The notions of int-soft semigroups and int-soft left  ideals are introduced, and several properties are investigated. Using these notions and the notion of inclusive set, characterizations of subsemigroups and left  ideals are considered. Using the notion of int-soft products, characterizations of int-soft semigroups and int-soft left  ideals are discussed. We prove that the soft intersection of int-soft left  ideals  is also int-soft left  ideals . The concept of int-soft quasi-ideals is also introduced, and characterization of a regular semigroup is discussed. Molodtsov  introducş and Çağman [Çağman and Enginoğlu  introducd Çağman  studied d Çağman –26). Recd Çağman  investigS be a semigroup. Let A and B be subsets of S. Then the multiplication of A and B is defined as follows:Let S is said to be regular if for every x ∈ S there exists a ∈ S such that xax = x.A semigroup A of S is calledS if AA⊆A, that is, ab ∈ A for all a, b ∈ A,a subsemigroup of S if SA⊆A , that is, xa ∈ A  for all x ∈ S and a ∈ A,a left  ideal of S if it is both a left and a right ideal of S,a two-sided ideal of S if AS∩SA⊆A.a quasi-ideal of A nonempty subset A soft set theory was introduced by Molodtsov , and ÇağU be an initial universe set and let E be a set of parameters. Let P(U) denote the power set of U and A, B, C,…⊆E.In what follows, let α, A) over U is defined to be the set of ordered pairsα : E → P(U) such that α(x) = ∅ if x ∉ A.A soft set . The subscript A in the notation α indicates that α is the approximate function of .The function E has a binary operation ↪. For any nonempty subset A of E, a soft set  over U is said to be intersectional over U if it satisfiesAssume that α, A) over U and a subset γ of U, the γ-inclusive set of , denoted by iA, is defined to be the setFor a soft set  over U is called an intersectional soft semigroup  over U if it satisfiesA soft set  over U is called an intersectional soft left  ideal  ideal) over U if it satisfiesA soft set  over U is both an int-soft left ideal and an int-soft right ideal over U, we say that  is an intersectional soft two-sided ideal  over U.If a soft set  be a soft set over U defined as follows:γ1, γ2, γ3, and γ4 are subsets of U with γ1⊋γ2⊋γ3⊋γ4. Then  is an int-soft two-sided ideal over U.Let U is an int-soft semigroup over U. But the converse is not true as seen in the following example.Obviously, every int-soft left  ideal over S = {0,1, 2,3, 4,5} be a semigroup with the following Cayley table:(1)α, S) be a soft set over U defined as follows:γ1, γ2, γ3, γ4, and γ5 are subsets of U with γ1⊋γ2⊋γ3⊋γ4⊋γ5. Then  is an int-soft semigroup over U. But it is not an int-soft left ideal over U since α(3 · 5) = α(3) = γ4⊉γ3 = α(5).Let ((2)β, S) be a soft set over U defined as follows:γ1, γ2, γ3, and γ4 are subsets of U with γ1⊋γ2⊋γ3⊋γ4. Then  is an int-soft semigroup over U. But it is not an int-soft right ideal over U since β(3 · 4) = β(2) = γ3⊉γ2 = β(3).Let  is a soft set over U, which is called the characteristic soft set. The soft set  is called the identity soft set over U.For a nonempty subset A of  S, the following are equivalent.A is a left  ideal of  S.χA, S) over  U is an int-soft left  ideal over  U.The characteristic soft set ⊇∅ = χA(y). If y ∈ A, then xy ∈ A since A is a left ideal of S. Hence χA(xy) = U = χA(y). Therefore  is an int-soft left ideal over U. Similarly,  is an int-soft right ideal over U when A is a right ideal of S.Assume that χA, S) is an int-soft left ideal over U. Let x ∈ S and y ∈ A. Then χA(y) = U, and so χA(xy)⊇χA(y) = U; that is, χA(xy) = U. Thus xy ∈ A and therefore A is a left ideal of S. Similarly, we can show that if  is an int-soft right ideal over U, then A is a right ideal of S.Conversely suppose that  over  U  is an int-soft two-sided ideal over  U.The characteristic soft set  over  U is an int-soft semigroup over U if and only if the nonempty γ-inclusive set of  is a subsemigroup of  S for all γ⊆U.A soft set  over U is an int-soft semigroup over U. Let γ⊆U be such that iS ≠ ∅. Let x, y ∈ iS. Then α(x)⊇γ and α(y)⊇γ. It follows from . Thus iS is a subsemigroup of S.Assume that  is a subsemigroup of S for all γ⊆U. Let x, y ∈ S be such that α(x) = γx and α(y) = γy. Taking γ = γx∩γy implies that x, y ∈ iS. Hence xy ∈ iS, and so α(xy)⊇γ = γx∩γy = α(x)∩α(y). Therefore  is an int-soft semigroup over U.Conversely, suppose that the nonempty α, S) over U is an int-soft left  ideal over U if and only if the nonempty γ-inclusive set of   is a left  ideal of  S for all γ⊆U.A soft set  over  U is an int-soft two-sided ideal over U if and only if the nonempty γ-inclusive set of   is a two-sided ideal of  S for all γ⊆U.A soft set  and  over U, we defineα, S) and  is defined to be the soft set U in which  soft intersection of  and  is defined to be the soft set U in which α, S) and  is defined to be the soft set U in which S to P(U) given byFor any soft sets , , , and   be soft sets over U. IfLet   and  be characteristic soft sets over  U where A and B are nonempty subsets of  S. Then the following properties hold:Let  Let x ∈ S, suppose x ∈ AB. Then there exist a ∈ A and b ∈ B such that x = ab. Thus we havex ∈ AB, we get χAB(x) = U. Suppose x ∉ AB. Then x ≠ ab for all a ∈ A and b ∈ B. If x = yz for some y, z ∈ S, then y ∉ A or z ∉ B. Hencex ≠ yz for all x, y ∈ S, then(2) For any α, S) over U is an int-soft semigroup over U if and only if A soft set  is an int-soft semigroup over U.Assume that α, S) is an int-soft semigroup over U. Then α(x)⊇α(y)∩α(z) for all x ∈ S with x = yz. Thusx ∈ S. Hence Conversely, suppose that  and a soft set  over  U, the following are equivalent: β, S) is an int-soft left ideal over U, is an int-soft left ideal over U. Let x ∈ S. If x = yz for some y, z ∈ S, thenSuppose that  is an int-soft left ideal over U.Conversely, assume that Similarly, we have the following theorem.χS, S) over U and a soft set  over U, the following assertions are equivalent: β, S) is an int-soft right ideal over U, over U and a soft set  over U, the following assertions are equivalent:β, S) is an int-soft two-sided ideal over U, and  are int-soft semigroups over  U, then so is the soft intersection If  and  are int-soft left ideals  over U, then so is the soft intersection If   and  are int-soft two-sided ideals over U, then so is the soft intersection If   and  be soft sets over U. If  is an int-soft left ideal over U, then so is the int-soft product Let   = (xa)b andy is not expressible as y = ab for a, b ∈ S, then x, y ∈ S, and so U.Let  Similarly, we have the following theorem.α, S) and   be soft sets over U. If  is an int-soft right ideal over U, then so is the int-soft product Let  be a soft set over U. For a subset γ of U with iS ≠ ∅, define a soft set  over U byδ is a subset of U with δ⊊α(x).Let  is an int-soft semigroup over U, then so is .If  , then xy ∈ iS since iS is a subsemigroup of S by x ∉ iS or y ∉ iS, then α*(x) = δ or α*(y) = δ. Thusα*, S) is an int-soft semigroup over U.Let By similar manner, we can prove the following theorem.α, S) is an int-soft left ideal  over U, then so is .If   is an int-soft two-sided ideal over U, then so is .If   is an int-soft right ideal over U and  is an int-soft left ideal over U, then If   is an int-soft right ideal over  U, then β, S) over  U.Let x ∈ S. Then there exists a ∈ S such that xax = x since S is regular. Thusα, S) is an int-soft right ideal over U. Since xax = x, we obtainLet In a similar way we prove the following.S be a regular semigroup. If   is an int-soft left ideal over  U, then α, S) over  U.Let S is regular, then α, S) and int-soft left ideal  over  U.If a semigroup S is a regular semigroup and let  and  be an int-soft right ideal and an int-soft left ideal, respectively, over U. By Assume that α, S) over U is called an int-soft quasi-ideal over U ifA soft set  ideal is an int-soft quasi-ideal over In fact, we have the following example.S = {0, a, b, c} be a semigroup with the following Cayley table:α, S) be a soft set over U defined as follows:γ is a subset of U. Then  is an int-soft quasi-ideal over U and is not an int-soft left  ideal over U.Let G be a nonempty subset of  S. Then G is a quasi-ideal of S if and only if the characteristic soft set  is an int-soft quasi-ideal over  U.Let G is a quasi-ideal of S. Let a be any element of S. If a ∈ G, thena ∉ G, then χG(a) = ∅. On the other hand, assume thatb, c, d, and e of S with a = bc = de such that χG(b) = U and χG(e) = U. Hence a = bc = de ∈ GS∩SG⊆G, which contradicts that a ∉ G. Thus we have χG, S) is an int-soft quasi-ideal over U.We first assume that χG, S) is an int-soft quasi-ideal over U. Let a be any element of GS∩SG. Then bx = a = yc for some b, c ∈ G and x, y ∈ S. It follows from  over  U.For a semigroup S is regular and let a ∈ S. Then a = axa for some x ∈ S. Henceα, S) is an int-soft quasi-ideal over U,Assume that A be a quasi-ideal of S. Then ASA⊆AS∩SA⊆A and  is an int-soft quasi-ideal over U. For any a ∈ A, we haveb, c ∈ S such that χA(c) = U. Thenb = st,  χA(s) = U = χS(t) for some s, t ∈ S. It follows that c, s ∈ A and t ∈ S so that a = bc = (st)c ∈ ASA. Hence A⊆ASA, and thus A = ASA. Therefore S is regular.Conversely, suppose that (2) is valid and let


Also, the necessary and sufficient conditions of efficient solution for fractional programming are established and a parameterization technique is used to establish duality results under generalized second order  A fractional programming problem arises in many types of optimization problems such as portfolio selection, production, information theory, and numerous decision making problems in management science. More specifically, it can be used in engineering and economics to minimize a ratio of physical or economical function or both, such as cost/time, cost/volume, and cost/benefit, in order to measure the efficiency or productivity of the system. Many economic, noneconomic, and indirect applications of fractional programming problem have also been given by Bector , Bector The central concept in optimization is known as the duality theory which asserts that, given a  minimization problem, the infimum value of the primal problem cannot be smaller than the supermom value of the associated  maximization problem and the optimal values of primal and dual problems are equal. Duality in fractional programming is an important class of duality theory and several contributions have been made in the past , 10–14. V, ρ) invexity assumption.Multiobjective fractional programming duality has been of much interest in the recent past. Schaible  and Bectρ-univexity.Duality for various forms of mathematical problems involving square roots of positive semidefinite quadratic forms has been discussed by many authors , 23–25. z et al.  to charaz et al.  introducz et al.  and Jagaz et al.  for bothρ-univexity assumption.To relax convexity assumption imposed on the function in theorems on optimality and duality, various generalized convexity concepts have been proposed. Hanson  introducρ-univexity assumption.Motivated by the earlier authors in this paper, we have introduced three approaches given by Dinklebaeh  and Jagaℝn be the n-dimensional Euclidean space and ℝ+n its nonnegative orthant. The following conventions for inequality will be used throughout this paper. For any x = , y = , we denote the following.(i)x > y⇔xi > yi, for all i = 1,2,…, n.(ii)x ≥ y⇔xi ≥ yi, for all i = 1,2,…, n.Throughout the paper, let X be a nonempty open subset of ℝn.Let Consider the following nondifferentiable multiobjective fractional programming problem.(i) MFP0. Minimize  where (ii) MFP1. Minimize  where  νi are fixed parameters.(iii)MFPλ. Minimize λF(x); λ is k-dimensional strictly positive vector, all subject to same constraintfi : ℝn → ℝ, gi : ℝn → ℝ, i = 1,2,…, k and h = ; hj : ℝn → ℝ, j = 1,2,…, m, are differentiable functions, Bi and Ci, i = 1,2,…, k are positive semidefinite matrices of order n. In the sequel, we assume that fi(x) ≥ 0 and gi(x) > 0 on ℝn for i = 1,2,…, k.X0 = {x ∈ X⊆ℝn : hj(x) ≤ 0, j = 1,2,…, m} for all feasible solutions of MFP0, MFP1, and MFPλ and denote I = {1,2, 3,…, k}, M = {1,2, 3,…, m}, J1 = {j ∈ M : hj(x) = 0}, and J2 = {j ∈ M : hj(x) < 0}. It is obvious that J1 ∪ J2 = M.Let fi : X → ℝ, η : X × X → ℝn, p ∈ ℝn, ρ ∈ ℝ.Throughout the paper, consider ψ : ℝ → ℝ satisfying a ≤ 0⇒ψ(a) ≤ 0 or ψ(a) ≤ 0⇒a ≤ 0 and ψ(−a) = −ψ(a), K : X × X → ℝ+. For Assume that fi is said to be second order ρ-univex at η, ψ, and K, if The real differentiable function fi is said to be second order ρ-pseudounivex at η, ψ, and K, if The real differentiable function fi is said to be second order ρ-quasiunivex at η, ψ, and K, if The real differentiable function p = 0, the above definitions reduce to the definitions of ρ-univex, ρ-pseudounivex, and ρ-quasiunivex as introduced in  + ∑j=1myjhj(x) is second order ρ-pseudounivex with respect to η,  ψ, and K at 2P(x))p = 0, where fi : X → ℝ, gi : X → ℝ, hj : X → ℝ, i = 1,2,…, k; j = 1,2,…, m, p ∈ ℝn, η : X × X → ℝn, K : X × X → ℝ+, and ψ : ℝ → ℝ satisfying ψ(a) ≤ 0⇒a ≤ 0.(ii)ρ ≥ 0. Let Then Suppose that the hypothesis holds.Since the conditions of So we can write Now for ρ ≥ 0, we have For P(x) is second order ρ-pseudounivex with respect to η,  ψ, and K at K,  ψ, it gives t ∈ {1,2,…, k}.Since λi > 0, implies that The above relation, together with the relation elations , 11), a, aλi > elations , we get ies that ∑i=1kλi[flations Q(x) = ∑i=1kλi[fi(x) + (xTBix)1/2 − vi{gi(x) − (xTCix)1/2}] is second order ρ-pseudounivex with respect to η, ψ0, and K at H(x) = ∑j=1myjhj(x) is second order σ-quasiunivex with respect to η, ψ1, and K at fi : X → ℝ, gi : X → ℝ, hj : X → ℝ, i = 1,2,…, k; j = 1,2,…, m, p ∈ ℝn, η : X × X → ℝn, K : X × X → ℝ+, and ψ0, ψ1 : ℝ → ℝ satisfying ψ0(a) ≥ 0⇒a ≥ 0 and ψ1(a) ≤ 0⇒a ≤ 0.(ii)ρ + σ ≥ 0. Then MFP1.Let Suppose hypothesis holds.From the relations , 11), a, a11), aσ-quasiunivexity of H(x) with respect to η,  ψ1, and K implies the following: ρ + σ ≥ 0, we get Q(x) is ρ-pseudounivex with respect to η,  ψ0, and K, we obtained K and ψ0, we get So we have the following: ≤0. From , we get llowing: ηT{0. From −Q MMFD0. Maximize  where (ii) MMFD1. Maximize Gi(u) = fi(u) + yJ1ThJ1(u) + uTBiw − νi{gi(u) − uTCiz}, i = 1,2,…, k; νi are fixed parameters.where (iii)MMFDλ. Maximize λG(u); λ is k-dimensional strictly positive vector, all subject to same constraintsfi : X → ℝ, gi : X → ℝ, hj : X → ℝ, i = 1,2,…, k; j = 1,2,…, m are differentiable functions, w, z ∈ ℝn, p ∈ ℝn. Bi, and Ci, i = 1,2,…, k are positive semidefinite matrices of order n.η : X × X → ℝn, K : X × X → ℝ+, and ψ0, ψ1 : ℝ → ℝ satisfying ψ0(a) ≥ 0⇒a ≥ 0 and b ≤ 0⇒ψ1(b) ≤ 0 and ρ, σ ∈ ℝ.For the following theorems, we assume that x be a feasible solution for the primal MFPλ and let  be feasible for dual SMMFDλ. If (i)i=1kλiGi(·) is second order ρ-pseudounivex with respect to η, ψ0, K, and for yJ2 ∈ ℝm−|J1|, yJ2ThJ2(·) is second order σ-quasiunivex with respect to η, ψ1, and K along with∑(ii)ρ + σ ≥ 0, then Inf(λF(x)) ≥ Sup(λG(u)).Let yJ2ThJ2 is second order σ-quasiunivex with respect to η,  ψ1, and K and in view of p≥0. So yJ2ThJ2 ∈ ℝn, we have ηT{∑i=1kλi[∇Gi(u) + ∇2Gi(u)p]} − σ||x−u||2 ≥ 0.Again from the dual constraint , we have0. Using  in aboveρ + σ ≥ 0, we get ρ||x−u||2 ≥ −σ||x−u||2.Since i=1kλiGi(u) is second order ρ-pseudounivex with respect to η, ψ0, and K, by Kψ0{∑i=1kλiGi(x) − ∑i=1kλiGi(u) + (1/2)pT∑i=1kλiGi(u)p} ≥ 0.So, we have ηT{∑ψ0 and K, we get h(x) ≤ 0⇒yJ1ThJ1(x) ≤ 0, for yJ1 ≥ 0.Using the property of Equation  gives h is second order ρ-pseudounivex with respect to η,  ψ0, and K and for yJ2 ∈ ℝm−|J1|, yJ1ThJ2(·) is second order σ-quasiunivex with respect to η,  ψ1, and K along with (iii) ρ + σ ≥ 0, then Let λ), by λi ∈ ℝ+; w, z ∈ ℝn, vi ∈ ℝ+, and y ∈ ℝm such that Since So λ and SMMFDλ are equal to zero. It follows from And the objective values of MFPλ. Then applying Lemmas So p = 0, Ci = 0, i = 1,2,…, k, then our dual programming reduces to the dual programming proposed by Tripathy [If Tripathy .ρ-univexity assumption. The results developed in this paper can be further extended to higher order mixed type fractional problem containing square root term. Also the present work can be further extended to a class of nondifferentiable minimax mixed fractional programming problems.In this paper, three approaches given by Dinklebaeh  and Jaga


L in the reproducing kernel space W2m(D). Then we show spectral analysis of L and derive several property theorems.We first introduce some related definitions of the bounded linear operator  It is well known that spectral analysis of linear operators  is an imSo far, the spectral decomposition method  has becoW2m(D) and derive some useful conclusions.To our knowledge, reproducing kernel space has been applied in many fields, such as linear systems –8, nonliL in the reproducing kernel space W2m(D). In L in W2m(D) are given. In L and also establish several theorems. The paper is organized as follows. In D be an abstract set, W2m(D) the reproducing kernel space, and BL[W2m(D) → W2n(D)] the bounded linear operator space. ∀L ∈ BL[W2m(D) → W2n(D)] with m, n ∈ N, if there exists nonvanishing vector u ∈ W2m(D), such thatλ is called an eigenvalue of L and u is called the eigenvector of L according to λ, where I denotes the identity operator.Let L ∈ BL[W2m(D) → W2n(D)] and all eigenvectors and zero vector of L compose the eigenvector space which is denoted by Eλ.∀Eλ is a linear closed subspace of W2m(D).Obviously, Eλ by dimEλ; it is called the multiplicity of eigenvalue λ. That is, dimEλ is the number of vectors of maximum linear independence.Denote the dimension of K be a binary function on D = {∣a ≤ s ≤ b, a ≤ t ≤ b}, L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N withλ is the eigenvalue of L if and only if the following integral equation has nonzero solution:K = ∑i=1nfi(s)gi(t), {fi}i=1n is a linear independence vector system, then ((a)λ = 0, then (u ∈ W2m(D) and ∫abgi(t)u(t)dt = 0, i = 1,2,…, n. It follows that, for the eigenvalue λ = 0 of L, the eigenvector space is infinite-dimensional.If  0, then  has nonz(b)λ ≠ 0, then the solution of  are constants.If ution of  can be dLet olution:λu(s)−∫abfi}i=1n in (Ci must satisfy the following linear equation system:Combine  with 4)4) and, ii=1n in , Ci musCi  are undetermined coefficients. In addition, in order to solve the eigenvector, we just need to solve Ci  in   5) are eEλ of L for the homogeneous equation  → W2n(D)], m, n ∈ N, D(L)⊆W2m(D), and R(L)⊆W2n(D), where D(L) denotes the domain of L and R(L) denotes the range of values of L. If the inverse operator L−1 of L exists and is linearly bounded, then L is called a regular operator.Let L be a linear operator;  D(L)⊆W2m(D) and R(L)⊆W2n(D); if L−1 exists, then L−1L = ID(L) and LL−1 = IR(L), where ID(L) and IR(L) are, respectively, identity operators of subspace D(L) and R(L). Inversely, if there exists a linear operator C : W2n(D) → W2m(D), such that CL = ID(L) and LC = ID(C), then L−1 exists and I−1 = C. In fact, ∀u1, u2 ∈ D(L); if Lu1 = Lu2, then u1 = CLu1 = CLu2 = u2. Hence, L is invertible. Since LC = ID(C), then ∀v ∈ D(C); we have u = Cv such that Lu = v. That is, D(C)⊆R(L).Let R(L) = D(C). Hence, we have L−1 = (CL)L−1 = CLL−1 = C. Particularly, when D(C) = W2n(D) and C is a bounded linear operator, we can derive the following results.Summing up the above disscusion, L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N; then L is a regular operator if and only if ∃C ∈ BL[W2n(D) → W2m(D)], such that CL = ID(L) and LC = ID(C).Let L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N; if L is a regular operator, then L* is also a regular operator and (L*)−1 = (L−1)*.Let L−1 ∈ BL[W2n(D) → W2m(D)], m, n ∈ N, andL* is a regular operator and (L*)−1 = (L−1)*. The proof is complete.Since D(L)⊆W2m(D), m ∈ N, λ ∈ C; C denotes the complex number field.λI − L is a regular operator, that is, λI − L is a one-to-one linear operator from D(L) to W2m(D). In addition, the inverse operator (λI − L)−1 is a linear bounded operator. Then λ is called a regular point of L. All regular points compose the regular set of L, which is denoted by ρ(L).λ is not a regular point, then λ is called a spectral point of L. All spectral points compose the spectral set of L, which is denoted by σ(L).If Let ρ(L)⋃σ(L) = C. Then we have the following property results.In view of L be a bounded linear operator in reproducing kernel space W2m(D), m ∈ N; then λ is a regular point of L if and only if ∀f ∈ W2m(D), there exists a solution g of (λI − L)g = f, which satisfies ||g|| ≤ m||f||, where m is a positive constant.Let R(λI − L) = W2m(D), then ∀f ∈ W2m(D), ∃g ∈ W2m(D) such that (λI − L)g = f. In addition, in view of the boundedness of (λI − L)−1 and the Cauchy-Schwartz inequality, we havem = ||(λI−L)−1|| > 0; then ||g|| ≤ m||f||.⇒ Since λI − L)g = f, we have R(λI − L) = W2m(D). Next, we will prove that λI − L is one-to-one. In fact, ∀f ∈ W2m(D); if (λI − L)g1 = f, (λI − L)g2 = f, theng1 − g2 is 0. Hence, ||g1 − g2|| ≤ m||0||; that is, g1 = g2. Therefore, λI − L is one-to-one and (λI − L)−1 exists. Furthermore, since ||g|| ≤ m||f||, we haveλI − L)−1 exists and is a bounded linear operator. Summing up the above, λI − L is a regular operator, where λ is a regular point of L.⇐ Since (f ∈ W2m(D); when L is a continuous linear operator and λ is the regular point of L, (λI − L)g = f has a unique solution g. Furthermore, the continuity of g depends on the right term. In other words, if {fi}i=1n are column vectors and fn → f, then gn → g.L be a bounded linear operator in the reproducing kernel space W2m(D), m ∈ N. If λ is not the eigenvalue of L and (λI − L)g1 = (λI − L)g2, one has L(g1 − g2) = λ(g1 − g2), g1 = g2. That is, λI − L is invertible.Let g ≠ 0 such that (λI − L)g = 0. That is, λ is not the eigenvalue of L.Otherwise, the invertible operator can convert the nonvanishing vector to nonvanishing vector. Hence, there exists W2m(D) is a finite dimension space and λ is not the eigenvalue of L, we can derive that C = λI − L is an invertible mapping. Obviously, R(C) = W2m(D). In fact, let {ei}i=1n be the basis of W2m(D); then {(λI − L)ei}i=1n is a linear independent system in W2n(D) and also a basis of W2n(D). Therefore, R(L) = W2n(D). In view of the inverse operator Theorem, (λI − L)−1 is bounded. It follows that λ ∈ ρ(L).When So, the proof of the theorem is complete.L can only be an eigenvalue in finite dimension normed space. This is entirely consistent with the conclusion of the theory of linear algebra. But if W2m(D) is an infinite-dimensional space and λ is not the eigenvalue of L, then λ may not be a regular point of L, so far as λI − L is not a map from W2m(D) to W2n(D).λ ∈ C, ∫abu(t)dt = λu(t) has only zero solutions. Hence, L has not eigenvalue. That is, zero is not the eigenvalue. However, the range of values is all functions of the from ∫abu(t)dt for (0I − L). This shows that the spectral point is complex in infinite-dimensional space for the operator L.For example, letλI − L is not one-to-one, then λ is called point spectral of L; the set of point spectral is denoted by σp(L).If λI − L is one-to-one and R(λI − L) is dense in W2m(D), then λ is called continuous spectral of L; the set of continuous spectral is denoted by σc(L).If λI − L is one-to-one and R(λI − L) is not dense in W2m(D), then λ is called residual spectral of L; the set of residual spectral is denoted by σr(L).If Now, we will classify the spectral set by three situations.σp(L), σc(L), and σr(L) are mutually disjoint sets and σ(L) = σp(L) ∪ σc(L) ∪ σr(L).Obviously, L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N, ε > 0, ∃N ∈ N*, ∀n > N, such that Ln||<(r + ε)n. In view of the completeness of W2m(D), there exists m > N, such thatn=0∞Ln converges in the sense of ||·|| and the limit is denoted by C = ∑n=0∞Ln.Let Cm = ∑n=0mLn; thenCm − C|| → 0, m ≥ N, we havem → ∞, thenρ(L) and (I−L)−1 = ∑n=0∞Ln.Let L|| < 1, one obtainsFor ||Summing up the above parts, we have the following theorems.L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N, then one has the following.ρ(L).Consider 1 ∈ I − L)−1 = ∑n=0∞Ln.Consider −1|| ≤ 1/(1 − ||L||).When ||Let L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N; if λ| > r if and only if λ is a regular point of L.|λ| > r, (λI−L)−1 = ∑n=0∞(Ln/λn+1).When |λ| > ||L||, ||(λI−L)−1|| ≤ (|λ|−||L||)−1.When |Let λ ≠ 0, since (λI − L) = λ(I − L/λ), λ ∈ ρ(L) if and only if 1 ∈ ρ(L/λ). Replacing L by L/λ in λ is a regular point of L and ||(λI−L)−1|| ≤ (|λ|−||L||)−1 with |λ| > r.∀λ| > ||L||, we have ||L/λ|| < 1. In view of (2) of In addition, when |L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N; then one has the following.ρ(L) is an open set.ρ(L) is nonempty, ∀λ0 ∈ ρ(L); if λ is a regular point of L and (λI−L)−1 = ∑n=0∞(−1)n(λ0I−L)n+1)−((λ−λ0)n, where |λ − λ0| < 1/rλ0.When σ(L) is a closed set.Consider Let ρ(L) = ∅, the conclusion is obvious. If ρ(L) ≠ ∅, thenλ0I−L)−1 is a bounded linear operator in the reproducing kernel space W2m(D). We use (λ − λ0)(λ0I−L)−1 instead of L in λ − λ0| < 1/rλ0, [I+(λ−λ0)(λ0I−L)−1]−1 exists and is bounded. Hence, when |λ − λ0| < 1/rλ0, λ ∈ ρ(L); that is, ρ(L) is an open set.(1) If ρ(L) is nonempty, ∀λ0 ∈ ρ(L), let (2) If ρ(L)⋃σ(L) = C and (1), σ(L) is a closed set.(3) Since σ(L)⊆{λ∣|λ| ≤ r}, which means that (4) In view of (1) of The proof is complete.L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N, r(L) = maxλ∈σ(L)|λ|; r(L) is called the spectral radius of L.Let λ| > r(L), due to the fact that λ is a regular point of L, then for any f ∈ W2m(D), (λI − L)g = f has a unique solution g.For |λ| ≤ r(L), it cannot guarantee this equation has a solution for any f ∈ W2m(D). In many practical problems, in order to calculate the spectral range, one needs to estimate the spectral radius. In terms of (4) of r(L) ≤ ||L||. In practical terms, this estimate is convenient, but it is imprecise.For |From the purpose of solving equations, spectral radius has the following meanings.L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N; then Let λ| > ||L||, one obtainsf ∈ W2m(D), f((λI−L)−1) = ∑n=0∞(f(Ln)/λn+1). If |λ| > r(L), then λ is a regular point of L. In addition, since {λ∣|λ| > r(L)}, then Laurent expansions of f((λI−L)−1) are established, where |λ| > r(L).In terms of (4) of a = r(L), ∀ε > 0; we haveBn = Ln/(a+ε)n, ∀f ∈ W2m*(D); thenLet Bn} must be bounded. It follows that there exists a positive constant M, such that ||Bn|| ≤ M and ||Ln|| ≤ (a+ε)n||Bn|| ≤ (a+ε)nM. Namely,ε → 0; thenIn terms of the resonance Theorem, {L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N; then σ(L) ≠ ∅.Let σ(L) = ∅, in view of the properties of the reproducing kernel space, W2m(D)≠{0}; hence, I ≠ {0}, where the unit element is denoted by I. In terms of the functional extension Theorem, ∃f ∈ W2m*(D), such that f(I) ≠ 0. In addition, ∀λ0 ∈ ρ(L) and ∃rλ0 ∈ R+; when |λ − λ0| < 1/rλ0, we haveσ(L) = ∅ and λ| > ||L||, one obtainsλ| ≥ ||L|| + 1, we havef((λI−L)−1) is bounded. In terms of the Liouville Theorem, f((λI−L)−1) must be a constant, so we have σ(L) ≠ ∅.If The proof is complete.L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N, L is called a generalized nilpotent operator.If L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N, ⊆D,L, one obtainsL is a generalized nilpotent operator; spectral point λ = 0 is not the eigenvalue of L.In terms of L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N, λ ∈ C; if there exists {un}n=1∞ ∈ W2m(D), such that (λI − L)un → 0, then λ is called an approximate spectral point. All approximate spectral points are denoted by σa(L); the other spectral point is called remainder spectral point. All the remainder spectral points are denoted by σr(L).Let L ∈ BL[W2m(D) → W2n(D)], m, n ∈ N; thenσp(L)⊆σa(L),σa(L)⋂σr(L) = ∅, and σa(L)⋃σr(L) = σ(L),σr(L) is an open set,σ(L)⊆σa(L), where ∂σ(L) denotes the boundary of σ(L),∂σa(L) is a nonempty closed set.Let λ ∈ σp(L), then there exists nonzero element u of W2m(D), such thatu|| = 1; we choose un = u, n = 1,2,…, and then ||un|| = 1 and (L − λI)un → 0; namely, λ ∈ σa(L); this shows that σp(L)⊆σa(L).(1) If σa(L)⋂σr(L) = ∅ and σa(L)⋃σr(L) = σ(L).(2) In terms of λ ∈ σr(L), then λ ∉ σa(L). Hence, ∃α ∈ N*, such thatλ′ − λ| < α/2, ∀u ∈ W2m(D), we haveλ′ which satisfies |λ′ − λ| < α/2 it is impossible to be an approximate spectral point of L. Hence, if one can prove that when |λ′ − λ| < α/2, λ′ is not a regular point of L, then λ′ ∈ σr(L). That is, λ is an inner point of σr(L), so σr(L) is an open set.(3) If λ′ − λ| < α/2, λ′ ∉ ρ(L). But not vice versa, ∃λ0 ∈ C, |λ0 − λ| < α/2; then λ0 ∈ ρ(L). Note that ||(L − λ′I)u|| ≥ ||(L − λI)u|| − |λ − λ′|||u|| ≥ (α/2)||u||; if λ′ = λ0, thenμ be a regular point of L; if |μ − λ0| < 1/rλ0, thenμ = λ; note thatλ ∈ ρ(L). This is a contradiction with λ ∈ σr(L). It follows that σr(L) is an open set.Now, we prove that when |σ(L) is a closed set, when λ ∈ ∂σ(L), λ ∈ σ(L). In addition, λ ∉ σr(L), ∂σ(L)⊈σr(L), so we have ∂σ(L)⊆σa(L).(4) Since σa(L) = σ(L) − σr(L), then σa(L) is a closed set. Furthermore, since ∂σ(L)⊆σa(L), σa(L) ≠ ∅, this shows that ∂σ(L) and σa(L) are all nonempty sets.(5) Since The proof is complete.Eλ of the bounded linear operator L in the reproducing kernel space W2m(D). Then we show some definitions and properties of the regular operator. The regular set and spectral set of bounded linear operator are also introduced. From the solvability of the equation, we show the spectral classification and give three conditions. Finally, we introduce the spectral analysis of the bounded linear operator L. It includes the definitions of spectral radius, nilpotent operator, approximate spectral point, and remainder spectral point. We also establish some property theorems of the bounded linear operator in the reproducing kernel space W2m(D).This paper first introduces the eigenvalue, eigenvector, eigenvector space, and dim


In particular, we give so far the smallest number of locally indistinguishable states of a completable orthogonal product basis in arbitrary quantum systems. Furthermore, we construct a series of small and locally indistinguishable orthogonal product bases in m ⊗ n (m ≥ 3 and n ≥ 3). All the results lead to a better understanding of the structures of locally indistinguishable product bases in arbitrary bipartite quantum system.As we know, unextendible product basis (UPB) is an incomplete basis whose members cannot be perfectly distinguished by local operations and classical communication. However, very little is known about those incomplete and locally indistinguishable product bases that are not UPBs. In this paper, we first construct a series of orthogonal product bases that are completable but not locally distinguishable in a general  Furthermore, they proved that a UPB cannot be perfectly distinguished by LOCC. DiVincenzo et al.et al.et al.d − 1 orthogonal states that are locally indistinguishable in d ⊗ d (d ≥ 3) and conjectured that any set of no more than 2(d − 1) product states is locally distinguishable in a d ⊗ d (d ≥ 3) quantum system. Wang et al.m + n) − 9 orthogonal product states and proved the local indistinguishability of these states in an m ⊗ n quantum system, where m ≥ 3 and n ≥ 3. Recently, Zhang et al.n + m − 4 locally indistinguishable orthogonal product states that do not constitute a UPB and presented a smaller set with 2n − 1 orthogonal product states that cannot be perfectly distinguished by LOCC in m ⊗ n (3 ≤ m ≤ n). All the results show it is a meaningful work to research the structure of the locally indistinguishable product basis and the smallest number of locally indistinguishable orthogonal product states in arbitrary high-dimensional quantum systems.Unextendible product basis (UPB) is a set of orthogonal product states that spans a subspace whose complementary subspace contains no product state. As an object with rich mathematical structure, it was introduced by Bennett m, n) − 4 members, respectively, in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system. Our results show that Yu et al.’s conjectured − 1) product states is locally distinguishable in a d ⊗ d (d ≥ 3) quantum system, is not true. In fact, eight is so far the smallest number of locally indistinguishable states of a completable orthogonal product basis2223m, n) − 1 members respectively, in m ⊗ n (m ≥ 3 and n ≥ 3). It should be pointed out that five is so far the smallest number of locally indistinguishable states of an orthogonal product basis by Refs. In this paper, we construct a series of completable and locally indistinguishable orthogonal product bases, which have eight members, twelve members, ···, 4 min is a set S of pure orthogonal product states spanning a subspace HSof H. An uncompletable orthogonal product basis is a PB whose complementary subspace, i.e., the subspace in H spanned by vectors that are orthogonal to all the vectors in HS, contains fewer mutually orthogonal product states than its dimension. An unextendible product basis (UPB) is an uncompletable product basis for whichcontains no product state20. We call a PB is completable if it is not an uncompletable orthogonal product basis.Definition 2Consider a multipartite quantum systemwith q parties. A strongly uncompletable product basis (SUCPB) is a PB spanning a subspace Hsin a locally extended Hilbert space Hextsuch that for all Hextthe subspacecontains fewer mutually orthogonal product states than its dimension.Definition 3. Suppose that {Mt} is a set of measurement operators, which can act on the measured Hilbert space. And t denote one of the possible measurement outcomes. If the measured state is |ϕ〉 before it is measured, the probability of the measurement outcome t is given byand the postmeasurement state is. Furthermore, the measurement operators {Mt} satisfy the completeness, i.e.,. If we denoteas Et, it is easy to see that Etis a positive semidefinite operator. We will say that the measurement is a positive operator-valued measure (POVM) and the objects Etare the POVM elements corresponding to each measurement outcome tDIt is easy to see that POVM is a general measurement to a measured quantum state according to the definition 3. As we mentioned in the preceding part, LOCC denote local operations and classical communication. When it comes to identify a given state that is chosen from a known set of orthogonal states by LOCC, the local operations are local POVMs or local unitary operations. That is, if a measured state is a bipartite (or multipartite) quantum system, each party that holds one particle of the bipartite (or multipartite) quantum system can only perform POVM or unitary operations on his (or her) own particle. For simplicity, we usually say a set of orthogonal states is locally distinguishable if it can be distinguished by LOCC.efinition 4We will say that a POVM is trivial if all the POVM elements are proportional to the identity operator since such a measurement yields no information about the measured state. Any measurement not of this type will be called nontrivial.DDifferent from Definition 4, we give a new definition about trivial measurement here. It should be noted that we say a measurement is trivial if it satisfies our new definition.efinition 5. A POVM is trivial to a set of orthogonal states,, if and only if we cannot get any useful information about the measured state that is arbitrarily selected from the set by the POVM, i.e., for each of the POVM elements,, we haveDefinition 6Alice goes first if Alice is the first person to perform a nontrivial measurement upon the system.Demma 1Given a PBon a Hilbert spaceof total dimension D. If the set S is completable in H or a locally extended Hilbert space Hext, then the density matrixis separable, where I is the identity matrix of rank D.Lp − 4 members that cannot be locally distinguished in a general m ⊗ n (m ≥ 3 and n ≥ 3) quantum system and give a proof for its indistinguishability.Now we construct a completable product basis with 4theorem 1. In an m ⊗ n quantum system, the 4p − 4 orthogonal product statescannot be perfectly distinguished by LOCC, where m ≥ 3, n ≥ 3, p is an arbitrary integer from 3 to min, j = i + 1 when i = 1, ···, p − 2 and j = 1 while i = p − 1.Proof. Many proof techniques are borrowed from Ref. m × m POVM elements Mt ⊗ IB|ψi〉: i = 1, ···, 4p − 4} should be mutually orthogonal. For the states |ψi〉 and |ψj〉, where 1 ≤ i ≤ p − 1, 1 ≤ j ≤ p − 1 and i ≠ j, we have i ≤ p − 1, 1 ≤ j ≤ p − 1 and i ≠ j. For the states j = i + 1 when i = 1, ···, p − 2 and j = 1 when i = p − 1, we can get j = 1, 2, ···, p − 1. For the states j = i + 1 when i = 1, ···, p − 2 and j = 1 while i = p − 1, we have i = 1, ···, p − 1.The post measurement states {Therefore, we haveMt for each of the 4p − 4 states. It is easy to see that t = 1, 2, ···, l, where p − 4 states can lead to the outcome that is corresponding to Mt with the same probability Mt ⊗ IB | t = 1, 2, ···, l} is trivial to the 4p − 4 states. In other words, Alice cannot get any information about which the measured state will be by the measurement {Mt}. Thus Alice cannot go first. In fact, a similar argument can be used to exhibit that Bob faces the same dilemma, i.e., he cannot gain any useful information by a nondisturbing measurement, either. Therefore, they cannot perfectly distinguish these states by LOCC. In other words, the 4p − 4 states cannot be perfectly distinguished by LOCC. This completes the proof.Now we consider the probability of the measurement outcome corresponding to the measurement operator In general, the local indistinguishability of an incomplete PB are proved by Definition 4422p − 4 states of (1) can become a completed orthogonal product basis in m ⊗ n (m ≥ 3 and n ≥ 3) by adding the following mn − 4p + 4 states:It is noted that the product basis (1) is completable since the 4p can be an arbitrary integer from 3 to min. That is, we actually construct a series of orthogonal product bases that are locally indistinguishable in m ⊗ n (m ≥ 3 and n ≥ 3). In particular, we have the following corollary by Theorem 1 when p = 3.From Theorem 1, we know that the parameter corollary 1. In an m ⊗ n quantum system, the eight orthogonal product statescannot be perfectly distinguished by LOCC, where m ≥ 3 and n ≥ 3.d ⊗ d quantum system when m = n = d ≥ 3. This fact shows that the conjectured − 1) product states is locally distinguishable in a d ⊗ d (d ≥ 3) quantum system, is not true. By Refs m ⊗ n (m ≥ 3 and n ≥ 3).As a special case, we can get eight states (2) that cannot be perfectly distinguished by LOCC in a p − 1 members that cannot be perfectly distinguished by LOCC in an m ⊗ n quantum system, where m ≥ 3, n ≥ 3 and 3 ≤ p ≤ min. Then we give a simple proof for its local indistinguishability.Now we give a small orthogonal product basis with 2theorem 2. In an m ⊗ n quantum system, the 2p − 1 orthogonal product statescannot be perfectly distinguished by LOCC, where m ≥ 3, n ≥ 3, p is an arbitrary integer from 3 to min, j = i + 1 when i = 1, ···, p − 2 and j = 1 while i = p − 1.Proof. Similar to the proof of Theorem 1, one of the two parties (Alice and Bob) has to start with a nondisturbing measurement to distinguish these states, i.e., the postmeasurement states should be mutually orthogonal. Without loss of generality, suppose that Alice goes first with a set of general m × m POVM elements t = 1, ···, l), wherei ≤ p − 1, 1 ≤ j ≤ p − 1 and i ≠ j, and j = 1, 2, ···, p − 1 by the same way as the proof of Theorem 1 since the postmeasurement states should be mutually orthogonal. For the states ψp−12〉, where j = i + 1 when i = 1, ···, p − 2 and j = 1 while i = p − 1, we have i = 1, 2, ···, p − 1. Therefore, we haveWe can get Mt for each of the 2p − 1 states. It is easy to see that t = 1, 2, ···, l, where p − 1 states can lead to the outcome that is corresponding to Mt with the same probability Mt ⊗ IB | t = 1, 2, ···, l} is trivial to the 2p − 1 states. That is, Alice cannot get any useful information about which the measured state will be by the measurement {Mt}. In fact, if Bob goes first with a nondisturbing measurement, they cannot distinguish the 2p − 1 states, either. Therefore, the 2p − 1 states cannot be perfectly distinguished by LOCC. This completes the proof. □Now we consider the probability of the measurement outcome corresponding to the measurement operator p can be an arbitrary integer from 3 to min in Theorem 2. That means that we construct a series of orthogonal product bases that are locally indistinguishable in m ⊗ n (m ≥ 3 and n ≥ 3). We have the following corollary directly by Theorem 2 when p = 3.The parameter corollary 2. In an m ⊗ n quantum system, the five orthogonal product statesare locally indistinguishable, where m ≥ 3 and n ≥ 3.m = n = 3, the five states of (4) form a UPB. In Ref. et al. exhibits two results. One is that a UPB is not completable even in a locally extended Hilbert space. The other is that if a set of orthogonal product states is exactly measurable by LOCC, then the set can be completed in some extended space. Thus it is obvious that the five states of (4) in m ⊗ n are locally indistinguishable by the two results, which is coincident with Corollary 2. Since any four orthogonal product states are shown to be locally distinguishableWhen p − 1 states of Theorem 2 are uncompletable in the m ⊗ n quantum system, where m ≥ 3, n ≥ 3 and p is an arbitrary integer from 3 to min. By the analysis of the last paragraph, we know that the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 3 and n ≥ 3) quantum system when p = 3. Then we prove that the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 4 and n ≥ 4) quantum system when p = 4. It is noted that some proof techniques are borrowed from Ref. Now we consider whether or not the 2S = {|ψ1〉, |ψ2〉, ···, |ψ7〉}. The density matrix S, which are not all mutually orthogonal:Let S is a SUCPB because p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 4 and n ≥ 4) quantum system when p = 4. On the other hand, we can prove that the 2p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 5 and n ≥ 5) quantum system by the same method, where 5 ≤ p ≤ min. By Theorem 2, we get the following 2p − 1 states in the p ⊗ p (p ≥ 5) quantum system.These six vectors are not enough to span the full Hilbert space j = i + 1 when i = 1, ···, p − 2 and j = 1 while i = p − 1. Let S′ = {|ψ1〉, |ψ2〉, ···, |ψp−12〉}. The density matrix p2 − (2p − 1) = (p − 1)2. We can enumerate the product states that are orthogonal to the members of S′, which are not all mutually orthogonal:where i = 1, 2, ···, p − 1; j = i + 1 for i = 1, 2, ···, p − 2 while j = 1 for i = p − 1; and j + 1 = i + 2 for i = 1, 2, ···, p − 3 while j + 1 = 1 for i = p − 2 and j + 1 = 2 for i = p − 1. These 2p − 2 vectors are not enough to span the full Hilbert space p − 2 product states, whereas p − 1)2. Therefore S′ is a SUCPB because p − 1 states of Theorem 2 are uncompletable in the m ⊗ n (m ≥ 5 and n ≥ 5) quantum system when 5 ≤ p ≤ min. Therefore, the 2p − 1 orthogonal product states of (3) are uncompletable in the m ⊗ n quantum system, where m ≥ 3, n ≥ 3 and p is an arbitrary integer from 3 to min.where p − 4 members that cannot be perfectly distinguished by LOCC in an m ⊗ n quantum system, where m ≥ 3, n ≥ 3 and p is an arbitrary integer from 3 to min, and give a simple but quite effective proof. As a special case, we get eight orthogonal product states that can be completable but cannot be locally distinguished in m ⊗ n (m ≥ 3 and n ≥ 3). On the other hand, we construct a samll locally indistinguishable orthogonal product basis with 2p − 1 members in m ⊗ n, which are uncompletable, where m ≥ 3, n ≥ 3 and p is an arbitrary integer from 3 to min. Our work is useful to understand the structures both of completable and uncompletable product bases that cannot be distinguished by LOCC in arbitrary bipartite quantum system.In this paper, we construct a completable orthogonal product basis with 4How to cite this article: Xu, G.-B. et al. Locally indistinguishable orthogonal product bases in arbitrary bipartite quantum system. Sci. Rep.6, 31048; doi: 10.1038/srep31048 (2016).


By using Tsuji's characteristic, we investigate uniqueness of meromorphic functions in an angular domain dealing with the shared set, which is different from the set of the paper  and obtain a series of results about the unique range set of meromorphic functions in angular domain. The purpose of this paper is to deal with the uniqueness problem of meromorphic functions sharing one set in an angular domain by using Tsuji's characteristic. Thus, the notation and theory of Nevanlinna see , 2) abou abou2]) C to denote the open complex plane, Ω(⊂C) to denote an angular domain.We use IM theorem of two meromorphic functions sharing five distinct values.In 1929, Nevanlinna (see ) first if and g are two nonconstant meromorphic functions that share five distinct values a1, a2, a3, a4, and a5  IM in C, then f(z) ≡ g(z).If C. In recent years, there are many results on the uniqueness of meromorphic function in an angular domain sharing values and sets . In 200ets see –16). Zha. ZhaC. Iets ..C. In reS be a set of distinct elements in Ω⊆C. Definefa(z) = f(z) − a if a ∈ C and f∞(z) = 1/f(z). We also defineLet f and g be two nonconstant meromorphic functions in C. If E = E, we say f and g share the set S  CM (counting multiplicities) in Ω. If f and g share the set S  IM (ignoring multiplicities) in Ω. In particular, when S = {a}, where f and g share the value a  CM in Ω if E = E, and we say f and g share the value a  IM in Ω if Ω = C, we give the simple notation as before, Let  on (see ).In 2006, Lin et al.  dealt wiS1 = {∞}, S2 = {ω | ωn−1(ω + a) − b = 0}, and S3 = {0}, where n(≥4) is an integer and a, b are two nonzero constants, such that the algebraic equation ωn−1(ω + a) − b = 0 has no multiple roots. Assume that f is a meromorphic function of lower order μ(f) ∈  in δ : = δ > 0 for some σ < ∞ with μ(f) ≤ σ ≤ λ(f), there exists an angular domain Ω = Ω : = {z : α < argz < β} with 0 ≤ α < β ≤ 2π andE = E and E = E   hold for a meromorphic function g of finite order or, more generally, with the growth satisfying either logT = O) orE1 is a set of finite linear measures, then f ≡ g.Let In 2011, Chen and Lin  further S1 and S2 be defined as in n ≥ 8 be an integer. Assume that f is a meromorphic function of lower order μ(f) ∈  in C and Θ > 2/(n − 1) and that g is a meromorphic function of finite order or, more generally, with the growth satisfying either logT = O) or condition (σ < ∞ with μ(f) ≤ σ ≤ λ(f), there exists an angular domain Ω = Ω with 0 ≤ α < β ≤ 2π and condition  = E  , then f ≡ g.Let ondition . Then, fondition , such thIM theorem of Nevanlinna's to an angular domain. Tsuji's characteristic will be introduced in In 2010, Zheng  proved tf(z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < argz < β} with 0 ≤ α < β ≤ 2π, and let f(z) be transcendental in Tsuji's sense. Assume that aj   are 5 distinct complex numbers. If f(z) ≡ g(z).Let S = {w ∈ A : P1(w) = 0}, wherec be a complex number satisfying c ≠ 0,1, and obtain the following results.In this paper, we will focus on the uniqueness problem of shared set of meromorphic functions in an angular domain by using Tsuji's characteristic. In fact, we will study the uniqueness of meromorphic functions in an angular domain sharing one set of the form f(z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg  z < β} with 0 ≤ α < β ≤ 2π, and let f(z) be transcendental in Tsuji's sense. If E = E and n is an integer ≥11, then f ≡ g.Let S is called a unique range set for meromorphic functions in an angular domain Ω, if for any two nonconstant meromorphic functions f and g the condition E = E implies f ≡ g. We denote by ♯S the cardinality of a set S. Thus, from A set S with ♯S = 11, such that any two meromorphic functions f and g in an angular domain Ω which are transcendental in Tsuji's sense must be identical if E = E.There exists one finite set f(z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < arg  z < β} with 0 ≤ α < β ≤ 2π, and let f(z) be transcendental in Tsuji's sense. If E = E, ΘT > 3/4, ΘT > 3/4, and n is an integer ≥7, then f ≡ g.Let S with ♯S = 7, such that any two analytic functions f and g in Ω which are transcendental in Tsuji sense must be identical if E1 = E1.There exists one finite set f(z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < argz < β} with 0 ≤ α < β ≤ 2π, and let f(z) be transcendental in Tsuji's sense. If E1 = E1 and n is an integer ≥15, then f ≡ g.Let S is called a unique range set with weight 1 for meromorphic functions in Ω, if for any two nonconstant meromorphic functions f and g the condition E1 = E1 implies f ≡ g. Thus, from A set S with ♯S = 15, such that any two meromorphic functions f and g in an angular domain Ω which are transcendental in Tsuji's sense must be identical if E1 = E1.There exists one finite set f(z) and g(z) be both meromorphic functions in an angular domain Ω = {z : α < argz < β} with 0 ≤ α < β ≤ 2π, and let f(z) be transcendental in Tsuji's sense. If E1 = E1, ΘT > 5/6, ΘT > 5/6, and n is an integer ≥9, then f ≡ g.Let From S with ♯S = 9, such that any two analytic functions f and g in Ω which are transcendental in Tsuji sense must be identical if E1 = E1.There exists one finite set f in Tsuji sense.We found that the conclusions of Theorems Thus, a question arises naturally, whether the conclusions of these theorems and corollaries hold for a general function in an angular domain.For the above question, we can get the following theorem.f(z) is transcendental in Tsuji sense. Assume that, for some ε > 0,ω = π/(β − α), N = ∫1r/t)dt, and n is the number of poles of f(z) in Ω∩{z : 1 < |z| ≤ t}. Then f(z) ≡ g(z).Let the assumptions of Theorems f in an angular domain Ω and ω = π/(β − α), we definebn = |bn|eiβn are the poles of f(z) in Ξ = {z = reiθ : α < θ < β, 1 < t ≤ r(sin(ω(βn − α)))ω−1} appearing often according to their multiplicities and then Tsuji characteristic of f isnα,β the number of poles of f(z) in Ξ, thenbn occurs in the sum ∑bn|<r(sin(ω(βn−α)))ω−11<| only once, and we denote it by f in Ω and for all complex numbers a, iff is called transcendental with respect to the Tsuji characteristic ).teristic , and we on, from , we haveq distinct points Qα,β satisfies  is the counting function of the zeros of f′ in Ω where f does not take any of the values aj  .In fact, from  and 16]16], we catisfies  and Nα,f(z) is a meromorphic function in Ω. Then for 0 < r < R, one hasK is a constant independent of r and R.Assume that M, N, Q, T, and N0 instead of Mα,β, Nα,β, Qα,β, Tα,β, and Nα,β0, respectively.For sake of simplicity, we omit the subscript in all notations and use Ω as follows.By a similar discussion as in , one canf(z) be a meromorphic function in Ω. Then for all irreducible rational function R in f with coefficients meromorphic and small with respect to f in Ω, one hasQ is stated as in  in f.Let ed as in  and d isf is a nonconstant meromorphic function in Ω. ThenQ is stated as in , we haveSincethat is,N≤).SinceT=MNext, we will give two main lemmas of this paper as follows.F and G be transcendental meromorphic functions in Ω in Tsuji sence satisfying E = E, and let c1, c2,…, cq be q(≥2) distinct nonzero complex numbers. IfN1) is the counting function which only counts simple zeros of the function · in Ξ, and I is some set of r of infinite linear measure, thena, b, c, d ∈ C are constants with ad − bc ≠ 0.Let SetH≢0; from Q(r) : = o{T(r)} and T(r) = max{T, T}. Since E = E and by an elementary calculation, we can conclude that if z0 is a common simple zero of F and G in Ω, then H(z0) = 0. Thus, from (N1)(r) = N1) = N1). The poles of H in Ω can only occur at zeros of F′ and G′ in Ω or poles of F and G in Ω. Moreover, H only has simple zeros in Ω. Hence, from (r)≤N ≡ 0; that is,a, b, c, d ∈ C and ad − bc ≠ 0.Sincehen from  and 28)(28)N¯+r).From qT≤N∉ E thatq{T+),since∑j=1qN¯(2r).From  and 33)(28)N¯ = E1, and let c1, c2,…, cq be q(≥2) distinct nonzero complex numbers. IfI are stated as in a, b, c, d ∈ C are constants with ad − bc ≠ 0.Let H be stated as in the proof of E1 = E1, it follows thatr ∉ E,f, g are transcendental in Tsuji sense in Ω, it follows thatH(z) ≡ 0; that is,a, b, c, d ∈ C and ad − bc ≠ 0.Let r).From  and 40)H be statThe following result can be derived from the proof of Frank-Reinders' theorem in .n ≥ 6 andP(w) is a unique polynomial for transcendental meromorphic functions; that is, for any two transcendental meromorphic functions f and g in Tsuji sense, P(f) ≡ P(g) implies f ≡ g.Let f(z) be a meromorphic function in Ω, for any real number ε > 0, Ωε = Ω. Then for ε > 0, one hasc < 1 is a constant depending on ε, ω = (π/(β − α)), N = ∫1r/t)dt, and n is the number of poles of f(z) in Ω∩{z : 1 < |z| ≤ t}.Let P1(w), we have P1(1) = 1 − c : = c1 ≠ 0, P1(0) = −c : = c2 ≠ 0, andQ1, Q2 are polynomials of degrees n − 3 and 2, respectively. We also see that Qi   and P1 have only simple zeros.From the definition of F and G be defined as F = P1(f) and G = P1(g). Since E = E, we have E = E. From  and bj   are the zeros of Q1(w) and Q2(w) in Ω, respectively.Let G). From  and 48)F and G bT = nT + Q(r). Thus, combining  is nonempty and E = E, we have b = 0, a ≠ 0. HenceA = c/a, B = d/a ≠ 0.From , we haveombining  and 50)(50)N¯ and (P1(g) + B/A in Ω has a multiplicity of at least n. Here, three following subcases will be discussed.From the definition of 1(w) and , we can ai ≠ 0,1 are distinct values. It follows thatn − 2 values satisfying the above inequality. Thus, from . It follows that every zero of g in Ω has a multiplicity of at least 2 and every zero of g − bi   in Ω has a multiplicity of at least n. Then, by g is transcendental in Tsuji sense in Ω and n ≥ 11, we can get a contradiction.From , we haveBy using the same argument as in B ≠ 1, from  − c2/B has at least n − 2 distinct zeros  e1, e2,…, en−2. Then, by If  1, from , we haveg).From  and 60)B ≠ 1, fr2, from , it folln ≥ 11 and f is transcendental in Tsuji sense in Ω, we can get a contradiction.By applying A = 0 and B = 1; that is, P1(f) = P1(g). Notting the form of P1(w), we can get that P(f) = P(g). Then, by f ≡ g.Thus, we have Therefore, the proof of T > 3/4 and ΘT > 3/4, it follows thatn ≥ 7, we getF ≡ (aG + b)/(cG + d), where a, b, c, d ∈ C and ad − bc ≠ 0. Thus, by using the same argument as in Since Θapplying , from (5applying  and 52)T >E1 = E1, we have E1 = E1. From  are the distinct zeros of P1(w). And from  = nT + Q(r) and n ≥ 15, we havea, b, c, d ∈ C and ad − bc ≠ 0. By using arguments similar to that in proof of f ≡ g.Since G). From –48), we, weE1(SAnd from  and 65)E1E1 > 5/6 and ΘT > 5/6, it follows thatn ≥ 9, we getF ≡ (aG + b)/(cG + d), where a, b, c, d ∈ C and ad − bc ≠ 0. Thus, by using the same argument as in Since Θows thatlimsupr→applying  and 68)T >Hence, the proof of f is transcendental in Tsuji sense, then the conclusions of Since condition  implies


X2 constructed as finite  linear combinations of functions denoted as , where  is a groupoid (binary system) and μ is a fuzzy subset of X and where : = μ(x∗y) − min{μ(x), μ(y)}. Many properties, for example, μ being a fuzzy subgroupoid of X, ∗), can be restated as some properties of . Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin(X), □) for example.We discuss properties of a class of real-valued functions on a set  Fuzzy Commutative Algebra, presented a fuzzy ideal theory of commutative rings, and applied the results to the solution of fuzzy intersection equations. The book included all the important work that has been done on L-subspaces of a vector space and on L-subfields of a field.The notion of a fuzzy subset of a set was introduced by Zadeh . His semX) and obtained a semigroup structure. Fayoumi , where  is a groupoid (binary system) and μ is a fuzzy subset of X and where : = μ(x∗y) − min{μ(x), μ(y)}. Many properties, for example, μ being a fuzzy subgroupoid of , can be restated as some properties of . Thus, the context provided opens up ways to consider well-known concepts in a new light, with new ways to prove known results as well as to provide new questions and new results. Among these are identifications of many subsemigroups and left ideals of (Bin(X), □), for example.In this paper, we discuss properties of a class of real valued functions on a set X, we let Bin(X) denote the collection of all groupoids , where ∗ : X × X → X is a map and where ∗ is written in the usual product form. Given elements  and  of Bin(X), define a product “□” on these groupoids as follows:x, y ∈ X. Using that notion, Kim and Neggers proved the following theorem.Given a nonempty set X), □) is a semigroup; that is, operation “□” as defined in general is associative. Furthermore, the left zero semigroup is the identity for this operation. a fuzzy subset μ of X, that is, a mapping μ : X → , and (ii) a groupoid (binary system) , where ∗ : X2 → X is a mapping. It is thus possible to consider a fuzzy subgroupoid of a groupoid as a composition :→, where μ and  interact to satisfy the condition:x, y ∈ X.Given a set  the intersection between a fuzzy subset μ and a groupoid  as given by the function denoted by  : X2 → R, wherex, y ∈ X.One might easily conceive related notions such as “an almost fuzzy subgroupoid of a fuzzy groupoid,” which though here unspecified will be considered in what follows. Since “almost” itself is a typical “fuzzy math” notion, we will considerX, ∗); μ] need not be a fuzzy subset of .Note that intersection  = μ(x + y) − min{μ(x), μ(y)} = (x + y) − ⌊x + y⌋ − min{x − ⌊x⌋, y − ⌊y⌋}. If we let x = y = 1/2, then  = −1/2. Hence,  is not a fuzzy subset of .Let X be set R of all real numbers. Define a binary operation ∗ on R by x∗y : = x − ⌊x⌋. Then μ(x∗y) = μ(x − ⌊x⌋) = (x − ⌊x⌋) − ⌊x − ⌊x⌋⌋ = x − ⌊x⌋ = μ(x) for all x, y ∈ X. It follows that  = μ(x∗y) − min{μ(x), μ(y)} = μ(x) − min{μ(x), μ(y)} ≥ 0. Hence,  is a fuzzy subset of .Let X be set R of all real numbers, and let + be the usual addition on R and let x∗y : = x − ⌊x⌋ for all x, y ∈ R. Define : = □. Then x□y = (x∗y)+(y∗x) = x − ⌊x⌋ + y − ⌊y⌋. It follows that μ(x□y) = x + y − (⌊x⌋ + ⌊y⌋) − ⌊x + y − (⌊x⌋ + ⌊y⌋)⌋ for all x, y ∈ X. Assume that x − ⌊x⌋ = y − ⌊y⌋ = 0.6. Then μ(x□y) = 1.2 − 1 = 0.2 and hence  = μ(x□y) − min{μ(x), μ(y)} = −0.4. Hence,  is not a fuzzy subgroupoid of .Let X, ∗); μ], note that the smallest value obtainable is when μ(x∗y) = 0 and μ(x) = μ(y) = 1. If such a pair  exists, then  = μ(x∗y) − min{μ(x), μ(y)} = −1. We call such a pair  a μ-orthogonal pair on .Given an intersection  be the characteristic function of the rationals; that is, if x is rational, then μ(x) = χQ(x) = 1 and if x is irrational, then μ(x) = 0. Suppose that x and y ≠ 0 are both rational. Then X, ∗); μ] = μ(x∗y) − min{μ(x), μ(y)} = 0 − 1 = −1. Thus,  is an χQ-orthogonal pair on .Let X, ∗) ∈Bin(X), if μ is a fuzzy subalgebra of , then there is no pair  ∈ X2 such that  is a μ-orthogonal pair on .Given , then μ(x∗y) ≥ min{μ(x), μ(y)} for all x, y ∈ X. Assume that  is a μ-orthogonal pair on . Then μ(x∗y) − min{μ(x), μ(y)} = −1. It follows that 0 ≤ μ(x∗y) = min{μ(x), μ(y)} − 1 ≤ 0, which shows that min{μ(x), μ(y)} = 1. Since μ is a fuzzy subalgebra of , we obtain μ(x∗y) = μ(x) = μ(y) = 1. Hence, 0 = μ(x∗y) − min{μ(x), μ(y)} = −1, a contradiction.If X, ∗); μ], a pair  ∈ X2 is said to be a μ-parallel pair on  if  = 1. In μ-parallel pair on . In fact, Given an intersection , we define α byx, y ∈ X. Using these notions, we will consider that a generating set for a vector space of groupoids  ∈ Bin(X) can be expressed as finite sums ∑i=1nλi, where λi ∈ R for i = 1,…, n, and μi : → is a fuzzy subset of  and where ; μi]): = α as usual for real-valued functions. If we let S : = ∑i=1nλi, then −∑i=1n | λi | ≤S ≤ ∑i=1n | λi| for all x, y ∈ X. Given a fuzzy subset μ of , we define a map 1 − μ : X →  by (1 − μ)(x): = 1 − μ(x) for all x ∈ X.Given an element X, ∗) ∈Bin(X), if μ is a fuzzy subset of , thenx, y ∈ X.Given ; μ]+. Then, for all x, y ∈ X, we haveLet S in X, ∗) ∈ Bin(X).Note that such an X, ∗),  ∈Bin(X), we haveGiven  ∈Bin(X) and a fuzzy subset μ : X → , there exist a groupoid  ∈Bin(X) and a fuzzy subset ν : X →  such that  = max{μ(x), μ(y)} − min{ν(x), ν(y)} for all x, y ∈ X.Given  ∈ Bin(X) and a fuzzy subset μ : X → , we define a surjective map ν : X → Im(μ). Since max{μ(x), μ(y)} ∈ Im(μ), ν−1(max{μ(x), μ(y)}) ≠ ∅. Define a binary operation ⋆ on X as follows: for any x, y ∈ X, x⋆y : = z for some z ∈ ν−1(max{μ(x), μ(y)}. It follows that ν(x⋆y) = max{μ(x), μ(y)}. This means that  = ν(x⋆y) − min{ν(x), ν(y)} = max{μ(x), μ(y)} − min{ν(x), ν(y)}.Given  ∈Bin(X) and a fuzzy subset μ : X → , there exists a groupoid  ∈Bin(X) such that  = S.Given ; ν] = max{μ(x), μ(y)} − min{μ(x), μ(y)} = |μ(x) − μ(y)| = S by In the proof of X, ∗) ∈ Bin(X), we define a set F(X): = {μ | μ : X → :a  map}. We consider the composition μ□ν of fuzzy subsets μ, ν ∈ F(X) as a function φ(x) whose variables are μ(x) and ν(x); that is, (μ□ν)(x): = φ(μ(x), ν(x)). For example, (μ□ν)(x) = (1/2)(μ(x) + ν(x)),(μν)(x) = λμ(x) + (1 − λ)ν(x), λ ∈ , or Given a groupoid (μ□ν)□ρ ≠ μ□(ν□ρ). In fact, if we define μ□(ν□ρ)](x)μ ∈ F(X) is said to be □-idempotent if μ□μ = μ.In general, ; μ], , we define a new interaction as follows:X, □) = □ and (μ□ν)(x): = φ(μ(x), ν(x)) for all x ∈ X.Given interactions □)□ = □; ν]□ for any ,,  ∈ Bin(X).It is easy to see that if  ∈ Bin(X) and μ, ν ∈ F(X), we define μ□ν : = (1/2)(μ + ν). It follows thatGiven a groupoid ; μ], we have already seen that −1 ≤  ≤ 1. If 0 ≤  ≤ 1, then μ is a fuzzy subgroupoid of  and conversely. Thus, we will be interested in bounds α ≤  ≤ β as classification parameters of the . Hence, we usually take α : = inf{ | x, y ∈ X} and β : = sup{ | x, y ∈ X} for the “narrowest” fit.Given an interaction ,  have the following bounds: α ≤  ≤ β, γ ≤  ≤ δ, then the interaction  has the bound α + γ ≤  ≤ β + δ.Given x, y ∈ X, we let p : = min{μ(x∗y), μ(y∗x)} and q : = min{μ(x), μ(y)}. If α ≤  ≤ β, γ ≤  ≤ δ, then μ((x∗y) · (y∗x)) − min{μ(x∗y), μ(y∗x)}∈. Since μ(x∗y) − min{μ(x), μ(y)}∈ and μ(y∗x) − min{μ(x), μ(y)}∈, we have μ(x∗y), μ(y∗x)∈. It follows that p = min{μ(x∗y), μ(y∗x)}∈; that is, p − q ∈ . Thus, μ(x□y) − min{μ(x), μ(y)} = μ((x∗y) · (y∗x)) − q = μ((x∗y) · (y∗x)) − p + p − q ∈ + = .Given μ in X, ∗)□; μ) if β + δ > 1.Note that map α ≤  ≤ β and γ ≤  ≤ δ. If there exist σ, τ ∈ R such that |μ(x∗y) − μ(y∗x)|∈ for all x, y ∈ X, thenLet γ ≤  ≤ δ, thenx, y ∈ X. If α ≤  ≤ β, thenσ, τ ∈ R such that |μ(x∗y) − μ(y∗x)|∈ for all x, y ∈ X. Let μ(x∗y) + ρ = μ(y∗x) for some ρ ≥ 0. Then μ(x∗y) − min{μ(x), μ(y)} + ρ = μ(y∗x) − min{μ(x), μ(y)} ≤ β. It follows that μ(x∗y) − min{μ(x), μ(y)} ≤ β − ρ ≤ β − σ. By applying −miμ(y∗x) − min{μ(x), μ(y)} = μ(x∗y) − min{μ(x), μ(y)} + ρ, we obtainSimilarly, since ≤applying , we obta}.Using , 17), a, aμ(y∗x)}.Using , we obtaμ ∈ F(X). A groupoid  ∈ Bin(X) is said to be μ-commutative if μ(x∗y) = μ(y∗x) for all x, y ∈ X.Let X, ·) ∈Bin(X) is μ-commutative, then □; μ) is also μ-commutative for all  ∈Bin(X).If  = μ((x∗y) · (y∗x)) = μ((y∗x) · (x∗y)) = μ(y□x).For all μ-commutative groupoids forms a left ideal of the semigroup (Bin(X), □).Note that X, ∗) ∈Bin(X), if μi : X →  are fuzzy subsets of  and λ ∈ , thenGiven (μ : = λμ1 + (1 − λ)μ2. Given a, b, c, d ∈ , we have min{a + b, c + d} ≥ min{a, c} + min{c, d}. It follows that, for all x, y ∈ X,Let f : X2 →  is said to be representable if f can be represented as  for some groupoid  ∈ Bin(X) and a fuzzy subset μ : X → . We denote it by f = .A function an}n=1∞ be a sequence of real numbers such that, for any n and k,A, B > 0. Then {an}n=1∞ is called a special sequence of type .Let {A′ ≥ A and B′ ≥ B, then a special sequence of type  is also a special sequence of type .If an}n=1∞ is a special sequence of type  and {bn}n=1∞ is a special sequence of type , then, for real numbers λ, δ > 0, {λan + δbn}n=1∞ is a special sequence of type . If A = B = C = D = 1 and λ + δ = 1, then λA + δC = 1 and λB + δD = 1. Such a special sequence of type  is called a standard special sequence.If {bn = an+t, t ≥ 0 and if {an}n=1∞ is a special sequence of type , then {bn}n=1∞ is also a special sequence of type . It follows that if {an}n=1∞ is a special sequence, then, for all n ≥ 1, limk→∞(an+1 + ⋯+an+k)/k = 0.If an = α}n=1∞, then limk→∞(an+1 + ⋯+an+k)/k = limk→∞(kα/k) = α, whence, α ≠ 0 implies that {an = α}n=1∞ is not a special sequence.Notice that if {k→∞[1/(n + 1) + ⋯+1/(n + k)]/k = 0, since limk→∞(1/k)∫nn+k(dx/x) = limk→∞[ln(1 + k/n)/1/k] = 0. However, there is no B such that, for any n and k, [1/(n + 1) + ⋯+1/(n + k)]/k ≤ B/k; that is, 1/(n + 1) + ⋯+1/(n + k) ≤ B since ∑n=1∞(1/n) diverges. Hence, the sequence {an = 1/n}n=1∞ is not special for any pair , with A, B > 0, a pair of real numbers.Notice that limWe have the following observations.X, ∗) be a groupoid. For x1 ∈ X, determine a sequence d(x1): = {x1, x2,…, xn,…} as follows: x2 = x1∗x1,  x3 = x2∗x2,…, xn+1 = xn∗xn,….. We consider d(x1) to be a doubling sequence for x1 relative to .Let ; μ], then the doubling sequence must be a standard special sequence.If f =  for some groupoid  and a fuzzy subset μ : X → . If d(x1) is a doubling sequence in , thenμ(xn+1) − μ(x1) ≤ 1, we obtain −1/n ≤ (1/n)[∑i=1nf] ≤ 1/n. Replace x1 by xk+1,…, xn by xn+k, the same inequalities are obtained. Hence, we conclude that {d(x1)} is a standard special sequence.Let f ≥ α > 0 or f ≤ β < 0, then f is not representable as f = .If f ≥ α > 0, then, for any expression  for some groupoid  ∈ Bin(X) and a fuzzy subset μ : X → , then we may deduce many properties of the representation. Note that μ(π/4) = 0 and μ((π/4)∗(π/4)) = 1. In fact, Sin = sin(π/2) = μ((π/4)∗(π/4)) − μ(π/4) = 1. Much other information is available via the use of trigonometric properties. Thus, for example, Sin = sin(x + y + π) = −sin(x + y) = −Sin = − = μ(x∗(y + π)) − min{μ(x), μ(y + π)] so that μ(x∗(y + π)) + μ(x∗y) = min{μ(x), μ(y + π)} + min{μ(x), μ(y)}. Thus, for example, μ(x) = 1 implies μ(x∗(y + π)) + μ(x∗y) = μ(y + π) + μ(y) and μ(x) = 0 implies μ(x∗(y + π)) = −μ(x∗y), whence μ(x∗y) = 0 for any y whatsoever. Hence, if Kerμ = μ−1(0), then Kerμ is a right ideal of .If we assume that f+⋯+f]/k must behave properly to yield at least the possibility of a representation.In the sine function case, doubling sequences may generate sequences with positive and negative values whose average over sections  has a sign pattern on {f}n=1∞ of the following type {+, −, +, +, −, −, +, +, +, −, −, −, +, +, +, +,…} and that f ≥ α > 0 when it is positive. If f is representable, that is, f = , and if {f}n=1∞ is a doubling sequence, then for n = k(k + 1), there are k slots of positive value between i = n and i = n + k. Thus, summing over all these slots, one obtains an upper bound of value k (the number of slots). Another associated value is the average of the functional values of that segment from i = n to i = n + k, and thus at least α > 0. This violates the condition that the doubling sequence obtained from f =  must be a standard special sequence. Hence, f is not representable.Here is another example of a negative solution to the representation problem. Suppose that {, if it is believed/known that f = , develop a method of finding a groupoid  and a fuzzy subset μ such that f = . So far, the best results have been of the negative kind, but interesting nevertheless.Given f : X2 →  is quite enormous with common properties to be examined. The doubling sequence technique developed above is an example of such a common property which can be applied to the existence problem of representations for function of the type addressed here.As a closing comment, we observe that the class of representable functions


Res. Natl. Inst. Stand. Technol. Volume 100, Number 3, May–June 1995, p. Page 211, above Eq. (3).Replace “the magnetic field within the balance chamber is vertical with the form…” by “the vertical magnetic field strength within the balance chamber has the form…”.Page 217, Eq. (8).Ia” by “Ia”.Replace “Page 218, two paragraphs above Sec. 5.1.µmHmax” by “µ0Hmax”.Replace “Page 223, first paragraph of Sec. 11.1.Replace “: it must be less than that based on a sum…; it must be greater than that based on a sum…” by “: it must be greater than that based on a sum…; it must be less than that based on a sum…”.Page 224, Table 5, column 4, row 2.Replace “0.1116” by “−0.1116”.Page 224, Eq. (10) should be:Page 225, two paragraphs above the Acknowledgment.Replace “This again overestimates χ” by “This again underestimates χ”.


In this paper, wegeneralize the results of Azam et al. (2011), and Bhatt et al. (2011), by improving the conditionsof contraction to establish the existence and uniqueness of common fixed point for twoself-mappings on complex valued b-metric spaces. Some examples are given to illustratethe main results.Azam et al. (2011), introduce the notion of complex valued metric spaces and obtainedcommon fixed point result for mappings in the context of complex valued metric spaces. Rao et al. (2013) introduce the notion of complex valued  The concept of complex valued b-metric spaces was introduced in 2013 by Rao et al. .The pair  is called a b-metric space. The number s ≥ 1 is called the coefficient of .Let X, d) be a metric space and ρ = )p, where p > 1 is a real number. Then  is a b-metric space with s = 2p−1.Let  = Re(z2) and Im(z1) = Im(z2),Re(z1) < Re(z2) and Im(z1) = Im(z2),Re(z1) = Re(z2) and Im(z1) < Im(z2),Re(z1) < Re(z2) and Im(z1) < Im(z2).We will write z1⋨  z2 if z1 ≠ z2 and one of (2), (3), and (4) is satisfied; also we will write z1≺z2 if only (4) is satisfied.Let a, b ∈ R and a ≤ b, then az≾bz for all z ∈ C;if z1⋨z2, then |z1 | <|z2|;if 0≾z1≾z2 and z2≺z3, then z1≺z3.if We can easily check that the following statements are held:X be a nonempty set. A function d : X × X → C is called a complex valued metric on X if for all x, y, z ∈ X the following conditions are satisfied:d and d = 0 if and only if x = y;0≾d = d;d≾d + d.The pair  is called a complex valued metric space.Let X = C. Define the mapping d : X × X → C byX, d) is a complex valued metric space.Let X = C. Define the mapping d : X × X → C byX, d) is a complex valued metric space.Let X be a nonempty set and let s ≥ 1 be a given real number. A function d : X × X → C is called a complex valued b-metric on X if for all x, y, z ∈ X the following conditions are satisfied:d and d = 0 if and only if x = y;0≾d = d;d≾s[d + d].The pair  is called a complex valued b-metric space.Let X = . Define the mapping d : X × X → C byX, d) is a complex valued b-metric space with s = 2.Let X, d) be a complex valued b-metric space. Consider the following.x ∈ X is called interior point of a set A⊆X whenever there exists 0≺r ∈ C such that B: = {y ∈ X : d≺r}⊆A.A point x ∈ X is called a limit point of a set A whenever, for every 0≺r ∈ C,  B∩(A − X) ≠ ∅.A point A⊆X is called open whenever each element of A is an interior point of A.A subset A⊆X is called closed whenever each element of A belongs to A.A subset τ on X is a family F = {B : x ∈ X and 0≺r}.A subbasis for a Hausdorff topology Let  be a complex valued b-metric space and {xn} a sequence in X and x ∈ X. Consider the following.c ∈ C, with 0≺r, there is N ∈ N such that, for all n > N,  d≺c, then {xn} is said to be convergent, {xn} converges to x, and x is the limit point of {xn}. We denote this by limn→∞xn = x or {xn} → x  as  n   → ∞.If for every c ∈ C, with 0≺r, there is N ∈ N such that, for all n > N,  d≺c, where m ∈ N, then {xn} is said to be Cauchy sequence.If for every X is convergent, then  is said to be a complete complex valued b-metric space.If every Cauchy sequence in Let  be a complex valued b-metric space and let {xn} be a sequence in X. Then {xn} converges to x if and only if |d|→0  as  n → ∞.Let  be a complex valued b-metric space and let {xn} be a sequence in X. Then {xn} is a Cauchy sequence if and only if |d|→0  as  n → ∞, where m ∈ N.Let  be a complete complex valued metric space and let λ, μ be nonnegative real numbers such that λ + μ < 1. Suppose that S, T : X → X are mappings satisfyingx, y ∈ X. Then S, T have a unique common fixed point in X.Let  be a complete complex valued metric space and let S, T : X → X be mappings satisfyingx, y ∈ X, where a ∈ [0,1). Then S, T have a unique common fixed point in X.Let  be a complete complex valued b-metric space with the coefficient s ≥ 1 and let S, T : X → X be mappings satisfyingx, y ∈ X, where λ, μ are nonnegative reals with sλ + μ < 1. Then S, T have a unique common fixed point in X.Let |>|d|, we getsλ + μ < 1 and s ≥ 1, we get λ + μ < 1.For any arbitrary point, 2n+1 in ; we haveδ = λ/(1 − μ) < 1, and for all n ≥ 0, consequently, we havem > n,  m,  n ∈ N, and since sδ = sλ/(1 − μ) < 1, we getxn} is a Cauchy sequence in X.Therefore, with  we have|d|≤0, a contradiction with |S and T have unique common fixed point of S and T. To show this, assume that u* is another common fixed point of S and T. Thend|<|d|, a contradiction. So u = u* which proves the uniqueness of common fixed point in X. This completes the proof.Now we show that X, d) be a complete complex valued b-metric space with the coefficient s ≥ 1 and let T : X → X be a mapping satisfyingx, y ∈ X, where λ, μ are nonnegative reals with sλ + μ < 1. Then T has a unique fixed point in X.Let  be a complete complex valued b-metric space with the coefficient s ≥ 1 and let T : X → X be a mapping satisfyingx, y ∈ X, where λ, μ are nonnegative reals with sλ + μ < 1. Then T has a unique fixed point in X.Let |≤λ | d|<|d|, a contradiction. So, Tu = u. HenceT is unique. This completes the proof.From ows fromd=dX = C. Define a function d : X × X → C such thatz1 = x1 + iy1 and z2 = x2 + iy2.Let X, d) is a complete complex valued b-metric space with s = 2, it is enough to verify the triangular inequality condition.To verify that , we haveTnz = 0 for n > 1, sox, y ∈ X and λ, μ ≥ 0 with 2λ + μ < 1. So all conditions of T.Now, define two self-mappings b-metric spaces.Our next theorem is a generalization of X, d) be a complete complex valued b-metric space with the coefficient s ≥ 1 and let S, T : X → X be mappings satisfyingx, y ∈ X, where sa ∈ [0,1). Then S, T have a unique common fixed point in X.Let |≤0, a contradiction with |S and T have unique common fixed point of S and T. To show this, assume that u* is another common fixed point of S and T. Thend|≤0, and then u = u* which proves the uniqueness of common fixed point in X. This completes the proof.Now we show that X, d) be a complete complex valued b-metric space with the coefficient s ≥ 1 and let T : X → X be a mapping satisfyingx, y ∈ X, where sa ∈ [0,1). Then T has a unique fixed point in X.Let  be a complete complex valued b-metric space with the coefficient s ≥ 1 and let T : X → X be a mapping satisfyingx, y ∈ X, where sa ∈ [0,1) and n ∈ N. Then T has a unique fixed point in X.Let (


The authors wish to make the following corrections to this paper :Mar. Drugs [We have found eight inadvertent errors in our paper published in r. Drugs . Δ6,12,15” should be “18:3Δ6,9,12”. Line 8 and column 8 of Table 2, “20:4Δ5,8,11,14,17” should be “20:4Δ8,11,14,17”. Line 9 and column 7, “Δ5/cis” should be “Δ4/cis”. Therefore replace this incorrect table:On page 1327, Line 6 and column 9 of Table 2, “18:3With the following corrected table:Δ9,12 → 18:3Δ6,9,12 and 18:3Δ9,12,15 → 18:3Δ6,9,12,15)-Desaturation” should be “The -Desaturation”. On page 1331, line 6, in the title of Section 2.4.11, “The Question of the -Desaturation” should be “The Question of the -Desaturation”. On page 1332, line 1, “E Δ6FAD” should be “ERΔ6FAD”. On page 1335, lines 25–26, in the Conclusions Section, “ERD6FAD, and ERD5FAD” should be “ERΔ6FAD, and ERΔ5FAD”.On page 1330, line 9, “18:3 and 14” should be “18:3 and 18:4”. Line 29, in the title of Section 2.4.10, “The (18:2These changes have no material impact on the conclusions of our paper. We apologize to our readers for any inconvenience caused.


Based on He's variational iteration method idea, we modified the fractional variational iteration method and applied it to construct some approximate solutions of the generalized time-space fractional Schrödinger equation (GFNLS). The fractional derivatives are described in the sense of Caputo. With the help of symbolic computation, some approximate solutions and their iterative structure of the GFNLS are investigated. Furthermore, the approximate iterative series and numerical results show that the modified fractional variational iteration method is powerful, reliable, and effective when compared with some classic traditional methods such as homotopy analysis method, homotopy perturbation method, adomian decomposition method, and variational iteration method in searching for approximate solutions of the Schrödinger equations. In the past decades, due to the numerous applications of fractional differential equations (FDEs) in the areas of nonlinear science , many imThe variational iteration method (VIM) established in 1999 by He in  is thoroWe give some basic definitions and properties of the fractional calculus theory which are used further in this paper; we define the following fractional integral and derivatives , 21.f(x) is said to be in the space Cμ, where μ ∈ R, x > 0, if there exists a real number p(>μ) such that f(x) = xpf1(x), where f1(x) ∈ C[0, ∞) and it is said to be in the space Cμm if and only if fm) ∈ Cμ, μ ≥ −1, is defined as follows:The Riemann-Liouville fractional integral operator of order α > 0, x > 0, and f(x) ∈ C−1n, the Caputo fractional derivative operator of order α on the whole space is defined as follows:For Also one has the following properties:u = u, ∂αu/∂tα = Dtαu, ∂β2u/∂xβ2 = Dxβ(Dxβu), v(x) is the trapping potential, and a, γ are the slowly increasing dispersion coefficient and nonlinear coefficient, respectively. If we select α = β = 1, v(x) = 0, this equation turns to the famous nonlinear Schrödinger equations in optical fiber [Consider the following generalized time and space fractional nonlinear Schrödinger equation with variable coefficients , 23:(5)al fiber –26.u0 = u = f(x), where λ is a general Lagrange's multiplier which can be identified optimally with the variational theory. The function λ that will be identified optimally via integration by parts [un+1, n ≥ 0, of the solution u will be readily obtained through λ and any selective function u0. The initial values are usually used for choosing the zeroth approximation u0. With λ determined, then several approximations uk, k = 1,2,… follow immediately. Consequently, the exact solution may be procured by using u = limn→∞un. The convergence of FVIM has been proved in [u|2, as we all know, a complex function u(ξ) can be written as c(ξ)eiθ(ξ), where c(ξ) and θ(ξ) are real functions, noticed that |u(ξ)|2 = |c(ξ)|2, we can give some modification for the iteration formulation  = Asec hx; using the iteration (Eα(aitα) is the Mittag-Leffler function. If we let α = 1 in i∂αui∂αu(7)i∂follows:un+1=un+ition of  = sinx, using the iteration (ck(x) = ck,0sinx + ck,1sin(x + πβ) + ck,2sin(x + 2πβ)+⋯+ck,ksin(x + kπβ), ck,0 = (−1)k, ck,1 = (1/2)ck−1,1 − ck−1,0,…, ck,k−1 = (1/2)ck−1,k−2 − ck−1,k−1, and ck,k = (1/2)ck−1,k−1, k ≥ 2, c0,0 = 1; c1,0 = −1, c1,1 = (1/2); c2,0 = 1, c2,1 = −1, c2,2 = (1/4); ….Consider the following time-space fractional NLS equation :(20)i∂αfollows:un+1=un+iα = 1 and let β = 1 in  = cosx, with the same process, we can also obtain the following exact solution of (The exact solution of  is23)uu(23)u=liβ = 1 in  can be oution of :(25)u=iβ = 1, the solution (If one selects solution  is more solution  and 25)β = 1, thsolution  to our ku4, uabs, and u are plotted in Figures Comparisons between the real part of some numerical results and the exact solution  are summIn this paper, the MFVIM is used for finding approximate and exact solutions of the GFNLS equation with Caputo derivative. The obtained results indicate that the MFVIM is effective, convenient, and powerful method for solving nonlinear fractional complex differential equations when comparing it with some other traditional asymptotic decomposition methods such as HAM, VIM, and ADM. We believe that these methods should play an important role for finding exact and approximate solutions in the mathematical physics.


This review on recent research advances of the lipid peroxidation product 4-hydroxy-nonenal (HNE) has four major topics: I. the formation of HNE in various organs and tissues, II. the diverse biochemical reactions with Michael adduct formation as the most prominent one, III. the endogenous targets of HNE, primarily peptides and proteins  physiological consequences discussed), and IV. the metabolism of HNE leading to a great number of degradation products, some of which are excreted in urine and may serve as non-invasive biomarkers of oxidative stress. Table of ContentsPreface…………………………………………………………………………………………………………………………………22511. Lipid Peroxidation as a Free Radical Amplification Process…………………………………………………………………22522. Structure, Properties and Generation of HNE…………………………………………………………………………………22553. Major Reaction Mechanisms……………………………………………………………………………………………………2257 3.1. Reactions of the C=C Double Bond……………………………………………………………………………………………2257  3.1.1. Michael Additions…………………………………………………………………………………………………………2257  3.1.2. Reduction…………………………………………………………………………………………………………………2258  3.1.3. Epoxidation………………………………………………………………………………………………………………2259 3.2. Reactions of the Carbonyl Group……………………………………………………………………………………………2259  3.2.1. Acetal and Thio-Acetal Formation………………………………………………………………………………………2259  3.2.2. Schiff-Base Formation……………………………………………………………………………………………………2259  3.2.3. Oxidation…………………………………………………………………………………………………………………2259  3.2.4. Reduction…………………………………………………………………………………………………………………2260 3.3. Reactions of the Hydroxy Group………………………………………………………………………………………………22624. Biophysical Effects………………………………………………………………………………………………………………22625. Biochemical Targets of HNE……………………………………………………………………………………………………2262 5.1. Reactions with Peptides and Proteins…………………………………………………………………………………………2263  5.1.1. Substrates…………………………………………………………………………………………………………………2265  5.1.1.1. Glutathione………………………………………………………………………………………………………………2265  5.1.1.2. Carnosine………………………………………………………………………………………………………………2267  5.1.1.3. Thioredoxin……………………………………………………………………………………………………………2267  5.1.1.4. Cytochrome c……………………………………………………………………………………………………………2268  5.1.2. Enzymes…………………………………………………………………………………………………………………2268  5.1.2.1. Oxidoreductases…………………………………………………………………………………………………………2269  5.1.2.1.1. Lactate Dehydrogenase………………………………………………………………………………………………2269  5.1.2.1.2. Glyceraldehyde-3-Phosphate Dehydrogenase (GAPDH)…………………………………………………………2269  5.1.2.2. Transferases……………………………………………………………………………………………………………2270  5.1.2.2.1. Glutathione-S-Transferase (GST)……………………………………………………………………………………2270  5.1.2.2.2. Liver Kinase B1 (LKB1)………………………………………………………………………………………………2270  5.1.2.2.3. 5'-AMP-Activated Protein Kinase (AMPK)…………………………………………………………………………2271  5.1.2.2.4. ZAK Kinase (Sterile Alpha Motif and Leucine Zipper Containing Kinase AZK)…………………………………2271  5.1.2.2.5. Serine/Threonine-Protein Kinase AKT2 (Proteinkinase B2)………………………………………………………2271  5.1.2.3. Hydrolases………………………………………………………………………………………………………………2271  5.1.2.3.1. ATP Synthase…………………………………………………………………………………………………………2271  5.1.2.3.2. Phosphatase and Tensin Homolog Deleted on Chromosome 10 (PTEN)……………………………………………2272  5.1.2.3.3. Sirtuin 3 (SIRT3)……………………………………………………………………………………………………2272  5.1.2.3.4. Cathepsins……………………………………………………………………………………………………………2273  5.1.2.3.5. Neprilysin (NEP)……………………………………………………………………………………………………2273  5.1.2.4. Lyases……………………………………………………………………………………………………………………2273  5.1.2.4.1. Mitochondrial Aconitase (ACO2)……………………………………………………………………………………2273  5.1.2.4.2. α-Enolase………………………………………………………………………………………………………………2273  5.1.2.5. Isomerases………………………………………………………………………………………………………………2274  5.1.2.5.1. Protein Disulfide Isomerase (PDI)……………………………………………………………………………………2274Cis/Trans-Isomerase A1 (Pin1)………………………………………………………………………  5.1.2.5.2. Peptidyl-Prolyl 2274  5.1.2.6. Ligases: Glutamine Synthetase…………………………………………………………………………………………2274  5.1.3. Carriers……………………………………………………………………………………………………………………2274  5.1.3.1. Albumin…………………………………………………………………………………………………………………2274  5.1.3.2. Hemoglobin and Myoglobin……………………………………………………………………………………………2275  5.1.3.3. Liver and Adipocyte Fatty Acid-Binding Protein (FABP)……………………………………………………………2275  5.1.3.4. Apolipoprotein B-100 (ApoB)…………………………………………………………………………………………2275  5.1.3.5. β-Lactoglobulin…………………………………………………………………………………………………………2276  5.1.4. Transporters and Channels………………………………………………………………………………………………2276  5.1.4.1. Glutamate Transport Protein……………………………………………………………………………………………2276  5.1.4.2. α-Synuclein (α-Syn)……………………………………………………………………………………………………22762+-ATPase (SERCA1a)……………………………………………………………  5.1.4.3. Sarco/Endoplasmic Reticulum Ca2277  5.1.4.4. Transient Receptor Potential Vanilloid 1 (TRPV1)……………………………………………………………………2277  5.1.4.5. Dopamine Transporter…………………………………………………………………………………………………2278  5.1.5. Receptors…………………………………………………………………………………………………………………2278  5.1.5.1. Platelet-Derived Growth Factor Receptor-β (PDGFR-β)………………………………………………………………2278  5.1.5.2. Lectin-Like Oxidized Low-Density Lipoprotein Receptor-1 (LOX-1)………………………………………………2278  5.1.5.3. Toll-Like Receptor 4 (TLR4)……………………………………………………………………………………………2278  5.1.6. Cytoskeletal Proteins………………………………………………………………………………………………………2279  5.1.6.1. Tau Proteins……………………………………………………………………………………………………………2279  5.1.6.2. Ankyrin…………………………………………………………………………………………………………………2279  5.1.6.3. Spectrins…………………………………………………………………………………………………………………2280  5.1.7. Chaperones: Heat Shock Proteins 70 and 90……………………………………………………………………………2280  5.1.8. Uncoupling Proteins 2 and 3 (UCP2 and UCP3)…………………………………………………………………………2282  5.1.9. Growth Factors: Platelet-Derived Growth Factor (PDGF)………………………………………………………………2283  5.1.10. Peptide Hormones………………………………………………………………………………………………………2283  5.1.10.1. Insulin…………………………………………………………………………………………………………………2283  5.1.10.2. Angiotensin II…………………………………………………………………………………………………………2283  5.1.11. Extracellular Matrix Proteins: Collagen…………………………………………………………………………………2283  5.1.12. Histones: Histone-H2A…………………………………………………………………………………………………2284 5.2. Reactions with Lipids…………………………………………………………………………………………………………2284 5.3. Reactions with Cofactors and Vitamins………………………………………………………………………………………2284  5.3.1. Vitamin C (Ascorbic Acid)………………………………………………………………………………………………2284  5.3.2. Pyridoxamine……………………………………………………………………………………………………………2285  5.3.3. Lipoic Acid………………………………………………………………………………………………………………2285 5.4. Reactions with Nucleic Acids…………………………………………………………………………………………………22856. Formation of HNE in Mammalian Cells and Tissues…………………………………………………………………………2287 6.1. HNE Formation in Cellular and Organ Systems………………………………………………………………………………2287 6.2. HNE in the Whole Healthy Organism…………………………………………………………………………………………2289 6.3. Influence of Nutrition…………………………………………………………………………………………………………22907. Metabolism of HNE………………………………………………………………………………………………………………2291 7.1. HNE Metabolism in Mammalian Cells and Organs…………………………………………………………………………2293 7.2. HNE Metabolism in Subcellular Organelles………………………………………………………………………………… 2294 7.3. HNE Metabolism in Whole Animals and Interorgan Relationships…………………………………………………………2295 7.4. Primary HNE Intermediates—Enzymatic Reactions and Quantitative Results………………………………………………2295 7.5. Secondary HNE Intermediates—Enzymatic Reactions and Quantitative Results……………………………………………2301 7.6. HNE Metabolism as a Component of the Antioxidative Defense System……………………………………………………2306 7.7. HNE Intermediates as Potential Biomarkers of LPO…………………………………………………………………………2307 7.8. Further Medical Applications of HNE Metabolism…………………………………………………………………………23078. Conclusions………………………………………………………………………………………………………………………2309Conflicts of Interest…………………………………………………………………………………………………………………2309Abbreviations………………………………………………………………………………………………………………………2309References……………………………………………………………………………………………………………………………2313Tables and FiguresTable 1. HNE concentrations in cells, tissues and organs……………………………………………………………………………2289Table 2. Maximal velocity of total HNE degradation in cells, subcellular organelles, and perfused organs…………………………………………………………………………………………………………………………………2292Table 3. Primary HNE metabolites in different cells and tissues after the addition of 100 μM HNE to the biological system…………………………………………………………………………………………………………………………………2301Figure 1. Idealized representation of the initiation and propagation reactions of lipid peroxidation2253Figure 2. Formation of lipid hydroperoxides and cyclic peroxides from arachidonic acid.…………………………………………2254Figure 3. Chemical structure of 4-hydroxy-2-trans-nonenal (HNE)…………………………………………………………………2255Figure 4. Overview of the reactions of 4-hydroxy-nonenal with different biomolecules……………………………………………2257Figure 5. HNE plasma concentration in dependence on age of the blood donor (5 to 90 years)……………………………………2288Figure 6. Degradation/metabolism of 4-HNE in rat hepatocytes……………………………………………………………………2292Figure 7. Identification of HNE and 4-hydroxynonenoic acid (HNA) by isocratic HPLC separation………………………………2297Figure 8. Mass spectrum of dihydroxynonene urethane (HPLC plus MS) with fluorimetric detection……………………………2298Figure 9. HNE metabolites.……………………………………………………………………………………………………………2299Figure 10. HPLC chromatogram of the isoindol derivative of the HNE-GSH conjugate ………………………………………………………………………………………………………………………2306Due to its diverse biochemical reactions and far-reaching biological effects 4-hydroxy-nonenal (HNE) is a fascinating compound, the discovery of which is hidden deep in the scientific literature. When Hermann Esterbauer isolated this aldehyde for the first time a quantitative analysis of its chemical elements was made resulting in values in between those for 4-hydroxy-octenal and 4-hydroxy-nonenal . Since these values were closer to those of 4-hydroxy-octenal Esterbauer and his supervisor Schauenstein published it with this chemical formula  to produce phosphatidylinositol-4,5-bisphosphate.473 and Thr308 -HNE was used ,297,298. minutes ,302,303 Originally it was assumed that HNE metabolism is especially fast in hepatocytes ,305,306 + and lactate/pyruvate ratio of resting cells. The consequence of the high demand of reduced NADH by partially uncoupled mitochondria for ATP production is a lower capacity of resting cells to use NADH for HNE reduction (DHN or GSH-DHN formation). In support, addition of NADH to the homogenates of rat hepatoma cells (MH1C1) resulted in a 1.5-fold increase of aldehyde consumption HNE into rats, the majority of the dose appeared in urine (67.1% after 48 h) HNE to rats, besides DHN-MA 15 polar urinary metabolites accounting for about 50% of the urinary radioactivity were separated by HPLC by the same group HNE. Five metabolites were present in the bile, two of them corresponded to GSH-HNE and to GSH-DHN adducts. Two further metabolites were identified as DHN and HNA-lactone mercapturic acid conjugates, whereas the fifth metabolite in the bile remained unidentified . The kinet al., 2005 [in vitro studies using rat liver cytosolic incubations and HNE-glutathione conjugates as substrate and found large differences in the reduction and retro-Michael conversion steps of the metabolism between the conjugates originating from the two enantiomers [The metabolism of the two enantiomers of HNE, the (R)- and the (S)-enantiomer of its racemic mixture, was the aim of a study by Gueraud l., 2005 . The patntiomers . Studiesntiomers  , and the formation of the products was simply calculated using the radioactivity of the spot or peak and dividing it by the specific radioactivity of the added HNE which can be taken as constant taking into account the overshoot of added radioactive HNE in comparison with the “low” formation of new HNE. After addition of labelled HNE aliquots of suspensions were taken. The HNE metabolizing reactions were stopped by the addition of e.g., 0.5 mL suspension to an equal volume of acetonitrile/acetic acid mixture . After centrifugation two aliquots of the supernatant were eluted on TLC plates. The elution with hexane/diethylether  allows the separation of HNE, HNA, and DHN. The elution with butanol/acetic acid/water  allows the determination of GSH-adducts of HNE . For quaSometimes acetonitrile/acetic acid extracts were used for the determination of HNE and HNA by an HPLC-method ,293. TheThe GSH-HNE-adduct was measured as an isoindol derivative after the reaction of the adduct existing in neutralized perchloric acid extracts with o-phthaldialdehyde in the presence of mercaptoethanol .3H-labeled HNE [The reaction is started by addition of a sample aliquot to the o-phthaldiadehyde solution. The reaction mixture is transferred into a Hamilton syringe and the reaction is stopped by injection of the mixture into the HPLC system after 2 min. HPLC separation is carried out in an isocratic manner with 65% methanol-0.1 M sodium acetate buffer, pH 6.5 (v/v) in the presence of an ion-pair reagent (tetraethylammoniumhydroxide). Isoindol is detected by means of a fluorescence detector at wavelengths of 345 nm and 445 nm for excitation and emission, respectively. The formation of enantiomer products in the reaction of GSH-HNE adduct with o-phthaldialdehyde  led to teled HNE ,297.et al., 1998 was combined with different detection methods: electrospray ionization MS for the detection of HNE-GSH and DHN-GSH, GC-chemical ionization MS for HNA [Different HPLC separations were applied in tracer kinetic experiments and in measurements of non-labelled HNE intermediates ,312,319.Various enzyme inhibitors were useful in studies on HNE metabolism. In investigations of HNE metabolic pathways of vascular smooth muscle cells it was shown that the formation of HNA was inhibited by cyanamide; indicating that the acid is derived from an ALDH-catalyzed pathway; while the overall rate of HNE metabolism was insensitive to inhibition of aldose reductase or aldehyde dehydrogenase. Nevetheless; inhibition of HNA formation by cyanamide led to a corresponding increase in the fraction of HNE metabolized by the GSH-linked pathway; indicating that ALDH-catalyzed oxidation competes with glutathione conjugation .etc.The catabolism of HNE may be regarded as a very important part of the antioxidative defense system of cells and organisms. This is due to the fact, that HNE like other aldehydic products of LPO is able to exert cytotoxic, mutagenic, signal, and carcinogenic effects as described above. The degradation of HNE to less toxic intermediates diminished the reactions between HNE and biomolecules and therefore effectively contributes to the antioxidative protection of cells and organisms. This protective effect is the more effective the more rapidly the metabolism of HNE leads to stable HNE products, which can be easily excreted. The primary importance of HNE-degrading pathways which function both at physiological and pathophysiological HNE levels, seems to be the protection of proteins from modification by aldehydic LPO products. This should especially be valid in regions with high HNE formation rate such as in postischemic or reperfused tissues or in synovial tissue of joints during rheumatoid arthritis and other inflammatory diseases or in lymphedematous tissue, et al., 2005 favor the 1,4-dihydroxynonene mercapturic acid (DHN-MA) [3per os and LPO was estimated every day for a 4-day-period. All three parameters significantly increased in response to oxidative stress. DHN-MA showed the highest increase during the 24–48 h period after treatment.Two groups of HNE derivatives seem to be useful as possible biomarkers in clinical chemistry and laboratory medicine. The first are the mercapturic acid conjugates including the derivatives of HNE, HNA, and DHN. These conjugates are excreted with the urine and, therefore can be analyzed from urine. The second are HNE-protein adducts of blood serum or plasma. The HNE-modified plasma proteins are already used for evaluation of rheumatic and autoimmune diseases . Both bo(DHN-MA) . The autet al., 2006 applying the specific and very sensitive method of electrospray ionization triple quadrupole mass spectrometry (ESI-MS-MS) [The quantitation of the glutathione conjugate of HNE (HNE-GSH) in the human post-mortem brain was reported by Volkel I-MS-MS) . Levels I-MS-MS) ,9,350.Several studies indicate that the glutathione conjugates of HNE metabolism are mediators of cell signalling and growth. A modification of the metabolic pattern, possibly by pharmacological interventions, could be of therapeutical importance . Additioet al., 2006 also demonstrated a contribution of aldose reductase to diabetic hyperproliferation of vascular smooth muscle cells [The demonstration that aldose reductase mediates endotoxin-induced inflammation and cardiomyopathy suggests that inhibition of this enzyme may be useful to attenuate maladaptive host responses and to treat acute cardiovascular dysfunction associated with endotoxic shock ,352. Srile cells . The autet al. [Tammali et al.  showed tet al. .Another oncological application of modulation of HNE metabolism was suggested in relation to the inhibition of cytosolic class 3 aldehyde dehydrogenase (ALDH3) by antisense oligonucleotides in hepatoma cells . SeveralN-acetylcysteine (NAC) is a pharmaceutical drug and nutritional supplement used primarily as a mucolytic agent and in the management of paracetamol (acetaminophen) overdose. It was used in various investigations to attenuate HNE-mediated organ damage, such as in alcoholic liver damage [The carbonyl scavenger r damage . In thesr damage . An impor damage .Drosophila) slows the age-related and the oxidative damage-related HNE accumulation [Further studies showed that caloric restriction in mammals and in flies (mulation . Additiomulation . Additiomulation , and a dmulation .This review tries to demonstrate that there is a multitude of possible targets of HNE within a cell. Several factors can be envisaged which determine the actual reactants within a cell: The concentration of HNE which can differentially modulate cell death, growth and differentiation , the con