Arthrobotrys is the most complex genus of Orbiliaceae nematode-trapping fungi. Its members are widely distributed in various habitats worldwide due to their unique nematode-trapping survival strategies. During a survey of nematophagous fungi in Yunnan Province, China, twelve taxa were isolated from terrestrial soil and freshwater sediment habitats and were identified as six new species in Arthrobotrys based on evidence from morphological and multigene  phylogenetic analyses. These new species i.e., Arthrobotrys eryuanensis, A. jinpingensis, A. lanpingensis, A. luquanensis, A. shuifuensis, and A. zhaoyangensis are named in recognition of their places of origin. Morphological descriptions, illustrations, taxonomic notes, and a multilocus phylogenetic analysis are provided for all new taxa. In addition, a key to known species in Arthrobotrys is provided, and the inadequacies in the taxonomic study of nematode-trapping fungi are also discussed. Arthrobotrys superba Corda)   and bigger than those of A. sphaeroides .Notes: The phylogenetic analyses revealed that ) and 8% 7/589 bp  conidia ,39,40, wArthrobotrys shuifuensis F. Zhang & X.Y. Yang sp. nov.  long, 3–5 µm  wide at base, gradually tapering upwards to apex, 1.5–3.5 µm  wide at apex, erect, septate, unbranched or rarely branched, hyaline, producing several separate nodes by repeated elongation of conidiophores, with each node consisting of 2–8 papilliform bulges and bearing polyblastic conidia. Conidia 17–36 × 5–12.5 µm , oblong or capsule-shaped, narrower towards the lower and pointed base, 1-septate, median septum, hyaline, rough-walled. Chlamydospores 6–18 × 3–7.5µm , cylindrical, in chains, hyaline, rough-walled. Capturing nematodes with adhesive networks.Additional specimen examined: CHINA, Yunnan Province, Zhaotong City, Shuifu County, 28°32′31.80″ N, 104°19′9.50″ E, from terrestrial soil, 16 June 2017, F. Zhang. YXY48.Arthrobotrys shuifuensis is the closest species to A. arthrobotryoides, there are 9.6% (57/596 bp) differences in ITS sequence between them. Morphologically, this species is similar to A. arthrobotryoides in their capsule-shaped, 1-septate conidia, whereas the conidia of A. shuifuensis are significantly longer than those of A. arthrobotryoides . In addition, the conidiophores of A. arthrobotryoides are unbranched and produces a continuous irregularly swollen node at apex, while the conidiophores of A. shuifuensis are branched, producing several separate nodes with the repeated elongation of the conidiophores . In addition, these three species differ slightly in the number of septation on conidia; A. zhaoyangensis produces 1–3-septate conidia (mostly 3-septate), while the conidia of A. sinensis are 2-septate; A. sphaeroides sometimes produces aseptate conidia [Notes: Phylogenetic analysis revealed that the systematic position of  conidia ,79.1.Conidia 0–1-septate……………………………………………………………………………21.Conidia multi-septate…………………………………………………………………………302.Conidia mostly aseptate………………………………………………………………………32.Conidia mostly 1-septate………………………………………………………………………63.A. botryosporaConidiophores branched near apex, producing a node at each branch, or producing several separate nodes by repeated elongation; conidia ovate, with a papilliform bulge at the base………………………………………………………………3.Conidiophores unbranched……………………………………………………………………44.A. amerosporaConidiophores with a cluster short denticles at apex; conidia obovoid, 15–31 (23.5) × 10–20 (15.9) μm……………………………………………………………………4.Conidiophores producing several clusters of short denticles by repeated elongation.…55.A. yunnanensisConidia elongated, ellipsoid–cylindrical, 0–1-septate, mostly non-septate, 17.5–32.5 (22.6) × 2.75–7.5 (5.5) μm………………………………………………………5.A. nonseptataConidia elongated, ellipsoidal, non-septate, constricted at the base, 11–16.8 × 5–6.6 μm………………………………………………………………………………6.Conidia develop on short denticles…………………………………………………………76.Conidia develop on nodes.……………………………………………………………………137.Conidia curved…………………………………………………………………………………87.Conidia straight………………………………………………………………………………108.A. musiformisConidiophores unbranched, conidia in loose capitate arrangement at apex; conidia ellipsoid, mostly curved, 20–47.5 (30.9) × 7–12.5 (10.3) μm…………………8.Conidiophores branched, producing several clusters short denticles by repeated elongation…………………………………………………………………………………………99.A. shahriariConidiophores simple or occasionally branched; conidia elongate-obovoid or elongate-ellipsoidal, 1-septate, straight or curved, 33.5–57 × 11–15.5 µm………9.A. eryuanensisConidiophores branched; macroconidia 1-septate, straight or slightly curved, 18–44.5 (28.4) × 5–11.5 (8.7) µm, microconidia aseptate…………………………10.A. perpastaConidiophores producing short denticles by repeated elongation; conidia 1-septate near the base, obpyriform, sometimes constricted at the septum, 24–32.5 × 12.5–20 µm…………………………………………………………………………………10.Conidiophores with clustered short denticles at apex; conidia in loose capitate arrangement at apex…………………………………………………………………………1111.A. javanicaConidia clavate, 1-septate at median or submedian, slightly constricted at the septum, 20–37.5 (27.9) × 7.5–10 (8.8) µm…………………………………11.Conidia obovoid or obpyriform…………………………………………………………1212.A. obovataConidia obovoid, 1-septate near the base, apical cell much larger, smaller at basal cell, 28.5–32 (30) × 18–20.5 (20) µm…………………………………………………12.A. koreensisConidia obpyriform, 1-septate at submedian, slightly constricted at the septum, 21.4–26.9 × 11.6–15.6 µm……………………………………………13.Conidia develop on short denticles or obscure nodes of conidiophores…………1413.Conidia develop in clusters on swollen nodes of conidiophores…………………1714.A. chazaricaConidiophores branched, producing short denticles by repeated elongation; conidia obovate, elongate–obovate, 22.5–32 × 11–22.5 µm…………………………14.Conidiophores unbranched; conidia clavate or pyriform…………………………1515.A. pseudoclavataConidia develop on apical conidiophores, conidia clavate, 0 or 1-septate, constricted at the base, 30–45 × 8–11 µm……………………………… 15.Conidia pyriform, 1-septate near the basal, apical cell much larger, smaller at basal cell; conidiophores producing several short denticles by repeated elongation………1616.A. paucisporaConidia perceptibly constricted at the septum, 25–33.8 × 12.5–16.3 µm…16.A. cystosporiaConidia non-constricted, 25–35 × 18–24 µm………………………………17.Conidiophores branched………………………………………………………………1817.Conidiophores unbranched………………………………………………………………2418.Conidia 1-septate at median………………………………………………………………1918.Conidia 1-septate at submedian…………………………………………………………2119.A. superbaConidia elongate–elliptical or cylindrical, 7.5–27.5 (15.8) × 5–10.5 (6.6) µm…19.Conidia short elliptical to oblong or capsule-shaped……………………………………2020.A. arthrobotryoidesConidiophores occasionally branched, with distinct continuous swollen apical nodes; conidia ellipsoidal, 20–22 × 9–10 µm…………………………………20.A. shuifuensisConidiophores usually branched, bearing conidia on slightly swollen nodes; conidia capsule-shaped, 17–36 (27.2) × 5–12.5 (8.2) µm…………………………21.A. robustaConidiophores bearing conidia on apical nodes; conidia oblong–pyriform, 20–27.5 (24.4) × 7.5–12.5 (10.8) µm…………………………………………………………21.Conidiophores producing several separate nodes by repeated elongation…………2222.A. cladodesConidia ellipsoid, elongate–obovate, 10–20 (17.5) × 5–8 (6.2) µm……………22.Conidia obovoid, obpyriform, or ovoid…………………………………………………2323.A. latisporaConidia subglobose or elliptical, 14.8–21.5 (18.3) × 10.1–16.3 (13.5) µm…… 23.Orbilia jesu-lauraeConidia obovoid or obpyriform, 1 septum at submedian, slightly constricted at the septum, 14–26 × 7.5–13 µm………………………………………………24.Condia develop on apical node of conidiophores………………………………………2524.Conidiophores producing several separate nodes by repeated elongation…………2725.A. flagransConidia non-constricted at the septum, obconical or ellipsoidal, 25–50 × 10–15 µm…………………………………………………………………………………25.Conidia obconical or pyriform, constricted at the septum……………………………2626.A. apscheronicaConidia larger size, constricted at septum, 21–42 (30.5) × 8–15 (12.7) µm……………………………………………………………………………26.A. conoidesConidia small size, perceptibly constricted at the septum, 15–37.5 (28.4) × 7.5–14.5 (11.8) µm…………………………………………………………………………27.Conidia 1-septate at median……………………………………………………………2827.Conidia 1-septate at submedian………………………………………………………….2928.A. anomalaConidiophores producing continuously expanded node or several separate nodes by repeated elongation; conidia cylindric, long ellipsoid, larger size, 13–22 × 3–7 µm…………………………………………………………………………………28.A. dendroidesConidiophores producing several slightly swollen nodes by repeated elongation; conidia ovate, oblong, cylindric, smaller size, 10–20 (14.6) × 2.5–5 (4) µm…29.A. jinpingensisConidia obpyriform or drop-shaped, some with a bud-like projection at the base, smaller size, 11.2–26.4(18.6) × 6.6–14.4(10.8) µm………29.A. oligosporaConidia pyriform or obovoid, slightly constricted at the septum, larger size, 17–35 (23) × 8.5–16 (12) µm…………………………………………………………30.Conidia without largest cell, with several septa, uniformly distributed among conidial cells ……………………………………………………………………………………3130.Conidia with largest cell…………………………………………………………………3731.A. iridisY-shaped conidia develop on conidiophores………………………………………31.Conidia other type, never Y-shaped…………………………………………………3232.Conidiophores branched……………………………………………………………….3332.Conidiophores unbranched………………………………………………………………3433.A. dianchiensisMacroconidia spindle-shape or clavate, with 1–7-septate, mostly 2–5, 37.5–100 (70) × 10–17.5 (14.3) µm, microconidia spindle-shape, 0 or 1-septate…………33.A. tabrizicaConidia elongate–ovate to elongate–doliform or ellipsoidal, with 1–3-septate, 28.5–56 × 11.5–22.5 µm……………………………………………………….……34.Conidia bearing on apical conidiophores………………………………………………3534.Conidiophores producing several cluster conidia by repeated elongation…………3635.A. multiformisSeveral conidia develop on apical conidiophores, macroconidia elongate-fusiform, clavate, 4–12-septate; microconidia clavate, cylindric–clavate, 0 or 1-septate………………………………………………………………………35.A. shizishannaConidiophores bearing single conidium; conidia clavate, sometimes slightly curved, 2–9-septate, 22.5–73.8 (50.6) × 5–10 (6.6) µm………………………36.A. polycephalaConidiophores with inconspicuous short denticles; macroconidia fusoid-shaped, curved, 2–4-septate, mostly 3–4, 30–50 (45.1) × 8–16.5 (12.2) µm, microconidia ellispsoid, slightly curved, 1 or 2-septate……………………………………36.A. pyriformiConidiophores producing several short denticles by repeated elongation; conidia elongate–pyriform, 1–3-septate, mostly 2 or 3, 17–38 × 6.5–11.5 µm………37.Conidiophores branched…………………………………………………………………3837.Conidiophores unbranched……………………………………………………….….……4338.Conidiophores bear a single conidium………………………………………………3938.Conidiophores bear several conidia…………………………………………………4039.A. globosporaConidia globose or obpyriform, 1–2-septate, 25–37.5 × 15–22.5 µm……39.A. sinensisConidia subspherical or obovoid or subfusiform, 1–3-septate, 23.5–30 (27.6) × 17–25 (20) µm……………………………………………………………………………40.Conidia in capitate arrangement at apex of conidiophores…………………………4140.Conidia in non-capitate arrangement on conidiophores……………………………4241.A. azerbaijanicaConidia obovoid or ellipsoidal, 1–4-septate, mostly 1, 18–36 (28.1) × 12–20 (15.3) µm……………………………………………………………………………41.A. oviformisConidia pyriform, 1–2-septate, mostly 1, 7.5–22.5 (15.8) × 5–10 (6.6) µm…42.A. indicaConidiophores bearing 1 conidium, sometimes 2 conidia; conidia elliptic, top-shaped, 0–2-septate, 17.5–30 (23.2) × 12.5–20 (14.8) µm…………………………………42.A. oudemansiiConidiophores bearing several conidia; macroconidia subfusiform, 2–4-septate, 40–65 (52) × 17–23 (20) µm, microconidia obovoid, aseptate ………43.Conidiophores bear several conidia ………………………………………………4443.Conidiophores bear a single conidium ………………………………………………5244.Conidiophores bear several conidia near apex by repeated elongation…………4544.Conidiophores bear several conidia at apex…………………4745.A. vermicolaConidia elongate–ellipsoidal to broadly fusiform, 1–3-septate, mostly1 or 2, 25–50 × 17.5–25 µm……………………………………………………………………45.Conidiophores producing denticles by repeated elongation, conidia fusiform, elongate–fusoid or clavate ……………………………………………………………………4646.A. mangrovisporaConidia variable in shape, broadly turbinate to elongate–fusoid, ellipsoidal, fusiform, clavate, 1–3-septate, mostly 2, 25–50 (38.9) × 12–24 (17.3) µm…………46.A. scaphoidesConidia fusiform, sometimes slightly curved, 1–6-septate, mostly 2–3, 36.6–79.3 (57) × 11–17.5 (14) µm…………………………………………………………………47.A. gampsosporaConidia spindle-shaped, curved,1–4-septate, 25–76 × 7–16 µm………47.Conidia straight……………………………………………………………………………4848.Conidia 0–3-septate, mostly 1 or 2………………………………………………………4948.Conidia 1–4-septate, mostly 3 or 4………………………………………………………5049.A. microscaphoidesConidia cymbiform or fusiform, mostly 2-septate, 22.5–45 (27.2) × 10–20 (13.9) µm…………………………………………………………………………49.A. clavisporaConidia pyriform, clavate, mostly 1 or 2 septate, 25–40 (17.5) × 7.5–19 (15.4) µm………………………………………………………………………………50.A. thaumasiaConidia small, subfusiform, 1–4-septate, mostly 3, 30–60 (36.2) × 15–30(20.2) µm………………………………………………………………………………50.Conidia larger, ellipsoidal, fusoid–ellipsoidal……………………………………5151.A. psychrophilaConidia ellipsoidal, fusoid–ellipsoidal, 2–4-septate, mostly 4, 46–70 (62.3) × 21–29 (24.7) µm……………………………………………………………………………51.A. megalosporaConidia fusiform, elongate–ellipsoidal or obovoid, 2–4-septate, mostly 3 or 4, 40–75 × 18–35 µm……………………………………………………………………52.A. eudermataConidia spindle-shaped, globose, 1–3-septate, mostly 2 or 3, 37–55 (49) × 17.5–35 (28) µm, microconidia ellipsoid, aseptate……………52.Without microconidia……………………………………………………………………5353.A. janusConidia turbinate or napiform, 1–2-septate, mostly 1 near the base, the largest cell at the apex of conidia, 15–26 (22.5) × 17.5–37.5 (28.5) µm………………………53.Conidia with more than 2 septa; the largest cell is located in the apex or center of conidia………………………………………………………………………………………5454.The septum of the conidia is not more than 3…………………………………………5554.Conidia 1–5-septate………………………………………………………………………5855.Conidia clavate, obovoid, or subspherical, 0–3-septate……………………………5655.Conidia fusiform, 2–3-septate……………………………………………………………5756.A. cookedickinsonConidia clavate or obovoid, 1–3-septate, mostly 2–3, 30–52.5 (42) × 15–22.5 (17.6) µm………………………56.A. sphaeroidesConidia subspherical or obovoid, 0–3-septate, mostly 2–3, 20–40 (32) × 17–25 (20.4) µm………………………………………………………………………………57.A. rutgerienseConidia globose or subfusiform, 2–3-septate, 27–47.5 (32.2) × 17.5–27.5 (22) µm………………………………………………………………………………57.A. fusiformisConidia spindle-shaped, fusiform or ellipsoidal, 2–3-septate, 32.5–47.5 (41) × 12.5–17.5 (15.5) µm………………………………………………………………………58.Conidia 1–5-septate, mostly 3 or 4…………………………………………………….5958.Conidia 1–4-septate, mostly 2 or 3………………………………………………………6159.A. xiangyunensisConidia variable in shape, obpyriform, broadly turbinate, subfusiform, elongate-fusoid or clavate, 1–5-septate, 27–72 (55.8) × 14.5–28.5 (21.9) µm…59.Conidia ellipsoid, obpyriform or subfusiform, 2–5-septate, mostly 3 or 4……………6060.A. reticulataConidia ellipsoid, fusiform, 50–65 × 20–25 µm…………………………………60.A. longiphoraConidia obpyriform or subfusiform, 40–90 (54) × 15–27.5 (18) µm………… 61.Conidia mostly 2-septate……………………………………………………………………6261.Conidia mostly 3-septate……………………………………………………………………6362.A. luquanensisConidia obovate, obpyriform or drop-shaped; the distal cell is much smaller, the largest cell usually at the apex, 28–53.5 (40.9) × 17–32.5 (26.3) µm……………… 62.A. guizhouensisConidia obpyriform or subfusiform, the largest cell usually at the centre, 30.5–71.5 (52.7) × 18.5–28.5 (23.9) µm……………………………………………63.A. zhaoyangensisConidia, subglobose, obovoid to obpyriform, 25.5–52 (35.4) × 14–32 (22.9) µm……………………………………………………………………………63.Conidia fusiform or ellipsoid…………………………………6464.A. salinaConidia fusiform to ellipsoid, 32.5–52.5 × 12.5–17.5 µm………………………… 64.A. lanpingensisConidia mostly subfusiform, 31.1–55.2 (45.4) × 13.5–24.3 (19.7) µm………Arthrobotrys, which contains the greatest number of species. Morphologically, 61 species of Arthrobotrys can be divided into different groups according to the morphologies of their conidiophores and conidia [Arthrobotrys species are unclear. The reason for this dilemma is the lack of molecular data for many species, and the existing data cannot provide a stable phylogenetic placement. Therefore, to thoroughly analyse the taxonomy of nematode-trapping fungi, we should use more comprehensive molecular data in future studies.In this phylogenetic analysis, 65 species of nematode-trapping fungi used in this study were clustered into two large clades according to their mechanisms of catching nematodes. Clade I contained species that catch nematodes with adhesive trapping devices (adhesive nets and knobs). Clade II contained species that catch nematodes with active traps (constricting rings). Within clade I, species were clustered into two clades according to their trap types: one clade contained all species that produce adhesive nets, and the other contained those species that produce adhesive knobs. The results were consistent with previous studies ,15,57,80 conidia ; howeverA. eryuanensis and A. shuifuensis could be easily distinguished from known species based on their distinct morphological characteristics. The remaining four species required more detailed characteristics (such as the size of conidia) to be identified from known species. When mycologists measure the size of conidia, they are accustomed to uniformly calculating the size data of conidia with different shapes and septate numbers, and the sizes of these conidia usually show significant differences. This causes the size range of conidia to be too extensive for effective comparisons of different species [The emergence of molecular phylogenetic methods has led to unprecedented breakthroughs in the study of fungal taxonomy. Phylogenetic studies based on only a few molecular barcodes cannot provide sufficient and reliable information for the definition of fungal species; therefore, morphological descriptions of each species are still extremely important ,82. Howe species ,19. (2)  species ,19. As ahttp://www.speciesfungorum.org (accessed on 6 March 2022)). These data indicated that the excavation of nematode-trapping fungi seems to have reached a plateau, and over time, it is unlikely that many new species will be discovered. However, in recent years, we have investigated nematode-trapping fungi in Yunnan Province and collected 10 new species (four previously published and six reported in this study) [After the first nematode-trapping fungus was established in 1839 , the hiss study) ,84,85. H


Scientific Reports 10.1038/s41598-022-23991-3, published online 15 November 2022Correction to: The original version of this Article contained an error in the y-axis labels of Figure “100000”now reads:“100”“10000”now reads:“10”“1000”now reads:“1”The original Figure The original Article has been corrected.


Scientific Reportshttps://doi.org/10.1038/s41598-021-02511-9, published online 29 July 2022Correction to: The original version of this Article contained errors in Table 1.The values for “Above secondary school education” in section “Educational status” in column “Frequency” were incorrectly given as “56” instead of “57”, and in column “Percentage” the values were incorrectly given as “23.4” instead of “23.8”. Moreover, in column “Percentage”, the value for “No education” was rounded down from “31.4” to “31.2”, and the value for “Primary school education” was rounded down from “21.3” to “21.2”. The correct and incorrect values appear below.†” was rounded down from “1.3” to “1.2”. The correct and incorrect values appear below.Additionally, the data for “Protestant” in section “Religion” was omitted. The value “50” was added in column “Frequency” and the value “20.8” was added in column “Percentage”. Moreover, in column “Percentage”, the value for “OtherIncorrect:Correct:The original Article has been corrected.


We employ some results about continued fraction expansions of Herglotz–Nevanlinna functions to characterize the spectral data of generalized indefinite strings of Stieltjes type. In particular, this solves the corresponding inverse spectral problem through explicit formulas. Stieltjes continued fractions played a decisive role in the solution of the inverse spectral problem for Krein strings , 27, 28.1m be a Herglotz–Nevanlinna function, that is, the function m is defined and analytic on C∖R, maps the upper complex half‐plane into the closure of the upper complex half‐plane and satisfies the symmetry relationa is a real constant, b is a non‐negative constant and ρ is a non‐negative Borel measure on R which is subject to the growth restrictiona and b, as well as the measure ρ in this integral representation are uniquely determined by the function m and may be recovered explicitly.Let s0,s1,… the moments of the measure ρ, that is, we setK∈N0, the function m allows the asymptotic expansion|z|→∞ along the imaginary axis, provided the moments of the measure ρ exist up to order 2K. Here, the real constants s−2 and s−1 are defined byTheorem 1.1K∈N0, the moments of the measure ρ exist up to order 2K if and only if the function m allows the asymptotic expansion|z|→∞ along the imaginary axis for some real constants s−2,…,s2K.For any fixed As long as they exist, we will denote with entation  that form. To this end, we first introduce the Hankel determinantsc∈Ck one computes thatk is zero. For future reference, we also state the useful relationsρ exist up to order 2K, then the relations  is the smallest integer k∈N0 for which the sequence Δ1,0,…,Δ1,k has precisely n non‐zero elements. One observes that the increasing function κ is defined in such a way that Δ1,κ(1),…,Δ1,κ(N+1) enumerates all non‐zero members of the sequence Δ1,0,…,Δ1,D. As it follows from relation =0, this determines κ recursively.Proposition 1.2m is a rational Herglotz–Nevanlinna function, then it admits the continued fraction expansionυ0,…,υN and the real constants ω0,…,ωN are given byl1,…,lN and the non‐negative constant r are given byIf Let us suppose for now that ows from  that therelation  that them is a rational Herglotz–Nevanlinna function. For each t∈R, we define the rational Herglotz–Nevanlinna function mt bymt will be denoted in a natural way with an additional subscript t. In particular, note that m0 coincides with our initial function m and hence so do the associated quantities . As each of the Hankel determinants depends analytically on t, we may conclude that the determinants Δt,1,0,…,Δt,1,κ(N+1) are all non‐zero as long as t≠0 is small enough. Indeed, even if Δ1,k does vanish for some k∈{1,…,κ(N+1)−1}  is certainly non‐zero), then this holds because the derivativeΔ1,k=0.Suppose that t≠0 is small enough, the function mt admits a Stieltjes continued fraction expansion [K=κ(N+1), the non‐zero real constants ωt,1,…,ωt,K are given bylt,1,…,lt,K as well as the non‐negative constant rt are given byK is zero and that the alternative expression in =qt,K clearly converge pointwise to the rational Herglotz–Nevanlinna function qκ(N+1) defined byr is given by  converge pointwise to some rational Herglotz–Nevanlinna function qκ(n+1). If Δ1,κ(n)+1 does not vanish, then κ(n+1)=κ(n)+1 and we infer from  converge pointwise to the Herglotz–Nevanlinna function qκ(n) defined viaυn, ωn and ln are given by +1 vanishes, one has κ(n+1)=κ(n)+2 and using the second expression in (ωn is given by (k=κ(n)+1 as well as k=κ(n)+2, in which the Δ1,κ(n)+1 terms vanish, we infer that the last limit is equal tok=κ(n)+2, this becomesυn is given by (υn is positive as relation (k=κ(n)+1 shows that the numerator on the left‐hand side is not zero. Now observe that in the current case, the functions qt,κ(n) satisfyz∈C∖R. With the help of the limits above, a computation then shows that the functions qt,κ(n) converge pointwise to the rational Herglotz–Nevanlinna function qκ(n) defined by , then the expansion in Proposition κ(n)=n−1 for all n∈{1,…,N+1} and thus it follows from the formulas in +1 is not zero. On the other side, if Δ1,κ(n)+1 is zero, then relation +1 vanishes and also that υn+|ωn|>0 for all n∈{1,…,N}.The constants in the continued fraction in Proposition relation  shows threlation  allows uIt is not difficult to see that any continued fraction of the form in Proposition m1 and m2 be Herglotz–Nevanlinna functions and denote with ρ1 and ρ2 the corresponding measures in the respective integral representations.Lemma 1.4K∈N0 and suppose(1)m2(z)/z→β for some positive β as |z|→∞ along the imaginary axis. Then the moments of the measure ρ2 exist up to order 2K if and only if the moments of the measure ρ1 exist up to order 2K+2.Assume that (2)m1(z)/z→0 and m2(z)/z→0 as |z|→∞ along the imaginary axis. Then the moments of the measure ρ2 exist up to order 2K if and only if the moments of the measure ρ1 exist up to order 2K.Assume that Let Before we proceed to non‐rational Herglotz–Nevanlinna functions, let us first provide two auxiliary results. In order to state them, let m2(z)/z→β for some positive β as |z|→∞ along the imaginary axis. If the moments of the measure ρ2 exist up to order 2K, then from Theorem |z|→∞ along the imaginary axis for some real constants s2,−1,…,s2,2K. This implies|z|→∞ along the imaginary axis. Since the first term on the right‐hand side is a rational function, we conclude from Theorem ρ1 exist up to order 2K+2. Conversely, if the moments of the measure ρ1 exist up to order 2K+2, then we have|z|→∞ along the imaginary axis for some real constants s1,1,…,s1,2K+2 and thus|z|→∞ along the imaginary axis. Like before, we may conclude again that the moments of the measure ρ2 exist up to order 2K by invoking Theorem We begin with the case when m1(z)/z→0 and m2(z)/z→0 as |z|→∞ along the imaginary axis. If the moments of the measure ρ2 exist up to order 2K, then from Theorem |z|→∞ along the imaginary axis for some real constants s2,−1,…,s2,2K. Since m1(z)/z→0 as |z|→∞ along the imaginary axis, we conclude that s2,−1 is not zero and thus we have|z|→∞ along the imaginary axis. As the first term on the right‐hand side is a rational function, we infer from Theorem ρ1 exist up to order 2K. The converse direction follows by symmetry.□Now let us assume that Lemma 1.5m∼1 and m∼2 withmi for i∈{1,2} admit the continued fraction expansionN∈N, non‐negative constants υ0,…,υN−1, real constants ω0,…,ωN−1 and positive constants l1,…,lN with υn+|ωn|>0 for all n∈{1,…,N−1}, then|z|→∞ along the imaginary axis, where the integer K∈N0 is given byj is equal to one whenIf there are Herglotz–Nevanlinna functions i∈{1,2}, let us define Herglotz–Nevanlinna functions qi,1,…,qi,N byn∈{1,…,N−1} recursively viami=υ0z+ω0+qi,1 by assumption. A computation then shows that|z|→∞ along the imaginary axis for all n∈{1,…,N−1}, which yields|z|→∞ along the imaginary axis. Since this entails that|z|→∞ along the imaginary axis, it remains to note that|z|→∞ along the imaginary axis.□For m.Theorem 1.6m is a non‐rational Herglotz–Nevanlinna function and let K∈N0. Then the moments of the measure ρ exist up to order 2K if and only if there is a Herglotz–Nevanlinna function m∼, an integer N∈N, non‐negative constants υ0,…,υN, real constants ω0,…,ωN−1 and positive constants l1,…,lN with υn+|ωn|>0 for all n∈{1,…,N−1} andm admits the continued fraction expansionSuppose that With the help of these two lemmas, we are now ready to add another item to the equivalence in Theorem m admits such a continued fraction expansion. We define the rational Herglotz–Nevanlinna function m1 by replacing m∼ in the continued fraction on the right‐hand side of /z→0 as |z|→∞ along the imaginary axis. This allows us to writeυ1 and a Herglotz–Nevanlinna function m∼, which is the claimed expansion. Now let K∈N and suppose that the assertion holds for all lesser integers. As before, we may write m like in (υ1 and a Herglotz–Nevanlinna function m∼ with m∼(z)/z→0 as |z|→∞ along the imaginary axis. In the case when K=1 and υ1≠0, this is the required expansion. Otherwise, we conclude from Lemma ρ∼ corresponding to the function m∼ exist up to orderm∼ then admits a continued fraction expansion of the claimed form, we readily see that so does m (it only remains to note that υ1+|ω1|>0 in this expansion since q(z)/z→0 as |z|→∞ along the imaginary axis).□For the converse direction, we will use induction. To this end, let us first consider the case when o writem(z)=−s−o write1m(z)=−sm is a non‐rational Herglotz–Nevanlinna function such that the moments of the measure ρ exist up to order 2K for some K∈N0. Under these assumptions, the determinants Δ0,k are well defined for all k∈{0,…,K+1} and positive in view of  is the smallest integer k∈{0,…,K} for which the sequence Δ1,0,…,Δ1,k has precisely n non‐zero elements. One observes that the increasing function κ is defined in such a way that Δ1,κ(1),…,Δ1,κ(N) enumerates all non‐zero members of the sequence Δ1,0,…,Δ1,K. As it follows from relation =0, this determines κ recursively.Corollary 1.7m is a non‐rational Herglotz–Nevanlinna function such that the moments of the measure ρ exist up to order 2K for some K∈N0, then there is a Herglotz–Nevanlinna function m∼ withm admits the continued fraction expansionυ0,…,υN−1 and the real constants ω0,…,ωN−1 are given by =K−1 and the function m∼ satisfiesIf given by  and the given by . FurtherAs in the rational case, we are able to provide explicit formulas for the constants in the continued fraction expansion in terms of the Hankel determinants. To this end, let us suppose that  view of , as the relation  that them is a non‐rational Herglotz–Nevanlinna function such that the moments of the measure ρ exist up to order 2K for some K∈N0. According to Theorem m admits a continued fraction expansion of the form  the corresponding function , we may obtain expressions for the constants in this expansion in terms of the Hankel determinants by comparison with Proposition N ,Δ1,κ(2),… enumerates all non‐zero elements of the sequence Δ1,0,Δ1,1,… (of which there are infinitely many as the sequence does not contain any consecutive zeros).Corollary 1.8m is a non‐rational Herglotz–Nevanlinna function. Then all moments of the measure ρ exist if and only if there are Herglotz–Nevanlinna functions m∼1,m∼2,…, non‐negative constants υ0,υ1,…, real constants ω0,ω1,… and positive constants l1,l2,… with υn+|ωn|>0 for all n∈N such that for every N∈N the function m admits the continued fraction expansionυ0,υ1,… and the real constants ω0,ω1,… are given by  be a generalized indefinite string so that L∈ and υ is a non‐negative Borel measure on [0,L). We consider the corresponding spectral problem of the formz is a complex spectral parameter. Of course, this differential equation has to be understood in a distributional sense  such thatdf. In this case, the constant df is uniquely determined and will henceforth always be denoted with f′(0−) for apparent reasons.A solution of  is a funLet refer to  for more is the corresponding Weyl–Titchmarsh function m defined on C∖R byψ is a non‐trivial solution of the differential equation ↦m establishes a one‐to‐one correspondence between generalized indefinite strings and Herglotz–Nevanlinna functions. In the following, we are going to use our findings from the last section to characterize those Herglotz–Nevanlinna functions that correspond in this way to generalized indefinite strings that begin with a discrete part. To be more precise, we consider generalized indefinite strings  of the formN∈N0, increasing points x1,…,xN+1 in , real weights ω0,…,ωN and non‐negative weights υ0,…,υN, where δx denotes the unit Dirac measure centered at a point x∈[0,L). For such coefficients, the solutions ψ of the differential equation  has the formL, the increasing points x1,…,xN in , the real weights ω0,…,ωN and the non‐negative weights υ0,…,υN are given byIf the Weyl–Titchmarsh function It was shown in [equation  are piecelationψ′ such thatx1,…,xN are given bym∼ corresponding to  admits the same continued fraction expansion as m in view of  has the claimed form and the explicit formulas follow readily from the ones in Proposition □Suppose that the Weyl–Titchmarsh function  view of , we concRemark 2.3w∈Lloc2[0,L) is the normalized anti‐derivative of the distribution ω so thatw is piecewise constant. After plugging in the expressions from +1 is zero or not.For generalized indefinite strings as in Proposition ∑j=0n−1ωidentity . For the,κ(n),∫0xnw(x)ons from , it remaelations  and , real weights ω0,…,ωN−1 and non‐negative weights υ0,…,υN with υn+|ωn|>0 for all n∈{1,…,N−1} and has the formSuppose that the Weyl–Titchmarsh function We now proceed to add another item to the equivalences in Theorem ρ exist up to order 2K so that the function m admits the continued fraction expansion  that satisfyx1,…,xN are given by , which guarantees that  has the claimed form.Assume first that the moments of the measure xpansion  by Theorgiven by  and it irding to , the Weyrding to . It thenl1,…,lN given bym∼  has the formx1,…,xN in , the real weights ω0,…,ωN−1 and the non‐negative weights υ0,…,υN−1 are given by =K−1 and one hasIf the Weyl–Titchmarsh function given by . FurtherIt is again possible to find explicit expressions for the weights and their positions in Theorem □In view of Corollary Remark 2.6|z|→∞ along the imaginary axis, of the Weyl–Titchmarsh function m uniquely determines the generalized indefinite string  near the left endpoint. This can be viewed as a variant of local inverse uniqueness results as in [Theorem ts as in , 33, 38 ts as in , 2439.One can infer from Lemma Corollary 2.7m is not rational. Then all moments of the spectral measure ρ exist if and only if there are increasing points x1,x2,… in , real weights ω0,ω1,… and non‐negative weights υ0,υ1,… with υn+|ωn|>0 for all n∈N such that the generalized indefinite string  has the formx1,x2,… in , the real weights ω0,ω1,… and the non‐negative weights υ0,υ1,… are given by  whose coefficients are supported on discrete sets, we just need to introduce one additional condition on the spectral data.Corollary 2.8m is not rational. Then all moments of the spectral measure ρ exist andx1,x2,… in  with xn→L, real weights ω0,ω1,… and non‐negative weights υ0,υ1,… with υn+|ωn|>0 for all n∈N such that the generalized indefinite string  has the formx1,x2,… in , the real weights ω0,ω1,… and the non‐negative weights υ0,υ1,… are given by (Suppose that the Weyl–Titchmarsh function given by .In order to make sure that the Weyl–Titchmarsh function ρ({0})=L−1, which holds for arbitrary generalized indefinite strings; see [□This is a consequence of Corollary Remark 2.9ω is supported on a discrete set and the measure υ vanishes identically. More precisely, these indefinite strings are determined by the additional conditions thatΔ1,1,Δ1,2,… is zero.In conjunction with the solution of the inverse spectral problem for generalized indefinite strings in [As already indicated before, this last result can be used to prove Corollary Bulletin of the London Mathematical Society is wholly owned and managed by the London Mathematical Society, a not‐for‐profit Charity registered with the UK Charity Commission. All surplus income from its publishing programme is used to support mathematicians and mathematics research in the form of research grants, conference grants, prizes, initiatives for early career researchers and the promotion of mathematics.The


Counterfeiting has become a prevalent business worldwide, resulting in high losses for many businesses. Considerable attention has been paid to research an individual attitude toward purchasing luxury counterfeit products in the offline context. However, there is currently lesser-known literature on the given phenomenon in the context of social commerce. Moreover, researchers observed that counterfeiting consumption is associated with consumer ethical values or beliefs. Practitioners and researchers are keen to find those factors that affect consumers’ ethical consumption behavior to reduce pirated products’ demand. However, the role of religion in shaping ethical behavior is less documented in the counterfeiting context. Therefore, this study investigated the effect of religiosity on the counterfeiting of luxury products in Pakistan. A five-dimensional Islamic religiosity model was adopted to understand the consumption phenomena. For quantitative research, cross-sectional data were collected from the generation M of Pakistan through self-administrative questionnaires. A total of 394 valid responses from active online users were collected to empirically examine the conceptual model by employing the partial least square structural equation model (PLS-SEM). The results reveal that all five dimensions of religiosity negatively affect the attitude of generation M. Moreover, it is found that knowledge has the highest negative effect on attitude, followed by orthopraxis, experience, central duties, and basic duties. The study also explains the theoretical and practical implications of the research. Finally, limitations and future research were also discussed. The virulence of the COVID-19 coronavirus pandemic has shaken the world . This paReligion plays an extremely important role in guiding human behavior. Although, the association between religion and buying behavior was already established . DespiteSecond, past research has found various significant predictors accountable for luxury counterfeit buying intention in the offline context but few in the online context . The intThird, the research mainly focused on generation M’s attitude toward counterfeiting products. Generation M refers to millennial Muslims of the new generation . This MuReligion is an essential cultural value because it is unique, extensive, and dictates social institutes that profoundly affect the individual and society’s perception, belief, and behavior . TheoretNevertheless, there is still limited research addressing the role of religiousness in counterfeiting and digital piracy phenomena . PreviouAccording to the Quran verses, counterfeiting is prohibitive in Islam because Islam said any activity leading to such action is considered a fraud. Islam forbids all kinds of cheating and all deceiving acts, whether in buying and selling fraud or between people in any other matter. All Muslims are urged to be honest and trustworthy in everything they do in all situations.يٰۤـاَيُّهَا الَّذِيۡنَ اٰمَنُوۡا لَا تَخُوۡنُوا اللّٰهَ وَالرَّسُوۡلَ وَتَخُوۡنُوۡۤا اَمٰنٰتِكُمۡ وَاَنۡـتُمۡ تَعۡلَمُوۡنَ“O you, who have believed, do not deceive Allah and the Messenger or deceive your trusts while you know (the consequences)” (Quran 8:27).وَلَا تُجَادِلْ عَنِ الَّذِيْنَ يَخْتَانُوْنَ اَنْفُسَهُمْ ۗ اِنَّ اللّٰهَ لَا يُحِبُّ مَنْ كَانَ خَوَّانًا اَثِيْمًاۙ“And do not argue on behalf of those who deceive themselves. Indeed, Allah loves not one who is a habitually sinful deceiver” (Quran 4:107).إِنَّ ٱ یَّأْمُرُكُمْ أَن تُؤَدُّوا۟ ٱلْأَمَنَتِ إِلَىٰٓ أَھْلِھَاوَإِذَا حَكَمْتُم بَیْنَ ٱلنَّاسِ أَن تَحْكُمُوا۟ بِٱلْعَدْلِ إِۚنَّ ٱ نَِّعِمَّا یَعِظُكُم بِھِ إۦِٓۗنَّ ٱ كَّانَ سَمِیعًۢا بَصِیرًۭا“Indeed, Allah orders you to render the trust to their owners, and when you judge between people to judge with justice. Excellent is that which Allah instructs you. Indeed, Allah is ever Hearing and seeing” (Quran 4:58)إِنَّ ٱ یَُّدَفِعُ عَنِ ٱلَّذِینَ ءَامَنُوٓا۟ إِۗنَّ ٱ لَّا یُحِبُّ كُلَّ خَوَّانٍۢ كَفُورٍ“Lo! Allah defended those who are true. Lo! Allah loved not each deceitful ingrate” (Quran 22:38).وَأَوْفُوا۟ ٱلْكَیْلَ إِذَا كِلْتُمْ وَزِنُوا۟ بِٱلْقِسْطَاسِ ٱلْمُسْتَقِیمِ ذَۚلِكَ خَیْرٌۭ وَأَحْسَنُ تَأْوِیلًۭ“And measure full when you measure. And weigh with an even balance. This is better, and its end is good” (Quran 17:35).وَیْلٌ لِلْمُطَفِّفِینَ الَّذِینَ إِذَا اكْتَالُوا عَلَى النَّاسِ یَسْتَوْفُونَ وَإِذَا كَالُوھُمْ أَوْ وَزَنُوھُمْ یُخْسِر“Woe to the defrauders who use short measures, who, when they measure [a commodity bought] from the people, take the full Measure, but diminish when they measure or weigh for them”.یَٰٓأَیُّھَا ٱلَّذِینَ ءَامَنُوا۟ لَا تَأْكُلُوٓا۟ أَمْوَٰلَكُم بَیْنَكُم بِٱلْبَٰطِلِ إِلَّآ أَن تَكُونَ تِجَٰرَةً عَنتَرَاضٍۢ مِّنكُمْ وَۚلَا تَقْتُلُوٓا۟ أَنفُسَكُمْ إِۚنَّ ٱ كَّانَ بِكُمْ رَحِیمًۭا“O you who have believed, do not consume one another’s wealth unjustly but only (lawful) business by mutual consent. And do not kill yourselves (or one another). Indeed, Allah is to you ever merciful” (Quran 4:29).وَلَا تَأْكُلُوٓا۟ أَمْوَلَكُمبَیْنَكُم بِٱلْبَطِلِ وَتُدْلُوا۟ بِھَآ إِلَى ٱلْحُكَّامِ لِتَأْكُلُوا۟ فَرِیقًۭا مِّنْ أَمْوَلِ ٱلنَّاسِ بِٱلْإِثْمِ وَأَنتُمْ تَعْلَمُونَ“And do not consume one another’s wealth unjustly or send it [in bribery] to the rulers in order that [they might aid] you [to] consume a portion of the wealth of the people in sin, while you know [it is unlawful]” (Quran 2:188).Hadith is the saying of the Prophet Muhammad (PBUH). Hadith and Sunnah are both critical aspects of Islam. Hadith and Sunnah are two of Islam’s key aspects. Hadith plays an essential role in everyone’s life because it forms the person you are and who you have become along this life’s journey. Following the Prophet Muhammad’s hadith or Sunnah (PBUH), Allah’s ways and the messages he sent down from the heavens will be followed to help us achieve our goals.Prophet Muhammad’s (PBUH) saying!“The one who deceives is not one of us” “Don’t deceive someone who trusts you so you can’t cheat him” (Wasa’ il ul-Shia Vol. 12 pages 364).He (PBUH) also said: “An honest and truthful Muslim trader shall be held with the martyrs on the Day of Resurrection.” .“If both parties were to speak the truth and explain the faults and attributes (of the goods), then they would be blessed in their transaction, and if they said lies and concealed anything, then their transaction’s blessings would be lost” (Bukhari and Muslim Resolution No. (101).The Prophet has been recorded to state that it is not permitted to sell goods without making clear about everything, nor is it allowed for anybody who knows about the goods’ defects, not to mention them. An act of dishonesty is the same as cheating. One of the worst forms of fraud is dishonesty. A dishonest person is always likely to defraud others as often as possible such as misuse, false claim, intimate good, and so on.The primary duties of Muslims contain all religious beliefs and attitudes. The Muslims believe that there is only one God (Allah), “Indeed, your God is one” (37:4), and Muhammad is the Prophet, “o Muhammad! They say ‘we testify that you are the last Messenger of Allah” (63:1). Religious teachings are based on three things, the “Quran” and the “Hadith” (Muhammad’s documented saying and acts), and the shari’a (Religious law) offers answers to all ethical questions . The Mus“Rather, to Allah belong the hereafter and the first life” (Quran 53:25).The second dimension is the fulfillment of the core religious duties. These consist of following more or less the “five pillars of Islam” duties which consist of religious beliefs and practices : (1) “ShThe experience dimension includes people’s perceptions and practices of their faith . ExperieThe knowledge aspect involves knowledge of the person about religion. The contents of the Quran and Sunnah are usually the primary source of Islamic knowledge. Believers are expected to know a minimum of these contents. Muslims call for the in-depth understanding of equality for all human beings, a strong sense of brotherhood, good or bad deeds, including morality, modesty, humility, trustworthiness, duty, justice, patience, fairness, and tolerance, ethics in dealing, care, empathy, and compassion .“He also says, we have revealed to you the book which clarifies every matter” (Quran 16:89).“Indeed this Quran guides to the path which is clearer and straighter than any other” .The Orthopraxis dimension of religiosity plays an important and distinct role in Muslim religiosity . OrthoprH1(a): Basic Duties dimension of religiosity negatively affect consumers’ attitude toward purchasing luxury counterfeit products.H1(b): Central Duties dimension of religiosity negatively affects consumers’ attitude toward purchasing luxury counterfeit products.H1(c): The experience dimension of religiosity negatively affects consumers’ attitudes toward purchasing luxury counterfeit products.H1(d): The knowledge dimension of religiosity negatively affects consumers’ attitudes toward purchasing luxury counterfeit products.H1(e): Orthopraxis dimension of religiosity negatively affect consumers’ attitude toward purchasing luxury counterfeit products.A survey method was employed to validate the conceptual model because the quantitative research method predicts individual responses and examines the relationship between constructs . Many reA 7-point Likert scale was used to measure all these constructs that ranged from strongly disagree = 1 to strongly agree = 7. A quantitative approach using the survey questionnaire method was used to examine the influence of religiosity on Pakistani Muslim youth’s attitude toward counterfeit products. The questionnaire comprises two parts. Section A will be the attitude (5 items) , basic dIn the current study, the non-probability sampling technique of purposive sampling was employed. It is suitable and more appropriate when the sampling frame is not available and population is unknown . MoreoveStructural Equation Modeling (SEM) has an advantage for statistical analysis in terms of efficiency, accuracy, and convenience over traditional multivariate statistical techniques . MoreoveThe measurement model is used to observe the relationship between observed data and latent variables, and it also explains the calculation of variables. The advantage of this model was to assess the valuation of validity and reliability test. Construct reliability is measured using outer loading, and internal consistency of reliability was measured through composite reliability. Moreover, convergent reliability is measured through the average variance extracted (AVE) . To asceNext discriminating validity, Heterotrait-Monotrait correlation ratio (HTMT) TEST, was used to measure discriminating validity because it is more potent than other methods. According to t-values, coefficient of determination (R2), effect size (f2), and predictive relevance (Q2). Using the bootstrapping method , path coefficient significance was measured. The results indicate that all hypotheses are accepted and significant toward individuals’ attitudes to purchasing counterfeit products. However, findings suggested that the knowledge dimension has the most significant negative impact on attitude to use luxury counter fitting products (47%), followed by orthopraxis dimension (19%), experience dimension (13%), central duties (8%), and basic religiosity (7%)  see . The R2 (7%) see .Counterfeiting is a centuries-old crime, the proliferation of counterfeit products has become a global phenomenon, and indeed is a serious business issue all over the nation . It’s fap > 0.05). The highest impact of the knowledge dimension is in line with the expectations that more knowledge and study of basic tenets and scriptures of Islam influence the followers to act accordingly. This finding also collaborates with the previous studies . The results are also in agreement with previous research . One plausible explanation for the significant results is that when the individual is more afraid of the creator, they abide by what is considered good  and avoid what is deemed bad (haram), which can be seen in their practices. Muslims follow the ALLAH instructions because they know that their excellent behavior rewards them and their bad behavior punished them in the hereafter  show a negative association of religious duties with attitude toward counterfeit products with a weak effect. The results are consistent with the earlier study. Previous studies also suggest that a person with strong religious commitment is less likely to engage in unethical behavior such as illegal use of the drug  have a weak negative relationship with attitude toward the purchase of counterfeit products. Some previous researches also validate the finding of the study. Basic duties are interconnected with the core or fundamental values of Islam, which differentiate Muslims from non-Muslims. The main reasons for the results are the previous author characterized these beliefs into three types. The first type of belief explains the exitance of the divine, the second type of belief explains the purpose of the divine, and the third describes the ethical structure of the whole region  on specific counterfeiting products like cosmetics, digital piracy, and online books. Third, many other factors or variables like culture, income, and education should be added to conduct further research. Fourth, this study only focuses on young Muslim consumers. Further studies can consider other religions like Christians, Hindus, Jews, and Buddhism.The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.Ethical review and approval was not required for the study on human participants in accordance with the local legislation and institutional requirements. Written informed consent from the patients/participants or patients/participants legal guardian/next of kin was not required to participate in this study in accordance with the national legislation and the institutional requirements.SA and HZ: conceptualization and writing of original manuscript. MA and NK: review and improve. PP: data collection and analysis. All authors contributed to the article and approved the submitted version.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


The industry-based internet of things (IIoT) describes how IIoT devices enhance and extend their capabilities for production amenities, security, and efficacy. IIoT establishes an enterprise-to-enterprise setup that means industries have several factories and manufacturing units that are dependent on other sectors for their services and products. In this context, individual industries need to share their information with other external sectors in a shared environment which may not be secure. The capability to examine and inspect such large-scale information and perform analytical protection over the large volumes of personal and organizational information demands authentication and confidentiality so that the total data are not endangered after illegal access by hackers and other unauthorized persons. In parallel, these large volumes of confidential industrial data need to be processed within reasonable time for effective deliverables. Currently, there are many mathematical-based symmetric and asymmetric key cryptographic approaches and identity- and attribute-based public key cryptographic approaches that exist to address the abovementioned concerns and limitations such as computational overheads and taking more time for crucial generation as part of the encipherment and decipherment process for large-scale data privacy and security. In addition, the required key for the encipherment and decipherment process may be generated by a third party which may be compromised and lead to man-in-the-middle attacks, brute force attacks, etc. In parallel, there are some other quantum key distribution approaches available to produce keys for the encipherment and decipherment process without the need for a third party. However, there are still some attacks such as photon number splitting attacks and faked state attacks that may be possible with these existing QKD approaches. The primary motivation of our work is to address and avoid such abovementioned existing problems with better and optimal computational overhead for key generation, encipherment, and the decipherment process compared to the existing conventional models. To overcome the existing problems, we proposed a novel dynamic quantum key distribution (QKD) algorithm for critical public infrastructure, which will secure all cyber–physical systems as part of IIoT. In this paper, we used novel multi-state qubit representation to support enhanced dynamic, chaotic quantum key generation with high efficiency and low computational overhead. Our proposed QKD algorithm can create a chaotic set of qubits that act as a part of session-wise dynamic keys used to encipher the IIoT-based large scales of information for secure communication and distribution of sensitive information. We applied the class “Random” in Java for the chaotic production of qubit variant states. We used octa-qubit bases to capture the matched patterns at the side of the receiver. The octal states are represented as {, , , , , , , } to signify the states of the qubits over the randomly directed transmission qubits as shown in The originator produces the range of values from 0 to 1, which is mapped with the full direction of the qubit; at the receiver side, we need to select the matched base according to our octal state representation. The considerable equivalent random numbers generated as per the movement of the qubit are {0, 0.125, 0.25, 0.375, 0.5, 0.625, 0.75, 0.875, 1} as represented in public String createRandomBase{double base_num = qubit.randomGenerator;    if (base_num <0 && base_num <0.25)        base = “±”;else if (base_num >0.25 && base_num<0.5))         base = “┼”;else if (base_num >0.5 && base_num<0.75)         base = “Ӿ”;else if (base_num >0.75 && base_num<1))        base = “X”;return base;    }According to the selected base, the SOP sequence is chaotic. The base values and the SOPs of the above snippet are represented as the polarizer angle base “┼” means to capture only the precise vertical and the horizontally directed matched qubits {, }. The polarizer angle base “X” means to capture only the precise left and right orthogonal-directed matched qubits {, }. Here, we added two more variants to also capture other directions of the qubit. The polarizer angle base “±” means to capture different directions of the qubit in between the straight rectilinear and straight orthogonal positions that are to capture the directed matched qubits { } of the top half circle. The polarizer angle base “Ӿ” means to capture other directions of the qubit in between the straight rectilinear and straight orthogonal positions that are to capture the directed matched qubits { } of the bottom half-circle. The entire process is given as:public String CreateSop(String base) {RL1 = “┼”; SRL1 = “±”;RL2 = “┼”; SRL2 = “±”;LOG1 = “X”; SLOG1 = “Ӿ”;ROG2 = “X”; SROG2 = “Ӿ”;if || base.equals(RL2))    {double base_num = qubit.randomGenerator;     if (base_num >0.25 && base_num <0.5)          sop = “|”;else          sop = “—”;  }else if || base.equals(ROG2))    {double base_num = qubit.randomGenerator;     if (base_num > 0.75 && base_num <1)          sop = “/”;else          sop = “\\”;  }else if || base.equals(SRL2))    {double base_num = qubit.randomGenerator;     if (base_num >0 && base_num < 0.25)     ±_|”;     sop = “else     ±_—”;     sop = “  }else if || base.equals(SROG2))    {double base_num = qubit.randomGenerator;     if (base_num > 0.5 && base_num < 0.75)          sop = “ Ӿ_/”;else          sop = “ Ӿ_\\”;  }return sop;   }The central portion of secure end-to-end transmission by using the proposed QKD technique is the measurement of the multi-qubits of Singh by King. This means, we need to address how King can produce his set of bits; these bits may be a part of the private key. The calculation of the qubits of Singh correspondingly abolishes the generated bits, letting Singh transmit his base set to King only, and access or capture by third-party users or intruders is impossible. The states of the multi-qubits calculation process function through predicting the base of Singh first alongside the base of King. If the chaotic grounds of Singh and King are accorded, the matched pattern bit sets will be used as part of the secret key for the encipherment and decipherment process. In a counter case, the chaotic bases of Singh and King are not accorded. Afterward, the restrained bit sets are discarded. In both cases, the multi-qubit set should be destroyed after matched bits are generated. The finally produced bits approach yields the bit 1 if the SOP is “|” or “⁄” or “±_|” or “Ӿ_/”;\ and 0 if the sop is “—“ or “\” or “±_—” or “Ӿ_\\”. The qubit measurement and bitrate production functions are specified as:public int bitCreate (String sop){if )        return 1;     else if )       return 1;     else if )     return 1;     else if )     return 1;     else if )       return 0;     else if )      return 0;     else if )   return 0;     else if ) return 0;     else return null;     }public int qubitMeasure {int bitset;     if )          {null;     bitset = bitCreate(myqubit.sop); singqubit =      }else          {     singqubit.base= base_;     singqubit.sop = singqubit.createSop(base_);     bitset = bitCreate(singqubit.sop);null;     singqubit =      }return bitset;     }To provide more security to encrypt the sensitive information of users, we need to produce a secret quantum key with a reasonable count of qubits. Here, we show the list of methods used to generate a good number of qubits; from these, we extract the final secret key for the encipherment and decipherment process. The first step of the algorithm is that Singh needs to pass a random integer, RI. Here, RI is the count of qubits he plans to produce for the transmission. Recall that we should have a base and SOP for the construction of qubits. Singh initially makes a list of commands by invoking the random base method, and as per that list, he produces his list of SOPs mapped with the list of qubits. Afterward, Singh utilizes both lists to make the final list of mapped qubits with their respective bases, which he will send to King. Later, King will obtain the qubit list of Singh; he utilizes the RI supplied by the count to produce his base list as Singh already has done. King also creates his bit set list with the help of using the qubit measurement process. Now that both Singh and King have their base and bit set lists, both shall interchange their base lists to construe their lists of bit sets. If their occurred bases are matched, then the resultant bit list is used to generate a secret key. Otherwise, it is discarded. The result is a shared secret key, whose size is approximately half of RI. To evaluate this, we maintain the approximate size of the secret key related to RI. For every running instance, the proportion varies and progressively touches just about half of the actual length. In existing models, due to the production of shorter key length and after filtration, the final key is much shorter and is vulnerable to attacks specifically in Social Internet of Things (SIoT) and related applications, blockchain based mobile applications, wireless body area network based applications, computer vision based applications , fuzzy oThe proposed model is highly chaotic and dynamic. Despite this, it works fine to exhibit the notion and how it performs compared to previous QKD models. In this paper, we only demonstrated how keys are exchanged between IIoT devices in the absence of attacks and theoretically proved that this model provides more security from all types of quantum attacks compared to traditional models. In addition, this model uses less qubit transmission rate and key exchange rate along with minimal computational overhead and shows more efficacy with regards to error-rate measurement to detect whether snooping has happened or not. In future work, we can add an unauthorized person or intruder and demonstrate how to address different advanced quantum attacks at one time while sensitive data are transmitted among multiple IIoT devices. It could be possible to examine various privacy and security parameters of the proposed standard, such as the minimum bit size needed to detect intruders. At the same time, with the communication of qubits among IoT and IIoT devices, we can enhance their security by strengthening the projected model with a multi-photon multi-state entanglement approach using Qiskit for working with quantum computers at the level of pulses, circuits, and IoT and IIoT application modules.


This article has been withdrawn by the authors. The authors and thejournal conclude that the images between and within Figures 2D and 3D were reused.Specifically, the image “pcDNA” in Figure 3D was a duplication of the image“GW9662 + Vehicle” in Figure 2D. Similarly, the image “p300WT + 15(S)-HETE” inFigure 3D was captured from the well “PFS1YF + 15(S)-HETE” in Figure 2D. The image“p300WT” in Figure 3D was captured from the well “pcDNA + 15(S)-HETE” in Figure 3Ditself. Additional potential internal reuse is in Figure 2D, STAT1(WT) + 15(S)-HETE”and Figure 2D, “PFS1YF + 15(S)-HETE”. The authors state that these imageduplications do not affect the conclusions of the paper. The authors apologize forthis oversight.


This study gives a critical examination of the performances of three top key opinion leaders (KOLs) on social media in China to explore whether gender serves as an important social factor in their interaction with their followers. A critical analysis of their Weibo posts has revealed that they construct different gender identities through their preferential choices of personality traits, speech acts, and addressing terms. This can be explained in terms of their target consumers and the products they promote. It concludes that although gender can serve as an important factor in top KOLs’ performances on social media, it can be appropriated and exploited in varied ways to serve different communicative purposes. China has the world’s largest live-streaming market in terms of the scale of participation and revenue generation . The COVSince KOLs play important roles in diffusing information, setting agenda, and influencing others’ decision-making or behaviour , how KOLPrevious studies on gendered interaction have identified different personality traits associated with males and females . For exaNevertheless, disputes remain over whether gender differences in language use will dissolve on social media. While the anonymity of social media might possibly lead to the hiding or invisibility of gender disparity, some previous studies have demonstrated that gender differences are still evident even when communication moves from offline to online . Some otBesides, studies on gendered interaction “reached a deadlock partly because of the research question itself (looking for differences between women and men) and its presumptions (taking ‘gender’ for granted)” , p. 411.A KOL is defined as “an individual who has a great amount of influence on the decision making, attitudes and behaviors of other people” , p. 511 This study focuses on three top KOLs, namely Li Jiaqi (Li), Viya, and Luo Yonghao (Luo). They were selected because of their influences as well as their distinctive gender performances. Li was once a makeup seller at a L’Oréal counter in Nanchang (a second-tier city in China). It is quite rare in China for a man selling cosmetics to women . HoweverViya was the first female KOL of Taobao Live , and sheLuo was the “first generation internet influencer” in China , p. 88, st 2021 to 30 September 2021 (6 months). They are all “V-users” (verified users on Weibo) with millions of followers. As a hybridized form of Twitter and Facebook, Sina Weibo is one of the most popular social media platforms in China. As a social networking platform, Weibo serves as a crucial channel for interpersonal communication . It is aThis study gives a critical discourse analysis (CDA) of gender construction in the Weibo posts of three top KOLs in Chinese mainland. CDA views language use as a social practice and underlines the significance of examining language use in its social contexts . It viewThe constitutive effects of language can be identified in three aspects: (1) social knowledge; (2) social identities; and (3) social relations . While sHow do the three KOLs vary in their choice of linguistic features?How do they construct relations between themselves and their followers?What gender images do they construct for themselves on their social media?How do they contribute to their respective communicative purposes?In order to reveal the dynamic relations between gender and interaction, this study conducts a top-down analysis of these posts at three levels: (1) personality traits; (2) speech acts; and (3) addressing terms. Previous studies have demonstrated that men and women tend to display different personality traits. Men tend to be “determined,” “effective,” “fighting,” “forceful,” “confident,” “champion,” and “rational” (masculine traits), whereas women tend to be “caring,” “warm,” “compassionate,” “understanding,” “congenial,” “humble,” and “empathetic” (feminine traits) . This stSpeech acts refer to the linguistic acts associated with the speaker’s utterance . Zhang aThey are followed by a close analysis of the self- and other-addressing terms used by the three KOLs on their social media. All the pronouns and nominal forms used to address themselves and others are examined . They arThe coding of personality traits and speech acts was performed by the first and second authors. The two authors coded the data independently after the initial training and discussion of the coding criteria and procedure. The discrepancies in some results were resolved through further discussion. The inter-rater reliability was measured with Cohen’s Kappa. The average percent agreement of all coded categories was greater than 90%, and the Cohen’s Kappa was 0.91, indicating a strong agreement between the two coders .zhinan”  image.They also show distinct differences in their preferences for specific personality traits. Luo is noted for his highest preferences for the masculine traits of “forceful” (50%) and “rational” (16%). Li shows the highest preferences for the feminine traits of “caring” (28%) and “congenial” (20%) and the masculine trait of “forceful” (23%). Viya shows the highest preferences for the feminine trait of “understanding” (20%) and the masculine traits of “forceful” (26%) and “confident” (14%). It shows that the personality trait “forceful” is valued by all three KOLs, which can be attributed to the primary function of their Weibo posts, i.e., to promote their products and engage the followers to buy their products (see Example 1). However, it is much more highlighted in Luo’s Weibo posts than in the other two KOLs’, suggesting that Luo prefers to deliver information about his products in a more straightforward way than the other two KOLs. This contributes to the construction of his “Besides, Luo shows a “rational” trait by questioning, challenging or criticizing some social phenomenon or popular ideas (see Example 2). He disaligns himself with the public and the popular ideas, thus further consolidating his image of being the “spiritual leader” of some youngsters or middle-aged men. In other words, Luo does two things on his Weibo posts: promoting the products and constructing a critical image for himself. More often than not, they are achieved on separate posts. Examples are as follows:(1) Forceful:全场平均5折!全场平均5折!全场平均5折! .Average 50% off for all items! Average 50% off for all items! Average 50% off for all items!(2) Rational:虽然 Chord 是人类做过的最丑的解码器，但音质确实好…你看，仅从商业角度来说，电子产品的设计没那么重要。.Although Chord is the ugliest decoder made by humans, its sound quality is indeed good…You see, just considering business, the design of electronic products is not that important.guimi”  of his overwhelming female followers. Usually, they are achieved on the same Weibo post. That means Li promotes his products through constructing himself as a congenial and understanding friend of his followers. As Example 5 shows, Li shows a “forceful” trait by announcing the news on the one hand and a “caring” trait by asking the followers whether they are ready or not on the other hand. In example 6, Li shows a “forceful” trait by announcing the news and a “congenial” trait by showing he and his followers share the same hobby at the same time.By contrast, although Li also promotes his products by underlining the “forceful” trait, he shows equal, if not more, preferences for the “congenial” and “caring” traits (see Examples 3 and 4). In other words, Li shows alignment with the followers by constructing himself as a “(3) Caring:夏天了，有没有想吃的零食呀特别是那种私藏零食或者家乡美食，给我抄抄作业呀 .Summer is coming, have any snacks wanna try? Especially those personal treasures or hometown specialties, let me copy your homework.(4) Congenial:我的宵夜和最近困困的Never胖死我算了 .My night meal and the recent sleepy Never will become too fat(5) Forceful + Caring:正式官宣!#所有女生的offer#倒计时2天咯。准备好了吗，9月27号12:00见!.th!Official announcement! # All Girls’ Offer # Countdown to 2 days. Ready? See you on 12:00 Sep. 27(6) Forceful + Congenial:今晚10点，他来咯!我就差报身份证号啦!.Tonight 10 O’clock, he’s coming! Only the ID number has not been disclosed!Viya is different in that she promotes her products by constructing herself as an expert in the field. On the one hand, she highlights that she understands what the followers need; on the other hand, she underlines her confidence in the quality and prices of her products. This is often achieved on the same posts (see Examples 7 and 8). In other words, she promotes her products because she believes that it is the best choice for the followers. The image she constructs for herself is a professional image rather than a gendered image. Examples are as follows:(7) Forceful + understanding:99爆品日!vivo、雅诗兰黛新品首发!玉泽、雅萌ACE、OLAY超红瓶、笔记本电脑、百威啤酒…全家人都需要的好物就在今晚! .今晚#薇娅直播间# 99 Top-sellers’ Day! Vivo, Estée Lauder debut! Dr. Yu, Yaman ACE, Olay Regenerist, laptop, Budweiser…Handy products for the whole family are here tonight!Tonight #Viya’s Live Room# (8) Forceful + Confident:零食节开吃啦!文和友、好利来、星巴克、自嗨锅、马迭尔冰棍、好利来甜品、北海牧场新品酸奶..0.60 + 无限回购的爆款零食，不来亏炸! .今晚 #薇娅直播间# Snack’s Day Start eating! Wenheyou, Hollyland, Starbucks, self-heating hotpot, Modern ice cream, Hollyland dessert, Beihai Ranch new arrival yogurt…60 + repurchase top-selling snacks, you’ll suffer great losses if not come!Tonight #Viya’s Live Room# They highlight different personality traits to construct different gender images for themselves on their Weibo posts. Luo prefers to construct a “straight male” image for himself, Li a “ladybro” image while Viya a “professional” image. This can be attributed to the types of products they promote and the target followers of their Weibo posts.Besides, the three KOLs also show distinct differences in the choice of specific speech acts. Among all speech acts, “directive” and “promoting self and others” are most frequently used by three KOLs. “Promoting self and others” is related to the primary function of their Weibo posts, i.e., to promote their products (see Example 10). “Directive” is used to ask the followers to buy their products (see Example 9).(9) Directive:现在就来我直播间抢购!.你没看错，今天周五，全场平均五折，now come and snap up in my live room!You did not misread it, today’s Friday, average 50% off for all items, (10) Promoting self and others:今晚#薇娅零食节# 来啦!文和友、肯德基、必胜客、星巴克..超多美食让爱爱爱不完!.Tonight #Viya’s Snack Day# Comes! Wenheyou, KFC, Pizza Hut, Starbucks…too many snacks to enjoy them all!As Among the three KOLs, Li shows the lowest preference for “promoting self and others,” which suggests that he values soft sell over hard sell. This can be further seen from Li’s highest preferences for “self-reporting moment and information” and “eliciting response.” They suggest that Li prefers to interact with the followers by sharing his private life and engaging the public. Hereby he constructs himself as a congenial and caring ladybro of his followers. Examples are as follows:(11) Self-reporting moment and information:到上海了，开始工作用着贵妇面膜回回血我困，我感觉今晚直播会胡言乱语 .Arrive in Shanghai, start working, refresh myself with a lady’s facemask, I’m sleepy, feeling that I’ll wander in my talk tonight.(12) Eliciting Response:今晚刷酸小课堂哦 你们有什么问题要问吗#李佳琦直播##李佳琦小课堂# .Tonight peeling class Do you have any questions # Li Jiaqi’s Live Room# #Li Jiaqi’s small class#.By contrast, Luo shows his highest preference for “showing judgement and appreciation.” That means that Luo is never afraid of expressing his own opinions towards some social phenomena or some things, thus constructing himself a key opinion leader or spiritual leader of youngsters and middle-aged men, as in the following:(13) Showing judgement and appreciation:卖了一年多的货，有些同事已经渐渐不说人话了，比如某个选品现货少，不够卖，说成“库存深度不足” 不说人话的吸引力究竟在哪里呢?.Have been selling for over 1 year, some colleagues have gradually not been able to speak like a normal man, for example “some product is in short supply, insufficient for sale,” they’ll say “stock in deep short supply” what’s the appeal in saying unlike human?Old Luo, 6), “罗老师” , and “这个星球上最优秀的软件产品经理” . Viya prefers to highlight her authority in her profession, such as “红色推荐官” , “品鉴团团长” , “品牌星鉴官” , “天猫小黑盒‘开新官’” , “杭州市反诈骗宣传大使” , and “微博电商号首席好物优选官” . She prefers to highlight her title as an official to construct her social esteem in her profession. In other words, she aligns with the established social recognition. Therefore, both Luo and Viya construct themselves as having a higher social status than the followers. However, Luo prefers to use self-designated labels, while Viya prefers to highlight social recognition. By contrast, Li prefers to locate himself at a lower power position to his followers by addressing himself as “佳琦” , “小李” , “小李老师” , “李老头” , “李时髦” , “古风小李” , “古装小李” , “李姓经纪人” , “国风小李” . Through using self-depreciating terms, nicknames, and first names, Li seeks to construct solidarity with his followers.In their choice of nominal addressing terms, Luo prefers to highlight his expertise and achievements in his field. In other words, he prefers to brag about his prestige and expertise, as can be seen from such expressions as “老罗” , “神经病” , “真老赖” , “流氓” , “土豪大哥洋财主大姐” , “各路善长仁翁” . In other words, he prefers to attack or satirize some people, thus constructing an aggressive or cynical image for himself. However, Viya prefers to use some nicknames or internet buzzwords to address others, such as “小伙伴们” , “薇娅的女人骑士们” , “盆友们” , “时尚达人们” , “体验官们” . These terms help to either construct intimacy between herself and her followers or show respect to her followers.Their different preferences can also be seen from the nominal terms they used to address others. Luo uses not only some ironic terms but also some abusive terms to address others, including “XX” (7), “贱人们” , “(所有)女生男生” , “MM们”, “美眉们” , “宝妈宝爸们” . This further demonstrates that Li prefers to use gender as a specific strategy in his promotion.Although Li also uses some internet buzzwords to address his followers, he also uses some intimate addressing terms to address his followers by their genders, such as “(所有)女生”  image. He prefers to highlight the masculine traits of “forceful” and “rational,” the speech acts of “directive” and “showing judgement and appreciation,” and the use of positive self-addressing terms and negative other-addressing terms. Li tends to construct a “guimi” (ladybro) image. He prefers to underline the feminine personality traits of “caring” and “congenial,” the speech acts of “self-reporting moment and information” and “eliciting response,” and the self-denigration self-addressing terms and respectful other-addressing terms. Viya tends to construct a “professional” image. She prefers to underscore both the masculine traits of “forceful” and “confident” and the feminine trait of “understanding,” the speech acts of “promoting self and others” and “self-reporting mood,” and the respectful self- and other-addressing terms.An analysis of the posts of the three top KOLs at three different levels finds that they tend to construct different images. Luo tends to construct a “zhinan” who is rational, critical, and brave enough to show his judgment towards some social phenomena. In other words, he attracts his followers by distinguishing himself from his followers. By contrast, since the products Li promotes are primarily female products and most of his followers are female. He has to cater to the interest of his followers by constructing himself as a congenial and caring “guimi” (ladybro) of his followers. In order to establish solidarity with female followers, he has to resort to “self-reporting moment and information” and “eliciting response” to attract his followers. In between is Viya. Although she is a female, she promotes products to both males and females. She attracts her followers by her expertise in selecting high-quality products with the lowest prices. Therefore, gender does not serve as a prominent promoting strategy because she sells products to both males and females.Their different ways of identity constructions can be attributed to their target consumers and the products they promote. Since the majority of Luo’s followers are males, they support him because of his sharp words and entrepreneurial spirits. Therefore, Luo prefers to construct himself as a “Therefore, different KOLs tend to construct different images on their social media. Although previous studies show that feminine traits are gaining advantage over time  as it caSocial media provides a platform for people to construct distinguishing personalities . AlthougThis study reveals not only the different strategies by different top KOLs to interact with their followers on social media but, more importantly, the dynamic relations between gender and interaction. It demonstrates that gender serves as not only a social constraint but also an important social resource which can be appropriated and explored for branding on social media . TherefoThe raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.ML contributed to the research design, data description and interpretation, paper writing and revising, and funding support. RZ analyzed the data, and contributed to the literature review and paper drafting. JF was responsible for the quality control and proofreading. All authors contributed to the article and approved the submitted version.This work was funded by the Hong Kong Polytechnic University .The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.


Current methods of the conversion between a rotation quaternion and Euler angles are either a complicated set of multiple sequence-specific implementations, or a complicated method relying on multiple matrix multiplications. In this paper a general formula is presented for extracting the Euler angles in any desired sequence from a unit quaternion. This is a direct method, in that no intermediate conversion step is required  and it is general because it works with all 12 possible sequences of rotations. A closed formula was first developed for extracting angles in any of the 12 possible sequences, both “Proper Euler angles” and “Tait-Bryan angles”. The resulting algorithm was compared with a popular implementation of the matrix-to-Euler angle algorithm, which involves a quaternion-to-matrix conversion in the first computational step. Lastly, a single-page pseudo-code implementation of this algorithm is presented, illustrating its conciseness and straightforward implementation. With an execution speed 30 times faster than the classical method, our algorithm can be of great interest in every aspect. R ∈ SO(3), where SO(3) is the group of invertible 3 × 3 matrices such that det(R) = 1 and When dealing with 3D orientation problems, many different formalisms can be used to describe a given rotation , each ofR = R2R1 is the rotation matrix corresponding to a rotation by R1 followed by a rotation by R2. 3D rotation matrices have some numerical shortcomings, however. For example, as many as 9 numbers (and 6 constraints) are required to represent a 3 degree of freedom rotation, and it can be difficult and computationally costly to orthogonalize a rotation matrix numerically  or  = [323] is equivalent to the sequence ZYZ). A Python implementation can be found on  − q[j] b ← q[i] + q[k] × ε c ← q[j] + q[0] d ← q[k] × ε − q[i] elsea ← q[0] b ← q[i] c ← q[j] d ← q[k] × ε endθ+ ← atan2θ− ← atan2switchvalue of θ2docase 0 do θ1 ← 0 // For simplicity, we are setting   θ3 ← 2 × θ+ − θ1  caseπ/2 do θ1 ← 0  θ3 ← 2 × θ− + θ1  otherwise do θ1 ← θ+ − θ−  θ3 ← θ+ + θ−  end endifnot_properthenθ3 ← ε × θ3 θ2 ← θ2 − π/2 endθ1, θ3 ←  // “wrap” assures θ1, θ3 ∈ 2 = 16 floating point multiplications , 4 × 3 = 12 multiplications by 2 and 3 × 3 + 6 = 15 additions/subtractions. This conversion step alone is more than enough to make an algorithm based on [Many operations are required to convert a quaternion into a rotation matrix. Using the homogeneous formula from −12. The execution times required in our tests for each sequence (and their ratios) are presented in the In this section, a performance comparison between our method and the Shuster method is presented. We adapted the SciPy library in order to compile the algorithm as described in Section 4. A real data set comprising the orientation of a spinning object with 3284 data points was used to compare the efficiency of the two algorithms. The full implementation and data set can be downloaded from [The Euler angles are still a useful intuitive 3D orientation parametrization. A fast method of conversion to/from any other set of parameters can therefore be of great interest for displaying or analyzing data, for instance. In this study, we therefore developed a general formula for this conversion which is concise, easy to implement and easy to debug. In addition, the fact that our method is about 30 times faster than the method proposed by , which r


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The article was published without the keywords “Drug prescription, aged 80 and over, primary care”.In “Abstract” section, “Conclusions” is corrected as “Conclusion”.The heading sizes for “Aim” and “Ethics approval” are corrected.The sentence “In summary, this study examined medication…” till “… prescription might be helpful in the very old.” should be placed under the “Conclusion” section.The section “Conflict of interest” is corrected as “Conflicts of interest”.The original article has been corrected.