Global Journal of Pure and Applied Mathematics. ISSN 0973-1768 Volume 13, Number 7 (2017), pp. 4075--4085 © Research India Publications http://www.ripublication.com/gjpam.htm

Soft topological vector spaces in the view of soft filters

M. Sheik John Department of Mathematics, NGM College, Pollachi-642 001, TamilNadu, India.

M. Suraiya Begum Research Scholar, Department of Mathematics, NGM College, Pollachi-642 001, TamilNadu, India.

Abstract In our present work, we have introduced the concept of soft topological vector spaces in the view of soft filters. We also discuss some of the properties of soft filters in soft topological vector spaces.

AMS subject classification: 03E72, 54A40. Keywords: Soft open set, Soft topological space, Soft topological , Soft balanced neighborhood, Soft filter.

1. Introduction Classical tools of Mathematics [26] cannot solve the problems which are vague rather than precise. To overcome these difficulties Molodtsov [16] initiated the concept of soft set theory which doesn’t require the specification of parameters. He applied soft set theory successfully in smoothness of functions, game theory, operation research and so on. Thereafter so many research works [3] [5] have been done on this concept in different disciplines of Mathematics. Research on soft sets based decision making has received much attention in recent years. Later, in 2003, Maji et al. [14] [15] made a theoretical study on soft set theory. They introduced several operations on soft sets and applied soft sets to decision making problems. In 2007, Aktas and Cagman [1] introduced a basic version of soft group theory, which extends the notion of a group to include the 2 M. Sheik John and M. Suraiya Begum algebraic structures of soft sets. Jun [9] [10] investigated soft BCK/BCI- algebras and its application in ideal theory. Kharal and Ahmed [13] defined soft mappings. In 2011, Shabir and Naz [24] came up with an idea of soft topological spaces. Later Aygun et al. [2], Zorlutuna et al. [26], Cagman et al. [4], Hussain et al. [8] studied on soft topological spaces. Nazmul and Samanta studied topological group structures in soft set approaching from different perspectives in [18] [19] [20]. In 2013 S. Das et al. [6] introduced soft linear spaces and soft normed linear spaces. Ozturk [21] [22] [23] introduced the concept of soft uniform spaces. In 2014, by developing the idea of , Chiney et al. [17] introduced vector soft topology. In our present work, we have introduced the concept of soft topological vector spaces in the view of soft filters. We also discuss some of the properties of soft filters in soft topological vector spaces.

2. Preliminaries

2.1. Preliminaries Throughout the paper, let X be an initial universe set and E be the set of parameters. P(X)denote the power set of X and A ⊆ E.

Definition 2.1. [6] A soft set FA over X is a set defined by the function fA representing a mapping fA : A → P(X)such that fA =∅if x/∈ A. Here, fA is called the approximate function of the soft set FA. A soft set over X can be represented by the set of ordered pairs FA ={(x, fA(x)) : x ∈ A, fA(x) ∈ P(X)}.

Definition 2.2. [2] Let FA and GA be two soft sets over X. The parallel product ˜ ˜ ˜ ˜ of FA and GA is defined as FA×GA = (F ×G)A where [F ×G](α) = F(α)×G(α), ˜ ˜ ∀α ∈ A ⊆ E. It is clear that (F ×G)A is a soft set over X×X. Definition 2.3. [24] Let τ be the collection of soft sets over X, then τ is said to be a soft topology on X if

i) φ, X˜ are belongs to τ.

ii) The union of any number of soft sets in τ belongs to τ.

iii) The intersection of any two soft sets in τ belongs to τ.

In this case the triplet (X,τ,A)is called a soft topological space over X, and any member of τ is known as soft open set in X. The complement of a soft open set is called soft closed set over X.

Definition 2.4. [18] A crisp element x ∈ X is said to be in the soft set FA over X, denoted by x∈˜ FA iff x ∈ F(α), ∀α ∈ A. Soft topological vector spaces in the view of soft filters 3

Definition 2.5. [18] A soft set FA is said to be τ soft nbd of an element x ∈ X if ∃GA ∈ τ such that x ∈ GA ⊆ FA. Definition 2.6. [20] [21] Let (X,τ,A) and (Y,υ,A) be soft topological spaces. The mapping f : (X,τ,A)→ (Y,υ,A)is said to be

−1 1. soft continuous if f [FA]∈τ,∀FA ∈ υ. − 2. soft homeomorphism if f is bijective and f, f 1 are soft continuous.

3. soft open if FA ∈ τ ⇒ f [FA]∈υ.

4. soft closed if FA is soft closed in (X,τ,A)⇒ f [FA] is soft closed in (Y,υ,A).

Proposition 2.7. [17] Let (X,τ,A), (Y,υ,A)and (Z,ϑ,A)be soft topological spaces. If f : (X,τ,A) → (Y,υ,A) and g : (Y,υ,A) → (Z,ϑ,A) are soft continuous and f(X)⊆ Y , then the mapping gf : (X,τ,A)→ (Z,ϑ,A)is soft continuous.

Definition 2.8. [20] Let τ be a soft topology on X. Then a soft set FA is said to be a x τ soft neighborhood (shortly soft nbd) of the soft element Eα if there exists a soft set ∈ x ∈ ⊆ x GA τ such that Eα GA FA. The soft nbd system of a soft element Eα in (X,τ,A) x is denoted by Nτ (Eα). { x : x ∈} Proposition 2.9. [20] If Nτ (Eα) Eα be the system of soft nbds then x = ∀ x ∈ 1. Nτ (Eα) φ, Eα . x ∈ ∀ ∈ x 2. Eα FA, FA Nτ (Eα). ∈ x ⊆ ⇒ ∈ x 3. FA Nτ (Eα), FA GA GA Nτ (Eα). ∈ x ⇒ ∩ ∈ x 4. FA,GA Nτ (Eα) FA GA Nτ (Eα). ∈ x ⇒∃ ∈ x ⊆ ∈ x 5. FA Nτ (Eα) GA Nτ (Eα) such that FA GA and GA Nτ (Eα), ∀ x ∈ Eα GA.

Proposition 2.10. [18] Let (X,τ,A)and (Y,υ,A)be two soft topological spaces. A mapping f : (X,τ,A) → (Y,υ,A) is soft continuous iff ∀x ∈ X and ∀VA ∈ υ such f(x) ∈ ∃ ∈ x ∈ [ ]⊆ that Eα VA, UA τ such that Eα UA and f UA VA. Definition 2.11. [18] Let (X,τ,A)be a soft topological space. A sub-collection B of τ is said to be an open base of τ if every member of τ can be expressed as the union of some members of B.

Definition 2.12. [20] The soft topology in X × Y induced by the open base F = {FA × GA : FA ∈ τ,GA ∈ υ} is said to be the product soft topology of the soft topologies τ and υ. It is denoted by τ × υ. The soft topological space(X × Y, τ × υ,A) 4 M. Sheik John and M. Suraiya Begum is said to be the soft topological product of the soft topological spaces (X,τ,A) and (Y,υ,A).

Proposition 2.13. [17] Let (X,τ,A) be the product space of two soft topological spaces (X1,τ1,A) and (X2,τ2,A) respectively. Then the projection mappings i : (X,τ,A)→ (Xi,τi,A), i = 1, 2 are soft continuous and soft open. Also τ 1 × τ 2 is the smallest soft topology in X ×Y for which the projection mappings are soft continuous.If further, (Y,υ,A) be any soft topological space then the mapping f : (Y,υ,A) → (X,τ,A)is soft continuous iff the mappings if : (Y,υ,A) → (Xi,τi,A), i = 1, 2 are soft continuous.

Definition 2.14. [17] Let FA and GA be two soft sets over the vector space V over the field K, the field of real and complex numbers. Then

1. FA + GA = (F + G)A where (F + G)(α) = F(α)+ G(α), ∀α ∈ A.

2. k(FA) = (kF )A where (kF )(α) ={kx : x ∈ F(α)}, ∀α ∈ A,∀k ∈ K.

3. x + FA = (x + F)A where (x + F )(α) ={x + y : y ∈ F(α)}, ∀α ∈ A,∀x ∈ V .

4. If GA be any soft set over K then GA · FA = (G · F)A where (G · F )(α) = G(α)·F(α), ∀α ∈ A.

Definition 2.15. [17] A soft set FA over a vector space V is said to be

1. convex if kFA + (1 − k)FA ⊆ FA, ∀k ∈[0, 1].

2. balanced if kFA) ⊆ FA for all scalar k with |k|≤1.

3. absolutely convex if it is balanced and convex.

B x ∀ ∈ Definition 2.16. [17] A collection of soft neighborhoods of a soft element Eα, α A x is said to be a fundamental soft nbd system or soft nbd base of Eα if for any soft nbd FA x ∃ ∈ ⊆ of Eα, HA B such that HA FA. Definition 2.17. [17] Let V be a vector space over the scalar field K endowed with the soft usual topology ν, A be the parameter set and τ be a soft topology on V . Then τ is said to be a vector soft topology on V if the mappings: f : (V × V,τט τ,A) → (V,τ,A), defined by f(x,y) = x + y. g : (K × V,νט τ,A) → (V,τ,A), defined by g(k,x) = kx are soft continuous ∀x,y ∈ V and ∀k ∈ K.

Proposition 2.18. [17] Let (X,˜ τ, A) be a soft . For a˜ ∈ (X,˜ A) ˜ ˜ ˜ and k ∈ (K,A) with k(λ) = 0, for each λ ∈ A then soft translation operator Ta˜ and soft ˜ ˜ multiplication Mk˜ are soft homeomorphism operator from (X, τ, A) to (X, τ, A). Soft topological vector spaces in the view of soft filters 5

3. Soft topological vector space in the view of soft filters

Definition 3.1. Let (X,˜ τ, A) be a soft topological space. Then F˜ is called a soft filter on X˜ if F˜ satisfies the following properties:

i. ∅ ∈/ F˜ . ˜ ˜ ii. ∀FA,GB ∈ F, FA ∩ GB ∈ F. ˜ ˜ iii. ∀FA ∈ F and FA ⊆ GB, GB ∈ F.

Definition 3.2. Let (X,˜ τ, A) be a soft topological vector space. For each x˜ ∈ (X,˜ A), ˜ ˜ ˜ the soft filter F is defined as F ={FA +˜x(λ)|FA ∈ F,λ∈ A}. Definition 3.3. Let (X,˜ τ, A) and (Y,ν,A)˜ be two soft topological vector spaces over K or C.

1. A soft topological f : X˜ → Y˜ is soft linear which is also soft continuous and soft open.

2. A soft topological monomorphism f : X˜ → Y˜ is an injective soft topological homomorphism.

3. A soft topological isomorphism f : X˜ → Y˜ is a bijective soft topological homo- morphism.

4. A soft topological automorphism of X˜ is a soft topological isomorphism from X˜ into itself.

Theorem 3.4. In a soft topological vector space (X,˜ τ, A), for k˜ ∈ (K,A)˜ with k(λ)˜ = 0, ∈ ˜ for each λ A the multiplication operator Mk˜ is a soft topological automorphism of X. : → = ˜ · ∈ Proof. Since the mapping Mk˜ X X defined by (Mk˜)λ(ξ) k(λ) ξ, for all λ A and ξ ∈ X is bijective on SE(X)˜ , the inverse exist. By the continuity of the function g ˜ ˜ ¯ at (k,x)˜ , for any soft nbd of VA of k ·˜x, there exist rˆ>˜ 0 and a soft nbd UA of x˜ such that ˆ ˜∈˜ ˆ∈˜ ˜ |ˆ − ˆ|˜ˆ ˜∈ ˜ ⊆ sy VA, for all s (K,A) with s k

Theorem 3.5. Any soft nbd of in a soft topological vector space (X,˜ τ, A) is soft absorbing.

Theorem 3.6. Let UA be a soft balanced subset of a soft topological vector space. Then UA is soft balanced and, if is an interior point of UA, the interior of UA is soft balanced. 6 M. Sheik John and M. Suraiya Begum

= Proof. Since the soft multiplication Mk˜ is a soft homeomorphism we have Mk˜UA tUA ⊆ UA, |t|≤1. Hence UA is soft balanced. Let is an interior point of UA, for |t|≤1, tint(UA) = int(tUA) ⊆ tUA ⊆ UA, and so interior of UA is soft balanced. Definition 3.7. A soft filter F˜ in a soft topological space X˜ is said to converges to a soft ˜ point x˜λ ∈ X if every soft neighborhood of x˜λ belongs to F for each λ ∈ A. Theorem 3.8. Let F˜ be a soft filter of a soft topological Hausdorff space X.˜ If F˜ converges to x(λ)˜ ∈ X˜ also to y(λ)˜ ∈ X˜ , for each λ ∈ A then x˜ =˜y.

Proof. Let X˜ is soft Hausdorff space and let x˜ =˜y. Since X˜ is a Hausdorff, there exists ˜ ˜ ˜ ˜ FA ∈ F(x)˜ and GB ∈ F(y)˜ such that FA∩GB =∅. By the assumption, FA,GB ∈ F ˜ ˜ and so FA∩GB ∈ F as soft filters are closed under finite intersections, which is a contradiction to the fact that ∅ ∈/ F˜ . ˜ ˜ Theorem 3.9. Let (X, τ, A) be a soft topological space and let FA ⊆ X. Then x˜λ ∈ FA ˜ ˜ ˜ ˜ if and only if ∃ a soft filter F of subsets of X such that FA ∈ F and F converges to x˜λ, ∀λ ∈ A.

Proof. Obvious.

Remark 3.10. The translation invariance of a soft topology τ on a absolute soft vector space (X, A) is not sufficient to conclude that (X,˜ τ, A) is a soft topological vector space. ˜ Proposition 3.11. For any soft open set FA in a soft topological vector space X, the soft ˜ ˜ ˜ ˜ ¯ sets a˜ + FA, GA + FA and kFA are soft open, for any a˜ ∈ X, k ∈ K and k = 0. In ¯ particular, if HA is a soft nbd of , then so is αH˜ A for any α ∈ K with α = 0. ˜ ˜ ˜ Proof. Let FA is soft open in X. For any a˜ ∈ X and k ∈ K,wehavea˜ + FA = Ta˜ (FA) ˜ = and kFA Mk˜(FA). By the proposition 2.21, Ta˜ and Mk˜(FA) are soft homeomorphisms. + Hence Ta˜ (FA) and Mk˜(FA) are soft open sets. Since GA FA can be written as the union of soft open sets,it is soft open. Let HA be any soft nbd of . By continuity of ¯ the mapping g(α,˜ x)˜ =˜αx˜ at (0, ), for any soft nbd HA of , there exists a soft nbd ¯ DA of and γ>0 such that α˜ y˜ ∈ HA for all soft scalar |˜α| <γand for all y˜ ∈ DA. Then αD˜ A ⊆ HA. Corollary 3.12. The soft filter F˜ of neighborhoods of x˜ ∈ X˜ coincides with the family of the sets +˜x for all ∈ F˜ (θ) is the soft filter of neighborhoods of the zero element θ.

Theorem 3.13. A soft filter F˜ of a soft topological vector space (X,˜ τ, A) over R(A), the field of real numbers is the soft filter of neighborhoods of if ˜ 1. belongs to every soft open set FA ∈ F. ˜ ˜ 2. ∀FA ∈ F, there exists GA ∈ F such that GA + GA ⊆ FA. Soft topological vector spaces in the view of soft filters 7

˜ ˜ 3. ∀FA ∈ F with α(λ) = 0, for each λ ∈ A we have α(λ)(FA)λ ∈ F. ˜ ˜ Proof. Let (X, τ, A) be the soft topological vector space over R(A) and FA ∈ F.

1. Since every soft set is a neighborhood of ,wehave ∈ FA.

2. Clearly the mapping f : SE(X)˜ × SE(X)˜ → R(A) is soft continuous. Therefore ¯ the pre image of FA is must also be a soft nbd of (0, ). Hence there exists a soft × ∈ F˜ = ∩˜ nbd HA HA where HA,HA . Let GA HA HA. Proof follows.

3. Since the g : SE(K)˜ × SE(X)˜ → R(A) is continuous, the pre image of every soft nbd FA of is also a soft nbd of , which is clearly αFˆ A, ∀α ∈ A. Hence ˜ αFˆ A ∈ F.

Theorem 3.14. The only soft open linear subspace in a soft topological vector space (X,˜ τ, A) is X˜ itself.

Proof. Suppose that (Y,A)˜ is an open soft linear subspace in a soft topological vector ˜ ˜ ˜ space X. Then belongs to (Y,A) so there is a soft nbd UA of in X such that ˜ ˜ ˆ UA ⊆ (Y,A). Let x˜ ∈ X. Since UA is absorbing, there is t>0 with t(λ) > 0, for ˆ−1 ˆ ˜ ˜ ˜ each λ ∈ A such that t x˜ ∈ UA. In particular, t(λ)x(λ)˜ ∈ (Y , A). Thus Y = X, since (Y,A)˜ is linear.

Theorem 3.15. Let (X,˜ τ, A) be soft topological vector space.

1. Every soft linear subspace of (X,˜ τ, A) endowed with the correspondent soft sub- space topology is itself a soft topological vector space.

2. The soft closure of a soft linear subspace of (X,˜ τ, A) is also a soft linear subspace of (X,˜ τ, A).

˜ ˜ | | Proof. Let Z be the soft linear subspace of X. The mappings f Z˜ and g Z˜ restricted to the subspace are the composition of continuous maps. Hence Z˜ is also a soft topological vector space. 2. Let (Z,A)˜ be the soft linear subspace of X˜ and (Z,A)˜ be the soft closure of (Z,˜ A). ˜ ˜ ˜ Let x˜1, y˜1 ∈ (Z,A) and FA ∈ F(θ). Then there exists GA ∈ F(θ) such that GA + ˜ ˜ GA ⊆ FA. Since (Z,A) be a soft linear subspace for any x(λ),˜ y(λ)˜ ∈ (Z,A),we ˜ ˜ ˜ have x(λ)˜ +˜y(λ) ∈ (Z,A) where x(λ)˜ ∈ GA + x1(λ) and y(λ)˜ ∈ GA + y1(λ). Hence x(λ)˜ +˜y(λ) ∈ (GA +˜x1)(λ) + (GA +˜y1)(λ) ⊆ (FA +˜x1 +˜y1)(λ). ˜ ˜ Let x˜ ∈ (Z,A) and for each λ ∈ A. Let FA ∈ F(θ). Then there exists αˆ ∈ (K, A) with ¯ α(λ)ˆ = 0, such that (αFˆ A)(λ) ∈ ZA. Theorem 3.16. Any maximal proper soft subspace of a soft topological vector space (X,˜ τ, A) is either dense or soft closed. 8 M. Sheik John and M. Suraiya Begum

Proof. Let (Z,τ,A)˜ is a maximal proper soft subspace of X˜ . Then the inclusion (Z,A)˜ ⊆ (Z,A)˜ implies that either (Z,A)˜ = (Z,A)˜ or (Z,A)˜ = X˜ .

Theorem 3.17. Let f˜ : SE(X)˜ → K be a soft linear functional on a soft topological vector space X˜ . Then either f˜ is soft continuous or Kerf˜ is a soft dense proper subspace of X˜ . ˜ ˜ Proof. If f(x˜λ) = 0, λ ∈ A, it is continuous and its kernel is the whole of X. Otherwise, Kerf˜ is a maximal proper linear subspace of X˜ which is either soft closed or dense. However, f˜ is continuous if and only if its kernel is soft closed, so if f˜ is not continuous its kernel is a proper dense soft subspace.

Theorem 3.18. Let f˜ be a soft linear mapping between soft topological vector spaces X˜ and Y˜ . The soft map f˜ is soft continuous if and only if f˜ is soft continuous at . ˜ Proof. Suppose f is soft continuous at and fix x(λ)˜ = θ, for each λ ∈ A. Let FA be the ˜ ˜ ˜ soft nbd of f(x(λ))˜ ∈ Y.Since by Remark 3.9, FA = f(x(λ))˜ + GA where GA is a soft ˜ ˜ ˜−1 ˜ ˜−1 nbd of θ ∈ Y.Given that f is soft linear, therefore we have f (FA)⊃˜x(λ) + f (GA) ˜−1 ˜ also x(λ)˜ + f (GA) is a soft nbd of x(λ)˜ ∈ X, for each λ ∈ A.

Theorem 3.19. A soft topological vector space X˜ is Hausdorff if and only if ∀θ = ˜ ˜ x(λ) ∈ X, there exists UA ∈ F(θ) such that x/˜ ∈ UA. ˜ ˜ Proof. Necessity Part: Let (X, τ, A) be soft Hausdorff. Then there exists UA ∈ F(θ) ˜ and VA ∈ F(x) such that UA ∩ VA =∅. Hence x˜ ∈ UA. ˜ ˜ Sufficient Part: Let x,˜ y˜ ∈ X with x˜ =˜y. Then there exists UA ∈ F(θ) such that x˜ −˜y/∈ UA.

Definition 3.20. Let (X, A) and (Y, A) be two absolute soft vector spaces over K (the field of real or complex numbers) and f : (X,˜ τ, A) → (Y,ν,A)˜ be a soft . The Range of the function f is defined by Range(f) ={f(x)˜ |˜x ∈ X˜ }, which is the soft sub vector space of (Y, A). The Kernel of f is defined by Ker(f) ={˜x ∈ X˜ | f(x)˜ = , being the zero element}, and Ker(f) is a soft sub vector space of (X, A).

Theorem 3.21. Let f˜ be a soft linear mapping between soft topological vector spaces X˜ and Y˜ .IfY˜ is soft Hausdorff topological vector space and f˜ is continuous and bijective then Ker(f) is soft closed in X.˜

− Proof. Since the function is bijective, Ker(f) = f 1({ }). Given that the space Y˜ is soft Hausdorff topological vector space is closed in Y˜ . Since f˜ is continuous, Ker(f) is soft closed in X.˜

Theorem 3.22. Let f˜ be a soft linear mapping between soft topological vector spaces X˜ ¯ and Y˜ . The soft map f˜ is soft continuous if and only if the soft map f˜ is soft continuous. Soft topological vector spaces in the view of soft filters 9

˜ ˜−1 Proof. Let f continuous and UA be soft open subset in Im(f ). Then f (UA) is soft ˜ ˜¯−1 ˜−1 ˜ open in X. Clearly, f (UA) = (f (UA)). We know that the quotient map : X → ˜ ˜ ˜−1 ˜ ˜ ˜¯−1 X \ Ker(f) is soft open, (f (UA)) is soft open in X \ Ker(f). Hence f (UA) ¯ ¯ is soft open in X˜ \ Ker(f)˜ , the soft map f˜ is soft continuous. Suppose that f˜ is soft ¯ continuous. Since f˜ = f˜ ◦ , is soft continuous, f˜ is also soft continuous.

Theorem 3.23. A soft Hausdorff topological vector space (X,˜ τ, A) is separated if and only if every one-point set is soft closed.

Proof. Since the space is Hausdorff, all one-point sets are soft closed. Conversely, let ˜ x˜λ =˜yλ ∈ X and let z˜λ =˜xλ −˜yλ, λ ∈ A so that z˜λ = θ, θ being the zero element ˜ ˜ of X.If{˜zλ} is closed, X \{˜zλ} is open and this is an soft open neighborhood of . ˜ Since the addition is continuous on X, there are two soft neighborhoods FA and GA of ˜ ˜ such that (F +G)A ⊆ X \{˜zλ}. Since GA is a soft neighborhood of ,sois−GA (∵ ˜ the multiplication operator Mk˜ is a soft topological automorphism of X.) and therefore ˜ z˜λ − GA is a soft neighborhood of z˜λ (∵ the translation Ta is a soft homeomorphism). Any w˜ λ ∈˜zλ − GA has the form w˜ λ =˜zλ −˜pλ , p˜λ ∈ GA and so w˜ λ = (z˜ +˜p)λ. ˜ Since z˜λ ∈/ (F +G)A , we must have that w/˜ ∈ FA. Therefore FA ∩ (z˜λ − GA) =∅, also ˜ (y˜λ + FA) ∩ ((y˜ +˜z)λ − GA) =∅, that is, (y˜λ + FA) ∩ (x˜λ − GA) =∅. Thus X is soft Hausdorff topological vector space.

Definition 3.24. A soft subset FA of a soft topological vector space is said to be bounded ¯ ˆ if for each soft nbd UA of there is s>ˆ 0 such that FA ⊆ˆsUA for every sˆ ≥ t. ˜ Theorem 3.25. Let X, τ, A be a soft topological vector space and if {˜zλ}, for each λ ∈ A ˜ and FA and GA are bounded soft subsets of X, then the sets {˜zλ}, FA ∪ GA and FA + GA are bounded.

4. Conclusion

There is an ample of scope of further research on soft vector spaces.

Acknowledgement The authors express their sincere thanks to the anonymous referees for their valuable and constructive suggestions which have improved the presentation. The authors are also thankful to the Editors-in-Chief and the Managing Editors for their valuable advice. The research work of the second author is supported by the [University Grants Com- mission, New Delhi], under grant [number: MANF-2015-17-TAM-56849] from gov- ernment of India.

Competing Interests Authors have declared that no competing interests exist. 10 M. Sheik John and M. Suraiya Begum

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