In this study, we used a natural resource, yarosite minerals, as a Fe2O3 precursor. Yarosite minerals were used for the synthesis of LaFeO3/Fe2O3 doped with ZnO via a co-precipitation method using ammonium hydroxide, which produced a light brown powder. Then, an ethanol gas sensor was prepared using a screen-printing technique and characterized using gas chamber tools at 100,200, and 300 ppm of ethanol gas to investigate the sensor's performance. Several factors that substantiate electrical properties such as crystal and morphological structures were also studied using X-Ray Diffraction (XRD) and Scanning Electron Microscopy (SEM), respectively. The crystallite size decreased from about 61.4 nm to 28.8 nm after 0.5 mol% ZnO was added. The SEM characterization images informed that the modified LaFeO3 was relatively the same but not uniform. Lastly, the sensor's electrical properties exhibited a high response of about 257% to 309% at an operating temperature that decreased from 205 ℃ to 180 ℃. This finding showed that these natural resources have the potential to be applied in the development of ethanol gas sensors in the future. Hence, yarosite minerals can be considered a good natural resource that can be further explored to produce an ethanol gas sensor with more sensitive response. In addition, this method reduces the cost of material purchase.
Citation: Endi Suhendi, Andini Eka Putri, Muhamad Taufik Ulhakim, Andhy Setiawan, Syarif Dani Gustaman. Investigation of ZnO doping on LaFeO3/Fe2O3 prepared from yarosite mineral extraction for ethanol gas sensor applications[J]. AIMS Materials Science, 2022, 9(1): 105-118. doi: 10.3934/matersci.2022007
[1] | Lina Liu, Huiting Zhang, Yinlan Chen . The generalized inverse eigenvalue problem of Hamiltonian matrices and its approximation. AIMS Mathematics, 2021, 6(9): 9886-9898. doi: 10.3934/math.2021574 |
[2] | Shixian Ren, Yu Zhang, Ziqiang Wang . An efficient spectral-Galerkin method for a new Steklov eigenvalue problem in inverse scattering. AIMS Mathematics, 2022, 7(5): 7528-7551. doi: 10.3934/math.2022423 |
[3] | Yalçın Güldü, Ebru Mişe . On Dirac operator with boundary and transmission conditions depending Herglotz-Nevanlinna type function. AIMS Mathematics, 2021, 6(4): 3686-3702. doi: 10.3934/math.2021219 |
[4] | Batirkhan Turmetov, Valery Karachik . On solvability of some inverse problems for a nonlocal fourth-order parabolic equation with multiple involution. AIMS Mathematics, 2024, 9(3): 6832-6849. doi: 10.3934/math.2024333 |
[5] | Wei Ma, Zhenhao Li, Yuxin Zhang . A two-step Ulm-Chebyshev-like Cayley transform method for inverse eigenvalue problems with multiple eigenvalues. AIMS Mathematics, 2024, 9(8): 22986-23011. doi: 10.3934/math.20241117 |
[6] | Liangkun Xu, Hai Bi . A multigrid discretization scheme of discontinuous Galerkin method for the Steklov-Lamé eigenproblem. AIMS Mathematics, 2023, 8(6): 14207-14231. doi: 10.3934/math.2023727 |
[7] | Lingling Sun, Hai Bi, Yidu Yang . A posteriori error estimates of mixed discontinuous Galerkin method for a class of Stokes eigenvalue problems. AIMS Mathematics, 2023, 8(9): 21270-21297. doi: 10.3934/math.20231084 |
[8] | Jia Tang, Yajun Xie . The generalized conjugate direction method for solving quadratic inverse eigenvalue problems over generalized skew Hamiltonian matrices with a submatrix constraint. AIMS Mathematics, 2020, 5(4): 3664-3681. doi: 10.3934/math.2020237 |
[9] | Phakhinkon Phunphayap, Prapanpong Pongsriiam . Extremal orders and races between palindromes in different bases. AIMS Mathematics, 2022, 7(2): 2237-2254. doi: 10.3934/math.2022127 |
[10] | Hannah Blasiyus, D. K. Sheena Christy . Two-dimensional array grammars in palindromic languages. AIMS Mathematics, 2024, 9(7): 17305-17318. doi: 10.3934/math.2024841 |
In this study, we used a natural resource, yarosite minerals, as a Fe2O3 precursor. Yarosite minerals were used for the synthesis of LaFeO3/Fe2O3 doped with ZnO via a co-precipitation method using ammonium hydroxide, which produced a light brown powder. Then, an ethanol gas sensor was prepared using a screen-printing technique and characterized using gas chamber tools at 100,200, and 300 ppm of ethanol gas to investigate the sensor's performance. Several factors that substantiate electrical properties such as crystal and morphological structures were also studied using X-Ray Diffraction (XRD) and Scanning Electron Microscopy (SEM), respectively. The crystallite size decreased from about 61.4 nm to 28.8 nm after 0.5 mol% ZnO was added. The SEM characterization images informed that the modified LaFeO3 was relatively the same but not uniform. Lastly, the sensor's electrical properties exhibited a high response of about 257% to 309% at an operating temperature that decreased from 205 ℃ to 180 ℃. This finding showed that these natural resources have the potential to be applied in the development of ethanol gas sensors in the future. Hence, yarosite minerals can be considered a good natural resource that can be further explored to produce an ethanol gas sensor with more sensitive response. In addition, this method reduces the cost of material purchase.
Since the 1960s, the rapid development of high-speed rail has made it a very important means of transportation. However, the vibration will be caused because of the contact between the wheels of the train and the train tracks during the operation of the high-speed train. Therefore, the analytical vibration model can be mathematically summarized as a quadratic palindromic eigenvalue problem (QPEP) (see [1,2])
(λ2A1+λA0+A⊤1)x=0, |
with Ai∈Rn×n, i=0,1 and A⊤0=A0. The eigenvalues λ, the corresponding eigenvectors x are relevant to the vibration frequencies and the shapes of the vibration, respectively. Many scholars have put forward many effective methods to solve QPEP [3,4,5,6,7]. In addition, under mild assumptions, the quadratic palindromic eigenvalue problem can be converted to the following linear palindromic eigenvalue problem (see [8])
Ax=λA⊤x, | (1) |
with A∈Rn×n is a given matrix, λ∈C and nonzero vectors x∈Cn are the wanted eigenvalues and eigenvectors of the vibration model. We can obtain 1λx⊤A⊤=x⊤A by transposing the equation (1). Thus, λ and 1λ always come in pairs. Many methods have been proposed to solve the palindromic eigenvalue problem such as URV-decomposition based structured method [9], QR-like algorithm [10], structure-preserving methods [11], and palindromic doubling algorithm [12].
On the other hand, the modal data obtained by the mathematical model are often evidently different from the relevant experimental ones because of the complexity of the structure and inevitable factors of the actual model. Therefore, the coefficient matrices need to be modified so that the updated model satisfies the dynamic equation and closely matches the experimental data. Al-Ammari [13] considered the inverse quadratic palindromic eigenvalue problem. Batzke and Mehl [14] studied the inverse eigenvalue problem for T-palindromic matrix polynomials excluding the case that both +1 and −1 are eigenvalues. Zhao et al. [15] updated ∗-palindromic quadratic systems with no spill-over. However, the linear inverse palindromic eigenvalue problem has not been extensively considered in recent years.
In this work, we just consider the linear inverse palindromic eigenvalue problem (IPEP). It can be stated as the following problem:
Problem IPEP. Given a pair of matrices (Λ,X) in the form
Λ=diag{λ1,⋯,λp}∈Cp×p, |
and
X=[x1,⋯,xp]∈Cn×p, |
where diagonal elements of Λ are all distinct, X is of full column rank p, and both Λ and X are closed under complex conjugation in the sense that λ2i=ˉλ2i−1∈C, x2i=ˉx2i−1∈Cn for i=1,⋯,m, and λj∈R, xj∈Rn for j=2m+1,⋯,p, find a real-valued matrix A that satisfy the equation
AX=A⊤XΛ. | (2) |
Namely, each pair (λt,xt), t=1,⋯,p, is an eigenpair of the matrix pencil
P(λ)=Ax−λA⊤x. |
It is known that the mathematical model is a "good" representation of the system, we hope to find a model that is closest to the original model. Therefore, we consider the following best approximation problem:
Problem BAP. Given ˜A∈Rn×n, find ˆA∈SA such that
‖ˆA−˜A‖=minA∈SA‖A−˜A‖, | (3) |
where ‖⋅‖ is the Frobenius norm, and SA is the solution set of Problem IPEP.
In this paper, we will put forward a new direct method to solve Problem IPEP and Problem BAP. By partitioning the matrix Λ and using the QR-decomposition, the expression of the general solution of Problem IPEP is derived. Also, we show that the best approximation solution ˆA of Problem BAP is unique and derive an explicit formula for it.
We first rearrange the matrix Λ as
Λ=[10 00Λ1 000 Λ2]t2s2(k+2l) t 2s2(k+2l), | (4) |
where t+2s+2(k+2l)=p, t=0 or 1,
Λ1=diag{λ1,λ2,⋯,λ2s−1,λ2s},λi∈R, λ−12i−1=λ2i, 1≤i≤s,Λ2=diag{δ1,⋯,δk,δk+1,δk+2,⋯,δk+2l−1,δk+2l}, δj∈C2×2, |
with
δj=[αj+βji00αj−βji], i=√−1, 1≤j≤k+2l,δ−1j=ˉδj, 1≤j≤k,δ−1k+2j−1=δk+2j, 1≤j≤l, |
and the adjustment of the column vectors of X corresponds to those of Λ.
Define Tp as
Tp=diag{It+2s,1√2[1−i1i],⋯,1√2[1−i1i]}∈Cp×p, | (5) |
where i=√−1. It is easy to verify that THpTp=Ip. Using this matrix of (5), we obtain
˜Λ=THpΛTp=[1000Λ1000˜Λ2], | (6) |
˜X=XTp=[xt,⋯,xt+2s,√2yt+2s+1,√2zt+2s+1,⋯,√2yp−1,√2zp−1], | (7) |
where
˜Λ2=diag{[α1β1−β1α1],⋯,[αk+2lβk+2l−βk+2lαk+2l]}≜ |
and \tilde{\Lambda}_2 \in {\bf \mathbb{R}}^{2(k+2l) \times 2(k+2l)} , \tilde{X} \in {\bf \mathbb{R}}^{n \times p} . y_{t+2s+j} and z_{t+2s+j} are, respectively, the real part and imaginary part of the complex vector x_{t+2s+j} for j = 1, 3, \cdots, 2(k+2l)-1 . Using (6) and (7), the matrix equation (2) is equivalent to
\begin{equation} A\tilde{X} = A^\top \tilde{X}\tilde{\Lambda}. \end{equation} | (8) |
Since \text{rank}(X) = \text{rank}(\tilde{X}) = p . Now, let the QR-decomposition of \tilde{X} be
\begin{equation} \tilde{X} = Q\left[ \begin{array}{c} R \\ 0 \\ \end{array} \right], \end{equation} | (9) |
where Q = [Q_1, Q_2]\in \mathbb{R}^{n \times n} is an orthogonal matrix and R \in \mathbb{R}^{p \times p} is nonsingular. Let
\begin{equation} \begin{array}{cc} Q^\top AQ = \left[ \begin{array}{cc} A_{11}& A_{12} \\ A_{21}& A_{22} \\ \end{array}\right]&\begin{array}{c} p \\ n-p \\ \end{array} \\ \begin{array}{cc} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p & \ n-p \\ \end{array}& \end{array}. \end{equation} | (10) |
Using (9) and (10), then the equation of (8) is equivalent to
\begin{eqnarray} && A_{11}R = A_{11}^\top R\tilde{\Lambda}, \end{eqnarray} | (11) |
\begin{eqnarray} && A_{21}R = A_{12}^\top R\tilde{\Lambda}. \end{eqnarray} | (12) |
Write
\begin{equation} \begin{array}{ccc} R^\top A_{11}R\triangleq F = \left[ \begin{array}{ccc} f_{11} & F_{12} & \ \ \ F_{13} \\ F_{21} & F_{22} & \ \ \ F_{23} \\ F_{31} & F_{32} & \ \ \ F_{33} \\ \end{array} \right]&\begin{array}{c} t \\ 2s \\ 2(k+2l) \\ \end{array} \\ \begin{array}{ccc} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ t & \ \ \ 2s & \ 2(k+2l) \\ \end{array}& \end{array}, \end{equation} | (13) |
then the equation of (11) is equivalent to
\begin{eqnarray} && F_{12} = F_{21}^\top \Lambda_1, \ F_{21} = F_{12}^\top, \end{eqnarray} | (14) |
\begin{eqnarray} && F_{13} = F_{31}^\top \tilde{\Lambda}_2, \ F_{31} = F_{13}^\top, \end{eqnarray} | (15) |
\begin{eqnarray} && F_{23} = F_{32}^\top \tilde{\Lambda}_2, \ F_{32} = F_{23}^\top \Lambda_1, \end{eqnarray} | (16) |
\begin{eqnarray} && F_{22} = F_{22}^\top \Lambda_1, \end{eqnarray} | (17) |
\begin{eqnarray} && F_{33} = F_{33}^\top \tilde{\Lambda}_2. \end{eqnarray} | (18) |
Because the elements of \Lambda_1, \tilde{\Lambda}_2 are distinct, we can obtain the following relations by Eqs (14)-(18)
\begin{eqnarray} &&F_{12} = 0, \ F_{21} = 0, \ F_{13} = 0, \ F_{31} = 0, \ F_{23} = 0, \ F_{32} = 0, \end{eqnarray} | (19) |
\begin{eqnarray} &&F_{22} = \text{diag} \left\{ \left[\begin{array}{cc} 0 & h_{1} \\ \lambda_1 h_{1} & 0 \\ \end{array}\right], \cdots, \left[\begin{array}{cc} 0 & h_{s} \\ \lambda_{2s-1} h_{s} & 0 \\ \end{array}\right] \right\}, \end{eqnarray} | (20) |
\begin{eqnarray} &&F_{33} = \text{diag}\left\{G_{1}, \cdots, G_{k}, \left[\begin{array}{cc} 0 & G_{k+1} \\ G_{k+1}^\top\tilde{\delta}_{k+1} & 0 \\ \end{array}\right], \cdots, \left[\begin{array}{cc} 0 & G_{k+l} \\ G_{k+l}^\top\tilde{\delta}_{k+2l-1} & 0 \\ \end{array}\right]\right\}, \end{eqnarray} | (21) |
where
\begin{eqnarray*} && G_i = a_iB_i, \ G_{k+j} = a_{k+2j-1}D_1+a_{k+2j}D_2, \ G_{k+j}^\top = G_{k+j}, \\ && B_i = \left[\begin{array}{cc} 1 & \frac{1-\alpha_{i}}{\beta_{i}} \\ -\frac{1-\alpha_{i}}{\beta_{i}} & 1 \\ \end{array}\right], \ D_1 = \left[\begin{array}{cc} 1 & 0 \\ 0 & -1 \\ \end{array}\right], \ D_2 = \left[\begin{array}{cc} 0 & 1 \\ 1 & 0 \\ \end{array}\right], \end{eqnarray*} |
and 1\leq i\leq k, 1\leq j\leq l . h_{1}, \cdots, h_{s}, a_{1}, \cdots, a_{k+2l} are arbitrary real numbers. It follows from Eq (12) that
\begin{equation} A_{21} = A_{12}^\top E, \end{equation} | (22) |
where E = R\tilde{\Lambda}R^{-1} .
Theorem 1. Suppose that \Lambda = \mathit{\text{diag}} \{\lambda_{1}, \cdots, \lambda_{p}\} \in {\mathbb{C}}^{p \times p} , X = [x_{1}, \cdots, x_{p}] \in {\mathbb{C}}^{n \times p} , where diagonal elements of \Lambda are all distinct, X is of full column rank p , and both \Lambda and X are closed under complex conjugation in the sense that \lambda_{2i} = \bar{\lambda}_{2i-1} \in {\mathbb{C}} , x_{2i} = \bar{x}_{2i-1} \in {\mathbb{C}}^{n} for i = 1, \cdots, m , and \lambda_{j} \in {\mathbb{R}} , x_{j} \in {\mathbb{R}}^{n} for j = 2m+1, \cdots, p . Rearrange the matrix \Lambda as (4) , and adjust the column vectors of X with corresponding to those of \Lambda . Let \Lambda, X transform into \tilde{\Lambda}, \tilde{X} by (6)-(7) and QR-decomposition of the matrix \tilde{X} be given by (9) . Then the general solution of (2) can be expressed as
\begin{equation} {\mathcal{S}}_{A} = \left\{A \left| A = Q\left[ \begin{array}{cc} R^{-\top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1} & A_{12} \\ A_{12}^\top E & A_{22} \\ \end{array} \right]Q^\top \right. \right\}, \end{equation} | (23) |
where E = R\tilde{\Lambda}R^{-1}, f_{11} is arbitrary real number, A_{12} \in {\mathbb{R}}^{p \times (n-p)}, A_{22}\in {\mathbb{R}}^{(n-p) \times (n-p)} are arbitrary real-valued matrices and F_{22}, F_{33} are given by (20)-(21) .
In order to solve Problem BAP, we need the following lemma.
Lemma 1. [16] Let A, B be two real matrices, and X be an unknown variable matrix. Then
\begin{eqnarray*} && \frac{\partial tr(BX)}{\partial X} = B^\top, \ \frac{\partial tr(X^\top B^\top)}{\partial X} = B^\top, \ \frac{\partial tr(AXBX)}{\partial X} = (BXA+AXB)^\top, \\ && \frac{\partial tr(AX^\top BX^\top)}{\partial X} = BX^\top A+AX^\top B, \ \frac{\partial tr(AXBX^\top)}{\partial X} = AXB+A^\top XB^\top. \end{eqnarray*} |
By Theorem 1 , we can obtain the explicit representation of the solution set {\mathcal{S}}_{A} . It is easy to verify that \mathcal{S}_A is a closed convex subset of {\bf \mathbb{R}}^{n \times n}\times {\bf \mathbb{R}}^{n \times n}. By the best approximation theorem (see Ref. [17]), we know that there exists a unique solution of Problem BAP. In the following we will seek the unique solution \hat{A} in \mathcal{S}_A. For the given matrix \tilde{A} \in {\bf \mathbb{R}}^{n \times n}, write
\begin{equation} \begin{array}{cc} Q^\top \tilde{A}Q = \left[ \begin{array}{cc} \tilde{A}_{11}& \tilde{A}_{12} \\ \tilde{A}_{21}& \tilde{A}_{22} \\ \end{array}\right]&\begin{array}{c} p \\ n-p \\ \end{array} \\ \begin{array}{cc} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p & \ n-p \\ \end{array}& \end{array}, \end{equation} | (24) |
then
\begin{eqnarray*} \|A-\tilde{A}\|^2 & = & \left\|\left[ \begin{array}{cc} R^{- \top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1}-\tilde{A}_{11} & A_{12}-\tilde{A}_{12} \\ A_{12}^\top E-\tilde{A}_{21} & A_{22}-\tilde{A}_{22} \\ \end{array} \right]\right\|^2\\ & = &\left\| R^{- \top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1}-\tilde{A}_{11}\right\|^2\\ &+& \| A_{12}-\tilde{A}_{12}\|^2+\|A_{12}^\top E-\tilde{A}_{21} \|^2+\|A_{22}-\tilde{A}_{22}\|^2. \end{eqnarray*} |
Therefore, \|A-\tilde{A}\| = \min if and only if
\begin{eqnarray} && \left\|R^{-\top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1}-\tilde{A}_{11}\right\|^2 = \min, \end{eqnarray} | (25) |
\begin{eqnarray} && \| A_{12}-\tilde{A}_{12}\|^2+\|A_{12}^\top E-\tilde{A}_{21} \|^2 = \min, \end{eqnarray} | (26) |
\begin{eqnarray} &&A_{22} = \tilde{A}_{22}. \end{eqnarray} | (27) |
Let
\begin{equation} \begin{array}{c} R^{-1} = \left[ \begin{array}{c} R_1 \\ R_2 \\ R_3 \\ \end{array} \right] \end{array}, \end{equation} | (28) |
then the relation of (25) is equivalent to
\begin{equation} \|R_1^\top f_{11}R_1+ R_2^\top F_{22}R_2 + R_3^\top F_{33} R_3- \tilde{A}_{11} \|^2 = \min. \end{equation} | (29) |
Write
\begin{equation} R_1 = \left[r_{1, t}\right], \ R_2 = \left[ \begin{array}{c} r_{2, 1} \\ \vdots \\ r_{2, 2s}\\ \end{array} \right], \ R_3 = \left[ \begin{array}{c} r_{3, 1} \\ \vdots \\ r_{3, k+2l} \\ \end{array} \right], \end{equation} | (30) |
where r_{1, t} \in {\mathbb{R}}^{t \times p}, r_{2, i} \in {\mathbb{R}}^{1 \times p}, r_{3, j} \in {\mathbb{R}}^{2 \times p}, \ i = 1, \cdots, 2s, \ j = 1, \cdots, k+2l .
Let
\begin{equation} \left\{ \begin{array}{rcl} && J_t = r_{1, t}^\top r_{1, t}, \\ && J_{t+i} = \lambda_{2i-1}r_{2, 2i}^\top r_{2, 2i-1}+r_{2, 2i-1}^\top r_{2, 2i} \ (1 \leq i \leq s), \\ && J_{r+i} = r_{3, i}^\top B_i r_{3, i} \ (1 \leq i \leq k), \\ && J_{r+k+2i-1} = r_{3, k+2i}^\top D_1 \tilde{\delta}_{k+2i-1}r_{3, k+2i-1}+r_{3, k+2i-1}^\top D_1 r_{3, k+2i} \ (1 \leq i \leq l), \\ && J_{r+k+2i} = r_{3, k+2i}^\top D_2 \tilde{\delta}_{k+2i-1}r_{3, k+2i-1}+r_{3, k+2i-1}^\top D_2 r_{3, k+2i} \ (1 \leq i \leq l), \end{array} \right. \end{equation} | (31) |
with r = t+s, q = t+s+k+2l. Then the relation of (29) is equivalent to
\begin{eqnarray*} && g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}) = \\ &&\|f_{11}J_t+h_1J_{t+1}+\cdots+h_sJ_{r}+a_1J_{r+1}+\cdots+a_{k+2l}J_{q}-\tilde{A}_{11}\|^2 = \text{min}, \end{eqnarray*} |
that is,
\begin{eqnarray*} && g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}) \\ && = \text{tr} [(f_{11}J_t+h_1J_{t+1}+\cdots+h_sJ_{r}+a_1J_{r+1}+\cdots+a_{k+2l}J_{q}-\tilde{A}_{11})^\top \\ && (f_{11}J_t+h_1J_{t+1}+\cdots+h_sJ_{r}+a_1J_{r+1}+\cdots+a_{k+2l}J_{q}-\tilde{A}_{11})]\\ && = f_{11}^2c_{t, t}+2f_{11}h_1c_{t, t+1}+\cdots+2f_{11}h_sc_{t, r}+2f_{11}a_1c_{t, r+1}+\cdots+2f_{11}a_{k+2l}c_{t, q}-2f_{11}e_t\\ && +h_1^2c_{t+1, t+1}+\cdots+2h_1h_sc_{t+1, r}+2h_1a_1c_{t+1, r+1}+\cdots+2h_1a_{k+2l}c_{t+1, q}-2h_1e_{t+1} \\ && +\cdots \\ && +h_s^2c_{r, r}+2h_sa_1c_{r, r+1}+\cdots+2h_sa_{k+2l}c_{r, q}-2h_se_{r}\\ && +a_1^2c_{r+1, r+1}+\cdots+2a_1a_{k+2l}c_{r+1, q}-2a_1e_{r+1}\\ && +\cdots \\ && +a_{k+2l}^2c_{q, q}-2a_{k+2l}e_{q}+\text{tr} (\tilde{A}_{11}^\top \tilde{A}_{11}), \end{eqnarray*} |
where c_{i, j} = \text{tr} (J_i^\top J_j), e_i = \text{tr} (J_i^\top\tilde{A}_{11}) (i, j = t, \cdots, t+s+k+2l) and c_{i, j} = c_{j, i} .
Consequently,
\begin{eqnarray*} \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial f_{11}}&& = 2f_{11}c_{t, t}+2h_1c_{t, t+1}+\cdots+2h_sc_{t, r}+2a_1c_{t, r+1}\\ &&+\cdots+2a_{k+2l}c_{t, q}-2e_t, \\ \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial h_{1}}&& = 2f_{11}c_{t+1, t}+2h_1c_{t+1, t+1}+\cdots+2h_sc_{t+1, r}+2a_1c_{t+1, r+1}\\ &&+\cdots+2a_{k+2l}c_{t+1, q}-2e_{t+1}, \\ &&\cdots\\ \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial h_{s}}&& = 2f_{11}c_{r, t}+2h_1c_{r, t+1}+\cdots+2h_sc_{r, r}+2a_1c_{r, r+1}\\ &&+\cdots+2a_{k+2l}c_{r, q}-2e_{r}, \\ \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial a_{1}}&& = 2f_{11}c_{r+1, t}+2h_1c_{r+1, t+1}+\cdots+2h_sc_{r+1, r}+2a_1c_{r+1, r+1}\\ &&+\cdots+2a_{k+2l}c_{r+1, q}-2e_{r+1}, \\ &&\cdots\\ \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial a_{k+2l}}&& = 2f_{11}c_{q, t}+2h_1c_{q, t+1}+\cdots+2h_sc_{q, r}+2a_1c_{q, r+1}\\ &&+\cdots+2a_{k+2l}c_{q, q}-2e_{q}. \end{eqnarray*} |
Clearly, g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l}) = \text{min} if and only if
\frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial f_{11}} = 0, \cdots, \frac{\partial g(f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l})}{\partial a_{k+2l}} = 0. |
Therefore,
\begin{equation} \begin{split} &f_{11}c_{t, t}+h_1c_{t, t+1}+\cdots+h_sc_{t, r}+a_1c_{t, r+1}+\cdots+a_{k+2l}c_{t, q} = e_t, \\ &f_{11}c_{t+1, t}+h_1c_{t+1, t+1}+\cdots+h_sc_{t+1, r}+a_1c_{t+1, r+1}+\cdots+a_{k+2l}c_{t+1, q} = e_{t+1}, \\ &\cdots\\ &f_{11}c_{r, t}+h_1c_{r, t+1}+\cdots+h_sc_{r, r}+a_1c_{r, r+1}+\cdots+a_{k+2l}c_{r, q} = e_{r}, \\ &f_{11}c_{r+1, t}+h_1c_{r+1, t+1}+\cdots+h_sc_{r+1, r}+a_1c_{r+1, r+1}+\cdots+a_{k+2l}c_{r+1, q} = e_{r+1}, \\ &\cdots\\ &f_{11}c_{q, t}+h_1c_{q, t+1}+\cdots+h_sc_{q, r}+a_1c_{q, r+1}+\cdots+a_{k+2l}c_{q, q} = e_{q}. \end{split} \end{equation} | (32) |
If let
C = \left[ \begin{array}{ccccccc} c_{t, t}&c_{t, t+1}&\cdots&c_{t, r}&c_{t, r+1}&\cdots&c_{t, q}\\ c_{t+1, t}&c_{t+1, t+1}&\cdots&c_{t+1, r}&c_{t+1, r+1}&\cdots&c_{t+1, q}\\ \vdots&\vdots& &\vdots&\vdots& &\vdots\\ c_{r, t}&c_{r, t+1}&\cdots&c_{r, r}&c_{r, r+1}&\cdots&c_{r, q}\\ c_{r+1, t}&c_{r+1, t+1}&\cdots&c_{r+1, r}&c_{r+1, r+1}&\cdots&c_{r+1, q}\\ \vdots&\vdots& &\vdots&\vdots& &\vdots\\ c_{q, t}&c_{q, t+1}&\cdots&c_{q, r}&c_{q, r+1}&\cdots&c_{q, q}\\ \end{array} \right], \ h = \left[ \begin{array}{c} f_{11}\\ h_1\\ \vdots\\ h_s\\ a_1\\ \vdots\\ a_{k+2l}\\ \end{array} \right], \ e = \left[ \begin{array}{c} e_t\\ e_{t+1}\\ \vdots\\ e_{r}\\ e_{r+1}\\ \vdots\\ e_{q}\\ \end{array} \right], |
where C is symmetric matrix. Then the equation (32) is equivalent to
\begin{equation} C h = e, \end{equation} | (33) |
and the solution of the equation (33) is
\begin{equation} h = C^{-1} e. \end{equation} | (34) |
Substituting (34) into (20)-(21), we can obtain f_{11}, F_{22} and F_{33} explicitly. Similarly, the equation of (26) is equivalent to
\begin{eqnarray*} &&g(A_{12}) = \text{tr} (A_{12}^\top A_{12})+\text{tr} (\tilde{A}_{12}^\top \tilde{A}_{12})-2\text{tr} (A_{12}^\top \tilde{A}_{12})\\ &&+\text{tr} (E^\top A_{12}A_{12}^\top E)+\text{tr} (\tilde{A}_{21}^\top \tilde{A}_{21})-2\text{tr} (E^\top A_{12}\tilde{A}_{21}). \end{eqnarray*} |
Applying Lemma 1, we obtain
\frac{\partial g(A_{12})}{\partial A_{12}} = 2A_{12}-2\tilde{A}_{12}+2EE^\top A_{12}-2E\tilde{A}_{21}^\top, |
setting \frac{\partial g(A_{12})}{\partial A_{12}} = 0 , we obtain
\begin{equation} A_{12} = (I_p+EE^\top)^{-1}(\tilde{A}_{12}+E\tilde{A}_{21}^\top), \end{equation} | (35) |
Theorem 2. Given \tilde{A} \in {\mathbb{R}}^{n \times n} , then the Problem BAP has a unique solution and the unique solution of Problem BAP is
\begin{equation} \hat{A} = Q\left[ \begin{array}{cc} R^{-\top}\left[ \begin{array}{ccc} f_{11} & 0 & 0 \\ 0 & F_{22} & 0 \\ 0 & 0 & F_{33} \\ \end{array} \right]R^{-1} & A_{12} \\ A_{12}^\top E & \tilde{A}_{22} \\ \end{array} \right]Q^\top, \end{equation} | (36) |
where E = R\tilde{\Lambda} R^{-1} , F_{22}, F_{33}, A_{12}, \tilde{A}_{22} are given by (20), (21), (35), (24) and f_{11}, h_{1}, \cdots, h_s, a_1, \cdots, a_{k+2l} are given by (34) .
Based on Theorems 1 and 2 , we can describe an algorithm for solving Problem BAP as follows.
Algorithm 1.
1) Input matrices \Lambda , X and \tilde{A} ;
2) Rearrange \Lambda as (4), and adjust the column vectors of X with corresponding to those of \Lambda ;
3) Form the unitary transformation matrix T_p by (5);
4) Compute real-valued matrices \tilde{\Lambda}, \tilde{X} by (6) and (7);
5) Compute the QR-decomposition of \tilde{X} by (9);
6) F_{12} = 0, F_{21} = 0, F_{13} = 0, F_{31} = 0, F_{23} = 0, F_{32} = 0 by (19) and E = R\tilde{\Lambda} R^{-1} ;
7) Compute \tilde{A}_{ij} = Q_i^\top \tilde{A}Q_j, i, j = 1, 2 ;
8) Compute R^{-1} by (28) to form R_1, R_2, R_3 ;
9) Divide matrices R_1, R_2, R_3 by (30) to form r_{1, t}, r_{2, i}, r_{3, j}, i = 1, \cdots, 2s, j = 1, \cdots, k+2l ;
10) Compute J_i, i = t, \cdots, t+s+k+2l, by (31);
11) Compute c_{i, j} = \mbox{tr} (J_i^\top J_j), e_i = \mbox{tr} (J_i^\top\tilde{A}_{11}), i, j = t, \cdots, t+s+k+2l ;
12) Compute f_{11}, h_1, \cdots, h_s, a_1, \cdots, a_{k+2l} by (34);
13) Compute F_{22}, F_{33} by (20), (21) and A_{22} = \tilde{A}_{22} ;
14) Compute A_{12} by (35) and A_{21} by (22);
15) Compute the matrix \hat{A} by (36).
Example 1. Consider a 11 -DOF system, where
\tilde{A} = \left[ {\begin{array}{rrrrrrrrrrr} 96.1898 & 18.1847 & 51.3250 & 49.0864 & 13.1973 & 64.9115 & 62.5619 & 81.7628 & 58.7045 & 31.1102 & 26.2212 \\ 0.4634 & 26.3803 & 40.1808 & 48.9253 & 94.2051 & 73.1722 & 78.0227 & 79.4831 & 20.7742 & 92.3380 & 60.2843 \\ 77.4910 & 14.5539 & 7.5967 & 33.7719 & 95.6135 & 64.7746 & 8.1126 & 64.4318 & 30.1246 & 43.0207 & 71.1216 \\ 81.7303 & 13.6069 & 23.9916 & 90.0054 & 57.5209 & 45.0924 & 92.9386 & 37.8609 & 47.0923 & 18.4816 & 22.1747 \\ 86.8695 & 86.9292 & 12.3319 & 36.9247 & 5.9780 & 54.7009 & 77.5713 & 81.1580 & 23.0488 & 90.4881 & 11.7418 \\ 8.4436 & 57.9705 & 18.3908 & 11.1203 & 23.4780 & 29.6321 & 48.6792 & 53.2826 & 84.4309 & 97.9748 & 29.6676 \\ 39.9783 & 54.9860 & 23.9953 & 78.0252 & 35.3159 & 74.4693 & 43.5859 & 35.0727 & 19.4764 & 43.8870 & 31.8778 \\ 25.9870 & 14.4955 & 41.7267 & 38.9739 & 82.1194 & 18.8955 & 44.6784 & 93.9002 & 22.5922 & 11.1119 & 42.4167 \\ 80.0068 & 85.3031 & 4.9654 & 24.1691 & 1.5403 & 68.6775 & 30.6349 & 87.5943 & 17.0708 & 25.8065 & 50.7858 \\ 43.1414 & 62.2055 & 90.2716 & 40.3912 & 4.3024 & 18.3511 & 50.8509 & 55.0156 & 22.7664 & 40.8720 & 8.5516 \\ 91.0648 & 35.0952 & 94.4787 & 9.6455 & 16.8990 & 36.8485 & 51.0772 & 62.2475 & 43.5699 & 59.4896 & 26.2482 \\ \end{array}} \right], |
the measured eigenvalue and eigenvector matrices \Lambda and X are given by
\begin{eqnarray*} &&\Lambda = \mbox{diag} \{1.0000, \ -1.8969, \ -0.5272, \ -0.1131+0.9936i, -0.1131-0.9936i, \\ &&1.9228+2.7256i, \ 1.9228-2.7256i, \ 0.1728-0.2450i, \ 0.1728+0.2450i\}, \end{eqnarray*} |
and
\begin{eqnarray*} &&X = \left[ {\begin{array}{rrrrr} -0.0132& -1.0000& 0.1753& 0.0840 + 0.4722i& 0.0840 - 0.4722i \\ -0.0955& 0.3937& 0.1196& -0.3302 - 0.1892i& -0.3302 + 0.1892i \\ -0.1992& 0.5220& -0.0401& 0.3930 - 0.2908i& 0.3930 + 0.2908i \\ 0.0740& 0.0287& 0.6295& -0.3587 - 0.3507i& -0.3587 + 0.3507i \\ 0.4425& -0.3609& -0.5745& 0.4544 - 0.3119i& 0.4544 + 0.3119i \\ 0.4544& -0.3192& -0.2461& -0.3002 - 0.1267i& -0.3002 + 0.1267i \\ 0.2597& 0.3363& 0.9046& -0.2398 - 0.0134i& -0.2398 + 0.0134i \\ 0.1140& 0.0966& 0.0871& 0.1508 + 0.0275i& 0.1508 - 0.0275i \\ -0.0914& -0.0356& -0.2387& -0.1890 - 0.0492i& -0.1890 + 0.0492i \\ 0.2431& 0.5428& -1.0000& 0.6652 + 0.3348i& 0.6652 - 0.3348i \\ 1.0000& -0.2458& 0.2430& -0.2434 + 0.6061i& -0.2434 - 0.6061i \\ \end{array}} \right. \\ &&\left. {\begin{array}{rrrr} 0.6669 + 0.2418i& 0.6669 - 0.2418i& 0.2556 - 0.1080i& 0.2556 + 0.1080i \\ -0.1172 - 0.0674i& -0.1172 + 0.0674i& -0.5506 - 0.1209i& -0.5506 + 0.1209i \\ 0.5597 - 0.2765i& 0.5597 + 0.2765i& -0.3308 + 0.1936i& -0.3308 - 0.1936i \\ -0.7217 - 0.0566i& -0.7217 + 0.0566i& -0.7306 - 0.2136i& -0.7306 + 0.2136i \\ 0.0909 + 0.0713i& 0.0909 - 0.0713i& 0.5577 + 0.1291i& 0.5577 - 0.1291i \\ 0.1867 + 0.0254i& 0.1867 - 0.0254i& 0.2866 + 0.1427i& 0.2866 - 0.1427i \\ -0.5311 - 0.1165i& -0.5311 + 0.1165i& -0.3873 - 0.1096i& -0.3873 + 0.1096i \\ 0.2624 + 0.0114i& 0.2624 - 0.0114i& -0.6438 + 0.2188i& -0.6438 - 0.2188i \\ -0.0619 - 0.1504i& -0.0619 + 0.1504i& 0.2787 - 0.2166i& 0.2787 + 0.2166i \\ 0.3294 - 0.1718i& 0.3294 + 0.1718i& 0.9333 + 0.0667i& 0.9333 - 0.0667i \\ -0.4812 + 0.5188i& -0.4812 - 0.5188i& 0.6483 - 0.1950i& 0.6483 + 0.1950i \\ \end{array}} \right]. \end{eqnarray*} |
Using Algorithm 1, we obtain the unique solution of Problem BAP as follows:
\begin{eqnarray*} \hat{A} = \left[ {\begin{array}{rrrrrrrrrrr} 34.2563 & 41.7824 & 33.3573 & 33.6298 & 23.8064 & 42.0770 & 50.0641 & 37.5705 & 31.0908 & 48.6169 & 19.0972\\ 18.8561 & 35.2252 & 35.9592 & 44.3502 & 31.9918 & 55.2920 & 55.3052 & 54.3793 & 31.3909 & 60.8345 & 16.9540\\ 29.6359 & 7.6805 & 19.1249 & 17.7183 & 16.7082 & 40.0636 & 18.2916 & 49.9437 & 37.6913 & 15.6027 & 4.9603\\ 58.8782 & 51.4906 & 47.8974 & 35.6985 & 45.6889 & 56.0434 & 53.0908 & 56.5402 & 55.5120 & 38.3447 & 35.8894\\ 33.4087 & 46.9635 & 9.7767 & 41.4215 & 51.4466 & 52.1058 & 65.6724 & 60.1293 & 5.8061 & 62.0139 & 16.5231\\ 31.6580 & 51.2359 & 24.7978 & 65.5567 & 61.7840 & 62.5494 & 58.9363 & 74.7099 & 52.2105 & 55.8532 & 44.3925\\ 19.2961 & 51.2333 & 22.4280 & 56.9340 & 42.6348 & 45.8453 & 56.3729 & 61.5555 & 31.6836 & 67.9525 & 40.2012\\ 41.2796 & 71.3821 & 34.4140 & 33.2817 & 77.4393 & 60.8944 & 32.1411 & 108.5056 & 49.6078 & 19.8351 & 85.7434\\ 64.0890 & 57.6524 & 19.1280 & 25.0394 & 39.0524 & 66.7740 & 20.9023 & 48.8512 & 14.4695 & 18.9284 & 24.8348\\ 37.2550 & 32.3254 & 38.3534 & 59.7358 & 33.5902 & 54.0265 & 50.7770 & 70.2011 & 65.4159 & 58.0720 & 40.0652\\ 28.1301 & 14.7638 & 8.9507 & 20.0963 & 25.5907 & 59.6940 & 30.8558 & 66.8781 & 30.4807 & 23.6107 & 12.9984\\ \end{array}} \right], \end{eqnarray*} |
and
\|\hat{A}X-\hat{A}^\top X\Lambda\| = 8.2431\times 10^{-13}. |
Therefore, the new model \hat{A}X = \hat{A}^\top X\Lambda reproduces the prescribed eigenvalues (the diagonal elements of the matrix \Lambda ) and eigenvectors (the column vectors of the matrix X ).
Example 2. (Example 4.1 of [12]) Given \alpha = \cos(\theta) , \beta = \sin(\theta) with \theta = 0.62 and \lambda_1 = 0.2, \lambda_2 = 0.3, \lambda_3 = 0.4 . Let
\begin{equation*} J_0 = \left[ {\begin{array}{rr} 0_2 & \Gamma\\ I_2 & I_2\\ \end{array}} \right], \ J_s = \left[ {\begin{array}{cc} 0_3 & \mbox{diag} \{\lambda_1, \lambda_2, \lambda_3 \}\\ I_3 & 0_3\\ \end{array}} \right], \end{equation*} |
where \Gamma = \left[{\begin{array}{cc} \alpha & -\beta\\ \beta & \alpha\\ \end{array}} \right]. We construct
\begin{equation*} \tilde{A} = \left[ {\begin{array}{rr} J_0 & 0\\ 0 & J_s\\ \end{array}} \right], \end{equation*} |
the measured eigenvalue and eigenvector matrices \Lambda and X are given by
\begin{eqnarray*} \Lambda = \mbox{diag} \{5, 0.2, 0.8139+0.5810i, 0.8139-0.5810i\}, \end{eqnarray*} |
and
\begin{eqnarray*} X = \left[ {\begin{array}{rrrr} -0.4155& 0.6875& -0.2157 - 0.4824i& -0.2157 + 0.4824i\\ -0.4224& -0.3148& -0.3752 + 0.1610i& -0.3752 - 0.1610i\\ -0.0703& -0.6302& -0.5950 - 0.4050i& -0.5950 + 0.4050i\\ -1.0000& -0.4667& 0.2293 - 0.1045i& 0.2293 + 0.1045i\\ 0.2650& 0.3051& -0.2253 + 0.7115i& -0.2253 - 0.7115i\\ 0.9030& -0.2327& 0.4862 - 0.3311i& 0.4862 + 0.3311i\\ -0.6742& 0.3132& 0.5521 - 0.0430i& 0.5521 + 0.0430i\\ 0.6358& 0.1172& -0.0623 - 0.0341i& -0.0623 + 0.0341i\\ -0.4119& -0.2768& 0.1575 + 0.4333i& 0.1575 - 0.4333i\\ -0.2062& 1.0000& -0.1779 - 0.0784i& -0.1779 + 0.0784i\\ \end{array}} \right]. \end{eqnarray*} |
Using Algorithm 1, we obtain the unique solution of Problem BAP as follows:
\begin{equation*} \hat{A} = \left[ {\begin{array}{rrrrrrrrrr} -0.1169& -0.2366& 0.6172& -0.7195& -0.0836& 0.2884& 0.0092& -0.0490& -0.0202& 0.0171\\ -0.0114& -0.0957& 0.1462& 0.6194& 0.3738& -0.1637& 0.1291& -0.0071& 0.0972& 0.1247\\ 0.7607& -0.0497& 0.5803& -0.0346& 0.0979& 0.2959& 0.0937& -0.1060& 0.1323& -0.0339\\ -0.0109& 0.6740& -0.3013& 0.7340& 0.1942& -0.0872& 0.0054& 0.0051& 0.0297& 0.0814\\ 0.1783& 0.2283& 0.2643& 0.0387& 0.0986& -0.3125& -0.0292& 0.2926& -0.0717& -0.0546\\ 0.0953& 0.1027& 0.0360& 0.2668& -0.2418& 0.1206& 0.1406& -0.0551& 0.3071& 0.2097\\ -0.0106& -0.2319& 0.1946& -0.0298& -0.1935& 0.0158& -0.0886& 0.0216& -0.0560& 0.2484\\ 0.1044& 0.1285& 0.1902& 0.2277& 0.6961& 0.1657& 0.0728& -0.0262& -0.0831& -0.0001\\ 0.0906& 0.0021& 0.0764& -0.1264& 0.2144& 0.6703& -0.0850& 0.0764& -0.0104& -0.0149\\ -0.1245& 0.0813& 0.1952& -0.0784& 0.0760& -0.0875& 0.7978& -0.0093& 0.0206& -0.1182\\ \end{array}} \right], \end{equation*} |
and
\|\hat{A}X-\hat{A}^\top X\Lambda\| = 1.7538\times 10^{-8}. |
Therefore, the new model \hat{A}X = \hat{A}^\top X\Lambda reproduces the prescribed eigenvalues (the diagonal elements of the matrix \Lambda ) and eigenvectors (the column vectors of the matrix X ).
In this paper, we have developed a direct method to solve the linear inverse palindromic eigenvalue problem by partitioning the matrix \Lambda and using the QR-decomposition. The explicit best approximation solution is given. The numerical examples show that the proposed method is straightforward and easy to implement.
The authors declare no conflict of interest.
[1] |
Zhang X, Lan W, Xu J, et al. (2019) ZIF-8 derived hierarchical hollow ZnO nanocages with quantum dots for sensitive ethanol gas detection. Sens Actuators B Chem 289: 144-152. https://doi.org/10.1016/j.snb.2019.03.090 doi: 10.1016/j.snb.2019.03.090
![]() |
[2] |
Wang C, Wang ZG, Xi R, et al. (2019) In situ synthesis of flower-like ZnO on GaN using electrodeposition and its application as ethanol gas sensor at room temperature. Sens Actuators B Chem 292: 270-276. https://doi.org/10.1016/j.snb.2019.04.140 doi: 10.1016/j.snb.2019.04.140
![]() |
[3] |
Zhang S, Wang C, Qu F, et al. (2019) ZnO nanoflowers modified with RuO2 for enhancing acetone sensing performance. Nanotechnology 31: 115502. https://doi.org/10.1088/1361-6528/ab5cd9 doi: 10.1088/1361-6528/ab5cd9
![]() |
[4] |
Wang X, Liu F, Chen X, et al. (2020) SnO2 core-shell hollow microspheres co-modification with Au and NiO nanoparticles for acetone sensing. Powder Technol 364: 159-166. https://doi.org/10.1016/j.powtec.2020.02.006 doi: 10.1016/j.powtec.2020.02.006
![]() |
[5] |
Yin M, Wang Y, Yu L, et al. (2020) Ag nanoparticles-modified Fe2O3@MoS2 core-shell micro/nanocomposites for high-performance NO2 gas detection at low temperature. J Alloys Compd 829: 153371. https://doi.org/10.1016/j.jallcom.2020.154471 doi: 10.1016/j.jallcom.2020.154471
![]() |
[6] |
Tan J, Hu J, Ren J, et al. (2020) Fast response speed of mechanically exfoliated MoS2 modified by PbS in detecting NO2. Chinese Chem Lett 31: 2103-2108. https://doi.org/10.1016/j.cclet.2020.03.060 doi: 10.1016/j.cclet.2020.03.060
![]() |
[7] |
Luo N, Zhang D, Xu J (2020) Enhanced CO sensing properties of Pd modified ZnO porous nanosheets. Chinese Chem Lett 31: 2033-2036. https://doi.org/10.1016/j.cclet.2020.01.002 doi: 10.1016/j.cclet.2020.01.002
![]() |
[8] |
Zhu K, Ma S, Tie Y, et al. (2019) Highly sensitive formaldehyde gas sensors based on Y-doped SnO2 hierarchical flower-shaped nanostructures. J Alloys Compd 792: 938-944. https://doi.org/10.1016/j.jallcom.2019.04.102 doi: 10.1016/j.jallcom.2019.04.102
![]() |
[9] |
Cao J, Wang S, Zhao X, et al. (2020) Facile synthesis and enhanced toluene gas sensing performances of Co3O4 hollow nanosheets. Mater Lett 263: 127215. https://doi.org/10.1016/j.matlet.2019.127215 doi: 10.1016/j.matlet.2019.127215
![]() |
[10] |
Zhang C, Luo Y, Xu J, et al. (2019) Room temperature conductive type metal oxide semiconductor gas sensors for NO2 detection. Sens Actuator A Phys 289: 118-133. https://doi.org/10.1016/j.sna.2019.02.027 doi: 10.1016/j.sna.2019.02.027
![]() |
[11] |
Liu W, Xie Y, Chen T, et al. (2019) Rationally designed mesoporous In2O3 nanofibers functionalized Pt catalyst for high-performance acetone gas sensors. Sens Actuators B Chem 298: 126871. https://doi.org/10.1016/j.snb.2019.126871 doi: 10.1016/j.snb.2019.126871
![]() |
[12] |
Song Z, Zhang J, Jiang J (2020) Morphological evolution, luminescence properties and a high-sensitivity ethanol gas sensor based on 3D flower-like MoS2-ZnO micro/nanosphere arrays Ceram Int 46: 6634-6640. https://doi.org/10.1016/j.ceramint.2019.11.151 doi: 10.1016/j.ceramint.2019.11.151
![]() |
[13] |
Zhang K, Qin S, Tang P, et al. (2020) Ultra-sensitive ethanol gas sensors based on nanosheet-assembled hierarchical ZnO-In2O3 heterostructures. J Hazard Mater 391: 122191. https://doi.org/10.1016/j.jhazmat.2020.122191 doi: 10.1016/j.jhazmat.2020.122191
![]() |
[14] |
Cao P, Yang Z, Navale ST, et al. (2019) Ethanol sensing behavior of Pd-nanoparticles decorated ZnO-nanorod based chemiresistive gas sensors. Sensor Actuat B Chem 298: 126850. https://doi.org/10.1016/j.snb.2019.126850 doi: 10.1016/j.snb.2019.126850
![]() |
[15] |
Zeng Q, Cui Y, Zhu L, et al. (2020) Increasing oxygen vacancies at room temperature in SnO2 for enhancing ethanol gas sensing. Mat Sci Semicon Proc 111: 104962. https://doi.org/10.1016/j.mssp.2020.104962 doi: 10.1016/j.mssp.2020.104962
![]() |
[16] |
Yu HL, Wang J, Zheng B, et al. (2020) Fabrication of single crystalline WO3 nano-belts based photoelectric gas sensor for detection of high concentration ethanol gas at room temperature. Sens Actuator A Phys 303: 111865. https://doi.org/10.1016/j.sna.2020.111865 doi: 10.1016/j.sna.2020.111865
![]() |
[17] |
Mao JN, Hong B, Chen HD, et al. (2020) Highly improved ethanol gas response of n-type α‑Fe2O3 bunched nanowires sensor with high-valence donor-doping. J Alloys Compd 827: 154248. https://doi.org/10.1016/j.jallcom.2020.154248 doi: 10.1016/j.jallcom.2020.154248
![]() |
[18] |
Cao E, Wu A, Wang H, et al. (2019) Enhanced ethanol sensing performance of Au and Cl comodified LaFeO3 nanoparticles. ACS Appl Nano Mater 2: 1541-1551. https://doi.org/10.1021/acsanm.9b00024 doi: 10.1021/acsanm.9b00024
![]() |
[19] |
Shingange K, Swart HC, Mhlongo, GH (2020) Design of porous p-type LaCoO3 nanofibers with remarkable response and selectivity to ethanol at low operating temperature. Sensor Actuat B Chem 308: 127670. https://doi.org/10.1016/j.snb.2020.127670 doi: 10.1016/j.snb.2020.127670
![]() |
[20] |
Cao E, Feng Y, Guo Z, et al. (2020) Ethanol sensing characteristics of (La, Ba)(Fe, Ti)O3 nanoparticles with impurity phases of BaTiO3 and BaCO3. J Sol‑Gel Sci Techn 96: 431-440. https://doi.org/10.1007/s10971-020-05369-x doi: 10.1007/s10971-020-05369-x
![]() |
[21] |
Xiang J, Chen X, Zhang X, et al. (2018) Preparation and characterization of Ba-doped LaFeO3 nanofibers by electrospinning and their ethanol sensing properties. Mater Chem Phys 213: 122-129. https://doi.org/10.1016/j.matchemphys.2018.04.024 doi: 10.1016/j.matchemphys.2018.04.024
![]() |
[22] | Nga PTT, My DTT, Duc NM, et al. (2021) Characteristics of porous spherical LaFeO3 as ammonia gas sensing material. Vietnam J Chem 59: 676-683. |
[23] |
Zhang Y, Zhao J, Sun H, et al. (2018) B, N, S, Cl doped graphene quantum dots and their effect on gas-sensing properties of Ag-LaFeO3. Sensor Actuat B Chem 266: 364-374. https://doi.org/10.1016/j.snb.2018.03.109 doi: 10.1016/j.snb.2018.03.109
![]() |
[24] |
Zhang W, Yang B, Liu J, et al. (2017) Highly sensitive and low operating temperature SnO2 gas sensor doped by Cu and Zn two elements. Sensor Actuat B Chem 243: 982-989. https://doi.org/10.1016/j.snb.2016.12.095 doi: 10.1016/j.snb.2016.12.095
![]() |
[25] |
Qin J, Cui Z, Yang X, et al. (2015) Three-dimensionally ordered macroporous La1-xMgxFeO3 as high performance gas sensor to methanol. J Alloys Compd 635: 194-202. https://doi.org/10.1016/j.jallcom.2015.01.226 doi: 10.1016/j.jallcom.2015.01.226
![]() |
[26] |
Chen M, Wang H, Hu J, et al. (2018) Near-room-temperature ethanol gas sensor based on mesoporous Ag/Zn-LaFeO3 nanocomposite. Adv Mater Interfaces 6: 1801453. https://doi.org/10.1002/admi.201801453 doi: 10.1002/admi.201801453
![]() |
[27] |
Wang C, Rong Q, Zhang Y, et al. (2019) Molecular imprinting Ag-LaFeO3 spheres for highly sensitive acetone gas detection. Mater Res Bull 109: 265-272. https://doi.org/10.1016/j.materresbull.2018.09.040 doi: 10.1016/j.materresbull.2018.09.040
![]() |
[28] |
Koli PB, Kapadnis KH, Deshpande UG, et al. (2020) Sol-gel fabricated transition metal Cr3+, Co2+ doped lanthanum ferric oxide (LFO-LaFeO3) thin film sensors for the detection of toxic, flammable gases: A comparative study. Mat Sci Res India 17: 70-83. https://doi.org/10.13005/msri/170110 doi: 10.13005/msri/170110
![]() |
[29] |
Ma L, Ma SY, Qiang Z, et al. (2017) Preparation of Co-doped LaFeO3 nanofibers with enhanced acetic acid sensing properties. Mater Lett 200: 47-50. https://doi.org/10.1016/j.matlet.2017.04.096 doi: 10.1016/j.matlet.2017.04.096
![]() |
[30] |
Manzoor S, Husain S, Somvanshi A, et al. (2019) Impurity induced dielectric relaxor behavior in Zn doped LaFeO3. J Mater Sci Mater Electron 30: 19227-19238. https://doi.org/10.1007/s10854-019-02281-1 doi: 10.1007/s10854-019-02281-1
![]() |
[31] |
Tumberphale UB, Jadhav SS, Raut SD, et al. (2020) Tailoring ammonia gas sensing performance of La3+-doped copper cadmium ferrite nanostructures. Solid State Sci 100: 106089. https://doi.org/10.1016/j.solidstatesciences.2019.106089 doi: 10.1016/j.solidstatesciences.2019.106089
![]() |
[32] | Suhendi E, Amanda ZA, Ulhakim MT, et al. (2021) The enhancement of ethanol gas sensors response based on calcium and zinc co-doped LaFeO3/Fe2O3 thick film ceramics utilizing yarosite minerals extraction as Fe2O3 precursor. J Met Mater Miner 31: 71-77. |
[33] |
Lai T, Fang T, Hsiao Y, et al (2019) Characteristics of Au-doped SnO2-ZnO heteronanostructures for gas sensing applications. Vacuum 166: 155-161. https://doi.org/10.1016/j.vacuum.2019.04.061 doi: 10.1016/j.vacuum.2019.04.061
![]() |
[34] |
Ehsani M, Hamidon MN, Toudeshki A, et al. (2016) CO2 gas sensing properties of screen-printed La2O3/SnO2 thick film. IEEE Sens J 16: 6839-6845. https://doi.org/10.1109/JSEN.2016.2587779 doi: 10.1109/JSEN.2016.2587779
![]() |
[35] |
Isabel RTM, Onuma S, Angkana P, et al. (2018) Printed PZT thick films implemented for functionalized gas sensors. Key Eng Mater 777: 158-162. https://doi.org/10.4028/www.scientific.net/KEM.777.158 doi: 10.4028/www.scientific.net/KEM.777.158
![]() |
[36] |
Moschos A, Syrovy T, Syrova L, et al. (2017) A screen-printed flexible flow sensor. Meas Sci Technol 28: 055105. https://doi.org/10.1088/1361-6501/aa5fa0 doi: 10.1088/1361-6501/aa5fa0
![]() |
[37] | Suhendi E, Ulhakim MT, Setiawan A, et al. (2019) The effect of SrO doping on LaFeO3 using yarosite extraction based ethanol gas sensors performance fabricated by coprecipitation method. Int J Nanoelectron Mater 12: 185-192. |
[38] |
Kou X, Wang C, Ding M, et al. (2016) Synthesis of Co-doped SnO2 nanofibers and their enhanced gas-sensing properties. Sensor Actuat B Chem 236: 425-432. https://doi.org/10.1016/j.snb.2016.06.006 doi: 10.1016/j.snb.2016.06.006
![]() |
[39] |
Wang H, Wei S, Zhang F, et al. (2017) Sea urchin-like SnO2/Fe2O3 microspheres for an ethanol gas sensor with high sensitivity and fast response/recovery. J Mater Sci Mater Electron 28: 9969-9973. https://doi.org/10.1007/s10854-017-6755-3 doi: 10.1007/s10854-017-6755-3
![]() |
[40] |
Li Z, Yi J (2017) Enhanced ethanol sensing of Ni-doped SnO2 hollow spheres synthesized by a one-pot hydrothermal method. Sensor Actuat B Chem 243: 96-103. https://doi.org/10.1016/j.snb.2016.11.136 doi: 10.1016/j.snb.2016.11.136
![]() |
[41] |
Cheng Y, Guo H, Wang Y, et al. (2018) Low cost fabrication of highly sensitive ethanol sensor based on Pd-doped α-Fe2O3 porous nanotubes. Mater Res Bull 105: 21-27. https://doi.org/10.1016/j.materresbull.2018.04.025 doi: 10.1016/j.materresbull.2018.04.025
![]() |
[42] |
Zhang J, Liu L, Sun C, et al. (2020) Sr-doped α-Fe2O3 3D layered microflowers have high sensitivity and fast response to ethanol gas at low temperature. Chem Phys Lett 750: 137495. https://doi.org/10.1016/j.cplett.2020.137495 doi: 10.1016/j.cplett.2020.137495
![]() |
[43] |
Wei W, Guo S, Chen C, et al. (2017) High sensitive and fast formaldehyde gas sensor based on Ag-doped LaFeO3 nanofibers. J Alloys Compd 695: 1122-1127. https://doi.org/10.1016/j.jallcom.2016.10.238 doi: 10.1016/j.jallcom.2016.10.238
![]() |
[44] |
Cao E, Yang Y, Cui T, et al. (2017) Effect of synthesis route on electrical and ethanol sensing characteristics for LaFeO3-δ nanoparticles by citric sol-gel method. Appl Surf Sci 393: 134-143. https://doi.org/10.1016/j.apsusc.2016.10.013 doi: 10.1016/j.apsusc.2016.10.013
![]() |
[45] |
Cao E, Wang H, Wang X, et al. (2017) Enhanced ethanol sensing performance for chlorine doped nanocrystalline LaFeO3-δ powders by citric sol-gel method. Sensor Actuat B Chem 251: 885-893. https://doi.org/10.1016/j.snb.2017.05.153 doi: 10.1016/j.snb.2017.05.153
![]() |
[46] |
Zhang Y, Duan Z, Zou H, et al. (2018) Fabrication of electrospun LaFeO3 nanotubes via annealing technique for fast ethanol detection. Mater Lett 215: 58-61. https://doi.org/10.1016/j.matlet.2017.12.062 doi: 10.1016/j.matlet.2017.12.062
![]() |
[47] |
Ariyani NI, Syarif DG, Suhendi E (2017) Fabrication and characterization of thick film ceramics La0.9Ca0.1FeO3 for ethanol gas sensor using extraction of Fe2O3 from yarosite mineral. IOP Conf Ser Mater Sci Eng 384: 012037. https://doi.org/10.1088/1757-899X/384/1/012037 doi: 10.1088/1757-899X/384/1/012037
![]() |
[48] |
Zhou Q, Chen W, Xu L, et al. (2018) Highly sensitive carbon monoxide (CO) gas sensors based on Ni and Zn doped SnO2 nanomaterials. Ceram Int 44: 4392-4399. https://doi.org/10.1016/j.ceramint.2017.12.038 doi: 10.1016/j.ceramint.2017.12.038
![]() |
[49] |
Hao P, Qiu G, Song P, et al. (2020) Construction of porous LaFeO3 microspheres decorated with NiO nanosheets for high response ethanol gas sensors. Appl Surf Sci 515: 146025. https://doi.org/10.1016/j.apsusc.2020.146025 doi: 10.1016/j.apsusc.2020.146025
![]() |
[50] |
Srinivasan P, Ezhilan M, Kulandaisamy AJ, et al. (2019) Room temperature chemiresistive gas sensors: challenges and strategies-a mini review. J Mater Sci Mater Electron 30: 15852-15847. https://doi.org/10.1007/s10854-019-02025-1 doi: 10.1007/s10854-019-02025-1
![]() |
[51] |
Gao H, Wei D, Lin P, et al. (2017) The design of excellent xylene gas sensor using Sn-doped NiO hierarchical nanostructure. Sensor Actuat B Chem 253: 1152-1162. https://doi.org/10.1016/j.snb.2017.06.177 doi: 10.1016/j.snb.2017.06.177
![]() |
[52] |
Hu X, Zhu Z, Chen C, et al. (2017) Highly sensitive H2S gas sensors based on Pd-doped CuO nanoflowers with low operating temperature. Sensors Actuat B Chem 253: 809-817. https://doi.org/10.1016/j.snb.2017.06.183 doi: 10.1016/j.snb.2017.06.183
![]() |
[53] |
Singh G, Virpal, Singh RC (2019) Highly sensitive gas sensor based on Er-doped SnO2 nanostructures and its temperature dependent selectivity towards hydrogen and ethanol. Sensor Actuat B Chem 282: 373-383. https://doi.org/10.1016/j.snb.2018.11.086 doi: 10.1016/j.snb.2018.11.086
![]() |
[54] |
Phan TTN, Dinh TTM, Nguyen MD, et al. (2022) Hierarchically structured LaFeO3 with hollow core and porous shell as efficient sensing material for ethanol detection. Sensor Actuat B Chem 354: 131195. https://doi.org/10.1016/j.snb.2021.131195 doi: 10.1016/j.snb.2021.131195
![]() |
[55] |
Montazeri A, Jamali-Sheini F (2017) Enhanced ethanol gas-sensing performance of Pb-doped In2O3 nanosructures prepared by sonochemical method. Sensor Actuat B Chem 242: 778-791. https://doi.org/10.1016/j.snb.2016.09.181 doi: 10.1016/j.snb.2016.09.181
![]() |
[56] |
Mirzael A, Lee J, Majhi SM, et al. (2019) Resistive gas sensors based on metal-oxide nanowires. J Appl Phys 126: 241102. https://doi.org/10.1063/1.5118805 doi: 10.1063/1.5118805
![]() |
[57] |
Yang K, Ma J, Qiao X, et al. (2020) Hierarchical porous LaFeO3 nanostructure for efficient trace detection of formaldehyde. Sensor Actuat B Chem 313: 128022. https://doi.org/10.1016/j.snb.2020.128022 doi: 10.1016/j.snb.2020.128022
![]() |
[58] |
Liu C, Navale ST, Yang ZB, et al. (2017) Ethanol gas-sensing properties of hydrothermally grown α-MnO2 nanorods. J Alloys Compd 727: 362-369. https://doi.org/10.1016/j.jallcom.2017.08.150 doi: 10.1016/j.jallcom.2017.08.150
![]() |
1. | Jiajie Luo, Lina Liu, Sisi Li, Yongxin Yuan, A direct method for the simultaneous updating of finite element mass, damping and stiffness matrices, 2022, 0308-1087, 1, 10.1080/03081087.2022.2092047 |