# Some Generalized Difference Sequence Spaces Defined by a Sequence of Moduli inn-Normed Spaces

Authors: Abdullah Alotaibi; Kuldip Raj; S. A. Mohiuddine
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We introduce some new generalized difference sequence spaces by means of ideal convergence, infinite matrix, and a sequence of modulus functions overn-normed spaces. We also make an effort to study several properties relevant to topological, algebraic, and inclusion relations between these spaces.


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## Body

## 1. Introduction and Preliminaries

The concept of 2-normed spaces was initially developed by Gähler [1] in the middle of 1960s, while that of n-normed spaces one can see in Misiak [2]. Since then, many others have studied this concept and obtained various results; see Gunawan [3, 4] and Gunawan and Mashadi [5]. Let n∈N and X be a linear space over the field R of reals of dimension d, where d≥n≥2. A real valued function ·,…,· on Xn satisfying the following four conditions:(1)
x1,x2,…,xn=0 if and only if x1,x2,…,xn are linearly dependent in X,(2)
x1,x2,…,xn is invariant under permutation,(3)
αx1,x2,…,xn=|α|x1,x2,…,xn for any α∈R,(4)
x+x′,x2,…,xn≤x,x2,…,xn+x′,x2,…,xn,is called a n-norm on X and the pair (X,·,…,·) is called a n-normed space over the field R. For more details about n-normed spaces, see [6, 7] and references therein.The notion of difference sequence spaces was introduced by Kızmaz [8], who studied the difference sequence spaces l∞(Δ), c(Δ), and c0(Δ). The notion was further generalized by Et and Çolak [9] by introducing the spaces l∞(Δm), c(Δm), and c0(Δm). Later the concept has been studied by Bektaş et al. [10] and Et and Esi [11]. Another type of generalization of the difference sequence spaces is due to Tripathy and Esi [12] who studied the spaces l∞(Δv), c(Δv), and c0(Δv). Recently, Esi et al. [13] and Tripathy et al. [14] have introduced a new type of generalized difference operators and unified those as follows.Letm and v be nonnegative integers; then for Z, a given sequence space, we have(1)ZΔvm=x=xk∈w:Δvmxk∈Z,for Z=c, c0, and l∞ where Δvmx=(Δvmxk)=(Δvm-1xk-Δvm-1xk+v) and Δv0xk=xk, for all k∈N, which is equivalent to the following binomial representation:(2)Δvmxk=∑ν=0m‍-1νmνxk+vν.Taking v=1, we get the spaces l∞(Δm), c(Δm), and c0(Δm) studied by Et and Çolak [9]. Taking v=m=1, we get the spaces l∞(Δ), c(Δ), and c0(Δ) introduced and studied by Kızmaz [8]. For more details about difference sequence spaces, see [15–19].A modulus function is a functionf:[0,∞)→[0,∞) such that(1)
f(x)=0 if and only if x=0,(2)
f(x+y)≤f(x)+f(y), for all x,y≥0,(3)
f is increasing,(4)
f is continuous from the right at 0.It follows that f must be continuous everywhere on [0,∞). The modulus function may be bounded or unbounded. For example, if we take f(x)=x/x+1, then f(x) is bounded. If f(x)=xp, 0<p<1, then the modulus function f(x) is unbounded. Subsequently, modulus function has been discussed in [20] and references therein.In [21], Mursaleen introduced the idea of λ-statistical convergence by extending the concept of [V,λ] summability. Further, Savaş and Das [22] unified the two approaches and gave new concepts of I-statistical convergence, I-Sλ-convergence, and I-[V,λ] convergence. Later some pioneer works have been extended in this direction by numerous authors such as Belen and Mohiuddine [23], Das et al. [24], Gürdal and Sarí [25], and references therein. Quite recently, many authors including Et et al. [26] and Maddox [27] have constructed some sequence spaces by using modulus function and difference sequences and investigate their properties.LetX and Y be two sequence spaces and A=(ank) be an infinite matrix of real or complex numbers ank, where n,k∈N. Then we say that A defines a matrix mapping from X into Y if, for every sequence x=(xk)k=0∞∈X, the sequence Ax={An(x)}n=0∞, the A-transform of x, is in Y, where(3)Anx=∑k=0∞‍ankxkn∈N.By (X,Y), we denote the class of all matrices A such that A:X→Y. Thus, A∈(X,Y) if and only if the series on the right-hand side of (3) converges for each n∈N and every x∈X.The matrix domainXA of an infinite matrix A in a sequence space X is defined by(4)XA=x=xk:Ax∈X.The approach constructing a new sequence space by means of the matrix domain of a particular limitation method has recently been employed by several authors (see [28]).Letλ=(λn) be a nondecreasing sequence of positive numbers such that λn+1≤λn+1, λ1=1, λn→∞ as n→∞, and In=[n-λn+1,n].LetN be a nonempty set. Then a family of sets I⊆2N (power set of N) is said to be an ideal if I is additive; that is, A,B∈I⇒A∪B∈I and A∈I, B⊆A⇒B∈I. A nonempty family of sets £(I)⊆2N is said to be filtered on N if and only if Φ∉£(I); for A,B∈£(I) we have A∩B∈£(I) and for each A∈£(I) and A⊆B implying B∈£(I).An idealI⊆2N is called nontrivial if I≠2N. A nontrivial ideal I⊆2N is called admissible if {{x}:x∈N}⊆I. A nontrivial ideal is maximal if there cannot exist any nontrivial ideal J≠I containing I as a subset. For each ideal I, there exist a filter £(I) corresponding to I; that is, £(I)={K⊆N:Kc∈I}, where Kc=N∖K.Definition 1.
A sequencex=(xk) in a n-normed space (X,·,…,·) is said to be statistically convergent to some L∈X if, for each ε>0, the set A(ε)={k∈N:xk-L,z1,…,zn-1≥ε} has its natural density zero.Definition 2 (see [26]).
A sequencex=(xk) is said to be λXm-statistically convergent to the number L, if, for every ε>0, limn→∞1/λn|{k∈In:Δmxk-L≥ε}|=0. In this case one writes Sλ(Δm,X)-limk→∞xk=L.Definition 3 (see [22]).
A sequencex=(xk) is said to be I-[V,λ] summable to L, if, for any δ>0, {n∈N:1/λn∑k∈In‍xk-L≥δ}∈I, where In=[n-λn+1,n].Definition 4 (see [22]).
A sequencex=(xk) is said to be I-λ-statistically convergent or I-Sλ convergent to L if, for every ε>0 and δ>0, {n∈N:1/λn|{k∈In:xk-L≥δ}|∈I. In this case one writes xk→L(I-Sλ) or I-Sλ-limk→∞xk=L.The following well-known lemma is required for establishing some important results in this paper.Lemma 5.
LetF=(fk) be a sequence of modulus functions and 0<δ<1. Then for each x>δ one has fk(x)≤2fk(1)x/δ.Throughout the paper,N will denote the set of all positive integers. By SXn we denote the space of all sequences defined over n-normed space (X,·,…,·).LetI⊆2N be an admissible ideal, F=(fk) a sequence modulus functions, p=(pk) a bounded sequence of positive (strictly) real numbers, u=(uk) a sequence of positive real numbers, A=(ank) an infinite matrix, and (X,·,…,·) a n-normed space. Then, for every δ>0 and z1,…,zn-1∈X, we define the following sequence spaces:(5)VFIΔvm,λ,A,u,p,·,…,·0=x=xk∈SXn:∑k∈In‍=iiin∈N:∑k∈In‍=iiii1λn∑k∈In‍ankfkukΔvmxk,z1,…,zn-1pk≥δ∈I,VFIΔvm,λ,A,u,p,·,…,·=x=xk∈SXn:∑k∈In‍=i=n∈N:∑k∈In‍=iiii=1λn∑k∈In‍ankfkukΔvmxk-L,z1,…,zn-1pk≥δaaa∑k∈In‍∈I,for some L>0, and(6)VFIΔvm,λ,A,u,p,·,…,·∞=x=xk∈SXn:∑k∈In‍aaaan∈N:∑k∈In‍aaaaaa1λn∑k∈In‍ankfkukΔvmxk-L,z1,…,zn-1pk≥Kaaa∑k∈In‍∈I,for some K>0.If we takeF(x)=fk(x)=x, u=(uk)=1 for all k∈N, v=1, and A=(ank)=I(identitymatrix) and n-normed space is replaced by 2-normed space then we get the sequence spaces defined by Kumar et al. [29].The following inequality will be used throughout the paper. Letp=(pk) be a sequence of positive real numbers with 0<pk≤sup⁡kpk=H, and let D=max⁡{1,2H-1}. Then, for the factorable sequences (ak) and (bk) in the complex plane, we have(7)ak+bkpk≤Dakpk+bkpk.


## 2. Main Results

The main purpose of this section is to study some topological properties and some inclusion relations between the sequence spaces which we have defined above.Theorem 6.
SupposeF=(fk) is a sequence of modulus functions, p=(pk) is a bounded sequence of strictly positive real numbers, u=(uk) is a sequence of positive real numbers, and A=ank  is an infinite matrix. Then the spaces VFI[Δvm,λ,A,u,p,·,…,·]0, VFI[Δvm,λ,A,u,p,·,…,·], and VFI[Δvm,λ,A,u,p,·,…,·]∞ are linear spaces over the real field R.Proof.
The proof of the theorem is easy so we omit it.For the next result we shall define the following sequence space:(8)VFΔvm,λ,A,u,p,·,…,·∞=x=xk∈SXn:∑k∈InankfkukΔvmxk-L,z1,…,zn-1pkaaaaaasup⁡n1λn∑k∈InankfkukΔvmxk-L,z1,…,zn-1pkaaaaaa∑k∈InankfkukΔvmxk-L,z1,…,zn-1pk<∞.Theorem 7.
LetF=(fk) be a sequence of modulus functions and p=(pk) a bounded sequence of strictly positive real numbers. Then the space VF[Δvm,λ,A,u,p,·,…,·]0 is a paranormed space with the paranorm:(9)gx=sup⁡n1λn∑k∈InankfkukΔvmxk,z1,…,zn-1pk1/G,where G=max⁡(1,H) and 0<pk≤sup⁡kpk=H.Proof.
Clearlyg(x)=g(-x) for x∈VFI[Δvm,λ,A,u,p,·,…,·]0. It is trivial that ukΔvmxk=0 for x=0. Since fk(0)=0, we get g(x)=0 for x=0. Since pk/G≤1, using Minkowski’s inequality, we have(10)1λn∑k∈InankfkukΔvmxk+yk,z1,…,zn-1pk1/G≤1λn∑k∈InankfkukΔvmxk,z1,…,zn-1aaaaaaaaaaaaaaa∑k∈Inank+ukΔvmyk,z1,…,zn-1pk1/G≤1λn∑k∈In‍ankfkukΔvmxk,z1,…,zn-1pk1/G+1λn∑k∈In‍ankfkukΔvmyk,z1,…,zn-1pk1/G.Hence g(x+y)≤g(x)+g(y). Finally, to check the continuity of scalar multiplication, let α be any complex number; therefore by definition(11)gαx=sup⁡n1λn∑k∈In‍ankfkukΔvmαxk,z1,…,zn-1pk1/G≤CαH/Ggx,where Cα is a positive integer such that α≤Cα. Let α→0 for any fixed x with g(x)≠0. By definition, for |α|<1, we have(12)1λn∑k∈In‍ankfkukΔvmαxk,z1,…,zn-1pk<ϵ,∑k∈In‍aaaafkukΔvmαxk,z1,pkforn>N(ϵ).Also, for 1≤n≤N, taking α small enough, since fk is coninuous, we have(13)1λn∑k∈In‍ankfkukΔvmαxk,z1,…,zn-1pk<ϵ.Now (12) and (13) imply that g(αx)→0 as α→0. This completes the proof.Theorem 8.
Letu=(uk) be a sequence of positive real numbers. Then for m≥1 the following inclusions:(i)
VFI[Δvm-1,λ,A,u,·,…,·]0⊂VFI[Δvm,λ,A,u,·,…,·]0,(ii)
VFI[Δvm-1,λ,A,u,·,…,·]∞⊂VFI[Δvm,λ,A,u,·,…,·]∞,are strict.Proof.
We will prove the result forVFI[Δvm-1,λ,A,u,·,…,·]0 only. The others can be proved similarly.
Supposex∈VFI[Δvm-1,λ,A,u,·,…,·]0; by definition, for every δ>0 and z1,…,zn-1∈X, we have(14)n∈N:1λn∑k∈In‍ankfkukΔvm-1xk,z1,…,zn-1≥δ∈I.By the property of modulus function, we have(15)1λn∑k∈In‍ankfkukΔvmxk,z1,…,zn-1≤1λn∑k∈In‍ankfkukΔvm-1xk,z1,…,zn-1≤1λn∑k∈In‍ankaa+fkukΔvm-1xk+1,z1,…,zn-1≤D1λn∑k∈In‍ankfkukΔvm-1xk,z1,…,zn-1+D1λn∑k∈In‍ankfkukΔvm-1xk+1,z1,…,zn-1by7.Now, for given δ>0, we have(16)n∈N:1λn∑k∈In‍ankfkukΔvmxk,z1,…,zn-1≥δ⊆n∈N:1λn∑k∈In‍ankfkukΔvm-1xk,z1,…,zn-1aaaaaa∑k∈In‍≥δ2D∪n∈N:1λn∑k∈In‍ankfkukΔvm-1xk+1,z1,…,zn-1aaaaaa∑k∈In‍≥δ2D,for each z1,…,zn-1∈X. Since x∈VFI[Δvm-1,λ,A,u,·,…,·]0, it follows that the sets on the right-hand side in the above containment belong to I. Hence x∈VFI[Δvm-1,λ,A,u,·,…,·]0. To show that the inclusion is strict, we give the following example.
We takefk(x)=x, λn=n, and A=(ank)=I, for all n,k∈N, v=1, and consider a sequence x=(xk)=km-1; then x∈VFI[Δvm,λ,A,u,·,…,·]0 but does not belong to VFI[Δvm-1,λ,A,u,·,…,·]0 for u=(uk)=1,k∈N. This shows that the inclusion is strict.Theorem 9.
LetF′=(fk′) and F′′=(fk′′) be sequences of modulus functions. If  limsup⁡t→∞fk′(t)/fk′′(t) = P>0, then VF′I[Δvm,λ,A,u,p,·,…,·]⊂VF′′I[Δvm,λ,A,u,p,·,…,·].Proof.
Letlimsup⁡t→∞fk′(t)/fk′′(t)=P; then there exists a constant M>0 such that fk′(t)≥Mfk′′(t) for all t≥0. Therefore, for each z1,…,zn-1∈X, we have(17)1λn∑k∈In‍ankfk′ukΔvmxk-L,z1,…,zn-1pk≥MH1λn∑k∈In‍ankfk′′ukΔvmxk-L,z1,…,zn-1pk.Then, for every δ>0 and z1,…,zn-1∈X, we have the following relationship:(18)n∈N:∑k∈In‍aii1λn∑k∈In‍ankfk′′ukΔvmxk-L,z1,…,zn-1pk≥δ⊆n∈N:∑k∈InankMHaaaaaa1λn∑k∈Inankfk′ukΔvmxk-L,z1,…,zn-1pkaaaaaa∑k∈Inank≥δMH.
Sincex∈VF'I[Δvm,λ,A,u,p,·,…,·], it follows that the set on left-side of the above containment belongs to I, which gives x∈VF′′I[Δvm,λ,A,u,p,·,…,·].Theorem 10.
LetF=(fk), F′=(fk′), and F′′=(fk′′) be sequences of modulus functions. Then(i)
VF′I[Δvm,λ,A,u,p,·,…,·]⊂VF∘F′I[Δvm,λ,A,u,p,·,…,·],(ii)
VF′I[Δvm,λ,A,u,p,·,…,·]∪VF′′I[Δvm,λ,A,u,p,·,…,·]⊂VF′+F′′I[Δvm,λ,A,u,p,·,…,·].Proof.
(i) Letx=(xk)∈VF′I[Δvm,λ,A,u,p,·,…,·], for every ε>0 and for some L>0 such that (19)n∈N:1λn∑k∈In‍ankfk′ukΔvmxk-L,z1,…,zn-1pk≥ε∈I,for each z1,…,zn-1∈X. For given ε>0, we choose δ∈(0,1) such that fk(t)<ε for all 0<t<δ. On the other hand, we have(20)1λn∑k∈In‍ankfk∘fk′ukΔvmxk-L,z1,…,zn-1pk=1λn∑k∈In,fk′ukΔvmxk-L,z1,…,zn-1pk<δankfk∘fk′1λn∑k∈In‍ank1λn∑k∈In‍aaaaaaaaaaaaaaaaaaaa·ukΔvmxk-L,1λn∑k∈In‍ank1λn∑k∈In‍aaaaaaaaaaaaaaaaaaaaaaafk∘fk′z1,…,zn-1pk+1λn∑k∈In,fk′ukΔvmxk-L,z1,…,zn-1pk≥δank‍+1λn∑k∈In,fk′ukΔvmxk-L,z1,…,zn-1pk≥δank‍·fk∘fk′+1λn∑k∈In,fk′ukΔvmxk-L,z1,…,zn-1pk≥δaaank‍·ukΔvmxk-L,+1λn∑k∈In,fk′ukΔvmxk-L,z1,…,zn-1pk≥δaaaaank‍fk∘fk′z1,…,zn-1pk≤εH+max⁡1,2fk1δH1λn·∑k∈In‍ankfk′ukΔvmxk-L,z1,…,zn-1pk,by Lemma 5.
By using (19), we obtain x∈VF∘F′IΔvm,λ,A,u,p,·,…,·.
(ii) The result of this part is proved by using the following inequality:(21)1λn∑k∈In‍ankfk′+fk′′ukΔvmxk-L,z1,…,zn-1pk≤Dλn∑k∈In‍ankfk′ukΔvmxk-L,z1,…,zn-1pk+Dλn∑k∈In‍ankfk′′ukΔvmxk-L,z1,…,zn-1pk,where sup⁡kpk=H and D=max⁡(1,2H-1).Theorem 11.
LetF=(fk) be a sequence of modulus functions and p=(pk) a bounded sequence of strictly positive real numbers. Then VI[Δvm,λ,A,u,p,·,…,·]⊆VFI[Δvm,λ,A,u,p,·,…,·].Proof.
This can be proved by using the same techniques as in Theorem10.Theorem 12.
LetF=(fk) be a sequence of modulus functions. If limsup⁡t→∞fk(t)/t=M>0, then VFI[Δvm,λ,A,u,p,·,…,·]⊆VI[Δvm,λ,A,u,p,·,…,·].Proof.
Supposex=(xk)∈VFI[Δvm,λ,A,u,p,·,…,·] and limsup⁡t→∞fk(t)/t=M>0, then there exists a constant K>0 such that fk(t)≥Kt for all t≥0. Thus, we have(22)1λn∑k∈In‍ankfkukΔvmxk-L,z1,…,zn-1pk≥KH1λn∑k∈In‍ankukΔvmxk-L,z1,…,zn-1pk,for each z1,…,zn-1∈X. This completes the proof.Theorem 13.
If0<pk≤qk and (qk/pk) be bounded. Then VFI[Δvm,λ,A,u,q,·,…,·] ⊂ VFI[Δvm,λ,A,u,p,·,…,·].Proof.
It is easy to prove so we omit the detail.


## 3. Statistical Convergence

The notion of statistical convergence introduced by Fast [30] in 1951 and later developed by Fridy [31], Šalát [32] and many others. Furthermore, Kostyrko et al. [33] presented a very interesting generalization of statistical convergence called as I-convergence. Some recent developments in this regard can be found in [34–37] and many others.In this section, we define a new class of generalized statistical convergent sequences with the help of an ideal, modulus functions and infinite matrix. We also made an effort to establish a strong connection between this convergence and the sequence spaceVFI[Δvm,λ,A,u,p,·,…,·].Definition 14.
LetI⊆P(N) be a non-trivial ideal and λ=(λn) be a non-decreasing sequence. A sequence x=(xk)∈X is said to be SλΔvm(I,u,A,·,…,·)-convergent to a number L provided that for every ε>0, δ>0 and z1,…,zn-1∈X, the set(23)n∈N:1λnk∈In:ankukΔvmxk-L,z1,…,zn-1≥ε1λn≥δ∈I.In this case, we write SλΔvm(I,u,A)-limk→∞ukΔvmxk,z1,…,zn-1=L/ρ,z1,…,zn-1. Let SλΔvm(I,A,u,·,…,·) denotes the set of all SλΔvm(I,u,A,·,…,·)-convergent sequences in X.Theorem 15.
LetF=(fk) be a sequence of modulus functions and 0<inf⁡kpk=h≤pk≤sup⁡kpk=H<∞. Then VFI[Δvm,λ,A,u,p,·,…,·]⊂SλΔvm[I,A,u,·,…,·].Proof.
Supposex=(xk)∈VFI[Δvm,λ,A,u,p,·,…,·] and ε>0 be given. Then for each z1,…,zn-1∈X, we obtain(24)1λn∑k∈In‍ankfkukΔvmxk-L,z1,…,zn-1pk=1λn∑k∈In,∥ukΔvmxk-L,z1,…,zn-1∥≥ε‍ankfkukΔvmxk-L,=1λn∑k∈In,∥ukΔvmxk-L,z1,⋯,zn-1∥≥ε‍ankaaaaaz1,…,zn-1pk+1λn∑k∈In,∥ukΔvmxk-L,z1,…,zn-1∥<ε‍ankfkukΔvmxk-L,+1λn∑k∈In,∥ukΔvmxk-L,z1,…,zn-1∥<ε‍aaaankaz1,…,zn-1pk≥1λn∑k∈In,∥ukΔvmxk-L,z1,…,zn-1∥≥ε‍ankfkukΔvmxk-L,≥aiaaai1λn∑k∈In,∥ukΔvmxk-L,z1,…,zn-1∥≥ε‍ankz1,…,zn-1pk≥1λn∑k∈In‍fkεpk≥∑k∈In‍min⁡fkεh,fkεH≥K1λnk∈In:ankukΔvmxk-L,z1,…,zn-1≥ε,where K=min⁡([fk(ε)]h,[fk(ε)]H). Then for every δ>0 and z1,…,zn-1∈X, we have(25)n∈N:1λnk∈In:ankukΔvmxk-L,z1,…,zn-1≥ε1λn≥δ⊆n∈N:1λn∑k∈In‍ankfkukΔvmxk-L,z1,…,zn-1pkaaiaa∑k∈In‍≥Kδ.Since xk→L(VFI[Δvm,λ,A,u,p,·,…,·])  so that SλΔvm(I,u,A)-limk→∞ukΔvmxk,z1,…,zn-1 = L,z1,…,zn-1.Theorem 16.
LetF=(fk) be a sequence of modulus functions and p=(pk) be a bounded sequence of strictly positive real numbers. If 0<inf⁡kpk=h≤pk≤sup⁡kpk=H<∞ then SλΔvm[I,A,u,·,…,·] ⊂ VFI[Δvm,λ,A,u,p,·,…,·].Proof.
By using ([11, Theorem 3.5]) we can prove easily.Theorem 17.
LetF=(fk) be a bounded sequence of modulus functions and p=(pk) be a bounded sequence of strictly positive real numbers. If  0<inf⁡kpk=h≤pk≤sup⁡kpk=H<∞. Then SλΔvm[I,A,u,·,…,·] = VFI[Δvm,λ,A,u,p,·,…,·] if and only if F=(fk) is a bounded.Proof.
This part can be obtained by combining Theorems15 and 16. Conversely suppose F=(fk) be unbounded defined by fk(k)=k for all k∈N. We take a fixed set B∈I, where I is an admissible ideal and define x=(xk) as follows(26)xk=km+1,forn-λn+1≤k≤n,n∉Bkm+1,forn-λn+1≤k≤n,n∈B0,otherwise.For given ε>0 and for each z1,…,zn-1∈X we have,(27)limn→∞1λnk∈In:ankukΔvmxk-0,z1,…,zn-1≥ε<λnλn⟶0,for n∉B. Hence for δ>0, there exists a positive integer n0 such that 1/λn|{k∈In:ankukΔvmxk-0,z1,…,zn-1≥ε}|<δ for n∉B and n≥n0. Now we have 1/λn|{k∈In:ankukΔvmxk-0,z1,…,zn-1≥ε}|≥δ⊂{B∪(1,2,…,n0-1)}. Since I be an admissible ideal it follows that SλΔvm(I,u,A)-limk→∞ukxk,z1,…,zn-1→0 for each z1,…,zn-1∈X.
On the other hand, if we takep=(pk)=1, for all k∈N then xk∉VFI[Δvm,λ,A,u,p,·,…,·]. This contradicts the fact SλΔvm[I,A,u,·,…,·]=VFI[Δvm,λ,A,u,p,·,…,·], so our supposition is wrong.



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*Source: 413850-2015-05-18.xml*