The aim of this research article is to derive a new relation between rough sets and soft sets with an algebraic structure quantale by using soft binary relations. The aftersets and foresets are utilized to define lower approximation and upper approximation of soft subsets of quantales. As a consequence of this new relation, different characterization of rough soft substructures of quantales is obtained. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Moreover, this concept generalizes the concept of rough soft substructures of other structures. Furthermore, the algebraic relations between the upper (lower) approximation of soft substructures of quantales and the upper (lower) approximation of their homomorphic images with the help of soft quantales homomorphism are examined. In comparison with the different type of approximations in different type of algebraic structures, it is concluded that this new study is much better. Quantale theory was proposed by Mulvey. It is bIn 1982, Pawlak developed the famous rough set theory , which iThere are many problems that arise in different fields such as engineering, economics, and social sciences in which data have some sort of uncertainty. Well-known mathematical tools have so many limitations because these tools are introduced for particular circumstances. There are many theories to overcome uncertainty such as fuzzy set theory, probability theory, rough sets, and vague sets, but these are limited due to its design.In 1998, Molodtsov present the idea of soft set theory, which is a mathematical tool to overcome the adversities affecting the above theories . Many auBy using aftersets and foresets notions associated with SBR, a new approximation space is widely utilized these days. By using generalized approximation space based on SBR, different soft substructures in semigroups were approximated by Kanawal and Shabir . MotivatQ- module [There are several authors who introduced rough sets theory in algebraic structures and soft algebraic structures. Iwinski analyzes algebraic properties of rough sets . Qurashi- module . Idea of- module . Rough p- module . Fuzzy i- module . General- module . In 27]Q- module- module , 29. Rou- module .The following scheme is designed for the rest of the paper. Some essential explanations related to quantales, its substructures, soft substructures, and their corresponding sequels are connected in q]Ψ denotes the equivalence class of the relation containing q. Any definable set in Θ would be written as finite union of equivalence classes of Θ. Let R⊆Θ in general R is not a definable set in Θ. However, the set R can be approximated by two definable sets in Θ. The first one is called Ψ-lower approximation (Ψ − Lappr) of R, and the second is called Ψ-upper approximation (Ψ − Uappr). They are defined as follows:Let Θ be a nonempty finite set called the universe set and Ψ be an E.R  over Θ. Let [Lappr of R in Θ is the greatest definable in Θ contained in R. The Ψ − Uappr of R in Θ is the least definable set in Θ containing R. For any nonempty subset R in Θ, P(Θ) × P(Θ) if P(Θ) denotes the set of all subsets of Θ.The Ψ − l,  w,  li,  wi ∈ Θ. Then,  is called quantale.Let Θ be a complete lattice. Define an associative binary relation ∘ on Θ satisfyingT1,  T2,  TI ⊂ Θ, i ∈ I. We define some notions as follows:Let 1 and Θ2.Throughout the paper, quantales are denoted by ΘW⊆Θ. Then, W is called a subquantale of Θ if the following holds:w1°w2 ∈ W, ∀w1, w2 ∈ W.i∈Iwi, ∈W, ∀wi, ∈W.∨Let ∅≠That is, Θ closed under ∘ and arbitrary supremum.I⊆Θ is called left (right) ideal if the following satisfied:u,  v ∈ I implies u∨v ∈ 1p ∈ Θ, u ∈ I such that p ≤ u implies p ∈ Iq ∈ Θ and u ∈ I implies q°u ∈ I(u°q ∈ I)Let Θ be a quantale, ∅≠I⊆Θ is called ideal of Θ if it is left as well as right ideal.A nonempty subset ={0,  p,  q,  r,  1} complete lattices are shown in Let Θp}, {0, q}, {0,  p, q,  r}, and Θ are all I of quantale Θ.Then, Θ is a quantale. Then, {0}, {0, I⊆Θ be an ideal. I is called prime ideal if, ∀u,  v ∈ Θ, u°v ∈ I⇒u ∈ I or v ∈ I. I is called semiprime I if, ∀ u ∈ Θ, u°u ∈ I⇒u ∈ II is called primary I if, ∀u,  v ∈ Θ, u°v ∈ I and u ∉ I implies vn ∈ I for some n ∈ Ν.Let ∅≠C) is called a soft set over Θ if Ψ : C⟶P(Θ) where C is a subset of E (the set of parameters).A pair  and  be two soft sets over Θ. Then,  soft subset  if the following conditions are fulfilled:C1⊆C2F(c)⊆H(c), ∀ c ∈ C1Let (C) be a soft set over Θ × Θ, that is, Ψ : C⟶P(Θ × Θ). Then,  is called a soft binary relation (SBR) over Θ × Θ. A SBR over Θ1 × Θ2 is a soft set  over Θ1 × Θ2. That is, Ψ : C⟶P(Θ1 × Θ2).Let  be a soft set over quantale Θ. Then,C) is called soft subquantale over Θ iff Ψ(c) is a subquantale of Θ, ∀ c ∈ C is called soft ideal over Θ iff Ψ(c) is an ideal of Θ, ∀ c ∈ C is called soft prime ideal over Θ iff Ψ(c) is a prime ideal of Θ, ∀ c ∈ C is called soft semiprime ideal over Θ iff Ψ(c) is a semiprime ideal of Θ, ∀ c ∈ C is called soft primary ideal over Θ iff Ψ(c) is a primary ideal of Θ, ∀ c ∈ C be a SBR over Θ1 × Θ2, where C⊆E (parametric set). Then, Ψ : C⟶P(Θ1 × Θ2). For a soft set  over Θ2, the LapprUapprF, C) w.r.t the afterset are essentially two soft sets over Θ1, which is defined asLet  over Θ1, the LapprUapprH, C) w.r.t the foreset are actually two soft sets over Θ2, which is defined asAnd for a soft set ={q2 ∈ Θ2 :  ∈ Ψ(c)} is called the afterset of q1 and Ψ(c)q2={q1 ∈ Θ1 :  ∈ Ψ(c)} is called the foreset of q2.For all c ∈ F, C) over Θ2, For each soft set  over Θ1, For each soft set (C) be a SBR over Θ1 × Θ2, that is, Ψ : C⟶P(Θ1 × Θ2). Then,  is called soft compatible relation (SCPR) if for all p,  r,  ji ∈ Θ1 and q,  s,  ki ∈ Θ2(i ∈ I), we havep, q),  ∈ Ψ(c)⇒∈Ψ(c) ∈ Ψ(c)⇒∈Ψ(c) over Θ1 × Θ2 is called soft complete relation (SCTR) with respect to the afterset if, for all p,  r,   ∈ Θ1, we havepΨ(c)∨rΨ(c)=(p∨r)Ψ(c)pΨ(c)∘2rΨ(c)=(p∘1r)Ψ(c)for all c ∈ C.A SCPR  is called ∨-complete w.r.t the aftersets if it satisfies only condition (1). A SCPR  is called °-complete w.r.t the aftersets if it satisfies only condition (2).A SCPR  over Θ1 × Θ2 is called soft complete relation (SCTR) with respect to the foreset if for all q,  s ∈ Θ2, and we havec)q∨Λ(c)s=Λ(c)(q∨s)Λ(c)q∘1Λ(c)s=Λ(c)(q∘2s)Λ is called ∨-complete w.r.t the foresets if it satisfies only condition (1).A SCPR  is called °-complete w.r.t the foresets if it satisfies only condition (2).A SCPR  be a SCPR with respect to the afterset over Θ1 × Θ2. Then, for any two soft sets  and  over Θ2, we haveLet ∩F1(c) ≠ ∅ and y2Ψ(c)∩F2(c) ≠ ∅, so there exist elements l,  m ∈ Θ2 such that l ∈ y1Ψ(c)∩F1(c) and m ∈ y2Ψ(c)∩F2(c). Thus, l ∈ y1Ψ(c),  m ∈ y2Ψ(c),  l ∈ F1(c) and m ∈ F2(c). So  ∈ Ψ(c) and  ∈ Ψ(c) imply  ∈ Ψ(c); that is, (l∘2m) ∈ (y1∘1y2)Ψ(c). Also, l∘2m∈F1(c)∘2F2(c); therefore, l∘2m ∈ y1∘1y2Ψ(c)∩F1(c)∘2F2(c). This shows that For arbitrary c ∈ C, let x=y1∨y2 for some y1Ψ(c)∩F1(c) ≠ ∅ and y2Ψ(c)∩F2(c) ≠ ∅, so there exist elements l,  m ∈ Θ2 such that l ∈ y1Ψ(c)∩F1(c) and m ∈ y2Ψ(c)∩F2(c). Thus, l ∈ y1Ψ(c),  m ∈ y2Ψ(c),  l ∈ F1(c), and m ∈ F2(c). So  ∈ Ψ(c) and  ∈ Ψ(c) imply  ∈ Ψ(c); that is, (l∨m) ∈ (y1∨y2)Ψ(c). Also, l∨m ∈ F1(c)∨F2(c); therefore, l∨m ∈ y1∨y2Ψ(c)∩F1(c)∨F2(c). This shows that Now, for arbitrary C) be a SCPR with respect to the foreset over Θ1 × Θ2. Then, for any two soft sets  and  over Θ1, we haveLet  be a SCTR w.r.t the afterset over Θ1 × Θ2. Then, for any two soft sets  and  over Θ2, we haveLet  and ΨF2(c) is empty, then (1) is obvious. Now, for arbitrary c ∈ C, consider that ΨF1(c) ≠ ∅ and ΨF2(c) ≠ ∅. Then, ΨF1(c)°1ΨF2(c) ≠ ∅. So, let x ∈ ΨF1(c)°1ΨF2(c). Then, x=y1°1y2 for some y1 ∈ ΨF1(c) and y2 ∈ ΨF2(c). This implies that ∅≠y1Ψ(c)⊆F1(c) and ∅≠y2Ψ(c)⊆F2(c). As (y1°1y2)Ψ(c)=y1Ψ(c)°2Ψ(c)⊆F1(c)°2F2(c). This shows that x=y1°1y2 ∈ ΨF1F2(c). Hence, (1) is proved.For arbitrary c ∈ C, if at least one of ΨF1(c) and ΨF2(c) is empty, then (2) is obvious. Now, for arbitrary c ∈ C, consider that ΨF1(c) ≠ ∅ and ΨF2(c) ≠ ∅. Then, ΨF1(c)∨ΨF2(c) ≠ ∅. So, let x ∈ ΨF1(c)∨ΨF2(c). Then, x=y1∨y2 for some y1 ∈ ΨF1(c) and y2 ∈ ΨF2(c). This implies that ∅≠y1Ψ(c)⊆F1(c) and ∅≠y2Ψ(c)⊆F2(c). As (y1∨y2)Ψ(c)=y1Ψ(c)∨y2Ψ(c)⊆F1(c)∨F2(c). This shows that x=y1∨y2 ∈ ΨFF21∨(c). Hence, (2) is proved.For arbitrary C) be a SCTR with respect to the foreset over Θ1 × Θ2. Then, for any two soft sets  and  over Θ1, we haveLet  to be the SBR over Θ1 × Θ2 and xΨ(c) ≠ ∅ for all x ∈ Θ1, c ∈ C, and Ψ(c)y ≠ ∅ for all y ∈ Θ2, c ∈ C unless otherwise specified.Throughout this section, we consider  be a SBR over Θ1 × Θ2 and  be a soft set over Θ2. If Uappr. 1, then  is called generalized upper soft (GUpS) subquantale of Θ1 w.r.t the aftersets. If Uappr1, then  is called GUpS ideal  of Θ1 w.r.t the aftersets.Let  be a SBR over Θ1 × Θ2 and  be a soft set over Θ1. If Uappr2, then  is called generalized upper soft (GUpS) subquantale of Θ2 w.r.t the foresets. If Uappr2, then  is called GUpS ideal  of Θ2 w.r.t the foresets.Let  be a SCPR over Θ1 × Θ2. If  is a soft subquantale of Θ2, then  is a GUpS subquantale of Θ1 w.r.t the aftersets.Let  is a soft subquantale, then c ∈ C. Let i ∈ I. Then, piΨ(c)∩F(c) ≠ ∅. So, there exists qi ∈ piΨ(c)∩F(c). Thus, qi ∈ piΨ(c) and qi ∈ F(c) since  is a SCPR. Therefore,  ∈ Ψ(c), i ∈ I implies  ∈ Ψ(c). This implies that ∨i∈Iqi ∈ ∨i∈IpiΨ(c). Also, ∨i∈Iqi ∈ F(c)  is a soft subquantale). So, ∨i∈Iqi ∈ ∨i∈Ipi ∈ Ψ(c)∩F(c). Hence, Suppose that (p1Ψ(c)∩F(c) ≠ ∅ and q1 ∈ p1Ψ(c)∩F(c) and q2 ∈ p2Ψ(c)∩F(c). Thus, q1 ∈ p1Ψ(c), q1 ∈ F(c), q2 ∈ p2Ψ(c), and q2 ∈ F(c) since  is a SCPR. Therefore, ,  ∈ Ψ(c) implies (p1°1q1), (p2°2q2) ∈ Ψ(c). This implies that q1°2q2 ∈ p1°p2Ψ(c). Also, q1°2q2 ∈ (c)  is a soft subquantale). So, q1°2q2 ∈ p1°p2Ψ(c)∩F(c). Hence, Let With the same arguments, the next C) be a SCPR over Θ1 × Θ2. If  is a soft subquantale of Θ1, then  is a GUpS subquantale of Θ2 w.r.t the foresets.Let  be a soft ∨-complete relation over Θ1 × Θ2 w.r.t the aftersets. If  is a soft left (right) ideal of Θ2, then  is a GUpS left (right) ideal of Θ1 w.r.t the aftersets.Let  is a soft left ideal of Θ2, then c ∈ C. Let u1Ψ(c)∩F(c) ≠ ∅ and u2Ψ(c)∩F(c) ≠ ∅. So, there exists v1 ∈ u1Ψ(c)∩F(c) and v2 ∈ u2Ψ(c)∩F(c). Thus, v1 ∈ u1Ψ(c), v1 ∈ F(c), v2 ∈ u2Ψ(c), and v2 ∈ F(c) since  is a SCPR. Therefore,  ∈ Ψ(c); that is, v1∨v2 ∈ (u1∨u2)Ψ(c). Also, v1∨v2 ∈ F(c)  is a soft left ideal). So, v1∨v2 ∈ (u1∨u2)Ψ(c)∩F(c). Hence, Suppose that ∩F(c). Thus, v2 ∈ u2Ψ(c) and v2 ∈ F(c). Since  is a soft ∨-complete relation, therefore, v2 ∈ u2Ψ(c)=u1∨u2Ψ(c)=u1Ψ(c)∨u2Ψ(c). This implies that v2=s∨t, for some s ∈ u1Ψ(c) and t ∈ u2Ψ(c). Thus, s ≤ v2 and v2 ∈ F(c) imply s ∈ F(c) (as F(c) is ideal). So, s ∈ u1Ψ(c)∩F(c). Hence, Now, let p,  x ∈ Θ1 and xΨ(c)∩F(c) ≠ ∅. So, there exist q ∈ xΨ(c)∩F(c). Thus, q ∈ xΨ(c) and q ∈ F(c). Since  is a soft left ideal so, y∘2q ∈ F(c) for any y ∈ pΨ(c)⊆Θ2. This implies that  ∈ Ψ(c). So,  ∈ Ψ(c); that is, y∘2q ∈ p∘1xΨ(c). So, y∘2q ∈ p∘1xΨ(c)∩F(c). Hence, Let C) be a SCTR over Θ1 × Θ2 w.r.t the aftersets. If  is a soft prime ideal of Θ2, then  is a GUpS prime ideal of Θ1 w.r.t the aftersets.Let  is a soft prime ideal of Θ2, then c ∈ C. Then, by F, C) is generalized upper soft ideal of Θ1. Let p1,  p2 ∈ Θ1 such that p1∘1p2)Ψ(c)∩F(c) ≠ ∅. So, there exist q ∈ (p1∘1p2)Ψ(c)∩F(c). This implies that q ∈ (p1∘1p2)Ψ(c) and q ∈ F(c). Since  is a SCTR, q ∈ (p1∘1p2)Ψ(c)=p1Ψ(c)∘2p2Ψ(c). Thus, q=c∘2d for some c ∈ p1Ψ(c) and d ∈ p2Ψ(c). Thus, c∘2d ∈ F(c) and  is a soft prime ideal of Θ2 so, c ∈ F(c) or d ∈ F(c). Thus, c ∈ p1Ψ(c)∩F(c) or d ∈ p2Ψ(c)∩F(c). Hence, Assume that (C) be a SCTR over Θ1 × Θ2 w.r.t the aftersets. If  is a soft semiprime ideal of Θ2, then  is a GUpS semiprime ideal of Θ1 w.r.t the aftersets.Let  is a soft semiprime ideal of Θ2, then c ∈ C. Then, by F, C) is generalized upper soft ideal of Θ1. Let p1 ∈ Θ1 such that p1∘1p1)Ψ(c)∩F(c) ≠ ∅. So, there exist q ∈ (p1∘1p1)Ψ(c)∩F(c). This implies that q ∈ (p1∘1p1)Ψ(c) and q ∈ F(c). Since  is a SCTR, q ∈ (p1∘1p1)Ψ(c)=p1Ψ(c)∘2 p1Ψ(c). Thus, q=c ∘2 c for some c ∈ p1Ψ(c). Thus, c∘2c ∈ F(c) and  is a soft semiprime ideal of Θ2 so, c ∈ F(c). Thus, c ∈ p1Ψ(c)∩F(c). Hence, Assume that (C) be a SCTR over Θ1 × Θ2 w.r.t the aftersets. If  is a soft primary ideal of Θ2, then  is a GUpS primary ideal of Θ1 w.r.t the aftersets.Let  is a soft primary ideal of Θ2, then C. Then, by F, C) is generalized upper soft ideal of Θ1. Let p1,  p2 ∈ Θ1 such that p1∘1p2∈p1∘1p2)Ψ(c)∩F(c) ≠ ∅. So, there exist q ∈ (p1∘1p2)Ψ(c)∩F(c). This implies that q ∈ (p1∘1p2)Ψ(c) and q ∈ F(c). Since  is a SCTR, q ∈ (p1∘1p2)Ψ(c)=p1Ψ(c)∘2p2Ψ(c). Thus, q=c∘2d for some c ∈ p1Ψ(c) and d ∈ p2Ψ(c). Thus, c∘2d ∈ F(c) and  is a soft primary ideal of Θ2 so dn ∈ F(c) for some n ∈ ℕ. Also, dn ∈ p2nΨ(c) for n ∈ ℕ. Thus, dn ∈ p2nΨ(c)∩F(c). Hence, Assume that C={c1, c2} and define SBR  over Θ1 × Θ2 by the ruleLet C) is SCPR. The aftersets with respect to Ψ(c1) and Ψ(c2) are given as follows:Then,  over Θ2 by the ruleDefine soft set  is not a soft subquantale of Θ2. But 1. So  is a GUpSSΘ of Θ1 w.r.t the aftersets.Then, (c1) and Ψ(c2) are given as follows:Foresets with respect to Ψ over Θ1 by the ruleDefine soft set  is not a soft subquantale of Θ1. But 2. So,  is a GUpS subquantale of Θ2 w.r.t the foresets.Then, ((2)C={c1, c2} and define SBR  over Θ1 × Θ2 by the ruleNow, let We define ∘c1) and Ψ(c2) are given as follows:Aftersets with respect to Ψ(C) is ∨-complete relation over Θ1 × Θ2 w.r.t the aftersets. Define soft set  over Θ2 by the ruleThen,  is not a soft ideal of Θ2. But 1. So,  is a GUpS ideal of Θ1 w.r.t the aftersets.Then, (C) over Θ1 × Θ2 by the ruleNow, define SBR  and Ψ(c2) are given as follows:Foresets with respect to Ψ(C) is soft ∨-complete relation over Θ1 × Θ2 w.r.t the foresets. Define soft set  over Θ1 by the ruleThen,  is not a soft ideal of Θ1. But 2. So,  is a GUpS ideal of Θ2 w.r.t the foresets.Then, (Similar examples can be presented to justify that converse of Theorems 11C) be a SBR over Θ1 × Θ2. Consider the soft set  over Θ2, if Lappr1, then  is called generalized lower soft (GLWS) subquantale of Θ1 w.r.t the aftersets. If Lappr1, then  is called GUpS ideal  of Θ1 w.r.t the aftersets.Let  be a SBR over Θ1 × Θ2. Consider the soft set  over Θ1, if Lappr2, then  is called GLWS subquantale of Θ2 w.r.t the foresets. If Lappr2, then  is called GUpS ideal  of Θ2 w.r.t the foresets.Let  be a SCTR over Θ1 × Θ2 w.r.t the aftersets. If  is a soft subquantale of Θ2, then  is a GLWS subquantale of Θ1 w.r.t the aftersets.Let  is a soft subquantale of Θ2 and C. Let uiΨ(c)⊆M(c). Since  is a SCTR, therefore, ∨i∈I(uiΨ(c))=(∨i∈Iui)Ψ(c)⊆M(c). Hence, Suppose that (u1Ψ(c)⊆M(c) and u2Ψ(c)⊆M(c). Since  is a SCTR and  is a soft subquantale, therefore, u1Ψ(c)∘2u2Ψ(c)⊆M(c)∘2M(c) implies (u1∘1u2)Ψ(c)⊆M(c). Hence, Now, let With the same arguments, next C) be a SCTR over Θ1 × Θ2 w.r.t the foresets. If  is a soft subquantale of Θ1, then  is a GLWS subquantale of Θ2 w.r.t the foresets.Let  be a SCTR over Θ1 × Θ2 w.r.t the aftersets. If  is a soft ideal of Θ2, then  is a GLWS ideal of Θ1 w.r.t the aftersets.Let  is a soft ideal of Θ2 and C. Let u1Ψ(c)⊆M(c) and u2Ψ(c)⊆M(c). Since  is a SCTR and  is a soft ideal of Θ2 so u1Ψ(c)∨u2Ψ(c)=(u1∨u2)Ψ(c)⊆M(c)∨M(c); that is, (u1∨u2)Ψ(c)⊆M(c) Hence, Suppose that  and v2 ∈ u2Ψ(c)⊆M(c). So, v1∨v2 ∈ (u1∨u2)Ψ(c), that is, v1∨v2 ∈ u2Ψ(c)⊆M(c). Since M(c) is ideal so v1 ≤ v1∨v2 ∈ M(c) implies v1 ∈ M(c). Thus, u1Ψ(c)⊆M(c). Hence, Now, let u,  y ∈ Θ1 and yΨ(c)⊆M(c). Consider v1∈(u∘1y)Ψ(c) since  is a SCTR so v1 ∈ u Ψ(c)∘2yΨ(c). Thus, v1=c∘2d for some c ∈ uΨ(c) and d ∈ yΨ(c). But yΨ(c)⊆M(c) so d ∈ M(c) and  is a soft ideal of Θ2; therefore, c∘2d ∈ M(c), that is, v1 ∈ M(c). Thus, (u∘1y)Ψ(c)⊆M(c). Hence, Now, let C) be a SCTR over Θ1 × Θ2 w.r.t the aftersets. If  is a soft prime ideal of Θ2, then  is a GLWS prime ideal of Θ1 w.r.t the aftersets.Let  is a soft prime ideal of Θ2 and C. Then, by Theorem 4.19,  is GLWS ideal of Θ1. Let u1,  u2 ∈ Θ1 such that u1∘1u2)Ψ(c)⊆M(c). Consider v ∈ (u1∘1u2)Ψ(c)⊆M(c). Since  is a SCTR, v ∈ (u1∘1u2)Ψ(c)=u1Ψ(c)∘2u2Ψ(c). Thus, v=c∘2d for some c ∈ u1Ψ(c) and d ∈ u2Ψ(c). This implies that v=c∘2d ∈ M(c). As  is a soft prime ideal so, c ∈ M(c) or d ∈ M(c). Thus, c ∈ u1Ψ(c)⊆M(c) or d ∈ u2Ψ(c)⊆M(c). Hence, Assume that (C) be a SCTR over Θ1 × Θ2 w.r.t the aftersets. If  is a soft semiprime ideal of Θ2, then  is a GLWS semiprime ideal of Θ1 w.r.t the aftersets.Let  is a soft semiprime ideal of Θ2 and C. Then, by M, C) is GLWS ideal of Θ1. Let u ∈ Θ1 such that u∘1u)Ψ(c)⊆M(c). Let v ∈ uΨ(c). As  is a SCTR so v∘2v ∈ (u∘1u)Ψ(c)⊆M(c). Since  is a soft semiprime ideal, v∘2v ∈ M(c) implies v ∈ M(c). Thus, uΨ(c)⊆M(c). Hence, Assume that (C) be a SCTR over Θ1 × Θ2 w.r.t the aftersets. If  is a soft primary ideal of Θ2, then  is a GLWS primary ideal of Θ1 w.r.t the aftersets.Let  is a soft primary ideal of Θ2 and C. Then, by Theorem 4.19.,  is a GLWS ideal of Θ1. Let u1,  u2 ∈ Θ1 such that u1∘1u2)Ψ(c)⊆M(c). Let v ∈ (u1∘1u2)Ψ(c). Since  is a SCTR, v ∈ u1Ψ(c)∘2u2Ψ(c). Thus, v=c∘2d for some c ∈ u1Ψ(c) and d ∈ u2Ψ(c). Thus, dn ∈ u2nΨ(c) for some n ∈ ℕ. Also, c∘2d ∈ M(c). As  is a soft primary ideal, c ∉ M(c) and dn ∈ M(c). Thus, u2nΨ(c)⊆M(c). Hence, n ∈ ℕ.Suppose that , and then, we established the relationship between homomorphic images and their approximation by SBR.η : Θ1⟶Θ2 is called weak quantale homomorphism (WQH) if η(p∘1q)=η(p)∘2η(q) and η(p∨q)=η(p)∨η(q), where  and  are quantales. If η is one-one, then η is monomorphism. If η is onto, then η is called epimorphism, and if η is bijective, then η is called isomorphism between  and .A function H, C1) be a soft quantale over Θ1 and  be a soft quantale over Θ2. Then,  is said to soft weak homomorphic to  if there exist ordered pair of functions  satisfies the followingη : Θ1⟶Θ2 is onto WQH, that is, η(p∘1q)=η(p)∘2η(q) and η(p∨q)=η(p)∨η(q)ζ : C1⟶C2 is surjectiveη(H(c1))=F(ζ(c1)), ∀c1 ∈ C1Let  of functions is SWQH. If in ordered pair  both η and ζ are one-to-one functions, then  is said to soft weak isomorphic to  and  is called SWQI.The ordered pair  be soft weak homomorphic to  with SWQH . Let  be a SBR over Θ2 and ⊆. Define Ψ1(c3)={}∈Θ1 × Θ1:(η(x), η(y)) ∈ Ψ2(c3){} be a SBR over Θ1. Then, the following holds:1, C3) is SCPR if  is SCPR is SWQI and  is SCPR w.r.t the aftersets (w.r.t the foresets), then  is SCPR w.r.t the aftersets (w.r.t the foresets)If  is SWQI, then η, ζ) be a SWQI. Then, Let (Let (and (2) are obviousH1, C1′)⊆ and for any c3 ∈ C3, z ∈ Θ2. Then, there exist a ∈ Θ1 such that η(a)=z. Thus, x ∈ aΨ1(c3)∩H1(c1′). So,  ∈ Ψ1(c3) and x ∈ H1(c1′). Thus, (η(a), η(x)) ∈ Ψ2(c3), that is, η(x) ∈ η(a)Ψ2(c3). Also, η(x) ∈ η(H1(c1′)). So, η(a)Ψ2(c3)∩η(H1(c1′))≠∅. This implies that Suppose (wΨ2(c3)∩η(H1(c1′)) ≠ ∅. This implies that y ∈ wΨ2(c3)∩η(H1(c1′)). Thus, y ∈ wΨ2(c3) and y ∈ η(H1(c1′)). This implies that there exists x ∈ H1(c1′)⊆Θ1 and x1 ∈ Θ1 such that η(x)=y and η(x1)=w. So, =(η(x1), η(x))∈Ψ2(c3). This implies that  ∈ Ψ1(c3). So, x ∈ x1Ψ1(c3)∩H1(c1′). Thus, Now, let H1, C1′)⊆ and for any c3 ∈ C3, z ∈ Θ2. Then, there exist a ∈ Θ1 such that η(a)=z. Thus, aΨ1(c3)⊆H1(c1′). Let x ∈ zΨ2(c3). Then, there exist y ∈ Θ1 such that η(y)=x. So, η(y) ∈ η(a)Ψ2(c3), that is, (η(a),  η(y)) ∈ Ψ2(c3). So,  ∈ Ψ1(c3), that is, y ∈ aΨ1(c3)⊆H1(c1′). Thus, η(y)∈η(H1(c1′)). So, η(a)Ψ2(c3)⊆η(H1(c1′)). Thus, Suppose (a ∈ Θ1 such that η(a)=z and η(a)Ψ2(c3)⊆η(H1(c1′)). Let x ∈ aΨ1(c3), that is,  ∈ Ψ1(c3). Then, (η(a), η(x)) ∈ Ψ2(c3). Then, η(x) ∈ η(a)Ψ2(c3)⊆η(H1(c1′)). So, η(x) ∈ η(H1(c1′)). This implies that x ∈ H1(c1′). So, aΨ1(c3)⊆H1(c1′). Then, Now, let c3 ∈ C3. Then, η is bijection so Let With a similar technique, H, C1) be soft weak isomorphic to  with SWQI . Let  be a SCPR over Θ2 and ⊆. Define Ψ1(c3)={}∈Θ1 × Θ1:(η(x), η(y))∈Ψ2(c3){} for any c3 ∈ C3. Then, the following holds:1 iff 2 for all c3 ∈ C31 iff 2 for all c3 ∈ C31 iff 2 for all c3 ∈ C31 iff 2 for all c3 ∈ C31 iff 2 for all c3 ∈ C3Let (1 for any c3 ∈ C3. We will show that 2. By Let η(u)=p and η(v)=q. Since η, ζ) is SWQI so p∨q=η(u)∨η(v)=η(u∨v)∈Let p, q ∈ Θ2 such that p ≤ q and u ∈ Θ1 and η(u)=p and η(v)=q. So, η(u) ≤ η(v) implies η(u∨v)=η(u)u ≤ v and Now, let p ∈ Θ2 and u ∈ Θ1 and η(u)=p and η(v)=q. Since η(u∘1v)2.Finally, let 2 for any c3 ∈ C3. We will show that 1.Conversely, suppose that η(u), η(v)∈Let u, v ∈ Θ1 such that u ≤ v and η(u∨v)=η(u)∨η(v)=η(v)∈η(u) ≤ η(v). Since u ∈ Θ1 and η(u) ∈ Θ2 and u∘1v∈Now, let The proof of (2)–(5) is similar to the proof of (1).With the same arguments, the next H, C1) be soft weak isomorphic to  with SWQI . Let  be a SCTR over Θ2 and ⊆. Define Ψ1(c3)={}∈Θ1 × Θ1:η(x), ηy∈Ψ2(c3){} for any c3 ∈ C3. Then, the following holds:1 iff 2 for all c3 ∈ C31 iff 2 for all c3 ∈ C31 iff 2 for all c3 ∈ C31 iff 2 for all c3 ∈ C31 iff 2 for all c3 ∈ C3Let (Yang and Xu  introduc1 × Θ2 is employed. This new relation over Θ1 × Θ2 enables us to use the concept of aftersets and foresets to express the lower and upper approximation. Important results regarding to the approximation of soft substructures of quantales under SBR with some essential algebraic conditions such as compatible and complete relations are introduced. To emphasize and make a clear understanding, soft compatible and soft complete relations are focused, and these are interpreted by aftersets and foresets. Particularly, in our work, soft compatible and soft complete relations play an important role. Crux of these results is that whenever we approximate a soft algebraic structure of quantale, corresponding upper and lower approximations, are again the same kind of soft algebraic structure. Furthermore, we presented the soft quantale homomorphism and established the relationship of soft homomorphic images with their approximation under SBR.The new combined effect of an algebraic structure quantale with rough and soft sets is presented by using soft binary relation, in this paper. The soft substructures of quantales like soft subquantale and soft ideal are discussed. The approximation w.r.t aftersets and foresets of these substructures by SBR which is an extended notion of Pawlak's rough approximation space are presented. The more generalized version of approximation space implied from SBR over ΘIn future, one can use this work and generalize it to different soft algebraic structures such as soft quantale modules, soft hypergroups, soft hyperquantales, and soft hyperrings. One can take motivation from our generalized approximation space and define new approximation spaces.


We study steady axisymmetric water waves with general vorticity and swirl, subject to the influence of surface tension. This can be formulated as an elliptic free boundary problem in terms of Stokes' stream function. A change of variables allows us to overcome the generic coordinate‐induced singularities and to cast the problem in the form “identity plus compact,” which is amenable to Rabinowitz's global bifurcation theorem, whereas no restrictions regarding the absence of stagnation points in the flow have to be made. Within the scope of this new formulation, local curves and global families of solutions, bifurcating from laminar flows with a flat surface, are constructed. In the irrotational and swirl‐free setting, such waves were studied numerically by Vanden‐Broeck et al.In this paper, we study axisymmetric water waves with surface tension, modeled by assuming that the domain is bounded by a free surface on which capillary forces are acting, and that in cylindrical coordinates z‐direction and ignore viscosity and gravity. In the irrotational swirl‐free case, the problem can be formulated in terms of a harmonic velocity potential. In contrast, we formulate the problem in terms of Stokes' stream function, which satisfies a second‐order semilinear elliptic equation known alternatively in the literature as the Hicks equation, the Bragg–Hawthorne equation, or the Squire–Long equation  and p=p(x⃗) denote the velocity and the pressure, respectively, and Ω is the fluid domain. In cylindrical coordinates , that is, x=rcosϑ, y=rsinϑ, and z=z, the velocity field u⃗ is expressed asuϑ≠0. From the incompressibility and the axisymmetry of the flow, it follows that we can introduce Stokes' stream function Ψ, such thatruϑ is constant along streamlines, which we express as ruϑ=F(Ψ) where F is an arbitrary function. The steady Euler equations are then equivalent to the Bragg–Hawthorne equationWe consider periodic axisymmetric capillary waves traveling at constant speed along the Ω={∈R2:0<r<d+η(z)} and its boundaries by ∂ΩS={∈R2:r=d+η(z)} (free surface) and ∂ΩC={∈R2:r=0} (center line). Although the latter could be considered as part of the domain, it is sometimes convenient to consider it as a boundary due to the appearance of inverse powers of r in the equations. On the free surface r=d+η(z), we have the kinematic boundary condition u⃗·n⃗=0, where n⃗=e⃗r−η′(z)e⃗z denotes a normal vector. Expressed in terms of Ψ, this takes the form Ψz+ηzΨr=0 on ∂ΩS. In addition, we have the dynamic boundary condition p=−σκ on ∂ΩS, where∂ΩS and σ>0 is the coefficient of surface tension. Using Bernoulli's law, we can eliminate the pressure and express this as∂ΩS, where Q is the Bernoulli constant. At the center line ∂ΩC, the identity Ψz=rur shows that Ψz=0. Summarizing, we have following boundary value problem:F and γ are arbitrary functions of Ψ.We next consider the boundary conditions. Assume that the fluid domain is given by 33.1∂ΩS and ∂ΩC. We normalize Ψ such that it vanishes on ∂ΩC and assign the name m to its value on ∂ΩS. Thus, we deal with the equationsQ and m are constants. The fluid velocity is given byr=0, provided that ψ is continuous at r=0, since then . Written in terms of ψ, Equation /r is of class C1 and rψ is of class C2, both viewed as functions on {∈ the inverse map. Then, with ψ¯=ψ, that is,As for Bernoulli's equation , we do nIψ¯=ψ∼, is trans3.5ϕ=0 on s=1 instead of functions ψ¯ with variable boundary condition at s=1. Thus, we introduce, for any λ∈R, the functionm=m(λ):=d2ψλ(1), read of ϕ=0 on s=ms of ϕ,  and m=m, readaLηIϕ=−γ1, andϕs+d2=M with M compact. Meanwhile, we also clarify what Q exactly is, namely, we define it as an expression in . An advantage of rewriting the problem in this form is that it facilitates the use of Rabinowitz' global bifurcation theorem, which was originally formulated for such problems. Although it is probably possible to apply some later version adapted to problems that are not of the form “identity plus compact” (Ω0), Q=Q, and m=m(λ); andthe tuple1solves  with Ω=H(iii)ψ∈Cper,e2,α(Ω¯) and satisfies  and Iϕ∈C2,α(Ω0I¯), provided F=0. Indeed, we have Iψ∈C2,α(ΩI¯) in this case since Iψλ∈C2,1(Ω0I¯); in particular, ψ satisfies  are solutions of —they makticular,  has the M being smooth, the property that M is of class C2 follows from the property that A is of class C2; this, in turn, is guaranteed by assumption ∈R×Uε be arbitrary. In the following, the quantities C can change from line to line, but are always shorthand for a certain expression in its arguments that remains bounded for bounded arguments. Moreover, let R>0 and suppose ∥∥R×X≤R. Since ψλ is of class C1 with respect to λ and Lη is elliptic uniformly in η due to η+d≥ε, we see thatM2 is compact on R×Uε because of the compact embedding of Cper2,α(Ω0I¯) in Hper1(Ω0I) and in Cper0,α(Ω0I¯) combined withM1, we immediately find, in view of the obtained estimates for A,M1 is compact on R×Uε because C3,α is compactly embedded in C2,α.▪The other operations in the definition of sumption . Now let44.1F and, in particular, its evaluation at a trivial solution. For simplicity, we always write Aη for Aηδη, that is, the partial derivative of A with respect to η evaluated at  and applied to a direction δη. The same applies similarly to expressions such as Aϕ, Lηη, and so forth.We now want to calculate the partial derivative Lη, which only depends on η and not on ϕ, leads toLφ=f gives Lδφ+δLφ=δf, we see that IAη is the unique solution ofIAϕ is the unique solution ofA=0 here, we haveLinearizing the operator M1. After a lengthy computation, we get the following results for the partial derivatives of M1 evaluated at a trivial solution , noticing that SAη=SAϕ=0 at such points:P is the projection onto the space of functions with zero average.Next, we turn to −c(λ) is the z‐component of the velocity at the surface of the trivial laminar flow corresponding to λ in view of δΦ. Hence, the “good unknown” for the linearized problem in the new variables is actually δw−(v′∘Φ)δΦ. In particular, (δw−(v′∘Φ)δΦ)∘Φ−1 satisfies the linearized problem in the original variables. In our setting, the latter linearization is formal due to the free boundary, but it has a simpler structure than the linearization in the flattened variables.Lemma 3c(λ)≠0. ThenLetBefore we proceed with the investigation of local bifurcation, we first introduce an isomorphism, which facilitates the computations later and is sometimes called T(λ) and [T(λ)]−1 are well defined, and a simple computation shows that they are inverse to each other.▪Both T‐isomorphism, we introducec(λ)≠0. Now recall thatV=V[η] the unique solution off:=−2ψλ+sψsλc(λ)Sθ+dc(λ)AηSθ+Aϕ(2ψλ+sψsλc(λ)Sθ)+V[Sθ] satisfiesIf=0 at |y|=1. Thus, recalling , L2(λ)θ is the unique solution ofL1(λ)θ is (in the set of L‐periodic functions with zero average) uniquely determined byLet us now consider a trivial solution . In view of the ecalling , 55), , w, wT‐isom4.3F. Clearly, in view of the T‐isomorphism, it suffices to study the kernel of L; here and in the following, we will suppress the dependency of L on λ. From , provided Lθ=0. Thus, combining =∑k=0∞θk(s)cos(kνz) as a Fourier series. Then we easily see thatθ0(1)=0 is already included in the definition of Y, andμ∈R, let us now introduce the function β∼=β∼μ,λ, which is defined to be the unique solution of the singular Cauchy problemβ∼∈C2,α by the same argument as in Section We now turn to the investigation of the kernel of  λ. From , we infeombining  and 74)F, and therefore, ∂sθ0(0)=0. Hence, θ0 is a multiple of β∼0,λ. However, since necessarily θ0(1)=0, θ0 has to vanish identically in view of k≥1 and notice as above that ∂sθk(0)=0 provided (θk is a multiple of β∼−(kν)2,λ if and only if (β∼−(kν)2,λ(1)=0 and that (θk(1)=0. Since therefore also ∂sθk(1)=0 by virtue of 2,λ(1)≠0 and define β−(kν)2,λ:=β∼−(kν)2,λ/β∼−(kν)2,λ(1). Hence, Equation (θk if and only if the dispersion relationθk is a multiple of β−(kν)2,λ. We summarize our results concerning the kernel.Lemma 4λ∈R with c(λ)≠0 and under the assumption =∑k=0∞θk(s)cos(kνz), is in the kernel of L(λ) if and only if θ0=0 and for each k≥1(a)θk=0, or(b)β∼−(kν)2,λ(1)≠0, θk is a multiple of β∼−(kν)2,λ, and the dispersion relationD given in  is at first only defined if β∼μ,λ(1)≠0. If this property fails to hold, we set D:=∞ in the following.Clearly, 4.4L can be written as an orthogonal complement with respect to a suitable inner product. This will be helpful later. To this end, we introduce the inner productf1,f2∈H0,per1(R), g1,g2:Ω∼0→R with Ig1,Ig2∈Hper1(Ω∼0I), where Ω∼0:=xK is even self‐adjoint, we first note that obviously, H admits the fundamental decomposition H=×{0})+˙({0}×C) into a positive and negative subspace. Associated with this decomposition is the fundamental symmetry⟨u1,u2⟩J==⟨d2φ1,φ2⟩Ls32+g(λ)−1b1b2¯. The operator JK is self‐adjoint with respect to ⟨·,·⟩J, since now the assumptions of Ref. J‐adjoint by an upper index ⟨∗⟩, we haveJ⟨∗⟩=J and JK⟨∗⟩J=K∗ >0 if u≠0. Thus, there exists exactly one negative eigenvector of K0; cf. again Ref. K0 has exactly one negative eigenvalue μ0 and its other eigenvalues are positive. With the same proof as in Ref. C>0, the estimateF are arbitrary, we define the perturbation A via D(A)={u∈H:φ∈D(T),b=φ(1)} andA is densely defined and bounded, and we have K=K0+A. Now consider a real μ<μ0−C∥A∥J. Because ofK−μI is invertible in view of the Neumann series. This completes the proof.▪The first assertion is clear because K, as we see in what follows.Proposition 3q−λ denotes the negative part of qλ. Then the operator K has only real eigenvalues, K has exactly one eigenvalue μ<−∥q−λ∥∞, and all its other eigenvalues satisfy μ>−∥q−λ∥∞. Moreover, all eigenvalues are algebraically simple.Assume thatUnder a certain condition, we can infer even more properties of the spectrum of K and u=) an associated eigenvector. Due to Lemma φ(1)≠0 =0 and thus φ≡0), it follows thatu cannot be neutral and μ has to be real. Since, additionally, by Ref. H is a π1‐space—there exists exactly one nonpositive eigenvector of K, the first assertion follows immediately. The second statement is a direct consequence of the fact that all eigenvalues are real and no eigenvectors are neutral.▪Let μ be an eigenvalue of Remark 2c± are not real). In particular, if γ and FF′ are bounded, this condition is satisfied if “c(λ) is sufficiently large” or, provided that additionally, F is bounded, if simply “|c(λ)| is sufficiently large.”If Equation  holds, t5.2F.We now turn to a more detailed investigation of the conditions for local bifurcation for specific examples of γ and 5.2.1γ=F=0. By (c(λ)≠0 if and only if λ≠0. Moreover, β∼=β∼−(kν)2,λ solvesI1 and K1 are modified Bessel functions of the first and second kinds. Since K1(x)→∞ as x→0, we necessarily have c2=0. Determining the remaining constant c1 yieldsdI1/dx=I0−I1/x (cf. Ref. (kν)2d2−1>0 if D(−(kν)2,λ)=0, the dispersion relation D(−(kν)2,λ)=0 can hence be written asν>0, k∈N with kν>1/d and then λ such that ≠0, Equation  is one‐dimensional if this relation is satisfied for some k∈N and is trivial if it fails to hold for all k. Indeed, Equation ; see Ref. As a first example, we consider the case without vorticity and swirl, that is, =F=0. By  and 24)γ=F=0. Bycf. Ref. ,(136)tten asσc(λ)2=ktten asσc(λ)2=ktten asσc(λ)2=ktten asσc(λ)2=kten as1σc(λ)2=kDλ(−(kν)2,λ)≠0 always holds in view of cλ(λ)≠0.Furthermore, it is therefore clear that the transversality condition qλ=0 and h(λ)=d−2+σ−1d−1c(λ)2>0.Moreover, it is easy to see that  is alway5.2.2γ≠0 is a constant and F=0. By Equations (c(λ)≠0 if and only if λ≠γd24. Noticing that β−(kν)2,λ is the same as in the previous example without vorticity, we, moreover, haveD(−(kν)2,λ)=0 for 1/c(λ), we see that necessarily1/(νd)∈N. We now want to reformulate the second case. Clearly, Equation  in view of 36+116<224. Therefore and because of I0(0)=1, I1(0)=0, and I1′(0)=1/2, the function χ:→ is strictly monotonically increasing and onto. Hence,  such that χ(x1)=4σd−5γ−2 and x0:=x1/d. Thus, we have the equivalenceD(−(kν)2,λ)=0 for c(λ) yieldsD(−(kν)2,λ) cannot vanish. Next, we search for solutions of (k and c(λ) (and not k and λ) since R∋λ↦c(λ)∈R is bijective. Both for the first case (for which 1/(νd)∈N is necessary) and for the second case, we can easily first choose an appropriate k and then c(λ) via =b±(x) withb−(1) and possibly b±(0) are to be interpreted as the limit of the above expression as x tends to 1 or 0; the limit for x→1 exists because x=1 is a simple root of both nominator and denominator, and the limit for x→0 also as I1(0)=0 and I1′(0)=1/2. Having clarified this, we see that b± is smooth on  and continuous on .Let us now turn to the case  x=0. By , 161), , ξ<1/2 azero. By , we therξ=16/81. Differentiating (bx−(0)=bxx−(0)=bxxx−(0)=0 yieldbxxxx−(0)<0; hence, b− is strictly monotonically decreasing.Let us now consider ntiating  twice mo1/(νd)∉N; in Figure ξ≥16/81:If D(−(kν)2,λ)=0 can have at most one root k∈N.The dispersion relation 16/81≤ξ<1/2 andIf ξ<16/81:If IfIfIfTo summarize, for fixed λ, we have therefore proved the following, provided 1/(νd)∈N andk=1/(νd).If γ<0, these statements remain true after reversing all inequalities in the conditions for c(λ) and changing maxb− to minb−.If, however, D(−(kν)2,λ)=0 has exactly one solution k∈N. Since cλ(λ)≠0, it holds that Dλ(−(kν)2,λ)≠0 if and only if ξ≤1/2 or kνd>x1 otherwise.Next, let us turn to the transversality condition, fix λ, and assume that qλ=0 and h(λ)=d−2+2σ−1d−1λ(4λ−γd2) by (h(λ)>0 for all λ∈R if γ2<8σd−5⇔ξ>12, and in the case ξ≤12, h(λ)>0 if and only if λ∉ whereFinally, we have a look at . Here, q−γd2) by . Therefo5.2.3γ=0 and F(x)=ax for some a∈R =0). Having a look at, for example, ≠0 has to hold to allow for c(λ)≠0. Moreover, the dispersion relation can be written asζ:=μ/a2, b:=ad>0. Due to this relation and under the assumption J0(b)≠0, which yields cλ(λ)≠0, it is clear that the transversality condition is always satisfied, provided that the corresponding kernel is one‐dimensional. Since A depends in a rather complicated way on ζ, we do not further study the dispersion relation and the possibility of multidimensional kernels rigorously; it is, however, interesting to notice that, due to  for which A>0 and J0(b)≠0, that is, exactly the points that allow for solutions of the dispersion relation (c(λ)≠0.The easiest case to include nonzero swirl is to take example, , we findtten asdσ−1c(λ)tten asdσ−1c(λ)6Theorem 3X be a Banach space, U⊂R×X open, and F∈C. Assume that F admits the form F=x+f with f compact, and that Fx∈C). Moreover, suppose that F=0 and that Fx has an odd crossing number at λ=λ0. Let S denote the closure of the set of nontrivial solutions of F=0 in R×X and C denote the connected component of S to which  belongs. Then one of the following alternatives occurs:(i)C is unbounded;(ii)C contains a point  with λ1≠λ0;(iii)C contains a point on the boundary of U.Let The theory for local bifurcation having setup, we now turn to global bifurcation, which is, of course, the main motivation of our formulation “identity plus compact.” To this end, we first state the global bifurcation theorem by Rabinowitz.U=X can be found in Ref. U.The proof of this theorem in the case Theorem 4λ0≠0 such that the dispersion relationD given by =0 in R×X and C denote the connected component of S to which  belongs. Then one of the following alternatives occurs:(i)C is unbounded in the sense that there exists a sequence ∈C such that(a)|λn|→∞, or(b)∥ηn∥C2,α→∞, or(c)∥r2/pγ(Ψn)+r2/p−2F(Ψn)F′(Ψn)∥Lp(Zηn)→∞ with p:=52−α, where Zηn denotes a L‐periodic instance of the axially symmetric fluid domain in R3 corresponding to ηn and Ψn=r2((ϕn+d2(d+ηn)2ψλn)∘H[ηn]−1) is the corresponding original Stokes stream function,n→∞; as (ii)C contains a point  with λ1≠λ0;(iii)C contains a sequence  such that ηn converges to some η in C0,per,e2,β(R) for any β∈ and such that there exists z∈ withAssume  and thatlation1D(−(kν)2Now we can prove the following result.F is of class C2 and admits the form “identity plus compact” on each R×Uε, ε>0. Moreover, it is well known that F has an odd crossing number at , provided that F is a Fredholm operator with index zero‐ and one‐dimensional kernel, and the transversality condition holds. These properties, in turn, are consequences of the hypotheses of the theorem in view of Lemmas F coincides with L(λ0) up to an isomorphism. For each ε>0, we can thus apply Theorem U chosen to be the interior of R×Uε. Thus, on each R×Uε, C coincides with its counterpart obtained from Theorem ε>0 is arbitrary and R×U=⋃ε>0(R×Uε), it is evident that necessarilyC is bounded in R×X and (ii) fails to hold.As was already observed in Lemma C is bounded in R×X if (i)(a)–(c) and  and, since  can be left out because of unique solvability of the Dirichlet problem associated with Lη), and changes of variables via H[η] and via cylindrical coordinates in R5 and R3. In the above, Ω∼η denotes a periodic instance of Ωη=H[η](Ω0) and Ψ, Zη are analogously defined as in the statement of (c). Moreover, the constant C>0 can change in each step.Let us investigate alternative (i) further. To show that it can be as stated above, we show that, in view of alternative (i) of Theorem –(c) and  fail to d, since  does notC0,per,e2,α(R) in C0,per,e2,β(R).▪Finally, we turn to alternative (iii). If Equation  holds, bRemark 3ωϑ=ω⃗·e⃗ϑ=−rγ(Ψ)−(FF′)(Ψ)/r, satisfies ∥r2/p−1ωnϑ∥Lp(Zηn)→∞ as n→∞.Alternative (i)(c) says that the angular component of the vorticity, in general, given by Remark 4In a two‐dimensional situation without surface tension (and with gravity), sometimes, an alternative such as (ii) above can be eliminated. The strategy to this end typically relies on maximum principle arguments, which, however, appear to be unavailable when capillary effects are taken into account. Therefore, it is unclear whether and, if so, how (ii) can be eliminated in the present situation.Proposition 4(i)(b')(α)∥ηn∥C1,α→∞, or(β)∥|u⃗n|2∥C0,α(Sn)→∞ , or(γ)|Q|→∞ (the Bernoulli constant is unbounded).In Theorem We also have the following.▪This follows easily from the Bernoulli equation


Hs(ℝ). If D > 0, the local well-posedness is given for s > 1/2 and for s > 3/2 if D=0.In this paper, we show the local well-posedness for the Cauchy problem for the equation of the Nagumo type in this equation (1) in the Sobolev spaces  D > 0 is a constant diffusion coefficient, α ∈  and ϵ > 0 is a small positive quantity. In . Thus, using the dominated convergence theorem, we have as follows:However, side of  is a intc on the interval  such thath⟶0+(1/h)∫tt+h‖V(t+h − τ)F(u(τ)) − F(u(t))‖s−2dτ=0.Now, from the mean value theorem for integrals, there exists a value ∂t+η(t)=Qη(t) − F(u(t)) in Hs−2(ℝ). where ∂t+ is the right derivative. In similar way, we can conclude that the left derivative is ∂t−η(t)=Qη(t) − F(u(t)) in Hs−2(ℝ). So, η ∈ C1) and ∂tη(t)=Qη(t) − F(u(t)). As V(t)ψ(x) is the solution of the linear problem (u(t)=V(t)ψ+η(t) ∈ C1) and satisfies  is locally integrable on  for all n. It follows that u ∈ 𝒳 for all n and satisfies ‖un(t)‖s ≤ ‖ψn‖s+M ≤ γ+M where γ=supn‖ψn‖s. Therefore, supt∈‖un(t)‖s ≤ γ+M for all n and supt∈‖u(t)‖s ≤ γ+M. Now, similar to the proof of the previous proposition, we have as follows:As E1/2,1((bΓ(1/2))2T) is finite .Let D=0 using a priori estimate and the parabolic regularization method, the so-called vanishing viscosity method .η(t), a(t) and b(t) be real valued positive continuous functions defined on ⊆ : Ω(A(τ))+τBτΩ(limr⟶∞G(r))}.Let This is a particular case of the theorem given in  [pp. 78]s > 3/2 be fixed. Then, F(u) satisfies the estimateu, w ∈ Hs(ℝ), where L0=ϵx∑k=0n−1xkyn−1−k+(ϵn/2)yn+x2+xy+y2+(α+1)(x+y).Let q=∑k=0n−1ukwn−1−k. As s > 3/2 thus Hs(ℝ) and Hs−1(ℝ) are Banach algebras. Moreover, we have that Hs(ℝ)⟶Hs−1(ℝ) and Hs(ℝ)⟶L∞(ℝ). Thus, using the Cauchy–Schwartz inequality, we have as follows:We define r ≥ 1 and s > 3/2 be fixed and h, v are real valued functions. Then, there exists a constant C=C such that(T. Kato). Let v, h∂xv)s| ≤ C‖∂xh‖s−1‖v‖s2.In particular, |) be the corresponding solution of  > 0, depending on ‖ψ‖s, such that uD can be extended to the interval , and there is a function ρ ∈ C]; .Let  problem  with iniution of  for someHs(ℝ) and Using the inner product in uD(t)‖s2 ≤ ρ(t) for all t ∈ , and therefore we have thatt ∈ .From n=1, from (ρ(t) ≤ (1/(‖ψ‖s2+2Cs(α+ϵ+2)t)−1 − 8Cst), for 0 ≤ t ≤ T∗ where Ts(ψ) such that 0 < Ts(ψ) < T∗, and therefore, we conclude thatt ∈ .For the case =1, from  we have uD(t)‖s2 ≤ ρ(t) for all t ∈ .Let u0‖s2 ≤ ρ(t) for all t ∈ , where ρ(t) is as in Moreover, ‖Ts=Ts(ψ) be as in Let uD(t))D>0 is a net which converges to a function u0 ∈ C) in the L2− norm, uniformly over .First we will show that . Then,Let ρ is the function defined in the proof of Let In order to bound the first term, we have as follows:We can bound the second term by the following:uD1 − uD2|uD12 − uD22|)0| ≤ 2M‖uD1 − uD2‖02.The third term is bounded by | − uD2(t)‖02 ≤ C|D1 − D2| for all t ∈ , and since L2(ℝ) is complete there exists the limit u0(t)=limD⟶0uD(t) in L2(ℝ) uniformly with respect to t ∈ , i.e.u0 ∈ C).Applying Gronwall's inequality to the last relation, we show that there is a constant u0 ∈ Hs(ℝ). Let t ∈ . Since uD⟶u0 in L2(ℝ), as D⟶0+, then there exists a subsequence {Dnj)(} such thatNow we show that We obtain by Fatou's Lemma as follows:uD⇀u0 in Hs(ℝ) for all t ∈  as D⟶0+.We must show that uD(t))D>0 is a weak Cauchy net in Hs(ℝ), uniformly with respect to t ∈ . In fact, given φ ∈ Hs(ℝ) and ϵ > 0, choosing φϵ ∈ Hs(ℝ) such that ‖φ − φϵ‖s ≤ ϵ, thenD1,D2⟶0+supt∈(uD1(t) − uD2(t)|φ|)s=0.First of all, we will show that . Moreover, since the convergence is uniform for all φ ∈ Hs(ℝ), we can conclude that u0 ∈ Cw).Thus, we have that u0 ∈ Cw1).Finally, we show that φ ∈ Hs−2(ℝ). Then,t ∈ . Since uD⟶u0 in L2(ℝ) and uD⇀u0 in Hs(ℝ), we have ∂xuD⇀∂xu0 in Hs−1(ℝ) and ∂x2uD⇀∂x2u0 in Hs−2(ℝ) uniformly on . Observe that if r > 1/2, fn⇀f in Hr(ℝ) and gn⇀g in Hr(ℝ) then fngn⇀fg in Hr(ℝ). After, we haveTs]. Thereby, taking the limit as D⟶0+ in (Let ℝ). Then,uDtφs−2=ψu0 be as in the preceding theorem, then u0 ∈ AC).Let t ∈  ↦ u(u − α)(u − 1)+ϵunux is weakly continuous in Hs−2(ℝ) and the Sobolev space is separable, then applying the Bochner–Pettis theorem, it is a strongly measurable function in Hs−2(ℝ). Therefore,u0 ∈ AC).Since So, from  we concls > 3/2 and T > 0 be fixed, ψj ∈ Hs(ℝ), j=1,2, and vj ∈ C)∩Cw)∩AC) two weak sense solutions to (D=0 such that vj(0)=ψj, j=1,2. Then,L0 is as in the Let w(t)=v1(t) − v2(t). Since s > 3/2, we have s − 2 > −1/2 > 1 − s and 1 − s > −s, and also Hs−2(ℝ)⟶Hs−(ℝ). Using the fact that w(t) is real valued we havet ∈  is fixed, h is such that t+h ∈ , and 〈·|·|〉s is the Hs duality bracket. As t ∈  ↦ w(t) ∈ Hs(ℝ) is bounded andHs−2(ℝ)⟶Hs−(ℝ), from (Let d we havewt+hwt+h0unded and∂twt=limhR is given by (From given by . Applyin∂twt02≤L0ψ ∈ Hs(ℝ) with s > 3/2. Then, there exists a Ts=Ts(ψ) > 0 and a unique u0 ∈ C) such thatLet u0 ∈ C).From the previous results, there exists a unique solution of  in the cφ ∈ Hs(ℝ) be such that ‖φ‖s=1. Then, we have |(u0|φ|)s| ≤ ‖u0(t)‖s ≤ ρ(t)1/2, for all φ ∈ Hs(ℝ) and for all t ∈ . Additionally,φ ∈ Hs(ℝ). As we have ‖ψ‖s=supφ‖s‖=1|(ψ|φ|)s|, then taking supremum over ‖φ‖s=1 in (t⟶0+‖u0(t)‖s=limsupt⟶0+‖u0(t)‖s=‖ψ‖s, i.e., the limit of ‖u0(t)‖s exists as t⟶0+ and limt⟶0+‖u0(t)‖s=‖ψ‖s. Since u(t)⟶ψ weakly in Hs(ℝ) as t⟶0+, it follows that limt⟶0+u0(t)=ψ in the norm of Hs(ℝ). Let t′ ∈ ; Hs+1(ℝ))∩C1]; Hs−1(ℝ)) be the corresponding solution of the problem ] which is independent of D. Then, uD can be extended, if necessary, to the interval , with ψ viewed as an element of Hs(ℝ).Let r=s+1 to obtainApplying  with r=sNow, from inequality (3.12) and Theorem 3.2 in , we obtat, we have as follows:Therefore, using , 67) in in67) int ∈  and therefore we can extend (if necessary) u=u(t) to  as a solution in Hs+1(ℝ). Thus, we conclude that Ts(ψ) ≤ Ts+1(ψ). So, uD ∈ C]; Hs+1(ℝ)) for D > 0. From , then we have u0 ∈ C]; Hs+1(ℝ)).Observe that the last inequality is independent of Following Lemma 5 in  we have s > 3/2. For ψ ∈ Hs(ℝ) and τ > 0, we defineLet τ⟶0+‖ψτ − ψ‖s=0, and there exists a constant C=C(s) such thatThen, limτ⟶0+‖ψτ − ψ‖s=0 uniformly on compact subsets of Hs(ℝ).Moreover, limτ⟶0+‖ψτ − ψ‖s=0. Now, to prove the uniformity on compact subsets, it is enough to show that ψn⟶ψ in Hs(ℝ) implies limτ⟶0+‖ψnτ − ψn‖s=0 uniformly for n=1,2,…, since sequential compactness is equivalent to compactness in metric spaces. Thus, observe thatNotice that ϵ > 0 be given and choose N such that if n ≥ N, then ‖ψn − ψ‖s < (1/3)ϵ. Thus, for τ0 > 0 small enough that 0 < τ < τ0 we haven ≤ N. Now, if n ≥ N then we haveLet n.Hence  holds foOn the other hand, we haveeτ(1+ξ2)s/2− − eτ(1+ξ2)s/2−| ≤ |τ − θ|(1+ξ2)s/2, and then we haveFinally, using the mean value theorem, |The proof is complete.s > 3/2, ψ ∈ Hs(ℝ), ψτ (for τ > 0) be as in the preceding lemma. If u0τ is solution of the problem (u0τ(0)=ψτ, for all τ > 0, then there are constants C=C > 0 and η=η(s) ∈  such thatτ sufficiently small and 0 ≤ θ ≤ 2τ.Let  problem  with u0ττ0 > 0 be such that u0τ(t) is well-define in  for all 0 < τ ≤ τ0. Then,Let Now, the right-hand side of the inequality  will be First, we will estimate . ApplyinNow, we will estimate . Observes > 3/2, there is s0 such that 3/2 < s0+1 < s. From the Cauchy–Schwartz inequality, we obtainFinally, we estimate . As s > q is defined in the proof of Now, we will estimate each term on the right-hand side of the last inequality. First, observe thatu0τ)n − (u0θ)n‖s0‖u0τ‖s+1. From τ ≤ τ0.We also estimate ‖(s0/s). To estimate the term ‖u0τ − u0θ‖0, observe thatFrom θ ≤ τ. Therefore, we haveFrom the term  is boundOf the bounds that were found for , 80b), , 80b),  we concle obtain .The following corollary follows immediately from F be a compact subset in Hs(ℝ). Suppose that ψ ∈ F, ψτ and u0τ are defined as in the preceding result. Then, u0τ converges uniformly to u0, for all t ∈ , as τ⟶0+.Let ψ ↦ u0 is continuous in the following sense: let ψj ∈ Hs(ℝ), j=1,2,3,… such that ψj⟶ψ in Hs(ℝ) and uj0, ∈ C; Hs(ℝ))∩C1) are the corresponding solutions of the problem =ψj. Let T ∈ . Then, there exists a positive integer N0=N0 such that Ts,j ≥ T for all j ≥ N0 andThe map  problem  with iniψ ∈ Hs(ℝ) and let {ψj}j∈ℕ be a sequence in Hs(ℝ) such that converges to ψ. Suppose that u0, uj0,, u0τ, uj0,τ are the corresponding solutions of .Let  problem  is local


Page 3, Table 1, column 4: “−£441” should read “−£681,” and “£680” should read “£227.”Volume 10, no. 3, e00425-22, 2022,


In particular, we show that such mappings are homeomorphisms when n=3 or when the branch set is empty. This gives a positive answer to the corresponding cases of a question of Vuorinen.We study global injectivity of proper branched coverings from the open Euclidean  Notice that throughout the paper by domain, we mean a connected open subset of Rn. The structure of this set is tied to the topology and geometry of the mapping itself, but in general the structure of the branch set is not well understood. Even for the important special class of continuous, open, and discrete maps called quasiregular mappings, many properties of the branch set remain largely unknown, but the topic garners great interest. In his ICM address →f(Ω) and any x∈Ω∩f−1(β(0)) there exists a path α:→Ω for which f∘α=β and α(0)=x. Such a path is called a lift of β(under f).Let Finally, a fundamental technique in the study of branched covers is the path‐lifting. For the terminology and basic theory of this technique, we refer to  there exists a maximal lift γ:I→Ω of β such that γ(0)=x, where I is a subinterval of  of type  or  since by the continuity of f we have f(γ(b))=β(b). This is again a contradiction with the maximality of γ, and so the original claim holds true; I= and we may choose α=γ.□Note first that since 3f:Bn→f(Bn)⊂Rn is in fact a covering map defined on a contractible n‐manifold. This observation can be used to show that the image f(Bn) is actually an Eilenberg–MacLane spaceK, that is, a path‐connected space whose fundamental group is isomorphic to a group G and which has contractible universal covering space, see, for instance, →X from x1 to xi. With the help of these paths define a function[f∘γi] denotes the homotopy class of the loop f∘γi in Y. Since X is simply connected, any two paths joining x1 and xi are homotopic and thus their images under f are also homotopic. This shows that the function Ψ is well defined. The function Ψ is also surjective. To see this, notice that for any loop γ in Y that starts and ends at y lifts to a path γ∼i that joins x1 to some xi∈f−1(y). Thus it follows thatΨ. Therefore□Take a point Proof of Theorem 1.2f:Bn→f(Bn)⊂Rn is a proper branched covering such that Bf=∅. Denotef is a covering map, and since Bn is simply connected, it is the universal cover of Y. This means, by definition, that Y is an Eilenberg–MacLane space K for G=π1(Y), see  of the path γ=γ0 that keeps the endpoints fixed at x1 and x2. On the other hand γ1 is a constant path as a lift of a constant path. Therefore we have x1=x2 and thus f is injective. Especially, f:Bn→Y is then a global homeomorphism as a continuous and open bijection.□In order to see that 4E^ of a set E⊂Ω is called a boundary component ofE if its closure in Ω is not compact. Note that a set E can have several boundary components.Lemma 4.1K⊂f(Ω) be a non‐empty compact set. DenoteV⊂Ω is a domain such that C⊂V⊂V¯⊂Ω. Then for the set U≔f−1(f(V)) we have the following:(a)U is a path‐connected open set such that f(U)=f(V),(b)U is a normal domain for f,(c)f|U:U→f(U) is a proper branched cover,(d)f|Ω∖U¯:Ω∖U¯→f(Ω)∖f(U¯) is a proper branched cover and Ω∖U¯=f−1(f(Ω∖U¯)), and(e)E⊂Ω is a boundary component of Ω∖U¯, then f(E) is a boundary component of f(Ω)∖f(U¯). Moreover, if a point y is contained in a boundary component of f(Ω)∖f(U¯), then all its preimages are contained in boundary components of Ω∖U¯.if a set LetWe start by proving a lemma which provides a useful collection of large normal domains in the setting of the Vuorinen question. In what follows a connected component V is a domain and f an open map, f(V) is open and so is its preimage U under the continuous map f.(a)U we first note that f(V) is a domain containing the compact set K. Therefore, for any point x∈U we may connect f(x) and K with a path α:→f(V). By Lemma α has a lift α∼:→Ω with α∼(0)=x and by the definition of U we have |α∼|⊂U. On the other hand α∼(1)∈C⊂V, and so each point x∈U can be connected with a path to an interior point of the connected set V⊂U. This implies that U is path‐connected.For the second claim of (a) we simply note that(b)∂f(U)=f(∂U). Openness of f gives the inclusiony∈f(∂U). If U∩f−1(y)≠∅, then U is a neighborhood of one of the preimages of y. Then by the openness of f we see that f(U)=f(V) is a neighborhood of y. This implies that U=f−1(f(V)) is a neighborhood of all the points in the preimage of y. Therefore, we havey∈f(∂U). Thus, we have proved that U∩f−1(y)=∅ and so y∈∂f(U) since y∈f(U)¯. This gives usBy (a) it is enough to show that (c)U is a domain by (a) and the restriction of a branched covering to a domain is a branched covering. To show that f|U:U→f(U) is proper, we fix a compact set A⊂f(U) and note that f−1(A)⊂Ω is compact since f is proper. Now as U=f−1(f(U)), we have that f−1(A)⊂U, and so we see that (f|U)−1(A) is compact. Thus f|U is proper.The set (d)∂f(U)=f(∂U), we see that alsof(Ω). As in part (c), the properness will follow after we show thatx∈f−1(f(Ω∖U¯)). Suppose, toward a contradiction, that x∉Ω∖U¯. Then either x∈∂U or x∈U. In the first case we have by applying (f(x)∈∂f(Ω∖U¯), which is not possible because by the choice of x we have f(x)∈f(Ω∖U¯). In the second case we get by the definition of U that f−1(f(x))⊂U. Thus, we have x∉Ω∖U¯ which again goes against our assumptions. Therefore, we conclude that x∈Ω∖U¯ and soFirst we note that the restriction of a branched covering to an open set is a branched covering. Since by part (b) we have at alsof(∂(Ω∖U¯(e)E be first some (not necessarily a boundary) component of Ω∖U¯. First we show that E is mapped onto some component of f(Ω)∖f(U¯). It is clear that f(E) is contained in some component C1 of f(Ω)∖f(U¯). If C1≠f(E), we find a point y∈C1∖f(E) and a sequence {yj}j∈N in f(E) converging to y. Notice that by the definition of U we have f−1(y)⊂Ω∖U¯. Choose a sequence {xj}j∈N in E such that f(xj)=yj. Since f−1 is compact, we see that {xj} has a convergent subsequence , which converges to a point x. By continuity we see that x∈f−1(y) and thus x∉E but since y∉f(U¯) we have x∈Ω∖U¯. Thus x is in a different component than all xj. This is a contradiction and therefore every component of Ω∖U¯ is mapped onto some component of f(Ω)∖f(U¯) by the mapping f.Let E is a boundary component of Ω∖U¯ and its image is not a boundary component of f(Ω)∖f(U¯), then since f is proper, f−1(f(E)) would have a compact closure which is not possible as E is a boundary component. This proves the first claim. Continuity now implies that other components are not mapped to boundary components, which implies the second claim.□If Note that since π1(X) of a space X.Definition 4.2Ω⊂Rn has torsion‐free fundamental group at infinity if for any compact set K⊂Ω there exists a domain V⊃K with V¯ being compact in Ω and such that π1(Ω∖V¯) is torsion‐free; recall that a group is torsion‐free if no element g≠e has the property that gj=e for some j∈N, where e is the neutral element of the group.We say that a domain The proof of Theorem simply connected at infinity, see, for example, →C with α(0)=x0 and α(1)=x1. The image of this path, β≔f∘α:→f(ER), is a loop based at y0≔f(x0). If β was zero‐homotopic in f(ER), we could lift the homotopy with the covering map f|ER:ER→f(ER) into a homotopy in ER contracting the path α to a point without changing the endpoints of the path, see [1.30]. This is not possible when k⩾2, and so we must have [β]≠0 in π1(f(ER),y0). Likewise, since f is a proper map, its restriction to B∖f−1(f(Bf)) is also a covering map and so β is not zero‐homotopic also in f(B)∖K⊂f(B)∖f(Bf).Let γ:→ER, see again Figure γ1=α. Then, when γk:→ER has been defined and if γk(0)≠γk(1), we define γk+1 by lifting the path β from the point γk(1) with Lemma γk. Since the covering map is a local homeomorphism, this procedure is well defined and since f−1(f(x0)) is finite, it terminates after at most k steps.Next we construct a loop n⩾3 the loop γ can be contracted to a point in the spherical shell B∖B¯R and thus in Es. This contracting homotopy can then be pushed with the covering map f|Es into f(Es), and so we see that[β] is a non‐trivial torsion element in π1. Since, as noted before, [β]≠0 also in π1(f(B)∖K) and clearly [β]m=0 in π1(f(B)∖K), we see that [β] is also a non‐trivial torsion element in f(B)∖K. This is a contradiction and so the original claim holds.□But now we note thatProposition 4.4Ω⊂R3 be a domain. Then π1(Ω) is torsion‐free.Let Our proof in dimension three relies on the following result of Papakyriakopoulos, see [R3 has torsion‐free fundamental group, in particular, it has torsion‐free fundamental group at infinity. This yields the proof of Theorem By Proposition Proof of Theorem 1.1f:B3→f(B3)⊂R3 be a proper branched covering and denote Y≔f(B3). By Proposition K⊂Y, the fundamental group of Y∖K is torsion‐free. Thus Y has torsion‐free fundamental group at infinity, and the claim follows from Proposition □Let Remark 4.5f restricted to the open ball B3 is a proper branched covering onto a space which is an open 3‐manifold outside one singular point. Furthermore Bf={0}, so in particular the branch is non‐empty but compact. Similar examples appear from universal covers of homology spheres. Thus we must assume that the image of f is a manifold. We do remark that we do not know if the Vuorinen question holds for mappings f:B3→N where N is a 3‐manifold not necessarily embeddable into R3.As mentioned in the introduction e, e.g.,  for the Remark 4.6essentially properf:B3→R3 with Bf=∅ that are not homeomorphisms. In the same paper it is claimed that one can obtain a proper branched covering by restricting the above mapping f to a ball Br with radius r<1 arbitrarily close to 1. However, no detailed argument is provided and the claim is not true for essentially proper branched coverings in general. It turns out that every open continuous map g:Bn→R3 is essentially proper. This can be seen as follows: Let K⊂g(Bn) be an arbitrary compact set. Since g is open, the sets {g(Br)}r∈ form an open covering of K. Thus, by compactness we find a ball Br0 such that B¯r0∩g−1(K) is compact and K=g(B¯r0∩g−1(K)). Especially, the mapping mentioned in (In  it was pioned in  is essenBulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not‐for‐profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.The


Experimental & Molecular Medicine 10.1038/s12276-023-00925-1, published online 13 January 2023Correction to: After online publication of this article, the authors noticed few errors in the Fig. 1b and Fig. 1c that were inadvertently introduced owing to a technical error. The unit for melatonin’s concentration is (pg/mg brain) rather than (ng/mg brain). The concentrations of all groups is divided by 100 to report the exact values as 16.52 ± 0.60  rather than 1652 ± 60 ; 20.31 ± 0.87  rather than 2031 ± 87 ; 13.41 ± 0.88  rather than 1341 ± 88 ; 33.44 ± 0.32  rather than 3344 ± 32 ; 18.79 ± 1.12  rather than 1879 ± 112 ; 24.70 ± 0.99  rather than 2470 ± 99 . Figure 1C, the value of all the groups is multiplied by 0.05, a missed factor in the calculation, to report the result as 0.36 ± 0.01 rather than 7.2 ± 0.3 ; 0.38 ± 0.02 rather than 7.6 ± 0.4 ; 0.69 ± 0.08 rather than 13.8 ± 1.7 ; 0.18 ± 0.01 rather than 3.6 ± 0.2 ; 0.39 ± 0.02 rather than 7.9 ± 0.4; 0.26 ± 0.02 rather than 5.4 ± 0.5; . The Fig. 1b and 1c are replaced with correct versions.p < 0.01) reduced by constant darkness (29 ± 8) but not by melatonin (112 ± 27) in comparison to that in the vehicle group .” Is replaced by “Immature OPCs were significantly (**p < 0.01) reduced by constant darkness (29 ± 8) but not by melatonin (112 ± 27) in comparison to that in the vehicle group (264 ± 48).”Authors also corrected the following typo in section of results: “Immature OPCs were significantly (**The authors apologize for any inconvenience caused.The original article has been corrected.


The bivariate or multivariate distribution can be used to account for the dependence structure between different failure modes. This paper considers two dependent competing failure modes from Gompertz distribution, and the dependence structure of these two failure modes is handled by the Marshall–Olkin bivariate distribution. We obtain the maximum likelihood estimates (MLEs) based on classical likelihood theory and the associated bootstrap confidence intervals (CIs). The posterior density function based on the conjugate prior and noninformative (Jeffreys and Reference) priors are studied; we obtain the Bayesian estimates in explicit forms and construct the associated highest posterior density (HPD) CIs. The performance of the proposed methods is assessed by numerical illustration. It is extremely common that the failure of a product or a system contains several competing failure modes in reliability engineering; any failure mode will lead to the failure result. Competing risks' data contain the failure time and the corresponding failure mode, which can be modeled by the competing risks' model and has been commonly performed in many research fields, such as engineering and medical statistics. Previous studies have mostly assumed the competing failure modes to be independent; Wang et al. , Ren andIn addition to using copula function to handle the relationship between different competing failure modes, the bivariate or multivariate distribution also can be used to account for the correlation between different failure modes. The Marshall–Olkin distribution , which hThe Gompertz distribution is a widely used growth model which has been studied extensively; Ismail  studied In the rest of this paper, we will present the model description and some properties. f is a Gompertz distribution; the density function and reliability function of it areλ is shape parameter and θ is scale parameter.Suppose that U0,  U1,  and  U2 are three independent Gompertz variables with different scale parameters, that is, U0 ~ GP, U1 ~ GP, and U2 ~ GP. Let T1=min and T2=min; we obtain T1 ~ GP and T2 ~ GP. Then, the pair of variables  follows the MOGP distribution denoted by  ~ MOGP. When θ0=0, the two variables T1 and T2 are independent and T1 and T2 will be dependent when θ0 > 0; hence, θ0 can be regarded as a correlation coefficient between T1 and T2.Suppose T1,  T2) can be written asThe joint PDF of  are presented in fT1, T2 is a unimodal function.The surface plots of n identical products into test, and each product has two dependent failure modes with lifetimes T1,  T2,  ~ MOGP. Then, the system lifetime is X=min ~ MOGP. Let δl0=I(Tl1=Tl2), δl1=I(Tl1 < Tl2), and δl2=I(Tl1 > Tl2), for l=1, ⋯, n, where I(·) is an indicator function. Then, we can compute n0=∑lδl0, n1=∑lδl1,  n2=∑lδl2, and n=n0+n1+n2.Put l=1, ⋯, n, δl0=I(Tl1=Tl2), δl1=I(Tl1 < Tl2), and δl2=I(Tl1 > Tl2), We haveFor l=1, ⋯, n, we have δl0+δl1+δl2=1,For δl0, δl1, δl2) ~ Multinomial, θ1/(θ0+θ1+θ2), θ2/(θ0+θ1+θ2)).Therefore, . Set the first partial derivation of log  L about θ0,  θ1,  θ2,  λ to 0, i.e.,The MLEs of θ0,  θ1,  and θ2 asFrom , we get L, we obtainλ.Substituting ∂2h(λ)/∂λ2 < 0, which implies that h(λ) is concave. Some iterative schemes can be used to find the MLE for λ, such as Newton–Raphson algorithm.We can show that θ0,  θ1,  θ2,  and λ. The Bootstrap method is a resampling method to estimate some statistical characteristics for the unknown parameters by taking samples from the original samples repeatedly; the obtained samples are called Bootstrap samples. This method has a great practical value since it does not need to assume the overall distribution or construct the pivot quantity. We generate the Bootstrap sample by the following three steps:n and observed data , we get the estimates Step 1: for the fixed value of n, x1∗, x2∗, ⋯, xn∗). Then, get the MLEs Step 2: for the values of M times to obtain M sets of the values Step 3: repeat Step 2 Since it is hard to construct the exact CIs for the unknown parameters, we consider the Bootstrap method to construct CIs for parameters θ0,  θ1,  θ2,  λ at 1 − γ confidence level asBased on the Bootstrap sample and by percentile Bootstrap (Boot-P) method, we construct the Boot-P CIs for λ is known. Denote θ=θ0+θ1+θ2, which has a Gamma prior with hyperparameters a and b asIn this section, we suppose the shape parameter θ0/θ+θ1/θ+θ2/θ=1, so given θ,  follows a Dirichlet prior with hyper parameters c0,  c1, and c2, that is,Due to θ0, θ1,  and θ2 becomesc=c0+c1+c2.Therefore, the joint prior of θ0, θ1,  θ2) asAccording to Jeffreys , Jeffreyni=n · θi/(θ0+θ1+θ2),  i=0,  1,  2, so I can be written asFrom Thus, the Jeffreys prior is given byπ2, the joint posterior distribution of  is proper.Based on the Jeffreys prior θ0, θ1,  θ2) based on π2 asFrom  and 7),,7), we oπ2 with respect to θ0, θ1,  and θ2, we obtainA=∑i=1n(eλxi − 1) and B is a beta function.Integrating θ0, θ1,  θ2) based on π2 is proper.Thus, the joint posterior distribution of  to  is one-to-one with the inverse transformation θ0=μ0μ1, θ1=μ0μ2, and θ2=μ0(1 − μ1 − μ2). The Jacobian matrix of the transformation has the formBernardo  and BergThe likelihood function  becomes can be written asThe Fisher information matrix of } and {, μ0}, the reference priors are the same, which is given by θ0, θ1,  θ2) is Under the ordering groups {μ0,  μ1,  μ2}, {μ0,  μ2, μ1}, {μ1,  μ0,  μ2}, and {μ1,  μ2,  μ0}, the reference priors are the same, which is given by θ0, θ1,  θ2) is Under the ordering groups {μ2,  μ0,  μ1} and {μ2,  μ1, μ0}, the reference priors are the same, which is given by θ0, θ1,  θ2) is Under the ordering groups {(i)μ0,  μ1,  μ2) isThe Fisher information matrix of } and {, μ0} is the same as in [The reference prior for the ordering groups {me as in , which i(ii)I1 isThe inverse of (iii)h1=1/μ02, h2=1/μ1(1 − μ1), and h3=(1 − μ1)/(μ2(1 − μ1 − μ2)).According the notations in , we obtak={|ak0 < μ0 < bk0,  ak1 < μ1,  ak2 < μ2,  μ1+μ2 < dk}, such that ak0,  ak1,  ak2⟶0, bk0⟶∞, and dk⟶1, as k⟶∞. Then, we haveChoose the compact sets Ωwhere Then, we get the reference prior asμ0∗,  μ1∗,  μ2∗) is an inner point of Ωk.where .Similarly, under the ordering group {μ1,  μ0,  μ2} isThe Fisher information matrix of {I2 isThe inverse of h1=1/μ1(1 − μ1), h2=1/μ02, and h3=(1 − μ1)/(μ2(1 − μ1 − μ2)).Similarly, we obtain k={|ak0 < μ1,  ak1 < μ0 < bk1,  ak2 < μ2,  μ1+μ2 < dk}, such that ak0,  ak1,  ak2⟶0, bk1⟶∞, and dk⟶1, as k⟶∞. Then, we haveChoose the compact sets Ωwhere μ1∗,  μ0∗,  μ2∗) be an inner point of Ωk; we get the reference prior asLet .Similarly, under the ordering group {(v)μ2,  μ1,  μ0} isThe Fisher information matrix of {I3 isThe inverse of h1=1/μ2(1 − μ2), h2=(1 − μ2)/(μ1(1 − μ1 − μ2)), and h3=1/μ02.Then, we obtain k={|ak0 < μ2,  ak1 < μ1,  μ2+μ1 < dk, ak2 < μ0 < bk2}, such that ak0,  ak1,  ak2⟶0, bk2⟶∞, and dk⟶1, as k⟶∞. Then, we haveChoose the compact sets Ωμ2∗,  μ1∗,  μ0∗) be an inner point of Ωk, we obtain the reference prior asLet . According to the one-to-one transformation from  to , we can obtain the reference priors π2, π3, π4 from ωR1, ωR2, and ωR3, respectively.Similarly, under the ordering group {π3 and π4, the posterior distributions of  are proper.Based on the reference priors θ0, θ1,  θ2) based on reference prior π3 and π4 are, respectively, asThe joint posterior distributions of  and π4 with respect to θ0, θ1,  and θ2, respectively, we obtainIntegrating θ0, θ1,  θ2) based on π3 and π4 are proper.Thus, the posterior distributions of  based on π1,  π2,  π3,  and π4 are, respectively, asw1=Γ(∑i=02ci)ba−c0−c1−c2/Γ(a) and w2=∏i=02bci/Γ(ci).The joint posterior distributions of , we, we12), θ0,  θ1,  θ2,  and θ can be constructed by the Monte Carlo method studied by Chen and Shao [The HPD credible intervals of parameters and Shao .n and the observed data , compute the Bayesian estimates of π1,  π2,  π3,  and π4, respectively.Step 1: given the value of M times; we obtain M sets of the values π1,  π2,  π3,  and π4, respectively. Arrange them in the ascending order, we obtainStep 2: repeat Step 1 γ confidence level asStep 3: compute the CIs at 1 −   Step 4θv, v=0,1,2, and θ are the shortest intervals among w=1,  2, ⋯, M − (1 − γ)M, respectively.: the HPD CIs for λ is known. The initial values for parameters  are . The initial values for the hyperparameters a,  b,  c0,  c1,  and c2 are all 0.001. Take the sample size n = 10, 20, 30, and 50. Generate the random samples  from MOGP by the following steps:n, generate n samples u01, u02, ⋯, un0 from GP, u11, u12, ⋯, un1 from GP, and u21, u22, ⋯, un2 from GP. Then, we obtain Tl1=min and Tl2=min,  l=1,2, ⋯, n.  Step 1: for a fixed value xl, δl0, δl1, δl2),  l=1,2, ⋯, n, where xl=min, δl0=I(Tl1=Tl2), δl1=I(Tl1 < Tl2), and δl2=I(Tl1 > Tl2).  Step 2: compute  of the MLEs, the average lengths (ALs), and coverage probabilities (CPs) of the 95% Boot-P CIs, and the MSEs of the Bayesian estimates, the ALs, and CPs of the 95% HPD CIs, which are shown in Table 2Table 2n, the Bayesian estimates based on π1, π2,  and π4 are smaller than the MSEs of MLEs. The MSEs of Bayesian estimates of θ0 and θ2 based on π4 are smaller than that based on π1, π2,  and π3. The MSEs of Bayesian estimates of θ1 based on π3 are smaller than that based on π1, π2,  and π4. The MSEs of Bayesian estimates of θ based on π1 are smaller than that based on π2, π3,  and π4.The MSEs of MLEs and Bayesian estimates decrease as the sample size increases. For given sample size The CPs of Boot-P and HPD CIs are all close to 0.95. The ALs of Boot-P and HPD CIs decrease; the associated CPs increase when the sample size increases. The CPs of HPD CIs based on Bayesian estimates are larger than the CPs of Boot-P CIs based on MLEs.λ, θ0, θ1, θ2) as , we use the procedures mentioned above to generate U0,  U1,  and U2 from GP, GP, and GP, respectively. We then get T1=min and T2=min; the latent lifetime of the system is min. The simulated data are listed in θ0,  θ1,  θ2,  and θ are shown in θ0,  θ1,  θ2, θ) are close to the true value.For illustrative purposes, with initial value for parameters .θ0,  θ1,  θ2,  and θ are shown in The MLEs, Bayesian estimates, and associated 95% CIs for parameters This paper discussed the point estimates and CIs for the parameters of the dependent competing risks' model from MOGP distribution. We studied the appropriateness of the posteriors based on conjugate prior and Jeffreys and Reference priors, obtained the Bayesian estimates in closed forms, and constructed the associated HPD CIs. From the simulations results, the use of the Bayesian method can be recommended if the priors are available. The results of the illustrative analysis show that the proposed methods work well; from the lengths of CIs, we can conclude the Bayesian estimates are better than MLEs in general.


In the digital era, the health information presented on virtual platforms plays a pivotal role in supporting people’s active and healthy life. The ageing, especially ageing women, are more likely to seek and accept health information through online media platforms. The study shows that short health videos on social media platforms are extremely popular among ageing women in China for the accessing of virtual coaching. Adopting the qualitative methodology of in-depth interview and discourse analysis, the study investigates virtual coaching with short health video practice among 39 Chinese ageing women in different fields, who are all over sixty years old. Specifically, with the analytical tools of transitivity and generic structure analysis, the study explores the impacts of short health videos on Chinese ageing women’s cognition, behavior and interpersonal relationships. The result shows that virtual practice and coaching via short health videos can build health awareness and a dynamic new lifestyle, and motivate women to positively practice physical activity and maintain positive interpersonal relationships. Factors affecting the effectiveness of short health videos are discussed for future research in the field of modeling and intervention. As a convenient and effective means of conveying health and medical information to the public, short health videos on social media platforms have been popular among the ageing in China, penetrating into their daily life and behavioral interactions. Exposed to digital platforms, ageing women, who are reported as more relationship-oriented or more vulnerable in some specific health areas in the existing research ,2,3, areRegarding the topics of short health videos and ageing groups’ virtual health behavior, scholars at home and abroad have studied the topic from different perspectives.The research on short health videos has been conducted from multiple perspectives. From the perspective of current and developmental situations, scholars ,5 have fWith the penetration of information technology, the ageing are unconsciously involved in the digital wave. Especially, the sense of familiarity and presence created by short video platforms further strengthens their participation. This positive and convenient network interaction is beneficial not only to their individual health but also to their emotional connection with others. Specifically, the viewing and posting of short videos can create opportunities to alleviate social isolation and loneliness among the ageing. Especially when participating in a common activity with family members, friends and other network members, the ageing can meet their needs for information, interpersonal relationships, and social life, which reduces their loneliness and has a positive impact on their individual behavioral tendencies ,14,15,16In terms of acceptance behaviors, a survey finds that the ageing are mainly affected by four factors when accepting information technology: needs satisfaction, perceived usability, support availability, and public acceptance, among which needs satisfaction and support availability are relatively more important . ManaginThe social and emotional support from family members and friends, as well as the extension of provided network communication, further affects the social relations of the ageing group . In termIn the existing literature, research on short video users mainly focuses on the youth group, and those on the ageing mainly concern the digital gap, intergenerational compensation, life improvement, etc., among which health videos are briefly overviewed as a tool element. In addition, current research on the health of ageing women is mainly concentrated on the fields of sports, medicine and others, focusing on the impact of physiological functions on their body, psychology and behavior, focusing on the specific topics of cancer, sex, suicide and depression ,25. In aTherefore, based on the existing literature, the study, taking Chinese ageing women as an important group in the video-based information audience, investigates the impacts of virtual practice and coaching using short videos from the perspective of cognition, behavior and interpersonal relationships.In order to gain a firsthand understanding of Chinese ageing women’s cognition, behavior and interpersonal relationship with virtual practice and coaching via short videos, 47 participants were recruited by 40 postgraduates at the School of Journalism and Communication, Shanxi University during the winter vacation of 2022. These postgraduates have professional knowledge and skills in qualitative methods in social studies and were informed of the research objectives before the interview. Most of the participants are the postgraduates’ family members or relatives or neighbors, who are mostly from the urban areas or the different counties of Shanxi Province in China. The inclusion criteria for selection were: (1) female over sixty years; and (2) possessing a smart phone and having had the experience of viewing the short health videos for at least half a year.The exploration of social media interaction can be approached by various methods, among which qualitative research has been widely and continuously chosen by researchers . Two typIn-depth interview is a widely employed and valued qualitative research methodology which involves comprehensive conversations between the researcher and interviewee, with a task-based exploration of the interviewee’s subjective experiences, meaning-making, accounting processes, and unspoken assumptions about life and the social world in general . An in-dIn-depth interviews provide access to the context of people’s behavior, opening a way for the researcher to understand the meaning of that behavior. A basic assumption in in-depth interviewing research is that the meaning people make of their experience affects the way they live that experience .The information from the interviewees is very useful for understanding the current situation regarding Chinese ageing women’s engagement with short health videos. The questions specifically asked during the interviews are: (1) What are the reasons for watching short videos? (2) Do you think short videos are beneficial? Has there been any change in your health awareness? (3) Have you made any adjustment to your living or eating habits after watching the short videos? (4) Will you follow the content or advice of the short videos? If not, what factors affect this? (5) Would you like to share or recommend the short videos with/to your family members or friends? Who do you usually share with? By forwarding the links or by recounting the information? (6) Has watching short videos influenced or improved your family life? Has there been any change or improvement in building your family relationships?Discourse analysis is the discipline devoted to the investigation of the relationship between form and function in verbal communication , in whicRegarding the theoretical and methodological framework for analyzing the interviewees’ narratives, the study takes a discourse-oriented perspective based on Systemic Functional Linguistics ,32 and CThe central idea in Systemic Functional Linguistics is that language is a semiotic system of choice making. Transitivity analysis carries out the description of the representation of social events and social actors and construes the experiential meaning through clauses that are constituted of participants, processes and circumstances as is displayed in Genre is defined as “a socially ratified way of using language in connection with a particular type of social activity”  (p. 14).Starting by examining the transitivity patterns and generic structures of the interviewees’ utterances based on the discourse analytic process, an interpretation and explanation of the cognitive, behavioral and interpersonal impacts of the virtual practice and coaching of short health videos on Chinese ageing women will be categorized and summarized. The detailed procedures for the study are displayed in During the data collection stage, the data of eight participants were excluded due to missing crucial information for the analysis. Thus, the final sample comprises 39 Chinese women, whose age ranges from 60 to 80 years (Mean = 64.5). The basic information on the participants is listed in Educational level and job experience. The participants have varied educational degrees: 18 of them have high school education, six have a college degree, 10 have middle school educational level and five only have the experience of primary education. Most of them have job experience, as is shown in Health situation. Living conditions. Most of the participants live with family, except that No. 15 lives alone and No. 37 lives with three room mates in a dormitory of a hotel where she works as a cook.Practice with short health videos. In general, the participants have an intensive and varied use of smart phones, including viewing short videos for at least between half a year to five or six years. The daily frequency for practicing with short videos is relatively high, from half an hour to three or four hours a day. The social media platforms the participants usually rely on include Kwai, Tik Tok, WeChat Short Video, Toutiao, etc., among which Kwai and Tik Tok are much more popular. At the beginning of their practice with social media platforms, most of them report that their children or grandchildren offered coaching or help with downloading the APPs. The contents of the short videos they mainly focus on include senile diseases, such as high blood pressure and diabetes, aerobics or dancing, and healthy eating. The providers of the videos include two rough categories: official medical institutions, and individuals and relevant groups. The former is mainly composed of health agencies and hospital accounts, and the latter includes both professional and non-professional health practitioners.a little bit)”, No. 39 responds “méiyǒu duōdàde biànhuà BA (not much has been changed BA)”.The analysis of the sample to assess the cognitive change of the interviewees shows that 32 give the definitely positive answer that virtual practice and coaching via short videos are beneficial, and their cognition regarding health problems has changed after watching the videos. One interviewee (No. 12), a doctor in a village, gives the negative answer that there is no change in her cognition. The rest of the interviewees respond positively but in a partial degree or hesitation; for example, No. 27 answers “duōshǎo yǒu yìdiǎn (As is exemplified in think) or “rènwéi” (think) and the attributive relational process “shì” (be) are the most recurring and important meaning resources that enable the interviewees to depict their cognitive state. The circumstances are mainly the positive comments on the effect of watching short videos, such as “tǐnghǎo” (very good), “yǒu dàolǐ” (reasonable), ”yǒuyì”, “yǒuhǎochù”, “shòuyìfěiqiǎn” , “yǒubāngzhù” (helpful), “yǒusuǒgǎibiàn” (changed). More specifically, the interviewees construe their cognitive change by using the mental process of “zhùyì” (pay attention to), “guānzhù” (pay attention to), “zhǎngwòle” (acquire), “zhīdào” (know), “bāngzhù wǒ liǎojiě le” (help me understand). The behavioral processes, such as “cǎinà” (adopt) and “fǎngzhào” (imitate) are graded ones in the interviewees’ cognitive change, i.e., they begin to accept useful information and take actions after virtual coaching on the short video health platforms.The cognitive mental process “juéde” , eating habits and other aspects in the interviewees’ narration. In this part of the statement, as is best exemplified in the following two transcriptions of No. 20 and No. 30, the main type of generic structure is exemplification, in which the contrasts between “yǐqián” or “zhīqián” (before) and “xiànzài” (now) or after watching a short video are made.Yǐqián BA, búài chūqù, ài zàijiālǐ dāizhe. Xiànzàile kànle yǐhòu, rènwéi rénjiā shuōde hái tǐngduì de, yǒuxiē fánxīnshì, chūqù hé biérén duō jiāoliú jiāoliú, duō wánwan, duō duànliàn duànliàn, huíjiāhòu zhège fánxīnshì jiù wàngle, tǐnghǎode. Xiànzài wǒ měitiān zǎoshàng qǐlái shǒuxiān hē yībēi wēnkāishuǐ, ránhòu chuāndàihǎo jiù hé línjūmen qīdiǎnláizhōng qù cūnwài mànzǒu. Wǒ yǐqián chīfàn bǐjiào kǒuwèizhòng, xiànzài bùgǎn kǒuwèi zhòngle, gào jiālǐderén yěshì, chīfàn yào shǎochīyán, kǒuwèi chīde qīng dàn yīdiǎn, píngshí zài yǐnshí shàng shǎochī yóunìde. Píngshí chīxìliáng, chīxiē shànshí xiānwéi, búyào lǎochī báimiàn, dàmǐ; duōchī qiáomàimiàn, yùmǐmiàn, fǎnzhèngshì duōchī cūliáng BA. (No. 20)BeforeAfter watching it, I think what they say is quite right. When there are some annoying things, go out to communicate more with others, play more exercises, I will forget the annoying things after returning home, which is good. Now after I wake up every morning, first I will drink a glass of warm water, then get dressed and go out for a slow walk outside the village at seven o’clock with my neighbors. I used to have a heavy taste in eating, but now I don’t dare to have a heavy taste. I also tell my family that we should eat less salt and less greasy food. Usually we eat refined grains, we should eat some dietary fiber, such as more buckwheat noodles, cornmeal, anyway, eat more coarse grains, instead of white noodles or rice all the time., I didn’t like going out, I liked staying at home. zhīqián kǒuwèi piānxián, chǎocài zuò fàn shíyán huì fàngdeduō, xiànzài yǐjīng shǎochī shíyán, háiyǒu xīnlà cìjì de shíwù, yóunì de ròulèi chīde yěshǎole, chīde shūcài, húluóbo, dòuzhìpǐn huì duōyīxiē. (No. 30)Bǐrú zài shēnghuó xíguàn fāngmiàn, shuìjiào zhīqián yòng rèshuǐ pàojiǎo, yībān huì pào bàngèduō xiǎoshí, zuòyizuò shǒubù yǐjí tóubù de ànmó duànliàn, yǒuzhùyú bāngzhù shuìmián; zài yǐnshí xíguàn fāngmiàn, before I prefer the salty taste and like putting more salt in cooking. Now I eat less salty, spicy food and less greasy meat, and eat more vegetables, carrots and soy products.For example, in terms of living habits, I wash my feet in hot water before going to bed, usually for more than half an hour, and do a massage on hands and head to help sleep; in terms of eating habits, The changes in living habits narrated by the ageing women are displayed as different aspects regarding their health, such as time rescheduling, as in No. 14′s narration “wǎnshàng yào zǎoshuì, bùyào chāoguò shíyīdiǎn” , “gēnzhe shìpín zuò jǐngzhuīcāo” . The reasons for virtual coaching are mainly because they have diseases  or they are not strong (as No. 39). Most of them can follow virtual coaching to exercise for a long term . In the interviewees’ narration, the effects of exercise through virtual coaching are obvious, such as “yǒu xiàoguǒ” (effective), “tǐng shíjì” , “tǐng guǎnyòng” (very helpful), “tǐng yǒu jīngshén” (very energetic), “hǎole” (improved), “qíngkuàng hǎoduōle” (much better), ”huǎnjiě” (relieved). Expectations are optional in their statement, but generally they expect that virtual coaching will be effective and they can be healthier.By observing the sample, as to the question whether they can follow the virtual coaching in order to exercise, the study finds that most of the interviewees can do so, as is indicated in I should make a change, but I can’t control myself. I can’t change it. It seems that I am still too young); No. 16, who is 72, says “Lǎnde zuò, yuányīn jiùshì shēntǐ háikěyǐ, yàoshì jīngcháng shēngbìng, kěndìng jiù gèng zhùzhòng yīxiē” .As for the interviewees who don’t follow virtual coaching; the reasons can be attributed to their relatively young age and strong and healthy body, for example, No. 15, who is 63, states “Ànlǐ yīnggāi gǎibiàn, dànshì zìjǐ kòngzhì bùzhù, gǎibiàn bùliǎo, kànlái háishì niánlíng xiǎo deguò”  share by forwarding the link, 8 (22%) use oral narration, 16 (44%) use both of the above, and 32 interviewees believe that viewing or following these videos has a positive impact on improving family life.The observation of transitivity finds the processes adopted by participants are mainly material processes, that is, the action of sharing, or verbal processes, the action of telling. The analysis of process types, Goal or Receiver (the one to which/whom the process is extended), and adverbial elements can best display the interviewees’ sharing activities.jiǎnjié yìdiǎn jiùshì, yǒuyòng de, zhòngyào de, fēnxiǎng gěi qítā rén” , “gēn biérén shìpín, tàntǎo duǎn shìpín lǐ de yǎngshēng zhīshi” , “gěi tāmen jiǎngshù yíbiàn, huòzhě gěi tāmen shìfàn yíxià zhège dòngzuò” , tōngcháng huì zhuǎnfā dào qúnlǐ, tóngshí yěhuì shuōjǐjù .36 of the interviewees would like to share useful videos to others. As to the processes, besides unilaterally recommending, sharing or forwarding the information, 53% would take part in more communication, for example, “tóngxuéqún, chūzhōng tóngxué, shīfàn tóngxué, tóngshì” , “yígè wéichíle sìnián de dàyuē, 50 rén de fěnsīqún” , “yīkuài dǎpái de péngyǒumen” . Among the Goals or Receivers, the participants also target their information to certain groups, for example, “yóuqí shì bú shàngbān de rén” ,” “Wǒ mèimei ma, tā búshì tángniàobìng…. Duì tā yǒuyì, wǒ jiù gěi tā fāguòqù le.” .The Goals or Receivers are not limited to their family members or relatives. The interviewees would share or communicate with a wide range of people, for example, “fāgěi dōuyào fāchūqù ne” “Yóuqí” used by No. 4 (a teacher) and “kěài zhuǎnfā YA” “tèbié ài fā”, “měitiān dōuyào fā, méiyǒu yìtiān bù fā de” used by No. 10  indicate their great desire and willingness to share the videos which they think useful and helpful to their family members, relatives and friends.3.dōuyào fāchūqù NE. Yóuqí shì bú shàngbān de rén wǒ jiù gěi duō fāxiē liànjiē, yěràng tāmen zàijiā biān kānháizi biān duànliàn shēntǐ. (No. 4)Fāzài péngyǒuquān le hé qīnyǒumen de qúnlǐ, háiyǒu huì fāgěi tóngxuéqún, chūzhōng tóngxué, shīfàn tóngxué, tóngshì, I send  to the moments (on Wechat) and all the groups, including the groups of relatives and friends, junior middle school classmates, normal college classmates, and colleagues. Especially for those who are not working, I send more links, and let them exercise at home while taking care of their grandchildren.4.ài fābù YA. Jiùshì nà shíhòu gānggāng názhe shǒujī wán MA HA, AIYA kě xiǎng zhuǎnfā NE, hǎoxiàng biérén dōu kàn bùjiàn, wǒ gǎnjǐn bǎ zhège zhuǎnfā chūqù HA…. hǎoxiàng yuánlái fǎnzhèng tèbié ài fā, měitiān dōuyào fā, méiyǒu yìtiān bù fā de, yǒushíhòu hái fā hǎojǐtiáo NE. En fēnxiǎng dào péngyǒu quān HA, fāchūqù. … jiùshì gāng náshàng shǒujī wánde shíhòu kě xiǎng fā NE. AIYA, kě yào fā NE, fā gěi zhège fā gěi nàge. (No. 10)… wǒ yuánlái yěshì, … I liked forwarding so much, and loved posting. When I was first playing with my mobile phone, I loved forwarding so much. It seemed that no one else could see it, so I quickly forwarded. …It seemed that I really loved posting, every day. Sometimes I posted several pieces in one day. Well, I shared them on moments, sending all of them out. …When I first got my phone, I just wanted to post. I wanted to forward links, to this one or to that one.The adverbs of degree or the mood particles used by the interviewees can clearly display their attitudes towards interpersonal behavior. Referring to their sharing activities, in the following transcriptions, the expressions “Everyone feels very happy. No. 23), “Mànman péngyǒu yě biànduōle” , “Gēn tāde jiāoliú duōle yǐhòu, xiǎoxífuer zhīdàole wǒde yòngyì, duì wǒ yěyǒule gǎiguān.” .The analysis of the sample also shows that sharing short health videos improve the interviewees’ family life, and 32 of the interviewees give an affirmation and positive comments, for example, “Dàjiā dōu juéde tǐng kāixīn de.”  use the expressions “shāowēi yǒuxiē” and “shāowēi gǎishànle yìdiǎn BA” to indicate very few or slight improvements in their family relationship. Four indicate that the sharing of the videos doesn’t have any or much impact on family life or relationship: “méiyǒu shénme biànhuà” , “biànhuà búdà” , “duì jiātíng shēnghuó yǐngxiǎng búdà BA” , and “méiyǒu tàidàde yǐngxiǎng” . However, No. 26 adds that the change mainly lies in diet (yǐnshí fāngmiàn huì yǒu yǐngxiǎng) and No. 33 admits that the change is mainly reflected in the individual (zhǔyào shì duì gèrén de yǐngxiǎng).Sometimes I get addicted to watching videos and spend a long time, which affects my family members. No. 21), “gēn jiārén de jiāoliú biàn shǎole, xiánxiá shíjiān jīběn huāzàile kànshìpín shàng” .While reporting the positive impacts, six of the interviewees (15%) refer to negative impacts, for example, “yǒu shíhòu kàn shìpín kàn shàngyǐn, kànde shíjiān tài cháng, yǐngxiǎngle jiārén de xiūxí”  or “rènwéi” (think) and the attributive relational process “shì” (be) with positive comments, such as ”yǒuyì” , “yǒubāngzhù” (helpful) about their cognitive change after adopting the virtual health information by expressing “zhǎngwòle” (acquire), “bāngzhù wǒ liǎojiě le” (help me understand), and even graded behavioral processes, such as “cǎinà” (adopt), “fǎngzhào” (imitate) or “gēnzhe shìpín zuò...” (follow the video to do...). The main generic arrangements, i.e., the contrast between their cognition before and after their watching the short videos with optional examples, the sequence of cognitive changes and actions to be taken, and the affirmation that these videos are definitely beneficial, along with some optional comments, uncover the representation of modification by contrasting, sequence and affirmation patterns.At the individual level, according to the knowledge, attitude and practice (KAP) theory ,39, peopforward), or verbal processes, the action of telling, such as “tàntǎo” (discuss), “jiǎngshù” (retell), shuōjǐjù (add a few words). The Goals or Receivers with whom the participants share or communicate are also not limited to their family members or relatives, but to a wide range of people, especially to those who need the specific health information. The adverbs of degree or the mood particles used by the participants vividly indicate their desire and willingness to share the short videos with their family members, relatives and friends. As to whether sharing short health videos improves the interviewees’ family life, most of them display an affirmative attitude by unveiling their representations with positive comments and evaluations.At the interpersonal level, in the Chinese context of collectivist culture, the ageing need more information technology especially regarding health information to connect with others, due to their closer interpersonal relationships . They maAs is indicated in many studies ,42, percI now cook with less oil and less salt, which is good for blood vessels... get more sun exposure, because some experts say that calcium in food is just a part of it and cannot be fully absorbed. It is good to go out and get some sun exposure.” In contrast, for those without any disease, their willingness and motivation towards behavioral change are not very strong. For example, No. 16, who is 72 years old, says that she is still healthy and too lazy, but she would pay more attention if she often got sick. Therefore, ageing women with diseases are more inclined to accept health information in short videos, and make adjustments to their lifestyle habits and eating habits or physical exercises.The analysis in According to the interviews, there is also a strong correlation between age and changes in cognition and behavior. Some ageing women in their early 60s make no major changes in acquired behaviors after watching health videos; on the contrary, elder women are more willing to receive virtual health coaching and adjust their health behaviors after viewing the short videos. For example, the 61-year-old No. 19 responds that there has not much change in her cognition and exercise behavior; and the 63-year-old No. 15 also mentions in the interview that she can’t control herself and make changes because she thinks she is still young.Interpersonal communication plays an important role in changing attitudes and plays a key role in persuading individuals to adopt healthy behaviors ,44. At tmy husband has high blood lipids. At the beginning, I imitated the short videos to make some light meals... Gradually, he began to accept it. And now my husband often comes home for dinner, and the family relationship is more harmonious”; and No. 30 often shares health information on eating habits to family members because “the granddaughter has bad eating habits, likes fried foods such as hamburgers, and rarely eats vegetables, which leads to her weak physical condition and often being ill”. Besides family members and relatives, the interviewees also target their information to a certain group, who have a great need for specific health information, for example, those who are not working (No. 4) and those who are in their eighties (No. 11).The interpersonal relationship structure is determined by the needs of human beings . Some inChildren are young and they have their own opinions. The views and living habits between the ageing and young are also different. My sharing health knowledge with them will be opposed,... so later on I will not communicate with them about this. No longer share!”.In addition, the study also finds that interactive communication between the interviewee and the communicator has an important impact on the willingness to share. According to Altman & Taylor , communiAlthough the rapid development of transportation and information makes co-presence face-to-face interaction between people increasingly unnecessary , to the Maybe it has something to do with my previous job. If I think it’s good, I want to promote it to everyone so that they can enjoy it”; No. 6 accounts the partial reason why she doesn’t like sharing health information, “I don’t know how to forward it, I just watch it myself”.Besides the above mentioned essential factors, others such as occupation and technology skill have less impact on interpersonal improvement, but are also mentioned by the interviewees. For example, No. 13, a retired doctor, says “The study provides clear evidence that short health videos on social media platforms can build ageing women’s health awareness, motivate behavioral modification and improve interpersonal relationships. Factors affecting these changes can be taken into consideration for future research in the field of short health videos in terms of modeling and intervention. At the same time, the study also attempts to apply and extend the notion of transitivity analysis and generic structure analysis to health communication research. It shows the great potential of discourse analysis, as a complementary investigation tool in analyzing qualitative material, especially on the effectiveness of short health videos in modifying cognition, behavior and interpersonal relationship.