The situation concerning adolescent mental health is a global public health concern, and the concept includes the ability to cope with problems of everyday life. A person’s approach and attitude towards themselves, i.e., their self-esteem, affects mental health. The study aimed to appraise and deepen the scientific understanding of adolescents’ self-reported self-esteem at age 12−13 from a resource perspective and test its ability to predict subsequent perceived mental well-being at age 17.n = 654) were analysed using ANCOVA. The outcome variable, perceived mental well-being (MWB), covers the aspects of mental well-being inspired by the “Mental Health Continuum,” representing positive mental health. Covariates were self-esteem (SE) and reported initially perceived MWB at age 12−13. Other independent explanatory variables were gender, the family’s economy, and the mother’s educational level.Data from the Longitudinal Research on Development in Adolescence (LoRDIA) prospective follow-up study of adolescents aged 12−13, and 17 . There was a significant but inverted U – shaped association between SE at age 12–13 and perceived MWB at age 17 [F  = 19.02, β-0.057; CI -0.08−-0.03, Eta = 0.03, Good self-esteem in early adolescence increases the likelihood of an unchanged favourable development of self-esteem and the probability of good perceived mental well-being. SE explained 18 per cent of the variation of MWB, and even more among girls. However, normal SE rather than high SE at 12 and 13 years is predictive of later mental well-being. Girls reported low self-esteem more often. Therefore, supporting self-esteem early in life can promote mental well-being in adolescence.The online version contains supplementary material available at 10.1186/s40359-023-01450-6. Adolescent mental health is a global public health concern and needs prioritisation . BesidesSelf-esteem involves an approach and attitude towards oneself that is either global or more specific . Global The theoretical base for the variable of mental well-being is the mental health continuum , consistAdolescence is the period in life when most developmental changes take place within a relatively short period. During this period, self-esteem generally fluctuates to a low level in childhood but increases throughout adolescence and young adulthood . TherefoWell-being as a concept relates to mental health. Mental health as well as self-esteem shows a gender bias: girls tend to report more emotional problems  and lowen = 1884) consisted of almost equal shares of boys  and girls . The research team visited the four municipalities to distribute and collect the survey responses in class.The Longitudinal Research in Development in Adolescence (LoRDIA) study started in 2013, following adolescents aged 12−13 to 17 in four municipalities in Southwestern Sweden and inviting the total population of sixth- and seventh-grade adolescents to participate. The participants (Data collection started in 2013 (Wave 1 = W1); adolescents aged 12 and 13 agreed to participate, a total of 1515 children. At 17 years of age, the fifth wave of LoRDIA was collected, and 949 children participated. A total of 654 individuals responded to both waves for this particular study’s relevant waves. A published report describes the LoRDIA planning and participants on p. 44–45 .A scientifically well-known and frequently used instrument  validateA variable assessing perceived mental well-being was created to test the relationship between positive mental health and self-esteem. The scale for perceived mental well-being (MWB) comprises a sum score from three questions: “How healthy do you think you are?”, “How do you feel about life right now?”, and “I think my life has purpose and meaning.” In W1, the response scale was a 3-grade rating scale from “very healthy to not healthy at all”, but in W5, a 4-grade rating scale ranging from “totally healthy, quite healthy, not particularly healthy, and not healthy at all” was used. For the second question: “how do you feel about life right now?“ in W1, the response scale started from “very good and ranged to not particularly good”. In W5, the response scale varied from “very good, pretty good, not particularly good, and not good at all”. For the third question “I think my life has purpose and meaning”, the response scale in W1 ranged from “yes to no” and finally in W5 as follows: “totally agree, partially agree, takes a partial distance, and takes distance completely”. When adding the three questions as a sum score, Cronbach’s α was .715 in W1 and .719 in W5, indicating high internal consistency in measuring positive mental health. The variables were recoded into high and low MWB in W1 as follows: 1 = very healthy, 0 = quite healthy, and not healthy. Corresponding for W5 was: 1 = very healthy, and 0 = all other alternatives. By handling each question this way, the sum score for MWB using the three questions varied from 0 to 3, indicating low, normal, or high positive mental health. Higher scores stand for better mental health. Categories described the groups, but the total sum was used in the forthcoming ANCOVA analyses as the outcome. In Wave 1, the range of the total sum was 0−6, and in Wave 5, 0−9.Gender, the mother and father’s educational levels (secondary vs. post-secondary education), and the family’s economy are relevant background variables. The variable of the family’s economy comprised three questions: “The family’s income in comparison with other families”, “The family’s income in comparison to other classmates”, and “I can afford things that are expensive to me.” To every question, three levels of answers were applicable: “worse”, “medium”, or “high” (0–2). The Cronbach’s α for the sum score of the family’s economy was .617, its range was 0–6, and it was divided into three categories: worse (0–2), medium (3–4), and high (5–6). All of these variables are fixed factors in the ANCOVA.Self-rated SE and perceived MWB at 12–13 and 17 years of age were tested using an independent t-test between groups divided into gender and a paired sample t-test for the dependent association over time. Categorical descriptions were made using Chi-square and cross-tables, and gender differences in the MWB and SE categories were investigated.p = .098) and using a random independent sample were fulfilled. The association was also checked for independent background variables, such as gender, the mother’s educational level, and the family’s economy.A general linear analysis of Covariance (ANCOVA) was conducted using the continuous variable self-esteem (SE) at age 17 as the outcome and concomitantly including the continuous sum scores SE and perceived MWB as covariates at age 12–13. The criteria for modelling normality and homogeneity via Levene’s test (p = .319), and using a random independent sample were fulfilled. Background variables  which were gender, mother’s educational level, and family´s economy, were used as categorical variables at age 12−13, chosen from previous research [p < .05.To analyse the relation between SE and subsequent perceived MWB, a general linear model, ANCOVA was used in several steps. The dependent continuous variable perceived MWB at 17 years as an outcome. First (a), the association of initial perceived MWB at age 12−13 (W1) with the outcome of MWB four years later (W5) was calculated in a univariate model. In a second univariate model (b), initial perceived MWB was replaced with initial SE at age 12−13 (W1) as a continuous variable. In a third model (c), these two covariates (SE and perceived MWB at age 12−13) were used in a multivariable model. The criteria for conducting the variance models, ANCOVA, normality, homogeneity of variance via Levene’s test p = .3, and usiresearch . The datn = 1472, vs. those participating in both waves, n = 936, and perceived MWB with the corresponding figures n = 1414 vs. n = 948 was compared. No significant difference was detected when comparing the mean values using an independent t-test for those participating in one versus two waves. The results are presented using descriptive statistics and variance models, including the β and F-values. SPSS version 27 was used for all statistical analyses.SE in those participating only in Wave 1, n = 654, 57% girls), indicating significantly more attrition in boys. Both SE and perceived MWB were stable over time. However, there was a slight and significant decrease in SE .The correlation between SE at 12−13 and 17 years) showed a stable trend during adolescence  at age 17 (W5), separately for girls and boys.Re-analysing the hierarchical ANCOVA separately for boys and girls showed differences between girls and boys. The outcome of MWB at the age of 17 depended more for girls on their mother’s education than for boys. For boys, a difference was present regarding SE as the sole covariate with a significant association for families’ economy, which was in line with results from the total sample in the ANCOVA models a and b (see Table The study aimed to appraise and deepen the scientific understanding of adolescents’ self-reported self-esteem (SE) at age 12−13 from a resource perspective and test its ability to predict subsequent perceived mental well-being (MWB) at age 17. SE was significantly negatively associated with perceived MWB, explaining 18 per cent of the variance in a hierarchical ANCOVA model, including initial MWB as a covariate. For every additional point on the SE scale at age 12−13, adolescents rated their MWB at 17 years by 0.06 units less. Arguably, normal SE rather than high SE at 12−13 years is predictive of later mental well-being. The negative correlation may also be connected to girls’ lower initial SE than boys. The association was not surprising, though earlier research has reported associations between self-esteem and mental disorders [Girls often report more internalising mental health problems, and gender differences have previously been shown . In thisMental well-being is important for boys as well. In a school-based survey in Norway, a similar result was found regarding self-esteem, gender, and life satisfaction, where girls reported lower SE and boys had higher quality of life . In our In this study, self-esteem was stable with a slight decrease, while previous research has indicated that self-esteem increases with age . HoweverThirty per cent of the adolescents with high self-esteem reported good mental well-being, while 16 per cent had low mental well-being. However, most had good mental well-being when looking at those with low self-esteem. However, in another study, boys seemed to have a higher mental well-being in both the young and older age groups . A revien = 74501) following students aged 12 to 19, Romano et al. [Global self-esteem is connected to psychological well-being , and thio et al.  found a o et al. , arguingo et al.  which ago et al.  with betA mother’s educational level was significantly associated with initial SE and MWB for girls, which aligns with  findingsp = .269 vs. p = .346) could be detected, pointing at a random dropout.In all longitudinal studies, participant drop-out is an almost inevitable event, and at the same time, a longitudinal study design is a strength. In this study, the total population of 12−13-year-olds from four municipalities was invited, and the majority participated. However, in senior high school, at age 17, adolescents were more spread out and more difficult to contact, increasing the dropout rate. While 1472 participated in the first wave, only 779 participated in the fifth regarding SE vs. 1414 in the first wave and 746 in the fifth regarding mental well-being. Of these, only 654 responded to both waves and questionnaires. Looking at those participating in Wave 1 but not in Wave 5 vs. those participating in both, we see a somewhat higher attrition rate among boys. Regarding SE and MWB, a slight non-significant difference in the mean scores . The strength of obtaining ready-collected longitudinal data is hampered by what variables were included and when the data related to them were collected. An outcome variable had to be developed, which was not validated as a scale in other studies. Theoretically and also according to a statistical evaluation, these variables fit together. The questionnaire lacked a specific item measuring social well-being as the third aspect of the mental health continuum, which may, besides the regression to the mean phenomenon, have impacted the negative association between initial self-esteem and subsequent mental well-being in the ANCOVA models. However, when creating the variable, the aim was to assess similar content, as is the case concerning the well-validated scale health continuum created by Westerhof and Keyes . AnotherGood self-esteem in early adolescence predicts a continuation of a favourable situation and increases the probability of subsequent good mental well-being. Adolescents’ mental well-being explains their self-esteem by 18 per cent, especially among girls. This highlights the question of sophisticated measuring instruments from a gender and resource perspective. Other researchers have also focused on these methodological aspects, which supports further research on handling the complex mental health situation. Accordingly, supporting self-esteem early in life can promote mental well-being.Additional file 1. Table of Flowchart within LoRDIA regarding self-esteem (SE) and mental well-being (MWB)


R-modules to finitely generated R-modules, for any commutative ring R whose spectrum is Noetherian. As Erman–Sam–Snowden pointed out, when applying this with R-module M we associate a topological space, which we show is Noetherian when In a previous paper, the third author proved that finite-degree polynomial functors over infinite fields are topologically Noetherian. In this paper, we prove that the same holds for polynomial functors from free  K is a functor P from the category of finite-dimensional K-vector spaces to itself such that for any two finite-dimensional vector spaces V, W the map P is a polynomial functor and U a constant vector space, then the functor V the vector space P has finite degree, which we will always require, then—much like a univariate polynomial and its shift by a constant—P and A polynomial functor over an infinite field P is a rule X that assigns to a vector space V a Zariski-closed subset X(V) of P(V) in such a manner that for each X(U) into X(V). In earlier work   is that if R has a Noetherian spectrum, then so does the polynomial ring R[x]. This special case, a topological version of Hilbert’s basis theorem, is easy and well-known; e.g., it also follows from -module.Let K yields that Note that tensoring with K replaced by R[1/r] for some nonzero r.The vector space R-module M. Then for each R such thatR[1/r] since their images in K. It follows that R[1/r]-module spanned by v is a linear combination of the first m basis elements and the basis elements do not satisfy any linear relation there. Hence To see this, extend M, N be R-modules. Denote by R-algebras.We follow  is just the subring of R[N] consisting of all polynomials that are constant on fibres of the projection When R is Noetherian and M is finitely generated, R[M] need not be Noetherian.The following example shows that, even when K is a field of characteristic zero, and let R-module generated by a single element R[M] is the subring of R[x] spanned by all homogeneous polynomials R[M] is the vector space over K spanned by R[M] is not Noetherian, since the ideal R[M] by its ideal of nilpotent elements is K.Let M is finitely generated, then a certain topological space R[M] is also Noetherian. In Example However, we will see later that if K of characteristic 2 and set oddi can only be in c is in (t). But for eveni, R[M] regardless of R[M] is the K-vector space with basisNow consider a field B is an R-algebra, then the base change functor from Definition R-algebra homomorphism B-algebra homomorphism If R[M] is the subring of R[x] spanned by all homogeneous univariate polynomials f such that Let B is a localisation of a domain R, then the map is an isomorphism:However, when R is a domain. Let M be a finitely generated R-module and let S be a multiplicative subset of R not containing 0. Set Suppose that M and embed R[M] as a graded R-subalgebra A of B of O be the kernel of the R-module homomorphism s gives R is a domain, the same holds over R and hence R-module M. Hence The first and last isomorphisms are standard. For the middle isomorphism, we choose generators Like in ordinary algebraic geometry, the coordinate ring of a direct sum is the tensor product of the coordinate rings.M, N be finitely generated R-modules. ThenLet R[M] and R[N] induce elements of R-linear map R-algebras. Denote by R-submodule of d, e). It suffices to show that M, N are free. In this case, we get R-linear maps from finitely generated free R-modules. Then we see thatElements of M is free. In that case, by Lemma R[M] is just a polynomial ring over R.Example M a finitely generated R-module. To be precise, R-domains with R-algebra monomorphisms, in the sense of the following definition.We now construct the topological space element of F is an element of F(C) for some subset of F is a subfunctor of X that assigns to each X(C) into X(D) for all morphisms topological space overF is a functor as above and F including the subsets Let C to the set F(C) with the collection We note that all definitions that can be stated in terms of elements and (closed) subsets of a topological space carry over to topological spaces over “injections" to refer to R-algebra monomorphisms.In what follows, we use the term X that assigns to each X(D) of X(D) into X(E) for all injections closed if Define D is an R-domain, then we can make R) on D is a domain. For any R-algebra homomorphism R-domains (not necessarily injective), the induced map If M. Note that in the definition of closed subsets of S to be independent of D, i.e., not every rule assigning to We think of Base change). If B is an R-algebra, and D is a B-domain, then B, coming from closed sets defined by subsets of R. If X is a closed subset of B-domain D to X(D).. Clearly, the inclusion X of A closed subset This follows from the previous proposition since The proof of Theorem R be a ring with Noetherian spectrum and r an element of R. Let R/(r). Then two closed subsets Let K be an R-field and let r in K is zero, then K is a r in K is nonzero, then K naturally is an R[1/r]-field. In this case, we have Suppose that Let The extension R be a ring.We now prove Proposition If This is an application of  define the same closed subset in not define the same closed subset in If two ideals M be a finitely generated R-module. Is R[M] to closed subsets of Suppose that R be a domain, let M be a finitely generated R-module and let X be a closed subset of Let R[1/r], the R[1/r]-module M is free, say of rank m, and so X is a closed subset of monic polynomials in r such that f in the latter ideal whose leading monomial is not divisible by any of the leading monomials of the f we would find that By Lemma R be a domain, M a finitely generated R-module, and X a closed subset of f vanishes identically on f vanishes identically on Let M is free of rank m. Let X. This time, for each nonzero monic polynomials in some power of which lies in r such that g be the image of f in g reduces to zero modulo g vanishes identically on As in the previous proof, it suffices to prove the statement in the case that R-modules into either For reasons that will become clear later, we will only be interested in polynomial functors from the category  commutes for every P has finite degree, i.e., there is a uniform bound U, V the polynomial law d.A polynomial functor R-linear map  commutes for all U, V. Note that post-composing with R-linear maps and hence, indeed, (linear) polynomial laws.Polynomial functors R-algebra A and R-modules U, V, W, let A-bilinear extension of the R-bilinear composition maps R-algebras A and For every R-modules U, V, there is a natural A-linear mapA”. Viewing  commutes; here A-linear extension of A is a polynomial ring over R, the mapRecall that for all mentclass2pt{minimP, Q be polynomial functors. We define a polynomial transformationR-algebras A and Let R-module homomorphisms, and the latter are precisely the linear polynomial laws, polynomial transformations generalise morphisms of polynomial laws, and the latter are precisely the linear polynomial transformations.Just like polynomial laws generalise R is an infinite field, then a polynomial functorR-vector spaces to itself such that for all R is a field but not necessarily infinite, then a polynomial functor If et-up in . If R isr–Suslin .Many of our proofs will involve passing to the case of (infinite) fields and invoking arguments from . This isBase change). Let B be an R-algebra and let P induces a polynomial functor B-module U fix a B-module isomorphism R-module of the same R-rank as the B-rank of U. Then, set B-algebra A, we need to assign to every  and the outermost Let P has the same degree as P and will play a role in Sect. P takes values in P.The dual functor U be a finitely generated free R-module.Let shift of P by U.We define the shift functor P.The composition R is an infinite field) carries over to the current more general setting. The proof in -modules for all R[1/r]-module.Let R[1/r].Note that we do not claim that the complement is itself the evaluation of another subobject; i.e., P. If P has degree 0, then so does S and then the statement is just Lemma P is d.Again, we proceed by induction on the degree of n we havem, P, S. By Lemma Schur Algebras. To introduce these, let R-modules, we obtain a mapR-module. Indeed, if U is free with basis The Friedlander–Suslin lemma relates polynomial functors of bounded degree to representations of certain associative algebras called R-module Schur algebra of degreeonU, and (given a basis of U), the basis distinguished basis.The The Schur algebra is associative  naturally carries the structure of an P(U) via U.Now suppose that :A-algebra A-module To conclude this section, we observe that Schur algebras behave well under base change: if R be a domain and let R. Recall that the base change functor yields a polynomial functor Let K and assume that Q of P in the category of polynomial functors Let Q is a functor into R is not a Noetherian ring.Note that we don’t require that In order to prove this proposition, we use the following lemma.A be a (not necessarily commutative) associative R-algebra and N an A-module that is, as an R-module, finitely generated and free. Suppose that Let R-basis of N. For each structure constants determined byR whose set of k-dimensional k consider the J are a R and whose other entries are variables R on which the coordinates are precisely these Let k-dimensional k and any k-set Let R  M.So let R[1/r]-module with basis R equal to R[1/r] and A equal to By Lemma P is a fixed polynomial functor Closed subsets of a polynomial functors play the role of affine varieties in finite-dimensional algebraic geometry. In this subsection, For any P. A subset of X that assigns to each X(U) of closed if X(U) is a closed subset of closure of X is the closed subset U for all We define U, V be finitely generated free R-modules, let D be an R-domain and let X(V)(D). In the particular case where X(U) is preserved under the polynomial action of X be a subset of Q. Then P.It is worth spelling out what this means. Let X to be the rule assigning U for all R-algebra over the category For Base change). If B is an R-algebra, then we obtain a closed subset R-module such that (We will use the following lemmas very frequently in our proof of Theorem R be a ring with Noetherian spectrum and r an element of R. Let R/(r). Then two closed subsets Let X(U), Y(U) for every This follows from Lemma Let X(U), Y(U) for every This follows from Lemma R-domains D defines a closed subset Y is the biggest closed subset of g is in the ideal of Y(U).Let Y(V) is a subset of Y is a subset of Y is a closed subset of Y(V) is a closed subset of It is easy to check that R-algebra A, consider the mapD be an R-domain and take p as an element of D, we find thatLet p while the left hand side also has the requirement that D-domain E. So It is not true in general thatR-algebra R[P(U)] has two natural gradings: first, the ordinary grading that each coordinate ring R[M] of a module M has ] is the tensor product of the i and use these to define a grading on R[P(U)], called the standard grading. The standard grading has an alternative characterisation, as follows: j if f of standard degree j.Let For any closed subset D be an R-domain. Thenf are also contained in Take R be a ring whose spectrum is Noetherian and let In this section we prove Theorem R has finitely many minimal primes R is a domain, we write Since We will prove the following stronger statement which clearly implies Theorem R, P, X) be a triple consisting of a domain R with Noetherian spectrum, a polynomial functor R, P, X) satisfies the following conditions: X eventually stabilises.Every descending chain f vanishes identically on There exists a nonzero Let (Condition (2) of the theorem means that P, Noetherian induction on X—for details, see below.The proof of Theorem R, P, X), we denote conditions (1) and (2) of Theorem For any fixed triple R; or(ii)R[1/r] of R, e such that the homogeneous parts In both cases, we write R, P). The reflexive and transitive closure of the relation To prove the theorem for The partial order on Suppose that we had an infinite sequenceEach step Let By Lemma R/(r). Assume that Let R, P, X) follows from condition (1) for R, P, X) follows from condition (2) for By Lemma R, P, X) by Combining this lemma with our induction hypothesis, we see that in order to prove R with Noetherian spectrum. Recall that K is the fraction field of R.Now let R-module, there exists a nonzero R, P, X) by d times, we may assume that the base change P.Suppose first that the base change  Theorem , we thenM of the top-degree part P whose base change with M exists after passing from R to a suitable finite extension of one of its localisations.We want a polynomial subfunctor R[1/r] of R and a polynomial subfunctor M of the top-degree part of the polynomial functor There exist a finite extension N of K of R. Let R[1/r] of R since the R[1/r]. Consider the submodule M of The R, P, X) by R. We now prove a version of Lemma Let Assume that R, P, X). Let R. Pick any nonzero R, P, X) with this particular r.By Lemma f be an element of f is naturally induces an element of f vanishes on R, for any r, then the prime ideal f vanishes identically on Indeed, let R, P, X) by M of the top-degree part P such that the base change We replace (R-module P(U) is the direct sum of a finitely generated free R-module and the  R-module M(U). In particular, both P and Proposition R by a further localisation R[1/r] which ensures that, if Let P(U) is the direct sum of the R-modules M(U) and X(U) but does not lie in the vanishing ideal of the pre-image in R[M(U)] have positive degree, so that a degree-0 homogeneous element of R[P(U)] that lies in the ideal of X(U) is an element of X(U) is the Cartesian product of U. We order closed subsets of We perform the same inner induction as in , §2.9. LY of X, we have R, P, X). Condition (2) for  follows from condition (2) for f as a polynomial in the coordinates on M(U) with coefficients in First suppose that X(U) but not on the preimage in f as a polynomial in the coordinates on R[M(U)] with coefficients in f does not vanish identically on the pre-image of R-basis of M(U) and extend this with R-basis of P(U), inducing an isomorphism t turns this expression intoNext, suppose that P(U), we may write K so that R[1/r] containing all R by R by R-domains D. Define Y to be the biggest closed subset of X where h does vanish.Let R-domains D.We haveY is the intersection of X with the biggest closed subset of h vanishes. So the lemma follows from Lemma The closed subset X. Since Y stabilise. This settles part of condition (1) of Y. This will afterwards be used to settle both condition (2) for Let M(V) is a submodule of R-modules M(V) and N(V) does not depend on the choice of section.In , gives that then V. Recall that the degree of the polynomial functor R by the localisation R[1/r] and may henceforth assume that We now replace P, R, X), we will first prove an analogous statement for the localised shift.To establish condition (2) for  with coefficients in Z be the closure of the projection of g and V, one concludes that g itself, vanish identically on Assume that Now we can establish condition (2) of g vanishes identically on There exists a nonzero V, such an r exists by Proposition r’s, the same applies to a finite number of V’s, so we may restrict our attention to all V of sufficiently large rank; we will do this in the proof.For each fixed R by a localisation R[1/r], we know that if By the inner induction hypothesis, after replacing V has rank at least that of U. Hence we may replace V by For any g that vanishes identically on R by a localisation that does not depend on g or V), g also vanishes identically on g then vanishes identically on Such a Y, the Z defined in the proof is not a subset of X in the sense of Definition Note that, unlike X. Recall from Sect. Y stabilise. Now, consider again the projection P, we have Noetherianity for Finally, we prove condition (1) of Let Since X is the following result.By the lemma, the fact that the sequence of The sequence m be the rank of U. As in equation Let t}(∗) in , §2.9, wmentclass2pt{minimX is Noetherian. This concludes the proof of condition (1) for  and hence the proof of Theorem Since both the sequence of entclass1pt{minimaTo illustrate that the proof method for Theorem n with integral coefficients for all For each (Proof sketch) Both statements follow by inductions identical to the one for Theorem X be the closed subset defined as the image closure of the polynomial transformation n for n for Take


The union-closed sets conjecture states that, in any nonempty union-closed family  For that purpose, every set A and denote i among all sets in Let Conjecture 1 (Frankl Union-Closed Sets Conjecture). This conjecture equivalently states that, for any union-closed family e, e.g., ,4,5. We e, e.g.,  for moreearchers ,9,10,11.earchers  further earchers ,13,14. Bearchers  showed tgiven in .The present paper further develops Gilmer’s (or Sawin’s) technique to derive new constants (or bounds) in the optimization form for the union-closed sets conjecture. These bounds include Sawin’s improvement as a special case. By providing cardinality bounds on auxiliary random variables, we make Sawin’s improvement computable and then evaluate it numerically which yields a bound approximately x. That is, the PMF of this measure takes the value 1 at x and takes the value 0 at other points.To state our result, we need to introduce some notations. Since we only consider distributions on finite alphabets, we do not distinguish between the terms “distributions” and “probability mass functions”. For a pair of distributions For a joint distribution e, e.g., . ClearlyA. We regard the set in >1}.If we choose bound in  since, fby Sawin , we can Substituting Proposition 1. For where the infimum is taken over all distributions and 1 at 0 at other points).with As a consequence of the two results above, we have the following corollary.Corollary 1. If The proof of Corollary 1 is given in https://leiyudotscholar.wordpress.com/ (accessed on 1 May 2023.) More decimal places of Sawin’s bound . i-th distribution i-th objective function For a conditional distribution  follows j in the last line is the optimal i attaining the minimum in HB be a finite subset of Let  in  and 7)..B be a fTherefore,f a,b in ). □A a.s. The union-closed sets conjecture states that such a code exists if and only if The breakthrough made by Gilmer  shows thduced in ). It is Furthermore, as an intermediate result, it has been shown that given in , expressFor simplicity, in this paper, we only consider the maximal correlation coefficient as the constraint function. In fact, the maximal correlation coefficient used here can be replaced by other functionals. The key property of the maximal correlation coefficient we used in this paper is the “tensorization” property, i.e.,  in fact was usedons, see ,21,22. (Γn(t) in  (i.e., i


M, g) with horizon boundary M, g) has non-negative scalar curvature, and the boundary M, g) that increase the scalar curvature. As a consequence, we show that equality holds in the above inequality if and only if the exterior region of  is isometric to a Schwarzschild half-space. Previously, these results were only known in the case where Building on previous works of Bray, of Miao, and of Almaraz, Barbosa, and de Lima, we develop a doubling procedure for asymptotically flat half-spaces ( M, g) be a connected, complete Riemannian manifold of dimension R(g) and non-compact boundary M.Let  is an asymptotically flat half-space if there is a number M whose complement is diffeomorphic to We say that  satisfies suitable energy conditions. As noted in [Escobar has studied asymptotically flat half-spaces in the context of the Yamabe problem for compact Riemannian manifolds with boundary; see  and alsothe work  of Schoethe work . AlmarazM, g) be an asymptotically flat half-space of dimension M, g) is isometric to ([R(g) across As explained in , §2, thesee also , p. 207.Theorem  Bray in  in the c Bray in  in the cAlmaraz, de Lima, and Mari  have stuTo describe recent results in this direction, we recall the following definitions from ; see FigM, g) has horizon boundary The connected components of Every minimal free boundary hypersurfaces or minimal closed hypersurfaces in We say that an asymptotically flat half-space  of dimension The proof of [The region entclass1pt{minima-2}; see , §2.The second-named author has recently proved the following Riemannian Penrose-type inequality for asymptotically flat half-spaces whose horizon boundary is a connected free boundary hypersurface.M, g) be an asymptotically flat half-space of dimension Then be an asymptotically flat half-space of dimension Let Let  be an asymptotically flat half-space of dimension ThenLet  can be locally perturbed to increase the scalar curvature near non-umbilical points of the boundary The approximation argument used to prove Corollary  and Yau  to charaM, g) be an asymptotically flat half-space of dimension Assume thatLet  be an asymptotically flat half-space with horizon boundary M, g) obtained by reflection across Let  unless R(g), preserves the condition m(g) nor To prove Theorem ality in  holds, tmentclasspt{minimaM, g) is an asymptotically flat half-space of dimension M, g) has horizon boundary In this section, we assume that , and  holds.i is sufficiently large. We consider the class of all embedded hypersurfaces of g in g in By , there iM, g) is an asymptotically flat half-space of dimension M, g) is conformally flat outside of a compact set. by reflection across In this section, we assume that  is conformally flat at infinity, that The assertions follow from the above construction, using that  be an asymptotically flat half-space of dimension Let , Proposition n and a sequence The assertion follows. Using Proposition M, g) be an asymptotically flat half-space of dimension Let . By  such that M whereLet Next, we compute the linearization of t led to  also shoThere holds M from Lemma Suppose, for a contradiction, that there is entclass1pt{minimaentclass1pt{minimaOn the one hand, using that The assertion follows. M, g) is an asymptotically flat half-space with horizon boundary m(g). Suppose that  and  of the pld space  with maslf-space  of mass In this section, we give an argument alternative to that in  to show For the statement of Theorem Let ld space .Following the argument given in , §6, we andThe argument presented in Lemma


Nature Neuroscience 10.1038/s41593-023-01452-y, published online 5 October 2023.Correction to: In the version of this article initially published, there was an error in the title: “changes in amygdala GABAergic inhibition” first appeared as “changes in gene amygdala GABAergic inhibition”. The error has been corrected in the HTML and PDF versions of the article.


Directed Feedback Arc Set, Directed Cutwidth and Optimal Linear Arrangement, also admit such algorithms on digraphs of bounded independence number. Towards this, we rely on the generic approach of Fomin and Pilipczuk , where to get the desired algorithms, it is enough to bound the number of k-cuts in digraphs of bounded independence number by a sub-exponential FPT function (Fomin and Pilipczuk bounded the number of k-cuts in transitive tournaments). Specifically, our main technical contribution is a combinatorial result that proves that the yes-instances of the problems (defined above) have a sub-exponential number of k-cuts. We prove this bound by using a combination of chromatic coding, inductive reasoning and exploiting the structural properties of these digraphs.Fradkin and Seymour  defined the class of digraphs of bounded independence number as a generalization of the class of tournaments. They argued that the class of digraphs of bounded independence number is structured enough to be exploited algorithmically. In this paper, we further strengthen this belief by showing that several cut problems that admit sub-exponential time parameterized algorithms  on tournaments, including  Directed Feedback Vertex Set (DFVS) and Directed Feedback Arc Set (DFAS).Tournaments form one of the most well studied families of digraphs, both algorithmically and structurally. In particular, whenever we try to generalize results that hold for undirected graphs to digraphs, arguably, one of the first families to consider is that of tournaments. Indeed, this has been the case when designing parameterized algorithms or approximation algorithms. Two problems that have been extensively studied on tournaments are DFVS admits a 7/2-approximation algorithm on tournaments , we denote the directed subgraph induced by the vertices of X. A set independent set of D if for any X is an independent set in the underlying undirected graph of D. The independence number of a digraph is equal to the size of the maximum independent set it contains. A directed Hamiltonian path in D is a directed simple path on all vertices in D. For a set of vertices ith vertex of the ordering.For any k-cuts in any digraph on n vertices with bounded independence number is k-cut in a directed graph D is a partition of the vertex set of D into two parts, D is a partition  of the vertex set V(D) such that there are no arcs from R to L in D.In this section, we prove that the number of k-cuts is easier. This simple observation is then exploited to its fullest using two main tools - (1) the Gallai-Milgram’s Theorem and (2) chromatic coding. Let us state them formally. We begin by stating this key observation, followed by formally defining both these ideas.At the heart of the proof of Lemma D be a digraph and V(D). For any k-cuts in k-cuts in D is at most Let k-cut  in D, there exists k-cuts k-cut in L, R) is not a k-cut in D. To prove the lemma, observe that, it is enough to prove that for any D into o(k) parts such that it is “easier” to bound the number of k-cuts in each of these parts, then we are done. At a high level, we will first partition the vertex set of D using chromatic coding, and then further partition each part of this partition using Gallai-Milgram’s Theorem. We will then conclude by proving that the number of k-cuts in each of the sub-parts is linear in the number of vertices. We now state the Gallai-Milgram’s Theorem formally.Thus, if we can partition the vertex set of -coloring family and then state the known results about the existence of such a families of bounded size. This result is called the chromatic coding lemma. For any graph G, a proper vertex coloring of G is a function Next, we state the technique of chromatic coding in its derandomized form. To this end, we first define Universal -Coloring Family) For integers n, k and r, a family n] to [r] is called a universal -coloring family, if for any graph G on the vertex set [n] with at most k edges, there exists an G. into at most k-cut  in D, there exists a partition u, v) of , there exists for any For any digraph V(D) from the family i, Let entclass1pt{minimaL, R) be some k-cut in D. Consider the digraph, say D with only the cut-arcs of . Note that h is a proper vertex coloring of h is a proper coloring of L, R) are in u, v) of , We will now show that D and integer k. For any arc  of a digraph and a partition u, v) goes across the parts of this partitionifandFor any partitionof the vertex set of the digraphD, we say that ak-cut  inDrespectsL, R) go across the parts of k-cuts in D is at most the sum of the number of k-cuts that respect a partition k-cuts that respect an arbitrary partition in D, an integer k and k-cuts in D that respect For the rest of this section, let mentclass2pt{minimk-cuts in D is at most The total number of k-cut  in D, there exists L, R) respects To prove the lemma, we need to prove that for any mentclass2pt{minimk-cuts in D that respect k-cuts in D, it is sufficient to bound the number of k-cuts in k-cuts in D that respect Henceforth, let us fix D, let D. For any k-cuts in D that respect For any digraph k-cut  of D that respects L, R) be some k-cut in D that respects u, v), such that L, R) respects Observe that to prove the lemma it is enough to prove that for any k-cuts that we want to bound. More precisely, we are now interested in bounding the number of 0-cuts in Thus, we have further narrowed down the class of D be a digraph on n vertices that has a directed Hamiltonian path. Then the number of 0-cuts in D is at most Let D has a directed Hamiltonian path, let D such that for each L, R) in D. Let i be the smallest integer such that i, for all D. Since L, R), which contradicts that  is a 0-cut.Since D where D, either does not contain any vertex in its right part (there is only one such cut) or contains some vertex, the total number of 0-cuts in D is at most Thus, for any We are now ready to prove Lemma N be the total number of k-cuts in D. Consider the family D and integer k. From Corollary k-cuts in D that respect Let mentclass2pt{minimmentclass2pt{minimmentclass2pt{minimD is said to be d-out-degenerate, if for every subgraph H of D, there exists a vertex Dd-out degenerate if and only if it has a d-out-degeneracy sequence.In this section we give the proof of Lemma D is a digraph on n vertices and d-out-degeneracy sequence of D. For any k-cut  in D is of type-i, if k-cut  in D is of type-0 if i cuts for all k-cuts. Observe that there is exactly 1 type-0 cut in any digraph.Throughout this section, k-cuts in D of type-i. Then the number of k-cuts in D is at most For any k-cuts in D of type-i, for an arbitrary D, then to bound the number of k-cuts in D, it is enough to bound the number of k-cuts in each D, where the number of parts of this partition is 4, compared to o(k) in Lemma Henceforth, our goal is to bound the number of k, for any fixed If k-cut  in D of type-i, For any k-cut  in D of type-i, For any For a digraph D can be partitioned into 4 parts with the following properties. The digraph induced on the first part is either empty or belongs to k-cuts in such a digraph we will use an induction on second part of this partition, we prove that for any k-cut  of type-i, all the vertices of this part belong to L. Similarly, for the third part of this partition, we prove that for any k-cut  of type-i, all the vertices of this part belong to R. Therefore, there is a unique k-cut of type-i in the digraph induced by the second and third part. The last part of the partition has the property that the number of vertices in this part is “small”. For the digraph induced by this part, we will get the desired bound by using Lemma 1.4 on this digraph.Lemma k-cuts of type-i in the digraph induced on each part of the partition, thereby bounding the number of k-cuts in D because of Observation The proof of Lemma mentclass2pt{minimk-cuts in c is the absolute constant hidden in the k-cuts in D for any digraph n vertices. We claim that for any positive integers n, d and k-cuts in D.We prove the lemma using induction on entclass1pt{miniman and d, D is 1, then from Lemma D such that for any k-cut  in D of type-i, k-cuts of type-i in D is at most the number of k-cuts in D, the independence number of k-cuts in D of type-i is at most k-cuts in D is at most Let us first prove that for any positive integers mentclass2pt{minimn, d and for all k-cuts in any digraph n vertices is k-cuts in the digraph k-cut  in D of type-i, k-cuts of type-i in D is at most the product of the number of k-cuts in k-cuts in k-cuts in k-cuts in k-cuts of type-i in D is at most k-cuts in D is at most By induction hypothesis, let us assume that for any positive integers mentclass2pt{minimProof of Partitioning Lemma. We start by a lemma that gives an upper bound on the size of a digraph in k, if for all For any digraph Let G and integer r, if G does not contain a clique of size G be the underlying undirected graph of D. Let G. Since G, G is the underlying undirected graph of D, To prove the above-mentioned claim, we invoke Turan’s Theorem , which Intuitive Ideas for the proof Lemmad-out-degeneracy sequence of D. Also recall that, the aim of proving Lemma k-cuts in D of type-i. Consider any k-cut in D of type-i. By definition, D. Thus, they go to d-out-degeneracy sequence of D, the number of out-neighbors of d. This set of neighbors goes to the set X, of vertices that appear before D[X], then there are at least v to D. Thus, such a vertex v should always belong to same part as k-cut. Thus, such vertices goes to X has out-degree at most k. By Lemma d-out-degeneracy sequence of D. Consider the partition of V(D) into three parts: Let D is a tournament, D and X, of size D of size D is bounded by i cut, for any k-cut  of type-i in D, L, R) is a k-cut in D and i cut, i cut .d-out-degeneracy sequence of D, D, This concludes the proof. We now claim that the desired partition entclass1pt{minimaDFAS, Directed Cutwidth and Optimal Linear Arrangement when the input graph belongs to first is observing that the digraphs that are Yes-instances of these problems have sub-exponential FPT many k-cuts. The proofs for DFAS and Directed Cutwidth are based on showing that the digraph in the Yes-instances of the problems are k-out-degenerate, and hence, the bounds follow from Lemma OLA, we show that if there is an ordering of the vertex set of a digraph of cost at most k then the cutwidth of this digraph is Directed Cutwidth, the number of k-cuts in the Yes-instances of OLA is also bounded. The second step is a procedure to enumerate all k-cuts of the input digraph. And the third is to do some dynamic programming procedure over these enumerated cuts to solve the respective problems. The last part of the algorithm (doing dynamic programming over k-cuts) is standard and is identical to the algorithm given by Fomin and Pilipczuk  has a dfas of size i, and L witnessing that D[L] has a dfas of size i, and We now prove that for any D[L] has a dfas of size at most i.Since D[L] has a dfas of size at most i and We will now prove that if D be a digraph. For an ordering V(D), the width of this ordering is cutwidth of D, denoted by V(D).Let k-cuts in the Yes-instances of DFAS.Next we bound the number of D, k) is a Yes-instance of Directed Cutwidth and k-cuts in D is at most c is a fixed absolute constant.If  is a Yes-instance of DFAS, then there is an ordering, say V(D) of width at most k. Recall that, the width of an ordering D is k-out-degenerate. Thus, the bound follows from Lemma If  is a No-instance of Directed Cutwidth. Otherwise, from Lemma k-cuts in D. Let us denote this set by Directed Cutwidth problem by doing a dynamic programming over the set k-cuts. Let T be the dynamic programming table indexed by a k-cut T) to store the following information.T) for any Using the algorithm of Lemma L, say L, say L, say L, for all We now prove that for any L, say We will now prove that if there exists an ordering of D be a digraph. For an ordering V(D), the cost of Let The following proposition gives an alternate definition of the cost of an ordering.D and an ordering V(D), the cost of this ordering is equal to  For a diLemma D, if there is an ordering say V(D), of cost at most k, then For any digraph k, from Proposition r, the number of arcs of length exactly r with tail in r. Thus, for any r, the total number of arcs of length at most r, with tail in Since k-cuts in the Yes-instances of OLA.Next we bound the number of D, k) is a Yes-instance of OLA and k-cuts in D is at most c is a fixed absolute constant.If  is a No-instance of OLA. Otherwise, from Lemma k-cuts in D. Let us denote this set by k-cuts. Let T be the dynamic programming table indexed by a k-cut T, i) to store the following information.D has an ordering of cost at most k. We now describe how we compute T, i) for any Using the algorithm of Lemma L, say L, say L, say L, say L, We now prove that for any L, say i-cut in D. Since We will now prove that if there exists an ordering of DFAS, Directed Cutwidth and OLA on digraphs of bounded independence number. We thus significantly generalized known results for the restricted case of input digraphs that are tournaments. Towards this, we obtained an upper bound on the number of k-cuts in digraphs in DFAS, Directed Cutwidth and OLA admit polynomial kernels on In this paper, we designed sub-exponential time parameterized algorithms for


Cell Death Discovery 10.1038/s41420-023-01298-0, published online 25 January 2023Correction to: The original version of this article contained a mistake. Due to a typesetting error in the “Introduction” section the references are out of order. We apologize for the error. The original article has been corrected.


The proposed solution is scaled and allows the use of many trusted status authorities that issue explicit short-term certificates confirming the validity of explicit long-term certificates. Furthermore, we demonstrate that our signature scheme has a short-term non-repudiation property for the shell validity model.This paper addresses the certificate revocation problem and proposes the first revocable pairing-based signature scheme with implicit and explicit certificates (IE-RCBS-kCAA). We should no longer discuss whether to revoke certificates but how to do it effectively, ensuring both the scalability of the revocation operation and the non-repudiation of the signature in the short or long term. Under the computational difficulty assumptions of the modified collusion attack algorithm with  Digital signatures are a critical component that ensure the integrity, authenticity and non-repudiation of electronic documents. In public key cryptography (PKC), a digital signature is considered valid if it is mathematically correct and a related certificate is valid. The certificates are issued for a limited period , they can be revoked  and in most cases, a user certificate is issued by an intermediate CA (certificate authority), and other certificates are issued by another intermediate CA or by a root CA with a self-signed certificate. The root CA is considered a trust anchor. All certificates between an end-entity certificate and a trust anchor certificate form the trusted certificate path. The validation of this chain is a challenging task.shell model, the certificate is valid as long as all the certificates in the certification path are valid when the signature is verified. The second one is the modified shell model. In this model, the signing time is the basis for decision making about the signature validity (this requires a timestamp). The third model is the chain model: once a signature is valid at signing time, it remains valid all the time. The chain model can only provide non-repudiation of property over a long period of time [Three models for certificate validation exist . In the  of time ,2.The purpose of certificate validation is to confirm the authenticity of the public key, i.e., to provide proof that the public key belongs to the signer and that the associated private key was under the sole control of an owner at the moment of signature creation. The process of verifying certificates for signatures that need to be valid for a long period of time is one of the main difficulties when implementing and managing a public key infrastructure. The authenticity of a public key in a public key cryptosystem can be achieved in two ways: either explicitly or implicitly. During explicit authentication, the public key authenticity can be verified directly using the certificate issued by a CA. In implicit authentication, the secret certificate issued by a CA  can be verified indirectly during signature verification or decryption operation. Combining both approaches and obtaining cryptographic schemes based on explicit and implicit certificates is another possible solution .It seemed that the certificate validation problem had been solved in 1984 when Shamir [The great interest shown in the ID-PKC, CL-PKC and IC-PKC schemes is because they eliminate explicit certificates from the encryption or signature schemes and allow the need to manage the status of certificates to be dispensed with. In practice, this is not the case, as the problem of certificate status management is shifted to the user identity management level ,6. In adThe digital signature schemes must include a revocation mechanism to support non-repudiation and achieve Girault trust level 3 . The reIn the online mediator  technique ,9, a useIn 2001, Boneh and Franklin  proposedIn 2014, Sun et al.  proposedIn 2017, Jia et al.  proposedmodified k-CAA problem and the discrete logarithm (DL) problem. When an explicit long-term certificate is revoked, its status information (in the form of an explicit short-term certificate) is available online at the moment of signature generation. The method used for revoking and providing certificate status is similar to that presented by Yum et al. [This paper proposes a new revocable signature IE-RCBS-kCAA scheme with implicit and two explicit (short- and long-term) certificates, which is secure in the random oracle model under the hardness assumption of the m et al. .The implicit and short-term explicit certificates are two components of the signature key that can be used to sign documents. A signatory who wishes to reject the signatures he or she has generated may intentionally compromise his or her signature key and falsely claim that the signatures have been forged by someone else. Such a scenario is impossible with an explicit short-term certificate because the signer cannot revoke the explicit short-term certificate used as part of the full signature key.Using explicit short-term certificates ensures that revocation latency  is irrelevant, as the digital signature is valid only until the associated short-term explicit certificate expires. Because of the proposed signature validation approach based on the certificate validation shell model see , the accIn our approach, a user’s private key consists of a secret value, a long-term partial private key and a short-term explicit certificate (a time key). The first two partial keys are kept secret by the user, and their authenticity is confirmed by a long-term certificate issued by the trusted authority (TA). An explicit short-term certificate is periodically updated and sent over a public channel. To perform such an operation  problem, kCAA problem, k-mCAA problem) can be found in Hyla et al.  and MitsThe architecture framework of the IE-RCBS-kCAA signature scheme is shown in t by a trusted authority, TA, is denoted Next, the TA issues the partial secret key for each registered user using its master system key. The TSA is an authentication service that decides the validity of an explicit long-term certificate at the current time and issues an explicit short-term certificate if the TSA answer is positive. Therefore, on request, the TSA checks the signer’s long-term explicit certificate status according to the signed revocation list from the TA. The revocation list issued at time The trust model (understood as building trust relationships between cryptographic keys and their owner’s identity) with the TSA, and separating its role from the TA role, allows various business models to be obtained that can provide certificate issuing services (explicit or implicit) and determine their status. In a particular case, the TSA may act as an independent third party and provide certificate status verification services to multiple TA authorities. This type of scenario aligns with the model that has long been used in PKI systems: CAs issue certificates, and the OCSP server issues certificates of their status .Let us assume that the earlier mentioned trust authorities and trusted status authorities are part of the trust model based on a common set of system parameters. These parameters may include, among others, the same algebraic groups The IE-RCBS-kCAA consists of eleven polynomial-time algorithms : the syDefinition 1 (IE-RCBS-kCAA scheme).An implicit and explicit certificates-based signature against k-traitors collusion attack scheme consists of the following eleven polynomial-time algorithms:TA-Setup The TA runs the algorithm and, in secret, keeps the master private key s, while the TSA-Setup Create-User Implicit-Cert-Gen LongTerm-Explicit-Cert-Gen ShortTerm-Explicit-Cert-Gen Set-Private-Key Set-Public-Key Cert-Revoke Sign algorithm that generates a signature σ for the given input: the Sign Verify m, m, It is required that if Remark 1.Implicit-Cert-Gen and Explicit-Cert-Gen algorithms are successful when the TA positively verifies the identity and certificate information of The security proof of the proposed IE-RCBS-kCAA signature scheme is based on the commonly accepted standard security notion EUF-CMA . The EUF-CMA notation guarantees the highest security level of the signature scheme and thus the resistance of the signature scheme against the strongest attacks of the adversary.The security proofs is a claim made within the random oracle model, where a hash function finally replaces the random oracle. It is easy to see that the last step is heuristic in nature. In practice, the heuristics are successfully used for problem solving An adversary does not know the TA and TSA master keys;(b)An adversary knows the TA and TSA master keys;(c)An adversary knows the TA master private key and does not know the TSA master private status key;(d)An adversary does not know the TA master private key TA and knows the TSA master private status key TSA.For the IE-RCBS-kCAA signature scheme, four cases of access or lack of access by an adversary to TA and TSA master keys should be considered:Access to or lack of access to TA and TSA keys may depend on the adversary’s knowledge or ignorance of different user keys. Consequently, this allows us to define five different types of adversaries, the capabilities of which are shown in Note that even if the A thorough analysis of the adversary types and their capabilities in n et al.  and Y. Hn et al. ).C and adversary A, assuming that the adversary chooses which game to play. In all cases, adversary Based on the above comments, the security model of the proposed IE-RCBS-kCAA scheme, from now on referred to as EUF-IERCBS-kCAA-CMA, is defined by three games between challenger A=A1,A2,A is tryine, e.g.,  and Y. He, e.g.,  that is e, e.g.,  and thei, et al. ,41,42 an, et al. ): NormalDefinition 2.An implicit and explicit certificate revocable signature scheme IE-RCBS-kCAA has existential unforgeability against chosen message attacks (EUF-IERCBS-kCAA-CMA) if no probabilistic polynomial-time adversary has a non-negligible probability of winning Game I, Game II and Game III.The IE-RCBS-kCAA scheme consists of eleven polynomial-time algorithms: TA-Setup, TSA-Setup, Create-User, Implicit-Cert-Gen, LongTerm-Explicit-Cert-Gen, Set-Private-Key, Cert-Revoke, Get-Cert-Status, Sign and Verify. The algorithms are as follows.1.TA-Setup: The system parameters are p, P, Q, k is the system security number), 2.TSA-Setup (3.Create-User (4.Implicit-Cert-Gen ((a)Randomly selects (b)Composes the user’s certificate information (c)For (d)Generates S’s partial private key (an implicit certificate):5.LongTerm-Explicit-Cert-Gen Implicit-Cert-Gen algorithm:(a)The TA creates the explicit certificate that links S’s identity with the public key components:(b)The TA sends 6.ShortTerm-Explicit-Cert-Gen ((a)Randomly selects secret key (b)Composes the certificate status information (c)For (d)S via a public (open) channel; in addition, the TSA sends Generates the explicit short-term certificate (the certificate status evidence) as:7.Set-Private-Key quations  and 4)))Set-Priv8.Set-Public-Key 9.Cert-Revoke (10.Sign S performs the following steps:(a)Picks two random numbers (b)Computes the hash value (c)Generates a short-term explicit certificate by calling the ShortTerm-Explicit-Cert-Gen ((d)Generates the signature (e)If in  k1h+sIDsNote. Each time a signature is generated, a fresh short-term explicit certificate is retrieved from the TSA .11.Verify (a)Computes Equation ) and the(b)Computes Equation );(c)accept, otherwise reject.If the status of the certificate rect and  is validRemark 2.Note that during the indirect signature verification, the long- and short-term explicit certificates are validated (If the conditions formulated in Equations –9) are  are 9) aSuppose we use a shell model and the verifier received the signature at time (a) (b) When the above conditions are successful, the signature will be accepted as valid short-period non-repudiation evidence in whole period Remark 3.Based on the properties of the asymmetric bilinear map groups:LongTerm-Explicit-Cert-Gen algorithm can be entrusted to the signatory S, who, after receiving the implicit certificate from the TA will use Equation  problem with non-negligible probability. To this end, we used a general forking lemma (Bellare and Neven ), similaC with the help of adversary In both cases, after creating a valid forged signature, the adversaries Lemma 1.Suppose the hash functions where e is the base of the natural logarithm, Proof. k-mCAA problem to be obtained.(sketch) According to the approach given in  (b)else, (a)(b)i.if ii.C = else, if else, (c)c, C, c as the answer.(a)if the query is made explicitly by (b)i.if ii.otherwise, if otherwise, (c)t, t is output as the answer.U, c. For each request made on c, Public-Key-Replacement-Query(a)(b)Else, Corruption-Query (ID). Implicit-Cert-Gen-Query ((a)Implicit-Cert-Gen-Query for On the running (b)i.failure i.Implicit-Cert-Gen algorithm in creates ii.runs Otherwise, LongTerm-Explicit-Cert-Gen-Query ((a)(b)i.failure Otherwise, ShortTerm-Explicit-Cert-Gen-Query ((a)(b)t, i.failure i.ShortTerm-Explicit-Cert-Gen algorithm in composes a certificate status information ii.runs Otherwise, Super-Sign-Query (Else:(a)if there is no a tuple (b)t, if tuple (c)failure i.sets ii.selects iii.calculates iv.v.vi.otherwise Super-Sign oracle.          Output. A successful adversary returns a valid forgery U = Create-User-Query. Moreover, ShortTerm-Explicit-Cert-Gen-Query or Super-Sign-Query queries, respectively.          Let C, U, C,           Note that side output U, C,  and the tuple           When an adversary returns an invalid forgery, failure .Now we can show how to build a reduction algorithm Algorithm 2 3R(if (b == 0) then return 0 // Event E3 3.Correctness of the k-mCAA Problem SolutionC, failure  with a single thread. The scheme was implemented using an IAIK ECCelerate library for Java. Type 2 asymmetric pairing (ate pairing) based on the Barreto–Naehrig curve was used with different field sizes . The results . In The run times of the  results  average In the IE-RCBS-kCAA scheme, a trusted status authority (TSA) directly performs the revocation procedure and, as a result, alleviates the load of the trusted authority (TA). Moreover, the architecture framework proposed in this paper can contain many TSA authorities, effectively solving the scalability problem. This property is unique and is not available in other similar solutions. Unlike other proposals, e.g., Jia et al. , obtainik-mCAA and the We also formalize the security model for an adversary that can obtain a valid signature in the super sign query phase relevant to the public key selected by itself without providing the corresponding secret key. This security model proves that our IE-RCBS-kCAA signature scheme is semantically secure against Super Type I/II/III attackers under the The digital signature schemes must include a revocation mechanism to support non-repudiation and achieve Girault trust level 3. In order to achieve such properties, the paper proposes an architecture framework adapted to the needs of the IE-RCBS-kCAA scheme. In this scheme, an important role from the point of view of practical applications is played by the trusted status authority TSA, see , which pSign and Verify algorithms are significantly below one second. The signature size in our scheme is approximately 1 KB. Such a size will be too large a value for some practical applications, e.g., the transmission of short messages from IoT sensors. However, this value will be within an acceptable range for large documents and for signing transactions that are sent over high-speed internet connections. The performance analysis shows that the scheme is suitable for further practical work when signing large documents is needed. In this case, the signature is always decidedly shorter than the message. In other practical applications, one must carefully analyse if the signature size is acceptable.The performance comparisons with two other revocable signature schemes show that our signature scheme has a similar number of time-intensive operations. The empirical execution times of the Future research includes deploying the signature scheme with the certificates to be validated according to the chain model see . Such a


Nature Genetics 10.1038/s41588-023-01343-9. Published online 3 April 2023.Correction to: y-axis units shown below 0, where “10” and “20” were shown as negative numbers. The errors have been corrected in the HTML and PDF versions of the article.In the version of this article originally published, the surname of author Stefan Johansson was misspelled as Johanson. In the abstract and “Genome-wide association analyses” section of the Results, the number of loci associated with preterm birth was shown as six instead of seven. In addition, there was a production error in the Figure 1a


Correction to: Graefe’s Archive for Clinical and Experimental Ophthalmology10.1007/s00417-023-05979-5In the published version of this article, the given name of the second author has been misspelled.Shengtao Liu”.The correct author name should be “This is being corrected in this publication.


T < 360 K; 0 < P < 100 MPa), density and self-diffusion obey a special type of dynamic scaling, similar to the “τTV” scaling of Casalini and Roland, but with the negative exponent γ. The model predictions are consistent with experimental data. The new equation of state can be used for various process models and generalized to include multicomponent mixtures.Water is one of the most important compounds on Earth, yet its material properties are still poorly understood. Here, we use a recently developed two-state, two-(time)scale (TS2) dynamic mean-field model combined with the two-state Sanchez–Lacombe (SL) thermodynamic theory in order to describe the equation of state (density as a function of temperature and pressure) and diffusivity of liquid water. In particular, it is shown that in a relatively wide temperature and pressure range (160 K <  Water is one of the most anomalous substances on Earth ,2 as welT ≈ 277 K), often understood in terms of competition between two or more competing local environments with different density and symmetry whose relative concentration is indeed determined by thermodynamic conditions ..SH,H=SH,XT) is a decreasing function of pressure, while in most conventional glass-formers the glass transition temperature (gT) and the melting temperature (mT) are increasing functions of pressure. This anomaly is connected to the fact that the L-state has a lower density and the H-state has a higher density; thus, increasing the pressure makes the H-state more favorable (at constant temperature) and causes the L-H transition to shift towards lower temperatures. The scaling exponent g is estimated to be ~(−1.7), while for most other materials obeying the “τTV” scaling, g is usually positive. There are several theories relating g to the effective interaction between the molecules or clusters; however, those theories apply for positive values of g. A consistent molecular-theory-based understanding of the “τTV” scaling for water is still a task for future research.We also demonstrated that water exhibits dynamic scaling, similar to the “τTV” scaling described by Casalini, Roland, and other authors for multiple glass-forming materials. One interesting distinction for water is as follows: The transition temperature from the low-T state to the high-T state  ,17,20—thg/TXT , it would be very close to 1 for polymers and organics, but only ~0.5–0.6 for water.It is important to note that while SL-TS2 was originally proposed to describe the glass transition in amorphous polymers and organics, here it is applied to events that are well separated from the “true” glass transition (~130–150 K for water). This is due to the low activation energy of the low-temperature state of water (~20–30 kJ/mol) as compared to polymers (~100–300 kJ/mol). Thus, if we consider the ratio Finally, we should emphasize that the Casalini–Roland-type scaling (Equation (15) in the Overall, the proposed simple model successfully describes both the equation of state and the transport properties of liquid water in a wide temperature (~180–360 K) and pressure (~0 to 100 MPa) range. The framework, in principle, is flexible enough to allow the model to be extended to higher pressures closer to the LLCP. Another, more challenging problem is extending the proposed approach to integrate the crystalline phase (“ice”), at least in its most common form. These will be topics for future research. N clusters, including X being in the L-state and (N-X) being in the H-state, is written as follows:We adopted the two-state SL-TS2 formalism that combines the description of the density and relaxation time of an amorphous liquid or glassy material . Lφ is the volume fraction of the L-clusters, Hφ is the volume fraction of the H-clusters, and Vφ is the volume fraction of the voids or empty spaces  represents the van der Waals energies between the neighboring sites. The “internal” entropy of an L-cluster is labeled H,LS, and that of an H-cluster is labeled H,HS; the first index “H” refers to the origin of this entropy term as arising from the combinatorics of hydrogen bonds within the cluster . Finally, It is convenient to rewrite the free energy in a non-dimensional way, as follows:ψ and ν leads to the following equations:Minimizing the free energy with respect to H < rLr; this means that the parameter Equations (7) through (12) are similar to Equations (6) and (7) in Ref.  (the indgT~130–160 K) [T and P after solving Equations (7) and (8) for y and ν. Given that the glass transition temperature of water is fairly low (0–160 K) ,38,40,68α-relaxation time is given byHE is the activation energy for the H-state, and LE is the activation energy for the L-state. Similar to the case of conventional (non-hydrogen-bonding) amorphous materials, LE > HE.Within the TS2 framework (similar to the Shi-Tanaka-Russo model ), the α-a0 is some characteristic length that is assumed to be temperature-independent but pressure-dependent. Strictly speaking, there is a linear temperature dependence of D on T in addition to the Arrhenius or super-Arrhenius dependence built into the relaxation time, but here we neglect this weaker dependence. Thus, log(D) and −log(ατ) differ only by an additive constant, to be determined if both the diffusion coefficient and relaxation time are known at any given temperature.The diffusion coefficient can be estimated as follows:f(x) is some material-specific monotonic function. This scaling indicates that the relaxation time depends not on T and P separately, but on some “state variable” Z. The exponent γ is usually a positive number, ranging from ~0.5 to ~9 [P, at least within some reasonably broad range. For water, however, this scaling obviously fails because of the non-monotonic temperature dependence of the specific volume at constant pressure. One can imagine two different temperatures T1 and T2 such that the right-hand sides are the same, but the left-hand sides are drastically different. The question is whether the corresponding states concept applies to water at all.It has been shown that for many amorphous materials, there is a relationship between the relaxation times for various “corresponding states”, described by the following formula:.5 to ~9 ,50,51,52*, EH, ELε) depend on the pressure, as follows:Y == , and Τα =1/PT0, [Recently, it was demonstrated that the Casalini-Roland scaling (Equation (15)) emerges naturally within the SL-TS2 framework (at least within some reasonably broad pressure range) ,48. Hererα =1/P0,v is the isothermal compressibility. Finally, we further hypothesize thatDα. Here, and below, we set a0(0) = 0.14 nm = 1.4 Å.Next, we consider the density scale and assume thatXT is defined as the temperature for which ψ = 0.5 . The pressure dependence of XT is given by Equation (16), such that XT decreases as P is increased. The state variables y and ν then depend on the combined variable X(P)T/T—not on T and P independently.The “corresponding states” hypothesis can be expressed as follows:T < 373 K, from Price et al. [T < 298 K, and from Xu et al. [T < 262 K. The first three works consider the nuclear magnetic resonance (NMR) pulsed-field-gradient spin-echo method, allowing the evaluation of the self-diffusion coefficient from the trend of the spin-echo decay as a function of the intensity (or duration) of the perturbing magnetic gradient pulse [T > 273 K) measured by Weingärtner [To test our models, we used the published experimental data from multiple sources; these data were collected using standard techniques and instruments and were found to reproduce one another fairly well when the measurements were done under similar environmental conditions . The self-diffusion data were obtained from a series of different experiments from the literature. In detail, at atmospheric pressure, we considered data from Yoshida et al.  in the 3e et al.  for 238 u et al.  for 180 nt pulse  used a pnt pulse . At highnt pulse , includingärtner  and by Tngärtner . All of T < 423 K, from Kell and Whalley [T < 373 K, from Hare and Sorensen [T < 260 K, and from Erko et al. [T < 220 K (the latter two were measured in a confining environment to prevent the water from freezing). In detail, Grindley et al. measured the water’s specific volume by means of a piezometer [The density data were also obtained from different experiments; in particular, we considered atmospheric-pressure data from Grindley et al.  for 298  Whalley  for 273 Sorensen  in the 2Sorensen  for 183 o et al.  for 170 ezometer ; the datezometer  measuredezometer  were obtezometer  were obtezometer ) at the ezometer  and Grinezometer  at the hezometer , and we We applied the “two-state, two-(time)scale” Sanchez–Lacombe model (SL-TS2) to describe the equation of state and diffusion coefficient of both equilibrium and supercooled liquid water. The model showed a good agreement with the experimental data for the ambient-pressure measurements in a broad temperature range (180–360 K). We also discovered non-trivial scaling rules similar to the Casalini-Roland “τTV” scaling in “conventional” glass-formers; using this scaling, we successfully described the density and diffusion coefficient at multiple pressures (0–100 MPa). The results could provide a new description of both the equilibrium and transport properties of liquid water and, thus, may be of interest in many applications in the chemical and other industries.


Designing reasonable MAC scheduling strategies is an important means to ensure transmission quality in wireless sensor networks (WSNs). When there exist multiple available routes from the source to the destination, it is necessary to combine a data traffic allocation mechanism and design a multi-path MAC scheduling scheme in order to ensure QoS. This paper develops a multi-path resource allocation method for multi-channel wireless sensor networks, which uses random-access technology to complete MAC scheduling and selects the transmission path for each packet according to the probability. Through theoretical analysis and simulation experiments, it can be found that the proposed strategy can provide a reliable throughput capacity region. Meanwhile, due to the use of random-access technology, the computational complexity of the proposed algorithm can be independent of the number of links and channels. In wireless sensor networks (WSNs), the use of scheduling policies as the MAC layer protocol can bring better QoS than the traditional competition-based CSMA/CD protocol . AccordiIn a single-hop or single-route network scenario, the MAC scheduling algorithm determines whether each link queue transmits packets in the current time slot. In actual network scenarios, if there exist multiple alternative paths available between the source node and the destination node, it is necessary to allocate data traffic  to different paths reasonably based on MAC scheduling, so as to better balance the network load and obtain better transmission performance . In a muResearchers have made a series of research progress in addressing the distributed MAC scheduling problem. We organize and summarize the current research status from two aspects: single-channel scheduling and multi-channel scheduling and introduce the ideas and innovations of this paper.With regard to the distributed MAC scheduling problem in single-channel networks, some researchers have proposed a maximal scheduling (MS) strategy for various interference models ,5 and deIn wireless networks, wireless spectrum resources can be divided into multiple different channels, each occupying a certain frequency range. Wireless network nodes can choose a certain channel to transmit data on their own. Thus, links within the same interference area can transmit data without interference by using different channels, which increases throughput and bandwidth . If multFor distributed MAC scheduling in multi-channel networks, due to capacity differences between different channels, directly extending the single-channel MAC scheduling strategies to a multi-channel scenario may result in extremely poor throughput performance ,13. FollL| links and |C| channels, performing a complete maximal match requires at least O(|L||C|) iterations  is located within a certain region Ω, then the region Ω is named as the maximum throughput capacity region. In the multi-path scenario used in this paper, the maximum throughput capacity region Ω can be defined as a set of all input loads  that meet the following conditions:kΩ, then the strategy is called throughput guaranteed, and the efficiency ratio is k (0 ≤ k ≤ 1). Obviously, a larger value of k means that the MAC scheduling strategy has better throughput performance. The multi-path MAC scheduling scheme proposed in this paper is throughput guaranteed. Moreover, the implementation complexity of the proposed scheme is independent of the network size. The following sections will provide a detailed explanation and analysis of the execution process and the performance of the proposed algorithm.In practical network systems, queue-stable is an important state measuring network throughput performance, which refers to the network being able to bear the current input load. If there exists a MAC scheduling strategy that can maintain the queue-stable state of the network when the input load vector . Such a proportion vector indicates the proportion of data packets sent by the source node to each path and its calculation is completed by solving the following maximum value problem:sϕ and ν are positive constants. After solving the problem of (4), the vector sT(n) can be obtained. In detail, the element in the vector sT(n) represents the probability of the source node sending a data packet to the corresponding path. For example, if siT(n) = 0.2, the source node s sends data to path i with a probability of 20% in time slot n. In this way, sA(n)siT(n) represents the total number of data packets sent to path i by source node s in time slot n.Step 1: Data traffic allocation. In each time slot n into two parts: a calculation time slot and a transmission time slot, as shown in P mini time slots. Within time slot n, each link selects one of the P mini-time-slots according to the mechanism proposed in step 3 as the starting time for data transmission.Step 2: Time slot segmentation. In order to make the implementation complexity of the MAC scheduling strategy independent of the network size, it is necessary to divide the time length of each time slot n, each link l selects a certain mini time slot in the set {1, …, P + 1} with probability:lY denotes the mini time slot picked by link l, and the parameter lψ is calculated using the following formula:α = ln P. When link l picks P + 1, it means that it will not send data in the current time slot. If two neighboring links that interfere with each other (interference may come from a shared channel or interface conflict) choose the same mini time slot, then both will not send data in this time slot. Meanwhile, if link l detects other links in its interference set sending data when it is about to start sending, it will remain silent during this time slot. From Formula (6), it can be seen that the calculation of parameter lψ requires the neighbor queue information of link l, which can be assisted by using the RTS/CTS mechanism in the MAC protocol.Step 3: Scheduling. In each time slot The performance of the M-Scheduler scheme is analyzed here from two aspects: throughput and complexity. With regard to throughput performance, we demonstrate that the proposed M-Scheduler is throughput guaranteed, and the throughput efficiency ratio of the algorithm is further obtained. In terms of complexity, we will explain that the implementation complexity of M-Scheduler is independent of the network size. Based on this characteristic, the M-Scheduler strategy has significant advantages in complexity compared with the existing maximal-matching-based multi-path MAC scheduling schemes.For the purpose of achieving the throughput performance of M-Scheduler, the following lemma is needed.Lemma 1.0 > 0 such that when τ ≥ τ0, we have:For any positive constant μ > 0, there exists τProof of Lemma 1.j regarding link j, denoted as:jb is 0 j ≤≤ b 1. At the same time, it can be found that there must be a link j in the network that makes jb = 1. Without loss of generality, let b1 = 1. Hence, if the following inequality holds for every jb (j ∈ L and j ≠ 1), Lemma 1 is then obtained:τ > 0, if jb ≥ eμ−, we have (jb)τ1+ ≥ (jb)τ eμ−; if j ≤ e−μb, then we have (jb)τ1+ ≥ 0. Hence, by combining the two cases, the following inequality can be obtained:L| denotes the cardinality of the set L. Hence, when the parameter μ > 0 is given, we can appropriately choose a positive number τ0 to make 1 − (|L| − 1) eμ(1+−τ0) ≤ eμ−. Thus, for arbitrary τ > τ0, it is not difficult to deduce (9). Therefore, Lemma 1 was proven. After obtaining Lemma 1, the following theorem is given to describe the throughput performance of M-Scheduler. □Firstly, define a parameter bTheorem 1.λ =  satisfiesEmploying the M-Scheduler strategy, for any μ > 0, if the input load vector denoted by then the network system can maintain a queue-stable state.Proof of Theorem 1.lx(n) and jΥ as:E[·] represents the mathematical expectation. Due to E[sA(n)siT(n)] = sTsiλ(n), it can be then obtained that:q(n) = , lZ indicates the event that link l is scheduled, and o[·] denotes the higher order infinitesimal. Furthermore, according to Equations (5) and (6), it is not difficult to obtain:j. Thus, according to Lemma 1, for any positive constant μ, there exists τ0 > 0 such that when τ ≥ τ0, we have:Firstly, construct the following Lyapunov function:tability , and Theλ meeting:j [μ is small enough and P is large enough, the right half of inequality Equation (12) approaches 1, i.e.:It should be emphasized that if the interference degree of the entire network is K, then the maximum throughput capacity region Ω can be denoted by the set of all input load vectors h link j . Hence, λ ∈ Ω/K. Thus, the proposed scheme is throughput guaranteed. According to the above analysis, the larger the number of mini time slots P, the closer the throughput efficiency ratio of M-Scheduler is to 1/K. In the future, the hardware performance of the communication terminal can be improved to ensure that the value of P is large enough. In theory, one can compare the throughput performance of M-Scheduler with other existing comparable strategies. The MP algorithm proposed in [ER) of the three schemes has a relationship of ERM-Scheduler > ERLDCS > ERMP. Among them, LDCS adopts a relay-forwarding mechanism in the process of traffic allocation, which brings better throughput performance than the MP algorithm. Furthermore, M-Scheduler utilizes the tuple-based model to virtualize multi-channel networks into single-channel networks, which improves the transmission performance significantly. Therefore, the efficiency ratio of M-Scheduler is greater than that of LDCS. In By comparing Inequalities (12) and 21), it can be concluded that the throughput efficiency ratio of the M-Scheduler strategy is close to 1/K. That is to say, M-Scheduler can ensure the queue-stable state of the network system when posed in  and the , it can L| links and |C| channels, the MAC scheduling based on the maximal matching method needs to perform O(|L||C|) iterations in each time slot. Obviously, such implementation complexity is not conducive to the deployment of the corresponding scheduling strategies in large-scale dense wireless networks. Under the proposed M-Scheduler policy, each link selects its own data transmission timestamp based on probability in each time slot. Therefore, the implementation complexity of M-Scheduler does not increase with the number of available channels and links in the network. Hence, it is said that the complexity of M-Scheduler is independent of the network scale and can be better applied to large-scale dense wireless networks.From the execution process of M-Scheduler, it can be seen that it adopts a distributed implementation mechanism. Compared to centralized implementation, distributed implementation has lower complexity and is easier to deploy in distributed networks such as WSNs. Meanwhile, compared to existing multi-path distributed MAC scheduling schemes applied to wireless multi-channel networks, M-Scheduler does not need to execute the maximal matching processes and does not require iterative operations. The research shows that if the multi-channel network contains |P = 103. The simulation results in this scenario are shown in P = 105. The simulation results of scenario 2 are shown in To verify the effectiveness of the multi-path MAC scheduling strategy proposed in this paper in practical network scenarios, the throughput performance of M-Scheduler is compared with the existing MP and LDCS schemes using the NS-2.31 tool. We use a grid topology as shown in As shown in Moreover, by comparing This paper put forward a novel MAC scheduling scheme for wireless multi-channel networks, named M-Scheduler. The scheme adopts a mathematical model based on tuples, which models nodes, interfaces, and channels as tuple nodes and models the link between a pair of tuple nodes as a tuple link. In this way, a multi-interface multi-channel network can be modeled as a single-channel network. At the same time, when there are multiple transmission paths from the source to the destination in the network, the proposed M-Scheduler first allocates the data traffic to each path reasonably in each time slot and adopts a random-access-based scheduling strategy to complete packet forwarding during the MAC scheduling stage. The theoretical basis for designing a data-traffic-allocation algorithm and random-access probabilities is to ensure the throughput performance of M-Scheduler. By using the Lyapunov analysis method, we have demonstrated that M-Scheduler can achieve a stable throughput capacity region. When the input load is located within the capacity region, the network system maintains a queue-stable state. In addition, due to the utilization of random-access technology to complete MAC scheduling, the implementation complexity of M-Scheduler is independent of the network size. Such a feature makes M-Scheduler perform better in terms of complexity than the existing multi-path MAC scheduling schemes and also enhances its applicability in practical networks.It should be noted that other types of distributed networks (such as Ad hoc) can also consider using the solution proposed in this paper. The difference lies in the relatively fixed position of nodes in WSNs and, hence, our proposed algorithm is easier to promote and apply in WSNs. In future work, we plan to transplant the M-Scheduler algorithm to the communication protocol stack of WSN terminals and collaborate with IoT companies to conduct practical experiments to verify the effectiveness of M-Scheduler and gradually promote its application in new products.