Inner product of fuzzy vectors

The inner product of vectors of non-normal fuzzy intervals will be studied in this paper by using extension principle and expression in decomposition theorem. The membership functions of inner product will be different with respect to these two different methodologies. Since the non-normal fuzzy interval is more general than the normal fuzzy interval, the corresponding membership functions will become more complicated. Therefore, we shall establish their relationship including the equivalence and fuzziness based on the α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}-level sets. The potential application of inner product of fuzzy vectors is to study the fuzzy linear programming problems.


Introduction
A fuzzy interval in R is a fuzzy set in R such that its α-level sets are bounded closed intervals. The purpose of this paper is to study the inner product of vectors of fuzzy intervals using extension principle and expression in decomposition theorem. Since the fuzzy linear programming problems can be formulated as the form of inner product of fuzzy vectors, the results obtained in this paper can be useful for studying the fuzzy linear programming problems.
There are two types of inner product that will be studied in this paper. The first type of inner product of fuzzy vectors is directly based on the inner product of vectors x and y given by the following expression x • y = x 1 y 1 + · · · + x n y n , where x and y are two vectors in R n . The extension principle and expression in decomposition theorem will directly apply to the (conventional) inner product x • y given above without considering the addition and multiplication of fuzzy intervals.
The second type of inner product of fuzzy vectors will be based on the addition and multiplication of fuzzy intervals B Hsien-Chung Wu hcwu@nknucc.nknu.edu.tw 1 Department of Mathematics, National Kaohsiung Normal University, Kaohsiung 802, Taiwan by considering the following expression ã (1) ⊗b (1) ⊕ · · · ⊕ ã (n) ⊗b (n) , whereã (i) andb (i) are fuzzy intervals in R for i = 1, . . . , n. The main issue of second type is the addition and multiplication of fuzzy intervals. In this paper, the addition and multiplication of fuzzy intervals will also be formulated based on the extension principle and expression in decomposition theorem. Therefore, the different combinations of using different addition and multiplication will generate many different second type of inner product of fuzzy vectors. Their relationship will be established. Moreover, the relationship between the first type and second type of inner product will also be studied.
The second type of inner product of vector of fuzzy intervals needs to consider the arithmetic operations ⊕ and ⊗. The original arithmetic operations are based on the minimum function and maximum function. The general t-norms and s-norms instead of minimum function and maximum function, respectively, are used by referring to Bede and Stefanini (2013), Dubois and Prade (1985), Gebhardt (1995), Gomes and Barros (2015), Fullér and Keresztfalvi (1990), Mesiar (1997), Ralescu (1992), Weber (1983), Wu (2013Wu ( , 2018Wu ( , 2019 and Yager (1986). More detailed properties regarding these arithmetic operations ⊕ and ⊗ can refer to the monographs Dubois and Prade (1988) and Klir and Yuan (1995). In this paper, we shall consider the general aggregation function rather than using t-norms and s-norms.
In Sect. 2, we shall present the basic properties of nonnormal fuzzy sets. In Sect. 3, using the general aggregation functions, the inner product of vectors of fuzzy intervals will be studied. On the other hand, expression in decomposition theorem will be used to study the first type inner product based on three different families. The equivalence and comparison of fuzziness will also be studied. In Sect. 4, the second type of inner product of vectors of fuzzy intervals will be proposed by using the addition and multiplication of fuzzy intervals. The relationship between the first type and second type of inner product will also be studied. Based on the fuzziness, the suitable appropriation for using the first type or second type is also suggested.

Non-normal fuzzy sets
LetÃ be a fuzzy set in R with membership function ξÃ. For α ∈ (0, 1], the α-level set ofÃ is denoted and defined bỹ (1) It is clear to see that if α ≥ sup x∈R ξÃ(x) thenÃ α = ∅. In this paper, we shall carefully avoid to be trapped in the empty α-level sets.
The support of fuzzy setÃ is a crisp set defined bỹ When R is endowed with a topology, the 0-level setÃ 0 is defined to be the closure of the support ofÃ, i.e.,Ã 0 = cl(Ã 0+ ). Let f : R → R be a real-valued function defined on R, and let S be a subset of R. We say that the supremum sup x∈S f (x) is attained when there exists x * ∈ S satisfying f (x) ≤ f (x * ) for all x ∈ S with x = x * . When the supremum sup x∈S f (x) is attained, we have (2) Then,Ã α = ∅ for all α ∈ IÃ andÃ α = ∅ for all α / ∈ IÃ. It is clear to see R(ξÃ) ⊆ IÃ. By referring to Wu (2019Wu ( , 2020b, we also havẽ The interval IÃ presented in (2) is called an interval range ofÃ. In general, we see that R(ξÃ) = IÃ. The role of interval range IÃ can be used to checkÃ α = ∅ for all α ∈ IÃ and A α = ∅ for all α / ∈ IÃ. The range R(ξÃ) is not helpful for identifying the α-level sets.
Recall thatÃ is called a normal fuzzy set in R when there exists x ∈ R satisfying ξÃ(x) = 1. In this case, the interval range ofÃ is given by IÃ = [0, 1]. However, the range R(ξÃ) is not necessarily equal to the whole unit interval [0, 1] even thoughÃ is normal.
The characteristic function χ A of a crisp set A is defined by The well-known decomposition theorem is based on the normal fuzzy sets in R, which says that the membership function ξÃ can be expressed as ξÃ(x) = sup α∈ [0,1] α · χÃ α (x) = sup α∈ (0,1] α · χÃ α (x), IfÃ is not normal, we can similarly obtain the following form.
Theorem 2.1 (Wu 2020a) (Decomposition Theorem) LetÃ be a fuzzy set in R. Then, the membership function ξÃ can be expressed as where the supremum is attained and the interval range IÃ is given in (2).

Definition 2.2
We say thatã is a fuzzy interval in R when the following conditions are satisfied: • The supremum sup x∈R ξÃ(x) is attained in the following sense sup x∈R ξÃ(x) = max x∈R ξÃ(x).
• The membership function ξã is upper semi-continuous and quasi-concave on R. • The 0-level setã 0 is a closed and bounded set in R.
It is well-known that the α-level sets of fuzzy intervalã are all bounded closed intervals denoted byã α = [ã L α ,ã U α ] for α ∈ Iã. When the fuzzy intervalã is normal and the 1level setã 1 is a singleton set {a} for some a ∈ R, the fuzzy intervalã is then called a fuzzy number with core value a.

The first type of inner product
Recall that the inner product of vectors x and y in R n is given by x • y = x 1 y 1 + · · · + x n y n , where x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) are two vectors in R n .
Given any fuzzy intervalsã (1) , . . . ,ã (n) andb (1) , . . . ,b (n) in R, we define and for i = 1, . . . , n. The interval ranges Iã(i) ofã (i) and Ib (i) of b (i) can be realized from (2). More precisely, we have and Let By referring to (5) and (6), we see that I * = ∅. Therefore, for each α ∈ I * , the α-level sets ofã (i) andb (i) are nonempty and denoted bỹ For convenience, we write We also writẽ Letã andb be two vectors of fuzzy intervals in R given bỹ We shall study the inner productã E Pb using extension principle, and the inner productã DTb using expression in decomposition theorem.

Using extension principle
We are going to use the extension principle to define the membership function of inner productã E Pb . Given two vectorsã andb of fuzzy intervals in (11) for each z ∈ R. If the aggregation function A is taken to be the minimum function, it recovers the conventional form of extension principle. In order to obtain the nonempty α-level sets ofã E Pb , we need to consider the interval range I (E P) of inner product a E Pb , which says that We also write R i ≡ R(ξã(i)) to denote the ranges of membership functions ξã(i) for i = 1, . . . , n, and write R n+i ≡ R(ξb (i) ) to denote the ranges of membership functions ξb (i) for i = 1, . . . , n. Then, we have By referring to (2), we have We also assume that the supremum sup R(ξã(i)) and sup R(ξb (i) ) are attained for i = 1, . . . , n. Then, the following supremum In particular, suppose that min α * 1 , . . . , α * n , β * 1 , . . . , β * n = A α * 1 , . . . , α * n , β * 1 , . . . , β * n .

Definition 3.2
We say thatã is a canonical fuzzy interval when the following conditions are satisfied.
•ã is a fuzzy interval.
In order to simplify the expression, we consider the nonnegativity of fuzzy sets. Definition 3.3 LetÃ be a fuzzy set in R with membership function ξÃ. We say thatÃ is nonnegative when ξÃ(x) = 0 for each x < 0.

Using expression in decomposition theorem
Letã (i) andb (i) be fuzzy intervals for i = 1, . . . , n. Now, we are going to use expression in decomposition theorem to define three different inner product by considering three different families.
Using expression in decomposition theorem, for DT ∈ { DT , DT , † DT }, the membership function ofã DTb is defined by where M • α corresponds to the above three cases (22), (23) and (24). We also have In order to consider the nonempty α-level sets. The interval ranges of ξã and and Therefore, the nonempty α-level sets can be realized below: Then, we have the following useful results regarding the interval ranges.
Proposition 3.7 Letã (1) , . . . ,ã (n) andb (1) , . . . ,b (n) be fuzzy intervals. Suppose that the supremum sup I * is attained. Then, the interval ranges are all identical given by Proof Recall the definition Iã(i) and Ib (i) in (5) and (6), respectively, for i = 1, . . . , n. It is clear to see that Since sup I * is assumed to be attained, it follows that I * = [0, α * ]. By referring to (26), we can take z ∈ M • α * , which says that the supremum α * is attained for the range R(ξã DTB ). Therefore, from (27) The other cases can be similarly obtained. This completes the proof.

The inner productã DTb
We shall study the inner productã DTb considering the family given in (22).
According to expression in decomposition theorem, the membership function ofã DTb is defined by We have the following interesting results.
we have Whenã (i) andb (i) are taken to be nonnegative canonical fuzzy intervals for i = 1, . . . , n, we simply have Proof The equality I ( DT ) = I * = [0, α * ] can be realized from Proposition 3.7. It is clear to see that {M • α : α ∈ I * for α > 0} is a nested family in the sense of M • α ⊆ M • β for β < α. Using the continuities regarding the canonical fuzzy intervals, we see that the family {M • α : α ∈ I * for α > 0} will continuously shrink when α increases on I * . Therefore, for α ∈ I * with α > 0, we have The equality I ( DT ) = I * can be realized from Proposition 3.7. Next, we are going to show that M • α = (ã DTb ) α for α ∈ I * . For α ∈ I * with α > 0 and any z ∈ M • α , the expression (31) says that ξã We consider the following cases.
Example 3.9 By referring to Example 3.5, Theorems 3.4 and 3.8 say that By referring to Example 3.6, we also have for α ≥ 2/3.

The inner productã DTb
We shall study the inner productã DTb considering the family given in (23). According to expression in decomposition theorem, the membership function of inner product a DTb is defined by We have the following interesting results.

Theorem 3.10 Letã
where M β is a bounded closed interval given by Suppose that the supremum sup I * is attained. Then, I ( DT ) = I * = [0, α * ], and, for α ∈ I * , we have Whenã (i) andb (i) are taken to be nonnegative canonical fuzzy intervals for i = 1, . . . , n, we simply have Proof The equality I ( DT ) = I * = [0, α * ] can be realized from Proposition 3.7. Next, we are going to show that M • α = (ã DTb ) α for α ∈ I * . By using (33) and the proof of Theorem 3.8, we can similarly obtain the inclusion M On the other hand, we can see that {M • α : α ∈ I * with α > 0} is a nested family in the sense of M • α ⊆ M • β for β < α. We define two functions ζ L and ζ U on I * as follows: It is clear to see that the functions ζ L and ζ U are continuous on I * by the continuities regarding the canonical fuzzy intervals. We also see that The continuities say that the family {M • α : α ∈ I * for α > 0} will continuously shrink when α increases on I * . For α ∈ I * with α > 0, it follows that Using the proof of Theorem 3.8, we can similarly obtain the inclusion For the 0-level set, we also have Suppose that min {β∈I * :β≥α} for some β * ∈ I * with β * ≥ α by part (vi) which says that min min On the other hand, since This completes the proof.
Example 3.11 Continued from Example 3.5, we can obtain Using (34) Therefore, we have min min Now, we can also obtain max {β∈I * :β≥α} Therefore, we have max max Using (34), we obtain the α-level sets

DTb
We shall study the inner productã † DTb considering the family given in (24). According to expression in decomposition theorem, the membership function of inner product a † DTb is defined by We need two useful lemmas.
Lemma 3.13 (Royden 1968, p.161) Let f be a real-valued function defined on R, and let K be a bounded and closed set in R. Suppose that f is upper semi-continuous. Then, f assumes its maximum on K ; that is, the supremum is attained in the following sense Lemma 3.14 Let I = [0, γ ] be a closed subinterval of [0, 1] for some 0 < γ ≤ 1. Suppose that the bounded real-valued functions ζ L : I → R and ζ U : I → R satisfying the following conditions: • ζ L is an increasing function and ζ U is a decreasing function on I ; • ζ L and ζ U are left-continuous on I \{0} = (0, γ ].
for α ∈ I . Then, for any fixed x ∈ R, the following function is upper semi-continuous on I .
Theorem 3.15 Letã (i) andb (i) be canonical fuzzy intervals for i = 1, . . . , n. The family {M α : α ∈ I * for α > 0} is taken by Suppose that the supremum sup I * is attained. Then, I ( †DT ) = I * = [0, α * ], and, for α ∈ I * , we have Whenã (i) andb (i) are taken to be nonnegative canonical fuzzy intervals for i = 1, . . . , n, we have The continuities regarding the canonical fuzzy intervals show that the functions ζ L and ζ U are continuous on I * . Using Lemma 3.14, given any fixed x ∈ R, the following function is upper semi-continuous on I * as described above, Lemma 3.13 says that the supremum of the function ζ is attained. Using (36), we have for some β * ∈ I * , which shows that z / ∈ (ã † DTb ) α . This contradiction says that there exists β 0 ∈ I * with β 0 ≥ α satisfying z ∈ M • β 0 . Therefore, we have the following inclusion On the other hand, the following inclusion is obvious. This shows the equality (38). Using the continuities regarding the canonical fuzzy intervals, we can also obtain the equality (37).
This completes the proof.

The equivalences and fuzziness
The equivalences amongã E Pb andã DTb for DT ∈ { DT , DT , † DT } will be presented below.
Theorem 3.17 Letã ( j) andb ( j) be canonical fuzzy intervals for j = 1, . . . , n. Suppose that the different inner productsã E Pb andã DTb are obtained from Theorems 3.4 and 3.8, respectively. Assume that the supremum sup R(ξÃ (i) ) and sup R(ξB (i) ) are attained for i = 1, . . . , n. Then, the supremum sup I * is attained, and we have Moreover, for α ∈ I * , we have Theorem 3.18 Letã ( j) andb ( j) be canonical fuzzy intervals for j = 1, . . . , n. Suppose that the different inner products a DTb andã † DTb are obtained from Theorems 3.10 and 3.15, respectively. Assume that the supremum sup I * is attained. Then, Moreover, for α ∈ I * , we have Theorem 3.19 Letã ( j) andb ( j) be nonnegative canonical fuzzy intervals for j = 1, . . . , n. Suppose that the different inner productsã E Pb ,ã DTb ,ã DTb andã † DTb are obtained from Theorems 3.4, 3.8, 3.10 and 3.15, respectively. Assume that the supremum sup R(ξÃ (i) ) and sup R(ξB (i) ) are attained for i = 1, . . . , n. Then, the supremum sup I * is attained, and we have Moreover, for α ∈ I * , we have The equivalence betweenã DTb andã DTb cannot be guaranteed. However, based on the α-level sets, we can compare their fuzziness.

The second type of inner product
The first type of inner product is directly based on the inner product of real vectors. Now, the second type of inner product will be based on the form of conventional inner product. First of all, we recall the addition and multiplication of fuzzy intervals. Letã andb be two fuzzy intervals with membership functions ξã and ξb, respectively. Given an aggregation function and for each z ∈ R. By referring to (22), (23) and (24), we can define the multiplication ofã andb according to expression in decomposition theorem by considering three different families as follows.
Using expression in decomposition theorem, given where M ⊗ α corresponds to the above three families. We also have Given any fuzzy intervalsc (1) , . . . ,c (n) , we shall also define the addition ofc (1) , . . . ,c (n) based on expression in decomposition theorem by considering three different families as follows.
Then, we see that In this case, we simply writec (1) ⊕ DT · · · ⊕ DTc (n) , and its membership function is defined by We also have Now, we are in a position to define the second type of inner product ofã andb as follows.
Definition 4.1 Letã ( j) andb ( j) be fuzzy intervals for j = 1, . . . , n. The inner product betweenã andb is defined bỹ where the addition and the multiplication The inner productã b depends on the choice of addition and multiplication according to (49) and (50), respectively. Therefore, it is completely different from the first type of inner productã b for ∈ { E P , DT , DT , † DT }. The membership function ξã b is in a very general situation, since the addition ⊕ i for i = 1, . . . , n − 1 and multiplication ⊗ j for j = 1, . . . , n can be any operations in (49) and (50), respectively. However, we can use the Decomposition Theorem 2.1 to rewrite the membership function ξã b using its α-level sets.
The interval range ofã b is denoted by I . Let (ã b ) α be the α-level set ofã b . According to the Decomposition Theorem 2.1, the membership function is given by The purpose is to obtain the α-level set (ã b ) α . We can see that The definition of interval range says that By referring to (2), we have (ã b ) α = ∅ for α ∈ I , and (ã b ) α = ∅ for α / ∈ I .
Example 4.4 Continued from Example 3.6, for 0.5 ≤ α < 2/3, we have and, for 0 ≤ α < 0.5, we have From (63), we obtain Let A, D, E be bounded closed intervals in R. Then, we have the subdistributivity We further assume that d · e ≥ 0 for every d ∈ D and e ∈ E. Then, we have the distributivity Letã andb be fuzzy intervals. We writeã ⊆b when their α-level sets satisfy the inclusionã α ⊆b α for all α ∈ Iã ∩ Ib, where Iã and Ib denotes the interval ranges ofã andb, respectively.
(i) We have the following subdistributivitỹ (ii) For each fixed j, suppose thatd ( j) andẽ ( j) are simultaneously nonnegative or nonpositive. Then, we have the following distributivitỹ Proof Since the fuzzy intervals are assumed to be normal, their interval ranges are all the unit interval [0, 1]. In this case, we can simply consider α ∈ [0, 1] for taking care of their α-level sets. Letb =d ⊕ E Pẽ . Then, regarding the components for j = 1, . . . , n, we havẽ α denote the α-level sets for α ∈ [0, 1]. From (61), we see that Therefore, for α ∈ [0, 1], we obtain (67) and (68)) This proves part (i). Applying (66) to the above argument, we can similarly obtain part (ii), and the proof is complete.
The above properties can be similarly extended for the case of non-normal fuzzy intervals.

Comparison of fuzziness
By referring to Definition 3.20, we are going to compare the fuzziness between the first type of inner productã b for ∈ { E P , DT , DT , † DT } and the second type of inner productã b . Theorem 4.9 Letã ( j) andb ( j) be canonical fuzzy intervals for j = 1, . . . , n such that the supremum sup I * is attained. Suppose that the first type of inner productsã E Pb and a DTb are obtained from Theorems 3.4 and 3.8, respectively. Then, we have the following results.
(i) Assume that the second type of inner product is taken bỹ Then, the first type of inner productsã E Pb andã DTb is fuzzier than the second type of inner productã E Pb in the sense of Assume that the second type of inner product is taken bỹ Then, the first type of inner productsã E Pb andã DTb are fuzzier than the second type of inner productã b in the sense of Proof It is clear to see that min (x,y)∈(ã α ,b α ) (x 1 y 1 + · · · + x n y n ) = n j=1 min (x,y)∈(ã α ,b α ) x j y j and max (x,y)∈(ã α ,b α ) (x 1 y 1 + · · · + x n y n ) = n j=1 max (x,y)∈(ã α ,b α ) x j y j , and max (x,y)∈(ã α ,b α ) (x 1 y 1 + · · · + x n y n ) To prove part (i), we obtain x • y (using Theorem 3.17) (61), (64), (81) and (85)).
Theorem 4.10 Letã ( j) andb ( j) be canonical fuzzy intervals for j = 1, . . . , n such that the supremum sup I * is attained. Suppose that the first type of inner productsã DTb and a † DTb is obtained from Theorems 3.10 and 3.15, respectively. We also assume that the second type of inner product is taken byã Then, the second type of inner productã DTb is fuzzier than the first type of inner productsã DTb andã † DTb in the sense of We can similarly obtain (1) L α + · · · + ã (n) ⊗ DTb (n) L α (using (76) and (78)) = ã (1) ⊗ †

Conclusions
Two different types of inner product of fuzzy vectors have been completely studied in this paper. Also, each type of inner product will be established using extension principle and expression in decomposition theorem. In other words, we can establish four different kinds of inner product.
• Regarding the first type inner product, the inner product a E Pb using extension principle has been established in (12). On the other hand, using expression in decomposition theorem, three inner productsã DTb ,ã DTb andã † DTb have been established in (31), (33) and (36), respectively, based on three different families of sets in R.
• Regarding the second type inner product, the inner productã E Pb using extension principle has been established in (54). On the other hand, the inner products a DTb using expression in decomposition theorem have been established in (70) in which three different families (44), (45) and (46) are implicitly considered.
The equivalences among these different inner products have also been investigated. For the cases of nonequivalences, the relations regarding their fuzziness have also been established. In real situation, we prefer to take the inner product that has less fuzziness.
Theorem 3.18 shows the equivalenceã DTb =ã † DTb . The equivalence betweenã DTb andã DTb cannot be established. Theorem 3.21 says thatã DTb is fuzzier thanã DTb . In this case, we prefer to takeã DTb rather thanã DTb because of the issue of fuzziness.
• Theorem 4.9 says that when the second type inner product a E Pb is taken by (81) orã DTb is taken by (82), we prefer to take this second type of inner products rather than the first type of inner productã E Pb andã DTb because of the issue of fuzziness. • Theorem 4.10 says that when the second type inner productã DTb is taken by (87) or (88), we prefer to take the first type of inner productsã DTb andã † DTb rather than the second type of inner productã b because of the issue of fuzziness.
Recall that a fuzzy intervalã is nonnegative whenã L α and a U α are nonnegative real numbers for all α ∈ [0, 1], and a fuzzy intervalã is nonpositive whenã L α andã U α are nonpositive real numbers for all α ∈ [0, 1]. The fuzzy interval that is neither nonnegative nor nonpositive is called a mixed type of fuzzy interval, which means that its membership function will be across the y-axis. The fuzzy intervals that are taken to be nonnegative or nonpositive will simplify the calculation of inner product. For the case of mixed type of fuzzy intervals, Examples 3.6, 3.12, 4.4 and 4.8 have demonstrated the essence of calculation. In real applications, the fuzzy data can be strongly nonnegative or nonpositive, which says that the fuzzy intervals can be taken to be nonnegative or nonpositive. The mixed type of fuzzy interval means that its corresponding fuzzy data are around zero. When the concrete problem has less numbers of fuzzy data that are around zero, which seems frequently occur in real applications, the less numbers of corresponding mixed type of fuzzy intervals can be taken. This situation will also simplify the calculation.
The potential application considering the inner product of fuzzy vector is to solve the fuzzy linear programming problem as shown below (FLP) max ( a 1 ⊗ x 1 ) ⊕ ( a 2 ⊗ x 2 ) ⊕ · · · ⊕ ( a n ⊗ x n ) subject to b i1 ⊗ x 1 ⊕ b i2 ⊗ x 2 ⊕ · · · ⊕ b in ⊗ x n c i for i = 1, . . . , m; each x j is a nonnegative fuzzy interval for j = 1, . . . , n, where the coefficientsã j ,b i j andc i for i = 1, . . . , m and j = 1, . . . , n are taken to be fuzzy intervals. Most of the cases, those fuzzy intervals can be assumed to be nonnegative, since the coefficients of linear programming problem frequently represent the prices of items, budget of project and benefit of corporation, which are always positive. Considering nonnegative fuzzy intervals in fuzzy linear programming problem will simplify the calculation of inner product of fuzzy vectors as explained above, which will not diminish the motivation of this paper.

Author Contributions
The sole author of this article is responsible for all the academic results.
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Data availability This article contains theoretical results without using the experimental data.

Conflict of interest
The author declares that he has no conflict of interests.
Ethical approval This article does not contain any studies with human participants or animals performed by the author.