T is sufficiently large, then uniformly for all positive integers ℓ⩽(logT)/(log2T), we have|ζ(ℓ)(σ+it)| when ℓ∈N and σ∈[1/2,1) are fixed.It is proved that if  Recall that Littlewood . In this case we obtain maxT/2⩽t⩽T|ζ(ℓ)(12+it)|⩾exp{(1+o(1))logT/log2T}, losing a log3T factor compared to the above result on the longer interval .By Soundararajan's original resonance method , we can ζ(1+it)=Ω(log2t) (see  denotes the least common multiple of m and n.Let σ=1 was studied by Gál , σ∈,Let ζ(s) and Cauchy's integral formula.□It follows from classical convex estimates for In the following, we will derive a “double version” convolution formula, similar to Lemma of 5.3 of de la Bretèche and Tenenbaum . The proK^ of K asLemma 3ℓ∈N and σ∈, satisfying the growth conditiont∈R∖{0}, thenLet whereandDefine the Fourier transform h(z):=Fℓ(z+it)Fℓ(z−it)K(iσ−iz). h(z) is a meromorphic function in the vertical strip σ⩽ℜ(z)⩽2, with two poles, namely, at z=1+it and z=1−it. Let Y be large and consider straight line integrals for h(z). Set J1=∫σ−iY2−iYh(z)dz,J2=∫2−iY2+iYh(z)dz,J3=∫2+iYσ+iYh(z)dz,J4=∫σ+iYσ−iYh(z)dz.Define Fℓ(s)=ℓ!/(s−1)ℓ+1+E(s), where E(s) is an entire function. The residue theorem gives thatNote that By (2) and (9) and applying Cauchy's theorem term by term,Clearly,Fℓ(s)≪1+|ζ(ℓ)(s)|, estimates for ζ(ℓ)(s)  Lemma  and 9),,Fℓ(s)≪1+ε=1/6, then J1≪1/(YlogY)→0, as Y→∞. Similarly for J3.□Take Lemma 4 Let μ,ν∈N and α1,α2∈ be fixed. Suppose α1+α2>1, thenThe following results are due to Hadamard, Landau, and Schnee .Lemma Remark 4o” are under the condition when the corresponding variable tends to infinity. Namely, as T→∞, t→∞, or N→∞.In this paper, if it is not stated, the limit notations “∼” and “ℓ∈N and σ∈ are fixed, one hasIn particular, when σ=12, Ingham [A″] has proved the following result on second moments of ζ(ℓ)(s).Lemma 5(Ingham) Let ℓ∈N be fixed. ThenFor 3We will use the construction of Bondarenko and Seip in .δ=ℓ·(ℓ+1)−1. Given a positive number y and a positive integer b, defineLet x and an integer b later to make P⩽T. Let M be the set of divisors of P and Mδ be the set of divisors of P. Let Mδ¯ be the complement of Mδ in M. Note that both M and Mδ are divisor‐closed which means k|n,n∈M⇒k∈M and k|n,n∈Mδ⇒k∈Mδ. Define the function r:N→{0,1} to be the characteristic function of M, thenWe will choose a number As showed in ,1|M|∑nAlso in . By the prime number theorem, P⩽T when T is sufficiently large. Take the choices of x,b and δ=ℓ·(ℓ+1)−1 into the above inequality, then we are done.□By the definition of 4N=[T12] and let R(t):=∑n⩽Nr(n)n−it. Define the moments as follows:Set Φ:R→R denotes a smooth function, compactly supported in , with 0⩽Φ(y)⩽1 for all y, and Φ(y)=1 for 5/4⩽y⩽7/4. Partial integration gives that Φ^(y)≪ν|y|−ν for any positive integer ν.As in , Φ:R→R dAlso in , SoundarSince Φ is compactly supported in , we deduce thatN⩽T12, for the off‐diagonal terms km≠n we have Φ^(Tlog(km/n))≪T−2, by the rapid decay of Φ^  N=[T12], we obtainAgain, by O(·) term is absolute:By Lemma M2, the big O(·) term above contributes at mostIn the integral of Combining this with , we haveFinally, the above formula together with (18) gives thatε=(log2T)−1. By Stirling's formula, if T is sufficiently large, then for all positive integers ℓ⩽(logT)(log2T)−1, we have  ℓ!·ε−ℓ·T−1+ε⩽(log2T)ℓ. Other big O(·) terms can be easily bounded. Together with Proposition □Now let 55.1M of positive integers and a parameter T, we will construct a resonator R(t), following ideas from , an, anM of eas from . DefineBy Cauchy's inequality, one has the following trivial estimates :R(0)2⩽Φ(t):=e−t2/2. Its Fourier transform satisfies Φ^(ξ)=2πΦ(ξ).As in  , set Φ. Choose κ∈ and set N:=[Tκ]. Fix ε>0 such that κ+4ε<1.Let As in , chooseDefine2Tβ⩽|t|⩽T2 and |y|⩽|t|2 gives the main term for I(T). We will frequently use the following trivial estimates |⩽R(0) and Φ(·)⩽1,A simple computation givesThe fast decay of Φ and (23) give thatUsing (21) and (23), one can computeCombining the above estimates, one gets2Tβ⩽|t|⩽T2 and |y|⩽|t|2 give Tβ⩽|t±y|⩽T.  Again, by (21)I2(T),I3(T) as follows:K, we have the following estimates for all 0⩽n⩽l:Note that 0⩽m⩽l, one hasAnd trivially, for all  satisfying m+n=ℓ, soI2(T), so we get (Note that there are finitely many non‐negative integer pairs o we get .I1(T) to the GCD sums, we would like to use Fourier transform on the whole real line. So setK^(lognm)=0 if mn⩾T2ε. Clearly, al(n)/nσ≪1. So one can getI1(T)=I1∼(T)+|M|·O(Tκ+4ε). Thus we haveNext, in order to relate I1∼(T) by expanding the product of the resonator and the infinite series of Gσ(t), and then integrate term by term, as in , soIn this case, let M,Also, in , p. 128,o(1). And clearly, the big O(·) terms in (32) can be ignored. ThusSo the second term on the right‐hand side of (32) is Case 2: σ∈.M be the set in (6) with |M|=N. Again, N=[Tκ], soIn this case, let Similarly as (31), we haveAnd by (4), we can get the following estimates:Hence,B).□Make κ slightly larger in the beginning then one can get (6(m⊗n) for the ordered pair of m and n.The idea of the proof is basically the same as in the proof of Proposition P∼=p1b−1·…·prb−1 , where pn denotes the nth prime. Define M to be the set of divisors of P∼, then |M|=br. By Gál's identity , then pr∼logN by the prime number theorem. Let b be the integer satisfying thatbr∼N, as N→∞. Choose a set M′⊂N such that M⊂M′ and |M′|=N.Let (eγlogpr)2∼(eγloglogN)2 as N→∞. The second product converges as N→∞ toFollowing Lewko–Radziwiłł  in , we use δ=ℓ·(2ℓ+1)−1 and define the sets Mδ(1),Mδ(2) as follows:Next, let Mδ(1):={(m⊗n)∈M×M|∀i>rδ,αi=min{αi,βi},   where m and n have prime factorizations as     m=p1α1p2α2⋯prαr,andn=p1β1p2β2⋯prβr}.Mδ(2):={(m⊗n)∈M×M|∀i>rδ,βi=min{αi,βi},   where m and n have prime factorizations as     m=p1α1p2α2⋯prαr,andn=p1β1p2β2⋯prβr}.Mδ to be the union of the above two sets and Mδ¯ to be the complement of Mδ in M×M:Then define Now we split the GCD sum into two parts:By symmetry, we haveMδ(1) and Gál's identity, we haveBy the definition of br∼N, the first product converges to 6/π2, and the third product is asymptotically equal to (eγlogpr)2∼(eγloglogN)2 as N→∞.Again, we have For the second product, it can be bounded asAs a result, we obtain thatHence by , 39), a, a∑∈Mδ¯, thenBy the construction of Thusδ=ℓ·(2ℓ+1)−1, we are done.□By our choice of 7ζ(ℓ)(1+it)=Ω((log2t)ℓ+1), when ℓ∈N is fixed.One can use the method of Bohr–Landau ⩾cos(2π/q) for all integers n∈. HenceWrite q=8 to getTake One can compute thatN thatand for large A (only depending on ℓ) such that (ℓ+1)Aℓ·e−A<1/12 and let σ−1=A/logN. Combining with (41) gives thatNow fix a positive constant ζ(ℓ)(1+it)≠Ω((loglogt)ℓ+1). So f(1+it)=o(1). Clearly, f(2+it)=o(1). Then we get a contradiction with  and σ∈R, define the following normalized log‐type GCD sums as:Problem 1Γσ(ℓ)(N).Given σ and ℓ, optimize Let Remark 5σ=1. Given ℓ, what is the optimal constant Cℓ such that Γ1(ℓ)(N)⩽Cℓ(log2N)2ℓ+2 ? (See , or ℓ=[2T]?In our Theorem Problem 6Can one find some range for ℓ, such that the results in Theorem Remark 9aℓ(n)⩾1 for all n and ℓ. It is not clear about the moments of derivatives of the zeta function if ℓ can depend on T. For instance, if we let ℓ=[(logT)(log2T)−1], then what can we say about the second moments as T→∞,T, it also seems difficult to bound the contributions of Δ++Δ− .The main terms always satisfy since we have |ζ(ℓ)(σ+it)| can be if ℓ can be taken arbitrary large with respect to the length T of the interval .Problem 7σ0∈, decide which one of following four properties can be true.Given Moreover, we have the following general problem, which asks how large or how small the extreme values of (A) Given any function V:→, there always exists some function fV:→N such that if ℓ=fV(T), then for sufficiently large T, we have(B) Given any function V:→, there always exists some function fV:→N such that if ℓ=fV(T), then for sufficiently large T, we have(C) There exists some function V:→, such that for all function f:→N, if ℓ=fV(T), then for sufficiently large T, we have(D) There exists some function V:→, such that for all function f:→N, if ℓ=fV(T), then for sufficiently large T, we haveMathematika is owned by University College London and published by the London Mathematical Society. All surplus income from the publication of Mathematika is returned to mathematicians and mathematics research via the Society's research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.


C is a closed, convex subset of a separable Banach space, C is Haar null if and only if C is Haar meagre. We then use this fact to improve a theorem of Matoušková and to solve a conjecture proposed by Esterle, Matheron and Moreau. Finally, we apply the main theorem to find a characterisation of separable Banach lattices whose positive cone is not Haar null.Haar null sets were introduced by Christensen in 1972 to extend the notion of sets with zero Haar measure to nonlocally compact Polish groups. In 2013, Darji defined a categorical version of Haar null sets, namely Haar meagre sets. The present paper aims to show that, whenever  A Borel set E⊂G is Haar null if one can find a Borel probability measure μ on G such that μ(gEh)=0 for all g,h∈G. The measure μ is said to witness that E is Haar null and can always be assumed to have compact support, as every Borel probability measure on a Polish space is inner regular. This definition was given for the first time in Abelian Polish groups by Christensen in  and andG be E in a Polish group G is Haar positive and which can be expressed as follows: for every compact set K⊂G, there are an open set V⊆G and g,h∈G such that K∩V≠⌀ and K∩V⊂gEh. Indeed, suppose that E enjoys this property, let μ be any Borel probability measure on G with compact support K and find an open set V⊆G and g,h∈G such that K∩V≠⌀ and K∩V⊂gEh. Clearly, μ(K∩V)>0, because K is the support of μ, hence μ(gEh)>0. Sets with this property are called compactivorous, can be defined in any Hausdorff topological group and were mentioned for the first time in . We havex∨y∈DX+ and, by property (2), (1−t)x+ty∈DX+. This shows that DX+ is convex.□Pick X is called an AM‐space if, for every x,y∈X+, the equality ||x∨y||X=max{||x||X,||y||X} holds. Our last result states that, up to lattice isomorphisms, only separable AM‐spaces have nonnegligible positive cones.Theorem 4.2X be a separable Banach lattice. The following assertions are equivalent.(1)X+ is Haar positive.(2)x=(xn)n=1∞∈c0(X), there exists z∈X+ such that z⩾|xn| for each n.For every sequence (3)k>0  such that for every integer n⩾1 and any x1,⋯,xn∈X+, one hasThere is (4)Y and a lattice isomorphism T:X→Y.There are an AM‐space Let Recall that a Banach lattice ⇔ (2) Suppose that X+ is Haar positive and pick x=(xn)n=1∞∈c0(X). By Theorem X+ is not Haar meagre and therefore compactivorous, hence there must be z∈X such that z+Σ(x)⊂X+. This implies that, for each n, we have z⩾xn and z⩾−xn, hence z⩾xn∨(−xn)=|xn|. Conversely, suppose that (2) holds and pick again x=(xn)n=1∞∈c0(X). Find z∈X+ such that z⩾|xn| for each n. Then z+Σ(x)⊂X+. This shows that X+ is compactivorous, therefore Haar positive.(1) ⇔ (3) Suppose that X fulfils assertion (3) and let k>0 be such thatn⩾1 and any x1,⋯,xn∈X+. Fix x=(xn)n=1∞∈c0(X). Define the sequence y=(yn)n=1∞ recursively in the following way: y1=|x1| and yn+1=yn∨|xn+1| for every n. Observe that y is an increasing sequence in X+. Choose ε>0 and let n0 be such that ||xn||X<ε/k for all n⩾n0. For every n and m with n>m⩾n0, we havey is a Cauchy sequence and it has therefore a limit z∈X+. Clearly, z⩾yn for every n, thus z⩾|xn| for every n. On the other hand, assume by contradiction that for each positive integer k there are xk,1,⋯,xk,n(k)∈BX∩X+ such that ||xk,1∨⋯∨xk,n(k)||X⩾k2. Let y=(yn)n=1∞∈c0(X) be the sequence whose elements are x1,1,⋯,x1,n(1), followed by 2−1x2,1,⋯,2−1x2,n(2), followed by 3−1x3,1,⋯,3−1x3,n(3) and so on. If there were z∈X+ such that z⩾yn for all n, then for each k it would hold that z⩾k−1xk,1∨⋯∨k−1xk,n(k). Thus, for each k,(2) ⇔ (4) Assume that (4) holds, that is, that there are an AM‐space Y and a lattice isomorphism T:X→Y. Set k=||T−1||BL||T||BL and choose an integer n⩾1 and x1,⋯,xn∈X+. ThenX which turns X into an AM‐space. If (3) holds, then it follows from the definition of DX+ and from the properties of lattice norms that ||x||X⩽k for every x∈DX, which implies BX⊆DX⊆kBX. DX is symmetric, as |x|=|−x| for every x∈X. Moreover, DX is convex. Indeed, if x,y∈DX and t∈, then |x|,|y|∈DX+ and |(1−t)x+ty|⩽(1−t)|x|+t|y|. By the convexity of DX+ and by (2) in Lemma |(1−t)x+ty|∈DX+, hence (1−t)x+ty∈DX. Let ν be the Minkowski functional associated to the set DX. That is, ν(x)=inf{λ>0:x∈λDX} for every x∈X. Since BX⊆DX⊆kBX, ν defines an equivalent norm on X. Moreover, one notices easily that ν(x)=ν(|x|) for every x∈X. To see that ν is indeed a lattice norm, pick x,y∈X such that |x|⩽|y|. For every ε>0, we have|x|∈ν(y)+εDX+. As ε is arbitrary, it follows that ν(x)=ν(|x|)⩽ν(y), as wished. Now, let x,y∈X+ and set m=max{ν(x),ν(y)}. For every ε>0, we have x,y∈(m+ε)DX+. By (1) in Lemma x∨y∈(m+ε)DX+. Since ε is arbitrary, this allows to conclude that ν(x∨y)⩽m. The opposite inequality is true in every Banach lattice, therefore ν turns X into an AM‐space.□(3) Remark: The proof of the equivalence between (2), (3) and (4) in Theorem Bulletin 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


Correction: Fluids Barriers CNS 20, 20 (2023)10.1186/s12987-023-00421-8Following publication of this article , the autThese errors were introduced during typesetting.The correct values are given below and the original article has been corrected.The publisher would like to apologise for any inconvenience caused.p = 2 × 10− 11 m Pa− 1 s − 1 to 3 × 10− 10 m Pa− 1 s − 1, diffusion membrane coefficients for small solutes in the range CM = 5 × 102 m− 1 to 6 × 103 m− 1 and gap area fractions in the range 0.2–0.6%, based on a inter-endfoot gap width of 20 nm.We provide structural-functional relationships between vessel radius and resistance that can be directly used in flow and transport simulations. We estimate endfoot sheath filtration coefficients in the range L


In the published article, there were errors in affiliations 1 and 2. Instead of1Positive Psychology Research Group, Institute of Psychology, Faculty of Education and Psychology, Eötvös Loránd University, Budapest, Hungary” and““2Doctoral School of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary”,it should be1Doctoral School of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary” and“2Institute of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary”.“Attila Oláh should be affiliated to “Institute of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary”.András Vargha should be affiliated to “Institute of Psychology, ELTE Eötvös Loránd University, Budapest, Hungary” and “Person- and Family-oriented Health Science Research Group, Institute of Psychology, Faculty of Humanities and Social Sciences, Károli Gáspár University of the Reformed Church in Hungary, Budapest, Hungary”.Additionally, in the published article, there was an error in the Appendix/The Mental Health Test (MHT). Item 16 of the questionnaire was incorrectly written as “I easily become impatient./SR”. The corrected Item 16 appears below.(16) I become frustrated when something does not happen the way I planned it./SRThe authors apologize for these errors and state that they do not change the scientific conclusions of the article in any way. The original article has been updated.


Multiword expressions (MWEs) are sequences of words that pose a challenge to the computational processing of human languages due to their idiosyncrasies and the mismatch between their phrasal structure and their semantics. These idiosyncrasies are of lexical, morphosyntactic and semantic 11 nature, namely: non-compositionality, i.e., the meaning of the expression cannot be computed from the meanings of its constituents; discontinuity, i.e., alien elements may intervene; non-13 substitutability, i.e., at least one of the expression constituents is lexicalized and therefore, does not enter in alternations at the paradigmatic axis; and non-modifiability, in that they enter in syntactically 15 rigid structures, posing further constraints over modification, transformations, etc. The paper presents a model for representing MWEs at the level of semantics by taking into account all these inherent idiosyncrasies. The model assumes the form of a linguistic ontology and is applied to Greek verbal multi-word expressions (VMWEs); moreover, the semantics of the lexical entries under scrutiny is also represented via the semantics of their arguments based on corpus evidence. In this regard, modeling the semantics of VMWEs is placed in the lexicon-corpus interface. MWEs are highly idiosyncratic structures Gross,  and thusvia encoding the lexical semantic relations between a VMWE and other single- or multi-word entries; and (d) at the syntagmatic axis, by modeling the semantics of their arguments based on corpus evidence. In the latter case, the VMWE is taken as a whole, that is, as a complex predicate. Our goal is to treat both single- and multi-word entries in a comparable way that would be useful for Natural Language Processing (NLP) applications.Our work seeks to fill this gap by proposing a model for encoding the semantic properties of VMWEs into a lexical resource by considering all the idiosyncrasies they exhibit. The semantics of VMWEs are thus defined along the following axes: (a) the type of VMWE in terms of the degree of compositionality, (b) their mapping onto concepts or word senses already existing in an inventory, that is, a semantic lexical resource already available; (c) at the paradigmatic axis, Most Lexical Resources (LRs) dedicated to MWEs give an account only of their lexical, morphological, and syntactic idiosyncrasies. Within the Lexicon-Grammar framework, the pioneering work of Gross  toward tQualia Structure, SIMPLE lexica encode structured semantic types and semantic (subcategorization) frames. A few years later, the Brandeis Semantic Ontology (BSO) seeks to extend the English SIMPLE lexicon  is one of the earliest corpora annotated with semantic roles , marked also for fixedness. Possible syntactic properties  are also encoded at this level. In the next sections, we elaborate on the encoding at the level of semantics. Our model provides mechanisms for encoding diathesis alternations, register, and for signaling MWEs that have a literal  besides their idiomatic one, as defined in Savary et al. , and syntactic properties at the SIGNIFIER level has been extensively presented in Fotopoulou et al. . Accorditopoulou  and Minitopoulou . Accordiy et al. .via semantic relations, and (d) identifying their arguments and the roles they assume. We will elaborate on the model itself in the next paragraphs.The semantic representation of lexical items—both single- and multi-word ones—is achieved at the SIGNIFIED level, taking into account the following aspects: (a) coarse classification that reflects their degree of compositionality; (b) mapping onto word senses or concepts; (c) linking with other entries verbal idiomatic expressions (VIDs), that bear a meaning that cannot be computed based on the meaning of their constituents and the rules used to combine them; (b) light verb constructions (LVCs), i.e., expressions with a rather transparent meaning; (c) multi-verb constructions (MVCs), that is, expressions with coordinated lexicalised head verbs ; and (d) verb-particle constructions (VPCs) comprising a verb and one of the adverbs μπρoστ ά (=in front), πíσω (=back), π άνω (up), κ άτω (=down), μ έσα (=in), έξω  in Greek; these adverbs are not morphologically derived from adjectives and exhibit most - if not all - of the properties particles in other languages have Giouli et al.  π έϕτω μ έσαlit. fall1−sg in (2) π έϕτω έξωlit. fall1−sg out (=to get bankrupt)In terms of meaning, the classification in the afore-mentioned classes is a first step toward defining their semantics: VIDs and MVCs are non-compositional, LVCs are semi-compositional, in that they have a transparent meaning which is retained by the predicative noun, whereas VPCs present semantic ambiguity. Of course, other dimensions exist along which these different types of VMWEs can also be compared, namely, non-modifiability, and non-substitutability. In this regard, VIDs, VPCs and MVCs are syntactically rigid structures posing constraints with respect to modification, syntactic transformations, or other alternations at the paradigmatic axis etc., as opposed to the more flexible LVCs.At the next level, the semantic representation of VMWEs makes use of the SIGNIFIED branch of our ontology, and each VMWE  is mapped onto a concept. In our model lexicon, concepts are treated as instances under hierarchically organized (sub-)classes; these sub-classes are themselves subsumed under a set of top-level classes (or top ontology) and roughly correspond to the notion of semantic or lexical fields Lyons, , p. 268.via lexical semantic relations: synonymy, near-synonymy, antonymy; similarly, concepts are also linked together via semantic relations as appropriate: hypernymy-hyponymy or is_a relation, meronymy, etc. Apart from the standard lexical semantic relations, other relations are also included: entailment (entails), causation (causes), temporal order (happens_before), etc. Moreover, relations that link together words and/or concepts that belong to different grammatical categories have been defined in the resource. For example, relations of the type is_the_agent_of, feels_emotion, is_cogniser, etc. link together concepts instantiated by verbs denoting an activity, an emotion or a cognitive state and concepts instantiated by nouns denoting the actor, the experiences of the cognitive agent, etc. Similarly, relations that link together adjectives with adverbs have been used. In total, more than 100 relations have been employed so far; some of them are generic in that they are relative to more than one semantic field , whereas others are domain-specific. Examples of the latter category include – but are not limited to – the following: Is_made_of, Is_located_in, Works_in, Is_workplace_of, Has_Habitat, Is_the_Inhabitant_of, Causes, Is_the_result_of, Has_color, Is_the_color_of, Is_payment_to, Is_payment_for, Wears_garment, etc. Contrary to resources like SIMPLE  in the lexicon.Lexical entries (words) are then linked together emotion event, relative to the emotion love. It is mapped, therefore, onto the concept prototypically defined for the single-word entry “ερωτ ε ύoμαι” .However, mapping words to concepts already defined in the lexicon is not an easy task. This is especially true for VMWEs. Moreover, in many cases, concepts already defined for single-word entries in the lexicon are perceived of as more general or neutral and only roughly correspond to the meaning load that VMWEs bear. For example, the VMWE δαγκώνω τη λαμαρíνα in (3) denotes an (3) δαγκώνω τη λαμαρíναlit. bite-1SG the panel−SG.ACC has_troponym links a concept that bears a more “grounded” or neutral sense with another one that signifies a shift in terms of quantity, intensity, quality, etc. For example, the verbs γ νωρíζω (=to know) and ξ έρω (=to know) are both lexicalizations of the concept [to know]; on the contrary, the VMWE παíζω στα δ άχτυλα (= γνωρíζω πoλ ύ καλ ά) is mapped onto the concept [to know well]. The two concepts are then linked via the troponymy relation:Note, however, that the two lexical instances are not absolute synonyms in the sense that there are subtle differences in terms of the intensity of the emotion experienced. As a matter of fact, VMWEs are rarely exact synonyms of a single verbal predicate. Within this context, the major challenge faced was to account for these fuzzy cases and find out ways for capturing the semantic distance. To overcome this issue and represent differences in meaning, near synonymous entries are also linked using relations, both generic and specific for each semantic class. More precisely, the generic relation has_troponym(4) has_troponym relation does not reflect this difference. To remedy this shortcoming, a list of attributes (or semantic features) with either binary or scalar values have also been defined for better representing the underlying meaning. In most cases, these attributes are specific to semantic fields. For example, lexical units that belong to the semantic field emotion are assigned values for the following attributes: (a) emotion polarity, (b) emotion intensity and (c) aspect of the emotion event. In effect, these features better account for capturing the semantic distinction between near synonyms, as for example the single word verbal predicate ϕoβ άμαι (=to be scared), and the VMWE in (5).Troponymy, however, is not a semantically homogenous relation  μoυ κóπηκαν τα ήπαταlit.me.01SG.GEN were-cut.03 the livers.PL.NOM fear emotion event that is more intense than the emotion conveyed by the single word; thus, the two predicates can hardly be encoded as being synonyms in the lexicon. Their semantic distance is captured by encoding them as related via the has_troponym relation, and the semantic distinction is highlighted by assigning the attribute high to the feature Intensity. This brings in mind the mechanism of Lexical Functions proposed by Mel'čuk , comprises the lexicalised elements παíρνω.v (=to take) and χαμπ άρι.n (=notice). Since the semantic load of the expression is on the noun, the expression is classified as LVC. The underlying syntactic configuration of the expression is that of a verb head that is light, and its complement (direct object). This configuration is compatible with the argument structure of the verb παíρνω.v . However, the whole expression as a lexical unit assumes the meaning of a cognitive event, and as such, it is conceived of as a predicate with two arguments: the first assumes the role of the CCOGNISER π ήρε χαμπ άρι [την αλλαγ ή]THEME(6) [O Γιάννηϛ]lit. The-NOM.SG John−NOM.SG took−3SG notice−ACC.SG the ACC.SG change ACC.SGJohn realized the changeThe semantic representation of the VMWE is expected to be similar to the representation of its single-word verbal counterpart καταλαβαíνω.v  as shown in (7):COGNISER κατ άλαβε [την αλλαγ ή]THEME(7) [O Γιάννηϛ]lit.The-NOM.SG John-NOM.SG noticed the-ACC.SG change-ACC.SGJohn realized the change.oργíζω.v (=make furious) in Greek is an Object Experiencer verb that is, a verb in which the Experiencer of the denoted emotion event is realized as a noun phrase in accusative in Object position. The cause of the event is realized as an argument, that functions as the Subject of the verb. On the contrary, in the case of the idiomatic expression (VID) ανεβ άζω τo αíμα στo κεϕ άλι (=make furious), the Experiencer is realized as a nominal complement (in genitive case), whereas the cause of the emotion is realized in Subject position:However, this is not always the case, and the argument structure of complex predicates is not realized in a uniform way. This is particularly true about VIDs. For example, the verb εξCAUSE μoυEXPERIENCER αν έβασε τo αíμα στo κεϕάλι(8) [O Γιάννηϛ]lit.The.nom John.nom me.gen raised3−sg the.acc blood.acc to-the headJohn made me furious.oύχα in (9) and βγαíνω απó τα ρoύχα μoυ in (10) correspond to the transitive and unaccusative usage of the verb θυμώνω.v (=to make angry) depicted in (11) and (12) respectively.There is no doubt that SRL is of major importance to computational systems since it provides a shallow meaning representation that is prerequisite of inferences that are not possible from the pure surface form or even from the parse tree. This is especially true for VMWEs  [O Γιάννηϛ]lit. The.nom John.nom took-out3−sg the.acc Maria.acc from the clothes hersJohn made Maria very angry)EXPERIENCER θ ύμωσε(10) [H Mαρíα]lit. The.nom Maria.nom got-angry3−sgNotice that θυμώνω.v (=to make angry) is an Object-Experiencer predicate that enters the choative-inchoative alternation as shown below:CAUSE/AGENT θ ύμωσε [τη Mαρíα]EXPERIENCER(11) [O Γιάννηϛ]lit. The.nom John.nom made-angry3−sg the.acc Maria.accJohn made Maria angry.EXPERIENCER βγ ήκε απó τα ρoύχα της(12) [H Mαρíα]lit. The.nom Maria.nom went-out3−sg from the.acc clothes.acc hers.possMaria got very angry.In this regard, our model seeks to address these issues by assigning semantic roles to the arguments of the VMWEs. The encoding of semantic roles was based on empirical data retrieved from annotation.Annotation at the level of semantics has been applied manually on top of an existing Greek (EL) corpus that has already been annotated for VMWEs. More precisely, we used the latest version (edition 1.2) of the Greek (EL) section of the PARSEME corpus  have been encoded under the SIGNIFIED branch of our linguistic ontology and have been mapped onto concepts. However, the annotation has been performed only on a subset, that is, on c. 800 instances of VMWEs, that is, the ones that were also found in the PARSEME corpus. Of these, 379 instances were identified as VIDs, 7 as VPCs, and 425 as LVCs: in total, 811 VMWEs. Circa 10% of the VMMEs (80 VMWEs) were annotated by a second student annotator in view of calculating the inter-annotator agreement (IAA) between the two. Prior to the annotation proper, extensive discussions took place. A pilot annotation of c. 20 VMWEs identified problematic cases and discrepancies. After reaching a consensus in annotation, the second annotator worked alone annotating c. 80 VMWEs in 120 sentences. IAA was then calculated (Cohen-κ) with respect to the number of arguments identified in each sentence, and the labels assigned to them, reaching an agreement of 0.80 and 0.75 respectively. In fact, VMWEs denoting As expected, SRL on VIDs was the most challenging. In fact, depending on the meaning SRL is occasionally straightforward:EXPERIENCER έχει ϕάει χυλóπιτα(13) [o Πέρεθ]lit. The−NOM.SG Perez-NOM−SG has−3SG eaten chilopita-ACC.SGPerez has been disappointed.oíγω τoυς ασκoύς τoυ Aιóλoυ (=to open Aeolus bag) is semantically equivalent to the phrase “create problems”. However, only the AGENT is realized in the sentence:Problematic cases are related to a shift in meaning and the incorporation of one or more arguments into the VMWE, diathesis alternations, or difficulties in word sense identification and/or sense mapping. For example, the VID ανAGENT άνoιξε τoυς ασκoύς τoυ Aιóλoυ στη [Mέση Aνατoλ ή]LOC(14) [O Tραμπ]lit. The-NM.SG Trump-NM.SG opened-3.SG the-ACC.SG bag-ACC.SG of-the-GEN.SG Aeolos-GEN.SG in the Mid-EastTrump created problems in the Mid-Eastoλ ύω πυρ ά (=unleash fire) to a single-word verb predicate proved to be difficult; as a result, disambiguation of their arguments proved to be problematic:Similarly, mapping the sense of VIDs like εξαπAGENT εξαπoλ ύει πυρά [κατ ά της κυβέρνησης]THEME/GOAL [για τoυς χειρισμoύς]CAUSE της(15) [H αντιπoλíτευση]lit. The opposition−NM.SG unleash-3.SG fire-ACC.SG against the-GEN.SG government-GEN.SG for the−ACC.PL handlings-ACC.PL it's to-the issueThe opposition accuses/attacks the government for handling the issue.This process revealed pairs of VIDs usually with shared lexicalised elements that differ only in their fixed verb heads; these are conceived of as diathesis alternations and are encoded accordingly:AGENT βγ άζει στo σϕυρí [τo ιστoρικó ξενoδoχεío]THEME(16) [H τράπεζα]lit. The bank−NOM.SG takes-3.SG to-the-ACC.SG hammer-ACC.SG the-ACC.SG hotel-ACC.SGThe bank auctions the historic hotelTHEME θα βγoυν στo σϕυρí(17) [χιλιάδες σπíτια]lit. thousands houses−NOM.PL will go-out-3.PL to-the-ACC.SG hammer-ACC.SGthousands of houses will be sold at auctionWe have presented a model for representing the semantics of VMWEs by taking into account their inherent idiosyncrasies: lexical, syntactic and semantic. The model entails a holistic approach to VMWE representation and touches upon the lexicon-corpus interface beyond providing lexical semantic relations. In contrast to dictionary models that try to model the internal structure of the MWE, our approach models argument structure taking the whole MWE as a semantic predicate. We seek to provide a shallow semantic representation for VMWEs that is similar to the semantic representation of single-word verb predicates. The model assumes the form of a linguistic ontology and has already been used to encode Greek VMWEs. The encoding of semantic properties is based on empirical data drawn from a corpus annotated at the level of semantic role labeling. Future work is underway toward enriching the lexicon with more instances of VMWEs also taking into account MWEs that belong to other grammatical categories. Moreover, inter-linking entries with other lexical resources, as for example, WordNet synsets, would be the next step. Additionally, SRL on the PARSEME corpus is still ongoing with a view to training a tool for the automatic SLR that takes VMWEs into account. Moreover, the quality of the annotation will be further ensured by obtaining more annotations to calculate inter-annotator agreement.https://www.clarin.si/repository/xmlui/handle/11356/1555.Publicly available datasets were analyzed in this study. This data can be found here: The author confirms being the sole contributor of this work and has approved it for publication.


Arthrobotrys and Drehslerella based on morphological and multigene  phylogenetic analyses. A. hengjiangensis sp. nov. and A. weixiensis sp. nov. are characterized by producing adhesive networks to catch nematodes. Dr. pengdangensis sp. nov., Dr. tianchiensis sp. nov., and Dr. yunlongensis sp. nov. are characterized by producing constricting rings. Morphological descriptions, illustrations, taxonomic notes, and phylogenetic analysis are provided for all new taxa; a key for Drechslerella species is listed; and some deficiencies in the taxonomy and evolution study of nematode-trapping fungi are also discussed herein.Nematode-trapping fungi are widely studied due to their unique morphological structure, survival strategy, and potential value in the biological control of harmful nematodes. During the identification of carnivorous fungi preserved in our laboratory, five novel nematode-trapping fungi were established and placed in the genera  Arthrobotrys superba Corda) because of their unique survival strategy, excellent application potential in nematode control, and significance of maintaining the balance of nematode populations in the ecosystem , 2-septate conidia with tapering base, and branched conidiophores . In addition, the conidiophore of A. globospora bears only a single conidium, while the conidiophore of A. weixiensis bears 1–3 conidia [Notes: Phylogenetically, Drechslerella pengdangensis F. Zhang & X.Y. Yang sp. nov.  long, 2.5–5 µm (n = 50) wide at the base, gradually tapering upwards to the apex 2.5–4 µm (n = 50) wide, erect, septate, unbranched, and bearing a single conidium at the knob-like apex. Conidia: 30–45 × 17–27 µm (n = 50), ellipsoidal to subfusiform, rounded at the apex, tapering towards narrow with truncate at the base, 1–2-septate (mostly 2-septate), hyaline, with the largest cell located at the middle or apex of the conidia, where the base cell is tiny. Chlamydospore: not observed. Nematodes were captured with constricting rings; in the non-constricted state, the outer diameter is 19–28.5 µm (n = 50), and the inner diameter is 13–22.5 µm (n = 50).Additional specimen examined: CHINA, Yunnan Province, Nujiang City, Pengdang County, N 27°56′16.88″, E 98°39′8.71″, from terrestrial soil, 4 May 2018, F. Zhang. Living culture DL53.Drechslerella pengdangensis forms a sister lineage with another new species (Drechslerella tianchiensis) reported in this study, with 89% MLBS support. There are 15% (128/853 bp) differences in ITS sequence between them. Morphologically, Dr. pengdangensis can be easily distinguished from Dr. tianchiensis in the shape of the conidia and single conidiophore. Dr. pengdangensis is similar to Dr. doedycoides in their ellipsoidal to sub-fusiform conidia and simple conidiophore with knob-like apex [Dr. doedycoides produces 3-septate conidia, while Dr. pengdangensis never. Moreover, the base cell of conidia produced by Dr. pengdangensis is significantly smaller than those of Dr. Doedycoides [Notes: Phylogenetically, are 15% 18/853 bp are 15% 18/853 bp Drechslerella tianchiensis F. Zhang & X.Y. Yang sp. nov.  long, 2.5–5 µm (n = 50) wide at the base, gradually tapering upwards to the apex with 1.5–3 µm (n = 50) wide, erect, septate, hyaline, unbranched or producing 1–2 short branches near the apex, each branch bearing a single conidium. Microconidiophores 137.5–245.5 µm (n = 50) long, 2–4 µm (n = 50) wide at the base, gradually tapering upwards to the apex with 1.5–3 µm (n = 50) wide, erect, septate, hyaline, unbranched, producing 3–12 short denticles near the apex, each denticles bearing a single conidium. Conidia two types: Maroconidia 30–41 × 14.5–24 µm (n = 50), ellipsoidal, rounded at the apex, tapering towards narrow with truncate base, 1–2-septate (mostly 2-septate), hyaline, with a largest cell located at the middle or apex of the conidia. Miroconidia 16–26.5 × 4.5–11.5 µm (n = 50), clavate or cylindrical, rounded at the apex, tapering towards narrow with truncate base, 0–1-septate (mostly 1-septate), hyaline. Chlamydospore not observed. Capturing nematodes with constricting rings, in the non-constricted state, the outer diameter is 20.5–27.5 µm (n = 50), the inner diameter is 14.5–22 µm (n = 50).Additional specimen examined: CHINA, Yunnan Province, Dali City, Yunlong County, Tianchi Nature Reserve, N 25°51′22.50″, E 99°13′38.43″, from burned forest soil, 28 May 2018, F. Zhang. Living culture XJ353.Drechslerella tianchiensis formed a sister lineage with Dr. pengdangensis (89% MLBS). Morphologically, Dr. tianchiensis is similar to Dr. hainanensis and the asexual morph of Orbilia pseudopolybrocha in their shape of macroconidia and microconidia. The difference between Dr. tianchiensis and Orbilia pseudopolybrocha is that the macro-conidiophore of the latter is simple and bears a single conidium, while some macro-conidiophore of Dr. tianchiensis produces 1–2 short branches near the apex and bears 1–2 conidia. The conidia of Dr. tianchiensis are significantly larger than those of O. pseudopolybrocha  × 14.5–24 (18.7) µm versus O. pseudopolybrocha, 26–30 × 16–22.2 µm) [Dr. tianchiensis can be easily distinguished from Dr. hainanensis by its 1–2-branch macro-conidiophore and wider microconidia  × 4.5–11.5 (6) µm versus Dr. hainanensis, 18.2–22.8 × 4.2–5.3 µm) [Notes: Phylogenetically, 22.2 µm) . Dr. tia–5.3 µm) .Drechslerella yunlongensis F. Zhang & X.Y. Yang sp. nov.  long, 2.5–5 µm (n = 50) wide at the base, gradually tapering upwards to the apex 1.5–3µm (n = 50) wide, erect, septate, unbranched, hyaline, bearing a single conidium at the apex. Conidia: 36–54 × 17–27 µm (n = 50), drop-shaped or fusiform, rounded at the apex, tapering towards narrow with truncate base, 1–4-septate (mostly 4-septate), hyaline, with the largest cell located at the apex or middle of the conidia. Chlamydospore: 5–14 × 5.5–10 µm (n = 50), cylindrical, globose or ellipsoidal, hyaline, and in chains when present. Nematodes were captured with constricting rings; in the non-constricted state, the outer diameter was 19.5–27 µm (n = 50), the inner diameter was 15–21.5 µm (n = 50).Additional specimen examined: CHINA, Yunnan Province, Dali City, Yunlong County, N 25°52′27.91″, E 99°22′19″, from terrestrial soil, 3 June 2018, F. Zhang. Living culture YL402.Drechslerella yunlongensis with the other four fusiform conidia-producing species . Dr. yunlongensis was 9.8% (55/559 bp), 8.1% (40/496 bp), 9.1% (51/559 bp), and 7.9% (47/596 bp) different from Dr. aphrobrocha, Dr. bembicodes, Dr. coelobrocha, and Dr. xiaguanensis in ITS, respectively. Morphologically, Dr. yunlongensis is also similar to these four species. However, the conidia of Dr. yunlongensis are bigger than those of Dr. bembicodes and Dr. xiaguanensis  × 17–27 (23.6) µm versus Dr. bembicodes, 36–43.2 (40) × 16.8–21.6 (20.5) µm versus Dr. xiaguanensis, 33–52 (42.5) × 9.5–28 (15.5) µm); moreover, Dr. bembicodes produces obovoid, 1-septate microconidia, while Dr. yunlongensis does not; the conidia of Dr. xiaguanensis are mostly 3-septate, while the conidia produced by Dr. yunlongensis are mostly 4-septate [Dr. yunlongensis and Dr. aphrobrocha is that Dr. aphrobrocha produces mostly 3-septate conidia, while Dr. yunlongensis produces mostly 4-septate conidia; the conidia of Dr. yunlongensis are smaller than that of Dr. aphrobrocha due to its smaller apical cell  × 17–27 (23.6) µm versus Dr. aphrobrocha, 40–57.5 (51) × 15.5–35 (24.6) µm) [Dr. yunlongensis can be distinguished from Dr. coelobrocha by its wider conidia  µm versus Dr. coelobrocha, 16.8–21.6 (19.8) µm), and shorter base and apical cells [Dr. yunlongensis produces cylindrical or ellipsoidal chlamydospores, while none of the four closely related species produces chlamydospores [Notes: The phylogenetic analysis clustered s, 33–52 .5 × 9.5–s, 33–52 .5 × 9.5–Arthrobotrys in this study because it has been updated in Zhang et al. [We do not update the species key of g et al. , and no 1. Conidia without super-cell…………………………………...………………………………21. Conidia with a super-cell………………………………………………………….…….……52. Conidia 1–3-septate…………………………………………………………………….……...32. Conidia 0–1-septate…………………………………………………………………….…...…4O. tonghaiensis3. 5–10 conidia cluster arrangement on a cluster of short denticles (5–10) at the apex of conidiophore, conidia 28.5–39.0 × 6.0–8.5 µm, microconidia cylindrical……………………………………………………………………………..Dr. brochapaga3. Conidiophore produce 3–8 short denticles by repeated elongation, conidia are cylindrical, botuliform, 20–45 (30) × 5–12.5 (6) µm, and do not produce microconidia…………………………………………………………………………….....Dr. dactyleoids4. Conidia digitiform are mostly curved, 1-septate, 35–51.5 (42.1) × 6.5–8 (7.5) µm, with 3–13 conidia capitate arrangement at the apex of conidiophore…………..Dr. yunnanensis4. Conidia are elongated and ellipsoidal, straight, 0-1-septate, 7.8–12.9 × 3.3–4.2 µm………………………………………………………………..………….5. Conidia are sub-fusiform to fusiform……………………………………………………..…65. Conidia are ellipsoidal, elongate ellipsoidal, subellipsoidal, or obovate……………….12Dr. acrochaetum6. Conidia are 1–2-septate…………………………………………………….....6. Conidia are 1–5-septate………………………………………………………………….…....77. Conidia are 1–4-septate, mostly 3-septate……………………………………………….…..87. Conidia are 1–5-septate, mostly 4-septate…………………………………………………10Dr. xiaguanensis8. Conidia are smaller in size, 33–52 (42.5) × 9.5–28 (15.5) µm, swollen at both ends of cells………………….…………………………………………………..…..8. Conidia are bigger, sometimes more than 52 µm in length and usually greater than 15 µm in width; the cells at both ends are not enlarged……………………………….…..9Dr. aphrobrocha9. Conidia are wider, 40–57.5 (51) × 15.5–35 (24.6) µm, 2–4-septate, and conidiophore occasionally bear two conidia……………………………………....….…….Dr. inquisitor9. Conidia are narrower, 42.5–62.5 (47) × 15–22.5 (16.9) µm, 1–4-septate, sub-fusiform, and conidiophore bear a single conidium……………………………………..…Dr. bembicodes10. Conidia are 3–4-septate, smaller in size, 36–43.2 (40) × 16.8–21.6 (20.5) µm, producing obovoid, 1-septate microconidia………………………………………..….10. Conidia are bigger, do not produce microconidia………………………………………11Dr. yunlongensis11. Conidia are 1–4-septate, 36–54 (47) × 17–27 (23.6) µm, producing cylindrical, globose, or ellipsoidal chlamydospore…………………………………………....Dr. coelobrocha11. Conidia are 2–5-septate, 45.6–55.2 (49.5) × 16.8–21.6 (19.8) µm, both ends cells are slender, and do not produce chlamydospore……………………………..12. Conidia are obovate and 1-septate…………………………………………………….….1312. Conidia are ellipsoidal, elongate ellipsoidal, and 0–3-septate………………………....1413. Conidia are obovate, 29–43 (35) × 15–19 (16.8) µm, base cells are pyramidal, with 3–8 conidia capitate arrangement at the apex of conidiophore….....................Dr. anchoniaDr. polybrocha13. Conidia are obovate or sub-ellipsoidal, 35 × 24 µm, single conidium bear at the apex of conidiophore……………………………………………………………..…...14. Conidiophore is branched or bears more than 1 conidium………………………….…1514. Conidiophore is unbranched, bears a single conidium……………………………..…..16Dr. tianchiensis15. Conidiophore is unbranched or produces 1–2 short branches near the apex, each branch bearing a single conidium, with conidia 30–41 (36.2) × 14.5–24 (18.7) µm, 1–2-septate………………………………………………………………...……Dr. effusa15. Conidiophore is unbranched, bearing a loose head consisting of 2–12 conidia, with conidia 32.5–45 (38.9) × 17.5–25 (21.4) µm, 1–2-septate……………………..….16. Conidiophore produces a swollen, knob-like apex………………………………..……1716. Conidiophore produces a truncated, non-swelling apex……………………………....2017. Produces cylindrical, clavate, or bottle-shaped, 1-septate microconidia……………...1817. Does not produce microconidia…………………………………………………………..19Dr. heterospora18. Macroconidia are bigger, 17.5–45 (34) × 17.5–25 (20.4) µm, 1–2-septate, mostly 1-septate, and microconidia are bigger, 23–40 (31.3)× 5–8 (6.8) µm……………………………………………………………………….....…O. pseudopolybrocha18. Macroconidia are smaller, 26–30 × 16–22.2 µm, 0–2-septate, mostly 2-septate, and microconidia smaller, 14.7–23 × 3.3–6 µm…………………….………...Dr. pengdangensis19. Conidia are bigger, 30–45 (38) × 17–27 (22.4) µm, 1–2-septate, and basal cells are tiny…………………………………………………………………...…...…Dr. doedycoides19. Conidia are 25–52.5 (33.2) × 12.5–29 (17.3) µm, and 1–3-septate………….Dr. stenobrocha20. Conidia are elongated and ellipsoidal, 1–3-septate, mostly 3-septate, 34–56.5 × 12.5–16.5 µm, and do not produce microconidia………………………………20. Conidia are ellipsoidal, 0–2-septate, and produce clavate or bottle-shaped microconidia……………………………………………………………………………………...21Dr. daliensis21. Macroconidia are thinner, 20–49.5 (38.5) × 8.5–15 (12) µm, 1–2-septate, mostly 2-septate, and microconidia wider, 6.5–22 (15.5) × 3.5–7 (5) µm……….……Dr. hainanensis21. Macroconidia are 32.5–43 × 17–25 µm, 0–2-septate, mostly 1 or 2-septate, and microconidia are 18.2–22.8 × 4.2–5.3 µm…………………………..…..…………Drechslerella produces constricting rings to capture nematodes with mechanical force, and the genera Arthrobotrys and Dactylellina catch nematodes with adhesive traps) [Dactylellina species into a stable cluster, possibly due to insufficient DNA data. We believe that as more DNA data are used, we will find more morphological or physiological features that match phylogenetic studies.Both the phylogenetic analysis in this study and previous studies divided NTF into two main clades based on the mechanisms by which they catch nematodes (the genus e traps) ,12,13,14Drechslerella in this study showed that some species with similar conidia morphology cluster stably into one branch, such as species in clade I producing fusiform conidia and species in clade II producing ellipsoidal conidia  is one crucial node to understanding the history of fungal evolution because of its unique morphological characteristics and survival strategy ,3,4,5. C conidia . Moreoveia shape . In addiDrechslerella species is that species are first roughly classified by those features that can be used to identify species and are easily distinguishable, such as whether the conidia produce a super-cell or not, the shape of the conidia , whether the conidiophore is branched or not, and the number of conidia on the conidiophore. Then, species are further classified by those features that can be used for species identification but require further measurement and observation, such as the detailed feature of macroconidia (number and position of the septum and the size of the macroconidia). Finally, morphologically similar species are distinguished by those characteristics that are uncertain whether they can be used for species identification but are differences between different species, such as the presence and features of microconidia, the detailed feature of the apex of the conidiophore, and the features of the chlamydospore. Even identifying Drechslerella species requires those morphological features that are not known to be valid, so how difficult would it be to identify the more complex Arthrobotrys and Dactylellina species based on these features alone? Therefore, follow-up research needs to systematically study all potential morphological characteristics to find more reliable characteristics for species identification.The compilation logic of the key of Orbilia [Orbilia sp.). This results in these different asexual species sharing the same sexual species name [https://www.ncbi.nlm.nih.gov/nuccore/?term=Orbilia+auricolor (accessed on 3 April 2023)). For this reason, we suggest that when reporting a pair of sexual and asexual species, it is necessary to discuss the difference between the sexual generation and known sexual species and, more importantly, consider the distinction between the asexual generation and known asexual generation. The naming of this pair of sexual and asexual species should be carefully evaluated separately, giving sexual and asexual generations different species names if necessary.Most of the sexual generations of Orbiliomycetes nematode-trapping fungi are members of Orbilia . HoweverOrbilia . AdditioOrbilia ,46, asexies name , which f


Fermatean fuzzy sets (FFSs) have piqued the interest of researchers in a wide range of domains. The striking framework of the FFS is keen to provide the larger preference domain for the modeling of ambiguous information deploying the degrees of membership and nonmembership. Furthermore, FFSs prevail over the theories of intuitionistic fuzzy sets and Pythagorean fuzzy sets owing to their broader space, adjustable parameter, flexible structure, and influential design. The information measures, being a significant part of the literature, are crucial and beneficial tools that are widely applied in decision-making, data mining, medical diagnosis, and pattern recognition. This paper aims to expand the literature on FFSs by proposing many innovative Fermatean fuzzy sets-based information measures, namely, distance measure, similarity measure, entropy measure, and inclusion measure. We investigate the relationship between distance, similarity, entropy, and inclusion measures for FFSs. Another achievement of this research is to establish a systematic transformation of information measures  for the FFSs. To accomplish this aim, new formulae for information measures of FFSs have been presented. To demonstrate the validity of the measures, we employ them in pattern recognition, building materials, and medical diagnosis. Additionally, a comparison between traditional and novel similarity measures is described in terms of counter-intuitive cases. The findings demonstrate that the innovative information measures do not include any absurd cases. The idea of the fuzzy set (FS) was developed by Zadeh  in 1965,Atanassov  establisIn many situations, it is conceivable that the sum of the MG and NMG will be greater than 1. To overcome these challenges, Yager  introducIn the field of PyFS, there are various approaches for solving real-life multiattribute decision-making (MADM) situations. A number of researchers have also suggested real-world applications in a Pythagorean fuzzy environment. However, if orthopair FSs as 〈0.9,0.5〉, where 0.9 is the MG of specific criteria of a parameter and 0.5 is the NMG, it does not fulfill the IFS and PFS requirements. However, the cubic sum of the MG and NMG is equal to or less than one. In this context, Senapati and Yager  known to be MG.(see ). Let 𝒵=𝒵={ϰ1, ϰ2, ϰ3,…, ϰn} be a finite set. An IFS Ϝ over ϰ is follows:i ∈ 𝒵 the functions ℑϜ : 𝒵⟶, 1 and ðϜ : ϰ⟶ denotes the MG and NMG, respectively, which must satisfy the property 0 ≤ ℑϜ(ϰi)+ðϜ(ϰi) ≤ 1.(see ). Let 𝒵=1, ϰ2, ϰ3,…, ϰn} be a finite set. The HFS H on ϰ is defined as follows:h(ϰi) is a set of values contain in , which shows the MG of ϰi ∈ H. The element of h(ϰi) is known as the HF element.(see ). Let ϰ=𝒵={ϰ1, ϰ2, ϰ3,…, ϰn} be a finite set, a Fermatean fuzzy sets (FFSs) ℱ over 𝒵 is defined as follows:i ∈ ϰ the functions ℑℱ : ϰ⟶ and ðℱ : ϰ⟶ denote the MG and NMG, respectively, which must satisfy (ðℱ(ϰi))3+(ℑℱ(ϰi))3 ≤ 1. The degree of indeterminacy is given as follows:(see ). Let 𝒵=𝒴 and 𝒫 be two FFSs, then the operations can be defined as follows:Addition: Multiplication: Scalar multiplication: Exponent: (see ). If 𝒴 a𝒴 and 𝒫 be two FFSs, then the operations can be defined as follows:𝒴c={〈ð𝒴(ϰi), ℑ𝒴(ϰi)〉|ϰi ∈ 𝒵};Complement: 𝒴=𝒫 iff for all 〈ϰi ∈ 𝒵, ℑ𝒴(ϰi)=ℑ𝒫(ϰi)and ð𝒴(ϰi)=ð𝒫(ϰi)〉;Equality: 𝒴∩𝒫=〈min (ℑ𝒴(ϰi), ℑ𝒫(ϰi)), max (ð𝒴(ϰi), ð𝒫(ϰi))|ϰi ∈ 𝒵〉;Intersection: 𝒴 ∪ 𝒫=〈max (ℑ𝒴(ϰi), ℑ𝒫(ϰi)), min (ð𝒴(ϰi), ð𝒫(ϰi))|ϰi ∈ 𝒵〉.Union: (see ). If 𝒴 a𝒴, 𝒫, and ℳ be three FFSs, then the following characteristics are held:𝒴 ∪ 𝒫=𝒫 ∪ 𝒴;𝒴∩𝒫=𝒫∩𝒴;𝒴 ∪ (𝒫 ∪ ℳ)=(𝒴 ∪ 𝒫) ∪ ℳ;𝒴∩(𝒫∩ℳ)=(𝒴∩𝒫)∩ℳ;α ∪ (𝒴 ∪ 𝒫)=α𝒴 ∪ α𝒫;𝒴 ∪ 𝒫)α=𝒴α ∪ 𝒫α.((see ). If 𝒴, This section explains the axiomatic framework of FFSs information measures , as well as their related formulations. Simultaneously, their transformation relationships are thoroughly examined.This section introduces the idea of a distance measures for FFSs. A number that is assigned to a pair of points in a space which indicates how far those points are from one another. A distance measure is called a metric if it is always positive and also it is always symmetric.𝒴, 𝒫, and ℳ be three FFSs on 𝒵. A distance measure 𝒟 is a mapping 𝒟: FFS (𝒵)× FFS (𝒵)⟶, carrying the following features:𝒟 ≤ 1;0 ≤ 𝒟=𝒟;𝒟=0 iff 𝒴=𝒫;𝒟=1 iff 𝒴 is a crisp set;𝒴⊆𝒫⊆ℳ, then 𝒟 ≤ 𝒟 and 𝒟 ≤ 𝒟.If Let 𝒴 and 𝒫 be two FFS s, then 𝒟i is a distance measure.𝒟=(1/2|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|+|ð𝒴3(ϰi) − ð𝒫3(ϰi)|+|π𝒴3(ϰi) − π𝒫3(ϰi)|);𝒟2=(1/2|𝒵|)∑i∈𝒵ϰ(|(ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi))+(ð𝒴3(ϰi) − ð𝒫3(ϰi))|);𝒟3=(1/4|X|)(∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|+|ð𝒴3(ϰi) − ð𝒫3(ϰi)|+|π𝒴3(ϰi) − π𝒫3(ϰi)|)+∑i∈𝒵ϰ(|(ℑ𝒴3(ϰi) − ð𝒴3(ϰi)) − (ℑ𝒫3(ϰi) − ð𝒫3(ϰi))|))𝒟4=(1/|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|(ð𝒴3(ϰi) − ð𝒫3(ϰi))|);𝒟5=(2/|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|(ð𝒴3(ϰi) − ð𝒫3(ϰi))|/1+|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|(ð𝒴3(ϰi) − ð𝒫3(ϰi))|);𝒟6=(2∑i∈𝒵ϰ|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|(ð𝒴3(ϰi) − ð𝒫3(ϰi))|/∑i∈𝒵ϰ(1+|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|ð𝒴3(ϰi) − ð𝒫3(ϰi)|));𝒟7=1 − α(∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))/∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))) − β(∑i∈𝒵ϰ(ð𝒴3(ϰi)∧ð𝒫3(ϰi))/∑i∈𝒵ϰ(ð𝒴3(ϰi)∨ð𝒫3(ϰi))), α+β=1, α, β ∈ ;𝒟8=1 − (α/|𝒵|)∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))/(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi)) − (β/|𝒵|)∑i∈𝒵ϰ(ð𝒴3(ϰi)∧ð𝒫3(ϰi))/(ð𝒴3(ϰi)∨ð𝒫3(ϰi)), α+β=1, α, β ∈ ;𝒟9=1 − (1/|𝒵|)∑i∈𝒵ϰ((ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))+(ð𝒴3(ϰi)∧ð𝒫3(ϰi))/(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))+(ð𝒴3(ϰi)∨ð𝒫3(ϰi)));𝒟10=1 − (∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))+(ð𝒴3(ϰi)∧ð𝒫3(ϰi))/∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))+(ð𝒴3(ϰi)∨ð𝒫3(ϰi)));𝒟11=1 − (1/|𝒵|)∑i∈𝒵ϰ((ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))+(1 − ð𝒴3(ϰi))∧(1 − ð𝒫3(ϰi)))/((ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))+(1 − ð𝒴3(ϰi))∨(1 − ð𝒫3(ϰi)));𝒟12=1 − (∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))+((1 − ð𝒴3(ϰi))∧(1 − ð𝒫3(ϰi))))/(∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))+((1 − ð𝒴3(ϰi))∨(1 − ð𝒫3(ϰi))));Let This section introduces the idea of similarity measures for FFSs. Similarity functions take a pair of points and return a large similarity value for nearby points, and a small similarity value for distant points. One way to transform between a distance function and a similarity measure is to take the reciprocal.𝒴, 𝒫, and ℳ be three FFSs on 𝒵. A similarity measure 𝒮(𝒴 and 𝒫) is a mapping 𝒮 FFS (𝒵) × FFS (𝒵) ⟶, possessing the following properties:𝒮 (𝒴 and 𝒫) ≤ 1;0 ≤𝒮 (𝒴 and 𝒫) = 𝒮 (𝒫 and 𝒴);𝒮 (𝒴 and 𝒫) = 1 iff 𝒴 = 𝒫;𝒮 (𝒴 and 𝒴c) = 0 iff 𝒴 is a crisp set;𝒴⊆𝒫⊆ℳ, then 𝒮 (𝒴 and 𝒫) ≤𝒮 (𝒴 and ℳ) and 𝒮 (𝒫 and ℳ) ≤𝒮 (𝒴 and ℳ).If Let 𝒴 and 𝒫 be two FFSs, then 𝒮i is a distance measure.𝒮2=1 − (1/2|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi) − (ð𝒴3(ϰi) − ð𝒫3(ϰi))|);𝒮3=1 − (1/4|X|)(∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|+|ð𝒴3(ϰi) − ð𝒫3(ϰi)|+|π𝒴3(ϰi) − π𝒫3(ϰi)|)+∑i∈𝒵ϰ(|(ℑ𝒴3(ϰi) − ð𝒴3(ϰi))+(ℑ𝒫3(ϰi) − ð𝒫3(ϰi))|))𝒮4=1 − (1/|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|(ð𝒴3(ϰi) − ð𝒫3(ϰi))|);𝒮5=(1/|𝒵|)∑i∈𝒵ϰ(1 − |ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|ð𝒴3(ϰi) − ð𝒫3(ϰi)|)/(1+|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|ð𝒴3(ϰi) − ð𝒫3(ϰi)|);𝒮6=(∑i∈𝒵ϰ(1 − |ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|ð𝒴3(ϰi) − ð𝒫3(ϰi)|))/(∑i∈𝒵ϰ(1+|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|∨|ð𝒴3(ϰi) − ð𝒫3(ϰi)|));𝒮7=α(∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))/∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi)))+β(∑i∈𝒵ϰ(ð𝒴3(ϰi)∧ð𝒫3(ϰi))/∑i∈𝒵ϰ(ð𝒴3(ϰi)∨ð𝒫3(ϰi))), α+β=1, α, β ∈ ;𝒮8=(α/|𝒵|)∑i∈𝒵ϰ((ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))/(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi)))+(β/|𝒵|)∑i∈𝒵ϰ((ð𝒴3(ϰi)∧ð𝒫3(ϰi))/(ð𝒴3(ϰi)∨ð𝒫3(ϰi))), α+β=1, α, β ∈ ;𝒮9=(1/|𝒵|)∑i∈𝒵ϰ((ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))+(ð𝒴3(ϰi)∧ð𝒫3(ϰi)))/((ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))+(ð𝒴3(ϰi)∨ð𝒫3(ϰi)));𝒮10=(∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))+(ð𝒴3(ϰi)∧ð𝒫3(ϰi)))/(∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))+(ð𝒴3(ϰi)∨ð𝒫3(ϰi)));𝒮11=(1/|𝒵|)∑i∈𝒵ϰ((ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))+((1 − ð𝒴3(ϰi))∧(1 − ð𝒫3(ϰi))))/((ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))+((1 − ð𝒴3(ϰi))∨(1 − ð𝒫3(ϰi))));𝒮12=(∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi))+((1 − ð𝒴3(ϰi))∧(1 − ð𝒫3(ϰi))))/(∑i∈𝒵ϰ(ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))+((1 − ð𝒴3(ϰi))∨(1 − ð𝒫3(ϰi))));Let i=1,2,3,…, 13, if α=β=1/2, then we have𝒮i=𝒮i;𝒮i=𝒮i;𝒮i=𝒮i;𝒮i=𝒮i.For i=1,2,…, 6, we have𝒮i=𝒮i;𝒮i=𝒮i;For i=1,4,5,6 and for all ϰi ∈ 𝒵, (ℑ𝒴3(ϰi)+ℑ𝒫3(ϰi))=1, and ð𝒴3(ϰi)+ð𝒫3(ϰi)=1, we have𝒮i=𝒮i,ℑ𝒴3(ϰi) ≤ ℑ𝒫3(ϰi), and(ð𝒴3(ϰi) ≥ ð𝒫3(ϰi))𝒮i=𝒮i,ℑ𝒴3(ϰi) ≥ ℑ𝒫3(ϰi), and(ð𝒴3(ϰi) ≤ ð𝒫3(ϰi))For 𝒴 and 𝒫 two FFSs on 𝒵. An entropy measure E(𝒴) is a mapping E: FFS (𝒵)⟶, carrying the following features:(i)E(𝒴) ≤ 1;0 ≤ (ii)E(𝒴)=0 iff 𝒴 is a crisp set;(iii)E(𝒴)=1 iff ℑ𝒴3(ϰi)=ð𝒴t(ϰi);(iv)E(𝒴)=E(𝒴c);(v)E(𝒴) ≤ E(𝒫) if 𝒴 is less fuzzy than 𝒫, that isIf Let Let𝒴be two FFSs, thenEiis an entropy.𝒴, 𝒫, and ℳ be three FFSs on 𝒵. An inclusion measure ℐ is a mapping ℐ: FFS (𝒵)× FFS (𝒵)⟶, carrying the following features:ℐ ≤ 1;0 ≤ ℐ=1 iff 𝒴⊆𝒫;ℐ=0 iff 𝒴=Φ and 𝒫=∅;𝒴⊆𝒫⊆ℳ, then ℐ ≤ ℐ and ℐ ≤ ℐ.If Let 𝒴 and 𝒫 be two FFS s, then ℐi is an inclusion measure.ℐ1=1 − (1/2|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi)|+|ð𝒴3(ϰi) − ð𝒴3(ϰi)∨ð𝒫3(ϰi)|);ℐ2={1,𝒴=∅(∑i∈𝒵ϰ(1+ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi) − ð𝒴3(ϰi)∨ð𝒫3(ϰi)))/(∑i∈𝒵ϰ(1+ℑ𝒴3(ϰi) − ð𝒴3(ϰi))),𝒴 ≠ ∅;ℐ3={1,𝒴=𝒫=∅(∑i∈𝒵ϰ(1+ℑ𝒴3(ϰi) − ð𝒴3(ϰi)))/(∑i∈𝒵ϰ(1+ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi) − ð𝒴3(ϰi)∧ð𝒫3(ϰi))),others;ℐ4={1,𝒴=𝒫=∅(∑i∈𝒵ϰ(1+ℑ𝒴3(ϰi) − ð𝒴3(ϰi)))/(∑i∈𝒵ϰ(1+ð𝒴3(ϰi)∧ð𝒫3(ϰi) − ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi))),others;ℐ5={1,𝒴=𝒫=∅(1/|𝒵|)∑i∈𝒵ϰ(1+ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi) − ð𝒴3(ϰi)∨ð𝒫3(ϰi))/(1+ℑ𝒴3(ϰi) − ð𝒴3(ϰi)),othersℐ6={1,𝒴=𝒫=∅(1/|𝒵|)∑i∈𝒵ϰ(1+ℑ𝒴3(ϰi) − ð𝒴3(ϰi))/(1+ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi) − ð𝒴3(ϰi)∧ð𝒫3(ϰi)),othersℐ7={1,𝒴=𝒫=∅(1/|𝒵|)∑i∈𝒵ϰ(1+ℑ𝒴3(ϰi) − ð𝒴3(ϰi))/(1+ð𝒴3(ϰi)∧ð𝒫3(ϰi) − ℑ𝒴3(ϰi)∨ℑ𝒫3(ϰi)),othersLet In this section, we study the relations between inclusion, entropy, similarity measure, and distance measure of Fermatean fuzzy sets. First, according to the definitions of similarity measure and distance measure of Fermatean fuzzy sets, one should note that they are all used for estimating the degree of similarity between two Fermatean fuzzy sets. The main difference is as follows: for the similarity measure, a greater value means that the two Fermatean fuzzy sets are more similar than are a pair with a lower value. The situation for the distance measure is just the opposite, that is, the smaller the value is, the more similar these two Fermatean fuzzy sets are. So, we can obtain the following theorem.𝒟 be the Fermatean fuzzy distance measure for 𝒴, 𝒫∈ FFS s, then 𝒮=1 − 𝒟 is the similarity measure of FFS s𝒴 and 𝒫. The proof is straightforward.Suppose 𝒴, 𝒫 ∈ FFSs, and we order 𝒟1=(1/|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|+|ð𝒴3(ϰi) − ð𝒫3(ϰi)|+|π𝒴3(ϰi) − π𝒫3(ϰi)|), then we have𝒮=1 − (1/2|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|+|ð𝒴3(ϰi) − ð𝒫3(ϰi)|+|π𝒴3(ϰi) − π𝒫3(ϰi)|)=𝒮1.For 𝒮i=1 − 𝒟i.Also, 𝒟 and 𝒮 be the distance and similarity measures of FFSs, for 𝒴∈ FFSs, then E(𝒴)=1 − 𝒮=1 − 𝒮 is the entropy of FFSs.Let E1) It is straightforward.(E2) If 𝒴 is a crisp set, then 𝒴=∅ or 𝒴=Φ, we have 𝒮=0. Therefore, E(𝒴)=0.(E3) E(𝒴)=1⇔𝒮=𝒮⇔ℑ𝒴3(ϰi)=ℐ𝒴3(ϰi)=ð𝒴3(ϰi) for ϰi ∈ 𝒵.(E4) E(𝒴)=𝒮=𝒮=E(𝒴c).(E5) Since ℑ𝒴3(ϰi) ≤ ℑ𝒫3(ϰi) ≤ ð𝒫3(ϰi) ≤ ð𝒴3(ϰi) implies 𝒴⊆𝒫⊆𝒫c⊆𝒴c. Therefore, according the definition of similarity measure of FFS s, we have 𝒮 ≤ 𝒮 ≤ E, that is, E(𝒴) ≤ E(𝒫). Similarly, if ð𝒴3(ϰi) ≤ ð𝒫3(ϰi) ≤ ℑ𝒫3(ϰi) ≤ ℑ𝒴3(ϰi), then we have 𝒮 ≤ 𝒮 ≤ 𝒮, that is, E(𝒴) ≤ E(𝒫). This completes the proof.=𝒮1=1 − (1/2|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)|+|ð𝒴3(ϰi) − ð𝒫3(ϰi)|+|π𝒴3(ϰi) − π𝒫3(ϰi)|), then we have E(𝒴)=𝒮1=1 − (1/|𝒵|)∑i∈𝒵ϰ|ℑ𝒴3(ϰi) − ð𝒴3(ϰi)|=E4(𝒴) Also, E4(𝒴)=𝒮2=𝒮3=𝒮4=𝒮13 E5(𝒴)=𝒮7=𝒮10, E6(𝒴)=𝒮8=𝒮9, E7(𝒴)=𝒮12E8(𝒴)=𝒮11. For 𝒴 be an FFS, m(𝒴), n(𝒴)∈ FFSs, ∀ϰi ∈ 𝒵, m(𝒴)(ϰi)=(ℑm(𝒴)(ϰi), ðm(𝒴)(ϰi)), and n(𝒴)(ϰi)=(ℑn(𝒴)(ϰi), ðn(𝒴)(ϰi)) their membership and nonmembership functions are defined as follows:    Let 𝒟 be the distance measure and 𝒮 be the similarity measure of FFSs, for 𝒴 in FFSs, then E(𝒴)=𝒮(m(𝒴), n(𝒴))=1 − 𝒟=1 − 𝒮(m(𝒴), n(𝒴)) is the entropy of FFS 𝒴.Let (i)E1) It is straightforward.((ii)E2) If 𝒴 is a crisp set, then ∀ϰi ∈ 𝒵, we have ℑ𝒴(ϰi)=1, ð𝒴(ϰi)=0, or ℑ𝒴(ϰi)=0, ð𝒴(ϰi)=1. Therefore, we can achieve(m(𝒴)=Φ, n(𝒴)=∅, consequently, 𝒮(m(𝒴), n(𝒴))=0.It implies that (iii)E3) ((iv)E4) Using the definitions of m(𝒴) and n(𝒴), we have m(𝒴)=m(𝒴c), n(𝒴)=n(𝒴c), hence 𝒮(m(𝒴), n(𝒴))=𝒮(m(𝒴c), n(𝒴c)).((v)E5) Since ℑ𝒴(ϰi) ≤ ℑ𝒫(ϰi) ≤ ð𝒴(ϰi) ≤ ð𝒫(ϰi) implies 𝒴⊆𝒫⊆𝒫c⊆𝒴c. Therefore, we have |ℑ𝒴3(ϰi) − ð𝒴3(ϰi)| ≥ |(ℑ𝒫3(ϰi) − ð𝒫3(ϰi))|. It means that n(A) ≤n(𝒫) ≤ m(𝒫) ≤ m(𝒴), so we have 𝒮(m(𝒴), n(𝒴)) ≤ 𝒮(m(𝒫), n(𝒴)) ≤ 𝒮(m(𝒫), n(𝒫)) that is, E(𝒴) ≤ E(𝒫).(𝒴(ϰi) ≤ ð𝒫(ϰi) ≤ ℑ𝒫(ϰi) ≤ ℑ𝒴(ϰi), then we haveSimilarly, if ðThis completes the proof.𝒟 and 𝒮 be the distance measure and similarity measures of FFSs, respectively, for 𝒴, 𝒫∈ FFSs, then ℐ=𝒮=1 − 𝒟 is the inclusion measure of FFSs 𝒴 and 𝒫.Suppose I1) It is straightforward.(I2) If 𝒴⊆𝒫, then 𝒮=𝒮=1=𝒮.(I3) ℐ=0⇔𝒮=0⇔𝒴=Φ, 𝒫=∅.(I4) If 𝒴⊆𝒫⊆𝒪, then ℐ=𝒮=𝒮 and ℐ=𝒮=𝒮. Known by the similarity measure of FFSs, we have ℐ ≤ ℐ. Similarly, ℐ ≤ ℐ. This completes the proof.=𝒮=1 − 𝒟 is the inclusion measure of FFSs 𝒴 and 𝒫.Suppose 𝒴 and 𝒫 be two FFS s, then we define g ∈ FFS s, ∀ϰi ∈ 𝒵,  Let E be the entropy measure of FFSs, for 𝒴, 𝒫∈ FFSs, then E) is the similarity measure of FFSs 𝒴 and 𝒫.Suppose S1)− ( S2) are straightforward.(S3) Known by the definition of entropy of FFS s, E)=1⇔ℑg(ϰi)=ðg(ϰi)⇔|ℑ𝒴3(ϰi) − ℑ𝒫t(ϰi)|=0, |ð𝒴3(ϰi) − ð𝒫3(ϰi)|=0⇔ℑ𝒴(ϰi)=ℑ𝒫(ϰi), ð𝒴(ϰi)=ð𝒫(ϰi)⇔𝒴=𝒫.(S4) If 𝒴 is a crisp set, then ℑ𝒴(ϰi)=1, ð𝒴(ϰi)=0 or ℑ𝒴(ϰi)=0, ð𝒴(ϰi)=1. Hence, ℑg(ϰi)=1, ℐ𝒴(ϰi)=0, ð𝒴(ϰi)=0, it implies g = Φ, so E)=0.(S5) Since 𝒴⊆𝒫⊆𝒪, then ∀ϰi ∈ 𝒵, we have ℑ𝒴(ϰi) ≤ ℑ𝒫(ϰi) ≤ ℑ𝒪(ϰi), ð𝒪(ϰi) ≤ ð𝒫(ϰi) ≤ ð𝒴(ϰi). Therefore, we have ℑ𝒴3(ϰi) − ℑ𝒪3(ϰi)| ≥ |ℑ𝒴3(ϰi) − ℑ𝒫3(ϰi)| and |ð𝒴3(ϰi) − ð𝒪3(ϰi)| ≥ |ð𝒴3(ϰi) − ð𝒫3(ϰi)|.) ≤ E).Also, we can knowE) ≤ E). This completes the proof. □Similarly, we can prove that ℐ be the inclusion measure of FFS s, for 𝒴∈ FFS s, then E(𝒴)=ℐ is the entropy of FFS s 𝒴.Suppose E1) It is straightforward.(E2) If 𝒴 is a crisp set, then 𝒴=Φ or 𝒴=∅, we have ℐ=ℐ=0. Therefore, E(𝒴)=0.(E3) E(𝒴) = 1 ⇔ℐ=1⇔𝒴 ∪ 𝒴c⊆𝒴∩𝒴c⇔𝒴 ∪ 𝒴c=𝒴∩𝒴c⇔ℑ𝒴(ϰi)=ℐ𝒴(ϰi)=ð𝒴(ϰi)(E4) E(𝒴)=ℐ=ℐ=E(𝒴c).(E5) Since ℑ𝒴(ϰi) ≤ ℑ𝒫(ϰi) ≤ ℐ𝒫(ϰi) ≤ ℐ𝒴(ϰi) ≤ ð𝒫(ϰi) ≤ ð𝒴(ϰi) implies 𝒴⊆𝒫⊆𝒫c⊆𝒴c. Furthermore, 𝒴∩𝒴c⊆𝒫∩𝒫c⊆𝒫 ∪ 𝒫c⊆𝒴 ∪ 𝒴c. ≤ ℐ ≤ ℐ, so E(𝒴) ≤ E(𝒫).According to the definition of inclusion measure, we have 𝒴(ϰi) ≤ ð𝒫(ϰi) ≤ ℑ𝒫(ϰi) ≤ ℑ𝒴(ϰi), then we can have ℐ ≤ ℐ ≤ ℐ, that is E(A) ≤E(𝒫). This completes the proof.Similarly, if ð𝒴, 𝒫∈ FFS s, and we order ℐ=ℐ1=1 − (1/2|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ℑ𝒴3(ϰi)∧ℑ𝒫3(ϰi)|+|ð𝒴3(ϰi) − ð𝒴3(ϰi)|), then we have E(𝒴)=ℐ1=1 − (1/2|𝒵|)∑i∈𝒵ϰ(|ℑ𝒴3(ϰi) − ð𝒴3(ϰi)|)=E4(𝒴)For 𝒮(ℑ𝒴(ϰi), ð𝒴(ϰi))is the entropy of FFS𝒴.(i)E1) It is straightforward.((ii)E2) If 𝒴 is a crisp set, then we have ℑ𝒴(ϰi)=Φ, and ð𝒴(ϰi)=∅ or ℑ𝒴(ϰi)=∅, and ð𝒴(ϰi)=Φ. Therefore, 𝒮(ℑ𝒴(ϰi), ð𝒴(ϰi)=0.((iii)E3) Known by the definition of similarity measure of FFSs, we have((iv)E4) E(𝒴)=𝒮(ℑ𝒴(ϰi), ð𝒴(ϰi))=𝒮(ð𝒴(ϰi), ℑ𝒴(ϰi))=E(𝒴c).((v)E5) Since ℑ𝒴(ϰi) ≤ ℑ𝒫(ϰi) ≤ ð𝒫(ϰi) ≤ ð𝒴(ϰi) implies 𝒴⊆𝒫⊆𝒫c⊆𝒴c. Namely, ℑ𝒴⊆ℑ𝒫⊆ð𝒫⊆ð𝒴. According to the definition of similarity measure, we can have 𝒮(ℑ𝒴(ϰi), ð𝒴(ϰi)) ≤ 𝒮(ℑ𝒫(ϰi), ð𝒫(ϰi)), that is, E(𝒴) ≤ E(𝒫).(𝒴(ϰi) ≤ ð𝒫(ϰi) ≤ ℑ𝒫(ϰi) ≤ ℑ𝒴(ϰi), then we can have E(𝒴) ≤ E(𝒫). This completes the proof.Similarly, if ðIn this part, we use numerical examples to show the feasibility and effectiveness of the innovative FFS distance measures to illustrate the applications of the established distance measures for specific FFSs in pattern recognition. Furthermore, we compare them to the existing distance measures.ℳ1, ℳ2, ℳ3, and ℳ4 be four known patterns that are illustrated by the following FFSs in X as follows:ℒ is an unknown pattern. Its aimed is to determine the class to which ℒ belongs. In order to do that, the distance between ℒ and classes ℳ1, ℳ2, ℳ3, and ℳ4 are measured, and ℒ is then allocated to the class ℳg specified as follows:Let 𝒟1 − 𝒟13) for FFS, the distance between 𝒟, 𝒟, 𝒟, and 𝒟 is determined and displayed in ℒ belongs to a class ℳ3 when 𝒟1 to 𝒟13 are used. It is clear that the cause for this difference is the first characteristic, i.e., (ϰ1). The FFNs of ϰ1 are as follows:ℳ1, ℳ2, ℳ3, ℳ4, and ℒ, respectively. It is predicted that the distance between  and  is larger than the distance between  and  is larger than the distance between  and  is larger than  and . As a conclusion, it appears that𝒟 to 𝒟13 as shown in For all the newly developed distance measures  for patients P:   with disease symptoms V: . The symptoms associated with the considered diagnosis are listed in Table 2Table 2𝒟13. We may conclude that all the patients suffer from viral fever.Assume that a doctor would like to diagnose the condition of To illustrate the effectiveness of the novel distance measure for specific FFSs in pattern recognition, we present a numerical example and compare the novel findings to those reported in the literature.ℳ={ℳ1, ℳ2, ℳ3, ℳ4} who have the symptoms temperature, headache, stomach pain, cough, and chest pain, which are represented by S={ϰ1, ϰ2, ϰ3, ϰ4, ϰ5}. Let ℒ={ℒ1, ℒ2, ℒ3, ℒ4, ℒ5} be a list of possible diagnoses. Possible disease are defined as follows: ℒ1: viral fever, ℒ2: malaria, ℒ3: typhoid, ℒ4: stomach problem, and ℒ5: chest problem. The FF relation ℳ⟶S is illustrated by FFS, as can be seen in S ⟶ℒ is denoted by the FFS, as seen in ℳj with regard to the diagnostic ℒi and the final diagnosis findings are given in Al has malaria 𝒟 ,,ℳ={ℳ1, ℳ 𝒟 ) ) ℒ belonasures 𝒮 , 𝒮 27],,ℒ belong 𝒮 ,,ℒ belong,,C:  , , }, ℳ2={, , }, ℳ3={, , }, ℳ4={, , }. Let, ℒ: ℒ={, , and}. Its purpose is to determine which class ℒ belongs to. To do this, the inclusion degrees between ℒ and classes ℳ1, ℳ2, ℳ3, and ℳ4 are measured, and ℒ is then allocated to the class ℳg specified as follows:The following is an unknown pattern ℐ1 − ℐ7) for FFS, the degree of inclusion between ℐ, ℐ, ℐ, and ℐ are determined and displayed in ℒ belongs to a class ℳ4 when ℐ1 to ℐ3 and ℐ5 to ℐ6 are used, and ℒ belongs to a class ℳ3 and ℳ2 when ℐ4 and ℐ7 are, respectively, used. It is clear that the cause for this difference is the first characteristic, i.e., (ϰ1). The FFNs of ϰ1 are , , , , and  for ℳ1, ℳ2, ℳ3, ℳ4, and ℒ, respectively. It is predicted that the inclusion degree between  and  is larger than the inclusion degree between  and  is larger than the inclusion degree between  and  is larger than  and . As a conclusion, it appears that ℐ >  ℐ > ℐ > ℐ is more acceptable. In a similar way, we can find the previously mentioned relations for ℐ2 to ℐ7.For all the established inclusion measures .We constructed various formulae for FFSs information measures and analyzed the associated transformation relationships in detail.𝒟1-𝒟13) to pattern recognition and medical diagnosis to demonstrate their efficacy. The applications substantiate the results and also illustrate the feasibility and effectiveness of the distance measures between FFSs information.We used the established distance measures (1-S13); several counterintuitive examples of existing similarity measures are shown. We employed them to pattern recognition, construction materials, and medical diagnosis. For pattern recognition problems, we conclude that the proposed similarity measures dominate existing similarity measures. In some special situations, it has been shown that many conventional similarity measures are incapable of providing reasonable findings. However, in these specific cases, the proposed similarity measure is proficient of discriminating FFSs. A comparison of the proposed similarity measures with conventional similarity measures is performed for medical diagnosis problems. The applications emphasise the results and also illustrate the significance and reliability of the established similarity measures.We demonstrated the efficacy of the novel similarity measures (SAdditionally, we illustrated the applicability of the suggested FFS inclusion measures to pattern recognition with an example. The findings demonstrate the feasibility and effectiveness of new inclusion measures.The main findings of this research are emphasized and encapsulated as follows:The experimental findings demonstrated that the proposed measures are more reliable and can avoid the counter-intuitive situation in dealing with practical applications based on Fermatean fuzzy environment. , 51, 52.The FFSs are inappropriate to deal with situations where the cube sum of membership and nonmembership grades of exceeds 1A near future target is to unfold the application of the proposed information measures in scientific investigations for decision-making, pattern recognition, linguistic summarization, and data miningWe have also a plan to apply the presented approach to procurement planning, water desalination station selection, wind power plant site selection, and many more domains of real world problemsAdditionally, we will be further interested to immerse them in a variety of fuzzy environmentsFurthermore, since this work presents an applicative analysis of the FFS information measures, we should develop an appropriate software to effectively apply the presented information measures in a realistic situation


Amanita sect. Vaginatae (Fr.) Quél. are challenging to delimitate due to the morphological similarity or morphostasis among different taxa. In this study, a multi-locus  phylogeny was employed to investigate the species diversity of the section in eastern China. Sixteen species were recognized, including four new species; namely, A. circulata, A. multicingulata, A. orientalis, and A. sinofulva. They were documented with illustrated descriptions, ecological evidence, and comparisons with similar species. A key to the species of the section from eastern China is provided.Species of  Amanita Pers. is a cosmopolitan genus with about 700 accepted species  ring-like zone at proximal end of marginal striations; volval remnants on pileus absent; margin striate (0.2–0.5 R), non-appendiculate; trama white (1A1), unchanging. Lamellae free, crowded, white (1A1); lamellar edges white (1A1); lamellulae truncate, plentiful. Stipe 9–18 cm long × 0.5–1.5 cm diam., slender, subcylindric, slightly tapering upwards, with apex slightly expanded, white (1A1), gray (1B1), brownish (2B2–4) to gray-brown (2C2–4); context white (1A1), hollow in center; basal bulb absent; volva saccate, membranous, both surfaces white (1A1). Annulus absent. Odor indistinct.Lamellar trama bilateral. Mediostratum 20–40 μm wide, composed of abundant, ellipsoid inflated cells (25–60 × 10–30 μm); filamentous hyphae abundant, 2–8 μm wide; vascular hyphae scarce. Lateral stratum composed of abundant, ellipsoid to fusiform inflated cells (20–40 × 10–25 μm), diverging at an angle of ca. 30° to 60° to mediostratum; filamentous hyphae abundant and 2–7 μm wide. Subhymenium 30–40 μm thick, with 2–3 layers of ellipsoid to fusiform or irregularly arranged cells, 5–10 × 5–10 μm. Basidia 45–60 × 15–20 μm, clavate, 4-spored; sterigmata 5–8 μm long; basal septa lacking clamps. Basidiospores [60/3/3] (10.5–) 11–13 (–13.5) × (9.5–) 10–12.5 (–13) μm, Q = 1–1.15 (–1.21), Qm = 1.08 ± 0.05, globose to subglobose, occasionally broadly ellipsoid, inamyloid, colorless, thin-walled, smooth; apiculus small. Lamellar edge appearing as a sterile strip, composed of subglobose to ellipsoid or sphaeropedunculate inflated cells (15–50 × 10–45 μm), single and terminal or in chains of 2–3, thin-walled, colorless; filamentous hyphae abundant, 2–6 μm wide, irregularly arranged or ± running parallel to lamellar edge. Pileipellis 50–90 μm thick; upper layer (15–40 μm thick) gelatinized, composed of radially arranged to interwoven, thin-walled, colorless, filamentous hyphae 2–5 μm wide; lower layer (35–50 μm thick) composed of radially arranged, filamentous hyphae 3–6 μm wide, colorless to brownish; vascular hyphae scarce. Interior of volval remnants on stipe base composed of longitudinally arranged elements: filamentous hyphae dominant and very abundant, 3–10 μm wide, colorless, thin-walled, branching, anastomosing; inflated cells rare, globose, subglobose, ellipsoid to fusiform, 50–80 × 40–50 μm, colorless, thin-walled, mostly terminal or sometimes in chains of 2–3. Outer and inner surface of volval remnants on stipe base similar to structure of interior part, but with inner surface gelatinized. Stipe trama composed of longitudinally arranged, clavate terminal cells, 80–250 × 15–40 μm; filamentous hyphae scattered to abundant, 2–10 μm wide; vascular hyphae scarce. Clamps absent in all parts of basidioma.Habitat: Solitary to scattered on soil in subtropical mixed forests with Fagaceae and Pinaceae.Distribution: known from eastern and southwestern China.Pinus, Quercus and Keteleeria, altitude 1900 m, 20 July 2009, Li-Ping Tang 858 (HKAS 56815); same county, in a forest with trees of Pinus armandii Franch. and Keteleeria fortune (A. Murray bis) Carrière, altitude 2010 m, 14 August 2010, Qing Cai 391 (HKAS 67955); same city, Changning County, forest type unknown, altitude 2000 m, 25 July 2009, Gang Wu 4 (HKAS 57535); Kunming, Panlong District, in a mixed forest with trees of Fagaceae and Pinaceae, altitude 1990 m, 21 August 2016, Xiao-Xia Ding 111 (HKAS 97054); same city, Wuhua District, in a mixed forest with trees of Fagaceae and Pinaceae, altitude 1990 m, 6 September 2012, Yan-Jia Hao 753 (HKAS 76411); Lincang, Fengqing County, in a mixed forest with trees of Fagaceae and Pinaceae, altitude 1800 m, 26 July 2009, Gang Wu 12 (HKAS 57543); Puer, Lancang Lahu Autonomous County, in a mixed forest with trees of Fagaceae and Pinaceae, altitude 1780 m, 29 September 2016, LC-LJW 280 (HKAS 97784); same county, in a forest dominated with trees of Fagaceae, altitude 1350 m, 31 August 2017, Zhu L. Yang 6049 (HKAS 101238).Additional specimens examined: CHINA. ANHUI PROVINCE: Huangshan, in a broad-leaved forest with trees of Fagaceae, altitude 620 m, 13 July 2018, Hong-Yu Chen 32 (HKAS 127629); same location, in a broad-leaved forest with trees of Fagaceae, altitude 610 m, 12 July 2018, Ting Guo 979 (HKAS 127639). YUNNAN PROVINCE: Baoshan, Tengchong, in a mixed forest with trees of Amanita circulata is somewhat related to A. flavidocerea   (9.5–) 10–12 (–12.5) × (9–) 9.5–11 (–11.5) μm, Q = 1–1.13 (–1.15), Qm = 1.06 ± 0.03, globose to subglobose, inamyloid, colorless, thin-walled, smooth; apiculus small. Lamellar edge appearing as a sterile strip, composed of subglobose, ellipsoid to clavate inflated cells (10–45 × 10–30 μm), single and terminal or in chains of 2–3, thin-walled, colorless; filamentous hyphae abundant, 2–8 μm wide, irregularly arranged or ± running parallel to lamellar edge. Pileipellis 50–100 μm thick; upper layer (30–50 μm thick) gelatinized, composed of radially arranged to interwoven, thin-walled, colorless filamentous hyphae 2–5 μm wide; lower layer (40–50 μm thick) composed of radially arranged filamentous hyphae 4–7 μm wide, colorless; vascular hyphae scarce. Volval remnants on pileus composed of more or less vertically arranged elements: inflated cells very abundant to dominant, globose, subglobose, ellipsoid to fusiform, 10–60 × 10–50 μm, brown to brownish or colorless, thin-walled, mostly terminal or sometimes in chains of 2–3; filamentous hyphae rare, 3–7 μm wide, brown to brownish or colorless, thin-walled, branching, anastomosing. Volval remnants on stipe base composed of longitudinally arranged elements, becoming horizontally arranged towards upper parts: inflated cells very abundant to nearly dominant, globose, subglobose, ellipsoid, fusiform to clavate, 20–80 × 10–50 μm, brown to brownish or colorless, thin-walled, mostly terminal or sometimes in chains of 2–3; filamentous hyphae rare to fairly abundant, 2–8 μm wide, brown to brownish or colorless, thin-walled, branching, anastomosing. Stipe trama composed of longitudinally arranged, clavate terminal cells, 100–400 × 15–40 μm; filamentous hyphae scattered to abundant, 2–10 μm wide; vascular hyphae scarce. Clamps absent in all parts of basidioma.Pinus plants.Habitat: Solitary to scattered on soil in subtropical broad-leaved forests dominated with Fagaceae, sometimes in mixed forests with fagaceous and Distribution: Known from eastern China.Additional specimens examined: CHINA. ANHUI PROVINCE: Huangshan, in a forest dominated with Fagaceae, altitude 1300 m, 13 July 2018, Ting Guo 1018 (HKAS 127631); same location, in a forest dominated with Fagaceae, altitude 670 m, 13 July 2018, Rui-Heng Yang 73 (HKAS 127632); same location, in a forest with Fagaceae and Pinaceae, altitude 940 m, 15 July 2018, Rui-Heng Yang 123 (HKAS 127633); same location, in a forest dominated with Fagaceae, altitude 760 m, 15 July 2018, Rui-Heng Yang 127 (HKAS 127634); same location, in a forest dominated with Fagaceae, altitude 1220 m, 17 July 2018, Rui-Heng Yang 184 (HKAS 127635); same location, in a forest dominated with Fagaceae, altitude 1200 m, 17 July 2018, Rui-Heng Yang 186 (HKAS 127636); same location, in a forest dominated with Fagaceae, altitude 940 m, 15 July 2018, Hong-Yu Chen 77 (HKAS 127637).A. multicingulata is closely related to A. liquii  and shorter striations on the pileal margin (0.3–0.4 R) distinguish A. cinctipes from A. multicingulata  (10–) 10.5–13 × 9–12 (–13) μm, Q = 1–1.26 (–1.31), Qm = 1.13 ± 0.07, subglobose to broadly ellipsoid, sometimes globose, inamyloid, colorless, thin-walled, smooth; apiculus small. Lamellar edge appearing as a sterile strip, composed of subglobose, ellipsoid to clavate inflated cells (10–50 × 10–30 μm), single and terminal or in chains of 2–3, thin-walled, colorless; filamentous hyphae abundant, 3–7 μm wide, irregularly arranged or ± running parallel to lamellar edge. Pileipellis 50–100 μm thick; upper layer (30–50 μm thick) gelatinized, composed of radially arranged to interwoven, thin-walled, colorless to brownish filamentous hyphae 2–6 μm wide; lower layer (30–40 μm thick) composed of radially arranged filamentous hyphae 3–8 μm wide, brownish to brown; vascular hyphae scarce. Volval remnants on stipe base composed of longitudinally arranged elements: inflated cells abundant, globose, subglobose, ellipsoid, fusiform to clavate, 20–40 × 10–40 μm, yellow-brown to gray-brown, thin-walled, mostly terminal or sometimes in chains of 2–3; filamentous hyphae abundant, 2–7 μm wide, yellow-brown to gray-brown, thin-walled, branching, anastomosing. Stipe trama composed of longitudinally arranged, clavate terminal cells, 80–300 × 15–40 μm; filamentous hyphae scattered to abundant, 2–10 μm wide; vascular hyphae scarce. Clamps absent in all parts of basidioma.Habitat: Solitary to scattered on soil in subtropical forests with Fagaceae and Pinaceae.Distribution: Known from eastern China.Additional specimens examined: CHINA. ANHUI PROVINCE: Huangshan, in a mixed forest with Fagaceae and Pinaceae, altitude 760 m, 14 July 2018, Rui-Heng Yang 117 (HKAS 127638).A. griseofolia can be distinguished from A. orientalis  (10.5–) 11–13.5 (–18) × (9–) 9.5–12 (–13) μm, Q = (1.01–) 1.05–1.23 (–1.34), Qm = 1.14 ± 0.08, subglobose to broadly ellipsoid, occasionally globose or ellipsoid, inamyloid, colorless, thin-walled, smooth; apiculus small. Lamellar edge appearing as a sterile strip, composed of subglobose, ellipsoid to clavate inflated cells (10–30 × 10–25 μm), single and terminal or in chains of 2–3, thin-walled, colorless; filamentous hyphae abundant, 2–8 μm wide, irregularly arranged or ± running parallel to lamellar edge. Pileipellis 50–90 μm thick; upper layer (30–50 μm thick) gelatinized, composed of radially arranged to interwoven, thin-walled, colorless to brownish filamentous hyphae 2–5 μm wide; lower layer (35–50 μm thick) composed of radially arranged filamentous hyphae 3–7 μm wide, colorless to brownish; vascular hyphae scarce. Interior of volval remnants on stipe base composed of longitudinally arranged elements: filamentous hyphae dominant and very abundant, 3–7 μm wide, colorless, thin-walled, branching, anastomosing; inflated cells rare to fairly abundant, subglobose, ellipsoid to fusiform, 40–65 × 15–50 μm, colorless, thin-walled, mostly terminal or sometimes in chains of 2–3. Stipe trama composed of longitudinally arranged, clavate terminal cells, 80–300 × 15–50 μm; filamentous hyphae scattered to abundant, 3–10 μm wide; vascular hyphae scarce. Clamps absent in all parts of basidioma.Pinus.Habitat: Solitary to scattered on soil in subtropical forests dominated with Fagaceae, sometimes mixed with Distribution: Known from eastern, central, and southwestern China. Based on the phylogenetic tree inferred from the ITS dataset, it also occurs in Tibet autonomous region and Hunan province . Castanea seguinii Dode and Pinus taiwanensis Hayata, altitude 840 m, Yan-Jia Hao 1609 (EFHAAU 207); same county, in a mixed forest with Fagaceae and Pinaceae, altitude 1000 m, Yan-Jia Hao 1715 (EFHAAU 313). YUNNAN PROVINCE: Nujiang Lisu Autonomous Prefecture, Lanping Bai and Pumi Autonomous County, in a subtropical forest dominated with Quercus, mixed with Pinus yunnanensis, altitude 2150 m, Gang Wu 743 (HKAS 75058).Additional specimens examined: CHINA. ANHUI PROVINCE: Liuan, Jinzhai County, in a forest dominated with Fagaceae, altitude 1110 m, 21 July 2017, Yan-Jia Hao 1520 (HKAS 100610); same county, in a mixed forest with Amanita orientifulva and A. suborientifulva can be confused with A. sinofulva. According to our multi-gene phylogenetic analysis  and is found in subalpine forests dominated by trees of Abies and Picea [A. suborientifulva set it apart from A. sinofulva [A. fulva is also similar to A. sinofulva, but differs in the globose to subglobose basidiospores and in the saccate volva, with inflated cells dominant in its outer part [Notes: analysis , the firnd Picea . The nonA. sect. Vaginatae. Five of them are most informative, viz. the color of the basidiomata, the striations on the pileal margin, the presence or absence of the annulus, the volval remnants on the stipe base, and the size of the basidiospores. In this study, A. cingulata is the only species with a white basidioma, while the other species from eastern China have basidiomata ranging from yellow, to gray, to brown. The striations on the pileal margin of A. zonata, A. pallidozonata, and A. circulata form a ring-like zone at the proximal end, while the remaining taxa in eastern China are without this zone [A. griseofolia, A. multicingulata, and A. orientalis. Species with an annulus, a ring-like zone at the proximal end, or with incomplete rings of volval remnants on the stipe base are clustered in non-monophyletic groups. Our data revealed that several macro- and microscopic characteristics could be useful for the delimitation of species in his zone . Most sphis zone ,21,50. TGiven that it is difficult to delimitate these species based solely on morphological studies, integrative taxonomy is indispensable in recognizing species of the section. This method, which delimits and describes taxa by integrating information from different types of data and methodologies , is proven to be useful for species recognition in plants, animals, and fungi ,62,63,64A. zonata, A. pallidozonata, and A. circulata are morphologically similar due to the pronounced ring-like zones at the proximal end of the marginal striations. However, they occupy different positions in the phylogenetic tree and are distantly related  [For example,  related . Followiipe base . Amanita5–10 μm) . Amanita sinofulva is phylogenetically close and morphologically similar to A. orientifulva. However, they are clustered into two independent lineages , and three species new to China . Thirty-nine of them are reported from the southern parts of China—namely, southwestern, central, eastern, and southern China. In this study, 40 taxa of own taxa ,2,21, foA. angustilamellata (Höhn.) Boedijn, A. brunneoprocera Thongbai et al., A. cinctipes, and A. pallidocarnea (Höhn.) Boedijn are typical tropical elements restricted in the tropical areas of China. The other three species, viz. A. brunneosquamata Thongbai et al., A. suborientifulva, and A. subovalispora, extend their distribution from Southeast Asia to subtropical China. In addition, several taxa found in subtropical or subalpine temperate areas in southern parts of China are phylogenetically close to species from Southeast Asia. For example, A. circulata and A. zonata, reported from the subtropical regions of China, are sister to A. flavidocerea and A. flavidogrisea Thongbai et al. from Southeast Asia, respectively , Qm = 1.13 ± 0.07……………………………….………………………………………3’.Basidiospores more rounded, globose to subglobose……………………………………..44.A. griseofoliaBasidioma more grayish; basidiospores slightly larger, 10–13.5 × 9.5–13 μm………………………………………………………………..……….…….……4’.A. multicingulataBasidioma more brownish; basidiospores slightly smaller, 10–12 × 9.5–11 μm………………………………………………………………….……………5.Pileal surface forming a distinctive ring-like zone at proximal end of marginal striations………………………………….…………………………………..………………………...65’.vPileal surface without a distinctive ring-like zone at proximal end of marginal striations………………………………….……………………………………………..……………...86.A. zonataPileal margin with relatively shorter striations, 0.15–0.3 R; basidiospores slightly smaller and rounder, globose to subglobose, 9–10.5 × 8.5–10 μm, Q = 1.00–1.11, Qm = 1.05 ± 0.04……………………………………...……………………………………..………….6’.Pileal margin with relatively longer striations; basidiospores slightly larger, globose, subglobose to broadly ellipsoid……………………….………………………………………...77.A. pallidozonataBasidiospores slightly smaller, 10–12 × 9–11 μm; volval remnants on stipe base with abundant inflated cells in inner part……………………………………………7’.A. circulataBasidiospores slightly larger, 11–13 × 10–12.5 μm; volval remnants on stipe base mainly with abundant filamentous hyphae………………………….……………………...8.Basidioma with distinctive yellow color…………………...…………………………..…….98’.Basidioma gray to brown, without yellow color…………………………………………..109.A. hamadaeStriations on pileal margin relatively longer, 0.2–0.5 R; basidiospores broadly ellipsoid to ellipsoid, 10–12 × 8–9 μm, Q = 1.22–1.37, Qm = 1.3 ± 0.07……………….………9’.A. croceaStriations on pileal margin relatively shorter, 0.2–0.3 R; basidiospores globose to subglobose, 9–11 × 9–11 μm, Q = 1–1.11, Qm = 1.05 ± 0.04……………………………….10.Basidioma orange-brown to yellow-brown…………………….…………….…………..1110’.Basidioma gray to gray-brown………………………………………………….………...1211.A. suborientifulvaPileus without an umbo……………………………...……….……………11’.A. sinofulvaPileus with a distinctive umbo at center……………...…………………...…...12.A. olivaceofuscaPileus gray-brown to yellow-brown, with distinct olivaceous tinge; basidiospores broadly ellipsoid to ellipsoid, relatively larger, 10.5–13 × 8.5–10 μm, Q = 1.05–1.45, Qm = 1.25 ± 0.09………………………….………………………………………………12’.Pileus gray to brown, without olivaceous tinge; basidiospores globose to subglobose, or broadly ellipsoid to ellipsoid, relatively smaller…………………………………………1313.Pileus gray, without umbo at the center; basidiospores broadly ellipsoid to ellipsoid……………………………………………………………………………….………………1413’.Pileus gray-brown to brown, umbonate at center; basidiospores globose to subglobose………………………………………………………………………………………….……1514.A. subovalisporaPileal margin with shorter striations, 0.36–0.4 R………………………….14’.A. ovalisporaPileal margin with longer striations, 0.4–0.6 R………...………………….......15.A. brunneoumbonataPileus dark brownish, with darker colored central disk; pileal margin with longer striations, 0.33–0.42 R…………………………………………………………..15’.A. brunneoproceraPileus grayish brown; pileal margin with shorter striations, 0.18–0.21 R………………………………………………………………………………..