Characterizing exposure-disease associations is a central issue in epidemiology, one which epidemiologists often approach by adopting the index of the odds ratio and presenting its point estimate, p-value and confidence interval. In this study, the parameter space of the odds ratio is partitioned into five mutually exclusive regions corresponding to ‘strong protective factor’, ‘weak protective factor’, ‘no association’, ‘weak risk factor’, and ‘strong risk factor’, respectively. The authors presented a suite of statistical methods tailored to such a five-region demarcation, including methods for hypothesis testing, confidence interval estimation and calculation of the sample size needed to obtain the desired level of statistical power. The authors show that the five-region methods can efficiently and informatively describe a putative exposure-disease association, including its presence or absence, as well as its direction and strength (if any association exists). Three published results were re-analyzed to demonstrate the methods. R code is provided for convenience as well. The five-region methods are recommended for routine use during the analysis of epidemiologic data. Epidemiologists often approach this issue by adopting the index of the odds ratio; presenting its point estimate, p-value and confidence interval; then declaring an exposure a risk factor for the disease if its odds ratio is statistically significantly greater than 1.0, or a protective factor if it is statistically significantly less than 1.00: OR = 1; H1: OR < 1 or OR > 1}, so can only be used to infer whether the exposure under study is a risk or protective factor for the disease. Further testing with regards to the cut-off points, such as {H0: OR ≥ 0.5; H1: OR < 0.5} or {H0: OR ≤ 2; H1: OR > 2}, incurs a multiple-testing penalty, and therefore, a power loss.Identifying a risk or protective factor for the disease may not be enough. We may also want to determine the strength of such an exposure-disease association. For example, we may want to know whether the odds ratio is statistically significantly greater than 2.0, indicating a strong risk factor; or between 0.5 and 1.0, indicating a weak protective factor. Conventional hypothesis testing for the odds ratio is for determining whether it is 1.0 or not: {Het al.2 proposed ‘three-sided testing’, a testing framework for simultaneous testing of ‘inferiority’, ‘equivalence’ and ‘superiority’ in clinical trials. A cardinal feature of Goeman et al.’s method is that the three hypotheses can be simultaneously performed at a significance level of α, while the family-wise error rate remains within α. The approach was later adopted outside of clinical trial settings to compare two digital communication systems3. However, our intended, more informative characterization of an exposure-disease association requires partitioning the parameter space of the odds ratio into a total of five ‘regions’, which is more than the three ‘sides’ that Goeman et al.2 had considered.Goeman et al.’s three-sided methods2 to the five-region characterization of exposure-disease associations. Our methods  are a reinterpretation of the results of the three-sided testing in terms of five regions. We will show that the ‘five-region methods’ are more statistically efficient than traditional methods in describing exposure-disease associations. Data from three published studies are re-analyzed to demonstrate the methods.In this paper, we expand the applicability of Goeman I, HII, …, HV as shown below:i is OR ∈Hi and the alternative hypothesis is OR ∈ i), for i = I, II, …, V.Assuming that two cut-off points for the odds ratio are chosen: 0.5 and 2. (Proper choices of the cut-off points will be discussed later.) For the lower cut-off point, OR < 0.5 indicates a strong protective factor, and 0.5 ≤ OR < 1 a weak protective factor. For the upper cut-off point, OR > 2 indicates a strong risk factor, and 1 < OR ≤ 2 a weak risk factor. The parameter space of the odds ratio is thus partitioned into five regions: H4 thus applies here, dictating that all five hypotheses can be simultaneously performed at a level of α, while still keeping the family-wise error rate within α. See Web Appendix 5 pointed out that the correction factor for a Bonferroni multiple testing procedure is not the total number of hypotheses tested, but the maximum number of hypotheses that can simultaneously be true. And in our case, the correction factor is 1. In other words, there is no need for multiple testing correction.Because the five regions are mutually exclusive, the true value of the odds ratio can lie in at most one of the regions, so that no two hypotheses can be simultaneously true. The ‘partitioning principle’α. Here, we propose to test HI with an α-level right-sided test , HII with one α/2-level left-sided test  and one α/2-level right-sided test  simultaneously; HIII with an α-level two-sided test , HIV with one α/2-level left-sided test  and one α/2-level right-sided test  simultaneously; and HV with an α-level left-sided test , respectively.The above partitioning principle says nothing about the types of the tests . For each hypothesis, any test can therefore be used as long as each of the five tests keeps its respective type I error rate within Zx is the x’th quantile of the standard normal distribution. As explained earlier, such a five-region test should keep the family-wise error rate within α.Assuming normality for the log odds ratio, the ‘five-region test’ is detailed below:α/2-level one-sided tests simultaneously’ that we used here for HII and HIV are very different from Schuirmann’s procedure for equivalence6. For the HII test, the null hypothesis (0.5 ≤ OR < 1) is to be rejected if either of the two null hypotheses  of the ‘α/2’-level one-sided tests is rejected. Similarly, for the HIV test, the null hypothesis (1 < OR ≤ 2) is to be rejected if either of the two null hypotheses  of the ‘α/2’-level one-sided tests is rejected. By contrast, in Schuirmann’s procedure, the null hypothesis  is to be rejected only if both null hypotheses  of the ‘α’-level one-sided tests are rejected.Note that the ‘two α.Alternatively, one can calculate the ‘five-region p-values’:i in {I, II, …, V}, the pi in (3) corresponds to the Hi test in (2), that is, an Hi will be rejected in (2) if and only if its pi ≤ α in (3).Web Appendix II, HIII, HIV, and HV, are rejected;(SP) a strong protective factor (the upper left non-shaded corner), if four hypotheses, HI, HIII, HIV, and HV, are rejected;(WP) a weak protective factor (the middle left non-shaded area), if four hypotheses, HIII, HIV, and HV, are rejected;(P) a protective factor without strength information (the non-shaded area nestled between the upper left non-shaded corner and the middle left non-shaded area), if three hypotheses, HI, HII, HIII, and HV, are rejected;(WR) a weak risk factor (the middle right non-shaded area), if four hypotheses, HI, HII, HIII, and HIV, are rejected;(SR) a strong risk factor (the upper right non-shaded corner), if four hypotheses, HI, HII, and HIII, are rejected;(R) a risk factor without strength information (the non-shaded area nestled between the middle right non-shaded area and the upper right non-shaded corner), if three hypotheses, HV, is rejected;(NSR) not a strong risk factor (the left lightly-shaded area), if one hypothesis, HI, is rejected;(NSP) not a strong protective factor (the right lightly-shaded area), if one hypothesis, HI and HV, are rejected.(NS) not a strong factor, risk or protective (the middle lightly-shaded area), if two hypotheses, HIf none of the five hypotheses are rejected, a five-region test is inconclusive  are possible. For example, it is not possible to have the middle three hypotheses, HII, HIII, and HIV, being rejected , we must also have III and HIV rejected) and V rejected), and if IV rejected), we must also have III and HII rejected) and 1−α/2 > Z1−α (HI rejected).Note that not all of the total 20: OR = 1; H1: OR < 1 or OR > 1}) leads to any conclusion at all , it only provides directional information about whether exposure is a protective or a risk factor , not a strong protective factor (HI rejected), or not a strong factor, risk or protective (both HI and HV rejected); at the same α level, the classical test leads to none conclusion at all.By comparison, if a classical hypothesis test  confidence interval for the odds ratio.The well-known duality between confidence intervals and hypothesis tests dictates that an α-level right-sided test for an odd ratio in HI, an α-level two-sided test for an odds ratio in {HII, HIII, HIV}, and an α-level left-sided test for an odds ratio in HV. This is essentially Goeman et al.’s three-sided testing2 . Therefore, the resulting ‘five-region confidence interval’ is also identical mathematically to Goeman et al.’s three-sided confidence interval2.Here, we propose using an et al.’s. Here, we explicitly check each and every one of the five regions, and reject a region if and only if none of its points fall within the confidence interval. Readers can verify that such use of the confidence interval leads to exactly the same conclusion of a five-region test in the previous section. For an example, see the solid lines  and the attached conclusions in a five-region test.However, our use of the confidence interval is different from Goeman Using the same principle that a region is rejected if none of its point fall within the confidence interval, a classical confidence interval can also be used to determine a five-region conclusion  We performed a total of 100,000 Monte-Carlo simulations to approximate the power for a given sample size and used the bisection method to calculate the sample size needed to reach the target power. For comparison, we used the same method to calculate the sample size needed for a classical confidence interval when used to make a five-region conclusion. Web Appendix As pointed out earlier, when a five-region test  does reach a conclusion it can lead to one of 9 different conclusions. Here we calculate the required sample size to reach a target power of 80% at Figures Figures a priori information is available) or for all possible odds ratio values  should help researchers better plan their studies under time and cost constraints.As many researchers often do not have a very precise aim for their studies at the outset, calculating and comparing the required sample sizes for different conclusions and specific odds ratio values  to examine the association between Parkinson’s disease and preceding nonfatal cancers. A conditional logistic regression was conducted for a matched-data analysis. Elbaz et al. reported an adjusted odds ratio of 0.22 [95%  confidence interval: 0.03–2.24] for bladder cancer occurring before the onset of Parkinson’s disease. We calculated a standard error from the confidence interval presented (and likewise for the following two examples). The fact that the value 1 is within the classical confidence interval indicates that the observed negative association between bladder cancer and Parkinson’s disease (OR = 0.22) is not statistically significant. The classical confidence interval itself is also inconclusive since it spans all five regions.Elbaz V = 0.031) of the HV region (a strong risk factor). Similarly, the 95% five-region confidence interval is 0.03–2.00, excluding the HV region entirely. In this example, while neither the classical test nor the classical confidence interval is conclusive about risk of exposure, the five-region test  at the same significance level of 0.05 can rule out the possibility that bladder cancer is a strong risk factor for Parkinson’s disease; it can only be one of the following four possibilities: a strong protective factor, a weak protective factor, no association, or a weak risk factor.We then used the proposed method to reanalyze the data. The five-region test leads to a rejection . Parent et al. reported an adjusted odds ratio of 1.04 [95%  confidence interval: 0.49–2.22] for the association between night work and melanoma. The fact that the value 1 is within the classical confidence interval indicates that the association is not statistically significant. Again, the classical confidence interval itself, which spans all five regions, is also inconclusive.Parent I (a strong protective factor) and HV (a strong risk factor) . Similarly, the 95% five-region confidence interval is 0.50–2.00, excluding both the HI and HV regions. We see that while neither the classical test nor the classical confidence interval is conclusive about the risk of exposure, the five-region test  at the same significance level of 0.05 can rule out the possibility that night work is a strong factor, risk or protective, for melanoma; it can only be one of the following three possibilities: a weak protective factor, no association, or a weak risk factor.Using the proposed method to reanalyze the data, the five-region test rejects Het al.9 recruited a total of 610 cases and 837 controls to examine the association between indoor air pollution from coal combustion and the risk of neural tube defects. Logistic regression was conducted adjusting for the matching variables  and a number of other potential confounders . Li et al. reported an adjusted odds ratio of 1.50 [95%  confidence interval: 1.10–2.10] for the association between indoor air pollution from cooking and neural tube defects. The fact that the value 1 is not within the classical confidence interval indicates that the observed positive association between indoor air pollution from cooking and neural tube defects (OR = 1.50) is statistically significant. However, the classical confidence interval encompasses the HIV (weak risk factor) and the HV (strong risk factor) regions, so we cannot deduce the strength of this risk factor.Li IV region (weak risk factor), all other 4 regions  are rejected by the five-region test. The 95% five-region confidence interval is 1.10–2.00, which is fully embedded within the HIV region. In this example, the classical method can only label indoor air pollution from cooking as a risk factor for neural tube defects. But the five-region method at the same significance level of 0.05 can be more specific in stating that it is a weak risk factor with an odds ratio no greater than 2.Using the proposed method to reanalyze the data, we find that except for the HIII). We then chose two cut-off points, an upper cut-off (cupper > 1) and a lower one (clower < 1) which could have been values other than the ‘2’ and ‘0.5’ used in this paper, to mark the boundaries between strong and weak effects. We thus demarcated two regions (using cupper) for strong and weak risk factors (HV and HIV) to the right of the HIII region, and to the left of that we demarcated two regions (using clower) for strong and weak protective factors (HI and HII). Ideally, if we let clower = 1/cupper, then the testing framework of the five-region test is completely symmetrical in the two directions of an exposure-disease association. By imposing symmetry, we avoid all appearance of prejudice on whether the study exposure is a risk or a protective factor. Also, note that the five-region demarcation should be done beforehand since a data-snooping demarcation will incur bias. The required sample sizes will be different with different demarcations of the five regions as well.The proposed five-region methods hinge on a proper demarcation of the five regions. For the odds ratio index used in this paper, the value ‘1’ was taken to be the center point, indicating no exposure-disease association. Therefore, we let a solitary {1} be a ‘region’  that should be performed in the region it belongs to. We pick one (or both) cut-off point(s), as appropriate, for the limit(s), if the previous step cannot find one or more suitable limit. Finally, we examine the span of this exact five-region confidence interval to make conclusions for the exact five-region test. Here, the coverage probability of the exact five-region confidence interval is at least (1 − α) and the type I error rate of the exact five-region test is at most α. Web Appendix In this paper, we invoked the normality assumption for the log odds ratio (large sample theory). For small sample sizes exact methods will be needed. One can calculate the four limits  using the classical V) or very small (in HI), can a five-region/three-sided confidence interval be wider than a classical one  odds ratio can produce at most two open regions (where the precisions of the odds ratios can be traded off). The two open regions plus the remaining one closed region constitute a total of three ‘sides’, and thus no further performance improvement can be expected beyond Goeman et al.’s three-sided confidence interval2.If a researcher opts for a more precise estimation, he/she will choose a method that produces a narrower confidence interval. In most circumstances, a five-region confidence interval  is at least as narrow as a classical one Fig. . Only whTo characterize the association between an exposure and a disease, the current epidemiologic paradigm calls for a trio of inferential statistics to be presented for the odds ratio: a point estimate, a  p-value, and a  confidence interval, respectively. In this paper, we show that the five-region methods can describe a putative association more efficiently and informatively, including its presence or absence, as well as its direction and strength (if any association exists). The five-region methods are recommended for routine use during the analysis of epidemiologic data.Supplementary information.


Although the Short Test of Functional Health Literacy in Adults (S-TOFHLA) is widely used, misidentification of individuals with low health literacy (HL) in specific HL dimensions, like numeracy, is a concern. We examined the degree to which individuals scored as “adequate” HL on the S-TOFHLA would be considered as having low HL by two additional numerical measures.English-speaking adults aged 45–75 years were recruited from a large, urban academic medical center and a community foodbank in the United States. Participants completed the S-TOFHLA, the Subjective Numeracy Scale (SNS), and the Graphical Literacy Measure (GL), an objective measure of a person’s ability to interpret numeric information presented graphically. Established cut-points or a median split classified participants and having high and low numeracy.n = 187), on average were: aged 58 years; 63% female; 70% Black/African American; and 45% had a high school degree or less. Of those who scored “adequate” on the S-TOFHLA, 50% scored low on the SNS and 40% scored low on GL. Correlation between the S-TOFHLA and the SNS Total was moderate , while correlation between the S-TOFHLA and the GL Total was large .Participants . Health literacy (HL) represents a complex intersection of skills needed to “obtain, process, understand, and communicate about health-related information needed to make informed health decisions.” –3 The 20HL measures are useful for evaluating and classifying patient abilities so that information can be presented in a way to meet patients’ skills and needs. Yet, existing measures of HL may lack the specificity to accurately assess patients’ ability to comprehend numeric information, providing a limited view of patients’ abilities , 12–14. The S-TOFHLA is the most frequently used measure of HL, used in over half of all published papers measuring HL , 11. Howclinicaltrials.gov: NCT02151032) designed to investigate the use of decision aids in colorectal cancer screening. Eligible participants were English-speaking, aged 45–75 years, and had no history of colorectal cancer. Participants were recruited in person between November 2012 and January 2013 in the Greater Houston Metropolitan area from: , 3210, 3Numeracy is a vital element of HL and is a priority for optimizing risk communication , 17. TheThe GL is a 13-item objective measure that assesses how individuals understand graphically-presented quantitative information . UnderstAs reported in the previous section, no optimal score cut-points exist for the SNS and the GL. Thus, we used three complementary scoring strategies. First, we used the median score from the total score to identify low and high literacy groups, based on a median split. This approach is justified based on the study setting, where more than half of the participants came from a community-based organization serving underserved groups. Second, a more conservative threshold was used, where the lowest quartile of participants from the total scores were classified as having low HL, an approach common in educational literature , 48. We Demographic characteristics, frequencies, and descriptive statistics were generated to summarize our participant population using IBM SPSS Version 23. The three S-TOFHLA HL categories  were used for scoring. For the other HL measures, we categorized the variables into high and low categories Table . These cOne hundred eighty-nine individuals enrolled in the study (67 from the cancer prevention center and 122 from the foodbank) and completed the questionnaires; however, two participants were excluded from the analyses due to missing data, for a total of 187 participants (Table n = 134) scored “adequate” on the S-TOFHLA, while only 10.7% (n = 20) scored in the “marginal” category, and 17.6% (n = 33) scored “inadequate”  of participants were in the bottom 25th percentile of the SNS Total score. Moreover, for the Ability and Preference Subscales, 30% (n = 40) and 35% (n = 47) of participants were categorized as “adequate” by the S-TOFHLA but in the bottom 25th percentile of their respective scale scores.Of those who scored “adequate” on the S-TOFHLA, 26% (n = 53) of those classified as “adequate” on the S-TOFHLA were low on GL Total score. Furthermore, 18% (n = 24) of those scoring in the bottom 25th percentile on GL Total, scored “adequate” on the S-TOFHLA. For the GL Subscales, 13% (n = 18) of those classified as “adequate” on the S-TOFHLA were in the bottom 25th percentile of the GL1 Subscale. For the GL2 subscale, 21% (n = 28) of those classified as “adequate” on the S-TOFHLA were in the bottom 25th percentile. Lastly, 30% (n = 40) were classified as “adequate” on the S-TOFHLA and in the bottom 25th percentile on the GL3 Subscale.Forty percent . The correlation between the S-TOFHLA and SNS Prefer scores was not significant . The S-TOFHLA and GL Total and GL 1 scores showed large, positive correlations . The correlation for the S-TOFHLA and GL3 was moderate in magnitude .Correlations between S-TOFHLA scores and SNS Total  to engage in these types of decisions. HL is a complex construct, and existing literature identifies and describes multiple ways of measuring it. For example, Duell et al. identified three levels for HL measurement: basic, communicative/interactive, and critical HL . These lThere are various approaches to help address the challenges associated with measuring HL. First, the assumption that a single HL measure is adequate may not be the case. The HL measures included in our investigation show the need for capturing the complex skills that make-up HL. While previous studies have provided evidence to push back against commonly-used measures, such as the S-TOFHLA and REALM, a continued effort to challenge the expected use of one of these tools as a way to definitively identify those with low HL is needed , 50. FurThis research study has potential limitations. This study was conducted in a large urban area using a convenience sample. Thus, the sample is diverse and matches the makeup of large urban centers, but rural patients may have not been well represented which may impact generalizability. Second, we used the 36-item S-TOFHLA measure, which does not assess numeracy. Although this measure is broadly accepted , includiThe S-TOFHLA is limited in its measure of HL. Though widely used, those with inadequate HL skills may be over-classified as having “adequate” HL, based on their S-TOFHLA scores. Use of HL measures that include numeracy, graphical, and preference questions is essential when considering measuring HL. Existing numeracy or graphical measures may be too cumbersome for the clinical setting but have higher utility when categorizing those with high and low numerical skills, which may be of particular interest to those developing decision support tools. Developing a brief tool with both subjective and objective quantitative questions may augment HL numeracy measurement.


Scientific Reports 10.1038/s41598-017-09769-y, published online 30 August 2017Correction to: The original version of this Article contained an error in the title of the paper, where the family name “Crassulaceae” was incorrectly given as “Crassulacaee”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supplementary Material.


G that together cover all nodes, or edges, of G.Reconstructing the genome of a species from short fragments is one of the oldest bioinformatics problems. Metagenomic assembly is a variant of the problem asking to reconstruct the circular genomes of all bacterial species present in a sequencing sample. This problem can be naturally formulated as finding a collection of circular walks of a directed graph safe if it returns only those walks  that appear as subwalk in all metagenomic assembly solutions for G. A safe algorithm is called complete if it returns all safe walks of G.We address this problem with the “safe and complete” framework of Tomescu and Medvedev (Research in computational Molecular biology—20th annual conference, RECOMB 9649:152–163, G, and a safe and complete algorithm finding all safe walks of G. In the node-covering case, our algorithm runs in time n and m denote the number of nodes and edges, respectively, of G. This algorithm constitutes the first theoretical tight upper bound on what can be safely assembled from metagenomic reads using this problem formulation.We give graph-theoretic characterizations of the safe walks of  In general, we do not know how many bacterial species are in the sample, so we cannot place any bound on the number of circular walks. Hence, in our above formulation they can be any arbitrary number. See the next section for formal definitions, and Fig. Because of the multiple genomes present in the sample, one can no longer formulate a solution for the metagenomic assembly problem as a It can be easily verified that the walks from , 20–22—wall walks w of a graph G such that for any metagenomic assembly solution R of G, w is a sub-walk of some circular walk in R. As opposed to the exhaustive search strategy from  be the subgraph of G induced by the edges of C. Clearly, C is a node-covering circular walk of G[C], and thus G[C] is strongly connected. Moreover, we can argue that w is a node-omnitig in G[C], as follows. By taking the shortest proper circular sub-walk of C passing through x we obtain a cycle x. From (b), we get that w is a sub-walk of G[C], then also all edges of w appear in G[C] and thus w is a walk in G[C]. The two conditions from the definition of node-omnitigs are preserved under removing edges from G, thus w is a node-omnitig also in G[C]. By applying Theorem G[C] we obtain that w is a sub-walk of all node-covering circular walks of G[C], and in particular, also of C. We have thus shown that for every node-covering metagenomic reconstruction R of G, there exists w is a sub-walk of C. Therefore, w is a node-safe walk for G. Otherwise, let mentclass2pt{minimThe following statement is a simple corollary of condition (b) from Theorem LetG be a strongly connected graph, and letw be a safe walk inG. Then w is either a path or a cycle.We now give the analogous characterization of edge-safe walks. We first recall the analogous definition of edge-omnitigs from . This isEdge-omnitig, [G be a graph and let G. We say that w is an edge-omnitig if for all  Let G bG that are sub-walks of all edge-covering circular walks of G. Our characterization of the edge-safe walks considered in this paper is:In , it was LetG be a strongly connected graph. A walk inG is an edge-safe walk inG if and only if the following conditions hold:w is an edge-omnitig, andthere exists such thatw is a sub-walk of all cycles passing through e.S(G) obtained from G by sub-dividing every edge once.Theorem G, we let S(G) denote the graph obtained from G by subdividing each edge once. Namely, each edge  of G is replaced by two edges u, and exactly one out-neighbor, v. We can analogously define this operation for a walk w in G, and then consider the walk S(w) in S(G).Given a graph w is an edge-safe walk in G if and only if S(w) is a node-safe walk in S(G). Indeed, observe that the edge-covering metagenomic reconstructions of G are in bijection with the node-covering metagenomic reconstructions of S(G), the bijection being w is a sub-walk of a walk C in G if and only if S(w) is a sub-walk of S(C) in S(G). Therefore, w is an edge-safe walk in G if and only if S(w) is a node-safe walk in S(G).The proof follows the outline given in Fig. w satisfies conditions (a) and (b) of Theorem G if and only if S(w) satisfies conditions (a) and (b) of Theorem S(G).It remains to show that Condition (a): It immediately follows from the definition that if S(w) is a node-omnitig in S(G) then w is an edge-omnitig in G. Assume now that w is an edge-omnitig in G. By the construction of S(G) and S(w), between any two consecutive nodes of S(w) there can be only one path in S(G) . Therefore, S(w) is a node-omnitig in S(G).Condition (b): Suppose that there exists an edge G passing through e contain w as sub-walk. Then by construction all cycles in S(G) passing through S(w) as sub-walk. Conversely, suppose that there exists a node S(G) passing through x contain S(w) as sub-walk. If x is a node of the type u, v) of G, then it also holds that all cycles in G passing through w as sub-walk. Otherwise, if x, y) be an arbitrary edge of G out-going from x; this exists because G is strongly connected. We claim that all cycles in G passing through w as sub-walk. Indeed, let S(G) corresponding to the edge . The set of cycles of S(G) passing through S(G) passing through x. Therefore, all cycles of S(G) passing through S(w) as sub-walk. We have now reduced this case to the previous one, when x is a node of the type u, v) of G, and the claim follows. In this section we give an algorithm for finding all node-safe walks of a strongly connected graph. In the next section we show how to implement this algorithm to run in We begin with an easy lemma stating a simple condition when a maximum overlap of two node-omnitigs is a node-omnitig.LetG be a graph, and let be a walk of length at least two inG. We have thatw is a node-omnitig if and only if and are node-omnitigs and there is no path with first edge different than and last edge different thanw is a node-omnitig. The forward implication is trivial, as by definition sub-walks of node-omnitigs are node-omnitigs. For the backward implication, since both w can also be a single node.The following definition captures condition (b) from Theorem Certificate) Let G be a graph and let w be a walk in G. A node w is a sub-walk of all cycles passing through x is called a certificate of w. The set of all certificates of w will be denoted  C first traverses C first traverses w is a node-omnitig and x-x path w is a node-omnitig.We now prove the reverse inclusion by induction on the length of w into the two walks x, once it passes through x passes also through w, and hence We now use the inductive hypothesis to show that if C the sub-walk of C starting at k, that is, Given a circular walk G (possibly with duplicates), but does not return each node-safe walk explicitly. Instead, it returns a node-covering circular walk C of G and the set of pairs  such that C is a node-safe walk.Algorithm 1 finds all node-safe walks of a strongly connected graph C and checking whether each sub-walk of C starting at index i and of length k is a node-omnitig and has at least one certificate. If so, then it stores the index i in a set k. The algorithm first deals with the case C is a node-omnitig (Line 7) and whether it has at least one certificate (Line 8). The case C has at least one certificate (Line 12).The algorithm works by scanning G and C to perform these verifications in constant time. This algorithm can be modified to output node-safe walks also without duplicates. For clarity, we explain this idea in the proof of Theorem maximal node-safe walks, i.e., those that are not sub-walks of any other node-safe walk.In the next section we show how to pre-process Given a strongly connected graphG, Algorithm 1 correctly computes all the node-safe walks ofG, possibly with duplicates.k that the set i for which C is a node-safe walk of length k. In the base case C is a node-omnitig (Line 7). We also check if C has at least one certificate, by checking  is a node-safe walk  is a node-omnitig if and only if We will first prove by induction on entclass1pt{minimaG are paths or cycles, thus of length at most n. By the definition of node-safe, they are also sub-walks of C. Thus for each node-safe walk w of G of length By Corollary In this section we describe the implementation of Algorithm 1. We first show how to compute the certificates of all nodes.G be a strongly connected graph withn nodes andm edges. We can compute the sets for all, in timeO(mn).Let x . We then construct the graph G once. That is, we replace every node x of G with two nodes y, z) of G, we add to We start by initializing O(m). We then iterate through all x belongs to all cycles of G passing through y, and thus we add y to O(mn) time. For every The following lemma shows how to check in constant time the first condition in the definition of node-omnitig.LetG be a graph withm edges. We can pre-processG in time  and space such that for every two distinct edges, we can answer inO(1) time if there is a path inG with first edge different than and last edge different thanWe show how to pre-compute a table G by removing z to compute how many of its in-neighbors are reachable from z, and for each in-neighbor w, we let w is reachable from z path in G with first edge different than w, z) if and only if O(m), and then we process each edge of We iterate through all edges Using e.g., the result of , we can LetG be a graph withm edges, we can pre-processG in timeO(m) such that for every edge we can answer inO(1) time whether  is the onlyx–y path .strong bridge is an edge whose removal increases the number of strongly connected components of a graph  is a strong bridge. In [A ee e.g., ). It is idge. In  it was si in C, the smallest  index C has a non-empty set of certificates if and only if k to i, that is, The following lemma shows how to check in constant time condition (b) from Theorem LetG be a graph withn nodes andm edges, and let be a circular walk inG, with We can pre-processG and C in time , such that for every and, we can answer inO(1) time ifi, by n telling in how many of To simplify the notation, given an integer t as long as t reaches 0, then all sets O(n), and there are O(d) intersections to compute, giving a total of We begin by computing O(n) time.For O(n), by scanning the current intersection and removing the elements of d times, because for each i we start this scan from index Otherwise, we keep increasing We are now ready to combine these lemmas into the main theorem of this section.Algorithm 1 can be implemented to run in time for any strongly connected graph withn nodes andm edges.G as G is strongly connected), and then continue from Any strongly connected graph admits a node-covering circular walk given in .G and C as indicated in Lemmas d. By Lemma In the next section we discuss how to optimize Algorithm 1 to start with a node-covering metagenomic reconstruction of minimum total length. However, there are graphs in which any node-covering metagenomic reconstruction has length maximal node-safe walks, that is, in node-safe walks that are not sub-walks of any other node-safe walk. A trivial way to obtain these is to take the output of Algorithm 1, convert it into the set of all node-safe walks of G, and run a suffix-tree-based algorithm for removing the non-maximal ones, in time linear in their total length. However, given a node-covering circular walk C of length In a practical genome assembly setting we want to reconstruct fragments of the genomes as long as possible. As such, we are interested only in k is maximal if it is not extended into a node-safe walk of length C to check for previous occurrences of each walk of length k.In the next theorem we show how to reduce this time complexity to Given a strongly connected graphG withn nodes andm edges, Algorithm 1 can be modified to output the maximal node-safe walks ofG explicitly and without duplicates, with a running time ofC of G, of length C there can start the occurrence of at most one maximal node-safe walk. By Corollary n, thus the sum of the lengths of all maximal node-safe walks of G is C of all maximal node-safe walks without duplicates, then we can output all of them explicitly within the stated time bound.Let w of length k is maximal if no occurrence C of w in C was extended left or right at step w in C, as follows. Initially, we build the suffix tree T of the (linear) string C for a length T to the node C. In A node-safe walk nt}O(n2) . As we si to suffix-link in T to move to the node T to C with its first character removed). For a detailed discussion on the properties of the suffix tree, see e.g., [T such that the path from the root to C at step k are maximal, we traverse C again to output these node-safe walk.As we advance from ee e.g., . We thenAfter building the suffix tree using , the chierations . Thus, tmentclasspt{minimaIn this section we adapt Algorithm 1 and its implementation to find edge-safe walks, as characterized by Theorem LetG be a strongly connected graph withn nodes andm edges. In time we can output an edge-covering circular walk C, and the set of all pairs  such thatC is an edge-safe walk ofG.w consists of the edges e such that all cycles passing through e contain w as sub-walk. Analogously to Lemma w equals the intersection of the sets of certificates of its individual edges. The algorithm for the edge-safe case is that same as Algorithm 1, with the difference that we now start with an edge-covering circular walk C and we do not check anymore that each C is the only The proof is analogous to the node-safe case, and thus we briefly sketch the differences. In the edge-covering case, the set of certificates of a walk C has length at most mn and can be found in time O(mn). The certificates of all edges can be similarly computed in time C of length O(mn). The only difference in its proof is that the sets of certificates now have size at most m, thus their intersection takes time O(m). This implies that we can pre-compute G and C in time By the same argument given in the proof of Theorem C has length O(mn). After this pre-processing phase, the algorithm itself works in time With a proof identical to the one of Theorem Given a strongly connected graphG withn nodes andm edges, we can output the maximal edge-safe walks ofG explicitly and without duplicates, in time ofG has a node-covering circular walk of length n if and only if G is Hamiltonian. Observe though that instead of a single node-covering circular walk, we can start with a node-covering metagenomic reconstruction possibly consisting of multiple circular walks, and apply Algorithm 1 to each walk in the reconstruction. This is correct by definition, since node-safe walks are sub-walks of some walk in any node-covering metagenomic reconstruction.A trivial way to optimize Algorithm 1 is to start with a node-covering circular walk of minimum length. However, this is an NP-hard problem, since G, we construct the graph G once . In terms of real graph instances, the worst-case complexity of our algorithm may be prohibitive, and thus improving it is an important problem.We also leave for future work an idealized experimental study similar to what was done for the single genome case in . This coThe ultimate goal of a “safe and complete” approach is to be adapted to the peculiarities of real data, such as sequencing errors, insufficient sequencing coverage, reverse complements. However, our belief is that we first need a clean and solid theoretical foundation, which can later be extended, or weakened, to account for such features.


An X-ray Talbot-Lau interferometer scanning setup consisting of three transmission gratings, a laboratory-based X-ray source that emits X-rays vertically, and an image detector on the top has been developed for the application of X-ray phase imaging to moving objects that cannot be tested clearly with conventional absorption contrast. The grating-based X-ray phase imaging method usually employs a phase-stepping (or fringe-scanning) technique by displacing one of the gratings step-by-step while the object stays still. Since this approach is not compatible with a scanner-type application for moving objects, we have developed a new algorithm for achieving the function of phase-stepping without grating displacement. By analyzing the movie of the moiré pattern as the object moves across the field of view, we obtain the absorption, differential phase, and visibility images. The feasibility of the X-ray phase imaging scanner has been successfully demonstrated for a long sample moving at 5 mm/s. This achievement is a breakthrough for the practical industrial application of X-ray phase imaging for screening objects carried on belt-conveyers such as those in factories. Among the methods of X-ray phase imaging reported so far, X-ray grating interferometry, such as X-ray Talbot interferometry5 and X-ray Talbot-Lau interferometry6, which is an advanced form of X-ray Talbot interferometry, is especially effective for practical X-ray phase imaging in clinical and industrial fields because they operate with conventional X-ray generators in hospitals and factories.X-ray phase imaging has been attracting attention because it produces significant contrast for weakly absorbing objects, such as polymers and soft biological tissues7, respectively. X-ray phase imaging separates these information quantitatively and generates absorption, refraction, and scattering  images.A Talbot or Talbot-Lau interferometer detects the slight X-ray refraction caused by an object. When the object is placed in the interferometer, the intensity change is detected through the gratings at each pixel of an image detector. This feature can be seen as a moiré pattern. X-ray phase imaging with interferometers is normally performed by analyzing moiré patterns acquired through a specific technique; that is, a fringe-scanning or phase-stepping technique by displacing one of the gratings. Moiré fringes contain the information of attenuation, phase shift (refraction), and scattering by an object; they appear as the changes of fringe intensity, fringe position, and fringe visibilityIn the fringe-scanning procedure in combination with the grating displacement, the object is assumed to stay still in the field of view (FOV) during the entire scan, which is typically 30 s in the clinical application of X-ray phase imaging reported in ref. This paper proposes an approach of a fringe-scanning method without any grating displacement for moving objects in the configuration of an X-ray Talbot-Lau interferometer. An apparatus (X-ray phase scanner) consisting of an X-ray source that emits X-rays vertically, X-ray transmission gratings, a sample stage that moves objects horizontally like a belt-conveyer, and a photon-counting image detector have been developed to demonstrate the approach. A pilot experiment of successful X-ray phase imaging for moving objects larger than the FOV of the apparatus is presented.An X-ray generator consisting of a tungsten rotating anode  was placed at the bottom of the scanner to generate cone-beam X-rays in the vertical direction. The generator can be operated at up to 60 kV tube voltage and 90 mA tube current (5.4 kW maximum load). The focus size on the anode surface was 0.3 mm × 3 mm and the effective X-ray source size was 0.3 mm × 0.3 mm, taking into account the take-off angle (6°). The X-ray source size does not satisfy the requirement of spatial coherence for a Talbot interferometer, and therefore Talbot-Lau configuration with a source grating (G0), π/2-type phase grating (G1) and absorption grating (G2)  was employed. The gratings were mounted and aligned using mechanical stages, as shown in Fig. 6 cps/pixel with a readout speed up to 174 fps frame rate.An image detector  was mounted on the top of the scanner to capture a movie of the moiré pattern while the sample is moved across the FOV. The detector is a photon-counting two-dimensional hybrid pixel array semiconductor detector having an active area of 77.5 mm × 38.5 mm with a pixel size of 100 μm × 100 μm. The detector has an X-ray photon count rate of greater than 1011. However, the actual gratings used for a Talbot-Lau interferometer have slight deformations due to the fabrication process and mounting, and the degradation in resultant image quality is not negligible. It is not easy to fabricate an ideal grating, especially when a large area is required. In this study, we propose an algorithm that is applicable even when the gratings are deformed. Thus, the degradation of image quality is avoided, which leads to reduction in the cost of the apparatus.An approach for the fringe-scanning method without grating movement has been reported by generating artificial moiré fringes with the assumption that the gratings are ideally fabricated and the spacing of the moiré fringes is constantA, refraction φ, and visibility V from a recorded moiré movie of sample scanning. Here, p and q are the image coordinates fixed to the moving sample in the x- and y-directions, respectively, as shown in Fig. p, q) on the detector plane, and therefore, the coordinates on the sample are calculated by considering the magnification due to a cone-beam projection. The movement of a sample is assumed to be parallel with the x-axis in this setup. Therefore, x = p−vt and y = q, where v is the nominal sample speed on the detector plane and t is time.The purpose of the algorithm is to calculate three resultant images, namely, absorption 12 with grating displacement without a sample. Given the moiré pattern observed at the j-th step of a fringe-scan is Ij0,d1 is the period of G1, z0 is the distance between G1 and G2 and M is the number of phase steps (M ≥ 3). A0 is illumination intensity, φ0 is the background differential phase caused by the deformation and alignment of gratings, and V0 is the visibility of moiré fringes. Note that φ0 is wrapped between −π to π, and phase unwrapping is not needed here.As a pre-processing, the effect of the built-in moiré pattern, which is normally observed without a sample because of incomplete grating alignment and slight grating deformation, was measured in advance as an instrumental function. For this purpose, we employed the conventional fringe-scanning technique or modified robust fringe-scanning technique explained elsewherep, q) of the sample must travel across at least one moiré fringe along the x-axis. Therefore, φ0 must have values spanning from −π to π at every y (Step 3). If this condition is not satisfied, gratings are re-aligned, for examply by inclining one of the gratings, so that more moiré fringes are introduced, and steps 1 and 2 are repeated. After φ0 is thus defined, the FOV is divided into n regions, so that the pixel  having the value of φ0 between −π + 2π(k − 1)/n and −π + 2πk/n belongs to the n-th region. Further, flag matrices Ak  (Step 5) are defined as,As explained later, this algorithm assumes that every coordinate point (Ak at every y is stored in Nk(y) (Step 6). The number n must be more than three and should be selected as large as possible, provided that sufficiently large values for Nk(y) are available.The number of flagged pixels in x-direction across the FOV (Step 7). Giving I as the movie frame at ti, where i indicates the frame number , Jk  is calculated (Step 8) byAfter this preparation, a movie is recorded while the sample moves linearly in the p, q) is fixed to the sample and relates to the coordinate  in the FOV with x = p−vti and y = q. Thus, Jk , which corresponds to the k-th step of fringe-scanning, is obtained without moving gratings but rather by utilizing the movement of the sample. Next, the resultant images are calculated with conventional formulae (Step 9):y (q) direction due to cone-beam illumination. The raw images with moiré fringes generated by the experimental setup and an example of the definition of Ak are shown in Fig. As mentioned above, the coordinate . These parameters were chosen so that the nominal speed of the sample at the detector plane was one pixel per frame. The mean visibility of moiré fringes  of the differential edge profile of the sample feature in the absorption image. The spatial resolution in the y-direction, which was independent of the motion blur, was evaluated by the FWHM of a sharp edge profile in the refraction image. They are shown as functions of v in Fig. x-direction. Actually, the spatial resolution near the Nyquist limit was attained below 5 mm/s, and clear degradation of the spatial resolution is found over that speed. Although the spatial resolution in the y-direction was almost constant as expected, the slight degradation shown in Fig. The sample speed is crucial to the spatial resolution in terms of motion blur. The spatial resolution at the detector plane in the 13. Thus, it is feasible to improve the apparatus design for a high-energy phase-imaging scanner.The apparatus presented here was developed mainly to demonstrate the function of the proposed algorithm to realize phase imaging for moving objects. In order to translate this result into the development of practical apparatuses for use in factories, various improvements are needed, such as increasing the sample speed and expanding the FOV. The spatial resolution due to vibration at higher speed could be improved, for example, by equipping a rigid and X-ray transparent stage, to which belt-conveyers are adjacent and products moving on the belt-conveyers are transferred smoothly. In order to increase the sample speed, detector sensitivity is essentially important. The detector employed in this study (Hypix-3000) was dedicated originally for low-energy X-ray experiments, which can be replaced for example with a CdTe-based photon counting image detector suitable for high-energy X-rays used for non-destructive testing. Fabrication of gratings with a high-aspect-ratio is also needed to enable the use of higher energy X-rays. While the height of the gold pattern of G2 was about 100 µm, a fabrication technique for realizing a height of 160 µm has been utilized for an experiment with 183-keV X-raysy-direction, larger gratings and a larger image detector are necessary. Fabrication of a large grating is a challenging task, but gratings of a 10 cm × 10 cm area14 have been fabricated. However, when a larger grating is employed with a cone-beam illumination, a curved form is required for a high-aspect-ratio grating, which creates another technical challenge. Instead, it is also possible to align multiple flat gratings and detectors to cover an area larger than the FOV of each grating-detector set.In order to expand the FOV especially in the Although the scanner was developed for imaging continuously moving objects, the concept can be applied for the relative movement between objects and the Talbot-Lau interferometer system. Therefore, scanning the Talbot-Lau interferometer with an X-ray source and detector against a stable object is feasible with the same concept of image formation. As a result, this study can be expanded to the examination of big structures that cannot be moved easily and to the human body for medical diagnoses.An X-ray phase-sensitive scanner with the Talbot-Lau interferometer configuration and a laboratory-based X-ray source was demonstrated. While conventional phase imaging is performed for an object stable in the FOV with the fringe-scanning technique, a special algorithm was proposed and implemented in this study for an object moving across the FOV. It should be noted that this algorithm allows the use of slightly deformed gratings. X-ray phase imaging was successfully performed for a sample moving at 5 mm/s with a spatial resolution of about 200 µm, which corresponded to the Nyquist frequency of the detector. Thus, the presented X-ray scanning setup is promising for practical use of X-ray phase imaging at factories for quality control of products moving on a belt-conveyor system.


In this work, we establish the maximal  A of an analytic semigroup on a Banach space X is said to have maximal u of the following parabolic differential equationX, every generator of a bounded analytic semigroup has maximal R-boundedness of a family of operators using the resolvent Maximal gularity , and Hilgularity . Beyond  by Weis  on UMD smentclass2pt{minimA-stable Runge–Kutta methods. Kemmochi and Saito ατα.If  is satisestimate  in the fThe constant in condition  is sharphe model , and forition in . Hence, S(A) is the closed interval spanned by the largest and smallest eigenvalues, but in general, the numerical range S(A) has to be approximated [The condition  covers bR-boundedness of operator-valued analytic functions) If the function R-bounded on X, and its R-bound is at most, and R-bound of the set of operators Let e scheme  satisfieR-bounded. In fact, for R-bounded. It suffices to prove that the family of operators mentclass2pt{minimLet the function fined in , and \doX be a UMD space, A be an R-sectorial operator on X of angle A), and R-bound of the set Let e scheme  satisfieR-bounded. Since R-bounded, by Lemma R-bounded for some R-bounded. However, C is some constant which may depend on A. By Lemma R-bounded and its R-bound is at mostR-bounded. If entclass1pt{minimaequation . Meanwhiθφ)τα.If  is satisIn this section, we consider maximal adrature .We shall need the scaled X be a Banach space, A be a sectorial operator on X of angle N, A.Let E scheme  has the n, we haveK(z) is defined byzR is bounded for zR is also bounded for K(z)v is analytic in z in the region enclosed by v to the piecewise constant functionBy multiplying both sides of  by \docuf, the maximal A, rather than the R-sectorial property. The general case  is a linear combination of  , for whiIn the proof of Theorem X be a Banach space, A be a sectorial operator on X of angle N, A.Let E scheme  has the In this section, we present a few examples of fractional evolution equations which possess the maximal Continuous problem) Consider the following time fractional parabolic equation in a bounded smooth domain R-sectorial operator in X with angle (Time discretization) Since the Dirichlet Laplacian R-sectorial of angle  Consider the following space–time nonlocal parabolic equation in R-sectorial of angle  Consider the following time-fractional PDE with a complex coefficient in a bounded Lipschitz domain R-sectorial of angle (estimate  when \doestimate  when \doAs an application of the maximal E scheme , with th}Lp-norm . Other tR-sectorial operator of angle u of (Let ion u of  be suffiion u of  satisfieWe denote by ‖En‖X≤cτ . By applu is smooth, and (If oth, and  gives thmentclass2pt{minimproblems . We will


L being a continuous lattice, two functors from the category of convex spaces (denoted by CS) to the category of stratified L-convex spaces (denoted by SL-CS) are defined. The first functor enables us to prove that the category CS can be embedded in the category SL-CS as a reflective subcategory. The second functor enables us to prove that the category CS can be embedded in the category SL-CS as a coreflective subcategory when L satisfying a multiplicative condition. By comparing the two functors and the well known Lowen functor , we exhibit the difference between (stratified L-)topological spaces and (stratified L-)convex spaces.Consider  Abstract convexity theory is an important branch of mathematics -convexity -convexity is precisely an M-fuzzifying convexity; when L, M)-convexity is precisely an L-convexity; and when L, M)-convexity is precisely a convexity. Similar to  topology, the categorical relationships between convexity and latticed-valued convexity is an important direction of research. When L being a completely distributive complete lattices with some additional conditions, Pang and Shi (L-convex spaces as a coreflective subcategory.It is easily seen that there is some similarity between convexity and topology (a family of subsets of a set closed for union and finite intersection). Thus, similar to lattice-valued topologies  to the category of stratified L-convex spaces (denoted by SL-CS). The first functor enables us to prove that the category CS can be embedded in the category SL-CS as a reflective subcategory when L being a continuous lattice. The second functor enables us to prove that the category CS can be embedded in the category SL-CS as a coreflective subcategory when the continuous lattice L satisfying a multiplicative condition. The second functor is an extension of Pang and Shi’s functor topological spaces and (stratified L-)convex spaces from the categorical sense.In this paper, we shall continue to study the categorical relationships between convex spaces and stratified  functor  from theCS in the category SL-CS. In “CS in the category SL-CS. Finally, we end this paper with a summary of conclusion.The contents are arranged as follows. In “a is way below b (2)(3)such that(4)for some: Maruyama  A subsetL-convex spaces is called L-convexity-preserving  provided that L-convex spaces and whose morphisms are L-CP mappings will be denoted by SL-CS.A mapping Finally, we recall some categorical notions from Adámek et al. .A and B are concrete categories; F, G) is called a Galois correspondence if B-morphism; and A-morphism. is a Galois correspondence, then F is a left adjoint of G , hence F and G form an adjunction If . It is obvious.(LC2). It is easily seen that N is the set of all choice functions For any Let (LC3) Let 2.The set It is easily seen that 3.For any We have proved that for any By a combination of (1)–(3), we have proved that Let be a convex space. Then iffThe sufficiency is obvious. We check the necessity. Let is CP iffisL-CP.L-CP. Let L-CP. Then for any Conversely, let It is easily seen that the correspondence Let be a stratifiedL-convex space. Then the setforms a convex structure onXand the correspondencedefines a concrete functor For any For any Let be directed. Thenfor anyLet L is called multiplicative  and (LC2) are obvious. We only check (LC3) below.C being closed for intersection and directed union. Let L-convex space. Then Pang and Shi convex space from (L-) topological spaces.At last, we give two interesting examples to distinguish –(2), we can define (stratified) L-convex structure on M.Let L being a continuous lattice, an embedding functor from the category CS to SL-CS is introduced, then it is used to prove that the category CS can be embedded in the category SL-CS as a reflective subcategory. When L being a continuous lattice with a multiplicative condition, Pang and Shi’s functor topological spaces and (stratified L-) convex spaces from categorical sense.When  functor  is gener


The results indicate that the band structure changes from semiconducting to metallic and a negative differential resistance effect occurs in the I-V curve. Interestingly, the current can be reduced to zero and this insulator-like state can be maintained as the bias increases. We find this unique property is originated from a peculiar band morphology, with only two subbands appearing around the Fermi level while others being far away. Furthermore the width and symmetry of the zigzag C2N-h2D nanoribbon can be used to tune the transport properties, such as cut-off bias and the maximum current. We also explore the electron transport property of an aperiodic model composed of two nanoribbons with different widths and obtain the same conclusion. This mechanism can be extended to other systems, e.g., hybrid BCN nanoribbons. Our discoveries suggest that the zigzag C2N-h2D nanoribbon has great potential in nanoelectronics applications.Motivated by the fabrication of layered two-dimensional material C Compared to graphene, its band opens a direct gap of 1.70 eV, and the field-effect-transistor fabricated thereon has a high on/off ratio of 107, suggesting great potential in device application. In previous studies on fullerences, carbon nanotubes and graphene-based structures, researchers have found the negative differential resistance (NDR) effect which can be applied to logic operators, molecular switch, memory, oscillator, etc refs et al.et al.p+/p junction based on zigzag-edged graphene. Khoo et al.2N-h2D)-based model, and investigate the effect of width and symmetry on its electronic and transport properties. By performing first-principles calculations, we find C2N-h2D changes from semiconducting to metallic after been cut into a nanoribbon. In order to explore the physical mechanism, we first set up a periodic model. A NDR effect appears in this periodic system, surprisingly, the current can reduce to zero and then this state can be maintained as the bias increases. In a few nano-sized systems, this kind of phenomenon is rarely reported. In-depth study shows that this unique phenomenon stems from the peculiar band morphology where there are only two subbands near the Fermi level and others are far away. In addition, the width and symmetry of the zigzag C2N-h2D nanoribbon can tune the cut-off bias and the maximum current. We also show that similar electron transport occurs in a heterogeneous model and other materials. These transport properties are quite useful for future nanoelectronic device applications.Two-dimensional 2D) materials have attracted tremendous attention for applications in future nanoelectronics owing to their physical, electrical and chemical propertiesD materia4562N-h2D into a zigzag nanoribbon due to the presence of holes and nitrogen atoms. Classification can be carried out according to the atomic species of nanoribbon edges, e.g., carbon and nitrogen-edged or carbon-edged. In our research, we select the structure whose edge atoms are all carbon atoms. To show the nanoribbons more intuitively and clearly, here we use W to denote the width and choose W = 1.0 to 3.0 models to study. The schematic illustration of zigzag C2N-h2D nanoribbons with different widths is shown in 2N-h2D nanoribbon electrode, the central scattering region and the right semi-finite zigzag C2N-h2D nanoribbon electrode  characteristic of the two-probe system with W = 1.0. Apparently, a NDR peak appears in our model. The current firstly increases as the bias increases and reaches the maximum when the bias is 0.2 V and then smoothly decreases to zero as the bias increases to 0.5 V. We define 0.5 V as the cut-off bias. Interestingly, as the bias continues to increase, the current maintains at zero. It indicates that the system is in an insulator-like state. This is contrary to its metallic state under a low bias. As far as we know, this phenomenon has not been reported in periodic systems with widths of several nanometers. A similar I-V characteristic can be seen in W = 1.5 model as shown in I-V characteristic of our models, we first calculate the transmission spectra of W = 1.5 model under different biases as shown in In order to investigate the physical mechanism of the peculiar 2N-h2D nanoribbon. For better illustration, we define the following variables, EAD = EA − ED, EAB = EA − EB and ECD = EC − ED where EA, EB, EC and ED are respectively the energy values of the four subbands near the Fermi level at the Γ-point. It should also be noted that EA is the minimum energy value of the second band above the Fermi level, ED is the maximum value of the second band below the Fermi level, EB is the maximum value of n = 1 subband and EC is the minimum value of n = 1′ subband. Within the energy range of the EAD, only two subbands cross each other and pass through the Fermi level. Both the left and right electrodes are semi-infinite zigzag C2N-h2D nanoribbons, so they also have such a band structure. If we apply a bias on the electrodes, an electric potential difference will appear between the two electrodes, and their band structures will shift downward or upward relatively. In the middle and right panels of V+) of 0.125 V on the left electrode and a negative bias (V−) of −0.125 V on the right electrode. As a result, there appears an energy window (the shadow region) due to the shift of the Fermi level. When the bias is low, the shadow region is small, and both n = 1 and n = 1′ subbands exist in the V+ and V− regions, that is to say the n = 1 and n = 1′ subbands in the V+ region overlaps those in the V− region. As the bias increases, the overlapping becomes smaller so that the transmission peak is getting lower. When the bias increases to 0.25 V, in the energy window, the n = 1 subband in the V+ region is connected to the n = 1′ subband in the V− region. Because the electrons can not hop from one subband to another, the transmission is forbidden. Interestingly, since there are sufficient gaps both in the EAB and ECD ranges, the tunneling channel does not exist and the insulator-like state can be maintained as the bias continues to increase. Meanwhile, we find that the insulator-like state will happen as long as EAB is greater than the absolute value of EC and ECD is greater than the value of EB. The greater EAB and ECD are, the greater bias range the insulator-like state can be kept in. 2N-h2D nanoribbon, whereas the same analysis is invalid for such models with an integer width. For example, the conduction should not be cut off at 0.5 V, so this abnormal phenomenon is worthy of further investigation. 2N-h2D nanoribbon. It is clear that W = 1.0 model has the same band morphology as W = 1.5 model, except that the n = 1 and n = 1′ subbands are more delocalized. σ mirror operation. Since these two subbands have opposite σ parity, an electron belonging to n = 1′ subband (blue) can not hop to n = 1 (red) subband18V+ and V− region, but the electron transport only comes from the coupling between the same color subbands. When the bias increases to 0.5 V, the n = 1 subbands in the V+ and V− region are connected in the energy window. However the n = 1 subband in the V+ region and the n = 1′ subband in the V− region have different parity, and the tunneling between these two subbands is forbidden, so that the current is reduced to zero. This state is held as the bias continues to increase. When the bias is high, we can find the insulator-like state is still derived from the presence of band gaps in the range of EAB and ECD. Consequently, the phenomenons of NDR effect and insulator-like state of W = 1.0 model originate from the peculiar band morphology too, except that the different parity between the subbands near the Fermi level prevents the electron transport and promotes the insulation-like state to happen in lower bias.In 2N-h2D nanoribbon, we calculated the band structures of other models with different widths shown in 202N-h2D nanoribbon to tune the cut-off bias in future applications.To explore the effect of width on the electron property of zigzag CI-V) characteristic of the device W2-1. It can be seen that the morphology of the I-V curve is similar to that of W = 1.0 model which is shown in In the above analysis, we focus on the investigation of theoretical physical mechanism. Taking into account the possibility of experimental operation and practical application, a finite bias voltage could not be applied in a homogeneous situation with discrete translational symmetry, in order to avoid this problem we design the device of W2-1 as shown in El and Er regions. The electrons can hop between them, so the current firstly increases. When the bias is larger than 0.15 V, n = 1 subbands do not overlap leaving only the n = 1′ subbands to contribute to the transmission, so the current starts to decrease. When the bias increases to 0.35 V, as show in El region is slightly far away from the n = 1′ subband in the Er region. According to our previous analysis, the current should be zero when these two n = 1′ subbands are connected under a bias less than 0.35 V. It is noteworthy that some disturbance appears in the center of the scattering region after optimization, so the energy levels of the central part are a little different from the electrodes. That is why the current becomes zero at a higher bias. When the bias continues to increase, the current can be maintained at zero. This insulator-like state still derives from the peculiar band morphology which possesses a sufficient band gap between these two n = 1′ subbands. The physics mechanism is consistent with what we discussed previously.It can be seen that when we apply a positive bias on the device, the band structures of the left and right electrodes will shift downward or upward relatively. If the bias is low, the shadow region becomes smaller, and there will be both n = 1 and n = 1′ subbands in the et al.I-V curve here.In our research, we find that this mechanism can be extended to other systems, that is to say, as long as the structure has the similar band morphology, this phenomenon that the current in NDR effect can be reduced to zero and the insulator-like can be maintained will happen. For example, Kan 2N-h2D nanoribbons with different widths are studied using first-principles calculations. By cutting into nanoribbon, C2N-h2D changes from the original semiconductor to metallic structure owing to the introduced two edge states near the Fermi level. The band morphologies for all the nanoribbons have the same special characteristic, i.e., there are only two subbands near the Fermi level while other subbands are far away which leads to the emergence of gaps. To explore the physical mechanism, we give some homogeneous models. An obvious NDR effect and an insulator-like state occur in the I-V curve which has not been reported in such a nano-sized periodic structure in previous works. For asymmetric structures (W = 1.5 and 2.5), this phenomenon originates from the peculiar band morphology. For symmetric structures , besides the peculiar band morphology, the opposite parity of the two edge states is also a factor as it prevents some electron transport and promotes the insulation-like state to happen in lower bias. In addition, as the width increases, the two edge states become more localized, and the cut-off bias decreases. Therefore we can tune the electron transport by changing the width of the zigzag C2N-h2D nanoribbon. We also show a heterogeneous model composed of two zigzag C2N-h2D nanoribbons with different widths. It has a similar I-V curve, the peculiar band morphology still plays a decisive role and the disturbance in the center of the scattering region leads to the occurrence of the insulator-like in a little higher bias. We can also predict that for other materials, as long as they have similar band morphologies, the NDR and insulation-like state will happen.In summary, the electronic and transport properties of the zigzag Ck-point mesh is 1 × 1 × 1002N-h2D nanoribbon are terminated with hydrogen (H) atoms to remove the dangling bonds.To investigate the transport property of our system accurately, the Atomistix Toolkit (ATK) package which is based on the combination of density functional theory (DFT) and non-equilibrium Green’s function (NEGF) technique is performedIn NEGF theory, the current of the two-probe system is calculated based on the Landauer-Büttiker formulaμL(R) is the chemical potential of the left (right) electrode, f(E − μL(R)) is the Fermi distribution of the left (right) electrode, and V = (μL − μR)/e represents the bias window. T is the transmission coefficient obtained by using Green’s function. Clearly, the current is determined by the area of the integral region in the bias window.where How to cite this article: He, J.-J. et al. Negative differential resistance and bias-modulated metal-to-insulator transition in zigzag C2N-h2D nanoribbon. Sci. Rep.7, 43922; doi: 10.1038/srep43922 (2017).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


D be a digraph on D. For any given sequence of pairs of integers k-arc strongly connected digraph D such that d is the degree sequence of D, then d is realizable and D is a realization of d. In this paper, characterizations for k-arc-connected realizable sequences and realizable sequences with arc-connectivity exactly k are given.Let  D is called a multiple digraph (or multi-digraph for short) if it has no loops. Furthermore, if D has parallel arcs neither, then D is strict. We follow ..D, if fodigraphic  if there exists a digraph  D with degree sequence d, where D is called a d-realization. Let d-realizations. Frank [d is required, then Hong et al. [A sequence of integer pairs entclass1pt{minimas. Frank   showed g et al.  gave theLetbe a sequence of non-negative integer pairs. Thendis multi-digraphic if and only if each of the following holds: (i)(ii)Furthermore, for a strict digraph, there is a similar result. The following theorem, which can be found in –10 among(Fulkerson-Ryser)Letbe a sequence of non-negative integer pairs withThendis strict digraphic if and only if each of the following holds: (i)(ii)(iii)D be a digraph and k be an integer. If for any arc set S of D with D is said to be k-arc strongly connected (or k-arc-connected for short). Clearly, 1-arc connected digraph is also a strongly connected digraph and vice versa. The arc-connectivity of D, denoted by k such that D is k-arc-connected. In [et al. characterized the sequence of pairs of integers d so that there is a strongly connected digraph k-arc-connected digraph in k-arc-connected multi-digraphs. Furthermore, we also give a characterization for multi-digraphs with arc-connectivity exactly k.Let cted. In , Hong etk-arc-connected realization. In Section k. In Section In the next section, we will give some tools and methods used in this paper. In Section D be a digraph and D. The 2-switch of D is an operation to obtain a new digraph In this section, we give a special notation used in this paper that is also the main tool. Let k-arc-connected digraph or a digraph with arc-connectivity exactly k from an arbitrary digraph in Note that in the operation of 2-switch, the two arcs d to have a k-arc-connected realization  of D if D. An out-fragment  F is minimal if any proper subset of F is no longer an out-fragment . Let D and D. As a vertex set F is an out-fragment if and only if its complement F̅ is an in-fragment, D. This observation can be used to prove the following theorem.Let Letbe a sequence of integer pairs. Thendhas ak-arc-connected realization if and only if each of the following holds: (i)(ii)for each(iii)for eachd has a k-arc-connected realization, then by Theorem k-arc-connectedness, (ii) holds. So, it suffices to prove the sufficiency. By (i), (iii) and by Theorem D is k-arc-connected. Suppose this is not true. Then If D. Let D and By the definition, mentclass2pt{minimClaim 1. If F is an out-fragment of (i)(ii)(iii)(iv)(v)By the definition of Claim 2. F with F is also an out-fragment of D. Since D, which contradicts the minimality of By contradiction, we assume that D and by Claim 2, F of D unless either F such that F is an out-fragment in D, then without loss of generality we may assume D. As D but not in D. Therefore, D is k-arc-connected, and this completes the proof. □By choice (a) of D D cannot exceed D is maximally arc-connected if the arc-connectivity of D equals By definition, the arc-connectivity of a digraph Letbe a multi-graphical sequence. Thendis also a degree sequence of some maximally arc-connected multi-digraph.k. Our method is to construct a new multi-digraph in k-arc-connected multi-digraph by reducing the arc-connectivity step by step. Moreover, by Corollary In this section, we consider the degree sequence of multi-digraphs with connectivity exactly Letbe two integers andbe a sequence of pairs of integers. DenoteandThendis a degree sequence of some multi-digraph with connectivity exactlykif and only if each of the following hold. (i)(ii)(iii)forwhereifandifd is the degree sequence of some multi-digraph D with connectivity exactly k. By Theorem D such that D has connectivity k, there exists First, we consider the necessity. Assume k-arc-connected multi-digraph D has arc-connectivity k, then we are done. So we may assume that k from D. First, we need some claims.Next, we consider the sufficiency. By Theorems Note thatClaim 1. Each of the following holds. (i)For any vertex disjoint two arcs (ii)For any (iii)For any  Denote byD. Now, by Theorem D that meets our requirements.D, D is required. So we may assume that By the choice of Claim 2. For any two arcs D.Suppose, to the contrary, that o choice  of D.Claim 3. For any two arcs D. Claim 3 is proved.Suppose, to the contrary, that X are incident with a vertex, say x. We only consider the case By Claim 2 and Claim 3, it is easy to see that all arcs leaving from or interring to Claim 4. We may assume that D.D. So X are incident with y. Let D. So Suppose, to the contrary, that there is an edge o choice  of D. Soo choice  of D. Som, all As k, then by Claim 3 there exists i such that k, and we are done. So we may assume that If u, with an arc between If j, j such that Moreover, by (ii) and (iii), for any k. In other words, all digraphs in If Theorem Letbe a sequence of integer pairs. DenoteandIf each of the following holds, then any digraphs inarek-arc-connected. (i)(ii)(iii)there exists somejsuch thatwhereifandifd to have a k-arc-connected realization by using some inequalities of these integers. As a consequence, we deduce a sufficient and necessary condition of d to have a max-arc-connected realization. Also, when d to have a realization D that has arc-connectivity exactly k. These results extend a similar result from undirect graphs into directed graphs.In this paper, sufficient and necessary conditions for a sequences of pairs of integers have been studied. For a sequence


Scientific Reports 10.1038/s41598-017-13236-z, published online 19 October 2017Correction to: The original version of this Article contained an error in the title of the paper, where the word “Phylogenetic” was incorrectly given as “Phylogentic”. This has now been corrected in the PDF and HTML versions of the Article, and in the accompanying Supplementary Information file.


One of our main results is a characterization of permutations that k-commute with a given permutation. Using this characterization, we obtain formulas for the number of permutations that k-commute with a permutation We introduce and study  H the Hamming metric between permutations. Also, Gorenstein et al. , and A053871 when In some cases, we need to know upper bounds for the number of permutations that almost commute with a given permutation, as in Păunescu . With thf Sloane : A208529The relationship between the number , Rivera  showed mk-commute with a given permutation k-commute with any n-cycle. We present explicit formulas for the number We review our results. We present a characterization of permutations that points. We write cycle decomposition (Dummit and Foote has cyclea cycle ofa is a point in cycle cycle type of a permutation i in its cycle decomposition. The Hamming metric between permutations good commuting point (resp. bad commuting point) of g.c.p. (resp. b.c.p.). In this work, we use the convention m.We first give some definitions and notation used throughout the work. The elements in the set nd Huang  for moreblockA in a cycle m equivalent expressions of m). The length |A| of block A, and the points first and the last element of the block, respectively. A proper block (resp. improper block) of an m-cycle is a block of length A and B are disjoint if they do not have points in common. The productAB of two disjoint blocks, A and B, not necessarily from the same cycle of AB is not necessarily a block in a cycle of block partition of cycle k pairwise disjoint blocks, not necessarily from the same cycle of block permutationP, induced by Our definition of blocks was motivated by the definitions presented in the work of Christie  and in tLet m ways to write block notation  with Let The following result is the key to relate commutation and blocks in cycles.LetLetbe integers, withLetbe a cycle ofIfcommutes withon the blockthenis a block in a cycle ofi. The base case It is enough to prove that The following result is an easy exercise.Letbe anm-cycle ofThencommutes withonif and only iftransformsinto anm-cycle ofcommutes with r points in Let We now present one of our main results.Letbe anm-cycle ofandkan integer such thatThencommutes withif and only ifwhere the blockssatisfy the following:ifthenis a proper block in a cycle ofifthenarekpairwise disjoint blocks, from one or more cycles ofsuch that the stringis not a block in any cycle offor every(1) Suppose that Conversely, we can assume, without lost of generality, that (2) Suppose that k points in i, Conversely, if The function k-commute with Using Theorem 1 we can characterize permutations that LetThenk-commutes withif and only if there existhcycles ofsaysuch thatcommutes withon each cycle ofnot inand for everywithwhere the blockssatisfy the following:if if Let As a first application of Theorem 1, we obtain the following result, that is used in the proof of Theorem 8Letbe a permutation whose maximum cycle length in its cycle decomposition ism. Ifcommutes withonpoints in anm-cycleofthencommutes withonP is a proper block in an l-cycle of m is the maximum cycle length of cycles in Suppose that Previous propositions is a generalization of Lemma 2(b) in Gorenstein et al. . The folLetandbe two permutations thatk-commute, Suppose thatdoes not commute withon the cyclesof lengthsrespectively, and that commutes withon the rest of cycles of(if any). Then, there exists exactlyrcycles ofsayof lengthsrespectively, such thatmore, suppose thatdoes not commute withon exactlyi-cycles ofand that commutes withon the rest of thei-cycles of(if any). Then there exists exactlyi-cycles ofsuch that each of them contains at least one point that is the image underof one b.c.p. ofandLet i-cycles of i-cycles of i-cycles of By a similar argument, we can show that if If all the points in ani-cycle ofare images underof g.c.p., then there exists ani-cycle ofsuch thati-cycle l-cycle, say x and y in different cycles of x is a b.c.p. because t, then First, we prove, by contradiction, that if all the points in the With this claim, every cycle in We finish this subsection with the following resultLetIf one cycle ofhas exactly one b.c.p. ofandthen there exist a cycle ofthat contains at least two b.c.p. ofandl of cycle x, such that B a block of length Let r-cycles of l-cycles of s-cycle of l-cycle of l-cycle, say m-cycle, say r be an integer between 1 and l such that Case I. If Case II. If Now we consider the case k-commutes with m-cycle, say m-cycle, say canonical cycle notation of a permutation transition function of Let ion Bóna , p. 97.Let Step 1m-cycles, say Choose two Step 2k-subset Choose a Step 3k-cycle permutation Choose a Step 4Construct Step 5Construct m cycles of m-cycles of m-cycles of l-cycles of l-cycles of Let The following two propositions shows that Algorithm 1 produces all the permutations with the desired properties.Any permutationconstructed with Algorithm 1 does not commute withon all points inand commutes withon all points ink points in Let LetLetbe any permutation thatk-commutes withand such that all the b.c.p. ofandare in exactly onem-cycle ofThencan be obtained with Algorithm 1.m cycle of m-cycle, k pairwise disjoint blocks in k, which is a contradiction.Let k expressions, choose the one that begins with block As k) is the greatest element in k-cycle in Now, we consider k-commute with In this section we present some results about the number Letbe a permutation of cycle typeThenandWhen k and any Now we show that for any nonnegative integer Ifthenfor anyFor The following result shows that LetSuppose thatis a non-empty set. ThenNow, let k-commutes with Let f(k) be the number of cyclic permutations (k-cycles) of Let Letbe a permutation of cycle typeLetkbe an integer, withThenf(k) ways to select the permutation t-cycles of t-cycles of As all the b.c.p. of T denote the number of n-permutations that k-commute with an n-cycle.Let Letnbe a positive integer andkand integer withThenT is now sequence A233440 in the OEIS database. With this corollary we can obtain, in an easy way, the binomial transform of sequence A000757. Let A. In Spivey and Steil .We have the following limit property for Letnbe a positive integer andmbe a fixed nonnegative integer withThenBy direct calculations we have thatLetn, kbe positive integers withThenf(k) is independent of n we haveP to obtainLet  in Wilf , p. 41 w in Wilf  to the sk, with k-commutes with h cycles, says k that consists of k ones.In this section we present formulas for the number Letbe anyn-permutation of cycle typeThenm-cycle), say m-cycle the strings The set of all points in the blocks From condition (a) to condition (c) we have that m ways to select the first point of block m ways to select the first point of block The number m-cycles m of m of t-cycles of t-cycles of m into cycles of length different than m. After summing over all possible values of m we have that There are In a similar way, but with many more cases to consider, we have obtained a formula for Letbe any permutation of cycle typeThenwhereIn this section we show formulas for Letand letthenIf The following theorem is a consequence of Proposition 7, Theorem 5, Theorem 6 and Proposition 11.Letbe a transposition. Thenn symbols, with Formulas (1), (2) and (3) in previous proposition coincide with the number of permutations of  Deutsch . Therefoa(n) be the “number of deranged matchings of 2n people with partners (of either sex) other than their spouse” (sequence A053871).Now we give a formula for Letbe a fixed-point free involution, Thenk and odd integer,m transpositions. From Proposition 3 we have that if k odd. Now, if k-commutes with j transpositions, of j transpositions of j transpositions, j transpositions of X onto j! ways to arrange the blocks X onto a(j) ways. Finally, there are By hypothesis, the cycle decomposition of Letbe a fixed-point free involution. Thena(n) [see the formula section of sequence A053871 in Sloane (We use the well-known EGF for n Sloane ]\documenk-commuting permutations. We present some formulas for the number of permutations that k-commute with n-cycle, or a fixed-point free involution, for any k. These results could be useful when we work in problems related with almost commuting permutations. Even more, these enumerative results could be useful to find relations between integer sequences in the OEIS database, as Rivera (In this paper we give some techniques to work with s Rivera  showed.The problem of computing in an exact way the number


Scientific Reports 10.1038/s41598-018-20591-y, published online 06 February 2018Correction to: In the original version of this Article, ‘Neuroscience’ was omitted from the category list.Additionally, in the original version of this Article, Figure 3b contained an error in the labelling of ‘Delay Walk Model’.Finally, the original version of this Article contained an error in Figure 4a, where the tau symbol was omitted from the legend.These errors have now been corrected in the PDF and HTML versions of the Article.