esa-sceva/satcom-chunk-collection huggingface dataset:
## 2 Preliminaries and Problem Description
### Problem Description  
In order to describe conveniently the control problem of this paper, the desired trajectory tracked by the hovercraft is defined as follows:  
\[\left[\begin{array}{c}\dot{x}_{d}\\ \dot{y}_{d}\\ \dot{y}_{d}\\ \end{array}\right]=\left[\begin{array}{ccc}\cos\psi_{d}&-\sin\psi_{d}&0\\ \sin\psi_{d}&\cos\psi_{d}&0\\ 0&0&1\\ \end{array}\right]\left[\begin{array}{c}u_{dest}\\ v_{dest}\\ r_{dest}\\ \end{array}\right] \tag{13}\]  
where \(x_{d}\), \(y_{d}\) and \(\psi_{d}\) denote desired position and heading angle, \(u_{dest}\), \(v_{dest}\), and \(r_{dest}\) denote setting speed and turn rate.  
The position tracking errors are defined as:  
\[\begin{array}{l}x_{e}=x-x_{d}\\ y_{e}=y-y_{d}\\ \end{array} \tag{14}\]  
Considering (12) and (13), the time derivative of position tracking errors is:  
\[\left[\begin{array}{c}\dot{x}_{e}\\ \dot{y}_{e}\\ \end{array}\right]=\left[\begin{array}{cc}\cos\psi&-\sin\psi\cos\phi\\ \sin\psi&\cos\psi\cos\phi\\ \end{array}\right]\left[\begin{array}{c}u\\ v\\ \end{array}\right]-\left[\begin{array}{c}\dot{x}_{d}\\ \dot{y}_{d}\\ \end{array}\right] \tag{15}\]  
In addition, the speed and turn rate tracking errors are defined as follows:  
\[\begin{array}{l}u_{e}=u-\alpha_{u}\\ v_{e}=v-\alpha_{v}\\ r_{e}=r-\alpha_{r}\\ \end{array} \tag{16}\]  
where \(\alpha_{u}\), \(\alpha_{v}\) and \(\alpha_{r}\) are the virtual control laws that will be designed later on.  
According to Equations (12) and (16), the derivatives of the tracking error of speed and turn rate are given by:  
\[\begin{array}{l}\dot{u}_{e}=vr+\frac{F_{xD0}}{m_{0}}+f_{u}+\frac{\tau_{u}}{ m_{0}}-\dot{\alpha}_{u}\\ \dot{v}_{e}=-ur+\frac{F_{xD0}}{m_{0}}+f_{v}-\dot{\alpha}_{v}\\ \dot{r}_{e}=\frac{M_{xD0}}{f_{x0}}+f_{r}+\frac{\tau_{r}}{f_{x0}}-\dot{\alpha}_{ r}\\ \end{array} \tag{17}\]  
**Assumption 1**.: _The initial tracking errors defined by (14) and (16) satisfy the following inequality:_  
\[\left|i_{e}(0)\right|<k_{i}\left(t\right),\quad i=x,y,u,v,r \tag{18}\]  
_where \(k_{i}(t)\) are appointed-time performance constraint functions that will be specified later on._  
The control objective in this paper can be formulated as follows:  
Considering the hovercraft model (12) in the presence of model uncertainties and multiple performance constraints, an adaptive appointed-time integral barrier Lyapunov function-based trajectory tracking controller is designed to ensure that all the tracking errors converge to the small region containing zero within user-appointed time while guaranteeing that the appointed-time performance constraint conditions on the tracking errors are never violated.
# Design of a Partially Grid-Connected Photovoltaic Microgrid Using IoT Technology
## 2 Microgrid Operation and Control
line impedance is \(Z\angle\theta=R+jX\), the AC bus voltage is \(V\angle-\delta\), and the load at the AC bus is \(P+jQ\), the current \(I\) flowing in this circuit is computed as follows:  
\[I=\frac{V_{s}\angle 0-V\angle-\delta}{R+jX} \tag{3}\]  
The active and reactive powers delivered by the source are calculated as follows:  
\[P_{S}+jQ_{S}=V_{s}\angle 0\frac{V_{s}\angle 0-V\angle\delta}{R-jX} \tag{4}\]  
Reducing (4) yields:  
\[P_{S}=\frac{V_{s}}{R^{2}+X^{2}}(R(V_{S}-V\cos\delta)+XV\sin\delta)) \tag{5}\]  
\[Q_{S}=\frac{V_{S}}{R^{2}+X^{2}}(-RV\sin\delta+X(V_{S}-V\cos\delta)) \tag{6}\]  
In high voltage power systems, usually \(R\ll X\), and considering a small value of the angle \(\delta\), hence the above equations can be reduced to the following:  
\[P_{S}=\frac{V_{s}V}{X}\delta,\text{ and }Q_{S}=\frac{V_{S}(V_{S}-V)}{X} \tag{7}\]  
The two last equations develop the classical active and reactive power droop controls, which are usually used to control voltage and frequency in modern power systems [44; 45; 46; 47]. The above equations show the dependency of both active and reactive powers on the power angle and voltage magnitude, respectively [48]. The droop control characteristics are expressed by the following equations [49]:  
\[f_{S}=f_{0}-k_{P}P_{s},\text{ \ \ \ \ where \ \ }k_{P}=\frac{f_{m}-f_{0}}{P_{m}} \tag{8}\]  
Similarly  
\[\sum_{G=1}^{N_{g}}Q_{G}=\sum_{L=1}^{N_{L}}Q_{L}+Q_{loss} \tag{2}\]0  
\[\sum_{G=1}^{N_{g}}Q_{G}=\sum_{L=1}^{N_{L}}Q_{L}+Q_{loss} \tag{2}\]1 **(a)** Power flow model of a line and (b) its represented phasor diagram. However, in low voltage networks, such as microgrids, \(R\gg X\), in this case, Equations (5) and (6) are expressed as follows:  
\[\sum_{G=1}^{N_{g}}Q_{G}=\sum_{L=1}^{N_{L}}Q_{L}+Q_{loss} \tag{2}\]2  
The last two equations and are usually referred to as reverse droop control characteristics which are expressed as follows:  
\[\sum_{G=1}^{N_{g}}Q_{G}=\sum_{L=1}^{N_{L}}Q_{L}+Q_{loss} \tag{2}\]3  
Similarity,  
\[\sum_{G=1}^{N_{g}}Q_{G}=\sum_{L=1}^{N_{L}}Q_{L}+Q_{loss} \tag{2}\]4  
Figure 3 shows plots of both conventional and reverse frequency and voltage droop characteristics. Droop characteristics must ensure that the generated power is distributed evenly throughout various DG units. This is accomplished mostly through an accurate adjustment of the droop characteristics, taking system stability into consideration. In the following section, a variety of essential components required for microgrid design and operation, including the controlling method of each component, are described and modeled. Simulation results are presented and discussed in Section 5.
# Efficient Knowledge Distillation for Brain Tumor Segmentation
## 2 Related Work
### Knowledge Distillation  
Knowledge distillation is a method that improves the performance of lightweight models by transferring knowledge from powerful, but heavy, networks without losing efficiency. [PERSON] et al. formally defined distillation and proposed corresponding training methods in 2014. After that, many distillation strategies were developed, which were roughly classified into three categories [28]. The first was logits-based knowledge, which extracted knowledge from the output layer of the teacher network. [PERSON] et al. [29] broke this predefined \"strong and weak relationship\" and proposed deep mutual learning (DML), which allowed a group of student networks to learn from each other in the training process, rather than a static predefined one-way transition path between teachers and students. In [30], the author observed that when the teacher network segmentation performance was very great, the distilled student network segmentation performance might not be very great, presumably due to capacity mismatch, which caused the student not to imitate the teacher and instead brought out the main loss. The author proposed an early-stop teacher regularization for distillation, which should be stopped early, when near convergence. Next was feature-based knowledge, which extracted knowledge from the middle-hidden layer of the teacher network. Compared with the first category, it could better learn deeper knowledge. [PERSON] et al. [31] included pairwise distillation techniques and GAN to extract knowledge. [PERSON] et al. [32] proposed the idea of computing the interclass contrast between different tissue regions under the guidance of the marker segmentation template. [PERSON] et al. [33] guided students' attentional maps to learn those of the teachers so that the students' attentional maps were similar to those of the teachers. Finally, relation-based knowledge, which learnt the relationship between layers to acquire knowledge, argued that learning the relationship between layers was similar to learning how to solve a problem. [PERSON] et al. [34] did not fit teacher output but fitted the relationship between the layers of the teacher model. The relationship was defined by the inner product (dot product) between layers. [PERSON] et al. [35] proposed a preserved similarity loss to motivate the student to mimic the teacher regarding the relational expressions within the data. Our proposed approach distilled knowledge from both the output and the intermediate hidden layers because it was easier to learn the answers directly than to learn the solution ideas.
## 1 Introduction
So far, SrF\({}_{2}\) nanoparticles have been synthesized in B\({}_{2}\)O\({}_{3}\)-Bi\({}_{2}\)O\({}_{3}\)-Eu\({}^{3+}\) glass using the heat treatment method, leading to enhanced luminescence of Eu\({}^{3+}\) ions [19]. The enhancement of rare-earth ion luminescence was also achieved via modification of the basic borate-bismuth glass with AlF\({}_{3}\) while maintaining the transparency of the glass [20; 21]. However, doping with Ag NPs has been successfully employed in fluoroborate B\({}_{2}\)O\({}_{3}\)-CaF\({}_{2}\) glasses with Eu\({}^{3+}\) ions [22]. An enhanced luminescence caused by plasmon resonance was then observed. Silver nanoparticles were also successfully synthesized using the heat treatment method in sodium borate glasses doped with Er\({}^{3+}\) ions [23]. The presence of Ag nanoparticles resulted in a significant enhancement of photoluminescence properties owing to the local field generated by Ag NPs. The suitability of borate glasses as a matrix for Ag nanoparticles was further confirmed by the results obtained for alkali borate-based glasses embedded with Nd\({}^{3+}\) and silver nanoparticles [24]. The photoluminescence intensity spectra were enhanced due to the presence of silver nanoparticles. The aim of this study was to investigate the possibility of Ag nanoparticle crystallization in B\({}_{2}\)O\({}_{3}\)-Bi\({}_{2}\)O\({}_{3}\) glass using a heat treatment method and to investigate the possible influence of the obtained nanoparticles on the emission intensity of Eu\({}^{3+}\) ions. Studies on the synthesis of Ag nanoparticles refer to matrices where B\({}_{2}\)O\({}_{3}\) is either one of the components in multicomponent glass or is associated with Ge\({}_{2}\)O\({}_{3}\) or Si\({}_{2}\)O\({}_{3}\). In this work, bismuth oxide was used as a component of borate glass. Due to the existence of eutectic between Bi\({}_{2}\)O\({}_{3}\) and Ag, it should have a positive effect on the formation of nanoparticles in the matrix. To the best of our knowledge, there is no information on the synthesis of Ag nanoparticles in two-component B\({}_{2}\)O\({}_{3}\)-Bi\({}_{2}\)O\({}_{3}\) glass. Due to the interesting properties of the proposed glass ceramics, borate-bismuth glasses with Ag nanoparticles are promising materials for Re ion luminescence enhancement and therefore for optical applications of such a material.
## 3 Signal Processing While Drilling
### Signal Filtering and Denoising
To further optimize the noise reduction effect, the noise reduction performance of a variety of wavelet basis functions under different decomposition levels is compared, and the optimal combination of wavelet basis functions and decomposition levels is determined, which makes the noise reduction effect of the vibration signal reach the best level, effectively improves the signal quality, and provides a more reliable data basis for subsequent feature extraction and model training. Multi-level wavelet coefficient decomposition is shown in Figure 4. Following the preprocessing of the original vibration signal, the noise-reduced signal is compared and analyzed. The changes before and after noise reduction are depicted in Figure 5.
## 2 Materials and Methods
### SPMESH Sheet-DART-MS  
The general approach for SPMESH sheet extraction from a multi-well plate followed by DART-MS analyses was adapted from a previous report (Figure 1) [18]. Briefly, the SPMESH sheet was baked-out at 250 \({}^{\circ}\)C for 15 min in a GC oven immediately prior to extraction. Samples and/or calibration standards were loaded into a 24-well plate. A Teflon gasket was laid over the plate, followed by a stainless-steel spacer, the SPMESH sheet, a second stainless steel spacer and finally, a cover. The assembly was then clamped tightly and incubated at 50 \({}^{\circ}\)C for 30 min, with agitation (230 RPM). After extraction, spacers, gaskets, well plates and clamps were rinsed with ethanol and allowed to dry prior to the next use.  
The SPMESH sheet was then removed from the assembly and transferred to a 3 DS-XZ automated positioning stage in front of a DART standardized voltage and pressure (SVP) ion source (Ionsense Inc.--Saugus, MA, USA). The DART source was coupled to a Q Exactive mass spectrometer (Thermo Fisher Scientific--Waltham, MA, USA) via a Vapur(tm) interface (Ionsense, Inc.). The DART source was operated in transmission mode and movement of the positioning stage in front of the MS  
Figure 1: Schematic overview of sample preparation, SPMESH (solid phase mesh enhanced sorption from headspace) extraction and DART (Direct Analysis in Real Time)-Orbitrap-MS analysis.  
inlet was controlled by the DART web-based software. The DART desorption temperature was set to 450 \({}^{\circ}\)C, with a scan speed of 0.5 mm/s, based on previous work [18].  
Both the DART and MS were operated in positive ion mode with high purity He as the carrier gas. The Orbitrap was operated with 140.000 scan resolution, 1 microscan, 250 ms maximum injection time and automatic gain control (AGC) of \(1\times 10^{6}\) IBMP and \(d_{3}\)-IBMP were measured by MS-MS using the parallel reaction monitoring (PRM) mode at 5 ppm mass tolerance: 167 \(>\) 124.0632 _m/z_ for IBMP, 170 \(>\) 127.0818 _m/z_ for \(d_{3}\)-IBMP, over a mass range 124-128 _m/z_. Transitions were identified previously [20], but exact mass was confirmed on the Orbitrap. Collision energy (CE) was set at 25 eV. Linalool and \(d_{3}\)-linalool were measured using extracted ions from full scans over a mass range of 137-141 _m/z_ at 5 ppm mass tolerance: 137.1326 _m/z_ for linalool, 140.1512 _m/z_ for \(d_{3}\)-linalool. For MS/MS, the following transitions were measured: 137 \(>\) 67.0549 _m/z_ for linalool and 140 \(>\) 70.0736 _m/z_ for \(d_{3}\)-linalool over a mass range 67-71 _m/z_; transitions were identified by dropping standard solutions with a Pasteur pipette in front of the DART source while running a product ion scan on the Orbitrap.
## Abbreviations  
The following abbreviations are used in this manuscript:  
\begin{tabular}{l l} DPD & Direct Position Determination \\ DOA & Direction of Arrival \\ NC & Non-circular \\ SNR & Signal-to-noise ratio \\ ESPRT & Estimating Signal Parameters Viarotational Invariance Techniques \\ SDF & Subspace Data Fusion \\ ML & Maximum Likelihood \\ QPSK & Quadrature Phase Shift Keying \\ AM & Amplitude Modulation \\ CRLB & Cramer Rao lower bound \\ ULA & Uniform Linear Array \\ EVD & Eigenvalue Decomposition \\ RMSE & Root Mean Square Error \\ DOF & Degree of Freedom \\ \end{tabular}
## 4 Uniform Load Test Results and Discussion
### Influence of Interlayer Thickness on Coupling Stress  
When comparing the maximum coupling stress of specimens in Table 3 with the same cold-bending curvatures but different interlayer thicknesses it was found that the maximum coupling stress increased with greater interlayer thicknesses. This phenomenon occurs because greater interlayer thicknesses can enhance the cold-bending stress and the coupling stress. When the interlayer thickness increased from 1.14 mm to 1.52 mm, the maximum coupling stress of the glass panels increased by 3.46% on average. Therefore, interlayer thickness is a less relevant factor affecting the coupling stress as compared to the cold-bending curvature.  
Moreover, it can be seen from Figure 10 that the two relation curves were almost parallel, and the curve with larger interlayer thickness was located above the other one with smaller interlayer thickness, suggesting that the glass panel with a large interlayer thickness has greater coupling stress. Besides, the glass is a linear elastic material, and the influence of interlayer material on the coupling stress was linear since the relationship between the coupling stress and uniform load was revealed as linear.  
Figure 10: Maximum coupling stress versus cold-bending curvature curves.  
Figure 9: Relationship between coupling stress and uniform load. (a) S1 to S4; (b) S5 to S8.
# 6G Network Architecture Using FSO-PDM/PV-OCDMA System with Weather Performance Analysis
## 4 Performance Analysis  
The received current of the desired channel at PD is expressed as [25]  
\[I_{Ch}=\frac{Rr_{P}C_{W}}{C_{L}} \tag{16}\]  
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Channels/Wavelengths** & **1550** & **1550.8** & **1551.6** & **1552.4** & **1553.2** & **1554** & **1554.8** & **1555.6** \\ \hline Channels 1, 4 & 0 & 0 & \(\lambda_{3}\) & 0 & 0 & 0 & \(\lambda_{7}\) & 0 \\ Channels 2, 5 & 0 & 0 & 0 & 0 & \(\lambda_{5}\) & \(\lambda_{6}\) & 0 & 0 \\ Channels 3, 6 &where \(R\) indicates the responsivity of PD and \(r_{P}\) is the received power which is given as [16]  
\[r_{P}=t_{P}\bigg{(}\frac{d_{r}}{d_{t}+P_{r}}\bigg{)}^{2}10^{\frac{-aP_{r}}{10}} \tag{17}\]  
where \(t_{P}\) is the transmitted power, \(d_{t}\) and \(d_{r}\), respectively are transmitter and receiver aperture diameters, represents the beam divergence angle of the laser, and \(P_{r}\) looks for propagation range. The signal-to-noise ratio, \(S/N\), is [25]  
\[S/N=\frac{\left(I_{Ch}\right)^{2}}{\sigma_{SN}+\sigma_{Th}} \tag{18}\]  
where \(\sigma_{SN}\) and \(\sigma_{Th}\), respectively are the powers of shot noise and thermal noise. \(\sigma_{SN}\) is given as [25]  
\[\sigma_{SN}=2 eB_{electric}(I_{Ch}) \tag{19}\]  
where \(e\) and \(B_{electric}\) are electron charge and electrical bandwidth, respectively.  
While the thermal noise, \(\sigma_{Th}\), is given as [26]  
\[\sigma_{Th}=\frac{4k_{B}TB_{electric}}{R_{L}} \tag{20}\]  
Here, \(k_{B}\) is Boltzmann constant, \(R_{L}\) and \(T\) are the load resistance of receiver and absolute temperature of receiver noise, respectively.  
Therefore, the \(S/N\) will be  
\[S/N=\frac{\left(I_{Ch}\right)^{2}}{2 eB_{electric}(I_{Ch})+\frac{4k_{B}TB_{ electric}}{R_{L}}} \tag{21}\]  
Finally, \(BER\) in terms of \(S/N\) will be [25]  
\[BER=\frac{1}{2}erfc\bigg{(}\sqrt{\frac{S/N}{8}}\bigg{)} \tag{22}\]  
While \(BER\) in terms of Q-factor can be written as [27]  
\[BER=\frac{1}{2}erfc\bigg{(}\frac{Q}{\sqrt{2}}\bigg{)} \tag{23}\]
# Ad Creative Discontinuation Prediction with Multi-Modal Multi-Task Neural Survival Networks
## 4 Methodology
### Learning the Hazard Function for Ad Creative Discontinuation  
We describe the problem settings of the hazard function in discrete-time survival prediction and how it can be learned for ad discontinuation. The hazard function can model the main variable of interest as the time to an event. To predict the timing of discontinuation, we are interested in discrete problems because currently discontinuation is done manually in real-world scenarios.  
In discrete-time survival prediction, the follow-up time is broken up into time intervals, which are often right-closed and left-open. Let \(t^{\prime}\) denote the time at which the event occurs, \(l=1,2,\cdots,L\) index the time intervals, and \(t_{l-1}\) and \(t_{l}\) denote the lower and upper bounds of the \(l\)-th time interval, respectively. The hazard probability within the \(l\)-th time interval is defined as follows:  
\[h_{l}=Pr(t^{\prime}\in(t_{l-1},t_{l}]|t^{\prime}>t_{l-1}). \tag{3}\]  
Here, the time interval is the same setting defined in Section 4.1.  
The hazard probability can be modeled as a function of covariates to assess the effect of covariates on the time to the event of interest. It is our belief that this property is appropriate for the discontinuation of ad creatives. The hazard probability of the observation surviving the \(l\)-th time interval \(h_{l}\) is equal to the \(l\)-th output of the hazard function, which is equivalent to the following:  
\[h_{l}=\sigma(f_{l}(t_{l}|\mathbf{x})). \tag{4}\]  
where \(f_{l}\) is the \(l\)-th output of hazard function \(f(\cdot|\cdot)\).  
The parameters are estimated through the maximization of the likelihood function for the discrete survival prediction strategy [28]. The likelihood is written as follows:  
\[\mathcal{L}(\mathcal{T},\mathbf{x})=\prod_{l=1}^{t^{\prime}}h_{l}^{\delta_{l}}(1- h_{l})^{(1-\delta_{l})}. \tag{5}\]  
where \(l^{\prime}\) is the number of time intervals that was observed, and \(\delta_{l}\) is the event indicator within the \(l\)-th time interval, which is equal to 1 if the event occurs within that interval, and 0 otherwise. To maximize the likelihood function, we minimized the following negative log-likelihood:  
\[\ell(\mathcal{T},\mathbf{x})=\sum_{l=1}^{t^{\prime}}\delta_{l}\log h_{l}+(1- \delta_{l})\log(1-h_{l}). \tag{6}\]  
The full log-likelihood is the sum of the log-likelihood for each data and we minimize the log-likelihood by mini-batch gradient decent.
## 5 Numerical Results  
The accuracy of the proposed model is evaluated by considering the slender three-dimensional solid shown in Figure 2 with length \(L=2\) cm, cross section \(A=H\times B\), \(H=B=L/10\). The element is clamped at one end and subjected to load control at the free end. An interface at \(L/2\) is placed. Meshes of hexahedra are considered consisting of at least \(5\times 5\) elements in the cross section and increasing number of elements along the loading direction (see Figure 3). Enriched hexahedra possess a quadrature grid of \(8^{3}\) Gauss points, while non-enriched hexahedra contain \(2^{3}\) Gauss points. The analytical solution predicts a uniform state of stress along the loading direction. The uniform norm of the error, namely the maximal error on the stress component \(\sigma_{xx}\)  
\[\mathcal{E}=\frac{\|\sigma_{xx}-\sigma_{xx}^{h}\|_{\infty}}{\sigma_{xx}} \tag{34}\]  
is evaluated [23]. The parameter \(\beta\) characterizing the ratio between the Young's modulus of the bulk and that of the interface is set equal to \(10^{-4}\).  
Tables 3 and 4 illustrate the errors obtained in the case of vertical interface with \(\alpha=0^{\circ}\) for fixed \(\rho\) and fixed \(h\), respectively. In both cases, the support width \(\ell_{\rho}=20\rho\) was adopted. The ratio \(\rho/h\) is incremented, in the former case by decreasing \(h\), and in the latter case by increasing \(\rho\). Figure 4 displays the error in percentage for the case of \(\rho=L/100\), \(\rho/h=5\) and the support w  
Figure 3: Typical mesh.  
Figure 2: Geometry of the uniaxial test.  
\begin{table}
\begin{tabular}{c c} \hline \hline \(\boldsymbol{\rho/h}\) & \(\boldsymbol{\mathcal{E}}\) \\ \hline \(4.9\times 10^{-2}\) & \(4.0\times 10^{-1}\) \\ \(1.5\times 10^{-1}\) & \(8.7\times 10^{-2}\) \\ \(3.5\times 10^{-1}\) & \(2.8\times 10^{-2}\) \\ \(5.4\times 10^{-1}\) & \(1.7\times 10^{-2}\) \\ \(7.4\times 10^{-1}\) & \(1.2\times 10^{-2}\) \\ \(1\) & \(8.4\times 10^{-3}\) \\ \(2.5\) & \(3.5\times 10^{-3}\) \\ \(4.8\) & \(1.8\times 10^{-3}\) \\ \(7.8\) & \(1.2\times 10^{-3}\) \\ \(17.3\) & \(5.5\times 10^{-4}\) \\ \(37\) & \(2.9\times 10^{-4}\) \\ \hline \hline \end{tabular}
\end{table}
Table 4: Error for increasing \(\rho\), fixed \(h=L/749\) corresponding to the \(749\times 5\times 5\) mesh, inclination of the vertical discontinuity \(\alpha=0^{\circ}\); the support width is \(\ell_{\rho}=20\rho\).  
We have obtained a level-set dependent delta function approximation \(\delta_{\rho}\). The function \(\delta_{\rho}\) satisfies three moment conditions up to a precision \(C\) when \(\hat{\varphi}(\kappa)\leq C\) for any \(\kappa\geq C\). This circumstance occurs if the ratio \(\rho/h\) is greater than \(20\). Moreover, the width of the support width \(\ell_{\rho}\) should be chosen at least equal to \(40\rho\) in order to contain the truncation error. Hence, we can conclude that, generally, the support width should be larger than \(40\rho\).