A generalisation of the Jaynes-Cummings-Hubbard model for coupled-cavity arrays is introduced, where the embedded two-level system in each cavity is replaced by a Ξ-type three-level system. We demonstrate that the resulting effective polariton-polariton interactions at each site are both two-body and three-body. By tuning the ratio of the two transition dipole matrix elements, we show that the strength and sign of the two-body interaction can be controlled whilst maintaining a three-body repulsion. We then proceed to demonstrate how different two-body and three-body interactions alter the mean field superfluid-Mott insulator phase diagram, with the possible emergence of a pair superfluid phase in the two-body attractive regime. Almost all of these studies have focused on the case where each lattice element of the system is comprised of a two-level system interacting with a single photonic mode, resulting in the emergence of quasiparticles, referred to as polaritons.Recently, there has been considerable effort applied to the study of solid-state phenomena in Jaynes-Cummings-Hubbard (JCH) systems. These studies have predicted a superfluid-Mott insulator phase transition15. For example, with a view to better understanding these Pfaffian states, methods of introducing effective three-body interactions in dilute gas Bose-Einstein condensates have been proposed19. In the context of the Bose-Hubbard model20, it has been predicted that three-body interactions can significantly change the superfluid-Mott insulator phase diagram enabling a pair superfluid phase to manifest itself25, for attractive (repulsive) two-body (three-body) interactions. Accordingly, here, we consider a three-level JCH system which supports both two-body and three-body effective contact interactions, on each site of the JCH lattice, and analyse the zero-temperature phases that this system exhibits.In this work we consider the properties of a generalised JCH system where each lattice element is comprised of a three-level system interacting with a single photon mode. Such a model provides an opportunity to construct a Hamiltonian in which three-body interactions are dominant. The effects of such interactions have been of great interest for some time, in part due to their connection to the Pfaffian state that plays a key role in the theory of the fractional quantum Hall effect26 an effective two-body repulsion, for the polariton quasiparticles, is provided by the interactions between the two-level atom and the photons. In this work we wish to consider the impact of three-body interactions and the interplay between two-body and three-body interactions on the properties of the mean-field superfluid-Mott insulator phase diagram. As such we introduce an additional, local, three-body interaction. To achieve this, we generalise the two-level system on each site to a three-level system. There exist a number of possible configurations for a cavity containing a three-level system. Parity considerations restrict the number of viable transition configurations to three - the cascade (Ξ), Λ and V-type27. However, the restriction to a single cavity mode and the use of the rotating wave approximation28 necessitates the use of solely the cascade configuration, represented in Fig. In the standard two-level coupled atom cavity Jaynes-Cummings systemε, see Fig. ω. This results in a coupled-cavity array identical to one that has previously been proposed as a platform for the emulation of Pfaffian states via cavity QED14. As we assume that the three-level system is in the Ξ-configuration, we allow for photon-mediated atomic transitions only between the |0a〉 → |1a〉 and |1a〉 → |2a〉 atomic levels; we denote the corresponding transition dipole moments as β01 and β12 respectively, with β02 = 0. We also introduce the photonic creation and annihilation operators a〉 → |1a〉 and |1a〉 → |2a〉 respectively. Subsequently, with na + nph = n.For simplicity, we model an ideal system with three atomic energy levels equally spaced by n-particle subspace, we work in the ordered basis i = 0 for n = 0, i = 1 for n = 1 and i = 2 for n ≥ 2. For the vacuum, where n = 0, n = 1), 26. However, for n ≥ 2, these new atomic operators are representable via the Gell-Mann matrices, generators of su (3)29, given byFor each n, given byThis yields the matrix blocks to be diagonalised for each n = 0, 1 are necessarily identical to those in the Jaynes-Cummings model27 with the existence of two distinct eigenstates for n = 1. However, it is readily seen from Eq.  has the lowest energy for a given polariton number n. If the arccosine in Eq. .Therefore, the ground state for any polariton number 33. In the three-level system coupled to a single cavity mode, the existence of the third atomic level, and its interaction with the cavity mode, induces an additional three-body nonlinearity14. Furthermore, in this case, the sign of neither the two-body or three-body nonlinearities are immediately evident, but may nonetheless be discerned from inspecting the eigenenergies of the lowest energy polariton branches for each occupation n. For each n, we initially define the quantity Eg(n) to be the eigenenergy of the respective lowest polariton branch, i.e. E−(1) for n = 1 and E(0) for n = 0. The quantity n polaritons in a cavity. Let us initially consider the case where there is a single polariton in the cavity. If the two-body nonlinearity is repulsive, there must be an energy cost associated with adding a second polariton to the cavity, i.e. An effective two-body repulsion for photons is provided in a Jaynes-Cummings cavity by the interactions between the two-level atom and the photons; this is the cause of the photon blockade effectn = 1, 2 yields the critical value 14. That the two-body repulsion-attraction crossover is independent of n = 1 and n = 2 regardless of In the Methods section, we show that taking the difference of the proportional energy 34z is the number of nearest-neighbour sites, a quantity that is dependent on the lattice geometry . This yields an energy approximation consisting of a sum over all sites of the HamiltonianWith the exact single-site solutions in hand we now consider a lattice of such systems with photon hopping permissible between nearest neighbours. For such a system, we wish to compute the resulting zero-temperature mean-field phase diagram. A reasonable first guess would be that the superfluid and Mott insulator phases also exist in this modified system, and thus we proceed to apply the mean-field approximation to the hopping terms to yield a purely local Hamiltonian. Explicitly, this involves the substitution of the expressionself-consistent field approach, which we shall briefly detail here. Initially, we truncate the single-site basis to Fock states |na, nph〉 such that Nmax = 16. With an initial guess for ψ, we diagonalise ψ, and the process is repeatedly iterated until adequate convergence is achieved.To numerically diagonalise the Hamiltonian in Eq. , we emplψ in a given region of parameter space. However, analytical expressions for the phase boundaries may also be found via Landau theory, provided that the transitions are second-order continuous. Following Landau’s prescription, the free energy (which reduces to the many-body ground state energy in the zero temperature limit) may be expressed as a power series in the order parameter ψ, i.e34.Subsequently, the phase diagram may be obtained from the solution space by plotting the self-consistently determined values of ψ yields the criterion that the phase boundary is demarcated by solutions to the curve35a2 = 0. Solutions to this are especially simple to find if the order parameter ψ is manifest in a perturbative expansion of the ground state energy as a power series, which one finds to be the case in Hubbard-type systems34. To construct this expansion, we note that Eq. n〉 is cleThe calculation of the second-order perturbative correction is somewhat tedious; we emphasise that in the summation over accessible virtual states in Eq. , the conWe first investigate the regime in which the two-body interaction is predicted to be repulsive, i.e. We now focus on the regime in which we predict the existence of an attractive two-body interaction, that is, when 34, or the JCH system5, or even the system in consideration for ψ is continuous across the transition. A discontinuous, first-order phase transition features a jump in ψ from zero in the disordered phase to a distinct non-zero value in the ordered phase, and the condition a2 = 0 does not describe the value of the phase boundary for such transitions37.The discrepancy between the numeric and analytical predictions, for n = 1 Mott lobe for pair superfluid phase, in which superfluidity occurs due to the condensation of (quasi-)particle pairs25. The true pair superfluid phase occurs when ψ = 0 and φ ≠ 0, and is associated with the partial breaking of the system’s 24.To understand the nature of the transition it is instructive to recall from Fig. see Fig. , between38. The relevant Hamiltonian, scaled appropriately such that the interaction strengths are dimensionless, is provided byThis anomalous behaviour in the phase transitions of the three-level JCH system when 19 or a hardcore three-body constraint17. We note that in the Hamiltonian described by Eq. , a sophisticated procedure of spin-model mapping followed by the derivation of an effective field theory has provided a rich understanding of both the phase diagram and the dynamics of the system in each phase22. These studies demonstrate that in the presence of an additional three-body repulsion, the two-body-attractive Bose-Hubbard system exhibits both superfluid  and pair superfluid  phases, with the phase transition between the pair superfluid and the Mott phase being first-order25. Noting the similarities between this system and the three-level JCH system studied here, we predict that when This model has been studied in the context of experimental proposals which suggest that accounting for the three-body loss rate in a Bose-Hubbard model with attractive onsite two-body interactions may potentially result in either an effective finite three-body repulsion mechanismd by Eq. , a hardcvis-a-vis spatially off-diagonal correlations, inherent in the mean-field approximation. It would therefore seem that utilising one or more of the methods employed for the three-body-repulsive, two-body-attractive Bose-Hubbard model–which include an effective potential mechanism with both single and paired source fields23, and quantum Monte Carlo methods24 – would serve to elucidate the possible pair superfluid phase in the three-level JCH system.However, it is also noted that in these previous studies of similar systems, the phase boundary between the zeroth and second Mott phases is resolved in such a manner that the pair superfluid phase is not merely confined to a straight line at The addition of a third energy level to each ‘atom’ in a Jaynes-Cummings cavity array is seen here to result in an additional effective three-body repulsion between the resonant cavity photons, analogous to existing modified Bose-Hubbard models with an additional onsite three-body repulsion. By uniformly adjusting the ratio of the two transition dipole moments of each atom, we predict that a crossover manifests itself, between the regimes of two-body repulsion and attraction. Strikingly, this is seen to result in a possible first-order phase transition between the superfluid and Mott vacuum phases, and the coexistence of first- and second-order superfluid-Mott insulator phase transitions across the second Mott lobe. Furthermore, the first Mott lobe vanishes in the regime of two-body attraction.However, the presence of a pair superfluid phase, which our results seem to suggest, is not provable via the methods employed here. Its existence in the three-body constrained Bose-Hubbard model, demonstrated via exact diagonalisation, effective potential and quantum Monte Carlo simulations, motivated our hypothesis that it is also present in this modified Jaynes-Cummings-Hubbard system. Thus it would seem prudent to employ one or more of these to verify the existence of a pair superfluid of polaritons in such a coupled-cavity array.Below we show that the crossover from two-body repulsion to two-body attraction in the modified cavity system occurs when From Eqs , 8, thisThe left-hand side of Eq.  is equalfrom Eq.  that \doFrom Eqs , 5, 24 wa0 is not required for the determination of the phase boundaries at second-order phase transitions, we include them for the sake of completeness. However, we do require the second-order Landau coefficients, Ek(n) represents the unperturbed energy eigenvalues. We note here that a solution to a2(n) = 0, for a given value of While


In this paper, we investigate the existence of common fixed points of a certain mapping in the frame of an extended b-metric space. The given results cover a number of well-known fixed point theorems in the literature. We state some examples to illustrate our results. Throughout the manuscript, we denote b-metric, a number of authors have reported several interesting results in this direction X be a nonempty set and  The function d is called a b-metric and the space b-metric space, in short, bMS.Let b-metric is the following.The immediate example of Let b-metric space. Notice also that if It is evident that b-metric becomes a usual metric.It is clear that for Recently, Kamran  introducX be a nonempty set and b-metric if, for all  The pair b-metric space, in short extended-bMS.Let bMS.If Let (i)Let (ii)For b-metric space.In conclusion, for any Obviously, bMS, are defined as follows [Some fundamental concepts, like convergence, Cauchy sequence, and completeness in a extended- follows .bMS. (i)X is said to converge to A sequence (ii)X is said to be Cauchy if, for every A sequence Let b-metric space X is convergent.An extended-Letbe a complete extended-bMS. Ifis continuous, then every convergent sequence has a unique limit.Letbe an extended-bMS such thatis a continuous functional. Letsatisfyfor allwhereis such that, for eachhereThenThas precisely one fixed pointu. Moreover, for eachFor our purposes, we need to recall the following definition which is proposed by Popescu .T is an α-orbital admissible if, for all Let X is regular with respect to mapping X such that n and n.A set bMS Suppose that We start with the following lemma which is essential in our main results.Letbe an extendedb-metric space.If there existssuch that the sequencefor an arbitrarysatisfiesand alsofor anythen the sequenceis Cauchy inX.Let equality  recursivOn the other hand, by b-metric space On the other hand, by employing the ratio test, we conclude that the series Letbe two self-mappings on a complete extended-bMSsuch that the pairforms anorbital-cyclic admissible pair. Suppose that(i)for eachwherefor each(ii)there existssuch that(iii)eitherSandTare continuous, or(iii*)ifis a sequence inXsuch thatthenandMoreover, if for allandthen the pair of the mappingspossesses a common fixed pointu, that is, By assumption (ii), there exists a point Recursively, we obtainS and T, which finalizes the proof. More precisely, to see that u is the common fixed point of S and T, we shall examine the following two cases. First, we assume that T. Now we shall prove that T and S. Analogously, one can derive the same conclusion for the case Without loss of generality, we assume that x2k+1 in  and 12)\documentThus, throughout the proof, we suppose thatCase (a): Let equality , 12), w, w\documCase (b): Let equality , 12), w, w\documT and S, we getT and S have a common fixed point u. □Combining  and 17)17), we cmentclasspt{minimaegarding  and 22)18\documeLet bMS : For Case (b): Let Case (c): Let Case (d): Let T has a unique fixed point, Therefore, all the conditions of Theorem X be a nonempty set, T is an Let for all LetTbe a self-mapping on a complete extended-bMSsuch that the mappingTforms anorbital-cyclic admissible mapping. Suppose that(i)for eachwhere(ii)there existssuch thatand(iii)eitherTis continuous, or(iii*)ifis a sequence inXsuch thatthenandMoreover, if for allandthen the pair of the mappingsTpossesses a fixed pointu, that is, It is sufficient to take Let Define also T is X such that We show that For (a): For laced in  we get\(b): For his case  is obvio(c): For ent}y=2,  becomesT has a fixed point, (d): For all other cases, equality  holds. TLetTbe a self-mapping on a complete extended-bMSsuch thatTis anα-orbital admissible mapping. Suppose that(i)for eachwhere(ii)there existssuch that(iii)eitherTis continuous, or(iii*)ifis a sequence inXsuch thatthenMoreover, if for allandthen the pair of the mappingsTpossesses a fixed pointu, that is, It is sufficient to take b-metric Let T for T. We show that T is an α-orbital admissible mapping. If T is α-orbital admissible. In reason of the above arguments, Define also T has a unique fixed point, Note that, for each Let (H)For all  Here, T and S.Notice that in this section we investigate the existence of (common) fixed points of certain operators. For the uniqueness of a fixed point of the observed results, we will consider the following hypothesis. Adding condition (H) to the hypotheses of Theorem we obtain thatuis the unique fixed point ofT.v is another fixed point of T. From (H), there exists T satisfies , one can suggest different conditions, see, e.g., .b-metric space.It is clear that one can list several consequences from our results. By letting On the other hand, regarding the techniques used in , one can


Materials and Methods section, subsection 3D Microfluidic Co-Culture and Blocking Experiments, where it was written that 100 µg/mL of anti-PD-L1-blocking or anti-PD-1-blocking antibody or their respective isotype control was used. The correct concentration is 10 µg/mL. The authors apologize for this error and state that this does not change the scientific conclusions of the article in any way.In our original research article, there was an error in the The original article has been updated.The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


Correction to: BMC Genomicshttps://doi.org/10.1186/s12864-019-5947-zFollowing the publication of the original article , the autIn the section “PhasiRNA analysis” there was an error in the first equation in the PDF version of the article. A fraction bar was erroneously introduced between (20 m and n-x), (m and x), and (21 m and n).The incorrect version was:The correct version is:Furthermore, due to the formatting of Table The publisher would like to apologize to the authors and readers for any inconvenience caused. The original article has been corrected.


The main results are the identification of several shape-preserving properties of these operators, including their monotonicity- and convexity-preserving properties with respect to n and q of In this paper, we introduce a new family of generalized Bernstein operators based on  In 1997, Phillips [q-Bernstein polynomials, and for decades thereafter the application of q integers in positive linear operators became a hot topic in approximation theory, such as generalized q-Bernstein polynomials [q-Bernstein operators [q-Bernstein operators [q integers play important roles not only in approximation theory, but also in CAGD. Based on the Phillips q-Bernstein polynomials [q-Bézier curves using the basis functions of Phillips q-Bernstein polynomials. Dişibüyük and Oruç [q generalization of rational Bernstein–Bézier curves and tensor product q-Bernstein–Bézier surfaces. Moreover, Simeonov et al. [q blossom, which is specifically adapted to developing identities and algorithms for q-Bernstein bases and q-Bézier curves. In 2014, Han et al. [q-analog Bézier curves with one shape parameter, and established degree evaluation and de Casteljau algorithms and some other properties. In 2016, Han et al. [q integers, and investigated the generalized rational Bézier curve from a geometric point of view, obtaining degree evaluation and de Casteljau algorithms, etc.A generalization of Bernstein polynomials based on Phillips  proposedynomials –6, Durrmperators –13, etc.ynomials , which aynomials –16. In [ynomials , Oruç anand Oruç , 16 defiv et al.  introducn et al.  proposedet al. [α-Bernstein operators, and investigated some approximation properties, such as the rate of convergence, Voronovskaja-type asymptotic formulas, etc. They also obtained the monotonic and convex properties. For α, the α-Bernstein operators they introduced are defined asα-Bernstein polynomial n is defined by Recently, Chen et al.  introducq analogue of α-Bernstein operators, called Motivated by above research, in this paper we propose the q integers, the details of which can be found in [k, we denote q-integers by q-factorial and q-binomial coefficients are defined as follows:q-analog of q derivative and q derivative of the product are defined as Here, we mention some definitions based on found in , 22. Forn and q of The rest of this paper is organized as follows. In the next section, we give some basic properties of the operators For proving the main results, we require the following lemmas.We have the following equalities:By , we havea and b.From Lemma We immediately obtain Lemma For all functionsfandgdefined inreal numbersλ, μdefined inandthe following statements hold true. (i)Endpoint interpolation: and(ii)Linearity: (iii)Non-negative: Forandiffis non-negative onso isBernstein operators(iv)Monotone: For fixedandifthen(i)TheBernstein operators may be expressed in the formwherewith(ii)The higher-order forward difference ofmay be expressed in the formwherewhich is defined in , we notn.Finally, if Ifis convex onfor fixedandwe havef and g are convex on Combining  and 18)18), and ;x) (see ). The deLetting the convex function n and q of In this paper, we proposed a new family of generalized Bernstein operators, named


The employment of mobile vehicles to charge sensors via wireless energy transfer is a promising technology to maintain the perpetual operation of wireless sensor networks (WSNs). Most existing studies assumed that sensors are powered with off-the-shelf batteries, e.g., Lithium batteries, which are cheap, but it takes some non-trivial time to fully charge such a battery . The long charging time may incur long sensor dead durations, especially when there are many lifetime-critical sensors to be charged. On the other hand, other studies assumed that every sensor is powered with an ultra-fast charging battery, where it only takes some trivial time to replenish such a battery, e.g., 1 min, but the adoption of many ultra-fast sensors will bring about high purchasing cost. In this paper, we propose a novel heterogeneous sensor network model, in which there are only a few ultra-fast sensors and many low-cost off-the-shelf sensors. The deployment cost of the network in the model is low, as the number of ultra-fast sensors is limited. We also have an important observation that we can significantly shorten sensor dead durations by enabling the ultra-fast sensors to relay more data for lifetime-critical off-the-shelf sensors. We then propose a joint charging scheduling and routing allocation algorithm, such that the longest sensor dead duration is minimized. We finally evaluate the performance of the proposed algorithm through extensive simulation experiments. Experimental results show that the proposed algorithm is very promising and the longest sensor dead duration by it is only about 10% of those by existing algorithms. Wireless sensor networks are widely used in many Internet of Things (IoTs) applications, including video monitoring, traffic control, structural health monitoring, radiation detection, forest fire and volcanor monitoing, etc. ,2,3,4,5.Many researchers proposed to prolong sensor lifetimes by enabling them to harvest energy from their surrounding environments, such as solar energy, wind energy, etc. ,10. HoweA lot of attention has been paid to the vehicle charging scheduling for wireless sensor networks. Most of these studies assumed that sensors are powered with off-the-shelf batteries ,23,24,25Although excellent studies have been conducted for shorten sensor dead durations, some sensors still expire their energy for some durations, as the charging time of a sensor is non-trivial . For example, in ing time . It can On the other hand, other researchers ,28,29,30It is obvious that the adoption of ultra-fast sensors can significantly shorten sensor dead durations or even avoid their energy expirations. For example, in It can be seen that, although the adoption of off-the-shelf sensors is cheap, but the dead durations of sensors may be long due to the long charging time of the sensors. Contrarily, the adoption of many ultra-fast sensors can significantly shorten sensor dead durations, but the adoption will incur a high purchasing cost. For example, the price of an ultra-fast charging battery is usually about 30–40 dollars, which is about ten times the cost of an off-the-shelf battery, e.g., a Lithium battery with the same capacity costs only 2–3 dollars ,31. TherIn this paper, we propose a novel heterogeneous sensor network model, in which there are only a few ultra-fast sensors and many low-cost off-the-shelf sensors in a sensor network. The ultra-fast sensors are deployed at some strategic locations near to the base station, as the sensors close to the base station need to relay a large amount of data from other remote sensors.We here illustrate a heterogeneous sensor network. In We, however, observe that we can further reduce sensor dead durations by jointly considering charging scheduling and routing allocation in the heterogeneous sensor network. We can enable ultra-fast sensors to relay more data than off-the-shelf sensors, since the former can be quickly charged. For example, sensor n v3 see c.Then, the residual lifetime In this paper, we propose a novel heterogeneous network model, where a sensor network consists of only a few ultra-fast sensors and many low-cost off-the-shelf sensors. Then, the cost of deploying such a sensor network is low, due to the limited number of deployed ultra-fast sensors. On the other hand, we can significantly shorten sensor dead durations by not only scheduling the charging vehicle but also allocating routing among sensors smartly.Unlike existing studies that either assumed that all sensors are powered with low-cost off-the-shelf batteries, or assumed that every sensor is equipped with a high-cost ultra-fast charging battery, in this paper, we propose a novel heterogeneous sensor network model, where a sensor network consists of a few ultra-fast sensors and many low-cost off-the-shelf sensors. The deployment cost of such a network then is very low.Under this novel network model, we study a fundamental problem of joint charging scheduling and routing allocation, such that the longest sensor dead duration is minimized. We also propose an efficient algorithm for this problem.We finally evaluate the performance of the proposed algorithm through extensive simulation experiments. The experimental results show that the proposed algorithm is very promising, and the longest sensor dead duration by it is only 10% of those by existing algorithms.The main contributions in this paper are highlighted as follows:The rest of this paper is organized as follows: In this section, we first introduce a novel heterogeneous network model and then present the charging model. We finally define the problem.b is a base station for collecting data from all sensors.We consider a wireless sensor network We assume that there are two types of sensors in the network. The first type of sensors are powered by off-the-shelf batteries, e.g., Lithium battery, and it usually takes a while to fully charge such a sensor, e.g., 30–80 min . The othWe now consider the placement of sensors. We assume that the off-the-shelf sensors are randomly deployed. On the other hand, since the communication range of a sensor is limited, sensors close to the base station have to relay sensing data from other remote sensors. Therefore, the former sensors consume more their energy on relaying data than the latter ones. We thus assume that the ultra-fast sensors are deployed at some strategic locations near to the base station, as they can be quickly charged. For example, the ultra-fast sensor can be co-located with the most energy-consuming off-the-shelf sensors.We assume that there is an edge t. On the other hand, following [t is t, respectively, t is Assume that each sensor  routing ,33. Eachollowing , the enes m/s.As the energy stored in every sensor battery is limited, it will run out of its energy due to data sensing, data transmission and data reception. To provide controllable and perpetual energy to sensors, we employ a charging vehicle equipped with a recharging device to replenish sensor energy. Denote by The energy consumptions of different sensors vary significantly, i.e., some sensors may have little energy left while the others consume only a small fraction of their energy. Then, it is unnecessary for the vehicle to charge all sensors in each round, and sensors should be charged in an on-demand manner. To this end, each sensor V be the set of to-be-charged sensors in the current charging round. There are two types of sensors in set V, the lifetime-critical off-the-shelf sensors in V, as they are deployed at strategic locations where data traffics are extremely heavy. Moreover, it only takes a very short time to fully charge an ultra-fast sensor, and the number of ultra-fast sensors Let C of the charging vehicle for the sensors in V and allocate routing for the network V along tour C one by one  shows the flow reservation of each sensor In this section, we devise a joint charging scheduling and routing allocation algorithm for the dead duration minimization problem. We first present the framework of the algorithm, which invokes two algorithms for two subproblems of the original problem. We then detail the two algorithms.C for the charging vehicle to replenish sensors in V, such that the longest dead duration of sensors is minimized. It can be seen that the allocation of routing C are tightly coupled. On one hand, the residual lifetime of each sensor is highly related to the routing C is delivered, the start charging time that each sensor will be charged by the vehicle can be derived, by following Constraint (7). The routing C, such that the longest dead duration of sensors in V is minimized, and this subproblem is referred to as the charging scheduling subproblem. In the second subproblem, we assume the charging tour C is given, and the subproblem is to allocate the routing the routing allocation subproblem. We will propose two algorithms for the two subproblems in two later subsections, respectively. Having the two algorithms, we devise a joint charging scheduling and routing allocation algorithm as follows.Recall that the dead duration minimization problem is to allocate the routing Kth iteration, respectively. Similarly, denote by Kth iteration, respectively, where K, the algorithm consists of two steps. In the first step, with the given routing The algorithm iteratively finds the routing and charging tour. Denote by Algorithm 1 Joint charging scheduling and routing allocation algorithm (CSRA).Input: a set V of to-be-charged sensors, the residual energy Output: routing C.   1:Obtain a routing   2:fordo  3: Find a charging tour   4: Obtain a routing   5: Let   6:ifthen   7:    8:  break;  9:else 10:  11:end if 12:ifthen 13:  /* the difference of the longest sensor dead durations 14:  break;15:end if 16:end for17:return charging tour C, and routing The algorithm for the dead duration minimization problem is presented in Algorithm 1.V is minimized, i.e.,Given a set Assume that the charging time of each lifetime-critical off-the-shelf sensor in nstant δ , e.g., δ vs. 1 h . Let Δ=δSince it takes only a very short time to fully charge ultra-fast sensors in Algorithm 2 Algorithm for the charging scheduling subproblem.Input: a set V of to-be-charged sensors, routing Output: charging tour 1:V; Calculate the residual lifetime 2:Find a charging sub-tour 3:Sort sensors in 4:Find a charging sub-tour 5:Let 6:return charging tour The algorithm for the charging scheduling subproblem is presented in Algorithm 2. We later will showthat Algorithm 2 delivers an optimal solution to the charging scheduling subproblem see .AlgoritV of to-be-charged sensors, and the charging tour V, the routing allocation subproblem is to find the routing V is minimized, i.e.,V, Given a sensor network To obtain the optimal flow rates, we rewrite constraint  as:(10)tWe note that the objective function in problem nstraint  is not aV. The basic idea behind is to guess the optimal value To solve problem When P P4 see . OtherwiAlgorithm 3 Algorithm for the routing allocation subproblem.Input: a set V of to-be-charged sensors, the residual energy Output: routing   1:V by Constraint (7);Calculate the start charging time   2:Let   3:whiledo   4: Let   5:if there are no feasible solutions then   6:  /* the guess   7:  Let   8:else   9:  /* the guess 10:  Let 11:end if 12:end while13:Let 14:Find a feasible solution 15:return routing The algorithm for the routing allocation subproblem is given in Algorithm 3.In this section, we analyze the proposed algorithms for the charging scheduling subproblem and routing allocation subproblem in V, such that the longest dead duration of sensors in V is minimized, given routing Recall that the objective of the charging scheduling subproblem is to find a charging tour Theorem 1.Given a set V of to-be-charged sensors, Algorithm 2 delivers an optimal solution to the charging scheduling subproblem.Proof of Theorem 1.For the sake of convenience, assume that n in In the following, we show that sub-tour i) Consider  and (17), we have thatTherefore, we have We finally analyze the time complexity of Algorithm 2. We first obtain a sub-tour In this subsection, we show that Algorithm 3 delivers an optimal solution to the routing allocation subproblem by Theorem 2.Theorem 2.Algorithm 3 delivers an optimal solution to the routing allocation subproblem.Proof of Theorem 2.i) there are no feasible solutions to the Linear Programming (LP) In i) there are no feasible solutions to LP We first show , by increasing the number of iterations In the following, we study the impact of the network size, the maximum data sensing rate, the off-the-shelf sensor charging rate, the vehicle traveling speed, and the number of ultra-fast sensors.We first evaluate the performance of the proposed algorithm CSRA against existing algorithms TSP, NETWRAP, AA, TSCA, by varying the number We then investigate the performance of the algorithms, by increasing the maximum data sensing rate We also evaluate the algorithm performance, by varying the charging rate s on the algorithm performance, by varying the speed s from 1 m/s to 10 m/s. We further study the impact of the vehicle traveling speed We finally investigate the performance of different algorithms, by increasing the number of ultra-fast sensors In summary, it can be seen from the experimental results that the longest and average sensor dead durations by the proposed algorithm CSRA are much shorter than those by existing algorithms TSP, EDF, NETWRAP, AA and TSCA.The employment of mobile charging vehicles is a promising technique to replenish sensor energy, and the vehicle scheduling for prolonging network lifetimes has been studied in literature. Most of these studies assumed that every sensor is powered with a low-cost off-the-shelf battery ,38,39,40Some studies employed only one charging vehicle to charge sensors in a sensor network ,19,20,23On the other hand, some other studies ,38,39,40It can be seen that, although excellent studies have been conducted for replenishing sensor energy and shortening their dead durations, some sensors may still expire their energy, as the charging time of a sensor is non-trivial, e.g., 30–80 min .We note that other researchers ,28,29,30Different from these mentioned existing studies, in this paper, we propose a novel heterogeneous network model by deploying a few ultra-fast sensors at some strategic locations near the base station. The network cost is low, since we deploy only a few expensive ultra-fast sensors. On the other hand, we show that the network performance can be significantly improved with even such a small number of ultra-fast sensors, by enabling ultra-fast sensors to relay more data for other lifetime-critical sensors.Unlike exiting studies that deployed only off-the-shelf sensors or ultra-fast sensors, in this paper, we proposed a novel heterogeneous sensor network model, in which a sensor network consists of a few ultra-fast sensors and many low-cost off-the-shelf sensors. Then, the deployment cost of the network is low as the number of ultra-fast sensors is limited. Under the novel network model, we studied a problem of finding a charging tour and allocating routing, such that the longest sensor dead duration is minimized. We further devised an efficient algorithm for the problem, by enabling ultra-fast sensors to relay more data from lifetime-critical off-the-shelf sensors. We finally evaluated the proposed algorithm through extensive simulations. In addition, experimental results showed that the longest sensor dead duration by the proposed algorithm is much shorter than those by existing algorithms, e.g., 90% shorter.


Recently, Abadie et al.  have expanded synthetic control methods by the so-called cross-validation technique. We find that their results are not being reproduced when alternative software packages are used or when the variables’ ordering within the dataset is changed. We show that this failure stems from the cross-validation technique relying on non-uniquely defined predictor weights. While the amount of the resulting ambiguity is negligible for the main application of Abadie et al. , we find it to be substantial for several of their robustness analyses. Applying well-defined, standard synthetic control methods reveals that the authors’ results are particularly driven by a specific control country, the USA. As a tool for policy evaluation, cross-validation: the predictor weights, whose data in the training period (first part of the pre-treatment period) are used to find optimal donor weights for synthesizing the treated unit, are selected such that the out-of-sample error in the validation period (second part of the pre-treatment period) is minimized.Over the last few years, many studies have applied SCM across several fields, e.g., When measuring the effect of the 1990 reunification on Germany’s GDP per capita using the software package R, another set of predictor weights: 71.0% (GDP), 11.1% (investment), 7.9% (trade), 6.4% (schooling), 2.7% (inflation), and 0.9% (industry). The corresponding weights for the countries synthesizing Germany were much closer to, but still different from the values found by Abadie et al. (2015)When trying to replicate these results using the software package Stata, however, we found different predictor weights: 84.5% (GDP), 4.5% (investment), 5.1% (trade), 4.2% (schooling), 0.5% (inflation), and 1.2% (industry). The corresponding synthetic Germany was slightly different from the one obtained by Closer inspection shows that the failure to reproduce the results of We therefore investigate the corresponding ambiguity by conducting large-scaled Monte Carlo studies. The variation of the estimated post-treatment development of West German GDP is very small, with all estimates being significantly above Germany’s actual GDP. Concerning several robustness studies of The remainder of the paper unfolds as follows: the “In the following, we describe how synthetic control methods work both with and without the cross-validation technique. Many additional explanations, in particular on how to select potential comparison units and predictor variables, are provided in Abadie et al. (2015)Y, and predictor variables, usually denoted by X. These are considered both for a unit that has at some point in time been “treated,” usually denoted by the subscript “1,” and for so-called donor units. The latter are units not too different from the first one, but unaffected from the treatment, and denoted by the subscript “0.” In the example discussed throughout this paper, the treated unit is Germany which has been reunified in 1990, the variable of interest is GDP per capita, and predictors are (pre-treatment) GDP per capita, a measure for trade openness, the inflation rate, the industry share of value added, the amount of schooling attained, and the investment rate. The donor units consist of sixteen OECD countriesW: these weights describe to what extent each donor country is used to produce a “synthetic”  Germany. Thereby, the weights should be such that synthetic Germany mimics actual Germany as well as possible with respect to the (pre-treatment) predictor variables. For the example at hand, this means that the differences between actual and synthetic Germany with respect to GDP per capita, trade openness, inflation rate, industry share, schooling, and investment rate should be as small as possible. As we have six predictors (k=6), operationalizing the last statement requires introducing some weighting scheme. These non-negative so-called predictor weights are usually denoted by vm or V, and the cross-validation technique introduced in For the synthetic control method, we have two types of data: the variable of interest, often denoted by the letter k×J)-matrix k-dimensional vector V=, the donor weights J-dimensional non-negative donor weights W summing to unity, i.e., as the solution of In the training period, one makes use of the -matrix L-dimensional vector V, i.e., V∗ is a minimizer of8In the validation period, one uses the -matrix k-dimensional vector where the  by Eq. . These “mentclass2pt{minimIn contrast, the standard synthetic control method consists of only one step, not distinguishing between a training and validation period.k×J)-matrix k-dimensional vector V=, the standard SCM approach defines the donor weights W∗(V) as the minimizer of Instead, all pre-treatment data are used, in our application those from 1971 to 1990, to build the following quantities: the  are then used for synthesizing the treated unit. For a schematic overview of standard SCM, see Fig. The donor weights unique minimizer of the out-of-sample error V∗ unambiguously. The reason is that the mapping V are different after scaling. Less formally, it is often the case that different predictor weights lead to the same “training” weights. The problem of the cross-validation approach is that such different predictor weights V, although scaled and entailing identical A crucial insight as to why the cross-validation technique of entclass1pt{minimap of Eq. .W weights training” rows of columns “ADH,” “Orig.,” and “Alph.” of Table V are equivalent with respect to Eq.  are used, but only five donor units obtain positive weights in the training period  although V are different after scaling. However, in contrast to the cross-validation method, this is not a problem, as predictor weights are not used as input for a second step with different predictor data. Instead, the unique donor weights W∗(V∗) are the only quantity needed to synthesize the treated unit and estimate treatment effects, and therefore the standard synthetic control method leads to well-defined estimators, in contrast to the cross-validation approach. This can also be seen from Table Eventually, the standard synthetic control method does ∗ to Eq. , in comp∗ to Eq. . Again, d by Eq.  is oftenhttps://dataverse.harvard.edu/dataverse/ajps) both the data and all code of For the upcoming analysis, we retrieved from the AJPS Data Archive on Dataverse =0 for all j with For any given predictor weights ∗ of Eq.  must fulj, consider fj at δ=0, δ)W∗+δej would for small positive δ yield a smaller value in (W∗. For j with δ)W∗+δej will have non-negative components summing to unity even for negative δ that are small enough in absolute value. Therefore, δ)W∗+δej for small negative δ would yield a smaller value than W∗ in Eq. ≥0 for all j with dj=0 for all j with dj is a linear function of v1,…,vk, the conditions for V to belong to k unknown quantities v1,…,vk, we have α linear equations and J−α linear inequalities, with J denoting the number of donor units, and Fixing k−α between the number of economic predictors (k) and the number of donor units with positive W weight in the “training” period (α) is positive. The larger the difference k−α, the larger is typically the ambiguity induced by the cross-validation technique.The cross-validation method is typically not well-defined if the difference V weights. In particular, this entails that as soon as k−α.Finally, notice that


A dataset of measurements of ETSI ITS-G5 Dedicated Short Range Communications (DSRC) is presented. Our dataset consists of network interactions happening between two On-Board Units (OBUs) and four Road Side Units (RSUs). Each OBU was fitted onto a vehicle driven across the FLOURISH Test Track in Bristol, UK. Each RSU and OBU was equipped with two transceivers operating at different frequencies. During our experiments, each transceiver broadcasts Cooperative Awareness Messages (CAMs) over the licensed DSRC band, and over the unlicensed Industrial, Scientific, and Medical radio (ISM) bands 2.4 GHz-2.5 GHz and 5.725 GHz-5.875 GHz.Each transmitted and received CAM is logged along with its Received Signal Strength Indicator (RSSI) value and accurate positioning information. The Media Access Control layer (MAC) layer Packet Delivery Rates (PDRs) and RSSI values are also empirically calculated across the whole length of the track for any transceiver. The dataset can be used to derive realistic approximations of the PDR as a function of RSSI values under urban environments and for both the DSRC and ISM bands – thus, the dataset is suitable to calibrate (simplified) physical layers of full-stack vehicular simulators where the MAC layer PDR is a direct function of the RSSI. The dataset is not intended to be used for signal propagation modelling.The dataset can be found at https://doi.org/10.5523/bris.eupowp7h3jl525yxhm3521f57, and it has been analyzed in the following paper: I. Mavromatis, A. Tassi, and R. J. Piechocki, “Operating ITS-G5 DSRC over Unlicensed Bands: A City-Scale Performance Evaluation,” IEEE PIMRC 2019. [Online]. Available: https://arxiv.org/abs/1904.00464. When two vehicles were present, one was driven in a clockwise direction while the other in an anticlockwise fashion. Each vehicle was equipped with an OBU operating the same set of transceivers that we employed in each RSU. Further details about both the hardware and software pertaining to the RSUs and OBUs can be found in All the RSUs and OBUs communicate using ETSI's ITS-G5 standard Each transmitted and received CAM has been recorded both in the CSV and PCAP traces. When a packet appears in the transmission or reception queue of a transceiver, a decision is made whether it is going to be recorded or not. The flowchart describing that, can be seen in In particular, for what concerns an RSU, raw data recorded during a specific day of trial and pertaining to a specific transceiver, is recorded under the following filename and directory naming convention:./rawData/ddmmyyyy/RSUs-ddmmyyyy/log_XX_YY.txt and./rawData/ddmmyyyy/RSUs-ddmmyyyy/log_XX_TCP_YY.pcap where “ddmmyyyy” signifies the date when the experiment took place, “XX” can either be “HP” or “LP” depending on the transceiver under consideration, and “YY” refers to one of the four RSUs . Thus, for instance, files “./rawData/11022019/RSUs-11022019/log_HP_MVB.txt” and “./rawData/11022019/RSUs- 11022019/log_HP_TCP_MVB.txt” refer to the HP transceiver of the RSU located on the MVB site during the 11th February 2019. A similar conversion has been adopted for recording raw data associated with the OBUs, which is defined as follows:./rawData/ddmmyyyy/OBU-ZZ-ddmmyyyy-TT/log_XX.txt and./rawData/ddmmyyyy/OBU-ZZ-ddmmyyyy-TT/log_XX_tcpd.pcap where “ZZ” can be either equal to “00” or “01” depending of the fact the data refers to the first or the second OBU, respectively. Then, if data refers to an experimental session that took place in the morning then “TT” is equal to “Morning”, otherwise, “TT” is set equal to “Afternoon”. For instance, files “./rawData/11022019/OBU-00-11022019-Morning/log_HP.txt” and “./rawData/11022019/OBU-00-11022019-Morning/log_HP_tcpd.pcap” refer to the experimental session recorded in the morning of 11th February 2019 and referring to the HP transceiver of the first OBU.The raw data part of our dataset has been filtered by means of our MATLAB framework •Heatmaps for the PDR values associated with the HP/LP transceivers across the roads forming the FLOURISH Testing Track. A sample of the sixteen resulting figures is shown in •Awareness horizons for both HP and LP transceivers showing the PDR as a function of the distance to a given RSU. A sample of the sixteen resulting figures is shown in •RSSI value for both HP and LP transceivers as a function of the distance to a given RSU. A sample of the thirty-two resulting figures is shown in •Our framework can be used to generate the entire set of figures for all the above-mentioned KPIs. It can also be further extended to accommodate different or more specific KPIs per-taining to the network interactions at specific road junctions at specific time of the day, or the network interactions among moving vehicles. As for the latter, for convenience, our MATLAB framework Our MATLAB framework 1.Modifying the two main loops of the script, namely, the loops associated with the day and the transmission power. The user can then choose to evaluate the results for a specific day or TX power. Based on that, our framework will load only the network interactions related to the chosen day and TX power.2.Modifying the internal loops of our script, namely, the ones related to the OBUs and the RSUs. One or more specific pairs of OBU/RSU can be chosen to be processed.3.Similarly to the “v2vInteractions.m” script, the unique sequence number for the transmitted packets per session, as well as the MAC address of the transmitter can be parsed at first.4.Using the above information, it can be identified whether a packet is successfully delivered or not, i.e., checking whether a packet with the given MAC address and sequence number exists in the list of received packets.5.Having this pairwise correlation between transmitted and received packets, a table can be later generated by incorporating all the given interactions between a set of devices (as in “v2vInteractions.m”).6.The above steps can be replicated for the requested number of OBU-RSU pairs generating a list of all the desired network interactions.Our MATLAB framework also gives the reader the option of filtering the entire dataset to retrieve all the network interactions between the pair of OBUs considered in our experiments. In particular, when the script “v2vInteractions/v2vInteractions.m” is executed, it produces a table-formatted MAT file (for each day and transceiver) defined by the columns summarized in


N = 101) ASD traits (measured using the Autism-spectrum Quotient) were associated negatively and significantly with the strength of both explicit gender self-concept  and implicit gender self-concept (measured using an Implicit Association Task). Further analyses showed that a subgroup with high/clinically significant ASD traits showed significantly weaker explicit and implicit gender self-concepts than a subgroup with low ASD traits. Results were similar in both males and females, although there was some evidence of a selective influence of ASD traits on implicit gender self-concept among females only.A link between autism spectrum disorder (ASD) and gender identity difficulties has been suggested. In this study, we found that, among adults from the general population (The online version of this article (10.1007/s10803-019-04262-z) contains supplementary material, which is available to authorized users. Gender describes a constellation of traits, behaviours, and roles attributed to males and females within society .Despite their extensive use in research, these self-report measures rely on accurate self-awareness of one’s own traits  and are considered susceptible to the effects of self-presentation . ASD is a neurodevelopmental disorder diagnosed on the basis of significant behavioural difficulties with social-communication and a restricted, repetitive pattern of interests and behaviour  took part in the current online experiment. Their average age was 36.93  years. Ninety-four percent of participants reported English as their first language and all were cisgender. Thirteen of the 101 participants had a formal diagnosis of autism, according to self-report. Participants were recruited through the Amazon’s online crowdsourcing platform MTurk and received compensation for their time. Informed consent was obtained from all individual participants included in the study. Ethical approval for this study was obtained from Kent School Psychology Research Ethics Committee.To measure implicit gender self-concept, we employed the IAT described by Greenwald and Farnham . The tasWords appeared on the middle of a computer screen in a random order and category labels were always presented in the upper right and left corners of the screen.All stimuli were presented to participants before the beginning of the task, and they were instructed to study the items that belonged to each category for 25 s. In block 1, the “self” category label was presented in the upper left corner of the screen and the “other” category label in the upper right corner. Participants had to sort words that belonged either to the “self” or the “other” category, by pressing the “z” key of a keyboard for words related to “self” and the “m” key for words related to “other”. In block 2, categories referred to “feminine” and “masculine” attributes. The “feminine” category label was presented in the upper left corner and the “masculine” category in the upper right corner. Participants were instructed to press the “z” key for words belonged to “feminine” category and the “m” key for items belonged to “masculine” category.In block 3, the four categories were presented combined  and participants practiced the sorting task, by pressing the key that was assigned to each category in the preceding two blocks . In block 4, they completed the first experimental condition of the sorting task. In block 5, the “self” category label was presented in the upper right corner and the “other” category label in the upper left corner, subsequently the assignment of the key for each category was reversed compared to the first block. Participants had to press “m” for items belonged to the “self” category and “z” for items belonged to the “other” category.In block 6, they practiced the combined task using the switched key assignments. That is, participants were instructed to press the “z” key to categorise items that belonged either to the “other” or to the “feminine” category and the “m” key for words that belonged either to the “self” or “masculine” category. In the last block, they completed the second experimental condition of the combined sorting task. The order of the blocks was counterbalanced across participants. Figure D) in response latencies between the two practical blocks  and the two experimental blocks . The resulting D scores formulate a bipolar scale that ranges from masculinity to femininity. The faster the responses in the “self-masculine vs other-feminine” combined sorting condition, than in the “self-feminine vs other-masculine” condition, the stronger the implicit masculine gender self-concept. Whereas, the faster the responses in the “self-feminine vs other-masculine” combined sorting condition than in, the “self-masculine vs other-feminine” condition, the stronger the implicit feminine gender self-concept. However, in the current study, the key dependent variable on the IAT related to the strength of the implicit gender self-concept, regardless of whether the participant was male or female. Therefore, all D scores were transformed to positive values with higher scores denoting stronger implicit gender self-concept.The dependent measure for the IAT was the strength of the automatic associations calculated with the scoring algorithm recommended by Greenwald et al. . That isThe PAQ . Therefore, only the dimensional scales of PAQ and the transformed D scores of the IAT were included in the analyses described below. The bipolar scales of these measures were used only in an initial analysis to explore the association with untransformed D score, for comparison with the equivalent result reported by Greenwald and Farnham  in the critical blocks was above chance, t(100) = 62.12, p < .001, d = 6.18, BF10 = 6.99. Likewise, participants showed the expected effect on the IAT, producing a mean D-score  that was significantly above zero, t(100) = 12.13, p < .001, d = 1.21, BF10 = 2.03.The average AQ score across participants was 19.65 (SD = 7.31). The average score on PAQ Masculinity and PAQ Femininity scale was 2.43 (SD = 0.73) and 2.85 (SD = 0.72), respectively. Participants were accurate when sorting items on the IAT. The mean proportion of correct item categorisation  and r = − .35, p < .001, BF10 > 100 (one-tailed), respectively. These results indicate that the higher the number of ASD traits manifested, the lower the explicitly reported strength of gender self-concept (for both feminine and masculine traits). Next, we examined the relation between AQ score and implicit gender self-concept (indexed by D score). As predicted, AQ was negatively and significantly associated with D score, r = − .25, p = .006, BF10 = 5.45 (one-tailed), indicating a negative association between the number of self-reported ASD traits and the strength of the implicit gender self-concept (high AQ score = low D score).A series of correlation analyses was conducted to explore the relation between ASD traits, on the one hand, and the explicit and implicit gender self-concept, on the other hand. In line with our predictions, AQ was negatively and significantly associated with both PAQ Femininity and PAQ Masculinity scale scores, SD) scores on PAQ Femininity and PAQ Masculinity scale and on the IAT among participants who scored above and below the median of the sample on the AQ.Table t-tests was conducted to examine differences between the low-AQ group and the high-AQ group in the explicit and implicit gender self-concept. Groups were equated for age and sex. The average age was 38.61 (SD = 9.02) in the low-AQ group and 35.22 (SD = 10.93) in the high-AQ group, a difference that was non-significant, t(94.85) = 1.70, p = .093, d = 0.38, BF10 = 0.75. The low-AQ group included 23 birth-assigned-males and 28 birth-assigned-females, whereas the high-AQ group included 28 birth-assigned-males and 22 birth-assigned-females. The difference between the high and low AQ groups in ratio of birth-assigned-males to birth-assigned-females was non-significant, χ2(1) = 1.20, p = .273, Cramer’s V = .11. As predicted, relative to participants in the low-AQ group, those in the high-AQ group had significantly lower scores on both PAQ scales, as well as significantly lower D scores on the IAT .In a set of exploratory analyses, we checked whether there were any of the effects reported above differed significantly between birth-assigned males and birth-assigned females. Fisher’s D score on the gender IAT and total score on the AQ, Z = 2.49, p = .013. The association between AQ and D score was significant within birth-assigned-females, indicating that the higher the number of ASD traits, the lower the strength of the implicit gender self-concept, r = − .41, p = .003, BF10 = 11.30. However, this association was non-significant among birth-assigned-males, r = .08, p = .584, BF10 = 0.20.First, there was a significant difference between males and females in the size of the association between F = 11.02, p = .001, D score among birth-assigned-females in the low AQ subgroup  was significantly higher than among birth-assigned-females in the high AQ subgroup , t(46.73) = 3.44, p = .001, d = 0.96, BF10 = 17.73. This indicates that females with a high number of ASD traits had a weaker gender self-concept than females who had a low number of ASD traits. In contrast, the IAT D score among birth-assigned-males in the low AQ subgroup  was non-significantly different from the IAT D score among birth-assigned-males in the high AQ subgroup , t(49) = − 0.90, p = .379, d = 0.25, BF10 = 0.39.Second, in a 2  × 2 (AQ subgroup: High-AQ/low-AQ) ANOVA on the gender IAT data, there was a significant group × sex interaction, The first notable set of findings of the current study was about the link between ASD traits and the explicit gender self-concept. As predicted, AQ score was significantly negatively associated with both Femininity and Masculinity scale scores of PAQ, indicating that as the number of self-reported ASD traits increases the strength of the explicit gender self-concept decreases. The associations were moderate in magnitude and Bayesian analyses consistently suggested that the data supported the alternative hypothesis. This is the first study, to our knowledge, to show a link between explicit gender self-concept and number of ASD traits among members of the general population. We also addressed this issue by dividing our sample into a high AQ group and a low AQ group. In keeping with our predictions, the results complemented the findings from the correlation analyses. That is, relative to the group of participants with low ASD traits, the group with high ASD traits scored significantly lower on both scales of PAQ. These results indicate that people with high ASD traits explicitly/consciously identify themselves with less strong masculine and feminine attributes relative to individuals with low ASD traits; they have weaker gender self-concepts, on average. Effects were moderate and Bayesian analyses again suggested that the data supported the alternative hypotheses.feminine gender identity/self-concept. The studies by Bejerot and Eriksson (n = 9) and birth-assigned-females without ASD (n = 9) in Gender Feminine scale, it is possible that a Type 2 error was made. Also, the Bem Sex Role Inventory, used by Bejerot and Eriksson  was significantly negatively associated with number of self-reported ASD traits, indicating that as the number of reported ASD traits increased the strength of the implicit gender self-concept decreased. The size of the association was relatively modest , but it was highly statistically significant and Bayesian analysis suggested that the data supported the alternative hypothesis. Complementary to this, relative to the group of participants with low ASD traits, the group with high ASD traits scored significantly lower (D-score) on the IAT, indicating that among people with high ASD traits the automatic identification of self with either masculine or feminine attributes was weaker relative to individuals with low ASD traits. Again, group differences were moderate-to-large in statistical magnitude and associated with Bayes factors that strongly favoured the alternative hypothesis. These results suggest that high ASD traits in the general population might signify an implicit/unconscious gender self-concept that is neither strongly masculine nor strongly feminine, among individuals who otherwise report themselves to be cisgender.Arguably more notable, is the second set of findings from the current study, regarding the relation between number of ASD traits and implicit gender self-concept. To our knowledge, the current study is the first to investigate the potential link between ASD and gender self-concept at a level deeper than that tapped by self-report. In keeping with our predictions, we found that the strength of the implicit gender self-concept (indexed by r = − .26 in our study and r = .20 in their study). Moreover, these associations are of an order of magnitude suggested by Hofmann et al.  and birth-assigned females (n = 50). That is, in the association analyses, the strength of explicit gender self-concept was negatively associated with number of ASD traits in both males and females. Likewise, in the case–control analyses, those with high ASD traits had a weaker gender self-concept than those with low ASD traits, regardless of whether they were male or female. However, results appeared to be not so straightforward when considering the effects of ASD traits on implicit gender self-concept. Here, in the association analyses, number of ASD traits was negatively associated with strength of implicit gender self-concept in birth-assigned females only. Likewise, in the case–control analyses, only females with high ASD traits had a weaker gender self-concept than females with low ASD traits . What should be made of these apparent sex differences in the relation between ASD and implicit gender self-concept? On the one hand, the existence of gender differences might be treated with some scepticism, given that the analyses were entirely post hoc and between-sex differences were not predicted. On the other hand, sex differences were reported by Nobili et al.  observed in the whole sample of 101 participants held in i et al.  in theiri et al.  found thOverall, the results from the explicit and implicit measures of gender self-concept imply that people with high ASD traits  have a weaker propensity to identify with, and incorporate into their self-concept gender differentiated traits. As levels of ASD increase, so too may be susceptible to gender identity difficulties. If this interpretation is correct, then this could explain gender non-conforming feelings and gender identity difficulties in ASD  that ASD is characterised by a diminished general tendency to identify with, or represent, the perspectives and attitudes of others  to incorporate these into their self-concept. Given that such a difficulty could provide an explanation for the high co-occurrence of gender identity difficulties in ASD, the current study makes an important contribution to our still limited understanding of this phenomenon.Supplementary material 1 (DOCX 18 kb)Below is the link to the electronic supplementary material.


In this paper, we propose computationally efficient parametric and semiparametric tests based on a set of specialized matrix techniques dubbed as the PB-transformation. The PB-transformation has two advantages: 1. The PB-transformed data will have a scalar variance-covariance matrix. 2. The original H-T problem will be reduced to an equivalent one-sample H-T problem. The transformed problem can then be approached by either the one-sample Student’s t-test or Wilcoxon signed rank test.For many practical hypothesis testing (H-T) applications, the data are correlated and/or with heterogeneous variance structure. The regression t-test produces notably better results than the weighted LMER test, especially in the high correlation case, using only a small fraction of computational cost (3 versus 933 s). We apply these two methods to a set of RNA-seq gene expression data collected in a breast cancer study. Pathway analyses show that the PB-transformed t-test reveals more biologically relevant findings in relation to breast cancer than the weighted LMER test.In simulation studies, the proposed methods outperform commonly used alternative methods under both normal and double exponential distributions. In particular, the PB-transformed https://github.com/yunzhang813/PBtest-R-Package.As fast and numerically stable replacements for the weighted LMER test, the PB-transformed tests are especially suitable for “messy” high-throughput data that include both independent and matched/repeated samples. By using our method, the practitioners no longer have to choose between using partial data (applying paired tests to only the matched samples) or ignoring the correlation in the data (applying two sample tests to data with some correlated samples). Our method is implemented as an R package ‘PBtest’ and is available at The online version of this article (10.1186/s12859-019-2783-8) contains supplementary material, which is available to authorized users. As proof, R packages genefilter [t-tests, respectively, both of which are hundreds times faster than the stock R function t.test. However, it is common to observe heterogeneous variabilities between high-throughput samples, which violates the assumption of the Student’s t-test. For example, samples processed by a skillful technician usually have less variability than those processed by an inexperienced person. For two-group comparisons, a special case of the heterogeneity of variance, i.e., samples in different groups have different variances, is well studied and commonly referred to as the Behrens-Fisher problem. The best known (approximate) parametric solution for this problem is the Welch’s t-test, which adjusts the degrees of freedom  associated with the t-distribution to compensate for the heteroscedasticity in the data. Unfortunately, the Welch’s t-test is not appropriate when the data have even more complicated variance structure. As an example, it is well known that the quality and variation of the RNA-seq sample is largely affected by the total number of reads in the sequencing specimen [sequencing depth or library size, which may vary widely from sample to sample. Fortunately, such information is available a priori to data analyses. Several weighted methods [Modern statistical applications are typically characterized by three major challenges: (a) high-dimensionality; (b) heterogeneous variability of the data; and (c) correlation among observations. For example, numerous data sets are routinely produced by high-throughput technologies, such as microarray and next-generation sequencing, and it has become a common practice to investigate tens of thousands of hypotheses simultaneously for those data. When the classical nefilter  and Rfasnefilter  implemenspecimen , 4. This methods –7 are prbias, the heterogeneity of variance remains to be an issue. Besides, the clustering nature of these data may induce correlation among observations within one center/batch. Correlation may arise due to other reasons such as paired samples. For example, we downloaded a set of data for a comprehensive breast cancer study [t-test and paired t-test are not ideal for comparing the gene expression patterns between normal and cancerous samples, because they either ignore the correlations of the paired subjects or waste information contained in the unpaired subjects. To ignore the correlation and use a two-sample test imprudently is harmful because it may increase the type I error rate extensively [As the technology advances and the unit cost drops, immense amount of data are produced with even more complex variance-covariance structures. In multi-site studies for big data consortium projects, investigators sometimes need to integrate omics-data from different platforms (e.g. microarray or RNA-seq for gene expression) and/or processed in different batches. Although many normalization –10 and ber study , which censively . On the p-values for the fitted models. The R package lmerTest [t- and F-tests for lme4 outputs with appropriate DFs. A fast implementation of LMER is available in the Rfast package, which is based on highly optimized code in C++ [One of the most versatile tools in statistics, the linear mixed-effects regression (LMER), provides an alternative inferential framework that accounts both unequal variances and certain practical correlation structures. The standard LMER can model the correlation by means of random effects. By adding weights to the model, the weighted LMER is able to capture very complex covariance structures in real applications. Although LMER has many nice theoretical properties, fitting it is computationally intensive. Currently, the best implementation is the R package lme4 , which ilmerTest  is the ce in C++ ; howevert-tests, have their corresponding rank-based counterparts, i.e. the Wilcoxon rank-sum test and the Wilcoxon signed rank test. A rank-based solution to the Behrens-Fisher problem can be derived based on the adaptive rank approach [t-distribution is more appropriate than the normal approximation for the Wilcoxon test when the design has low DFs. Their method is implemented in the svyranktest function in R package survey. But most of the rank-based tests are designed for group comparisons; rank-based approaches for testing associations between two continuous variables with complex covariance structure are underdeveloped.Many classical parametric tests, such as two-sample and paired approach , but it approach  derived approach  derived approach  proved tone-group testing problem. After the PB-transformation, classical parametric and rank-based tests with adjusted DFs are directly applicable. We also provide a moment estimator for the correlation coefficient for repeated measurements, which can be used to obtain an estimated covariance structure if it is not provided a priori. In “t-test to an RNA-seq data for breast cancer. Utilizing the information of the paired samples and sequencing depths, our method selects more cancer-specific genes and fewer falsely significant genes (i.e. genes specific to other diseases) than the major competing method based on weighted LMER.Based on a linear regression model, we propose two H-T procedures (one parametric and one semiparametric) that utilize a priori information of the variance (weights) and correlation structure of the data. In “https://github.com/yunzhang813/PBtest-R-Package.Lastly, computational efficiency is an important assessment of modern statistical methods. Depending on the number of hypotheses to be tested, our method can perform about 200 to 300 times faster than the weighted LMER approach in simulation studies and real data analyses. This efficiency makes our methods especially suitable for fast feature selection in high-throughput data analysis. We implement our methods in an R package called ’PBtest’, which is available at For clarity, we first present our main methodology development for a univariate regression problem. We will extend it to multiple regression problems in “Consider the following regression-type H-T problem: y is the response variable, x is the covariate, and ε is the error term that follows an n-dimensional multivariate normal distribution Σ. By considering a random variable Y in the n-dimensional space, the above problem can also be stated as Here, μ is the intercept or grand mean that is a nuisance parameter, and β is the parameter of interest that quantifies the effect size. We express the variance-covariance matrix of ε in the form In this model, σ2 is a nonzero scalar that quantifies the magnitude of the covariance structure, and S is a symmetric, positive-definite matrix that captures the shape of the covariance structure. Additional constraints are needed to determine σ2 and S; here, we choose a special form that can subsequently simplify our mathematical derivations. For any given Σ, define where From the above definition, we have the following nice property S the standardized structure matrix satisfying Eq. Hereinafter, we refer to S is proportional to I, the identity matrix, it is well-known that regression t-test is a valid solution to this H-T problem. If S≠I, e.g. the observed data are correlated and/or have heterogeneous variance structure, the assumptions of the standard t-test are violated. In this paper, we propose a linear transformation, namely As a special case of Model , if S isweighted mean of the original data), and subtract n−1)-dimensional subspace of Y(1) (i.e. the centered data). It’s clear that H0 is true.Estimate Y(1) to reshape its “elliptical” distribution to a “spherical” distribution. The intermediate data from this step is Y(2).Use the eigen-decomposition of the covariance matrix of unique rotation that transforms the original H-T problem to an equivalent problem of testing for a constant deviation along the unit vector. The equivalent data generated from this step is Use the QR-decomposition technique to find a To shed more lights on the proposed method, we first provide a graphical illustration in Fig. In the proposed PB-transformation, B-map performs both transformations in Step 1 and 2; P-map from Step 3 is designed to improve the power of the proposed semiparametric test to be described in “J=1·1′ . With some mathematical derivations  is a semi-orthogonal matrix containing the first n−1 eigenvectors and Λn−1∈Mn−1)×(n−1)×2×2×(n−1)(−Q(I2×2−Rot)Q′ is the unique orthogonal transformation that satisfies the following properties: Matrix See Additional file P defined by Theorem 1 the “P-map”. Equation z is mapped to 1n−1 scaled by a factor ζ. Equation 1n−1 and z, i.e. Lz=span. This property defines a uniqueminimum map that only transforms the components of data in Lz and leaves the components in We call the linear transformation  used in .B and P, we define the final transformed data as With both i.i.d. normal distribution, for i=1,⋯,n−1. The location parameter of the common marginal distribution is to be tested with unknown σ2. Therefore, we can approach this equivalent H-T problem with the classical one-sample t-test and Wilcoxon signed rank test , the above estimator coincides with the maximum likelihood estimator of ρ, which has the form Moreover, if See Additional file Standard correlation estimates are known to have downward bias , which ct-test and the paired t-test. Suppose there are n observations in each group, the former test has DF=2n−2 for i.i.d. observations, and the latter only has DF=n−1 because the observations are perfectly paired. These trivial examples indicate that we need to adjust the DF according to the correlation structure in our testing procedures.The degree of freedom (DF) can have nontrivial impact on hypothesis testing when sample size is relatively small. Intuitively, a correlated observation carries “less information” than that of an independent observation. In such case, the effective DF is smaller than the apparent sample size. Simple examples include the two-sample We adopt the degrees of freedom approximation proposed by  , g(−y)=g(y). Assume that Y∗ be a random number that is independent of σ2. For every symmetric semi-definite u,v, such that Let n-dimensional random vector with Y, the result is a sequence of (n−1) equal variance and uncorrelated random variables with zero mean if and only if β=0.is an See Additional file i.i.d. sequence of random variables with a symmetric common p.d.f., we can use linear transformations to generate a family of distributions that is expressive enough to include a non-normal distribution with an arbitrary covariance matrix and a mean vector specified by the effect to be tested. This distribution family is semiparametric because: a) the “shape” of the density function, g(y), has infinite degrees of freedom; b) the “transformation”  has only finite parameters.The essence of this Proposition is that, starting with an t-distribution with an adjusted DF. Note that Wilcoxon signed rank test is only valid when the underlying distribution is symmetric; therefore, the symmetry assumption in Proposition 2 is necessary. In summary, this PB-transformed Wilcoxon test provides an approximate test (up to the second order moment) for data that follow a flexible semiparametric distributional model.As mentioned before, applying both the B- and P-maps enables us to use the Wilcoxon signed rank test for the hypotheses with this semiparametric distribution family. This approach has better power than the test with only the B-map as shown in “In this section, we present an extension of the proposed methods for the following multiple regression ε is assumed to have zero mean but does not need to have scalar covariance matrix. For example, ε can be the summation of random effects and measurement errors in a typical LMER model with a form specified in Eq. Here the error term βk,k=1,…,p, we need to specify two regression models, the null and alternative models. Here the alternative model is just the full Model  from XTo test the significance of ll Model , and thenuisance covariates in the multiple regression case are Xk−βk− instead of 1μ in Eq. Xk−E)=σ2(S−Xk−X−k′). The P-transformation is derived the same as before, but with the new B matrix.Compared with the original univariate problem, we see that the ρ=0.2, ρ=0.4,ρ=0.6, and ρ=0.8). We compare the parametric and rank-based PB-transformed test with oracle and estimated correlation to an incomplete survey of alternative methods. Each scenario was repeated 20 times and the results of ρ=0.2 and 0.8 for normal and double exponential distributions are summarized in Figs. ρ=0.4 and 0.6, and results for logistic distribution.We design two simulation scenarios for this study: SIM1 for completely paired group comparison, and SIM2 for regression-type test with a continuous covariate. For both scenarios we consider three underlying distributions  and four correlation levels . Of note, lmer may fail to converge occasionally, e.g. 0 – 25 failures  in each repetition of our simulations. We resort to a try/catch structure in the R script to prevent these convergence issues from terminating the main loop.We record the system time for testing 2000 simulated hypotheses using our method and lmer, since they are the most appropriate methods for the simulated data with the best statistical performance. Our method takes less than 0.3 s with given n=40 with nA=nB=20. It is straightforward to calculate the DFs used in the two-sample t-test and the paired t-test, which are 38 and 19, respectively. Using lmerTest (weighted LMER) with default parameters, it returns the mean DF = 35.51 with a large range  from the simulated data with ρ=0.2. Using the oracle ΣSIM, our method returns the adjusted DF = 14.35; if the covariance matrix is estimated, our method returns the mean DF = 14.38 with high consistency . When ρ=0.8, the adjusted DFs become smaller. The weighted LMER returns the mean DF = 20.63 . Our method returns DF = 12.48 for the oracle covariance, and mean DF = 12.56  for the estimated covariance. Also, the rank-based test svyranktest returns a DF for its t-distribution approximation, which is 18 for both small and large correlations.We also check the degrees of freedom in all applicable tests. In this section, we report the DFs used/adjusted in SIM1, i.e. the completely paired group comparison. Recall that We download a set of RNA-seq gene expression data from The Cancer Genome Atlas (TCGA)   level combined with a fold-change (FC) criterion (FC<0.5 or FC>2), the PB-transformed t-test selected 3,340 DEGs and the weighted LMER selected 3,485 DEGs  for the weighted LMER. Expression level of these genes are plotted in Fig. To make the comparison between these two methods more fair and meaningful, we focus on studying the biological annotations of the top 2,000 genes from each DEG list. Specifically, we apply the gene set analysis tool DAVID  to the 1t-test are known to be involved in cell survival, proliferation and migration. The CXCR4-CXCL12 chemokine signaling pathway is one of the deregulated signaling pathway uniquely identified by PB-transformed t-test in HER2+ breast cancer cells. This pathway is known to play a crucial role in promoting breast cancer metastasis and has been reported to be associated with poor prognosis [t-test identifies more genes whose protein products can be targeted by pharmaceutical inhibitors. CXCR4 inhibitors have already demonstrated promising anti-tumor activities against breast [Those 11 genes uniquely identified by the PB-transformed rognosis , 34. Comt breast , 36, prot breast  and lungt breast  cancers.t-type test and rank-based test. In the simulation studies, our proposed tests outperform the classical tests (e.g. two-sample/regreesion t-test and Wilcoxon rank-sum test) by a large margin. In a sense, this superiority is expected, because the classical methods do not consider the correlation nor heteroscedasticity of the data.In this paper, we present a data transformation technique that can be used in conjunction with both the Student’s t-test and the weighted LMER. The fact that the PB-transformed t-test outperforms the weighted LMER, and this advantage is more pronounced for data with higher correlation , hypothesis testing, among other things; whereas our methods are only designed for the hypothesis testing. We envision that in a typical high-throughput data application, an investigator may quickly run PB-transformed t-test for individual covariates, not the ANOVA F-test for the significance of several covariates simultaneously. In fact, the B-map can be defined in this case so we can define a transformed parametric test easily; but there is no clear counterpart for the P-map, which is needed to overcome the identifiability issue for the semiparametric generalization. 2. The performance of PB-transformations depends on a good estimation of S, the shape of the covariance matrix of the observations. Currently, our moment-based estimator only works for problems with just one random intercept, which is only appropriate for relatively simple longitudinal experiments. It is a challenging problem to estimate the complex covariance structure for general LMER models , and we think it can be a nice and ambitious research project for us in the near future.In “t-test provides the same test statistic and degrees of freedom as those from the paired t-test for perfectly paired data and the regression t-test for i.i.d. data. In this sense, the PB-transformed t-test is a legitimate generalization of these two classical tests. The rank-based test is slightly different from the classical ones, since we used a t-distribution approximation instead of a normal approximation for the rank-based statistic. The t-distribution approximation is preferred for correlated data because the effective sample size may be small even in a large dataset [Numerically, the PB-transformed  dataset .Recall that the PB-transformation is designed in a way that the transformed data have the desired first and second order moments. For non-normal distributions, the transformed samples may not have the same higher order moments. Note that, the P-map is currently defined in part by Eq. , the minIn this study, we presented an example in RNA-seq data analysis. In recent bioinformatics research, advanced methods such as normalization and batch-effect correction were developed to deal with data heterogeneities in bio-assays. While most of these approaches are focused on the first moment , our approach provides a different perspective based on the second order moments (i.e. the covariance structure). The dramatic computational efficiency boost of our method also opens the door for investigators to use the PB-transformed tests for ultra-high-dimensional data analysis, such as longitudinal studies of diffusion tensor imaging data at the voxel-level –41, in wAdditional file 1This file contains: a) proofs of the main theorems; b) details of the moment-based correlation estimator; c) details of simulation design; and d) additional information about the real data analysis. (PDF 3523 kb)


Nature Communications 10.1038/s41467-019-12017-8, published online 18 September 2019.Correction to: +GrzB+ cells in PbAluc-infected mice was incorrectly given as ‘0’ and should have been ‘69.6’. This error is now corrected in both the PDF and HTML versions of the Article.The original version of this Article contained an error in one of the dot plots in Fig. 2d. The percentage of Pb1


G. We investigate the question “When is G and internally via structural properties of the group G.Let  G has been the topic of several previous investigations  is a nonzero cyclic subgroup of p of x, G]. Therefore, for Let Define We actually have a little more.G is a nonabelian p-group, If From the proof of Theorem G be a p-group, G is 2-generated (generated by 2 elements) then Let If We remark that Example G is abelian by cyclic, or G is said to be an extension of an abelian group by a cyclic group if there exists an abelian normal subgroup A of G such that G / A is cyclic. For finite G one may always take A to be a maximal abelian normal subgroup.Recall that a group G be a nonabelian p-group, Let A be a maximal abelian normal subgroup such that G is nonabelian there exists G is of class 2 so We let Using this decomposition we define G be a nonabelian p-group, Let Let As we did following Theorem G be a p-group, nilpotent of class 2, which is abelian by cyclic. Then Let A be a maximal abelian normal subgroup with G. Let As above we let c is a sum of commutators. Since A is a maximal abelian normal subgroup we have G is nilpotent of class 2, p, a contradiction. Thus Further Now let G has a partition In Theorems p-groups, G be a group of order G is abelian so We apply the above results to G be a nonabelian p-group, G is nilpotent of class 2. Then Let (i)g is nonabelian we have (ii)G. We know G is generated by k elements [G is generated by 2 elements. Using Theorem G has a unique maximal normal subgroup which cuts G. The result now follows from Theorem elements . If \doc(iii)G is of class 2. For G is of class 2, G and G is abelian by cyclic so the result follows from Theorem Z(G) cuts G is a finite abelian p-group then G is 1-affine complete. For nonabelian p-groups, p-group for which G is 1-affine complete. We thus close with the following.In conclusion we have found that when p-groups G is 1-affine complete.For finite nonabelian


Correction to: npj Digital Medicine 10.1038/s41746-017-0007-z, Published online: 12 February 2018The original version of the published Article contained an error in the spelling of the fifth Author’s name. “Carly Winakor” has been changed to “Carly Winokur”. This has been corrected in the HTML and PDF version of the Article.


Correction to:npj Digital Medicines 10.1038/s41746-019-0189-7, published online 18 November 2019The original version of the published Article omitted of the last Author’s middle initial. “Matthew Lungren” has been changed to “Matthew P. Lungren”. This has been corrected in the HTML and PDF version of the Article.


Scientific Reports 10.1038/s41598-018-20106-9, published online 26 January 2018Correction to: The original version of this Article contained an error in the title of the paper, where the word “diverse” was incorrectly given as “divers”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supplementary Information file.


Nature Communications 10.1038/s41467-019-10303-z, published online 14 June 2019.Correction to: 2’ was inadvertently added. This has been corrected in both the PDF and HTML versions of the Article.The original version of this Article contained an error in Fig. 4a, in which an extra label ‘HThe correct version is:It replaces the incorrect version:


In the Materials and Methods, there is an error in the equation for the calculation of an asynchronous gait. Please view the complete, correct equation here: Dg-a = Dm-p+3/4SL


Scientific Reports 10.1038/s41598-018-37997-3, published online 12 February 2019Correction to: The original version of this Article contained an error in the title of the paper, where the phrase “cells overexpressing” was incorrectly given as “cellsoverexpressing”. This has now been corrected in the PDF and HTML versions of the Article.


Nature Communications, 10.1038/s41467-018-06915-6; published online 31 October 2018Correction to: In the original version of this Article, the author name ‘Shengzhong Liu’ was incorrectly given as ‘Liu Shengzhong’. This has been corrected in both the PDF and HTML versions of the Article.


Scientific Reports 10.1038/s41598-018-32296-3, published online 20 September 2018Correction to: The original version of this Article contained an error in the title of the paper, where the word “Reservoir” was incorrectly given as “Reservior”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supporting Information file.


With this approach, a limit of 10−7 mol L−1 was clearly detected. To improve the detection to lower p-MBA concentration, as 10−8 mol L−1, the SDME technique was applied. The p-MBA Raman signature was detected in two performed extractions and its new concentration was determined to be ~4.6 × 10−5 mol L−1. This work showed that coupling SDME with SERS allowed a rapid (5 min) and efficient pre-concentration (from 10−8 mol L−1 to 10−5 mol L−1), detection, and quantification of the analyte of interest, proving to be an interesting analytical tool for SERS applications.Single-drop microextraction (SDME) was coupled with surface-enhanced Raman scattering (SERS) to provide sample extraction and pre-concentration for detection of analyte at low concentrations. A gold nanohole array substrate (AuNHAS), fabricated by interference lithography, was used as SERS substrate and para-mercaptobenzoic acid (p-MBA) was tested as a probe molecule, in the concentration range 10 Theolecules .−6 mol L−1, many nanostructured materials may not be sensitive enough for an efficient detection. To overcome this, a pre-concentration technique, such as the single-drop microextraction (SDME), can be introduced as an analytical strategy to improve low concentration detection.Moreover, several nanofabrication strategies have been explored for the fabrication of SERS substrates with enhanced electric fields (hot spots), and three main classes of SERS substrates have been developed: Metallic rough surfaces, colloidal nanoparticles, and periodic nanostructures . A reproSDME is a liquid-phase microextraction technique, where a micro drop of an organic solvent is used to transfer a dissolved substance from one liquid phase to another liquid phase  in contact with it . SDME haIn this context, we proposed the application of a single-drop microextraction technique as a pretreatment step before SERS analysis of low concentration analytes. For this purpose, p-MBA was used as a Raman probe molecule and gold nanohole array was used as SERS substrate. The off-line coupling of SDME with SERS was applied only when direct p-MBA detection was no longer possible.2 each, the photoresist was developed for 50 s with the Microposit AZ 351 developer diluted 1:3 in water. The photoresist template was coated with an 80-nm gold film by e-beam evaporation, and then the photoresist was removed using acetone. The AuNHAS were characterized by FEG  with 20 kV of acceleration voltage, and energy-dispersive X-ray spectroscopy (EDS) for elemental analysis; the sample presented a homogeneous gold surface and periodic holes over an area of 2 cm2, as can be seen in 2). The gold nanohole array substrates (AuNHAS), with period ʌ = 500 nm, were fabricated using a combination of two-beam interference lithography (IL) and a lift-off step. Details of the experimental process can be found in Menezes et al. . Briefly2-cooled charge-coupled device (CCD) detector. The excitation source was a He-Ne laser at 633 nm. The laser power at the sample surface was about 2.5 mW. The laser was focused with a 100x focal-lens objective, to obtain a diameter of 1 µm at the surface, for an exposure time of 10 s. All the measurements were done in triplicate. After that, aliquots of 25 μL from each solution of p-MBA were dropped onto the surface and dried. The surface Raman mapping data was carried out using the p-MBA solution of 10−6 mol L−1, collecting a total of 169 SERS spectra.SERS spectra were acquired using a confocal Jobin-Yvon T64000 Raman spectrometer system, equipped with a liquid-N−8 mol L−1. A toluene drop of 2 μL was generated inside the p-MBA solution, which was kept at 22 °C and stirred at 100 rpm for 5 min. After this time, the drop was collected back and dropped onto the AuNHAS. The procedure was repeated one more time, dropping the extracted solution (p-MBA in toluene) on a different area of the same AuNHAS. After solvent evaporation, the SERS spectra were collected.The experimental setup shown in −1 can be related to the δ(CSH) bending mode, and the band at 1176 cm−1 was associated with the δ(CH) deformation vibration. Moreover, the intensity of the band at 1074 cm−1, assigned to the p-MBA aromatic ring breathing [−1 was selected as an indicator for the SERS data. Moreover, in −7 mol L−1 was considered the lowest concentration with a distinctly detected SERS signal on the fabricated AuNHAS.Thiophenol-based molecules, such as p-MBA, are a class of compounds commonly used as Raman probe for SERS studies due to the large Raman signal of the benzene ring, offering great sensitivity and specificity . Figure −1 for each p-MBA concentration were analyzed , leading to a coefficient of determination (R-squared) of 0.956. This result suggests that the SERS intensities are proportional to the molecular amount of p-MBA on AuNHAS. A similar result was reported by Saleh et al. [−1 band from the 169 SERS spectra of the Raman mapping is plotted as a histogram: The black line in h et al. ; therefop-MBA at 10−8 mol L−1 resulted in a very weak signal  spectra.In terms of sensitivity, the SERS detection of k signal , which c studies . As can −4 mol L−1 and 10−5 mol L−1. Moreover, the value of RSD 20%, obtained with the combination of SDME and SERS, perfectly matches the performance of the substrate; thus, the new p-MBA concentration of ~4.6 × 10−5 mol L−1 can be extrapolated from the calibration curve  during the SDME procedure. This value is comparable with the average SERS intensity of the p-MBA solution between the concentrations 10on curve a. This rmolecule ,22. Alth−4 mol L−1 up to the concentration of 10−7 mol L−1. To improve the analytical applicability of SERS, and lower the detection limit of this analyte, the SDME technique was successfully applied for the pre-concentration of the 10−8 mol L−1 p-MBA solution. In only five minutes, the SDME method allowed us to obtain a solution of analyte almost a thousand times more concentrated, from 10−8 to ~4.6 × 10−5 mol L−1. Moreover, the tailored choice of extraction solvent, time, and temperature can largely improve the detection of specific analytes in dense or rich matrices, greatly expanding the versatility and applicability of SDME-SERS analysis. Ultimately, for the first time, this work demonstrated that the coupling of the SDME technique with SERS analysis is a very promising method for a rapid and easy-to-apply quantitative detection of molecules at very low concentrations.AuNHAS with a periodicity of 500 nm was fabricated by interference lithography and its SERS performance was investigated using p-MBA as Raman probe molecule. Different concentrations of p-MBA were easily detected by SERS with a good reproducibility (RSD 21%), from 10