This notion was introduced independently by Kadison   d on an  Sourour  proved t𝒜 is a local derivation. But the converse is in general not true. Kadison [Of course, every derivation on an algebra  Kadison  construc𝒜 which is generated by its idempotents into any 𝒜-bimodule is a derivation. This motivated us to study local generalized -derivations on algebras generated by their idempotents.In the last few decades a lot of work has been done on local mappings on some algebras. The results show that, in many important cases, local mappings of some certain class of transformations on a given algebra are global (see ). In the𝒜 will be an algebra and ℳ will be an 𝒜-bimodule. A linear mapping d : 𝒜 → ℳ is called a derivation ifg : 𝒜 → ℳ is a generalized derivation if there exists a derivation d : 𝒜 → ℳ such thatg is a generalized derivation associated with a derivation d . On the other hand, Nakajima [g : 𝒜 → ℳ be a linear mapping and m an element of ℳ. A pair  is called a generalized derivation if𝒜 has a unit 1 ∈ 𝒜, then m = −g(1) and g is a generalized derivation if it satisfiesg : 𝒜 → ℳ is a generalized derivation if and only if it satisfies the above condition. We refer the readers to [Before continuing, let us fix the notation and write some basic definitions which we will need in our further investigation. Throughout the paper, ,b∈𝒜.In , Brešar Nakajima  defined aders to , where tα, β be endomorphisms of a unital algebra 𝒜 and I𝒜 the identity map on 𝒜. Motivated by the above notions, we define -derivations as linear mappings d : 𝒜 → ℳ satisfyingg : 𝒜 → ℳ is called a generalized -derivation ifI𝒜, I𝒜)-derivation -derivation, resp.) is just a derivation . The next example will show that there exist -derivations which are not derivations.Let 𝒜 be an algebra with a nontrivial central idempotent e. Let us define d(a) = ea, α(a) = a − ea for all a ∈ 𝒜, and β = I𝒜. Then d is an -derivation which is not a derivation since d(ee) = eee ≠ 2eee = (ee)e + e(ee) = d(e)e  +  ed(e). Moreover, if 𝒜 is a semiprime algebra and α ≠ I𝒜 is an endomorphism of 𝒜, then d = I𝒜 − α is an -derivation but not a derivation.Let α, β)-derivations -derivations, resp.) can be self-explanatory, A linear mapping g : 𝒜 → ℳ is called a local generalized -derivation -derivation, resp.) if for every a ∈ 𝒜 there exists a generalized -derivation -derivation, resp.) ga : 𝒜 → ℳ depending on a such that g(a) = ga(a).The definition of local generalized )g(eaf)(1 − α(f)) = 0,(1 − g(eaf) = g(ea)α(f) + β(e)g(af) − β(e)g(a)α(f). Let a ∈ 𝒜 and all idempotents e, f ∈ 𝒜. Thena ∈ 𝒜 and all idempotents e, f ∈ 𝒜. Therefore,a ∈ 𝒜 and all idempotents e, f ∈ 𝒜.Obviously, (ii) implies (i). So, assume that (i) holds for every g is a linear mapping from an algebra 𝒜 into an 𝒜-bimodule ℳ such that g(eaf) = g(ea)α(f) + β(e)g(af) − β(e)g(a)α(f) for every a ∈ 𝒜 and all idempotents e, f ∈ 𝒜, thena ∈ 𝒜 and all idempotents e1,…, em, f1,…, fn ∈ 𝒜.If n, we first prove that for every a ∈ 𝒜 and all idempotents e, f1,…, fn ∈ 𝒜,n = 1 is clear. Assume now that  = ℐ, β(ℐ) = ℐ, and let g : 𝒜 → ℳ be a linear mapping. If for every a ∈ 𝒜 and all idempotents e, f ∈ 𝒜,g is a generalized -derivation. In particular, if g(1) = 0, then g is an -derivation.Let The idea of the proof comes from [p, q ∈ ℐ be arbitrary elements. Then, according to  = ℐ, it follows thata ∈ 𝒜. Furthermore, if a, b ∈ 𝒜 and q ∈ ℐ, thena, b ∈ 𝒜. If g(1) = 0, then g is, obviously, an -derivation.Let rding to  and Lemmhand, by , we haveg be a local generalized -derivation from an algebra 𝒜 into an 𝒜-bimodule ℳ. Suppose that a ∈ 𝒜 is an arbitrary element and suppose that e, f ∈ 𝒜 are idempotents. Then there exists a generalized -derivation geaf : 𝒜 → ℳ such that g(eaf) = geaf(eaf). It is also easy to see thata ∈ 𝒜 and all idempotents e, f ∈ 𝒜. Thus, by g satisfies the condition  = ℐ, β(ℐ) = ℐ. Then every local generalized -derivation -derivation, resp.) from an algebra 𝒜 into an 𝒜-bimodule ℳ is a generalized -derivation -derivation, resp.).Let α, β = I𝒜, we have the next direct consequence of Taking ℐ be a separating set of an 𝒜-bimodule ℳ contained in the algebra generated by all idempotents in 𝒜. Then every local generalized derivation  from an algebra 𝒜 into an 𝒜-bimodule ℳ is a generalized derivation .Let α = I𝒜, then we have the next result for local generalized skew derivations, that is; linear mappings g : 𝒜 → ℳ with the propertyAt the end, if ℐ be a separating set of an 𝒜-bimodule ℳ contained in the algebra generated by all idempotents in 𝒜 and let β be an endomorphism of 𝒜 such that β(ℐ) = ℐ. Then every local generalized skew derivation  from an algebra 𝒜 into an 𝒜-bimodule ℳ is a generalized skew derivation .Let 𝒜 is a unital algebra and n ≥ 2, a positive integer, then Mn(𝒜), that is, the algebra of all n × n matrices over 𝒜, belongs to this class . Th. Th𝒜 is α, β be automorphisms of Mn(𝒜). Then every local generalized -derivation -derivation, resp.) from an algebra Mn(𝒜) into any Mn(𝒜)-bimodule is a generalized -derivation -derivation, resp.).Let ℬ is called a local matrix algebra if any finite subset of ℬ can be embedded in a subalgebra which is a matrix algebra Mn(𝒜), n ≥ 2.The next result involves local matrix algebras: an algebra α, β be automorphisms of 𝒜. If, for any a1, a2 ∈ 𝒜, there exists a unital subalgebra ℬ of 𝒜 which contains a1, a2 and is isomorphic to a matrix algebra, then every local generalized -derivation -derivation, resp.) from an algebra 𝒜 into any 𝒜-bimodule is a generalized -derivation -derivation, resp.).Let X and Y be complex Hausdorff topological linear spaces and let ℬ be the algebra of all continuous linear mappings from X into Y. We say that a subset 𝒮 of ℬ is reflexive if T ∈ 𝒮 whenever T ∈ ℬ and x ∈ X, where 𝒮x. By a subspace lattice on X we mean a collection ℒ of closed subspaces of X containing {0} and X such that, for each family {Lϵ} of elements of ℒ, both ⋂Lϵ and ⋁Lϵ belong to ℒ, where ⋁ denotes the closed linear span of {Lϵ}. If ℒ is a subspace lattice, then we denote the algebra of all operators on X, that leave invariant each element of ℒ by algℒ. A totally ordered subspace lattice 𝒩 is called a nest and the associated reflexive algebra alg𝒩 is called a nest algebra.Let α, β)-derivations on a reflexive subalgebra in a factor von Neumann algebra. The proof of the following corollaries uses Now we consider local generalized -derivation -derivation, resp.) from ℳ∩algℒ into ℳ is a generalized -derivation -derivation, resp.).Let 𝒩 be a nest in a factor von Neumann algebra ℳ on H and let α, β be automorphisms of ℳ∩alg𝒩. Then every local generalized -derivation -derivation, resp.) from ℳ∩alg𝒩 into ℳ is a generalized -derivation -derivation, resp.).Let 𝒜 is topologically generated by its idempotents  and suppose that g : 𝒜 → ℳ is a continuous local generalized -derivation, where α, β are automorphisms of 𝒜. Let a = ∑i=1mλi∏j=1tieji)( for some idempotents eji) = g(a)α(b) + β(a)g(b) − α(a)g(1)β(b) and since g is continuous and 𝒜 is generated by its idempotents, we have the following proposition.Suppose that α, β be automorphisms of 𝒜. If 𝒜 is topologically generated by its idempotents and ℳ is a topological 𝒜-bimodule, then every continuous local generalized -derivation -derivation, resp.) from 𝒜 into ℳ is a generalized -derivation -derivation, resp.).Let 𝒩 is a nest in a von Neumann algebra ℳ and 𝒜 = ℳ∩alg𝒩, then the linear span of all idempotents in 𝒜 is w*-dense in 𝒜 -derivation -derivation) from 𝒜 into ℳ is a generalized -derivation -derivation).Let α, β)-derivations from 𝒜 into ℳ, denoted by GDerα,β, is reflexive.At the end, let us prove that the set of all continuous generalized  is reflexive.Let g : 𝒜 → ℳ be a continuous linear mapping such that for any x ∈ 𝒜,e, f ∈ 𝒜 be arbitrary idempotents, a any element from 𝒜, and x = eaf. Then, according to above observations, there exists a sequence {gn}n=1∞ ⊂ GDerα,β such thatg is a generalized -derivation; that is, g ∈ GDerα,β. The proof is completed.Let


In this paper, we introduce a new iterative scheme that converges strongly to a common fixed point of a countable family of strictly pseudo-contractive mappings in a real Hilbert space which is also a solution of variational inequality problem related to quadratic minimization problems. Our results extend ones of Yao et al. , Gu et al.  and some authors. H is a real Hilbert space with inner product and norm denoted by 〈·,·〉 and ∥·∥, respectively and let C is a nonempty closed and convex subset of H. A mapping f:C→H is called a contraction on C if there exists a constant ρ ∈ , such thatwhere Therefore, fixed point algorithms can be applied to solve variational inequalities.hierarchical fixed point problem: Find x∗∈F(T) such thatThe following problem is called a S:C→H be a mapping. It is known that the hierarchical fixed point problem (5) links with some monotone variational inequalities and convex programming problems; see Gu et al. , Moudafi  intoduceS,T:C→C are two nonexpansive mappings, {αn} and {βn} are two sequences in . Then he showed that {xn} converges weakly to a fixed point of T which is a solution of problem (5). For obtaining a strong convergence result, in Mainge and Moudafi . Then they showed that {xn} converges strongly to a fixed point of T which is a solution of problem (5).where On the other hand, Cianciaruso et al.  introducf:C→C is a contraction mapping, S and T:C→C are two nonexpansive mappings, {αn} and {βn} are two sequences in . Under some certain restrictions on parameters, the authors proved the sequence {xn} generated by (8) converges strongly to x∗∈F(T), which is a unique solution of the following variational inequality:where xn} generated by (8) converges strongly to x∗∈F(T), which is a unique solution of the following variational inequality:By changing the restrictions on parameters, the authors obtained another result on the iterative scheme (8), i.e., the sequence {τ ∈  is a constant.where f from C to H by using the metric projection of H onto C. They introduced the following iterative scheme:In 2010, Yao et al. . The authors proved the sequence {xn} generated by (11) converges strongly to x∗ ∈ F(T), which is a unique solution of one of the variational inequalities (9) and (10).where In 2011, Gu et al. . The authors proved the sequence {xn} generated by (12) converges strongly to x∗ ∈ F(T), which is a unique solution of one of the variational inequalities (9) and (10).where In this paper, motivated and inspired by the results of Gu et al. , we intrVi=kiI+(1−ki)Ti and ki-strict pseudo-contraction mappings. Under some certain condition on parameters, we first prove that the sequence {xn} generated by (13) converges strongly to where xn} generated by (13) converges strongly to By changing the restrictions on parameters, we also prove that the sequence {τ ∈  is a constant. It is easy to see that, if ki=0 for each i≥1, then our algorithm (13) is reduced to algorithm (12) of Gu et al. Also our results extend the corresponding one of Yao et al. → for strong convergence and ⇀ for weak convergence.(ii)ω-limit set of {xn}.We will use the following notation: H be a Hilbert space, C is a closed convex subset of H and T:C→C be a nonexpansive mapping with F(T)≠∅. If {xn} is a sequence in C weakly converging to x and if {(I−T)xn} converges strongly to y, then (I−T)x=y; in particular, if y=0 then x ∈ F(T).Browder  Let H beC be a nonempty closed convex subset of a real Hilbert space H. If T:C→C is a k-strict pseudo-contraction, then the mapping I−T is demiclosed at 0. That is, if {xn} is a sequence in C weakly converging to x and {(I−T)xn} converges strongly to 0, then (I−T)x=0.Acedo and Xu  Let C bex ∈ H and z ∈ C be any points. Then we have the following: z=PC[ x] if and only if there holds the relation:That z=PC[ x] if and only if there holds the relation:That There holds the relation:Let PC is nonexpansive and monotone.Consequently, H be a Hilbert space, C be a closed convex subset of H, f:C→H be a contraction with coefficient 0 < ρ < 1 and T : C → C be a nonexpansive mapping. Then, for 0 < γ < γ̄/ρ, for x,y∈C, I−f) is strongly monotone with coefficient (1−ρ) that isthe mapping (I−T) is monotone, that isthe mapping  and {δn} is a sequence in where {or n→∞an=0.Then limC be a closed convex subset of H. Let {xn} be a bounded sequence in H. Assume that ω-limit set ωw(xn)⊂C,The weak z ∈ C, limn→∞∥xn−z∥ exists.For each Acedo and Xu  Let C bexn} is weakly convergent to a point in C.Then {H be a real Hilbert space, C be a closed and convex subset of H, and T be a k-strict pseudo-contraction mapping on C, then F(T) is closed convex, so that the projection PF(T) is well defined.Zhou  Let H beH be a Hilbert space, C be a closed and convex subset of H, and T:C→H be a k-strict pseudo-contraction mapping. Define a mapping V:C→H by Vx=λx+(1−λ)Tx for all x ∈ C. Then, as λ ∈ /αnβn=0;● (H8) limK > 0 such that● ● (H9) there exists a constant where xn} and {yn} are bounded.Assume that (H1) holds. Then {Let So, by induction, one can obtain thatxn} is bounded. Of course {yn} is bounded too. □Hence {(i)xn} is asymptotically regular, that is,{(ii)the weak cluster points set Suppose that (H1) and (H3) hold. Also, assume that either (H4) and (H5) hold, or (H6) and (H7) hold. Then PC is a nonexpansive mapping, we haveSet yn one obtain thatBy definition of So, substituting (22) in (21), we obtainBy Proposition 1, we saySo, we haveSo, if (H4) and (H5) hold, we obtain the asymptotic regularity by Lemma 5, if instead, (H6) and (H7) hold, from (H1), we can writeBy Lemma 5, we obtain the asymptotics regularity.Vixn ∈ C for each i ≥ 1 and In order to prove (2), since Now, fixing It follows that Now, from Lemma 9 and (27), we get βn→0, as n→∞, so thatBy (H1) and (H3), it follows that i≥1 and {αn} is strictly decreasing, one hasSince Hence, we obtainxn} is asymptotically regular and demiclosedness principle, we obtain the proposition. □Since {(i)n→∞∥xn−yn∥=0;lim(ii)n→∞∥xn−Viyn∥=0, ∀i≥1;lim(iii)n→∞∥yn−Viyn∥=0, ∀i≥1.limSuppose that the hypotheses of Proposition 2 hold. Then i), we can observe thatTo prove .To prove (ii), we observe thatSince andyn−xn∥→0 and ∥xn−Vixn∥→0 as n→∞, ∀i≥1, then ∥yn−Vixn∥→0, that is, we obtain (ii). To prove (iii), we can observe thatSince ∥i) and (ii), we obtain (iii). □By  and {αn} is a strictly decreasing sequence, Vi=kiI+(1−ki)Ti, {βn}⊂ and {αn} and {βn} are sequences satisfying the conditions (H2) with τ=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {xn} converges strongly to a point where {First of all, since xn} is bounded. Moreover, since either (H4) and (H5) or (H6) and (H7) then {xn} is asymptotically regular. Similarly, by Proposition 2, the weak cluster points set of xn, that is, ωw(xn), is a subset of Since (H2) implies (H1), thus {xn} such thatLet and Set By Lemma 3(1), we haveFrom (32) and (33), it follows thatxn+1 − z∥2∥n≥1. SinceLet xn→z as n→∞. □Hence, by Lemma 5, we conclude that f≡0, then we get In the iterative scheme (30), if we set That isz is the unique solution to the following quadratic minimization problem:Therefore, the point By changing the restrictions on parameters in Theorem 1, we obtain the following results.C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→C be a nonexpansive mapping and ki-strict pseudo-contraction mappings and α0 = 1, and x1 ∈ C and define the sequence {xn} byLet αn}⊂ and {αn} is a strictly decreasing sequence, Vi=kiI+(1−ki)Ti, {βn}⊂ and {αn} and {βn} are sequences satisfying the conditions (H2) with τ ∈ , (H3), (H8) and (H9). Then the sequence {xn} converges strongly to a point where {x′ and x∗ be two solutions. Then, since x′ is solution, for y=x∗ one hasFirst, we shows that (49) has the unique solution. Let andAdding (36) and (37), we obtainx′=x∗. Also now the condition (H2) with 0<τ<∞ implies (H1) so the sequence {xn} is bounded. Moreover, since (H8) implies (H6) and (H7), then {xn} is asymptotically regular. Similarly, by Proposition 2, the weak cluster points set of xn, i.e., ωw(xn), is a subset of so From (20)-(24), we observe thatγn=(1−ρ)αn and Let By Lemma 5, we obtainFrom (34), we haveIt follows thatLet By Lemma 4, we haveandBy Lemma 3(1), we obtainNow, from (38)-(42), it follows thatvn→0 and (I−Vi)yn→0, as n→∞, then every weak cluster point of {xn} is also a strong cluster point. By Proposition 2, {xn} is bounded, thus there exists a subsequence x∗. For all since vn→0,(I−Vi)yn→0 for all i≥1, and ∥un−un−1∥/αn→0, letting k→∞ in (44), we obtainSince ωw(xn)={x∗}. Since every weak cluster point of {xn} is also a strong cluster point, we conclude that xn→x∗ as n→∞. This completes the proof.Since (49) has the unique solution, it follows that Ti=T, for all i≥1, where T:C→C is a k-strict pseudo-contraction mapping in Theorem 1, then we get the following result: □If we take C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→H be a nonexpansive mapping and T:C→C be a k-strict pseudo-contraction mapping such that F(T)≠∅. Let x1 ∈ C and define the sequence {xn} byLet V=kI+(1−k)T,{αn}⊂ and {βn}⊂ are sequences satisfying the conditions (H2) with τ=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {xn} converges strongly to a point z ∈ F(T), which is the unique solution of the variational inequality:where ki=0, for all i≥1 in Theorem 1, then we get the following result:Taking C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→H be a nonexpansive mapping and α0 = 1, x1 ∈ C and define the sequence {xn} byLet αn}⊂ and {αn} is a strictly decreasing sequence, {βn}⊂ and {αn} and {βn} are sequences satisfying the conditions (H2) with τ=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {xn} converges strongly to a point where {k=0 in Corollary 2, then we get the following result:If we take C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→H be a nonexpansive mapping and T:C→C be a nonexpansive mapping such that F(T)≠∅. Let x1∈C and define the sequence {xn} byLet αn}⊂,{βn}⊂ and {αn} and {βn} are sequences satisfying the conditions (H2) with τ=0, (H3), either (H4) and (H5), or (H6) and (H7). Then the sequence {xn} converges strongly to a point z ∈ F(T), which is the unique solution of the variational inequality:where {Ti=T, for all i≥1, where T:C→C is a k-strict pseudo-contraction mapping in Theorem 2, then we obtain the following result:If we take C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→C be a nonexpansive mapping and T:C→C be a k-strict pseudo-contraction mapping and x1∈C and define the sequence {xn} byLet V=kI+(1−k)T, {αn}⊂, {βn}⊂ and {αn} and {βn} are sequences satisfying the conditions (H2) with τ ∈ , (H3), (H8) and (H9). Then the sequence {xn} converges strongly to a point where ki=0, for all i≥1 in Theorem 2, then we get the following result:If we take C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S:C→C be a nonexpansive mapping and α0 = 1, x1 ∈ C and define the sequence {xn} byLet αn}⊂ and {αn} is a strictly decreasing sequence, {βn}⊂ and {αn} and {βn} are sequences satisfying the conditions (H2) with τ ∈ , (H3), (H8) and (H9). Then the sequence {xn} converges strongly to a point where {k=0 in Corollary 5, then we get the following Corollary:If C be a nonempty closed and convex subset of a real Hilbert space H. Let f:C→H be a ρ-contraction mapping, S,T:C→C be nonexpansive mappings and x1∈C and define the sequence {xn} byLet αn}⊂, {βn}⊂ and {αn} and {βn} are sequences satisfying the conditions (H2) with τ ∈ , (H3), (H8) and (H9). Then the sequence {xn} converges strongly to a point where {αn=nθ− and βn=nω− . Since |αn−αn−1|≈nθ− and |βn−βn−1|≈nω−, it is not difficult to prove that (H8) is satisfied for 0<θ,ω<1and (H9) is satisfied if θ+ω≤1.Prototypes for the iterative parameters are, for example, Theorem 1 and Theorem 2 extend and improve the result of Gu et al.  from the


Then we introduce bifuzzy soft Lie subalgebras and investigate some of their properties.We introduce the concept of ( The concept of Lie groups was first introduced by Sophus Lie in nineteenth century through his studies in geometry and integration methods for differential equations. Lie algebras were also discovered by him when he attempted to classify certain smooth subgroups of a general linear group. The importance of Lie algebras in mathematics and physics has become increasingly evident in recent years. In applied mathematics, Lie theory remains a powerful tool for studying differential equations, special functions, and perturbation theory. It is noted that Lie theory has applications not only in mathematics and physics but also in diverse fields such as continuum mechanics, cosmology, and life sciences. A Lie algebra has nowadays even been applied by electrical engineers in solving problems in mobile robot control .bifuzzy sets. The elements of the bifuzzy sets are featured by an additional degree which is called the degree of uncertainty. This kind of fuzzy sets have now gained a wide recognition as a useful tool in the modeling of some uncertain phenomena. Bifuzzy sets have drawn the attention of many researchers in the last decades. This is mainly due to the fact that bifuzzy sets are consistent with human behavior, by reflecting and modeling the hesitancy present in real-life situations. In fact, the fuzzy sets give the degree of membership of an element in a given set, while bifuzzy sets give both a degree of membership and a degree of nonmembership. As for fuzzy sets, the degree of membership is a real number between 0 and 1. This is also the case for the degree of nonmembership, and furthermore the sum of these two degrees is not greater than 1.After introducing the concept of fuzzy sets by Zadeh  in 1965,γ, δ)-bifuzzy Lie algebra, and discuss some of its properties. Then we introduce bifuzzy soft Lie algebras and investigate some of their properties. We use standard definitions and terminologies in this paper.In 1999, Molodtsov  initiateIn this section, we review some known basic concepts that are necessary for this paper.Lie algebra is a vector space L over a field F  on which L × L → L denoted by → is defined satisfying the following axioms:(L1)x, y] is bilinear, = 0 for all x ∈ L,, z] + , x] + , y] = 0 for all x, y, z ∈ L (Jacobi identity)., z] = ]. But it is anticommutative; that is,  = −. A subspace H of L closed under  will be called a Lie subalgebra.Throughout this paper, μ be a fuzzy set on X; that is, a map μ : X → . As an important generalization of the notion of fuzzy sets in X, Atanassov introduced the concept of a bifuzzy set defined on a nonempty set X as objects having the formμ : X →  and ν : X →  denote the degree of membership ) and the degree of nonmembership ) of each element x ∈ X to the set A, respectively, and 0 ≤ μA(x) + νA(x) ≤ 1 for all x ∈ X.Let A =  be a bifuzzy set on X and let s, t ∈  be such that s + t ≤ 1. Then the sets, t)-level subset of A. As,t) on L is called a bifuzzy Lie subalgebra if the following conditions are satisfied: μA(x + y) ≥ min{μA(x), μA(y)},νA(x + y) ≤ max{νA(x), νA(y)},μA(αx) ≥ μA(x), νA(αx) ≤ νA(x),μ ≥ max{μ(x), μ(y)},ν ≤ min{ν(x), ν(y)}A bifuzzy set x, y ∈ L and α ∈ F.for all c be a point in a nonempty set X. If γ ∈  are two real numbers such that 0 ≤ γ + δ ≤ 1, then the bifuzzy set c = 〈x, cγ, 1 − cδ1−〉 is called a bifuzzy point (BP for short) in X, where γ  is the degree of membership  of c and c ∈ X is the support of c. Let c be a BP in X and let A = 〈x, μA, νA〉 be a bifuzzy set in X. Then c is said to belong to A, written c ∈ A, if μA(c) ≥ γ and νA(c) ≤ δ. We say that c is quasicoincident with A, written cqA, if μA(c) + γ > 1 and νA(c) + δ < 1. To say that c∈∨qA ∈∧qA) means that c ∈ A or cqA  ∈ A and cqA) and c∈∨qA does not hold.Let f and g be any two bifuzzy subsets of L. Then the sum f + g is a bifuzzy subset of L defined byLet U) denote the family of all bifuzzy sets in U.Let IF is called a bifuzzy soft set over U, where f is a mapping given by f : A → IF(U). A bifuzzy soft set is a parameterized family of bifuzzy subsets of U. For any ɛ ∈ A, fɛ is referred to as the set of ɛ-approximate elements of the bifuzzy soft set , which is actually a bifuzzy set on U and can beμfɛ(x) and νfɛ(x) are the membership degree and nonmembership degree that object x holds on parameter ɛ, respectively.Let f, A) and  be two bifuzzy soft sets over U. We say that  is a bifuzzy soft subset of  and write ⋐ if  (i)A⊆B, (ii)ɛ ∈ A, f(ɛ)⊆g(ɛ).for any Let  and  are said to be bifuzzy soft equal and write  =  if ⋐ and ⋐. and  be two bifuzzy soft sets over U. Then their extended intersection is a bifuzzy soft set denoted by , where C = A ∪ B andɛ ∈ C. This is denoted by Let  and  are two bifuzzy soft sets over the same universe U then “ AND ” is a bifuzzy soft set denoted by ∧ and is defined by ∧ = , where, h = h(a)∩g(b) for all  ∈ A × B. Here ∩ is the operation of a bifuzzy intersection.If  and  be two bifuzzy soft sets over U. Then their extended union is denoted by , where C = A ∪ B andɛ ∈ C. This is denoted by Let  and  be two fuzzy soft sets over a common universe U with A∩B ≠ ∅. Then their restricted intersection is a bifuzzy soft set  denoted by ⋓ = , where h(ɛ) = f(ɛ)∩g(ɛ) for all ɛ ∈ A∩B.Let  and  be two bifuzzy soft sets over a common universe U with A∩B ≠ ∅. Then their restricted union is denoted by ⋒ and is defined as ⋒ = , where C = A∩B and for all ɛ ∈ C, h(ɛ) = f(ɛ) ∪ g(ɛ).Let  and  over U is a fuzzy soft set, denoted by , where C = A ∪ B, and defined byɛ ∈ C. This is denoted by The γ, δ)-bifuzzy Lie subalgebra, and discuss some of its properties.In this section, we introduce a new kind of generalized bifuzzy Lie algebra, an  in L is called an -bifuzzy Lie subalgebra of L if it satisfies the following conditions: xγA,  yγA⇒(x + y), max)δA,xγA⇒(mx)δA,xγA,  yγA⇒, min)δAA bifuzzy set x, y ∈ L,  m ∈ F,  s, s1, s2 ∈ .for all xγA⇒(−x)δA,Consider (i) xγA⇒(0)δA.(ii) V be a vector space over a field F such that dim(V) = 5. Let V = {e1, e2,…, e5} be a basis of a vector space over a field F with Lie brackets as follows:ei, ej] = 0 for all i = j. Then V is a Lie algebra over F.Let A =  : V → × bys = 0.4 ∈ . By routine computations, it is easy to see that A is not an -bifuzzy Lie subalgebra of L.We define a bifuzzy set A in L, we denote L = {x ∈ L : μ(x) > 0 and ν(x) < 1}.For a bifuzzy set A =  be an -bifuzzy Lie subalgebra of L; then the nonzero set L is a Lie subalgebra of L.Let x, y ∈ L. Then μA(x) > 0 and νA(x) < 1, μA(y) > 0 and νA(y) < 1. Assume that μA(x + y) = 0 and νA(x + y) = 1. If γ ∈ {∈, ∈∨q}, then we can see that x(μA(x), νA(x))γA and y(μA(y), νA(y))γA, but δ ∈ {∈, ∈∨q, ∈∧q}, a contradiction. Also, xqA and yqA, but δ ∈ {∈, ∈∨q, ∈∧q}, a contradiction. Thus μA(x + y) > 0 and νA(x + y) < 1. Thus x + y ∈ L. For other conditions the verification is analogous. Consequently L is a Lie subalgebra of L.Let A =  in L is called an -bifuzzy Lie algebra of L if it satisfies the following conditions: (f)x ∈ A, y ∈ A⇒(x + y), max)∈∨qA,(g)x ∈ A⇒(mx)∈∨qA,(h)x ∈ A, y ∈ A⇒, min)∈∨qAA bifuzzy set x, y ∈ L, m ∈ F, s, s1, s2 ∈ .for all A =  be a bifuzzy set in a Lie algebra L. Then A is an -bifuzzy Lie subalgebra of L if and only ifμA(x + y)⩾min(μA(x), μA(y), 0.5), νA(x + y) ⩽ max(νA(x), νA(y), 0.5),μa(mx)⩾min(μA(x), 0.5), νa(mx) ⩽ max(νA(x), 0.5),μA⩾max(μA(x), μA(y), 0.5), νA ⩽ min(νA(x), νA(y), 0.5)Let x, y ∈ L, m ∈ F.hold for all x, y ∈ L. We consider the following two cases:μA(x), μA(y)) < 0.5, max(νA(x), νA(y)) > 0.5,min)⩾0.5, max) ⩽ 0.5.min((f)⇒(i): Let 1.Case  Assume that μA(x + y) < min(μA(x), μA(y), 0.5), νA(x + y) > max(νA(x), νA(y), 0.5). Then μA(x + y) < min(μA(x), μA(y)), νA(x + y) > max(νA(x), νA(y)). Take s, t such that μA(x + y) < s < min(μA(x), μA(y)), νA(x + y) > t > max(νA(x), νA(y)). Then xs, ys ∈ μA and xt, yt ∈ νA, but f).2.Case  Assume that μA(x + y) < 0.5, νA(x + y) > 0.5. Then x, y ∈ A but x, y ∈ A; then μA(x)⩾s1, μA(y)⩾s2, νA(x) ⩽ t1, νA(y) ⩽ t2. Now, we haves1, s2) > 0.5, max⩾0.5⇒μA(x + y) + min > 1, νA(x + y) ⩽ 0.5⇒νA(x + y) + max < 1. On the other hand, if min ⩽ 0.5, max⩾0.5, then μA(x + y)⩾min, νA(x + y) ⩽ max. Hence (x + y), max)∈∨qA. The verification of (g)⇔(j) and (h)⇔(k) is analogous and we omit the details. This completes the proof.A =  be a bifuzzy set of Lie algebra of L. Then A is an -bifuzzy Li subalgebra of L if and only if each nonempty As,t) is a Lie subalgebra of L.Let A =  is an -bifuzzy Lie subalgebra of L and let s ∈ . If x, y ∈ As,t) ≥ s and μA(y) ≥ s, νA(x) ≤ t and νA(y) ≤ t. Thus,x + y, mx,  ∈ As,t) are Lie subalgebras of L. The proof of converse part is obvious. This ends the proof.Assume that A be a bifuzzy set in a Lie algebra L. Then As,t), 0.5)⩾min(μA(x), μA(y)), min(νA(x + y), 0.5) ⩽ max(νA(x), νA(y)),max(μA(mx), 0.5)⩾μA(x), min(νA(mx), 0.5) ⩽ νA(x),max, 0.5))⩾min(μA(x), μA(y)), min, 0.5)) ⩽ max(νA(x), νA(y)) for all x, y ∈ L, m ∈ F.max, 0.5) < min(μA(x), μA(y)) = s, min(νA(x + y), 0.5) > max(νA(x), νA(y)) = t for some x, y ∈ L; then s ∈ , μA(x + y) < s, νA(x + y) > t, x, y ∈ As,t)( or μA(x + y)⩾s, νA(x + y) ⩽ t, which is contradiction with μA(x + y) < s, νA(x + y) > t. Hence (1) holds. For (2), (3) the verification is analogous.Suppose that s ∈ , x, y ∈ As,t)–(3) hold. Assume that q)-bifuzzy Lie subalgebras of L is an -bifuzzy Lie subalgebra.The intersection of any family of -bifuzzy Lie subalgebra of L and let A : = ⋂i∈ΛAi = . Let x, y ∈ L, then by μA(x + y)⩾min(μA(x), μA(y), 0.5), νA(x + y) ⩽ max(νA(x), νA(y), 0.5), and henceA is an -bifuzzy Lie subalgebra of L.Let {L0 ⊂ L1 ⊂ ⋯⊂Ln = L be a strictly increasing chain of -bifuzzy Lie subalgebras of a Lie algebra L; then there exists -bifuzzy Lie subalgebra A =  of L whose level subalgebras are precisely the members of the chain with A0.5 =  = L.Let r, s ∈  and fuzzy subset μ in L, denotes μ〉r = {x ∈ L | xrqμ}, [μ]r = {x ∈ L | xr ∈ ∨qμ}, r, s ∈ .For any We state here a nice characterization without proof.L be a Lie algebra and A a bifuzzy set in L. Then A is an -bifuzzy Lie subalgebra of L if and only if nonempty subsets L for all r ∈ r and L for all r ∈ -bifuzzy Lie subalgebra of L if and only if nonempty subsets [μA]r and L for all r, s ∈  be a bifuzzy soft set over L. Then  is said to be a bifuzzy soft Lie subalgebra over L if f(x) is a bifuzzy Lie subalgebra of L for all x ∈ A; that is, a bifuzzy soft set  on L is called a bifuzzy soft Lie subalgebra of L if μfɛ(x + y)⩾min{μfɛ(x), μfɛ(y)},νfɛ(x + y) ⩽ max{νfɛ(x), νfɛ(y)},μfɛ(mx)⩾μfɛ(x),νfɛ(mx) ⩽ μfɛ(x),μfɛ⩾min{μfɛ(x), μfɛ(y)},νfɛ ⩽ max{νfɛ(x), νfɛ(y)}Let x, y ∈ L and m ∈ K.hold for all ℜ2 = { : x, y ∈ ℝ} be the set of all 2-dimensional real vectors. Then ℜ2 with  = x × y is a real Lie algebra. Let ℕ and ℤ denote the set of all natural numbers and the set of all integers, respectively. Define f : ℤ → ℜ2 by f(n) = fn : ℜ2 →  ×  for all n ∈ ℤ,f, ℤ) is a bifuzzy soft Lie subalgebra of ℜ2.Let The following proposition is obvious.f, A) be a bifuzzy soft Lie subalgebra of L; then Let  be a bifuzzy soft set over U. For each s, t ∈ , the set s,t) is called an -level soft set of , where fɛs,t)( = {x ∈ U | μfɛ(x) ≥ s, νfɛ(x) ≤ t} for all ɛ ∈ A.Let  be a bifuzzy soft set over L.  is a bifuzzy soft Lie subalgebra if and only if s,t) is a bifuzzy soft Lie subalgebra. For each s, t ∈ , ɛ ∈ A, and x1, x2 ∈ ɛs,t) ≥ s, μfɛ(x2) ≥ s and νfɛ(x1) ≤ t, νfɛ(x2) ≤ t. From f, A)ɛs,t) ≥ min(μfɛ(x1), μfɛ(x2)), μfɛ(x1 + x2) ≥ s, νfɛ(x1 + x2) ≤ max(νfɛ(x1), νfɛ(x2)), νfɛ(x1 + x2) ≤ t. This implies that x1 + x2 ∈ ɛs,t)ɛs,t)s,t)s,t), μfɛ(x2)} and let t = max{νfɛ(x1), νfɛ(x2)}; then x1, x2 ∈ ɛs,t)ɛs,t)ɛs,t)(. This means that μfɛ(x1 + x2) ≥ min(μfɛ(x1), μfɛ(x2)), νfɛ(x1 + x2) ≤ max(νfɛ(x1), νfɛ(x2)). Verification for other conditions is similar. Hence we omit the details. Thus, ɛs,t) is a bifuzzy soft Lie subalgebra over L. This completes the proof.Conversely, assume that  is called a bifuzzy soft function from X to Y.Let f, A) and  be two bifuzzy soft sets over L1 and L2, respectively, and let  be a bifuzzy soft function from L1 to L2.(1)f, A) under the bifuzzy soft function , denoted by , is the bifuzzy soft set on L2 defined by  = (ϕ(f), ψ(A)), where for all k ∈ ψ(A), y ∈ L2The image of ((2)g, B) under the bifuzzy soft function , denoted by −1, is the bifuzzy soft set over L1 defined by −1 = (ϕ−1(g), ψ−1(B)), where for all a ∈ ψ−1(A), for all x ∈ L1,The preimage of  be a bifuzzy soft function from L1 to L2. If ϕ is a homomorphism from L1 to L2, then  is said to be bifuzzy soft homomorphism. If ϕ is a isomorphism from L1 to L2 and ψ is one-to-one mapping from A onto B then  is said to be bifuzzy soft isomorphism.Let  be a bifuzzy soft Lie subalgebra over L2 and let  be a bifuzzy soft homomorphism from L1 to L2. Then −1 is a bifuzzy soft Lie subalgebra over L1.Let −1 is a bifuzzy soft Lie subalgebra over L1.Let f, A) be a bifuzzy soft Lie subalgebra over L1 and let  be a bifuzzy soft homomorphism from L1 to L2. Then  may not be a bifuzzy soft Lie subalgebra over L2.Let  be a bifuzzy soft Lie subalgebra over L and let { | i ∈ I} be a nonempty family of bifuzzy soft Lie subalgebras of . Then f, A);i∈I is a bifuzzy soft Lie subalgebra of ⋀i∈I;⋀Bi∩Bj = ∅ for all i, j ∈ I, then if Let  be an -bifuzzy soft Lie subalgebra over L and let { | i ∈ I} be a nonempty family of -bifuzzy soft K-subalgebras of ; then q)-bifuzzy soft Lie subalgebra of ;i∈I is an -bifuzzy soft Lie subalgebra of ⋀i∈I;⋀Bi∩Bj = ∅ for all i, j ∈ I, then q)- bifuzzy soft Lie subalgebra of if Let  and  be two -bifuzzy soft K-subalgebras over a Lie algebra L. Then q)-bifuzzy soft Lie subalgebra over L.Let . Then h(α) = f(α) is an -bifuzzy Lie subalgebra of L since  is an -bifuzzy soft Lie subalgebra over L.2.Case  Consider (α ∈ B − A). Then h(α) = g(α) is an -bifuzzy Lie subalgebra of L since  is an -bifuzzy soft Lie subalgebra over L.3.Case  Consider (α ∈ A∩B). Then h(α) = f(α)∩g(α) is an -bifuzzy Lie subalgebra of L by the assumption. Thus, in any case, h(α) is an -bifuzzy Lie subalgebra of L. Therefore, q)-bifuzzy soft Lie subalgebra over L.f, A) and  be two -bifuzzy soft K-subalgebras over a Lie algebra L. If A and B are disjoint, then q)-bifuzzy softLie subalgebra over L.Let  and  intuitionistic bifuzzy soft sets on G. If  and  are -intuitionistic bifuzzy soft Lie subalgebra on L, then so are ∩ and f, A) and  are an -bifuzzy soft Lie subalgebra on L and an -bifuzzy soft Lie subalgebra on L, then Let L be a Lie algebra and  and  bifuzzy soft sets on L. If  and  are -bifuzzy soft Lie subalgebra on L, then so are ∪ and Let 𝕊𝕀 the set of all -bifuzzy soft Lie subalgebras on L.Denote by f, A),  ∈ 𝔽𝕊𝕀, by above Lemmas, f, A)∩ ∈ 𝔽𝕊𝕀. It is obvious that f, A)∩ are the least upper bound and the greatest lower bound of  and , respectively. There is no difficulty in replacing {, } with an arbitrary family of 𝔽𝕊𝕀 and so f, A), ,  ∈ 𝔽𝕊𝕀. Suppose thatɛ ∈ A∩(B ∪ C), it follows that ɛ ∈ A and ɛ ∈ B ∪ C. We consider the following cases.For any . Then I(ɛ) = f(ɛ)∩h(ɛ) = J(ɛ).2.Case  Consider . Then I(ɛ) = f(ɛ)∩G(ɛ) = J(ɛ).3.Case  Consider . Then I(ɛ) = f(ɛ)∩(g(ɛ) ∪ h(ɛ)) = (f(ɛ)∩g(ɛ)) ∪ (f(ɛ)∩h(ɛ)) = J(ɛ).I and J are the same operators, and so Therefore, f, A) and  over a Lie algebra is a bifuzzy soft set over G, denoted by , where C = A ∪ B andɛ ∈ C. This is denoted by  = ⊙.The product of two bifuzzy soft sets , , , and  be bifuzzy soft sets over a Lie algebra L such that ⊂ and ⊂. Then f1, A)⊙⊂⊙,∩⊂∩ and ∪⊂∪ and , , and  be bifuzzy soft sets over a Lie algebra L. Then ⊙⊙) = ⊙)⊙.Let ,  ∈ 𝔽𝕊A(G).Now we consider the bifuzzy soft sets over a definite parameter set. Let f, A) and  be -bifuzzy soft Lie subalgebra over a Lie algebra L. Then so is ⊙.Let , ⊙, ∩) is a complete lattice under the relation ⊂.Let Presently, science and technology are featured with complex processes and phenomena for which complete information is not always available. For such cases, mathematical models are developed to handle various types of systems containing elements of uncertainty. A large number of these models are based on an extension of the ordinary set theory such as bifuzzy sets and soft sets. In the current paper, we have presented the basic properties on bifuzzy soft Lie subalgebras. The most of these properties can be simply extended to bifuzzy soft Lie ideals. A Lie algebra is known algebraic structure and there are still many unsolved problems in it. In our opinion the future study of Lie algebras can be extended with the study of (i) roughness in Lie algebras and (ii) fuzzy rough Lie algebras.


Gm×n (m ≥ n) be a complex random matrix and W = Gm×nHGm×n which is the complex Wishart matrix. Let λ1 > λ2 > …>λn > 0 and σ1 > σ2 > …>σn > 0 denote the eigenvalues of the W and singular values of Gm×n, respectively. The 2-norm condition number of Gm×n is Let  W = Gm×nHGm×n, where Gm×n is a complex Gaussian matrix and W is known to follow a complex Wishart distribution. Over the past decade, multiple-input and multiple-output (MIMO) systems have been at the forefront of wireless communications research and development, due to their huge potential for delivering significant capacity compared with conventional systems –13. The In recent years, the statistical properties of Wishart matrices have been extensively studied and applied to a large number of MIMO applications. In statistics, the random eigenvalues are used in hypothesis testing, principal component analysis, canonical correlation analysis, multiple discriminant analysis, and so forth (see ). In nucGm×n (m ≥ n) be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. As we know, the n × n complex random matrix W = Gm×nHGm×n is a complex Wishart matrix. Its distribution is denoted by W∼W, Σ = σ2I. In addition, W is a positive definite Hermitian matrix with real eigenvalues; let λ1 > λ2 > ⋯>λn > 0 and σ1 > σ2 > ⋯>σn > 0 denote the eigenvalues of the W and singular values of Gm×n, respectively. The 2-norm condition number of Gm×n is κ2(W) = κ2(Gm×n)2.Let n matrix whose elements are independent and identically distributed standard normal real or complex random variables are given in [σ2I.The exact distributions of the condition number of a 2 × given in  by Edelmgiven in  for realκ2(Wn×n) for complex case, is proved.This paper is arranged as follows. W and W∼W, Σ = σ2I. The determinant, trace, and norm of a square matrix B are denoted by |B|, tr(B), and ||B||, respectively. In this section, we give some results on joint density of the eigenvalues of a complex Wishart matrix p, a portion of p is a multiple , where a1 ≥ a2 ≥ ⋯≥an ≥ 0 such that ∑i=1nai = p, 𝒟p is the set of all portions of p, and the symbol ∑paκ means the summation over 𝒟p; that is, ∑paκ = ∑κ∈𝒟paκ.For any nonnegative integer κ be any portion of p, and let μ1 > μ2 > ⋯>μn be the eigenvalues of an n × n matrix B as follows:B are defined byLet ynomials :(3)[trThe zonal polynomial of the identity matrix is defined byW = Gm×nHGm×n, where W is positive definite Hermitian matrix with real eigenvalues, let Gm×n be a complex random matrix whose elements are independent and identically distributed (i.i.d.) standard normal random variables. Let λ1 > λ2 > ⋯>λn > 0 be the eigenvalues of W and Λ = diag. If W∼W, Σ = σ2I with m ≥ n, then the joint density of its eigenvalues is defined [For n × n matrix B, the product of two zonal polynomials can be expressed in terms of a weighted combination of another zonal polynomial [ν ∈ 𝒟s and ∀κ ∈ 𝒟p, one hast = s + p and gν,κτ is a constant coefficient.For an lynomial ; that ism × n matrix B and for all κ ∈ 𝒟p, one has [t = s + p and gν,κτ is a constant coefficient.For an  one has (9)|I−B|Z = diag, Z1 = diag and D = {1 > ξ2 > ⋯>ξn > 0}, then [Let = diagξ,…, ξn, W is derived by the following. The exact distribution of the 2-norm condition number of the Wishart matrix Gm×n be a complex random matrix and W = Gm×nHGm×n. λ1 and λn are the maximum and minimum eigenvalues of W, κ2(W) = λ1/λn. Then, the exact distribution of κ2(W) is given byt = s + p. Let λ1 = λ1, γi = 1 − λi/λ1, i = 2,…, n, where 1 > γn > ⋯>γ2 > 0, we obtain the joint distribution of λ1 and γ2,…, γn, as follows:By making the transformation By using Taylor's formula,A = diag.By using property , we haveξi = γi/γn, i = 2,…, n − 1, using ξn−1 > ⋯>ξ2 > 0, we haveZ = diag and Z1 = diag.By making the transformation n be replaced by n − 1, let κ be replaced by τ, and let b = n + 1.By using Then,λ1, γn) is given byIt follows that the distribution of (κ2(W) = λ1/λn = (1 − γn)−1; then, the distribution of κ2(W) = (1 − γn)−1 is given byNote that  In this paper, the exact distribution of the condition number of complex Wishart matrices is derived. The distribution is expressed in terms of complex zonal polynomials. This distribution plays an important role in numerical analysis and statistical hypothesis testing.


C over an alphabet A is called synchronized if there exist x, y ∈ C* such that xA*∩A*y⊆C*. In this paper we describe the syntactic monoid Syn(C+) of C+ for a complete synchronized code C over A such that C+, the semigroup generated by C, is a single class of its syntactic congruence PC+. In particular, we prove that, for such a code C, either C = A or Syn(C+) is isomorphic to a special submonoid of 𝒯l(I) × 𝒯r(Λ), where 𝒯l(I) and 𝒯r(Λ) are the full transformation semigroups on the nonempty sets I and Λ, respectively.A complete code  Theory of codes is an important branch in the field of information science. Many methods, including combinatorics methods, analysis methods, and algebraic methods, are applied to study codes. As a kind of algebraic methods, it is effective to study some kinds of codes by considering syntactic monoids of the semigroups and monoids generated by these codes.C+) of C+, the semigroup generated by a maximal prefix code C for which C+ is a single class of the syntactic congruence PC+.As we have known, prefix codes have fundamental importance in theory of codes. Many authors are devoted to the investigation of prefix codes by using several methods cf. –4). In p. In p4])On the other hand, synchronized codes are also important both in theory and in applications. Many interesting results are obtained on this class of codes in the text of Berstel et al. . RecentlC+) of C+ for a complete synchronized code C over an alphabet A such that C+ is a single class of its syntactic congruence PC+.In this paper, by using the algebraic characterization of complete synchronized codes obtained in Liu  and someS is a left zero semigroup if ab = a for any a, b ∈ S. Dually, we have right zero semigroups. A rectangular band is a semigroup which is isomorphic to a direct product of a left zero semigroup and a right zero semigroup. For rectangular bands, we have the following obvious result.A semigroup S be a rectangular band. Then axa = a for any a, x ∈ S. As a consequence, aSa = {a} for any a ∈ S. In particular, if S has an identity e, then S = {e}.Let ideal of a semigroup S is a nonempty subset I of S satisfying that the union of IS and SI is contained in I. Recall that the unique minimum ideal (with respect to set inclusion) of a semigroup S (if exists) is called the kernel of S. For the kernel of a semigroup, we have the following.An S only consists of idempotents, then this kernel is a rectangular band.If the kernel of a semigroup S be a semigroup. A function λ  on S is a left translation  of S if λ(xy) = (λx)y ρ = x(yρ)) for all x, y ∈ S. A left translation λ and a right translation ρ are linked if x(λy) = (xρ)y in which case the pair  is a bitranslation of S. Denote the set of left translations and that of right translations on S by Λ(S) and P(S), respectively. Clearly, Λ(S) forms a monoid under the usual composition of functions: (λλ′)x = λ(λ′x) for all x ∈ S. Dually, P(S) forms a monoid under the usual composition of functions: x(ρρ′) = (xρ)ρ′ for all x ∈ S. The set of all bitranslations of S forms a submonoid of the direct product Λ(S) × P(S), which is called the translation hull of S, to be denoted by Ω(S).Let s be an element of a semigroup S. Then the function λs defined by λsx = sx for all x ∈ S is the inner left translation induced by s. Dually, we have inner right translation ρs induced by s. Finally, the pair πs =  is the inner bitranslation induced by s. The set Π(S) of all inner bitranslations is the inner part of Ω(S). From Corollary III.1.7 in Petrich ψ = (xPC*)ψ(yPC*)ψ.(b)x = 1, y ≠ 1. In this case,PC*α, yPC*αy. This implies that yPC+αy by item (1) in xPC*)(yPC*)]ψ = (xPC*)ψ(yPC*)ψ.(c)x ≠ 1, y = 1. This is the dual of case (b).(d)x ≠ 1, y ≠ 1. This is obvious.On the other hand, for any M is a monoid with identity 1 and M∖{1} is a subsemigroup of M, then M has a kernel if and only if M∖{1} has a kernel. If this is the case, the two kernels are equal.If Observe thatCombining Lemmas C over A is a synchronized code if and only if the kernel of Syn(C+) is a rectangular band.A complete code I and Λ be two nonempty sets and ∅ ≠ K⊆𝒯l(I) × 𝒯r(Λ). Assume that p1 and p2 are the projections onto the first and second components of 𝒯l(I) × 𝒯r(Λ), respectively. For ,  ∈ I × Λ, we denoteLet C be a complete synchronized code over an alphabet A such that C ≠ A and C+ is a PC+-class. Then Syn(C+) is isomorphic to a submonoid K of 𝒯l(I) × 𝒯r(Λ) for some nonempty sets I and Λ with ≠, and the following conditions hold:𝒯0r(I) × 𝒯0r(Λ)⊆K,K∖{} is a subsemigroup of K,i0, λ0) ∈ I × Λ such that the identity Ωi0,λ0) =  for all ,  ∈ I × Λ and the identity  =  implies that ∈{, } for all  ∈ K.there exists -submonoid of 𝒯l(I) × 𝒯r(Λ).The above submonoid M = Syn(C+) and denote the set of PC+-classes with representatives from T by T⊆A*. Since C+ is a PC+-class, we can also let e is an idempotent in M.Let e is disjunctive in M. Let w ∈ A* and c ∈ C. Then PC+-class containing 1 is {1} by C+ is a single PC+-class, we have wc, cw, c ∈ C+. Because C* is stable, it follows that w ∈ C*. This implies that p, q ∈ A* and M. Then for all s, t ∈ A*, C+ is a single PC+-class, it follows that spt ∈ C+ if and only if sqt ∈ C+, and so pPC+q. Thus We first assert that J = MeM. We assert that J is the kernel of M. In fact, J is an ideal of M clearly. Moreover, since C is complete, there exist u, v ∈ A* such that uwv ∈ C+ for all w ∈ A*. Therefore, there exist m, n ∈ M such that mxn = e for all x ∈ M. Now, let I be an ideal of M and x ∈ I. Then uxv = e for some u, v ∈ M whence J = MeM = MuxvM⊆I. Thus, J is the least ideal of M and so is the kernel of M. By J is a rectangular band.Now, let J = {e}, then we have em = me = e for any m ∈ M. This implies that c ∈ C and w ∈ A*. Since C+ is a single PC+-class, it follows that wc, cw, c ∈ C+. Because C* is stable, we have w ∈ C*. Therefore, C* = A* and hence C = A. A contradiction. Thus, J ≠ {e}. Now let ρ be a congruence on M whose restriction to J is the identity relation on J. Assume that x ∈ M and xρe. Then xeρeρex, where e, xe, ex ∈ J and thus e = xe = ex. Since M, it follows that y ∈ J, we obtain yρye, yρey, and ye, y, ey ∈ J, whence y = ye = ey. Thus, J is a rectangular band with the identity e. And hence, J = {e} by x = e and {e} is a ρ-class in M. Now, let x, y ∈ M and xρy. Then for any u, v ∈ M, we have uxvρuyv. Observe that {e} is a ρ-class, it follows that uxv = e if and only if uyv = e. Therefore, xPey in M. By the disjunctiveness of e in M, we have x = y. We conclude that ρ is the equality relation on M. Thus, M is a dense extension of J.If M is isomorphic to a subsemigroup of Ω(J) containing the inner part Π(J) as an ideal. Since J is a rectangular band, J is isomorphic to the product I × Λ of a left zero semigroup I and a right zero semigroup Λ. Observe that J ≠ {e}, |J | = |I × Λ | >1, whence ≠. By Ω(I × Λ)≅𝒯l(I) × 𝒯r(Λ), and in this isomorphism Π(I × Λ)≅𝒯0l(I) × 𝒯0r(Λ). Therefore, M is isomorphic to a subsemigroup K of 𝒯l(I) × 𝒯r(Λ) and K contains 𝒯0l(I) × 𝒯0r(Λ) as an ideal. Furthermore, since M is a monoid, K has an identity. Let  be the identity of K. Then for any  ∈ 𝒯0l(I) × 𝒯0r(Λ),Fi, λΦ) = . This implies that  = . Thus, K is a submonoid of 𝒯l(I) × 𝒯r(Λ). Since M∖{1} is a subsemigroup of M, it follows that K∖{} is a subsemigroup of K. Thus, Conditions (1) and (2) hold.By J is the kernel of M and e is an idempotent in J, it follows that the image of e in K is the form a =  for some i0 ∈ I and λ0 ∈ Λ. Since e is disjunctive in M, a is disjunctive in K. Let ,  ∈ I × Λ and Ωi0,λ0) ∈ p1(M) × p2(M). This implies thatF, Φ) ∈ p1(M) × p2(M). Thus,F, Ψ),  ∈ K. This shows that Pa and so  =  since a is disjunctive in K. Thus  = .Since F, Φ) ∈ K and  = . Then,i0〉, 〈λ0〉) = . Since e is stable in M, it follows that a is stable in K. Hence, ∈{, }. Therefore, Condition (3) is also satisfied.Finally, let -submonoid of 𝒯l(I) × 𝒯r(Λ). In fact, observe that α2 = 〈2〉, M1 is submonoid of 𝒯l(I) × 𝒯r(Λ), and M1∖{} is a subsemigroup of M1. Let λ, μ ∈ Λ. If λ = 1, μ = 2 or λ = 1, μ = 3, then there exists Φ = ιΛ ∈ p2(M1) such that λΦ = 1, μΦ = μ. If λ = 2, μ = 3, then there exists Φ = α ∈ p2(M1) such that λΦ = 2, μΦ = 1. Furthermore, if  ∈ M1 with F1 = 1, 1Φ = 1, then F = 〈1〉 = ιI and Φ = 〈1〉 or Φ = ιΛ. Thus, M1 is a -submonoid of 𝒯l(I) × 𝒯r(Λ). It is easy to see thatC+) onto M1.On the other hand, let


We discuss the existence and uniqueness of solutions for initial value problems of nonlinear singular multiterm impulsive Caputo type fractional differential equations on the half line. Our study includes the cases for a single base point fractional differential equation as well as multiple base points fractional differential equation. The asymptotic behavior of solutions for the problems is also investigated. We demonstrate the utility of our work by applying the main results to fractional-order logistic models. The theory of impulsive differential equations describes processes which experience a sudden change of their state at certain moments. Processes with such a characteristic arise naturally and are often, for example, studied in physics, chemical technology, population dynamics, biotechnology, and economics. These processes are modeled by impulsive differential equations. In 1960, Milman and Myshkis introduced the concept of impulsive differential equations , LakshmiFractional differential equations (FDEs for short), regarded as the generalizations of ordinary differential equations to an arbitrary noninteger order, find their genesis in the work of Newton and Leibniz in the seventieth century. Recent investigations indicate that many physical systems can be modeled more accurately with the help of fractional derivatives . FractioSome recent work on the existence of solutions for initial value problems of Caputo type impulsive fractional differential equations can be found in a series of papers –16, wherDa+αx and Db−αx, a is called a left base point and b right base point. Both a and b are called base points of fractional derivatives. A fractional differential equation (FDE) containing more than one base points is called a multiple base points FDE while an FDE containing only one base point is called a single base point FDE.In the left and right fractional derivatives t1 < t2 < ⋯<tm < b, b > 0 is a fixed real number, f :  × R → R is continuous, Ik, Jk : R → R   are continuous functions, u(tk+) = limt→tk+u(t) and u(tk−) = limt→tk−u(t) and u′(tk+) = limt→tk+u′(t). One can see that both fractional differential equations in =t1, t2,…, tm (impulse points).In , the autDcαu(t)=t1, t2,…, tm (impulse points).In , the autt1 < t2 < ⋯<tm < b, b > 0 is a fixed real number, f :  × R → R is jointly continuous, Ik : R → R   are continuous functions, u(tk+) = limt→tk+u(t) and u(tk−) = limt→tk−u(t) and u′(tk+) = limt→tk+u′(t). Observe that the fractional differential equation in α,  β ≤ 1,  0 < t1 < t2 < ⋯ are fixed points, D+0 is the Riemann-Liouville fractional derivative, Φ : R → R is a sup-multiplicative function, f, g, h are impulsive Caratheodory functions, m, q, n, ρ : → are continuous functions, and Ik, Jk are impulse functions. In  IVPs of one base point FDEs , 29, 30 x0 ∈ R, α ∈  → R satisfies that there exists l > −α such that |q(t)|≤tl for all t ∈ , q may be singular at t = 0, cD∗ is the standard Caputo fractional derivative at the base points t = tk  ; that is, cD∗α|tk,tk+1](u(t) =  cDtk+αu(t) for all t ∈  × R2 → R is a Caratheodory function, Ik :  × R → R   and {Ik} is a Caratheodory function sequence, and Δx(tk) = limt→tk+x(t) − limt→tk−x(t),  k = 1,2,….In this paper, we study the following two initial value problems (IVPs for short) of nonlinear multi-term FDEs with impulses on half lines:t → ∞ is established; (iii) the method of proof relies on the Schauder fixed point theorem; (iv) our approach for dealing with impulsive problems at hand is different from the ones employed in earlier work on the topic and thus opens a new avenue for studying impulsive fractional differential equations; (v) as an application, we apply our results to fractional-order logistic models and present sufficient conditions for the existence and asymptotic behavior of solutions of these logistic models.The salient features of the present work include the following: (i) to establish sufficient conditions for the existence of solutions for the IVP  with a sThe paper is organized as follows: the auxiliary material is given in We recall some basic concepts of fractional calculus , 10 and Define the Gamma function and Beta function, respectively, asα > 0 of a continuous function f :  → R is given byRiemann-Liouville fractional integral of order α for a function f ∈ ACn−1), R) is defined byn − 1 < α ≤ n, n ∈ N. If 0 < α ≤ 1, thenCaputo's derivative of fractional-order α > 0, the general solution of fractional differential equation cD+0αx(t) = 0 is given by x(t) = c0 + c1t + c2t2 + ⋯+cn−1tn−1, where ci ∈ R, i = 0,1, 2,…, n − 1,  n − 1 < α ≤ n.For x :  are continuous, x satisfies the differential equation cD+0αx(t) = q(t)f, cD+0px(t)) a.e. on ∖{t1, t2, t3,…}, and the limits limt→tk+x(t) and limt→tk+cD+0px(t)   exist and the following conditions are satisfied:A function  the IVP  if both x :  are continuous, x satisfies the differential equation cDtk+αx(t) = q(t)f, cDtk+px(t)) on  and limt→tk+cD+0px(t)   exist and the following conditions are satisfied:A function  the IVP  if both σ > max{0, α + l} and μ > max{σ, σ − α − l}. Letx ∈ X, define the norm on X asX is a real Banach space.Choose f : , there exists δ > 0 such thatx1, x2 ∈  with |x1 − x2 | <δ, k = 1,2,…, N − 1. From  In order to show that the operator (53)Since ff is a Caratheodory function, there exists δ1 > 0 such thatt ∈  and u1, u2, v1, v2 ∈  with |u1 − u2 | <δ1, |v1 − v2 | <δ1. From , we getAs 1. From , there et ∈ (1 + tμ))Ω and (tp+σ−α−l/(1 + tμ)) cD+0pΩ are equicontinuous on any closed subinterval  of  and equiconvergent at t = tk  , and t = ∞.(b) Let us recall that W ⊂ X be a nonempty bounded set. To prove that T is completely continuous, we need to prove that TW is bounded, TW is equicontinuous on finite closed sub-interval on , TW is equiconvergent at t = tk  , and TW is equiconvergent at t = ∞.Let W is bounded, therefore, , a, aW is berefore,  hold forTW is equicontinuous on finite closed sub-interval on .Next we show that a, b]⊂. For μ > σ > 0, we find thatTW is equiconvergent as t → 0+.Now wee prove that nd thattσ−α−l, we haveTW is equiconvergent as t → tk+  .For TW is equiconvergent as t → ∞. Observe thatTW is equiconvergent as t → ∞.Our next task is to show that T is completely continuous. This completes the proof.From the above steps, it follows that H1), C ≥ 0 are real numbers;where 0 < H2) is a Caratheodory sequence and there exist numbers Aki ≥ 0  , Dk ≥ 0  , δi ≥ 0   such that{In the sequel, we need the following assumption:Furthermore, we setH1) and (H2) hold. Then IVP (x ∈ X ifSuppose that (Then IVP  has at lX be the Banach space as defined in T : X → X be an operator given by (H1) and (H2) that T is well defined and is completely continuous. Thus, we seek solutions of IVP  and (H2), we find thatLet us introduceThus, by , it follδm < 1, we can choose r0 > 0 sufficiently large such that [r0 + ||Ψ||]δmM0 < r0. Let Ωr0 = {x ∈ X : ||x|| < r0}. It is easy to see that T has a fixed point (i) For n of IVP .δm = 1, we chooseΩr0 = {x ∈ X : ||x|| < r0}. Then it can easily be shown that T has a fixed point (ii) In case n of IVP .δm > 1, we choose r = r0 = ||Ψ||/(δm − 1) such thatΩr0 = {x ∈ X : ||x|| < r0}. As before, it is easy to show that T has a fixed point (iii) For n of IVP . This coH1) and (H2) hold with δm = 1. Then IVP  are constants, ∑k=1∞|ak| is convergent, and f is a Caratheodory function; there exists l ∈  such that |q(t)|≤tl for all t ∈ .Next, consider the following IVP:H1) and (H2) hold. Then every solution of (x0 + ∑k=1∞ak as t → ∞ provided that (Assume that the conditions (ution of  tends toded that  is satisf is a Caratheodory function by (H1), therefore, there exists Mr > 0 such thatBy Tm on Y asIn this section, we show the existence for solutions for IVP  with mulf is a Caratheodory function and {Ik} is a Caratheodory function sequence and λ0 = :infk=1,2,3,…(tk − tk−1) > 0. ThenTm : Y → Y is well defined;Tm coincides with the solution of IVP  such thatTmx ∈ Y. This implies that Tm : Y → Y is well defined.(i) For Tm coincides with the solution of IVP ((ii) It follows from n of IVP .Tm is completely continuous, we split the proof into several steps.(iii) To show that Step 1.  Tm is continuous.xn ∈ Y with xn → x0 as n → ∞. We will prove that Tmxn → Tmx0 as n → ∞. It is easy to see that there exists r > 0 such thatλ0 = infk=1,2,…(tk − tk−1) > 0, we get tk > kλ0 for all k = 0,1, 2,….Let j=K+1∞(1/jμ+1−σ) is convergent, there exists K > 0 such thatSince ∑f is a Caratheodory function, there exists δ1 > 0 such thatt ∈  and u1, u2 ∈  with |u1 − u2 | <δ1, |v1 − v2 | <δ1. From  with n > N2, we haven→∞Tmxn = Tmx0 which implies that Tm is continuous.Since 1. From , there equence,supk=0,ow thatsupk=0,W ⊂ X be a nonempty bounded set. To prove that Tm is completely continuous, we need to prove that TmW is bounded, TmW is equicontinuous on finite closed sub-interval on , TmW is equiconvergent at t = tk  , and TmW is equiconvergent at t = ∞.Let Step 2. As in the proof of TmW is bounded.Step 3. We prove that TmW is equicontinuous on finite closed sub-interval on . For ⊂.TmW is equiconvergent as t → 0+. For t → tk+, we haveTmW is equiconvergent as t → tk+  .As in Step 5.  TmW is equiconvergent as t → ∞. Notice thatTmW is equiconvergent as t → ∞. This completes the proof in which Tm is completely continuous.H1) and (H2) hold. Then IVP  and (H2), we find thatδm.For δm < 1, we can choose Tm has a fixed point (i) For n of IVP .δm = 1, we select(ii) For Tm has a fixed point Let n of IVP .δm > 1, we set (iii) For Tm has a fixed point Let n of IVP . This coH1) and (H2) hold with δm = 1. Then IVP (x ∈ Y if N0 < 1.Suppose that (Then IVP  has a unThe proof is similar to that of N′(t) = rN(t), where N(t) is the population at time t and r is the proportionality constant. When the growth of the population in any environment is stopped due to the density of the population, this model modifies to a nonlinear logistic model of the form N′(t) = rN(t)(1 − N(t)/π). The generalization of the nonlinear logistic model is represented by N′(t) = rN(t)[1 − (N(t)/π)α]/α. For α → 0, the model is known as the Gompertz model and can be found in the literature on actuarial science and mortality analysis of elderly person , and b(t)∈ with a∗ > 0, b∗ > 0.In , Das et β≤1.In , the autt1 < t2 < t3 < ⋯, α ∈  → R are continuous functions, and ak ∈ R → R   are constants.As an application of the main results established in the paper, we discuss the sufficient conditions for the existence and asymptotic behavior of solutions for the logistic models:Dk ∈ R, Ak1, Ak2 ≥ 0 such thatSuppose thatThen IVP  has at lf = u[a(t) − b(t)uδ]. ThenC = 0, A1 = a0, A1 = a1, δ1 = 1, δ2 = 2, B1 = B2 = 0. Then the conditions (H1) and (H2) hold. By Let Dk ∈ R, Ak1, Ak2 ≥ 0 such thatSuppose thatThen IVP  has at lThe proof immediately follows from


We consider an interval-valued multiobjective problem. Some necessary and sufficient optimality conditions for weak efficient solutions are established under new generalized convexities with the tool-right upper-Dini-derivative, which is an extension of directional derivative. Also some duality results are proved for Wolfe and Mond-Weir duals. Recently, Yuan and Liu  considerIn many real-life situations data suffer from inexactness. The interval-valued optimization problems are closely related to optimization problems with inexact data. Recently, Wu –12 derivIn this paper we consider an interval multiobjective optimization problem. Some new optimality conditions and duality results are stated under new generalized convexities with the tool-right upper-Dini-derivative. The paper is organized as follows. In n be the n-dimensional Euclidean space and let ℝ+n be its nonnegative orthant. For x =  ∈ ℝn and y =  ∈ ℝn we consider the following conventions:X ⊂ ℝn be an arcwise connected set in Avriel and Zang , there exists z ∈ X such that f(z)≦tf(x)+(1 − t)f(u).A k-dimensional vector-valued function f : X → ℝk is called ρ-generalized (strong) pseudoright upper-Dini-derivative arcwise connected (with respect to H) at u, if there exists vector-valued function ρ such that f(x)<(≦)f(u)⇒(dφ)+) < ρ, for x ∈ X; f is called ρ-generalized (weak) quasi-right upper-Dini-derivative arcwise connected (with respect to H) at u, if there exists vector-valued function ρ such that f(x)≦(<)f(u)⇒(dφ)+)≦ρ, for x ∈ X, where ρ = T.A S ⊂ ℝn be a nonempty set and let Ψ : S → ℝm be a convex-like vector-valued function on S. Then either Ψ(x) < 0 has a solution x ∈ S, or there exists λ ∈ ℝ+m such that the system λTΨ(x)≧0 holds for all x ∈ S, but both are never true at the same time.Let a = ∈ CBI(ℝ), we haveaL and aU mean lower and upper bounds of a. If aL = aU, then a =  is a real number. Also, let b = . Then, by definition we haveLet CBI(ℝ) be the class of all closed and bounded intervals in ℝ. Thus if α, we haveFor a real number Using , 18, we a =   and b = ∈ CBI(ℝ). We say that a is less than b and write a≺b if ai≺bi, Let a, b∈ CBI(ℝn). We say that a is less than or equal to b and write a≼b if aL≦bL and aU≦bU.Let X be a nonempty subset of ℝn. A function φ : X→CBI(ℝ) is called an interval-valued function. In this case,φL, φU : X⟶ℝ, φL(x)≦φU(x), ∀x ∈ X.Let g = T, fL = T, fU = T, where fiL, fiU : ℝn → ℝ, fi(x) =  for gj : ℝn → ℝ, X0 = {x ∈ X : g(x)≦0} be the set of all feasible points of +), (dfU)+), and (dgJ(x*))+ ) are convex-like on X with respect to the variable x and  gj is upper semicontinuous at x* for j ∈ J∖J(x*), then there exist ξL,   ξU ∈ ℝ+k, μ ∈ ℝ+m,  ≠ 0 such thatAssume that tion for . If   there exists δj > 0 such thatgj, for j ∉ J(x*), is semicontinuous at x*, then gj) is semicontinuous at t = 0. Finally we gett ∈ , where δ0 = min{mini  δiL, mini  δiU, minj  δj} > 0. These inequalities contradict that x* is a weak efficient solution for (x ∈ X. Now we can apply dfL)+, (dfU)+, and (dgJ(x*))+) are convex-like on X, we obtain that (11)(dfα)ain that  and (10)dfL)+), (dfU)+), and (dgJ(x*))+) be convex-like on X with respect to the variables x and gj, for j ∉ J(x*), be upper semicontinuous at x*. If there exists x′ ∈ X0 such thatx* is a weak efficient solution for the problem T ∈ ℝ+k, α ∈ {L, U}, and μ ∈ ℝ+m satisfying  ≠ 0.Let , a, adfL)+ξL, ξU) = 0. Then, by (μj = 0 for j ∈ J∖J(x*), by (j ∈ J(x*) such that μj > 0, by (μT(dg)+) < 0, which contradicts , by , and the > 0, by , it resutradicts  and theoIn this section we give some Karush-Kuhn-Tucker type sufficient optimality conditions under generalized convexity with upper-Dini-derivative concept.x* be a feasible solution of  ≠ 0 and μ*≧0, i ∈ J(x*), such that fis   and gj  (j ∈ J) are  and -right upper-Dini-derivative locally arcwise connected at x* with respect to H, respectively. Also one assumes α≧0 andx ∈ X0 feasible. Then x* is a weak efficient solution for (Let ution of . Assume tion for .x* is not a weak efficient solution for (x′ ∈ D such that f(x′)≺f(x*). Now, since  ≥ 0, μJ(x*)*≧0, and gJ(x*)(x′)≦0 = gJ(x*)(x*), we getWe suppose to the contrary that tion for . Then thfis   and gj  (j ∈ J) are  and -right upper-Dini-derivative locally arcwise connected at x*, by ≧0 and  we get aThe next sufficient optimality condition is given in the case of generalized pseudo- and quasi-right upper-Dini-derivative arcwise connected type, where the proof is on the line of the above theorem.x* be a feasible solution of  ≥ 0 and μ*≧0, i ∈ J(x*), such that fs   is generalized pseudoright upper-Dini-derivative locally arcwise connected at x* with respect to H and g is ρ′-generalized quasi-right upper-Dini-derivative locally arcwise connected at x* with respect to H, respectively. Also one assumes that there exist ξL*, ξU*,  ≥ 0 and μ* ≥ 0 such that (ξL*)TρL + (ξU*)TρU + μT∗ρ  ′≧0 for any x ∈ X0. Then x* is a weak efficient solution for (Let ution of . Assume tion for .f(y) =  and e =  such that  =  = 1. Let FD denote the set of all feasible solutions of  be a feasible solution for  and g are  and -right upper-Dini-derivative arcwise connected at y on X0 ∪prXFD with respect to H, respectively. Moreover, one assumes α≧0 and (ξL)TρL + (ξU)TρU + μTρ′≧0, for all x ∈ X0. Then the following cannot hold: f(x)≺Ψ.Let tion for  and  ≥ 0 and μ* ∈ ℝm such that  ∈ FD and the objective values of  for  is an efficient solution of  for , then  be a weak efficient solution of  and g are  and -right upper Dini derivative arcwise connected at y on X0 ∪  prXFD with respect to H, respectively. If α≧0 and (ξL*)TρL + (ξU*)TρU + (μ*)Tρ′≧0, for all x ∈ X0, then y* is a weak efficient solution of  ∈ ℝk, with  =  = 1. Let FD denote the set of all feasible solutions of As in x and  be a feasible solution for  and g are  and -right upper-Dini-derivative arcwise connected at y on X0 ∪prXFD with respect to H, respectively. Also one supposes that α≧0 and (ξL)TρL + (ξU)TρU + μTρ′≧0, for all x ∈ X0. Then the following cannot hold: f(x)⪯f(y).Let tion for  and  ≥ 0, and μ* ∈ ℝm, μ*≧0, such that  is a feasible solution for (IMWVD) and the objective values of  is a weak efficient solution for  be a weak efficient solution of  and g are  and -right upper-Dini-derivative arcwise connected at y on X0 ∪prXFD with respect to H, respectively. Further, if α≧0 and (ξL*)TρL+(ξU*)TρU+(μ*)Tρ′≧0, for all x ∈ X0, then y* is a weak efficient solution of (Let (ution of . Supposeution of .In this paper we considered an interval multiobjective optimization problem. Some new optimality conditions and duality results were studied under the generalized convexity considered by Yuan and Liu . Necessa


Reference is missing a book title. The complete Reference 10 is: 10. Gannushkin MS (1961) (Book in Russian). Общая эпизоотология  Moscow: State Publishing House of Agricultural Literature


DC0+αu(t) = f, u′(t)), 0 < t < 1, u(1) = u′(1) = u′′(0) = 0, where 2 < α ≤ 3 is a real number, DC0+α is the Caputo fractional derivative, and f : ×[0, +∞) × R → [0, +∞) is continuous. Firstly, by constructing a special cone, applying Guo-Krasnoselskii's fixed point theorem and Leggett-Williams fixed point theorem, some new existence criteria for fractional boundary value problem are established; secondly, by applying a new extension of Krasnoselskii's fixed point theorem, a sufficient condition is obtained for the existence of multiple positive solutions to the considered boundary value problem from its auxiliary problem. Finally, as applications, some illustrative examples are presented to support the main results.This paper is devoted to the existence of multiple positive solutions for fractional boundary value problem  Then,a, b].Let α > β > 0. If one assumes that u(t) ∈ C∩L, then D0+βI0+αu(t) = I0+α-βu(t).Let n − 1 < α ≤ n  (n ∈ N). The fractional differential equation D0+αx(t) = 0 has solutionci ∈ R, i = 0,1,…, n − 1, n = [α] + 1.Let g ∈ C and 2 < α ≤ 3, the unique solution of problemFor any G is said to be the Green function of BVP  = u′(1) = u′′(0) = 0, one hasFrom the boundary condition Therefore, by The following properties of the Green function play an important role in this paper. G defined as ((i)G ∈ C and G > 0 for any t, s ∈ ×. For 0 ≤ s ≤ t < 1, we have(i) It is obvious that G/∂t < 0 for all s, t ∈ × be endowed with the normFirst, let P ⊂ X byDefine the set P is a cone in the space X.It is easy to verify that θ on the cone P be defined byT on P by the formulaLet the nonnegative continuous concave function T : P → P is completely continuous. The operator u ∈ P, we have that Tu(t) ≥ 0 in view of nonnegativeness of G and f. It is obvious thatFor any t ∈  and τ ∈ , we obtain thatT(P) ⊂ P.By T : P → P is continuous in view of continuity of G and f.The operator U ⊂ P be bounded; that is, there exists a positive constant K > 0 such that ||u|| ≤ K for all u ∈ U. By means of the definition of ||u||, we have |u(t)| ≤ K and |u′(t)| ≤ K for t ∈ . Let C = maxt∈,u∈U|f, u′(t))| + 1. Then, for u ∈ U, by T(U) is bounded.Let u ∈ U, t1, t2 ∈  with t1 < t2, thenT is completely continuous. The proof is completed. For each Now, we are in a position to state the main results.For convenience, denoteσ2 > σ1 > 0 such that H1) ≤ Mσ2 for ∈××;H2) ≥ Nσ1 for ∈××.Assume that there exist two positive constants u such that σ1 ≤ ||u|| ≤ σ2.Then, BVP  has at lT : P → P defined by  ≤ σ2, −σ2 ≤ u′(t) ≤ σ2 for all t ∈ . It follows from condition (H1) and t ∈ ,Tu|| ≤ ||u||, u ∈ P∩∂Ω2.(i) Let Ω1 = {u ∈ P : ||u|| < σ1}. For any u ∈ P∩∂Ω1, we have 0 ≤ u(t) ≤ σ1, −σ1 ≤ u′(t) ≤ σ1 for all t ∈ . It follows from condition (H2) and t ∈ ,Tu|| ≥ ||u||, u ∈ P∩∂Ω1.(ii) Let T has a fixed point in In view of n P∩Ω¯2∖Ω which isa < b < c such that the following assumptions hold:A1) < Ma for ∈××;A2) ≥ Nb for ∈××;A3) ≤ Mc for ∈××.Suppose that there exist constants 0 < u1, u2, and u3 withThen, BVP  has at lWe will show that all the conditions of u|| ≤ c. By condition (A3) and Tu|| ≤ c for First, if A1) that ||Tu|| < a for Next, by using the analogous argument, it follows from condition (u(t) = (b + c)/2 for t ∈ . It is easy to see thatu ∈ P | θ(u) > b} ≠ ∅. Hence, if u ∈ P, then b ≤ u(t) ≤ c for t ∈ . By condition (A2), we have f, u′(t)) ≥ Nb for t ∈ . So,θ(Tu) > b for u ∈ P.Choose u1, u2, and u3 withBy L > d > ηd > q > 0 such thatB1) < q/M′ for ∈××;B2) ≥ d/N′ for ∈××;B3) < L/Q for ∈××,whereAssume that there exist constants u(t) satisfying q < α(u) < d, |u′(t)| < L for t ∈ .Then, BVP  has at lf* : ×, and condition (B2) thatStep 3. In view of condition (B3), for u ∈ P∩Λ2, we haveβ(T*u) < L. By u(t) = (T*u)(t). Consequently, u(t) is a positive solution for the auxiliary BVP (q < α(u) < d, |u′(t)| < L. In addition, by virtue of the definition of f*, we know that f*, u′(t)) = f, u′(t)), t ∈ . Therefore, u is a positive solution of BVP (iary BVP  satisfyiwe have5β(T∗u) =mConsider the following fractional BVP:η = 1/6, N ≈ 33.333. Choosing σ1 = 1/104, σ2 = 1/10, we havet, u, u′) ∈ ×× andt, u, u′) ∈  × ×.By a simple calculation, one can obtain that u such that 1/104 ≤ ||u|| ≤ 1/10.With the use of Consider the following fractional BVP:a = 1/10, b = 1/50, and c = 4, then, there holdt, u, u′) ∈ ××t, u, u′) ∈ ××; andt, u, u′) ∈ ××. Hence, all the conditions of u1, u2, and u3 such thatChoosing Consider the following fractional BVP:After a simple computation, one can find thatd = 36, q = 1/3, and L = 59 = 1953125, we know thatt, u, u′) ∈ ××;t, u, u′) ∈ ××; andt, u, u′) ∈ ××. So, all the conditions of u(t) satisfying 1/3 < maxt∈|u(t)| < 36, |u′(t)| < 59.Choosing In this paper, we study the existence of multiple positive solutions for the nonlinear fractional differential equation boundary value problem  in the C


The problem of absolute stability of Lur'e systems with sector and slope restricted nonlinearities is revisited. Novel time-domain and frequency-domain criteria are established by using the Lyapunov method and the well-known Kalman-Yakubovich-Popov (KYP) lemma. The criteria strengthen some existing results. Simulations are given to illustrate the efficiency of the results. Absolute stability of nonlinear systems has been investigated comprehensively for the past several decades –12. It iIn this paper, both time-domain criterion and frequency-domain criterion for absolute stability of Lur'e systems with sector and slope restricted nonlinearities are presented based on the Lyapunov method and the KYP lemma. Some mathematical tools are used through the derivation of the absolute stability criterion. Compared with some existing results, the proposed results are less conservative. This should be owed to the effect of the slope restricted conditions on the nonlinearities. The rest of the paper is organized as follows. In W, W > 0 (W ≥ 0) denotes that W is a positive definite (semidefinite) matrix and W < 0 denotes that W is a negative definite matrix. Re{Y} means (1/2)(Y + Y*) for any real or complex square matrix Y.Throughout this paper, the superscript ∗ means transpose of real matrices and conjugate transpose of complex matrices. For a Hermitian matrix A ∈ ℝn×n, B ∈ ℝn×m, and C ∈ ℝn×m are real matrices, φ(0) = 0, m, and φi(σi(t))   are assumed to satisfyγi2 ≤ γi1, δi2 ≥ δi1, γi2 ≤ 0, and δi2 ≥ 0. The inequalities (φ(σ(t)), respectively. Let Γ1 = diag, Δ1 = diag, Γ2 = diag, Δ2 = diag. Then Γ2 − Γ1 ≤ 0, Δ2 − Δ1 ≥ 0, Γ2 ≤ 0, and Δ2 ≥ 0. Setting ψi(σi(t)) = dφi(σi(t))/dt, (φ(σ(t)) to −σ(t) is denoted as χ(s) = C*(A − sI)−1B.Consider the following multi-input and multioutput Lur'e system)  i = 1,,…, m areiσi2(t),γ2i≤dφi(σiσi2(t),γ2i≤dφi(σx(t) = 0 is globally asymptotically stable for all nonlinear vector valued functions  φ(σ(t)) satisfying ..x(t) = 0tisfying  are giveA ∈ ℝn×n, B ∈ ℝn×m, and symmetric matrix Σ ∈ ℝn+m)×(n+m) ≠ 0 for ω ∈ ℝ, and the pair  is controllable, the following two statements are equivalent.ω ∈ ℝ.P = P* such that A, B) is not controllable.There exists a matrix Given that S22 > 0 and S11 + S12S22−1S12* < 0;S11 < 0 and S22 + S12*S11−1S12 > 0.The LMI P = P* and λi ∈ ℝ   are necessary to be determined. It should be pointed out that P is not necessary to be positive definite and λi   are not necessary to be nonnegative.We choose the following Lur'e-Postnikov function:φ(σ(t)) satisfying , T1 ≥ 0, T2 > 0, and symmetric matrices P such that the LMI is feasible:System  is absoltisfying  and 3)  φ(σ(t)) V(x(t)).We will demonstrate that the given conditions imply the negative definiteness of V(x(t)) along the trajectory of (φi(σi(t)) are equivalent toti1 ≥ 0 and ti2 > 0, i = 1,2,…, m, it followsT1 = diag ≥ 0 and T2 = diag > 0. ThenTaking the derivative of ctory of , we havenditions  and 4)  V(x(t)) ondition  guarante= diagt,…, t1m V(x(t)) is positive definite. In ) is a little difficult and complex. Without loss of generality, letting λi < 0   and λi ≥ 0    (0 ≤ k ≤ m), then V(x(t)) has the following form:k1 = diag and Γk1 = diag. Since (σi(t)(φi(σi(t)) − γi1σi(t)) ≥ 0, ∑i=k+1mλi∫0σi(t)(φi(s) − γi1s) ds ≥ 0 is satisfied. Then V(x(t)) is positive definite if P + (1/2)CΛΓ1C* + (1/2)CΛ(Δk1 − Γk1)C* is positive definite, which is proved in what follows.Now we are only left to demonstrate that nite. In , P is on0  i = 1,,…, k andA1 = A + BΓ1C*, P1 = P + (1/2)CΛΓ1C*, Ak = A1 + B(Δk1 − Γk1)C*, and Pk = P1 + (1/2)CΛ(Δk1 − Γk1)C*. Firstly, the given conditions imply that α satisfying 0 < α ≤ 1 can be found such thatω0 ∈ ℝ. Since A1 is Hurwitzian, det(jω0I − A1) ≠ 0 and ν ≠ 0 such thatG(jω0) = C*(jω0I − A1)−1B. Then we deriveW1*, we haveG(jω) = C*(jωI − A1)−1B and jωG(jω) = C*A1(jωI − A1)−1B + C*B. Letting ω = ω0 in (Ak = A + BΓ1C* + B(Δk1 − Γk1)C* is Hurwitzian. Secondly, the given conditions imply that P + (1/2)CΛΓ1C* + (1/2)CΛ(Δk1 − Γk1)C* is positive definite. Actually, pre- and postmultiplying both sides of ). This completes the proof.Denote sides of  by W1=[I we have[A1∗P1+P1lement, ([A1∗P1+P1ve that ([(jωI−A1)quality (−T1+12(Δ1e deriveν∗Δ~{−T1+ows thatν∗Δ~{−T1+ we have[A1∗P1+P1A + BΓ1C* is Hurwitzian if and only if P + (1/2)CΛΓ1C* > 0.It is found in the proof of φ(σ(t)) satisfying CΛΓ1C* > 0 and the LMI (System  is absoltisfying  and 3)  φ(σ(t))  the LMI  holds.The LMI  can be tφ(σ(t)) satisfying   φ(σ(t)) χ(jω) = C*(A−jωI)−1B. Substituting − lemma, ([(jωI−A−),where2Ω11=[Σ11Σtituting  is deriv1 = 0, the FDI (For the case Γ the FDI  reduces .15.1 in . Howeverφ(σ(t)) are removed, another absolute stability criterion is derived by choosing (If the slope restrictions on choosing  as the Lφ(σ(t)) satisfying A*CΛ + (1/2)CT(Γ1 + Δ1), Ω22 = (1/2)ΛC*B + (1/2)B*CΛ − T.System  is absoltisfying  if the mThe proof is similar to that of Similar to φ(σ(t)) satisfying  = C*(A−jωI)−1B and C*A(A−jωI)−1B = C*B + jωχ(jω).From the KYP lemma,  is equiv between  and 31)(32)[) = m1x1(t) + (1/2)(m0 − m1)(|x1(t) + 1| − |x1(t) − 1|), α, β, γ, m0, and m1 are numbers. System (σ(t) = x1(t), and φ(σ(t)) = m1σ(t) + (1/2)(m0 − m1)(|σ(t) + 1| − |σ(t) − 1|). The nonlinearity φ(σ(t)) satisfies1 = Γ2 = min{m0, m1} and Δ1 = Δ2 = max{m0, m1}.Consider Chua's oscillator  with thex˙1(t)=α[α = −0.8018, β = 0.136, γ = 0.1097, and m0 = −2.96 are taken, system (m1 ≤ 2.009 by applying m1 ≤ 1.81 and m1 ≤ 1.51, respectively, by m1 = 2 at the initial value When , system  is absolWe have proposed new absolute stability criteria for Lur'e systems with sector and slope restricted nonlinearities from time-domain and frequency-domain points of view. The slope restrictions on nonlinearities improve the absolute stability conditions. We have shown that the criteria are less conservative than some existing results.


The exponential inequality for weighted sums of a class of linearly negative quadrant dependent random variables is established, which extends and improves the corresponding ones obtained by Ko et al. (2007) and Jabbari et al. (2009). In addition, we also give the relevant precise asymptotics. X and Y are said to be negative quadrant dependent  if P ≤ P(X > x)P(Y > y) for all x, y ∈ R. Based on the concept of NQD, another notion of negative dependence was formulated by Newman  denotes the integral part of x, Sn = ∑i=1nXi, σn2 = ESn2, and u(n) = supi≥1∑j:|j−i|≥n|Cov| and denote logn = ln(n∨e). This paper is organized as follows. Throughout this paper, we always suppose that (A1)Xi, i ≥ 1} be a sequence of stationary LNQD random variables with |Xi | ≤M and let {ani : 1 ≤ i ≤ n, n ≥ 1} be a triangular array of numbers satisfying |ani | ≤M1, where M and M1 are generic positive constants.Let {(A2)pn, n ≥ 1} be a positive integer sequence satisfying pn ≤ n/2 and pn → ∞ as n → ∞.Let {In this section, our main results will be given. For formulation of the theorems obtained, some assumptions are needed, which are listed below.ε < 1, one hasC1 and C are positive constants.Suppose that the assumption (A1) holds. Then for any 0 < Takingα > 1. Suppose that ε is as in (ani = 1 for 1 ≤ i ≤ n.Assume that the assumptions (A1) and (A2) hold. Let is as in . Then oni=1n(Xi − EXi)/n is O(1)n−1/2log1/2n, which is obviously faster than the corresponding one n−1/2(pnlogn)1/2 that Ko et al.  + 1 and then definej = 1,2,…, rn andi=1nani(Xi − EXi) = Sn1 + Sn2 and n < 2rnpn ≤ 2n.Next, we give some notations used later. Define Now, we can obtain the following lemma.Xi, i ≥ 1} be a sequence of stationary LNQD random variables with |Xi| ≤ M and let {ani : 1 ≤ i ≤ n, n ≥ 1} be a triangular array of numbers satisfying |ani| ≤ M1, where M and M1 are generic positive constants. If 0 < λpnMM1 ≤ 1/2 for some λ > 0, then on account of Definitions (Let {initions  and 10)Xi, i ≥ ani ≥ 0. Applying EYnj = 0 and λpnMM1 ≤ 1/2, we have by |λYnj| ≤ 2λpnMM1 thatani ≥ 0 and λ > 0, {λYnj, 1 ≤ j ≤ rn} is LNQD. Therefore, from Lemma 3.1 in Ko et al. [Sn2. The proof is completed.Without loss of generality, assume that of LNQD,∏j=1rnE, so that this exponent becomes equal to −nε2/(16C) as desired. The proof is completed.Applying Markov inequality and N is a standard normal random variable.Under the conditions of x ∈ Rx) denotes the standard normal distribution function.Under the conditions of Based on the above lemmas, the proofs of Theorems λ > 0 which satisfies λpnMM1 ≤ 1/2; then it follows from Let σ = 1 in what follows. Sinceβ > −1.Without loss of generality, set P(|N | ≥x) = 2P(N ≥ x) for any x > 0. It is easy to observe thatβ > −1, which implies  = exp(M/ϵ2), where M > 4 and 0 < ϵ < 1/4. Obviously,I1. Set n → 0 as n → ∞. It follows thatϵ↘0ϵβ+1)2(I1 = 0. Turn to I2. By ϵ. Thus we obtainϵ↘0ϵβ+1)22(I3 → 0, when M → ∞. Combining the earlier results together yields (Next, we will prove . Let J(ϵove thatlimϵ↘0ϵ2


By applying techniques in the theory of convex functions and Schur-geometrically convex functions, the author investigates a conjecture of Satnoianu on an algebraic inequality and generalizes some known results in recent years. For more information on the theory of means, see the monograph  ≥ ⋯≥xn][ and y[1] ≥ ⋯≥yn].The set φ : Ω → ℝ+ is said to be Schur-geometrically convex on Ω if lnx = ≺lny =  implies φ(x) ≤ φ(y) for every x, y ∈ Ω.A function φ : Ω → ℝ+ is said to be Schur-geometrically concave on Ω if lnx = ≺lny =  implies φ(x) ≥ φ(y) for every x, y ∈ Ω.A function Let Ω⊆ℝ+n be a symmetric and geometrically convex set with inner points and φ : Ω → ℝ+ a symmetric and differentiable function in Ω°. Then φ is a Schur-geometrically convex  function on Ω if and only ifLet r ≥ 1 and t ≥ −1 or for r ≤ 0 and t > −1. If  0 < r < 1 and t ≥ −1, inequality r≥1+Now we start off to demonstrate our main results.m, M be defined as in mβ ≥ 2pMα, thenp)Mβ ≤ 2pmα, inequality )r is convex on ;for r ≤ 1, or for r > 1 and r − 1 − 2m′ < 0, the function (t/(1 + t))r is concave on .for 0 < By Jensen's inequality, when r < 0, or when r > 1 and r − 1 − 2M′ > 0, we haveu = ∈n. When 0 < r ≤ 1, or when r > 1 and r − 1 − 2m′ < 0, inequality (For  we have∑i=1n(ui1ui = αxiq/βAn(xq) for i = 1,2,…, n, r = 1/p, m′ = αm/βM, and M′ = αM/βm. Then it is easy to show that An(u) = α/β. Making use of m, if (1 − p)β > 2p(n − 1)α, thenp)(n − 1)β < 2pα, inequality (Under the conditions of )α, then∑i=1nn;if r < 0 and 1 + rM′ > 0, the function f(u) is Schur-geometrically concave on n.if By the fact thatr > 0, or if r < 0 and 1 + rm′ < 0, we haver < 0 and 1 + rM′ > 0, inequality (Using  we have∑i=1n(1+uui = βGn(xq)/αxiq for i = 1,2,…, n leads to Gn(u) = β/α. Putting r = −1/p, m′ = βm/αM and M′ = βM/αm in inequality m, if β ≥ p(n − 1)α, thenn − 1)β ≤ pα, inequality (Under the conditions of )α, then∑i=1n[xiqn ≥ 2 and p > 1, from When n ≥ 2, α, β, xi ∈ ℝ+ for i = 1,2,…, n, and p, q ∈ ℝ with p ≠ 0. If p ≥ −1, thenp ≤ −1, inequality r is a convex  function on ℝ+ for r ≥ 1 or r < 0 , by Jensen's inequality, if r ≥ 1 or r < 0, we haver ≤ 1, inequality (Since (1 +  we have∑i=1n(1+uui = βHn(xq)/αxiq and r = −1/p shows An(u) = β/α. Further from 22) both n ≥ 2, α, β, xi ∈ ℝ+ for i = 1,2,…, n, p, q ∈ ℝ with p ≠ 0, and m, M defined as in ((1)p < 0, one hasWhen  −1 ≤ (2)p < 1 and (1 − p)mβ > 2pMα, one hasWhen 0 < (3)p > 0 and mβ > pMα, one hasWhen (4)p > 0 and Mβ < mpα, one hasWhen Let This follows from utilizing the well-known harmonic-geometric-arithmetic mean inequalityM = (n − 1)m,(1)p < 1 and (1 − p)β > 2p(n − 1)α, one hasif  0 < (2)p > 0 and β > p(n − 1)α, one hasif (3)p > 0 and (n − 1)β < pα, one hasif Under the conditions of


We consider fluctuation relations between the transport coefficients of a spintronic system where magnetic interactions play a crucial role. We investigate a prototypical spintronic device - a spin-diode - which consists of an interacting resonant level coupled to two ferromagnetic electrodes. We thereby obtain the cumulant generating function for the spin transport in the sequential tunnelling regime. We demonstrate the fulfilment of the nonlinear fluctuation relations when up and down spin currents are correlated in the presence of both spin-flip processes and external magnetic fields. Nonequilibrium fluctuation relations overcome the limitations of linear response theory and yield a complete set of relations that connect different transport coefficients out of equilibrium using higher-order response functions -7. Even We recently proved nonequilibrium fluctuation relations valid for spintronic systems , fully tα=L,R, as shown in Figure ρα↑(ω)≠ρα↓(ω) /(ρα↑+ρα↓). In the limit of ε is the dot level spacing, kB is the Boltzmann constant, and T is the temperature) effectively only a single energy level εσ  in the dot contributes to the transport and can be occupied by 0, 1, or 2 electron charges. In the presence of an external magnetic field B, the Zeeman splitting is ε↑−ε↓=gμBB (g is the Landé factor and q as the electron charge). Tunneling between lead α and the dot yields a level broadening given by Γασ(ω)=Πρασ|Vα|2 (Vα is the lead-dot tunneling amplitude). Notice that the level width is then spin-dependent due to the spin asymmetry of the density of states: Γασ=(Γ/2)(1+spα), with Γ=ΓL=ΓR and s=+(−) for ↑(↓).Consider a quantum dot coupled via tunnel barriers to two ferromagnetic leads Γ≪kBT, tunneling occurs sequentially, and transport is thus dominated by first-order tunnelling processes. The dynamics of the system is governed by the time evolution of the occupation probabilities calculated from the master equation P≡{P0,P↑,P↓,P2} denoting the probabilities associated to states with 0 electrons on the dot, 1 electron with spin ↑ or ↓ and 2 electrons. We also take into account spin-flip relaxation mechanisms possibly present in our system due to magnetic interactions with a spin-fluctuating environment  or spin-orbit interactions in the dot: χ={χL↑,χL↓,χR↑,χR↓} the counting fields:In the limit of weak dot-lead coupling, f±(ε)=1/[ exp(±ε/kBT)+1]. Here, Vασ is a spin-dependent voltage bias, and μiσ is the dot electrochemical potential to be determined from the electrostatic model. i=0,1 is an index that takes into account the charge state of the dot. Then, the cumulant generating function in the long time limit is given by λ0(χ) denotes the minimum eigenvalue of χ. From the generating function, all transport cumulants are obtained [where obtained .Q↑ and Q↓: Q↑=Cu1(ϕ↑−VL↑)+Cu2(ϕ↑−VL↓)+Cu3(ϕ↑−VR↑)+Cu4(ϕ↑−VR↓)+C(ϕ↑−ϕ↓) and Q↓=Cd1(ϕ↓−VL↑)+Cd2(ϕ↓−VL↓)+Cd3(ϕ↓−VR↑)+Cd4(ϕ↓−VR↓)+C(ϕ↓−ϕ↑), where Cℓi represent capacitance couplings for ℓ=u/d and i=1⋯4. We then find the potential energies for both spin orientations, Nσ being the excess electrons in the dot. For an empty dot, i.e., N↑=N↓=0, its electrochemical potential for the spin ↑ or ↓ level can be written as μσ0=εσ+Uσ−Uσ. This is the energy required to add one electron into the spin ↑ or ↓ level when both spin levels are empty.We consider a gauge-invariant electrostatic model that treats interactions within a mean-field approach . For theN↑=1 or N↓=1, and we find Importantly, our results are gauge invariant since they depend on potential differences  between fluctuations ΔGα,β  to the equilibrium noise Gα,βγ, and the noise susceptibilities, Sαβ,γ, by means of a fluctuation relation,Small fluctuations around equilibrium and their responses are related through the fluctuation-dissipation theorem. In particular, the Kubo formula for the electrical transport relates the linear conductance  systems . Thus, ipL=p and a normal lead with polarization pR=0. We take into account the presence of spin-flip processes described by γsf. In Figure VL↑=V1, VL↓=V2, VR↑=V3, VR↓=V4. When the dot is subjected to an externally applied magnetic field, one must consider the antisymmetrized version of Equation 4 using A−=A(B)−A(−B), where A can be G, S, or higher order correlation functions  and zero magnetic field (ε↑=ε↓). Then, we are able to obtain an analytical expression for the cross correlations between ↑ and ↓ currents in the left terminal:We now discuss the analytical expressions for the spin noises of our spin diode. We consider that the system is biased with a source-drain voltage εeff=ε+e2/2CΣ, with p increases independently of γsf. Moreover, SL↑L↓ is always negative due to the antibunching behavior of fermions [where fermions . The shoFL↑L↑=SL↑L↑/IL↑,with an associated Fano factor εeff lies inside the transport window. This is due to correlations induced by Coulomb interactions [Notably, the Fano factor is always sub-Poissonian whenever ractions .Nonequilibrium fluctuation relations nicely connect nonlinear conductances with noise susceptibilities. We have derived spintronic fluctuation relations for a prototypical spintronic system: a spin diode consisting of a quantum dot attached to two ferromagnetic contacts. We have additionally investigated the fulfilment of such relations when both spin-flip processes inside the dot and an external magnetic field are present in the sample. We have also inferred exact analytical expressions for the spin noise current correlations and the Fano factor. Further extensions of our work might consider noncollinear magnetizations and energy dependent tunneling rates.The authors declare that they have no competing interests.RL and DS defined the research subject. JSL and RL performed the calculations. All authors discussed the results and co-wrote the paper. All authors read and approved the final manuscript.


Tα1 can stimulate T cell proliferation and differentiation from bone marrow stem cells, augment cell-mediated immune responses, and regulate homeostasis of immune system. In this study, we developed a novel strategy to produce Tα1 concatemer (Tα1③) in Escherichia coli and compared its activity with chemically synthesized Tα1. Results showed that Tα1③ can more effectively stimulate T cell proliferation and significantly upregulate IL-2 receptor expression. We concluded that the expression system for Tα1 concatemer was constructed successfully, which could serve as an efficient tool for the production of large quantities of the active protein.Thymosin alpha 1 (T Trypanosoma cruzi transsialidase stabilized the catalytic activity. In addition, repeats present on T. cruzi shed proteins increased trans-sialidase half-life in blood from 7 to almost 35 h [The tandem repeats of proteins and peptides are studied widely and formidable progress has been made in this field. It was reported that tandem amino acid repeats have many functions of stabilizing proteins , maintaiost 35 h , 5.α1) is a heat-stable, acidic polypeptide composed of 28 amino acid residues blocked at the N-terminus by an acetyl group [α1 could be a useful restorative therapeutic agent in the treatment of immunodeficiency diseases and immunosuppressed conditions [Thymosin alpha1  in E. coli TOP10 strain and purified by heat treatment and Q-Sepharose Fast Flow ion-exchange chromatography. Then, Tα1③ was released by treatment with 0.5 M Cyanogen bromide (CNBr) and purified by SP-Sepharose Fast Flow chromatography. In our strategy, trx acts as a chaperon to help Tα1③ folding and CNBr treatment removed any exogenous amino acid (such as Met at the N-terminus for translation start) from Tα1③ molecule. So we can get the “natural” Tα1③. Finally, the biological activity of Tα1③ on T lymphocyte proliferation and IL-2R expression was assessed.In this study, TE. coli strain TOP10 (F-mcrAΔ(mrr-hsd RMS-mcrBC) φ80 lacZΔM15 ΔlacX74 recA1 araΔ139Δ(ara-leu)7697 galU galK rpsL (Strr)endA1 nupG) were from Invitrogen. DNA fragments were synthesized in BIOASIA. Synthetic Tα1 (ZADAXIN) was from Sciclone Pharmaceuticals, USA. The anti-Tα1 antibody (ab55635) was purchased from Abcam and FITC-anti-IL-2Rβ (18344D 554452) was from BD Pharmingen. Restriction enzymes, Taq DNA polymerase, and T4 DNA ligase were purchased from TaKaRa. Expression vector pThioHisA and α1③ gene was cloned by gene synthesis and PCR (EcoR I site) was p1: 5′-GGAATTCATGTCTGATGCAGCCGTGGAC ACCAGCAGCG-3′ and the reverse primer (with an introduced Pst I site) was p2: 5′-GCACTGCAGTCAGTTCTGGGCCTCCTCCACCACCT-3′. The template for PCR was annealing products of 4 synthesized fragments listed in α1③ for identification.T and PCR . The forα1③ was digested with EcoR I and Pst I and cloned into expression vector pThioHisA digested with the same enzymes. The candidate plasmid pThioHisA-Tα1③ was then confirmed by restriction enzymes digestion and DNA sequencing. The vector pGEM-Tα1③ was transformed into E. coli TOP10 strain. A single colony was inoculated into 10 mL Luria-Bertani (LB) medium supplemented with ampicillin (100 μg/mL) and grown at 200 rpm and 37°C overnight. Then it was inoculated into 300 mL fresh LB medium in a 500 mL shake flask and cultured until the OD600 reached 0.5. Trx-Tα1③ expression was induced by 1 mM IPTG  for 4 h. Large scale fed-batch culture was performed in a 5-L fermentor as previously described [The plasmid pThioHisA-Tescribed .α1③ was then cleaved by CNBr (0.5 M) in 70% formic acid for 24 h. The cleavage reaction was stopped by addition of ten times amount of H2O [α1③ was purified by SP-Sepharose Fast Flow chromatography. Purified Tα1③ was dialyzed against PBS for later use. Cell pellet was suspended in 20 mM Tris/HCl buffer (pH 8.0) in proportion of 200 g/L and disrupted by sonication. Then, the lysate was incubated at 80°C for 10 min (shaken once every 2-3 min) and cooled quickly. Samples were centrifuged at 12 000 g for 20 min and the supernatant was loaded onto a Q-Sepharose Fast Flow chromatography column and eluted with linear NaCl gradient. The purified Trx-Tμm, Invitrogen) after SDS-PAGE using a Bio-Rad Semi-Dry electrophoretic cell. Western blot analyses were carried out using a Tα1 specific antibody and followed by a phosphatase-conjugated goat anti-mouse IgG . Western Blue Stabilized Substrate (Promega) for alkaline phosphatase was used for visualization. Proteins were transferred to nitrocellulose membranes  and cultured in the presence of 2.5 μg/mL concanavalin A (ConA) at 37°C in 5% CO2 in humid air. Six h later, 90 μL of Tα1③ diluted with RPMI 1640 media was added to all but the control wells. The synthetic Tα1 and media were used as positive and negative controls. After 66 h incubation, 20 μL of MTT (0.5 mg/mL) solution was added and the plates were centrifuged  4 h later. Supernatants were discarded, and 100 μL of DMSO was added. After incubated at room temperature for 10 min, the solubilized reduced MTT was measured at 570 nm using a Bio-Rad plate reader and the optical densities were used for calculate growth rate with the formulaThe proliferation response of splenocytes was determined by MTT assay. Spleens from C57BL6 mice were dispersed through nylon mesh to generate a single-cell suspension. Then lymphocytes were separated by EZ-Sep 1× Lymphocyte Separation Medium  and suspended at 4 × 10α1③ on the expression level of IL-2R on T lymphocytes, cells were isolated as before and cultured in the presence of ConA and Tα1③. The synthetic Tα1 and a recombinant Tα1 monomer prepared in our lab were used as positive controls. Cells were collected and stained 48 h later according to standard protocol. In brief, 5 × 105 cells were washed with PBS and stainedin “FACS buffer”  with FITC-anti-mIL-2Rβ for 10 min at room temperature. After washing, cells were fixed for 30 minutes on ice with 4% paraformaldehyde and analyzed on a FACSCalibur flow cytometer (BD Biosciences). To evaluate the effect of Tα1 has been successfully applied in clinical trials for immunodeficiency diseases therapy, the high costs is still a hard nut to crack. Fortunately, molecular biology techniques allowed us to produce recombinant Tα1 in E. coli. Considering that Tα1 is too small to be directly expressed in E. coli, it was usually assembled as concatemers. But some exogenous amino acid residues such as His6 tag or methionine (Met) introduced by the initiation codon AUG usually affects the effect of concatemers [ Although synthetic Tcatemers . α1, we put forward a new strategy as showed in α1 concatemers in which Tα1③ that was assembled by three repeated copies of Tα1 gene owned highest proportion. After cloning into pGEM-3Zf vector, the gene was proven by enzyme digestion and DNA sequencing. The sequence of Tα1③ gene was consistent with our design as follows: 5′-atgagcgacgccgccgtggacaccagcagcgagatcaccaccaaggaccggaaggagaagaaggaggtggtggaggaggccgagaacagcgacgccgccgtggacaccagcagcgagatcaccaccaaggaccggaaggagaagaaggaggtggtggaggaggccgagaacagcgacgccgccgtggacaccagcagcgagatcaccaccaaggaccggaaggagaagaaggaggtggtggaggaggccgagaactga-3′. In order to produce the real “natural” concatemers of Tα1③/TOP10 showed that a new 31 kDa protein which can be specifically recognized by Tα1 antibody was produced. It suggested that trx-Tα1③ was successfully expressed. Trx was used as a chaperon to guarantee the correct folding of Tα1③ and trx-Tα1③ was expressed as a soluble fusion protein. Both SDS-PAGE  and Westα1 are heat-stable proteins, so trx-Tα1③ was easily purified by one-step Q-Sepharose Fast Flow chromatography after the lysate of recombinant bacterial cells was heated at 80°C for 10 min , whereas 5 μg/mL Tα1③ could induce significant proliferation (P < 0.05). Furthermore, the effect of 10 μg/mL Tα1③ was stronger than that of 40 μg/mL synthetic Tα1  of Tα1③ apparently stimulated the proliferation of T lymphocytes compared with that of ZADAXIN (40 μg/mL). In addition, Tα1③ significantly upregulated IL-2R on T cell, which is very important for T cell activation and proliferation in vivo. The detailed mechanism for stronger effect of Tα1③ and the pharmacokinetics of different tandem repeats are still under investigation. Trx-T