R is stable if every non-zero ideal I of R is projective over its ring of endomorphisms. Motivated by a paper of Bass in the 1960s, stable rings have received wide attention in the literature ever since then. Much is known on the algebraic structure of stable rings and on the relationship of stability with other algebraic properties such as divisoriality and the 2-generator property. In the present paper, we study the arithmetic of stable integral domains, with a focus on arithmetic properties of semigroups of ideals of stable orders in Dedekind domains.A commutative ring  A non-zero ideal R). The domain R is called (finitely) stable if every non-zero (finitely generated) ideal of R is stable. By definition, invertible ideals are stable and this implies that Dedekind domains are stable and Prüfer domains are finitely stable. On the other hand, stable domains need neither be noetherian, nor one-dimensional, nor integrally closed. For background on stable rings, their applications, and for results till 2000 we refer to the survey [Motivated by a paper of Bass , Lipman,e survey  by Olbere survey , 44–47.R is a BF-domain  if a = bc with set of atoms of H and we say that H is atomic if every non-unit is a finite product of atoms. Two elements associated if a = bc for some k is a factorization length and the set a is called the set of lengths of a. For convenience, we set system of sets of lengths of H,Let elasticity is accepted if H ishalf-factorial if it is atomic and monoid if it is atomic and each element of H is divisible by only finitely many non-associated atoms, andan FF-monoid if it is atomic and all sets of lengths are finite.a BF-We say that the H is atomic but not half-factorial, then we have By definition, an atomic monoid is half-factorial if and only if H is cancellative, H. We denote byseminormal closure of H, and bycomplete integral closure of H.Suppose that H is seminormal  if A is a divisorial ideal  if A is an s-ideal if A = AH. If s-ideal of H, then prime s-ideal of H if for all s-ideal I of H let radical of I. The monoid H is said to beMori if it satisfies the ascending chain condition on divisorial ideals,Krull if it is a completely integrally closed Mori monoid,monoid if the intersection of all non-empty prime s-ideals is non-empty,a G-primary if strongly primary if finitely primary (of rank s and exponent α) if H is a submonoid of a factorial monoid Then H be a primary Mori monoid, H is strongly primary.) Mori monoids and strongly primary monoids are BF-monoids.Finitely primary monoids and primary Mori monoids are strongly primary.  primary, seminormal, completely integrally closed) if its monoid R is primary if and only if R is one-dimensional and local. The domain Rfinite character if every non-zero element is contained in only finitely many maximal ideals.has divisorial if every non-zero ideal is divisorial.is h-local if R has finite character and every non-zero prime ideal of R is contained in a unique maximal ideal of R.is By a One-dimensional Mori domains have finite character by [S be an integral domain such that R is an order in S if S is a finitely generated R-module. Moreover, the following statements are equivalent if R is not a field [R is an order in a Dedekind domain.R is one-dimensional noetherian and the integral closure R is a finitely generated R-module.Let quadratic if R-module between R and S is a ring. If R. Thus every quadratic extension is an integral extension.The extension 3.In this section, we first gather main properties of stable domains –3.5. TheR be a domain. A non-zero ideal stable if it is invertible as an ideal of the overring R. The domain is called finitelystable if every non-zero (finitely generated) ideal of R is stable. Since invertible ideals are obviously stable, Dedekind domains are stable and Prüfer domains are finitely stable. Conversely, if R is completely integrally closed and stable, then R and R is a Dedekind domain. Recall that R is an almost Dedekind domain if R is a Dedekind domain if and only if R is a stable almost Dedekind domain. In particular, every almost Dedekind domain that is not a Dedekind domain is not stable. For an example of an almost Dedekind domain that is not a Dedekind domain we refer to [Let Proposition 3.1.Let R be a domain that is not a field. Then the following statements are equivalent.R is a one-dimensional stable domain.R is a finitely stable Mori domain.R is a stable Mori domain.Proof.This is due to Gabelli and Roitman. More precisely, the equivalence of (a) and (b) is proved in [R is a Prüfer domain. Next we consider the local case.Examples given by Olberding in , 46 showCorollary 3.2.Let R be a local domain that is not a field.The following statements are equivalent.R is a one-dimensional stable domain.R is a primary stable domain.R is a stable Mori domain.R is a strongly primary stable domain.If these conditions hold and (b) (c) age 2 of ).(d) R be a domain. By Let R is a local domain whose ideals are 2-generated, then R is finitely stable with stable maximal ideal (e.g. see R is noetherian). Nevertheless, such a domain is  neither half-factorial nor an FF-domain. In what follows, we present suitable counterexamples.Note that if e.g. see  and the K be a quadratic number field with maximal order p be a prime number such that p is split (i.e. p = 2.) Let K with conductor S is a local domain whose ideals are 2-generated and there are precisely two maximal ideals of S. It follows from [S is not half-factorial.  R is noetherian, and thus R is one-dimensional by R is not integrally closed. Since R is not integrally closed, we have that (b) R-module by R is finitely stable.Consequently, R is a one-dimensional stable domain by R is 2-generated by (c) R is said to be weakly Krull ifA domain afrullah , 3, and Theorem 3.7.Let R be a domain with, and suppose that R is either weakly Krull or Mori. Then R is stable if and only if every ideal of R is 2-generated. If this holds, then R is an order in a Dedekind domain.Proof.R is 2-generated, then R is stable by R be stable.If every ideal of R is weakly Krull. Then, for every R is weakly Krull, this implies that R is Mori by [Let us first suppose that R is Mori in both cases. Using R is one-dimensional. Therefore, R is 2-generated. □Thus Corollary 3.8.Let R be a seminormal G-domain and suppose that R is either Mori or one-dimensional. Then R is stable if and only if every ideal of R is 2-generated. If this holds, then R is an order in a Dedekind domain.Proof.R is a seminormal G-domain, Since Example 3.9.K be a field, Y an indeterminate over K, and X an indeterminate over K(Y). Then R is not a valuation domain. By There exist integrally closed one-dimensional local Mori domains which are neither valuation domains nor finitely stable. Let R is a local two-dimensional domain with maximal ideal R is stable as well by [R is not Mori by There exists a seminormal two-dimensional local stable domain. Let R is seminormal. Indeed, we know D is seminormal. Now clearly R is seminormal in D. Therefore, R is seminormal.R which is a BF-domain, whence R is a finitely stable BF-domain that is not Mori . In SecH be a cancellative monoid and K a quotient group of H. An ideal system on H is a map Let r be an ideal system on H. A subset r-ideal if rI = I. Furthermore, a subset r-ideal of H if there is some cJ is an r-ideal of H. We denote by r-ideals, and we define r-multiplication by setting r-multiplication is a reduced semigroup with identity element H. Let r-ideals, r-invertible fractional r-ideals, and r-invertible r-ideals of H with r-multiplication. We denote by s-ideals of H, by r-r-ideals of H, and by r-r-ideals of H. We say that r is finitary if We refer to , 32 for s-system, the v-system, and the t-system. For every ideal system r, we have r is finitary, then H has finite r-character if each r-ideals of H.We will use the R be a domain with quotient field K and r an ideal system on R  for all  (b) for all  (c) The following statements are equivalent.2. is half-factorial.  (a) for all  (b) for all  (c) Ifandfor every, then  (d) H is half-factorial and for every nonprincipalit follows that  (e) Proof.H is finitely primary of rank one, there is some Since 1.(a) mq is a product of a atoms of H and b atoms of H. We infer that 1.(b) 1.(c) H is strongly primary and r-H is r-local, we have that I is a non-empty s-ideal of H, then 2. Note that r-ideal of H, there is some 2.(a) mHq is an r-product of a atoms of r-product of b atoms of 2.(b) 2.(c) 2.(c) H is half-factorial. Let 2.(d) 2.(e) A is principal. Then A = uH for some Case 1. A is not principal. Then Case 2. Observe that some of the semigroups . Thencatenary degree and the monotone catenary degree of H. By definition, we have H is factorial if and only if H is cancellative but not factorial, then, by [H is half-factorial and that Let transfer homomorphism if the following two properties are satisfied.A monoid homomorphism (T 1) u = vw, (T 2) If H is said to be a transfer Krull monoid if it allows a transfer homomorphism θ to a Krull monoid B. Since the identity map is a transfer homomorphism, Krull monoids are transfer Krull, but transfer Krull monoids need neither be commutative (though here we restrict to the commutative setting), nor Mori, nor completely integrally closed. The arithmetic of Krull monoids is best understood (compared with various other classes of monoids and domains), and a transfer homomorphism allows to pull back arithmetical properties of the Krull monoid B to the original monoid H. We refer to the surveys [A monoid  surveys , 28 for H is half-factorial, then All Dedekind domains are transfer Krull and stable. However, there are orders in Dedekind domains that are transfer Krull but not stable  and therWe start with a result on the finiteness of the catenary degree of weakly Krull Mori domains.Theorem 5.1.Let R be a weakly Krull Mori domain.For every, andif and only ifandis local.andIf, thenis a Mori monoid and it is half-factorial if and only if it is transfer Krull.Proof.R is a weakly Krull Mori domain, we have t-Let Since the catenary degree of a coproduct equals the supremum of the individual catenary degrees [Since See [Since H with There are primary Mori monoids H be a finitely primary monoid of rank H and some system H is said to bestrongly ring-like if Let The concept of strongly ring-like monoids was introduced by Hassler , and thenumerical monoid is a submonoid of value monoidA Proposition 5.2.Let R be a local stable Mori domain with. Thenis finitely primary of rankand it is strongly ring-like. If s = 2, thenand if s = 1 and, then the elasticityis accepted withProof.R is one-dimensional. By [R is 2-generated by Proposition 3.5.4, whence R is noetherian. If R, then s = 2, then s = 1. Since By R be a one-dimensional local Mori domain with R is stable, then, by Let Lemma 5.3.Let R be a Mori domain and a G-domain and let I be a divisorial stable ideal of R. Thenfor someProof.I is divisorial and R is a Mori domain, we have that Since every overring of a G-domain is a G-domain, Example 5.4(Stable orders in number fields). 1. Let R is 2-generated, R is a stable order in the Dedekind domain Let K be an algebraic number field, R is stable by a result of Greither [R and 2. Let s and exponent α, with all notation as in s = 1 and by R in a quadratic number field whose localization n but the catenary degree Next we discuss the catenary degree of finitely primary monoids, which has received a lot of attention in the literature. Let ure  holds. By (a) R is a one-dimensional Mori domain, we have that (b) R is a one-dimensional Mori domain, we have that R is of finite character. Furthermore, if R. Let R is a Mori domain and t-finitely generated and locally principal and R.(c) (d) R is a Mori domain, we have that (d) (e) (f) 2. If Thus the claim follows from R be as in Let R is a semilocal domain, then R is not semilocal, then the statements for R. If R is any order in an algebraic number field, then R can be transfer Krull without being half-factorial [By Theorem 5.10.Let R be a stable order in a Dedekind domain.The following statements are equivalent.is transfer Krull.is transfer Krull.is half-factorial.is half-factorial.cmon(I*(R))<∞.has finite elasticity if and only ifis bijective. If this holds, then the elasticity is accepted.Proof.1. This is an immediate consequence of R is an order in a Dedekind domain, R is a weakly Krull Mori domain with non-zero conductor. By 2. Since 3. By [Thus the assertion follows from In Remark 5.11(Structure of sets of lengths and of their unions).R is an order in a Dedekind domain, then sets of lengths of (Structure of sets of lengths) If H be an atomic monoid. For every (Structure of unions of sets of lengths) Let is the union of sets of lengths containing k. The structure theorem for unions of sets of lengths states that there is a bound M such that almost all sets Remark 5.12(On catenary degrees).Remark 5.13. We compare the arithmetic of stable orders with the arithmetic of seminormal orders in Dedekind domains. Note that stable orders need not be seminormal  and semR be a seminormal order in a Dedekind domain and let π is bijective, then π is not bijective, then π is bijective. For stable orders, only one implication is true .R in a Dedekind domain such that The Statements 1.(c) and 1.(d) of L be the field with 16 elements, let X be an indeterminate over L and let R is a local divisorial half-factorial Cohen-Kaplansky domain such that Let Proof.R is a half-factorial local Cohen-Kaplansky domain, K-basis of L and R. Note that R is divisorial by [E of R let S(E) is a K-subspace of V and T(E) is a K-subspace of L.By [A and B are proper ideals of R, then Claim: If A and B be proper ideals of R. First let Let Now let A and B of R such that I = AB. It follows by the claim that K-subspace W of S(A). We infer that Assume that a = 1. Then Case 1: a = y. Then Case 2: Case 3: Case 4: Case 5: Case 6: Case 7: Remark 5.15(Transfer Krull does not imply stability). If R is an order in a Dedekind domain with R such that R has ideals which are not 2-generated. Thus, R is not stable.We construct a seminormal one-dimensional local noetherian domain X be an indeterminate over L, and K-basis of L. Then R-module. Since R is noetherian, and hence Let R itself is local and one-dimensional. Moreover, R is transfer Krull by [Since x0 is the constant term of x), and thus if R is seminormal.Now if y0 is the constant term of y). If Note that x1, y1 be the linear coefficients of x respectively y. Then Assume to the contrary, that


The Editors-in-Chief have retracted this article because there appear to be anomalies in four of the figures, specifically Figure 5, Figure 6A, Figure 6C and Figure 8A. The Editors-in-Chief therefore no longer have confidence in the data presented. The authors have been invited to submit a new manuscript for peer review. Xialong Tang stated on behalf of all co-authors that they do not agree with this retraction.


We have studied one of the most common distributions, namely, Lindley distribution, which is an important continuous mixed distribution with great ability to represent different systems. We studied this distribution with three parameters because of its high flexibility in modelling life data. The parameters were estimated by five different methods, namely, maximum likelihood estimation, ordinary least squares, weighted least squares, maximum product of spacing, and Cramér-von Mises. Simulation experiments were performed with different sample sizes and different parameter values. The different methods were compared on the generated data by mean square error and mean absolute error. In addition, we compared the methods for real data, which represent COVID-19 data in Iraq/Anbar Province. The Lindley distribution was proposed in 1958 by Lindley  studied θ, and the second is gamma distribution with 2 and θ, that is,The Lindley distribution is made up of mixing two continuous distributions with different weights; the first is exponential distribution with Thus, the resulting function is as follows:w = θ/(θ + 1); hence, the probability density function (p.d.f.) is as follows:Lindley  used w =Ghitany et al.  introducParameter estimation of the two-parameter Lindley distribution was conducted by many researchers, such as Al-Bayati , Sharafiw = αθ/(αθ + β) was used, as follows:The three-parameter Lindley distribution (THPL) was proposed by Shanker et al. ; the weiFigures W−1(.) denotes the negative branch of the Lambert W function.The quantile function of the three-parameter Lindley distribution is given by the following:We present five well-known methods to estimate the parameters of the three-parameter Lindley distribution, including maximum-likelihood (ML), ordinary least-squares (OLS), weighted least-squares (WLS), maximum product of spacing (MPS), and Cramér-von Mises (CVM).x =  under the three-parameter Lindley distribution can be written as follows:The log-likelihood of the positive vector of observations θ, α, and β, by solving the following nonlinear equations:Shanker et al.  derived We can also obtain MLE by maximising  via fminX(1) ≤ X(2) ≤ ⋯≤Xn)( are the order statistics of a random sample from any probability distribution. The ith-order statistic has the mean and the variance as follows:Suppose that F(xi) represents the theoretical c.d.f. of the observation xi)( of the distribution under study and ℱ(i) represents the empirical c.d.f. which is usually estimated by OLS and WLS were proposed in 1988 by Swain et al. . We can F(xi) in the previous equation by its c.d.f. defined in equation  in the previous equation by its c.d.f. defined in equation  − F(xi − 1) and F(x(0)) = 0; F(xn + 1) = 1.MPS was derived by Cheng & Amin in 1979 ; the ideF(xi) in the previous equation by its c.d.f. defined in equation .Note that if there is a tie, we cannot find the natural logarithm of CVM was proposed by MacDonald in 1971 . The ideF(xi) in the previous equation by its c.d.f., which was defined in equation  values, p values are greater than 0.05, indicating that the data follow the three-parameter Lindley distribution.We note that all We draw p.d.f. and c.d.f. of the three-parameter Lindley distribution based on the following:According to equations  and 30)30), the We note that the behaviour of the estimated functions is relatively close to that of the empirical functions. This finding is a good indication that the estimated models can represent the COVID-19 data.The parameters of the three-parameter Lindley distribution (THPLD) were estimated by five different methods. A simulation study was performed, and these methods were compared using MSE and MAE. All estimators were consistent because their MSE and MAE values decrease as the sample size increases. The MPS and WLS methods were good in small samples. MPS, MLE, and WLS were good in medium samples. MLE and MPS were good in large samples.On the practical side, the results indicated that the COVID-19 data follow the three-parameter Lindley distribution. The p.d.f. and c.d.f. were estimated based on the five methods, and then, these functions were drawn. The graphics indicated that the behaviour of the estimated functions is close to the empirical functions.


The cost for a single test can be controlled to ~1 dollar. This strategy provides a competitive solution for real-time, sensitive, selective, and large-scale application in cold-chain food quarantine.Due to the friendly temperature for virus survival, SARS-CoV-2 is frequently found in cold-chain foods, posing a serious threat to public health. Utilizing an interdigitated microelectrode chip modified with an antibody probe and integrating dielectrophoresis enrichment with interfacial capacitance sensing, a strategy is presented for the detection of trace level spike-protein from SARS-CoV-2. It achieves a limit of detection as low as 2.29 × 10 More typical examples have been showing their destructive power, such as large outbreaks of hepatitis A3 and norovirus infection5. As expected, these viruses are detected in various foods as well as water7, which exacerbates the spread of the disease. Therefore, the viral contamination detection in food poses a more great challenge in the recent decade8.The contamination of pathogens in food and the environment has always been a serious problem worldwide for food safety, entry-exit quarantine, epidemic prevention, and public health management. Although viral epidemics are less reported for food safety compared with bacteria9. Although there is not yet a direct evidence to indicate the foodborne characteristic of SARS-CoV-2, food contamination of SARS-CoV-2 has presented a risk for virus spreading, which attracts considerable attention in many countries12. SARS-CoV-2 virions adhering on the solid surface are reported to be stable, with a viability up to longer than 72 h (on plastic)13. As a matter of fact, besides the wet markets providing a breeding ground for the virus14, positive identification of SARS-CoV-2 on cold-chain foods are frequently reported in China recently, including the confirmed SARS-CoV-2 virions on plastic packing. Due to the below 0 °C temperature during the cold-chain transportation, the viruses can have a longer survival time than in room-temperature environment. On the other hand, the virus concentration may be of trace level, while a large number of samples need to be tested. Therefore, sensitive, fast, and low-cost SARS-CoV-2 detection is urgently needed for cold-chain foods.Since winter 2019, the outbreak of a so-called coronavirus disease 19 (COVID-19) has become a global pandemic, with the culprit being a new SARS coronavirus (SARS-CoV-2)8. Because the culture process is very time-consuming, PCR-based methods have become a mainstream approach15. Numerous works on various virus detection by PCR or reverse transcription-PCR (RT-PCR) in food have been reported, including hepatitis A17, hepatitis E18, and norovirus17. Although PCR techniques have been developed over the past 30 years, the operation is still complicated, and the required turnaround time cannot be shorter than several hours15. Based on the good specificity and versatile sensing mechanisms, bio-probe-based sensors for virus detection are becoming more practicable in recent years as promising approaches21. Also, due to the high specificity, reproducibility, and stability, antibody-based detections of causative agents are continually adopted in food safety detection23.For virus detection in food, there are mainly two types of techniques, culture/counting methods and polymerase chain reaction (PCR)-based methods26. As a result, S-protein (S1 subunit) is selected as a preferred biomarker for SARS-CoV-2 detection29. Because the S-protein existing on the periphery of the virion is exposed on the food surface independent of body infection and viral assembly, at a quantity more than viral RNA itself, it can act as a good biomarker for food contamination of SARS-CoV-2. For specific recognition of certain antigens, the antibody is considered as a promising probe due to the usually high affinity between the antigen and antibody30.A SARS-CoV-2 virion binds directly with its receptor through spike (S-) protein on its surface, and the subunit of S1 serves as the receptor-binding domain−6 ng/mL, while the time from sample to result is as short as 20 s due to effective target enrichment embedded into the detection process. Meanwhile, the detection is specific against the interferences, demonstrating a satisfactory selectivity in S-protein recognition. This report constructs a cost-effective immunosensor as well as a simple detection strategy for real-time, selective, and large-scale screening of SARS-CoV-2 for cold-chain food quarantine.Here, a method for trace S-protein (S1 subunit) detection associated with cold-chain food is developed for real-time SARS-CoV-2 contamination screening. An anti-SARS-CoV-2 S-protein antibody is immobilized on a low-cost interdigitated microelectrode (IDME) chip. Then, an alternating current (AC) signal is applied to the IDME to induce a dielectrophoresis force on the protein particles. Thus, the S-proteins are rapidly driven toward the IDME surface in the nanofluid and captured by the antibodies. Utilizing the electrical double layer capacitance at the IDME surface as an ultra-sensitive indicator, the limit of detection (LOD) reaches 1031, which is abundant in the antibody. Sodium (Na1s) and phosphorus (P2p)  solvent. The presence of P2p at the bare IDME  should be first verified. Al (Al2p) is dominant at the IDME surface before functionalization Fig. , demonst2p) Fig.  are accoDME Fig.  could be2 to 105 Hz is acquired to reflect the change on the IDME surface during functionalization by an electrical method. The impedance distinctly increases after antibody modification, but changes little after blocking  = −2.66lg X (ng/mL), demonstrates an excellent semi-log linear relationship between the target concentration and the response from 10−5 ng/mL until 10−1 ng/mL, with a correlation coefficient (R2) of 0.994. With the cut-off line (Y = −2.05%) defined as a response value three standard deviations from the background, the LOD is calculated to be 2.29 × 10−6 ng/mL plugging the cut-off value into the calibration equation. This LOD is at an ultra-low level. The dose response as well as the calibration demonstrates a quantitative performance of the sensor, with a wide linear range.For calibration using dose response Fig. , the resFor a qualified sensor, selectivity for the target is the most important performance. In this work, the acquired effective response is verified in three ways: (1) comparing the response from functionalized sensors with that from dummy sensors when S-protein is the target of the test, (2) comparing the response from S-protein with that from the background of 0.1× PBS, and (3) comparing the response from S-protein with that from the interferences. Here, the dummy sensor is defined for an IDME chip without antibody functionalization to verify the probe’s effectiveness.−1 ng/mL is equivalent to the response from S-protein at 4.27 × 10−4 ng/mL calculated using the calibration line, then the selectivity ratio of this sensor is calculated to be 234:1 (10−1 ng/mL:4.27 × 10−4 ng/mL).According to the comparison result Fig. , the res−3 mg/mL, which most possibly exist at the food surface dissociated from various bacteria. In this test, the spiked S-protein is 10−4 ng/mL. Although the concentrations of nontarget molecules are 10-fold higher than that of S-protein, the response is only a bit larger than that in PBS without impurities , the interfacial capacitance will increase as illustrated in Supplementary Material  force, S-protein particles are attracted towards the sensor’s surface and captured by the antibodies in 20 s. Also, the response acquisition is completely integrated into the target enrichment process, thus the response time is shortened to 20 s, a duration meeting the real-time detection demand on site. Besides, the LOD of this detection reaches an ultra-low level, i.e., 10Another merit of this strategy is the high specificity. The selectivity for S-protein against the interference is calculated to be 234:1. Not only common organic molecules but also various matrices as backgrounds are verified to be nonresponsive, with known or unknown components. Therefore, the sensor has demonstrated the required specificity to identify the S-proteins from practical samples with complex backgrounds.The cost for a single sensor device is estimated to be ~1 dollar. As a result, this sensor is designed to be a disposable chip. Meanwhile, the impedance analyzer for capacitance detection is also inexpensive and lightweight. As discussed previously, the detection operation as well as the pretreatment is simple, only including dilution and centrifugation steps. Therefore, this sensor can be operated by a layperson. Based on the above characteristics, the platform and method for S-protein detection provide a promising solution for large-scale applications in cold-chain food quarantine requiring a quick response, low LOD, high specificity, user-friendly operation, and low cost as well.36. Compared with most techniques for preconcentration needing an extra incubation or equipment, AC electrokinetic (ACEK) effects can manipulate nanoparticles efficiently to realize target enrichment rapidly without extra processes or devices39. For relatively large molecules such as proteins, DEP force, an important ACEK effect, has been demonstrated as a dominant force applied on the particles41. In this work, an IDME chip is used as a physical device to induce the DEP effect. DEP force can be expressed as a function of electric field , directly reflects the S-protein adsorption level, by which the S-protein concentration in the solution can be indicated. Using an IDME of micron-scale, dC/dt is a competitive parameter to reflect the tiny change at the electrode interface44. Therefore, ultra-trace S-protein detection can be expected. In practice, the lab-prepared sensors may have inconsistency due to different effective electrode surfaces and total recognition sites. Here, the normalization by the initial capacitance  is purchased from Cellregen (Beijing) Life Science and Technology Co., Ltd, which is recombinant and expressed by the prokaryotic system with the host of 2  , a small piece of a certain sample of ~8×8 mm2 is first cut-off, and 10 μL S-protein of 1 ng/mL is spread over it. After the solid sample is incubated for 12 h in a closed tube at 4 °C, 990 μL of 0.1× PBS is added into the tube. Then, the solution is sufficiently agitated and centrifuged at 3000 r.p.m. for 10 min. The supernatant from this mixture with a theoretical S-protein concentration of 10−2 ng/mL is ready for further dilution to obtain the final samples with theoretical concentrations of 10−3 and 10−4 ng/mL. During pretreatment of the salmon slice, an additional centrifugation at 5000 r.p.m. for 10 min is necessary to remove the abundant interference, and the floating lipid layer should be excluded when the supernatant is collected. To obtain the corresponding backgrounds for these practical samples, a similar process is performed except for the S-protein spreading and incubation steps.The test procedures mainly include incubation, extraction, and centrifugation steps (as shown in Fig. S41, and the voltage is optimized to be 100 mV according to the optimization (described in Fig. SC/dt can be calculated and analyzed. To demonstrate the sensor’s repeatability and consistency, all the data are presented with error bars, which represent the standard deviation obtained from three sensors of the same batch.The frequency of the AC signal is set to be 100 kHz according to a previous studyFurther information on research design is available in the Supplementary InformationReporting Summary


X be an abelian variety of dimension g over a field k of characteristic g modules over p > 0 and k is perfect, we have also a p-adic analog of covariant Dieudonné module g modules over the ring of Witt vectors Let Definition 1.1:L is an R-lattice in L has an R-basis that is an Given rings u is given by A; equivalently, there exists a u-stable Zarhin There exists a u-stablelattice(b)There exists a u-stablelattice(c)If p > 0 and k is perfect, then there exists a u-stablelattice(d)If p > 0 and k is perfect, then there exists a u-stablelatticeThe following restatement of (b) answers a question implicit in ref. , Remark Corollary 1.3.LetThen there exists a matrixsuch that, for everythere is a-basis ofon which the action of u is given by A, and such that, if p > 0 and k is perfect, there is a W-basis ofon which the action of u is given by A.X, u) in characteristic p > 0 cannot always be lifted to characteristic 0 Themoduleis free of rank h.(ii)If p > 0 and k is perfect, then themoduleis free of rank h.(iii)For eachthemoduleis free of rank h.(iv)Themoduleis free of rank h.(v)If p > 0 and k is perfect, then themoduleis free of rank h.Proof:(i)This is ref. , Theorem(ii)jE of M induces an jE-linear and K-semilinear automorphism of jM. By jM is free.The following proof is essentially a combination of the proofs of ref. , Proposix on On the other hand, by ref.  (p. 96), of ref. , Theorem(iii)λ of λ. Since Fix These induce decompositionsh over h over By (i), (iv)We have (v)Similarly, this follows from (i), (ii), and (iii). __Proof of Theorem 1.2:(a)k up to isogeny. By ref. [u is contained in a subring of iE. Then X is isogenous to iY with iE-stable i, then we may take E. By P. Then u-stable We work in the category of abelian varieties over  By ref. , Theorem(b)V as in (a). We havep = 0. Then u-stable Given (c)E. By P. Then u-stable As in the proof of (a), we reduce to the case in which (d)V be as in (c). We have u-invariant Let u, we consider a R-stable r stable for every (1)R is contained in a subring of iE, then the proof of R-stable lattice exists.If (2)X is an elliptic curve such that R-stable R cannot act on a two-dimensional Serre observed that if (3)R is assumed to be commutative, then the conclusions of Y is an elliptic curve such that B, and If R is commutative. For each nonzero The ideal R-stable V in U, and hence equal to U. Thus we obtain a Suppose that there is an It is an isomorphism sincef in R-stable Since In


Howevet al. ,7. This et al. ,9 insighSenescence evolves because natural selection ‘cares less about’ late life than about early life ,11. Thisx; of a newborn's probability of still being alive at that age, xl; and of her instantaneous rate of production of offspring at that age given she is alive, xb; for simplicity assume asexually reproducing female populations. Life histories can also be modelled for discrete intervals, where the simplest case is a birth-pulse population with all reproduction occurring on the birthday; this assumption makes the following equations exact [x entering interval X of unit length; Xl = xl be a newborn's probability of survival to the beginning of interval X; and Xm be the number of daughters she produces at the beginning of interval X, given she is alive.An age-structured life history  is a schns exact , p. 16; ns exact , p. 403.x at time t surviving to be censused for the first time at time t + 1, for pre-birth censusing is r is the intrinsic rate of increase, the population's exponential growth rate once in stable age distribution, calculated as the unique real root of the Euler–Lotka equation:Further quantities can be derived: interval age-specific survival probability is notation , p. 814.X, add a bit to interval fecundity X at which the gene's effect occurs. This embodies the general approximationHamilton  considerr versus r and, as mentioned, considered age-specific interval survival and fecundity, parameterized as X and can be ignored for our purposes. The numerators are monotonically decreasing functions of the interval X at which the gene acts, vindicating the insight of Haldane, Medawar and Williams that natural selection cares less about late than about early life , can rise with age in rapidly shrinking populations; these tend to go extinct anyway [et al. used the sensitivity r, but genes that multiplicatively reduce interval average hazard rate, thus used the sensitivity The partial derivative on the right is a fitness sensitivity, or selection gradient, or indicator of the force of selection. Applying this approximation requires choosing the fitness measure yway [R0 ; thus Da∂lnμ¯X) ,9.Hamilton's assumption that genes multiply ntuitive , p. 191:a), can in fact rise with age under some conditions; however, it still must decrease with age in initially non-senescing age distribution is a right eigenvector w of the matrix A; the (st)age-specific individual reproductive values also form a left eigenvector v of the matrix A; the dominant eigenvalue is j; i; and 〈w,v〉 is the eigenvectors' dot product. Caswell summarizes: ‘The effect  assumes additive genetic effects on survival probability, while equation (1.6a) assumes multiplicative genetic effects on survival probability; see [et al. repeat a subtle mischaracterization by Caswell [a) and (1.9b) as a ‘reformulation of’ equations (1.6a) and (1.6b), and write, still implying equivalence, that ‘For any given (st)age of a life cycle, the force of selection on an increase in the reproduction or mortality of that (st)age is proportional to the product of two key components: (i) the stable age distribution of individuals at that (st)age and (ii) the reproductive value of individuals that the (st)age contributes to the population (new offspring or their surviving selves)’ [b) ‘shows why reproductive value is apparently missing from’ equation (1.6b): ‘reproductive value at birth … is scaled to equal 1’ [et al. observe in life-history data [r to proportional hazards (equation 1.7) equals Stage-structured life histories are modelled in discrete time using population projection matrices –5; discr Caswell  showed tation of ,1.8∂λ∂aiulation’ , p. 529. Caswell , p. 534,RVX≠VX+1 . Caswellequal 1’ , p. 540.Click here for additional data file.


Hinks TSC, Cureton L, Knight R, et al. Azithromycin versus standard care in patients with mild-to-moderate COVID-19 (ATOMIC2): an open-label, randomised trial. Lancet Respir Med 9: 1130–402021; —The appendix of this Article has been corrected and the data in the following Results text is now corrected: “…while systemic corticosteroids were co-prescribed in 13 (9%) receiving azithromycin”. These corrections have been made to the online version as of Sept 2, 2021, and will be made to the printed version.


Persons with autism spectrum disorder (ASD) have impaired mentalizing skills. In this study, a group of persons with ASD traits (high-AQ scores) initially received sham tDCS before completing a pre-test in two mentalizing tasks: false belief and self-other judgments. Over the next week, on four consecutive days, they received sessions of anodal electrical stimulation (a-tDCS) over the right temporo-parietal junction (rTPJ), a region frequently associated with the theory of mind. On the last day, after the stimulation session, they completed a new set of mentalizing tasks. A control group (with low-AQ scores) matched in age, education and intelligence received just sham stimulation and completed the same pre-test and post-test. The results showed that the high-AQ group improved their performance (faster responses), after a-tDCS, in the false belief and in the self-other judgments of mental features, whereas they did not change performance in the false photographs or the self-other judgments of physical features. These selective improvements cannot be attributed to increased familiarity with the tasks, because the performance of the low-AQ control group remained stable about one week later. Therefore, our study provides initial proof that tDCS could be used to improve mentalizing skills in persons with ASD traits. The human brain has developed extraordinary mentalistic capacities, far superior to those of other species of primates, to deal with our complex social world. The so-called “theory of mind” (herein ToM) is a cognitive skill that allows us to navigate the social world while continuously tracking our own and others’ mental states and engaging in social reasoning ,2. ToM iWe can distinguish two main neural networks involved in mentalizing processes: the affective ToM, which is mainly responsible for emotional empathy, and the cognitive ToM, which is assumed to compute others’ mental states, dissociating them from ours and enabling our complex social reasoning capabilities ,5,6. EspSince autism is characterized by impaired cognitive ToM, some studies have explored the brain functional organization of persons with ASD, compared to control persons, while dealing with ToM tasks. For instance, neuroimaging studies by ,21 askedThe study of the efficiency of non-invasive brain stimulation to improve social cognition both in control individuals and in persons with ASD is a growing field. According to a recent review , some stAn alternative technique is transcranial direct current stimulation (tDCS). Although tDCS has the significant drawback of offering much less focal stimulation than rTMS, it also has some advantages, such as having relatively long-lasting effects—about an hour after 12 min of stimulation —a generaAs we have seen, evidence of the therapeutic efficacy of non-invasive brain stimulation to improve social cognition in persons with autistic traits is still scarce. In this study, we developed for the first time a multisession intervention with anodal tDCS over the right TPJ in persons with high-AQ scores. We measured their performance in validated mentalizing tasks before and after the treatment. A group of low-AQ persons served as controls, receiving sham tDCS and performing the same mentalizing tasks in two sessions with the same time delay as the high-AQ group. For this study, we selected two well-known tasks that are representative of cognitive ToM skills, that is, the false belief task and the self-other judgment task, which are associated with activations in the rTPJ. Both tasks involve a dissociation between representations of the self and representations of the other. The false belief paradigm consists of presenting stories in which a protagonist performs an action based on a false belief because he/she is unaware of a critical event, which the reader/listener knows. For example, “Mary puts her chocolate in the top drawer of the table, and while she was at the school, her mother moved it to the bottom drawer of the table. When Mary comes back home, will she look for the chocolate in the top drawer?” To correctly answer this question, participants need to recreate the outdated belief of the protagonist to understand her actions (looking in the top drawer) even though they know the chocolate is actually in the bottom drawer. These stories are contrasted with stories of “false photos” that also require the representation of outdated content but do not involve monitoring the characters’ beliefs. For example, “An old photo showed a house with a front door and two windows on the sides. In 2010 one of the windows was replaced with a new door. Did the old photo show the two doors?” Therefore, false belief stories and false photographs are similar in their general difficulty, logical complexity and inhibitory demands, but differ in the need to think about someone’s thoughts  ,22,34. WThe second task involves self-reference or other-reference judgments of physical or mental features. The participants were asked to judge to what extent a physical characteristic  or a mental characteristic  applies to you (self-reference) or to a famous person (other-reference). This task requires that the participant know the other’s mental states , which frequently differ from our own . We predAs for the low-AQ group, it serves as a passive control (with participants receiving only placebo stimulation) to assess whether there is any improvement between the initial and the final test of the mentalizing tasks, as a simple consequence of familiarization or learning. The expectation is that their performance would remain stable in ToM tasks. p = 0.0001), and in the EQ scores  = 16.225; p = 0.0001). However, they did not differ in age  = 3.253; p = 0.079), or Raven scores  = 0.870; p = 0.357). Forty-two Spanish-speaking participants, 21 with high-AQ scores and 21 with low-AQ scores serving as the control group, were selected. All were right-handed, had normal or corrected-to-normal visual acuity and had signed an informed consent form. Participants were screened via safety guidelines for transcranial stimulation  and the 2 and the current intensity was 2 mA, which involves a current density of 0.08 mA/cm2. The sham sessions had identical electrode setup and timing, except that real stimulation only took place during the first 30 s. Afterward, participants completed a survey regarding the tolerability of stimulation and guessing the stimulation condition. None of the participants reported special discomfort, and all believed they had been stimulated.The active stimulation sessions consisted of 20 min of anodal tDCS over the right TPJ, with the cathode electrode placed at the vertex, that is, at the CP6 and Cz positions in the 10/20 international localization system, respectively . The elep = 0.602). Belief task. In this task, a repeated measures factorial design was used, with 2 Test (Pre-test vs. Post-test) × 2 Modality  × 2 Response polarity (Yes vs. No) as within-participant factors and 2 Group (high-AQ vs. low-AQ) as a between-participant factor. Two sets of 20 stories (A and B) were used in a counterbalanced way for the pre-test and the post-test sessions. Set A consisted of the original stories created by , translaSelf-other judgment task. In this task, the design included 2 Test (Pre-test vs. Post-test) × 2 Reference (You vs. Other) × 2 Feature  as within-participant factors and 2 Group (high-AQ vs. low-AQ) as between-participant factor. Two sets of self-other questions were adapted from an earlier version used by . ParticiIn each task set, there were 20 items in each type of question , which is a total of 80 items. Participants issued a judgment on a scale of 1–4 . To encourage participants to engage in the task, they were told that their judgments would be compared with the answers given by the famous persons in various interviews, and they would receive an ‘accuracy score’ at the end of the task. This ‘’score’’ was randomly generated and displayed on the screen at the end of the task. t-tests), to assess the effects of the test (pre-test vs. post-test) for each experimental condition in each group, and to contrast the two groups (high-AQ and low-AQ) for each experimental condition. Global analyses of variance (ANOVAs) were performed in each task for reaction times and accuracy (only in the belief task), including Group (high-AQ and low-AQ), Test (pre-test and post-test) and the specific variables of each task: Modality (belief vs. photo) in the belief task and Feature  and Reference (you vs. other) in the self-other task. To explore the resulting interactions, separate ANOVAs were applied to each group separately, followed by pairwise comparisons  which focused on the variables with theoretical interest: Group × Modality × Test. However, the statistics referring to response polarity are offered in the p = 0.032; η² = 0.110), suggesting certain familiarization with the task that was independent of the stimulation protocol. However, the crucial Test × Group interaction was not significant (F < 1), which means that the learning effect was similar in both high-AQ and low-AQ participants . There was a slight improvement from the pre-test  to the post-test , which was statistically significant  = 4.947; icipants . Furtherp = 0.232). Furthermore, the t-test contrasting high-AQ and low-AQ for each experimental condition substantiated a significant group difference for the false belief items in the pretest session  = 2.946, p < 0.005), while the two groups did not differ in the false photo items  = 0.878, p = 0.354). As for the RTs, the high-AQ group showed a nonsignificant trend to respond more slowly than the low-AQ group  = 1.47; p = 0.0001; η² = 0.295 (slower responses in the pre-test than the post-test), and Modality: F  = 6.092; p = 0.018; η² = 0.132 . Most important, the interaction Test × Group was also significant  = 7.597; p = 0.009; η² = 0.160), since the difference between pre- and post-test was much larger in high-AQ  than in low-AQ participants . To explore this interaction in detail, separate ANOVAs were performed for the high-AQ and the low-AQ group. There were significant main effects of Test: F  = 16.765; p = 0.005; η² = 0.532) and Modality  = 5.785; p = 0.026; η² = 0.224). However, these effects were qualified by the crucial Test × Modality interaction  = 6.265; p = 0.021; η² = 0.239), illustrated in p < 0.0001), but not for false photo stories (t (20) = 1.972, p = 0.063). As for the low-AQ group, neither the main effects of Test or Modality nor their interaction were statistically significant .For the high-AQ group, there was a main effect of Test  = 22,713; p = 0.075). However, the between-group t-tests for each experimental condition obtained a significant difference: in the first session, high-AQ participants responded more slowly to you-mental questions  than low-AQ participants  = 2.173, p < 0.036). Significant main effects were obtained for Test: F  = 21.295; p = 0.0001; η² = 0.347 (faster responses in the post-test than in the pre-test), Reference: F  = 106.926; p = 0.0001 η² = 0.728 (faster responses for you than for other) and Feature: F  = 11.553; p = 0.002; η² = 0.224 . In this task, only the reaction time was used as a dependent measure. The high-AQ group showed a nonsignificant trend to respond more slowly than the low-AQ group  = 3.33; p = 0.030; η² = 0.112, which indicates that high-AQ and low-AQ participants responded differentially in the pre-test and post-test sessions depending on the features of the task. Finally, there was a Group × Feature × Reference interaction: F  = 4.828; p = 0.034; η² = 0.108. To explore these interactions, new ANOVAs were performed for the high-AQ and the low-AQ groups separately. However, these main effects were modulated by important three-way interactions. Especially relevant was the Test × Group × Feature interaction: F  = 5.063; p < 0.0001; η² = 0.479 (post-test faster than pre-test), Feature: F  = 23. 419, p < 0.0001, η² = 0.539  and Reference: F  = 43.663, p < 0.0001, η² = 0.686 (you faster than other). However, a significant Test × Feature interaction was also obtained: F  = 4.503; p = 0.047, η² = 0.184, indicating that whereas for the mental features, responses were significantly faster in the post-test  than in the pre-test  = 4.239; p < 0.0001), for the physical features, the post-test and pre-test  did not differ significantly: t (20) = 1.483; p = 0.154. This means that the beneficial impact of stimulation was selective over those items that demand the evaluation of mental features. This was corroborated by the fact that the effect of Test in the high-AQ group was constrained to the combination of other-mental: t (20) = 4.239; p = 0.0001, while it did not reach significance in the combination of you-mental: t (20) = 1.692; p = 0.106. The analysis of repeated measures with the high-AQ group alone showed robust main effects of Test: F  = 18.381, p = 0.0001; η² = 0.783; that is, faster responses were found for you  than for other , although this effect was modulated by an interaction with Feature: F  = 4.544; p = 0.046; η² = 0.185, since responses to you-physical  were faster than responses to other-physical . Importantly, there was no significant effect of Test  = 3.648; p = 0.071) or its interaction with any other factor, which means that there was no appreciable familiarity or learning effect for low-AQ participants, which serve as controls  = 72.050; controls . We explored the relationship between self-report measures and performance measures. Specifically, the correlations between the scores in AQ and EQ and the reaction times in the tasks were computed for the pre-test session. This procedure is justified, because at this stage both the high-AQ and low-AQ groups passed exactly the same protocol, before undergoing any treatment with tDCS. As This study showed for the first time that a multisession excitatory stimulation on the social brain of persons with high levels of autistic traits (high-AQ scores) improves their performance on representative mentalizing tasks. Anodal tDCS was applied over the right temporo-parietal junction, a significant region with ToM functions, in four consecutive sessions, and participants’ performance in false belief and self-other distinction tasks was measured before and after the treatment. High-AQ participants improved after treatment (faster responses) exclusively in those tasks requiring mentalizing; that is, in false belief judgments and you-other distinction of mental features. Interestingly, persons with high AQ did not modify their performance in the tasks with little or no demand for ToM, such as the false photograph or you-other distinction of physical features. This suggests that the beneficial effects of a-tDCS in persons with ASD traits are specifically related to mentalizing processes and cannot be attributed to general learning or familiarity with the tasks. This was corroborated by the fact that in the low-AQ control group that received no active stimulation, there was no improvement in the ToM tasks.As expected, performance was better in the false photo than in false belief trials; that is, responses were faster and more accurate. This was true for both groups of participants and is consistent with previous results in the literature . It may The effect of a-tDCS on false belief performance in the high-AQ group is especially relevant here. It suggests that the stimulation of rTPJ has a very selective effect on mentalizing processes and does not affect the false photo task, despite being comparable in logical complexity  and inhibitory demands. The critical difference is that, unlike false belief, the false photo task does not require thinking about someone’s thoughts. Nonetheless, the impact of a-tDCS on performance was limited to response times and did not affect the accuracy of the false belief judgments. This could be due to the fact that the false belief task was simple enough to induce a ceiling effect that prevents further improvements with the treatment. In future studies with ASD participants, the use of an implicit false belief task embedded in naturalistic social contexts ,45 couldBoth participants with high AQ and participants with low AQ scores responded faster to self-referred questions than to other-referred questions. This could indicate the additional cost of mental perspective taking (putting myself in someone else’s shoes) or it could simply be a matter of familiarity (I know myself better than I know anyone else). More important are the differences observed between the two groups of participants. First, the a-tDCS applied to the high-AQ group selectively improved the response speed to questions about mental features referring to another person, while the control group’s performance did not change in any task. That is, stimulating rTPJ in persons with high levels of autistic traits have a selective effect on those questions that are more demanding of ToM inferences. We did not obtain a similar improvement in questions referred to oneself, despite the fact that previous studies have reported that persons with ASD also have impaired self-reference functions . HoweverAs stated above, this study is a basic demonstration that a-tDCS applied over rTPJ could improve social cognition in persons with autistic traits. However, as such the study has some limitations that we want to highlight for future research. First, in this study, the high-AQ cut-off score was 30, rather than the standard 32 usually reported in the literature. Second, there are general limitations to non-invasive brain stimulation; for instance, the stimulation is applied to a specific cortical region, whereas potentially relevant subcortical regions cannot be directly stimulated; in addition, the stimulation on the target region overlaps with neighboring regions . These lOur findings demonstrate the efficacy of tDCS applied to persons with ASD traits, to improve their performance on representative social cognition tasks. We can establish two general consequences of this study. First, it confirms that the rTPJ used as a target for stimulation is functionally involved in the attribution of mental states to others and oneself, which are basic skills of cognitive ToM, necessary to properly navigate the social world. Second, it serves as initial proof that multisession anodal tDCS could be used to improve social cognition in persons with ToM deficits.


X family of distributions. For the practical illustration, we introduced a new special sub-model, called the new generalized-Weibull distribution, to describe the new family’s significance. For the proposed family, we introduced some mathematical reliability properties. The maximum likelihood estimators for the parameters of the new generalized-X distributions are derived. For assessing the performance of these estimators, a comprehensive Monte Carlo simulation study is carried out. To assess the efficiency of the proposed model, the new generalized-Weibull model is applied to the coating machine failure time data. Finally, Bayesian analysis and performance of Gibbs sampling for the coating machine failure time data are also carried out. Furthermore, the measures such as Gelman-Rubin, Geweke and Raftery-Lewis are used to track algorithm convergence.As is already known, statistical models are very important for modeling data in applied fields, particularly in engineering, medicine, and many other disciplines. In this paper, we propose a new family to introduce new distributions suitable for modeling reliability engineering data. We called our proposed family a new generalized- Within the area of reliability engineering and other related fields, modeling of data related to lifetime events is very crucial. A range of probability models, such as Weibull, gamma, exponential, etc., are available for modeling lifetime data. However, in many cases, these classical models are not suitable for modeling lifetime data, and there is always a clear need for modified forms of these existing distributions. Therefore, the researchers have introduced new families that are more advantageous than the old ones. For more reading about statistical distributions using different approaches; see .In the literature, the Weibull distribution is the most outstanding one that has broadly been used in reliability engineering and in other various areas of research; see, for instance, . AlthougThe foremost goal of this research is to introduce a new flexible modification of the Weibull model by means of inducting one additional parameter. The induction of the additional parameter leads to the greater flexibility to enhance goodness-of-fit to reliability data. In fact, we show empirically that the new extension of the Weibull distribution offers the best fit to the coating machine failure time data than the two-parameter, three-parameter, and four-parameter competitive distributions (see section 5). The practical example really shows that the proposed distribution is a good alternative candidate for modeling reliability data.X (NG-X) family. A random variable X is said to follow a NG-X family, if its cumulative distribution function (cdf) is given byθ is an extra shape parameter, and F is the baseline cdf which may depend on the parameter vector X distributions can provide best fit to reliability engineering data. The corresponding density function isNow we introduce the proposed family of distributions called a new generalized-S and the hf h of NG-X distributions are given byThe reliability function -X distributions approach, we introduce a new form of the Weibull model called, a new generalized Weibull (NG-Weibull) distribution. Furthermore, we consider maximum likelihood (Non-Bayesian) and Bayesian procedures in order to estimate the unknown parameters of the NG-Weibull model. In the Bayesian discussion, we consider different types of symmetric and asymmetric loss functions such as squared error loss, weighted squared error, precautionary, K-loss, and modified squared error loss function to estimate the unknown parameters of the NG-Weibull model. Since all the parameters are positive, we use gamma prior distributions. Bayesian 95% credible and highest posterior density (HPD) intervals  are obtX family. Some mathematical properties of NG-X distributions are derived in Section 3. Section 4 is specified for obtaining the estimates using the maximum likelihood estimation, and the Monte Carlo simulation study is also provided in the same section. Section 5 is concerned with the goodness of fit of the proposed distribution. In this section, we showed that NG-Weibull model provides fit to reliability engineering data. Section 6 offers the Bayesian analysis. The future research directions are provided in Section 7. Some concluding comments are presented in Section 8.This paper is outlined in the following manner: In Section 2, we define a NG-ξ = . Then, a random variable say X is said to follow the NG-Weibull distribution, if its cdf is given byLet The density function corresponding to The reliability function and hf of the NG-Weibull distribution are given byIn the X distributions, which include identifiability, quantile function, random number generation, rth noncentral moments, and the Renyi entropy with numerical illustrations. A characterization theorem extending the NG-X class of distributions in terms of the hf is also provided.In this part of the paper, we derived the mathematical properties associated with the NG-X distributions. To prove the identifiability property of the NG-X distributions, we have to show that θ1 = θ2. Let θ1 and θ2 be the two parameters having the NG-X distributions with cdfs given by G and G, respectively. From the definition of identifiability, we haveThe identifiability is an important statistical property that a model must obey to make sure that the inference should be precise. In this subsection, we prove the identifiability property of the NG-Theorem 1.The quantile function of NG-X distributions is given bywhere F−1is the quantile of the distribution with cdf F(x), 0 < p < 1, and W(z) gives the principal solution for w in z = wewProof.p < 1, we must solve the following equation for Q(p)Let 0 < y = F(Q(p)). Using software like MATHEMATICA, if we solve the equation below for yLet ThusNotation:If the cdf of the Weibull distribution is given bywe writeif Y is a NG-Weibull random variable.Some numerical values of the quantile measure are provided in U ∼ Uniform, then random numbers from NG-X distributions can be obtained fromF−1 is the quantile of the distribution with cdf F(x), and W(z) gives the principal solution for w in z = wew.If Theorem 2.The rthNon Central Moments of NG-X distributions can be expressed aswhere Ωi,k, n, qis defined as in the proof of the theorem, E[⋅] denotes an expectation, and U ∼ Uniform.Proof. From subsection 3.3, we know if U ∼ Uniform, then the following random variableF−1 is the quantile of the distribution with cdf F(x), and W(z) gives the principal solution for w in z = wew, follows the NG-X family of distributions.F(x) distribution. For a power series raised to a positive integer r ≥ 1, we haveδr,i are obtained from i = 1, 2, …; see [E(⋅) is an expectation. By the binomial series, we can writeAccording to , we can , …; see . Thus weBy using integer powers of the Lambert W function we can writeBy the binomial series we can writePutIt follows thatSome numerical description of the ordinary moments are presented in Theorem 3.The Renyi entropy of the NG-X distributions, for δ ≠ 1, δ > 0, can be expressed aswhere X is a random variable with cdf F(x) and pdf f(x), and Ωk,q, ris defined as in the proof of the theorem.Proof. Recall the pdf of NG-X distributions is given byg(x)δ where δ ≠ 1 and δ > 0. By the binomial series we can writeWe first find an expansion for By the binomial theorem we can writeBy the power series representation for the exponential function, we can writePutIt follows thatTherefore, the Renyi entropy ishf, of a function, F, that can be differentiated twice satisfies the following differential equationIt is clear that X type distribution. The result here is inspired by [In this section, we present a Kumaraswamy NG-pired by . First, Definition:We say a random variable X follows a Kumaraswamy-G type distribution if its cdf is given bywhere G is some baseline distribution, x ∈ Supp(G), and ξ is a vector of parameters in the baseline distribution whose support depends on G.Remark:Note that if we take λ = 1 and φ = 2 inof [then we get the cdf in the above definition. 2 inof , then weG type distribution is given byg is the pdf of the baseline distribution. Clearly the hf of the Kumaraswamy-G type distribution is given byThe pdf of the Kumaraswamy-Theorem 4.Letbe a continuous random variable. The pdf of X isfor some baseline distribution with pdf g and cdf G, if and only if its hazard rate function h(x) satisfies the following differential equationwith boundary condition h(0) = 2g(0).Proof. If X has pdf as stated in the theorem, then the differential equation as stated holds. Now if the stated differential equation holds, thenG type distribution.X type distribution is obtained from the above theorem by letting the baseline pdf and cdf given in section 1.Clearly, a characterization of the Kumaraswamy NG-X parameters using the maximum likelihood estimation approach and providing a comprehensive Monte Carlo (MC) simulation study to assess the maximum likelihood estimators (MLEs) performance.This section is devoted to estimating the NG-x1, x2, …, xn be the observations of a random sample of size n taken from the NG-X distribution with the parameter vector Θ = T. For Θ, the log-likelihood function (LLF) is given byLet The partial derivatives of the LLF are given byθ and ξ of the NG-X distributions can be obtained by maximizing The MLEs of the unknown parameters optim R-function with the argument method = “L-BFGS-B”. We made 1000 MC-iterations using different sizes of the samples as follows, n = 25, 50,…,1000. We computed the average MLEs, the associated mean square errors (MSE), biases and absolute biases. For the first set of MC simulation results, the plots MLEs and MSE are provided in In the following sub-section, we assess the behavior of the MLEs of NG-Weibull distribution by means of the MC simulation study. The process is carried out by maximizing the LLF using the In this section, we evaluate the usefulness of the NG-Weibull distribution by means of analyzing reliability engineering data taken from . The datTo figure out about the goodness of fit amongst the competitive distributions, we consider certain goodness of fit measures such as Cramer-Von-Mises (CM) statistic, Anderson Darling (AD) statistic, and Kolmogorov-Smirnov (KS)statistics alongside with its p-values.Corresponding to the coating machine failure time data, the model parameters’ estimated values with standard errors in the parenthesis are presented in QQ) plots are presented in In support of the goodness of fit measures given in Furthermore, for the coating machine failure time data, we calculated the KS statistic values of the NG-Weibull and other considered models. Then, we utilized the parametric bootstrap approach , and booIn this section, we consider different types of symmetric and asymmetric loss functions such as squared error loss function (SELF), weighted squared error loss function (WSELF), modified squared error loss function (MSELF), precautionary loss function (PLF) and K-loss function (KLF). These functions associated with Bayesian estimators and posterior risks are reported in For more detail, we refer to –18. Nextα, γ and θ of NG-W distribution have independent prior distributions asα0, α1, γ0, γ1, θ0 and θ1 are positive. Consequently, the joint prior density function is formulated as follows:Assume that the parameters ζ asFor simplicity, let us define the function L(data) isThe joint posterior distribution defined from α, γ, θ) and K is given asTherefore, the joint posterior density function can be expressed byα,γ and θ assuming that Ψ =  = , can be given byi, j, k = 1, 2, 3,i ≠ j ≠ k and Ψi is the ith member of a vector Ψ.Moreover, the marginal posterior density functions of 1, Ψ2, Ψ3) =  is obtained via minimizing the expectation of loss function under the marginal posterior density as follows:Under the marginal posterior density function as in However, in practice, because of the intractable integral in relation 1, Ψ2, Ψ3) = , which is associated with the NG-Weibull distribution and j;  as in η ∈ , the (1 − η)100% credibility interval In the Bayesian framework, interval estimation can be done via credibility interval conception. Consider the parameter vector Ψ = 100%  HPD interval for Ψj, j = 1, 2, 3 is the simultaneous solution of the following integral equationsThe HPD interval is a kind of credibility interval with a specific restriction. The  and observed data k, are as follows:It is clear from Eqs Starting with an initial estimate Draw Draw Draw α, γ, θ) and initial parameter vector Ψ0 = c, the posterior samples are extracted by above Gibbs sampler where the full conditional distributions are given asIn case of the NG-Weibull distribution, by considering the parameter vector Ψ = , Geweke (G) and Raftery-Lewis (RL) for checking the convergence of the Gibbs algorithm are provided in The statistical decision theory helps to address the state of uncertaintyand provides potentially a sound framework for dealing with problems of bio-medical, reliability, actuarial, economic and financial decision-making. Among the applied fields, the reliability engineering has received a serious consideration. In the field of reliability theory, modeling of lifetime data is very crucial. The data related to the lifetime of electronic product or any entity, etc., are usually positive and skewed to the right. In lifetime analysis and reliability theory, the failure rate function  is a prominent reliability characteristic. Among the possible failure rate functions, the unimodal, modified unimodal or bathtub-shaped failure rate curves are well-known in reliability literature. The classical models are not so flexible to model such complex forms of data.Due to the importance and application of the statistical models in reliability, medical and financial sciences, a reasonable work has been done in the literature aiming to improve the characteristics of the classical models. Although the new improvement has achieved the respective goal, unfortunately, the numbers of parameters have been increased, and the estimation of parameters, statistical inference and derivation of mathematical problems become complicated.To provide a better description and best fitting to the reliability data, therefore, in the present study, a new family of distributions has been studied. The key goal of introducing the class of distributions is to improve the characteristics of the classical distributions.From the above theory and discussion, it is quite clear that the researchers are always in search of new flexible distributions. Therefore, to introduce new flexible distributions and bring further flexibility in the model proposed in this paper, we suggest to introduce modified forms of the proposed model. As a future research direction, we suggest to introduce new models useful for modeling data in reliability engineering and financial sciences.As we stated above that the statistical models with bathtub-shaped failure rate function are very useful in reliability engineering. Here, we suggest a new modification of the proposed distribution to model lifetime data with a bathtub-shaped failure function.F = 1 − eγxα − ηx−, be the cdf of the three-parameter modified Weibull (MW) distribution, where ξ = . Then, a random variable say X is said to follow the new generalized modified Weibull (NGM-Weibull) distribution, if its cdf is given byLet The corresponding pdf and hf are given byDifferent plots for the failure rate function of the NGM-Weibull distribution are provided in α = 1.2, γ = 1.2, θ = 0.01, η = 0.08 (red line), (ii) modified unimodal for α = 4.8, γ = 0.3, θ = 0.1, η = 0.1 (green line), and (iii) bathtub-shaped for α = 3.5, γ = 1, θ = 0.1, η = 4.6 (black line). Since, the NGM-Weibull captures all the important forms of the hf, therefore, the NGM-Weibull distribution can be a good candidate distribution to model complex form of the reliability engineering data. In the future, therefore, we are intended to model lifetime and reliability data having unimodal, modified unimodal and bathtub-shaped failure rate behavior.From the plots provided in Furthermore, in the practice of actuarial and financial sciences, the heavy-tailed distributions are useful for modeling heavy-tailed financial data sets. Heavy-tailed distributions are those, whose right tail probabilities are heavier than the exponential distribution, and satisfiesX family.Here, we suggest a new heavy-tailed distribution to provide the best description of the heavy-tailed financial data set. The new heavy-tailed distribution can obtained by using the Burr-XII (B-XII) distribution as a special case of the NG-F = 1 − (1 + xc)k−, be the cdf of the two-parameter B-XII distribution, where ξ = . Then, a random variable say X is said to follow the new generalized B-XII (NGB-XII) distribution, if its cdf is given byLet The corresponding pdf is given byc = 1.2, k = 1.5, θ = 1.2 (red line), (ii) c = 1.4, k = 0.8, θ = 1.4 (green line), and (iii) c = 1.6, k = 0.3 and θ = 1.6 (black line).Different plots for the pdf of the NGB-XII distribution are provided in c and θ, the NGB-XII distribution tends to a heavy-tailed distribution. Henceforth, the NGB-XII distribution can be a good candidate distribution to model heavy-tailed financial and other related data sets. In the future, therefore, we are intended to use the proposed method to introduce new heavy-tailed distributions.From the plots provided in X family. The MLEs of the model parameters along with some mathematical properties, are derived. Based on the proposed approach, a new modified form of the Weibull model called an NG-Weibull distribution is introduced and studied in detail. The NG-Weibull model is very versatile and is able to cater to the different patterns of failure rates. Due to the flexible behavior of the hf, the proposed model is capable to describe adequately the failure behavior of several lifetime datasets, particularly reliability engineering data. The usefulness of the NG-Weibull distribution is proved by analyzing the coating machine failure time data. We performed Bayesian estimation and estimated the model parameters using five different loss functions. The diagnostics measures such as the Gelman-Rubin, Geweke, and Raftery-Lewis are discussed to evaluate the MCMC procedure in the Bayesian analysis. As a future work, new models based on the proposed approach will be introduced to model complex form of reliability engineering and heavy-tailed financial data sets.Due to the great importance of statistical distributions in modeling data in reliability engineering, we introduced and studied a new family of distributions, called a NG-


Scientific Reports 10.1038/s41598-021-82611-8, published online 11 February 2021Correction to: The original version of this Article contained an error in the title of the paper, where “to measure” was incorrectly given as “to detect of”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supplementary Information file.


The Funding statement had an error in the reference of one project. Instead of PTDC/EMD-TLM/4644/2021, it should be PTDC/EMD-TLM/7289/2020.This work was supported by UID/BIM/50005/2019 and PTDC/EMD-TLM/7289/2020, a project funded by Fundação para a Ciência e a Tecnologia - Ministério da Ciência, Tecnologia e Ensino Superior (FCT-MCTES), through Fundos do Orçamento de Estado . CSL also acknowledges FCT-MCTES fellowship PD/BD/135045/2017.The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way. The original article has been updated.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.


Phylogenetic networks can represent evolutionary events that cannot be described by phylogenetic trees. These networks are able to incorporate reticulate evolutionary events such as hybridization, introgression, and lateral gene transfer. Recently, network-based Markov models of DNA sequence evolution have been introduced along with model-based methods for reconstructing phylogenetic networks. For these methods to be consistent, the network parameter needs to be identifiable from data generated under the model. Here, we show that the semi-directed network parameter of a triangle-free, level-1 network model with any fixed number of reticulation vertices is generically identifiable under the Jukes–Cantor, Kimura 2-parameter, or Kimura 3-parameter constraints. Typically, the goal of a phylogenetic analysis is to find a tree that describes the evolutionary relationships among a set of taxa. However, because trees, as directed graphs, have acyclic skeletons, they cannot represent reticulate evolutionary events, such as hybridization, introgression, and lateral gene transfer. Recognizing this limitation, it has become increasingly common to use phylogenetic networks in order to more accurately describe the history of some sets of taxa . In Gross and Long  to determine the unrooted skeleton of N as well as which vertices (edges) are hybrid vertices (edges).To illustrate the implications of Theorem Our proof is largely combinatorial, as we are able to use the algebraic results for small networks obtained in Gross and Long  and Holldistinguishable and precisely what it means for the network parameter of a class of models to be generically identifiable.We begin this section by defining the graph theoretic terminology that we will use throughout the paper. Then, in Sect. (rooted binary) phylogenetic networkN on a set of leaves skeleton of a phylogenetic network is the undirected graph that is obtained from the network by removing edge directions.A tree vertex if it has in-degree one and out-degree two. A vertex is a reticulation vertex if it has in-degree two and out-degree one, and the edges that are directed into a reticulation vertex are called reticulation edges. Let rooted phylogenetic tree.A vertex is a level of a phylogenetic network is the maximum number of reticulation vertices in a biconnected component of the network. Of particular interest in this paper are level-1 networks, which can also be characterized as phylogenetic networks where no vertex belongs to more than one cycle in the network’s skeleton .Note that deciding whether a mixed graph, a graph with some edges directed and others undirected, is a semi-directed network can be done in quadratic time in the number of edges [Corollary 4 of Huber et al. ]. The untriangle-free level-1 semi-directed network is a level-1 semi-directed network where every cycle in the unrooted skeleton has length greater than three. We will also refer to level-1 semi-directed networks with exactly one reticulation vertex as k -cycle networks, where k is the length of the unique cycle in the unrooted skeleton.In a semi-directed network, the reticulation vertices are the vertices of indegree two and the level is defined the same as for a rooted phylogenetic network. A A, B non-empty. An edge e in a network separatesA and B if every path (not necessarily directed) between any e. If e separates A and B then we call e a cut-edge and we say  split.We finish these preliminaries with one additional bit of graph theory terminology that will be useful throughout. Let n-leaf rooted binary phylogenetic network. For the description below, we assume that the network belongs to the set of tree-child networks  the underlying substitution process is known as the -Sánchez . Howeverconstraints. For example, we will refer to the image of model associated tounder the Jukes–Cantor constraints.In order to not overload the word “model," we will refer to these restrictions on the transition matrices as We end this section on network-based Markov models by noting that there exist other natural extensions of tree-based Markov models. For example, in Francis and Moulton , the autN for the group-based models considered in this paper. We will see that for a given set of constraints , if N is the semi-directed network attained from N does not depend on the location of the root. Then, we show that the associated model does not change if we suppress degree two vertices or remove parallel edges in the network. Thus, the phylogenetic semi-directed network N contains all of the information necessary to recover In this section, we show how to associate a model mentclass2pt{minime and f are incident to a vertex of degree two and that the Markov transition matrices For a tree-based phylogenetic model under the Jukes–Cantor, Kimura 2-parameter, or Kimura 3-parameter constraints, we may relocate the root and suppress vertices of degree two without changing the underlying model .n-leaf r-reticulation triangle-free level-1 semi-directed networks are distinguishable. In order to show this, we will require a number of results concerning 4-leaf networks which we prove in Lemma Our aim is to prove Theorem Q), a single triangle network (DT), or a 4-cycle network (4C), and compare the resulting varieties. We only need to consider four possibilities for each of toric fiber product argument that is described in the remark following Proposition 4.5 in Gross and Long If (ii)If (iii)If (iv)If (v)If (vi)If (vii)If See Table Let mentclass2pt{minimThe proof of Lemma In order to reduce the total number of invariants required to prove each part, we take advantage of the symmetry between networks. As an example, suppose that the statement in part (vii) is false. Then there must exist two double-triangle networks with distinct skeletons, In order to prove some parts of the lemma, we require two or more invariants to distinguish all of the relevant networks, though all parts can be proven using some combination of just the following six polynomial invariants:https://github.com/colbyelong/DistinguishingLevel1PhylogeneticNetworks,we provide the code to verify that these polynomials vanish or do not vanish on the referenced varieties as claimed.Small trees catalog  for the K2P and K3P constraints we require a set of invariants that vanishes on exactly one of the single-triangle networks. The set ullivant .To prove (v), we require a set of invariants that vanishes on one of the tree varieties, but on none of the double-triangle network varieties, and a set of invariants that vanishes on one of the single-triangle networks varieties, but on none of the double-triangle network varieties. The set For (vi), we must first show that there is a set of invariants that vanishes on one of the double-triangle network varieties but on none of the 4-cycle network varieties. The set The invariant We also need a result on 4-leaf networks that does not fit into Table common split if non-trivial if conflicting if For networks Let (i)when (ii)when The invariant Note that 4-cycles have no non-trivial splits, so we just need to compare trees, single-triangle networks, and double-triangle networks. Moreover, Table restriction of a network to a subset of leaves.Finally we require Lemma N be an n-leaf semi-directed network on restriction of N to A is the semi-directed network A (where undirected edges are treated as bidirected) and then suppressing all degree two vertices and removing parallel edges.Let Lemma n-leaf semi-directed networks on Let f contained in Let and Long , it is sLemma N/X the network x. Observe also that If n-leaf semi-directed networks and If The next lemma follows immediately from Lemma n-leaf semi-directed networks on Let n-leaf triangle-free level-1 semi-directed network on C a cycle in C. Then the partition induced by C is the partition x is separated from C by below the reticulation vertex if C. If C then we also say that x is below the reticulation vertex for any Let meet at a cycle C if each leaf in C. We say that they induce a cycle in t-cycle network. Note that if the set of leaves We say a set of three or more leaves N, i.e., this cycle is unique in N. Denote this cycle by Observe that if n-leaf triangle-free level-1 semi-directed networks on refines a, b that are contained in different sets in Let We recall a combinatorial result from Baños  on four-N be an n-leaf triangle-free level-1 semi-directed network on N such that (Lemmas 12 and 13 of Baños ) Let N bn-leaf triangle-free level-1 semi-directed networks on Let Let a, b, c, d all meet at a with d is below the reticulation vertex in To see this, first consider any We thus have that for Now consider any two leaves Suppose that every cycle in e in b such that e separates b from A. If e separates A from B, then Let u be the vertex in e nearer to A. If u is on a cycle, then denote this cycle by b (since C via e). If j, then the corresponding edge C is an edge that is further away from b than e and which separates A from b, contradicting the choice of e. So we may assume that the partition subdivideA  contains all of B and also none of A, which would imply that A and B. As every cycle in A and B. But this contradicts the fact that Furthermore See Fig. a.u is not on a cycle, let f and g be the other edges incident to u. By choice of e, neither f nor g can separate A from b. Thus there is at least one element u via f, and at least one element u via g. As e does not separate A from B, there is at least one u via either f or g, say  f. Then let f, with Y the set containing b. Observe that If See Fig. b. \documTheorem n-leaf triangle-free level-1 semi-directed networks on Let We prove the claim by induction on mentclass2pt{minimn. We first show that we may assume that any set of 4 leaves that induces a 4-cycle in So now assume that different 4-cycle or a double-triangle, then by Lemma Indeed, suppose this is not the case, and consider some arbitrary mentclass2pt{minimOtherwise, mentclass2pt{minimSo we may now assume that any set of 4 leaves that induces a 4-cycle in Firstly, if entclass1pt{miniman leaves. Thus by the induction hypothesis, if Secondly, suppose that n-cycle network, that is, Finally, suppose that mentclass2pt{minimWe are now ready to prove Theorem Theorem 2 The network parameter of a network-based Markov model under the Jukes–Cantor, Kimura 2-parameter, or Kimura 3-parameter constraints is generically identifiable with respect to the class of models where the network parameter is an n-leaf triangle-free, level-1 semi-directed network with n-leaf triangle-free level-1 semi-directed networks on Let We have shown that triangle-free level-1 semi-directed networks are generically identifiable under the Jukes–Cantor, Kimura 2-parameter, and Kimura 3-parameter constraints. This means that, given a long enough multiple sequence alignment that evolved on a network of this class under one of these models, this network is, with high probability, the only network from the class that coincides with the given data. Roughly speaking, this means that the data provide sufficient information to reconstruct the network. To prove this result, we employed a blend of algebraic and combinatorial methods to show that any pair of networks are geometrically distinguishable.Previously, it had been shown that networks cannot be identified from certain substructures. For example, networks cannot be inferred from their displayed trees since more than one network can display the same set of trees (Gambette and Huber On the algebraic front, the computations reveal differences between the relationships between the network ideals under the JC constraints and the relationships between the ideals under the K2P and K3P constraints that would be interesting directions for further exploration. In Hollering and Sullivant , the autAdditionally, from an algebraic perspective, we note that adapting the random search strategy described in Hollering and Sullivant  is what Finally, a major open problem, which is the larger setting for this paper, is to determine whether generic identifiability results such as these can be extended to higher level networks. We expect finding the necessary invariants for the increased number of non-unique induced 4-leaf subnetworks will be challenging. Furthermore, the complexity of the combinatorial part of the proof will explode for higher levels. This question is open not only for the group-based models studied in this paper, but also for the general Markov model, which has just started to be studied in the context of networks Casanellas and Fernández-Sánchez .


The findings suggests that respondents’ good self-appraisal of coping resources could moderate the impact of adversities on their future quality of life. Interventions for promoting positive psychological growth among middle-aged and older adults should cover four domains, i.e. event-related factors, environmental factors, personal factors, and cognitive and coping responses. Traditional Chinese wisdom emphasizes the importance of understanding the bad (‘Yin’—the shady side) and the good (‘Yang’—the sunny side) aspect of life events. Future research may explore the Yin Yang perspective on life-threatening experiences and its applications in cross-cultural quality of life studies in the era of globalization.The current study explores the interaction effect of adversities and self-efficacy at baseline on quality of life (QoL) at follow-up among middle-aged and older Chinese women. 531 women were interviewed in 2008 and 226 of them were re-interviewed a year later using Quality of Life Ladder (QoLL), General Self-Efficacy Scale (GSE), List of Threatening Experiences (LTE), Somatic Complaint Scale, and self-rated health. Respondents’ mean age at baseline was 55.7 . Over a year’s time, respondents had a decline in quality of life and self-rated health (p < .001), experienced more life-threatening events (p < .05) and somatic complaints. The hierarchical multiple regression model, employed in the study, identifies three predictors of future quality of life after adding the interaction term ‘Previous LTE × Previous GSE × Previous household income’ — previous quality of life , previous LTE , and the interaction term . This model explains 34.1% of the variance of future quality of life (Adjusted R Studies have suggested that emotional distress or post-traumatic stress disorder is ubiquitous in affected communities after disasters such as the current Covid-19 pandemic  Dependent variableThe Quality of Life Ladder (QoLL), which is a single item leader-shaped visual analogue scale, was adopted to assess a participant’s level of quality of life level. The score anchors from ‘(0) worst possible’ to ‘(10) best possible’, indicating that the higher the score, the better the quality of life Cantril, . BowlingB)Independent variablesThe List of Threatening Experiences (LTE) is a modified 6-item inventory of life events by reporting the number of threats that respondents have experienced in the last 12 months   : a single-item measure asking respondents to rate their health as ‘5 = Very Good’, ‘4 = Good’, ‘3 = Fair’, ‘2 = Poor’, or ‘1 = Very Poor’.Sociodemographic variables include age and monthly household income.All participants were sent a self-administered questionnaire, which was formulated in English and then translated into Chinese. The Chinese version was back-translated to English again to examine whether the contents were well-translated. The questionnaire used for data collection was the Chinese version, and these Chinese-translated measurement tools have been tested in a pilot study. Thirteen adult learners filled the Chinese-translated questionnaire to test the inter-correlation among items of the questionnaire. Results of the pilot study was satisfactory  scale, 0.512 for List of threatening experiences (LTE), and 0.857 for Somatic complaint scale (SCS)). As the LTE is just a checklist for calculating the number of critical life events rather than a measurement scale, the low Cronbach’s score did not affect its reliability. Details of the measurements are as follows:Using IBM SPSS version 25, the changes of score in measurement tools over time were examined by t-test, ANOVA, and Chi-square test. Hierarchical multiple regression analysis was applied to explore how independent variables and covariates at baseline predicted quality of life at follow-up. The regression model also examined the interaction effect of financial resources, self-efficacy, and life-threatening experiences at baseline on quality of life at follow-up.The research protocols were approved by the ethics approval committee of the University where the study was conducted (Project No.: 07/1.12). Informed consent was sought from all respondents who participated in this study.t = 2.733, df = 516, p < 0.01), the result of Little’s Missing Completely at Random (MCAR) test  = 5.073, p = 0.407). In other words, the missingness is completely unsystematic and unrelated to the study variables.A total of 531 middle-aged and older Chinese female learners participated in the baseline survey in 2008 and 42.7% (n = 226) of them returned their questionnaires at follow-up in 2009. Though the attrition for follow-up was high, there were no statistical differences in the key measures including LTE, GSE, SCS, and SRH between the remaining group and the drop-out group. Despite a relatively lower QoL mean score in the drop-out group compared with the remaining group . However, after adding LTE and GSE at baseline in the equation (Block 2), the quality of life and LTE at baseline were found to be strong predictors of future quality of life . The LTE factor can explain an additional 2.5% of variance of future quality of life. After adding the interaction term ‘LTE at baseline × GSE at baseline × Household income at baseline’ in Block 3, three key predictors of future quality of life were found: previous quality of life , previous LTE , and the moderation effect of ‘LTE at baseline × GSE at baseline × Household income at baseline’ . The model can explain 34.1% of the variance of future quality of life .From the results of hierarchical multiple regression analysis  and opportunity (jī 機)  is identical with the Western concept of crisis, and that is to tell a latent danger  indicates a critical moment for better or worse, and that is like the meaning of ‘crisis’ in English — ‘the moment in which a decision was about to occur’ (Graf, Maercker and Zoellner’s  Janus-Fa Zimmer, . For exa Zimmer, . The meaer Mair, . On the r’ Graf, : 599. Thr’ Graf,  because r’ Graf, . This inr’ Graf, . That shr’ Graf, . The Yinr’ Graf, . The conr’ Graf, , and by r’ Graf,  has applThere are several limitations to this study. Firstly, respondents of the current panel study were drawn from a large-scale lifelong educational program, instead of the whole population of Hong Kong. This may generate a biased result as active learners tend to perceive their psychological well-being and self-efficacy more positively. Even so, because the program is a government-funded territory-wide community project, which has a wide coverage in Hong Kong HKSAR, . The cur