The notions of an ideal and a fuzzy ideal in BN-algebras are introduced. The properties and characterizations of them are investigated. The concepts of normal ideals and normal congruences of a BN-algebra are also studied, the properties of them are displayed, and a one-to-one correspondence between them is presented. Conditions for a fuzzyset to be a fuzzy ideal are given. The relationships between ideals and fuzzy ideals of a BN-algebra are established. The homomorphic properties of fuzzy ideals of a BN-algebra are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals are obtained. The class of BM-algebras contains Coxeter algebras . Some oras  the class of all BCK-algebras . The interrelationships between some classes of algebras mentioned before are visualized in X and Y are classes of algebras, then X → Y means X ⊂ Y).We will denote by μ, we make the least fuzzy ideal containing μ. This leads us to show that the set of fuzzy ideals of a BN-algebra is a complete lattice. Moreover, the homomorphic properties of fuzzy ideals are provided. Finally, characterizations of Noetherian BN-algebras and Artinian BN-algebras in terms of fuzzy ideals are given in In this paper we consider ideals and fuzzy ideals in BN-algebras. In A; ∗, 0) of type  is called a Coxeter algebra θ we denote the congruence class containing x; that is, [x]θ = {y ∈ A : xθy}.Let θ be a congruence on 𝒜. Then θ ∈ CN(𝒜) if and only if [0]θ is a normal ideal of 𝒜.Let I = [0]θ and let θ ∈ CN(𝒜). It follows easily that 0 ∈ I. Let x and y be elements of A such that x∗y, y ∈ I. Then, (x∗y)θ0 and yθ0. Since yθ0, we obtain (x∗y)θ(x∗0) and hence x = x∗0 ∈ I. Consequently, I is an ideal. Now, suppose that x∗y, a∗b ∈ I, where x, y, a, b ∈ A. Then, (x∗y)θ0 and (a∗b)θ0. By the definition of a normal congruence, ((x∗a)∗(y∗b))θ0; that is, (x∗a)∗(y∗b) ∈ I. Thus, I = [0]θ is normal.Set The converse is obvious.There is a bijection between the set of normal ideals and the set of normal congruences of a BN-algebra.𝒜 be a BN-algebra. We consider functions f : IdN(𝒜) → CN(𝒜) and g : CN(𝒜) → IdN(𝒜) given as follows:I ∈ CN(𝒜) and [0]θ ∈ IdN(𝒜), we conclude that f and g are well-defined. Next, note that I = [0]I~. Indeed,g∘f = idN(𝒜)Id and f∘g = idCN(𝒜). Thus, f and g are inverse bijections between IdN(𝒜) and CN(𝒜).Let .Hence,g∘fI=g~I=e obtainf∘gθ=f0θ=𝒜 =  and ℬ =  be BN-algebras. A mapping f : A → B is called a homomorphism from 𝒜 into ℬ if f(x∗y) = f(x)∗f(y) for any x, y ∈ A.Let f(0A) = 0B. Indeed, f(0A) = f(0A∗0A) = f(0A)∗f(0A) = 0B. We denote by kerf the subset {x ∈ A : f(x) = 0B} of A (it is the kernel of the homomorphism f).Observe that f : A → B be a homomorphism from 𝒜 into ℬ. Then kerf is an ideal of 𝒜.Let A ∈ kerf; that is, (I1) holds. Let x∗y ∈ kerf and y ∈ kerf. Then 0B = f(x∗y) = f(x)∗f(y) = f(x)∗0B = f(x). Consequently, x ∈ kerf. Therefore, (I2) is satisfied. Thus I is an ideal of 𝒜.Obviously, 0A; ∗, 0) be the algebra given in A : A → A is a homomorphism and the ideal ker(idA) = {0} is not normal.The kernel of a homomorphism is not always a normal ideal. Let  be a BN-algebra and let ℬ =  be a BM-algebra. Let f : A → B be a homomorphism from 𝒜 into ℬ. Then kerf is a normal ideal.Let kerf is an ideal of 𝒜. Let x, y, a, b ∈ A and x∗y, a∗b ∈ kerf. Then 0B = f(x∗y) = f(x)∗f(y). From f(x) = f(y). Similarly, f(a) = f(b). Consequently, f((x∗a)∗(y∗b)) = (f(x)∗f(a))∗(f(y)∗f(b)) = (f(x)∗f(a))∗(f(x)∗f(a)) = 0B, and hence, (x∗a)∗(y∗b) ∈ kerf.By I be a normal ideal of 𝒜. For x ∈ A, we write x/I = {y ∈ A : x~Iy}; that is, x/I = [x]I~. We note thatA/I = {x/I : x ∈ A} and set x/I∗′y/I = x∗y/I. The operation ∗′ is well-defined, since ~I is a congruence of 𝒜. It is easy to see that 𝒜/I =  is a BN-algebra. The algebra 𝒜/I is called the quotient BN-algebra of 𝒜 modulo I.Let 𝒜 and θ be given as in 𝒜 is a BN-algebra and θ is a normal congruence of 𝒜. Since [0]θ = {0,1}, from I = {0,1} is a normal ideal of 𝒜. We have 0/I = 1/I = {0,1} and 2/I = 3/I = {2,3}. Then A/I = {0/I, 2/I}. Clearly, 0/I∗′0/I = 0/I = 2/I∗′2/I and 0/I∗′2/I = 2/I = 2/I∗′0/I.Let 𝒜 be a BN-algebra and let ℬ be a BM-algebra. Let f : A → B be a homomorphism from 𝒜 onto ℬ. Then 𝒜/kerf is isomorphic to ℬ.Let I = kerf is a normal ideal of 𝒜. Define a mapping g : A/I → B by g(x/I) = f(x) for all x ∈ I. Let x/I = y/I. Then, x~Iy; that is, x∗y ∈ I. Hence, f(x)∗f(y) = 0B. By f(x) = f(y). Consequently, g(x/I) = g(y/I). This means that g is well-defined. It is easy to check that g is a homomorphism from 𝒜/I onto ℬ. Observe that g is one-to-one. Let g(x/I) = g(y/I). Then f(x) = f(y) and hence f(x∗y) = 0B; that is, x∗y ∈ I. Therefore, x~Iy and consequently, x/I = y/I. Thus g is an isomorphism from 𝒜/I onto ℬ.By α, β}, then α∧β = min{α, β} and α∨β = max{α, β}. Recall that a fuzzy set in A is a function μ : A → .We now review some fuzzy logic concepts. First, for Γ⊆ we define ⋀Γ = infΓ and ⋁Γ = supΓ. Obviously, if Γ = {μ and ν in A, we defineA.For any fuzzy sets A and B be any two sets, μ any fuzzy set in A, and f : A → B any function. Set f←(y) = {x ∈ A : f(x) = y} for y ∈ B. The fuzzy set ν in B defined byy ∈ B, is called the image of μ under f and is denoted by f(μ).Let A and B be any two sets, f : A → B any function, and ν any fuzzy set in f(A). The fuzzy set μ in A defined by preimage of ν under f and is denoted by f←(ν).Let Now, we give the definition of a fuzzy ideal in a BN-algebra.μ in A is called a fuzzy ideal of a BN-algebra 𝒜 if it satisfies, for all x, y ∈ A, μ(0) ≥ μ(x),μ(x) ≥ μ(x∗y)∧μ(y).A fuzzy set μ be a fuzzy ideal of 𝒜. Then, for any x, y ∈ A, if x ≤ y, then μ(x) ≤ μ(y).Let x ≤ y, then x∗y = 0. From y∗x = 0. Hence, by (d2) and (d1), we have μ(y) ≥ μ(y∗x)∧μ(x) = μ(0)∧μ(x) = μ(x).If ℱId(𝒜) the set of all fuzzy ideals of a BN-algebra 𝒜.Denote by 𝒜 =  be the BN-algebra given in α3 < α2 < α1 ≤ 1. Define a fuzzy set μ in A byμ satisfies (d1) and (d2). Thus μ ∈ ℱId(𝒜).Let I be an ideal of a BN-algebra 𝒜 and let α, β ∈  with α ≥ β. Define μIα,β as follows:μIα,β = μ. Since 0 ∈ I, μ(0) = α ≥ μ(x) for all x ∈ A. To prove (d2), let x, y ∈ A. If x ∈ I, then μ(x) = α ≥ μ(x∗y)∧μ(y). Suppose now that x ∉ I. By the definition of an ideal, x∗y ∉ I or y ∉ I. Therefore, μ(x∗y)∧μ(y) = β = μ(x). Thus μ is a fuzzy ideal of 𝒜.Let χI of I:𝒜.In particular, the characteristic function μ in A is a fuzzy ideal of 𝒜 if and only if it satisfies (d1) and x, y, z ∈ A, if (z∗y)∗x = 0, then μ(z) ≥ μ(x)∧μ(y).for all A fuzzy set μ ∈ ℱId(𝒜) and let x, y, z ∈ A. Suppose that (z∗y)∗x = 0. Since μ is a fuzzy ideal, we have μ(z∗y) ≥ μ((z∗y)∗x)∧μ(x) = μ(0)∧μ(x) = μ(x) and μ(z) ≥ μ(z∗y)∧μ(y). Therefore, μ(z) ≥ μ(x)∧μ(y).Let μ satisfy (d3). From (B1) we have (x∗y)∗(x∗y) = 0. By (d3), μ(x) ≥ μ(x∗y)∧μ(y). Then μ satisfies (d2) and hence μ ∈ ℱId(𝒜).Conversely, let μ be a fuzzy set in A. Then μ ∈ ℱId(𝒜) if and only if its nonempty level subset𝒜 for all α ∈ .Let μ ∈ ℱId(𝒜). Let α ∈  and U ≠ ∅. Then μ(x0) ≥ α for some x0 ∈ A. Since μ(0) ≥ μ(x0), we have 0 ∈ U. Let x, y ∈ A such that x∗y, y ∈ U. Then μ(x∗y) ≥ α and μ(y) ≥ α. It follows from (d2) thatx ∈ U. Therefore U is an ideal of 𝒜.Assume that α ∈ , U = ∅ or U is an ideal of 𝒜. If (d1) is not valid, then there exists x0 ∈ A such that μ(0) < μ(x0) = β. Then U ≠ ∅ and, by assumption, U is an ideal of 𝒜. Hence 0 ∈ U and consequently μ(0) ≥ β. This is a contradiction and (d1) is valid. Now assume that (d2) does not hold. Then there are a, b ∈ A such that μ(a) < μ(a∗b)∧μ(b). Takingμ(a) < γ < μ(a∗b)∧μ(b) ≤ μ(a∗b) and γ < μ(b). Therefore a∗b, b ∈ U but a ∉ U. This is impossible, and μ is a fuzzy ideal of 𝒜.Conversely, suppose that, for each By μ is a fuzzy ideal of a BN-algebra 𝒜, then the set𝒜.If The following example shows that the converse of 𝒜 be a BN-algebra. Define a fuzzy set μ in A byAμ = {0} is the ideal of 𝒜 but μ ∉ ℱId(𝒜) (because μ does not satisfy (d1)).Let 𝒜 =  given in μ be defined as in α ∈  we haveA are ideals of 𝒜, this is another proof  be a strictly decreasing sequence in . Let μ be the fuzzy set in A defined byI0 = ∅. Then μ is a fuzzy ideal of 𝒜.Let I = ⋃n∈ℕIn. By I is an ideal of 𝒜. Obviously, μ(0) = t1 ≥ μ(x) for all x ∈ A; that is, (d1) holds. Now we show that μ satisfies (d2). Let x, y ∈ A. We have two cases.Let Case 1  (x ∉ I). Then x∗y ∉ I or y ∉ I. Therefore μ(x∗y)∧μ(y) = 0 = μ(x). Case 2  (x ∈ In − In−1 for some n ∈ ℕ). Then x∗y ∉ In−1 or y ∉ In−1. Hence μ(x∗y) ≤ tn or μ(y) ≤ tn. Therefore μ(x∗y)∧μ(y) ≤ tn = μ(x).μ is a fuzzy ideal of 𝒜.Thus (d2) is also satisfied and consequently T be a nonempty set of indexes. Let μt ∈ ℱId(𝒜) for t ∈ T. The meet ⋀t∈Tμt of fuzzy ideals μt of 𝒜 is defined as follows:Let μt ∈ ℱId(𝒜) for t ∈ T. Then ⋀t∈Tμt ∈ ℱId(𝒜).Let μ = ⋀t∈Tμt. Then, by (d1),x ∈ A. Let x, y ∈ A. Since μt ∈ ℱId(𝒜), we have μt(x) ≥ μt(x∗y)∧μt(y). Henceμ(x) ≥ μ(x∗y)∧μ(y) and therefore μ ∈ ℱId(𝒜).Let ν be a fuzzy set in A. A fuzzy ideal μ of 𝒜 is said to be generated by ν if ν ≤ μ and, for any fuzzy ideal ρ of 𝒜, ν ≤ ρ implies μ ≤ ρ. The fuzzy ideal generated by ν will be denoted by  let μ∨ν denote the join of μ and ν; that is, μ∨ν =  = μ(x)∨ν(x) for all x ∈ A.For From 𝒜 be a BN-algebra. Then (ℱId(𝒜); ∧, ∨) is a complete lattice.Let The following two theorems give the homomorphic properties of fuzzy ideals.𝒜 and ℬ be BN-algebras and let f : A → B be a homomorphism and ν ∈ ℱId(ℬ). Then f←(ν) ∈ ℱId(𝒜).Let x ∈ A. Since f(x) ∈ B and ν ∈ ℱId(ℬ), we have ν(0) ≥ ν(f(x)) = (f←(ν))(x), but ν(0) = ν(f(0)) = (f←(ν))(0). Thus we get (f←(ν))(0)≥(f←(ν))(x) for any x ∈ A; that is, f←(ν) satisfies (d1).Let x, y ∈ A. Since ν ∈ ℱId(ℬ), we obtainf←(ν))(x)≥(f←(ν))(x∗y)∧(f←(ν))(y). Consequently, f←(ν) ∈ ℱId(𝒜).Now let 𝒜 and ℬ be BN-algebras and let f : A → B be a homomorphism and μ ∈ ℱId(𝒜). Then, if μ is constant on kerf = f←(0), then f←(f(μ)) = μ.Let x ∈ A and f(x) = y. Hencea ∈ f←(y), we have f(x) = f(a). Hence f(a∗x) = 0; that is, a∗x ∈ kerf. Thus μ(a∗x) = μ(0). Therefore, μ(a) ≥ μ(a∗x)∧μ(x) = μ(0)∧μ(x) = μ(x). Similarly, μ(x) ≥ μ(a). Hence μ(x) = μ(a). Thusf←(f(μ)) = μ.Let 𝒜 and ℬ be BN-algebras and let f : A → B be a surjective homomorphism and μ ∈ ℱId(𝒜) such that Aμ⊇kerf. Then f(μ) ∈ ℱId(ℬ).Let μ is a fuzzy ideal of 𝒜 and 0 ∈ f←(0), we havex ∈ A. Hencey ∈ B. Thus f(μ) satisfies (d1). Suppose thatxB, yB ∈ B. Since f is surjective, there are xA, yA ∈ A such that f(xA) = xB and f(yA) = yB. HenceAμ⊇kerf, μ is constant on kerf. Then, by μ is a fuzzy ideal. Thus, we obtain f(μ) ∈ ℱId(ℬ).Since In this section we characterize Noetherian BN-algebras and Artinian BN-algebras using some fuzzy concepts, in particular, fuzzy ideals.𝒜 is called Noetherian if for every ascending sequence I1⊆I2⊆⋯ of ideals of 𝒜 there exists k ∈ ℕ such that In = Ik for all n ≥ k. A BN-algebra 𝒜 is called Artinian if for every descending sequence I1⊇I2⊇⋯ of ideals of 𝒜 there exists k ∈ ℕ such that In = Ik for all n ≥ k.A BN-algebra 𝒜 be a BN-algebra. The following statements are equivalent: 𝒜 is Noetherian,μ of 𝒜, Im(μ) = {μ(x) : x ∈ A} is a well-ordered set.for each fuzzy ideal Let 𝒜 is Noetherian and μ is a fuzzy ideal of 𝒜 such that Im(μ) is not a well-ordered subset of . Then there exists a strictly decreasing sequence (μ(xn)), where xn ∈ A. Let tn = μ(xn) and Un = U = {x ∈ A : μ(x) ≥ tn}. Then, by Un is an ideal of 𝒜 for every n ∈ ℕ. So U1 ⊂ U2 ⊂ ⋯ is a strictly ascending sequence of ideals of 𝒜. This is a contradiction with the assumption that 𝒜 is Noetherian. Therefore Im(μ) is a well-ordered set for each fuzzy ideal μ of 𝒜.(a) ⇒ (b): Assume that 𝒜 is not Noetherian. Then there exists a strictly ascending sequence I1 ⊂ I2 ⊂ ⋯⊂In ⊂ ⋯ of ideals of 𝒜. Let μ be a fuzzy set in A such thatI0 = ∅. By μ ∈ ℱId(𝒜), but Im(μ) is not a well-ordered set, which is impossible. Therefore 𝒜 is Noetherian.(b) ⇒ (a): Assume that (b) is true. Suppose that 𝒜 be a BN-algebra. If, for every fuzzy ideal μ of 𝒜, Im(μ) is a finite set, then 𝒜 is Noetherian.Let 𝒜 be a BN-algebra and let T = {t1, t2,…} ∪ {0}, where (tn) is a strictly decreasing sequence in . Then the following conditions are equivalent: 𝒜 is Noetherian,μ of 𝒜, if Im(μ)⊆T, then there exists k ∈ ℕ such that Im(μ)⊆{t1, t2,…, tk} ∪ {0}.for each fuzzy ideal Let 𝒜 is Noetherian. Let μ be a fuzzy ideal of 𝒜 such that Im(μ)⊆T. From Im(μ) is a well-ordered subset of . Then, since 1 > t1 > t2 > ⋯>tn > ⋯>0 and Im(μ)⊆{t1, t2,…} ∪ {0}, there exists k ∈ ℕ such that Im(μ)⊆{t1, t2,…, tk} ∪ {0}.(a) ⇒ (b): Assume that 𝒜 is not Noetherian. Then there exists a strictly ascending sequence I1 ⊂ I2 ⊂ ⋯ of ideals of 𝒜. Define a fuzzy set μ in A byI0 = ∅. By μ is a fuzzy ideal of 𝒜. This is a contradiction with our assumption. Thus 𝒜 is Noetherian.(b) ⇒ (a): Assume that (b) is true. Suppose that 𝒜 be a BN-algebra and let T = {t1, t2,…} ∪ {0,1}, where (tn) is a strictly increasing sequence in . Then the following conditions are equivalent: 𝒜 is Artinian,μ of 𝒜, if Im(μ)⊆T, then there exists k ∈ ℕ such that Im(μ)⊆{t1, t2,…, tk} ∪ {0,1}.for each fuzzy ideal Let ti1 < ti2 < ⋯<tim < ⋯ is a strictly increasing sequence of elements of Im(μ). Let Um = U for m ∈ ℕ. It is immediately seen that U1⊃U2⊃⋯⊃Um⊃⋯ is a strictly descending sequence of ideals of 𝒜. This contradicts the assumption that 𝒜 is Artinian.(a) ⇒ (b): Suppose that 𝒜 is not Artinian. Then there exists a strictly descending sequence I1⊃I2⊃⋯⊃In⊃⋯ of ideals of 𝒜. Define a fuzzy set μ in A byμ(0) = 1 ≥ μ(x) for all x ∈ A; that is, (d1) holds. Now we show that μ satisfies (d2). Let x, y ∈ A. We have three cases.(b) ⇒ (a): Assume that (b) is true. Suppose that Case 1  (x ∉ I1). Then x∗y ∉ I1 or y ∉ I1. Therefore μ(x∗y)∧μ(y) = 0 = μ(x).Case 2  (x ∈ In − In+1 for some n ∈ ℕ). Then x∗y ∉ In+1 or y ∉ In+1. Hence μ(x∗y) ≤ tn or μ(y) ≤ tn. Therefore μ(x∗y)∧μ(y) ≤ tn = μ(x). Case 3  (x ∈ ⋂{In : n ∈ ℕ}). It is obvious.μ is a fuzzy ideal of 𝒜. This contradicts our assumption. Thus 𝒜 is Artinian.Thus 𝒜 be a BN-algebra. If, for every fuzzy ideal μ of 𝒜, Im(μ) is a finite set, then 𝒜 is Artinian.Let The following example shows that the converse of p be a prime number. Set A = {z ∈ ℂ : zpn = 1 for some n ≥ 0}. It is known that  is the p-quasicyclic group. Define x∗y = xy−1 for all x, y ∈ A. By 𝒜 =  is a BN-algebra. Let In = {z ∈ ℂ : zpn = 1} for n ∈ ℕ ∪ {0}. It follows easily that I is an ideal of 𝒜 if and only if I = A or I = In for some n ≥ 0. We have I0 = {1} ⊂ I1 ⊂ ⋯⊂In ⊂ ⋯⊂A and hence 𝒜 is Artinian. Define μ byI−1 = ∅. Since A = ⋃n∈ℕIn, μ is a fuzzy set in A. By the proof of μ ∈ ℱId(𝒜). However, Im(μ) = {1/n : n ∈ ℕ} is not a finite set.Let This paper begins by considering the notion of ideals in BN-algebras. We give its characterizations and introduce the concept of normal ideals investigating its properties. We also define the notion of normal congruences proving that there is a one-to-one correspondence between normal ideals and normal congruences of a BN-algebra. Moreover we obtain the isomorphism theorem for BN-algebras. Next, we define the notion of fuzzy ideals of BN-algebras giving its characterizations and providing conditions for a fuzzy set to be a fuzzy ideal. We give the relationships between ideals and fuzzy ideals of a BN-algebra and also provide the homomorphic properties of fuzzy ideals. Finally, we display characterizations of Noetherian BN-algebras and Artinian BN-algebras via fuzzy ideals.The next step in studying fuzzy ideals in BN-algebras may be introducing and investigating the notions of fuzzy maximal ideals and fuzzy prime ideals of BN-algebras.


Application for the sequence { The sinc function is defined to beThis function plays a key role in many areas of mathematics and its applications –6.x = π/2.The following result that provides a lower bound for the sinc is well known as Jordan inequality :(2)sincThis inequality has been further refined by many authors in the past few years –35.In , it was We noticed that the lower bound in  is the fTo the best of the authors' knowledge, few results have been obtained on fractional lower bound for the sinc function. It is the first aim of the present paper to establish the following fractional lower bound for the sinc function.x ∈ , one hasFor any m, the following Carleman type inequality holds:an⩾0, n = 1,2, 3,…, with 0 < ∑n=1∞an < ∞, whereIn , Yang prbn}n=1∞ has very interesting properties. Yang [n,From a mathematical point of view, the sequence {es. Yang  and Gylles. Yang  have proIn , the autbn}n=1∞.As an application of n⩾2, one hasFor any positive integer The proof is not based on . We firsx ∈ ⩾0. Elementary calculations reveal thatt < 1/3, we haveSet equality  is equivquality =t2(−x) and 16π4/(3π2+x2)2, we find thatx2/π2 = t, 0 ⩽ t ⩽ 1. Consider the function f(t) defined byf(0) = 2/9 and f(1) = 0. n⩾4,n⩾4, we havec with  such that g(c) = 0. Using Maple, we find that on the open interval  the equation g(t) = 0 has a unique real root t1 ≈ 0.89⋯.By using the power series expansions of sin(ind that1+sinc(x)Hence, from  we get, a, a31)f(f((31)f = 2/π, H(1) = 1, and H is concave up, it follows thatTaking the natural log giveslnH(x)=(1.Using  from (42ows thatH(x)⩽2(1−


The inverse of (x + 1) enters into  = (1 + x)→(1 + x)−1;expansion, (1 + x)−1 = 1 − x + x2 − x3 + x4 − ⋯ (|x | <1); integration, ⇒x − x2/2 + x3/3 − x4/4 + ⋯;term-by-term limits x = 1 minus x = 0, ⇒(1 − 1/2)+(1/3 − 1/4)+(1/5 − 1/6)⋯.taking theFor ers into  and we pm = 2:f(n)∶ = (−1)n+1/n. Note that after the integration, the limit x = 1 can be taken. Equation (x2 + 1), separating the roots ±1 and ±i; now (1 + x2)−1 enters into .m = 2,4) for any integer m ∈ N (in principle) and include (when possible) the appropriate Dirichlet character.In this paper, as said, we will generalize the constructions above  ≡ 1 + 1/2 + 1/3 + ⋯+1/N + ⋯+1/(3N − 2) + 1/(3N − 1) + 1/3N; so these sums diverge, for N arbitrarily large, and indeed as  for i = 1,2,…, m as displaced sums:Also, we definemm = 1/m + 1/2m + ⋯+1/mN = (1/m)(1 + 1/2 + ⋯+1/N) soAnd finally, ∑m(N) (in (mi(N) with i : {1,2,…, m}:Therefore, ∑m(N) (in ) is the Note that these sums make sense, spite diverging, because all have only positive terms and they are finite sums.N): divergences are no higher than logarithmic. We will try to be careful in subtracting two divergent series. We use a philosophy close to physics : we first truncate the series, taking a fixed upper bound N ≫ 1 . Then we subtract other series, also divergent but with a similar type of divergence : the result should be convergent (radiative corrections); see, for example, the book by Schwinger [The convergence or divergence of the above series is usually self-explained: convergence occurs always for the decreasing alternative series (Leibniz), but as the convergence is not absolute, but conditional, the ordering in the series should be maintained. The higher divergence behaves, if at all, with a factor ∝log result for the case m = 2   by four (1) Redundance (Direct) Proof. ∑2(N) and ∑22(N), defined above, diverge in a known way, for fixed (and large) N and, as said, )((1) ReSo(2) Integration Proof. Trivial here, as it is already ∫01(3) Summation Proof. We have the convergent series (so now N = ∞) digamma function Ψ(z)  is the ordinary, Euler's gamma function), with help of the expression [z = −1/2. In total we get, as expected,(4) Use of Hansen Formula. This formula, again depending on the Digamma function Ψ(z), reads  N = ∞)1∑21−∑22=1n′ above in ∑1(1/n′(n′ − 1/2)) as (n′ = n + 1), because n runs now from zero; hence this is equivalent to  = −γ and Ψ(1/2) = −γ − 2log(2))We put our alent to  with the Mathematica Computer Programs give many integrals and double sums also directly.Of the four methods, the most common and “easy” is the second (integration); it can, in principle , always be used. Them = 2 case that what appears as result is the sign function f(n)∶ = ∑n=1∞((−1)n+1/n); later, for some m > 2, (including m = 4) this will become a Dirichlet character: here we have, as résumé (to repeat),Please note in this  m = 3 first. Now (x3 − 1) = (x − 1)(1 + x + x2) as the roots are +1, ω = exp(2πi/3) and conjugate ω is equivalent to a plane rotation by 120°. By direct integration, we obtain at onceWe discuss  positive integer m we obtain, after an easy calculation (ω is root of xm − 1 = 0),For a generalm = 3,4, and 6, yielding (This formula  can be uyielding  and(29)I = ∫01dx(1/(1 − Trωx + x2)) for ω = exp(2πir/m), for r natural between 1 and m − 1. The integral is nowlater, we shall need also m = 3, the specific series is 1 − S + S2 − S3 + S4 − ⋯, with S ≡ x + x2; after developing, we get Coming back to 31 − ∑32 finishes the calculation for m = 3 .Hence ∑χ2(3), we haveχ2(3) =  (so 3-periodic). We can even use Dirichlet's L-functions, which will include the 1/n factors, but we refrain from doing that, as it does not illuminate the matter any further.In terms of the  Dirichle recapitulation, the series are obtained from the polynomial of roots ≠1:x = 1. In principle, the result of the series summation can be also obtained from the Hansen formula : complete solution for the m = 3 case. We quote the following four divergent series for later use, still in this m = 3 case (the limit → meaning just N ≫ 1):Finally for this m = 4, we have first the natural cyclotomic expressionm = 3, our first final result is here:χ2m = 4) =  being periodic mod 4 (and restricted multiplicative). Note also why do we get i = 1 and 3 in ∑4i (not 2!) mod 4: the expansion is for 1/(1 + x2) ≈ 1 − x2 + x4 − x6, and so forth, so it is with even powers only, so with only odd powers after integration!Now, for 4, ∑40, and ∑42 are automatically obtainable.We are done with this, as ∑ second factorization of x4 − 1 is obtained by separating only the x = 1 root:TheB∶ = (1 + x + x2 + x3) contains the x = −1 root, one writes B = (x + 1)(1 + x2), where the integral can be computed at once . The final result for factorization (II) is redundant result, as defines ∑41 − ∑42 computable from the above calculation (I). In fact, we end up this m = 4 case by writing the analogy to ∑44m implies only a difference calculation, as it was also the case for m = 3, prime.The nonprime structure of m = 4 case has one redundancy (two factorizations), and it is done (solved) once a single calculation is made; for example, ∫01dx(1/(1 + x2)) = π/4.So the full solution for the m = 6. But here there are also several factorizations, as for the m = 4 case above. The simplest is (perhaps)Now it is the turn of x3) = (1 + x)(1 − x + x2), the integral is easy, with this factoring:As  = 1 + x3 yield the infinite but convergent sumThe operations INV, EXP, INT, and TAK  jump, now by three: it is due to the cubic xn3 terms in 1/(1 + x3).Notice again the redundant; we just write them:All other factorizations are therefore61, as ∑64 is directly computable , as ∑63 is again directly computable.Redundancy arises because from  one obtamputable ; so, in x6 − 1) = (x3 − 1)(x3 + 1) the simplest factorization (x3 + 1) = (x + 1)(1 − x + x2) yields, for P2(x) = 1 − x + x2, the double series expansion redundant calculation, as included in the (summed) ∫01dx(1/(1 + x3)).It is remarkable here that in the original expansion  = Rotation by 72° and ω2 = exp(4πi/5) = Rotation by 144°.Here the cyclotomic equation is, for We obtain easilyT ≡ ∫01(dx/(1 + x + x2 + x3 + x4)) can be done as the real denominator splits in two real ones and quadratic, but we abstain to write explicitly (ω = eπi/52)The integral x + x2 + x3 + x4) = ⋯ = (1 − 1/2)+(1/6 − 1/7)+⋯ = ∑n=1∞(1/(5n − 4) − 1/(5n − 3)) still computable  but still too long. The final result will beThe expression for the integral is too long to be written. In summation terms, it is 1/ function Mathematica) isThe simplest summation /2 pairs of complex conjugate roots; “a priori,” the only integral versus series is the simplest case, generalizing the above result:This is the general trend for prime numbers p − 1)/2 coincides also with the Euler number.The number  = 8 − 4 = 4:We deal now with some composite numbers. Compound numbers are easier; we just add a calculation for x8 − 1) = (x2 − 1)(1 + x2 + x4 + x6). It is equivalent toAnother factorization is (x8 − 1) = (x − 1)(1 + x + x2 + ⋯+x7), which yields ∑81 − ∑82, so the remaining ∑87 is obtained by difference . The case m = 8 is potentially resolved. Again, we refrain from elaborating.Still, a third factorization is  in the literature (like ∫dx(1/(1 + x)) = log(2)) to very complicated cases, still feasible: the integrals have denominators factoring in quadratic ones, and the series are of the type ∑(1/(an2 + bn + c)), computable, in principle, by means of the Hansen's formula.Asx + x2 + x3 + ⋯+xq) we identify it with the series ∑q+11; this is correct, but we have checked it “case by case,” offering no general proof, and so forth. Also we feel that some new series might perhaps appear, whenever the quadratic components offer an integer expansion: those are two questions for the future.There are, however, some questions left in our work: for example, the series for 1/(1 +


Tolypocladium are herein revised based on the most comprehensive dataset to date. Two species-level phylogenies of Tolypocladium were constructed: a single-gene phylogeny (ITS) of 35 accepted species and a multigene phylogeny  of 27 accepted species. Three new species, Tolypocladium pseudoalbum sp. nov., Tolypocladium subparadoxum sp. nov., and Tolypocladium yunnanense sp. nov., are described in the present study. The genetic divergences of four markers  among Tolypocladium species are also reported. The results indicated that species of Tolypocladium were best delimited by rpb1 sequence data, followed by the sequence data for the rpb2, tef-1α, and ITS provided regions. Finally, a key to the 48 accepted species of Tolypocladium worldwide is provided.The taxonomy and phylogeny of the genus  Tolypocladium was originally described as an anamorph genus by Gams in 1971 to accommodate three species collected from soil: T. cylindrosporum W. Gams, T. geodes W. Gams, and T. inflatum W. Gams [T. lignicola G.L. Barron, T. parasiticum G.L. Barron, and T. trigonosporum G.L. Barron, all of which were isolated from bdelloid rotifers, were added to this genus [T. nubicola and T. tundrense from soil in 1983 [Tolypocladium: T. balanoide , T. microsporum (basionym: Verticillium microsporum) and T. niveum (basionym: Pachybasium niveum). Additionally, Bissett [T. niveum were similar to those of T. inflatum. Because T. niveum precedes T. inflatum, Bissett proposed that T. inflatum be synonymized with T. niveum [T. inflatum produces cyclosporine and is the type species of the genus Tolypocladium. The name T. inflatum is also commonly accepted [T. inflatum with T. niveum [Tolypocladium is morphologically characterized by sparingly branched conidiophores, swollen phialides, and one-celled conidia borne in slimy heads. Approximately 20 species have been included in the Tolypocladium based on morphological characteristics.is genus ,3,4. Bis in 1983  and reas Bissett  noted th. niveum . Howeveraccepted . Therefo. niveum . The genTolypocladium has been discussed extensively for decades. Cordyceps sensu lato was recently reclassified into three families  and four genera  based on multigene phylogeny [Tolypocladium species fall within the Ophicordycipitaceae [Elaphocordyceps Sung and Spatafora 2007 was proposed for 23 species of the Cordyceps Fr. (1818: 316); these species parasitize the fungal genus Elaphomyces and some species of arthropods  [Elaphocordyceps species within the Ophiocordycipitaceae form a clade sister to those of the genus Ophiocordyceps. Gams established the Chaunopycnis to accommodate C. alba, which morphologically resembles Tolypocladium species in its conidiogenesis [Tolypocladium was chosen over Elaphocordyceps and Chaunopycnis as Tolypocladium is the oldest and most commonly used name [Chaunopycnis was integrated into the genus Tolypocladium. Accordingly, C. alba, C. ovalispora, and C. pustulata were renamed T. album, T. ovalisporum, and T. pustulatum, respectively [The taxonomy of hylogeny . Moleculpitaceae ,8. The gogenesis . With thTolypocladium records, including 5 varieties, are listed in the Index Fungorum . Tolypocladium balanoides, which was reassigned to Drechmeria , and Tolypocladium parasiticum, which was reassigned to Metapochonia (as Tolypocladium parasiticum), should be excluded from the Tolypocladium. However, some of these records are doubtful, because the original identifications were presumptive based on host associations or based on the morphology of only one or two ascospore stages of the asexual or sexual morph. For 16 species, no molecular data are available in the GenBank database [Tolypocladium species have a cosmopolitan distribution and a broad host range that includes bdelloid rotifers, mosquito larvae, nematodes, fireflies, beetles, cicada nymphs, batmoth larvae, macrocystic fungi, Ophiocordyceps sinensis, and even plants (as endophytes) [At present, 53 database . Tolypocophytes) ,18,19.Tolypocladium species have been widely studied due to their importance in the medicinal domain. These species can produce cyclosporine A, tolypoalbin, tolypin, cyclosporine D hydroperoxide, cylindromicin, and tolyprolinol [T. inflatum, is widely used in autoimmune disease treatment and to prevent allograft rejection [T. album [Tolypocladium sp. FKI-7981, contains a rare moiety prolinol and was the first natural product isolated from Tolypocladium species. Tolyprolinol exhibits moderate antimalarial activity without cytotoxicity or any other antimicrobial properties [prolinol ,21, all prolinol . Cyclospejection ,24,25. TT. album . TolypinT. album . Like koT. album . Tolyprooperties .Tolypocladium. Therefore, the diversity of Tolypocladium may be underestimated. In the present study, we aimed first to investigate and document the worldwide diversity of Tolypocladium fungi using our current collection of specimens and data collected over the last several years. We used comprehensive morphological and molecular phylogenetic reconstructions to identify and reevaluate our specimens. Based on these reconstructions, we herein describe and illustrate three new taxa. We then clarify the phylogenetic affinities of these new taxa using rDNA sequence analyses.Recent investigations and phylogenetic analyses have ascribed many new taxa to Tolypocladium species were collected in Kunming, Pu’er, Yunnan, China. Voucher specimens and the corresponding isolated strains were deposited in the Yunnan Herbal Herbarium (YHH) and the Yunnan Fungal Culture Collection (YFCC), respectively, of Yunnan University, Kunming, China.Tolypocladium strains were isolated from soil samples, as described in our previous publication [lication . In brieMorphological studies were performed as described in our previous study . MicromoSSU-CoF and nrSSU-CoR [SSU, the primer pair LR5 and LR0R [LSU, and the primer pair EF1α-EF and EF1α-ER [tef-1α). The primer pair RPB1-5′F and RPB1-5′R and the primer pair RPB2-5′F and RPB2-5′R [rpb1 and rpb2), respectively. The ITS fragment was amplified using the primer pair ITS5 and ITS4 [Total DNA was extracted from the fungal mycelia on PDA plates or from herbarium materials using the modified CTAB procedure . The prirSSU-CoR  was usedand LR0R ,35 was u EF1α-ER ,36 was uRPB2-5′R ,36 were and ITS4 .2+) , 1 μL forward primer (10 µmol/L), 1 μL reverse primer (10 µmol/L), 0.25 μL Taq DNA polymerase , 2 μL dNTP (2.5 mmol/L), 1 μL DNA template (500 ng/μL), and 17.25 μL sterile ddH2O. Amplification reactions were performed in a Bio-Rad T100 thermal cycler . The PCR cycling conditions for the amplification of nrSSU were as follows: 95 °C for 4 min; eight cycles of 94 °C for 50 s, 56 °C for 50 s, and 72 °C for 2 min, with the annealing temperature decreasing 0.5 °C/cycle; 25 cycles of 94 °C for 50 s, 52 °C for 50 s, and 72 °C for 2 min; and 72 °C for 10 min. The nucleotide sequences of ITS, nrLSU, tef-1α, rpb1, and rpb2 were amplified using the following cycling conditions: 95 °C for 4 min; eight cycles of 94 °C for 50 s, 56 °C for 50 s, and 72 °C for 70 s, with the annealing temperature decreasing 0.5 °C/cycle; 25 cycles of 94 °C for 50 s, 52 °C for 50 s, and 72 °C for 70 s; and 72 °C for 10 min. PCR products were purified using a gel extraction and PCR purification combo kit  and sequenced on an automatic sequence analyzer  using the amplification primers.The matrix for the polymerase chain reaction (PCR) was comprised of 2.5 μL PCR 10× buffer  were aligned and manually checked using Bioedit v7.0.9 [http://paup.phylosolutions.com, accessed on 28 August 2022) [To investigate the placement of our samples within  GenBank . Individt v7.0.9 . To idenst 2022) . The resSSU, nrLSU, tef-1α, rpb1, rpb2, and ITS sequences were retrieved from GenBank, and combined with those generated in this study. Taxon information and GenBank accession numbers are given in SSU sequences of some species were excluded from the phylogenetic analyses, and gaps were treated as missing data. After alignment of the five genes individually, the alignments were concatenated. A partition homogeneity test was conducted in PAUP* 4.0a166 [tef-1α, rpb1, and rpb2) and one each for nrLSU and nrSSU [p = 0.02).Phylogenetic analyses were based on a concatenated five-gene dataset and the ITS sequences alone. nr 4.0a166 , and thend nrSSU ,43. The Maximum likelihood (ML) phylogenetic analyses were conducted using RaxML 7.0.3  with theDrechmeria W. Gams and H.-B. Jansson, Harposporium Lohde, Ophiocordyceps Petch, Purpureocillium Luangsa-Ard, Hywel-Jones, Houbraken and Samson, and Tolypocladium. Two species of Polycephalomyces Kobayasi were used as outgroups. ITS analysis was performed on Tolypocladium taxa only. Phylogenetic trees were visualized with FigTree v1.4.0 [The following taxa were included in the five-gene concatenated dataset: e v1.4.0 , edited tef-1α, rpb1, and rpb2 to assess the species boundaries of the 10 Tolypocladium species , Tolypocladium , Purpureocillium , Drechmeria , Harposporium , and Polycephalomyces   were recognized in Tolypocladium   within Osp. nov. , while Taradoxum . T. yunnsp. nov. .Purpureocillium lilacinum CBS 284.36 and Purpureocillium lilacinum NHJ 3497 were chosen as outgroup sequences. The three phylogenetic algorithms  recovered trees with similar topologies  formed an independent lineage with Tolypocladium  the minimum thresholds (p-distances) required to distinguish species within the ectively ; and (2) and ITS .Tolypocladium W. Gams, Persoonia 6(2): 185 (1971). emend. C. A. Quandt et al. IMA Fungus 5: 125 (2014).Synonyms: Chaunopycnis W. Gams, Persoonia 11: 75 (1980).Elaphocordyceps G. H. Sung and Spatafora, Stud. Mycol. 57: 36 (2007).Sexual morph: Stromata are solitary or several, simple or branched. The stipe is tough, dark-brownish to greenish, cylindrical, and abruptly to enlarging in the fertile part. The fertile part is cylindrical to clavate. Perithecia are superficial, wholly or partially immersed, ordinal or oblique in arrangement. Asci are cylindrical with a thickened ascus apex. Ascospores are usually cylindrical, multiseptate, disarticulate into part spores, and are occasionally non-disarticulating. Part spores are cylindrical.Asexual morph: Tolypocladium-like, Chaunopycnis-like, or Verticillium-like. Conidiophores typically are short and bear whorls of phialides. Phialides often have bent necks and are usually swollen at the base. Conidia are ellipsoidal, globose, or reniform, and aggregate in small heads at the tips of the phialides.Tolypocladium pseudoalbum, H. Yu, Y. Wang and Q.Y. Dong, sp. nov., MycoBank: MB 845430.Etymology: Referring to the morphological resemblance of this species to Tolypocladium album, despite its phylogenetic dissimilarity.Type: China, Yunnan Province, Kunming City, Wild Duck Forest Park , from the soil on the forest floor, 10 August 2019, Yao Wang .Teleomorph: Unknown.Anamorph: Colonies on PDA are moderately fast-growing, attaining a diameter of 42–44 mm in 21 days at 22 °C. Colonies pulvinate, with high mycelial density, white or pale yellow, reverse deep yellow. Hyphae branched, smooth-walled, septate, hyaline, 1.1–2.7 μm wide. Cultures readily produce phialides and conidia on PDA after two weeks at room temperature. Phialides arising from aerial hyphae, solitary, 12.3–48.5 × 1.0–2.0 μm, cylindrical, tapering gradually toward the apex, neck 1.4–4.6 × 0.8–1.8 µm. Conidia hyaline, one-celled, globose to broadly ellipsoidal 1.8–3.4 × 1.3–1.9 μm. Chlamydospores present.Habitat: Soil.Known distribution: China.Additional specimens examined: China, Yunnan Province, Kunming City, Songming County, Dashao Village , from the soil on the forest floor, 12 August 2018, Yao Wang (living culture: YFCC 876).Comments: Five species are closely related to T. pseudoalbum sp. nov., i.e., T. pustulatum, T. tropicale, T. endophyticum, T. amazonense, and T. yunnanense sp. nov. This clade is characterized by cylindrical to lageniform phialides, globose to broadly ellipsoidal conidia, and primarily white colonies. The phialides of T. pseudoalbum sp. nov. (12.3–48.5 × 1.0–2.0 μm) are longer than those of T. album (3.5–10 × 1.0–1.5 µm).Tolypocladium subparadoxum H. Yu, Y. Wang and Q.Y. Dong, sp. nov., MycoBank: MB 845431.Etymology: Referring to the phylogenetic placement is closely related to T. paradoxum.Holotype: China, Yunnan Province, Pu’er City, Simao District , from soil on the forest floor, 27 August 2021, Yao Wang .Teleomorph: Not observed.Anamorph: Colonies on PDA are moderately fast-growing, attaining a diameter of 36–38 mm in 21 days at 22 °C. Colonies flocculent, fluffy, with low mycelial density, white or pale yellow, reverse deep yellow. Hyphae smooth-walled, branched, septate, hyaline, 0.8–2.2 μm wide. Cultures produce phialides and conidia on PDA after two weeks at room temperature. Phialides arising from aerial hyphae, solitary, or in verticils of two to four, 5.4–40.1 × 0.9–1.8 μm, cylindrical, tapering gradually toward the apex, neck 3.2–5 × 0.7–1.2 µm. Conidia hyaline, one-celled, ellipsoidal or globose, single or aggregating in heads at the apex of phialides, 2.6–6.5 × 1.0–2.9 μm. Chlamydospores not observed.Habitat: Soil, larvae of cicada.Known distribution: China, Japan.Additional specimens examined: NBRC 106958, Niryo, Takatsuki-shi, Osaka Prefecture.Comments: Our phylogenetic analysis indicates that Tolypocladium subparadoxum sp. nov. is closely related to Tolypocladium sp. and T. paradoxum. The two strains (YFCC 879 and NBRC 106958) formed a distinct lineage. NBRC 106958 was firstly isolated from cicada in Japan by S. Ban  and subsequently isolated from soil in China (YFCC 879). Since no significant morphological differences were found between the Chinese collections and that of Japan , and minor conidia (2.3–4.8 × 1.9–5.2 µm vs 2.6–6.5 × 1.0–2.9 μm)  .Tolypocladium subparadoxum similar to T. dujiaolongae and sharing cicada host, solitary, or verticillate, cylindrical or conical phialides, globose to ovoid conidia, and conidia aggregating mostly in small heads, but the latter differs by its relatively shorter phialides (11–35 × 1.0–2.7 μm vs 5.4–40.1 × 0.9–1.8 μm) [–1.8 μm) . Our phyTolypocladium geodes is also similar to T. subparadoxum in their soil habitats and ellipsoidal or globose conidia. However, T. geodes has relatively shorter phialides (5.6–12.4 × 1.4–2.4 µm) and somewhat minor conidia (1.9–2.4 × 1.6–2.0 µm) [Tolypocladium yunnanense H. Yu, Y. Wang and Q.Y. Dong, sp. nov., Figure 5MycoBank: MB 845432.Etymology:Yunnanense (Lat.) refers to the type locality .Holotype: China, Yunnan Province, Kunming City, Wild Duck Forest Park , from soil on the forest floor, 12 August 2018, Yao Wang .Teleomorph: Unknown.Anamorph: Colonies on PDA are moderately fast-growing, attaining a diameter of 44–46 mm in 21 days at 22 °C. Colonies pulvinate, with high mycelial density, whitish to orange-yellow, reverse deep yellow. Hyphae smooth-walled, branched, septate, hyaline, 1.0–2.4 μm wide. Cultures produce phialides and conidia on PDA after two weeks at room temperature. Phialides are usually curved, solitary, 7.6–62.6 × 0.9–2.3 μm, cylindrical, narrowing slightly or abruptly into a neck, 3–4.2 × 0.5–1 µm. Conidia hyaline, one-celled, elliptical to subglobose, 1.2–2.4 × 0.9–1.9 μm. Chlamydospores present.Habitat: Soil.Known distribution: China.Additional specimens examined: China, Yunnan Province, Pu’er City, Simao District , from soil on the forest floor, 7 October 2019, Yao Wang (living culture: YFCC 878).Comments:Tolypocladium yunnanense sp. nov. is characterized by its solitary cylindrical phialides (7.6–62.6 × 0.9–2.3 μm), elliptical to subglobose conidia (1.2–2.4 × 0.9–1.9 μm), and white colonies. The five-gene phylogenetic analysis suggested that T. yunnanense sp. nov. was closely related to five other species . Phylogenetic analyses of this clade using ITS sequences, for which more complete data were available, showed that T. yunnanense sp. nov. formed clade with T. album, T. pseudoalbum sp. nov., T. tropicale, T. amazonense, and T. endophyticum. Morphologically, Tolypocladium yunnanense sp. nov. has longer phialides than other species in this clade: Tolypocladium yunnanense sp. nov., 7.6–62.6 × 0.9–2.3 μm; T. pustulatum, 4–10 × 2–4 µm, T. tropicale, 4.6 × 1.5 µm; T. endophyticum, 4.1 × 1.6 µm; T. amazonense, 4.1 × 1.6 µm; T. pseudoalbum sp. nov., 12.3–48.5 × 1.0–2.0 μm, and T. album, 3.5–10 × 1.0–1.5 µm.Tolypocladium species worldwideKey to Sexual state observed…………………………………………………………………………………………………………………………..1Sexual state not observed…………………………………………………………………………………………………………………..27 1a. Perithecia superficial or half-immersed ……………………………………………………………………………………………..2 1b. Perithecia completely immersed………………………………………………………………………………………3 T. inegoense2a. Perithecia pyriform, relatively larger, 520–550 × 260–280 µm, asci relatively larger, 400–450 × 7–7.5 µm, part spores 2.5–3.0 × 3.0 µm, on cicada nymphs, stromata relatively longer, 14 cm long…………………………………………………………………………………………………………… Elaphomyces, stromata shorter, 3.5–4.5 cm long……………………………T. ramosum2b. Perithecia ovoid, relatively smaller, 320–380 × 220–280 µm, asci cylindrical, smaller, 240–250 × 6 µm, not dissociate into part spores, on  3a. Perithecia ellipsoid, subglobose to ovoid…………………………………………………………………………………….4 3b. Perithecia ampullaceous……………………………………………………………………………………………………………………………..26 Scarabaeidae ; soil, humus, Picea glauca, roots of Picea mariana, the surface of Mycobates sp. , the sclerotium of Ophiocordyceps gracilis ………………………………………………………………….………..T. inflatum4a. From multiple substrate/host ……………..………………………………… T. flavonigrum16a. Perithecia relatively narrower, 567–697 × 206–248 µm, part spores smaller, 2–5 × 1.5–2 µm, stromata 1.5–3 cm long…………………………………………………………………………………………………..…… T. japonicum16b. Perithecia relatively wider, 500–700 × 250–350 µm, part spores larger, 10–18 × 2.5–4 µm, stromata 2.5–7 cm long………………………………………………………………………………………………………..… 17a. Perithecia larger………………………………………………………………………………………………………18 T. virens17b. Perithecia smaller, 400 × 250 µm…………………………………………………………………….……… T. longisegmentatum18a. Stromata 12 cm long, part spores very long, 40–65 µm long.……………………………….. 18b. Stromata shorter than 12 cm, part spores < 40 µm long……………………………………………………………19 19a. Part spores ≤ 8 µm long……………………………………………………………………………………………….20 19b. Part spores > 8 µm long…………………………………………………………………………………………….…22 T. intermedium20a. Asci shorter than 300 µm (240–300 × 7–8 µm), perithecia relatively smaller (450–540 × 230–260 µm).…………………………………………………………………………………………………….… 20b. Asci longer than 300 µm, perithecia larger……………………………………………………………………..…21 T. fractum21a. Stipe slender, 0.5–1.0 mm thick, yellowish green to olivaceous, stromata shorter, 1.5–2.5 cm long, part spores, 2–5 × 1.5–2 µm………………………………………………………………………………………………… T. valliforme21b. Stipe 1–5 mm thick, dark brown, smooth or furfuraceous, stromata 5–7 cm long, part spores longer, 3–8 × 2 µm…………………………………………………………………………………………………………… T. delicatistipitatum22a. Perithecia < 550 µm long (480–540 µm)………………………………………………………. 22b. Perithecia > 550 µm long……………………………………………………………………………………………..23 T. miomoteanum23a. Part spores < 15 µm long (8–11 µm)……………………………………………………………….. 23b. Part spores ≥ 15 µm long…………………………………………………………………………….……………….24 24a. Part spores < 3 µm wide……………………………………………………………………………………………..25 T. inusitaticapitatum24b. Part spores ≥ 3 µm wide (3.0–4.5 µm)………………………………………………………… T. capitatum25a. Fertile part olive-brown to olive-black, perithecia relatively larger, 650–950 × 250–420 µm, asci wider, 350–540 × 10–12 μm, part spores cylindrical or somewhat fusoid, 8–25 × 2.5–3 µm……………..…………… T. rouxii25b. Fertile part purple-brown, blacker when older, perithecia smaller, 600–750 × 200–300 µm, asci slender, 350–500 × 8–10 µm, part spores filiform, spindle-shaped, 15–20 × 2–3 µm………………………………. T. dujiaolongae26a. Perithecia relatively shorter, 520–740 × 300–330 μm, part spores cylindrical, 3–7 × 2–3 μm, on cicada nymphs…………………………………………………………………………………………………… Elaphomyces…………………………………………………………………………………………………………T. minazukiense26b. Perithecia relatively longer, 900–930 × 220–250 µm, part spores fusoid, 16–18 × 3 µm, on  27a. From multiple substrate/host…………………………………………………………………………………..……28 27b. From only a type of substrate/host………………………………………………………………………………….31 28a. Phialides cylindrical…….……………………………………………………………………………………………29 T. cylindrosporum28b. Phialides ellipsoidal to subglobos…………………………………………………………….… T. album29a. Colonies white, conidia globose to ovoid ……………………………………………………………………………………………………….………… 29b. Colonies white to pale yellow, conidia ellipsoidal, globose or broadly ellipsoidal……………………………30 T. pustulatum30a. Phialides 4–10 × 2–4 µm, conidia 2–3 × 1.5–2.5 µm………………………………………………… T. subparadoxum30b. Phialides 5.4–40.1 × 0.9–1.8 μm, conidia larger, 2.6–6.5 × 1–2.9 μm…………………………… 31a. From substrate…………………………………………………………………………………………………….….32 31b. On insects……………………………………………………………………………………………………………..45 32a. Substrate is not fungus………………………………………………………………………………………………33 32b. Substrate is fungus……………………………………………………………………………………………………43 33a. From plant tissue……………………………………………………………………………………..………………34  33b. From soil…………………………………………………………………………………………………………………………37 T. endophyticum34a. Conidia relatively more minor ………………………………………………………. 34b. Conidia larger, diam > 1.3 µm……………………………………………………………………………….………35 T. ovalisporum35a. Conidia > 4 µm long (4.5–9.0 × 2.5–3.5 µm)…………..……………………………………………… 35b. Conidia < 4 µm long……………………………………………………………………………………………….…36 T. amazonense36a. Phialides 4.6 ± 1.2 × 1.5 ± 0.3µm, conidia spherical, larger, 2.1–2.2 µm diam………………………. T. tropicale36b. Phialide 4.6 × 1.5 µm, conidia spherical, relatively smaller, 1.5 ± 0.1 µm diam……………………………… 37a. Phialides cylindrical………………………………………………………………………………………………….38 37b. Phialides subglobose or ellipsoidal…………………………………………………………………………………41 38a. Conidia ellipsoidal, globose or broadly ellipsoidal………………………………………………….……………39 T. microsporum38b. Conidia asymmetrically flattened, with a minute apical……………………………………….… 39a. Colonies white………………………………………………………………………………………………………..40 T. pseudoalbum39b. Colonies white or pale yellow ………………………………………………………….……………………………………………. T. geodes40a. Phialides shorter, 5.6–12.4 × 1.4–2.4 µm, conidia 1.9–2.4 × 1.6–2.0 µm……………………………………………….. T. yunnanense40b. Phialides longer, 7.6–62.6 × 0.9–2.3 μm, conidia 1.2–2.4 × 0.9–1.9 μm…………………………………….. 41a. Conidia only one type…………………………………………………………………………..……………………42 T. tundrense41b. Conidia two types  …………………………………………………………..…………………………………………… T. nubicola42a. Phialides relatively longer, 4.4–7.8 × 1.5–2.7 µm, conidia cylindrical, 2.6–4.1 × 0.8–1.3 µm, colonies white to pale cream…………………………………………………………………………………………………….. T. terricola42b. Phialides shorter, 2.8–3.5 × 2.0–3.0 µm, conidia broadly oval, 2.5–3 × 2.0–2.5 µm, colonies white…. Elaphomyces………………………………………………………………………………………T. guangdongense43a. On  Ophiocordyceps sinensis……………………………………………………………………………………………………4443b. From  T. reniformisporum44a. Conidia reniform, 1.0–3.2 × 0.7–1.6 µm, phialides 3.4–10.6 × 1.1–3.8 µm………………………. T. sinense44b. Conidia spherical, 1.4–3.6 µm diam, phialides 7.6–19.4 × 2.9–3.6 µm.…………………………………… T. extinguens45a. On mosquito larvae, conidia two types ……………………………………………………………………………..  45b. On bdelloid rotifers, conidia only one type……………………………………………………..…………………46 T. lignicola46a. Phialides thicker, 4–8 × 3–4.5 µm, conidia circular, 2.5–3.2 × 1.5–2.0 µm, colonies pure white………T. trigonosporum46b. Phialides slender, 4.8–9.8 × 1.4–3.5 µm, conidia like an equilateral triangle or less ellipsoidal, 2–3 × 1.3–1.7 µm, colonies white or pale yellow……………………………………………………………………………………Tolypocladium is one of the most diverse fungal groups in terms of shape, substrate or host, and habitat range. Many new species have recently been added to Tolypocladium [T. pseudoalbum sp. nov., T. subparadoxum sp. nov., and T. yunnanense sp. nov.) based on phylogenetic analyses and morphological characteristics. Phylogenetically, these three species fell within the Tolypocladium clade, while morphologically all three species possessed cylindrical phialides and ellipsoidal or globose conidia. It is challenging to distinguish species of Tolypocladium based only on morphological characteristics, because several species in this genus are morphologically cryptic [ocladium ,13,14,73 cryptic ,11. Sexu cryptic . HoweverTolypocladium play a significant role in a variety of artificial and wild ecosystems and may participate in antifungal, host–fungi, and insecticidal interactions [Tolypocladium based on host associations or morphology [Tolypocladium was developed based on morphological characteristics. However, the advent of molecular biology, which was an important scientific milestone, revolutionized the taxonomic characterization of this genus. Over the last few decades, the number of accepted species in Tolypocladium has doubled.Species of ractions ,77. Manyrphology ,12. Overrphology . In mostTolypocladium were included in the key developed in this study. However, because the sequence loci for many of these taxa were incomplete, only 27 species were included in the multigene phylogenetic analyses , Tolypocladium species are mainly known from insects [Tolypocladium species in the soil and in plant roots. Recently, Tolypocladium species in Chinese soils were surveyed, but no new species were identified.athogens ,12. Beca insects , and few


We also introduce and study a new type of generalized open sets in GTSs, called β-open sets, and apply them to obtain more properties of μ-β-Lindelöf sets.We introduce and study  The primary purpose of this article is to introduce and study μ on a non-empty set X is a collection of subsets of X such that μ is closed under non-empty arbitrary unions. For a GT μ on X, the pair μ will be called μ-open sets, and a subset A of μ-closed if μ-open. A space A is a subset of a space μ-closure of Aμ-closed sets containing A, and the μ-interior of Aμ-open sets contained in A. A GT μ on X is called a strong GT μ-space μ is a strong GT. Strong GTs were introduced by Lugojan A generalized topology (GT) A will stand, respectively, for the closure of A in X and the interior of A in X. A subset A of a topological space β-open α-open β-open, α-open) subsets of a topological space μ is a strong GT on X.If A, A.Throughout this article, the set-theoretic framework is ZFC, that is the Zermelo-Fraenkel system of axioms (ZF) together with the Axiom of Choice (AC); for a set 2We recall the following definitions and facts for their importance in the material of our article.Definition 2.1A be a subset of a space A is calledμ-semi-open if (i) μ-preopen if (ii) μ-β-open if (iii) μ-α-open if (iv) μ-semi-open sets by σ, the class of μ-preopen sets by π, the class of μ-β-open sets by β, and the class of μ-α-open sets by α. It was pointed out in σ, π, α, and β is a GT. However, it is easy to see that for a space σ and β is a strong GT, while π and α need not.For a space Proposition 2.2Letbe a space. Then(i);(ii);(iii);(iv).Definition 2.3A of a μ-space μ-Lindelöf if any cover of A by μ-open subsets of X has a countable subcover.μ-space μ-Lindelöf if any cover of X by μ-open sets has a countable subcover.(ii) A Definition 2.4A of a topological space A by preopen subsets of X has a countable subcover.(i) A subset X by preopen subsets of X has a countable subcover.(ii) A topological space Definition 2.5A of a μ-space μ-Lindelöf if any cover of A by μ-preopen subsets of X has a countable subcover, that is A is μ-space μ-Lindelöf if any cover of X by μ-preopen sets has a countable subcover, that is (ii) A Definition 2.6A of a topological space XA by semi-open subsets of X has a countable subcover.(i) A subset X by semi-open subsets of X has a countable subcover.(ii) A topological space Definition 2.7A of a space μ-semi-Lindelöf relative to XA by μ-semi-open subsets of X has a countable subcover, or equivalently A is A is μ-semi Lindelöf to mean A is μ-semi Lindelöf relative to X.(i) A subset μ-semi-Lindelöf X by μ-semi-open sets has a countable subcover, or equivalently (ii) A space Proposition 2.8Let A be a subset of a topological space X. Then(i)A is semi-open if and only if there exists an open set U such that.(ii)A is preopen if and only if, where U is open and D is dense.(iii)A is β-open if and only if, where S is semi-open and D is dense.3μ-β-Lindelöf sets in GTSs.This section is mainly devoted to introducing and studying Definition 3.1A of a topological space β-Lindelöf relative to A by β-open subsets of X has a countable subcover.β-Lindelöf if any cover of X by β-open subsets of X has a countable subcover.(ii) A topological space Definition 3.2A of a space μ-β-Lindelöf if any cover of A by μ-β-open subsets of X has a countable subcover, that is A is (i) A subset μ-β-Lindelöf if any cover of X by μ-β-open sets has a countable subcover, that is (ii) A space Remark 3.3μ-β-Lindelöf subsets of a space μ-β-Lindelöf.It is easy to see that the countable union of The following two remarks follow from Remark 3.4A be a subset of a space Let A is μ-β-Lindelöf if and only if A is β-Lindelöf;(i) A is μ-β-Lindelöf if and only if A is β-Lindelöf;(ii) A is μ-β-Lindelöf if and only if A is β-Lindelöf.(iii) Remark 3.5A is a subset of a topological space If The next remark follows from Remark 3.6A be a subset of a space Let A is μ-β-Lindelöf if and only if A is strongly (i) A is μ-β-Lindelöf if and only if A is strongly (ii) μ-β-Lindelöf sets and some other related types.Now, we will study the relationships between Remark 3.7A of a space μ-β-Lindelöf, then A is μ-semi-Lindelöf; in particular, if a space μ-β-Lindelöf, then μ-semi-Lindelöf. However, the converse need not be true even for topological spaces as it can be easily seen from each of the following two examples.Clearly, if a subset Example 3.8X be an uncountable set equipped with the co-countable topology. Then it follows from X is open, but X is Lindelöf, so X is semi-Lindelöf. On the other hand, since the dense subsets of X are the uncountable subsets of X, it follows from X are the uncountable subsets of X. As X is uncountable, there exists a disjoint family X by preopen sets and X is not strongly Lindelöf, and thus, X is not β-Lindelöf.Let Example 3.9X be an uncountable set equipped with the indiscrete topology. Then it follows from X is open, but X is Lindelöf, so X is semi-Lindelöf. On the other hand, since every singleton subset of X is preopen, it is clear that X is not strongly Lindelöf, and thus, X is not β-Lindelöf.Let Remark 3.10A of a μ-space μ-β-Lindelöf, then A is strongly μ-Lindelöf, in particular, if a μ-space μ-β-Lindelöf, then μ-Lindelöf. However, the converse need not true even for topological spaces as it can be easily seen from the following example.Clearly, if a subset Example 3.11X be an uncountable discrete space, X are unchanged and a basic open neighborhood of p has the form A is a countable subset of X. Then it is easy to see that X and A is an uncountable subset of X, is semi-open in X is uncountable, there exists a disjoint family β-Lindelöf.Let A of a space μ-β-closed μ-β-open. The following proposition tells that a μ-β-closed subset of a μ-β-Lindelöf space is μ-β-Lindelöf; the straightforward proof is omitted.Recall that a subset Proposition 3.12Let A be a μ-β-Lindelöf subset of a spaceand B be a μ-β-closed subset of X. Thenis μ-β-Lindelöf. In particular, a μ-β-closed subset A of a μ-β-Lindelöf spaceis μ-β-Lindelöf.Proposition 3.13If every proper μ-β-closed subset of a spaceis μ-β-Lindelöf, thenis μ-β-Lindelöf.ProofX by μ-β-open subsets of X. Choose μ-β-closed subset of X, thus by assumption, μ-β-Lindelöf, so there exist X is μ-β-Lindelöf. □Let μ-β-Lindelöf sets in a subspace; to proceed, we recall the following definition.As for the next issue, we characterize Definition 3.14A be a non-empty subset of a space A is the collection A is the GTS Proposition 3.15Let B be a non-empty subset of a spaceand. Then A is μ-β-Lindelöf if and only if A is-Lindelöf.ProofNecessity. Suppose that A by A by μ-β-open sets, but A is μ-β-Lindelöf, so there exist A is Sufficiency. Suppose that A by μ-β-open sets. Then A, but A is A is μ-β-Lindelöf. □Corollary 3.16Let A be a non-empty subset of a space. Then A is μ-β-Lindelöf if and only if A is-Lindelöf.The proof of the next proposition is straightforward, and thus omitted.Proposition 3.17A non-empty subset A of a spaceis μ-β-Lindelöf if and only if for every familyof μ-β-closed sets having the property that for every non-empty countable subfamilyof,, then.μ-β-Lindelöf sets using filter bases; to proceed, we recall the following definition.Now, we will characterize Definition 3.18X is called a strong filter base on X if whenever X is called a maximal strong filter base on X if whenever X with (ii) A strong filter base X is contained in a maximal strong filter base on X.It was pointed out in Definition 3.19μ-β-open subset U of X such that μ-β-open subset U of X such that A filter base Remark 3.20Let x, then x;(i) If x if and only if x.(ii) If μ-β-Lindelöf sets using filter bases, the straightforward proof is omitted.The next proposition meets our goal for characterizing Proposition 3.21For a non-empty subset A of a space, the following are equivalent:(i) A is μ-β-Lindelöf;(ii) Every maximal strong filter base on X, each of whose members meet A,-converges to some point of A;(iii) Every strong filter base on X, each of whose members meet A,-accumulates at some point of A.μ-β-Lindelöf spaces; to proceed, we recall the following.At the end of this section, we study a sum property of Definition 3.22μ of subsets of Remark 3.23μ be as in μ is a GT on Remark 3.24μ-open and μ-closed for each Proposition 3.25Letbe a-space for each, and letbe the generalized topological sum of. Thenis μ-β-Lindelöf if and only ifis μ-β-Lindelöf for eachand Λ is countable.ProofNecessity. By μ-β-closed for each μ-β-Lindelöf, it follows by μ-β-Lindelöf for each μ-β-open for each μ-β-open sets, but μ-β-Lindelöf, so, Λ is countable.Sufficiency. Follows from 4β-open sets. This study will be useful later to obtain further properties of μ-β-Lindelöf sets in GTSs.In this section, we introduce and study a new type of generalized open sets in GTSs, called Definition 4.1A be a subset of X. Then A is called β-open if whenever μ-β-open set x such that A is called β-closed if β-open. The collection of all β-open subsets of X will be denoted by Let Proposition 4.2(i) If A is a μ-β-open subset of a space, then A is-β-open, that is.(ii) Ifis a space, thenis an-space.(iii) Let A be a subset of a space. Then A is-β-open if and only if A is-β-open.ProofA be μ-β-open and A is β-open.(i) Let β-open, that is β-open for each β-open, there exists a μ-β-open set x such that β-open, that is X is μ-β-open, it follows by (i) that X is β-open, that is (ii) Clearly, ∅ is A is β-open and let μ-β-open set U containing x such that U is β-open. By U is β-open. Thus, A is β-open. Conversely, suppose that A is β-open and let β-open set U containing x such that U is β-open. Thus, there exists a μ-β-open set V containing x such that A is β-open. □(iii) Suppose that The following example shows that if Example 4.3X is an uncountable set and μ is the co-countable topology on X. Since the dense subsets of X are the uncountable subsets of X and since every semi-open subset of X is open, it follows from μ-β-open subsets of X are the uncountable subsets of X. We will show now that X. Let A and B be any uncountable subsets of X such that A and B are μ-β-open, and thus β-open by β-open because if X such that X.Let Remark 4.4A is a countable subset of a space A is β-closed. However, the converse need not be true, in general, even for topological spaces as the following example tells.It is easy to see that if Example 4.5X be an uncountable set equipped with the discrete topology μ. Then clearly every subset of X is μ-β-open; thus by X is β-open. Now let A be a subset of X such that β-closed which is uncountable.Let The next example shows that the converse of Example 4.6X be an uncountable set and A is a non-empty countable subset of X. Then by β-open; however, μ-β-open becauseLet β-open sets in terms of μ-β-open sets and countable sets.The following proposition characterizes Proposition 4.7Letbe a space and A be a subset of X. Then A is-β-open if and only if whenever, there exists a μ-β-open setcontaining x and a countable subsetof X such that.ProofNecessity. Let A be β-open and μ-β-open set x such that Sufficiency. Let μ-β-open set x and a countable subset X such that A is β-open. □Corollary 4.8Letbe a space and A be an-β-closed subset of X. Thenfor some μ-β-closed subset B of X and some countable subset C of X.ProofA be an β-closed subset of X. Then β-open. If μ-β-open set x and a countable subset X such that B is μ-β-closed, and Let β-open μ-β-Lindelöf set.The next proposition describes the shape of an Proposition 4.9Let A be an-β-open subset of a space. If A is μ-β-Lindelöf, thenfor some μ-β-open subset B of X and some countable subset C of X.ProofA is β-open, it follows that for each μ-β-open set x such that A is μ-β-Lindelöf, so there exist B is μ-β-open and C is countable. □Since 5μ-β-Lindelöf sets in GTSs.The primary purpose of this section is to provide more properties and mapping properties of Proposition 5.1Let A be a subset of a space. Then A is μ-β-Lindelöf if and only if A is-Lindelöf.ProofNecessity. Suppose that A by μ-β-open set x such that A by μ-β-open sets, but A is μ-β-Lindelöf, so there exist A, it is covered by a countable subcollection A is Sufficiency. Follows from Proposition 5.2Let A be a μ-β-Lindelöf subset of a spaceand B be an-β-closed subset of X. Thenis μ-β-Lindelöf. In particular, an-β-closed subset A of a μ-β-Lindelöf spaceis μ-β-Lindelöf.Proofμ-β-open sets. Then A by β-open sets, but A is μ-β-Lindelöf, so it follows from μ-β-Lindelöf. □Suppose that μ-β-Lindelöf sets in GTSs.The next issue is to study several mapping properties of κ-open set is μ-open, and called μ-closed set is κ-closed.Recall that a function Proposition 5.3Letbe a-continuous function, whereandare spaces. If A is μ-β-Lindelöf, thenis κ-β-Lindelöf.Proofκ-β-open sets. Then A, but f is β-open for each A is μ-β-Lindelöf, it follows from κ-β-Lindelöf. □Suppose that Corollary 5.4Letbe a-continuous function, whereandare spaces. If A is μ-β-Lindelöf, thenis κ-β-Lindelöf.μ-β-Lindelöf spaces; to proceed, we recall the following.Now, we are ready to study a product property of Proposition 5.5Letandbe GTSs, and let. Thenis a GT on.Remark 5.6λ from (i) (ii) Lemma 5.7Letandbe spaces, and λ be the generalized product topology on. Then the projection(resp.) is-continuous (resp.-continuous).ProofA be a μ-β-open subset of X. Then λ-β-open. Now by λ-β-open. □We will show that the projection Corollary 5.8Letandbe spaces, and λ be the generalized product topology on. Ifis λ-β-Lindelöf, thenis μ-β-Lindelöf andis κ-β-Lindelöf.ProofFollows from Proposition 5.9Letbe a-closed function, whereandare spaces. If for each,is μ-β-Lindelöf, thenis μ-β-Lindelöf whenever A is κ-β-Lindelöf.Proofμ-β-open sets. Then it follows by assumption that for each μ-β-open. Let β-open as f is A by β-open sets, but A is κ-β-Lindelöf, so it follows from μ-β-Lindelöf. □Suppose that Corollary 5.10Letbe a-closed function, whereandare spaces. If for each,is μ-β-Lindelöf, thenis μ-β-Lindelöf whenever A is κ-β-Lindelöf.This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors.Mohammad Shawqi Sarsak: Conceived and designed the experiments; Performed the experiments; Analyzed and interpreted the data; Contributed reagents, materials, analysis tools or data; Wrote the paper.The authors declare no conflict of interest.


Correction to: Microchimica Acta (2021) 189:14https://doi.org/10.1007/s00604-021-05113-4− 1 instead of 100 μg⋅mL− 1. The corrected line should read as: "Briefly, using the lateral flow reagent dispenser, control lines were drawn using 200 μg⋅mL− 1 of mouse IgG and test lines were drawn using a solution consisting of Ab01680 antibodies (1000 μg⋅mL− 1) and ACE2 protein (100 μg⋅mL− 1) at a rate of 0.125 mL⋅min− 1 on nitrocellulose membrane".In this article, there is an error in the concentration of Ab01680 antibodies used. It should be 1000 μg⋅mLThe original article has been corrected.


At the end, an example is given to illustrate the theoretical result.This paper describes the existence and uniqueness of the solution,  Many physical problems can be expressed in mathematical models using differential equations. Differential equations enable us to study the rapid changes in physical problems, for example, blood flows, river flows, biological systems, control theory, and mechanical systems with impact. A system of differential equations with impulses can be used to model several above-listed problems. A few existing results for a general class of impulsive systems were discussed by Ahmad . The theH1 the group and by H2 the metric group with a metric δ and a constant ν > 0. The problem is to study if there exists λ > 0 satisfies for every h : H1⟶H2 such thatf : H1⟶H2 that satisfiesAt the University of Wisconsin, Ulam  proposedf(x+y)=f(x)+f(y), and their solutions have been discussed in several spaces. A linear transformation is a solution of a linear functional equation. By considering the H1 and H2 as Banach spaces, Hyers [The linear functional equations, of the form s, Hyers  discusses, Hyers  and Rasss, Hyers  extendedIn 2012, Wang et al.  studied β-Hyers–Ulam–Rassias stability and generalized β-Hyers–Ulam–Rassias stability of the impulsive difference system of the formH, B ∈ ℝn×n, f ∈ ℂ and Θn ∈ B space of bounded and convergent sequences, ℤ+={0,1,2,…} and 𝕏=ℝn, I={0,1,2,…, n}. In fact, we are presenting a discrete version of the work given in [β-Hyers–Ulam–Rassias stability was discussed for differential equations. With the help of [β-Hyers–Ulam–Rassias stability of the difference equation.In this paper, we will explain the given in , in whic help of , 18, we n-dimensional Euclidean space will be denoted by ℝn along with the vector norm ‖·‖, and n × n matrices with real-valued entries will be denoted by ℝn×n. The vector infinite norm is defined as ‖v‖=maxi≤n1≤|vi|, and the matrix infinite-norm is given as ‖A‖=maxi≤n1≤∑j=1n|aij| where v ∈ ℝn and A ∈ ℝn×n, also vi and aij are the elements of the vector v and the matrix A, respectively. ℂ will be the space of all convergent sequences from I to 𝕏 with norm ‖v‖=supn∈I‖vn‖. We will use ℝ, ℤ, and ℤ+ for the set of all real, integer, and nonnegative integer numbers, respectively. The next lemma is a basic result about the solution of the difference system  is called β-norm, with 0 < β ≤ 1, where 𝕍 is a vector space over the field K, if the function satisfied the following properties:β=0 if and only ℋ=0‖ℋ‖κℋ‖β=|κ|β‖ℋ‖β,  foreachk κ ∈ K and ℋ ∈ 𝕍‖1‖β ≤ ‖ℋ‖β+‖ℋ1‖β,  forall ℋ,  ℋ ∈ 𝕍‖ℋ+ℋA function ‖·‖𝕍, ‖·‖β) is said to be β-norm space.And . A sequence Θn will be an ϵ-approximate solution of Θnβ-Hyers–Ulam–Rassias stable if for every ϵ-approximate solution Yn of system  satisfies  and a sequence hk, k ∈ M satisfyingFrom , it is catisfies  if and oThe solution of n ≥ 0 withfor any γk by γkn, thenIf we replace G1: for f, Ik : ℤX×X+×⟶𝕏, k ∈ I, there exist constants ℒf > 0 and ℒIk > 0, such thatG2:G2∗:To describe the uniqueness and existence of the solution of system , we willG1, G2, and G2∗ are held, then system .If assumptions 𝒜 : ℂ⟶ℂ byDefine ℂ, we haveNow, for Θ, Θ′ ∈ This implies that𝒜 is a contraction map using the Banach contraction principle, we say that system  such thatβ-Hyers–Ulam–Rassias stable over discrete bounded interval, if G1, G2 and G3 are satisfied.System  is β-HyeThe solution of system  is as foYn be the solution of inequality r ≤ (xr+yr), x, y ≥ 0,  for any r ∈ {0,1}, whereNow,β-Hyers–Ulam–Rassaias stable. □Hence, system  is β-Hyeβ-Hyers–Ulam–Rassias stability on an unbounded discrete interval, we must need the following assumptions:To explain G4: the operators family ‖Hn−i‖ ≤ Meω(n − i) and ‖T(n − nk)‖ ≤ Meω(n − nk).G1 and G3 − G8 are holds, then system ΘnYn be the solution of inequality  is 1/2− Hyers–Ulam–Rassias stable with respect to Z+ and Also, ùβ-Hyers–Ulam–Rassias stability of difference equations has been considered as one of the important topics of the literature, in which different types of conditions have been used in the form of inequalities, and most results have been obtained through discrete Gronwall inequality. In this paper, we have investigated the existence and uniqueness of the solution through the Banach contraction principle and β-Hyers–Ulam–Rassias stability of the impulsive difference system with the help of Gronwall inequality.Nowadays, studies on the qualitative behavior of impulsive difference equations have a significant contribution to the literature. In particular, the discussion regarding the


Scientific Reportshttps://doi.org/10.1038/s41598-022-18138-3, published online 30 September 2022Correction to: The original version of this Article contained an error in Figure 2, where the y-axis label “Range mm” in panel (A) was incorrectly described as “Range nm”, and the y-axis units in panel (B) “2.5”, “2.0”, “1.5”, “1.0”, “0.5” and “0.0” were incorrectly described as “25”, “20”, “15”, “10”, “5” and “0”, respectively. The original Figure The original Article has been corrected.