Based on the light relation between a normal subgroup and a complete congruence relation of a group, we consider the homomorphism problem of rough groups and rough quotient groups and investigate their operational properties. Some new results are obtained. Rough set theory, proposed by Pawlak , is an en-ary hypergroups based on a complete residuated lattice. Xiao and Zhang ρ indicates congruence class of a about ρ. Furthermore, if [a]ρ[b]ρ = [ab]ρ, then ρ is called complete.Let G  be a groρ be a congruence relation of G. Then, Nρ = {a ∈ G |  ∈ ρ} is a normal subgroup of G and  ∈ ρ⇔a ∈ bNρ; contrarily, if N is a normal subgroup of G, one can define a congruence relation ρN, where  ∈ ρN⇔a ∈ bNρ; then ρN is a congruence relation of G, and Nρ = {a ∈ G |  ∈ ρN}.Let G and the normal subgroup of G is one to one correspondence.N be a normal subgroup of G; then ρN is a complete congruence relation.Let a, b ∈ G[a]ρN[b]ρN = (aN)(bN) = abN = [ab]ρN. Therefore, ρN is a complete congruence relation.Consider ∀G and the normal subgroup of G is also one to one correspondence.According to ρ be an equivalence relation on G and A a nonempty subset of G. Then, the setsρ-lower and ρ-upper approximations of the set A. And ρ.Let ρ be an equivalence relation on G and A a nonempty subset of G. ThenA.Let ρ be an equivalence relation on G and, for all A, B⊆G, one has the following.; ; If A⊆B, then Let ρ be an equivalence relation on G and ∀A, B⊆G one has the following.; ; If A⊆B, then Let ρ be a complete congruence relation on G and A⊆G; if G, then A is called the lower rough group  of G and G, then A is called the upper rough group  of G and G, then A is called rough group  of G.Let ρ be a complete congruence relation on G, A⊆G, and Nρ⊆A. If A is a subgroup  of G, then A is rough group  of G and Let ρ be a complete congruence relation on G, A⊆G, and Nρ⊆A. Then, G⇔G/ρ.Let ρ be a complete congruence relation on G, A⊆G, and Nρ⊆A. If A is a subgroup  of G, thenLet ρ be a complete congruence relation on G and let A be a subgroup of G and Nρ⊆A. Then, Let Based on the third isomorphism theorem of group, it is easy to prove this corollary.G and T be two groups; f : G → T is a homomorphism. If ρ is a congruence relation on G and Nρ⊇Kerf, then f(ρ) is a congruence relation on T. Further, if f is an injective, then (1)f(ρ) = ρf(Nρ); (2)f(Nρ) = Nf(ρ),where f(ρ) = {(f(a), f(b)) |  ∈ ρ}.Let f(ρ) is a congruence relation on T. Nρ is a normal subgroup on G, then f(Nρ) is the normal subgroup on T. So ∀(f(a), f(b)) ∈ f(ρ)⇔ ∈ ρ⇔a ∈ bNρ⇔f(a) ∈ f(b)f(Nρ)⇔(f(a), f(b)) ∈ ρf(Nρ), so f(ρ) = ρf(Nρ).Because It follows immediately from (1).It is easy to prove f : G → T be an injective homomorphism. If ρ is a complete congruence relation on G and Nρ⊇Kerf, then f(ρ) is a complete congruence relation on T.Let f(ρ) is a congruence relation on T. Now we prove it is complete; ∀a, b ∈ G, we have [f(a)]f(ρ)[f(b)]f(ρ) = (f(a)f(Nρ))(f(b)f(Nρ)) = f(aNρbNρ) = f(ab)f(Nρ) = [f(ab)]f(ρ) = [f(a)f(b)]f(ρ).According to f(ρ) is complete.Therefore, f : G → T be a surjective homomorphism. If ρ is a congruence relation on T, then f−1(ρ) is a congruence relation on G. Further, if f is an injective, thenf−1(ρ) = ρf−1(Nρ);f−1(Nρ) = Nf−1(ρ),where f−1(ρ) = { | (f(a), f(b)) ∈ f(ρ)}.Let f−1(ρ) is a congruence relation on G. Nρ is a normal subgroup on T, then f−1(Nρ) is the normal subgroup on G. So ∀ ∈ f−1(ρ)⇒(f(a), f(b)) ∈ ρ⇒f(a) ∈ f(b)Nρ, and because f is injective, then a ∈ bf−1(Nρ)⇒ ∈ ρf−1(Nρ); that is, f−1(ρ)⊆ρf−1(Nρ). On the contrary  ∈ ρf−1(Nρ)⇒a ∈ bf−1(Nρ)⇒f(a) ∈ f(b)Nρ⇒(f(a), f(b)) ∈ ρ⇒ ∈ f−1(ρ); that is, ρf−1(Nρ)⊆f−1(ρ); hence, f−1(ρ) = ρf−1(Nρ).Because It follows immediately from (1).It is easy to prove f : G → T be an injective homomorphism. If ρ is a complete congruence relation on T, then f−1(ρ) is a complete congruence relation on G.Let f−1(ρ) is a congruence relation on G. Now we prove it is complete; ∀a, b ∈ G, we have [a]f−1(ρ)[b]f−1(ρ) = (af−1(Nρ))(bf−1(Nρ)) = abf−1(Nρ) = [ab]f−1(Nρ).According to f−1(ρ) is complete.Therefore, f : G → T be a surjective homomorphism. If ρ is a congruence relation on G, A⊆G, and Kerf⊆Nρ⊆A, thenf is injective, then if Let f(a) = b⇒[a]ρ∩A ≠ ∅, f(a) = b⇒∃a′ ∈ [a]ρ, a′ ∈ A⇒ ∈ ρ, a′ ∈ A⇒(f(a), f(a′)) ∈ f(ρ), f(a′) ∈ f(A)⇒f(a′) ∈ [f(a)]f(ρ), f(a) = b, [b]f(ρ)∩f(A) ≠ ∅⇒∃b′ ∈ f(A), b′ ∈ [b]f(ρ)⇒∃a′ ∈ A, f(a′) = b′, f(a′) ∈ [b]f(ρ)⇒(f(a′), f(a)) ∈ f(ρ), a′ ∈ A⇒ ∈ ρ, a′ ∈ A⇒a′ ∈ [a]ρ, (1) Consider f(a) = b⇔[a]ρ = aNρ⊆A⇔f([a]ρ) = f(a)f(Nρ)⊆f(A)⇔bNf(ρ) = (2) Consider f : G → T be a surjective homomorphism, let ρ be a congruence relation on G, and let A be a subgroup of G and Kerf⊆Nρ⊆A. Then,f is injective, then if Let A is a subgroup of G; then G and T; according to (1) Suppose that f is injective, according to ≅(2) If f : G → T be a surjective homomorphism, let ρ be a complete congruence relation of G based on Kerf, and let A be a subgroup of G and A⊇Kerf. Then,f is injective, then if Let f : G → T be a surjective homomorphism and let ρ be a congruence relation on T and B⊆T. Thenf is an injective, then if Let f(a′) ∈ B, f(a′) ∈ [f(a)]ρ⇔a′ ∈ f−1(B), (f(a′), f(a)) ∈ ρ⇔a′ ∈ f−1(B), (1) Consider (2) Consider f : G → T be a surjective homomorphism and let ρ be a congruence relation on T and B is a subgroup of T. Then,f is an injective, then if Let   By ρ, ϱ be the congruence relations on G. Then Nρ∩ϱ = Nρ∩Nϱ.Let a ∈ Nρ∩ϱ⇔ ∈ ρ∩ϱ⇔ ∈ ρ,  ∈ ϱ⇔a ∈ Nρ, a ∈ Nϱ⇔a ∈ Nρ∩Nϱ, and, therefore, Nρ∩ϱ = Nρ∩Nϱ.Consider ∀ρ, ϱ be two congruence relations on G and a ∈ G. Then a(Nρ∩Nϱ) = aNρ∩aNϱ.Let a(Nρ∩Nϱ)⊆aNρ∩aNϱ. On the contrary, ∀b ∈ aNρ∩aNϱ⇒b ∈ aNρ, b ∈ aNϱ⇒ ∈ ρ,  ∈ ϱ⇒ ∈ ρ∩ϱ⇒b ∈ aNρ∩ϱ = a(Nρ∩Nϱ); that is, aNρ∩aNϱ⊆a(Nρ∩Nϱ); therefore, aNρ∩aNϱ = a(Nρ∩Nϱ).It is easy to prove that ρ, ϱ be two congruence relations on G and a ∈ G. Then [a]ρ∩ϱ = [a]ρ∩[a]ϱ.Let b ∈ [a]ρ∩ϱ⇔b ∈ aNρ∩ϱ = a(Nρ∩Nϱ) = aNρ∩aNϱ⇔b ∈ aNρ, b ∈ aNϱ⇔b ∈ [a]ρ∩[a]ϱ; hence [a]ρ∩ϱ = [a]ρ∩[a]ϱ.Consider ∀ρ, ϱ be two complete congruence relations on G. Then ρ∩ϱ is a complete congruence relation on G.Let Nρ, Nϱ are both the normal subgroups of G, then Nρ∩ϱ = Nρ∩Nϱ is a normal subgroup of G, so Nρ∩ϱ is a complete congruence relation.Because ρ, ϱ be two congruence relations on G and A⊆G. ThenLet a]ρ∩ϱ = [a]ρ∩[a]ϱ, then [a]ρ∩ϱ⊆[a]ρ([a]ϱ)⇒[a]ρ∩A ≠ ∅, (1) Consider (2) Consider ρ, ϱ be two congruence relations on G. Then ρ∘ϱ is a congruence relation on G⇔ρ∘ϱ = ϱ∘ρ.Let ρ, ϱ be two complete congruence relations on G and ρ∘ϱ = ϱ∘ρ. Then ρ∘ϱ is a complete congruence relation on G.Let ρ∘ϱ is the congruence relation and [a]ρ∘ϱ[b]ρ∘ϱ = (aNρ∘ϱ)(bNρ∘ϱ) = abNρ∘ϱ = [ab]ρ∘ϱ; hence, ρ∘ϱ is the complete congruence relation.By ρ, ϱ be two congruence relations on G. Then Nρ∘ϱ = NρNϱ.Let a ∈ Nρ∩ϱ⇒ ∈ ρ∩ϱ⇒∃b ∈ G,  ∈ ρ,  ∈ ϱ⇒a ∈ bNρ, b ∈ Nϱ⇒ab ∈ bNρNϱ⇒a ∈ NρNϱ; that is, Nρ∘ϱ⊆NρNϱ. On the contrary, ∀a ∈ NρNϱ⇒∃b ∈ Nρ, c ∈ Nϱ, a = bc⇒ ∈ ρ,  ∈ ϱ⇒ ∈ ρ,  ∈ ϱ⇒ ∈ ρ∘ϱ⇒bc = a ∈ Nρ∘ϱ; that is, Nρ∘ϱ⊇NρNϱ; hence, Nρ∘ϱ = NρNϱ.Consider ∀ρ, ϱ be two congruence relations on G and a, b ∈ G. Then [ab]ρ∘ϱ = [a]ρ[b]ϱ.Let ab]ρ∘ϱ = abNρ∘ϱ = abNρNϱ = (aNρ)(bNϱ) = [a]ρ[b]ϱ.Consider ρ∩A ≠ ∅, [c]ϱ∩A ≠ ∅, a = bc⇒∃b′ ∈ [b]ρ, b′ ∈ A, c′ ∈ [c]ϱ, c′ ∈ A⇒b′c′ ∈ [b]ρ[c]ϱ = [bc]ρ∘ϱ = [a]ρ∘ϱ, (1) Consider a]ρ = [a]ρ[e]⊆[a]ρ[e]ϱ⊆A, and a = bc⇒[b]ρ⊆A, [c]ϱ⊆A, a = bc⇒[a]ρ∘ϱ = [bc]ρ∘ϱ = [b]ρ[c]ϱ⊆AA = (2) Consider ρ be complete congruence relation on G and A, B⊆G. ThenLet b ∈ AB⇒a ∈ bNρ, ∃c ∈ A, d ∈ B, b = cd⇒a ∈ cdNρ = (cNρ)(dNρ) and [c]ρ∩A ≠ ∅, [d]ρ∩B ≠ ∅⇒∃c′ ∈ Nρ, d′ ∈ dNρ, a = c′d′, because [c′]ρ = [c]ρ, [d′]ρ = [d]ρ⇒[c′]ρ∩A ≠ ∅, [d′]ρ∩B ≠ ∅, a = c′d′⇒a = a = bc⇒[b]ρ∩A ≠ ∅, [c]ρ∩B ≠ ∅⇒∃b′ ∈ [b]ρ, b′ ∈ A, c′ ∈ [c]ρ, c′ ∈ B⇒b′c′ ∈ [b]ρ[c]ρ = [bc]ρ = [a]ρ, (1) Consider b′ ∈ [b]ρ, b′∋A, c′ ∈ [c]ρ, c′∋B, then b′c′ ∈ [b]ρ[c]ρ, b′c′∋AB, Contradiction, that is, [b]ρ⊆A, [c]ρ⊆B, so, a = bc⇒[b]ρ⊆A, [c]ρ⊆B, (2) ρ be a complete congruence relation on G and A⊆G, n ∈ N. ThenLet ρ be a complete congruence relation on G and let A be a subgroup of G and n ∈ N. ThenLet A is a subgroup, then G, so An = A, and ρ is an equivalence relation; then ρn = ρ; hence, we can get (1) and (2).Because ρ be a complete congruence relation on G and let A, B be two subgroups of G. ThenLet A = A(e)⊆AB, and B⊆AB, so A ∪ B⊆AB; hence, A ∪ B⊆AB, so (1) Because


Scientific Reports5: Article number: 1316110.1038/srep13161; published online 08122015; updated on 01122016In this Article, the images depicting the Wiring Diagarms and Numerical solutions for models 1 to 5 have been omitted. The correct ′M → M + 1′ and Reaction 2 ′M → M − 1′ were incorrectly given as ′M → M − 1′ and ′M → M + 1′ respectively.In Table S5, the transition values for Reaction 1


Furthermore, judgement theory and discernibility matrix associated with δ-confidence reduct are also introduced, from which we can obtain the approach to knowledge reduction in multilabel decision tables.Owing to the high dimensionality of multilabel data, feature selection in multilabel learning will be necessary in order to reduce the redundant features and improve the performance of multilabel classification. Rough set theory, as a valid mathematical tool for data analysis, has been widely applied to feature selection . In this study, we propose a variable precision attribute reduct for multilabel data based on rough set theory, called  Conventional supervised learning deals with the single-label data, where each instance is associated with a single class label. However, in many real-world tasks, one instance may simultaneously belong to multiple class lultilabel decision tablabels, for example, in text categorization problems, where every document may be labeled as several predefined topics, such as religion and political topics ; in imagOwing to the high dimensionality of multilabel data, feature selection in multilabel learning will be necessary in order to reduce the redundant features and improve the performance of multilabel classification. Among various feature selection approaches, rough set theory, proposed by Pawlak , has attβ-reduct to improve the ability of modeling uncertain information B : x ∈ U}, where [x]B denotes the equivalence class determined by x with respect to B; that is,A decision table is an information system X⊆U and B⊆A. One can define a lower approximation of X and an upper approximation of X byX and denoted alternatively as POSB(X). If X is called a rough set.Let Attribute reduct is one of the most important topics in rough set theory, which aims to delete the irrelevant or redundant attributes while retaining the discernible ability of original attributes. Among many attribute reduction methods, the positive region reduct , 8 is a S =  be a decision table and B⊆A. B is a positive region reduct of S if and only if B satisfies the following conditions:B(D) = POSA(D),POSB′(D) ≠ POSA(D) for any B′⊆B,POSLet B(D) = ⋃i=1rPOSB(Di) and Di   are the equivalence classes, called decision classes generated by the indiscernibility relation RD = { ∈ U × U : d(x) = d(y), ∀d ∈ D}.where POSIn this section, we first introduce the multilabel decision table and then analyze the limitations of existing attribute reduction approaches to multilabel data.S =  with A∩L = ∅, where U = {x1, x2,…, xn} is a nonempty finite set of objects, called universe; A = {a1, a2,…, ap} is a nonempty finite set of condition attributes, called condition attribute set; F = {ak : U → Vk, k = 1,2,…, p} is a set of information functions with respect to condition attributes and Vk is the domain of ak; L = {l1, l2,…, lq} is a nonempty finite set of q possible labels called label set; G = {lk : U → Vk′, k = 1,2,…, q} is a set of information functions with respect to labels and Vk′ = {0,1} is the domain of the label lk. If the object x is associated with label lk, then lk(x) = 1; otherwise lk(x) = 0. The 5-tuple  can be expressed more simply as  if F and G are understood.Multilabel data can be characterized by a multilabel decision table The object having no labels is irrelevant to multilabel learning and thus is not taken into account in the setting , 16. NotL associates with at least one object in U , one defines a δ-confidence label function ηBδ : U → P(L), as follows:Let δ-confidence label function ηBδ(x) is the collection of the labels that associate with at least δ% objects in [x]B. In other words, ηBδ(x) is the collection of the labels which associate with each object in [x]B by at least δ% confidence level.The S =  given by δ = 0.6, then the 0.6-confidence label function with respect to attribute set A can be calculated thatConsider the multilabel decision table We have the following property.S =  be a multilabel decision table, B, C⊆A. Thenδ1 ≤ δ2 ≤ 1, then ηBδ2(x)⊆ηBδ1(x);if 0 ≤ B⊆C, then ηB1(x)⊆ηC1(x)⊆ηC0(x)⊆ηB0(x);if x ∈ U, ηB0(x) ≠ ∅;for any x]B = [y]B, then ηBδ(x) = ηBδ(y).if [Let li ∈ ηBδ2(x). Then we have |[x]B∩Hi|/[x]B ≥ δ2. Note that δ2 ≥ δ1; thus |[x]B∩Hi|/[x]B ≥ δ1. It means that li ∈ ηBδ1(x). Therefore ηBδ2(x)⊆ηBδ1(x).(1) Let B⊆C, we have [x]C⊆[x]B.(2) Since li ∈ ηB1(x), then [x]B⊆Hi. Since [x]C⊆[x]B, we have [x]C⊆Hi. Thus li ∈ ηC1(x). Therefore ηB1(x)⊆ηC1(x).If ηC1(x)⊆ηC0(x).According to lj ∈ ηC0(x), then [x]C∩Hj ≠ ∅. Since [x]C⊆[x]B, we have [x]B∩Hj ≠ ∅. That means lj ∈ ηB0(x). Therefore ηC0(x)⊆ηB0(x).If x ∈ U such that ηB0(x) = ∅, then [x]B∩Hi = ∅,  i = 1,2,…, q. Thus [x]B∩(H1 ∪ H2 ∪ ⋯Hq) = ∅. Since ∪i=1qHi = U, we have [x]B∩(H1 ∪ H2 ∪ ⋯Hq) = [x]B∩U ≠ ∅. It is a contradiction.(3) If there exists ηBδ(x) and [xB] = [y]B.(4) It is straightforward by the definition of δ-confidence set using δ-confidence label function. Furthermore, we present the definition of new attribute reduct.Now we define the consistent S =  be a multilabel decision table and B⊆A. If ηBδ(x) = ηAδ(x), for all x ∈ U, one says that B is a consistent δ-confidence set of S. If B is a consistent δ-confidence set and no proper subset of B is a consistent δ-confidence set, then B is called a δ-confidence reduct of S.Let δ-confidence reduct is the minimal set of condition attributes that preserves the invariances of the δ-confidence label function of all objects in U.A S =  given by a, b}.For the multilabel decision table a, c} and {b, c} are two positive region reducts for the same multilabel decision table. We think δ-confidence reduct is more appropriate for multilabel data than positive region reduct. This is because δ-confidence label function can more reasonably characterize the uncertainty implied among labels than the indiscernibility relation RL.Considering RL is also considered by the other existing attribute reduction methods. Therefore, for multilabel data, δ-confidence reduct has significant advantages when compared with existing attribute reduction methods.Note that the uncertainty characterized by δ-confidence reducts. Firstly, we present the judgement theorem of consistent δ-confidence set.This section provides a discernibility matrice approach  to obtaiS =  be a multilabel decision table, B⊆A and δ ∈ . Then the following conditions are equivalent:B is a consistent δ-confidence set;x, y ∈ U, if ηAδ(x) ≠ ηAδ(y), then [x]B∩[y]B = ∅.for any Let x, y ∈ U such that [x]B∩[y]B ≠ ∅, then [x]B = [y]B. By ηBδ(x) = ηBδ(y). Note that B is a consistent δ-confidence set; we have ηBδ(x) = ηAδ(x) and ηBδ(y) = ηAδ(y). Therefore ηAδ(x) = ηAδ(y).(1)⇒(2). If there exist B⊆A, it is easy to verify that I([x]B) = {[y]A : [y]A⊆[x]B} forms a partition of [x]B.(2)⇒(1). Since x ∈ U, if [y]A⊆[x]B, then [x]B∩[y]B ≠ ∅. By the assumption we obtain ηAδ(x) = ηAδ(y).For any li ∈ ηAδ(x). Then for all [y]A ∈ I([x]B), we have li ∈ ηAδ(y); that is to say, |[y]A∩Hi|/|[y]A| ≥ δ.Let li ∈ ηBδ(x).Therefore we have thatli ∈ ηBδ(x); however, li ∉ ηAδ(x). For any [y]A ∈ I([x]B), we have ηAδ(x) = ηAδ(y); hence li ∉ ηAδ(y); that is to say, |[y]A∩Hi|/|[y]A| < δ. Since [x]B = ∪{[y]A : [y]A ∈ I([x]B)}, we haveli ∉ ηBδ(x), which is a contradiction.On the other hand, we assume that ηAδ(x) = ηBδ(x) for any x ∈ U. According to B is a consistent δ-confidence set.Thus we conclude that δ-confidence set in multilabel decision tables. Now we present a method for computing all δ-confidence reducts. First, we give the following notion.S =  be a multilabel decision table and U/RA = {X1, X2,…, Xm}. One denotesLet ak(Xi) the value of ak with respect to the objects in Xi. DefineBy Mδ is called δ-confidence discernibility attribute sets. And Mδ = , i, j ≤ m) is called the δ-confidence discernibility matrix.Then δ-confidence discernibility matrix, we have the following property.For the Mδ = , i, j ≤ m) satisfies the following properties:Mδ is a symmetric matrix; that is, for any i, j ≤ m, Mδ = Mδ;A; that is, for any i ≤ m, Mδ = A;elements in the main diagonals are all i, s, j ≤ m, Mδ⊆Mδ ∪ Mδ.for any The discernibility matrix ak ∈ A such that ak ∈ Mδ but ak ∉ Mδ ∪ Mδ, that is, ak ∉ Mδ and ak ∉ Mδ, then according to ak(Xi) = ak(Xs) and ak(Xs) = ak(Xj). Thus ak(Xi) = ak(Xj); that is, ak ∉ Mδ, a contradiction.The proofs of (1) and (2) are straightforward. We only need to prove (3). If there exists δ-confidence set and discernibility matrix.In the following, we establish some connections between consistent S =  be a multilabel decision table, B⊆A and δ ∈ . Then, B is a consistent δ-confidence set if and only if B∩Mδ ≠ ∅ for all  ∈ Δδ.Let Xi, Xj) ∈ Δδ, there exist x, y ∈ U, such that Xi = [x]A and Xj = [y]A. From the definition of Δδ, we have ηAδ(x) ≠ ηAδ(y). Since B is a consistent δ-confidence set, we have [x]B∩[y]B = ∅ from ak ∈ B such that ak(x) ≠ ak(y); that is, ak(Xi) ≠ ak(Xj). Hence ak ∈ Mδ; that is, B∩Mδ ≠ ∅.“⇒” For any  ∈ Δδ. Since B∩Mδ ≠ ∅, for all  ∈ Δδ, there exists al ∈ B such that al ∈ Mδ. Then we have al(Xi) ≠ al(Xj); that is, al(x) ≠ al(y) for [x]A = Xi and [y]A = Xj. It means [x]B∩[y]B = ∅. We then conclude that if  ∈ Δδ, that is, ηAδ(x) ≠ ηAδ(y), then [x]B∩[y]B = ∅. It follows from B is a consistent δ-confidence set.“⇐” Let  be a multilabel decision table, let δ ∈ , and let Mδ = , i, j ≤ m) be the δ-confidence discernibility matrix, where A = {a1,…, ap}. A δ-confidence discernibility function FSδ for a multilabel decision table S is a boolean function of p boolean variables a1,…, ap, respectively, and is defined as follows:Mδ is the disjunction of all variables a ∈ Mδ.Let ai instead of In the sequel we will write S =  be a multilabel decision table. Then an attribute subset B of A is a δ-confidence reduct of S if and only if ∧B is a prime implicant of S.Let δ-confidence reducts. The following example illustrates the validity of the approach.U/RA = {X1,…, X5}, whereηA0.6(x) in ηA0.6(x1) = ηA0.6(x2) = ηA0.6(x3). Therefore  ∉ Δ0.6.Consider the multilabel decision table given by δ-confidence discernibility matrix shown in We can calculate the Consequently, we havea, b} is the unique 0.6-confidence reduct which accords with the results in By δ-confidence reduct presented in this paper is an attribute reduction method designed for multilabel decision tables. Compared with the existing attribute reduction methods, the δ-confidence reduct accurately characterizes uncertainty implied among labels; thus it is more appropriate for multilabel data. Moreover we proposed the corresponding discernibility matrix based method to compute δ-confidence reduct, which is significant in both the theoretic and applied perspectives. In further research, the property of δ-confidence reduct and corresponding heuristic algorithm will be considered.The


By the use of the way of real analysis, we estimate the weight functions and givesome new Hilbert-type integral inequalities in the whole plane with nonhomogeneouskernels and multiparameters. The constant factors related to the hypergeometric functionand the beta function are proved to be the best possible. We also consider theequivalent forms, the reverses, and some particular cases in the homogeneous kernels. If 0 < λ < 1, p > 1, and (1/p) + (1/q) = 1, Yang [In 2007, Yang  first ga 1, Yang  gave anoe et al. –20 also In this paper, by the use of the way of real analysis, we estimate the weight functions and give some new Hilbert-type integral inequalities in the whole plane with nonhomogeneous kernels and multiparameters, which are extensions of . The conα > 0, we have Γ(α) = ∫0∞xα−1ex−dx, where Γ(α) is the Γ function . For β β > −1, min{μ, σ} > −α, μ + σ = λ < 1 + β, and δ ∈ {−1,1}, define the weight functions ωδ and ϖδ as follows:y, x ∈ ∪, we haveIf δ = 1, setting u = xy, we find, for y ∈ ∪,(i) For rem (cf. ), in vie∪,ω1=∫δ = −1, setting y/x, we still can obtain ω−1 = Kβ(σ). Setting u = xδy, we findβ > −1,  0 < θ0 < min{μ + α, σ + α},L, such that ((−lnu)/(1 − u))βuθ0 ≤ L;   then, by (Kβ(σ) ∈ R+. Hence we have ((ii) For then, by , it foll we have .F (cf. [Re(γ) > Re(θ) > 0, |arg(1 − z)|<π, thenz = −1, γ = θ + 1  (θ > 0), it follows thatWe have the following formula of the hypergeometric function F (cf. ). If Re(β = 0  (λ < 1), in view of +(1/q) = 1, β > −1,  min{μ, σ} > −α, μ + σ = λ < 1 + β, δ ∈ {−1,1}, Kβ(σ) is indicated by (f(x) is a nonnegative measurable function in , then one hasIf cated by , and f, we hav1[∫−∞∞|ln1[∫−∞∞|ln view of  follows.p > 1,  (1/p)+(1/q) = 1,  β > −1,  min{μ, σ}>−α,  μ + σ = λ < 1 + β, and δ ∈ {−1,1},  f(x),  g(y) ≥ 0, satisfying 0 < ∫∞−∞|x|p(1−δσ)−1fp(x)dx < ∞ and 0 < ∫∞−∞ | y|q(1−σ)−1gq(y)dy < ∞, then one hasKβ(σ) and Kβp(σ) are the best possible and Kβ(σ) is defined by dy}1/q,J≔∫−∞∞|y|δ = 1, we have the following equivalent inequalities:In particular, for y ∈ ∪, then there exist constants A and B, such that they are not all zero, andA ≠ 0 (otherwise B = A = 0). Then it follows that∞−∞|x|p(1−δσ)−1fp(x)dx < ∞. Hence (J = ∫∞−∞ | y|q(1−σ)−1gq(y)dy. By  is an even function, then it follows thatv = xδ in the above integral, by Fubini theorem (cf. [Kβ(σ) in (k with Kβ(σ) < k, such that (Kβ(σ) by k. Then we have k < Kβ(σ). Hence the constant factor Kβ(σ) in dy. By , we have 0, then  is obviothen, by , we obta we have , which i we have . We indiero, andA|x|(1−δσβ(σ) in  is not tuch that  is valid obtain2L~≔{∫−∞∞|ero, andA|x|(1−δσβ(σ) in  is the bβ(σ) in  is not t. Hence (I=∫−∞∞[|yβ(σ) in  is not tp > 1 by 0 < p < 1, one has the equivalent reverses of (As the assumptions of erses of  and 20)p > 1 by g(y) as  in the reverse of , such that the reverse of (Kβ(σ) by k. By the reverse of δ0 > −α, and then 0 < Kβ(σ − (δ0/2)) < ∞. For 0 < ɛ < δ0|q|/4  (q < 0), since uσ+α+(2ɛ/q)−1 ≤ uσ+α−(δ0/2)−1, u ∈  ≥ k, which contradicts the fact that k > Kβ(σ). Hence, the constant factor Kβ(σ) in the reverse of , we havg(y) as  in Theorg(y) as , we haveverse of  is obvioverse of , we obtaverse of . Hence wverse of , which iverse of . If the verse of  is not tverse of  is stillverse of , we haverem (cf. ), we finrem (cf. ), for ɛ erse of  by f(x), we obtain the following equivalent inequalities with a homogeneous kernel and the best possible constant factors:(i) For  = −1 in  and 20)δ = −1 inβ = 0  (λ < 1) in (K0(σ) is indicated by ((ii) For  < 1) in  and 20)β = 0   in ((iii) For  < 1) in , we findand then  follows.and then  and 19)α = 0,  σand then .


Previously, we suggested prototypal models that describe some clinical states based on group postulates. Here, we demonstrate a group/category theory-like model for molecular/genetic biology as an alternative application of our previous model. Specifically, we focus on deoxyribonucleic acid (DNA) base sequences.5 = {r, u, d, l, n} of operations is defined for the wallpaper pattern, with which a sequence of points can be generated corresponding to changes of a base in a DNA sequence by following the orbit of a point of the pattern under operations in group Z5. Other manipulations of DNA sequence can be treated using a vector-like notation ‘Dj’ corresponding to a DNA sequence but based on the five-letter base set; also, ‘Dj’s are expressed graphically. Insertions and deletions of a series of letters ‘E’ are admitted to assist in describing DNA recombination. Likewise, a vector-like notation Rj can be constructed for sequences of ribonucleic acid (RNA). The wallpaper group B = {Z5×∞, ●} (an ∞-fold Cartesian product of Z5) acts on Dj (or Rj) yielding changes to Dj (or Rj) denoted by ‘Dj◦B(j→k) = Dk’ (or ‘Rj◦B(j→k) = Rk’). Based on the operations of this group, two types of groups—a modulo 5 linear group and a rotational group over the Gaussian plane, acting on the five bases—are linked as parts of the wallpaper group for broader applications. As a result, changes, insertions/deletions and DNA (RNA) recombination  are described. As an exploratory study, a notation for the canonical “central dogma” via a category theory-like way is presented for future developments.We construct a wallpaper pattern based on a five-letter cruciform motif with letters C, A, T, G, and E. Whereas the first four letters represent the standard DNA bases, the fifth is introduced for ease in formulating group operations that reproduce insertions and deletions of DNA base sequences. A basic group ZDespite the large incompleteness of our methodology, there is fertile ground to consider a symmetry model for genetic coding based on our specific wallpaper group. A more integrated formulation containing “central dogma” for future molecular/genetic biology remains to be explored. Group theory is the cornerstone in classifying and studying abstract concepts involving symmetry ,2. In geDeoxyribonucleic acid (DNA) is a nucleic acid containing genetic instructions coded in ordered sequences of four bases located in genes that determine specific genetic characteristics of an organism. In the canonical Watson-Crick DNA base pairing, adenine (A) forms a base pair with thymine (T) and guanine (G) forms a base pair with cytosine (C) -26. SimiOver the latter half of the 20th century, the nature of the genetic code became fairly well established. As for the coding sequences of DNA into nucleotide units, one needs to build up more general, sophisticated, rationally functionalized systematics concerning DNA base sequences that will enable genes to be understood at the molecular biology level in more optimized form. Indeed, many approaches have been undertaken to describe gene characteristics from various viewpoints within the participating disciplines -42. In pHowever, each has its advantages and disadvantages in terms of utility and convenience in applications. To our knowledge, so far, if we intend to incorporate a sequence of bases into another sequence and/or exclude certain bases from that substitution, we need to look further afield because normally, sequencing and inserting-deleting operations cannot help in distinguishing one from the other. That means that multiple types of operations are necessary if features of DNA containing exceptional sequences are to be treated.Previously, we suggested prototypal models that describe some clinical states based on group postulates . In thisFirst, we consider a certain wallpaper pattern that helps us to visualize the operations of the present model 5);1) Associativity: x●(y●z) = (x●y)●z,  Identity: ‘n’ is an identity element such that x●n = n●x = x;−1 exists such that x● x−1 = x−1●x = n ;3) Inverse: a unique element x4) Commutativity: x●y = y●x,5.5) Closure: any combination of operations between x●y belongs to Z5 is an Abelian group , then < Dj > changes into<Dj1 > = [C1|T2(E3)A4|T5|A6(E7)C8|E9|E10|E11|E12|E13|…], and subsequently into<Dj1>> = [C1|T2|A3|T4|A5|C6|E7|E8|E9|E10|E11|…]. The sequence < Dj1 > contains implicit ‘E’s aside from the trailing ‘E’s, and can be written as<<Dj1>} = [C1|T2|A4|T5|A6|C8|E9|E10|E11|E12|E13|…]. Naturally, {<Dj1>} and < Dj1 > are equivalent, but < <Dj1> > and < Dj1 > differ. Moreover, as long as place numbers are recognized/traced precisely, combinations of manipulations ‘{ }’ and ‘< >’ are allowed; e.g., {<{{<Dj1>}}>}. Hence, with appropriate use, we could treat  conventional sequences of DNA via ‘{Dj}’ or ‘<Dj>’. However, below we shall focus on simple sequences ‘Dj’.{<DLooking at the beginning of a base sequence as in the following:j, N: the number bases Dj)’ and ‘(3 → 5)’, is simply an additional label representing the two possible types of endings of single-stranded DNA. Nonetheless, when the number of bases is finite, two sequences can be equivalent, as for exampleandunless the prime endings <5’(five prime) → 3’(three prime) > or <3’ → 5’ > accompanies the sequence designation.j | Dj ∈ C5 × C5 × C5 × … } as the set of all possible sequences of recognized N-tuple single-stranded DNA bases. We can regard N to be a positive integer or infinity.In accordance with these postulates, we now can define the set D = {D5, are similarly definable because all results obtained for DNA pertain to RNA under the base substitution. Thus, set R = {Rj | Rj ∈ C5 × C5 × C5 × …} is the set of all possible sequences of recognized N-tuple single-stranded RNA bases with C5 = {C, A, U, G, E}.An analogous definition is clearly possible for the set R of RNA sequences; with ‘T’ substituted by ‘U’, operations of group Zm  | Bm ∈ Z5× Z5× Z5 × …} = {Z5×N, ●}, where elements of B act on any Dj. This means that Dj covers all possible sequences of the DNA bases, and this situation is the same for Rj of sequences of RNA bases.Next, we can consider B = {B5×N, ●} is confirmed.Because B is a Cartesian product of the same Abelian group, it is also Abelian, where composition of any two elements of B is denoted by ‘●’ . Details(j→k)’ that effects the change Dj into Dk is definable in the following way:In a more general context, a Cartesian vector that is composed of the respective operators ‘bHence,(j→k) = Bm ’; despite the difference in notation, the two are identical in practice.Clearly, for arbitrary ‘j’ and ‘k’, there exists a unique ‘m’ such that ‘B(j→k)’s. Consider the scenario that a certain sequence of a single strand (or one side of a double-strand) of DNA transitions from D1 to D3, in stepwise fashion,Here, we present a simple example that consists of a multiple product of ‘B1 → D2’. There exists an operator ‘B(1→2) = [n1|n2|n3(r4|u5)l6|d7|n8|n9|…]’ that is able to produce this change, specifically, the insertion of two ‘E’s between ‘C3’ and ‘G4’ yields the change ‘D1 → D2’. However, this sort of manipulation can be troublesome. Hence, in our model, insertion/deletion of ‘E’s are instead ascribed to the way the vector Dj is interpreted. This is preferable as this avoids easier manipulations. Next, we construct the operator ‘B(2→3)’ that maps ‘D2 → D3’ (the details are shown in Appendix C). With reference to Figures We next consider the change ‘DIn a similar manner,4 is obtained from D1 recursively,Naturally, the final DFrom the decompositionwheredenotes the identity element of D.j irrespective of whether the ‘E’s are explicit or implicit as defined in §1. Moreover, any sequence ‘Dj’ can be presented as a polygonal line; as an example, the evolution of changes ‘D1 → D3’ is displayed in Figure Note that the group operations can act on D5, D, and B, another approach is possible. The five bases can be represented by five equispaced phasors with a ‘2π/5’ angular phase separation located on the unit circle on the Gaussian plane, as depicted in Figure Looking at the definitions of groups ZHerein, in the Gaussian plane, if ‘ω’ is defined to be the counterclockwise rotational angle ‘ω = 2π/5 (rad)’ and composition of ‘ω’ is denoted ‘●’, then assuming ‘ω’ obeys the ‘right translation rule’, we havem ↔ Exp(m · ω · i)’ , m = {0, 1, 2, 3, 4, 5}). With {*} meaning one of the bases among ‘C, A, T, G and E’, we construct the following map. Denoting composition by ‘◦’, ωm acts on the identity trivially and hence yields the correspondencesThe general form of an arbitrary base is expressed as ‘X1, ω2, ω3, …’ on bases ‘C, A, T, G and E’, we establish for instance:Expanding the operations for ‘ωω = {ω1, ω2, ω3, ω4, ω0 (= ω5)} is readily confirmed to form group {Pω, ●} where the identity element is ‘ω0’ and the inverse of ‘ωm’ is ‘ωm−1’:In continuance, the set PClosure and associativity follow from (9) and (10).Here, if we turn our attention to the wallpaper pattern, a further bijection obeying the postulates of the wallpaper group can be confirmed. Corresponding to Figures 2−1 = ω3) are preserved in accordance with the inverses for ‘r, u, d, l, and n’. Any right translation of the horizontal line in Figures 1 = T = E◦d’ and ‘A◦l = A◦ω4 = C = E◦r’ can be confirmed. All possible one-step changes between ‘A, C, T, G and E’ and ‘ω1, ω2, ω3, ω4 and ω0,’ are shown in Figure Naturally, inverses e.g., ω2− = ω3 are5, and group B.Therefore, this rule for ‘E’ does not break the postulates for set D, group Zm’ is given; its complementary base ‘Xm†’ to ‘Xm’ is defined as follows; for ‘Xm = {Exp(m · ω · i)}, m = {0, 1, 2, 3, 4, 5}, then ‘Xm†’ is obtained by ‘Xm† = {Exp((5 – m) · ω · i)}’, where ‘{Exp(5 · ω · i)} = {1} = E’. In this regard,Suppose, from among ‘C, A, T, G and E’, a base ‘X† = T’ and ‘C† = G’.The procedure yields specifically ‘A† = E’.Clearly, the complement of ‘E’ is ‘E’ itself; ‘Em’ expressed as a base can be given. We introduce the one-value function ‘ωX(m)’ that provides the same results,Another notation for the ‘XAs for ‘m’ in (14), both positive and negative integers are permissible. Thus,m†’ is expressible as‘XA simple example is illustrated below.j’ = [A1|T2|C3|E4|G5|T6|…] = [ωX(2)|ωX(3)|ωX(1)|ωX(0)|ωX(4)|ωX(3)|…], then,Suppose ‘Dr that contains only {r, r2, r3, r4, re (= r0 = r5 = n)}. This group is isomorphic with group Pω = {ω1, ω2, ω3, ω4, ω0 (= ω5)}, as is the group similarly generated over a vertical line.In accordance with the wallpaper group in Figure ωX(m)’, ‘Xm’ can be expressed using another one-value function rX(m) = E◦rm’:Similar to ‘m†’ (the complementary base of ‘Xm’) is written asHence, ‘XExtension to vertical translations is straightforward;m†’ can be identified similarly although the order of letters are somewhat different.and its complementary base ‘Xj†’ using ‘rX(m)’s;Consider the following simple example in identifying ‘DωX(m)’ by ‘rX(m)’ in formula (15), the same result is obtained.by replacing ‘1◦b[X1→ X4] = X1◦r3 = E◦r4 = G’. In general, when the i-th component ‘b[Xm1→ Xm2]i’ of ‘B(m1→m2)’ changes Xm1 (= E◦rm1 = E◦(m1ω)) to Xm2 (= E◦rm2 = E◦(m2ω)).According to these rules, ‘XHence, the highlighted form of the operator vector is expressed as(j→k)’ that changes Dj to Dk,For a further example, given the operator ‘B(j→k)’ takes the formWith details shown in Appendix D, ‘Bk is obtained through recursively applying the operations, Dj ◦B(j→k) = Dk. (Details are presented in Appendix D).Naturally, the state Dj†’s might have components in reverse order in terms of sense (5’ or 3’), there exists however certain ‘Dk’ such that ‘Dk = Dj†, ’. With this, ‘Dj†’ is one of the ordinal elements belonging to the same set D. Thus, the symbol ‘†’ need only be present when elements are distinct.Whereas ‘Dr, (or ω ∈ group Pr) is ‘a’ and the number of up translations ‘u’ ∈ group P (or 2ω (= ω2) ∈ group Pr) is ‘b’ with a, b = …,-2, −1, 0, 1, 2,…. Similarly, with ‘d ↔ 3ω’ ‘l ↔ 4ω’, the total change can be summarized as ‘x’. We can confirm that there exists at least a pair of ‘a, b’ that satisfiesConsider Figure a●ub’ constitutes a multiple composition of elements of group Z5. In addition, ‘x = r-a●u-b = la●db’ or ↔ ‘(−a)ω●(−b)(2ω) = (−a-2b)ω’. E.g., ‘x means ‘r−3●u−2 = l3●d2’ or ↔ ‘(−3)ω●(−2 · 2)(ω) = (−3 -4)ω = (−7)ω = (−2)ω = 3ω = ω3’.because any base in Figure For the wallpaper group, the ‘a’ and ‘b’ should be interpreted in modulo 5 addition. The Cayley table for the wallpaper group are presented in Appendix A.†’ is obtained from ‘X’ using ‘X = E◦x’, ‘X†’ can be determined asWithin ‘the square unit cell’ in Figure †’ are symmetrically disposed with respect to ‘E’ over the wallpaper pattern that would be selected as a standard for the definition of ‘a, b’. In practice, for an arbitrary ‘X’, ‘X†’ can be obtained via (22) or (23) by making use of an arbitrary ‘E’ as the reference point for the symmetry.because ‘X’ and ‘X† = E◦x−1’s asm’, with ‘Dj’ expressed as ‘Dj =  i|…]’, one of the candidates of the appropriate ‘B(j→j†)’s that produces ‘Dj◦B(j→j†) = Dj†’ is identified:we can confirm descriptions (20), (22) and (23). As for the operators ‘Bi -4bi’ in (25b) are permitted to take positive or negative integer values.The exponents ‘-2aIn these expressions, the rules for the wallpaper group (25a) can also be expressed as either for the linear group or for the rotational group (25b or 25c).More generally,j =  i|…]’ is changed into ‘Dk =  i|…]’, and‘D(j→k)’s that provides ‘Dj◦B(j→k) = Dj†’ is identified as‘B†’ of a certain sequence ‘X’ is reversed to <3’ → 5’ > .As mentioned in §1, if a certain sequence ‘X’ has sense <5’ → 3’>, the complementary sequence ‘XTo aid understanding, we present the following examples: Givenj†’ is simplythen, according to (22), ‘Dj◦B(j→j†) = Dj†’ is derived. Details are given in Appendix E.Apart from these examples, additional identities for the wallpaper group can be verified using Figure If we use the optional formula (25a − c), the relation ‘DWe develop various general formulas:Other unknown rules might underlie the wallpaper pattern.m = rX(m) in , and ωX(m) in  could be regarded as a specific combination that are displayed asConcerning style in treating the wallpaper group, examples ‘XBelow, we demonstrate, using several examples containing ‘E’s, changes and inclusion/exclusion of DNA bases using a more generalized scheme.j’ be the sequence ‘CGTAT…C…TA’, we consider the change of its ‘1–3’ components ‘C1G2T3’ into ‘G1T2A3’, and moreover the insertion of two bases ‘GC’ between ‘T3’ and ‘A4’ denoted ‘’:For definiteness, let ‘Dk,We denote the result of this transformation as Dj to Dk is described recursively to find operator ‘B(j→h).The procedure from Dj → Dh’) in preparation for insertion of ‘GC’;First, two ‘E’s are inserted after the 3rd component (this change is denoted ‘D(j→h)Thus, ‘Bj’s dependent upon ‘E’s.This change is in accordance with those rules for vector-like ‘D(h→k) that produces the change from Dh to Dk is described as:Hence, the operator BThereby,With reference to Figure j → Dh (inserting two ‘E’s after the ‘3rd’ component), and 2) Dh ◦B(h→k) = Dk. Note that the exclusion of the ‘4–5’ components ‘GC’ from Dk and the transformation of the ‘1–3’ components from ‘GTA’ to ‘CGT’ constitute the recursive procedure for the inverse operatorThis indicates a code change of the ‘1–3’ components and a ‘GC’ insertion after the 3rd as described via the two steps: 1) Dh → Dj’ is obtained by deleting the two ‘E’s from the ‘4–5’ components of Dh to yield the initial state ‘Dj’ in accordance with the characteristics of the vector-like ‘Dj’s.Alternatively, ‘D(… →… ) ∈ group B.In summary, essentially, all transitions (changes and inclusion/exclusion) of a certain sequence within the same single-stranded DNA, whether it has finite or infinite length, can be described in principle within a single operation using only the unique operator Bc1)’ and ‘ATAGCTA (= Dd1)’. These have vector expressionsAs a further development, to demonstrate recombination, take two finite sequences ‘GETAGT -th component of ‘Dc1’ and the ‘AGCTA’ of the (3–7)-th component of ‘Dd1’ at the same instant.To illustrate for the pair Dc1’ just before the sequence to be converted, and in ‘Dd1’ just after the sequence to be converted. For example, for ‘Dc1’, five ‘E’s, ‘EEEEE’, of size equivalent to that of ‘A3G4C5T6A7’ of ‘Dd1’, are inserted just before ‘T3’ in ‘Dc1’ where ‘A3G4C5T6A7’ is to be located, that is, the interval between ‘the 2nd ‘E2’ and 3rd ‘T3’ within ‘Dc1’. Under this procedure, Dc1 changes into Dc2:First, in the pair of sequences, a series of ‘E’s of complementary size is inserted in ‘DEEEEE’ is changed into ‘A3G4C5T6A7’ -th component of ‘Dd1’). In addition, ‘T8A9G10T11’ is transformed into the same number of ‘E’s, ‘EEEE’, at the same time. By this process, ‘Dc2’ changes in ‘Dc3’:Here, we assume that ‘EEEE’ equivalent in size to ‘T3A4G5T6’ of ‘Dc1’ would be inserted after ‘A7’ of ‘Dd1’ where ‘T3A4G5T6’ of ‘Dc1’ is to be located within ‘Dd1’. That is, ‘T3A4G5T6’ is inserted into the interval between the 7th ‘A7’ and 8th ‘E8’ within ‘Dd1’. In this procedure, Dd1 changes into Dd2:Note that bold type and underline are here merely pedagogical aids to help identify sequence changes. Meanwhile, four ‘E’s ‘T8A9G10T11’ -th components of ‘Dc1’) while ‘A3G4C5T6A7’ is transformed into the equivalent-sized ‘EEEEE’. Through this procedure, ‘Dd2’ changes in ‘Dd3’:Furthermore, we change ‘EEEE’ into the equivalent-sized ‘c1, Dd1’) with ‘Dc1’ = ‘G1E2T3A4G5T6’ being transformed into ‘Dc3’ = ‘G1E2A3G4C5T6A7’ and ‘Dd1’ = ‘A1T2A3G4C5T6A7’ being transformed into ‘Dd3’ = ‘A1T2T3A4G5T6’. We define the manipulation of the recombination  between ‘Dc1, Dd1’ in this way.As a result, if we omit the infinite series of ‘E’s from right end, we have the recombination ) and this rule depends upon the characteristics of these vectors .In the initial stage in the previous illustration, we inserted different sizes of ‘E’ sequences in each line; however, processes ‘DAs previously explained, the operations can be performed in any of the three equivalent linear group, rotational group, and wallpaper group. Choosing the wallpaper group,Also,With respect to (42) and (44), the inverse identities are confirmed:(_→_) giving transition ‘Dc2 → Dc3’ automatically produces an inverse change for ‘Dd2 → Dd3’, as stated in (46) and reduces troublesome manipulations, even if only partially.Generally, BWe next comment on other possible applications of the model. The category theory-like construction for treating DNA transcription to RNA might be conceivable, and the combination of the set and the group can comprise a category when these satisfy category theory postulates ,49. Thatj.To begin, according to our description for handling ‘E’s, it seems difficult to define inverse elements in a group theoretical way when there are deletions of ‘E’s from any place in a sequence because we cannot find sufficient numbers of ‘E’s in the target component of DThus, we consider the ‘morphism f’ that transforms the sequence of DNA bases within set D as follows ,49.j, cod(f) = Dk. Object ‘X’ is the set of ‘Dj’s. There exists a morphism ‘1X’ such that ‘1X●f = f = f●1X’ for every ‘morphism f’, when ‘1X’ = [n1|n2|n3|…|ni|…|nN-1|nN|…] (∈ group B). If supplemented, the ‘morphisms f’ comprise ‘group B’  = Dj) to those of its complement (Dj†), and ΙΙ) alternation from ‘C, A, T, and G’ to ‘C, A, U, and G’ (Dj† → Rj†)  transcription from the original DNA sequences  (for transcription to pre-messenger RNA (pre-mRNA) before splicing) can be performed via the manipulation  in §3 and §4. ‘Dj†’ can be obtained via the linear group , the rotational group  and also the wallpaper group . Thereby, morphism ρ : X → Y, dom(ρ) = Dj, cod(ρ) = Dj†. Object ‘Y’ is the set of ‘Dj†’s .The transformation from the original DNA bases ‘DX’ and ‘1Y’ such that ‘1Y●ρ = ρ = ρ●1X’ for ‘morphism ρ’, whereThere exist morphisms ‘1m’s ∈ group B  [A|G|T|A|E|T|C|G|A|C|T|…] → (Rj†=) [A|G|U|A|E|U|C|G|A|C|U|…]’.Next, we define manipulations that change the above ‘DThis process can also be expressed in a similar way as transcription.j†, cod(ρ) = Rj†. Object ‘Z’ is the set of ‘Rj†’s. There exist morphisms ‘1Y’ and ‘1Z’ such that ‘1Z●τ = τ = τ●1Y’ for every ‘morphism τ’, wheremorphism τ : Y → Z, dom(τ) = D4‘1Y=1Z=n1Evidently, morphism τ does not satisfy the group postulates because the source object ‘Y’ and target object ‘Z’ are different and a single set of operations cannot be defined at this stage.Additionally, as for Steps І and ΙΙ, the resultant process for morphisms ρ and τ can be expressed as:j, cod(g) = Rj†  = Dj† and Rj is the appearance ‘T’ and ‘U’ in the sequences.The only difference between DNaturally, for RNA base sequences, similar treatments are possible in the single group B:j†, cod(h) = Rk†  = R Figures  and 8.Ordinarily, in prokaryotic cells, the DNA sequences are transcribed along their entire length. For eukaryotic cell, a splicing process is needed using nascent pre-messenger RNA (pre-mRNA) where introns of DNA bases are removed and exons are joined before producing a correct protein through translation, resulting in the mature messenger RNA (mRNA). Thus, the previous procedure was about the prokaryotic cell or the pre-translation of pre-mRNA in the eukaryotic cell. Therefore, to treat the products after this RNA splicing procedure in the eukaryotic cell, the following approach might be possible. The removal of introns can be regarded as changes from a certain series of bases to ‘E’s as follows.GUA’ is removed from ‘A(GUA)EUCGACU…’ to become ‘AEUCGACU…’, this procedure can be described as; ‘Rj† → Rsj†’,If ‘j†’ form a set Rs = {Rsj† } that is a part of set R  = Rj†, cod(j) = Rsj†. There exists a morphism ‘1Zs’ such that ‘1Zs●j = j = j●1Zs’.Hereon, we admit ‘E’s in the sequences of RNAs (as elements of set Rs) during the operations before morphism ‘f’ and after morphism ‘j’ to maintain theoretical consistency. Thus, if the result of a series of these maps is ‘Rs‘j’ changes some series of bases from ‘C, A, U, G, E’ to an equivalent-sized series of ‘E’s within the partial operations of the group B. However, morphism ‘j’ fails the group axioms, as inverse might not be definable.Finally, as in §1, we apply the simultaneous deletions of all explicit ‘E’s of mRNA other than the trailing ‘E’s, the state after these deletions being denoted with ‘< >’; forj† > = [A1U2C3G4A5C6U7…]’ is specified without explicit non-trailing ‘E’s. In this regard, as in §1, if some indels (insertions/deletions) occur at certain bases of < Rsj†>, as for ‘<Rsj1† > = [A1U2(E3)G4(E5)C6U7…]’ (with the deletion of ‘C3’ and ‘A5’), we state the result as ‘<<Rsj1†>> = [A1U2G3C4U5…]’. ‘Rj†’s include < Rsj† > s and ‘<<Rsj†> > s from the set R and both still satisfy the postulates of group B. This rule is a relative postulate and explicit ‘E’s are not absolutely forbidden in ‘<Rsj†>’s or ‘<<Rsj†>>’s, hence further indels of ‘E’s into ‘<Rsj†>’s or ‘<<Rsj†>>’s are not forbidden.the description ‘<Rsj1†>} = [A1U2G4C6U7…], where place numbers ‘3’ and ‘5’ are absent indicating implicitly their presence in the vector.  Similarly, t-tuples of ‘< >’s are denoted ‘<<<<Rsj†>>>> (t-tuple) = <Rsj†> t’ representing multiple deletions of ‘E’s (t-times). Combinations of symbols ‘{ }’ and ‘< >’ are also allowed when necessary, as for example {<{{<Rsj1>}}>}, as long as the subscripted place numbers are adequately recognized/traced.Also, omissions of explicit ‘E’s are considered as in ‘{<Rsj†’ should have a unique meaning with regard to protein synthesis. As a result, the subsequent reading/translation in line with ‘codon’-like ‘[AUC|GAC|U…]’, ‘[A|UCG|ACU|…]’ or ‘[AU|CGA|CU…]’ leads in an ordinal way to a description of protein synthesis. Through the use of ‘{ }’ and/or ‘< >’, the concept ‘E’ may have benefits, although this may need to be intensely explored in future studies.Nevertheless, the multiple use of ‘< >’ to remove all ‘E’s in the vector ‘RsThe procedure reversing transcription, found for example in retrovirus, is also describable if additional options are added to the scheme. However, these options are omitted at this stage to keep the model simple.In summary, suppose we have a ‘category C’ with objects ‘X’, ‘Y’, ‘Z’, ‘Zs’ and morphisms ‘f’ , ‘ρ’ , ‘τ’ , ‘g’ , ‘h’ and ‘j’. We affirm that these definitions satisfy the postulates of category. A list is given in Figure The expression ‘hom’ denotes all morphisms f: ‘from X to X’. Likewise, ‘hom’ denotes all morphisms ρ: ‘from X to Y’. In addition, ‘hom’ denotes all morphisms τ: ‘from Y to Z, and hom denotes all morphism g: from X to Z. Then, hom denotes all morphisms h: ‘from Z to Z’. Finally, hom denotes all morphism h: from Y to Z. (Details are displayed in Appendix F)1 and C2 are linked.As is explained in §3 and §4, the rotational group can be regarded as a specific bijection of the wallpaper group ,44-47, s1 and C2 with a ‘functor F’ from C1 to C2 written ‘F: C1 → C2’. For example, the pre-category C is denoted C1 and the product of functor F on category C1 is denoted C2’ and ‘D2 = [A1C2C3(E4E5)G6T7E8E9E10…]’ are different over the wallpaper pattern despite being equivalent as real sequences. Although by use of electronic tools, these graphics might be of versatility for detection or identification of DNA sequences, these might produce other confusions in the present form. We hope that more appropriate devices would be performed in future study.Third, as for the graphical displays of ‘D’s in Figure Fourth, DNA transcription to RNA and/or mRNA and translation of RNA and/or mRNA into proteins at the ribosomes are performed using a grammar rule based on a three-base set called a ‘codon’. Codons have information to synthesize twenty types of proteins; for example, ‘CAG’ codes for ‘glutamine’. As mentioned before, a number of approaches have been proposed exploit group-theoretic methods. These cover the rules for composition of triplet of bases ‘XXX’, the ways of reading codons, and models to compose geometric solids such as the tetrahedron and hexahedron, ,33,34,50Fifth, the traditional symmetry model of DNA bases often is based on the chemical types ‘purine/pyrimidine’, ‘amino/keto’, and ‘strong/weak hydrogen binding’ using biomolecular characteristics, which often have advantages for their treatments where three-dimensional graphics aid the imagination, and `matricized’ expressions are possible ,35,36. ISixth, there might be too many speculative conjectures with hypothetical situations those should be used to prove scientific facts using verified methods. Thus, a more rigorous examination for a rational style with a more effective methodology is necessary.Our model is far from a complete systematization. However, we believe that it is necessary that some principal breakthrough should be pursued if we intend to systemize a descriptive model, and that if appropriate definitions are devised, that might help to systemize biomolecular/genetic biology in a more optimized manner with greater sophistication to make a significant contribution to the field.Within the large limitations of our methodology, it is considered that there is fertile ground where variants of the symmetry model for genetic coding based upon a specific wallpaper group are constructible. By integrating the linear group and rotational group over a specific wallpaper pattern, a more integrated formulation based on a group/category theory-like description is open to exploration in applications to a number of topics from molecular/genetic biology.According to Figures Here the symbol ‘↔’ signifies ‘bijection’ and the meaning of ‘x’ is explained in §4. Hence, operators that are regarded to effect changes from one base to another can be re-expressed as illustrated in the following examples for various types of component operations:As for B,j●Bk)●Bl = Bj●(Bk●Bl)’ holds for all positive integers j, k and l.1) Associativity: ‘(B0 = [n1|n2|n3| … |ni| … |n(n ‒ 1)|nn|nn + 1|nn + 2|nn + 3| …]’ is an identity element that satisfies ‘ B0●Bm = Bm●B0 = Bm ’.  ’ is an element of Z5 (no movement of the point P))2) Identity: ‘Bm−1’ that satisfies ‘Bm‒ 1●Bm = Bm●Bm‒ 1 = B0’. Actually, the components of the inverse are the inverses of each individual component.3) Inverses: there exists a unique ‘Bj●Bk = Bk●Bj’.4) Commutativity: ‘Bj●Bk’ belongs to the set B.5) Closure law: any ‘BAgain, with reference to Figure k is generated through the following sequence of operations:Naturally, the series DThen,The axioms are:Ι) A binary operation and closure law: the combination of two morphisms satisfies hom × hom → hom. Moreover, hom × hom → mor  and hom × hom → mor  both hold.II) Associativity: If f: X → X, ρ: X → Y, τ: Y → Z, g: X → Z, h: Z → Z, and j: Z → Zs. Then, ‘f●(ρ●τ) = (f●ρ)●τ’, ‘ρ●(τ●h) = (ρ●τ)●h’, ‘f●(g●h) = (f●g)●h’, and ‘τ●(h●j) = (τ●h)●j’ hold.X, 1Y, 1Z, 1Zs’ such that ‘1X●f = f = f●1X’, and ‘1Y●ρ = ρ = ρ●1X’, ‘1Z●τ = τ = τ●1Y’, ‘1Z●g = g = g●1X’, ‘1Z●h = h = h●1Z’. ‘1Zs●j = j = j●1Z’. In practice,III) Identity: there exist morphisms ‘11,For Category C2, for each object F(X1) = X2, F(Y1) = Y2, F(Z1) = Z2, F(Zs1) = Zs2 (∈C2), the following relationships also hold:Similarly for category C1●ρ1) = F(f1)●F(ρ1), F(ρ1●τ1) = F(ρ1)●F(τ1), F(τ1●h1) = F(τ1)●F(h1), F(f1●g1) = F(f1)●F(g1), F(g1●h1) = F(g1)●F(h1), and F(h1●j1) = F(h1)●F(j1) are satisfied, the composition of C1 and C2 linked with ‘functor F’ is possible although the proof is omitted here.Other than these, if relationships F(f1), ‘F(1X) = 1F(X) (∈C2)’ is true, for object Y (∈C1), ‘F(1Y) = 1F(Y) (∈C2)’ and for object Z (∈C1), ‘F(1Z) = 1F(Z) (∈C2)’, is true under the condition:Furthermore, the following postulates hold: for object X (∈CThe authors declare that they have no competing interests.JS conceived of the main idea of this article and wrote the manuscript. SM revised the manuscript. JI gave advice on potential versatilities of the model to the biological science. In addition, all authors read and approved the final version of the manuscript.


AbstractPorcellanidae, which comprises 283 species, larval development remains to be described. Full development has been only described for 52 species, while part of the larval cycle has been described for 45 species. The importance of knowing the complete larval development of a species goes beyond allowing the identification of larval specimens collected in the plankton. Morphological larval data also constitute a support to cladistic techniques used in the establishment of the phylogenetic status (see Porcellanidae proposed by For most of the family atus see . Neverth Porcellanidae, commonly known as porcelain crabs, is a family of decapods belonging to the infraorder Anomura . The group comprises 283 species according to the classification proposed by Cancer pagurus, described as Cancer germanicus by Linnaeus, 1767), the morphology of a porcellanid larva was not described until 1835, when J. Vaughan Thompson published a brief description of a larva of Porcellana reared from eggs of females collected in British waters. Eight years later, Pisidialongicornis . Numerous descriptions of the larval stages have been published during more than 170 years. The number of published descriptions of the larval morphology of porcellanids, and of other groups of decapods, has grown exponentially since the 1960’s , it is complicated to access it using common bibliographic search engines. As a consequence, in studies requiring the identification of planktonic organisms , or in morphological studies in which new larval stages are described, where it becomes necessary to compare results with those reported in previous publications of larval descriptions, the researcher has a difficult task in compiling the available information for the target taxon. Although this situation has yielded publication of several bibliographic compilations for brachyurans, like those of PageBreakMany larval publications first appeared more than 30 years ago; for example according to Porcellanidae.Therefore, the objective of this study is three-fold: 1) to compile the available literature on porcellanid larval morphologies; 2) to record the possible changes in the nomenclature of species, or synonymies; and 3) to describe the state-of-the-art on the larval development of species belonging to the family Google Scholar, Scopus, Science Direct and Web of Science have been used for the bibliographic compilation. The current total number of porcellanid species and the taxonomic classification used for the present checklist follow those of World Register of Marine Species (http://www.marinespecies.org).The data set of this study comprises a total of 133 entries obtained from 83 published papers (from 1835 to 2012). Search engines and scientific databases such as prezoeal stage (PR), first to fifth zoeal stage (Z1-5), and megalopal stage (M); iii) the method used to obtain the larvae, according to the following designations: from plankton samples (Pl), larvae reared under laboratory conditions from an identified ovigerous females (Lab) and larvae obtained from plankton and by instar-to-instar laboratory rearing, from unknown parentage, but often a species recognizable from its postlarval or juvenile stages (P+L). Entries marked with asterisk mean that the larval description available, in our opinion, is accurate enough to establish comparisons with other species and have all stages fully described and illustrated. In the checklist, if the taxonomical name of the species described does not match the current taxonomic name according to as’ followed by the name of the species cited in the description.In the checklist, the status of current knowledge of the larval development is specified for each species as follows: i) the author(s) and the date of publication of the larval description; ii) the specific larval stages described, using the following classification: Petrochelesspinosus, which has five zoeal stages.The larval development of porcellanids usually consists of two zoeal stages and one megalopal stage, with the exception of Pisidialongicornis, referred PageBreakto as Porcellanalongicornis. The larval descriptions available were poor in number until the 1960’s and 1970’s, when an increasing trend in the number of publications is observed; this was possibly due to the increased number of scientists specializing in this area, to the increased facilities for cultivating larvae in laboratory conditions, and to the advances in microscope technology  and Pachycheles (44 species); however, the complete larval development has been described for only 21 species of Petrolisthes (19.8%) and only nine species of Pachycheles (20.4%).The current knowledge of larval development by genus (percentages) and the number of species in each genus are shown in Figure Porcellanidae Haworth, 1825Family Porcellana sp; Z1: LabThompson (1935) as Porcellana sp; M: PlPlAliaporcellanakikuchii Nakasone & Miyake, 1969: larvae undescribedPageBreakAliaporcellanapygmaea : larvae undescribedAliaporcellanataiwanensis Dong, Li & Chan, 2011: larvae undescribedAliaporcellanasuluensis : larvae undescribedAliaporcellanatelestophila : larvae undescribedAllopetrolisthesangulosus : full larval descriptionPR, Z1, Z2, M: Lab*Allopetrolisthespunctatus : larvae undescribedAncylochelesgravelei : larvae undescribedCapilliporcellanamurakamii : larvae undescribedCapilliporcellanawolffi Haig, 1981: larvae undescribedClastotoechusdiffractus : larvae undescribedClastotoechusgorgonensis Werding & Haig, 1983: larvae undescribedClastotoechushickmani Harvey, 1999: larvae undescribedClastotoechuslasios Harvey, 1999: larvae undescribedClastotoechusnodosus : full larval descriptionM: Lab*Enosteoideslobatus Osawa, 2009: larvae undescribedEnosteoidesmelissa : larvae undescribedEnosteoidesornatus : partial larval descriptionPorcellanaornata; PR, Z1: LabLabEnosteoidespalauensis : larvae undescribedEnosteoidesphilippinesnsis Dolorosa & Werding, 2014: larvae undescribedEuceramuspanatelus Glassell, 1938: larvae undescribedEuceramuspraelongus Stimpson, 1860: full larval descriptionM: LabPlEuceramustransversilineatus : larvae undescribedEulenaioscometes : partial larval descriptionLabHeteropolyonyxbiforma Osawa, 2001: larvae undescribedHeteroporcellanacorbicola : larvae undescribedLiopetrolisthesmitra : larvae undescribedLissoporcellanademani Dong & Li, 2014: larvae undescribedLissoporcellanaflagellicola Osawa & Fujita, 2005: larvae undescribedLissoporcellanafurcillata : larvae undescribedLissoporcellanamiyakei Haig, 1981: larvae undescribedLissoporcellanamonodi Osawa, 2007: larvae undescribedLissoporcellananakasonei : larvae undescribedLissoporcellananitida : larvae undescribedLissoporcellanapectinata Haig, 1981: larvae undescribedLissoporcellanaquadrilobata : larvae undescribedLissoporcellanaspinuligera : larvae undescribedPageBreakMegalobrachiumerosum : larvae undescribedMegalobrachiumfestae : larvae undescribedMegalobrachiumgarthi Haig, 1957: larvae undescribedMegalobrachiummortenseni Haig, 1962: partial larval descriptionPlMegalobrachiumpacificum Gore & Abele, 1974: larvae undescribedMegalobrachiumperuvianum Haig, 1960: larvae undescribedMegalobrachiumpoeyi : full larval descriptionM: Lab*Megalobrachiumroseum : full larval descriptionM: Lab*Megalobrachiumsinuimanus : larvae undescribedMegalobrachiumsmithi : larvae undescribedMegalobrachiumsoriatum : full larval descriptionM: Lab*Megalobrachiumtuberculipes : larvae undescribedMinyocerusangustus : partial larval descriptionLabMinyoceruskirki Glassell, 1938: larvae undescribedNeopetrolisthesalobatus : larvae undescribedNeopetrolisthesmaculatus : partial larval descriptionLabPlNeopetrolisthesspinatus Osawa & Fujita, 2001: partial larval descriptionLabNeopisosomaangustifrons : full larval descriptionM: Lab*Neopisosomabicapillatum Haig, 1960: larvae undescribedNeopisosomacuracaoense : larvae undescribedNeopisosomadohenyi Haig, 1960: larvae undescribedNeopisosomamexicanum : larvae undescribedNeopisosomaneglectum Werding, 1986: full larval descriptionM: Lab*Neopisosomaorientale Werding, 1986: larvae undescribedNovorostrumdecorocrus Osawa, 1998: full larval descriptionM: Lab*Novorostrumindicum : partial larval descriptionLabNovorostrumphuketense Osawa, 1998: larvae undescribedNovorostrumsecuriger : larvae undescribedOrthochelapumila Glassell, 1936: larvae undescribedPachycheles sp.Pachycheles nrs39; Z2: PlPageBreakPachychelesackleianus A. Milne-Edwards, 1880: larvae undescribedPachychelesattaragos Harvey & de Santo, 1997: larvae undescribedPachychelesbarbatus A. Milne-Edwards, 1878: larvae undescribedPachychelesbellus : larvae undescribedPachychelesbiocellatus : larvae undescribedPachychelescalculosus Haig, 1960: larvae undescribedPachycheleschacei Haig, 1956: partial larval descriptionM: PlPachycheleschubutensis Boschi, 1963: partial larval descriptionLabGonzález et al. (2006); Z1: Pachychelescrassus : larvae undescribedPachychelescrinimanus Haig, 1960: larvae undescribedPachychelescristobalensis Gore, 1970: larvae undescribedPachychelesgarciaensis : partial larval descriptionLabPachychelesgranti Haig, 1965: larvae undescribedPachychelesgreeleyi : larvae undescribedPachychelesgrossimanus : larvae undescribedPachycheleshertwigi Balss, 1913: partial larval descriptionLabPachychelesholosericus Schmitt, 1921: larvae undescribedPachychelesjohnsoni Haig, 1965: larvae undescribedPachycheleslaevidactylus Ortmann, 1892: full larval descriptionPachycheleshaigae; Z1, Z2, M: Lab*Pachychelesmarcortezensis Glassell, 1936: larvae undescribedPachychelesmonilifer : full larval descriptionM: Lab*Pachychelesnatalensis : full larval descriptionLabM: Lab*M: Lab*Pachychelespanamensis Faxon, 1893: larvae undescribedPachychelespectinicarpus Stimpson, 1858: larvae undescribedPachychelespilosus : full larval descriptionM: Lab*PlPachychelespisoides : larvae undescribedPachychelespubescens Holmes, 1900: full larval descriptionM: Lab*McMillan (1972); Z1, Z2, PR, Z1, Z2, M: P+L*Pachychelesriisei : partial larval descriptionPlPachychelesrudis Stimpson, 1859: full larval descriptionPageBreakLabPR, Z1, Z2, M: Lab*Pachychelesrugimanus A. Milne-Edwards, 1880: larvae undescribedPachychelessahariensis Monod, 1933: larvae undescribedPachychelessculptus : partial larval descriptionLabPachychelesserratus : full larval descriptionM: Lab*PlPachychelessetiferous Yang, 1996: larvae undescribedPachychelessetimanus : larvae undescribedPachychelesspinidactylus Haig, 1957: larvae undescribedPachychelesspinipes : larvae undescribedPachychelesstevensii Stimpson, 1858: full larval descriptionPlM: Lab*Pachychelessubsetosus Haig, 1960: larvae undescribedPachychelessusanae Gore & Abele, 1974: partial larval descriptionPlPachychelestomentosus Hendersson, 1893: full larval descriptionM: Lab* Pachychelestrichotus Haig, 1960: larvae undescribedPachychelesvelerae Haig, 1960: larvae undescribedPachychelesvicarius Nobili, 1901: larvae undescribedParapetrolisthestortugensis : partial larval descriptionPlPetrochelesaustraliensis : larvae undescribedPetrochelesspinosus : full larval descriptionPlPl; M: P+L*PR: LabPetrolisthesaegyptiacus Werding & Hiller, 2007: larvae undescribedPetrolisthesagassizii Faxon, 1893: larvae undescribedPetrolisthesamoenus : larvae undescribedPetrolisthesarmatus : full larval descriptionLabPlM: Lab*M: Lab*PlPetrolisthesartifrons Haig, 1960: larvae undescribedPetrolisthesasiaticus : partial larval descriptionLabPageBreakPetrolisthesbifidus Werding & Hiller, 2004: larvae undescribedPetrolisthesbispinosus Borradaile, 1900: larvae undescribedPetrolisthesbolivarensis Werding & Kraus, 2003: full larval descriptionM: PlPetrolisthesborradailei Kropp, 1984: larvae undescribedPetrolisthesboscii : full larval descriptionM: Lab*Petrolisthesbrachycarpus Sivertsen, 1933: larvae undescribedPetrolisthescabrilloi Glassell, 1945: larvae undescribedPetrolisthescaribensis Werding, 1983: partial larval descriptionLabPlPetrolisthescarinipes : larvae undescribedPetrolisthescelebesensis Haig, 1981: larvae undescribedPetrolisthescinctipes : full larval descriptionPR, Z1, Z2, M: Lab*Petrolisthescoccineus : partial larval descriptionLabPetrolisthescocoensis Haig, 1960: larvae undescribedPetrolisthescolumbiensis Werding, 1983: partial larval descriptionPlPetrolisthescrenulatus Lockington, 1878: larvae undescribedPetrolisthesdecacanthus Ortmann, 1897: larvae undescribedPetrolisthesdesmarestii : larvae undescribedPetrolisthesdissimulatus Gore, 1983: partial larval descriptionPlPetrolisthesdonadio Hiller & Werding, 2007: larvae undescribedPetrolisthesdonanensis Osawa, 1997: larvae undescribedPetrolisthesedwardsii : partial larval descriptionPlPetrolistheseldredgei Haig & Kropp 1987: larvae undescribedPetrolistheselegans Haig, 1981: larvae undescribedPetrolistheselegantissimus Werding & Hiller, 2015: larvae undescribedPetrolistheselongatus : full larval descriptionLab; Z2, M: P+L*PR, Z1: Lab; Z2, M: P+L*Petrolistheseriomerus Stimpson, 1871: full larval descriptionM: LabPR, Z1, Z2, M: Lab*Petrolisthesextremus Kropp & Haig, 1994: larvae undescribedPetrolisthesfimbriatus Borradaile, 1898: larvae undescribedPetrolisthesgalapagensis Haig, 1960: larvae undescribedPetrolisthesgalathinus : partial larval descriptionPageBreakPlPetrolisthesgertrudae Werding, 1996: larvae undescribedPetrolisthesglasselli Haig, 1957: larvae undescribedPetrolisthesgracilis Stimpson, 1859: larvae undescribedPetrolisthesgranulosus : full larval descriptionPR, Z1, Z2, M: Lab*Petrolistheshaigae Chace, 1962: partial larval descriptionLabPlPetrolistheshaplodactylus Haig, 1988: larvae undescribedPetrolistheshastatus Stimpson, 1858: partial larval descriptionLabPetrolistheshaswelli Miers, 1884: larvae undescribedPetrolisthesheterochrous Kropp, 1986: larvae undescribedPetrolistheshians Nobili, 1901: larvae undescribedPetrolistheshirtipes Lockington, 1878: larvae undescribedPetrolistheshirtispinosus Lockington, 1878: larvae undescribedPetrolistheshispaniolensis Werding & Hiller, 2005: larvae undescribedPetrolisthesholotrichus Nobili, 1901: larvae undescribedPetrolisthesinermis : larvae undescribedPetrolisthesjaponicus : partial larval descriptionLabLabPetrolisthesjugosus Streets, 1872: partial larval descriptionPlPetrolistheskranjiensis Johnson, 1970: larvae undescribedPetrolistheslaevigatus : full larval descriptionPR, Z1, Z2, M: Lab*Petrolistheslamarckii : full larval descriptionLabM: Lab*M: Lab*Petrolisthesleptocheles : larvae undescribedPetrolistheslewisi : partial larval descriptionPlPetrolistheslimicola Haig, 1988: larvae undescribedPetrolistheslindae Gore & Abele, 1974: larvae undescribedPetrolisthesmagdalenensis Werding, 1978: full larval descriptionM: LabM: Lab*Hernández and Magan (2012); Z1, Z2, Petrolisthesmanimaculis Glassell, 1945: larvae undescribedPetrolisthesmarginatus Stimpson, 1859: partial larval descriptionPlPageBreakPetrolisthesmasakii Miyake, 1943: larvae undescribedPetrolisthesmelini Miyake & Nakasone, 1966: partial larval descriptionPetrolisthescarinipes; Z1, Z2: LabPetrolisthesmesodactylon Kropp, 1984: larvae undescribedPetrolisthesmilitaris : larvae undescribedPetrolisthesmiyakei Kropp, 1984: larvae undescribedPetrolisthesmoluccensis : partial larval descriptionLabPetrolisthesmonodi Chace, 1956: partial larval descriptionM: PlPetrolisthesnanshensis Yang, 1996: larvae undescribedPetrolisthesnigrunguiculatus Glassell, 1936: larvae undescribedPetrolisthesnobilii Haig, 1960: partial larval descriptionLabPetrolisthesnovaezelandiae Filhol, 1885: full larval descriptionM: P+L*PR, Z1: Lab; Z2, M: P+L*Petrolisthesobtusifrons Miyake, 1937: larvae undescribedPetrolisthesornatus Paulson, 1875: full larval descriptionM: Lab*Petrolisthesortmanni Nobili, 1901: partial larval descriptionPlPetrolisthesperdecorus Haig, 1981: larvae undescribedPetrolisthesplatymerus Haig, 1960: full larval descriptionM: Lab*Petrolisthespolitus : full larval descriptionM: Lab*Petrolisthespolymitus Glassell, 1937: larvae undescribedPetrolisthespubescens Stimpson, 1858: partial larval descriptionLabPetrolisthesquadratus Benedict, 1901: partial larval descriptionPlPetrolisthesrathbunae Schmitt, 1921: larvae undescribedPetrolisthesrobsonae Glassell, 1945: full larval descriptionM: Lab*García-Guerrero et al. (2005); Z1, Z2, Petrolisthesrosariensis Werding, 1982: partial larval descriptionPlPetrolisthesrufescens : full larval descriptionM: Lab*Petrolisthessanfelipensis Glassell, 1936: larvae undescribedPetrolisthessanmartini Werding & Hiller, 2002: larvae undescribedPetrolisthesscabriculus : larvae undescribedPetrolisthesschmitti Glassell, 1936: larvae undescribedPageBreakPetrolisthessquamanus Osawa, 1996: larvae undescribedPetrolisthesteres Melin, 1939: larvae undescribedPetrolisthestiburonensis Glassell, 1936: larvae undescribedPetrolisthestomentosus : partial larval descriptionLabPetrolisthestonsorius Haig, 1960: full larval descriptionM: Lab*PlPetrolisthestridentatus Stimpson, 1859: full larval descriptionM: Lab*PlPetrolisthestrilobatus Osawa, 1996: partial larval descriptionLabPetrolisthestuberculatus : larvae undescribedPetrolisthestuberculosus : larvae undescribedPetrolisthestuerkayi Naderloo & Apel, 2014: larvae undescribedPetrolisthesunilobatus Henderson, 1888: full larval descriptionM: Lab*Petrolisthesuruma Osawa & Uyeno, 2013: larvae undescribedPetrolisthesviolaceus : full larval descriptionPorcellanamacrocheles; Z2: PlPR, Z1, Z2, M: Lab*Petrolisthesvirgatus Paulson, 1875: larvae undescribedPetrolistheszacae Haig, 1968: full larval descriptionM: Lab*Pisidia sp.PlPisidia sp asm10; Z1, Z2, M: PlPisidiabluteli : full larval descriptionPorcellanabluteli; Z1, Z2, M: PlPorcellanabluteli; M: PlM: PlPisidiabrasiliensis Haig, in Rodrigues da Costa, 1968: partial larval descriptionLabPlPisidiadehaanii : full larval descriptionM: Lab*Pisidiadelagoae : larvae undescribedPisidiadispar : full larval descriptionPR, Z1: Lab; Z2, M: P+LPisidiagordoni : larvae undescribedPisidiainaequalis : partial larval descriptionPorcellanainaequalis; PR, Z1, Z2: PlPageBreakPisidialongicornis : full larval descriptionPorcellanalongicornis; Z1: LabGalathea; Z2: PlPorcellanalongicornis; Z1: PlPorcellanaplatycheles; Z1: PlPorcellanalongicornis; Z1, Z2: P+LPorcellanalongicornis; Z1, M: PlPorcellanalongicornis; Z1, Z2: PlPorcellana sp; Z2: PlPorcellanalongicornis; Z1, Z2, M: P+L*Porcellanalongicornis; Z1, Z2, M: PlPorcellanalongicornis; M: PlPorcellanalongicornis; Z1, Z2, M: P+LPisidialongimana : partial larval descriptionPlPisidiamagdalenensis : larvae undescribedPisidiaserratifrons : partial larval descriptionPisidiaspinulifrons; PR, Z1: LabLabPisidiastreptocheles : larvae undescribedPisidiastreptochiroides : partial larval descriptionPR, Z1: Lab; Z2: P+LPisidiastriata Yang and Sun, 1990: larvae undescribedPisidiavanderhorsti : partial larval descriptionClastotoechusvanderhorsti; PR, Z1, Z2: Lab*Pisidiavariabilis : larvae undescribedPolyonyxbiunguiculatus : larvae undescribedPolyonyxboucheti Osawa, 2007: larvae undescribedPolyonyxbouvieri Saint Joseph, 1900: larvae undescribedPolyonyxconfinis Haig, 1960: larvae undescribedPolyonyxgibbesi Haig, 1956: partial larval descriptionPR, Z1, Z2: LabPlPolyonyxhaigae McNeil, 1968: larvae undescribedPolyonyxhendersoni Southwell, 1909: full larval descriptionPR, Z1: LabM: Lab*Polyonyxloimicola Sankolli, 1965: full larval descriptionM: Lab*Polyonyxmaccullochi Haig, 1965: larvae undescribedPolyonyxnitidus Lockington, 1878: larvae undescribedPolyonyxobesulus Miers, 1884: larvae undescribedPolyonyxpedalis Nobili, 1905: larvae undescribedPageBreakPolyonyxplumatus Yang & Xu, 1994: larvae undescribedPolyonyxquadratus Chace, 1956: larvae undescribedPolyonyxquadriungulatus Glassell, 1935: full larval descriptionM: Lab*Polyonyxsenegalensis Chace, 1956: larvae undescribedPolyonyxsinensis Stimpson, 1858: larvae undescribedPolyonyxspina Osawa, 2007: larvae undescribedPolyonyxsplendidus Sankolli, 1963: larvae undescribedPolyonyxthai Werding, 2001: larvae undescribedPolyonyxtransversus : full larval descriptionPR, Z1: Lab; Z2, M: P+LPolyonyxtriunguiculatus Zehntner, 1894: larvae undescribedPolyonyxtulearis Werding, 2001: larvae undescribedPolyonyxutinomii Miyake, 1943: larvae undescribedPolyonyxvermicola Ng & Sasekumar, 1993: larvae undescribedPorcellanaafricana Chace, 1956: larvae undescribedPorcellanacancrisocialis Glassell, 1936: full larval descriptionM: Lab*García-Guerrero et al. (2006); Z1, Z2, Porcellanacaparti Chace, 1956: larvae undescribedPorcellanacorbicola Haig, 1960: larvae undescribedPorcellanacurvifrons Yang and Sun, 1990: larvae undescribedPorcellanaelegans Chace, 1956: larvae undescribedPorcellanaforesti Chace, 1956: larvae undescribedPorcellanahabei Miyake, 1961: larvae undescribedPorcellanahancocki Glassell, 1938: larvae undescribedPorcellanalillyae Lemaitre & Campos, 2000: larvae undescribedPorcellanapaguriconviva Glassell, 1936: larvae undescribedPorcellanapersica Haig, 1966: larvae undescribedPorcellanaplatycheles : full larval descriptionLabPorcellana (Polyonyx) macrocheles; Z2: PlM: PlPlM: LabPR, Z1, Z2, M: LabM: PlPlM: Lab*Porcellanapulchra Stimpson, 1858: larvae undescribedPorcellanasayana : full larval descriptionPorcellanaocellata; PR, Z1: LabM: LabPorcellanasigsbeiana A. Milne-Edwards, 1880: full larval descriptionPageBreakM: Lab*PlPorcellanellahaigae Sankarankutty, 1963: larvae undescribedPorcellanellatriloba White, 1852: larvae undescribedPseudoporcellanellamanoliensis Sankarankutty, 1961: larvae undescribedRaphidopusciliatus Stimpson, 1858: larvae undescribedRaphidopusindicus Henderson, 1893: larvae undescribedRaphidopusjohnsoni Ng & Nakasone, 1994: larvae undescribedUlloaiaperpusillia Glassell, 1938: larvae undescribed


U∣t ≥ τ}, we introduce a new concept, called the weak 𝒟-pullback exponential attractor, which is a family of sets {ℳ(t)∣t ≤ T}, for any T ∈ ℝ, satisfying the following: (i) ℳ(t) is compact, (ii) ℳ(t) is positively invariant, that is, Uℳ(τ) ⊂ ℳ(t), and (iii) there exist k, l > 0 such that distB(τ), ℳ(t)) ≤ ket−τ)− pullback exponential attracts B(τ). Then we give a method to obtain the existence of weak 𝒟-pullback exponential attractors for a process. As an application, we obtain the existence of weak 𝒟-pullback exponential attractor for reaction diffusion equation in H01 with exponential growth of the external force.First, for a process { 𝒟-pullback attractor for nonautonomous dynamical systems and gave a general method to prove the existence of 𝒟-pullback attractor. However, pullback attractors or 𝒟-pullback attractors attract any bounded set of phase space, but the attraction to it may be arbitrarily slow. In order to describe the attracting speed, the concept of pullback exponential attractor is put forward  satisfies |f − f| ≤ ξ(t)|u − v|. In fact, these conditions are relatively strict; for general conditions, we can not get the result.Pullback attractor is a suitable concept to describe the long time behavior of infinite dimensional nonautonomous dynamical systems or process generated by nonautonomous partial differential equations. There are many references concerned with the existence of pullback attractors for nonautonomous PDEs see –5). In [. In [5])ard (see ), which ard see , 8) and  and 𝒟-puard }, for any T ∈ ℝ, there exists a family of sets {ℳ(t)∣t ≤ T} satisfying the following:(i)ℳ(t) is compact.(ii)ℳ(t) is positively invariant; that is, Uℳ(τ) ⊂ ℳ(t).(iii)t ∈ B(τ), ℳ(t)) ≤ ket−τ)− pullback exponential attracts B(τ) for all {B(t)} ∈ 𝒟.∀Motivated by these problems and some ideas in –6, we in𝒟-pullback exponential attractor is not necessarily uniformly bounded or even unbounded, and the positively invariant only holds for any t ∈  be the set of all bounded subsets of X; 𝒟 is a nonempty class of parameterised setsU∣t ≥ τ} = {U∣t ≥ τ, t, τ ∈ ℝ} act on X, that is, U : X → X, ∀t ≥ τ.Let U} is said to be a process in X, if(1)UU = U, ∀t ≥ s ≥ τ,(2)U = Id is the identity operator, τ ∈ ℝ.The pair , X) is generally referred to as a nonautonomous dynamical system, and , X)  is called a nonautonomous discrete dynamical system generated by , X). If x → Ux is continuous in X, we say that the process is continuous process; if Uxn⇀Ux as xn → x, we say that the process is the norm-to-weak continuous process. Obviously, continuous process is also a norm-to-weak continuous process.A two-parameter family of mappings {B(t)∣t ∈ ℝ} ∈ 𝒟 is called 𝒟-pullback bounded absorbing sets for the process {U} if, for any t ∈ ℝ and any bounded sets {D(t)∣t ∈ ℝ} ∈ 𝒟, there exists τ0) ≤ t such that UD(τ) ⊂ B(t) for all τ ≤ τ0.A family of sets {𝒜 = {𝒜(t)∣t ∈ ℝ} ⊂ B(X) is said to be a 𝒟-pullback attractor for U if the following hold: (1)𝒜(t) is compact for all t ∈ ℝ;(2)𝒜 is invariant; that is, U𝒜(τ) = 𝒜(t)  ∀t ≥ τ;(3)𝒜 is 𝒟-pullback attracting; that is, limτ→−∞dist B(τ), 𝒜(t)) = 0, ∀{B(t)} ∈ 𝒟, and t ∈ ℝ;(4)C(t)}t∈ℝ is another family of closed attracting sets, then 𝒜(t) ⊂ C(t)  ∀t ∈ ℝ.if {Here dist denotes the nonsymmetric Hausdorff distance between sets in X; that is, dist = supa∈Ainfb∈B‖a − b‖.The family α(B) of B ⊂ X is defined by The Kuratowski measure of noncompactness The following summarizes some of the basic properties of the measure of noncompactness.B, B1, B2 ⊂ X. Then α(B) = 0 if, and only if,α(B1 + B2) ≤ α(B1) + α(B2);α(B1) ≤ α(B2) for B1 ⊂ B2;α(B1 ∪ B2) ≤ max{α(B1), α(B2)};F1⊃F2… are nonempty closed sets in X such that α(Fn) → 0 as n → ∞, then F = ⋂n=1∞Fn is nonempty and compact.if In addition, let X be an infinite dimensional Banach space with a decomposition X = X1 ⊕ X2 and let P : X → X1, Q : X → X2 be projectors with dimX1 < ∞. Then α(B(ε)) = 2ε, where B(ε) is a ball of radius ε;α(B) < ε for any bounded subset B of X for which the diameter of QB is less than ε.Let U} is called 𝒟-pullback ω-limit compact for {B(t)∣t ∈ ℝ} if, for any ε > 0, there exists a τ0, ε) ≤ t such that αB(τ)) ≤ ε.A process {U∣t ≥ τ} is 𝒟-pullback ω-limit compact forτn} ⊂ , there exists a convergence subsequence of {Uxn} whose limit lies inAssume that the process {U∣t ≥ τ} be a continuous or norm-to-weak continuous process and {U∣t ≥ τ} is 𝒟-pullback ω-limit compact; let {B(t)∣t ∈ ℝ} ⊂ B(X) be a family of 𝒟-pullback bounded absorbing sets for the process. Then the process {U∣t ≥ τ} has a 𝒟-pullback attractor 𝒜 = {𝒜(t)∣t ∈ ℝ}, and Let {U∣n, m ∈ ℤ, n ≥ m}, the above conclusions also hold true.For a discrete process {X be a Banach space; ‖·‖ denotes the norm of X, 𝒟 is a nonempty class of parameterised setsU} is a continuous process on X.Let 𝒟-pullback exponential attractor.Now, we give our main theorems which describe the relationship between the measure of noncompactness and the weak B(n)} ∈ 𝒟 is positively invariant 𝒟-pullback bounded absorbing sets of {U}; that is, for any {D(n)} ∈ 𝒟, N ∈ ℤ, there exists T ∈ ℕ, such that UD(m) ⊂ B(n) for any n − m ≥ T, and UB(m) ⊂ B(n) for any m ≤ n ≤ N; then the following are equivalent: 𝒟-pullback decays exponentially for the discrete process {U}; that is, there exist k, l > 0 such thatThe measure of noncompactness U} has a weak 𝒟-pullback exponential attractor; that is, there exists a family of sets {ℳ(n)∣n ≤ N} satisfying the following:The process {ℳ(n) is compact;ℳ(n) is positively invariant; that is, Uℳ(m) ⊂ ℳ(n);ℳ(n)∣n ∈ ℤ} attracts {D(n)} exponentially in a 𝒟-pullback sense; more precisely, {Assume that {𝒟-pullback decays exponentially for {U}, from U} is 𝒟-pullback ω-limit compact. By 𝒟-pullback attractor of {U}. Using (3) of n ≥ m, there exist finite points xn,im ∈ B(n) such that UB(m) ⊂ ⋃i=1nmB−). Letting Wnm = {xn,im∣i = 1,2,…, nm} and M(k) = ⋃n=0∞+⋃i=0nUWk−in−i, we getn ∈ ℤ, the family {M(n)∣n ≤ N} is positively invariant.((I)⇒(II)) Since the measure of noncompactness ℳ(n) = M(n) ∪ 𝒜(n); we claim that {ℳ(n)∣n ≤ N} satisfies (II).Let xk ∈ ℳ(n), there exist mk and yk such that xk = Uyk. By (I), we get that the process {U} is pullback 𝒟-ω-limit compact; we deduce from xk has subsequence convergent in ℳ(n). We get that ℳ(n) is compact.(Compactness) for any sequence UM(n) ⊂ M(n + 1), U𝒜(n) = 𝒜(n + 1), we get (Positively invariant) since D(m)} ∈ 𝒟, there exists T ∈ ℕ, such that B(n)} is positively invariant, we getWnm ⊂ M(n), we get n ≥ m. for any {x ∈ B(m), we have x ∈ B(m), there exists yx ∈ ℳ(n), such that ℳ(n) is a compact set, we get that there exist y1, y2,…, yl ∈ ℳ(n) such that yx, there exists yix ∈ {y1, y2,…, yl} such that 𝒟-pullback decays exponentially.((II)⇒(I)) By the definition of dist, we get B(t)} ∈ 𝒟 is positively invariant 𝒟-pullback bounded absorbing sets of {U}; that is, for any {D(t)} ∈ 𝒟, R ∈ ℝ, there exists T ≥ 0, such that UD(τ) ⊂ B(t) for any t − τ ≥ T, and UB(τ) ⊂ B(t) for any t ≤ R, and there exists a continuous function r(t) that satisfies ‖Ux − Uy‖ ≤ r(t)‖x − y‖ for any x, y ∈ B(τ), t − τ ≤ 1; then the following are equivalent:𝒟-pullback decays exponentially for the process {U}; that is, there exist k, l > 0 such thatThe measure of noncompactness U} has a weak 𝒟-pullback attractor; that is, there exists a family of sets {ℳ(t)∣t ≤ R} satisfying the following:The process {ℳ(t) is compact;ℳ(t) is positively invariant; that is, Uℳ(τ) ⊂ 𝒜(t);ℳ(t)∣t ≤ R} attracts {D(t)} exponentially in 𝒟-pullback sense; more precisely,{Assume that {U} generated by {U} has a weak 𝒟-pullback exponential attractor {ℳ(n)}, that is, ℳ(n) is compact and positively invariant and 𝒟-pullback exponentially attracts {D(n)} ∈ 𝒟. We set ℳ(t) = Uℳ(k), t ∈ [k, k + 1), for all k ≤ R. As proof of ℳ(t) is compact and positively invariant. Next, we will prove that ℳ(t) attracts {D(n)} ∈ 𝒟 exponentially in 𝒟-pullback sense.((I)⇒(II)) By D(t)} ∈ 𝒟, there exists T ∈ ℕ such that UD(τ) ⊂ B(t) for any t − τ ≥ T. For discrete process {U}, by k0, l0 > 0 such that t, τ ∈ ℝ, there exist t0, τ0 ∈ [0,1) such that t = n + t0, τ = m + τ0; thereforeℳ(t)∣t ≤ R} attracts {D(t)} exponentially in a 𝒟-pullback sense.For any {((II)⇒(I)) The proof is the same as that of 𝒟-pullback decays exponentially for the process {U}.We now present a method to verify that the measure of noncompactness X be a uniformly convex Banach space; that is, for all ε > 0, there exists δ > 0 such that, given x, y ∈ X, ‖x‖ ≤ 1, ‖y‖ ≤ 1, ‖x − y‖ > ε; then ‖x + y‖/2 < 1 − δ. Requiring a space to be uniformly convex is not a severe restriction in application, since this property is satisfied by all Hilbert spaces, the Lp space with 1 < p < ∞, and most Sobolev spaces Wk,p with 1 < p < ∞.Let X be a uniformly convex Banach space; for a family of bounded sets {B(t)} ⊂ X, there exist k, l, T > 0, and for any finite dimension subspace X1 of X, such that(i)PmB(τ)) is bounded;(ii)I − Pm)⋃t−τ≥sUx‖ ≤ kel(t−τ)− + k, ∀x ∈ B(τ),‖ is real-valued function satisfyingLet U} satisfies the enhanced flattening property; then the measure of noncompactness 𝒟-pullback decays exponentially for {U}.Assume that the process {B(t)} ∈ 𝒟, from (2) and (7) of k → 0, for ε0 = kel(t−τ)−, there exists M > 0, for any m > M; we haveαB(t)) ≤ 2kel(t−τ)−; that is, the measure of noncompactness of {U}𝒟-pullback decays exponentially.For any {ℛ be the set of all functions r(t) : R →  such that limt→−∞tβeλtr2(t) = 0 for some β ≥ 0, λ > 0, and denote by 𝒟 the class of all families 𝒟 = {D(t)∣t ∈ ℝ} ⊂ B(X) such thatr(t) ∈ ℛ,X with radius r(t).Let U} satisfies β ≥ 0, 0 < α < λ, and t − τ ≥ T′ and for any t ≤ R; then the process {U} has a family of positively invariant 𝒟-pullback bounded absorbing sets {B(t)∣t ≤ R}; that is, for any D ∈ 𝒟, there exists T > 0 such that UD(τ) ⊂ B(t) for any t − τ ≥ T and UB(τ) ⊂ B(t).Assume that the process {D0(t)} ∈ 𝒟, there exists T0 > 0 such that D(t)} is a family of 𝒟-pullback bounded absorbing sets. Moreover, there exists T > 0 such that t ∈ ℝ. Let B(t) ⊂ D(t) and {B(t)} is also a family of 𝒟-pullback bounded absorbing sets. We also haveLet us define By Theorems X be a uniformly convex Banach space; {U} is a process on X, and the process {U} satisfies the following:Uuτ‖2 ≤ K0(t − τ)βeλ(t−τ)−‖uτ‖2 + K1 + K2eα|t| for some β ≥ 0, 0 < α < λ, and t − τ ≥ T and any t ≤ R.‖I − Pm)x)‖ ≤ kel(t−τ)− + k, ∀x ∈ B(τ) = {x : ‖x‖ ≤ 2K1 + K2eα|τ|}, for all s ≥ T. Here m is the dimension of subspace X1 of X, and k is real-valued function that satisfies‖x − Uy‖ ≤ r(t)‖x − y‖, for any t − τ < 1, x, y ∈ B(τ); then the process {U} has a weak 𝒟-pullback exponential attractor; that is, for any R ∈ ℝ, there exists a family of sets {ℳ(t)∣t ≤ R} satisfying the following:‖ℳ(t) is compact.ℳ(t) is positively invariant; that is, Uℳ(τ) ⊂ M(t).ℳ(t) attracts {D(t)} exponentially in a 𝒟-pullback sense; more precisely,Let 𝒟-pullback exponential attractor in H01(Ω) for the process generated by the solution of the following nonautonomous reaction diffusion equation: f ∈ C1, g(·) ∈ Lloc2), Ω is a bounded open subset of ℝn, and there exist p ≥ 2,  ci > 0,  i = 1,…, 5, l > 0 such thatu ∈ ℝ.As an application of A : = −▵, naming λ the first eigenvalue of A, and denote H = L2(Ω) by scalar product  and norm |·|; let ) and ‖·‖ denote the scalar product and norm of H01(Ω) and ) = ∫Ω∇u∇v dx for all u, v ∈ H01(Ω). Moreover, we suppose for any t ∈ ℝ that there exist M ≥ 0 and 0 ≤ α < λ such thatWe set τ, T ∈ ℝ,  T > τ, there exists a unique solution u(·) ∈ C∩L2)∩Lp).For this initial boundary value problem, we know from , 8 that,U}t≥τ defined byu(t) is the solution of  wit witU is a weak solution associated with (t > τ:Assume that  satisfy –41) and andf andted with . Then thf and g satisfy (p < +∞(n ≤ 2), 2 ≤ p ≤ (2n − 2)/(n − 2) (n ≥ 3). Then the process defined by , wh, whf andfined by  has a weNext, we will prove that the process defined by  satisfy t ≤ 0, t > 0,t ∈ ℝ, we have T > 0, for any t − τ ≥ T; we have By , for t ≤1+eαt.By  and usin we have e−λt∫−∞te can get e−λt∫−∞t∫R ∈ ℝ, the process {Ut, τ}} generated by (𝒟-pullback bounded absorbing sets {B(t)∣t ≤ R} and for any x ∈ B(t), ‖x‖ ≤ 2K1 + K2eα|t|.By rated by  is a famℛ be the set of all functions r : R →  such that limt→−∞teλtr2(t) = 0 and denote by 𝒟 the class of all families 𝒟 = {D(t)∣t ∈ R} ⊂ B(H) such thatr(t) ∈ ℛ,H with radius r(t).Let A−1 is a continuous compact operator in H, by the classical spectral theorem, there exist a sequence {λj}j=1∞, ej}j=1∞ of H01(Ω) which are orthogonal in H such thatHm = span{e1, e2,…, em} in H and P : H → Hm is an orthogonal projector. For any u ∈ H we writeSince u1(t) = Uuτ1 and u2(t) = Uuτ2 to be solutions associated with . Let w(t) = u1(t) − u2(t); by (We set ted with  with ini2(t); by  we get , 2 ≤ p ≤ n/(n − 2) + 1(n ≥ 3), using Sobolev embedding theorem, we obtain u1(t), u2(t) ∈ B(t), we get Taking into account  and Höld1,2.From –56), we, we(54)fu ∈ H we writeu2, we haveu(t) ∈ B(t), hence eλ(t−τ)−‖uτ‖2 → 0 for any uτ ∈ B(τ) and λm → +∞, which imply that there exists T > 0, for any t − τ ≥ T, we have k = 1/λm + 1/(λm − α(p − 1)) + 1/(λm − α) + eα(p−1)|t|/(λm − α(p − 1)) + eα|t|/(λm − α); obviously k → 0 as m → +∞.For any oduct of  with −▵ux+gt2.By , we findαp−1t.By , we get}t≥τ generated by , a, a59), arated by  satisfie


Philometra Costa, 1845  are reported: Philometra inexpectata n. sp. from the mottled grouper Mycteroperca rubra and P. jordanoi  from the dusky grouper Epinephelus marginatus. Identification of both fish species was confirmed by molecular barcoding. The new species is mainly characterized by the length of equally long spicules (147–165 μm), the gubernaculum (63–93 μm long) bearing at the tip two dorsolateral lamellar parts separated from each other by a smooth median field, a V-shaped mound on the male caudal extremity, the presence of a pair of large caudal papillae located posterior to the cloaca and by the body length of the males (1.97–2.43 mm). Philometra inexpectata n. sp. is the fifth known gonad-infecting philometrid species parasitizing serranid fishes in the Mediterranean region. The males of P. jordanoi were examined by scanning electron microscopy for the first time; this detailed study revealed some new taxonomically important morphological features, such as the number and arrangement of cephalic and caudal papillae, presence of amphids and phasmids and mainly the lamellate structures at the posterior end of the gubernaculum. A key to gonad-infecting species of Philometra parasitic in serranid fishes is provided.Based on light and scanning electron microscopical studies of nematode specimens  collected from the ovary of groupers  in the Mediterranean Sea off Tunisia (near Tunis and Sfax), two gonad-infecting species of  Gonad-infecting species of philometrid nematodes (Philometridae) are widely distributed in marine fishes of the Atlantic, Indian and Pacific Oceans, and sometimes occur in brackish-water environments , 26. ThePhilometra Costa, 1845 have been described from a variety of marine fishes belonging to different families and their number is quickly increasing  obtained from the market in Tétouan, Morocco (probably caught in the Mediterranean Sea). Later the species was transferred to Philometra as P. jordanoi [Sanguinofilaria Yamaguti, 1941 was subsequently synonymized with Philometra [P. jordanoi [E. marginatus collected in the Mediterranean Sea near the Balearic Islands, Spain and those from wild and cultured E. marginatus in the Tyrrhenian Sea off Sicily, Italy, provided a somewhat more detailed description of philometrid gravid females, which was, more or less, in agreement with that of P. jordanoi. Nevertheless, the authors identified this material as P. lateolabracis, a species described from females collected in three species of perciform fishes off Japan [P. jordanoi to be its junior synonym.This species was originally described by López-Neyra  as Sangujordanoi . The genilometra . The orijordanoi , based off Japan , and desE. marginatus in the Mediterranean Sea off Turkey (Iskenderun Bay), described the female anterior end including cephalic structures of these nematodes and the first-stage larva from the uterus. They again identified these nematodes as P. lateolabracis. From the same locality (Iskenderun Bay off Turkey) and the same host species, P. lateolabracis was also reported by Genc et al. [Subsequently, Moravec and Genc , based oc et al. .E. marginatus in waters near Majorca, Spain, which was identified as P. lateolabracis. But in their subsequent paper [P. jordanoi, to which they assigned the above-mentioned male specimen.Merella et al.  were thent paper , they reP. lateolabracis from the type-host in Japan and provided their detailed description based on LM and SEM examinations, which enabled a comparison with other gonad-infecting Philometra spp. with described males. Their study showed that P. lateolabracis is a specific parasite of Lateolabrax japonicus (Cuvier) (Lateolabracidae) and that all previous records of this parasite from many other fish species were apparently based on misidentifications. Comparison of the males of P. lateolabracis and those of P. jordanoi confirmed the validity of the latter species [However, Quiazon et al.  were the species .P. lateolabracis in E. marginatus in the Mediterranean region [P. jordanoi. Also, the nematodes designated as Philometra sp. from the ovary of E. marginatus in Iskenderun Bay off Turkey [It is apparent that all the previous records of n region , 18, 24 P. jordanoi, including the first use of SEM, made it possible to describe some new taxonomically important morphological features in this species, such as the presence of lamellate structures on the distal end of the gubernaculum, the number and distribution of cephalic and caudal papillae, the character of the male caudal mound, the structure of the male oesophagus, the location of the nerve ring and excretory pore, and the morphology of the mature female. As compared with other gonad-infecting congeneric species parasitizing serranid fishes, P. jordanoi is remarkable for its rather long spicules, which may attain up to 265 μm [P. jordanoi with other species from the gonads of serranids is more apparent from the following key.The present detailed study of the males of o 265 μm . The comSerranus cabrilla (Linnaeus); Mediterranean Sea region ………………………....................…………….. P. serranellicabrillaeMales unknown. Body length of gravid female 40–60 mm. In Both males and females known …………………………….........................……………..………….. 2Epinephelus morio ; Gulf of Mexico ……………..............................………………..…. P. margolisiSpicules 432–468 μm long. Gravid female 65–85 mm long. In Spicules shorter than 300 μm ……………………………….........................………………...………. 3Spicules longer than 160 μm ……………………………….........................………………...…..…… 4Spicules shorter than 160 μm ……………………………..........................…………………………… 9Epinephelus bleekeri (Vaillant); Indian Ocean  ………………………..............…………...… P. tropicaSpicules conspicuously distended between second and fourth quarter of their length; spicules 168–186 μm long, length of gubernaculum 120–138 μm. In Spicules slender, not markedly distended ……………..................………….……….…………… 5Length of gubernaculum at most 87 μm ……………………...................………………………… 6Gubernaculum longer than 90 μm ………………………………….....................……….…………. 8Epinephelus costae (Steindachner); Mediterranean Sea ……....……………... P. tunisiensisPair of large caudal papillae posterior to cloaca absent. Length of spicules 201–219 μm, gubernaculum 78–87 μm long; length ratio of gubernaculum and spicules 1:2.52–2.77. In Pair of large caudal papillae posterior to cloaca present ………………….........……………. 7Epinephelus merra Bloch; Indian Ocean  ……………… …………………………………………………………………………………………………………………….. P. indicaLength of spicules 192–195 μm, gubernaculum 84 μm long; length ratio of gubernaculum and spicules 1:2.32. In Epinephelus marginatus; Mediterranean Sea ……………………………………………………………………...............................………....… P. jordanoiLength of spicules 213–265 μm, gubernaculum 81–84 μm long; length ratio of gubernaculum and spicules 1:2.61–3.15. In Epinephelus coioides (Hamilton); South Pacific  and Indian Ocean (Persian Gulf) ………………………………………............................…………..…………... P. piscariaLength of spicules 171–180 μm, representing 4–5% of male body length; gubernaculum 126–144 μm long. Male caudal mound V-shaped. Body length of male 3.67–4.19 mm. In Epinephelus cyanopodus (Richardson); South Pacific  ... P. cyanopodiLength of spicules 183–228 μm, representing 6–8% of male body length; gubernaculum 129–162 μm long. Male caudal mound U-shaped. Body length of male 2.72–3.59 mm. In Epinephelus aeneus (Geoffroy Saint-Hilaire); Mediterranean Sea …………………………………………………………………………………………… P. aeneiGubernaculum with distinct dorsal barb situated postequatorially and conspicuous dorsal protuberance at posterior end in lateral view. Length of spicules 108–123 μm, gubernaculum 96–108 μm long. In Gubernaculum without dorsal barb situated postequatorially and dorsal protuberance at its end ………………………………………………...........................…………………………….……… 10Epinephelus adscensionis (Osbeck); Gulf of Mexico ………………...........................…………..…………….. P. mexicanaBody length of male 1.63–1.86 mm. Length of spicules 90–120 μm, gubernaculum 57–66 μm long. Body length of gravid female 178–230 mm. In Body length of male longer than 1.90 mm ……………………..............……………………… 11Epinephelus fasciatus (Forsskål); South Pacific  …………………………………………………………................................………………..……. P. fasciatiSpicules 147–156 μm long, length of gubernaculum 69–84 μm. Testis reaching anteriorly only to posterior end of oesophagus. Body length of males 2.75–3.32 mm, of gravid female 387 mm. In Spicules 117–165 μm long. Testis reaching anteriorly at least to level of oesophageal nucleus ……………………………………………………………...................................…………..….... 12Caudal mound of male simple, V-shaped, not dorsally divided …………….……...…...… 13Caudal mound of male consisting of two lateral reniform parts widely separated from each other dorsally ………………………………………………………….........................…………… 14Cephalopholis sonnerati ; South Pacific  ……………………………………………………………………………………………………. P. cephalopholidisLength of spicules 129–147 μm, gubernaculum 96–111 μm long. Testis reaching anteriorly to region of oesophageal nucleus. Body length of male 2.53–2.91 mm. In Mycteroperca rubra; Mediterranean Sea ………………………………………….........................……P. inexpectata n. sp.Length of spicules 147–165 μm, gubernaculum 63–93 μm long. Testis reaching anteriorly to level of nerve ring. Body length of male 1.97–2.43 mm. In Hyporthodus flavolimbatus (Poey); Gulf of Mexico ………………..........…….. P. hyporthodiBody length of male 3.62–4.07 mm, of gravid female 105 mm. Maximum body width/length ratio of gravid female 1:59. Length of spicules 135–138 μm, gubernaculum 84 μm long. Caudal projections in females absent. Larvae in uterus 618–648 μm long. In Mycteroperca phenax; western Atlantic Ocean (off USA) including Gulf of Mexico ……...…. P. charlestonensisBody length of male 2.01–3.14 mm, of gravid females 178–230 mm. Maximum body width/length ratio of gravid females 1:131–163. Length of spicules 123–141 μm, gubernaculum 54–93 μm long. Gravid and subgravid females with pair of small papilla-like caudal projections. Larvae from uterus 544–648 μm long. In http://publicationethics.org), to which Parasite adheres, advises special treatment in these cases. In this case, the peer-review process was handled by an Invited Editor, Dominique Vuitton.The Editor-in-Chief of Parasite is one of the authors of this manuscript. COPE (Committee on Publication Ethics,