While for s polarization, the G-H shift is a constant value for any incident angle.Light reflection and refraction at an interface between two homogeneous media is analytically described by Snell's law. For a beam with a finite waist, it turns out that the reflected wave experiences a lateral displacement from its position predicted by geometric optics. Such Goos-Hänchen (G-H) effect has been extensively investigated among all kinds of optical media, such as dielectrics, metals, photonic crystals and metamaterials. As a fundamental physics phenomenon, the G-H effect has been extended to acoustics and quantum mechanics. Here we report the unusual G-H effect in zero index metamaterials. We show that when linearly polarized light is obliquely incident from air to epsilon-near-zero metamaterials, no G-H effect could be observed for  Such a lateral displacement is called the Goos-Hänchen (G-H) shift2357841416e.g, the plasmonic materials at optical frequencyi.e. the s polarized light with its electric-field vector perpendicular to the incident plane, and the p polarized light with its electric-field vector parallel to the incident plane. It is shown that there is no G-H effect for p polarized light, while for s polarization, the G-H shift is a constant value, independent of the incident angle. Such a unique behavior has not yet been revealed in previous study on the G-H effect.Recently, zero index metamaterials (ZIMs) have drawn much attention for their intriguing electromagnetic (EM) properties181920212223242526x-y plane separates two semi-infinite media. Region 1 is air, and region 2 is ENZ metamaterial with parameters μ2 = 1 and ε2 ≈ 0 . A linearly polarized light with incident angle inθis incident from air onto ENZ metamaterial. As known, the ENZ metamaterial can be regarded as optically thinner medium compared to air, and the critical angle for total reflection to occur is given by s and p polarizations, will be totally reflected by the ENZ metamaterial for any incident angles excluding the normal one.Our configuration is shown the schematically in s polarization. Suppose the incident linearly polarized light with an angular frequency ω is given as xk = k0 sin inθ, zk = k0 cos inθ, and k0 = ω/c is the wave number in air, c is the velocity of light in air. For simplicity, the time dependence exp(−iωt) will be omitted throughout this work. After reflection by ENZ metamaterial, the reflected light is z = 0, the reflected coefficient is calculated as ε2 is close to zero, i.e.ε2 ≈ 0, the G-H shift is sd = λ/π, a constant value for any incident angle. Meanwhile, we can work out that the phase of reflected light is sϕ = −2inθ from For G-H effect, however, both polarizations have different responses during such a total reflection process. First, let us consider the p polarized light. In this case, electric-field in y direction in the above equations should be changed into magnetic field. In region 1 of air, the incident and reflected light can be written as p polarized light is given as ε2 ≈ 0, the G-H shift is pd = 0 for any incident angle. From π, which means there is a half-wave phase lag for p polarized light after total reflection. Such a feature is analogous to the propagation of the s polarized light reflected by the perfect electric metal. With that in mind, the ENZ metamaterial could be regarded as a perfect magnetic conductor (PMC) for p polarized light. From the perspective of intrinsic property of ENZ metamaterial, we know that as the permittivity ε2 tends to zero, the right term H2 = Const. Then, following the continuous conditions of the magnetic fields at the boundary z = 0, we get (rH + H)exp(xxik) = Const. For the p polarized light obliquely incident onto the ENZ material, we have xk ≠ 0. To keep the above boundary condition valid for arbitrary coordinate x, the only satisfaction is that rH + H = 0. It also means that the magnetic field inside ENZ metamaterial is exactly zero. Consequently, the reflected coefficient is p = Hrρ/H = −1 and its reflected phase is −π. Such understanding from the intrinsic property itself is more straightforward, and gives the same results on G-H effect for p polarized light.A similar procedure can be implemented for e.g. the ENZ metamaterial). The time averaged energy flow can be calculated by applying s polarized incident light, we have l = 1/κ. It implies that the incident EM energy will enter into ENZ metamaterial, and the energy will go back into air when it arrives at z = l. Moreover, the G-H shift from the view of geometric optics, is d ≈ 2l sin inθ = λ/π when ε2 ≈ 0. For p polarized light, however, the corresponding averaged energy flow is The G-H effect can also be interpreted from the point of view of an energy flow along the interface in the optically thinner mediums and p polarized light. For an absorptive medium, the m = s and p)inθ. Combining ε2] ≠ 0, where Re[ε2] and Im[ε2] represent, respectively, the real part and imaginary part of the permittivity of loss ENZ metamaterial. As usual, the imaginary part indicates the loss.We considered a lossless ENZ metamaterial in above theoretical analysis. However, for a real ENZ metamaterial, such as the noble metal at plasma frequency or doped semiconductors at terahertz28ε2] = 10−4, and the solid and dashed lines are corresponding to s and p polarizations, respectively. As shown by the blue lines, the G-H shifts for both polarized light of s and p are sd = λ/π and pd = 0, respectively, for any incident angle except the vicinity of the critical angle. It is noted that when the incident angles vary across the critical one, the reflection phase will experience an abrupt change, usually leading to quite an obvious G-H effecte.g. Im[ε2] = 0.01, for s polarized light, as shown by the black solid line, there is a region where the G-H shift gradually changes from zero to λ/π. Such a property stems from the fact that the added losses make the reflection phases deviate from the linear relationship of sϕ = −2inθ. In particular, the deviation is quite notable at small incident angle. For p polarized light, however, the situation is more complicated. Generally, for an absorptive medium, there is an abrupt change of phase across the (pseudo) Brewster angle when a p polarized light is reflected by it. As a result, such an abruptly changed phase induces a negative G-H shift around the (pseudo) Brewster angleε2] = 0.01, as shown by the black dashed line, a resonance for G-H shift could be observed. Further, we use a larger loss with Im[ε2] = 0.1, as shown by the red lines. The gradually changing region of G-H shift for s polarized light is enlarged (see the red solid line), while for p polarized light, the resonance position is shifted to a larger angle (see the red dashed line). Nevertheless, for an incident angle large enough, the G-H shifts for the loss cases will approach those of the situation without loss, i.e., for s polarized light, it is almost λ/π, while for p polarized light, it is close to zero. The larger the loss is, the narrower the range of the working incident angle is. In particular, the working incident angle range for Im[ε2] = 0.01 is at about  = 0.1, it changes into  to the G-H shift, ε2] increases to 10−2. For the lossless case, the critical angle becomes larger, then the working angle for both polarizations is accordingly shifted to a larger angle, as shown by the blue lines. When the same levels of losses are involved, the black and red lines are corresponding to those of Im[ε2] = 0.01 and 0.1, respectively. Similar results can be observed including the gradually changed regions of G-H shifts for s polarized light and the resonances of G-H shifts for p polarized light. Although the real part of the permittivity is larger, the working incident angle ranges do not vary that much if we compare ε2] increases from 10−4 to 10−2. However, the amplitudes of resonances change a little bit. Generally, for an absorptive dielectric, the amplitudes of resonances decrease as the ratio of Im[ε2]/Re[ε2] increasesIn order to examine the sensitivity of the small values of Re[λ. Because of the small G-H shift, it is easier to observe the lateral shift L = sd/cos(inθ). s polarization, the lateral shift L is about 0.5λ, as shown in p polarization, the shift is zero, as shown in λ for s-polarization. Considering the fact that the incident beam is not a plane wave, such a deviation is acceptable. Moreover, p polarization, the symmetry axis of the field distribution (the red line) is located at the original point x = 0, as shown by the dotted line. While for s polarization its symmetric center is shifted to the left. Such a contrast further verifies our theoretical prediction that the G-H shift exists only for s polarization. Lastly, we compare the numerical results L for different incident angles with the analyzed results, as shown in L = sd/cos(inθ), while the five blue points are the numerical results for incident angles with 15°, 30°, 45°, 60° and 75°, respectively. In the plot, the numerical results fit well with the analytical curve.To give a comprehensive understanding and to verify the above theoretical analysis, we perform the numerical simulations of G-H shifts, where we consider a beam with a spatial Gaussian profile incident from air to ENZ metamaterial, and the half-width of the beam is 7.5e.g., mu-near-zero (MNZ) metamaterials, matched impedance zero-index metamaterials (MIZIMs), and anisotropic ZIMs. For MNZ metamaterials, we predict based on current results that the G-H shift is a constant for p polarized light, independent of the incident angle. Yet, there is no G-H shift for s polarization. For MIZIMs, there are no G-H shifts for both polarizations of s and p because the oblique incident light cannot enter into MIZIMs at all.Finally, it is worth mentioning that we only focus on the ENZ metamaterials here. In fact, the zero index metamaetrials can be further categorized, s or p polarization is obliquely incident onto the ENZ metamaterial from air. We have shown that for lossless ENZ material, there is no G-H shift for p polarization, which indicates that it obeys exactly the prediction by the geometrical optics. While for s polarization, the G-H shift is a constant value, independent of the incident angle. We also have explored analytically the influence of loss of ENZ metamaterials on the G-H shift for both polarizations and we found that the novel features still hold for a broad range of angles. Analogous deviations of G-H effect are expected for acoustic waves, quantum matter waves and so on. In addition, we have shown that the reflected phase of p polarized light is pϕ = −π, while for p polarized light, the reflected phase obeys that linear relationship sϕ = −2inθ. Due to such different responses, we believe that the ENZ metamaterial can be utilized to manipulate the polarization states of reflected light, by only adjusting the incident angle. We expect our work to contribute some new notions to the manipulation of the polarization states of light at the THz region.We have investigated the G-H effect of reflected light, when a linearly polarized light of either x axis from −8λ to 8λ. Due to the interference effect, the field amplitudes are oscillating along z direction. The line plots are located at the first peak close to the interface between air and ENZ metamaterial. Meanwhile, we zoom in the line plot of |yE| enough to get the distance between its symmetric axis and x = 0, which indicates the lateral shift L. The numerical results in The numerical simulation results shown in Y.X. and H.C. conceived the idea. Y.X. did the theoretical calculations and the numerical simulations. H.C. and C.T.C. helped with the theoretical analysis. H.C. supervised the whole project. All the authors wrote the manuscript.


We apply the well-known Condorcet criterion from voting theory outside of its classical framework and link it with spanning trees of an undirected graph. In situations in which a network, represented by a spanning tree of an undirected graph, needs to be installed, decision-makers typically do not agree on the network to be implemented. Instead, each of these decision-makers has her own ideal conception of the network. In order to derive a group decision, i.e., a single spanning tree for the entire group of decision-makers, the goal would be a spanning tree that beats each other spanning tree in a simple majority comparison. When comparing two dedicated spanning trees, a decision-maker will be considered to be more satisfied with the one that is “closer” to her proposal. In this context, the most basic and natural measure of distance is the usual set difference: we simply count the number of edges the spanning tree has in common with the proposal of the decision-maker. In this work, we show that it is computationally intractable to decide (1) if such a spanning tree exists, and (2) if a given spanning tree satisfies the Condorcet criterion. Dating back more than two centuries  or a crucial supply network such as a gas or oil pipeline. These networks are generally designed as spanning trees connecting all countries represented by vertices in a graph. The decision-makers could be the countries themselves, or board members of some EU institution. For political and historical reasons involving trust, self-interest, or stability concerns, different decision-makers have different perceptions which links  between the countries should be used in order to establish the whole network. In particular, the decision-makers could have different conceptions of the entire network because of political relations and alliances , or due to concerns regarding the security of supply and political stability. In addition, different decision-makers could base their conceptions on different criteria, which typically results in differing networks. Finding a plausible solution which takes into account the interests of all the decision-makers in such a setting is clearly an important aggregation task.T such that for each other spanning tree T beats (is not beaten by) In principle, we are thus concerned with the aggregation of spanning trees of an undirected graph into a single spanning tree. Hence, this work is somewhat related to the axiomatically motivated work of Endriss and Grandi , where, The first question is related to the work of Darmann , where iThe second question is in the spirit of the seminal paper by Bartholdi et al. , who sho Darmann  focuses  Darmann  providesentclass1pt{minima Darmann  is close Darmann  the agen Darmann  it follo Darmann  and, to  Darmann  by showiV and a set E of two-element subsets of V. We call the elements of V vertices and the elements of E edges. A subset spanning tree of G, if  contains no cycle. Note that for any two spanning trees G we have We briefly introduce the formal framework of this paper. An undirected graph A be a set of agents. For each agent G. Then, a spanning tree T of G is called a Condorcet tree if for all spanning trees G, T of G is called a weak Condorcet tree, if for all spanning trees G, Let In the first decision problem considered in this paper we ask for the existence of a Condorcet tree.Condorcet-Tree)  Condorcet tree.Condorcet-Tree Winner)  clauses made up of exactly 3 literals ; the task is to decide if C is unsatisfiable, i.e., if there is no truth assignment for X that satisfies all the clauses in C. 3-Unsat remains Sat is known to be 3-Unsat is We provide a reduction from the ns Tovey ; obvious3-Unsat, with Z be the set of literals, i.e., Let Step I: Constructing instanceof Condorcet-Tree. From Condorcet-Tree as follows.Step Ia: Defining the graph. Let t and q respectively. Identifying a literal with an edge of the same label, we formally defineG consists ofmentclass2pt{minimStep Ib: Introducing the agents. We introduce a set G. Note thatH is contained in the spanning tree T proposed by each edge in the set B is contained in all but each edge in D is contained in at most one proposed spanning tree andeach edge in Z is contained in at most each edge in From these observations, it is easy to conclude that T is the unique maximum spanning tree w.r.t. the edge weights for each for each for each for each Step II: Proving the reduction. We complete the proof by showing that the following claim holds:The agent set mentclass2pt{minim3-Unsat if and only if Condorcet-Tree.Condorcet-Tree. As a consequence, T is the unique Condorcet tree. Assume that 3-Unsat, i.e., there is a truth assignment C. We show that this leads to the conclusion that T is not a Condorcet tree in .“A:agents in t. Hence, for each agents in agents in On the other hand, the agents in T. Hence, we getT is not a Condorcet tree. Hence there is no Condorcet tree in 3-Unsat.Recall that Condorcet-Tree. Thus, T is not a Condorcet tree, i.e., there must be a spanning tree “First, we show that we can assume thatG from G. Obviously, H, we have(i) Second, we show thatholds. By construction, (ii) If entclass1pt{minimatated in .n from the sets Hence, mentclass2pt{minimtradicts . Thus, \Third, we show thatholds for eacht, n from the sets (iii) mentclass2pt{minimn from the sets n of the voters Analogously, tradicts .Finally, fromwe derive a truth assignmentthat satisfies all clauses inC. Let for each agent for each agent for each agent As a result, due to C, i.e., 3-Unsat. (iv) mentclass2pt{minim satisfy  each ageWeak-Condorcet-Tree.In the following theorem, we state an analogous result for Weak-Condorcet-Tree is The proof follows from the one of Theorem We now turn to the problems of deciding whether a given spanning tree is a (weak) Condorcet tree. These problems turn out to be Condorcet-Tree Winner is T. Thus, in instance T a Condorcet tree?” is “Yes”. It is hence straightforward that Condorcet-Tree Winner is In the proof of Theorem Condorcet-Tree Winner, where we ask if Finally, membership in Analogously, with (the proof of) Theorem Weak-Condorcet-Tree Winner is Condorcet-Tree and Weak-Condorcet-Tree only provide lower bounds for their computational complexity. A finer placement of these problems in terms of their computational complexity is an interesting open question.We have shown that both deciding if a (weak) Condorcet tree exists and deciding if a given spanning tree is a (weak) Condorcet tree, are computationally difficult problems. While the latter problems turn out to be


Large reorientations of the line of sight, involving combined rotations of the eyes, head, trunk and lower extremities, are executed either as fast single-step or as slow multiple-step gaze transfers. In order to obtain more insight into the mechanisms of gaze and multisegmental movement control, we have investigated time-optimal gaze shifts (i.e. with the instruction to move as fast as possible) during voluntary whole-body rotations to remembered targets up to 180° eccentricity performed by standing healthy humans in darkness. Fast, accurate, single-step movement patterns occurred in approximately 70 % of trials, i.e. considerably more frequently than in previous studies with the instruction to turn at freely chosen speed (30 %). Head-in-space velocity in these cases was significantly higher than during multiple-step transfers and displayed a conspicuously regular bell-shaped profile, increasing smoothly to a peak and then decreasing slowly until realignment with the target. Head-in-space acceleration was on average not different during reorientations to the different target eccentricities. In contrast, head-in-space velocity increased with target eccentricity due to the longer duration of the acceleration phase implemented during trials to more distant targets. Eye saccade amplitude approached the eye-in-orbit mechanical limit and was unrelated to eye/head velocity, duration or target eccentricity. Overall, the combined movement was stereotyped such that the first two principal components accounted for data variance almost up to gaze shift end, suggesting that the three mechanical degrees of freedom under consideration  are on average reduced to two kinematic degrees of freedom . Synchronous EMG activity in the anterior tibial and gastrocnemius muscles preceded the onset of eye rotation. Since the magnitude and timing of peak head-in-space velocity were scaled with target eccentricity and because head-on-trunk and trunk-in-space displacements were on average linearly correlated, we propose a separate controller for head-in-space movement, whereas the movement of the eye-in-space may be, in contrast, governed by global, i.e. gaze feedback. The rapid progression of the line of sight can be sustained, and the reactivation of the vestibulo-ocular reflex would be postponed, until gaze error approaches zero only in association with a strong head-in-space neural control signal. Upright human subjects reorient to angularly distant targets (≥90°) by assembling fairly stereotypical, simultaneous, combined movements of the eye, head, trunk and lower extremities Land . Such laDuring the less time-efficient multiple-step gaze transfers, target acquisition takes at least 100–300 ms longer. It has been suggested that the higher the head-in-space velocity during this task, the more likely that a single-step gaze shift pattern will be elicited  of such gaze reorientations is characterized by tight segmental coupling, as the three mechanical degrees of freedom available  are reduced to two kinematic degrees of freedom , turning at natural, freely chosen speeds , mean age 27 ± 1.2 years. No one wore spectacles, and selection ensured excellent physical condition. The study was approved by the Imperial College Riverside Ethics Committee in accordance with the Declaration of Helsinki, and all subjects provided informed consent.Details have been given previously .In contrast to all our previous studies with this paradigm, where Ss were given no instruction as to the speed of the movement required, here Ss were instructed to accomplish turns as soon and as fast as possible.Head-in-space, upper trunk and feet horizontal (yaw plane) movements were recorded using a Polhemus Fastrak motion analysis system . Markers were placed on a light, adjustable helmet, on the spinous process of the seventh cervical vertebra and dorsally on each foot. Horizontal eye-in-orbit rotations were recorded using bi-temporal DC electrooculography (EOG) with a flat response to 90 Hz. EMG activity was collected by adhesive surface disc electrodes  placed at a distance of 3 cm near the motor point of the following muscles on the right side: anterior tibial, gastrocnemius, biceps femoris and external oblique.sgolayfilt), while head and trunk velocities were calculated after subjecting the position signals to moving mean linear filtering . Movement onset/offset for the various segments was calculated from the time derivative. Gaze (eye-in-space) displacement was obtained by adding EOG and head signals after appropriate scaling. Subtracting the trunk from the head signal resulted in an estimate of head-on-trunk movement. Biphasic deflections in the eye velocity trace were thought to represent blinks , so that the first and the last PCs, respectively, account for maximal and minimal amount of variance in the original data. The PCs combine linearly with the original  variables, and if the three elemental variables are totally uncovaried, three statistically significant PCs will account for data set variability. In contrast, if two of the variables covary, then only two significant PCs account for data set variability and the space defined by the three elemental variables is reduced by one dimension, i.e. variation in the data set under consideration can be described by a plane . Similarly, if all three variables covary, then only one significant PC accounts for data set variability and the space defined by the three variables is reduced by one further dimension (i.e. to a straight line). Thus, the number of significant PCs and the percentage of data variance they account for stand for the degree of covariation among the variables.We applied PC analysis to examine and quantify statistically the relationships among a set of 3-covarying variables (channels), i.e. eye-in-orbit, trunk-in-space and head-on-trunk, of mth sample of the trial, m = 1, …, n. For the purpose of PC analysis, raw position data were used (i.e. before filtering), and EOG was resampled at 30 Hz. Increasing time intervals, starting always with the onset of gaze, was analysed (‘sliding’ approach). Data centring and auto-scaling were carried out by subtracting the arithmetic means Percentages of the variation in the original set of variables, which the principal components account for, equal the normalized eigenvalues eig. For each data set , the eigenvalue k = 1, …, N, are calculated. Further, by averaging across all trials, the mean eigenvalue Eigenvalues Bartlett’s test of sphericity was used to validate the hypothesis that the first principal component was significantly different from all those of lower importance, i.e. both second and third. This has to be accepted if the test statistic function j = 1, 2), a test for proportion of total variance was applied post hoc. A desired proportion of total variance at 99.0 % was selected and excluded all principal components, which accounted for <1 % of the total variance; one-sample independent t test was performed for the arithmetic means of variance proportion of second and third PCs and tested whether they were significantly >1 %. When normality failed according to a one-sample Kolmogorov–Smirnov nonparametric test, a binomial nonparametric test for the median value was carried out instead.N-trials, separately for various target eccentricities and time intervals from gaze onset . Outbound turns and visually guided gaze shifts  will not be considered further here.Inbound or ‘return’ reorientations to targets ≥90° consisted predominantly of single-step displacements of the visual axis covering at least 85 % of target eccentricity Fig. . Here, wAs already reported in the literature for large saccades with the head free to move , and the line of sight was stabilized, in darkness, quite near the memorized centrally located target. Ss sometimes made a corrective small saccade after the LED was turned on.F = 26.1 p < 0.0001 and F = 25.1 p < 0.0001, respectively). In contrast to head-in-space, the dynamic trajectories of trunk-in-space, and most particularly head-on-trunk motion, were more variable , confirming that the Ss generally tried to comply with the instruction ‘go as fast as you can’. Noteworthy, in reorientations made by means of the multiple-step pattern . In many instances, the premature termination of primary gaze shift long before reaching the position of the centrally located remembered target was signalled by an abrupt deceleration break in the, up to this point, monotonically increasing head-in-space velocity trace . However, even if interrupted by variable periods of stationary gaze, the displacement of the visual axis was terminated again very close to the remembered target.Head-in-space movement resulted from the sum of head-on-trunk and trunk rotation in space. The dynamic trajectory of head-in-space velocity was conspicuously monophasic, increasing smoothly to a peak and then decreasing slowly until realignment with the memorized gaze direction . Single-step transfers were approximately 100–300 ms faster than multiple-step gaze displacements . Finally, head-in-space oscillations  did not increase with the amplitude of head-on-trunk movement (R2 = 0.24).As already reported in the literature , we examined how the timing of peak head-in-space velocity was associated with the end of the gaze shift in single-step movement patterns Fig. . If headN > 15) of trials for statistical analysis were available. The first PC was significant  and accounted for almost 90 % of variance in all analysis time windows. The second PC was also significant overall and accounted approximately for about 10 % of variance (p < 0.001). In contrast, the third PC was non-significant in all time windows and target eccentricities, accounting for <1 % of variance. Variance up to half a second after movement initiation could be thus accounted for by essentially the first two PCs (i.e. the sum of their percentages was approximately 100 %), suggesting that the three mechanical degrees of freedom under consideration are reduced to two kinematic degrees of freedom. Visualization of these relationships is exemplified in Fig. x and y axes give mean PC scores, units are normalized as a consequence of auto-scaling, c.f. “Mean ± standard deviations of percentages of variance accounted for by individual principal components (PCs) are shown in Table θ and φ. Note from Table θ is often close to 90° while φ is mostly larger than 45°. Because θ is mostly close to 90°, the covariation plane is approximately perpendicular  to the plane defined by the axes head-on-trunk and trunk-in-space. In consequence, head-on-trunk and trunk-in-space displacements are linearly correlated, i.e. they lay approximately upon a regression line and can be replaced by one variable, for example, head-in-space. The slope of this relationship is larger than unity as φ is mostly >than 45°. In this way, trunk-in-space is on average larger than head-on-trunk displacement throughout the gaze transfers. Of note, these relationships do not apply during ‘multistep’ (nystagmic) transfers. Here, three derived significant PCs account for the data variance . There was not any statistical difference in the onset of EMG activation from the right tibial and gastrocnemius muscles, both for inbound and for outbound reorientations, i.e. they were activated on average synchronously. Not unexpectedly, EMG activity in these muscles decreased as the right foot started to rotate rightwards.Ss activated the right anterior tibial and gastrocnemius muscles approximately 30 ms on average before starting to rotate gaze (i.e. the eye) rightwards to the centrally located remembered target Fig. . Biceps Ss were instructed to move as fast as possible and were thus able to attain high head and trunk velocities. In the remaining 30 % of trials, however, the progression of the line of sight was interrupted by one or more variable intervals of stable gaze . All the same, gaze displacement ended again close to the remembered target. Multiple-step gaze shifts were characterized by significantly low head-in-space velocity. High head-in-space velocity during task execution thus increases the probability of eliciting single-step gaze shifts and quickly acquiring a predetermined direction of the visual axis, as hypothesized in previous studies . This presumably presents the CNS with a simpler control task.A conspicuous, central feature of large, single-step reorientations to remembered targets was the high peaked monophasic head-in-space velocity traces Fig. , suggestA further salient finding of our study in favour of separate control for head-in-space is that command timing does not relate to gaze displacement but, instead, head-in-space displacement Fig. . If the Why did Ss frequently choose to move head-on-trunk far below 60–70° (i.e. the limit of the so-called neutral zone, McClure et al. Separate control signals for the eye and head have been proposed by several researchers because of the variability in the relative timing of the start of eye and head movements and the effect of initial eye-in-orbit position on the contribution of the eyes and head to gaze displacement (for a review, see Freedman The fact that high head-in-space velocity and single-step gaze shifts are interconnected makes sense biologically when target acquisition occurs under ‘as soon as you can’ conditions, but how are these two variables associated mechanistically? It is clear from our findings that reemergence of the VOR and gaze stabilization does not depend solely on gaze error, as most gaze models propose (cf. Galiana and Guitton Foot rotations in this experimental paradigm are extremely variable and start mostly considerably later than movements of the eye, head and trunk (on average 350 ms later than the eye, Anastasopoulos et al.


Statistics in Medicine published by John Wiley & Sons, Ltd.Estimating the size of hidden or difficult to reach populations is often of interest for economic, sociological or public health reasons. In order to estimate such populations, administrative data lists are often collated to form multi-list cross-counts and displayed in the form of an incomplete contingency table. Log-linear models are typically fitted to such data to obtain an estimate of the total population size by estimating the number of individuals not observed by any of the data-sources. This approach has been taken to estimate the current number of people who inject drugs (PWID) in Scotland, with the Hepatitis C virus diagnosis database used as one of the data-sources to identify PWID. However, the Hepatitis C virus diagnosis data-source does not distinguish between current and former PWID, which, if ignored, will lead to overestimation of the total population size of current PWID. We extend the standard model-fitting approach to allow for a data-source, which contains a mixture of target and non-target individuals . We apply the proposed approach to data for PWID in Scotland in 2003, 2006 and 2009 and compare with the results from standard log-linear models. © 2013 The Authors.  Estimating population sizes using two (or more) data lists has a long history Traditionally, log-linear models Log-linear models have been extended to allow additional (discrete) covariates to be incorporated The fact that a cell count of the table is unknown, or missing, may motivate us to consider the missing data mechanism only observe members of the target population or are able to identify whether an individual is a member of the target population. However, this may not always be the case. For example, for the data that we introduce in Section 1.1, we are interested in the current number of people who inject drugs (PWID) in Scotland. Four data-sources are used to observe such individuals, including the Hepatitis C virus (HCV) diagnosis database. Individuals are identified as being PWID on this database, but their status as ‘current’ or ‘former’ injector is not recorded. Thus, this data-source records not only current PWID but also former PWID who do not belong to the target current PWID population of interest. A previous analysis of one year of these data by King et al. In this paper, we relax the standard assumption that all individuals recorded by the given data-sources belong to the population of interest or ‘target’ population. Typically, data-sources are chosen so that they et al. In the last decade, data from multiple administrative data-sources have been repeatedly used to estimate the total number of current PWID in Scotland. We shall consider the data collected in the last three studies, corresponding to years 2003, 2006 and 2009. Four data-sources were used corresponding to social enquiry reports S1), hospital records (S2), Scottish Drug Misuse Database  and HCV diagnosis database (S4). The HCV diagnosis database contains epidemiological data (including risk group and date of first diagnosis) on all individuals who have been newly diagnosed with HCV in Scotland. In addition, a range of covariates was recorded for each individual, including age (C1), gender (C2) and region (C3). For each study, age is categorised into two levels . For 2003, region was categorised into the two levels, Greater Glasgow and Rest of Scotland, whereas in 2006 and 2009, the two levels were Greater Glasgow and Clyde, and Rest of Scotland. This change was to conform with the National Health Service board regions, which altered between the 2003 and 2006 studies. See King , hospita7 contingency table, with 8 ( = 23) unknown cell counts. However, as discussed by et al. et al. The number of individuals observed by each combination of data-sources and cross-classified across covariates can be presented in the form of an incomplete 2In this paper, we investigate, in further detail, the issue of individuals observed by the HCV diagnosis data-source including both target and non-target individuals. The main contribution of the paper is the development of a new modelling strategy to incorporate this complexity, without excluding the HCV diagnosis data-source, and hence, potentially, discarding a significant amount of information. The additional advantages of this strategy over the approach of The outline of the paper is as follows. Section 2 describes the new modelling strategy in detail and includes a simulation study to test the strategy's efficacy. In Section 3, we apply the strategy to the Scotland PWID multi-list data, for 2003, 2006 and 2009, to provide the aforementioned estimates. We end with Section 4 discussing future work and epidemiological conclusions of the results in Section 3.In this section, we discuss the assumptions relating to the modelling approach before describing the notation that will be used and the new model-fitting approach to deal with the HCV diagnosis data-source's potential for recording both former and current PWID.only the HCV diagnosis data-source  for each cross-classification of covariate values may contain non-target individuals. The corresponding cell entries can then be regarded as an upper bound on the true cell count (i.e. true number of current PWID observed by only the HCV diagnosis data-source), leading to left censored cells.We assume that the HCV diagnosis data-source may record individuals from both current and former PWID  but that each of the other data-sources only records current PWID. This implies that individuals recorded by the HCV diagnosis data-source and any other data-source are a current injector. Thus, only the observed cells corresponding to being recorded by n cells in the incomplete contingency table, of which nU are unobserved and nC are censored. For each year of these data, n = 27 = 128, nU = 23 = 8 (corresponding to being unobserved by each data-source for the 8 covariate cross-classifications) and nC = 23 = 8 (corresponding to being recorded by only the HCV diagnosis data-source for the 8 covariate cross-classifications). We label each cell from 1, … ,n, such that cells i = 1, … ,nC correspond to the censored cells and cells i = nC + 1, … ,nC + nU to the unobserved cells. For i = 1, … ,n, we let yi denote the true number of current PWID in cell i. Further, we let yO,zC}are the observed data, whereas {yU,yC}are parameters to be estimated so that the total population size β denote the set of log-linear parameters in the model (we return to the issue of model uncertainty later).Notationally, we assume that there yi, has an independent Poisson distribution, that is, for i = 1, … ,n, μi has the form,β denotes the (q + 1) × 1 vector of (identifiable) log-linear parameters, β0,β1, … ,βq, and xi the (q + 1) × 1 design vector relating to cell i = 1, … ,n. The term β0 corresponds to the intercept term, whereas β1, … ,βq correspond to the main-effect and log-linear interaction terms. For parameter identifiability, we use the standard sum-to-zero constraints for the log-linear main-effect and interaction parameters. Because each data-source and covariate only has two levels, each element of xi is equal to ± 1. Finally, we let X denote the n × (q + 1) design matrix with ith row, xi.We use a special case of a generalised linear model (GLM), where we assume that each (true) cell count, y = {y1, … ,yn}is to set, An alternative  model specification for ith element of p is given by μi given by Equation 2. This is the model specification considered by N, to be directly incorporated through a prior distribution on N.where the yU, yC and all model parameters, denoted by θ, given the observed data yO and zC, can be written in the form, y = {yO,yC,yU}, and π( · ) and π( · | · ) represent marginal and conditional probability density/mass functions, respectively. This result is derived because we assume that the observed cell entries yO are independent of zC, yU and yC, given θ; and the observed censored cells zC are independent of yU, given the log-linear parameters θ and true cell entries yC. The term π(y | θ) corresponds to the complete data likelihood for the full set of cell entries, y, as specified by Equation 1 and π the distribution of the observed upper bound for the censored cells, given the log-linear parameters and true cell entries. For these data, we assume non-informative censoring. In other words, zi | yi ∼ U[yi, ∞ ), independently, for i = 1, … ,nC  = π(zC | yC)). In Section 4, we discuss an alternative distribution for zi, which can be used to incorporate individual prior information on the proportion of individuals observed by the HCV diagnosis data-source who are current PWID.Given a log-linear model (in terms of the interaction terms present in the model), the posterior distribution of θm denote the parameters present within model π(m) denotes the prior model probability for model To allow for additional model uncertainty, in terms of the log-linear interaction terms that are present in the model, we extend the posterior distribution given in Equation 5. We treat the model indicator as a discrete parameter and form the joint posterior distribution over both parameters and model space. Let θm and m, into the analysis via the prior distributions, π(θm | m) and π(m). However, we assume there is weak prior information on the model parameters and so need to specify a prior distribution, which reflects this position. In general, care must be taken when specifying prior distributions on the model parameters under weak prior information in the case of model uncertainty due to Lindley's paradox βm. To counter this issue, we use the idea of ‘default prior distributions’ βm, we drop the subscript m, for notational simplicity. Because the intercept, β0, is present in all models, we assume π(β0) ∝ 1. Let the matrix Y be formed from X by removing the first column of ones corresponding to β0. The GHGP for the remaining elements, α = , of β is given by α | σ2 ∼ N , where σ2 > 0 is an unknown hyperparameter. The GHGP extends the Zellner g-prior α and an imaginary sample of size 1 / σ2, which is referred to as the ‘prior sample size’ by σ2 = 1, thus giving a prior sample size of one. This is intuitively appealing because the amount of information the prior provides is directly interpretable. Sabanes-Bové and Held σ2 be unknown by assuming a gamma hyperprior distribution with parameters a / 2 and b / 2, so that, on the inverse scale Within the Bayesian approach, we can incorporate any available prior information on the model parameters, a = b = 10 − 3 so that the hyperprior on 1 / σ2 is vague but centred at one. However, in Section 3, we investigate the sensitivity of the resulting analysis to these values of a and b. The inverse-gamma hyperprior distribution assumed earlier also has the minor computational advantage of being conditionally conjugate (see step 1 in Section 2.4) meaning the full conditional distribution of σ2 is also an inverse-gamma distribution. Because, in our case, all of the data-sources and covariates have two levels, under sum-to-zero constraints, the GHGP corresponds to the elements of α being independent with variance given by σ2.where IG denotes the inverse-gamma distribution. We specify m is given by θm = {βm,σ2}.Finally, we specify a uniform prior over all admissible models, that is, yU, yC, θm and m under the Poisson specification is identical to that obtained using the multinomial model formulation when π(N) ∝ N − 1, corresponding to Jeffreys’ prior The joint posterior distribution of N (obtained by combining the estimated number of injecting drug related death rates with an estimated associated death rate). Thus, the posterior distributions for the total population size ignoring the censored cells for 2003 and 2006 will be slightly different to those published in The aim of this paper is to consider the impact of censored contingency cells, and so, to maximise consistency across years, we specify the same default prior from above on the parameters for each dataset. The previous analyses of the Scotland PWID multi-list data have used the alternative multinomial model specification in order to specify an informative prior on the total population size, N. Under the multinomial model formulation, and for the subset of decomposable models, Madigan and York p, which gives a closed form expression for the marginal posterior distribution of the total population size. However, as Dellaportas and Forster The priors we have specified earlier are not conjugate. In fact, for hierarchical log-linear models, conjugate priors do not exist. This means it is necessary to employ the computational methods, described in Section 2.4, to evaluate the marginal posterior distribution of the total population size, yU, yC, θm and m, given by Equation 5. The algorithm employed is an example of data-augmentation and, as such, can be viewed as iterative multiple imputation yU and yC, update the parameters and model, θm and m.Step 1 Conditional on the current values of the unknown and censored cell counts, θm and m, update the unknown and censored cell counts, yU and yC.Step 2 Conditional on the current parameters and model, In this section, we describe the computational algorithm used to generate an MCMC sample from the posterior distribution of We consider each step in turn.βm, and model, m, are updated using the reversible jump (RJ) algorithm m remains unchanged, and we update only the log-linear parameters, βm. Previous approaches, including the study of King and Brooks m. We use an alternative algorithm that utilises the iterated weighted least squares implementation of the Metropolis–Hastings algorithm for GLMs The log-linear parameters, σ2. The inverse gamma prior distribution for σ2 is conditionally conjugate so that we update the parameter using the closed form of the full conditional distribution, that is, βm = .Now consider the hyperparameter, yi | βm ∼ Poisson(μi). Similarly, to update the true cell entries for the censored cells, yi | βm,zi ∼ Poisson(μi)I(yi⩽ zi), where I denotes the indicator function.To update the unobserved cell entries The computational approach described in this section is implemented in the R In this section, we conduct a small simulation study to assess the efficacy of the proposed modelling strategy and to compare this against other methods. The simulation study attempts to match the observed properties of the Scotland PWID multi-list data introduced in Section 1.1 and its inferred properties from Section 3.yU, is completely unknown; one cell count, yC, is unknown but bounded from above by zC; and the remaining 14 cell counts, yO, are observed. We consider three different true total population sizes N = 10000,15000 and 20000. Additionally, we consider two different forms for the true data-generating model: (i) a model with non-zero interactions for S1 : S2, S1 : S4 and S2 : S4 and (ii) a model with a non-zero interaction for S1 : S2. Note that model (ii) is decomposable but model (i) is not decomposable . Table β under each of the true data-generating models. These values were chosen so that the simulated data were consistent with the observed Scotland PWID multi-list data. For example, the negative values of the true parameters associated with the main-effect terms mean that the probability of observing an individual is less than that of not observing an individual.The simulation study is set up as follows. We assume there are four data-sources: S1, S2, S3 and S4, and that S4 is the censored data-source observing a mixture of individuals from the target and non-target populations. For simplicity, we assume that there are no covariates. This means that there are 16 cells in the table. One cell count, N, we generate the true cell counts from the multinomial distribution. We remove the cell count, yU, corresponding to not being observed by any of the data-sources. We also replace yC (the cell count corresponding to just being observed by S4) by zC, where zC is generated from a negative binomial distribution (under the parameterisation of yC and probability 2 / 3 (so as to be consistent with the inferred properties from the observed data for PWID in Scotland). For each true data-generating model and each true total population size, we generate 1000 datasets using the aforementioned procedure. We analyse each dataset using the following four methods. INC-C The proposed modelling strategy, described in Sections 2.1 to 2.4, of incorporating the censoring mechanism, using hierarchical models.REM-C The approach of IGN-C The approach of MY-C The approach of Using the parameter values in Table yC, in the censored cell from a truncated posterior predictive distribution. Also, note that, to apply the approach of N. To maintain consistency with the three other approaches listed earlier, we assume Jeffreys’ prior for N, that is, π(N) ∝ N − 1 (Section 2.3).Note that the MY-C method of φ, of individuals observed by the S4 data-source who are actually members of the target population or identify interactions involving the S4 data-source. Additionally, for this method, by removing the censored data-source, we are discarding information. The INC-C and MY-C methods actually model the censoring mechanism and will be able to provide a posterior distribution of φ. Additionally, these two methods can identify interactions involving the S4 data-source. Finally, the INC-C method has the advantage over the MY-C method of considering the much more flexible class of hierarchical models. However, this advantage comes at the price of requiring more complex computational methods to evaluate the resulting posterior distributions.Before we present the results of the simulation study, we briefly contrast the relative advantages and disadvantages of the four methods. The REM-C and IGN-C methods are the simplest approaches because we do not model the censoring by either removing the data-source or ignoring the problem altogether, respectively. The REM-C method will not be able to evaluate the posterior distribution of the proportion, N, and φ. These rates and lengths, averaged over the 1000 datasets, are given for each of the three true values for N, each of the two true data-generating models and each of the four methods. An entry of NA indicates that this method cannot estimate the value of φ. Note that the true value of φ depends on the true cell counts generated for each individual dataset.Table N was less than the lower bound of the interval under the IGN-C method for every occurrence of the interval not covering N. This indicates that the IGN-C method over-estimates the population size under the chosen two true data-generating models. The remaining methods, INC-C, REM-C and MY-C, all perform fairly well. As expected, the MY-C method performs in a more satisfactory manner when the true data-generating model is a decomposable model, that is, for model (ii). The INC-C and REM-C methods perform very similarly. However, the INC-C method has the advantage of being able to estimate φ.In terms of coverage, the IGN-C method performs poorly. Further investigation shows that the true value of In terms of the mean lengths of the intervals, the INC-C and REM-C methods perform similarly again with, typically, the INC-C intervals being slightly shorter, that is, more precise. This supports our intuition, that by removing the censored data-source, we would be discarding information. However, the actual different in mean lengths between the INC-C and REM-C methods is small. Typically, the MY-C intervals are slightly wider than the INC-C and REM-C intervals, and this difference is more obvious under the decomposable model (ii).We further compare the INC-C, REM-C and IGN-C methods using the Scotland PWID multi-list data in Section 3.χ2 statistic and its associated Bayesian p-value (Section 3.1). In a similar analysis of multi-list data from England, King et al. We apply the proposed methodology (INC-C) from Section 2 to the Scotland PWID multi-list data for 2003, 2006 and 2009. For comparison, we also conduct two additional analyses for each dataset: (i) the approach of p-value χ2 statistic on the observed cell entries, yO, as the discrepancy function. We take a pragmatic approach whereby a Bayesian p-value in the interval  indicates no lack of a goodness-of-fit to the data of the model-averaging approach. For the INC-C analyses, we obtain Bayesian p-values of 0.15, 0.44 and 0.30 for years 2003, 2006 and 2009, respectively; for the IGN-C analyses, values of 0.09, 0.27 and 0.08, respectively; and for the REM-C analyses, values of 0.06, 0.28 and 0.12, respectively. Thus, for all analyses, the Bayesian p-value suggested an adequate goodness-of-fit for each method. Therefore, these do not suggest that the IGN-C modelling approach that ignores censoring is inadequate, but these assessments use the same false assumption as the model does, that is, that all individuals observed by the HCV diagnosis data-source are current PWID.In order to assess overall goodness-of-fit of the four data-source analyses, we calculate the (model-averaged) Bayesian The corresponding posterior estimates for the total population size for current PWID are provided in Table Figure Table Table For 2003 and 2006, the IGN-C analysis lends additional support for a further one interaction terms being present (i.e. posterior model probability > 0.75); for 2009, a further six interaction terms have strong positive support, whereas the interaction term S2 × S4  loses posterior support. For the interactions identified under the IGN-C method, the sign of the posterior mean of the data-source × data-source interactions are all positive, which leads to the increased population estimates.Finally, for all interactions where it is deemed that there is positive evidence that they are present, except the S1 × S3  interaction discussed earlier, the sign of the posterior means of the interactions were consistent across all years and analyses.As stated, an advantage of the INC-C analysis over the REM-C method is that it allows us now to consider the proportion of individuals identified by the HCV diagnosis data-source as having a risk factor of injecting who are actually current PWID (i.e. members of the target population).Because of the initiative (Section 1.1) to encourage former PWID, born in 1956–1975, to be tested for HCV diagnosis, we consider the posterior distribution of the proportion of current PWID for the two different age groups. Table a and b, which specify the prior distribution for σ2 (Section 2.3), we repeat the INC-C analysis for a range of values of these hyperparameters. To do this, we fix a = 0.001 and vary b according to the expectation of the prior sample size 1 / σ2. We choose b = 0.004,0.002 and 0.0005 corresponding to expectations of the prior sample size of 4, 2 and 0.5, respectively. Additionally, we also use the Gelman prior σ so that a = − 1 and b = 0 in the full conditional distribution of σ2 given by 8. We compare the resulting posterior mean of the total population size under each of the four priors to the corresponding posterior mean under the prior specified in Section 2.3 where a = 0.001 and b = 0.001, for each year. In each case, the resulting posterior mean is within 4% of the corresponding mean for when a = 0.001 and b = 0.001. Note that the posterior means under the four priors are given in Table S1 of Section 4 of the Supporting information, along with 95% HPDIs.To investigate the sensitivity of the analysis to the values of the hyperparameters, Within standard capture-recapture studies, it is assumed that all individuals that are observed by the data-sources are members of the target population of interest. If this is not the case, the results obtained can lead to biased population estimates. For the data studied within this analysis, the HCV diagnosis database identified as PWID those who had listed injecting as their HCV risk factor, thus including former PWID in addition to the target population of current PWID. In such instances, one possibility is simply to remove such data-sources from the capture-recapture analyses. However, this may remove a significant amount of information that is contained within the data-source(s). In this paper, we have accounted for the possibility of censoring occurring for a single data-source  by explicitly accounting for the censoring within the modelling structure for the HCV diagnosis database.In particular, we considered estimating the total number of current PWID in Scotland in the years 2003, 2006 and 2009 allowing for censoring for the HCV diagnosis database and compared this with the analyses that ignored the potential censoring issue or removed the HCV diagnosis database and considered only three data-sources. Consistently, the analyses that ignored the issue of the HCV diagnosis database observing a mixture of former and current PWID led to significantly larger estimates for the total population size than the other two approaches that accounted for this (either by directly modelling the censoring or simply removing the given data-source). This conclusion was confirmed by the simulation study in Section 2.5.The approach of removing the HCV diagnosis data-source produced remarkably similar posterior estimates to that of incorporating the censoring. However, the analysis incorporating censoring allowed us to estimate the proportion of those present on the HCV diagnosis database with a history of drug use who are actually current PWID and identify interactions involving the HCV diagnosis data-source. Both of these are of interest to epidemiologists.We divide the remainder of this concluding section into two parts. In Section 4.1, we discuss ways in which the proposed modelling approach that incorporates censoring can be extended by considering a larger model space, p-value. This may not always be the case and we may need to consider a larger set of models, In this section, we discuss two ways in which the modelling approach, given in Section 2, can be extended. First, in Section 3, the maximal model was specified as the model containing all of the two-way interactions. This gave sufficient complexity to obtain an adequate model according to the Bayesian his case . The MCMzi, on the censored cell count, yi, for each cross-classification of covariates, is uniform, namely, Second, in this paper, we considered uninformative censoring, that is, the distribution of the upper bound, ψi, of individuals, within each cross-classification, i, of covariates, observed solely by the HCV diagnosis data-source who are currently PWID. One way to incorporate this is to assume that We may actually have some prior information on the proportion, zi is the random variable of the number of trials given yi successes and a probability of success ψi. In this context, the number of trials is the number of individuals observed solely by the HCV diagnosis data-source and the number of ‘successes’ is the number of current PWID. The prior information on ψi may be in the form of a point estimate or encoded in a prior distribution, for example, a beta distribution.where NB denotes the negative binomial distribution We found that Scotland's number of current PWID had risen between 2003 and 2006 but decreased by 2009. This is good news in public health terms but celebration should be cautious because cessation from injecting need not mean that the individuals concerned have ceased opiate dependency. The dramatic decrease in Scotland's number of current PWID between 2006 and 2009, including among those aged 35 +  years .


The following information is missing from the Funding section: This study was supported by the A*MIDEX project (n° ANR-11-IDEX-0001-02) funded by the « Investissements d’Avenir » French Government program, managed by the French National Research Agency (ANR). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript.


Recently, the study of the fractional formal  has been increased. This study not only in mathematics but extended to another topics. In this effort, we investigate a generalized integro-differential operator  The subject of fractional calculus  has acquired significant popularity and major attention from several authors in various science due mainly to its direct involvement in the problems of differential equations in mathematics, physics, engineering and others for example Baskonus and Bulut , Yin et l-Oboudi . Severaljafzadeh  investigf et al.  gave soml-Ashwah  used S\dz et al.  investigz et al.  defined z et al.  defined In this paper, we define a generalized mixed integro-differential operator f of form (Let mentclass2pt{minima et al.  introduca et al. . Now, leSălăgean  and whenl-Oboudi . Afterwaf, which satisfies the followingf which satisfies the followingf is said to be in the class positive real numbers, Let ivastava , Dixit aivastava  and receivastava . Let a ft}k=0 in . For \dond Breaz . Also, tFor y Frasin .For nd Breaz .For z et al. . In partd Pescar .For r et al. . In partlexander .We start our first result.Letbe positive real numbers, Ifandthen the integral operatorgiven by (is in the classwhereOn successive differentiation of Let Letbe positive real numbers. Ifandthenis in the classwhereLetbe positive real numbers, We assume thatare starlike functions by orderand that isandIfthengiven by (is in the classwhereBy following same methods as in Theorem 1, we haveition in , we haveBy setting Letbepositive real numbers, We assume thatandIfIn geometric function theory, we defined and studied a new integro -differential operator


There are several methods of ITA available. The basic tool to compare such algorithms concerning their quality are large-scale simulation studies that are crucially set up on a large collection of quasi-orders. A serious problem is that all known ITA algorithms are sensitive to the structure of the underlying quasi-order. Thus, it is crucial to base any simulation study that tries to compare the algorithms upon samples of quasi-orders that are representative, meaning each quasi-order is included in a sample with the same probability. Up to now, no method to create representative quasi-orders on larger item sets is known. Non-optimal algorithms for quasi-order generation were used in previous studies, which caused misinterpretations and erroneous conclusions. In this paper, we present a method for creating representative random samples of quasi-orders. The basic idea is to consider random extensions of quasi-orders from lower to higher dimension and to discard extensions that do not satisfy the transitivity property.Dependencies between educational test items can be represented as quasi-orders on the item set of a knowledge domain and used for an efficient adaptive assessment of knowledge. One approach to uncovering such dependencies is by exploratory algorithms of  Orders play an important role in various formal theories of behavioral, social, economic, or computer sciences. Examples are in decision making and preference modeling  as a cornerstone concept in the educational or psychological theory of knowledge or learning spaces  i ≤ j is interpreted as stating “Each subject who is able to solve item j is also able to solve item i.” As an example, a plausible dependency to postulate between the two algebra items “a. Perform the multiplication 4x4y4· 2x · 5y2and simplify” and “b. Find the greatest common factor of the expressions 14t6y and 4tu5y8and simplify” is to assume that the mastery of problem a is a prerequisite for the mastery of problem b. Thus, the items a and b are in relation with respect to a quasi-order ≤, denoted by a ≤ b.Prerequisite relations can be modeled using quasi-orders. In knowledge or learning space theory, a quasi-order ≤ on a test or item set item tree analysis . We call X a representative subset or sample if P(≤1 ∈X) = P(≤2 ∈X) for any two quasi-orders ≤1 and ≤2 on Q, where P denotes probability. Since this definition is for arbitrary quasi-orders, as a corollary, we obtain that any such representative set X of quasi-orders must also be “proportionally representative” for the population distributions of quasi-order size, quasi-order width, and quasi-order height (defined below), in the sense that the corresponding sample distributions computed in X provide unbiased estimates of their population analogs. The latter distributional properties are the evaluation criteria used for assessing the extent of representativeness in this paper.Let In Ünlü and Schrepp , it was In the next section, we discuss existing approaches to addressing this problem and the critical issues related with these. In a next step, the description of the proposed algorithm is given, and then simulation results are presented demonstrating the usefulness of the sampling procedure. Finally, we conclude with a summary and further related remarks.Q of n test items. The goal is to create randomly a sample, PQO(n), of quasi-orders that is representative, in the sense that each quasi-order on Q is included in the sample with the same probability.Formally, the problem we address in this paper can be described as follows. Presuppose a set R on Q (not necessarily reflexive or transitive) is denoted by a matrix (rij) with rij∈ {0, 1} and rij = 1 ⇔ iRj.We use the relational matrix notation, i.e., a binary relation R mean that its corresponding relational matrix satisfies the following properties:rii = 1 for all i = 1, …, n (reflexive R),rij = 1 ∧ rjk = 1 → rik = 1) for all i,j,k ∈ {1, …, n} (transitive R)., where rii = 1 for all i = 1, …, n, and the other values rij are chosen with equal probability from {0, 1}.Create a relational matrix  quasi-orders, we can see in Table n the chance to pick a transitive relation using this random process is tiny.This method, directly operating on all items simultaneously, also fails even for moderate sizes of n = 6 items requires to select ~50,000,000 random reflexive relations, i.e., 1,500,000,000 random 0–1-numbers. If we would be able to generate and check 1000 reflexive relations per second, it would require ~50,000 s until the process stops. It is obvious that this will not work at all for larger item numbers, for example for n = 10. In contrast, the processing times of the newly proposed procedure are feasible and the results can be found in the section Creating Quasi-orders on Bigger Item Sets.Thus, finding for example 10,000 quasi-orders on ad-hoc procedures to draw random quasi-orders. Although these methods are very flexible and virtually work for any value of n, they lack representativeness in their generation processes. The simplest method is to create a random relation on n items and to compute the transitive closure of this relation. It was found in earlier publications  such that:rii = 1 for all i = 1, …, n.rij with i, j = 1, …, n and i ≠ j, the value rij is chosen randomly with probabilities P(rij = 1) = x and P(rij = 0) = 1−x. The probability x itself is a realization of a normal distribution with parameters μ = 0.16 and σ = 0.06. Values “< 0” or “> 0.3” are set to 0 or 0.3, respectivelyFor all rij) is the resulting random quasi-order.The transitive closure of the binary relation corresponding to  of m quasi-orders on n items and assume that PQO(n) is representative, in the sense that each quasi-order on the n items is contained in this set with the same probability. Thus, PQO(n) can be seen as a random sample drawn from the set of all quasi-orders on n items. In the induction step, we create for each quasi-order in PQO(n), subsequently represented by its relational matrix (qij), a number of z random extensions to n + 1—i.e., each random extension of (qij), denoted as the matrix . Duplicates are removed, i.e., each quasi-order occurs only once in PQO(n + 1). Regarding the choice of z, note that the chance a random extension of a quasi-order on n items again yields a quasi-order on n + 1 items decreases with increasing item number n. Thus, higher values for z must be used when n is getting larger, and in particular, this number must be adjusted depending on the number of quasi-orders that should be created during simulation. A general strategy easily possible is to specify z in a way such that slightly more quasi-orders than desired are generated. In a final step of the procedure, PQO(n + 1) is reduced to a required or reasonable size, for example by keeping only m randomly selected elements from PQO(n + 1) based on simple random sampling. This step is necessary to have the number of elements of the PQO(n)'s reasonably limited, if this process is repeated several times; for instance, when creating a sample consisting of 1000 quasi-orders on 10 items starting with the set of all four quasi-orders on two items.The This inductive algorithm produces representative samples of quasi-orders.Proposition. Let ≤1 and ≤2 be any two quasi-orders on n + 1 items. Denote by P1 and P2 the probabilities that ≤1 and ≤2 are contained in the set PQO(n + 1) that is generated according to the proposed inductive procedure, respectively. Then, P1 = P2.Proof. The proof is by induction. As its anchoring, the algorithm always starts with a representative set of quasi-orders for a sufficiently small item number l. We consider the inductive step from n≥l to n + 1. Let ≤i, n be the traces of ≤i  on the n items of the predecessor construction step, that is, the quasi-orders restricted to those n items. Since PQO(n) constructed in the predecessor step n is assumed to be representative in the sense we defined (the induction hypothesis), the probabilities for the two traces to belong to PQO(n), in respective order, Pn1,  and Pn2, , are the same. Moreover, all random extensions (to the n + 1 items) of any given trace quasi-order (on the n items) are equally likely, with the same probability of 2n−2 and independent of the trace quasi-order considered. Thus, Pn2,  = P2.To assess the quality of the algorithm used to create random quasi-orders we performed a first simulation study. In the study, we generated 100 random samples each of (sample sizes) 100, 1000, and 5000 quasi-orders on a set of six items, which were compared with the set of all possible quasi-orders concerning their distributions of important properties.Size: The size of a quasi-order ≤ is defined as the number of item pairs  with i ≤ j.Width: The size of a longest anti-chain in ≤ : An anti-chain is a subset of items {i1, …, ik} with ¬(ix ≤ iy) for all x, y ∈ {1, …, k} .Height: The size of a longest chain in ≤ : A chain is a subset of items {i1, …, ik} with i1 ≤ i2∧ …∧ ik−1 ≤ ik.We use the following properties as the criteria for representativeness of the samples.n = 6 items. The computations reported in this paper were done with a C program that implements the described algorithm on a computer with an Intel Core i5 2.50 GHz processor.The sampling process started with the set of all four quasi-orders on a set of two items. The inductive procedure described in the last section was then employed to create random quasi-orders on If we compare the mean values of the average sizes, widths, and heights taken over the 100 generated samples, we see that the results approximate the true values calculated for the set of all quasi-orders on six items very well . For example, linear orders are a special case of quasi-orders and, as pointed out by Augustin Kelava, may be useful in the analysis of item position or item order effects in large-scale assessments. Knowledge or learning space theory combinatorial structures or computational ITA analyses could be utilized to study these effects. Future research into this issue is needed.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.


The aim of the study was to verify if there is any association between exposure to Cu, Zn, Cd, Pb, As and the formation of malondialdehyde (MDA), 8-hydroxydeoxyguanosine (8-OHdG), advanced oxidation protein products (AOPP), and whether in this process cigarette smoking plays a role. The investigations were performed in the 352 smelters occupationally exposed to heavy metals and 73 persons of control group. Metals concentration was determined by atomic absorption spectrometry. MDA and AOPP concentrations were determined by spectrophotometric methods. The concentration of 8-OHdG was determined by ELISA method. It was demonstrated an increased Cu concentration in smoking smelters compared to non-smoking control group. It was noted no differences in Zn and Mg concentrations between the examined groups. Pb concentration was more than sixfold higher in the group of smoking smelters and about fivefold higher in the group of non-smoking smelters compared to the control groups (smokers and non-smokers). It was shown that Cd concentration in the blood was nearly fivefold higher in the smoking control group compared to the non-smoking control group and more than threefold higher in the group of smoking smelters compared to non-smoking. It was shown an increased As concentration (more than fourfold) and decreased Ca concentration in both groups of smelters compared to control groups. In groups of smelters (smokers and non-smokers), twofold higher MDA and AOPP concentrations, and AOPP/albumin index compared to control groups (smokers and non-smokers) were shown. Tobacco smoke is the major source of Cd in the blood of smelters. Occupational exposure causes accumulation of Pb in the blood. Occupational exposure to heavy metals causes raise of MDA concentration and causes greater increase in AOPP concentration than tobacco smoke. E (NO3−) , 5. Comp (NO3−) . These s. E , which is also generated in the process of biosynthesis of prostaglandins. MDA can be transmitted to distant tissues and, thanks to the possibility of forming covalent bonds with other molecules, modifies their structure, and consequently, changes its properties. MDA has high cytotoxicity, is mutagenic, and inhibits the activity of various enzymes, leading to inhibition of DNA replication, transcription, and breathing , 7. MDA Exposure of proteins to reactive oxygen species results in modification of amino acid residues, what alters protein structure and function . The chaAlbumin is a major protein of blood plasma, where it constitutes about 55 % of all proteins. This protein is also a major and predominant antioxidant in plasma . The abiThe oxidation of proteins, including albumin, leads to the formation of permanent products, which vary in terms of structure, function, and physicochemical properties. These are known as an advanced oxidation protein products (AOPP) . AlthougThe aim of the study was to verify if there is any association between exposure to metals  occurring in the work environment of smelters and the formation of MDA, 8-OHdG, and AOPP and whether in this process cigarette smoking plays a role. The influence of the exposure to heavy metals and tobacco smoke on the concentration of Ca and Mg also was analyzed.The investigations were performed in the whole blood, serum, plasma, and urine of 352 males working in the copper foundry and 73 healthy men, non-exposed occupationally to heavy metals, who were qualified as control group by clinician of primary medical care. The study protocol was approved by Local Bioethics Committee of Wroclaw Medical University (Nr KB: 469/2008). The study population was in similar age and similar BMI. Table 2 , immediately gently mixed, and centrifuged . The obtained serum, plasma, and whole blood were portioned and stored in sealed tubes . Blood samples of smelters and persons of control groups were stored at −80 °C until analysis.Venous blood was collected in the morning, after 12-h fasting. Serum was obtained according to the standard procedure by taking venous blood for disposable trace element-free tubes  with serum clotting activator, left at 25 °C to complete thrombosis, and centrifuged . In order to obtain plasma, blood was collected into trace element-free tubes containing EDTA-KThe urine was collected in the morning, according to the standard procedure. Subjects were asked to wash their hands before supplying the urine, in order to reduce contamination. Urine samples were stored in special container containing liquid nitric acid at −20 °C until determination.3 for acidification) was determined by FAAS method with snap Phillips PU 9360, used for hydride generation. In this method, wavelength λ = 193.7 nm deuterium background correction and quartz atomization chamber were used.Concentrations of metals were determined using Solaar M6 aparatus . The Pb concentration in whole blood (Pb-B) was determined by graphite furnace atomic absorption spectrometry (GFAAS) in the graphite Massman cuvette, the absorbance measurement at wavelength λ = 283.3 nm, with Zieeman background correction. The reference material was used BCR-194, -195, -196, IRMM, EU. The Cd concentration in whole blood (Cd-B) and in urine (Cd-U) was determined by the same method at wavelength λ = 228.8 nm, with Zieeman background correction. The concentration of Zn and Cu in serum (Zn-S and Cu-S) was determined by flame atomic absorption spectrometry (FAAS) in air-acetylene flame at wavelength λ = 324.8 nm. The reference material was used Single-Element Zinc (Copper), standard 1000 μg/ml, CPI International. The determination of Ca and Mg concentrations in serum (Ca-S and Mg-S) was also performed by this method at wavelength λ = 422.7 nm for Ca and λ = 285.2 nm for Mg, with deuterium correction of the glass. As a reference material for these metals was used Seronorm™ Trace Element Serum . The As concentration in urine (As-U) . The cotinine determination was carried out by the method described earlier , 29.5 M−1 cm−1). The MDA intensity was measured by spectrophotometric method at λ = 535 nm. The concentrations of MDA were expressed as micromoles per liter in plasma.Lipid peroxidation was measured in plasma by measuring the formation of thiobarbituric acid reactive substances (TBARS), quantified as malondialdehyde (MDA) equivalents, according to the method described earlier , 31. The2O. The next step, 200 μl blocking buffer was added and incubated at room temperature for 1 h. Fifty microliter of samples and 8-OHdG standards were added. After 10 min of incubation, monoclonal anti-8-OHdG was added , washed three times followed by addition of secondary antibody conjugated to horseradish peroxidase . The plate was washed three times with washing buffer and to each well 100 μl of substrate for peroxidase was added and incubated for 20 min. Followed by the addition of 100 μl of reaction stop solution. A spectrophotometric measurement of absorbance was performed at a wavelength λ = 450 nm. The content of 8-OHdG in the tested samples was calculated by comparison with a standard curve determined from standards treated similarly to the samples tested.To evaluate the concentration of 8-OHdG in the serum, a commercial test OxiSelect™ DNA Oxidative Damage ELISA Kit was used . To 96-well plate, 100 μl of conjugate 8-OHdG/BSA was added and incubated overnight at 4 °C, and then washed with HDetermination of AOPP-P concentration was performed by spectrophotometric method developed by Witko-Sarsat et al. , based o280 nm0.1 % = 0.5). To make the standard curve, the standard solutions at concentrations 0, 11.3, 22.6, 33.9, 45.2, 56.5 g/l were prepared, which were obtained by the dilution of albumin in PBS.Alb-S concentration was determined by using the bromocresol purple (BCP) , which reacts with albumin in phosphate buffer saline (PBS) at pH = 6.8. The concentration of albumin standard  was determined using the molar absorption coefficient , serum sample, or albumin standard were added and mixed. All samples were incubated for 5 min at 25 °C. Absorbances of the standards and of the samples were measured against the blank sample (working reagent) at wavelength λ = 603 nm. The increase in absorbance was proportional to the concentration of albumin in the sample.p < 0.05.Statistical analysis was carried out using the program Statistica 10.0. Data about examined groups, BMI, age, years of work in smeltery, and cotinine concentration, were analysed using ANOVA Kruskal−Wallis test. The concentration of metals , MDA-P, 8-OHdG, AOPP-P, albumin, and the value of AOPP/albumin index also using the same test were analyzed. In order to verify the association between parameters , the multiple linear regression models were performed. Statistical significance was accepted for In this study, the concentrations of Cu, Zn, Ca, and Mg in serum; Pb and Cd in whole blood; and Cd and As in urine were determined Table . In the In both smoking and non-smoking smelters, a significant increase in the concentration of Pb in the blood (Pb-B) compared to the control group (smokers and non-smokers) was observed. In the group of smoking smelters, 6-fold increase in Pb-B concentration in comparison to the smoking control group and 7.4-fold higher in relation to the non-smoking control group were observed. In the group of the non-smoking smelters, 5.7-fold higher Pb-B concentration compared to the non-smoking control group was observed. The concentration of Cd in the blood (Cd-B) was significantly higher at smokers (both the smelters and the control group) compared to non-smokers. In the blood of smokers (both the smelters and the control group), 3.3-fold higher Cd concentration compared to the non-smoking smelters and 4.8-fold higher compared to the non-smoking control group were observed.In the group of smoking smelters, 2.5-fold higher concentration of Cd in urine (Cd-U) compared to the non-smoking control group was observed. In the urine of smoking and non-smoking smelters, As concentration (As-U) was significantly higher compared to control groups (both smokers and non-smokers). In the smelters (smoking and non-smoking), more than 4-fold higher of As-U concentration in comparison to the smoking control group and more than 4.6-fold higher compared to the non-smoking control group were observed.In the groups of smelters (smokers and non-smokers), 2-fold higher MDA-P concentration compared to control groups  was shown Table . HoweverIn the plasma of smoking and non-smoking smelters, 2-fold higher of AOPP-P concentration in comparison to the smoking control group and 2.2-fold higher compared to the non-smoking control group was shown Table . There wAOPP/albumin index, which is expressed in micromole as the amount of AOPP per gram of albumin, was calculated. In the group of smoking smelters, 1.7-fold increase in the value AOPP/albumin index compared to the smoking control group and 3-fold increase of this parameter compared to the non-smoking control group was observed. 2.3-fold increase in the value of AOPP/albumin index in the non-smoking smelters compared to non-smoking control group was observed . The correlation of Pb-B and AOPP-P concentrations , and the AOPP/albumin index  in this group also were shown.In the serum of non-smoking smelters, the correlation was demonstrated between the concentration of Cu-S and Alb-S  compared to controls (99.7 ± 12.1 μg %) . Also, iBizoń et al. (2013) demonstrated that exposure to heavy metals causes an increase of Pb-B concentration . Higher Cd-B and Cd-U concentrations increase during exposure to tobacco smoke, that was observed by other scientists. Bizoń et al. (2013) demonstrated an increase in Cd-B concentration of smokers compared to non-smokers, both in the control group and the group of smelters  . In otheAn increase of As-U concentration in the group of persons occupationally exposed to heavy metals (18.9 ± 22.2 μg/g creatinine) in comparison to the control group (3.5 ± 3.0 μg/g creatinine) was observed . Bizoń eThe concentrations of Ca and Mg macroelements, which values have a significant impact on the reduction of the penetration capability of some toxic metals, such as Cd and Pb were also determined. The demonstrated differences in Ca-S concentration between the groups of smelters and control groups (both smokers and non-smokers) confirm that exposure to heavy metals causes a decrease Ca-S concentration. These results are similar to the study made by Kossowska et al., where was demonstrated a difference between Ca-S concentration in the group of smelters (90.4 ± 7.3 μg/ml) and the control group (99.3 ± 4.1 μg/ml) . In our Chronic exposure to heavy metals causes their accumulation in tissues and has the influence on the human health –50. It wThere was observed significantly higher plasma MDA levels (2.67 ± 0.69 μM) of workers exposed to lead than those in the control group (1.23 ± 0.61 μM) . HoweverBarbato et al. (2010) conducted a meta-analysis in environmental pollution exposed employees and in five studied groups, increase of 8-OHdG in the urine was observed, and in three studied group, there was no effect . Garçon It should be noted that AOPP concentration depends on many physiological and environmental factors, which may influence on the examined population  –59. The There are some studies demonstrating the impact of cigarette smoking on the formation the changes in the structure of proteins. It was shown an increase in the amount of carbonyl groups in the molecules of proteins, which are the component of AOPP. A difference in the amount of carbonyl groups was shown between the group of persons, which were intensive smokers (4.00 ± 1.31 nmol/mg protein) and the persons not exposed to tobacco smoke (0.80 ± 1.94 nmol/mg protein) . HoweverBecause of the fact that in the composition of AOPP is mainly the products of albumin’s oxidation; the determination of this protein in the serum of smelters was an important element of this study. On the concentration of AOPP, it has an influence not only in many factors discussed above, but also the concentration of albumin in the blood of the examined persons, which were in reference range (35–55 g/l) . For thiIn this study, it was demonstrated that an important factor influencing AOPP-P concentration is occupational exposure to heavy metals, especially Cu. The inverse correlation between Cu-S and Alb-S concentration may indicate the participation of Cu in the formation of AOPP and its oxidation effect on albumin. In the group of non-smoking smelters, it was demonstrated the correlation between the concentration of Pb-B and AOPP-P, and the value of AOPP/albumin index. This may indicate that high Pb-B concentration is an oxidation factor for proteins. These results suggest that Cu and Pb play an important role in protein oxidation. Cu can generate free radicals by Fenton reaction. Additionally, Pb can replace Cu in antioxidant enzymes that causes an increase in serum Cu concentration and intensifies an oxidative stress.In this study, it was confirmed the influence of heavy metals exposure on lipids peroxidation. It indicates that occupationally exposure can have an important role in lipid degradation and cell damage. It was not observed any effect of heavy metals exposure or cigarette smoking on oxidative modification of DNA. It was shown that cigarette smoking and occupational exposure to heavy metals have an influence on protein oxidation. These oxidizing factors of which the source is the environment seem to be important for destruction of protein molecule and limit its function.In this study, it was confirmed that tobacco smoke is the main source of Cd in the blood of smelters. The occupational exposure on heavy metals has no influence on Cd-B and Cd-U concentrations. With the increasing exposure to tobacco smoke and heavy metals, Ca concentration in serum was decreased.Exposure to heavy metals causes an increase of heavy metals concentrations in the blood, especially Pb-B, but also Cu-S and As-U of smelters. This oxidizing factor intensifies lipids peroxidation and causes an increase in MDA concentration, but they have no influence on DNA. The intensity of AOPP formation depends on the type of factor that it causes—for smelters, occupationally exposure to heavy metals is a stronger oxidation factor than tobacco smoke. The strongest oxidation effect on the protein has Cu and Pb.