize, Table 3 shows the number of constraints and optimization methods associated with each mode.
# Dealing with Deadlocks in Industrial Multi Agent Systems
## 6 Discussion  
Two methods (approaches) how to avoid deadlocks in industrial multi-agent systems were presented in this paper, namely: (i) the approach based on \(P\)-invariants, and (ii) the approach based on siphons and traps. Let us compare them now.  
Both approaches have their advantages and disadvantages.  
The former approach represents the procedure in exact analytical terms. The supervisor synthesis is clear and simple. It is suitable especially for the S\({}^{3}\)PR paradigm of AMS. Thorough analysis of RT makes possible to find conditions how to mutually eliminate certain states (markings). In such a way it is possible to eliminate deadlocks. However, in a more complicated structure of the PN model with very branched RT the choice of the restriction inequality (5) (especially the matrix \(\mathbf{L}\) and the vector \(\mathbf{b}\)) may be intricate or even impossible. Another disadvantage is that RT depends on the initial marking \(\mathbf{M}_{0}\) of the uncontrolled PN. For another \(\mathbf{M}_{0}\) RT is different.  
The latter approach has not so exact analytically expressed procedure. This is due to the fact that siphons and traps are structural entities of the PN model. Their computation (notwithstanding that it may be realized by different algorithms) may sometimes take a very long time, especially in case of structurally complicated PN models. An advantage of this approach is that calculation of siphons and traps does not depend on the initial state of the PN model. This approach is able to deal with both paradigms of AMS--S\({}^{3}\)PR and S\({}^{4}\)PR.  
As a summary it can be said that the weak point (shortcoming, weakness) in both approaches is computational complexity--at computing RT and handling it as well as at computation of siphons and traps--especially at large-scale and structurally complicated PN models.  
Despite what has been said, the approach based on siphons and traps seems to be upon the whole more advantageous than the approach based on \(P\)-invariants. Apart from the time-consuming calculation of siphons and traps, it is also simpler. However, as it was demonstrated by the Example 6 in the Section 5.4, computational problems may be huge also in relatively simple and not very large PN models.
# Study on Composite Fracture Characteristics and Hydraulic Fracturing Behavior of Hard Rock
## 3 Composite Fracture Characteristics in Hard Rock
### Verification of Fracture Criteria  
The maximum tangential strain criterion is utilized to evaluate the results of the fracture tests in this study, verifying the effectiveness of this criterion. The fracture angle \(\theta_{0}\) is calculated using Equation (3) with and without considering the T stress, as shown in Figure 4.  
\begin{table}
\begin{tabular}{c c c c} \hline \(\beta\) (\({}^{\circ}\)) & \(K_{\text{I}}\) (MPa\(\cdot\)m\({}^{\text{1/2}}\)) & \(K_{\text{II}}\) (MPa\(\cdot\)m\({}^{\text{1/2}}\)) & \(T\) (MPa) \\ \hline
0 & 0.5498 & 0 & \(-17.5319\) \\
5 & 0.5873 & 0.2192 & \(-19.0127\) \\
10 & 0.5106 & 0.4175 & \(-17.3881\) \\
15 & 0.4124 & 0.6038 & \(-15.6009\) \\
20 & 0.2858 & 0.7739 & \(-13.4206\) \\
28.7 & \(\approx\)0 & 0.9040 & \(-7.7515\) \\ \hline \end{tabular}
\end{table}
Table 2: \(K_{\text{I}}\), \(K_{\text{II}}\), and \(T\) values of different \(\beta\) specimens.  
Figure 4: The effect of T stress on the fracture angle of the CSTBD splitting test.  
Figure 3: Failure of the specimen.  
Combining the fracture angle \(\theta_{0}\), the theoretical values of \(K_{\rm I}/K_{\rm IC}\) and \(K_{\rm II}/K_{\rm IC}\) for the Copper Mine Gorge ore rock are calculated using Equation (17), as shown in Figure 5. Simultaneously, the stress intensity factors from the CSTBD indoor splitting tests are included in the graph for comparison. It can be observed that in the CSTBD splitting test of the Copper Mine Gorge ore rock, the theoretical predictions of the GMTS criterion, considering the influence of T stress, matched well with the experimental results compared to the MTS criterion.
###### Abstract  
Frequency-guided sequential demodulation (FSD) is a technique used to recover the phase of closed fringe patterns when only a single interferogram is available. FSD is a sequential method that follows simple and intuitive processing steps for interferogram demodulation, guided by the frequency of the fringes. The objective of this work is to apply an optimizer block that feeds from the first stages of the FSD and make feedback in such a manner as to optimize the search for guided frequencies. This approach significantly reduces demodulation errors, especially when dealing with high-frequency fringe patterns. The results of using the FSD with and without optimizers are compared. The optimizers used are the Harmonic Search Optimization (HSO) as well as the Particle Swarm Optimization (PSO). Results show that FSD with PSO delivers better results compared with other methods here analyzed.  
PSO; HSO; FSD; optical demodulation; wrapped phase Article
# Calibration Method of PFC3D Micro-Parameters under Impact Load
## 4 Numerical Results and Discussion
### Micro-Parameters Calibration of Peak Stress and Young's Modulus  
According to the test results in Section 4.2, six micro-parameters, including \(\mu\) and \(\overline{E}_{c}\), influence the samples' peak stress \(\sigma_{p}\) and Young's modulus \(E_{c}\). Since the influence of \(\mu\) on \(\sigma_{p}\), \(E_{c}\), and \(f_{\rm m}\) are smaller than that of the other five micro-parameters on the macroscopic physical and mechanical parameters, \(\mu\) is set at 0.15 in the subsequent study.  
\(\overline{E}_{c}\) was set at 10, 20, 30, 40, and 50 GPa, and \(\overline{\sigma}_{c}\) was set at 0.1, 0.2, 0.3, 0.4, and 0.5 GPa. The results of the 21 orthogonal experiments are shown in Figures 19 and 20. Since \(\overline{E}_{c}\) and \(\overline{\sigma}_{c}\) are mainly linearly related to sample \(\sigma_{p}\) and \(E_{c}\), \(\overline{E}_{c}\) and \(\overline{\sigma}_{c}\) are used as double independent variables to carry out linear fitting for \(\sigma_{p}\) and \(E_{c}\), respectively. The fitting formula is as follows:  
\[\sigma_{p}=10.590+0.091\overline{E}_{c}+197.059\overline{\sigma}_{c}\ \ R^{2}=0.962 \tag{7}\]  
\[E_{c}=15.606+0.458\overline{E}_{c}+15.190\overline{\sigma}_{c}  
The residuals of \(\sigma_{p}\) and \(E_{c}\) obtained by linear fitting are shown in Figures 19 and 20.
## 5 Case Study Analysis  
To validate the proposed methodology, a case study involving the adjustment of Line 543 in Guangzhou is conducted. Utilizing C# and Oracle for system development, we implemented bus model construction, path set calculations, and passenger assignments. The total operation time for this calculation process is about 6 h and meets engineering application timeliness requirements.  
Line 543 (highlighted in Figure 9), located in northwestern Guangzhou's central urban area, features 27 stops and ceased operations on 18 June 2023. A pre-evaluation of passenger travel changes following route adjustments was performed and subsequently verified against real travel data collected on 20 July 2023. Prior to the route's suspension, the daily passenger traffic volume was about 2290, with the line connected to 109 associated sub-network lines (depicted as gray solid lines in the figure), accounting for one-seventh of the bus lines in the central urban area of Guangzhou. The analysis reveals a complex network of 998 original stops and 978 destination stops within the sub-network, with calculations yielding 106,000 directly related OD pairs and a total of 2.77 million routes, corresponding to 560,000 passenger flows throughout the entire sub-network.
## 3 Results and Discussion
### Prediction of the Erosion Surface Geometry
#### 3.3.3 Determination of the Maximum Lateral Distance  
In this section, the AB straight-line section of the erosion surface is predicted and analysed. At this point, the maximum depth of the erosion surface can be determined with Equation (2). The ordinate of point B in the AB straight-line section is \(\alpha_{h}\cdot h_{max}\), so it is necessary to determine the slope of the AB section of the erosion surface to predict the linear equation.  
In Table 1, the slope of the straight line of the erosion surface is \(k_{\text{AB}}\), and the \(k_{\text{AB}}\) range is 1.79-6.69. As the nozzle speed \(u_{nz}\) increases, \(k_{\text{AB}}\) decreases. When \(p_{j}\) is small (5-10 MPa), \(u_{nz}\) is the main factor affecting \(k_{\text{AB}}\). When \(p_{j}\) is large (25-30 MPa), \(p_{j}\) has a great influence on \(k_{\text{AB}}\).  
\(k_{\text{AB}}\) is positively correlated with \(p_{j}\) and negatively correlated with \(u_{nz}\). When \(p_{j}\) is large, its increase has a more significant effect on \(k_{\text{AB}}\). The relationship between \(k_{\text{AB}}\) and \(p_{j}\), \(u_{nz}\) is as follows:  
\[k_{\text{AB}}=0.108p_{j}+2.79/u_{nz}+1.17 \tag{3}\]  
where \(k_{\text{AB}}\) is the slope of the starting point of the erosion surface, \(p_{j}\) is the jet pressure (MPa), and \(u_{nz}\) is the nozzle speed (cm/s).  
Figure 13 shows the relationship between the values modelled using Equation (3) and the experimental values. The correlation coefficient of 0.95 indicates a good fit.  
Thus, the abscissa \(x_{B}\) of point B in the straight line AB can be obtained as follows:  
\[x_{B}=\alpha_{h}h_{\text{max}}/k_{\text{AB}} \tag{4}\]  
The maximum lateral distance \(x_{\text{max}}\) can be obtained as follows:  
\[x_{\text{max}}=x_{B}/\alpha_{x}=\alpha_{h}h_{\text{max}}/\alpha_{x}k_{\text{AB}} \tag{5}\]  
Figure 12: Relationship among soil-breaking depth, jet pressure, and nozzle speed.
## 4 Materials and Methods  
Analytical grade sodium chloride, potassium chloride, potassium phosphate monobasic, sodium phosphate dibasic, hydrochloric acid, and sodium hydroxide, used for the phosphate-buffered saline solution (PBS pH 7.4) preparation, as well as the LC-MS grade chromatographic solvents water, methanol, acetonitrile, and formic acid, were purchased form Sigma-Aldrich (Poznan, Poland).  
Urine was obtained from advanced-stage bladder cancer patients and from healthy volunteers (control group). The urine samples were collected from the first urination in the morning. Samples were provided by the Department of Biopharmaceutics and Pharmacodynamics, Medical University of Gdansk.  
Solid phase microextraction (SPME) in its thin film format was used for preparation of samples, and each step of the process was carried out on a high throughput 96-semi-automated SPME system (Professional Analytical System (PAS) Technology, Magdala, Germany), which allowed for simultaneous analysis of all samples (Figure 3). Extraction was performed by using steel blades coated with a polystyrene divinylbenzene (PS-DVB) sorbent (Alchem, Torun, Poland). Coating preparation procedures were based on the spraying method described by [PERSON] et al. [39]. Steel blades were purchased form Professional Analytical System (PAS) Technology (Magdala, Germany) and polypropylene Nunc 96 DeepWell plates were purchased from Sigma-Aldrich (Poznan, Poland).  
Prior to the analysis, urine was diluted in PBS (1:1; \(v/v\)) so as to establish a matrix pH of 7.4. Before extraction, coatings were conditioned for 30 min with 1.0 mL of a methanol: water (1:1; \(v/v\)) solution in 96-well-plates with agitation set at 750 rpm to improve sorbent surface activation and prepare the sorbent to retain the analytes, then subsequently submitted to a 10 s wash step to remove residual methanol. The extraction process was executed from 0.5 mL of urine samples for 1 h at 37 \({}^{\circ}\)C. After extraction, the blades were placedin 1 mL of nanopure water for 10 s to remove particulates, salts, and other contaminants loosely attached on the coating surface by non-specific interactions, which could potentially cause matrix effects and instrument contamination. Following this wash step, desorption was conducted in 1 mL of acetonitrile: water (1:1; \(v/v\)) solution with agitation (750 rpm) for 90 min.
## 2 Method
In addition, based on Equation (30), we can easily form:  
\[\left[ \begin{array}{c}\frac{\vec{u}_{x}}{\vec{u}_{z}}\\ \frac{\vec{\sigma}_{z}}{\vec{\tau}_{xz}}\end{array} \right]_{i}^{-}=\left[ \begin{array}{c}\frac{\vec{u}_{x}}{\vec{u}_{z}}\\ \frac{\vec{\sigma}_{z}}{\vec{\tau}_{xz}}\end{array} \right]_{f}^{+}=P_{f}Q_{f} ^{+}\left[ \begin{array}{c}U_{p}\\ D_{p}\\ U_{s}\\ D_{s}\end{array} \right]_{f}^{-ik^{f}_{x}{x}} \tag{31}\]  
\[\left[ \begin{array}{c}\frac{\vec{u}_{x}}{\vec{u}_{z}}\\ \frac{\vec{\sigma}_{z}}{\vec{\tau}_{xz}}\end{array} \right]_{i+1}^{-}=P_{f}Q_{r} ^{-}\left[ \begin{array}{c}U_{p}\\ D_{p}\\ U_{s}\\ D_{s}\end{array} \right]_{i+1}^{-ik^{f}_{x}{x}} \tag{32}\]  
For the \((n+1)th\) layer, there only exists down-going P- and S- waves, therefore \(U_{\text{P}}=0\) and \(U_{\text{s}}=0\). Because only the P-wave impinges on the first layer, \(D_{\text{s}}\) is equal to 0. Based on Equation (40), the reflected and transmitted waves can be calculated. We can obtain reflection-transmission coefficients that are defined as ratios of reflected and transmitted wave amplitudes to the incident wave amplitude. Specifically, the reflection-transmission coefficients can be written as \(R_{\text{P}\to\text{P}}=\max(abs(U_{\text{P}}^{\text{1}}))\), \(R_{\text{P}\to\text{s}}=\max(abs(U_{\text{s}}^{\text{1}}))\), \(T_{\text{P}\to\text{P}}=\max(abs(D_{\text{P}}^{\text{n+1}}))\) and \(T_{\text{P}\to\text{s}}=\max(abs(D_{\text{s}}^{\text{n+1}}))\). For these symbols, the first subscript letter represents the incident wave and the second subscript letter stands for the transmitted or reflected wave. Details can be found in [21]. During calculation, the wave has to be first transferred into frequency domain signals by the fast Fourier transform (FFT) to calculate the reflected and transmitted waves. Finally, these waves in the frequency domain are transferred into signals in the time domain by the inverse fast Fourier transform (IFFT). In the following section, the experimental incident wave is first used as the input incident wave for model validation and then one cycle of sinusoidal wave is used as the input incident wave for parametric studies. According to [21], the start time to oscillate needed to have enough delay rather than from 0 to eliminate the aliasing effect as much as possible in the processing of FFT. Meanwhile, some amplitude compensations are also needed.
## 5 Simulation Results
### Scenario 2: Simulation of Controller for AHV
Improved adaptive laws are adopted to compensate for the adverse influence of uncertainties on AHVs. The tracking errors (caused by uncertainties) under the proposed method are observed to be smaller than those of BSMC and SMC.
## Appendix A Gradient: Divergence, and Curl in Cylindrical Coordinates
### The Nabla Operator in Cylindrical Coordinates  
To express _nabla_ operator \(\
abla\) in cylindrical coordinates, we must map the terms \(\hat{i}\frac{\partial}{\partial x}\), \(\hat{j}\frac{\partial}{\partial y}\), and \(\hat{k}\frac{\partial}{\partial z}\) onto the terms \(\hat{r}\frac{\partial}{\partial x}\), \(\hat{\phi}\frac{\partial}{\partial y}\), and \(\hat{z}\frac{\partial}{\partial z}\) that are defined in cylindrical coordinates \(U(r,\phi,z)\). A direct approach is to use the chain rule and transformation relationships introduced above, namely,  
\[\hat{\rho}\frac{\partial}{\partial\rho} =(\hat{i}\cos\phi+\hat{j}\sin\phi)(\frac{\partial x}{\partial \rho}\frac{\partial}{\partial x}+\frac{\partial y}{\partial\rho}\frac{ \partial}{\partial y}) \tag{13}\] \[=(\hat{i}\cos\phi+\hat{j}\sin\phi)(\cos\phi\frac{\partial}{ \partial x}+\sin\phi\frac{\partial}{\partial y})\] \[=\cos^{2}\phi\hat{i}\frac{\partial}{\partial x}+\sin\phi\cos\phi \hat{i}\frac{\partial}{\partial y}+\sin\phi\cos\phi\hat{j}\frac{\partial}{ \partial x}+\sin^{2}\phi\hat{j}\frac{\partial}{\partial y}\] \[\hat{\phi}\frac{\partial}{\partial\phi} =(-\hat{i}\sin\phi+\hat{j}\cos\phi)(\frac{\partial x}{\partial \phi}\frac{\partial}{\partial x}+\frac{\partial y}{\partial\phi}\frac{ \partial}{\partial y})\] \[=(-\hat{i}\sin\phi+\hat{j}\cos\phi)(-\rho\sin\phi\frac{\partial}{ \partial x}+\rho\cos\phi\frac{\partial}{\partial y})\] (14) \[=\rho\sin^{2}\phi\hat{i}\frac{\partial}{\partial x}-\rho\sin\phi \cos\phi\hat{i}\frac{\partial}{\partial y}-\rho\sin\phi\cos\phi\hat{j}\frac{ \partial}{\partial x}+\rho\cos^{2}\phi\hat{j}\frac{\partial}{\partial y}\]  
and  
\[z\frac{\partial}{\partial z}=\hat{k}\frac{\partial}{\partial z} \tag{15}\]  
Compare to Equation (12), an intuitive attempt is to divide Equation (14) by \(\rho\), and add it to Equations (13) and (15), then,  
\[\hat{\rho}\frac{\partial}{\partial\rho}+\frac{\hat{\phi}}{\rho} \frac{\partial}{\partial\phi}+\hat{z}\frac{\partial}{\partial z} =(\sin^{2}\phi+\cos^{2}\phi)\hat{i}\frac{\partial}{\partial x}+( \sin^{2}\phi+\cos^{2}\phi)\hat{j}\frac{\partial}{\partial y}+\hat{k}\frac{ \partial}{\partial z} \tag{16}\] \[=\hat{i}\frac{\partial}{\partial x}+\hat{j}\frac{\partial}{ \partial y}+\hat{k}\frac{\partial}{\partial z}\]  
Therefore, we have  
\[\
abla=\hat{\rho}\frac{\partial}{\partial\rho}+\frac{\hat{\phi}}{\rho}\frac{ \partial}{\partial\phi}+\hat{z}\frac{\partial}{\partial z} \tag{17}\]