L-fuzzy set by incorporating membership functions, nonmembership functions from a nonempty set X to any lattice L, and lattice homomorphism from L to the interval . In this article, lattice-valued intuitionistic fuzzy subgroup type-3 (LIFSG-3) is introduced. Lattice-valued intuitionistic fuzzy type-3 normal subgroups, cosets, and quotient groups are defined, and their group theocratic properties are compared with the concepts in classical group theory. LIFSG-3 homomorphism is established and examined in relation to group homomorphism. The research findings are supported by provided examples in each section.The lattice-valued intuitionistic fuzzy set was introduced by Gerstenkorn and Tepavcevi as a generalization of both the fuzzy set and the  U as described by Zadeh is based on the formulation of a function μ from U to the closed interval . The function is called a membership function, whereas the images of elements of U under this function are called membership grades. For instance, let U be a collection of finite groups, p(x) be the total number of subgroups in x ∈ U. If q(x) is the total number of normal subgroups computed by a student in x, then μ(x)=q(x)/p(x) defines a fuzzy membership grade to the normal subgroups in x. But there is a chance that if the group order is large and the student is unable to compute all the normal subgroups, then μ(x) will be greater than the one reported by the student. This leads us toward the concept of nonmembership grades first introduced by Atanassov  is replaced by a partially ordered set L. Atanassov ; and termed their finding lattice-valued intuitionistic fuzzy set of a second type (LIFS-2). Different properties, such as the decomposition theorem and synthesis, were established for these fuzzy sets. However, the choice of a linearization map makes LIFS-2 less capable of dealing with basic set operations. For instance, the union of two LIFS-2s need not be a LIFS-2. Thus, the map was replaced with lattice homomorphism α : L⟶, and the refinement is called a lattice-valued intuitionistic fuzzy set type-3 and abbreviated as LIFS-3.Over the years, several other generalizations of fuzzy sets have been introduced depending upon various parameters of uncertainty, vagueness, and imprecision by employing membership, nonmembership, hesitancy, and indeterminacy grades. In all these generalizations, the grades are real numbers ranging between 0 and 1. The interval  inherits the natural partial order from the set of real numbers and constitutes a lattice. Partial ordering and fuzzy uncertainties are key features of real-life problems with infinite solutions or no solution at all. So it is quite obvious to think about the replacement of  by any suitable lattice. Goguen  introductanassov  presenteˇ cev i´  refined The term group was first used by Évariste Galois in the 1830s for the set of roots of polynomial equations. However, the modern-day definition of the group was established in 1870. Since then, significant research has been carried out in this area, and now the group is one of the most important algebraic structures providing a basic structure for several mathematical branches including analysis, game theory, coding theory, and algebraic geometry. Groups have strong applications in different scientific fields, especially symmetric groups, which play a vital role in theocratical physics and quantum mechanics. In genetics, the four-codon basis constitutes a group isomorphic to the Klein four-group. Gene mutation can be identified by establishing group homomorphism on copies of the sixty-four codon system. The coset diagram depicting group action has a close link with the crystal structure in chemistry. After Zadeh's invention, many researchers attempted to use and replace the ordinary set with the fuzzy set in various theocratical and experimental areas.G is a fuzzy subgroup if and only if all the level sets are subgroups of G  and refinement is abbreviated as (LIFS-3).Gerstenkorn and Tepa ˇ cev i´  refined 𝒮, L, ζℑL, ξℑL, ℘), with 𝒮 a nonempty set, L is a complete lattice with top and bottom elements T and B, respectively, ζℑL : X⟶L and ξℑL : 𝒮⟶L are membership and nonmemberships functions. The map ℘ : L⟶ is a lattice homomorphism with ℘(T)=1, ℘(B)=0)+℘(ξℑL(s)) ≤ 1.Such that for every S={a, b, c, d, e, f} and the lattice L={B, r, s, t, τ} w.r.t partial orderConsider ℘ : L⟶, ζℑL : S⟶L and ξℑL : 𝒮⟶LDefine s ∈ S, ℘(ζℑL(s))+℘(ξℑL(s)) ≤ 1 imply that  is a LIFS-3 of 𝒮.Then, for every G, membership ζIL  :  G ⟶ L, nonmembership ξIL  :  G ⟶ L, and a lattice homomorphism ℘  :  L ⟶ L. The combination of LIFS − 3 and group will provide us a new refined algebraic structure which we can use effectively in real world problems.We will define lattice-valued intuitionistic fuzzy subgroups (LIFSG) by using a group G, lattice L with top and bottom elements T and B, respectively, and lattice homomorphism ℘ : L⟶. A LIFS-3 of G, that is, IL =  formulates a LIFSG-3 of G provided that for every x, y ∈ G,ζIL(xy) ≥ ζIL(x)∧ζIL(y) and ξIL(xy)  ≤  ξIL(x)∨ξIL(y)ζIL(x)=ζIL(x−1) and ξIL(x)=ξIL(x−1)For a group L be a lattice  and G be a group. Let IL =  be a LIFSG-3 of G. For t1  ∈  ImζIL and t2  ∈  ImξIL, the -cut set (or level set) is a subgroup of G, called -cut subgroup (or level subgroup).Let Recall the definition of cut sets in IFS where these sets are defined as follows:e be the identity element in G. Then, for any u ∈ G e ∈ uu−1Let ζIL(e) is an upper bound for ImζIL.By the property of LIFSG, we haveξIL(e) ≤ ξIL(u)∀ u ∈ G, this implies that ξIL(e) is an lower bound for Im ξIL. We get that e ∈ ILt1, t2) ≤ t2, this implies that uv−1 ∈ ILt1, t2) be a LIFS-3 such that for each t1, t2  ∈  L, the -cut set ILt1, t2)( andCase 1: Let ζIL(u)=ℓ1 and ζIL(v)=ℓ2 and ℓ3=ℓ1∧ℓ2. Then ℓ3 ≤ ℓ1 and ℓ2 and uv ∈ ILℓ3, t2)(. We get that  Suppose ξIL(v)=ℓ5 and ℓ6=ℓ4∨ℓ5. Then, ℓ6 ≥ ℓ4 and ℓ5 and uv ∈ ILt1, ℓ6)(. Now using the same argument as above it is easy to show that  If (ii)u ∈ ILt1, t2) ≥ t3 ≥ t1,  ξIL(u) ≤ t2 ≤ t5,  ξIL(v) ≤ t4 ≤ t5Now, (3)t1  ≥  t3,  t4  ≤  t2. As L is a lattice so t1 ∧ t3 exist. Suppose t6 = t1 ∧ t3 then t1, t3  ≥  t6,Suppose for all ζIL(v) ≥ t3 ≥ t6,  ζIL(u) ≥ t1 ≥ t6,  ξIL(v) ≤ t4 ≤ t2Now G sustain the axioms of LIFSG-3. Hence, IL is a LIFSG-3 of G.On the basis of previous discussion we conclude that the elements of G and lattice L with T and B as the top and bottom element, the LIFS-3 IL= is a LIFSG-3 of G if and only ifFor every group G and lattice L with T and B as the top and bottom element, let H  ≤  G. For t ∈ L, define ζHt and ξHt from G to L asFor a group ℘ : L⟶ be a lattice homomorphism. Then, ℘(T)=1 and ℘(B)=0. Suppose ℘(t)=0.2, then G.Let ℘ : L⟶ be a lattice homomorphism defined as ℘(T)=1 and ℘(B)=0 and ℘(t)=0.2. For g  ∈  G we have two cases:g  ∈  H, ζHt(g)=t,  ξHt(g)=tCase 1: Let g ∈ G/H, ζHt(g)=B,  ξHt(g)=T(2) Case 2: Let Let g1, g2  ∈  G, we have three cases:g1, g2  ∈  H, this implies that g1g2  ∈  Hg1 ∈ H, g2 ∈ G/H, this implies that g1g2  ∉  H(2)ζHt(g1)=t, ζHt(g2)=B, ζHt(g1g2)=B  Thus, ξHt(g1g2)=T=ξHt(g1)∨ξHt(g2).Similarly, (3)g1, g2 ∈ G/H, then two cases arises:g1g2 ∈ G/H(b)g1g2 ∈ HNow, for From previous discussion we get thatG.s1 ∈ Im ζHt, s2 ∈ Im ξHtNow, if t, t)− cut subgroupThen, . Then, using this definition, we proved some useful results in this section.G, lattice L with top element T and bottom element B and lattice homomorphism ℘ : L⟶. Let IL= be a LIFSG-3. Then, IL is called LIFNSG-3 G ifFor a group G, lattice L with top element T and bottom element B and lattice homomorphism ℘ : L⟶. If IL= is a LIFNSG-3 of G, then for t1 ∈ Im ζIL and t2 ∈ Im ξILFor a group IL is a LIFNSG-3 of G. Let y  ∈  ILt1,t2 and x  ∈  G. Then, ζIL(y) ≥ t1 and ξIL(y) ≤ t2 this implies thatSuppose xyx−1 ∈ ILt1, t2) is a LIFNSG-3 of G if and only ifFor a group IL is a LIFNSG-3 of G. Then, ζIL(xyx−1)=ζIL(y)∀ x, y ∈ G, implies that ζIL(xy)=ζIL(yx). Similarly we get that ξIL(xy)=ξIL(yx). Conversely, assume that ζIL(xy)=ζIL(yx) and ξIL(xy)=ξIL(yx)∀ x,  y ∈ G. Then, ζIL((xy)x−1)=ζIL(x−1(xy))=ζIL(y) and ξIL((xy)x−1)=ξIL(x−1(xy))=ξIL(y).Suppose IL= be a LIFSG-3 of G. Then IL is LIFNSG-3 if and only if for all x, y  ∈  G,Let IL is LIFNSG-3 of G. Then, ζIL=ζIL(xyx−1y−1) implies that ζIL ≥ ζIL(xyx−1)∧ζIL(y−1). We get that ζIL ≥ ζIL(y). Similarly, ξIL ≤ ξIL(y). Conversely, assume that ζIL ≥ ζIL(y) and ξIL ≤ ξIL(y). Then ζIL(xyx−1)=ζIL(xyx−1y−1y). We get that ζIL(xyx−1) ≥ ζIL∧ζIL(y).Suppose ζIL(y)=ζIL(x−1xyx−1x) implies ζIL(y) ≥ ζIL(x)∧{ζIL(xyx−1)∧ζIL(x)}. If ζIL(x)∧ζIL(xyx−1)=ζIL(x), then ζIL(y) ≥ ζIL(x)∀ x, y ∈ G. ⇒ ζIL is constant. In this case,As ζIL(x)∧ζIL(xyx−1)=ζIL(xyx−1)If ξIL(x)∨ξIL(xyx−1)=ξIL(x), then this implies that ξIL(y) ≤ ξIL(x)∀ x, y ∈ G. Then, ξIL is constant. In this case ξIL(y)=ξIL(xyx−1). If ξIL(x)∨ξIL(xyx−1)=ξIL(xyx−1)If ζIL(xyx−1)=ζIL(y). Similarly, from (ξIL(xyx−1)=ξIL(y). Thus, IL is LIFNSG-3 of G. Hence proved.From  and 38)38), we gly, from  and 39)ζIL be a LIFSG-3 of G, such that all -cut subgroups of IL are normal in G. Then, IL is a LIFNSG-3.Let G is finite this implies that ImζIL and Im ξIL are finite sets. Suppose Im ζIL={t1, t2,…, tk} with t1 < t2,…, <tk and Im ξIL={s1, s2,…, sm} with s1 > s2,…, >sm−1 > sm. Consider 𝒳={ILti, sj)-cut subgroup of IL. As each As G, it can be expressed as union of 𝒞={xyx−1 : x ∈ G} and y ∈ ILti, sj)=ζIL(y) and ξIL(xyx−1)=ξIL(y) for all x  ∈  G and y ∈ ILti, sj)=ζIL(y) and ξIL(xyx−1)=ξIL(y)∀ x, y ∈ G. This implies that IL is a LIFNSG-3 of G.It is normal in In the first section, we introduced the fundamental concepts of the factor group and group homomorphism. In this section, we will discuss these two important features of classical group theory for LIFSG-3.G, lattice L with top and bottom element T and B. Consider the lattice homomorphism ℘ : L⟶. Let IL= be a LIFSG-3 of G. For x  ∈  G define two mapsFor a group Then,ILx= is a LIFS-3 of G. The LIFS-3This implies that G is called the lattice-valued intuitionistic fuzzy coset type-3 (LIFC-3) of G induced by x and IL.where IL is a LIFNSG-3 of G, then for any x  ∈  G,If IL is a LIFNSG-3 of G. Then, for x in GSuppose ξIL. Hence proved.Similarly, we proved for IL= be a LIFNSG-3 of G. Let G/IL={ILx : x ∈ G} be the collection of all LIFC-3 of G induced by x  ∈  G and IL. Then, G/IL is a group under the binary operation ILxoILy= and G/IL, whereLet ILx1, ILx2, ILy1, ILy2 ∈ G/IL such that ILx1=ILy1 and ILx2=ILy2Let ζILx1x2(g)=ζIL(g(x1x2)−1) implies that ζILx1x2(g) ≥ ζIL(gy2−1y1−1)∧ζIL(y1y2x2−1x1−1). As ζILx1 = ζILy1 and ζILx2 = ζILy2Consider g = y1y2x2−1, then ζIL(y1y2x2−1x1−1)=ζIL(y2x2−1), because IL is a lattice-valued intuitionistic fuzzy normal subgroup type-3. As ζIL(gx2−1)=ζIL(gy2−1), implies that ζIL(y1y2x2−1x1−1)=ζIL(e). Thus, we get thatIf Similarly, we get℘(ζILx1x2(g))+℘(ξILx1x2(g))=℘(ζIL(g(x1x2)−1))+℘(ξIL(g(x1x2)−1))Now we have, ζILe(g)=ζIL(ge−1)=ζIL(g) and ξILe(g)=ξIL(ge−1)=ξIL(g)∀ g ∈ G, this implies that ILe = IL and for any ILx ∈ G/IL.Thus, the binary operation is well defined. The associativity of composition of functions implies that the given binary operation is associative. ConsiderILx o ILe = ILx this implies that ILe is the identity element in G/IL. Let ILx ∈ G/IL, where x  ∈  G. As G is a group so x−1  ∈  G, implies that ILx−1 ∈ G/IL andG/IL is a group.Similarly, G/IL. Hence proved.Consider IL be a LIFNSG-3 of G. Then, G denoted by LIFQSG − 3.Let G be a group and IL be a LIFNSG-3 of G. The map φ : G⟶G/IL defined by φ(x)=ILx∀ xnG is epimorphism withLet x, y  ∈  G, then φ(xy)=ILxy=ILxoILy=φ(x)oφ(y) this implies that φ is a group homomorphism. Clearly, φ is onto,Let st isomorphism theorem we get that G/GζILξIL≅G/IL. Hence proved.From 1G be a group and IL= be a lattice-valued intuitionistic fuzzy subgroup type-3 of G. Suppose Let is a LIFSG-3 of Let Similarly, ϑoξIL)(a−1)=ϑoξIL(a). Now,JL is a LIFSG-3 of Similarly,  be a LIFNSG-3 of G. If Let IL is a LIFNSG-3 of G. Let Suppose ϑoξIL(uvu−1)=ξIL(ϑ−1(v))=ϑoξIL(v)⇒JL is a LIFNSG-3 of Similarly, G from the LIFS-3 of In the following proposition we will induce LIFSG-3 of IL= is a LIFSG-3 of G. Where ∀ g ∈ GLet g1, g2  ∈  G. Then,Let ξJLo φ(g1g2) ≤ ξJLo φ(g2)∨ξJLo φ(g2). Now,Similarly, ξJLo φ(g−1)=ξJLo φ(g).Similarly, G, L, ζJLoφ, ξJLoφ, ℘) is a LIFSG-3 of G. Hence proved.⇒=ξJLoφ(v). ⇒ IL is a LIFNSG-3 of G. Hence proved.Similarly, G and L be a lattice with top element T and bottom element B and ℘ : L⟶ be a lattice homomorphism. Let IGLIFSG−3 and G and Let Defined byis bijective.From previous propositions we know that ifThen,Now, ifϑ is bijectiveAs Similarly,ϑ is bijective, this implies ξIL1 = ξIL2. Thus, we get that, ifAs then G, L, ζJLoϑ, ξJLoϑ, ℘) ∈ IGLIFSG−3 and then (ϑo(ξJLo ϑ)=ξJL. This implies that Similarly, G be a group, L and T and B and ℘ : L⟶ be lattice homomorphisms such that ℘(T)=1,  ℘(B)=0. If IL= be a LIFSG-3 of G, then G. WhereLet are defined asandg1, g2  ∈  G. Then,Let foξIL(g1g2) ≤ foξIL(g1)∨foξIL(g2). Let g ∈ G andSimilarly, M1 and M2 are of the formThen, Thus,Hence, we get the required result.G and L and T and B and ℘ : L⟶ be lattice homomorphisms such that ℘(T)=1, ℘(B)=0. Let IL= be a LIFSG-3 of G. Then,Let is a LIFSG-3 of IL= is a LIFSG-3 of G, then from Theorem 4If G. From Proposition 9is a LIFSG-3 of is a LIFSG-3 of G, lattice L with top and bottom elements T and B, respectively, lattice homomorphism ℘ : L⟶. Let IL= be a LIFNSG-3 of G. Then, the setFor a group G andG/GζILξIL.is normal in N ⊲ G and ILN= is a LIFNSG-3 of G/N, then INL= is a LIFNSG-3 of G, whereConversely, if are defined asIL be a LIFNSG-3 of G. Then, ζIL(x) ≤ ζIL(e) and ξIL(x) ≥ ξIL(e)∀ x ∈ G, implies that GζILξIL is a cut subgroup so it is normal in G and allow to construct factor group G/GζILξIL. Now, for the map Let Similarly, for the map AlsoG/GζILξIL. Conversely, supposeG/N. Define ζINL : G⟶L asbe a LIFNSG-3 of Then,ξINL : G⟶L asDefine Then,INL is a LIFNSG-3 of G.⇒G are exactly the subgroups of G, and conversely, any LIFS-3 of G whose level sets are subgroups of G is LIFSG-3. Lattice-valued intuitionistic fuzzy normal subgroups type-3 and lattice-valued intuitionistic fuzzy factor subgroups type-3 of G are governed by normal subgroups and factor groups of G. Structure preserving LIFSG-3 maps are also discussed, and it is observed that they can be derived by extending group homomorphism. In the future, the notion foundations laid in this article can be used to find the Abelian subgroups of finite p-groups [L. The research findings can be utilized for application in algebra [The article is about the study of group theocratic concepts in a lattice-valued intuitionistic fuzzy type-3 environment. The structure is established by introducing lattice homomorphism, membership and nonmembership grades obeying certain laws for the binary operation defined on the group and inverses of elements under that operation. It is concluded that the level sets of LIFSG-3 of a group p-groups , verify p-groups , 36, comp-groups , aggregap-groups , and funp-groups  for lattp-groups –42 where algebra , 44 and  algebra  to handl


We study the boundary continuity of solutions to fully nonlinear elliptic equations. We first define a capacity for operators in non-divergence form and derive several capacitary estimates. Secondly, we formulate the Wiener criterion, which characterizes a regular boundary point via potential theory. Our approach utilizes the asymptotic behavior of homogeneous solutions, together with Harnack inequality and the comparison principle. For the existence of a solution u (in a suitable sense) to the Dirichlet problemu = f on ∂Ω is satisfied by the upper Perron solution, in general. Instead, we are forced to discover an additional condition for the boundary ∂Ω, which enables us to capture the boundary behavior of Let Ω be an open and bounded subset in x0 ∈ ∂Ω is regular with respect to Ω, iff ∈ C(∂Ω). One simple characterization of a regular boundary point is to find a barrier function; see ∂Ω such as an exterior sphere condition or an exterior cone condition have been invoked to guarantee the boundary continuity at x0 ∈ ∂Ω for a variety of elliptic operators.To be precise, we say a boundary point x0 ∈ ∂Ω is regular if and only if the Wiener integral diverges, i.e.2 is defined by the variational capacity of the Laplacian operator. Surprisingly, the Wiener criterion becomes both a sufficient and necessary condition for the regularity of a boundary point. Here the notion of capacity is used to measure the ‘size’ of sets in view of given differential equations. Roughly speaking, x0 ∈ ∂Ω is regular if and only if Ωc is ‘thick’ enough at x0 in the potential theoretic sense.On the other hand, Wiener  developeDj(aijDi), where aij is bounded and measurable, and for the Laplacian operator. For the p-Laplacian operator , Maz’ya  function u in Ω is a (viscosity) F-supersolution [resp. (viscosity) F-subsolution] in Ω, when the following condition holds:x0 ∈ Ω, φ ∈ C2(Ω) and u − φ has a local minimum at x0, thenu − φ has a local maximum at x0, then F(D2φ(x0)) ≥ 0.]if u ∈ C(Ω) a (viscosity) F-solution if u is both an F-subsolution and an F-supersolution.We say that Lemma 2.2. Suppose that a lower semi-continuous function u is an F-supersolution in Ω. ThenProof. We argue by contradiction: suppose thatφ ∈ C2(Ω), it follows that u − φ has a local minimum at x0 and so we can test this function. Therefore,Then for any Theorem 2.3.(Stability) Let {uk}k≥1 ⊂ C(Ω) be a sequence of F-solutions in Ω. Assume that ukconverges uniformly in every compact set of Ω to u. Then u is an F-solution in Ω.(Compactness) Suppose that {uk}k≥1 ⊂ C(Ω) is a locally uniformly bounded sequence of F-solutions in Ω. Then it has a subsequence that converges locally uniformly in Ω to an F-solution.Theorem 2.4 (Harnack convergence theorem). Let {uk}k≥1 ⊂ C(Ω) be an increasing sequence of F-solutions in Ω. Then the function u = limk→∞ukis either an F-solution or identically +∞ in Ω.Proof. If u(x) < ∞ for some x ∈ Ω, it follows from Harnack inequality that u is locally bounded in Ω. The interior Cα-estimate yields that the sequence uk is equicontinuous in every compact subset of Ω. Thus, applying Arzela-Ascoli theorem and We demonstrate two essential tools for Perron’s method, namely, the comparison principle and the solvability of the Dirichlet problem in a ball.Theorem 2.5 . Lef, there exists an increasing sequence of continuous functions {fn} such that fn → f pointwise as n → ∞.In the previous theorem, Theorem 2.6 (The solvability of the Dirichlet problem). Let Ω satisfy a uniform exterior cone condition and f ∈ C(∂Ω). Then there exists a unique F-solutionof the Dirichlet problemProof. The existence depends on the construction of global barriers achieving given boundary data and the standard Perron’s method; see  = −ℳ[u]; for example, any linear operator L and p-Laplcian operators Δp possess this property. Then we havex0 ∈ ∂Ω is regular iff ∈ C(∂Ω). Nevertheless, for the general fully nonlinear operator F, we do not have this property. Therefore, it seems that we have to require both conditions simultaneously, when we define a regular point for F. To the best of our knowledge, it is unknown whether the two conditions in the definition are redundant. One possible approach to show that only one condition is essential is to prove that f is resolutive whenever f is continuous on ∂Ω; see Before we define a barrier function, which characterizes a regular boundary point, we shortly deal with the resolutivity of boundary data:Definition 2.17 (Resolutivity). We say that a bounded function f on ∂Ω is (F-)resolutive if the upper and the lower Perron solutions f is resolutive, we write Lemma 2.18. Let Ω be a bounded open set of, let f and g be bounded functions on ∂Ω, and let c be any real number.If f = c on ∂Ω, then f is resolutive and Hf = c in Ω.andIf f is resolutive, then f + c is resolutive and Hf+c = Hf + c.If c > 0, then cf = cHffor c ≥ 0.If f ≤ g, thenandf does not implyx ∈ ∂Ω. However, the converse is true in some sense:Note that the resolutivity of Lemma 2.19. Let Ω be an open and bounded subset of Ω. Suppose that there exists F-harmonic h in Ω such thatfor any x ∈ ∂Ω. Then. In particular, f is resolutive.Proof. Since h ∈ 𝒰f ∩ ℒf, we have Lemma 2.20. If u is a bounded F-superharmonic (or F-subharmonic) function on the bounded open set Ω such that f(x) = limΩ∋y→xu(y) exists for all x ∈ ∂Ω, then f is a resolutive boundary function.Proof. Obviously, we have u ∈ 𝒰f and so F-harmonic in Ω andf is resolutive. An analogous argument works for the F-subharmonic case. □2.3.Definition 2.21 (Barrier). Let x0 ∈ ∂Ω. A function upper barrier [resp. lower barrier] in Ω at the point x0 ifw+ [resp. w−] is F-superharmonic [resp. F-subharmonic] in Ω;y→xΩ∋w+(y) > 0 [resp. lim supy→xΩ∋w−(y) < 0] for each x ∈ ∂Ω \ {x0};lim infw+ is positive in Ω and a lower barrier w− is negative in Ω. Moreover, under the condition (F2), cw+ is still an upper barrier for any constant c > 0 and an upper barrier w+. See also  byWe define the Lemma 3.1. Suppose that ℱ is a family of F-superharmonic functions in Ω, locally uniformly bounded below. Then the lower semi-continuous regularization s of inf ℱ,is F-superharmonic in Ω.Proof. Since ℱ is locally uniformly bounded below, s is lower semi-continuous. Fix an open D ⊂⊂ Ω and let F-harmonic function satisfying h ≤ s on ∂D. Then h ≤ u in D whenever u ∈ ℱ. It follows from the continuity of h that h ≤ s in D. □Definition 3.2 .ψ : Ω → capacity potential of E in Ω.In particular, we call the function Remark 3.3. For an operator in divergence form, there exists an alternative method to define the capacity potential. For simplicity, suppose that the operator is given by the p-Laplacian. Let Ω be bounded and K ⊂ Ω be a compact set. For ψ ≡ 1 on K, the p-harmonic function u in Ω \ K with capacity potential of K in Ω and denoted by ℛ. Here note that ℛ is independent of the particular choice of ψ and the existence of the capacity potential is guaranteed by the variational method. Indeed, both definitions of capacity potentials coincide; see  part in F-superharmonic function Br2 such that U of u across K, which is defined byU is the lower semi-continuous regularization of the function v, whereU = u in Br2 \ K and so U is F-superharmonic in Br2 \ K.Now we consider a canonical lower semi-continuous extension  K.See  for detaF-superharmonic in Br2, for any ε > 0 and z0 ∈ Br2\K. Indeed, the convexity of F immediately guarantees that F-superharmonic in Br2 \ K. On the other hand, since K. In other words, for any φ ∈ C2(Ω), x0 ∈ K. Thus, recalling the equivalence of F-supersolution and F-superharmonic function , consider the upper Perron solution f(x0) = 0 and max∂Ω |f| ≤ 1. For ε > 0, we can choose a ball B with center x0 such that ∂(2B) ∩ Ω ≠ ∅ and |f| < ε in 2B ∩ ∂Ω. Then we defineB, F-harmonic in Ω ∩ 2B. On the other hand, by u is continuous in Ω and by the pasting lemma, u is F-superharmonic in Ω. Moreover, it can be easily checked thatu ∈ 𝒰f and so v ∈ ℒf and so,ε > 0 is arbitrary, we conclude thatx0 is regular. □Next, we exhibit a converse direction of the above lemma: i.e. a characterization of an irregular boundary point. We expect that this lemma may be employed to prove the necessity of the Wiener criterion for the general case.Lemma 3.22 (Characterization of an irregular boundary point). If there exists a constant ρ > 0 such that the capacity potential u = uρofwith respect to B2ρ(x0) satisfies the inequalitythen the boundary point x0 ∈ ∂Ω is irregular.Proof. Since the capacity potential u is the lower semi-continuous regularization, we haveuρ′ ≤ uρ when 0 < ρ′ < ρ. Thus, we can choose a sufficiently small ρ > 0 such that (∂Bρ2(x0) ≠ ∅.we haveu(x0)=lif on ∂(Ω ∩ Bρ2(x0)) such that f(x) = 3/2 if x ∈ ∂Ω ∩ Bρ/2(x0), 0 ≤ f(x) ≤ 3/2 if x ∈ ∂Ω ∩ (Bρ(x0) \ Bρ/2(x0)) and f(x) = 0 on the remaining part of ∂(Ω ∩ Bρ2(x0)). Then we consider the lower Perron solution Bρ2(x0). For this purpose, let v ∈ ℒf(Ω∩Bρ2(x0)) and g is given by ) First, for x ∈ ∂Ω ∩ Bρ(x0), we have(on x ∈ ∂Ω ∩ (Bρ2(x0) \ Bρ(x0)), we haveNext, for ∂Bρ2(x0)) Similarly, we obtain. Recalling x0 is irregular with respect to Ω. □Therefore, 4.In this section, we prove the sufficiency of the Wiener criterion and its sequential corollaries, via the potential estimates. More precisely, we first develop quantitative estimates for the capacity potential Definition 4.1. We say that a set E is F-thick at z if the Wiener integral diverges, i.e.Remark 4.2. Recalling c > 0 which is independent of t > 0 such that form of :∫01caparing in , 40.Now we can state an equivalent form of our main theorem, c is both F-thick and x0 ∈ ∂Ω, then x0 is regular. To prove this statement, we need several auxiliary lemmas regarding the capacity potential.If ΩLemma 4.3. Fix a ball B. Suppose that K ⊂ B′ is compact andIf 0 < γ < 1 and Kγ ≔ {x ∈ B : v(x) ≥ γ} ⊂ B′, thenProof. We write F-superharmonic in B and so is v/γ due to (F2). Since v ≥ γ in Kγ, we have v/γ ≥ 1 in Kγ. Thus, Clearly, B\Kγ whereRecalling x ∈ ∂Kγ. Since u is F-superharmonic and v/γ is F-harmonic in B \ Kγ, the comparison principle leads to u ≥ v/γ in B \ Kγ and soThen for Consequently, we conclude that□Lemma 4.4. Fix a ball B = B2r(x0). Let K ⊂ Br = Br(x0) be a compact set andThen there exists a constant c > 0 which is independent of K and r such thatfor any x ∈ Br.Proof. Denotev is a non-negative F-solution in B\K, Harnack inequality yields that there exists a constant c1 > 0 independent of r > 0 such thatSince B \ Br/56 implies thatMorevoer, the strong maximum principle in K and r.Here we applied KM ⊂ B′, we can apply Now since Finally, combining , 4.3) a a4.3) anWe may rewrite the previous lemma asLemma 4.5. Let x0 ∈ ∂Ω, ρ > 0 andThen for all 0 < r ≤ ρ, there exists a constant c > 0 such thatfor any x ∈ Br(x0).Proof. Denote r ≤ ρ and let k be the integer with 2k−ρ < r ≤ 2k1−ρ.i = 0, 1, 2, …Then write for et ≥ 1 + t, estimate  is given byFurthermore, we claim that Thus, for u ≥ ui+1 in Di+1 and so Bi+1.Therefore, by the comparison principle, v0 ≥ u1 ≥ ⋯ ≥ uk in Bk, which implies thatRepeating the argument above, we conclude that t ≤ s ≤ 2t, thenn, λ, Λ and these results also hold for Finally, the result follows fromNow we are ready to prove the sufficiency of the Wiener criterion, Proof of Theorem 1.2. Let x0 ∈ ∂Ω, ρ > 0 and definer ≤ ρ, there exist a constant c1, c2 > 0 such thatx ∈ Br(x0). Letting r → 0+, we conclude thatThen applying ρ > 0 can be arbitrarily chosen, an application of x0 ∈ ∂Ω is a regular boundary point.  at x0 ∈ ∂Ω, then we can deduce the continuity of the Perron solution at x0 under a relaxed condition:On the other hand, if additional information is imposed on the boundary data Corollary 4.6. Suppose that f ∈ C(∂Ω) attains its maximum [resp. minimum] at x0 ∈ ∂Ω. If Ωcis F-thick [resp. thick] at x0 ∈ ∂ Ω, thenProof. Similarly as in the proof of the previous theorem, this corollary is the consequence of f ∈ C(∂Ω) is resolutive, then we are able to obtain a quantitative estimate for the modulus of continuity.Furthermore, if the given boundary data Lemma 4.7 (The modulus of continuity). Suppose that Ω is an open and bounded subset ofLet f ∈ C(∂Ω).If x0 ∈ ∂Ω with f(x0) = 0, then for 0 < r ≤ ρ, we haveandwhere Ωr ≔ Ω ∩ Br(x0) and ∂Ωρ2 ≔ ∂Ω ∩ Bρ2(x0).Furthermore, if f is resolutive, then we have the quantitative estimate:whereProof. Let Bρ2.w ≔ 1 − v and writeThen let f(x0) = 0, we have max∂Ωf ≥ 0 and u is F-subharmonic and s is F-harmonic in Ωρ2. Moreover,Note that since we assumed s ≥ u in Ωρ2 and so ρ2.Thus, the comparison principle yields that On the other hand, let∂Ωρ2.By the same argument, we derive w  such that x0 ∈ ∂Ω, then Ω satisfies an exterior corkscrew condition at x0. Thus, the following corollary obtained from the  Wiener criterion is a generalized result of Note that if Ω satisfies an exterior cone condition at Corollary 4.9 (Exterior corkscrew condition). Suppose that Ω satisfies an exterior corkscrew condition at x0 ∈ ∂Ω. Then x0is a regular boundary point. Moreover, if f is Hölder continuous at x0and is resolutive, then Hfis Hölder continuous at x0.Proof. A small modification of n, λ, Λ and δ. Thus, if x0 satisfies an exterior corkscrew condition, then we havex0 is a regular boundary point by the Wiener criterion.f(x0) = 0 by adding a constant for f, if necessary. Since f is resolutive, we can apply the quantitative estimate obtained in Next, for the second statement, we may assume Heref is Hölder continuous at x0: there exists a constant C > 0 such that |f(x)| = |f(x) − f(x0)| ≤ C|x − x0|γ ≤ Cργ for x ∈ ∂Ωρ2.x0:Ω satisfies an exterior corkscrew condition at ρ = r/21, we conclude that the Perron solution Hf is Hölder continuous at x0. □Thus, choosing Remark 4.10 (Example). In this example, we suppose n = 2, λ < Λ. Then it immediately follows that∂Ω.We consider a domain α*(F) < 0, we know that a single point has non-zero capacity. More precisely, recalling the homogeneous solution for F is given byc = c > 0 such thatSince Therefore, we havec is F-thick at 0.In other words, ΩOn the other hand, since c is not In other words, Ωf1 ∈ C(∂Ω) is a boundarye data given byLet f1 is resolutive) andThen clearly the function f1 attains its maximum at 0 and Ωc is F-thick at 0.Alternatively, one can apply f2 ∈ C(∂Ω) is a boundary data given byLet ε > 0, we have ε → 0, we conclude Then since the zero function belongs to Therefore, we deduce that 5.F. Indeed, our strategy is to employ the argument made in [p-Wiener criterion for p-Laplacian operator with p > n − 1. Since the assumption p > n − 1 was essentially imposed to ensure the capacity of a line segment is non-zero in [In this section, we provide the necessity of the Wiener criterion, under additional structure on the operator  made in  which pr-zero in , we begiLemma 5.1. Suppose that F is convex and α*(F) > s for some s > 0. Let K be a compact subset in  ℋs(K) < ∞, where ℋsis the s-dimensional Hausdorff measure. ThenProof. For any δ > 0, defineK by balls Bi with diameter ri not exceeding δ. Then since K is compact, for each δ ∈ , there exist finitely many open balls ri < δ, F. Here we may assume minx|=1|V (x) = 1 by normalizing V. If we let Wi is non-negative and F-superharmonic in Wi(x) ≥ 1 on Bi.Now we consider the homogeneous solution F is convex, W is F-superharmonic in W ≥ 1 on W ≥ 1 on K. Therefore, α* > s. Letting δ → 0, we finish the proof. □Finally, we let  we used  and α* >K is given by a line segment L, whose Hausdorff dimension is exactly 1.Now we prove the partial converse statement of Lemma 5.2. Suppose that F is concave and α*(F) < 1. Let L = {x0 +se : ar ≤ s ≤ br} be a line segment in Br(x0), where e is an unit vector in  0 < a < b < 1 are constants satisfyingThenProof. Note that since L is a line segment, for any δ > 0, one can cover L by open balls Bi = Bδ3(xi), 1 ≤ i ≤ N(δ) where xi ∈ L, |xi − xj| ≥ 2δ whenever i ≠ j, and ε > 0, there exist a sufficiently small δ > 0 and corresponding cover Kδ such thatIf we denote V is given byα*(F) ∈ . Note that if α* < 0, then a single point has a positive capacity , writeOn the other hand, for simplicity, we suppose that the homogeneous solution capacity  and the F is concave, W is F-subharmonic in Since y ∈ ∂Bi for some i. Then for j ≠ i, we have(On α* < 1.Here we used the condition ∂Br2) For z ∈ ∂Br2,. Suppose that F is concave and α*(F) < 1. Let K be a compact subset in Br(x0) such that K meets  ∈ , where 0 < a < b < 1 are constants satisfyingThen there exists a constant c = c such thatProof. The proof is similar to the one of xt)( ∈ K∩S(t) for all t ∈ . In particular, for small δ > 0, we define xi ≔ xar+2δi) so thatδ > 0, we define a set Kδ byε > 0, there exists a sufficiently small δ > 0 such thatNote that V is given byα*(F) ∈ . For each i = 1, 2, ⋯ , N(δ), writeOn the other hand, for simplicity, we suppose that the homogeneous solution F is concave, W is F-subharmonic in Since ∂Kδ) For y ∈ ∂Kδ, let y ∈ ∂Bi for some i. Then for j ≠ i, we have(On α* < 1.Here we used the condition ∂Br2) For z ∈ ∂Br2,, letLet Lemma 5.4. Suppose that F is concave and α*(F) < 1. Then, there exists a constant c1 > 0 depending only on n, λ, Λ such that: ifthen the set Aγcontains a sphere  ∈ .Proof. For 0 < γ < 1, let Eγ ≔ {x ∈ Br2 : u(x) ≥ γ}. We argue by contradiction: suppose that Aγ does not contain any S(t) for t ∈ . Then the set Eγ meets S(t) for all t ∈  and we havea = 1/10 and b = 1/5.On the other hand, by Combining two estimates above, we obtainTherefore, by choosing Now we are ready to prove the necessity of the Wiener criterion, Proof of Theorem 1.3. For simplicity, we write Br = Br(x0). Suppose that Ωc is not F-thick at x0 ∈ ∂Ω, i.e.ε > 0 to be determined, choose r1 > 0 small enough so thatFor ri+1 = ri/2 andSet Applying i, choose a regular domain Ei such that Next, by Lemma 2.27 and bi ≤ (c0 +1)ε for i = 2, 3, ⋯ . Moreover, let γi = c1 · bi, the setS(ti) for some ti ∈ . Now by selecting γi < 1. In particular, since u2 = 1 on E2 and S(t2) ⊂ A2, we conclude that S(t2) ⊂ Ω.Then we havef ∈ C(∂Ω) be the boundary function defined byNext, let Then we have the following results for the lower Perron solution r > 0 large enough so that Ω ⊂ Br. Moreover, set a domain f0 ∈ C(∂Ω0) byBr is regular, we have v ∈ ℒf(Ω) and v ≤ w in Ω using the comparison principle.Then since Therefore, we conclude that F-harmonic in Ω and u ≤ 0 in S(t2), we claim thatFor E3 is a regular domain, we haveIndeed, since u ≤ u3 in S(t3) ⊂ A3, we observe thatThus, the comparison principle yields that u − γ3 instead of u), we conclude thatIterating this argument (for example, consider leads to .u, the estimate (x0 ∈ ∂Ω is an irregular boundary point. □Finally, recalling the definition of estimate  is equiv


Wireless sensor networks (WSNs) have been developed recently to support several applications, including environmental monitoring, traffic control, smart battlefield, home automation, etc. WSNs include numerous sensors that can be dispersed around a specific node to achieve the computing process. In WSNs, routing becomes a very significant task that should be managed prudently. The main purpose of a routing algorithm is to send data between sensor nodes (SNs) and base stations (BS) to accomplish communication. A good routing protocol should be adaptive and scalable to the variations in network topologies. Therefore, a scalable protocol has to execute well when the workload increases or the network grows larger. Many complexities in routing involve security, energy consumption, scalability, connectivity, node deployment, and coverage. This article introduces a wavelet mutation with Aquila optimization-based routing (WMAO-EAR) protocol for wireless communication. The presented WMAO-EAR technique aims to accomplish an energy-aware routing process in WSNs. To do this, the WMAO-EAR technique initially derives the WMAO algorithm for the integration of wavelet mutation with the Aquila optimization (AO) algorithm. A fitness function is derived using distinct constraints, such as delay, energy, distance, and security. By setting a mutation probability P, every individual next to the exploitation and exploration phase process has the probability of mutation using the wavelet mutation process. For demonstrating the enhanced performance of the WMAO-EAR technique, a comprehensive simulation analysis is made. The experimental outcomes establish the betterment of the WMAO-EAR method over other recent approaches. With the rapid advancement in Future Internet technologies, the mobile Internet, the Internet of Things (IoT), the physical world, and the sensor cloud are regularly getting more connected and moving faster toward the always-connected model . In thisIn WSNs, numerous types of research have involved routing strategy, corporeal design, procedure for sensing capability, power management, and security issues of SNs. The lifetime of SNs is the major problem for WSNs, since SNs have constrained energy resources. The routing algorithm has played a crucial role in the SN’s lifetime . RoutingThis article introduces a wavelet mutation with Aquila optimization-based routing (WMAO-EAR) protocol for wireless communication. The presented technique aims to accomplish energy-aware routing in WSNs. To do this, it initially derives the WMAO algorithm for the integration of wavelet mutation with the Aquila optimization (AO) algorithm. A fitness function (FF) is derived using distinct constraints, such as delay, energy, distance, and security. Once the mutation probability P has been set, every individual next to the exploitation and exploration process has the probability of mutation using the wavelet mutation process. For establishing the enhanced performance of the WMAO-EAR technique, a comprehensive simulation analysis is made.Jagadeesh and Muthulakshmi  examinedAl-Otaibi et al.  developeSrikanth et al.  establisIn this study, a new WMAO-EAR algorithm is proposed for effectual and energy-aware wireless communication. The presented WMAO-EAR technique aims to accomplish energy-aware routing in WSNs.Approaches utilized to carry out the presented routing protocol were the network model, energy model, and energy-dissipation model . In the α times more than that of the intermediate and normal node energy is β times more than normal node energy. Now, the α via WSNs involve link heterogeneity, energy heterogeneity, and computational heterogeneity. Energy heterogeneity is regarded as the most important to ensure optimal network performance. In the study, we discussed three levels of energy heterogeneity: advanced, normal, and intermediate nodes. Intermediate node initial energy is between advanced and normal node initial energy, given that m and b represent the percentage of advanced nodes and intermediate nodes correspondingly. Advanced node energy is d is formulated as follows.Reception and data transmission are two fundamental functions in WSNs. Usually, the data-communication method expends lots of energy when compared to data reception. Now, energy cost during In Equation (4), Furthermore, energy costs during d indicates a predetermined range Consider that the n sensor is positioned in the region of M size and the sensor is static. Every sensor is regarded as being aware of the identification and location of other SNs. Also, consider that the advanced node’s location is predetermined, while the normal and intermediate nodes are placed randomly. All the nodes transmit information to neighboring CH for data aggregation. The distance between the BS and nodes is Abualigah et al.  introducThe next strategy used to perform the Levy flight Levy, and the quality function rand rand shows a random number. Given that, An initial strategy is used to upgrade agents inside the exploitation stage based on Algorithm 1: Aquila optimizer (AO)while (The ending criteria are not satisfied) do                                                                                End if             End if                                                                                End if             End if         Else                                                                                end if             end if                                                                                end if             end if         End if    end forEnd whileAlgorithm 1 illustrates the fundamental steps of the AO.i-th individual location in t-th generation, and LB and UB are the lower and upper bounds of the present search space. Correspondingly, σ represents the wavelet mutating coefficient. Its formula is [φ signifies an arbitrary number within −2.5α and 2.5α. The a is the scaling parameter and its expression is [s indicates a given constant. The mutant individual In the WMAO algorithm, the mutation was considered a significant technique to support the technique’s jump from local optimum . In thisrmula is :(19)σ=1αssion is :(20)α=s⋅This procedure assures that an individual with superior fitness will be entering the next iteration, thereby enhancing the convergence speed and optimization capability of the algorithm.In the presented WMAO-EAR technique, an FF is derived using distinct constraints, such as delay, energy, distance, and security. The main concept of the proposed algorithm is to decrease the communication distance between the selected nodes and CH . It alsoEquation (25) illustrates the FF for distance, where Equation (28) illustrates the FF of energy. The value Equation (29) shows the FF of delay This section inspects the routing performance of the WMAO-EAR method on WSNs. The presented WMAO-EAR model is tested under two node counts (NC) of 100 and 300. A comparative NODN assessment of the WMAO-EAR model with other models is provided in Eventually, with 2500 rounds, the WMAO-EAR method attained a lower NODN of 2, whereas the CE-EC, SEED, NEH-CP, and HCEHUC approaches attained higher NODN of 18, 44, 97, and 158, respectively. These experimental values show the enhancements of the WMAO-EAR technique over other existing models.In this study, a new WMAO-EAR approach was devised for effective and energy-aware wireless communication. The presented WMAO-EAR technique aims to accomplish an energy-aware routing process in WSNs. To do this, the WMAO-EAR technique initially derives the WMAO algorithm for the integration of wavelet mutation with the AO algorithm. An FF is derived using distinct constraints, such as delay, energy, distance, and security. Once the mutation probability P is set, every individual next to the exploitation and exploration stage process has a probability of mutation using the wavelet mutation process. To demonstrate the enhanced performance of the WMAO-EAR technique, a comprehensive simulation analysis has been presented. The experimental results establish the superiority of the WMAO-EAR method over other recent approaches. In the future, lightweight cryptographic solutions can be applied to boost secure communication.


Here we demonstrate the piRNAi assay by quantifying acute and inherited silencing in the ribonucleotidyltransferase rde-3 (ne3370) mutant.In the absence ofrde-3,acute silencing was reduced but still detectable, whereas inherited silencing was abolished.We recently developed a piRNA-based silencing assay (piRNAi) to study small-RNA mediated epigenetic silencing: acute gene silencing is induced by synthetic piRNAs expressed from extra-chromosomal array and transgenerational inheritance can be quantified after array loss. The assay allows inheritance assays by injecting piRNAs directly into mutant animals and targeting endogenous genes ( C. elegans is a convenient model for studying small RNA-mediated inherited silencing due to the animal's short generation time (three days) and the ability to identify molecular pathways in genetic screens . Epigenetic silencing of an endogenous gene is often done by targeting a temperature-sensitive gain-of-function allele ofoma-1(zu405)with dsRNA, and silencing persists for up to three generations . For a visual read-out, single-copy transgenes with GFP expression in the germline  have been engineered to contain endogenous piRNA binding sites in the 3' UTR . For transgenes, piRNA-induced silencing persists longer and sometimes indefinitely. Genetic factors required for small RNA-mediated inherited silencing have primarily been identified by crossing silenced piRNAgfpsensor strains into mutant genetic backgrounds . However, introducing mutations by genetic crosses raises several concerns. First, there are several examples of mating causing changes in epigenetic inheritance. For example, the lack of transgene pairing during meiosis after a cross can lead to permanent transgene silencing via PRG-1-dependent mechanisms , and mating can induce multigenerational silencing inherited for over 300 generations . Moreover, Dodson and Kennedy (2019) characterized a transgenerational disconnect between the genotype and phenotype (sensitivity to exogenous RNAi) ofmeg-3/4mutants for more than seven generations after a genetic cross. Second, crosses frequently require molecular genotyping, which makes it cumbersome to perform many biological replicates. Third, there are some concerns about using transgenes as a proxy for endogenous gene silencing. For example, most piRNA sensor strains include synthetic piRNA binding sites in the 3' UTR , but endogenous genes are resistant to piRNA silencing when targeting their 3' UTRs . Moreover, transgene insertion site , non-coding DNA structures , coding sequence , and transgene structure  can influence epigenetic silencing. These observations suggest that transgenes may not fully recapitulate the balance between silencing foreign DNA and protecting endogenous gene expression . Finally, distinguishing between silencing initiation and maintenance phases is complicated using genetic crosses. Experiments require crossing mutant alleles to sensor strains, de-repress silencing, and outcrossing mutations to monitorde novoestablishment of silencing .et al., 2022; Gajicet al., 2022). Using piRNAi, we identified two endogenous targets,him-5 andhim-8, that inherit silencing for three and six generations, respectively .him-5 andhim-8 mutants are generally healthy but have a similar loss-of-function phenotype that is easy to score . We reasoned that piRNA-mediated silencing ofhim-5orhim-8might be useful as a tool to directly test the role of gene mutations in initiating and maintaining inherited silencing. Here, we show that piRNAi can be used to test acute and inherited silencing inrde-3, a gene also known asmut-2.We recently developed a method called piRNA interference (piRNAi) that can efficiently silence both transgenes and endogenous genes by expressing synthetic piRNAs from arrays generated by injection  and RNA interference (RNAi) .In vitro, RDE-3 has ribonucleotidyltransferase activity  and, in vivo, rde-3is required for the addition of non-templated poly (UG) tails to the 3’ end of mRNAs targeted by RNAi and repressed transposons . pUGylated mRNAs are templates for RNA-dependent RNA polymerases (RdRPs), resulting in small RNA amplification and inherited silencing . RDE-3 is required to maintain the silencing of piRNA transgene sensors . However, the role ofrde-3in initiating silencing is unclear; re-introducing RDE-3 led to rapid re-silencing of agfp::cdk-1transgene, but variable and incomplete re-silencing of agfp::csr-1transgene . Also,rde-3mutants are insensitive to the injection of dsRNA targetingunc-22but are sensitive to dsRNA expressed from transgenes . These conflicting results could be caused by differences between transgenes, the effects of mating, or the levels of the primary silencing dsRNA. We, therefore, decided to use piRNAi to test the role of RDE-3 in the initiation and maintenance of silencing of an endogenous gene. We targetedhim-5 with six synthetic guide piRNAs (sg-piRNAs)  in wild-type (N2) animals andrde-3(ne3370) mutants.rde-3 is a mutator strain and is relatively unhealthy, with a small brood size and infrequently produces males. To account for an elevated male frequency in the mutant population, we generated transgenicrde-3animals with non-targeting sg-piRNAs as a control. Targetinghim-5with piRNAi resulted in an increased frequency of males in N2 animals but a significantly lower male frequency inrde-3animals  . However, male frequency inrde-3animals was significantly increased compared to negative controls . We tested the role of RDE-3 in maintaining silencing by losing the piRNAi trigger (the piRNAi arrays with a Pmyo-2::mCherryfluorescent marker) and scoring male frequency in the following generations. In wild-type animals, the male frequency remains elevated for at least three generations after the primary piRNAs targetinghim-5are lost , consistent with prior observations . In contrast, we could not detect an inherited elevation of male frequency inrde-3mutants . The initial frequency of males was relatively low inrde-3animals, which limits our ability to make strong conclusions. However, our results support a model where primary piRNAs can post-transcriptionally silence a target transcript (him-5mRNA) at reduced efficiency, butrde-3is required for small RNA amplification and transcriptional silencing. These results support the observations by Chenet al. (2005) that persistently high somatic expression of dsRNA targetingunc-22from a plasmid causes a phenotype. In contrast, a single transient injection ofin vitrotranscribed dsRNA is inefficient. Presumably, RDE-3 amplifies the primary trigger by generatinghim-5pUG RNA templates for RdRP-mediated 22G amplification; these secondary RNAs are subsequently used to set up transcriptional silencing via repressive chromatin marks deposited by thehrde-1dependent nuclear RNAi pathway.him-5orhim-8) is easy to score in various genetic backgrounds and allows distinguishing between silencing initiation and maintenance of endogenous genes.More generally, we demonstrate that piRNAi can be used as a tool to directly test genetic factors required for acute and inherited silencing of endogenous genes. Elevated male frequency . The injection mix for all experiments consisted of ~15-20 ng/µl of synthetic dsDNA piRNA fragments (Twist Bioscience), 12.5 ng/µl of a plasmid encoding hygromycin resistance (pCFJ782), and 2 ng/µl of a fluorescent co-injection marker Pmyo-2::mCherry (pCFJ90). The total concentration of the injection mix was adjusted to 100 ng/µl with a 1kb DNA ladder . This mix was injected into young adult hermaphrodite animals and allowed to recover on standard NGM plates seed with OP50 bacteria. 36-48 hours post-injection, 500 µl of 4 mg/ml stock of Hygromycin solution  was topically added to the bacterial lawn of injection plates to select for transgenic (F1) progeny. A single healthy transgenic F2 adult was picked from each plate to generate a clonal strain, and pharyngeal mCherry fluorescence was visually confirmed.Quantification of male frequency.Quantification of male frequency was performed as previously reported by . Briefly, six virgin L4 hermaphrodites were picked to freshly seeded NGM plates with hygromycin selection to select for the piRNAi array. The frequency of males was determined using a dissection microscope and by visual inspection of 100 adult animals on plates incubated on ice for 30 minutes to immobilize animals.Inherited silencing assay.Six virgin L4 animals were picked to non-selective NGM plates to obtain a mixed progeny population with and without the piRNAi array. Males were not quantified in this mixed population; however, L4 animals carrying the sg-piRNAs were propagated in parallel, and their progeny were scored for males (G0). In the following generation, non-transgenic L4 animals were carefully picked from the mixed population based on the absence of pharyngeal mCherry expression (a marker for the piRNAi array). The progeny of these animals was quantified for male frequency (G1). Male frequency was quantified in all following generations by picking L4s until the male frequency was below 1%.Data quantification and statistics. Independently generated transgenic animals were treated as biological replicates. piRNA-mediated silencing is stochastic , and the data do not follow a normal distribution. We performed statistical tests using one-side parametric Mann-Whitney tests to account for this.Software. Statistical analysis was performed with GraphPad Prism (v 9.4.1), figures were generated with Adobe Illustrator (v 26.4.1), and the manuscript was written with Microsoft Word (v 16.63.1).List of strains, plasmids, and piRNAi fragments used in this study.StrainsN2 Standard wildtype strain (Brenner 1974)rde-3(ne3370) IWM286Plasmidsmyo-2::mCherry::unc-54 UTR pCFJ90 Prps-0::HygroRpCFJ782 PpiRNAi fragments him-5(six targeting piRNAs in upper-case):T288cgcgcttgacgcgctagtcaactaacataaaaaaggtgaaacattgcgaggatacatagaaaaaacaatacttcgaattcatttttcaattacaaatcctgaaatgtttcactgtgttcctataagaaaacattgaaacaaaatattaagTGAGTTAGCTTTCCGGAGCTTctaattttgattttgattttgaaatcgaatttgcaaatccaattaaaaatcattttctgataattagacagttccttatcgttaattttattatatctatcgagttagaaattgcaacgaagataatgtcttccaaatactgaaaatttgaaaatatgttTCCTCACGAAAAACCTGCCTAttGccagaactcaaaatatgaaatttttatagttttgttgaaacagtaagaaaatcttgtaattactgtaaactgtttgctttttttaaagtcaacctacttcaaatctacttcaaaaattataatgtttcaaattacataactgtgtATGCAGAGAGATCAGTAGGTActgtagagcttcaatgttgataagatttattaacacagtgaaacaggtaatagttgtttgttgcaaaatcggaaatctctacatttcatatggtttttaattacaggtttgttttataaaataattgtgtgatggatattattttcagacctcatactaatctgcaaaccttcaaacaatatgtgaagtctactctgtttcactcaaccattcatttcaatttggaaaaaaatcaaagaaatgttgaaaaattttcctgtttcaacattatgacaaaaatgttatgattttaataaaaacaaTCGATCACTGTTGACAATCACttctgtttttcttagaagtgttttccggaaacgcgtaattggttttatcacaaatcgaaaacaaacaaaaatttttttaattatttctttgctagttttgtagttgaaaattcactataatcatgaataagtgagctgcccaagtaaacaaagaaaatttggcagcggccgacaactaccgggttgcccgatttatcagtggaggaTAATCCGGCACGTAGAATGTAtctaatgtgatgtacacggttttcatttaaaaacaaattgaaacagaaatgactacattttcaaattgtctatttttgctgtgtttattttgccaccaacaaTCGATGCGACCAACTGTTTTTtcaatctagtaaactcacttaatgcaattcctccagccacatatgtaaacgttgtatacatgcagaaaacggttttttggttttaatgggaacttttgacaaattgttcgaaaatcttaagctgtcccatttcagttgggtgatcgatttT119 Control (non-targeting piRNAs):ctcggtcaattaaagaaagaCATTTTTCATCGGATTTGCTActaaaaaataattttaaAAACGATCATATGCAAATCCAgtgaaactttattcaaaccaaaacgtttaatcagctaattgaaacattaaaaattttatgattttgttagtttttctagcaatgtcaatgcaatcaaataattttcaagtaagatgtttaatgagttatagacttttttattaaatttttgaaaaaaaaaccgatttcagatttaagtaaaattatctctgcttctgctgcattgctgcgaaacaaaaattcctttctgtgcaaagtatagtATAAACGAGGAGCACAAATGAgtgacaattagaaatctcaccgggttttctagatcatctgaaacatataattttaaaaaattgacaccttgttcaacTGTTGCACATATCACTTTTGAtcgaaacattaaatgtctcatgatttttaaagctcttttagaacagtcgCCAATCCCCTTATCCAATTTAttgaaaacaattttctagcgagatgttaaatgagtttgttgaaacagtagattttcgtgtaaacttttgaaaacaaaaattacgttttaaataaaattatatccacttcagcagtgtgcccttgaaacaaaaaagctcgatcaaaaaatttattttttgtgaatggccaccaacttttcaggcaaaattacaaaaaaacataaaatttactgtttcaaaaagttaatataattttggcagcgcatatacctacacTGAATTTTGGCAGAGGCAATTacctctttttgaaaataaag


In the originally published version of this manuscript, there were errors within Table 4. In the line for variable “Alfieri stitch” the value should read: “3 (2.9)” instead of “3 (13.3)”. For variable “Posterior semi-annulus suture plasty” should read: “8 (7.8)” instead of “8 (6.7)”. These errors are now corrected in online.


Nature Neurology 10.1038/s41593-022-01107-4, published online 11 July 2022.y axis of Fig. 5d should have been “−20” and “−40”. Further, the scale values near the spike waveforms in Fig. 2a, now reading “100 μs” appeared originally as “10 μs.” The errors have been corrected in the HTML and PDF versions of the article.In the version of this article initially published, “−2” and “−4” on the


The authors wish to amend the sections of the manuscript detailed below. These corrections are due to inaccuracies in the adsorption activity, as well as in the surface area of the adsorbent.The corrections are as follows:2 g−1” should instead read as “4.55 nm and the surface area is 120.7 m2 g−1”.(1) On page 1, the sentence including “4.13 nm and the surface area is 112.9 m−1” should instead read as “483 to 2190 mg g−1”.(2) On page 1, in the sentence including “490 to 2210 mg g2 g−1” should instead read as “120.7 m2 g−1”, and “4.13 nm” should instead read as “4.55 nm”.(3) On page 3 (left column), the sentence including “16.00 m−1” should instead read as “790 mg g−1”.(4) On page 4 (left column), the sentence including “808 mg g−1” should instead read as “849 to 1000 mg g−1”.(5) Page 5 (right column), the sentence including “868 to 1000 mg g−1” should instead read as “417, 425, 476 and 483 mg g−1”.(6) Page 5 (right column), the sentence including “422, 431, 481 and 490 mg g−1” should instead read as “1800, 2043, 2110 and 2190 mg g−1”.(7) Page 5 (right column), the sentence including “1812, 2061, 2164 and 2210 mg g−1” should instead read as “849 to 1000 mg g−1”.(8) Page 5 (left column), the sentence including “868 to 1000 mg gThe Royal Society of Chemistry apologises for these errors and any consequent inconvenience to authors and readers.


In the aggregation of uncertain information, it is very important to consider the interrelationship of the input information. Hamy mean (HM) is one of the fine tools to deal with such scenarios. This paper aims to extend the idea of the HM operator and dual HM (DHM) operator in the framework of complex intuitionistic fuzzy sets (CIFSs). The main benefit of using the frame of complex intuitionistic fuzzy CIF information is that it handles two possibilities of the truth degree (TD) and falsity degree (FD) of the uncertain information. We proposed four types of HM operators: CIF Hamy mean (CIFHM), CIF weighted Hamy mean (CIFWHM), CIF dual Hamy mean (CIFDHM), and CIF weighted dual Hamy mean (CIFWDHM) operators. The validity of the proposed HM operators is numerically established. The proposed HM operators are utilized to assess a multiattribute decision-making (MADM) problem where the case study of tourism destination places is discussed. For this purpose, a MADM algorithm involving the proposed HM operators is proposed and applied to the numerical example. The effectiveness and flexibility of the proposed method are also discussed, and the sensitivity of the involved parameters is studied. The conclusive remarks, after a comparative study, show that the results obtained in the frame of CIFSs improve the accuracy of the results by using the proposed HM operators. MADM is an essential process of decision-making (DM) science whose objective is to see the best options from the arrangement of likely ones. In DM, a singular requirement is to evaluate the given choices by different classes, such as single, range, and so for appraisal purposes. Nevertheless, in various fanciful conditions, it is, for the most part, pursuing for the person to convey their choices as a new number. For this, the frame of the fuzzy set (FS)  was deverμeπiθμ2 where rμ, θμ ∈  and such kind of framework describes two different aspects of an uncertain phenomena. Moreover, the idea of CFS recently got a large number of attractions, and some useful work may be found in  and νÇ(x) : Y⟶ provided that 0 ≤ μÇ(x)+νÇ(x) ≤ 1 and hesitancy degree represent vc(x)=1 − (μÇ(x)+νÇ(x)), (x) ∈ . Furthermore, Ç=) denotes an intuitionistic fuzzy number CIFN.An IFS is of the form of Ç = {, νÇ(x))|x ∈ y} such that μÇ(x) = rμÇ(x)eπiθμÇ(x)2 and νÇ(x) = sνÇ(x)eπiΦνÇ(x)2 where rμÇ(x), ΦsνÇ(x) denote the amplitude terms and θμÇ(x)Φ,νÇ(x) denote the phase terms of μÇ(x) and νÇ(x), respectively, from  provided that 0 ≤ rμÇ(x) + sνÇ(x) ≤ 1 and 0 ≤ θμÇ(x) + ΦsνÇ(x) ≤ 1 and hesitancy degree represent ℏ(x) = 1 − (rμÇ(x)eπiθμÇ(x)2 + sνÇ(x)eπiνÇ(x)2), ℏ(x) ∈ . Furthermore, Ç = (rμÇ(x)eπiθμÇ(x)2, sνÇ(x)eπiΦνÇ(x)2) represents a complex intuitionistic fuzzy number CIFN.A CIFS is of the form of Ç = (rμÇ(x)eπiθμÇ(x)2, sνÇ(x)eπiνÇ(x)2) be a CIFN. The score function S is as follows:LetÇ = (rμÇ(x)eπiθμÇ(x)2, rνÇ(x)eπiΦνÇ(x)2) be a CIFN. The accuracy function ′H is as follows:LetWe gave Ç1=(0.3ei2π(0.45), 0.5ei2Π(0.25)) and Ç2=(0.6ei2Π(0.70), 0.2ei2Π(.25)) be two CIFNs. By using Definitions Ç1)=1/2((0.3 − 0.5)+(0.45 − 0.25))=1/2(−0.20+0.20)=0 ∈  š(Ç2)=1/2((0.6 − 0.2)+(0.70 − 0.25))=1/2(0.40+0.45)=0.48 ∈  š(H(Ç1)=1/2((0.3+0.5)+(0.45+0.25))=1/2(0.80+0.70)=0.75 ∈  H(Ç2)=1/2((0.6+0.2)+(0.70+0.25))=1/2(0.8+0.95)=0.88 ∈  Let 0.3ei2π0.5, 0.5ei2Ç1 = (rμ1(x)eπiθμ1(x)2, rν1(x)eπiΦν1(x)2) and Ç2 = (rμ2(x)eπiθμ2(x)2, rν2(x)eπiΦν2(x)2) be two CIFNs. ThenS(Ç1) < S(Ç2), then Ç1 < Ç2If S(Ç1) > S(Ç2), then Ç1 > Ç2If S(Ç1) = S(Ç2), then:If H(Ç1) > H(Ç2), thenÇ1 > Ç2H(Ç1) < H(Ç2), thenÇ1 < Ç2H(Ç1) = H(Ç2), thenÇ1 ≈ Ç2.Let Ç1 = (rμ1(x)eπiθμ1(x)2, rν1(x)eπiΦν1(x)2) and Ç2 = (rμ2(x)eπiθμ2(x)2, rν2(x)eπiΦν2(x)2) be two CIFNs. Then some fundamental operations are defined as follows:Ç1⊆Ç2⟺rμ1(x) ≤ rμ2(x),  θμ1(x) ≤ θμ2(x) and rν1(x) ≥ rν2(x),  Φν1(x) ≥ Φν2(x)Ç1 = Ç2⟺rμ1(x) = rμ2(x),  θμ1(x) = θμ2(x)rÇ1(x) = rÇ2(x), θÇ1(x) = ΔÇ2(x) and rν1(x) = rν2(x),  Φν1(x) = Φν2(x)Ç1′ = {(rν1(x)eiΦν1(x), rμ1(x)eiθμ1(x))}LetÇ1 = (rμ1(x)eπiθμ1(x)2, rν1(x)eπiΦν1(x)2) and Ç2 = (rμ2(x)eπiθμ2(x)2, rν2(x)eπiΦν2(x)2) be two CIFNs and λ > 0 be a real number. ThenλÇ1 = (1 − (1 − rμ1(x))λeπi(1 − (1 − θμ1(x))λ)2, (rν1λ(x) · eπi(Φν1(x))λ2))Ç1λ = ((rμ1λ(x)eπi(θμ1(x))λ2), 1 − (1 − rν1(x))λeπi(1 − (1 − Φν1(x))λ)2)LetIn this section, we recall the basic definition of HM operator. We also discuss the HM operators that are previously defined. Furthermore, we point out towards the limitations of such existing HM operators that lead us to propose some new HM operators.Consider the HM operators defined for real numbers.x is such that 1 ≤ x ≤ n and Ckx represent the binomial coefficient, that is, Cnx = x!/n!(x − n)!The HM operator is as follows:x)=Ç if Çi=Ç, HMx) ≤ HMx) if Çi ≤ ωi, HMÇi) ≤ HMx) ≤ maxÇimin(x)=(1/k)∑i=1kÇiFor arithmetic mean operator HMx)=(∏i=1kÇi)x1/For geometric mean operator HMThe HM operator is likely to satisfy the following properties:We recall the definition of the DHM operator for real numbers.The DHM operator is defined as follows:Ij = (rμj(x), sj(x)), j = 1,2,…, k be the collection of IFNs. Then intuitionistic fuzzy Hamy mean (IFHM) operator is defined as follows:Let Ij = (, ), j = 1,2,…, k be the collection of interval-valued IFNs (IVIFNs). Then interval-valued intuitionistic fuzzy Hamy mean operator is defined as follows:LetAll the above-discussed HM operators deal with two real values TD and FD. Consider a scenario with TD and FD having further two aspects, then the operators discussed above become unable to deal with such information. Therefore, we aim to propose the concept of HM operator in the framework of CIFS because such an operator can deal with two aspects of TD and FD at a time.In this section, we introduced the idea of HM operators in the framework of CIFSs. We also proved that the CIFHM operator satisfied the basic properties of AO. We give example to support the proposed operator. First, consider the HM operators based on CIFNs as follows.Çj=(rμj(x)eiθμj(x), svj(x)eiΦνj(x)), j=1,2,…, k, be the collection of CIFNs. Then, the CIFHM operator is defined as follows:Let Çj=(rμj(x)eπiθμj(x)2, sνjeπiΦνj(x)2), j=1,2,…, k be the collection of CIFNs. Then the aggregated value of the CIFHM operator is also a CIFN such thatLet This theorem has two parts: first, we derive the formula given in equation  as follorμ (x)+sν(x) ≤ 1 and 0 ≤ θμ(x)+Φν(x) ≤ 10 ≤ Now, we prove that equation  represenrμ(x) ≤ 1 and 0 ≤ θμ(x) ≤ 1 We haveWe know that 0 ≤ Similarly,rμ(x)eπiθμ(x)2 ≤ 1 and 0 ≤ sν(x)eπiΦν(x)2 ≤ 1 so we haveSince 0 ≤ Now, we prove that the CIFHM operator satisfies the properties of the aggregation function in Theorems Çj=(rμj(x)eiθ(μ/j)(x), sνj(x)eiΔνj(x)), j=1,2,…, k be the collection of all identical values of CIFNs. ThenLetÇj=(rμj(x)eiθ(μ/j)(x), sνj(x)eiΦνj(x))=(r(x)eiθ(x), s(x)eiΦ(x))=Ç, j=1,2,…, k. ThenWe know that Çj=(rμj(x)eiθμ/j(x), sνj(x)eiΦνj(x)), j=1,2,…, k and Dj(x)=(gμj(x)eiαμ/j(x), hν j(x)eiβν/j(x)), j=1,2,…, k be two sets of CIFNs. If Çj(x) ≤ Dj(x), that is, rμj(x) ≤ gμj(x), θμj(x) ≤ αμj(x) and sνj(x) ≤ hνj(x), Φνj(x) ≤ βνj(x) thenLet Çj ≤ Dj, that is, rμj(x) ≤ gμj(x), θμj(x) ≤ αj(x) and sj(x) ≤ hj(x), Φj(x) ≤ βj(x). ThenWe know that rμj(x)eπiθμ(x)2 ≤ geπiαμ(x)2. In a similar way, we can investigate the value of sνj(x)eπiΦν(x)2 ≥ hν(x)eπiβν(x)2.(1)rμj(x)eπiθμ(x)2 < gμj(x)eπiαμ(x)2 and sν(x)eπiΦν(x)2 > hν(x)eπiβν(x)2, thenIf (2)rμj(x)eπiθμ(x)2=gμj(x)eπiαμ(x)2 and sν(x)eπiΦν(x)2=hν(x)eπiβν(x)2, thenIf According to equation (19), Çj=(rμj(x)eiθμ j(x), sν j(x)eiΦν j(x)), j=1,2,…, k be the collection of CIFNs. If Çj−=min and Çj+=max, thenLet From boundedness property:CIFHMxHMx) = Ç+.From monotonicity property:In a decision-making problem, the weights of all attributes and the experts sometimes matter. So we discuss the influence of weights on the HM operator in this section and develop the weighted HM operator as follows.Çj=(rμj(x)eiθμj(x), sνj(x)eiΦνj(x)), j=1,2,…, k be the collection of CIFNs with weight vector wi=T,  wi ∈  and ∑i=1nwi=1. Then the CIFWHM operator is defined as follows:Let Çj=(rμj(x)(x)eπiθμj(x)2, sνj(x)eπiΦνj(x)2), j=1,2,…, k be the collection of CIFNs. Then the aggregated value of the CIFWHM operator is also a CIFN such thatLet rμ(x)+sν(x) ≤ 1 and 0 ≤ θμ(x)+Δν(x) ≤ 10 ≤ Now, we have to show that is a CIFN.rμ(x) ≤ 1 and 0 ≤ θμ(x) ≤ 1 We haveWe know that 0 ≤ Similarly,rμ(x)eπiθμ(x)2 ≤ 1 and 0 ≤ sν(x)eπiΦν(x)2 ≤ 1 so we haveSince 0 ≤ We gave Ç1=0.45eπi(0.3)2, 0.62eπi(0.41)2, Ç2=0.2eπi(0.7)2, 0.52eπi(0.6)2, Ç3=0.5eπi(0.6)2, 0.2eπi(0.91)2, Ç4=0.7eπi(0.8)2, 0.42eπi(0.15)2 be the CIFNs with weight vector w=. Then we use the proposed CIFWHM operator to aggregate the given CIFNs. Suppose that x=2.Let Çj=(rμj(x)eiθμj(x), sνj(x)eiΦνjj(x)),  j=1,2,…, k be the collection of all identical values of CIFNs. ThenLet Similar to Çj=(rμj(x)eiθμj(x), sνj(x)eiΦνj(x)),  j=1,2,…, k and Dj=(gμj(x)eiαμj(x), hνj(x)eiβνj(x)), j=1,2,…, k be two sets of CIFNs. Then rμj(x) < gμj(x), θμ j(x) < αμ j(x) and sνj(x) > hνj(x),  Φνj(x) > βνj(x). ThenLet Similar to Çj=(rμj(x)eiθμj(x), sνj(x)eiΦνj(x)),  j=1,2,…, k be the collection of CIFNs. IfLet thenFrom boundedness property:x = Ç+.CIFWHMFrom monotonicity property:From From property 4, we haveIn this section, we use the idea of the DHM operator in the framework of CIFSs. We prove the validity of the proposed AO. We also give a numerical example to support the proposed CIFDHM operator.Çj=(rμj(x)eiθμj, sνj(x)eiΦνj),  j=1,2,…, k be the collection of CIFNs. Then CIFDHM operator is defined as follows:Let Çj=(rμj(x)eiθμj, sνj(x)eiΦνj),  j=1,2,…, k be the collection of CIFNs. Then CIFDHM operator is defined as follows:Let The proof is analogous to the proof of The CIFDHM operator also satisfies the basic properties of aggregation as discussed in Theorems Now we will elaborate on the concepts of DHM operator in the framework of CIFSs by keeping the weight under observation.Çj=(rμj(x)eiθμj(x), sνj(x)eiΦνj(x)), j=1,2,…, k be the collection of CIFNs with weight vector wi=T,  wi ∈ , and ∑i=1nwi=1. Then the CIFWDHM operator is defined as follows:Let Çj=(rμj(x)eiθμj(x), sνj(x)eiΦνj(x)), j=1,2,…, k be the collection of CIFNs. Then the aggregated value of the CIFWDHM operator is also a CIFN and is given byLet The proof is similar to the proof of We gave Ç1=0.55eπi(0.45)2, 0.70eπi(0.85)2, Ç2=0.40eπi(0.70)2, 0.82eπi(0.77)2, Ç3=0.60eπi(0.40)2,  0.72eπi(0.60)2, Ç4=0.50eπi(0.75)2, 0.52eπi(0.35)2 be the CIFNs with the weight vector of the attributes be w=. Then we use the proposed CIFWDHM operator to investigate the CIFN. Suppose that x=2.Let The CIFWDHM operator also satisfies the basic properties of aggregation as discussed in Theorems A CIFS is an extension of an IFSs in which the sum of TD and FD lies on interval ; each TD and FD has two aspects: amplitude terms and phase terms. In this section, we developed a MADM method based on CIFWHM and CIFWDHM operators to solve a problem involving the development of the tourism industry. First, we proposed the MADM algorithm, and then we present a comprehensive example to utilize the proposed algorithm.  Step 1: Information are collected from the decision-maker about the tourism industry . All information are in the form of CIFNSs.  Step 2: This step involves the utilization of the proposed CIFWHM and CIFWDHM operator to aggregate the CIF information depicted in the decision matrix given in step 1.  Step 3: This step is about the analysis of the aggregated information based on score values of CIFNs by using   Step 4: In this step, we investigate the score values of all aggregated information for ordering and ranking the attributes.In this subsection, we describe the steps of the algorithm as follows:The average increase in the economy induces ever-greater competitive surroundings for tourism enterprises. The essential competition of the tourism industry depends on the tour destination. In the tourism environment, to assure the regular development of a visitor vacation spot, it is essential to take measures to increase tourism vacation spots to enhance competitiveness. The vital role of the tourism industry is to analyze and evaluate the vacation spot competitiveness, which can display the attraction strength of a vacation spot and imply the direction for the efficient allocation of assets. Therefore, it is a crucial manner for determining the vacation spot development mode and route to analyze the important elements of tourism destination and extract qualitative and quantitative assessment index, which can uphold long-term competitive benefit and the nonstop improvement of vacation spot. This is a problem of interest, and we try to investigate this problem using our proposed AOs and adopt the tourism destination problem from .Ai that have to be evaluated. These five destinations are to be examined based on four attributes where G1: is the attractiveness of tourism sources, G2 is the infrastructure and development of the tourism industry, G3 is supporting force of the tourism environment, and G4 is the tourist demand. The five possible vacationer locations are to be assessed with IVIFNs that are weighting vectors w= as shown in   Step 1: The decision matrix containing the information about the five alternatives by anonymous decision-makers is provided in x=3. All aggregated results are shown in   Step 2: Aggregate CIFNs are shown in the decision matrix in   Step 3: By utilizing Consider five possible tourist destinations A3 by using the CIFWHM operator and A1 by using the CIFWDHM operator at the parameter of x=3. Consider In Another view of the results of Tables x on the results. We see the effect on ordering and ranking of the results by the variation of the parameter x=1,2,3,4 obtained by using CIFWHM and CIFWDHM operators. We listed the analysis of the results after varying the parameter x in Tables In this subsection, we see the impact of parameter A3 as the best tourist destination when we take x=1,2,3,4 in the case of the CIFWDHM operator also. Variation of x is studied in both cases, and we reached the following results:x does not create any impact on ranking results.Variation of x may have an impact in some other cases depending on the data we dealt with must be analyzed each time in each case.Variation of We analyzed In this section, we aim to analyze the aggregated results obtained using the CIFHM operators with the aggregated results obtained using the AOs of Akram et al. , Wang etA3 is the suitable destination. Moreover, CIFHWG operator by Akram et al. [A5 and A1 as suitable destinations, respectively. The only reason for having different results is that the operators discussed in [From m et al.  and CIFMm et al.  gave A5 ussed in , 39, 43 CIFHM and CIFDHM operators are the generalizations of the results discussed in , 40 and CIFWHM and CIFWDHM operators deal with such information that has two aspects of the TD and FD denoted by amplitude term and phase termThe advantages of the proposed work are as follows:We proposed CIFHM and CIFDHM operators based on CIFSsWe also investigated the basic properties of the proposed work in the form of idempotency, monotonicity, and boundednessWe elaborated the concepts of the DHM operator in the framework of CIFSs by keeping the weight under observationWe analyzed the consequences of the proposed work through application and numerical examples to illustrate the CIFSs. In this paper, we utilized the idea of HM operators to elaborate the CIFSs in the framework of CIFHM operators to find the reliability of CIFSs. A CIFS has two aspects: TD and FD; TD has also two aspects amplitude terms and phase terms; and similarly, FD has two parts: amplitude terms and phase terms of FD that carried the more flexible information. The main contribution and key factors in this paper are as follows:


The concept of resolving set and metric basis has been very successful because of multi-purpose applications both in computer and mathematical sciences. A system in which failure of any single unit, another chain of units not containing the faulty unit can replace the originally used chain is called a fault-tolerant self-stable system. Recent research studies reveal that the problem of finding metric dimension is NP-hard for general graphs and the problem of computing the exact values of fault-tolerant metric dimension seems to be even harder although some bounds can be computed rather easily. In this article, we compute closed formulas for the fault-tolerant metric dimension of lattices of two types of boron nanotubes, namely triangular and alpha boron. These lattices are formed by cutting the tubes vertically. We conclude that both tubes have constant fault tolerance metric dimension 4. Computer networks are graphs with vertices representing hosts, servers, or hubs, and edges as connecting mediums between them. Vertex is actually a possible location to find faults or some damaged devices in a computer network. This idea somehow created an urge in Slater and independently in Harary and Melter  to uniquA moving point in a graph may be located by finding the distance from the point to the collection of sonar stations that have been properly positioned in the graph. Thus finding a minimal but sufficiently large set of labeled vertices such that a robot can find its position, is a well-established problem known as robot navigation. This sufficiently large set of labeled vertices is a resolving set of the graph space and the corresponding cardinality is the metric dimension. Similarly, on another node, a real-world problem is the study of networks whose structure has not been imposed by a central authority but has arisen from local and distributed processes. It is very difficult and expensive to obtain a map of all nodes and the links between them. A commonly used technique is to obtain a local view of the network from various locations and combine them to obtain a good approximation for the real network. Metric dimension also has some applications in this aspect as well.G, and metric dG : V(G) × V(G) → ℕ ∪ {0}, where ℕ is the set of positive integers and dG is the minimum number of edges in any path between x and y. Let W = {w1, w2, ..., wk} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v|W) of v with respect to W is the k−tuple , dG, ..., dG). If distinct vertices of G have a distinct representation with respect to W, then W is called a resolving set of G  is its cardinality.Consider a simple, connected graph Wn to be n ≥ 7. Caceres et al. (n ≥ 7. Tomescu and Javaid (Jn2 to be n ≥ 4. A particular metric-feature of the family of graphs is independent of metric dimension on the particular element of the family. A connected graph has a constant metric dimension if β(G) = k, where k ∈ ℕ is fixed. This feature has been presented in Imran et al. , is the minimum cardinality of such Ś. A family Recent development in this context has paved way for a new related concept known as fault-tolerance in the metric dimension. Suppose that, in a network, o Slater , 2002 ano Slater . A resols Hayes, . Slater s Hayes,  introducm×n and triangular boron nanotubes Tm×n. These lattices are formed by cutting both tubes vertically. Kwun et al.  = , if 1 ≤ q ≤ lFor k is even, a is odd, and If a is odd, k is even, and If k is even, a is even, and If k is even, a is even, and If a = 2l, r = , if 1 ≤ b ≤ lFor a = 2l − 1, r = , if 1 ≤ q ≤ lFor k is even, a is odd, and If k is even, a is odd, and If k is even, a is even, and If k is even, a is even, and If a = 2l, r = , if 1 ≤ b ≤ lFor Case II: Let k > 2l.y = rl + p − 1. If r ≥ 2, rl < k ≤ (r + 1)l, and k is odd.Let a = rl + p, where 1 ≤ p ≤ l and a is oddFor a is evenIf k is even, r ≥ 2, and rl < k ≤ (r + 1)l thenIf y = rl + p − 1. For a = rl + p where 1 ≤ p ≤ l and a is oddLet a is evenIf F is an FTRS for Tk,l. Therefore, These vectors are distinct in at least two coordinates. So Theorem 2.2. Let Tk,l denote the graph of k × l, 2D-lattice of triangular boron nano tubes. Then Proof. For the lower bound on Case I:k ≤ l.Tk,l. Let F = {vi,j, va,b, vr,s} where i < a < r and j < b < s be an FTRS for Tk,l. Then F1 = F\{v} is a R.S. for each v ∈ F. We discuss the following possibilities:We claim that any set of cardinality 3 is not an FTRS for Possibility 1: When all vertices in F lie on the same row.F lie in the first row, then i = a = r = 1. Let F1 = {vj1,, vb1,} then r = r, a contradiction.(i) If all vertices in F lie in p − th row and 1 < p. Let F1 = {vp,j, vp,b} then either r = r or r = r, a contradiction.(ii) If all vertices in Possibility 2: When two vertices in F lie on the same row.i = a = l and r ≠ l. Let F1 = {vl,j, vl,q} then either r = r or r = r, a contradiction.WLOG we may suppose that Possibility 3: When the vertices in F lie on three different rows.F lie on same column, let j = b. If j = b = 1 and F1 = {v1,1, va,1} then r = r, a contradiction.(i) Two vertices in F1 = {vi,1, vj,1}, 1 ≤ i < j < k then either r = r or r = r, a contradiction.(ii) If F1 = {vi,b, vj,b}, 1 ≤ i < j < k, and 1 < q < l then either r = r or r = r, a contradiction.(iii) If S lie on different columns. Let S1 = {Vi,j, Va,b}, i < a then(iv) If all vertices in j < b then either r = r or r = r, a contradiction.(a) If j > b then either r = r or r = r, a contradiction.(b) If F is not an FTRS for Tm,n. So From the above discussion, we conclude that Case II:k > lTk,l having cardinality 3, it is enough to prove that any set of vertices having two elements is not an R.S. for Tk,l. Let F1 = {vi,j, va,b} be an R.S. for Tk,l. There are three possibilitiesIn order to prove that there is no FTRS for Possibility 1: If vi,j, va,b are taken from same row, i.e., i = a thenF1 = {v1,1, vl1,} then l is even and l is odd, a contradiction.(i) If F1 = {vi,j, vi,b}, 1 ≤ i < k, and 1 ≤ j < b < l then r = r, a contradiction.(ii) If F1 = {vi,j, vi,b}, 1 ≤ i < k, and 1 < j < b = l then r = r, a contradiction.(iii) If F1 = {vk,j, vk,b} and 1 ≤ j < b < l then r = r or r = r, a contradiction.(iv) If F1 = {vk,j, vk,b} and 1 < j < b = l then r = r, a contradiction.(v) If F1 = {vi,1, vi,l} and 1 < i < k then r = r, a contradiction.(vi) If F1 = {vk,1, vk,l} then r = r, a contradiction.(vii) If Possibility 2: If vi,j, va,b are taken from same column, i.e., j = b thenF1 = {v1,1, vk,1} then r = r, a contradiction.(i) If F1 = {vi,1, vj,1} where 1 ≤ i < j < m then r = r or r = r, a contradiction.(ii) If F1 = {vi,1, vj,1} and 1 < i < j = k then r = r or r = r, a contradiction.(iii) If F1 = {vi,b, vj,b}, 1 ≤ i < j < k, and 1 < b < l then r = r or r = r, a contradiction.(iv) If F1 = {vi,b, vj,b}, 1 < i < j ≤ k, and 1 < b < l then r = r or r = r, a contradiction.(v) If F1 = {vi,b, vj,b}, i = 1, j = k, and 1 < q < l then r = r, a contradiction.(vi) If F1 = {vi,l−1, vj,l−1} and i < j < k then r = r or r = r, a contradiction.(vii) If F1 = {vl1,, vk,l} then r = r or r = r, a contradiction.(viii) If Possibility 3: If vi,j, va,b are taken from different rows and different columns, i.e., i ≠ p,j ≠ q. Let F1 = {vi,j, va,b} and i < aj < b then r = r or r = r, a contradiction.(i) If j > b then r = r or r = r, a contradiction.(ii) If a = k and i = 1 then r = r or r = r or r = r, a contradiction.(iii) If Tk,l. So Thus no set having two vertices is an R.S. for The next theorem gives Theorem 2.3. Let αk,l denote the graph of k × l 2D-lattice of α boron nano tubes. Then Proof. Consider k × l 2D-lattice of α-boron Nanotubes. We denote this graph by αk,l. In general, the tube is “cut” vertically and the upper left triangle is pointing up as shown in k,l is partitioned as {v1,1, v1,2, v1,3, ...., vl1,, v2,1, v2,2, ...., vl2,, v4,1, v4,2, ......, vl4,, v5,1, v5,2, ......, vl5,, v7,1, v7,2, ......, vl7,, ......, vk,1, vk,2, vk,3, ...., vk,l} ∪ {v3,1, v3,2, v3,4, v3,5, v3,7, v3,8, ...., vl3,, v6,1, v6,3, v6,4, v6,6, v6,7, v6,9, ...., vl6,, v9,1, v9,2, v94,, v9,5, v9,7, v9,8, ....,l9,, ......}.Case I: Whenk < l.F = {v1,1, vl1,, vk,1, vk,l}. We show that F is an FTRS for αk,l. We give the distance vectors of all vertices va,b of αk,l relative to F.Let k be odd and k ≠ 6q + 1, q ∈ Z+.Let a = 1For a = 2For a is odd, a ≠ 3p, p ∈ Z+, and In general, if a is odd, a ≠ 3p, p ∈ Z+, and If a is even, a ≠ 3p, p ∈ Z+, and If a is even, a ≠ 3p, p ∈ Z+, and If k = 6p + 1.Let a is odd, a ≠ 3p, k ∈ Z+, and If a is odd, a ≠ 3p, p ∈ Z+, and If a is even, a ≠ 3p, p ∈ Z+, and If a is even, a ≠ 3p, p ∈ Z+, and If a is odd, a = 3p, p ∈ Z+, b ≠ 3q thenIf a is odd, a = 3p, p ∈ Z+, b ≠ 3q thenIf a is even, a = 3p, p ∈ Z+, b ≠ 3q − 1 thenIf a is even, a = 3p, p ∈ Z+, b ≠ 3q − 1 thenIf k be even and k ≠ 6p+2, p ∈ Z+.Let a is odd, a ≠ 3p, p ∈ Z+, and If a is odd, a ≠ 3p, p ∈ Z+, and If a is even, a ≠ 3p, p ∈ Z+, and If a is even, a ≠ 3p, p ∈ Z+, and If m = 6k+2.Let a is odd, a ≠ 3p, p ∈ Z+, and If a is odd, a ≠ 3p, p ∈ Z+, and If a is even, a ≠ 3p, p ∈ Z+, and If a is even, a ≠ 3p, p ∈ Z+, and If a is odd, a = 3p, p ∈ Z+, b ≠ 3q, and q ∈ Z+ thenIf a is odd, a = 3p, p ∈ Z+, b ≠ 3q, and q ∈ Z+ thenIf a is even, a = 3p, p ∈ Z+, b ≠ 3q − 1, and q ∈ Z+ thenIf a is even, a = 3p, k ∈ Z+, b ≠ 3q − 1, and q ∈ Z+ thenIf Case II:k ≥ 2l.r ≥ 2, rl ≤ k ≤ (r + 1)l, and k is odd thenIf a = rn + p where 0 ≤ p ≤ l − 1, a ≠ 3p, and a is oddFor a is evenIf a = 3p, a is odd, b ≠ 3q, and q ∈ Z+If a = 3k, a is even, b ≠ 3q − 1, and q ∈ Z+If k is even, r ≥ 2, and rl ≤ k ≤ (r + 1)l thenIf a = rl + p where 0 ≤ p ≤ l − 1For a is odd, a ≠ 3pIf a is even, a ≠ 3pIf a = 3p, a is odd, b ≠ 3q, and q ∈ Z+If a = 3p, a is even, b ≠ 3q − 1, and q ∈ Z+If F is an FTRS for αk,l. Therefore These representations are distinct in at least two coordinates. So Case I: k ≥ l.k,l) = 3 [19], so Since β = r, a contradiction.(i) If all vertices in F lie in p − th row and 1 < p ≠ 3l, l ∈ Z+. Let F1 = {vp,j, vp,b} then either r = r or r = r, a contradiction.(ii) If all vertices in i = a = r = 3q, l ∈ Z+. Let F1 = {vq, j3, vq, b3} then either r = r or r = r, a contradiction.(iii) If Possibility 2: When two of vertices in F lie on the same row.i = a = p and r ≠ l. Let F1 = {vp,j, vp,b} then either r = r or r = r, a contradiction.WLOG let Possibility 3: If all the three vertices in F lie on three different rowsF lie on same column, let j = b. If j = b = 1 and F1 = {v1,1, va,1} then r = r, a contradiction.(i) Two vertices in F1 = {vi,1, vj,1}, 1 ≤ i < j < k then either r = r or r = r, a contradiction.(ii) If F1 = {vi,b, vj,b}, 1 ≤ i < j < k, and 1 < b < l then either r = r or r = r, a contradiction.(iii) If F lie on different columns. Let F1 = {vi,j, va,b}, i < a then(iv) If all vertices in j < b then either r = r or r = r, a contradiction.(I) If j > b then either r = r or r = r, a contradiction.(II) If F is not an FTRS for αk,l. So Thus In this article, we computed the fault-tolerant metric dimension of triangular and alpha boron nanotubes. In both cases, we proved that this dimension is 4. Hence, these tubes are families of a constant fault-tolerant metric dimension. These facts can be used in the networking of nano-devices using these tubes and nano-engineering.Formal analysis was done by MM. Investigation and draft are done by ZH. Both authors contributed to the article and approved the submitted version.


The author wishes to make the following correction to this paper :In the original article, there were following mistakes:k′+5/K′5K′6” should be replaced with “k′+5K′5K′6”. Parameters were incorrectly assigned in the legend for Figure 5. The incorrect parameters “K′5K′6” should be replaced with “1/(K′5K′6)”. Parameters were incorrectly assigned in the legend for Figure 6. The incorrect parameters “K′DPAk+4” should be replaced with “k+4K′DPA”. A common term was incorrectly used in the legend for Table 2. The incorrect parameters “Steady-state ATPase” should be replaced with “Steady-state ATPase Activity”. A parameter was incorrectly assigned in the legend for Table 4. The incorrect parameters “In the original article, there was a mistake in Table 3 as published. Rate and equilibrium constants, and corresponding values were incorrectly assigned. The corrected In the original article, there were mistakes in Tables 4 and 5 as published. Rate and equilibrium constants and a unit were incorrectly assigned. The corrected There were following errors in the original article. K′5K′6” should be replaced with “1/(K′5K′6)”.Paragraph 8. The incorrect constants “K′DPA” should be replaced with “1/K′DPA”. The incorrect constants “K′DPAk′+4” should be replaced with “k′+4K′DPA”.Paragraph 10. The incorrect constants “K′DPA” should be replaced with “1/K′DPA”. Paragraph 11. The incorrect constants “A correction has been made to 2. Results and Discussion: K′5K′6” should be replaced with “1/K′5K′6”.A correction has been made to 3. Conclusions, Paragraph 1. The incorrect constants “The authors state that the scientific conclusions are unaffected. This correction was approved by the Academic Editor. The original publication has also been updated.