esa-sceva/satcom-chunk-collection huggingface dataset:
## Materials and Methods
### Preference Test  
The preference test of toner base was conducted in 10 Thai volunteers, including 5 males and 5 female volunteers, aged 20-45 years old, by using a questionnaire with hedonic system scoring from 1 to 5 (dislike to prefer the most). The score was calculated as a percentage [26]. The stable toner base with the highest preference score was then used for the development of emblica toner.  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Ingredient**} & \multicolumn{4}{c}{**Toner Base (\%, \(w/w\))**} & \multicolumn{4}{c}{**Emblica Toner (\%, \(w/w\))**} \\ \cline{2-7}  & **A** & **B** & **C** & **B1** & **B2** & **B3** \\ \hline Glycerin (humectant) & 1 & 2 & 4 & 2 & 2 & 2 \\ Alantoin (anti-irritant), & & 3.1 & 3.1 & 3.1 & 3.1 & 3.1 & 3.1 \\ Trimethylglycine (anti-inflammatory), and & & & & & & \\ Glidant (preservative) & & & & & & \\ DI water (solvent) & 95.9 & 94.9 & 92.9 & 93.9 & 92.9 & 91.9 \\ Triethanolamine (pH adjuster) & q.s. & q.s. & q.s. & q.s. & q.s. & q.s. \\ Emblica extract (active substance) & - & - & - & 1 & 2 & 3 \\
**pH** & & & & & & \\ Intial & 6.02 \(\pm\) 0.03 & 6.00 \(\pm\) 0.02 & 6.01 \(\pm\) 0.02 & 5.68 \(\pm\) 0.02 & 5.63 \(\pm\) 0.02 & 5.59 \(\pm\) 0.03 \\ Heating and Cooling & 5.95 \(\pm\) 0.03 & 5.93 \(\pm\) 0.02 & 5.87 \(\pm\) 0.03 & 5.63 \(\pm\) 0.02 & 5.56 \(\pm\) 0.03 & 5.51 \(\pm\) 0.03 \\
**Preference (\%)** & & & & & & \\ Spreadability & 80 \(\pm\) 3.0 \({}^{\mathrm{a}}\) & 86 \(\pm\) 3.1 \({}^{\mathrm{a}}\) & 80 \(\pm\) 3.0 \({}^{\mathrm{a}}\) & & \\ Absorption & 86 \(\pm\) 3.1 \({}^{\mathrm{b}}\) & 90 \(\pm\) 4.5 \({}^{\mathrm{b}}\) & 70 \(\pm\) 4.5 \({}^{\mathrm{c}}\) & & \\ Greasiness & 84 \(\pm\) 4.0 \({}^{\mathrm{d}}\) & 88 \(\pm\) 3.3 \({}^{\mathrm{d}}\) & 72 \(\pm\) 3.3 \({}^{\mathrm{a}}\) & NA & NA & NA \\ Color & 82 \(\pm\) 3.6 \({}^{\mathrm{f}}\) & 86 \(\pm\) 3.1 \({}^{\mathrm{f}}\) & 78 \(\pm\) 3.6 \({}^{\mathrm{f}}\) & & \\ Odor & 70 \(\pm\) 3.3 \({}^{\mathrm{s}}\) & 78 \(\pm\) 5.5 \({}^{\mathrm{s}}\) & 68 \(\pm\) 3.3 \({}^{\mathrm{s}}\) & & \\ Overall preference (\%) & 80.4 \(\pm\) 1.7 \({}^{\mathrm{h}}\) & 85.6 \(\pm\) 1.8 \({}^{\mathrm{h}}\) & 73.6 \(\pm\) 1.7 \({}^{\mathrm{i}}\) & & \\ \hline \hline \end{tabular} Different letters in the same row indicate statistically significant differences, (\(p<0.05\)); NA: not available.  
\end{table}
Table 1: Development of the stable emblica toners.
# Rainy Environment Identification Based on Channel State Information for Autonomous Vehicles
## 1 Introduction
We use the Channel State Information (CSI), estimated from Cooperative Awareness Messages (CAM) exchanged in vehicle-to-vehicle (V2V) communication, as input features, which are then fed into the proposed Convolutional Neural Network (CNN) model. This enables the model to reliably identify the surrounding environment of the vehicle based on the channel characteristics of different rainy vehicular environments. The rest of this paper is organized as follows: Section 2 introduces related research. Section 3 describes our wireless communication vehicular network. Section 4 provides our rainy environment identification method for autonomous vehicles. Section 5 describes the performance evaluation of the proposed method. Finally, Section 6 presents the conclusions. Figure 1: Vehicular environment identification process.
# Targeted Sentiment Classification Based on Attentional Encoding and Graph Convolutional Networks
## 5 Conclusions and Future Work  
In this paper, a model based on attention encoding and a graph convolution network is proposed for targeted sentiment classification. In order to solve the problems of losing long-distance emotional words in semantic modeling and parallel computing for input data, this paper proposes a semantic encoding method combining bidirectional LSTM with a multi-head self-attention mechanism. To makeuse of the syntactic information that most models ignore, we developed a graph convolution neural network that integrated the point-wise convolution and builds the graph convolution network over the syntax dependency tree to encode the syntactic information. It is then interacted with the semantic information through multi-head interactive attention. The experimental results on five datasets--twitter, lap14, rest14, rest15, and rest16--show that the AEGCN model proposed in this paper was significantly better than the models based on traditional machine learning and other deep learning models, proving its effectiveness.  
Although the current model achieved good experimental results, there is still a great deal of work to be done. In our future work, we intend to reduce the training parameters of the model to make our model more lightweight. Second, extracting more original contextual semantic information will also be an important part of our future work. Finally, combining domain knowledge with syntactic information can be taken into consideration in future research.  
conceptualization, L.X. and X.H.; methodology, Y.X. and X.H.; investigation, L.X.; formal analysis, Y.X.; software, L.X. and B.C.; validation, D.G. and B.C.; writing, X.H., Y.C. and L.X.; resources, T.Z.; review, X.H., Y.C. and L.X. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the China Scholarship Council, Guangdong Science and Technology Department under Grant no. 2016A010101020, 2016A010101021, 2016A010101022, Guangzhou Science and Information Bureau under Grant no. 201802010033.  
The authors declare no conflict of interest.
## 1 Introduction
### Related Research
From the literature coverage, one can deduce that:  
* the influence of the information transmission between the agents has been identified as a crucial factor in performance and reliability of the control design;
* the presence of parameters' uncertainties and external disturbances may critically influence the performances of the prior;
* application of the DDS middleware technique augmented with the flocking approach has not been applied for heterogeneous systems, which is closer to the real-word robotic applications. This paper proposes an approach that can possibly address all of the above points. Indeed, navigation and control for formation and containment maintenance of a fleet of multiagent heterogeneous systems composed of AUVs and UAVs is considered. A distributed method has been developed using an L1 adaptive control technique taking into account actuator time delay to stabilize the agents and make them more robust towards parameters change. The flocking approach has been used for the navigation, and the DDS middleware has been utilized for the data transmission between the agents. The navigation ensures formation as well as containment control in the leader-follower context. Potential field-like methodology has been adopted to avoid obstacle while maintaining the formation and containment. The main contributions of this research are highlighted as follows:  
* A robust architecture for the formation control design of heterogeneous systems has been developed. * A robust navigation system for the containment control design of heterogeneous systems has been proposed and designed. * The navigation algorithm avoids obstacles while still maintaining the required formation or containment. * Reliable middleware for data transmission has been developed. The remainder of this manuscript is organized as follows. In Section 2, the preliminaries of DDS middleware, the dynamics of the quadrotor, and the dynamics of the underwater system are presented. In Section 3, the multi UAVs formation stability has been proved using the L1 technique with an actuator time delay and Lyapunov analysis. The simulation results are presented and discussed in Section 4. Finally, the findings and the concluding remarks have been highlighted at the end.
## 3 Results
### Mitigation of Odor and Targeted Gases as a Function of UV-A Dose
#### 3.1.2 Volatile Organic Compounds
Control temperature = 28.5 \(\pm\) 2.3 \({}^{\circ}\)C, control RH = 69.8 \(\pm\) 9.5%, treatment temperature: 31.5 \(\pm\) 1.2 \({}^{\circ}\)C, treatment RH = 66.0 \(\pm\) 4.3%, Values in the table are mean \(\pm\) S.D. Bold signifies statistical significance. \begin{table}
\begin{tabular}{c c c c c c} \hline
**UV Dose** & **Light Intensity** & **Treatment** & **Control** & **Treatment** & **\% Reduction** \\
**(m)**\(\cdot\)**cm\({}^{-2}\))** & **(mW\(\cdot\)**cm\({}^{-2}\))** & **Time (s)** & **(ppm)** & **(ppm)** & **(\(p\)-Value)** \\ \hline \multicolumn{5}{c}{**UV Dose Control with UV Light Intensity and Treatment Time**} \\ \hline
1.9 & 0.04 and 0.41 & 15.8 and 3.2 & 896 \(\pm\) 81 & 1016 \(\pm\) 108 & \(-\)13.5 (0.33) \\
2.9 & 0.04 and 0.41 & 23.8 and 4.8 & 737 \(\pm\) 12 & 892 \(\pm\) 18 & \(-\)**21.2 (0.01)** \\
2.9 & 0.04 and 0.14 & 43.3 and 8.7 & 1032 \(\pm\) 21 & 1380 \(\pm\) 104 & \(-\)**33.7 (0.03)** \\
4.0 & 0.04 and 0.26 & 43.3 and 8.7 & 1032 \(\pm\) 21 & 1251 \(\pm\) 40 & \(-\)**21.2 (0.02)** \\
5.3 & 0.04 and 0.41 & 43.3 and 8.7 & 1201 \(\pm\) 296 & 1534 \(\pm\) 105 & \(-\)27.8 (0.16) \\ \hline \end{tabular}
\end{table} 
Table 5: Mitigation of H\({}_{2}\)S with UV-A photocatalysis treatment. Control temperature = 28.5 \(\pm\) 2.3 \({}^{\circ}\)C, control RH = 69.8 \(\pm\) 9.5%, treatment temperature: 31.5 \(\pm\) 1.2 \({}^{\circ}\)C, treatment RH = 66.0 \(\pm\) 4.3%, Values in the table are mean \(\pm\) S.D. Bold signifies statistical significance. The mitigation of odorous VOCs was consistent with the results presented for odor (Table 1). A statistically significant odor reduction was found for higher UV doses in which several targeted VOCs were reduced, e.g., the phenolic compounds. It is important to highlight the generation of some targeted compounds for all UV doses. Generated compounds (several in the VFAs group, DMDS and phenol) are odorants that are considered slightly less impactful than _p_-cresol, skatole and indole [19]. Thus, it is feasible to hypothesize that the generated compounds offset the overall odor's mitigation (Table 1).
# Direct Multistep Wind Speed Forecasting Using LSTM Neural Network Combining EEMD and Fuzzy Entropy
## 5 Case Study
### Classification of Components Based on FuzzyEn  
The original wind speed series of the three cases are all decomposed into 13 components by EEMD. Then, the FuzzyEn values of each component were calculated respectively, and the results are shown in Figure 7.  
In the three cases, the FuzzyEn values of the 1 st component to the 13 th component gradually decrease firstly, and then become smooth. The FuzzyEn of the high-frequency component is large, which means it is more complicated. On the contrary, the FuzzyEn of the low-frequency component is small, indicating that its complexity is small. The components with similar FuzzyEn values can be classified into one group. In Figure 7, the FuzzyEn values of the components at the front descending curve segment are quite different, so each component can be regarded as a subsequence. The FuzzyEn values of the 7 th component to 13 th component at the final stationary curve segment are almost the same, so they can be grouped together and superimposed to form a new subsequence. After the classification was completed, 7 subsequences were finally obtained.
# Dual-Band MIMO Antenna with Enhanced Isolation for 5G NR Application
## 2 Antenna Design and Analysis
### Parametric Study  
The radiators can generate two resonances. The incomplete circular radiation patch generates the lower resonance of the antenna, and the L-shaped antenna branches can generate the higher resonance. Figure 4 describes the two resonance frequencies, which can be independently tuned by changing the values of R\({}_{2}\) and H\({}_{1}\). When the value of R\({}_{2}\) increases, the lower resonance shifts to the lower frequencies, as is shown in Figure 4a. When the  
Figure 3: Design stage responses of the intended two-port MIMO antenna: (**a**) S\({}_{11}\) responses and (**b**) S\({}_{21}\) responses.  
Figure 2: Design stages of the proposed design.  
value of H\({}_{1}\) increases, the ground current path increases, the lower resonance frequency remains at 2.5 GHz, the higher resonance frequency shifts to the lower frequencies, and the \(-10\) dB bandwidth of the higher frequency band is reduced. The optimum values of R\({}_{2}\) and H\({}_{1}\) are 9 mm and 2.5 mm, respectively.
## 2 Materials and Methods
### Analysis of the Model Structure  
The six big-leaf LUE-GPP models described in this paper consider the temperature and moisture constraints on \(GPP\) estimation. Two comparisons were additionally conducted to explore the effect of a single environmental limitation factor and their combination on estimating \(GPP\). The model comparison schemes are described in Table 2.  
Additionally, two simple functions (power and exponential) were used to construct the dynamic \(LUE\) model by introducing \(K_{\rm t}\) to improve the simulation accuracy under different weather conditions. For the optimized big-leaf LUE-GPP (OBL) model, the constructed \(K_{\rm t}\)-based dynamic \(LUE\) is expressed as below:  
Power function-based optimized model:  
\[GPP_{OBL-P}=PAR\times fPAR\times A_{1}K_{\rm f}{}^{B_{1}}\times T_{\rm s}\times W _{\rm s}\]  
Exponential function-based optimized model:  
\[GPP_{OBL-E}=PAR\times fPAR\times A_{1}\exp(B_{1}K_{\rm t})\times T_{\rm s} \times W_{\rm s}\]  
where \(A_{1}\) represents the dynamic \(LUE_{\rm max}\) based on \(K_{\rm t}\) and \(B_{1}\) represents the adjustment parameter related to \(K_{\rm t}\).  
For the two-leaf LUE-GPP model (TL), the constructed \(K_{\rm t}\)-based dynamic \(LUE\) is expressed as follows:  
Power function based optimized model:  
\[GPP_{TL-P}=(A_{1}K_{\rm t}{}^{B_{1}}APAR_{sh}+A_{2}K_{\rm t}{}^{B_{2}}APAR_{ su})\times T_{\rm s}\times W_{\rm s}\]  
Exponential function based optimized model:  
\[GPP_{TL-E}=(A_{1}exp(B_{1}K_{\rm t})APAR_{sh}+A_{2}exp(B_{2}K_{\rm t})APAR_{ su})\times T_{\rm s}\times W_{\rm s}\]  
where \(A_{1}\) and \(A_{2}\) represent the \(K_{\rm t}\)-based dynamic \(LUE_{\rm su-max}\) and \(LUE_{\rm sh-max}\) values, respectively, set at a range of 0 and 4, and \(B_{1}\) and \(B_{2}\) represent the \(K_{\rm t}\)-related adjustment parameters for shaded and sunlit leaves, respectively, set at a range of \(-3\) to 3. The parameters \(A_{1}\), \(A_{2}\), \(B_{1}\), and \(B_{2}\) in the optimized LUE-GPP models were determined by least-square nonlinear fitting (Table 3).  
\begin{table}
\begin{tabular}{l l} \hline \hline
**Model Description** & **Model Structure** \\ \hline Estimated \(GPP\) without any environmental constraints & \(GPP_{\rm Nos}=PAR\times fPAR\times LUE_{\rm max}\) \\ Estimated \(GPP\) with \(T_{\rm s}\) only & \(GPP_{\rm Ts}=PAR\times fPAR\times LUE_{\rm max}\times T_{\rm s}\) \\ Estimated \(GPP\) with \(W_{\rm s}\) only & \(GPP_{\rm Ps}=PAR\times fPAR\times LUE_{\rm max}\times W_{\rm s}\) \\ Estimated \(GPP\) with the maximum limitation combination & \(GPP_{\rm min(T_{\rm s},W_{\rm s})}=PAR\times fPAR\times LUE_{\rm max}\times \min(T_{\rm s},W_{\rm s})\) \\ Estimated \(GPP\) with common limitation combination of \(T_{\rm s}\) and \(W_{\rm s}\) & \(GPP_{\rm(T_{\rm s}\times W_{\rm s})}=PAR\times fPAR\times LUE_{\rm max}\times T _{\rm s}\times W_{\rm s}\) \\ \hline \hline \end{tabular}
\end{table}
Table 2: Model structures of big-leaf light use efficiency-based GPP models considering single limitation factor (temperature or moisture) and their combination form.
## 3 Results
### Dimensionality Reduction  
Dimensionality reduction is performed using a wavelet decomposition of the kinematic data in each plane separately, namely the flexion/extension angle, with respect to the sagittal plane (Figure 4a), the abduction/adduction angle, with respect to the frontal plane (Figure 4b), and the internal/external angle, with respect to the transverse plane (Figure 4c). The dimension of the data before feature extraction is 100, corresponding to the percentage of gait cycle (1% to 100%), for each of the three knee rotation angles (Figure 4, Line 1).  
Figure 4: Wavelet decomposition using Daubechies db1 of the (**a**) the flexion/extension angle, with respect to the sagittal plane, (**b**) the abduction/adduction angle, with respect to the frontal plane, (**c**) and internal/external angle, with respect to the transverse plane. Each line corresponds to a decomposition level and each column to a kinematic plane.  
Using the wavelet decomposition for dimensionality reduction, the dimension has been reduced to a fewer number of most relevant coefficients. We experimented with different wavelet families, namely Daubechies, Coiflet, and Symlet, and different levels of decomposition. The level of a wavelet representation, as well as the relevant planes of data, are chosen experimentally so as to maximize the recognition rate of the sample-encoding Kononen network. Following extensive testing, we were able to retain a subset of four coefficients of the Daubechies DB1 wavelet representation at level 3, which initially contained 13 coefficients (Figure 4, Line 4).
# A Novel Context-Aware Reliable Routing Protocol and SVM Implementation in Vehicular Area Networks
## 4 Mathematical Analysis for Proposed Protocol
Determine the routes with low PDR, high E2E delay, and low throughput, and also determine the corresponding nodes which occur frequently in these routes. 10. After determining the nodes which occur frequently in the non-optimal routes, the proposed approach eliminates the routes which consist of nodes from the BAD cluster and shifts the load of the malicious nodes to its nearby node to maintain reliability. ```
Input Parameters: (a) Set of Routes \(R=\{r_{1},r_{2},r_{3},\dots,r_{N}\}\) (b) Initial number of clusters \(K\) (c) Direction, Velocity, and Location of each Vehicle node Output Parameters: (a) Optimal clusters: GOOD and BAD 1. Randomly initialize \(K\) centroids in space 2. For\(i=1\) to \(N\)do
3. Calculate the cluster membership function \(l(x_{i},j)\) 4. Assign routes to convenient clusters according to \(l(x_{i},j)\)
5. End of 6. If all routes are assigned to a cluster, then
6. End of the algorithm
7. Else
9. \(K=K+1\)
10. End if
11. If MSE of Cluster = Low, then
12. Cluster = GOOD
13. Else
14. Cluster = BAD
15. End if
```  
**Algorithm 1**K-means Clustering-based VANET Routing
# Evolution of V2X Communication and Integration of Blockchain for Security Enhancements
## 7 Open Issues and Future Research Directions
### Quality of Service (QoS) Issue  
The integration of blockchain in 5G-based MEC environments can introduce novel QoS issues, which would adversely affect the overall system performance of V2X networks. The data traffic generated from the integrated technologies needs to be managed efficiently to reduce latency, and to handle network resources in order to achieve good QoS. One of the most critical goals of future C-V2X based blockchain is to provide high QoS and ultra-low latency for vehicles to satisfy the demands from increased traffic and new services. A solution based on 5G network virtualization was studied in [103] in order to solve blockchain scalability and improve system performance by separating blockchain management from transaction processing in order to reduce the latency and enhance the QoS of blockchain operations. Blockchain requires extensive resources in storage, processing, and network bandwidth to accomplish the consensus process. Without proper planning, the integrated blockchain in 5G-based MEC environments will lead to the deterioration of QoS due to high latency, high energy usage, high demands for bandwidth, and dense network congestion.  
Some techniques were presented to solve the QoS issues from two perspectives: (1) lightweight blockchain designs, as discussed in [104; 105; 106; 107; 108] reducing the consensus mechanism computation in blockchain and (2) computation offloading as mentioned in [109; 110; 111] that is based on integrated 5G technologies with edge computing along with SDN, NFV and D2D communications.
# An Unsupervised Image Denoising Method Using a Nonconvex Low-Rank Model with TV Regularization
## 2 Related Work
### TV Norm  
The total variation (TV) model first proposed by [PERSON] et al. measures the structural variations in the gradient domain and serves as a good sparsity regularization for image denoising [43]. This technique can keep the edge information and improve the smoothness of the denoised image. Additionally, it can save significant image information in addition to producing pleasing visual effects, which plays a constructive role in some areas. For instance, in the medical field, reducing noise from clinical computed tomography pictures while maintaining their fundamental diagnostic information is very beneficial in identifying the condition. Previous research suggested that the anisotropic TV methods could produce better denoising results than the isotropic TV method. The anisotropic TV is defined as follows:  
\[\begin{split}\|\mathbf{L}\|_{TV}=&\sum_{i=1}^{m-1} \sum_{j=1}^{n-1}\big{\{}\big{|}\mathbf{L}_{i,j}-\mathbf{L}_{i+1,j}\big{|}+\big{|} \mathbf{L}_{i,j}-\mathbf{L}_{i,j+1}\big{|}\big{\}}\\ &+\sum_{i=1}^{m-1}\big{|}\mathbf{L}_{i,n}-\mathbf{L}_{i+1,n}|+ \sum_{j=1}^{n-1}\big{|}\mathbf{L}_{m,j}-\mathbf{L}_{m,j+1}\big{|}\end{split} \tag{4}\]  
where \(\mathbf{L}_{i,j}\) is a pixel of the matrix \(\mathbf{L}\).
## 3 The Multi-Material Topology Optimization Problem
### Matrix Norms via Singular Values  
With reference to Figure 1, the (parameter-dependent) matrix \(\mathbf{G}(\mathbf{p})\) mapping the \(r\times 1\) input vector \(\mathbf{f}\) to the \(n_{o}\times 1\) output vector \(\mathbf{z}\) is given by  
\[\mathbf{G}(\mathbf{p})=\mathbf{CK}(\mathbf{p})^{-1}\mathbf{B} \tag{17}\]  
where \(\mathbf{B}\) is an \(n\times r\) matrix that distributes the loads to the appropriate degrees of freedom, \(\mathbf{C}\) is an \(n_{o}\times n\) output matrix that extracts the relevant components of the output vector \(\mathbf{z}\), and \(\mathbf{K}(\mathbf{p})\) is the (parameter-dependent) system stiffness matrix, and \(\mathbf{p}\) is a vector of design parameters. For any fixed \(\mathbf{p}\), the corresponding transfer matrix \(\mathbf{G}\) admits a Singular Value Decomposition (SVD) [19], which can be written as:  
\[\mathbf{G}=\mathbf{U}\mathbf{\Sigma}\mathbf{V}^{H}, \tag{18}\]  
where:  
* \(\mathbf{U}\) and \(\mathbf{V}\) are unitary matrices of dimensions \(l\times l\) and \(m\times m\), respectively.
* \(\mathbf{\Sigma}\) is an \(l\times m\) diagonal matrix with the form: \[\mathbf{\Sigma}=\left\{\begin{bmatrix}\mathbf{\Sigma}_{1}\\ \mathbf{0}\\ \mathbf{\Sigma}_{1}\end{bmatrix},\quad\text{if }l\geq m,\right.\] where: \[\mathbf{\Sigma}_{1}=\text{diag}\{\sigma_{1},\sigma_{2},\ldots,\sigma_{r}\},\] (19) with \(r=\min(l,m)\) denoting the rank of \(\mathbf{G}\). The singular values are sorted in descending order, \[\overline{\sigma}\equiv\sigma_{1}\geq\sigma_{2}\geq\cdots\geq\sigma_{r}\equiv \underline{\sigma}>0,\] (20)as indicated in Equation (20), in order to define the matrix norms introduced below, which will be used in the sequel:  
* 2-Norm: \[\|\mathbf{G}\|_{2}=\sigma_{1},\] (21)
* Nuclear Norm: \[\|\mathbf{G}\|_{N}=\sum_{k=1}^{r}\sigma_{k}.\] (22)
* Frobenius Norm: \[\|\mathbf{G}\|_{F}=\sqrt{\sum_{k=1}^{r}\sigma_{k}^{2}}.\] (23)  
The matrix norms introduced above lay the groundwork for the multi-material topology optimization problem formulated in the following section.