esa-sceva/satcom-chunk-collection huggingface dataset:
# Feature of Review Papers in Nanotechnology and Applied Nanosciences
## 2 Contributions  
The first publication in the special issue reviews the structural and physico-chemical properties of nanocellulose-based materials. The authors comment on recent applications of these eco-friendly bio-nanomaterials for the restoration and conservation of cultural heritage, and describe the advantages of nanocellulose for the recovery of degraded wood and the treatment of painting canvases and ancient papers.  
Water electrolysis by proton exchange membranes is a promising route for the production of hydrogen. In the second paper, it is explained that metal-based bipolar plates play an important role in this technology. However, the metal plates must be protected against corrosion through surface modification. In this contribution, different protective coatings are reviewed and compared. The authors demonstrate that hybrid nanoparticle-promoted polymers have several advantages over other coating solutions.  
Carbon dioxide electrocatalytic reduction reactions are investigated using experimental and computational tools in the third paper. The catalysts of interest in this study are transition metal sulfide nanomaterials. The proportion of each product in the reduction reaction, among which are carbon monoxide, methane, ethene, methanol and ethanol, is shown to depend on the particular transition metal the sulfide is made of.  
Pollution of the environment by micro- and nano-plastics is a subject of great concern. The detection of these plastic residues via plasmonic effects is investigated in paper four, where the authors compare the efficiency of different techniques that exploit visible and UV plasmon resonances in noble metal nanoparticles.  
The authors of the fifth paper review different models of perfect absorbers based on graphene, despite the fact that the light absorption coefficient of atomically thin graphene is only around 2%. As emphasized in this review, the application of graphene as a perfect absorber is particularly suited to devices that do not require a back-reflection mirror, due to their ease of fabrication. Patterned substrates, multilayers and metasurfaces offer interesting alternatives.  
The large-scale use of green hydrogen as a fuel, for instance, raises a lot of challenges. Some of its applications are under scrutiny from a nanotechnological point of view. In paper six, the properties of nano-titanium dioxide are reviewed in the context of hydrogen production and storage. In addition to exhibiting photocatalytic activity, the authors emphasize that TiO\({}_{2}\) may be used to enhance the performance of magnesium hydride for solid-state hydrogen storage.  
Aside from catalysis, metal-salen complexes have many promising applications. The seventh paper reports on the use of X-ray spectroscopy (XPS, valence band PES, EXAFS and NEXAFS). Some of the spectra and data presented in the review contribute to elucidating the atomic and electronic structures of salen-based complexes and derivatives, including polymers and their composites with carbon nanostructures, graphene and nanotubes.
## 2 Materials and Methods
### Study Design and Settings
#### 2.1.3 Sample Collection and Storage  
While designing this study, the frequency of sample collection was determined from previous studies [8; 9; 16; 21]. Collecting samples at three-time points was deemed necessary to capture dynamic changes during the study period. As multiple-time points are essential to assess temporal changes in bacterial flora, samples were collected at three different time points (T0, T1, and T2). Saliva and stool samples were collected three times each from every participant. The first collection (T0) of saliva and stool samples was performed before PMTC was initiated. The first PMTC (t0) was conducted after specimen collection. The second sampling (T1) was performed approximately one month after the first PMTC (t0). The second PMTC (t1) was performed after collecting the second specimen. One month after the second PMTC (t1), the third PMTC (t2) was performed. The final sample collection (T2) for the third time point was performed approximately one month after the third PMTC (t2). In total, 72 saliva and 72 stool samples were collected across these three-time points, thus providing a robust dataset for our analysis. A flowchart of this study is shown in Figure 1.  
The unstimulated saliva was collected [8] from each patient at fixed time points within an average of 90 min. In individuals, the circadian rhythm of the salivary microflora is linked to the human body clock [22]. Considering the diurnal variation in the oral microbiota, sample collection was conducted in such a way that changes in the oral microbiota were not affected by differences in collection time.  
\begin{table}
\begin{tabular}{l c} \hline
**Clinical Characteristics** & **Data** \\ \hline Age (year), mean \(\pm\) SD & 25.7 \(\pm\) 4.3 \\ Sex, n (\%) & \\ Male & 1 (4.2) \\ Female & 23 (95.8) \\ Treatment duration & 11.2 \(\pm\) 5.0 \\ (year), mean \(\pm\) SD & 11.2 \(\pm\) 5.0 \\ \hline \end{tabular} SD, standard deviation.  
\end{table}
Table 1: Clinical characteristics of the study participants.  
Figure 1: Study flow diagram. PMTC, professional mechanical tooth cleaning; PCR, plaque Control Record.  
Saliva was collected using an ORAL OM505 (DNA Genotek Inc., Kanata, ON, Canada) and stool was collected using FS-0014 (TechnoSuruga Laboratory Co., Ltd., Shizuoka, Japan). Participants were instructed not to eat or clean their mouths for at least 2 h before saliva collection [8; 15]. Saliva and stool were frozen at \(-20\)\({}^{\circ}\)C after collection [9; 23].
# Fast Simulation of Laser Heating Processes on Thin Metal Plates with FFT Using CPU/GPU Hardware
## Methodology
### Scheme 4--Odd-Symmetry 2D FFT  
Finally, in this scheme the original list of coefficients is duplicated and mirrored in each direction exactly as in Section 3.6. However, this scheme also takes advantage of the even symmetry of the cosine function at \(2k\pi\) (\(k\in\mathbb{N}\), see Figure 6). Similar to the 1D odd-symmetry approach, the duplicated coefficients are mirrored in each direction (rows and columns), and multiplied by \(-1\). The final temperature is retrieved from the real component of the 2D FFT, which considers the sine components and the cosine components (that become 0 due to the cosine symmetry). The mathematical proof of the scheme is presented in Appendix D.  
Figure 5: Matrix structure for the odd symmetry FFT (1D and 2D). The blue block contains the original coefficients and the remaining blocks contain their odd-symmetry counterpart. All blocks are separated by rows and columns of zeros.  
Algorithm 4 presents the method used to retrieve the temperature of Equation (A16) at any given time \(t\) using a 2D FFT. Line 1 initializes the extended matrix of Fourier coefficients with \(M,N\) trailing zeros. Similar to the 1D odd symmetry method, Lines 3-8 add the reversed sequences of Fourier coefficients (with negative sign) in each direction (see Figure 5). Line 9 computes the 2D FFT of the extended Fourier matrix. Line 11 extracts the real part of the FFT solution and removes the mirrored part. Finally, Line 12 applies the initial and boundary conditions. The complexity of the presented algorithm is \(\mathcal{O}(MN\log(MN))\).  
```
0:\(\Theta\in\mathbb{R}^{(M-2)\times(N-2)},\quad u_{\infty}\in\mathbb{R}\)
0:\(U\in\mathbb{R}^{M\times N}\)
1:\(\Theta_{ODD\_SYM}\leftarrow\text{zeros}(2M,2N)\)
2:\(\Theta_{ODD\_SYM}[1:M-1,1:M-1]\leftarrow\Theta\)
3:for\(m=M+1,m<2M-1,m\gets m+1\)do
4:\(\Theta_{ODD\_SYM}[m,:]\leftarrow-\Theta_{ODD\_SYM}[2M-m-1]\)
5:endfor
6:for\(n=N+1,n<2N-1,n\gets n+1\)do
7:\(\Theta_{ODD\_SYM}[n,:]\leftarrow-\Theta_{ODD\_SYM}[2N-n-1]\)
8:endfor
9:\(Mat\leftarrow\text{fft2d}(\Theta_{ODD\_SYM})\)
10:\(U\leftarrow\text{zeros}(M,N)\)
11:\(U\leftarrow\text{real}(Mat[0:M-1,0:N-1])\)
12:\(U\gets U+u_{\infty}\)
13:return\(U\).
```  
**Algorithm 4** Retrieve temperature using 2D FFTs by applying odd sine symmetry and even cosine symmetry to the original coefficients
# An Evolutionary-Based MPPT Algorithm for Photovoltaic Systems under Dynamic Partial Shading
## 5 Conclusions  
Solar energy remains one of the best choices among renewables, being an inexhaustible energy source available everywhere with zero carbon emissions, but optimization in the performance of PV systems has become a priority.  
This paper has proposed a comparison of traditional MPPT algorithms with an evolutionary based computational technique able to optimize extraction of solar power under dynamic environmental conditions. Partial shading conditions in different shape, size, and dynamics have been simulated, thus proving that dynamic conditions always represent a critical task for maximum power extraction. The effective use of a PSO-based algorithm has demonstrated that computational intelligence techniques can provide effective methods only if the population size increases, but under dynamic shading conditions computational time cannot be a negligible issue related to high conversion efficiency tasks. Moreover, this study has shown that PSO-based MPPT algorithm guarantees to effectively reach the global MPP, but the search of the global power point requires significant computational time, thus compromising conversion efficiency, especially in fast dynamic conditions.  
Future works will include further improvements in the efficiency of the algorithm by reducing the computational time taken for scanning, choosing a more efficient approach that also considers hybrid strategies, and a deep experimental test campaign.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline  & \multicolumn{3}{c}{**Efficiency**} \\ \cline{2-4}
**MPPT** & **2 \(\times\) 1** & **3 \(\times\) 1** & **4 \(\times\) 1** \\ \hline Perturb \& Observe—A & 95.06\% & 95.48\% & 95.64\% \\ Perturb \& Observe—B & 97.65\% & 94.10\% & 95.22\% \\ Perturb \& Observe—C & 93.35\% & 93.82\% & 93.19\% \\ Incremental Conductance & 95.34\% & 94.16\% & 94.31\% \\ Constant Voltage & 71.59\% & 67.37\% & 63.15\% \\ Open Voltage & 70.92\% & 73.81\% & 72.42\% \\ Short Current Pulse & 93.01\% & 92.81\% & 91.77\% \\ PSO—3 particles & 91.07\% & 92.62\% & 88.54\% \\ PSO—4 particles & 91.78\% & 92.02\% & 90.40\% \\ PSO—5 particles & 92.47\% & 91.61\% & 91.10\% \\ \hline \hline \end{tabular}
\end{table}
Table 6: Efficiency—Trapezoidal-C shapes (simulations).  
Figure 10: Trapezoidal-C shape—4 \(\times\) 1: (a) P&O—A; (b) PSO 5 particles.  
**Author Contributions:** [PERSON] has been the corresponding author and with [PERSON] contributed to the manuscript composition and test scenario developments. [PERSON], [PERSON] and [PERSON] also implemented and ran the algorithms in the Matlab environment. [PERSON] supervised the manuscript composition and the experimental validation, carried out at the Department of Energy, Politecnico di Milano (Italy).  
**Conflicts of Interest:** The authors declare no conflict of interest.
# Distributed Information Compression for Target Tracking in Cluster-Based Wireless Sensor Networks
## 5 Simulation
### Compressing Bound  
Figure 24 depicts that the error distance of DCTTP increases as the compressing bound increases. Because the sensed data of each active sensor node is compressed through the bitmap process, the distance between the active sensor node and the target may be represented by the average distance. Therefore, not only distortion may occur, but also the compressed data may influence the result of target tracking. The compressed data influences the weight of each active sensor node, so the reference value of the weight may decrease. Notice that in TCAT and CTCI, since the results of the positioning are not compressed through the bitmap process, the error distance is not influenced by the compressing bound. The bitmap process of the CTCI scheme is to obtain the information of measurement uncertainties in the original measurement data.  
Observe that in Figure 25 when the compressing bound is zero, the data of DCTTP and CTCI will be reserved completely. As expected, the data size of DCTTP and CTCI decreases as the compressing bound increases. When the compressing bound is larger than two, the data are almost fully represented  
Figure 24: Impact of the compressing bound on the average error distance.  
Figure 23: Impact of noise on the average error distance.
## 2 Previous Research
### Fuzzy Inference  
Fuzzy inference [15] is a method for estimating the comprehensive results using given datasets by approximate reasoning through the fuzzification of a general conditional proposition into a fuzzy conditional proposition defined as fuzzy sets. Among the fuzzy inference methods, the min-max method proposed by [PERSON] [16; 17] was adopted in this study. [PERSON]'s fuzzy inference system has been widely used to take advantages of its simple reasoning approach and easy graphical expression. Figure 2 shows an example for better understanding of the fuzzy inference system utilized in this study, in which two fuzzy rules (the fuzzy Rules 1 and 2) were introduced. In this example, \(\mu_{C_{j}}(z_{j})\), which is the membership degree of the output value \(z_{j}\) in the output fuzzy set \(C_{j}\), can be expressed as a function of the two membership degrees \(\mu_{A_{i}}(x_{i})\) and \(\mu_{B_{i}}(y_{i})\) of the input values \(x_{i}\) and \(y_{i}\) in the input fuzzy sets \(A_{i}\) and \(B_{i}\), respectively, based on the min-max method, as follows:  
\[\mu_{C_{j}}(z_{j})=\bigvee_{n=1}^{R}\big{(}\mu_{A_{i}}(x_{i})\wedge\mu_{B_{i}}( y_{i})\big{)}, \tag{2}\]  
where \(i\) and \(j\) ranges from \(1\) to \(q\) and \(k\), respectively, \(q\) and \(k\) are the number of the input and output fuzzy sets, respectively, \(R\) is the number of the fuzzy rules and \(\vee\) and \(\wedge\) are the maximum and minimum calculation operators, respectively. As expressed in Equation (2), the output fuzzy sets \(C_{j}\) are derived by taking a smaller value between the membership degrees of the input fuzzy sets \(A_{i}\) and \(B_{i}\) in each rule, and the maximum value of the output fuzzy sets \(C_{j}\) is taken as the combined output fuzzy set \(C\) that is the final result of the fuzzy inference. Then, the combined output fuzzy set \(C\) needs to be converted into a quantitative crisp value through the defuzzification process. In this study, the defuzzified output value (\(z_{c}\)) was determined by the centroid method, i.e., the centroid of the fuzzy set area shown in Figure 2, as follows:  
Figure 1: Visual inspection and experimental investigation.  
\[z_{c}=\frac{\sum\limits_{j=1}^{k}\mu_{C_{j}}(z_{j})\cdot z_{j}}{\sum\limits_{j=1} ^{k}\mu_{C_{j}}(z_{j})}, \tag{3}\]  
As expressed in Equations (2) and (3), the computational process of the fuzzy inference system simply consists of the min-max and the defuzzification processes.
## 3 Controller Design  
### ADRC for Multivariate Coupled Systems  
Without loss of generality, the multivariable coupling system can be considered as follows:  
\[\left\{\begin{array}{l}x_{s_{1}}^{(n_{1})}=f_{s_{1}}+b_{11}u_{1}+b_{12}u_{2}+ \cdots+b_{1k}u_{k}\\ y_{s_{1}}=x_{s_{1},1}\\ x_{s_{2}}^{(n_{2})}=f_{s_{2}}+b_{21}u_{1}+b_{22}u_{2}+\cdots+b_{2k}u_{k}\\ y_{s_{2}}=x_{s_{2},1}\\ \vdots\\ x_{s_{k}}^{(n_{k})}=f_{s_{k}}+b_{k1}u_{1}+b_{k2}u_{2}+\cdots+b_{kk}u_{k}\\ y_{s_{k}}=x_{s_{k},1}\\ \end{array}\right. \tag{7}\]  
where \(x_{s_{i}}(i=1,2,\cdots,k)\) represents the system states, \(b_{i\bar{\eta}_{i}}(i,j=1,2,\cdots,k)\) represents the coefficients of the control input, and \(f_{s_{i}}(i=1,2,\cdots,k)\) represents the whole disturbance of each channel, which can be estimated and compensated by ADRC. Defining \(x=\left[x_{s_{1}},x_{s_{2}},\cdots,x_{s_{k}}\right]^{\mathrm{T}}\), \(y=\left[y_{s_{1}},y_{s_{2}},\cdots,y_{s_{k}}\right]^{\mathrm{T}}\), \(f=\left[f_{s_{1}},f_{s_{2}},\cdots,f_{s_{k}}\right]^{\mathrm{T}}\), \(u=\left[u_{1},u_{2},\cdots,u_{k}\right]^{\mathrm{T}}\), and \(n_{1}=n_{2}=\cdots=n_{k}=n\), system (6) can be described in brief as  
\[\left\{\begin{array}{l}x^{(n)}=f+Bu=f+U\\ y=x\end{array}\right.,B=\begin{bmatrix}b_{11}&\cdots&b_{1k}\\ \vdots&\vdots&\vdots\\ b_{k1}&\cdots&b_{kk}\end{bmatrix} \tag{8}\]  
Furthermore, for any _i_th channel of system (8), it has  
\[\left\{\begin{array}{l}x_{i}^{(n)}=f_{i}+U_{i}\\ y_{i}=x_{i}\end{array}\right. \tag{9}\]  
Equation (9) shows that a single input single output relationship is formed between the virtual control quantity \(U_{i}\) and the output quantity \(y_{i}\), that is, complete decoupling between the output quantity \(y_{i}\) and the virtual control quantity \(U_{i}\) is realized. Therefore, if the controlled quantity \(y_{i}\) can be measured, the target \(y_{id}\) can be tracked by designing an ADRC. The block diagram of the ADRC for a multivariable coupling system is shown in Figure 2.  
It can be seen from Figure 1 that there are k ADRC systems embedded in parallel between the control quantity \(U_{i}\) and the output quantity \(y_{i}\), and each ADRC system is  
Figure 2: Block diagram of the ADRC for a multivariable coupling system.  
composed of n first-order ADRC controllers in series; thus, the cascade decoupled control for high-order multivariate coupled systems is realized.
# Synthesis of Hydrogel-Based Microgels and Nanogels Toward Therapeutic and Biomedical Applications
## 2 Hydrogel Materials
### Synthetic Hydrogels
#### 2.1.5 Polyurethane-Based Hydrogels
\begin{table}
\begin{tabular}{p{56.9 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt} p{113.8 pt}} \hline \hline \multicolumn{2}{p{56.9 pt}}{**Hydrogel Materials**} & \multicolumn{1}{p{113.8 pt}}{**Crosslinking Methods**} & \multicolumn{1}{p{113.8 pt}}{**Functional Features**} & \multicolumn{1}{p{113.8 pt}}{**Applications**} & \multicolumn{1}{p{113.8 pt}}{**Ref.**} \\ \hline \multirow{4}{*}{\begin{tabular}{} \end{tabular} } & \(\bullet\) & Free radical polymerization & \(\bullet\) & High water retention & \(\bullet\) & Controlled drug delivery & \\  & \(\bullet\) & Photopolymerization & \(\bullet\) & (body temperatures) & \(\bullet\) & Tissue scaffolds & [15; 19; 24] \\  & \(\bullet\) & Physical crosslinking & \(\bullet\) & pH-sensitive & \(\bullet\) & Biosensors & \\ \hline \multirow{4}{*}{\begin{tabular}{} \end{tabular} } & \multirow{4}{*}{\begin{tabular}{} \end{tabular} } & \(\bullet\) & Low cytotoxicity & \(\bullet\) & Tissue engineering & \\  & \(\bullet\) & Chemical crosslinking & \(\bullet\) & Tunable mechanical strength & \(\bullet\) & Bio-adhesive wound & \\  & \(\bullet\) & Photopolymerization & \(\bullet\) & Tunable water & \(\bullet\) & Controlled drug & \\  & & & retention level & & delivery & \\ \hline \multirow{4}{*}{\begin{tabular}{} \end{tabular} } & \multirow{4}{*}{\begin{tabular}{} \end{tabular} } & \(\bullet\) & Hydrogen bonds & \multirow{4}{*}{\begin{tabular}{} \end{tabular} } & \multirow{4}{*}{\begin{tabular}{} \end{tabular} } \\  & & & & High thermal & \(\bullet\) & Wound dressing and \\  & \(\bullet\) & Hydrolysis/alcoholysis & \(\bullet\) & Excellent & \(\bullet\) & Regenerative \\  & \(\bullet\) & Freeze–thaw cycles & \(\bullet\) & biocompatibility & \(\bullet\) & therapies & \\  & & & \(\bullet\) & Reversible & \(\bullet\) & Biomedical coatings & \\ \hline \multirow{4}{*}{\begin{tabular}{} \end{tabular} } & \multirow{4}{*}{\begin{tabular}{} \end{tabular} } & \(\bullet\) & pH-responsive & \(\bullet\) & Oral drug delivery & \\  & & & Strong electrostatic interactions & \(\bullet\) & systems & \\  & \(\bullet\) & Hydrogen bonding & \(\bullet\) & \(\bullet\) & interactions & \\  & & & High thermal & \(\bullet\) & diagnostics & \\  & & & stability & & \(\bullet\) & Anticancer & \\ \hline \multirow{4}{*}{\
\begin{tabular}{} \end{tabular} } & \multirow{4}{*}{
\begin{tabular}{} \end{tabular} } & \(\bullet\) & Excellent elasticity & \(\bullet\) & Wound healing & \\  & & \(\bullet\) & Self-healing & \(\bullet\) & Vascular targeted & \\  & \(\bullet\) & Physical crosslinking & \(\bullet\) & Controllable & \(\bullet\) & Soft tissue & \\  & & & hydrophilicity & \(\bullet\) & engineering & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Key features and biomedical applications of common synthetic hydrogels.
# Quantum Photonic Simulation of Spin-Magnetic Field Coupling and Atom-Optical Field Interaction
## 2 Mechanical Interacting Systems
### Spin-I/2 Particle Interacting with a Magnetic Field
\tag{5}\]  
This Hamiltonian \(\hat{H^{\prime}}\) can be rewritten as a product of the form \(-\frac{\hbar}{2}\gamma_{p}(\mathbf{\sigma}\cdot\mathbf{B}_{ef})\), with \(\mathbf{B}_{ef}\) an effective static magnetic field, that is,  
\[\hat{H^{\prime}}=-\frac{\hbar}{2}\gamma_{p}(\mathbf{\sigma}\cdot\mathbf{B }_{ef})=-\frac{\hbar}{2}\gamma_{p}B_{0}\Delta\cos\theta^{\prime}\sigma_{z}- \frac{\hbar}{2}\gamma_{p}B_{0}\Delta\sin\theta^{\prime}\sigma_{x}=-\frac{ \hbar}{2}\Delta_{o}\left( \begin{array}{cc}\cos\theta^{\prime}&\sin\theta^{ \prime}\\ \sin\theta^{\prime}&-\cos\theta^{\prime}\end{array} \right), \tag{6}\]  
with:  
\[\Delta_{o}=\gamma_{p}B_{0}\Delta=\gamma_{p}B_{o}^{\prime} \tag{7}\]  
and:  
\[\Delta=\gamma_{p}B_{0}\sqrt{1-\frac{2\omega}{\gamma_{p}B_{0}}\cos \theta+\frac{\omega^{2}}{\gamma_{p}^{2}B_{0}^{2}}}; \tag{8}\]  
therefore, the effective magnetic field is given by \(\mathbf{B}_{ef}=(B_{0}^{\prime}\sin\theta^{\prime},0,B_{0}^{\prime}\cos\theta^{ \prime})\), where \(B_{0}^{\prime}\) and \(\theta^{\prime}\) are related to \(B_{0}\) and \(\theta\) by \(B_{0}^{\prime}=B_{0}\Delta\), with \(\sin\theta^{\prime}=\gamma_{p}B_{0}\sin\theta/\Delta_{o}=\sin\theta/\Delta\) and \(\cos\theta^{\prime}=\gamma_{p}B_{0}[\cos\theta-(\omega/\gamma_{p}B_{0})]/\Delta_ {o}=[\cos\theta-(\omega/\gamma_{p}B_{0})]/\Delta\). As seen from the expression for \(\Delta\), it would seem that for this reparametrization to have physical meaning, restrictions on the parameters would appear, as \(\Delta\) contains a possible non-positive term under the square root. The worst case would be likely to happen if \(\cos\theta=1\), but one can find that the resulting quantity \(1-(2\hbar\omega/\mu B_{0})+(\hbar\omega/\mu B_{0})^{2}\) is never negative, for any value of the parameters. Next, it is interesting to compute the eigenstates \(|\eta\rangle\) and eigenvalues \(E_{\eta}\) of Hamiltonian \(\hat{H^{\prime}}\) given by Equation (6), that is \(|\eta(t)\rangle=\exp{\left(-\frac{i}{\hbar}E_{\eta}t\right)}|\eta(0)\rangle\). After a standard calculation, we have:  
\[|\eta_{+}(0)\rangle=\cos\frac{\theta^{\prime}}{2}|0\rangle+\sin \frac{\theta^{\prime}}{2}|1\rangle,\ \ \ |\eta_{-}(0)\rangle=\sin\frac{\theta^{\prime}}{2}|0\rangle-\cos \frac{\theta^{\prime}}{2}|1\rangle, \tag{9}\]  
where \(|0\rangle\equiv(1,0)^{T}\) and \(|1\rangle\equiv(0,1)^{T}\), with \(T\) denoting the transpose. The eigenvalues are \(E_{\pm}=\pm\frac{\hbar}{2}\Delta_{o}\), that is \(E_{\pm}=\pm\frac{\hbar}{2}\gamma_{p}B_{0}^{\prime}=\pm\frac{\mu}{2}B_{0}^{\prime}\). Therefore, by taking into account Equation (4), the full evolved states \(|\Psi(t)\rangle\) can be easily obtained.