Melanopus is a morphological group of Polyporus which contains species with a black cuticle on the stipe. In this article, taxonomic and phylogenetic studies on Melanopus group were carried out on the basis of morphological characters and phylogenetic evidence of DNA sequences of multiple loci including the internal transcribed spacer (ITS) regions, the large subunit nuclear ribosomal RNA gene (nLSU), the small subunit nuclear ribosomal RNA gene (nSSU), the small subunit mitochondrial rRNA gene sequences (mtSSU), the translation elongation factor 1-α gene (EF1-α), the largest subunit of RNA polymerase II (RPB1), the second largest subunit of RNA polymerase II (RPB2), and β-tubulin gene sequences (β-tubulin). The phylogenetic result confirmed that the previously so-called Melanopus group is not a monophyletic assemblage, and species in this group distribute into two distinct clades: the Picipes clade and the Squamosus clade. Four new species of Picipes are described, and nine new combinations are proposed. A key to species of Picipes is provided. Melanopus Pat. was established by Patouillard Polyporus admirabilis Peck, Bulletin of the Torrey Botanical Club 26: 69 (1899). [MB 157762]Basionym: Cerioporus admirabilis (Peck) Zmitr. et Kovalenko, International Journal of Medicinal Mushrooms 18 (1): 33 (2016). [MB 812034]≡ Picipes americanus (Vlasák & Y.C. Dai) J.L. Zhou & B.K. Cui, comb. nov. [MB 817137]Polyporus americanus Vlasák & Y.C. Dai, Fungal Diversity 64: 136 (2014). [MB 803796]Basionym: Picipes austroandinus (Rajchenb. & Y.C. Dai) J.L. Zhou & B.K. Cui, comb. nov. [MB 817138]Polyporus austroandinus Rajchenb. & Y.C. Dai, Fungal Diversity 64: 138 (2014). [MB 803797]Basionym: Picipes conifericola (H.J. Xue & L.W. Zhou) J.L. Zhou & B.K. Cui, comb. nov. [MB 817139]Polyporus conifericola H.J. Xue & L.W. Zhou, Mycological Progress 13(1): 139 (2014). [MB 801216]Basionym: Picipes fraxinicola (L.W. Zhou & Y.C. Dai) J.L. Zhou & B.K. Cui, comb. nov. [MB 817140]Piptoporus fraxineus Bondartsev & Ljub., Novosti Sistematiki Nizshikh Rastenii 2: 135 (1965). [MB 337052]Basionym: Polyporus fraxinicola L.W. Zhou & Y.C. Dai, Fungal Diversity 64: 141 (2014). [MB 803799]≡ Picipes rhizophilus (Pat.) J.L. Zhou & B.K. Cui, comb. nov. [MB 817141]Polyporus rhizophilus Pat., Journal de Botanique 8: 219 (1894). [MB 150169]Basionym: Cerioporus rhizophilus (Pat.) Zmitr. et Kovalenko, International Journal of Medicinal Mushrooms 18 (1): 33 (2016). [MB 812040]≡ Picipes submelanopus (H.J. Xue & L.W. Zhou) J.L. Zhou & B.K. Cui, comb. nov. [MB 817142]Polyporus submelanopus H.J. Xue & L.W. Zhou, Mycotaxon 122: 436 (2013). [MB 800237]Basionym: Picipes taibaiensis (Y.C. Dai) J.L. Zhou & B.K. Cui, comb. nov. [MB 817143]Polyporus rhododendri Y.C. Dai & H.S. Yuan, Annales Botanici Fennici 46 (1): 58 (2009). [MB 540894]Basionym: Polyporus taibaiensis Y.C. Dai, Fungal Diversity 64: 142 (2014). [MB 803798]≡ Picipes virgatus  J.L. Zhou & B.K. Cui comb. nov. [MB 817144]Polyporus virgatus Berk. & M.A. Curtis, Botanical Journal of the Linnean Society 10: 304 (1869). [MB 202513]Basionym: Leucoporus virgatus (Berk. & M.A. Curtis) Pat., Énumération Méthodique des Champignons Recueillis à la Guadeloupe et à la Martinique: 25 (1903). [MB 102236]≡ Picipes baishanzuensis was collected from subtropical area of China. It is characterized by its radially striped infundibuliform pileus with a slender black stipe. In phylogenetic analysis (Pi. virgatus (100/100/1.00). Morphologically, Pi. virgatus and Pi. baishanzuensis share infundibuliform pileus, similar pore size, decurrent tubes and wrinkled dark stipe; however, the basidiospores of Pi. virgatus are much larger (9–12.5 × 4–5 μm for Pi. virgatus and 6.6–7.9 × 2.5–3.1 μm for Pi. baishanzuensis) [Polyporus tuberaster also has depressed pileus and decurrent pores, but its pileus is covered with dark brown flecks, pores (0.5–2 per mm) and basidiospores (12–14.5 × 4.8–6 μm) [Pi. baishanzuensis; besides, P. tuberaster usually grows on the ground, arising from a black underground sclerotium [analysis , it stro.8–6 μm)  are muchlerotium .Picipes subtropicus was found in subtropical areas of China. It can be identified by a continuous variation in pore size, bright pileal surface color, short black stipe-like base and medium cylindrical basidiospores (5.1–6.2 × 2.2–2.7 μm). In phylogenetic analysis , larger basidiospores (7–8.5 × 2.5–4 μm) and pantropical distribution [Picipes badius share similar basidiocarps and pore size with P. subtropicus; but it differs in its larger basidiospores (7.5–9.5 × 3–3.5 μm), simple-septate generative hyphae and absence of cystidioles [Picipes baishanzuensis was also found in subtropical areas of China, but its infundibuliform pilei, slender stipe and lager basidiospores (6.6–7.9 × 2.5–3.1 μm) are quite different from Pi. subtropicus.analysis , it did Picipes subtubaeformis was described from temperate zone of China. It can be distinguished by the irregularly semicircular or elliptical pileus, terra-brown to black stipe, and oblong to cylindrical basidiospores (5.7–6.8 × 2.7–3.1 μm). In the phylogenetic analysis (Pi. subtubaeformis grouped together with Pi. tubaeformis (88/88/1.00); morphologically, both of them have orange to reddish-brown pileus and dark stipe, but Pi. tubaeformis differs in its slender stipe and basidiospores  [Pi. virgatus and Pi. subtubaeformis have reddish-brown or chestnut basidiocarps with centrally to laterally dark stipe, but the former one has both larger pores (3–4 per mm) and basidiospores [Pi. virgatus is absence of cystidioles [Picipes taibaiensis is another temperate species described from China. It has similar upper pileal surface color with Pi. subtubaeformis, but the flabelliform or spathulate pileus, larger basidiospores (7.5–10.5 × 3.2–3.8 μm) and fusoid cystidioles make it different from Pi. subtubaeformis [analysis , Pi. sub2.75 μm) . Both Piaeformis .Picipes tibeticus is a special species found from eastern Tibetan Plateau. it can be identified by its reddish-brown to blackish-brown fan-shaped or semicircular basidiocarps, small angular pores (6–9 per mm), oblong basidiospores (5–5.9 × 2.8–3.3 μm) and growth on coniferous trees. Phylogenetically, it grouped together with Pi. conifericola  and basidiospores (8–10 × 3–3.9 μm). In addition, Pi. submelanopus has both simple septate and clamped generative hyphae [e hyphae .Picipes admirabilis was was initially collected on wood of apple trees in northeastern United States [Pi. admirabilis is a variety of P. varius or belongs to group Melanopus for its black stipe [Pi. admirabilis, P. gayanus Lév. and P. pseudobetulinus (Murashk. ex Pilát) Thorn, Kotir. & Niemelä as members of Polyporus group Admirabilis [P. pseudobetulinus has recently been combined into Favolus as F. pseudobetulinus (Murashk. ex Pilát) Sotome & T. Hatt. [Pi. admirabilis as a member of genus Cerioporus Quél. But according to our phylogenetic analysis, Pi. admirabilis strongly clusters in the picipes clade (Pi. admirabilis has a long (up to 8 cm) and pale buff to black stipe, firm-corky basidiocarps and uninflated hyphae [Picipes.d States . Lloyd cck stipe ,40. Núñeirabilis . Among tT. Hatt.  regardedes clade . Morphold hyphae . Based oPicipes rhizophilus was treated as a member of group Polyporellus, and it is a special polypore merely grows on the grass roots [Pi. rhizophilus into Cerioporus as C. rhizophilus (Pat.) Zmitr. et Kovalenko. In our current phylogenetic analysis, Pi. rhizophilus strongly groups into the picipes clade . These above-mentioned features fit Picipes well, so we consider this species as a member of Picipes.ss roots ,7,41,42.ss roots  transferes clade . AccordiPicipes is showed to be a monophyletic group based on the 8-gen-squences data analysis, and sixteen species are included in this clade , cylindrical basidiospores and lignicolous habit. According to our study with more samples, we find several species with large pores, oblong basidiospores and terrestrial habit are also members of Picipes.is clade . Among tDe by De  for its . Corner  considerovalenko  considerPolyporus umbellatus is a particular species that merely grows on the ground from a sclerotium close to stumps of hardwoods, and characterized by numerous stipitate pilei arising from a common, strongly branched stipe [Polyporus group Dendropolyporus [P. umbellatus was reported as a distinctive species that could not cluster with any other species [P. umbellatus into genus Cladomeris Quél. But in our analysis, P. umbellatus strongly clusters with P. tuberaster and P. hapalopus in the core polyporus clade  Zmitr. et Kovalenko, but this name is illegitimate because of its earlier homonym N. suavissimus (Fr.) J.S. Seelan.Previous phylogenetic analyses showed that favolus clade and neofavolus clade did not gather together and favolus clade has closer relationships with beraster ,8. But on et al.  estimateovalenko  also treLentinus Fr. and Polyporus have been suspected for a long time. Both Pegler [Lentinus divides from polypores, and this assumption had been evidenced by Hibbett & Vilgalys [Polyporellus has a much closer relationship with Lentinus compared with other Polyporus spp. [Lentinus s. str. and group Polyporellus which have the inflated generative hyphae. Seelan et al. [Lentinus and Polyporellus group is circular pores, with independent transitions to angular pores and lamellae. Zmitrovich [Lentinus as L. arcularius (Batsch) Zmitr., L. brumalis (Pers.) Zmitr., L. crinitus (L.) Fr. and L. tricholoma (Mont.) Zmitr. But the last name is illegitimate because of its earlier homonym L. tricholoma Berk. & Cooke. Then Zmitrovich & Kovalenko [P. tricholoma to L. flexipes (Fr.) Zmitr. et Kovalenko. In this article, we prefer to treat species in polyporellus clade as members of Lentinus.The relationships of h Pegler  and Singh Pegler  believedVilgalys  and HibbVilgalys . Moleculrus spp. ,46,51–56rus spp.  proposeditrovich  combinedovalenko  renamed Melanopus distribute into two different clades: picipes clade and squamosus clade. This conclusion verified the view that Melanopus group is not a monophyletic assemblage of dark-stiped Polyporus species, and whether having black cuticle on stipe or not, is not a sufficient feature to define the natural Melanopus group. In our study, sixteen species including four new species of Picipes are recognized. A key to species of Picipes is provided.Our phylogenetic analysis based on multiple gene sequences data of ITS, nLSU, EF1-α, mtSSU, β-tubulin, RPB1, RPB2 and nSSU suggested that species of group 1aGrowing on woods or ground………………………………………… 21bPi. rhizophilusGrowing on grass roots…………………………………………………. 2aGenerative hyphae bearing simple septa……………………………… 32bGenerative hyphae only with clamps……………………………………….. 43aPi. submelanopusPores 2–3 per mm, generative hyphae bearing both simple septa and clamp connections, growing on ground or hardwoods, basidiospores 8–10 × 3–3.9 μm……………………………………………….. 3bPi. badiusPores 5–6 per mm, generative hyphae only with simple septa, growing on hardwoods, basidiospores 6.5–8 × 3–3.8 μm……………………………………………………………………. 4aStipe very short or attach to the substrate with a flattened base…………………………………… 54bStipe usually more than 1 cm long……………………………………………………………………….. 65aPi. subtropicusPores 8–9 per mm when young and becoming 5–7 per mm when mature, basidiospores 5.1–6.2 × 2.2–2.7 μm……………………………………………………………………………………… 5bRhododendron, basidiospores 7.5–10.5×3.2–3.8 μm……………………………………………………………………………………………Pi. taibaiensisPores 3–5 per mm, growing on 6aBasidiospores more than 9 μm long…………………………… 76bBasidiospores less than 9 μm long……………………………. 87aAustrocedrus or Lomatia woods, pores 4–5 per mm, basidiospores 9–11.5 × 3–3.8 μm………………………………………………………………………………………………………………………………….Pi. austroandinusGrowing on 7bPi. virgatusGrowing on hardwoods, pores 3–4 per mm, basidiospores 9–12.5 × 4–5 μm…………………………………………. 8aCystidioles present……………………………………………………..98bPi. fraxinicolaCystidioles absent……………………………………………………. 9aPores 3–6 per mm…………………………………………….. 109bPores 6–10 per mm………………………………………………. 1410aBasidiospores usually more than 3 μm wide…………………………………………. 1110bBasidiospores usually less than 3 μm wide…………………………………………. 1211aPi. admirabilisCystidioles subulate……………………………………….. 11bPi. melanopusCystidioles fusoid……………………………………….. 12aPilei not infundibuliform in shape……………………………………. 1312bPi. baishanzuensisPilei infundibuliform in shape……………………………………… 13aPi. americanusPilei nearly circular, basidiospores 7–9 × 2.5–3.1 μm…………………………………….. 13bPi. subtubaeformisPilei irregularly semicircular or elliptical, basidiospores 5.7–6.8 × 2.7–3.1 μm……………………………….14aBasidiospores more than 6 μm in length………………………………….. 1514bPi. tibeticusBasidiospores 5–5.9 × 2.8–3.3 μm………………………………………….. 15aPi. conifericolaGrowing on coniferous trees………………………………… 15bPi. tubaeformisGrowing on hardwoods……………………………………….


The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined. G =  be a simple connected graph with vertex set V = {v1, v2,…, vn} and edge set E. Its adjacency matrix A(G) = (aij) is defined as n × n matrix (aij), where aij = 1 if vi is adjacent to vj, and aij = 0, otherwise. Denote by d(vi) or dG(vi) the degree of the vertex vi. Let Q(G) = D(G) + A(G) be the signless Laplacian matrix of graph G, where D(G) = diag(d(v1), d(v2),…, d(vn)) denotes the diagonal matrix of vertex degrees of G. It is well known that A(G) is a real symmetric matrix and Q(G) is a positive semidefinite matrix. The largest eigenvalues of A(G) and Q(G) are called the spectral radius and the signless Laplacian spectral radius of G, denoted by ρ(G) and q(G), respectively. When G is connected, A(G) and Q(G) are a nonnegative irreducible matrix. By the well-known Perron-Frobenius theory, ρ(G) is simple and has a unique positive unit eigenvector and so does q(G). We refer to such an eigenvector corresponding to q(G) as the Perron vector of G.Let G are independent if they are not adjacent in G. A set of mutually independent edges of G is called a matching of G. A matching of maximum cardinality is a maximum matching in G. A matching M that satisfies 2 | M | = n = |V(G)| is called a perfect matching of the graph G. Denote by Cn and Pn the cycle and the path on n vertices, respectively.Two distinct edges in a graph A(G) is det(xI − A(G)), which is denoted by Φ(G) or Φ. The characteristic polynomial of Q(G) is det(xI − Q(G)), which is denoted by Ψ(G) or Ψ.The characteristic polynomial of Cp and Cq be two vertex-disjoint cycles. Suppose that v1 is a vertex of Cp and vl is a vertex of Cq. Joining v1 and vl by a path v1v2 ⋯ vl of length l − 1, where l ≥ 1 and l = 1 means identifying v1 with vl, denoted by B, is called an ∞-graph , is called a θ-graph  be the set of all bicyclic graphs on n = 2μ  (μ ≥ 2) vertices with perfect matchings. Obviously Bn(2μ) consists of two types of graphs: one type, denoted by Bn+(2μ), is a set of graphs each of which is an ∞-graph with trees attached; the other type, denoted by Bn++(2μ), is a set of graphs each of which is θ- graph with trees attached. Then we have Bn(2μ) = Bn+(2μ) ∪ Bn++(2μ).Let The investigation on the spectral radius of graphs is an important topic in the theory of graph spectra, in which some early results can go back to the very beginnings (see ). The reIn this paper, we deal with the extremal signless Laplacian spectral radius problems for the bicyclic graphs with perfect matchings. The graph with the largest signless Laplacian spectral radius among all bicyclic graphs with perfect matchings is determined.G − u or G − uv denote the graph obtained from G by deleting the vertex u ∈ V(G) or the edge uv ∈ E(G). A pendant vertex of G is a vertex with degree 1. A path P : vv1v2 ⋯ vk in G is called a pendant path if d(v1) = d(v2) = ⋯ = d(vk−1) = 2 and d(vk) = 1. If k = 1, then we say vv1 is a pendant edge of the graph G.Let In order to complete the proof of our main result, we need the following lemmas.G be a connected graph and u, v two vertices of G. Suppose that v1, v2,…, vs ∈ N(v)∖{N(u) ∪ u}  (1 ≤ s ≤ d(v)) and x =  is the Perron vector of G, where xi corresponds to the vertex vi  (1 ≤ i ≤ n). Let G* be the graph obtained from G by deleting the edges vvi and adding the edges uvi  (1 ≤ i ≤ s). If xu ≥ xv, then q(G) < q(G*).Let G is commonly known as its matching number, denoted by μ(G).The cardinality of a maximum matching of From w and v be two vertices in a connected graph G and suppose that s paths {ww1w1′, ww2w2′,…, wwsws′} of length 2 are attached to G at w and t paths {vv1v1′, vv2v2′,…, vvtvt′} of length 2 are attached to G at v to form Gs,t. Then either q > q  (1 ≤ i ≤ t) or q > q  (1 ≤ i ≤ s)or μ = μ = μ.Let u is a vertex of graph G with d(u) ≥ 2. Let G : uv be a graph obtained by attaching a pendant edge uv to G at u. Suppose t paths {vv1v1′,…, vvtvt′} of length 2 are attached to G : uv at v to form Lt0,. LetLt0, has a perfect matching, then we have that Mt1, has a perfect matching andSuppose G is a sequence of vertices v1, v2,…, vm with m ≥ 2 such thatv1 = vm);the vertices in the sequences are distinct ;d(vi) satisfy d(v1) ≥ 3, d(v2) = ⋯ = d(vm−1) = 2 (unless m = 2) and d(vm) ≥ 3.the vertex degrees An internal path of a graph G be a connected graph, and uv ∈ E(G). The graph Guv is obtained from G by subdividing the edge uv, that is, adding a new vertex w and edges uw, wv in G − uv. By similar reasoning as that of Theorem 3.1 of [Let m 3.1 of , we haveP : v1v2 ⋯ vk  (k ≥ 2) be an internal path of a connected graph G. Let G′ be a graph obtained from G by subdividing some edge of P. Then we have q(G′) < q(G).Let v1v2 ⋯ vk  (k ≥ 3) is an internal path of the graph G and v1vk ∉ E(G) for k = 3. Let G* be the graph obtained from G − vivi+1 − vi+1vi+2  (1 ≤ i ≤ k − 2) by amalgamating vi, vi+1, and vi+2 to form a new vertex w1 together with attaching a new pendant path w1w2w3 of length 2 at w1. Then q(G*) > q(G) and μ(G*) ≥ μ(G).Suppose that q(G*) > q(G). Next, we prove that μ(G*) ≥ μ(G). Let M be a maximum matching of G. If vivi+1 ∈ M or vi+1vi+2 ∈ M, then {M − {vivi+1}}∪{w2w3} or {M − {vi+1vi+2}}∪{w2w3} is a matching of G*. Thus, μ(G*) ≥ μ(G); If vivi+1 ∉ M and vi+1vi+2 ∉ M, then there exist two edges viu and vi+2v ∈ M. Thus, {M − {viu}}∪{w2w3} is a matching of G*. Hence, μ(G*) ≥ μ(G), completing the proof.From S(G) be the subdivision graph of G obtained by subdividing every edge of G.Let G be a graph on n vertices and m edges, Φ(G) = det(xI − A(G)), Ψ(G) = det(xI − Q(G)). Then Φ(S(G)) = xm−nΨ.Let u be a vertex of a connected graph G. Let Gk,l   be the graph obtained from G by attaching two pendant paths of lengths k and l at u, respectively. If k ≥ l ≥ 1, then q > q.Let v1v2 ⋯ vk  (k ≥ 3) is a pendant path of the graph G with d(v1) ≥ 3. Let G* be the graph obtained from G − v1v2 − v2v3 by amalgamating v1, v2, and v3 to form a new vertex w1 together with attaching a new pendant path w1w2w3 of length 2 at w1. Then q(G*) > q(G) and μ(G*) ≥ μ(G).Suppose that q(G*) > q(G). By the proof as that of μ(G*) ≥ μ(G).By e = uv be an edge of G, and let C(e) be the set of all circuits containing e. Then Φ(G) satisfiesZ ∈ C(e).Let v be a vertex of G, and let φ(v) be the collection of circuits containing v, and let V(Z) denote the set of vertices in the circuit Z. Then the characteristic polynomial Φ(G) satisfiesw adjacent to v, and the second summation extends over all Z ∈ φ(v).Let G be a connected graph, and let G′ be a proper spanning subgraph of G. Then ρ(G) > ρ(G′), and, for x ≥ ρ(G), Φ(G′) > Φ(G).Let G) denote the maximum degree of G. From Let Δ > q(G′).Let G =  be a connected graph with vertex set V = {v1, v2,…, vn}. Suppose that v1v2 ∈ E(G), v1v3 ∈ E(G), v1v4 ∈ E(G), d(v3) ≥ 2, d(v4) ≥ 2, d(v1) = 3, and d(v2) = 1. Let Gv1v3(Gv1v4) be the graph obtained from G − v1v3(G − v1v4) by amalgamating v1 and v3(v4) to form a new vertex w1(w3) together with subdivising the edge w1v2(w3v2) with a new vertex w2(w4). If q(Gv1v3) > q(G) or q(Gv1v4) > q(G);either μ(Gv1v3) ≥ μ(G) and μ(Gv1v4) ≥ μ(G).Let u is a vertex of the bicyclic graph G with dG(u) ≥ 2. Let G : uv be a graph obtained by attaching a pendant edge uv to G at u. Suppose that a pendant edge vw1 and t paths {vv1v1′,…, vvtvt′} of length 2 are attached to G : uv at v to form Lt1,. Let Mt+10, = Lt1, − vv1 − ⋯−vvt + uv1 + ⋯+uvt. Then we haveq > q, (t ≥ 1);μ ≤ μ.Suppose G1, G2,…, G6 be the graphs as μ ≥ 3, we have q(G1) > q(Gi), .Let μ ≥ 12, for S(G2)) − Φ(S(G1)) > 0. Hence, ρ(S(G1)) > ρ(S(G2)) for μ ≥ 12. When μ = 4,5,…, 11, by direct calculation, we also get ρ(S(G1)) > ρ(S(G2)), respectively. So, ρ(S(G1)) > ρ(S(G2)) for μ ≥ 4. By q(G1) > q(G2)  (μ ≥ 4). By similar method, the result is as follows.From Φ(S(G1)) G ∈ Bn(2μ)  (n ≥ 6), then q(G) ≤ q(G1), with equality if and only if G = G1.If X = T be the Perron vector of G. From μ ≥ 3, Let G* ∈ Bn(2μ) such that q(G*) is as large as possible. Then G* consists of a subgraph H which is one of graphs B, B, and P | is the number of vertices of T including the vertex z. In the following, we prove that tree T is formed by attaching at most one path of length 1 and other paths of length 2 at z.Let P : v0v1 ⋯ vk is a pendant path of G* and vk is a pendant vertex. If k ≥ 3, let H1 = G* − v2v3 + v0v3. From H1 ∈ Bn(2μ) and q(H1) > q(G*), which is a contradiction.Suppose u ∈ V(T − z), we prove that d(u) ≤ 2. Otherwise, there must exist some vertex u0 of T − z such that d = max{d | v ∈ V(T) − z, d(v) ≥ 3}. From the above proof, we have the pendant paths attached u0 which have length of at most 2. Obviously, there exists an internal path between u0 and some vertex w of G*, denoted by m ≥ 2, let H2 be the graph obtained from G* − u0w1 − w1w2 by amalgamating u0, w1, and w2 to form a new vertex s1 together with attaching a new pendant path s1s2s3 of length 2 at s1. From H2 ∈ Bn(2μ) and q(H2) > q(G*), which is a contradiction. If m = 1, by H3 such that H3 ∈ Bn(2μ) and q(H3) > q(G*), which is a contradiction.For each vertex T which is obtained by attaching some pendant paths of length 2 and at most one pendant path of length 1 at z.From the proof as above, we have the tree G* which must be attached at the same vertex of H.From G* is isomorphic to one of graphs G1, G2,…, G6 ). We prove that G* is isomorphic to one of graphs G1, G2, and G3.Cp of G* with length of at least 4. From G*, which is not a triangle, has length 1. Note that all the pendant paths of length 2 in G* must be attached at the same vertex, then there must exist edges v1v2 ∈ E(G*), v1v3 ∈ E(Cp), and v1v4 ∈ E(Cp) and d(v1) = 3, d(v2) = 1, d(v3) ≥ 3, and d(v4) ≥ 3. Let H4  (H5) be the graph obtained from G* − v1v3  (G* − v1v4) by amalgamating v1 and v3(v4) to form a new vertex y1(y3) together with subdividing the edge y1v2  (y3v2) with a new vertex y2  (y4). From Hi ∈ Bn+(2μ)   and either q(H4) > q(G*) or q(H5) > q(G*), which is a contradiction. Then for each cycle Cg of G*, we have g = 3.Assume that there exists some cycle l ≥ 4. If there exists an internal path G*. Then, by H6 such that q(H6) > q(G*) and H6 ∈ Bn+(2μ), which is a contradiction. Thus, d(vi) ≥ 3   and either d(v2) = 3 or d(v3) = 3. By H7 such that q(H7) > q(G*) and H7 ∈ Bn+(2μ), which is a contradiction. Hence, l ≤ 3.Assume that We distinguish the following three subcases:Subcase  1.1 (l = 1). Then G* is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of ,G¯5 see .xu ≥ xv, let H8 = G* − rv − sv + ru + su; if xv ≥ xu, let H9 = G* − ut + tv. Obviously, Hi ∈ Bn+(2μ)   and either q(H8) > q(G*) or q(H9) > q(G*) by Assume that Subcase  1.2 (l = 2). Then G* is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of G¯14 see .xv1 ≥ xv2, let H10 = G* − v2u + v1u; if xv2 ≥ xv1, let H11 = G* − v1r + v2r. Obviously, Hi ∈ Bn+(2μ)   and either q(H10) > q(G*) or q(H11) > q(G*) by Assume that Subcase  1.3 (l = 3). Then G* is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of G¯20 see .xv1 ≥ xv2, let H12 = G* − v2v3 + v1v3; if xv2 ≥ xv1, let H13 = G* − v1z1 + v2z1. Obviously, Hi ∈ Bn+(2μ)   and either q(H12) > q(G*) or q(H13) > q(G*) by Assume that G* is isomorphic to one of graphs G1, G2 and G3.Thus, G* is obtained by attaching all the pendant paths of length 2 at vertex y4 of xv1 ≥ xy4, let H14 be the graph obtained from μ − 3 pendant paths of length 2 at v1. If xy4 ≥ xv1, let H15 = G* − v1y3 − v1y1 − v1y2 + y4y3 + y4y1 + y4y2. Obviously, H14 = H15 = G1 and q(G1) > q(G*) by G* = G1. By similar reasoning, the result follows.Assume that Case  2 (G* ∈ Bn++(2μ)). By similar reasoning as that of Case  1, we have G* is the graph obtained by attaching all the pendant paths of length 2 at the same vertex of G¯24 see .G* is isomorphic to one of graphs G4, G5 and G6  > q(Gi), . Thus, G* = G1.So,


The aim of the present paper is to investigate coefficient estimates, Fekete-Szegő inequality, and upper bound of third Hankel determinant for some families of starlike and convex functions of reciprocal order. A denote the class of functions f(z) which are analytic in the open unit disk U = {z ∈ C : |z| < 1} and normalized byS*(α) and K(α) denote the usual classes of starlike and convex functions of order α, 0 ≤ α < 1, respectively. In 1975, Silverman , if it satisfies the inequalityA function Fλ(z) byqth Hankel determinant Hq(n), q ≥ 1, n ≥ 1, for a function f(z) ∈ A is studied by Noonan and Thomas  unless otherwise stated.Throughout in this paper we assume that For our results we will need the following Lemmas.q(z) is a function with Req(z) > 0 and is of the formIf q(z) is of the form (If the form  with posq(z) is of the form  ∈ L. Thenn = 3,4, 5,…Let q(z) byFλ(z) is given by (q(z) is analytic in U with q(0) = 1, Req(z) > 0.Let us define the function given by  with12), we hlity and 1−nAn≤2γ1lity and 1−nAn≤2γ1k,whereAk=1+λk−1λ = 0 and γ = 1 − α, we get the following result.If we take f(z) ∈ S∗(α). Then, for n = 3,4, 5,…, one hasa2| ≤ 2(1 − α).Let λ = 1 and γ = 1 − α, we get the following result.Making f(z) ∈ K∗(α). Then, for n = 3,4, 5,…, one hasa2| ≤ (1 − α).Let f(z) ∈ L and be of the form (Let the form . Then(3f(z) ∈ L. Then from  ∈ L. ThenLet f(z) ∈ L and be of the form (Let the form . Then(3f(z) ∈ L. Then, from /∂ρ > 0 for c ∈  and ρ ∈ , maximum of F will exist at ρ = 1 and let F = G(c). Thenc, we obtainG(c)/∂c > 0 for c ∈ , G(c) has a maximum value at c = 2 and henceLet en, from , we havef(z) ∈ L and be of the form  ∈ L and is of the form (If the form , then(4Since


We give further improvements of the Jensen inequality and its converse on time scales, allowing also negative weights. These results generalize the Jensen inequality and its converse for both discrete and continuous cases. Further, we investigate the exponential and logarithmic convexity of the differences between the left-hand side and the right-hand side of these inequalities and present several families of functions for which these results can be applied. By a time scale 𝕋 we mean any nonempty closed subset of real numbers. Using the delta and nabla derivatives, the notions of delta and nabla integrals were defined , was inf : 𝕋 → ℝ be continuous function and let a, b ∈ 𝕋. Then, the diamond-α integral of f from a to b is defined by Let α = 1 the diamond-α integral reduces to the standard delta integral and for α = 0 the diamond-α integral reduces to the standard nabla integral.From the above definition it is clear that for 𝕋 = ℝ, then 𝕋 = ℤ, then 𝕋 = hℤ, where h > 0, then 𝕋 = qℕ0, where q > 1, then Moreover, if Recently, the Jensen inequality, the improvement of the Jensen inequality, and their converses are given for time scale integrals see –4)..4]).𝕋 be a time scale, and let a, b ∈ 𝕋. Then the time scale interval is denoted and defined by 𝕋 = ∩𝕋.Let c, d ∈ ℝ. If g ∈ C), w ∈ C with ∫ab | w(t) | ⋄αt > 0 and Φ ∈ C, ℝ) is convex, then Let α-Steffensen-Popoviciu (α-SP) weight.Note that in this Jensen inequality we have nonnegative weights. In order to give a better version of this Jensen inequality on time scales, Dinu  gives thg ∈ C. Then, w ∈ C is an α-Steffensen-Popoviciu (α-SP) weight for g on 𝕋 if C, where Let α-SP weight for a nondecreasing function g on time scales.In the following lemma he gives a characterization for w ∈ C such that ∫abw(t)⋄αt > 0. Then, w is an α-SP weight for a nondecreasing function g ∈ C if and only if it verifies the following condition: s ∈ 𝕋.Let w is also an α-SP weight for the nondecreasing continuous function g.If the following stronger (but more suitable) condition holds α-SP weights, for any continuous function g and every α ∈ . But there are some α-SP weights that are allowed to take the negative values. The Jensen inequality on time scales, where it is allowed that the weight function takes some negative values, is given in the following theorem.As given in , all posg ∈ C and let w ∈ C such that ∫abw(t)⋄αt > 0. Then, the following two statements are equivalent: (i)w is an α-SP weight for g on 𝕋; (ii) for every convex function Φ ∈ C, it holds Let g be nondecreasing function. If 𝕋 = ℕ, then 𝕋 = ℝ in Let α-Hermite-Hadamard (α-HH) weight, its characterization for a nondecreasing function g on time scales, and the improvement of the converse of the Jensen inequality for some negative weights.Considering the converse of the Jensen inequality, Dinu  gives thg ∈ C. Then w ∈ C is an α-Hermite-Hadamard (α-HH) weight for g on 𝕋 if C, where Let w ∈ C be such that ∫abw(t)⋄αt > 0. Then w is an α-HH weight for a nondecreasing function g ∈ C if and only if it verifies the following condition: s ∈ 𝕋.Let In the next result Dinu  gives thg ∈ C. Then every α-SP weight for g on 𝕋 is an α-HH weight for g on 𝕋, for all α ∈ .Let In the following two sections of our paper we give some further generalizations of the Jensen-type inequalities on time scales allowing negative weights, and we also give the mean-value theorems of the Lagrange and Cauchy type for the functionals obtained by taking the difference of the left-hand side and right-hand side of these new inequalities. These results also generalize the results given in  for contm, M ∈ ℝ, where m ≠ M. Consider the Green function G : × → ℝ defined by G is convex and continuous with respect to both x and y.Let C2 can be represented by G is defined in  any fined in . Using (ented by Φ(x)=M−xMIn the following theorem, we give the generalization of the Jensen inequality on time scales, where negative weights are also allowed.g ∈ C be such that g⊆. Let w ∈ C be such that ∫abw(t)⋄αt ≠ 0 and ∫abg(t)w(t)⋄αt/∫abw(t)⋄αt ∈ . Then, the following two statements are equivalent: (i)Cfor every convex function Φ ∈ (ii)y ∈ G : × → ℝ is defined in  and (ii) are also equivalent if we change the sign of inequality in both Φ(∫abg(t) ∈ G(∫abg(t)G  is continuous and convex, it follows that ((i) ⇒ (ii): let (i) hold. As the function ows that  holds.C2. Then by using (y) ≥ 0 for all y ∈ , and hence it follows that for every convex function Φ ∈ C2 inequality ((ii) ⇒ (i): let (ii) hold. Let Φ ∈ by using  we get  the reverse inequality in  and (ii′) are also equivalent if we change the sign of inequality in both statements (i′) and (ii′).Let the conditions of g is nondecreasing and that it has the first derivative. Let m = g(a) and M = g(b), and make the substitution y = g(s). Then we get g is nondecreasing, we have that g′(s) ≥ 0 for all s ∈ 𝕋. If Φ ∈ C2 is convex, then Φ′′(g(s)) ≥ 0 for all s ∈ 𝕋. Hence, if and only if s ∈ 𝕋, then for every continuous convex function Φ inequality  and let w ∈ C be such that ∫abw(t)⋄αt > 0. Then w is an α-SP weight for g on 𝕋 if and only if y ∈ , where G is defined in  be nondecreasing function and w ∈ C such that ∫abw(t)⋄αt > 0. Then s ∈ 𝕋, if and only if y ∈ , where G is defined in : m, M]. Clearly, if Φ is continuous and convex, then 𝒥1 is nonnegative.Under the assumptions of g ∈ C and Φ ∈ C2. Let 𝒥1 be defined as in  = x2/2.Let ed as in . Then thm, M], applying the integral mean-value theorem on ϕ1 and ϕ2 are continuous and convex, we have ξ ∈  such that  and Φ, Ψ ∈ C2. Let 𝒥1 be defined as in J1χ be defined as the linear combination of functions Φ and Ψ by χ ∈ C2. By applying χ, it follows that there exists ξ ∈  such that 𝒥1 = 0. By hypothesis 𝒥1 ≠ 0  ≠ 0), so it follows that Let alent to .In x) = x2/2 in Note that setting the function Ψ as Ψ, Φ, Ψ :  → ℝ, and let w ∈ C be an α-SP weight for g. Let 𝒥1 be defined as in ((i)C2, then there exists ξ ∈  such that (If Φ ∈ uch that  holds.(ii)C2, then there exists ξ ∈  such that  (statement (ii), resp.,) directly follows from g ∈ C be monotone function, Φ, Ψ :  → ℝ, and w ∈ C such that ∫abw(t)⋄αt > 0. Let C2, then there exists ξ ∈  such that (If Φ ∈ uch that  holds.(ii)C2, then there exists ξ ∈  such that  (statement (ii), resp.,) directly follows from Using the similar method as in previous section, in the following theorem we obtain the generalization of the converse of the Jensen inequality on time scales, where negative weights are also allowed.g ∈ C be such that g⊆ and let c, d ∈   (c ≠ d) be such that c ≤ g(t) ≤ d for all t ∈ 𝕋. Let w ∈ C be such that ∫abw(t)⋄αt ≠ 0. Then, the following two statements are equivalent. (i)CFor every convex function Φ ∈ (ii)y ∈ G : × → ℝ is defined in  and (ii) are also equivalent if we change the sign of inequality in both ∫abΦ(g(t) ∈ ∫abG(g(t)The idea of the proof is very similar to the proof of G  is continuous and convex, it follows that ((i) ⇒ (ii): let (i) hold. As the function ows that  holds.C2. Then by using (y) ≥ 0 for all y ∈ , and hence it follows that for every convex function Φ ∈ C2 the inequality ((ii) ⇒ (i): let (ii) hold. Let Φ ∈ by using  we get , the reverse inequality in  and (ii′) are also equivalent if we change the sign of inequality in both statements (i′) and (ii′).Let the conditions of m, M], while in the results from the previous section we demanded that Note that in all the results in this section we allow that the mean value c = m and d = M in Setting g ∈ C be such that g⊆. Let w ∈ C be such that ∫abw(t)⋄αt ≠ 0. Then, the following two statements are equivalent. (i)CFor every convex function Φ ∈ (ii)y ∈ G : × → ℝ is defined as in  and (ii) are also equivalent if we change the sign of inequality in both ∫abΦ(g(t) ∈ ∫abG(g(t)As a consequence of c = m and d = M. Then (Let  M. Then  transforabw(t)⋄αt > 0, and suppose that g is nondecreasing and that it has the first derivative. Now, similarly as in 𝕋. If Φ ∈ C2 is convex, then Φ′′(g(s)) ≥ 0 for all s ∈ 𝕋. Hence, if and only if s ∈ 𝕋, then for every continuous convex function Φ inequality equality .g ∈ C and let w ∈ C be an α-SP weight for g on 𝕋. Then y ∈ .Let The proof follows directly from 𝒥2: m, M]. Clearly, if Φ is continuous and convex, then 𝒥2 is nonnegative.Under the assumptions of g ∈ C and Φ ∈ C2. Let 𝒥2 be defined as in  = x2/2.Let ed as in . Then thThe idea of the proof is very similar to the proof of m, M], applying the integral mean-value theorem on  and Φ, Ψ ∈ C2. Let 𝒥2 be defined as in J2The proof is very similar to the proof of First we recall some definitions and facts about exponentially convex and logarithmically convex functions  which wf : I → ℝ(I⊆ℝ) is n-exponentially convex in the Jensen sense on I, if ξi ∈ ℝ and xi ∈ I, i = 1,…, n.A function f : I → ℝ is n-exponentially convex if it is n-exponentially convex in the Jensen sense and continuous on I.A function n-exponentially convex functions in the Jensen sense are k-exponentially convex in the Jensen sense for every k ∈ ℕ, k ≤ n.It is clear from the definition that 1-exponentially convex functions in the Jensen sense are in fact nonnegative functions. Also, By definition of positive semidefinite matrices and some basic linear algebra, we have the following proposition.f is an n-exponentially convex function in the Jensen sense, then the matrix [f((xi + xj)/2)]i,j=1k is positive, semidefinite for all k ∈ ℕ, k ≤ n. Particularly, det[f((xi + xj)/2)]i,j=1k ≥ 0 for all k ∈ ℕ, k ≤ n.If f : I → ℝ is exponentially convex in the Jensen sense on I, if it is n-exponentially convex in the Jensen sense for all n ∈ ℕ.A function f : I → ℝ is exponentially convex if it is exponentially convex in the Jensen sense and continuous.A function (i)f : I → ℝ defined by f(x) = cekx, where c ≥ 0 and k ∈ ℝ;(ii)f : ℝ+ → ℝ defined by f(x) = xk−, where k > 0;(iii)f : ℝ+ → ℝ+ defined by k > 0.Some examples of exponentially convex functions are (see ) as follf : I → ℝ+ is log-convex in the Jensen sense on I, if and only if ρ, σ ∈ ℝ and x, y ∈ I. It follows that a positive function is log-convex in the Jensen sense if and only if it is 2-exponentially convex in the Jensen sense. Also, using basic convexity theory, it follows that a positive function is log-convex if and only if it is 2-exponentially convex.It is known (and easy to show) that The following lemma is equivalent to the definition of convex function x1, x2, x3 ∈ I are such that x1 < x2 < x3, then the function f : I → ℝ is convex if and only if the following inequality holds: If We will also need the following result .f : I → ℝ is a convex function and x1, x2, y1, y2 ∈ I are such that x1 ≤ y1, x2 ≤ y2, x1 ≠ x2, and y1 ≠ y2, then the following inequality is valid: If When dealing with functions with different degree of smoothness, divided differences are found to be very useful.f : I → ℝ at mutually different points x0,   x1,   x2 ∈ I is defined recursively by The second order divided difference of a function x0, x1, x2; f] is independent of the order of the points x0, x1, and x2. This definition may be extended to include the case in which some or all the points coincide  such that the function ρ ↦  is n-exponentially convex in the Jensen sense on J for every choice of three mutually different points x0, x1, x2 ∈ . Then ρ ↦ 𝒥i is an n-exponentially convex function in the Jensen sense on J. If the function ρ ↦ 𝒥i is also continuous on J, then it is n-exponentially convex on J.Let fined in  and 50)𝒥i, i = ν : I → ℝ by ξj, ξk ∈ ℝ, rj, rk ∈ J, 1 ≤ j, k ≤ n, rjk = (rj + rk)/2, and Φrjk ∈ Ω. Using the assumption that for every choice of three mutually different points x0, x1, x2 ∈ ρ ↦  is n-exponentially convex in the Jensen sense on J, we obtain ν is convex (and continuous) function on I. Hence 𝒥i ≥ 0, i = 1,2, which implies that ρ ↦ 𝒥i is n-exponentially convex on J in the Jensen sense.Define the function ρ ↦ 𝒥i is continuous on J, then ρ ↦ 𝒥i is n-exponentially convex by definition.If The following corollary is an immediate consequence of 𝒥i, i = 1,2, be linear functionals defined in  such that the function ρ ↦  is exponentially convex in the Jensen sense on J for every choice of three mutually different points x0, x1, x2 ∈ . Then ρ ↦ 𝒥i is an exponentially convex function in the Jensen sense on J. If the function ρ ↦ 𝒥i is also continuous on J, then it is exponentially convex on J.Let fined in  and 50)𝒥i, i = 𝒥i, i = 1,2, be linear functionals defined in  such that the function ρ ↦  is 2-exponentially convex in the Jensen sense on J for every choice of three mutually different points x0, x1, x2 ∈ . Then, the following statements hold. (i)ρ ↦ 𝒥i is a 2-exponentially convex function in the Jensen sense on J.(ii)ρ ↦ 𝒥i is continuous on J, then it is also 2-exponentially convex on J. If ρ ↦ 𝒥i is additionally strictly positive, then it is also log-convex on J, and for r, s, t ∈ J such that r < s < t, we have If (iii)ρ ↦ 𝒥i is strictly positive and differentiable function on J, then for every p, q, u, v ∈ J such that p ≤ u, q ≤ v, we have p, Φq ∈ Ω.If Let fined in  and 50)𝒥i, i = ρ ↦ 𝒥i is continuous and strictly positive, its log-convexity is an immediate consequence of f(x) = log𝒥i and r, s, t ∈ J (r < s < t), we get (i) and the first part of (ii) are immediate consequences of equality .ρ ↦ 𝒥i be strictly positive and differentiable and therefore continuous too. By (ii), the function ρ ↦ 𝒥i is log-convex on J; that is, the function ρ ↦ log𝒥i is convex on J, and by p ≤ u, q ≤ v, p ≠ q, and u ≠ v, concluding that p = q and u = v follow from (To prove (iii), let , and by logJi, and furthermore, they still hold when all three points coincide for a family of twice differentiable functions with the same property. The proofs are obtained by recalling Note that the results from Ω = {Φρ : ρ ∈ J} in order to construct different examples of exponentially convex functions and construct some means.In this section we will vary on choice of a family d2/dx2)κρ(x) = eρx > 0 which shows that κρ is convex on ℝ for every ρ ∈ ℝ. From ρ ↦ (d2/dx2)κρ(x) is exponentially convex. Therefore, ρ ↦  is exponentially convex ,  are exponentially convex in the Jensen sense. It is easy to verify that these mappings are continuous, so they are exponentially convex.Consider a family of functions vex (see )  from  from  becomes nd using  we have 𝒥i   are positive, using Theorems κp ∈ Ω1 and Ψ = κq ∈ Ω1, it follows that ℵp,q∈. If we set g =  then we have that ℵp,q are means (of the function g). Note that by (If  that by  they ared2/dx2)βρ(x) = xρ−2 = eρ−2)logxβρ(x) is exponentially convex. Therefore ρ ↦  is exponentially convex . Here we assume that ⊂, so our family Ω2 of βρ fulfills the conditions of ℳp,q from  are positive, by applying Theorems βp ∈ Ω2 and Ψ = βq ∈ Ω2, it follows that for i = 1,2 there exist ξi ∈  such that ξi ↦ ξip−q is invertible for p ≠ q, we have Consider a family of functions  Ω) from  becomes ℳp,q is continuous, symmetric, and monotonous  = , then we have that ℳp,q are means (of the function g).Also, nous (by ). If we r in case g = . For r ≠ 0 by substituting g ↦ gr, p ↦ p/r, and q ↦ q/r in γρ(x) = ρx− > 0, which shows that γρ is convex function for ρ > 0. Also, from ρ ↦ (d2/dx2)γρ(x) is exponentially convex. Therefore ρ ↦  is exponentially convex . Here we assume that ⊂, so our family Ω3 of γρ fulfills the conditions of ℳp,q from (p and q.Consider a family of functions  Ω) from  becomes , and by  it is mo𝒥i   are positive, using Theorems γp ∈ Ω3 and Ψ = γq ∈ Ω3, it follows that ℵp,q∈. Here L is the logarithmic mean defined by L = (p − q)/(logp − logq) for p ≠ q, L = p.If δρ is convex function for ρ > 0. Also, from ρ ↦ (d2/dx2)δρ(x) is exponentially convex. Therefore ρ ↦  is exponentially convex . Here we assume that ⊂, so our family Ω4 of δρ fulfills the conditions of ℳp,q from (p and q by (Consider a family of functions  Ω) from  becomes and q by .𝒥i   are positive, using Theorems δp ∈ Ω4 and Ψ = δq ∈ Ω4, it follows that ℵp,q∈.If


Under the new Hölder conditions, we consider the convergence analysis of the inverse-free Jarratt method in Banach space which is used to solve the nonlinear operator equation. We establish a new semilocal convergence theorem for the inverse-free Jarratt method and present an error estimate. Finally, three examples are provided to show the application of the theorem. C is the set of all continuous functions in ; k is the Green function:We consider the following boundary value problem:x′′=−λG(xion (see ):(2)x(sF(s) = 0, whereF is defined on an open convex Ω of a Banach space X with values in a Banach space Y.Instead of conomics , chemist0, whereF:Ω⊂Ca,b⟶conomics –8 are soF(x) = 0. Particularly iterative methods are often used to solve this problem ., 10.F(x)To improve the convergence order, many modified methods have been presented. The famous Halley's method and the supper-Halley method are the third-order convergence. References –22 give ω :  and z ∈ (1/(1 − f(a0)g)); hence, x1, y0 ∈ S. ConsiderF′(x1)−1 exists, andBy (A1)–(A6), F′(x2)−1F′(x0) exists, and ‖F′(x2)−1F′(x0)‖ ≤ f(a1)‖F′(x1)−1F′(x0)‖. By induction, we can prove that the following By n ≥ 1:F′(xn)−1F′(x0) exists and ‖F′(xn)−1F′(x0)‖  ≤f(an−1)‖F′(xn−1)−1F′(x0)‖;yn − xn‖ ≤ f(an−1)g‖yn−1 − xn−1‖;‖H ≤ M‖F′(xn)−1F′(x0)‖‖yn − xn‖ ≤ an;N‖F′(xn)−1F′(x0)‖‖yn − xn‖2 ≤ bn;F′(xn)−1F′(x0)‖ω(‖yn − xn‖)‖yn − xn‖2 ≤ cn;‖xn+1 − yn‖ ≤ (an/2)(1 + an)‖yn − xn‖;‖xn+1 − xn‖ ≤ [1 + (an/2)(1 + an)]‖yn − xn‖;‖xn+1 − x0‖ ≤ Rη, where R = [1 + (a0/2)(1 + a0)] (1/(1 − f(a0)g)).‖Under the hypotheses of X and Y be two Banach spaces and F : Ω ⊂ X → Y has continuous Fréchet derivative of the third-order on a nonempty open convex Ω. One supposes that Γ0 = F′(x0)−1 ∈ L exists for some x0 ∈ Ω and conditions (A1)–(A6) and xn} generated by x* of S − Rη)∩Ω. Furthermore, the following error estimate is obtained:γ = f2(a0)g = a1/a0 and Δ = 1/f(a0), R = (1 + (a0/2)(1 + a0)) (1/(1 − γΔ)).Let xn} is a Cauchy one. From (II) and by n ≥ 0, m ≥ 1,x)k − 1 > kx, so (3 + p)k − 1 > k(2 + p). Hence, we havexn} is a Cauchy sequence and x* = limn→∞xn. Obviously, xm ∈ B, for all m ≥ 1, as if n = 0 in yn ∈ B, for all n ≥ 0.Firstly, we prove that the sequence { we havexn+m−xn <n → ∞ in F′(xn)−1F(xn)‖ → 0. Besides ‖F(xn)‖ → 0, since ‖F(xn)‖ ≤ ‖F′(xn)‖‖F′(xn)−1F(xn)‖ and {‖F′(xn)‖} is a bounded sequence, therefore F(x*) = 0 by the continuity of F in Now, let m → ∞ in By letting y* of S − Rη)∩Ω. Then01F′(x* + t(y* − x*))dt exists andx* = y*.To show uniqueness, let us assume that there exists a second solution This completes the proof of In this section, we apply the convergence theorem and show three numerical examples.F(x) = x10/3 + x7/2 − x − 1 = 0 on x ∈ . Then, we easily get thatK, p) Hölder conditionp ∈  ≤ t1/3ω(z) for t ∈  and z ∈  → C,K, p) Hölder condition; for instance, the max-norm is considered. In this case,ω(z) = Σi=1mLizpi satisfy ω(tz) ≤ tqω(z), where q = min{pi, p2,…, pm}.Let us consider a particular case of F′′′ is Lipschitz continuous in Ω, we can choose ω(z) = Kz, K > 0, so that Jarratt's method is of R-order, at least four order. If F′′′ is  Hölder continuous in Ω, then we can choose ω(z) = Lzp, L < 0, p ∈  = 1 and we look for a domain in the forma0 = Mη = 0.00123353 < 1/2, b0 = 6.52127 × 10−6, c0 = 1.47132 × 10−6, γ = f2(a0)g = 3.48167 × 10−7 < 1, Δ = 0.998766⋯, and R = 1.00062⋯. This means that the hypothesis of n = 1,2, 3,4, we getThis equation arises in the theory of the radiative transfer and neutron transport and in the kinetic theory of gasses. Let us define the operator F = 0, wherex0 =  =  and Ω = {x∣‖x − x0‖ ≤ 1.5}. We take the max-norm in R2 and the norm ‖A‖ = max{|a11| + |a12|, |a21| + |a22|} for B on R2 byu = T and Let us consider the system of equations η = ‖F′(x0)−1F(x0)‖ = 0.09598⋯, M = 9.20456⋯, N = 10.7635⋯, and p = 1/3.Then, we get the following results: We get that the hypotheses of


Sharp radius of Janowski starlikeness is obtained for functions f whose nth coefficient satisfies |an | ≤ cn + d   or |an | ≤ c/n  (c > 0  and  n ≥ 3). Other radius constants are also obtained for these functions, and connections with earlier results are made.Let  A denote the class of analytic functions f defined in the open unit disk D≔{z ∈ C : |z | <1}, normalized by f(0) = 0 = f′(0) − 1, and let S denote its subclass consisting of univalent functions. If f(z) = z + ∑n=2∞anzn ∈ S, de Branges . O. OS poss (n ≥ 2) –7. The n 1)cosβ , and thecosβ  of Janowski starlike functions  for appropriate choices of the parameters A and B. For example, for 0 ≤ β < 1, ST(β)≔ST is the familiar class of starlike functions of order β. Denote by STβ the class STβ≔L0 = ST. Janowski .The class unctions  consistsJanowski  obtainedAb consisting of functions f(z) = z + ∑n=2∞anzn,  , in the disk D. The subclass of univalent functions in Ab have been studied in -radius are derived for these classes. Several known radius constants are shown to be special cases of the results obtained.This paper finds radius constants for functions f ∈ A to belong to the class L is given in the following lemma.A sufficient condition for functions β ∈ R∖{1} and α ≥ 0. If f(z) = z + ∑n=2∞anzn ∈ A satisfies the inequalityf ∈ L.Let L-radius is obtained for f ∈ Ab satisfying the coefficient inequality |an | ≤cn + d.Making use of this lemma, the sharp β ∈ R∖{1}, 6α + 3 − β ≥ 0, and α ≥ 0. The L-radius for f(z) = z + ∑n=2∞anzn ∈ Ab satisfying the coefficient inequality |an | ≤cn + d, c, d ≥ 0, n ≥ 3, is the real root in  of the equationβ < 1, this number is also the L0-radius of f ∈ Ab. The results are sharp.Let r0 is the L-radius for f ∈ Ab if and only if f(r0z)/r0 ∈ L. Therefore, by r0 is the real root in  of  of . Using tβ < 1, consider the functionz = r0 in  of -radius for f ∈ Ab. For β < 1, (N(r0)/D(r0) is positive, and therefore the equalityr0 is the sharp L0-radius for f ∈ Ab when β < 1.For  of , f0 satatisfiesRe-radius of f(z) = z + ∑n=2∞anzn ∈ Ab satisfying the coefficient inequality |an | ≤c/n for n ≥ 3 and c > 0 is the real root in  of the equationβ < 1, this number is also the L0-radius of f ∈ Ab. The results are sharp.Let r0 is the L-radius of functions f ∈ Ab when inequality  (r0 of  and 9)  r0 is tβ < 1, consider the functionz = r0 in  of -radius for f ∈ Ab. For β < 1, the rational expression in -radius for f ∈ Ab. For β > 1, sharpness of the result is demonstrated by the function f0 given byTo verify sharpness for  of , f0 satatisfiesRe = z + ∑n=2∞anzn ∈ A satisfies the inequalityf ∈ ST.Let −1 ≤ ST-radius for f ∈ Ab satisfying the coefficient inequality |an | ≤cn + d.The next result finds the sharp B < A ≤ 1. The ST-radius for f(z) = z + ∑n=2∞anzn ∈ Ab satisfying the coefficient inequality |an | ≤cn + d, n ≥ 3 and c, d ≥ 0, is the real root in  of the equationLet −1 ≤ r0 is the ST-radius of f ∈ Ab if and only if f(r0z)/r0 ∈ ST. Hence, by r0 is the root in  of  of . From  of , and  of  of , the fun1.Then,  yields = z + ∑n=2∞anzn ∈ Ab satisfying the coefficient inequality |an | ≤c/n, n ≥ 3 and c > 0, is the real root in  of the equationLet −1 ≤ r0 is the ST-radius of f ∈ Ab where r0 is the real root of  of (f0 satisfies (By  root of . Therefo root of  and 19)r0 is thgiven by . Indeed, of . Evidentatisfies , and hen


We focus our discussion on the uncertainty measures of vague soft sets. Wepropose axiomatic definitions of similarity measure and entropy for vague soft sets. Furthermore, wepresent a new category of similarity measures and entropies for vague soft sets. The basic properties ofthese measures are discussed and the relationships among these measures are analyzed. In 1999, Molodtsov –11.Accordingly, works on soft set theory are progressing rapidly. Maji et al.  defined The measurement of uncertainty is an important topic for the theories dealing with uncertainty. Majumdar and Samanta  initiateIn this section, we recall some fundamental notions of soft sets and vague soft sets. See especially , 33, 34 U be a nonempty set, called universe. A fuzzy set μ on U is defined by a membership function μ : U → . For x ∈ U, the membership value μ(x) essentially specifies the degree to which x belongs to the fuzzy set μ. We denote by F(U) the set of all fuzzy sets on U.The theory of fuzzy sets initiated by Zadeh  providesAmong the extensions of the classic fuzzy set, vague set is one of the most popular sets treating imprecision and uncertainty. It was proposed by Gau and Buehrer .A over the universe U can be expressed by the notion A = {, 1 − fA(x)]); x ∈ U}; that is, A(x) = , and the condition 0 ≤ tA(x) ≤ 1 − fA(x) should hold for any x ∈ U, where tA(x) is called the membership degree (true membership) of element x to the vague set A, while fA(x) is the degree of nonmembership  of the element x to the vague set A.A vague set tA(x) is a lower bound on the grade of membership of x to A derived from the evidence for x and fA(x) is a lower bound on the negation of x derived from the evidence against x. The vague value  indicates that the exact grade of membership of x to A may be unknown, but it is bounded by tA(x) and 1 − fA(x).In this definition, μ corresponds to the following vague set:Every fuzzy set A, B be two vague sets over the universe U. The union, intersection, and complement of vague sets are defined as follows:Let A, B be two vague sets over the universe U. If, for  all  x ∈ U, tA(x) ≤ tB(x), 1 − fA(x) ≤ 1 − fB(x), then A is called a vague subset of B, denoted by A⊆B.Let The operations on vague sets are natural generalizations of the corresponding operations on fuzzy sets. Also, the notion of vague subset is a generalization of the notion of fuzzy subset.U be the universe set and E the set of all possible parameters under consideration with respect to U. Usually, parameters are attributes, characteristics, or properties of objects in U.  will be called a soft space. Molodtsov defined the notion of a soft set in the following way.In 1999, Molodtsov  be a mapping. For  ∈ VSS(U), H is called the entropy of  if it satisfies the following conditions:H1) = 0⇔for  all  e ∈ E, x ∈ U, tF(e)(x) = 0, fF(e)(x) = 1, or tF(e)(x) = 1, fF(e)(x) = 0;H2) = 1⇔for  all  e ∈ E, x ∈ U, tF(e)(x) = fF(e)(x);H3) = Hc);H4)⊆ and tG(e)(x) ≤ fG(e)(x), or ⊇ and tG(e)(x) ≥ fG(e)(x), then H ≤ H.for  all  Let H1) means the entropy of a soft set is minimal. By (H2), the entropy of the most vague soft set is maximal. (H3) means the entropies of a vague soft set and its complement are equal. (H1), (H2), and (H3) are natural generalizations of the corresponding conditions needed for the entropy of vague set. Now we focus our discussion on the condition (H4). If ⊆, tG(e)(x) ≤ fG(e)(x) for any e ∈ E, x ∈ U, then tG(e)(x) ≤ 0.5 by tG(e)(x) ≤ 1 − fG(e)(x). In this case, ⊆ means F(e) is crisper than G(e) for any e ∈ E. Similarly, if ⊇, tG(e)(x) ≥ fG(e)(x) for any e ∈ E, x ∈ U, then F(e) is crisper than G(e) for any e ∈ E. Thus (H4) is reasonable, but it is not complete. In fact, let  be crisper that . There may exist parameters e1, e2 ∈ E such that F(e1)⊆G(e1) and F(e2)⊇G(e2). Also, for a specific parameter e ∈ E, there may exist some elements x ∈ U1 ⊂ U such that tG(e)(x) ≤ fG(e)(x), tF(e)(x) ≤ tG(e)(x), and fF(e)(x) ≥ fG(e)(x), and tG(e)(x) ≥ fG(e)(x), tF(e)(x) ≥ tG(e)(x), and fF(e)(x) ≤ fG(e)(x) for any x ∈ U − U1. In these cases, (H4) cannot guarantee H ≤ H because ⊈ and ⊈. As illustration, we consider the following example.In this definition,  and  are defined by(1) Let F(e1) and F(e2) are crisper than G(e1) and G(e2), respectively. We noticed that ⊈ and ⊈. Thus (H4) can not guarantee H ≤ H.By the interpretation of true membership and false membership, E = {e}, U = {x1, x2}. The vague soft sets  and  are defined by(2) Let F(e) is crisper than G(e). Also, ⊈ and ⊈. (H4) can not guarantee H ≤ H.Similarly, U) is the set of all vague soft sets over U, but (H1)~(H4) are all talking about the vague soft sets with the whole parameter set E. Thus, the entropy derived from this definition is actually a partial entropy.By the way, VSS→ be a mapping. For  ∈ VSS(U), H is called the entropy of  if it satisfies the following conditions:H1) = 0⇔for  all  e ∈ A, x ∈ U, tF(e)(x) = 0, fF(e)(x) = 1, or tF(e)(x) = 1, fF(e)(x) = 0;H2) = 1⇔for  all  e ∈ A, x ∈ U, tF(e)(x) = fF(e)(x);H3) = Hc);H4),  ∈ VSS(U). If for  all  e ∈ A, x ∈ U, tF(e)(x) ≤ tG(e)(x), fF(e)(x) ≥ fG(e)(x) whenever tG(e)(x) ≤ fG(e)(x), and tF(e)(x) ≥ tG(e)(x), fF(e)(x) ≤ fG(e)(x) whenever tG(e)(x) ≥ fG(e)(x), then H ≤ H.Let  × VSS(U)→ be a mapping. For  ∈ VSS(U) and  ∈ VSS(U), M, ) is called the degree of similarity between  and  if it satisfies the following conditions:M1), ) = M, );  M2), )∈;  M3), ) = 1⇔ = ;  M4), ) = 0⇔for  all  e ∈ E, x ∈ U, tF(e)(x) = 0, fF(e)(x) = 1, tG(e)(x) = 1, fG(e)(x) = 0, or tF(e)(x) = 1, fF(e)(x) = 0, tG(e)(x) = 0, fG(e)(x) = 1;  M5)⊆⊆ implies M, ) ≤ min, ), M, )).  (Let d : VSS(U) × VSS(U)→ be a mapping. For  ∈ VSS(U) and  ∈ VSS(U), d, ) is called the degree of distance between  and  if it satisfies the following conditions:d1), ) = d, );d2), )∈;d3), ) = 1⇔for  all  e ∈ E, x ∈ U, tF(e)(x) = 0, fF(e)(x) = 1, tG(e)(x) = 1, fG(e)(x) = 0, or tF(e)(x) = 1, fF(e)(x) = 0, tG(e)(x) = 0, fG(e)(x) = 1;d4), ) = 0⇔ = ;d5)⊆⊆ implies d, ) ≥ max, ), d, ))., ) and d, ) be the similarity measure and distance measure between two vague soft sets  and  as defined in Definitions M, ) and d, ) can be given as follows:Let M, ) + d, ) = 1 just by the definitions of similarity measure and distance measure. Furthermore, in Definitions E are compared. Thus the similarity measure and distance measure are all partial measures. By the way, conditions (M2) and (d2) are clearly not necessary because the codomain of M and d has already been restricted to .We notice that this theorem holds for the special similarity measure and distance measure presented in Theorems 3.2 and 3.3 . But it Similarity measure and distance measure are closely related. In what follows, we focus our discussion on entropy and similarity measure. Taking the above observations into account, we propose the following definition of similarity measure for vague soft sets.M : VSS(U) × VSS(U)→ be a mapping. For  ∈ VSS(U) and  ∈ VSS(U), M, ) is called the degree of similarity between  and  if it satisfies the following conditions:M1), ) = M, );M2), ) = 1⇔ = ;M3), ) = 0⇔for  all  e ∈ A∩B (A∩B ≠ ∅), x ∈ U, tF(e)(x) = 0, fF(e)(x) = 1, tG(e)(x) = 1, fG(e)(x) = 0, or tF(e)(x) = 1, fF(e)(x) = 0, tG(e)(x) = 0, fG(e)(x) = 1;M4)⊆⊆ implies M, ) ≤ min, ), M, )). bya ≥ 0, b ≥ 0, and a + b > 0; then, Nθ and Nδ are similarity measures for fuzzy sets in the sense thatNθ = 0, Nδ = 0 and Nθ = 1, Nδ = 1 whenever A ∈ F(U);Nθ = Nθ, Nδ = Nδ whenever A, B ∈ F(U);A, B, C ∈ F(U), Nθ ≤ min, Nθ), Nδ ≤ min, Nδ) whenever A⊆B⊆C.for all Suppose a and b, one can obtain some typical similarity measures for fuzzy sets  is a similarity measure, where, for any ,  ∈ VSS(U),M1) is trivial.(M2) If  = , then A = B and tF(e) = tG(e), fF(e) = fG(e) for any e ∈ A. It follows that Nθ(tF(e), tG(e)) = 1, Nθ(fF(e), fG(e)) = 1 and hence Nθ(F(e), G(e)) = 1. Thus Mθ, ) = (1/|A|)∑e∈ANθ(F(e), G(e)) = |A|/|A| = 1., ) = 1. By Nθ(F(e), G(e)) ≤ 1 we have 1 = Mθ, ) ≤ |A∩B|/|A ∪ B|. It follows that A = B and Nθ(tF(e), tG(e)) = 1 and Nθ(fF(e), fG(e)) = 1 for any e ∈ A. If there exist e ∈ A, x ∈ U such that tF(e)(x) ≠ tG(e)(x), thenNθ(tF(e), tG(e)) < 1. This is a contradiction. Thus tF(e)(x) = tG(e)(x) for each e ∈ A, x ∈ U. Similarly, we have fF(e)(x) = fG(e)(x) for each e ∈ A, x ∈ U. Consequently, we have  = .Conversely, we assume that M3) Let M, ) = 0. It follows that, for each e ∈ A∩B, Nθ(F(e), G(e)) = 0 and hence Nθ(tF(e), tG(e)) = 0, Nθ(fF(e), fG(e)) = 0. By Nθ(tF(e), tG(e)) = 0 we have, for each x ∈ U, tF(e)(x) = 0, tG(e)(x) = 1 or tF(e)(x) = 1, tG(e)(x) = 0.tF(e)(x) = 0, tG(e)(x) = 1, then fG(e)(x) = 0 by tG(e)(x) + fG(e)(x) ≤ 1. Thus tG(e)(x) = 1 by Nθ(fF(e), fG(e)) = 0.If tF(e)(x) = 1, tG(e)(x) = 0, then fF(e)(x) = 0 by tF(e)(x) + fF(e)(x) ≤ 1. Therefore fG(e)(x) = 1 by Nθ(fF(e), fG(e)) = 0.If  Let , ,  ∈ VSS(U) and ⊆⊆. It follows that A⊆B⊆C and F(e)⊆G(e)⊆P(e) for each e ∈ A. Consequently, tF(e)⊆tG(e)⊆tP(e), fP(e)⊆fG(e)⊆fF(e). Thus we have Nθ(tF(e), tP(e)) ≤ Nθ(tF(e), tG(e)), Nθ(tF(e), tP(e)) ≤ Nθ(tG(e), tP(e)), Nθ(fF(e), fP(e)) ≤ Nθ(fF(e), fG(e)), Nθ(fF(e), fP(e)) ≤ Nθ(fG(e), fP(e)), and hence Nθ(F(e), P(e)) ≤ Nθ(F(e), G(e)), Nθ(F(e), P(e)) ≤ Nθ(G(e), P(e)). Therefore we obtain × VSS(U)→ is a similarity measure, where, for any ,  ∈ VSS(U),It can be proved in the same manner with a = 0, b = 1; then, we havea = 0, b = 2; then, we haveIn , let a =U = {x1, x2}, E = {e1, e2, e3, e4}, A = {e1, e2, e3}, and B = {e1, e2, e4}. Suppose that  and  are vague soft sets over U given byLet By the definition, we haveIt follows thatSimilarly, we haveA, B ∈ F(U). We consider Nθ. For any x ∈ U, by 1 − |A(x) − B(x)| ≥ 0, min(A(x), B(x)) ≥ 0, a − a|A(x) − B(x)| + b · min(A(x), B(x)) = a(1 − |A(x) − B(x)|) + b · min(A(x), B(x)) we conclude that ∑x∈U(a − a | A(x) − B(x)|+b · min(A(x), B(x))) is increasing with respect to a and b. Thereforea and b. ByNθ is increasing with respect to a and b. Thus we have the following corollary.Let F, A),  ∈ VSS(U). Then M1, ) ≤ M2, ), M1, ) ≤ M6, );M3, ) ≤ M4, ) ≤ M5, );M4, ) ≤ M6, ).,  ∈ VSS(U). Then M7, ) ≤ M8, ).Let  is an entropy, whereF, A) ∈ VSS(U).H1) We note that Mθ is a similarity measure. For any  ∈ VSS(U), we have Hθ = 0⇔Mθ, ) = 0⇔for  all  e ∈ A, x ∈ U, tF(e)(x) = 0, fF(e)(x) = 1, tFc(e)(x) = 1, fFC(e)(x) = 0, or tF(e)(x) = 1, fF(e)(x) = 0, tFc(e)(x) = 0, fFc(e)(x) = 1⇔for  all  e ∈ A, x ∈ U, tF(e)(x) = 0, fF(e)(x) = 1, or tF(e)(x) = 1, fF(e)(x) = 0.(H2) For any  ∈ VSS(U), we have Hθ = 1⇔Mθ, ) = 1⇔ = ⇔for  all  e ∈ A, x ∈ U, tF(e)(x) = tFc(e)(x), fF(e)(x) = fFc(e)(x)⇔for  all  e ∈ A, x ∈ U, tF(e)(x) = fF(e)(x).(H3) is trivial.(H4) Let ,  ∈ VSS(U), and for  all  e ∈ A, x ∈ U, tF(e)(x) ≤ tG(e)(x), fF(e)(x) ≥ fG(e)(x) if tG(e)(x) ≤ fG(e)(x), and tF(e)(x) ≥ tG(e)(x), fF(e)(x) ≤ fG(e)(x) if tG(e)(x) ≥ fG(e)(x). We note that Nθ(tF(e), tFc(e)) = Nθ(fF(e), fFc(e)) = Nθ(tF(e), fF(e)) and hence Hθ = (1/|A|)∑e∈ANθ(tF(e), fF(e))., fF(e)) ≤ Nθ(tG(e), fG(e)) and consequently, Hθ ≤ Hθ.By the definition, for each Hδ is an entropy, whereF, A) ∈ VSS(U).The proof is similar to that of Mi  (1 ≤ i ≤ 8), we can obtain the corresponding entropies Hi  (1 ≤ i ≤ 8) as follows:Using similarity measures F, A) given in We consider the vague soft set (Soft set theory was originally proposed as a general mathematical tool for dealing with uncertainties. Wang and Qu  introduc


Later, we obtain the initial fuzzy soft proximity determined by a family of fuzzy soft proximities. Finally, we investigate relationship between fuzzy soft proximities and proximities.We study the fuzzy soft proximity spaces in Katsaras's sense. First, we show how a fuzzy soft topology is derived from a fuzzy soft proximity. Also, we define the notion of fuzzy soft  In 1999, Molodtsov –9.Fuzzy soft set which is a combination of fuzzy set and soft set was introduced by Maji et al. . Roy andX as a map δ : LX × LX → {0,1} satisfying certain conditions, where L is a completely distributive lattice with an order reversing involution. In connection with a fuzzy topology in [Proximity structure was introduced by Efremovic in 1951 , 26. It ology in , the difology in . In 2005ology in  presenteExtensions of proximity structures to the soft sets and also fuzzy soft sets have been studied by some authors. More recently, Hazra et al.  defined δ on X induces a fuzzy soft topology τ(δ) on the same set. Also, we define the notion of fuzzy soft δ-neighborhood in a fuzzy soft proximity space and obtain a few results analogous to the ones that hold for δ-neighborhood in proximity spaces. We prove the existences of initial fuzzy soft proximity structures. Based on this fact, we introduce products of fuzzy soft proximity spaces. The relation between a fuzzy soft proximity and a proximity is also investigated.Motivated by their works, we continue investigating the properties of fuzzy soft proximity spaces in Katsaras's sense. We show that each fuzzy soft proximity X be an initial universe, IX be the set of all fuzzy subsets of X and E be the set of all parameters for X.In this section, we recall some basic notions regarding fuzzy soft sets which will be used in the sequel. Throughout this work, let f on the universe X with the set E of parameters is defined by the set of ordered pairsf is a mapping given by f : E → IX.A fuzzy soft set X is denoted by FS.Throughout this paper, the family of all fuzzy soft sets over f, g ∈ FS. Then, we have the following.f is called null fuzzy soft set, denoted by f(e) = 0X for every e ∈ E.The fuzzy soft set f(e) = 1X, for all e ∈ E, then f is called absolute fuzzy soft set, denoted by If f is a fuzzy soft subset of g if f(e) ≤ g(e) for each e ∈ E. It is denoted by f⊑g.f and g are equal if f⊑g and g⊑f. It is denoted by f = g.f is denoted by fc, where fc : E → IX is a mapping defined by fc(e) = 1X − f(e) for all e ∈ E. Clearly, (fc)c = f.The complement of f and g is a fuzzy soft set h defined by h(e) = f(e)∨g(e) for all e ∈ E. h is denoted by f⊔g.The union of f and g is a fuzzy soft set h defined by h(e) = f(e)∧g(e) for all e ∈ E. h is denoted by f⊓g.The intersection of Let J be an arbitrary index set and let {fi}i∈J be a family of fuzzy soft sets over X. Then,h defined by h(e) = ∨i∈Jfi(e) for every e ∈ E and this fuzzy soft set is denoted by ⊔i∈Jfi;the union of these fuzzy soft sets is the fuzzy soft set h defined by h(e) = ∧i∈Jfi(e) for every e ∈ E and this fuzzy soft set is denoted by ⊓i∈Jfi.the intersection of these fuzzy soft sets is the fuzzy soft set Let J be an index set and f, g, fi, gi ∈ FS, for all i ∈ J. Then, the following statements are satisfied:f⊓(⊔i∈Jgi) = ⊔i∈J(f⊓gi);f⊔(⊓i∈Jgi) = ⊓i∈J(f⊔gi);i∈Jfi)c = ⊔i∈Jfic;(⊓i∈Jfi)c = ⊓i∈Jfic; is a fuzzy point in X (x) = α ∈ (x′) = 0 for all x′ ∈ X − {x}) and f(e′) = 0X for every e′ ∈ E∖{e}. It will be denoted by exα.A fuzzy soft set exα is said to belong to a fuzzy soft set f, denoted by α ≤ f(e)(x).The fuzzy soft point FS and FS be the families of all fuzzy soft sets over X and Y, respectively. Let φ : X → Y and ψ : E → K be two mappings. Then, the mapping φψ is called a fuzzy soft mapping from X to Y, denoted by φψ : FS → FS.(1)f ∈ FS. Then φψ(f) is the fuzzy soft set over Y defined as follows:Let  k ∈ K and all y ∈ Y.for all  φψ(f) is called an image of a fuzzy soft set f.(2)g ∈ FS. Then φψ−1(g) is the soft set over X defined as follows:Let  e ∈ E and all x ∈ X.for all  φψ−1(g) is called a preimage of a fuzzy soft set g.The fuzzy soft mapping φψ is called injective, if φ and ψ are injective. The fuzzy soft mapping φψ is called surjective, if φ and ψ are surjective.Let fi ∈ FS and gi ∈ FS for all i ∈ J, where J is an index set. Then, for a fuzzy soft mapping φψ : FS → FS, the following conditions are satisfied:f1⊑f2, then φψ(f1)⊑φψ(f2);if g1⊑g2, then φψ−1(g1)⊑φψ−1(g2);if φψ(⊔i∈Jfi) = ⊔i∈Jφψ(fi);φψ(⊓i∈Jfi)⊑⊓i∈Jφψ(fi);φψ−1(⊔i∈Jgi) = ⊔i∈Jφψ−1(gi);φψ−1(⊓i∈Jgi) = ⊓i∈Jφψ−1(gi);Let f, fi ∈ FS for all i ∈ J, where J is an index set, and let g ∈ FS. Then, for a fuzzy soft mapping φψ : FS → FS, the following conditions are satisfied:f⊑φψ−1(φψ(f)), and the equality holds if φψ is injective;φψ(φψ−1(g))⊑g, and the equality holds if φψ is surjective;φψ(⊓i∈Jfi) = ⊓i∈Jφψ(fi) if φψ is injective;φψ is surjective.Let f ∈ FS and g ∈ FS. The fuzzy soft product f × g is defined by the fuzzy soft set h where h : E × K → IX×Y and h = f(e) × g(k) for all  ∈ E × K.Let f ∈ FS and g ∈ FS and let pX : X × Y → X, qE : E × K → E and pY : X × Y → Y, qK : E × K → K be the projection mappings in classical meaning. The fuzzy soft mappings (pX)qE and (pY)qK are called fuzzy soft projection mappings from X × Y to X and from X × Y to Y, respectively, where (pX)qE(f × g) = f and (pY)qK(f × g) = g.Let X is a fuzzy soft set. Also, every fuzzy soft set is a parameterized collection of fuzzy subsets in some universe.Every parameterized collection of fuzzy subsets in μα : α ∈ Δ} of fuzzy subsets in X. Then, f : Δ → IX defined by f(α) = μα is a fuzzy soft set over X.Consider any parameterized collection {τ be the collection of fuzzy soft sets over X; then τ is said to be a fuzzy soft topology on X iffst1) is called a fuzzy soft topological space. The members of τ are called fuzzy soft open sets in X. A fuzzy soft set f over X is called a fuzzy soft closed in X if fc ∈ τ.. Let us consider the following fuzzy soft sets on X with the set E of parameters:X.Let X, τ) be a fuzzy soft topological space and f ∈ FS. The fuzzy soft interior of f is the fuzzy soft set fo = ⊔{g : g  is  a  fuzzy  soft  open  set  and  g⊑f}.Let (fst2) for fuzzy soft open sets, fo is fuzzy soft open. It is the largest fuzzy soft open set contained in f.By property  be a fuzzy soft topological space and f ∈ FS. The fuzzy soft closure of f is the fuzzy soft set Let (X which contains f.Clearly f on X another fuzzy soft set fo1)(fo2)(fo3)(fo4), the fuzzy soft set f in the fuzzy soft topological space .Then, the familyThis operator is called the fuzzy soft closure operator.X, τ1) and  be two fuzzy soft topological spaces. A fuzzy soft mapping φψ : → is called fuzzy soft continuous if φψ−1(g) ∈ τ1 for every g ∈ τ2.Let  be a fuzzy soft topological space, where τ = {fα : α ∈ Δ}. Then, the collection τe = {fα(e)∣α ∈ Δ} for every e ∈ E defines a fuzzy topology on X.Let  : e ∈ E} be a parameterized family of fuzzy topological spaces. Let us define a fuzzy soft topological space  as the following: let τ be the collection of all the mappings f, where f : E → IX such that f(e) ∈ τe for each e ∈ E. Then, τ is a fuzzy soft topology on X. Indeed,fst1) ∈ τe for each e ∈ E; therefore, ⊔i∈Jfi is a mapping ⊔i∈Jfi : E → IX such that (⊔i∈Jfi)(e) = ∨i∈Jfi(e) ∈ τe for each e ∈ E; consequently, ⊔i∈Jfi ∈ τ;let {fst3), g(e) ∈ τe for each e ∈ E and hence f⊓g is a mapping f⊓g : E → IX such that (f⊓g)(e) = f(e)∧g(e) ∈ τe for each e ∈ E; thus, f⊓g ∈ τ.let Let {(p2)(Aδ(B ∪ C) if and only if AδB or AδC;p5):(p1)∅δδ.where X, δ) is called a proximity space; two subsets A and B of the set X are close with respect to δ if AδB; otherwise, they are remote with respect to δ.The pair (fp2)(μδ(ρ∨σ) if and only if μδρ or μδσ;fp5) comprising a set X and a fuzzy proximity δ on the set X.A fuzzy proximity space is a pair ×FS is called a Katsaras -soft fuzzy proximity on a set X, where E and K are arbitrary nonempty sets viewed on the sets of parameters, if, for any f, g, h ∈ FS and k ∈ K, the following conditions are satisfied:sfp1)(sfp2)(if sfp3)(fδk(g⊔h) if and only if fδkg or fδkh;sfp5) such that if A mapping X, δ) is called a Katsaras -soft fuzzy proximity space, where, for every k ∈ K, δk ⊂ FS × FS is a relation on FS.The pair  × FS is called a fuzzy soft proximity on X if δ satisfies the following conditions:fsp1)(fsp2)(fδ(g⊔h) if and only if fδg or fδh;fsp5) such that if A binary relation X, δ) is called a fuzzy soft proximity space.The pair  is a fuzzy soft proximity space, then it satisfies the following properties:fδg and k⊒f, h⊒g, then kδh;if fδf for each If  be a fuzzy soft proximity space. For every f ∈ FS, we defineLet  be a fuzzy soft proximity space. Then, the mapping fo1)–(fo4). Therefore, the collectionX.Let (fo1)–(fo4).We will show that the mapping fo1) Suppose that e ∈ E and x ∈ X. Take any g ∈ FS such that fsp2), g(e)(x) = 0 or f(e)(x) = 0. In both cases, we obtain g(e)(x) ≤ 1 − f(e)(x). Therefore, (fo2) It is enough to show that gδf if and only if fsp5), there is an h ∈ FS such that e ∈ E and an x ∈ X such that a, where exa1− ∈ FS. Since 1 − a ≤ 1 − h(e)(x), we have exa1−δf, since, otherwise, we would have exa1−δf and (fo3) It is easy to verify that e ∈ E and an x ∈ X such that ϵ > 0 satisfyingh ∈ FS such that h(e)(x) < a − ϵ. By the inequality k ∈ FS such that h(e)(x) − ϵ/2 < k(e)(x). Now, since fsp2), it follows that h(e)(x) − ϵ/2 < (k⊓h)(e)(x). Thus,(fo4) Because of  be a fuzzy soft proximity space. For f, g ∈ IX, the fuzzy soft set g is said to be a fuzzy soft δ-neighborhood of f if f⋐g.Let  be a fuzzy soft proximity space. Then the relation ⋐ satisfies the following properties:fspn1)(fspn2)(f⋐g implies fspn3)(f⋐g implies fspn4)(f⋐(g⊓h) if and only if f⋐g and f⋐h;fspn5) such that f⋐h⋐g.Let (fspn1) is obvious.(fspn2) If f⋐g, then fsp3), (fspn3) Let f⋐g. Then, from (fsp2), it follows that (fspn4) Consider f⋐g⊓hf⋐g and f⋐h.(fspn5) If f⊑g and (fspn6) Consider that f⋐g implies fsp5), there exists an h ∈ FS such that f⋐h⋐g. be a fuzzy soft proximity space and f, g ∈ FS. Then, the following statements are satisfied:f⋐g if and only if f⋐g, then there is a k ∈ τ(δ) such that f⊑k⊑g;if h, k such that f⋐h, g⋐k, and if Let (gδf if and only if (i) It is clear by the fact that f⋐g. Then, k ∈ τ(δ) and f⊑k⊑g.(ii) Let fsp5), there is a fuzzy soft set k such that h such that h and k such that f⋐h, g⋐k, and (iii) If FS satisfying (fspn1)−(fspn6). Then, δ is a fuzzy soft proximity on X defined as follows:g is a fuzzy soft δ-neighbourhood of f if and only if f⋐g.Let ⋐ be a relation on fsp1)–(fsp5).fsp1). By (fspn1), we have Let fsp2)(f⋐gc and from (fspn3) it follows thatLet fsp3)(f⋐gc. By (fspn2), g⋐fc and hence If fsp4)(Consider fsp5)(fspn6), there is a fuzzy soft set h such that Let We first need to verify axioms  is a fuzzy soft proximity space and f ∈ FS, thenIf  we obtain e ∈ E and an x ∈ X such that g(e)(x) : f⋐g} = a. Then there exists an ϵ > 0 such thatk such that k(e)(x) < a − ϵ. Because Let us take a fuzzy soft set X, δ1) and  be two fuzzy soft proximity spaces. A fuzzy soft mapping φψ : → is a fuzzy soft proximity mapping if it satisfiesf, g ∈ FS.Let  and  be two fuzzy soft proximity spaces. A fuzzy soft mapping φψ : → is a fuzzy soft proximity mapping if and only ifh, k ∈ FS.Let → is fuzzy soft continuous with respect to τ(δ1) and τ(δ2).A fuzzy soft proximity mapping f ∈ τ(δ2). Now let us take any h ∈ FS such that φψ is a fuzzy soft proximity mapping, we obtain fsp2), it follows that e ∈ E and every x ∈ X, we haveφψ−1(f) ∈ τ(δ1).Let δ1 and δ2 are two fuzzy soft proximities on X, we defineδ2 is finer than δ1, or δ1 is coarser than δ2.If We prove the existences of initial fuzzy soft proximity structure. Based on this fact, we define the product of fuzzy soft proximity spaces.X be a set and { : α ∈ Δ} a family of fuzzy soft proximity spaces, and, for each α ∈ Δ, let (φψ)α : FS→ be a fuzzy soft mapping. The initial structure δ is the coarsest fuzzy soft proximity on X for which all mappings (φψ)α : →  (α ∈ Δ) are fuzzy soft proximity mapping.Let X be a set { : α ∈ Δ} a family of fuzzy soft proximity spaces, and, for each α ∈ Δ, let (φψ)α : FS→ be a fuzzy soft mapping. For any f, g ∈ FS, define fδg if and only if, for every finite families {fi : i = 1,…, n} and {gj : j = 1,…, m}, where f = ⊔i=1nfi and g = ⊔j=1mgj, there exist an fi and a gj such thatδ is the coarsest fuzzy soft proximity on X for which all mappings (φψ)α : →  (α ∈ Δ) are fuzzy soft proximity mapping.Let δ is a fuzzy soft proximity on X.fsp1)(is obvious.fsp2). Conversely, assume that f = ⊔i=1nfi and g = ⊔j=1mgj of f and g, respectively, such that α = sij ∈ Δ, where i = 1,…, n and j = 1,…, m. In the same way, there are finite covers f = ⊔p=1qkp and h = ⊔j=m+1m+lgj of f and h, respectively, such that α = tpj ∈ Δ, where p = 1,…, q and j = m + 1,…, m + l. Now, f = ⊔{fi⊓kp : i = 1,…, n; p = 1,…, q} and g⊔h = ⊔{gj : j = 1,…, m + l} are finite covers of f and g⊔h, respectively. Hence, from the fact that α = sij or α = tpj, it follows that It is easy to verify that if fsp5) such that fδh or h ∈ FS. The validity of (fsp5) will follow from the fact that Ω is empty. Suppose, on the contrary, that  ∈ Ω. Then, (φψ)α(f)δα(φψ)α(g) for each α ∈ Δ. Indeed, let h ∈ FS and k = (φψ)α−1(h). If fδk, then (φψ)α(f)δα(φψ)α(k). Because (φψ)α(k)⊑h, we have (φψ)α(f)δαh. Similarly, if δα is a fuzzy soft proximity on Xα, we obtain (φψ)α(f)δα(φψ)α(g). Also, we observe that for each  ∈ Ω there are positive integers n, m and covers f = ⊔i=1nfi and g = ⊔j=1mgj such that, for every pair ∈{1,…, n}×{1,…, m}, there exists an α ∈ Δ satisfying l = n + m. It easy to see that l > 2. Then, for each  ∈ Ω, let us choose such an integer l. But l is not uniquely determined by . Let κ be the set of all integers corresponding to members of Ω and let l be the smallest member of κ. Take a  ∈ Ω such that l is the integer corresponding to it. Then, there are covers f = ⊔i=1nfi and g = ⊔j=1mgj such that l = n + m and for every pair ∈{1,…, n}×{1,…, m} and there exists an α ∈ Δ satisfying n, m is greater than 1. Consider n > 1 and let f′ = f1⊔⋯⊔fn−1. In this case, one of the following conditions should be true:Let us define the set (i)h ∈ FS, either f′δh or for every (ii)h ∈ FS, either fnδh or for every We first prove that h1, h2 ∈ FS such that h = h1⊓h2, we obtain f, g) ∈ Ω.In fact, suppose that neither (i) nor (ii) holds. Then, there are f′⊑f and f′, g) ∈ Ω. But this is now a contradiction since (n − 1) + m = l − 1 ∈ κ, contrary to the choice of l. If (ii) holds, we get a contradiction in a similar way. Therefore, the set Ω is empty. Thus, δ is a fuzzy soft proximity on X.Suppose that (i) holds. Because φψ)α : → are fuzzy soft proximity mapping. Let δ* be another fuzzy soft proximity on X making each of the mappings (φψ)α : → fuzzy soft proximity mapping. We will show that δ < δ*, which will complete the proof. Let fδ*g and consider any covers f = ⊔i=1nfi and g = ⊔j=1mgj of f and g, respectively. Since f = (f1⊔⋯⊔fn)δ*g, by (fsp4), there is an i ∈ {1,…, n} such that fiδg. In the same way, since fiδ*g = (g1⊔⋯⊔gm), by (fsp4), there is a j ∈ {1,…, m} such that fiδgj. From the fact that all mappings (φψ)α : → are fuzzy soft proximity mapping, it follows that (φψ)α(fi)δα(φψ)α(gj) for each α ∈ Δ. Thus, we get fδg.It is easy to see that all mappings → is a fuzzy soft proximity mapping if and only if (φψ)α∘φψ : → is a fuzzy soft proximity mapping for every α ∈ Δ.A fuzzy soft mapping φψ)α∘φψ is a fuzzy soft proximity mapping for every α ∈ Δ. Let fδ*g and let φψ(f) = ⊔i=1nfi and φψ(g) = ⊔j=1mgj. Then, we havefδ*g, by (fsp4), there exist i, j such that φψ−1(fi)δ*φψ−1(gj). Becauseφψ)α(fi)δα(φψ)α(gj) for every α ∈ Δ. This shows that φψ(f)δφψ(g).The necessity is easy. We prove the sufficiency. Suppose that  : α ∈ Δ} be a family of fuzzy soft proximity spaces and let X = ∏α∈ΔXα and E = ∏α∈ΔEα be product sets. An initial fuzzy soft proximity structure δ = ∏α∈Δδα on X with respect to all the fuzzy soft projection mappings (pXα)qEα, where pXα : X → Xα and qEα : E → Eα, is called the product fuzzy soft proximity structure.Let { is said to be a product fuzzy soft proximity space. : α ∈ Δ} be a family of fuzzy soft proximity spaces. Let X = ∏α∈ΔXα and E = ∏α∈ΔEα be sets and for each α ∈ Δ let (pXα)qEα be a fuzzy soft mapping. For any f, g ∈ FS, define fδg if and only if, for every finite families {fi : i = 1,…, n} and {gj : j = 1,…, m}, where f = ⊔i=1nfi and g = ⊔j=1mgj, there exist an fi and a gj such that (pXα)qEα(fi)δα(pXα)qEα(gj) for each α ∈ Δ. Then,δ = ∏α∈Δδα is the coarsest fuzzy soft proximity on X for which all mappings (pXα)qEα  (α ∈ Δ) are fuzzy soft proximity mapping;φψ : → is a fuzzy soft proximity mapping if and only if (pXα)qEα∘φψ : → is a fuzzy soft proximity mapping for every α ∈ Δ.a fuzzy soft mapping Consider { be a proximity space. By letting, for f, g ∈ FS,Let –(fsp5).fsp1)(From fsp2)(It is clear because fsp4)(A and B of X such that C and D of X such that It is easy to see that if fsp5)(A and B of X such that p5), there is a C⊆X such that If We will show that X, δ*) be a fuzzy soft proximity space.δ on X such that δ* = δi.There is a proximity relation A and B of X such that If AδB if and only if X.The relation Let (a) and (b) are equivalent and they imply (c).Then, (The implication (a)⇒(b) is obvious.δ satisfies (p5). Let A, B of X. Because f ∈ FS such that C, D, G, H of X such that δ has the axiom (p5).To prove that (b)⇒(c), since the other axioms are readily verified, it is enough to show that δ on the power set of X as follows:δ is a proximity on X. To complete the proof, it will suffice to prove that δ* = δi, that is, that fδig if and only if fδ*g. Assume that A and B of X such that δi, we obtain subsets A, B of X such that In order to prove (b)⇒(a), let us define the binary relation δ-neighborhood. We believe that these results will help the researchers to advance and promote the further study on fuzzy soft topology to carry out a general framework for their applications in practical life.Each proximity space determines in a natural way a topological space with beneficial properties. Also, this theory possesses deep results, rich machinery, and tools. With the development of topology, the theory of proximity makes a great progress. Hence, the concept of proximity has been studied by many authors in both the fuzzy setting and the soft setting. In the present work, we mainly establish some properties of fuzzy soft proximity spaces in Katsaras's sense. We have shown that each fuzzy soft proximity determines a fuzzy soft topology by using fuzzy soft closure operator. Also, we present an alternative description of the concept of fuzzy soft proximity, which is called fuzzy soft


As an important tool for data analysis and knowledge processing, formal concept analysis (FCA) has been applied to many fields. In this paper, we introduce a new method to find all formal concepts based on formal contexts. The amount of intents calculation is reduced by the method. And the corresponding algorithm of our approach is proposed. The main theorems and the corresponding algorithm are examined by examples, respectively. At last, several real-life databases are analyzed to demonstrate the application of the proposed approach. Experimental results show that the proposed approach is simple and effective. Formal concept analysis (FCA), proposed by Wille in 1982 [Most of the researches on FCA concentrate on the following topics: construction and pruning algorithm of concept lattices , 4; relan objects, then we should calculate 2n times to obtain all intents. Obviously, the computational costing is very huge. To solve this problem, we give a new method to obtain all intents. And correspondingly, the formal concepts are determined.Formal concepts are very important notions of FCA. And intents and extents are also very important elements of formal concepts. The set of intents (extents) is isomorphic to the corresponding concept lattice under the order relationship “⊇” (“⊆”). So, if the set of intents is determined, the corresponding concept lattice is identified. Thus, obtaining all intents or extents is very important. Generally, the basic way to obtain all intents or extents is via their definitions. If there are This paper is organized as follows. In In this section, we recall some basic notions and properties in FCA.G, M, I) consists of two sets G and M and a relation I between G and M. The elements of G are called the objects and the elements of M are called the attributes of the context. In order to express that an object g is in a relation I with an attribute m, we write gIm or  ∈ I and read it as “the object g has the attribute m.”A formal context , Ganter and Wille [A⊆G and B⊆M byWith respect to a formal context  be a formal context. ∀A1, A2, A⊆G, ∀B1, B2, B⊆M; the following properties hold.A1⊆A2⇒A2*⊆A1*,   B1⊆B2⇒B2′⊆B1′.A⊆A∗′,  B⊆B′∗.A* = A∗′∗, B′ = B′∗′.A⊆B′⇔B⊆A*.A1∪A2)* = A1*∩A2*, (B1∪B2)′ = B1′∩B2′.(A1∩A2)*⊇A1* ∪ A2*, (B1∩B2)′⊇B1′ ∪ B2′. is called a formal concept, where A is called the extent of the formal concept and B is called the intent of the formal concept. For any g ∈ G, a pair  is a formal concept and is called an object concept. Similarly, for any m ∈ M, a pair  is a formal concept and is called an attribute concept. The family of all formal concepts of  forms a complete lattice that is called the concept lattice and is denoted by L. For any ,  ∈ L, the partial order is defined byA1, B1) and  are defined byIf respectively.G, M, I) be a formal context. |G| < ∞ and |M| < ∞. Denote LM = {Y∣Y⊆M and Y is an intent of }.Let . G = {1,2, 3,4, 5,6} is an object set and M = {a, b, c, d, e} is an attribute set. The corresponding concept lattice L is shown in G, M, and ∅.n objects, then we should calculate 2n times to get all intents. Obviously, the amount of computation is very large. So our paper presents a new approach to solve the problem. In this section, we give this new method and some theorems to explain its rationality and validity.The basic way to obtain all intents or extents is via their definitions. If there are Before giving the method, we firstly propose a related definition.G, M, I) be a formal context. |G | <∞, |M | <∞. Denote αn = {X∣X⊆G and |X | = n}, βn = {Y∣Y⊆M, Y = X* and X ∈ αn}, n = 1,2,…, |G|, where |·| presents the cardinal of a set.Let  such that βn+1⊆βn, then βn+2⊆βn+1.If there exists Y ∈ βn+2. By Xn+2 ∈ αn+2 such that Y = Xn+2*.Suppose Xn+2 ≠ ∅, there exists a ∈ Xn+2 such that Y = Xn+2* = (Xn+2∖{a}∪{a})* = (Xn+2∖{a})*∩{a}*. Noting that |Xn+2∖{a}| = n + 1, we have Xn+2∖{a} ∈ αn+1. Moreover, from βn+1⊆βn, we know that there exists Xn ∈ αn satisfying (Xn+2∖{a})* = Xn*; that is, Y = Xn*∩{a}* = (Xn∪{a})*.Since Y ∈ βn+1.Now, we discuss two cases to prove a ∉ Xn. In this case, |Xn ∪ {a}| = n + 1. Thus, Y = Xn*∩{a}* = (Xn∪{a})* ∈ βn+1.The one case is that a ∈ Xn. In this case, Y = Xn*∩{a}* = Xn*. Because Xn+2∖Xn ≠ ∅, there exists b ∈ Xn+2∖Xn such that Xn⊆Xn ∪ {b}⊆Xn ∪ Xn+2. Therefore, we have Xn*⊇(Xn∪{b})*⊇(Xn∪Xn+2)*; that is, Xn*⊇(Xn∪{b})*⊇Xn*∩Xn+2*. That means, Y⊇(Xn∪{b})*⊇Y. Therefore, we have Y = (Xn∪{b})*. Thereby, we can obtain Y ∈ βn+1.The other one is that βn+2⊆βn+1 holds.To sum up the above two cases, n  such that βn+1⊆βn, then for any m, m ≥ n, we have βm+1⊆βm⊆βn.If there exists βn+1⊆βn, we have βn+2⊆βn+1 by βm+1⊆βm⊆βm−1⊆βm−2⊆⋯⊆βn+1⊆βn.According to the condition Y ∈ βk, 1 ≤ k ≤ |G|, 1 < m < k; if Y ∉ βm, then we have Y ∉ βm−1.Suppose We will adopt the proof by contradiction.Y ∈ βm−1; there is Xm−1 ∈ αm−1 satisfying Y = Xm−1*; according to the condition and Xk ∈ αk satisfying Y = Xk*.Suppose Xk∖Xm−1 ≠ ∅, there exists a ∈ Xk∖Xm−1 such that |Xm−1 ∪ {a}| = m; that is, Xm−1 ∪ {a} ∈ αm. Obviously, Xm−1⊆Xm−1 ∪ {a}⊆Xm−1 ∪ Xk, and thus, Xm−1*⊇(Xm−1∪{a})*⊇(Xm−1∪Xn)*. That is, Xm−1*⊇(Xm−1∪{a})*⊇Xm−1*∩Xk*. That means Y⊇(Xm−1∪{a})*⊇Y. Therefore, we have Y = (Xm−1∪{a})*. From Y ∈ βm. It is a contradiction with Y ∉ βm.Because Y ∉ βm−1 holds.Therefore, Y ∈ βk, 1 ≤ k ≤ |G|, and Y ∉ βk−1,βk⊆βk−1; then for any m, 1 ≤ m < k, Y ∉ βm.Suppose Y ∈ βk, and Y ∉ βk−1, we have Y ∉ βk−2 by Y ∉ βm.Because G, M, I) is canonical; then βn+1⊆βn if and only if  ∪i=1nβi ∪ {M} = LM.Suppose . If Y ≠ {M}, then there exists n0 such that Y ∈ βn0. By Xn0 ∈ αn0 such that Y = Xn0*. Obviously, Y ∈ LM. Since Y is arbitrary and M ∈ LM, we have ∪i=1n βi ∪ {M}⊆LM.For any Y ∈ LM, by Y = (Y*)*, Y*⊆G. Without loss of the generality, we can suppose |Y* | = m. If m ≤ n, then Y ∈ βm⊆∪i=1nβi by m > n, from above Y ∈ βm⊆βn⊆∪i=1nβi. Since M ∈ ∪i=1nβi ∪ {M} and Y is arbitrary, we obtain ∪i=1nβi ∪ {M}⊇LM.For any i=1nβi ∪ {M} = LM.Therefore, ∪Sufficiency. We assume βn+1⊈βn and prove ∪i=1nβi ∪ {M} ≠ LM.βn+1⊈βn, then there is Y ∈ βn+1, but Y ∉ βn. From m, 1 ≤ m < n, we have Y ∉ βm. So, Y ∉ ∪i=1nβi.  is canonical and Y ∈ βn+1, so Y ≠ M. Thus, Y ∉ ∪i=1nβi ∪ {M}.If Y ∈ βn+1, by Xn+1 ∈ αn+1, such that Y = Xn+1*. By the definition of LM, Y ∈ LM.On the other hand, since Y such that Y ∉ ∪i=1nβi ∪ {M}, but Y ∈ LM. Therefor, ∪inβi ∪ {M} ≠ LM.That means there exists one set LM. Now, the process to calculate all intents is summarized as follows. Step  1. Calculate α1 and β1 by  Step  2. Calculate α2 and β2 by β2⊆β1, then the set of intents is β1 ∪ {M}. Otherwise, we proceed Step  3. Step  3. Calculate α3 and β3 by β3⊆β2, then the set of intents is β1 ∪ β2 ∪ {M}. Otherwise, calculate βi (1 ≤ i ≤ |G|) continuously. The computation needs to stop at βn+1 which exactly meets βn+1⊆βn. Meanwhile, the set of intents is ∪i=1nβi ∪ {M}.βi, 1 ≤ i ≤ |G| and the computation needs only to stop at βn+1 which exactly meets βn+1⊆βn. Now all the intents have been found and there is no extra computing.The merit of our method is that we do not need to calculate all In the following, we use an example in the literature  to examia (water) from the original formal context. The objects are living beings mentioned in the film and are denoted by G = {1,2, 3,4, 5,6, 7,8}, where 1 is leech, 2 is bream, 3 is frog, 4 is dog, 5 is spike-weed, 6 is reed, 7 is bean, and 8 is maize. And the attributes in M = {b, c, d, e, f, g, h, i} are the properties which the film emphasizes: b: lives in water, c: lives on land, d: needs chlorophyll to produce food, e: two seed leaves, f: one seed leaf, g: can move around, h: has limbs, and i: suckles its offspring.The formal context in L of this formal context is shown in The corresponding concept lattice αi and βi  firstly:We calculate β4, β5,…, β8 and find β2⊇β3⊇β4⊇β5⊇⋯⊇β8. And we can also know β2⊈β1. In fact, we only need to calculate β1, β2, β3. Once we have β3⊆β2, but β2⊈β1, the computation can be stopped.Similarly, we can calculate β1 ∪ β2 ∪ {M}; that is, LM = {{b, g}, {g}, {b}, ∅, {b, g, h}, {g, h}, {c, g, h}, {b, c}, {c}, {b, d, f}, {d}, {d, f}, {c, d}, {c, d, f}, {b, g, c, h}, {c, g, h, i}, {b, c, d, f}, {c, d, e}, M}. These results are easily examined from According to The time complexity of N = min{|G | , |M|}; by O(CNi). So we can get two matters as follows.O(2N).The time complexity of algorithm is kth step; then the time complexity of O(∑i=1i=k(CNi)) by O(∑i=1i=k(CNi)) ≤ O(2N).Suppose that Denote a,…, h, where a is headache, b is fever, c stands for painful limbs, d represents swollen glands in neck, e is cold, f is stiff neck, g is rash, and h is vomiting. Input the formal context and run the program; we obtain the set of all intents when n = 2:∅, {b}, {a, b}, {b, e}, {g}, {b, e, g}, {a, b, f, h}, {a, b, c, e}, {a, b, c, d, e, f, g, h}} {.We present an example demonstrating performance of Living beings and water  introducPatients and ill symptoms  introducBacterial Taxonomy [ Escherichia coli (ecoli), Salmonella typhi (styphi), Klebsiella pneumoniae (kpneu), Proteus vulgaris (pvul), Proteus morganii (pmor), and Serratia marcesens (smar). The phenotypic characters are H2S, MAN, LYS, IND, ORN, CIT, URE, ONP, VPT, INO, LIP, PHE, MAL, ADO, ARA, and RHA.Taxonomy . Data arMembership of Developing Countries in Supranational Group [al Group . In thisIn this section, we conduct some experiments to compare I| presents the number of intents and the efficiency is equivalent to (Time 1 − Time 2)/Time 1. It can be seen that I|.The results are shown in To find new methods to solve the difficult problems of the concept lattice construction is a hot problem. Constructing concept lattices is a novel research branch for data processing and data analysis. Different methods play essential roles in different problems. This paper first defines some basic notions. Based on the basic notion of intents, we obtain a new judgment method of finding all intents of formal concepts. Moreover, an example is given to explain the feasibility of this method. At last, we give the corresponding algorithm of this method and do the experiments to illustrate the effectiveness of this method.G| with |M| of a formal context. If |G | ≤|M|, then we use subsets of G to determine subsets of M and output the set of intents. Otherwise, according to the duality principle, the subsets of M can be used to determine subsets of G and output the set of extents. We will improve the corresponding algorithm of this method in the future.For


Finally to illustrate easy application and rich behavior of EP method, several examples are given.Firstly in this paper we introduce a new concept of the 2nd power of a fuzzy number. It is exponent to production (EP) method that provides an analytical and approximate solution for fully fuzzy quadratic equation (FFQE) :  The problem of finding the roots of equations like the quadratic equation has many applications in applied sciences like finance , 2, econThe rest of the paper is set out as follows. In the second section some related basic definitions of fuzzy mathematics for the analysis are recalled. In The basic definitions are given as follows.R;x0 ∈ R with A.A fuzzy number is a function x0 ∈ R with Note: in this paper we consider fuzzy numbers which have a unique u~(x0)=1 .F. Obviously, R⊆F. For 0 < r ≤ 1, we define r-cut of fuzzy number r ∈ . We denote the r-cut of fuzzy number The set of all these fuzzy numbers is denoted by ≥0}¯. In  from 4)F. Obvioux < 0  (x > 0).A fuzzy number r ≤ 1, which satisfy the following requirements:r ≤ 1.A fuzzy number r ≤ 1, and scalar k, we define addition, subtraction, and scalar product by k and multiplication is, respectively, as follows. Addition:  Subtraction:  Scalar product: Multiplication:For two important cases multiplication of two fuzzy numbers is defined by the following terms. If  If  If  If For arbitrary r-cuts is similar to arithmetics of the parametric form recalled previously .Two fuzzy numbers α in parametric form is r ≤ 1. A triangular fuzzy number is popular and represented by α > 0 and β > 0, which has the parametric form as follows:A crisp number D : F × F → R ∪ {0} and let k ∈ R and Therefore  is a complete metric space.Let α, as a real root of crisp 1-cut equation and then we get α1(r)⩾0, α2(r)⩾0, α1′(r) < 0, and α2′(r) < 0. To find fi(α1(r), α2(r)), for i = 1,2, we consider two cases:FFQE has analytical solution;FFQE does not have analytical solution.Let (r)) andF(X~)=D~,fi(α1(1), α2(1)) = 0, for i = 1,2;To find fi(α1(r), α2(r)), i = 1,2, at first we substitute α − f1(α1(r), α2(r)), α + f2(α1(r), α2(r))) and ≅(α − α1(r), α + α1(r))(α − α2(r), α + α2(r)), in parametric form of (r = 1. Using condition (1) and α ∈ R; that meansλ, μ, and χ(0) = α1(0) + α2(0). The third equation comes from the equality below, for r = 0,λ, μ, and χ(0). To construct solution of the new method, in the above parametric form, let χ(r) = α1(r) + α2(r) and ν(r) = α1(r) × α2(r); then, by solving a 2 × 2 system via χ(r) and ν(r), we obtain χ(r) and we setα1(r) + α2(r) ∈ R, because α1(r) and α2(r) are the roots of Z2 − χ(r)Z + ν(r) = 0.In this case we have  form of , and theution of . Notice χ(r).Using the proposed method we obtain the following set of expressions for Case (1). (1)if (2)if (3)if (4)if Case (2). (1)if (2)if (3)if (4)if whereξ ∈ {λ, μ}.λ = μ = 0.5 because of In this case we do not have analytical solution, and we do not have α is real core point of λ = μ = 0.5 is the best choice in EP method.If Without loss of generality suppose that λ and μ, we must solve the minimization problem as follows:To find the optimum parameters α ∈ R, we choose λ = μ = 0.5. This completes the proof.To delimitate maximum error for any λ = μ = 0.5 isNecessary condition for existence of EP solution with χ(r) is sum of α1(r) and α2(r) and we want α1(r) and α2(r) to be nonnegative, then necessary condition for existence of EP solution with λ = μ = 0.5 is χ(r) ≥ 0 and since ⊆, for all 0 ≤ r ≤ 1, it is sufficient to have χ(0) ≥ 0. Considering denominators of (χ(0) ≥ 0, and this completes the proof.Since ators of –20) we  we χ(r) ators of  and 25)χ(r) is sλ = μ = 0.5 does not have solution,The EP method with This lemma is conclusion of λ = μ = 0.5 in EP method and χ(0) ≥ 0, then sufficient condition for existence of solution is ν(0) ≥ 0.Suppose α1(r) and α2(r) are nonnegative if χ(r) ≥ 0 and ν(r) ≥ 0.We know λ = μ = 0.5 and ν(0) ≥ 0.Because of construction of EP method for χ(0) ≥ 0 and λ = μ = 0.5 in EP method, then we have solution with (1)if (2)if (3)if (4)if with  (1)if (2)if (3)if (4)if Suppose χ(r) and ν(r) in r = 0; we obtainν(0) ≥ 0, if the numerator is nonnegative and this completes the proof.We consider only one case to discuss. We consider nd solve  via χ(r)−4.In the next examples we use round numbers with approximation less than 10Let =(3/2/2) .We will look for a solution where Equation Therefore we have analytical solution and EP solution by  and 29)29).λ = 0.7067, μ = 0.3296, andBy  and 14)14), we fLetting λ = μ = 0.5 too.Therefore this example does not have analytical solution. We look for EP solution. By  we find Now we consider an example with Letting λ = μ = 0.5 as follows: Therefore, this example does not have analytical solution. We look for EP solution. By , 30), a, a30), aNotice that the numerical methods needed In this paper we introduced a new method to solve a fully fuzzy quadratic equation. To this purpose we found the optimum spreads to decrease maximum error. One of the advantages of this method is that complications do not depend on the sign of the coefficients and variable. It is possible that these equations do not have any analytical solution, but the proposed method gives us an approximate analytical solution.


In Gao et al. −3 mmol/kg.” The correct dosage was given to animals.In paragraph 2, letter (d) indicated that the animals were given the dosage “ … 2.46 mmol/kg once a day at P7, P11 …. This should be corrected to “2.46 x 10The authors apologise for the typographical error.


Scientific Reports6: Article number: 29863; 10.1038/srep29863published online: 07202016; updated: 08252016This Article contains typographical errors.In the Results and Discussion section under subheading ‘Properties of second-harmonic generation imaging with Bessel beam excitation’,ρz × 1/ρz2, i.e. should not exhibit the same drop as 2PEF from an extended sample (Table 1).”“Interestingly, in the case of SHG, if all contributions scattered by the sample along the optical axis interfere in a constructive manner, the signal should scale as should read:ρz2 × 1/ρz2, i.e. should not exhibit the same drop as 2PEF from an extended sample (Table 1).”“Interestingly, in the case of SHG, if all contributions scattered by the sample along the optical axis interfere in a constructive manner, the signal should scale as ρz2 × 1/ρz2 = 1’ is incorrectly given as ‘ρz × 1/ρz2’.In Table 1, ‘