Abstract
Recently, spaces of ideal convergent sequences of bounded linear operators were studied by Khan et al. (Numer. Funct. Anal. Optim. 39:1278-1290, 2018). This has motivated us to propose the intuitionistic fuzzy I-convergent double sequence spaces determined by the bounded linear operator. In this paper, we investigate the algebraic and topological properties. We also study the concept of the ideal Cauchy and ideal convergence on the said spaces.
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1 Introduction
Zadeh [27] introduced the concept of fuzzy sets in 1965 and Goguen [6] extended it to L-fuzzy sets. As far as the theme of the concept of fuzzy sets is concerned, the idea has been utilized by the researchers around the globe heavily. Fuzzy sets have been put on to the metric spaces and emerged as fuzzy metric spaces studied by George et al. [5] and Amini et al. [1]. Furthermore, the idea of I and \(I^{\ast }\) convergent sequences in fuzzy normed spaces was due to Kumar et al. [14]. In 1986, Atanassov [2] started the study of intuitionistic fuzzy sets which is a generalization of fuzzy sets. Park [20] initiated the notion of intuitionistic fuzzy metric spaces moreover, Saadati et al. [21] extended this idea to intuitionistic fuzzy topological spaces. The study of the convergence of sequences in a fuzzy normed space is vital to fuzzy functional analysis, we feel that I-convergence in intuitionistic fuzzy normed space would yield a more general foundation. Later on, statistical convergence and ideal convergence of sequences concerning intuitionistic fuzzy normed space were studied by Mursaleen et al. [16, 19]. Moreover, the contributions to the study of intuitionistic fuzzy metric spaces and intuitionistic fuzzy normed spaces can be found in [7, 10, 11, 24].
The notion of ideal convergence was initiated by Kostyrko et al. [12] using the concept of the ideal I as a subset of the set of positive integers which is a generalization of statistical convergence given by Fast [4] in 1951. Furthermore, it was examined from the sequence space viewpoint and connected with the summability theory by Šalát et al. [22, 23]; Tripathy et al. [25, 26] defined paranorm I-convergent sequence spaces; Khan et al. [9] studied the ideal convergent sequence of bounded linear operator. Later on, Das et al. [3] studied I and \(I^{\ast }\)-convergence of double sequences. Mursaleen et al. [15, 17, 18] analyzed ideal convergence in random 2-normed spaces and probabilistic normed spaces.
Our aim for the present paper is to discuss the concept of intuitionistic fuzzy ideal convergence of double sequence spaces defined by the bounded linear operator which would yield a more convenient structure to deal with the inexactness of the sequence spaces in some situations.
2 Preliminaries
Now, we present some notations and basic definitions.
Definition 2.1
([12])
A family of sets \(I\subseteq 2^{Y}\) is called an ideal in nonempty set Y, if
-
\(\emptyset \in I\);
-
I is additive; that is, \(C, D\in I \Rightarrow C\cup D\in I\);
-
I is hereditary that is, \(C\in I\), \(D\subseteq C\Rightarrow D\in I\).
\(I\subseteq 2^{Y}\) is said to be nontrivial if \(I\neq 2^{Y}\). If \(\{ \{y\}: y\in Y \} \subseteq I\), then a nontrivial ideal \(I\subseteq 2^{Y}\) is called admissible. I is maximal if there cannot exist any nontrivial ideal J containing I as a subset.
Definition 2.2
([12])
Suppose Y is a nonempty set. Then \(\mathcal{F} \subset 2^{Y}\) is called a filter on Y if and only if the following implications hold:
-
\(\emptyset \notin \mathcal{F}\);
-
for \(C, D \in \mathcal{F} \Rightarrow C \cap D \in \mathcal{F}\);
-
for each \(C \in \mathcal{F}\) and \(D \supset C \Rightarrow D \in \mathcal{F}\).
For every ideal I there corresponds a filter defined as
We take \(I_{2}\) as a nontrivial ideal in \(\mathbb{N} \times \mathbb{N}\) throughout the paper.
Definition 2.3
([26])
A double sequence \(y=(y_{ij}) \in {_{2}\omega }\) is said to be \(I_{2}\)-convergent to L if, for every \(\epsilon >0\), we have
We write \(I_{2}-\lim y_{ij} =L\).
Definition 2.4
([26])
A sequence \(y = (y_{ij})\) is said to be \(I_{2}\)-Cauchy, if for each \(\epsilon > 0\), there exist positive integers \(m = m(\epsilon )\) and \(n = n(\epsilon )\) such that the set
Definition 2.5
([19])
Let \((Y, \phi , \psi , \ast , \diamond )\) be an intuitionistic fuzzy normed space (IFNS). A sequence \(y = (y_{ij})\) is termed convergent to \(L \in Y\) under intuitionistic fuzzy norm \((\phi ,\psi )\) if, for every ϵ, \(t > 0\), there exist \(k_{0} \in \mathbb{N}\) such that \(\phi (y_{ij}- L, t) > 1- \epsilon \) and \(\psi (y_{ij} - L, t) < \epsilon \) for all \(i,j \geq k _{0}\).
Definition 2.6
([19])
Let \((Y, \phi , \psi , \ast , \diamond )\) be an IFNS. A sequence \(y = (y_{ij})\) is termed a Cauchy sequence with respect to the intuitionistic fuzzy norm \((\phi ,\psi )\), if for every ϵ, \(t > 0\), \(\exists ~k_{0} \in \mathbb{N}\) such that \(\phi (y_{ij} - y_{mn}, t) > 1- \epsilon \) and \(\psi (y_{ij} - y _{mn}, t) < \epsilon \), for all \(i,j,m,n \geq k_{0}\).
Remark
([21])
If ∗ is a continuous t-norm, ⋄ is a continuous t-conorm and \(p_{i} \in (0, 1)\), \(1 \leq i \leq 7\). Then:
-
for any \(p_{1}\), \(p_{2} \in (0, 1)\) with \(p_{1} > p_{2}\), there exist \(p_{3}\), \(p_{4} \in (0, 1)\) such that
$$ p_{1} \ast p_{3} \geq p_{2} \quad \text{and}\quad p_{1} \geq p_{4} \diamond p _{2}; $$ -
for any \(p_{5} \in (0, 1)\), there exist \(p_{6}, p_{7} \in (0, 1)\) such that \(p_{6} \ast p_{6} \geq p_{5}\) and \(p_{7} \diamond p_{7} \leq p _{5}\).
Definition 2.7
([19])
Let \(I_{2} \subset 2^{ \mathbb{N}\times \mathbb{N}}\) be a nontrivial ideal and \((Y, \phi , \psi , \ast , \diamond )\) be an IFNS. A sequence \(y = (y_{ij})\) in Y is called \(I_{2}\)-convergent to \(L\in Y\) with respect to the intuitionistic fuzzy norm \((\phi , \psi )\) if, for every ϵ, \(t > 0\), the set
We write \(I_{2}^{(\phi , \psi )}-\lim y_{ij} = L\).
Definition 2.8
([19])
Let \((Y, \phi , \psi , \ast , \diamond )\) be an IFNS. A sequence \(y = (y_{ij})\) in Y is called an \(I_{2}\)-Cauchy sequence with respect to the intuitionistic fuzzy norm \((\phi ,\psi )\), if, for every ϵ, \(t > 0\), the set
Definition 2.9
([13])
Let U and V be two normed linear spaces and \(B : \mathcal{D}(B) \rightarrow V\) be a linear operator, where \(\mathcal{D}(B) \subset U\). An operator B is bounded, if there exists \(k > 0\) such that
We denote by \(\mathcal{B}(U, V)\) the set of all bounded linear operators which is normed linear spaces normed by
and \(\mathcal{B}(U, V)\) is a Banach space if V is a Banach space.
3 Main results
In this section, we introduce the following new sequence spaces:
An open ball with center x and radius r with respect to t is defined as
Theorem 3.1
\({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) and \({}_{2}S^{I_{2}}_{0(\phi , \psi )}(B)\) are linear spaces.
Proof
Let \(x = (x_{ij})\), \(y = (y_{ij}) \in {}_{2}S^{I_{2}} _{(\phi , \psi )}(B)\) and α, β be scalars. For a given \(\epsilon > 0\), we obtain
Define \(A_{3} = A_{1} \cup A_{2}\), so that \(A_{3} \in I_{2}\). It implies that \(A^{c}_{3}\) is a nonempty set in \(\mathcal{F}(I_{2})\). Now, we have to show that, for each \((x_{ij}), (y_{ij}) \in {}_{2}S^{I_{2}}_{( \phi , \psi )}(B)\), \(A^{c}_{3} \subset \{(i,j): \phi ((\alpha B(x_{ij}) + \beta B(y_{ij})) - (\alpha L_{1} + \beta L_{2}), t) > 1 - \epsilon \mbox{ and }\psi (( \alpha B(x_{ij}) + \beta B(y_{ij})) - (\alpha L_{1} + \beta L_{2}), t) < \epsilon \}\). Let \(m,n \in A^{c}_{3}\). In this case
We have
Also
This implies
Hence \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) is a linear space.
In a similar way, we can prove that \({}_{2}S^{I_{2}}_{0(\phi , \psi )}(B)\) is linear space. □
Theorem 3.2
Every open ball \({}_{2}\mathcal{B}_{x}(r, t)(B)\) is an open set in \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\).
Proof
Suppose \(y \in {}_{2}\mathcal{B}_{x}(r, t)(B)\). Then, from the definition of \({}_{2}\mathcal{B}_{x}(r, t)(B)\), we have
From (3.1), there exists \(t_{0} \in (0, t)\) such that \(\phi (B(x _{ij}) - B(y_{ij}), t_{0}) > 1 - r\) and \(\psi (B(x_{ij}) - B(y_{ij}), t_{0}) < r\). Setting \(p_{0} = \phi (B(x_{ij}) - B(y_{ij}), t_{0})\). This implies that \(p_{0} > 1 - r\). Thus, there exists \(s \in (0, 1)\) with \(p_{0} > 1 - s > 1 - r\). For given \(p_{0}\), s with \(p_{0} > 1 - s\), there exist \(p_{1}\), \(p_{2} \in (0, 1)\) such that \(p_{0} \ast p_{1} > 1 - s\) and \((1 - p_{0}) \diamond (1 - p_{2}) \leq s\). Select \(p_{3} = \max \{{p_{1}, p_{2}}\}\) and consider the ball \({}_{2}\mathcal{B}_{y}(1 - p_{3}, t - t_{0})(B)\). Now, we show
Let \(z = (z_{ij}) \in {}_{2}\mathcal{B}_{y}(1 - p_{3}, t - t_{0})(B)\). Then \(\phi (B(y_{ij}) - B(z_{ij}), t - t_{0}) > p_{3}\) and \(\psi (B(y_{ij}) - B(z_{ij}), t - t_{0}) < 1 - p_{3}\).
Therefore
and
The above inequalities imply that \(z \in {}_{2}\mathcal{B}_{x}(r, t)(B)\). Thus \({}_{2}\mathcal{B}_{y}(1 - p_{3}, t - t_{0})(B) \subset {}_{2}\mathcal{B}_{x}(r, t)(B)\). □
Remark
Let \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) be an IFNS. Define \({}_{2}\tau ^{I_{2}}_{(\phi , \psi )}(B) = \{K \subset {}_{2}S^{I_{2}} _{(\phi , \psi )}(B) : \text{ for each } x \in K, \exists~ t > 0 \text{ and } r \in (0, 1) \text{ such that } {}_{2}\mathcal{B}_{x}(r, t)(B) \subset K\}\). Then \({}_{2}\tau ^{I}_{( \phi , \psi )}(B)\) is a topology on \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\).
Theorem 3.3
The spaces \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) and \({}_{2}S^{I_{2}}_{0(\phi , \psi )}(B)\) are Hausdorff.
Proof
Let x, \(y \in {}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) be two distinct points. Then \(0 < \phi (B(x) - B(y), t) < 1\) and \(0 < \psi (B(x) - B(y), t) < 1\). Putting \(p_{1} = \phi (B(x) - B(y), t)\), \(p_{2} = \psi (B(x) - B(y), t)\) and \(p = \max \{{p_{1}, 1 - p _{2}}\}\).
For each \(p_{0} \in (p, 1)\) there exist \(p_{3}\) and \(p_{4}\) such that \(p_{3} \ast p_{3} \geq p_{0}\) and \((1 - p_{4}) \diamond (1 - p_{4}) \leq (1 - p_{0})\).
Put \(p_{5}= \max \{{p_{3}, p_{4}}\}\) and consider the open balls \({}_{2}\mathcal{B}_{x}(1 - r_{p}, \frac{t}{2})\) and \({}_{2}\mathcal{B} _{y}(1 - p_{5}, \frac{t}{2})\).
Clearly \({}_{2}\mathcal{B}_{x}(1 - p_{5}, \frac{t}{2}) \cap{}_{2} \mathcal{B}_{y}(1 - p_{5}, \frac{t}{2}) = \emptyset \).
If there exists \(z \in {}_{2}\mathcal{B}_{x}(1 - p_{5}, \frac{t}{2}) \cap{}_{2}\mathcal{B}_{y}(1 - p_{5}, \frac{t}{2})\), then
and
which is a contradiction. Hence \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) is Hausdorff space. Similarly, one can show that \({}_{2}S^{I_{2}}_{0( \phi , \psi )}(B)\) is Hausdorff space. □
Theorem 3.4
\({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) is an IFNS and \({}_{2}\tau ^{I_{2}}_{(\phi , \psi )}(B)\) is a topology on \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\). Then a sequence \((x_{ij}) \in {}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\), \(x_{ij} \rightarrow x\) if and only if \(\phi (B(x_{ij}) - B(x), t) \rightarrow 1\) and \(\psi (B(x_{ij}) - B(x), t) \rightarrow 0\) as \(i,j \rightarrow \infty \).
Proof
Choose \(t_{0} > 0\). Suppose \(x_{ij} \rightarrow x\). Then for \(0 < r < 1\), \(\exists ~N_{0} \in \mathbb{N}\) such that \((x_{ij}) \in {}_{2}S^{I_{2}}_{(\phi , \psi )}(B) \) for all \(i, j \geq N _{0}\),
such that \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B) \in \mathcal{F}(I_{2})\). Then \(1 - \phi (B(x_{ij}) - B(x), t) < r\) and \(\psi (B(x_{ij}) - B(x), t) < r\).
Hence \(\phi (B(x_{ij}) - B(x), t) \rightarrow 1 \) and \(\psi (B(x_{ij}) - B(x), t) \rightarrow 0\) as \(i,j \rightarrow \infty \).
On the other way around, if for each \(t > 0\), \(\phi (B(x_{ij}) - B(x), t) \rightarrow 1\) and \(\psi (B(x_{ij}) - B(x), t) \rightarrow 0\) as \(i,j \rightarrow \infty \). Then for \(0 < r < 1\), \(\exists ~N_{0} \in \mathbb{N}\) such that \(1 - \phi (B(x_{ij}) - B(x), t) < r\) and \(\psi (B(x_{ij}) - B(x), t) < r\), for all \(i,j \geq N_{0}\). This implies that \(\phi (B(x_{ij}) - B(x), t) > 1 - r\) and \(\psi (B(x_{ij}) - B(x) , t) < r\) for all \(i,j \geq N_{0}\). Thus \((x_{ij}) \in {}_{2}S^{I_{2}} _{(\phi , \psi )}(B)\) for all \(i,j \geq N_{0}\) and hence \(x_{ij} \rightarrow x\). □
Theorem 3.5
Let \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) be an IFNS. If a double sequence \(x = (x_{ij})\) is \(I_{2}\)-convergent with respect to the intuitionistic fuzzy norms \((\phi , \psi )\), then the \(I_{2}^{(\phi , \psi )}\)-limit is unique.
Proof
Let \(I_{2}^{(\phi , \psi )}- \lim x = L_{1}\) and \(I_{2}^{(\phi , \psi )}- \lim x = L_{2}\). For a given \(\epsilon > 0\), select \(s > 0\) in such a way that \((1-s) \ast (1-s) > 1-\epsilon \) and \(s\diamond s < \epsilon \). Then, for any \(t > 0\), we define the following sets:
Since \(I_{2}^{(\phi , \psi )}- \lim x = L_{1}\), we obtain \(P_{\phi , 1}(s, t)\) and \(P_{\psi , 1}(s, t) \in I_{2}\).
Moreover, using \(I_{2}^{(\phi , \psi )}- \lim x = L_{2}\), we have \(P_{\phi , 2}(s, t)\) and \(P_{\psi , 2}(s, t) \in I_{2}\). Now, suppose that
Thus, \(P_{\phi , \psi }(s, t) \in I_{2}\), implies that \(P^{c}_{\phi , \psi }(s, t)\) is a nonempty set in \(\mathcal{F}(I_{2})\).
If \((i, j) \in P^{c}_{\phi , \psi }(s, t)\), then two possibilities arise:
Firstly, we consider that \((i, j) \in P^{c}_{\phi , 1}(s, t) \cap P ^{c}_{\phi , 2}(s, t)\). Then we get
Since \(\epsilon > 0\) was arbitrary, we obtain \(\phi (L_{1} - L_{2}, t) = 1\) for every \(t > 0\), which yields \(L_{1} = L_{2}\). Under other conditions, if \((i, j) \in P^{c}_{\psi , 1}(s, t) \cap P^{c}_{\psi , 2}(s, t)\), then we may write
Therefore, we obtain \(\psi (L_{1} - L_{2}, t) = 0\), for all \(t > 0\), which yields \(L_{1} = L_{2}\). Hence, in all cases, we find that the \(I_{2}^{(\phi , \psi )}\)-limit is unique. □
Theorem 3.6
A sequence \(x = (x_{ij}) \in {}_{2}S^{I_{2}} _{(\phi , \psi )}(B)\) is \(I_{2}\)-convergent with respect to the intuitionistic fuzzy norm \((\phi , \psi )\) if and only if it is \(I_{2}\)-Cauchy with respect to same norm.
Proof
Suppose that the sequence \(x = (x_{ij}) \in {}_{2}S ^{I_{2}}_{(\phi , \psi )}(B)\) is \(I_{2}\)-convergent, i.e., \(I_{2}^{( \phi , \psi )}- \lim x = L\). Take \(s > 0\), in such a way that \((1 - s) \ast (1 - s) > 1 - \epsilon \) and \(s\diamond s < \epsilon \). For all \(t > 0\), we get
This implies \(P^{c} = \{ (i, j) \in \mathbb{N} \times \mathbb{N} : \phi (B(x_{ij}) - L, t) > 1 - s \text{ and } \psi (B(x_{ij}) - L, t) < s\} \in \mathcal{F}(I_{2})\). Suppose \((m, n) \in P^{c}\). Then we obtain
Let \(Q = \{ (i, j) \in \mathbb{N} \times \mathbb{N} : \phi (B(x_{ij}) - B(x_{mn}), t) \leq 1 - \epsilon \text{ or } \psi (B(x_{ij}) - B(x _{mn}), t) \geq \epsilon \}\).
Furthermore, we prove the inclusion \(Q \subset P\). Let \((i, j) \in Q\), we have
There are two possible cases, firstly we consider \(\phi (B(x_{ij}) - B(x _{mn}), t) \leq 1 - \epsilon \). Then we have \(\phi (B(x_{ij}) - L, \frac{t}{2}) \leq 1 - s\), therefore \((i, j) \in P\). On the other hand, if \(\phi (B(x_{ij}) - L, \frac{t}{2}) > 1 - s\) then
which is impossible. Hence \(Q \subset P\).
Similarly, consider \(\psi (B(x_{ij}) - B(x_{mn}), t) \geq \epsilon \). Then we have \(\psi (B(x_{ij}) - L, \frac{t}{2}) \geq s\), hence \((i, j) \in P\). Otherwise, if \(\psi (B(x_{ij}) - L, \frac{t}{2}) < s\), then
which is impossible. Hence \(Q \subset P\). Thus in both cases we conclude that \(Q \subset P\). Therefore \(Q \in I\). Hence x is \(I_{2}\) Cauchy with respect to the intuitionistic fuzzy norm \((\phi , \psi )\).
Contrarily, suppose that \(x = (x_{ij})\) is \(I_{2}\)-Cauchy but not \(I_{2}\)-convergent with respect to the intuitionistic fuzzy norm \((\phi , \psi )\). Then there exist p and q such that
and
Equivalently, \(Q^{c}(\epsilon , t) \in \mathcal{F}(I_{2})\). Since
and
if \(\phi (B(x_{ij}) - L, \frac{t}{2}) > \frac{1 - \epsilon }{2}\) and \(\psi (B(x_{ij}) - L, \frac{t}{2}) < \frac{\epsilon }{2}\), respectively, we obtain \(P^{c}(\epsilon , t) \in I_{2}\) and so \(P(\epsilon , t) \in \mathcal{F}(I_{2})\), which contradicts our assumption. □
Theorem 3.7
Suppose \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) be an intuitionistic fuzzy normed space such that \({}_{2}S^{I_{2}}_{( \phi , \psi )}(B)\) has a convergent subsequence. Then \({}_{2}S^{I_{2}} _{(\phi , \psi )}(B)\) is complete.
Proof
Let \((x_{i_{m}j_{n}})\) be a subsequence of Cauchy sequence \((x_{ij})\) that converges to x. We show that \((x_{ij}) \rightarrow x\) as \((i,j) \rightarrow \infty \). Let \(t > 0\) and \(\epsilon \in (0, 1)\). Choose \(s \in (0, 1)\) in a such way that \(s\diamond s \leq \epsilon \) and \((1- s)\ast (1-s) \geq 1 - \epsilon \). Since \((x_{ij})\) is a Cauchy sequence, \(\exists ~N_{0} \in \mathbb{N}\) such that \(\phi (B(x_{ij}) - B(x_{pq}) , \frac{t}{2}) > 1 - s\) and \(\psi (B(x_{ij}) - B(x_{pq}) , \frac{t}{2}) < s\), for all \(i, j, p \text{ and } q \geq N_{0} \). Since \((x_{m_{i}n_{j}}) \rightarrow x\), there is positive integer \(i_{k}, j_{l} > N_{0}\) such that \(\phi (B(x_{i_{k}j_{l}}) - B(x) , \frac{t}{2}) > 1 - s\) and \(\psi (B(x_{i_{k}j_{l}}) -B( x) , \frac{t}{2}) < s\). Then, if \(i,j \geq N_{0}\),
or
Since B is a bounded linear operator, \(x_{ij} \rightarrow x\) as \((i,j) \rightarrow \infty \). Hence \({}_{2}S^{I_{2}}_{(\phi , \psi )}(B)\) is complete. □
4 Conclusion
The concept of intuitionistic fuzzy convergence of sequences has been studied by numerous researchers. In the present work, we introduced a more general type of convergence of sequence spaces, namely the intuitionistic fuzzy ideal convergence of double sequence spaces defined by a bounded linear operator. We investigated some algebraic and topological properties on these spaces. Furthermore, we contributed new tools to work with the convergence problems of sequences in the intuitionistic fuzzy settings, occurring in various fields of science and engineering.
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We would like to thank reviewer for his valuable suggestions and comments which were very helpful in improving the manuscript.
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This work is financially supported by Mohd Faisal Khan, Assistant Professor in Department of Mathematics, Saudi Electronic University, Riyadh, -602002, Saudi Arabia.
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Khan, V.A., Ahmad, M., Fatima, H. et al. On some results in intuitionistic fuzzy ideal convergence double sequence spaces. Adv Differ Equ 2019, 375 (2019). https://doi.org/10.1186/s13662-019-2306-y
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DOI: https://doi.org/10.1186/s13662-019-2306-y