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Selected articles from the International Conference on Intelligent Biology and Medicine (ICIBM) 2015: genomics Detection of high variability in gene expression from single-cell RNA-seq profiling Hung-I Harry Chen1,2, Yufang Jin2, Yufei Huang2 & Yidong Chen1,3 The advancement of the next-generation sequencing technology enables mapping gene expression at the single-cell level, capable of tracking cell heterogeneity and determination of cell subpopulations using single-cell RNA sequencing (scRNA-seq). Unlike the objectives of conventional RNA-seq where differential expression analysis is the integral component, the most important goal of scRNA-seq is to identify highly variable genes across a population of cells, to account for the discrete nature of single-cell gene expression and uniqueness of sequencing library preparation protocol for single-cell sequencing. However, there is lack of generic expression variation model for different scRNA-seq data sets. Hence, the objective of this study is to develop a gene expression variation model (GEVM), utilizing the relationship between coefficient of variation (CV) and average expression level to address the over-dispersion of single-cell data, and its corresponding statistical significance to quantify the variably expressed genes (VEGs). We have built a simulation framework that generated scRNA-seq data with different number of cells, model parameters, and variation levels. We implemented our GEVM and demonstrated the robustness by using a set of simulated scRNA-seq data under different conditions. We evaluated the regression robustness using root-mean-square error (RMSE) and assessed the parameter estimation process by varying initial model parameters that deviated from homogeneous cell population. We also applied the GEVM on real scRNA-seq data to test the performance under distinct cases. In this paper, we proposed a gene expression variation model that can be used to determine significant variably expressed genes. Applying the model to the simulated single-cell data, we observed robust parameter estimation under different conditions with minimal root mean square errors. We also examined the model on two distinct scRNA-seq data sets using different single-cell protocols and determined the VEGs. Obtaining VEGs allowed us to observe possible subpopulations, providing further evidences of cell heterogeneity. With the GEVM, we can easily find out significant variably expressed genes in different scRNA-seq data sets. Single-cell analysis has emerged a decade ago to understand the heterogeneity of a cell population, especially in biology contexts such as early embryonic development and tumor etiology [1]. Single-cell quantitative PCR (qPCR) [2–4] or single-molecule RNA fluorescence in situ hybridization (FISH) [5] have been widely used as low-throughput approaches to measure the expression of specific genes at a single-cell level. Although experiments using these methods can provide crucial information of cellular heterogeneity and the presence of distinct cell subpopulations, only a small number of genes can be monitored simultaneously. RNA sequencing (RNA-seq), a developed approach using next-generation sequencing (NGS) technology, can unbiasedly detect the genome-wide gene expression of a sample. Bulk RNA-seq experiments start with a large population of cells (> 105), and the gene expression levels are considered as the average expression across the population of a cell pool [6]. Bulk RNA-seq might be sufficient in many contexts such as revealing the aberration of mRNA expression between different treatments, conditions, or phenotypes. However, biological questions like diversity in early stage development embryonic cells, which each cell has distinct functions, can't be explained by bulk RNA-seq experiments. With recent introduction of Smart-seq protocol, the required volume of starting materials has been vastly reduced, making the single-cell RNA sequencing (scRNA-seq) achievable [7, 8]. There are already several protocols for sequencing of single cells, which allow researchers to assay high-throughput gene expression profiling at the single-cell level of a large number of cells. However, unlike the conventional RNA-seq where analysis tools are abundantly available, the lack of bioinformatics tools for single-cell RNA-seq limits its huge potential. Comparing with bulk RNA-seq measurements, single-cell RNA-seq data tend to have much lower read counts (~200,000 to 5 million reads per cell) [9], higher variability, and large number of outliers, and all these are poorly accommodated by conventional RNA-seq analysis methods [10]. Unlike the objectives of conventional RNA-seq where differential expression analysis and the detection of differentially expressed genes (DEGs) are integral components, the most important goal of scRNA-seq is to identify variably expressed genes (VEGs) across a population of cells to account for the discrete nature of single-cell gene expression and uniqueness of sequencing library preparation protocol for single-cell sequencing. As we observed, the transcriptional heterogeneity of the cell population can be assessed by the expression variation difference, whether they are lowly or highly expressed, which conventional RNA-seq analysis failed to identify due to the assumption of homogeneity within each cell subtype. In recent studies, gene expression variation models were proposed specifically for the corresponded scRNA-seq experiments in order to detect VEGs deviated from the Poisson model [11, 12]. However, different scRNA-seq data sets rendered different distributions and a common mathematical description is necessary. Hence, the purpose of this study is to provide a mathematical description of a gene expression variation model (GEVM) for scRNA-seq data. The model addresses the over-dispersion of single-cell data and the additional variability caused by different sources of variation. By exploiting existing statistical tools such as local regression and nonlinear least squares curve fitting, the parameters of gene variation model are estimated and statistical significant VEGs can be identified. To study the robustness of the model, we have built a simulation framework to generate single-cell RNA-seq data using different distributions in each step to imitate the dispersion of real data in different conditions. We demonstrated robustness of our method by applying it to the simulated data and test how precise we can estimate the parameters to the initial settings. Modeling of single-cell RNA-seq data To develop a generic GEVM, we exploited the over-dispersion concept from edgeR [13]. Assuming each gene's expression follows a negative binomial (NB) distribution with parameter NB(r i , p i ) for i th gene, we have $$ {\sigma}_i^2=\frac{\mu_i}{1-{p}_i}={\mu}_i+\frac{\mu_i^2}{r_i}, $$ where the μ and σ 2 are gene expression mean and variance, respectively. We further assume that in a given condition across a cell population, the model parameter r does not change (invariant to gene expression level), or $$ {\sigma}^2=\mu +\alpha {\mu}^2, $$ where α is defined as the dispersion, or α = 1/r. For simplicity, we omitted gene index from Eq. 2. Clearly, when α > 0, the data are from a NB distribution. If α = 0, the data can be represented by a Poisson distribution (or r → ∞), which follow the diagonal line with a slope of $ -\frac{1}{2} $ in a log-log CV-mean plot where σ 2 = μ in Eq. 2. However, there are many sources of technical variation that contribute to the variability of scRNA-seq data. For instance, single-molecule capture efficiency, 3』 end bias due to single-cell RNA library preparation protocol, and low expression of genes that are easily affected by noises [14]. In this respect, we assume σ 2 = μ + bμ = βμ, where β = 1 + b, and bμ is an additive noise component (proportional to the mean signal strength). Thus, the data deviate from the original diagonal line, following a line of $ { \log}_{10}(CV)=-\frac{1}{2}{ \log}_{10}\left(\mu \right)+\frac{1}{2}{ \log}_{10}\left(\beta \right) $. Consequently, we extended the relation between the mean and variance given in Eq. 2, by adding a model parameter β to represent the multiplicative effect of different sources of technical noises. $$ {\sigma}^2=\beta \mu +\alpha {\mu}^2, $$ where we also assumed β is invariant within each cell population. We further obtained, from Eq. 3, the relationships between the coefficient of variation (CV, defined as σ/μ) of each gene across the cell population and its average expression level as follows, $$ { \log}_{10}(CV)=\frac{1}{2}{ \log}_{10}\left(\frac{\beta }{\mu }+\alpha \right). $$ Therefore, by measuring the CV and mean abundance of gene expression μ from all genes, we can estimate the two parameters α and β and dissect the baseline of the cell population. Note that from Eq. 4 when the mean expression level μ becomes larger, $ CV\to \sqrt{\alpha } $, or a constant coefficient variation [15], and when μ ≪ 1 the σ 2 → βμ, or equivalently to a Poisson parameter λ ' = βμ. Estimation of model parameters and selection of significant VEGs In order to identify genes whose variation of gene expression are larger than those defined by Eq. 4, we need to estimate model parameter α and β from a scRNA-seq data set derived from a given cell population. The estimation procedure is as follows (Fig. 1): firstly we calculate the mean and coefficient of variation of each gene across a set of cells; afterwards, we perform a robust local regression implemented in locfit (R package) for fitting a robust CV-mean relationship. The nonlinear curve starts at the point with enough neighboring points (>0.5 % of total genes) to prevent overemphasizing the low expression section due to the subsampling in the next step. In addition, we also terminate the nonlinear curve at the smallest CV point to constrain to a flat line. As a typical phenomenon in scRNA-seq, only a few genes with high expression levels, results in an inaccurate local fitting at the right-tail side. On the other hand, a large proportion of genes locates in the middle section, leading to a bias during least-squares fitting in the next step. To remedy this bias, we subsample the fitted data points in a fixed interval (0.01 in log10 scale) from the start to the terminal point. Then we employ nonlinear least-squares fitting implemented in nls (R package) to estimate the two model parameters (α and β) of the GEVM. Now we can get the CV difference D i , which is the shortest distance of gene i to the ideal model with parameter α and β as a measure of variability. Workflow of identifying significantly variably expressed genes and the following analyses for single-cell RNA-seq data Determination of p-value of VEGs Instead of picking VEGs by the rank of CV difference D i , we hypothesize that under the assumption of a homogeneous cell population, the CV difference to the model curve (Eq. 4) possesses a normal distribution (around baseline). We further assume that majority of genes, in a heterogeneous cell population, do not deviate much. Therefore, we use the CV differences of the data points around the model curve (Eq. 4) to fit a normal distribution. Even though robust local regression is used to estimate the expression variation model, the model is still influenced by those outlier genes. Hence, we use kernel density to find the center of the normal distribution. Afterwards, we fit the normal distribution using the CV differences below the center point. We can calculate p-value of each data point from the normal distribution and determine the significance of VEGs comparing to initial homogenous cell population. The procedure of Benjamini and Hochberg [16] is also applied to obtain the false discovery rate (FDR). Fig. 1 shows the overall workflow for detecting VEGs in scRNA-seq data set. Simulation of scRNA-seq data from a homogeneous cell population In order to evaluate the robustness of our GEVM, we generated a set of simulated data where we could control the baseline parameters and the differential expression status for a set of genes in a random set of cells. First, we utilized exponential distribution (with 3 different mean values: 0.25, 1, and 10, respectively) to create a "master cell" and its genome-wide expression levels of a cell population. The two lower mean values were designed to reflect the nature of low expression events in scRNA-seq data. The master cell expression level M i would be the base expression value of gene i for the other single cells in the population (children cells derived from a single master cell). Given the master cell expression level M i , and the assigned parameters α, β, the children single cells x ij were simulated with a negative binomial distribution, $$ {x}_{ij} \sim NB\left({r}_{ij},\ {p}_{ij}\right) $$ where the two NB parameters r ij and p ij were further computed by, $$ {r}_{ij}=\frac{\mu_{ij}^2}{\sigma_{ij}^2-{\mu}_{ij}}=\frac{\mu_{ij}}{\beta -1+\alpha {\mu}_{ij}} $$ $$ {p}_{ij}=\frac{\mu_{ij}}{\sigma_{ij}^2}=\frac{1}{\beta +\alpha {\mu}_{ij}} $$ Equations 6 and 7 were obtained utilizing our model Eq. 3. The mean value of gene i in cell j, μ ij , was derived from the master cell expression level with a Gaussian distribution of μ ij = N(M i , max(0.2, 0.2 * M i )). Here we required standard deviation greater than 0.2 to avoid small or near 0 standard deviation. Simulation of scRNA-seq data from a heterogeneous cell population To generate a cell population with non-distinct grouping effects, we first select a percentage of cells to be deviated from its original homogeneous population governed by the master cell. To achieve that and with a set of selected cells, we determine a subset of genes (variable prct) whose expression levels to be altered, and we generate the log fold change of each selected gene from a normal distribution to simulate a gradual fold change, with majority of them with minimal alteration. The fold change of a selected gene k is generated as, $$ { \log}_2\left(F{C}_k\right) \sim Normal\left(\mathrm{mean}=0,\ s=2\right) $$ where the variation level can be controlled by modifying the standard deviation s of the normal distribution. To determine a subset of cells to be altered, the probability of each cell to be deviated is in a uniform distribution, uniform(0, 1) and a cell with probability larger than 0.9 is classified as a heterogeneous cell. By using different distributions for simulation, we are able to generate data close to real scRNA-seq data under different conditions by changing the assigned parameters. We also compare our model with the noise model (Eq. 9) from a previous study [12]. At last, we measure the root mean square error (RMSE) to test the robustness of both methods on the simulated data, where RMSE is evaluated against log10(CV) over μ at a fixed interval, between input and estimated models. $$ { \log}_{10}(CV)={ \log}_{10}\left({\mu}^{\gamma }+\delta \right) $$ Single-cell RNA-seq data set for testing Two mouse scRNA-seq data sets were obtained from Gene Expression Omnibus (GSE65525 and GSE60361) [11, 12]. GSE65525 is the mouse embryonic stem cells with 24,175 genes in 933 single cells, sequenced using CEL-seq protocol [17], and GSE60361 is the mouse cerebral cortex cells with 19,970 genes in 3,007 cells, sequenced using quantitative single-cell RNA-seq protocol [18]. Both data sets were counted using unique molecular identifiers (UMIs) to eliminate duplicated reads caused by library amplification. Following previous study [11], we also performed the same scaling normalization method on both UMI count data sets, $$ {\widehat{k}}_{ij}={k}_{ij}\overline{K}/{K}_j,\kern0.1em \mathrm{where}\kern0.2em {K}_j={\displaystyle \sum_i}{k}_{ij} $$ where k ij is the UMI count of gene i in cell j, K j is the total UMI count of cell j and $ \overline{K} $ is the average UMI count among the cell population. Genes that expressed in less than 1 % of the cell population were removed before applying to the model. As we shown later, the two data sets distribute differently. Under these two distinct cases, we will test the performance of the proposed method under different conditions. Implementation of noise model on simulation data To understand the robustness and limitation of the noise model, simulated data sets with different parameters compositions were generated by using R and then proceeded to identify the significantly VEGs following the flow chart in Fig. 1. Simulation modules implemented were: 1) Master cell gene expression generation; 2) homogeneous cell population gene expression generation (with model parameter α and β); 3) heterogeneous cell population generation (with model parameter prct for number of genes deviated from homogeneous cell population, and s for gene expression variation, Eq. 8). The VEG analysis algorithm will first estimate model parameter α and β described in Eq. 4 by using a cascade of regression (local fit, subsampling, and nonlinear least-squares). For single-cell gene expression data, in the ideal condition all genes should obey CV = μ − 1/2 [11], following a Poisson distribution as depicted by a black diagonal line in log(μ) vs log(CV) plot shown in Fig. 2. In reality, the variance typically exceeds the sample mean, justifying the negative binomial distribution in many NGS applications (and in our simulation example, Eq. 5. The cyan curve in Fig. 2 is the likelihood model of robust local regression using the function locfit.robust in R where outliers were iteratively identified and down-weighted, which allowed us to accurately fit a baseline for the data. The red line in Fig. 2 is the fitted homogeneous variation model and the orange line is the noise model in Eq. 9. With the estimated model parameters $ \hat{\alpha}\ \mathrm{and}\ \hat{\beta}, $ we will evaluate the regression robustness using RMSE. The parameter estimation process was evaluated by varying initial model parameters (α and β in Table 1, s and prct in Table 2, and then number of cells in Table 3) that deviated from master cell population. CV-mean plot of data under different α and β. Other parameters were fixed as gene number = 15,000, cell number = 1,000 cells, prct = 10 %, and s = 2 Table 1 Estimation of model parameters $ \hat{\alpha}\ and\ \hat{\beta} $ under different α and β with fixed number of cells, prct, s, and gene number = 15,000. Comparing with the noise model in Eq. 9, we have obtained fairly low RMSE in each condition Table 2 Estimation of model parameters $ \hat{\alpha}\ and\ \hat{\beta} $ under different prct and s with fixed number of cells, α, β, and gene number = 15,000 Table 3 Estimation of model parameters $ \hat{\alpha}\ and\ \hat{\beta} $ under number of cells with fixed α, β, prct, s, and gene number = 15,000 Estimation of model parameters (α and β) We firstly fixed the data set size 15,000 genes and 1,000 cells with prct = 10 % and s = 2, only the model parameters α and β were changed, and the fit results of simulation data are shown in Fig. 2. When α = 0 and β = 1, we simply simulated the data in a Poisson distribution, following a diagonal line in the figure. When α became larger, the curve angled more prominent, which indicated data deviated from Poisson distribution at the larger expression level. The increase of β resulted in the entire data shifting away from the diagonal line, which might be associated with different sources of technical noises. We observed the robust parameter estimation as shown in Table 1 in all initial model parameters (with RMSE less than 0.01 for all these simulated cases). We noted that sometimes the current model failed to fit a straight line when α = 0, which we will investigate further for regression procedures at higher expression level specifically. When the input parameter β became larger, the two estimated model parameters were deviated from the input parameters. However, even in the extreme case where α = 0.5 and β = 1.5, the RMSE still very consistent in our model (0.0054 ± 0.0005, see Table 1). On the other hand, the orange line - the simple noise model fitting using Eq. 9, can hardly fit the baseline of the simulated data, which results in high RMSE (~0.05, 10x larger than our proposed method) in most conditions. We further examined the number of significant VEGs under each condition. The pale green points in the log(μ)-log(CV) plots in Fig. 2 were the selected as significant VEGs with FDR < 0.05. In the ideal condition where α = 0 and β = 1, there are in average 940 genes changed by at least two fold change and we have detected around 700 VEGs. Along with the increase α and β, the number of significant VEGs decreased. In the condition where α = 0.5 and β = 1.5, there are only around 250 VEGs detected, where around 950 genes are altered by at least two fold change. It is reasonable since the data are more disperse when α and β become larger. The dispersion affects the fitted normal distribution of CV difference while determining the p-value for VEGs, which results in worse FDR when the model parameters are large. Test estimation robustness with varying degree of heterogeneity of cell population Next we tested the performance of model under different percentage of genes affected by random log2 fold change values, which were generated by a normal distribution with zero mean and standard deviation s (Eq. 8). The data set size was still set as 15,000 genes and 1,000 cells, and we fixed the model parameters where α = 0.15 and β = 1.2. From the results in Table 2, we could observe that model parameter β is mostly identical and remained close to 1.2 under different levels and numbers of variable genes. However, the model parameter α became larger (from 0.156 to 0.178) with the increments of s and prct. This is unavoidable because α represents the dispersion of the data set. With more genes deviated from the homogeneous population, the dispersion increased and estimated α biased from the input model parameter value. Due to the deviation of α, RMSE also increased when s and prct became larger. We concluded that the scale and number of variable genes influence the estimation of model parameter α, which results in the increase of RMSE. Nevertheless, this issue is solved during the determination of the distribution of CV difference, where we use kernel density to adjust the center of the normal distribution. Test estimation robustness with varying number of cells At last, we would like to know if the model could be properly fit with limited number of cells. We reduced the population size to 50, 100, or 500 cells. To test under a moderate variation condition, we set prct = 10 %, and s = 2, with model parameters remained as α = 0.15 and β = 1.2. The results in Table 3 show that reducing the number of cells slightly affected the estimation of α: α is larger when the number of cells is smaller, in which CV of genes are more disperse. The estimation of β also deviated a bit with the decrease of the population size. Under 50 and 100 cells conditions, the scattering of the data points around the diagonal line resulted in the estimation error of β and a higher RMSE in lower number of cells. Moreover, the two factors that influenced the estimation of α and β also played a role in calling significant VEGs. Under the same number of genes, we determined only about 355 VEGs in 500 cells condition, whereas about 596 VEGs were called in 50 cells condition. With only a small number of cells, the normal distribution of homogeneous genes is difficult to estimate accurately, which might result in the increase or decrease of detected VEGs. Hence, a sufficient number of cells is necessary to accurately determine VEGs among a cell population. In conclusion, the major factors that influence the robustness of the noise model are how data distributes and the number of cells. Fitting errors arise from two situations, 1) the estimated parameters are unusually large (especially β) in the simulated cases, which is unlikely in real scRNA-seq data, 2) the data distribute close to the diagonal line in the CV-mean plot, but with many variable genes at higher expression level, which results in the failure of fitting a straight line. The cell population size is also a concern; however, in reality a single-cell experiment should be designed with a large number of cells. Hence, the population size may not be a major factor for most single cell applications. From the simulation results we could find out that a simple fitting method is not enough. By fitting the model in Eq. 9 straightforwardly, we got much larger RMSE in every condition. In contrast, our expression variation model design with multiple layers of estimates can be fitted properly for most of the experiment condition. However, in some cases the fitted model curve (red) lay under the local fit curve (cyan) at the middle mean abundance interval, which it might be a potential problem occasionally. Application on real data sets We have identified the VEGs for the two scRNA-seq data sets, and the respective CV-mean plots are shown in Fig. 3. From Fig. 3a, we can see that most genes in the first data set (GSE65525) distribute nearby the diagonal line, inferring that the data were only affected slightly by technical noises. Part of the fitted model overlaps with the Poisson distribution line until the mean abundance is larger than 1. Foreseeably, the two model parameters are close to the ideal case, we estimated that α = 0.044 and β = 1.260. In Fig. 3b, the cyan line is the kernel density estimation of CV difference to find the peak of the normal distribution of homogeneous genes. Using the left side of the peak, the red line is the fitted normal distribution and we identified 883 VEGs with FDR less than 0.001. a CV-mean plot of data set GSE65525 and b the CV difference histogram The second data set, GSE60361 shown in Fig. 4a, is much more disperse and deviated away from the diagonal line. However, our method still fitted a reasonable noise model. Even though the local fit curve was terminated around μ = 10, the extension of the noise model at tail interval fitted well. The model parameters where α = 0.558 and β = 2.356 are much larger, and the histogram of CV difference is also widely distributed. Similar with the simulation case with high percentage of variable genes, the fitted model can't locate accurately on the center of the normal distribution of homogeneous genes. In Fig. 4b, we estimated the normal distribution where the peak is around −0.2. As a result, 3103 genes were defined as VEGs, which is a very large number. We found out that the average UMIs of each cell in the second data set is only around 14,000, which is far less than the first data set with around 29,500 UMIs. The small number of UMI counts results in large dispersion of data and detecting a large number of VEGs. Clearly, the total UMI reads per cell in this data is too small to obtain a precise estimation of model parameters. Additional simulation perhaps is needed to further evaluate the requirement of effect of number of UMIs for single cell study. Determination of single-cell subpopulations After the determination of VEGs, we can use different conventional bioinformatics tools to further study the heterogeneity and subpopulation of single-cell population. Principal component analysis (PCA) can be used to find out possible subpopulations among the entire single-cell population. Here we picked the first data set to demonstrate the subsequent scRNA-seq analysis. First, we used principal component analysis (PCA) on the log-transformed data of 883 selected genes to observe the heterogeneity among all cells, shown in Fig. 5. We could find some possible subpopulations at the left, top left, right, and bottom corners, which were labeled in different colors in Fig. 5 After we determined subpopulations from the PCA result, other methods can be applied to study the heterogeneity of the cell population: using the principal component (PC) loadings to classify the genes; or using Single-Cell Differential Expression (SCDE) [19] and/or DESeq [20] algorithms to identify differential expressed (DE) genes between different subpopulations. We can further perform functional annotation and pathway analyses on identified DE genes to understand the origins of cell heterogeneity. 3-D PCA plot of data set GSE65525 Even though the two scRNA-seq experiments obtained from GEO database used two different techniques to capture single cells with vastly different distributions in the CV-mean plots as shown in Figs. 3 and 4 , we could fit the expression variation models properly for both data. In the previous two studies [11, 12], it has been demonstrated that, using VEGs, cell heterogeneity has been detected along with associated biological functions of subpopulations. Clearly, finding the VEGs of a single-cell experiment is just the first step. The subsequent analyses that utilizing VEGs and their expression changes across the cell population are the key of single-cell RNA-seq analysis. In this paper, we proposed a single cell gene expression variation model, and demonstrated the method to regress the model parameters for a single-cell RNA-seq experiment by exploiting the relationship between the coefficient of variation and mean transcript abundance of all genes in the genome. A single-cell data simulation was also designed and used to determine the robustness of the model parameter estimation. In most condition the model parameters were estimated precisely, and resistant to the influence of factors such as population size, and dispersion of genes. The results of testing on two real scRNA-seq data sets further confirmed our simulation, while additional modeling requirement due to lower total UMI count per cell warrants further investigation. CV, Coefficient of Variation; DE, Differential Expression; DEG, Differentially Expressed Gene; GEVM, Gene Expression Variation Model; NB, Negative Binomial (distribution, model); NGS, Next Generation Sequencing; PC, Principal Component; PCA, Principal Component Analysis; RMSE, Root-mean Square Error; scRNA-seq, single-cell RNA-seq; UMI, Unique Molecular Identifier; VEGs, Variably Expressed Genes Stegle O, Teichmann SA, Marioni JC. Computational and analytical challenges in single-cell transcriptomics. Nat Rev Genet. 2015;16(3):133–45. Taniguchi K, Kajiyama T, Kambara H. Quantitative analysis of gene expression in a single cell by qPCR. Nat Methods. 2009;6(7):503–6. Livak KJ, Wills QF, Tipping AJ, Datta K, Mittal R, Goldson AJ, Sexton DW, Holmes CC. Methods for qPCR gene expression profiling applied to 1440 lymphoblastoid single cells. Methods. 2013;59(1):71–9. Sanchez-Freire V, Ebert AD, Kalisky T, Quake SR, Wu JC. 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This research was supported in part by the Genome Sequencing Facility of the Greehey Children's Cancer Research Institute, UTHSCSA. Fundings for this research were provided partially by the NCI Cancer Center Shared Resources (NIH-NCI P30CA54174), NIH (CTSA 1UL1RR025767-01 to YC and CPRIT (RP120685-C2) grants to YC and HHC, and NIGMS (R01GM113245) to YH and YC. The publication costs for this article were funded by the CPRIT grants to YC mentioned below. This article has been published as part of BMC Genomics Volume 17 Supplement 7, 2016: Selected articles from the International Conference on Intelligent Biology and Medicine (ICIBM) 2015: genomics. The full contents of the supplement are available online at http://bmcgenomics.biomedcentral.com/articles/supplements/volume-17-supplement-7. The R scripts of the algorithm and the UMI data for GSE65525 will be available from GitHub, https://github.com/hillas/scVEGs. All authors contribute to the manuscript. HHC, YJ, YH and YC conceived and designed the study. HHC carried out the simulation procedure and implemented the algorithm in R. All authors read and approved the final manuscript. Authors declare no competing interest in preparing the paper and developing the software associated to this paper. Greehey Children`s Cancer Research Institute, The University of Texas Health Science Center at San Antonio, San Antonio, TX, 78229, USA Hung-I Harry Chen & Yidong Chen Department of Electrical and Computer Engineering, The University of Texas at San Antonio, San Antonio, TX, 78249, USA Hung-I Harry Chen, Yufang Jin & Yufei Huang Department of Epidemiology and Biostatistics, The University of Texas Health Science Center at San Antonio, San Antonio, TX, 78229, USA Yidong Chen Hung-I Harry Chen Yufang Jin Yufei Huang Correspondence to Yufei Huang or Yidong Chen. Chen, HI.H., Jin, Y., Huang, Y. et al. Detection of high variability in gene expression from single-cell RNA-seq profiling. BMC Genomics 17, 508 (2016). https://doi.org/10.1186/s12864-016-2897-6 Single-cell Cell heterogeneity Negative binomial distribution Gene expression variation model Variably expressed genes	CommonCrawl
ANZAMP Meetings 2023 Transport and accommodation Attendees of the meeting are listed below. For attendees giving a talk, click on their talk title to see the abstract. Talk title (if applicable) Nezhla Aghaei Combinatorial Quantisation of Super Chern Simons theory GL(1\|1) Chern-Simons theories with gauge supergroups appear naturally in string theory and they possess interesting applications in mathematics, e.g. for the construction of knot and link invariants. In this talk we explain the combinatorial quantisation of Chern-Simons theories and also the GL(1\|1) generalisation of it, for punctured Riemann surfaces of arbitrary genus. We construct the algebra of observables, and study their representations and applications to the construction of 3-manifold invariants. This work has also an application to Topological Phases of Matter. John Baez (virtual) The Tenfold Way The importance of the tenfold way in physics was only recognized in this century. Simply put, it implies that there are ten fundamentally different kinds of matter. But it goes back to 1964, when the topologist C. T. C. Wall classified the associative real super division algebras and found ten of them. The three "purely even" examples were already familiar: the real numbers, complex numbers and quaternions. The rest become important when we classify representations of groups or supergroups on Z/2-graded vector spaces. We explain this classification, its connection to Clifford algebras, and some of its implications. Rinat Kashaev The quantum dilogarithm and its applications. The quantum dilogarithm is a special function of two variables that finds various applications in quantum theory. Although a special case of that function was introduced already in 1886 by Hölder, its deep connections to the quantum world were revealed only in the early 1990's after the discovery of the quantum five term identity by Ludwig Faddeev. I will review its properties and applications in spectral theory, quantum integrable systems, and quantum topology. Jeong-Hyuck Park Double Field Theory as Closed String Gravity What is the gravity that string theory predicts? While the conventional answer is General Relativity, this talk will introduce Double Field Theory as an alternative. The theory gravitises the whole closed string massless sector, possesses its own Einstein equation, and describes not only Riemannian geometry but also various non-Riemannian ones where some known (Riemannian) curvature singularities are to be identified as regular non-Riemannian backgrounds. Milena Radnovic Poncelet porism, integrable billiards, and extremal functions We present classical and new results inspired by XIX century works of Jean-Victor Poncelet. Different streams of his work have been recently connected in the study of resonance of ellipsoidal billiards. Remy Adderton Generalised Temperley-Lieb Algebras I will discuss a coupled Temperley-Lieb algebra featuring N 'coupled' copies of the standard Temperley-Lieb algebra. The generalised cubic relations and the diagrammatic interpretation will be introduced as well as the relation to the chiral Potts and staggered XX quantum spin chains. Alhanouf Almutairi Michael Assis Statistical mechanics and folliculogenesis – a review of the current state of modelling In this talk we will review the current state of modelling of folliculogenesis from the perspective of statistical mechanics. Women are born with between 0.5-2 million eggs encased in follicles, which decrease exponentially throughout life, leading to menopause when the number reaches around 1000 [1]. While this decrease is exponential, it is quite striking that the ovulation number is a constant, one egg a month. It is even more striking when one considers that there are two ovaries. Only a total of around 500 eggs are ovulated in a lifetime; the vast majority of eggs atrophy and die in a process called atresia. The initial pool of follicles is in a primordial, dormant state. Once they proceed to a growing state, they grow from primary, to secondary, and then antral follicle states, a process called folliculogenesis. Many eggs die at each stage, leading to fewer at each step. Nevertheless, this large pool of eggs contributes to a group dynamic before they die, marked by the appearance of order: the ovulation of one and only one egg each month. This group dynamic affects growing eggs in both ovaries simultaneously, since serum levels of hormones are shared between them. Mid-menstrual cycle, follicle stimulating hormone (FSH) is released by the pituitary, which protects follicles, helping them to keep growing. As they grow, the follicles develop receptors to luteinizing hormone (LH), which helps them express the hormone estradiol. As the follicles continue to grow, the levels of estradiol rise, which has the effect of decreasing the FSH levels, which in turn hinders further growth. In other words, follicle growth is dependent on a feedback system, several in fact. To first order, one could consider the growth of ovarian follicles as a many-body problem with spatially independent interactions. There have been several mathematical models developed based on hormone levels which show the emergence of order from straightforward assumptions (e.g. [2]). Recently multi-scale models have been developed that consider the growth of individual cells and their receptivity to hormones or their expression of hormones (see e.g. [3]). One motivation for studying these models is to understand disorders that possibly arise from a dysfunctional feedback system, for example polycystic ovarian syndrome (PCOS). Nevertheless, current models leave out much new research on the roles of various proteins in folliculogenesis, some having their own feedback systems. The roles of these proteins could be elucidated through exploration within more accurate models. It is the goal of this talk to summarize and review the current state of the understanding of folliculogenesis and its modelling from the perspective of statistical mechanics in the hope of involving more of the mathematical community, including students, into this fascinating endeavour. 1. Wallace, W. Hamish B., and Thomas W. Kelsey. "Human ovarian reserve from conception to the menopause." PloS one 5.1 (2010): e8772. 2. Michael Lacker, H., and Allon Percus. "How do ovarian follicles interact? A many-body problem with unusual symmetry and symmetry-breaking properties." Journal of statistical physics 63.5 (1991): 1133-1161. 3. Clément, Frédérique, and Danielle Monniaux. "Mathematical modeling of ovarian follicle development: A population dynamics viewpoint." Current Opinion in Endocrine and Metabolic Research 18 (2021): 54-61. Murray Batchelor The imaginary world of free parafermions Rodney Baxter Nicholas Beaton A solvable model of weighted SAWs in a box We consider a solvable model of self-avoiding walks (SAWs) which cross an $L \times L$ box, namely partially directed walks (PDWs). For SAWs, the number is conjectured to be asymptotic to $\Lambda^{L^2+bL+c}\cdot L^g$, for constants $\Lambda, b, c, g$. Moreover, when a Boltzmann weight $t$ is associated with the length of the walks, a phase transition occurs at $t=\mu^{-1}$, where $\mu$ is the connective constant for the lattice. For $t<\mu^{-1}$ the average walk is of length $O(L)$, while for $t>\mu^{-1}$ it is of length $O(L^2)$. Here we solve the PDW version of this model and compute the asymptotic behaviour for all $t$. The phase transition occurs at $t=1$, and we use quite different methodology for $t$ below, above and at the critical point. Lachlan Bennett Integrable Multi-well Bosonic Tunnelling Models For this presentation, I'll introduce a family of quantum integrable models. These models, characterised by a Hamiltonian, describe boson tunnelling in multi-well systems. After discussing the properties of these models, I will demonstrate how a Bethe Ansatz technique can be applied to find exact solutions. These solutions allow us to analyse the quantum dynamics and measurement outcomes at specified times. If physically realised, this family of Hamiltonians can be beneficial in studying entanglement. Jean-Emile Bourgine Algebraic engineering and integrable hierarchies The algebraic engineering consists in constructing observables of supersymmetric gauge theories within the representation theory of a quantum group. It is based on the branes system realization in string theory, this system being mapped to a network of modules on which act intertwining operators. The algebraic construction brings new perspectives on many important properties of gauge theories (e.g. AGT-correspondence, dualities, integrability,…). In this talk, I will briefly review the recent advances on this topic, and then use the underlying algebra to revisit the relation between topological strings and the KP integrable hierarchy. This talk will be based on the preprint arXiv:2101.09925. Peter Bouwknegt T-duality for toroidal orbifolds through group cohomology In this talk I will show how T-duality for circle bundles over tori with background H-flux can be reformulated in terms group cohomology. This will then be generalised to T-duality for circle bundles over toroidal orbifolds with background flux. The geometric counterpart of this T-duality will be discussed in a talk by my PhD student Jaklyn Crilly. Tony Bracken Eve Cheng Topological Data Analysis of the extended SSH models The Hermitian two-band SSH model proposed by Su, Schrieffer and Heeger is the simplest topological insulator model. It describes the single spinless non-interacting fermion Hamiltonian on a one-dimensional finite lattice with staggered hopping amplitudes. The topological properties of this system have been thoroughly researched, including bulk-boundary correspondence, the existence of edge states (also zero-energy states), and topological invariants calculated using Berry curvature. There have been numerous Hermitian and non-Hermitian extensions of the SSH model. The Hermitian extensions include long-range hoppings, extended unit cells and the inclusion of on-site potentials and spin-orbit interaction. The non-Hermitian SSH models can be roughly divided into two main classes: the ones with asymmetrical hopping (either long-range or short-range) and the ones with complex onsite potentials. There are also other higher dimensional SSH models proposed in the area of superconductivity. In this talk, I explore the possibility of using topological data analysis to detect the topological phases in Hermitian and non-Hermitian SSH models. I will review the current literature and introduce my in-house program. I will also sketch out some directions for future analysis with more complicated models using this method. Nathan Clisby Catherine Colbert Jaklyn Crilly T-duality on Orbifolds In this talk, I will focus on global aspects of T-duality applied to geometric backgrounds and explore how T-duality affects a group action on such background. This will naturally lead to exploring how T-duality applies to orbifolds, and backgrounds in the presence of a discrete torsion factor. Jan de Gier Chris Djelovic Norman Do The topological vertex and its symmetries The topological vertex is a beautiful theory that was inspired by topological strings and allows one to explicitly compute Gromov-Witten invariants of toric Calabi-Yau threefolds. In this talk, we briefly describe some of the rich algebraic, combinatorial, and geometric structures underlying the theory. Finally, we state a recent result obtained with Brett Parker, which presents symmetries for the topological vertex that are captured by a quantum torus Lie algebra. Allan Ernest Gravitational quantum theory and dark matter The accepted paradigm for understanding the nature of dark matter is based on the existence of an intrinsically weakly interacting, "as yet unknown", particle beyond the standard model. Calculations from gravitational quantum theory, however, show quite conclusively that ordinary protons and electrons can similarly exhibit reduced interaction cross sections in the weak gravity regions of large gravitational wells like galaxy halos, by virtue of their gravitational eigenspectral composition [1,2]. A galaxy halo consisting entirely of baryonic gas would appear largely invisible, its fraction of "dark matter" depending on the proximity to equilibrium, the halo particles' position and uncertainty in phase space, and the size and depth of the gravitational well they are in. This environmentally induced darkness is in some ways analogous to an electronic wavefunction in mixtures of dark atomic eigenstates. A halo's erroneously inferred dark matter fraction depends on the "quantum temperature". Over a narrow range of halo temperatures, a halo will be composed predominantly of atomic hydrogen, the correct gas mass being obtained from the internal 21 cm line, and no dark matter required to make up a mass shortfall. Such galaxies have already been observed [3]. Dark matter-rich, hot ionized halos have reduced scattering cross sections, causing their gas mass fraction to be underestimated, hence requiring a need for dark matter when it may not be necessary. The low quantum temperature halos of the super-dark matter dominated dwarf spheroidal galaxies have predominantly molecular hydrogen whose internal molecular transitions may soon be detectable with the James Webb Space Telescope. This presentation will elaborate further on weak-gravity quantum theory and its implications for dark matter. [1] A. D. Ernest, J. Phys. A: Math. Theor. 42, 115207 (2009). [3] P. E. M. Pina et al., arXiv:2112.00017v2 (2021) Justine Fasquel Zachary Fehily Free field realisations and W-algebras W-algebras are an important class of vertex operator algebras that appear frequently in both mathematics and physics. Understanding their structure and representation theory is therefore a fruitful endeavour. In this talk, I will discuss how free field realisations and screening operators can help. Ethan Fursman Alexandr Garbali Shuffle algebras and lattice models I will talk about some recent developments related to connections between integrable lattice models and shuffle algebras associated to quantum algebras. Tim Garoni Gregory Gold The Gauss-Bonnet Invariant in 5D, N=1 Gauged Supergravity To probe detailed phenomena predicted by the AdS/CFT correspondence, quantum corrections (i.e., higher-derivative corrections) to gauged supergravity must be constructed, the classification of which remains an open problem. For instance, only two curvature squared invariants are currently known in the presence of a cosmological constant (gauged supergravity) in five dimensions which is dual to four-dimensional quantum field theories. In 2014, a third invariant was constructed in superspace, but its component field structure has only now been constructed. Importantly, this third invariant is key to obtain the extension of the Gauss-Bonnet term which is expected to describe the first quantum correction of compactified string theory in five dimensions. In this talk, I review aspects of our new analysis of the 5D N=1 Gauss-Bonnet term and its applications. For example, recent studies of supersymmetric black-hole entropy by quantum corrected 5D gauged supergravity were performed using only two curvature-squared invariants. The off-shell supersymmetric extension of the Gauss-Bonnet term in gauged supergravity allows one to extend these results which has relevance in studying the entropy of asymptotically AdS5 black holes. Pinhas Grossman New examples of modular data Modular tensor categories arise as representation categories of rational conformal field theories, and in recent years have also attracted interest for their role in topological quantum computation. Given a modular tensor category, there is associated a pair of matrices $S$ and $T$ called the modular data. The $S$ and $T$ matrices generate a projective unitary representation of $SL(2,\mathbb{Z})$, and the fusion rules of the category can be recovered from the $S$ matrix via the Verlinde formula. In this talk we will discuss recent discoveries of large classes of modular data defined in terms of pairs of involutive metric groups. This is joint work with Masaki Izumi, generalizing work of Evans and Gannon. Tony Guttmann Lucas Hackl Volume-law entanglement entropy of typical pure quantum states In this talk, I will discuss the statistical properties of the entanglement entropy, which serves as a natural measure of quantum correlations between a subsystem and its complement. Entanglement is a defining feature of quantum theory and understanding its statistical properties has applications in many areas of physics. First, I will introduce the class of physical models and explain its relevance for practical applications. Second, I will explain how the statistical ensemble of quantum states can naturally be described through the methods of random matrix theory. Third and finally, I will discuss a number of new results describing the typical properties (e.g., average, variance) of the entanglement entropy for various ensembles of quantum states (general vs. Gaussian, arbitrary vs. fixed particle number). See PRX Quantum 3, 030201 for further details. Bolin Han Coupled free fermions and q-identities It has been demonstrated in the literature that combining techniques from number theory and mathematical physics can produce useful and interesting, or even unexpected results for both areas. During our study of Gepner's coupled free fermions from the coset construction and coupled free fermions constructed from scaled root lattices, we observe some q-identities, including some of Rogers-Ramanujan type, from their charaters and universal chiral partition functions, which are then rigorously proved using various techniques of q-series. These identities further motivate us to investigate more general q-identities and help reveal a connection between these two constructions. Daniel Hutchings Superprojectors in four-dimensional N=2 anti-de Sitter space (Super)spin projection operators have found numerous applications within the landscape of high energy physics. In particular, recent studies of these projectors in anti-de Sitter (AdS) space have revealed an innate connection to partially massless fields. This observation yields a novel method to derive the characteristics of these exotic fields. In this talk, we will explore this relationship in the context of four-dimensional N=2 AdS superspace. Jessica Hutomo Three-point functions of conserved higher-spin supercurrents in 4D N=1 superconformal field theory In (super)conformal field theory, two- and three-point correlation functions of conserved (super)currents are important physical observables. This talk will review the recent results of arXiv:2106.14498 and 2208.07057. I will first describe a formalism aimed at deriving all constraints imposed by N=1 superconformal symmetry and conservation laws on the three-point function of higher-spin supercurrents. This formalism is then applied to constrain several new mixed three-point functions involving higher-spin supercurrents and the flavour current multiplet. Phillip Isaac Vladimir Jakovljevic Confocal Families of Quadrics on Hyperboloids in Pseudo-Euclidean Spaces We study the geometry of confocal families on hyperboloids in pseudo-Euclidean spaces of dimension four in all signatures. The aim is to completely classify and describe them, and to prove Chasles' theorem in this ambient. The methodology we use includes concepts of pseudo-Euclidean and Euclidean geometry, and linear algebra as well. We also give a clue about the natural characteristics confocal families possess to be applied in a billiard theory. This research is done as a part of a Ph.D. project at the University of Sydney. Peter Jarvis Indecomposable representations of type I Lie superalgebras We study the class of indecomposable representations of superalgebras in which a given finite dimensional representation is repeated with arbitrary multiplicity. Such "replicant" or "matryoschka" modules have dimension rD, with r composition factors equivalent to the fixed D-dimensional module. For the case of the classical Lie superalgebras sl(m/n) and osp(2/2n) and r=2, we prove by a cohomology argument that for each Kac module there is a 1-parameter family of indecomposable doubles. For general r, we provide an explicit construction of the replicant Kac modules. In conclusion, we give some illustrative examples from physics, including a possible application to family generation structure in the standard model. Mitchell Jones On the gl(2\|1) Gaudin Algebra An examination on the interesting properties of the Gaudin Algebra derived from the gl(2\|1) Lie Superalgebra. Andrew Kels $\mathbb{C}^8\times Q(E^8)$ extension of the elliptic Painlevé equation The elliptic Painlevé is the top level equation that arises from Sakai's classification. Recently, Noumi has given the construction of hypergeometric tau-functions for the elliptic Painlevé equation by solving the appropriate forms of the Hirota bilinear equations. Such tau-functions are defined on an infinite sequence of hyperplanes parallel to the highest root in $E_8$ and their restriction to each individual hyperplane is invariant under the action of the Weyl group of type $E_7$. In this talk, I will present an extended version of such Hirota bilinear equations, and their solutions, which involve an additional 8 independent discrete parameters taking values from the $E_8$ root lattice. Based on A.P. Kels, M. Yamazaki, Int. Math. Res. Not. 1, 110-151 2021 Mario Kieburg Winding Number Statistics for Chiral Random Matrices Topological invariance is an extremely important concept in physics. On the one hand, it leads to a classification of systems that will share similar behaviour in some regimes. On the other hand, topological properties are especially robust against perturbations. In a system, where the Hamiltonian is chiral and shows a spectral gap about the origin, one particular quantity is the winding number of the determinant of the off-diagonal block matrix. The spectrum of this off-diagonal block is complex ad shows interesting and universal behaviour. We investigated the local statistic of this winding number with the help of a Gaussian random matrix field on a one dimensional Brillouin zone. I will report on these developments in my presentation. Johanna Knapp Nowar Koning Supertwistor realisation of N-Extended AdS superspace The most natural and efficient setting for analysing the properties of superconformal field theories is conformally compactified Minkowski superspace. Amongst the most powerful formulations of the latter are those utilising supertwistor techniques. These supertwistor methods have recently been extended to three and four dimensional AdS superspaces. In this talk I will discuss a supertwistor realisation of four dimensional N-extended AdS superspace, as well as the procedure to develop field theory on such a space using a variant of Cartan's coset construction. Jonathan Kress Algebra conditions for conformally superintegrable systems The Stäckel transform of natural Hamiltonian systems gives rise to a conformally invariant notion of second order superintegrability. Non-degenerate second order superintegrable systems have been classified on three-dimensional conformally flat spaces, but extending methods used were not easily extended to higher dimensions. In this talk, simple algebraic conditions describing these systems in arbitrary dimensions will be given. It is hoped that this formulation will lead to a similar classification of non-degenerate second order superintegrable systems in all dimensions. Sergei Kuzenko Jon Links Integrability-based NOON state protocol The study of integrable quantum systems has recently made inroads into the field of quantum technology. Examples include general results towards a deeper understanding of quantum circuits, and specific investigations such as simulation of the ground-state for the 1-d Heisenberg model on a quantum computer. NOON states are "all and nothing" examples of Schroedinger-cat states. They have been well-studied over the last 20 years, for both fundamental tests of quantum theory and potential applications in quantum metrology. In this presentation I will describe a simple protocol, designed around Hamiltonian time evolution of an integrable system and local measurement, to produce high-fidelity NOON states. Xilin Lu Vladimir Mangazeev CTM approach to the Lee-Yang singularity in the 2D Ising model We study the 2D Ising model in a complex magnetic field in the vicinity of the Lee-Yang edge singularity. Using Baxter's CTM method combined with analytic techniques, we obtain the scaling function together with an accurate estimate of the location of the Lee-Yang singularity. Our results are in excellent agreement with the Ising field theory calculations by Fonseca, Zamolodchikov (2001) and Zamolodchikov, Xu (2022). Ian Marquette Algebraic constructions of superintegrable systems from commutant It was discovered how polynomial algebras appear naturally as symmetry algebra of quantum superintegrable quantum systems. They provide insight into their degenerate spectrum, in particular for models involving Painlevé transcendents for which usual approaches of solving ODEs and PDEs cannot be applied. Those algebraic structures extend the scope of usual symmetries in context of quantum systems, but they also been connected to different areas of mathematics such as orthogonal polynomials. Among them, the well-known Racah algebra which also admit various generalisations. I will take a different perspective on those algebraic structures which is based on Lie algebras, their related enveloping algebras, partial Casimir and commutant. I will discuss how such approach differs from using differential operator realizations and why this framework offers advantages for their classification. I will point out as well how different methods from the study of Casimir invariant of non semisimple algebras which involves solving systems of PDEs can be applied in this context and greatly facilitate making calculations. The talk will present various explicit examples, and in particular the symmetry algebra of the generic superintegrable systems on the 2-sphere which can be understood in a purely algebraic manner using an underlying Lie algebra. The talk is based on following works: R Campoamor-Stursberg, I Marquette, Hidden symmetry algebra and construction of quadratic algebras of superintegrable systems, Annals of Physics 424 168378 (2021), arXiv 2020.168378 F Correa, MA del Olmo, I Marquette, J Negro, Polynomial algebras from su(3) and a quadratically superintegrable model on the two sphere, J.PhysA. Math. and Theor 54 015205 (2021) arXiv:2007.11163 R Campoamor-Stursberg, Ian Marquette, Quadratic algebras as commutants of algebraic Hamiltonians in the enveloping algebra of Schrodinger algebras, Annals of Physics. 437 168694 (2022) arXiv 2021.168694 D Latini, I Marquette, YZ Zhang, Construction of polynomial algebras from intermediate Casimir invariants of Lie algebras, J.PhysA Math. and Theor. (2022) arXiv:2204.06840 Rutwig Campoamor-Stursberg, Danilo Latini, Ian Marquette, Yao-Zhong Zhang, Algebraic (super-)integrability from commutants of subalgebras in universal enveloping algebras, arXiv:2211.04664 Daniel Mathews The geometry of spinors in Minkowski space Work of Penrose and Rindler in the 1908s developed a formalism for spinors in relativity theory. In their work they gave geometric interpretations of 2-component spinors in terms of Minkowski space. We present some extensions of this work, involving 3-dimensional hyperbolic geometry. William Mead Exclusion Process Dualities from Integrable Vertex Models We demonstrate a method for obtaining expectation values of duality observables in the asymmetric simple exclusion process. This approach is then mimicked using an integrable vertex model which allows for some generalisations of the duality observables for higher-rank exclusion processes. Paul Norbury Volumes of moduli spaces of super hyperbolic surfaces Mirzakhani produced recursion relations between polynomials that give Weil-Petersson volumes of moduli spaces of hyperbolic surfaces. Stanford and Witten described an analogous construction for moduli spaces of super hyperbolic surfaces producing Mirzakhani-like recursion relations between polynomials that give super volumes. This was achieved in the so-called Neveu-Schwarz case. Both of these stories have an algebro-geometric description, and in particular this led Mirzakhani to a new proof of Witten's conjecture on intersection numbers over the moduli space of stable curves. In this lecture, via the algebro-geometric description, I will describe what occurs in the Ramond case of the super construction. It produces deformations of the Neveu-Schwarz polynomials again satisfying Mirzakhani-like recursion relations. Jeremy Nugent Semi-degenerate superintegrable systems Superintegrable systems are physical systems with the maximal amount of symmetry. A large portion of the literature deals with non-degenerate systems, which are the 'nicest' superintegrable systems. In this talk we discuss previous and current research efforts into semi-degenerate superintegrable systems, which can be considered as the 'second-nicest' class of superintegrable systems. Jordan Orchard Scattering in right-angled polygonal billiard channels Polygonal billiard channels are examples of pseudo-chaotic dynamics, a combination of integrable evolution and sudden jumps due to conical singular points that arise from the corners of the polygon. Such pseudo-chaotic behaviour, often characterised by an algebraic separation of nearby trajectories, is believed to be linked to the wild dependence that particle transport has on the billiard geometry. We focus on a two-parameter family of right-angled parallel billiard channels having either finite or infinite horizon and consider a scattering problem defined on the periodic boundaries of an elementary cell. Through studying singular points of the billiard flow, we partition the phase space into eighteen subsets for the finite horizon and sixteen for the infinite horizon, where each subset is associated with a unique itinerary. The corresponding eighteen and sixteen branch scattering maps are presented in explicit form with natural extensions enabling the study of transport from the scattering dynamics. Aleks Owczarek SAW in a square and a proof from a Monte Carlo algorithm Self-avoiding walks (SAW) confined in a square admit a different window on the behaviour of SAW than the usual length scaling considered. Previously the endpoints of the walks have been fixed to the corners of the square or perhaps the sides of the square. A proof that the endpoints drive only subdominant behaviour can be made using an expanded set of moves of the endpoint that arises in a Monte Carlo algorithm for Hamiltonian walks. A separate Monte Carlo method and advanced series analysis confirm and expand on this result. Anthony Parr Symmetry Algebras of Superintegrable Systems We consider polynomial Lie algebras with two generators obtained from the set of symmetries of an exactly soluble Hamiltonian by the ladder operator approach. We develop the method for explicit computations of the algebra, obtain its Casimir and its spectrum. We show their realisations as differential operators and deformed oscillator algebras. Michael Ponds Conformal higher-spin supergravity as an induced action Conformal higher-spin (CHS) gravity is a rare example of a local Lagrangian theory involving bosonic fields of all spins interacting with one another. It arises as the logarithmically divergent part of the effective action associated with scalar matter coupled to background CHS fields, or, in other words, as an induced action. In this talk, I will give a brief overview of this theory, and discuss its N=1 supersymmetric generalisation which was proposed recently. Robert Pryor D-branes in B-twisted (2,2) Hybrid models B-twisted (2,2) Hybrid models are a class of superconformal field theories with string theoretic relevance. They can be understood as Landau-Ginzburg models fibred over non-linear sigma models with on a compact Kahler manifold. The bulk theory of these hybrid models is relatively well understood. In particular, the spectra and correlators for several examples of topologically B-twisted theories are known. On the other hand, the boundary theory and the associated D-branes are still relatively unexplored. In this talk, I will introduce these hybrid models, as well as some of their key features. I will then discuss D-branes in B-twisted (2,2) Landau-Ginzburg and non-linear sigma models, as well as some results about D-branes in B-twisted (2,2) Hybrid models. In particular, I will focus on how these D-branes arise physically, as well as their categorical interpretation and relevance. Cheng Kevin Qu Extended Criticality of Deep Neural Networks The recent and continuing success of deep learning in many real-world problems has motivated an intense effort to theoretically understand the dynamical principles of deep learning in the training and generalization of complex tasks. Although empirical data has suggested that the weights of deep neural networks acquire heavy-tailed statistics after training, most theoretical studies have based their analysis on random networks with coupling weights obeying Gaussian statistics. In this work, we investigate the phenomenon of heavy-tailed coupling weights in deep neural networks (DNNs) across fully- connected, convolutional and residual architectures. After verifying the emergence of heavy-tailed coupling across many common pretrained neural networks, we analyse the propagation of signals through DNNs with random, heavy-tailed weights using mean-field and random matrix theory. Importantly, we introduce a criterion for criticality using the entire set of Jacobian eigenvalues of random, heavy-tailed DNNs which extends the classical, edge-of-chaos notion in a consistent fashion while accounting for empirical observations on signal propagation. In this manner, we establish an extended heavy-tailed regime of criticality across those architectures which classically have a fixed critical point under Gaussian coupling. We show that this extended critical regime allows networks already initialised with random, heavy-tailed coupling to learn real-world tasks faster without fine-tuning the weight statistics. Surprisingly, our empirical simulations reveal that despite the fully-connected mean-field analysis, the predictions of our theory are largely shared by residual networks which continue to benefit from heavy-tailed initialisation from a classically chaotic regime. Thomas Quella Quantum group invariant spin chains, discrete symmetries and symmetry-protected topological phases We study the fate of certain discrete symmetries of spin chains under quantum group deformation of their continuous symmetry and report on the implications on the classification of symmetry-protected topological phases. Reinout Quispel Building superintegrable Lotka-Volterra systems using Darboux polynomials In this talk we show how to construct large classes of Lotka-Volterra ODEs in Rn with n-1 first integrals. The building blocks we use will be linear Darboux Polynomials of the ODE. In the talk these concepts will be defined, and the procedure explained. Emmanouil Raptakis Conformal (p,q) supergeometries in two dimensions Local superconformal symmetry has played a major role in string theory and supergravity in two dimensions (2D). In particular, the N=1 and N=2 spinning strings may be formulated as a 2D linear sigma model coupled to conformal supergravity. In this talk I will report on some recent results in constructing superspace formulations for 2D conformal (p,q) supergravity as the gauge theory of the superconformal group OSp_0(p\|2;R) x OSp_0(q\|2;R) and some applications of this formalism. Christopher Raymond Unifying Galilean W-algebras Galilean algebras are infinite-dimensional symmetry algebras for conformal field theories in two dimensions. One way of obtaining these algebras is through a parametric contraction of conformal symmetry algebras such as the Virasoro algebra or affine Lie algebras. However, it does not generally apply to algebras such as W-algebras, which are of great interest in the literature. We provide an introduction to the problems that arise and discuss how to give a uniform construction of Galilean W-algebras via quantum hamiltonian reduction. David Ridout Pieter Roffelsen Cubic surfaces, Segre surfaces and Painlevé equations A fundamental result due to M. Jimbo (1982), relates Painlevé VI to a family of affine cubic surfaces via the Riemann-Hilbert correspondence. In recent work with Nalini Joshi, a q-analog of this result was obtained, relating q-Painlevé VI to a family of affine Segre surfaces. I will explain this result and some of its consequences. Yang Shi Translations in affine Weyl groups and their applications in discrete integrable systems Recently, we reviewed [1] some properties of the affine Weyl group in the context of their applications to discrete integrable systems such as the discrete Painleve equations [2]. In particular, a dual representation is used to discuss translational elements of the Weyl groups. They are found to give rise to the dynamics of various discrete integrable equations. Liam Smith New deformations of quantum field theories Quantum field theory (QFT) is one of the most successful frameworks to describe a wide array of physical phenomena from particle physics to condensed matter systems. It is also the core description of models of (quantum) gravity. Despite its success, the understanding of strongly coupled, interacting QFTs remains an outstanding mathematical problem. One route to make progress is to study exactly-solvable models and deformations thereof, together with symmetries, to move within the set of QFTs. The TT deformation is an exciting tool which aids in this exploration. Defined as the determinant of the stress-energy tensor for a two-dimensional QFT, it has proven to preserve integrability, (super-)symmetries, and it has shed new light on various areas of research including: non-local QFT, string theory, and holographic (AdS/CFT) dualities. TTbar-like deformations have been proposed also in D>2 dimensions finding surprising relations with interesting effective actions, such as the Born-Infeld theory of non-linear Electrodynamics, that describe universal sectors of string theory at low-energy. A sqrt{TT} type of deformation have recently also been proven to lead to the ModMAX theory of non-linear Electrodynamics in D=4 that has attracted substantial attention in the last couple of years. This talk will summarise our results in finding theories in higher dimensions which obey a TT "like" flow equation, as well as pioneering work on understanding the new aforementioned sqrt{TT} deformation, which, in the D=2 case, has been shown to preserve classical integrability for a large class of theories. Supersymmetric extensions will also be presented. Yury Stepanyants Frequency downshifting of decaying NLS solitons in an ocean covered by ice floes We study a frequency downshifting in wavetrains propagating in an ocean covered by ice floes. Using the empirical model which suggests that small-amplitude surface waves in such an environment decay exponentially with the decay rate depending on frequency as ki ~ 3 [1, 2], we derive the frequency downshifting of a wavetrain within the framework of the linear theory. We show that the apparent downshifting appears due to the faster decay of high-frequency components compared to the low-frequency components with no energy flux along the spectrum. The alternative model is also considered within the framework of the nonlinear Schrödinger (NLS) equation [3] augment by the empirical dissipative terms. This model describes the propagation and decay of weakly nonlinear wavetrains and accounts for an energy flux along the spectrum down to lower frequencies. Assuming that the dissipation is relatively small compared to the nonlinear and dispersive terms in the NLS equation and using the asymptotic approach [4], we derive the frequency downshift for the decaying envelope soliton. The comparison of the downshifting obtained within the framework of linear and nonlinear models shows that in the latter case the frequency downshifting is much greater. In conclusion, estimates for the real oceanic conditions will be provided. [1] Meylan M.H., Bennetts L.G., Mosig J.E.M., Rogers W.E., Doble M.J., Peter M.A. Dispersion relations, power laws, and energy loss for 845 waves in the marginal ice zone. Journal of Geophysical Research: Oceans, 2018, v. 123, 3322–3335. DOI: 10.1002/2018JC013776 [2] Squire V.A. Ocean wave interactions with sea ice: a reappraisal. Annu. Rev. Fluid Mech., 2020, v. 52, 37–60. DOI: 10.1146/annurev-fluid-830 010719-060301 [3] Slunyaev A.V., Stepanyants Y.A. Modulation property of flexural-gravity waves on a water surface covered by a compressed ice sheet. Phys. Fluids, 2022, v. 34, 077121. DOI: 10.1063/5.0100179 [4] Fabrikant A.L., Stepanyants Yu.A. Propagation of waves in shear flows. World Scientific, Singapore, 1998, 287 p. DOI: 10.1142/2557 Martin Sticka On Special and Universal Geometry A simplistic overview of the moduli space of the heterotic string. In a physically unrealistic subset of heterotic vacuum solutions, we have special geometry, which is essentially the space of Calabi Yau manifolds. In a more general case, we have universal geometry, a fibration, which makes understanding the general case somewhat tractable. Benjamin Stone Correlation functions of conserved currents in 3D/4D conformal field theory It is a well known fact that the general structure of two- and three-point correlation functions of primary operators is fixed up to finitely many parameters by conformal symmetry. In particular, correlation functions of conserved currents, such as the energy-momentum tensor, vector current and more generally, higher-spin currents, are of fundamental importance as they possess properties associated with spacetime and internal symmetries. Deriving the explicit form of three-point functions of conserved currents for arbitrary spin remains an open problem. In this talk we will discuss a general formalism for constructing three-point functions of conserved currents for arbitrary spin in 3D and 4D (S)CFT. Gabriele Tartaglino-Mazzucchelli James Tener Kai Turner Embedding Formalism of Three-dimensional Anti-de Sitter Superspaces Anti-de Sitter (AdS) superspaces originate as maximally symmetric solutions of supergravity theories. In three dimensions, the isometry supergroup of N-extended AdS superspace is labeled by two positive integers (p,q) with N=p+q. When working with AdS supergroups, supertwistor techniques serve as an effective tool, and have recently been used to construct three-dimensional (p,q) AdS superspace. In this talk, I will discuss this supertwistor formulation, and also present the supercoset construction of three-dimensional (p,q) AdS superspace. It is shown that this formalism is ideal for extracting geometric information which is crucial for developing field theories on these superspaces. Willem van Tonder Integrability of the spin-1/2 XY central spin model Central spin models are closely related to Richardson-Gaudin models and have many present and potential physical applications. Recently the XX central spin model was shown to be integrable for arbitrary spin and the eigenstates and eigenvalues found using a Bethe-Ansatz method of solution. We have shown that integrability carries over to XY models albeit restricting to spin-1/2. The conserved charges were found and a quadratic relation in these charges have been used to obtain the Bethe-Ansatz equations. Luc Vinet Entanglement of free fermions on graphs The entanglement of free Fermions on graphs of the Hamming and Johnson schemes will be discussed. A parallel with time and band limiting problems will be made. The role of the Terwilliger algebra in the identification of a Heun type operator commuting with the truncated correlation matrix and the access it gives to the entanglement entropy will be explained. Ben Wootton Junze Zhang Algebraic approach and exact solutions of superintegrable systems in 2D Darboux spaces Superintegrable systems in 2D Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this talk, I will explore obtaining exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. Yao-zhong Zhang Zongzheng Zhou Geometric upper critical dimension of the Ising model The Ising model is one of the most fundamental models in statistical physics and condensed matter. It is well-known that, from the renormalization-group theory, the upper critical dimension of the Ising model is 4, above which the critical behaviour follows mean-field theory. However, under the geometric Fortuin-Kasteleyn random-cluster representation, we argue that the Ising model simultaneously exhibits two upper critical dimensions: 4 and 6. In this talk, we will show strong numerical evidence to support this argument.	CommonCrawl
Controllable skyrmion chirality in ferroelectrics Gate-controlled skyrmion and domain wall chirality Charles-Elie Fillion, Johanna Fischer, … Hélène Béa Switching of Skyrmion chirality by local heating Yoshinobu Nakatani, Keisuke Yamada & Atsufumi Hirohata Regioselective magnetization in semiconducting nanorods Tao-Tao Zhuang, Yi Li, … Edward H. Sargent Electric and antiferromagnetic chiral textures at multiferroic domain walls J.-Y. Chauleau, T. Chirac, … M. Viret Strain-mediated voltage-controlled switching of magnetic skyrmions in nanostructures Jia-Mian Hu, Tiannan Yang & Long-Qing Chen Switching magnon chirality in artificial ferrimagnet Yahui Liu, Zhengmeng Xu, … J. Li Observation of strong excitonic magneto-chiral anisotropy in twisted bilayer van der Waals crystals Shoufeng Lan, Xiaoze Liu, … Xiang Zhang Topological transitions among skyrmion- and hedgehog-lattice states in cubic chiral magnets Y. Fujishiro, N. Kanazawa, … Y. Tokura Stabilizing magnetic skyrmions in constricted nanowires Warda Al Saidi & Rachid Sbiaa Yu. Tikhonov1,2, S. Kondovych2,3, J. Mangeri4,5, M. Pavlenko1, L. Baudry6, A. Sené2, A. Galda7, S. Nakhmanson5,8, O. Heinonen9, A. Razumnaya1, I. Luk'yanchuk2,10 & V. M. Vinokur9,11 Scientific Reports volume 10, Article number: 8657 (2020) Cite this article Ferroelectrics and multiferroics Topological matter Chirality, an intrinsic handedness, is one of the most intriguing fundamental phenomena in nature. Materials composed of chiral molecules find broad applications in areas ranging from nonlinear optics and spintronics to biology and pharmaceuticals. However, chirality is usually an invariable inherent property of a given material that cannot be easily changed at will. Here, we demonstrate that ferroelectric nanodots support skyrmions the chirality of which can be controlled and switched. We devise protocols for realizing control and efficient manipulations of the different types of skyrmions. Our findings open the route for controlled chirality with potential applications in ferroelectric-based information technologies. Chirality, a fundamental asymmetry property describing systems that are distinguishable from their mirror images, remains in the focus of modern science1,2,3,4, and chiral materials find diverse applications5,6,7,8. Chiral topological textures set the stage for a new generation of chiral materials, where the chirality is extended over nano- and micro-scales. Nonuniform chiral states, helical, blue, and twist grain boundary (TGB) phases have been observed in cholesteric liquid crystals9,10. Skyrmions, which are the chiral texture of a vector order parameter, such as magnetization or polarization density, have been attracting considerable attention in magnetic materials during the past decade11,12,13 due to their potential applications in information technologies. However, a salient feature of these materials is the specific non-chiral symmetry, carried either by non-mirror-symmetric molecules in cholesterics or the antisymmetric spin exchange in magnetic systems leading to the Dzyaloshinskii-Moriya spin interaction. Recently, an extension of the class of magnets hosting skyrmions onto systems without Dzyaloshinskii-Moriya spin interaction has been reported14,15. However, the possibility of tuning the chirality of skyrmions in these systems remains an open question. Although a pre-defined chiral symmetry is absent in ferroelectric materials, they were recently found to host a wealth of chiral topological excitations, including Bloch domain walls16,17,18,19, coreless vortices with a skyrmion structure20,21,22, single skyrmions23,24, skyrmion lattices25, and Hopfions26. A distinct feature of ferroelectrics is that the chirality appears as a result of the spontaneous symmetry breaking due to specific interplay of confinement and depolarization effects when the depolarization charges $\rho =\nabla \cdot {\bf{P}}$ rearrange to reduce their interaction energy, leading to the chiral twisting of the polarization. Importantly, the different chiralities ("left" and "right" states) are energetically degenerate and hence inter-switchable. However, performing such chirality-switching poses a challenge because of the non-chiral nature of the fundamental fields that could serve as a control parameter. We find that ferroelectric nanodots can provide rich phase diagrams as depolarization effects lead to an abundance of topological excitations, and we demonstrate that ferroelectric nanodots harbor polarization skyrmions. In particular, we devise a system in which controlled switching between the opposite chiralities may be implemented by the applied electric field. Our target system is a ferroelectric nanodot in a shape of the disk deposited on a substrate (see Fig. 1a). We choose the lead titanate pseudo-cubic perovskite oxide, PbTiO3, as the model ferroelectric material. The typical nanodots that we use for observation of the skyrmion have diameter about 40 nm and thickness about 20 nm. For calculations we employ the phase field approach, described in details in the Methods section. The ferroelectric nanodot is located in a capacitor, the upper plate of which is separated from the sample either by vacuum or by a dielectric material with low dielectric constant. The thin lower electrode forms an interface between the nanodot and the substrate, which induces a weak strain caused by the ferroelectric lattice mismatch with the substrate. The strain results in an out-of-plain polarization anisotropy (see Methods for relevant parameters). This system enables the creation and manipulation of topological nonuniform textures confined in the nanodot by applying voltage $U$ to the electrodes. Ferroelectric switch circuit and skyrmion states. (a) The circuit is controlled by the external switching voltage $U$. The top electrode is separated from the ferroelectric nanodot carrying the polarization topological states. (b) Four types of the skyrmions, differing by their chirality and polarity. The hand pictograms define the classification of the skyrmions. The emergence of these topological excitations is related to the effect of depolarization charges that arise because of the abrupt termination of the spontaneous polarization at the nanodot surface. Induced depolarization fields destabilize the uniformly-polarized state, resulting in a polarization texture corresponding to a self-consistent local energy minimum. In thin-film geometry, the depolarization effect yields Kittel domains27, structured as soft polarization stripes28,29 (or parallel-anti-parallel vortices30,31), or bubble domains in a skyrmion lattice structure25. Confinement of these textures to within a nanodot gives rise to the competition between striped and cylindrical domains32. We demonstrate that the latter configuration generates chiral skyrmions, provided that the substrate-induced uniaxial anisotropy is weak and that the polarization draws up a Bloch twisting of the polarization inside the domain wall. The left and right chiral skyrmions, shown in Fig. 1b, are mnemonically visualized by left and right hands, where thumbs point in the direction of the polarization orientation in the core, which is referred to as skyrmion polarity, and the fingers curl along the orientations of the winding of the polarization around the skyrmion core corresponding to the skyrmion chirality. Accordingly, we introduce the notations ${L}_{+}$, ${L}_{-}$, ${R}_{+}$, and ${R}_{-}$, where the "$L$" and "$R$" correspond to the left and right-hand skyrmions and "+" and "$-$" subscript denote the "up" and "down" polarities, respectively. The structure of a skyrmion The polarization distribution inside the skyrmion confined within the nanodot is shown in Fig. 2a. The polarization texture preserves the structure of the polarization rotation inside the chiral Bloch-like circular domain wall over almost entire height of the sample. It is within these chiral domain walls where the chirality of the nanodot is concentrated. At the top near-surface layer the polarization configuration assumes the sinc-like shape to form a Néel-type non-chiral skyrmion to maintain the polarization tangential to the surface, see the top view in Fig. 2b, in order to prevent the formation of surface depolarization charges. The entire texture resembles the structure of the bubble domain in a double-periodic domain structure observed in ferroelectric superlattices25, and has a topology of the Hopf fibration similar to that in the ferroelectric nanoparticles26. In what follows we will construct protocols that enable formation of a skyrmion as well as electric field-tuned transitions between different polarization configurations. The switching between the different states will be described as switching between different mean chiralities, which we define as $$\bar{\chi }=\frac{1}{V}{\int }_{V}\,\chi (r)\,d{\bf{r}}\,,\,\chi ={\bf{P}}\cdot [\nabla \times {\bf{P}}],$$ where $\chi (r)$ is a chirality density, and integration is performed over the nanodot volume $V$, so that the $\bar{\chi } > 0$ for right- and $\bar{\chi } < 0$ for left skyrmions. It is important to clearly distinguish between chirality defined by Eq. (1) and identifying the objects that cannot be mapped to their mirror images by rotations and translations1, and another swirling characteristics of the vector fields, vorticity, quantified as $\nabla \times {\bf{P}}$ and toroidal moment $\int {\bf{r}}\times ({\bf{P}}-\bar{{\bf{P}}})\,dV$. The controlled manipulation by four distinct skyrmion states, ${R}_{\pm }$ and ${L}_{\pm }$, offers the opportunity for implementing a platform for ferroelectric-based multivalued logic units32,33. Field-tuned topological states in the nanodot. (a) Cross section of the nanodot displaying the polarization distribution, with white arrows showing the direction of the polarization. The polarization rotation over 180° in the plane of the Bloch domain walls results in the chirality distribution, $\chi (r)$, shown by the colour map. The legend to the map is given below in low-right corner of the figure. The crossed circle, $\otimes $, at the domain wall denotes the polarization vector going into the cross-section plane, and the circle with the central dot, ⊙, stands for the out–of–plane polarization. (b) The top view of the sinc-like distribution of the polarization at the near-surface layer of the nanodot. (c) Hysteresis behaviour of the polarization of the nanodot as a function of the applied field. The blue and red branches correspond to the up-down and down-up sweeps of the applied field. The numbers mark the different topological states of the polarization. (d) Hysteresis protocols of the chirality switching that allow to come to the ${L}_{\pm }$ and ${R}_{\pm }$ skyrmion states. The arrows show the direction of the sweep. The gray branch corresponds to the virgin curve of the poling of the nanodot reflecting both, positive and negative directions of the electric field variation. The blue and red branches again correspond to the up-down and down-up sweeps of the applied field. The violet and yellow branches correspond to the reversal of the field sweep from the blue and red branches, respectively. (e) The distribution of the polarization and chirality in the original polarization state (0) and in the sequence of the topological states arising during the re-polarization of the nanodot by the applied field (view from the bottom) that follows the blue branch of panels c and d. The yellow points mark the cross-section of the Bloch lines piercing the nanodot. The behaviour of a skyrmion in the electric field In order to elaborate on the protocols for controlling and tailoring skyrmions, we investigate the response of the polarization in the nanodot to applied electric field $U$. The mean polarization ($\bar{P}(U)$) and the chirality ($\bar{\chi }(U)$) hysteresis curves are shown in Fig. 2c,d respectively. The corresponding stages of the process are displayed in Fig. 2e. We first set the zero-field ferroelectric state at room temperature by quenching it from the paraelectric state with randomly oriented small-amplitude polarization. The resulting state, which we denote as 0-state, has in Fig. 2e a structure of four-band stable domain stripes32. The domains are separated by the Bloch domain walls in which the direction of the polarization rotations determines the domain wall chirality that can be either positive (shown in red) or negative (shown in blue). At the loci where the chiralities of the opposite sign meet, a linear topological defect, so-called Bloch line (denoted as yellow dot in the cross-section images) penetrating the nanodot forms. The Bloch lines were observed in ferromagnetic domain walls34 and predicted to appear in ferroelectrics35. It is the dynamics of the Bloch lines that eventually controls the chirality switch in the nanodot. The precise distribution of the chiralities in the Bloch domain walls in the initial state of course depends on which random paraelectric configuration was quenched, but does not affect the polarization evolution after the initial poling in a large electric field. To create and manipulate the skyrmion, we ramp up the electric field by applying a positive voltage $U$ to the circuit, necessary to uniformly polarize the sample. The virgin curve (gray color line) passes through the polarization stripe states differing by the structure of domain walls which move and interswitch their chiralities, see Supplementary Video. The similar effect is achieved by varying the field in the negative direction. The evolution concludes with the jump into the skyrmion state at the latest stage before complete poling is achieved (state 1 in Fig. 2d.). We then start reducing the field strength to zero and, subsequently, having reversed the field direction, increase its magnitude. By analyzing the evolution of the polarization, we establish a protocol that allows us to switch the chirality of the system, as shown in Fig. 2d,e. We consider first the evolution of the system under decreasing field from state 1, depicted by the blue line in Fig. 2d. The details of the transformation of the polarization distribution are visualized in the video in the SI. Upon decreasing the electric field below the threshold value, the polarization in the central region of the nanodot switches its orientation to the opposite one. As a result, the bubble domain (state 2) forms, for which the cylindrical domain wall partitions into two half-cylindrical segments with opposite chiralities separated by Bloch lines, the total chirality remaining zero. As the field strength is further reduced, the skyrmion ${L}_{-}$ (state 3) with negative chirality $\bar{\chi }$ forms as a result of merging the Bloch lines and a concomitant collapse of the positive-chirality segment, and a skyrmion appears (state 3). Further reducing the field strength, the thickness of the skyrmion core grows, leading to repartitioning of the up- and down-oriented polarization regions (state 4) and, hence in the $\bar{P}(U)$-dependence shown in Fig. 2c. The skyrmion texture remains as the field is completely removed (state 4), and even as the field direction is reversed, although the texture now becomes metastable. At some negative field (state 5) the skyrmion becomes unstable and decays into a multi-domain state (state 6) with zero chirality. The arc-shape domain walls form three pairs of segments with opposite chirality that connect to the nanodot sides. The two segments of each pair are separated by a Bloch line, and the total chirality of each pair is zero. As the field magnitude is further increased, one of the domain walls pairs is rearranged to favor the nucleation of a cylindrical domain at the surface of the nanodot (state 7) with the abrupt propagation into the interior of the sample. Two other pairs of domain walls disappear from the sample. The cylindrical bubble domain with two opposite-chirality domain walls settles at the the center of the nanodot (state 8) and further transforms to the ${R}_{+}$ skyrmion with $\bar{\chi } > 0$ and negative mean polarization (state 9) through the collapse of the negative-chirality domain wall. As the field magnitude continues to increase, the core of the skyrmion shrinks and the system arrives at state 10. Finally, the system jumps to the uniformly polarized state 11 with the negative polarization orientation. If, however, the field at point 9 is reversed with a decreasing magnitude to reach zero (the violet branch in Fig. 2d), the system remains in the ${R}_{+}$ skyrmion state at $U=0$. With the further field increase in the positive direction, the system repeats the sequence 3–11 but with the opposite polarization and chirality. In other words, the system finally returns into the uniform up-polarized state 1. One can make the system evolve having started with the negative fields and the polarization poled in the negative direction. In this case, the system will follow the branches denoted by the red-yellow traces in Fig. 2d. The emerging hysteresis branches are symmetric to the blue-violet ones with respect to $U\to -U$ reversal. The red branch corresponds to the blue one and the yellow branch corresponds to the violet one. The corresponding polarization states for the potential $U$ are obtained from their counterparts from the blue-violet branches at the potential $-U$ by the reversing the sign of the ${P}_{z}$ component of the polarization ${\bf{P}}$. By the proper sweep protocol of the electric field, one arrives therefore to the ${R}_{-}$ and ${L}_{+}$ skyrmions at $U=0$. Therefore, one sees that by the appropriate set of the protocol, one can obtain and switch between all of the four skyrmion states, ${L}_{\pm }$ and ${R}_{\pm }$ with different chirality and polarity orientations. The spontaneous chirality breaking It is important to note that the applied electric field does not possess its own chirality. This implies that the direction of the switch is determined rather by underlying local fluctuations in the chirality of the material which then serve as the nucleation centers of the emergent skyrmions. We describe this effect by introducing the fluctuating chirality field $\Lambda (r)$, which enters the system energy functional as the additional term $-\varLambda \,({\bf{P}}\cdot \nabla \times \,{\bf{P}})$. On the verge of the spontaneous symmetry breaking, even the slightest fluctuations would push the system into the either of degenerate, "left" and/or "right" free energy minima. In nature, one or another type of fluctuations would arise spontaneously but randomly. It is natural, in the case of numerical simulations, to quantify and control this effect, which is achieved by introducing the conjugate chiral field $\Lambda $ which impersonates random fluctuations of chirality that would appear in a real experimental system. In our numerical experiments, these spatial fluctuations are implemented via generating random tetrahedral configurations, maintaining the approximately constant mesh size. Altering a particular mesh changes the sign of emerging 1–2 and/or 6–8 jumps at the blue branch. The triggering effect of mesh fluctuations is verified by the mirror reflection of the discretization mesh leading to changing the sign of the chirality jump to the opposite one. Remarkably, even small fluctuations in $\Lambda $ lead to switching chirality. This implies an opportunity for laser-activated manipulation of the polarization36 employing the circular polarized irradiation of the optical tweezers for controlling the direction of the chirality switch. Functional and coefficients The polarization of the strained nanodot is obtained from the minimization of the free energy functional, depending on the polarization, ${\bf{P}}=({P}_{1},{P}_{2},{P}_{3})$, and the electrostatic potential, $\varphi $, $$F=\int \,({[{a}_{i}^{\ast }({u}_{m},T){P}_{i}^{2}+{a}_{ij}^{\ast }{P}_{i}^{2}{P}_{j}^{2}+{a}_{ijk}{P}_{i}^{2}{P}_{j}^{2}{P}_{k}^{2}]}_{i\le j\le k}+\frac{1}{2}{G}_{ijkl}({\partial }_{i}{P}_{j})({\partial }_{k}{P}_{l})+({\partial }_{i}\varphi ){P}_{i}-\frac{1}{2}{\varepsilon }_{0}{\varepsilon }_{b}{(\nabla \varphi )}^{2}-\Lambda ({\bf{P}}\cdot \nabla \times \,{\bf{P}})\,){d}^{3}r\,,$$ where the summation over the repeated indices $i,j,\mathrm{..}.=1,2,3$ (or $x,y,z$) is performed. The numerical parameters are specified for the PbTiO3 nanodot, strained by the substrate with the compressive misfit strain ${u}_{m}\simeq -0.002$. The first square brackets term of (2) stands for the Ginzburg-Landau energy of the strained ferroelectric film37, written in the form given in33. The 2nd-order coefficients depend on the misfit strain ${u}_{m}$ and temperature $T$ and are expressed as ${a}_{1}^{\ast }={a}_{2}^{\ast }=3.8\times {10}^{5}(T-479{}^{\circ }C)-11\times {10}^{9}\,{u}_{m}$ C−2 m2 N−1 and ${a}_{3}^{\ast }=3.8\times {10}^{5}(T-{479}^{\circ }C)+9.5\times {10}^{9}\,{u}_{m}$ C−2 m2 N−1. The strained renormalized 4th-order coefficients partially account for the elastic interactions, obey the tetragonal symmetry conditions and equal to ${a}_{11}^{\ast }={a}_{22}^{\ast }\simeq 0.42\times {10}^{9}$ C−4 m6 N, ${a}_{33}^{\ast }\simeq 0.05\times {10}^{9}$ C−4 m6 N, ${a}_{13}^{\ast }={a}_{23}^{\ast }\simeq 0.45\times {10}^{9}$ C−4 m6 N and ${a}_{12}^{\ast }\simeq 0.73\times {10}^{9}$ C−4 m6 N. The 6th-order coefficients conserve the cubic symmetry, ${a}_{111}={a}_{222}={a}_{333}\simeq 0.26\times {10}^{9}$ C−6 m10 N, ${a}_{112}={a}_{113}={a}_{223}\simeq 0.61\times {10}^{9}$ C−6 m10 N, and ${a}_{123}\simeq -\,3.7\times {10}^{9}$ C−6 m10 N. The second term of (2) corresponds to the gradient energy. The gradient energy coefficients ${G}_{ijkl}$ are obtained by the cubic symmetry permutations of the non-equivalent representatives ${G}_{1111}=2.77\times {10}^{-10}$ C−2 m4 N, ${G}_{1122}=0$, and ${G}_{1212}=1.38\times {10}^{-10}$ C−2 m4 N38. The next two terms in (2) correspond to the electrostatic energy, written in terms of the electrostatic potential $\varphi $39. Here, ${\varepsilon }_{0}=8.85\times {10}^{-12}$ C V−1 m−1 is the vacuum permittivity, and ${\varepsilon }_{b}\simeq 10$ is the background dielectric constant of the non-polar ions40. The last term of (2) emulates the interaction of the ferroelectric polarization with the material chirality fluctuation, described by the parameter $\Lambda $. In most simulations, we took $\Lambda =0$. The local fluctuation, defining the direction of the chirality jumps naturally arise due to the random mesh configuration. To calibrate the effect of the mesh fluctuations we swept the value of $\Lambda $ and found that at the threshold value ${\Lambda }_{c}\simeq 7\times {10}^{-5}$ C−2 m3 N, the direction of the chirality jump 2–3 at Fig. 2d changes to the opposite one. Computational techniques The simulations were performed using the FERRET package40, designed for the multi-physics simulation environment MOOSE41. In order to determine the minima of the free energy functional we use the standard technique of letting system to evolve ("relax") into these minima. This effective relaxation process is set by the time-dependent Ginzburg-Landau (TDGL) relaxation equation $-\gamma \,\partial {P}_{i}/\partial t=\delta F/\delta {P}_{i}$, where $F$ is the total static free energy functional (2) including the electrostatic potential $\varphi $. The latter is found at each step of the relaxation as a solution of the Poisson equation ${\varepsilon }_{0}{\varepsilon }_{b}{\nabla }^{2}\varphi ={\partial }_{i}{P}_{i}$. The TDGL equation includes the parameter $\gamma $ that sets a time scale for the dissipation of energy hence the rate of the motion of the system to a local free energy minimum. Since we are interested in the rapid computation of the sought local minima, but not in the real dynamics of the system, we choose $\gamma $ that provides the fast relaxation ensuring, at the same time, that the observation time scale remained longer than the relaxation time of the ferroelectric system. Typically, this is the case also in experimental situations since nanoscale ferroelectric systems can relax in a static field on times of the order of a few nanoseconds. Therefore, the TDGL equation gives indeed a good description of quasi-static properties, as has been demonstrated over the past few decades25,31,42,43,44,45. Hence, when talking about the "relaxation" and "time scale," we refer to the standard so-called Phase Field Relaxation numerical method designed for finding the minima of the free energy of complex systems. In our calculations we thus set the relaxation parameter $\gamma $ equal to unity. For more details on simulations see Supplementary Information. The geometry and conceptual setup of the simulated system are shown in Fig. 1a. We selected the diameter and the thickness of the nanodot as 40 and 20 nm respectively as the optimal geometry for the skyrmion's observation. According to estimates the modification of the geometrical parameters over 15–20% does not change the results much, although the evolution of the Bloch points may somewhat differ marginally and in ways that are inconsequential for our results. The driving field was controlled by the voltage $U$, applied to the electrodes. The upper electrode was separated from the nanodot by the empty, or low-$\varepsilon $ dielectric space of the thickness of $60$ nm. The initial paraelectric state with the small randomly-distributed polarization was used as an initial condition for the quench to the original (virgin) state. Then, the quasi-static field variation protocols were used with the polarization distribution at the previous stage taken as the initial condition. The different finite-element meshes were used to ensure the stability of the process. Computational scripts are available online at https://github.com/ferroelectrics/skyrmion. Kelvin, W. T. B. 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Philosophical Transactions of the Royal Society B: Biological Sciences 371, 20150403 (2016). Bahr, C. & Kitzerow, H.-S. Chirality in liquid crystals (Springer, 2001). Chandrasekhar, S. Chirality in liquid crystals (Springer Science & Business Media, 2006). Seki, S. & Mochizuki, M. Skyrmions in magnetic materials (Springer, 2016). Liu, J. P., Zhang, Z. & Zhao, G. Skyrmions: topological structures, properties, and applications (CRC Press, 2016). Zhang, S. Chiral and topological nature of magnetic skyrmions (Springer, 2018). Phatak, C., Heinonen, O., De Graef, M. & Petford-Long, A. Nanoscale skyrmions in a nonchiral metallic multiferroic: Ni2MnGa. Nano letters 16, 4141–4148 (2016). Navas, D. et al. Route to form skyrmions in soft magnetic films. APL Materials 7, 081114 (2019). Tagantsev, A. K., Courtens, E. & Arzel, L. Prediction of a low-temperature ferroelectric instability in antiphase domain boundaries of strontium titanate. Physical Review B 64, 224107 (2001). Lee, D. et al. Mixed Bloch-Néel-Ising character of 180° ferroelectric domain walls. Phys. Rev. B 80, 060102 (2009). Wojdeł, J. C. & Íñiguez, J. Ferroelectric transitions at ferroelectric domain walls found from first principles. Phys. Rev. Lett. 112, 247603 (2014). Cherifi-Hertel, S. et al. Non-ising and chiral ferroelectric domain walls revealed by nonlinear optical microscopy. Nature communications 8, 15768 (2017). Baudry, L., Luk'yanchuk, I. A. & Sené, A. Inhomogeneous polarization switching in finite-size cubic ferroelectrics. Ferroelectrics 427, 34–40 (2012). Baudry, L., Luk'yanchuk, I. A. & Sené, A. Switching properties of nano-scale multi-axial ferroelectrics: geometry and interface effects. Integr. Ferroelectr. 133, 96–102 (2012). Baudry, L., Sené, A., Luk'yanchuk, I. A., Lahoche, L. & Scott, J. F. Polarization vortex domains induced by switching electric field in ferroelectric films with circular electrodes. Phys. Rev. B 90, 024102 (2014). Nahas, Y. et al. Discovery of stable skyrmionic state in ferroelectric nanocomposites. Nat. Commun. 6 (2015). Gonçalves, M. P., Escorihuela-Sayalero, C., Garca-Fernández, P., Junquera, J. & Íñiguez, J. Theoretical guidelines to create and tune electric skyrmion bubbles. Science advances 5, eaau7023 (2019). Das, S. et al. Observation of room-temperature polar skyrmions. Nature 568, 368–372 (2019). Luk'yanchuk, I., Tikhonov, Y., Razumnaya, A. & Vinokur, V. Hopfions emerge in ferroelectrics. arXiv preprint arXiv:1907.03866 (2019). Bratkovsky, A. & Levanyuk, A. Abrupt appearance of the domain pattern and fatigue of thin ferroelectric films. Phys. Rev. Lett. 84, 3177 (2000). Stephanovich, V., Luk-yanchuk, I. & Karkut, M. Domain-enhanced interlayer coupling in ferroelectric/paraelectric superlattices. Phys. Rev. Lett. 94, 047601 (2005). De Guerville, F., Luk-yanchuk, I., Lahoche, L. & El Marssi, M. Modeling of ferroelectric domains in thin films and superlattices. Mater. Sci. Eng. B 120, 16–20 (2005). Zubko, P., Stucki, N., Lichtensteiger, C. & Triscone, J.-M. X-ray diffraction studies of 180° ferroelectric domains in PbTiO3/SrTiO3 superlattices under an applied electric field. Phys. Rev. Lett. 104, 187601 (2010). Yadav, A. et al. Observation of polar vortices in oxide superlattices. Nature 530, 198–201 (2016). Martelli, P.-W., Mefire, S. M. & Luk'yanchuk, I. A. Multidomain switching in the ferroelectric nanodots. EPL (Europhysics Letters) 111, 50001 (2015). Baudry, L., Lukyanchuk, I. & Vinokur, V. M. Ferroelectric symmetry-protected multibit memory cell. Scientific reports 7, 42196 (2017). Malozemoff, A. & Slonczewski, J. Magnetic Domain Walls in Bubble Materials: Advances in Materials and Device Research, vol. 1 (Academic press, 2016). Salje, E. & Scott, J. Ferroelectric bloch-line switching: A paradigm for memory devices? Appl. Phys. Lett. 105, 252904 (2014). Stoica, V. et al. Optical creation of a supercrystal with three-dimensional nanoscale periodicity. Nature materials 18, 377–383 (2019). Pertsev, N., Zembilgotov, A. & Tagantsev, A. Effect of mechanical boundary conditions on phase diagrams of epitaxial ferroelectric thin films. Phys. Rev. Lett. 80, 1988 (1998). Wang, J., Shi, S.-Q., Chen, L.-Q., Li, Y. & Zhang, T.-Y. Phase-field simulations of ferroelectric/ferroelastic polarization switching. Acta Materialia 52, 749–764 (2004). Landau, L., Lifshitz, E. & Pitaevskii, L. Eloectrodynamics of Continuous Media (Oxford: Pergamon, 1984). Mangeri, J. et al. Topological phase transformations and intrinsic size effects in ferroelectric nanoparticles. Nanoscale 9, 1616–1624 (2017). Gaston, D., Newman, C., Hansen, G. & Lebrun-Grandie, D. Moose: A parallel computational framework for coupled systems of nonlinear equations. Nucl. Eng. and Design 239, 1768–1778 (2009). Li, Y., Hu, S., Liu, Z. & Chen, L. Effect of substrate constraint on the stability and evolution of ferroelectric domain structures in thin films. Acta materialia 50, 395–411 (2002). Choudhury, S., Li, Y., Krill Iii, C. & Chen, L. Effect of grain orientation and grain size on ferroelectric domain switching and evolution: Phase field simulations. Acta materialia 55, 1415–1426 (2007). Chen, L.-Q. Phase-field method of phase transitions/domain structures in ferroelectric thin films: a review. Journal of the American Ceramic Society 91, 1835–1844 (2008). Zhang, J., Schlom, D., Chen, L. & Eom, C. Tuning the remanent polarization of epitaxial ferroelectric thin films with strain. Applied Physics Letters 95, 122904 (2009). This work was supported by the US Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division (O.H. V.M.V. and partly I.L. and A.G.); by the H2020 RISE-ENGIMA, RISE-MELON and ITN-MANIC actions (Y.T., A.R. and I.L.), By Bourse Ostrogradski of French Goverment (Y.T.), and by the Southern Federal University, Russia (A.R. and Y.T.). Faculty of Physics, Southern Federal University, 5 Zorge str., 344090, Rostov-on-Don, Russia Yu. Tikhonov, M. Pavlenko & A. Razumnaya University of Picardie, Laboratory of Condensed Matter Physics, Amiens, 80039, France Yu. Tikhonov, S. Kondovych, A. Sené & I. Luk'yanchuk Life Chemicals Inc., Murmanska st. 5, Kyiv, 02660, Ukraine S. Kondovych Institute of Physics, Academy of Sciences of the Czech Republic, Na Slovance 2, 18221, Praha 8, Czech Republic J. Mangeri Department of Physics, University of Connecticut, Storrs, CT, USA J. Mangeri & S. Nakhmanson Institute of Electronics, Microelectronics and Nanotechnology (IEMN)-DHS Départment, UMR CNRS 8520, Université des Sciences et Technologies de Lille, 59652, Villeneuve d'Ascq Cedex, France L. Baudry James Franck Institute, University of Chicago, Chicago, Illinois, 60637, USA A. Galda Department of Materials Science & Engineering and Institute of Material Science, University of Connecticut, Storrs, Connecticut, 06269, USA S. Nakhmanson Materials Science Division, Argonne National Laboratory, 9700S. Cass Avenue, Argonne, Illinois, 60637, USA O. Heinonen & V. M. Vinokur L. D. Landau Institute for Theoretical Physics, Akademika Semenova av., 1A9, Chernogolovka, 142432, Russia I. Luk'yanchuk Consortium for Advanced Science and Engineering (CASE) University of Chicago, 5801S Ellis Ave, Chicago, IL, 60637, USA V. M. Vinokur Yu. Tikhonov M. Pavlenko A. Sené O. Heinonen A. Razumnaya I.L., O.H. and V.M.V. conceived the work, S.K., A.S., O.H., I.L., and V.M.V. performed analytical calculations, J.M., O.H., and S.N. developed the Ferret code, Y.T. elaborated on the Ferret code adding new functionalities, Y.T., S.K., J.M., M.P., L.B., A.G., A.R. performed numerical calculations, Y.T. and A.R. visualized numerical data, Y.T., S.N., A.R., O.H., I.L. and V.M. analyzed the results, I.L., and V.M.V. wrote the manuscript, all the authors discussed the manuscript. Correspondence to V. M. Vinokur. Supplementary Information. Supplementary Video Caption. Tikhonov, Y., Kondovych, S., Mangeri, J. et al. Controllable skyrmion chirality in ferroelectrics. Sci Rep 10, 8657 (2020). https://doi.org/10.1038/s41598-020-65291-8 Phase-field simulations of vortex chirality manipulation in ferroelectric thin films Di Liu Houbing Huang npj Quantum Materials (2022)	CommonCrawl
Working with the Exponential Power Distribution Using gnorm Maryclare Griffin The exponential power distribution, also known as the generalized normal distribution, was first described in Subbotin (1923)1 and rediscovered as the generalized normal distribution in Nadarajah (2005)2. It generalizes the Laplace, normal and uniform distributions and is pretty easy to work with in many ways, so it can be very useful. Accordingly, I've made a little R package called gnorm that provides density, CDF and quantile functions for the exponential power distribution as well as random variate generation that are analogous to dnorm, pnorm, qnorm and rnorm. The package can be installed either via CRAN using install.packages(gnorm) or via Github using the devtools package and the command install_github("maryclare/gnorm"). We can load after installation as follows: library(gnorm) Since we're going to be generating some random variables, we should also set a seed: Exponential Power Density, dgnorm An exponential power distributed random variable, $x$, has the following density: \[ p(x \| \mu, \alpha, \beta) = \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{\|x - \mu\|}{\alpha})^\beta\}, \] where $\mu$ is the mean of $x$ and $\alpha > 0$ and $\beta > 0$ are scale and shape parameters. The function dgnorm gives the exponential power density given above, $p(x \| \mu, \alpha, \beta)$, evaluated at specified values of $x$, $\mu$, $\alpha$ and $\beta$. Below, we give an example of evaluating the exponential power density using dgnorm for $\mu = 0$, $\alpha = \sqrt{2}$ and $\beta = 2$. Note that this is in fact the standard normal density! xs <- seq(-1, 1, length.out = 100) plot(xs, dgnorm(xs, mu = 0, alpha = sqrt(2), beta = 2), type = "l", xlab = "x", ylab = expression(p(x))) Like dnorm, dgnorm has a log argument, with the default log=FALSE. When log=TRUE is specified, $\text{log}p(x \| \mu, \alpha, \beta)$ is returned. An Aside on Parametrizing the Exponential Power Distribution This is not necessarily the most natural way to parametrize the exponential power density. One can replace $\alpha$ and $\beta$ with the standard deviation of $x$, $\sigma$, and a shape parameter $q$: \[ p(x \| \mu, \sigma, q) = \frac{q}{2\tau}\sqrt{\frac{\Gamma(3/q)}{\Gamma(1/q)^3}}\text{exp}\{-(\frac{\Gamma(3/q)}{\Gamma(1/q)})^{q/2}(\frac{\|x - \mu\|}{\sigma})^q\} \] This is similar but not identically to the parametrization preferred by Box and Tiao (1973)3. That said, the parametrization using $\alpha$ and $\beta$ is the one featured on Wikipedia so realistically, it's the one everyone will find when they look up this distribution, so we'll use the $\alpha$ and $\beta$ parametrization for the rest of the vignette. Exponential Power CDF, pgnorm The exponential power CDF is derived as follows: \[ \begin{aligned} \text{Pr}(x \leq q \| \mu, \alpha, \beta) &= \int_{-\infty}^q \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{\|x - \mu\|}{\alpha})^\beta\} d x \\ &=\int_{-\infty}^{q - \mu} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta \|z\|^\beta \} d z \\ &=\Bigg\{\begin{array}{cc} \int_{-\infty}^{q - \mu} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta \left(-z\right)^\beta \} d z & q \leq \mu \\ \int_{-\infty}^{0} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta (-z)^\beta + \int_{0}^{q - \mu} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta \} d z & q > \mu \\ \end{array} \\ &=\Bigg\{\begin{array}{cc} \int_{-(q - \mu)}^{\infty} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta \} d z & q \leq \mu \\ \int_{0}^{\infty} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta dz + \int_{0}^{q - \mu} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta \} dz & q > \mu \\ \end{array} \\ &=\Bigg\{\begin{array}{cc} \int_{0}^{\infty} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta \} d z - \int_{0}^{-(q - \mu)} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta \} d z& q \leq \mu \\ \int_{0}^{\infty} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta\} dz+ \int_{0}^{q - \mu} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta \} d z & q > \mu \\ \end{array} \\ &= \int_{0}^{\infty} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta \}dz+ \text{sign}(q - \mu) \int_{0}^{\|q - \mu\|} \frac{\beta}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta z^\beta \} d z \\ &= \int_{0}^{\infty} \frac{w^{1/\beta - 1}}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta w \}dw+ \text{sign}(q - \mu) \int_{0}^{\|q - \mu\|^\beta} \frac{w^{1/\beta - 1}}{2\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta w \} d w && z^\beta = w \\ &= \frac{1}{2}+ \frac{\text{sign}(q - \mu)}{2} \underbrace{\int_{0}^{\|q - \mu\|^\beta} \frac{w^{1/\beta - 1}}{\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{1}{\alpha})^\beta w \} d w}_{\text{Gamma CDF, Shape=$\frac{1}{\beta}$, Rate=$(\frac{1}{\alpha})^\beta$}} \end{aligned} \] We can evaluate the exponential power CDF using the gamma CDF - the pgamma function evaluates the CDF of a gamma distribution with parameters $\frac{1}{\beta}$ and $(\frac{1}{\alpha})^\beta$ at $\|q - \mu\|^\beta$. The function pgnorm returns the value of the CDF for specified values of $q$, $\mu$, $\alpha$ and $\beta$. Like the corresponding pnorm function, it also has the following arguments: log.p, which returns the log probability when set to TRUE. The default is log.p=FALSE; lower.tail, which returns $\text{Pr}(x > q \| \mu, \alpha, \beta)$ when set to FALSE. The default is lower.tail=TRUE. Below, we give an example of evaluating the exponential power CDF using pgnorm for $\mu = 0$, $\alpha = \sqrt{2}$ and $\beta = 2$. As in the previous example - this is the standard normal CDF. plot(xs, pgnorm(xs, 0, sqrt(2), 2), type = "l", xlab = "q", ylab = expression(paste("Pr(", x<=q, ")", sep = ""))) Exponential Power Quantile Function (Inverse CDF), qgnorm A (relatively) straightforward way to compute the inverse CDF function for an exponential power random variable is to use its scale-sign representation (Gupta and Varga, 1993)4. We can write $x \stackrel{d}{=} su + \mu$, where $s$ and $u$ are independent random variables with $p(s \| \alpha, \beta) = \frac{\beta}{\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{s}{\alpha})^\beta\}$ and $u$ uniformly distributed on $\{-1, 1\}$. Given $p$, we want to find the value of $q$ that satisfies $p = \text{Pr}(x \leq q \| \mu, \alpha, \beta)$. We can rewrite the problem using $s$ and $u$ as finding $p$ and $q$ that satisfy: \[ \begin{aligned} \|p - 0.5\|&= \text{Pr}(s \leq \|q - \mu\| \| \alpha, \beta)\text{Pr}(u =\text{sign}(q - \mu)) \\ &= \frac{1}{2}\text{Pr}(s \leq \|q - \mu\| \| \alpha, \beta) \\ &= \frac{1}{2}\int_0^{\|q - \mu\|} \frac{\beta}{\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{s}{\alpha})^\beta\} ds \\ &= \frac{1}{2}\underbrace{\int_0^{\|q - \mu \|^\beta} \frac{1}{\alpha \Gamma(1/\beta)}t^{1/\beta - 1}\text{exp}\{-(\frac{1}{\alpha})^\beta t\} dt}_{\text{Gamma CDF, Shape=$\frac{1}{\beta}$, Rate=$(\frac{1}{\alpha})^\beta)$}} & s^\beta = t \end{aligned} \] Let $g(2\|p - 0.5\|; \frac{1}{\beta}, (\frac{1}{\alpha})^\beta)$ refer to the inverse CDF function for a gamma distribution with shape $\frac{1}{\beta}$ and rate $(\frac{1}{\alpha})^\beta$ evaluated at $2\|p - 0.5\|$. Then the inverse CDF of the exponential power distribution is given by: \[ \|q - \mu\| = g(2\|p - 0.5\|; \frac{1}{\beta}, (\frac{1}{\alpha})^\beta)^{1/\beta}. \] Noting that $q > \mu$ when $p > 0.5$ and $q \leq \mu$ when $p \leq 0.5$, we get: \[ q = \text{sign}(p - 0.5)g(2\|p - 0.5\|; \frac{1}{\beta}, (\frac{1}{\alpha})^\beta)^{1/\beta} + \mu. \] This is what is implemented by qgnorm for specified values of $q$, $\mu$, $\alpha$ and $\beta$. Like the corresponding qnorm function, it also has the following arguments: log.p, which indicates that the log probability has been provided when set to TRUE. The default is log.p=FALSE; lower.tail, which indicates that $q$ satisfies $p \text{Pr}(x > q \| \mu, \alpha, \beta)$ when set to FALSE. The default is lower.tail=TRUE. Below, we give an example of evaluating the exponential power inverse CDF using qgnorm for $\mu = 0$, $\alpha = \sqrt{2}$ and $\beta = 2$. As in the previous examples - this is the standard normal CDF. xs <- seq(0, 1, length.out = 100) plot(xs, qgnorm(xs, 0, sqrt(2), 2), type = "l", xlab = "p", ylab = expression(paste("q: p = Pr(", x<=q, ")", sep = ""))) Exponential Power Random Variate Generate, rgnorm The function rgnorm generates exponential power random variates using the same scale-sign stochastic representation of an exponential power random variable used to compute the inverse CDF. Recall that an exponential power distributed random variable, $x$, with parameters $\mu$, $\alpha$ and $\beta$ can be written as $x \stackrel{d}{=} su + \mu$, where $s$ and $u$ are independent random variables with $p(s \| \alpha, \beta) = \frac{\beta}{\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{s}{\alpha})^\beta\}$ and $u$ uniformly distributed on $\{-1, 1\}$ Drawing a value of $u$ that is uniformly distributed on $\{-1, 1\}$ is simple. We can draw a value $s$ according to $p(s \| \alpha, \beta)$ using the inverse CDF function of $s$. For fixed $p$, the inverse CDF function of $s$ finds the value $q$ that satisfies $p = \text{Pr}(s \leq q \| \alpha, \beta)$: \[ \begin{aligned} p &= \int_0^q \frac{\beta}{\alpha \Gamma(1/\beta)}\text{exp}\{-(\frac{s}{\alpha})^\beta\}ds \\ &= \int_0^{q^\beta} \frac{1}{\alpha \Gamma(1/\beta)}t^{1/\beta - 1}\text{exp}\{-(\frac{1}{\alpha})^\beta t\}dt & s^\beta = t. \end{aligned} \] As we've seen a few times before, this is a gamma CDF. Let $g(p; \frac{1}{\beta}, (\frac{1}{\alpha})^\beta)$ refer to the inverse CDF function for a gamma distribution with shape $\frac{1}{\beta}$ and rate $(\frac{1}{\alpha})^\beta$ evaluated at $p$. Then we can generate $s$ as follows by drawing a value of $p$ from a uniform distribution on $(0, 1)$ and setting $s = g(p; \frac{1}{\beta}, (\frac{1}{\alpha})^\beta)$. This is how rgnorm generates draws from the exponential power distribution. The function rgnorm is a parametrized like rnorm. The first argument is an integer n, which gives the number of draws to take, and the second through fourth arguments are values of $\mu$, $\alpha$ and $\beta$. Below, we give an example of using the rgnorm to draw exponential power random variates for $\mu = 0$, $\alpha = \sqrt{2}$ and $\beta = 2$. As in the previous examples - this is the standard normal CDF. xs <- rgnorm(100, 0, sqrt(2), 2) hist(xs, xlab = "x", freq = FALSE, main = "Histogram of Draws") An Aside on Alternative Approaches to Exponential Power Random Variate Generation There at least one other approach to generating exponential power random variables for any value of $q$. A uniform scale mixture representation of an exponential power distributed random variable, $x$, was originally given in Walker and Guttierez-Pena (1999)5 and was reprinted in Choy and Walker (2002)6. It can be used to generate an exponential power random variate, $x$, as follows: Draw $\gamma \sim \text{gamma}(\text{shape} = 1 + 1/\beta, \text{rate}=2^{-\beta/2})$; Set $\delta = \alpha\gamma^{1/\beta}/\sqrt{2}$; Draw $x \sim \text{uniform}(\mu -\delta, \mu + \delta)$. Subbotin, M. T. "On the Law of Frequency of Error." Mat. Sb. 31.2 (1923): 206-301.↩ Nadarajah, Saralees. "A generalized normal distribution." Journal of Applied Statistics 32.7 (2005): 685-694.↩ Box, G. E. P. and G. C. Tiao. "Bayesian inference in Statistical Analysis." Addison-Wesley Pub. Co., Reading, Mass (1973).↩ Gupta, A. K. and T. Vargas. "Elliptically Contoured Models in Statistics." Kluwer Academic Publishers (1993).↩ Walker, S. G. and E. Guttierez-Pena "Robustifying Bayesian Procedures". Bayesian Statistics 6, (1999): 685-710.↩ Choy, S. T. B. and S. G. Walker. "The extended exponential power distribution and Bayesian robustness." Statistics & Probability Letters 65.3 (2003): 227-232.↩	CommonCrawl
Physics in a nutshell hamburger menu iconthree horizontal bars on top of each other Solid State Physics Evidence for the Earth's Climate History Surface Temperature of the Earth $ \renewcommand{\D}[2][]{\,\text{d}^{#1} {#2}} $ $\DeclareMathOperator{\Tr}{Tr}$ Stop the war — peace for Ukraine! 1. Heat Exchange with the Environment 2. Black Body Radiation 2.1. Origin of Thermal Radiation 2.2. Planck's Law 2.3. Stefan-Boltzmann Law 3. Radiation Balance 3.1. Incident Radiation 3.1.1. The Solar Constant 3.1.2. The Role of Albedo 3.2. Emitted Radiation 3.3. Result The mean surface temperature of the earth is around $T = 288 \,\text{K}$.[1] Can this number be derived by means of a simple model? Heat Exchange with the Environment A glance at the recent history of the earth's climate shows that the mean surface temparature has been relatively constant. During the past 40 million years the global mean temperature has been varying by only 10 degrees Kelvin and in the past 10.000 years the variation was only of about 1 degree.[2] Thus, even on large time scales the earth's temperature can be regarded as constant. In other words: There is no net transfer of heat from or to the earth's surface. It is in equilibrium with its environment and it therefore exhibits a constant temperature. This implies that the incident heat flux $P_\text{in}$ has to be equal to the outgoing $P_\text{out}$: [3][4] \begin{align} P_\text{in} &= P_\text{out} \label{equilibriumCondition} \\ \left[ P \right] &= \text{W}\,\text{m}^{-2} \end{align} Since the earth is essentially surrounded by a vacuum, the only way of exchanging heat with its environment is through electromagnetic radiation because in contrast to conduction and convection this process does not require a propagation medium.[5][6] Indeed, there is a minor additional contribution through geothermal activity but this will be neglected in the following due to its relative weakness compared to the solar radiation. Black Body Radiation However, we have to specify the expressions for the incoming and outgoing radiant flux. Obviously, the incident radiation originates basically from the sun with its high surface temperature whereas the outgoing radiation has to be related somehow to the earth's temperature. Origin of Thermal Radiation What is the origin of thermal radiation? On a microscopic level an object's temperature is a measure of the kinetic energy of its constituents (eventually charged particles). Thus thermal radiation originates from the thermal motion of charged particles which implicates emission of electromagnetic waves. The kinetic energy of these particles is distributed statistically around a mean value and the same applies for its radiation spectrum. Planck's Law In general it is rather difficult to predict the exact emission spectrum. But Max Planck was able to derive such an expression for so called black bodies (objects that are in thermodynamic equilibrium with their environment and perfect absorbers and emitters of radiation). It can be obtained by applying the laws of statistical and quantum mechanics to the radiation (which is treated as a photon gas). The resulting Planck law relates the energy density per frequency $u(\nu)$ and the frequency $\nu$ itself by \begin{align} u(\nu) \D \nu = \left( \frac{8 \pi h}{c^3} \right) \cdot \frac{\nu^3 \D\nu}{e^{\frac{h\nu}{k_\text{B}T}}-1} \end{align} Stefan-Boltzmann Law The energy emitted from the surface of a black body can be obtained by integrating Planck's law wich yields \begin{align} J(T) &= \sigma \cdot T^4 \label{stefanBoltzmann} \end{align} where $J$ [W m$^{-2}$] is the radiated energy per time and per unit area (energy flux density), ${\sigma \approx 5.67 \cdot 10^{-8} \;\text{W}\,\text{m}^{-2}\,\text{K}^{-4}}$ is the so called Stefan-Boltzmann constant and $T$ is the surface temperature. This equation is known as Stefan-Boltzmann law. The total rate of energy transfer $P$ through any surface is the product of the perpendicular component of the surface and the energy flux density $J(T)$ as given in eq. \eqref{stefanBoltzmann}.[7] Radiation Balance The earth orbits the sun in a distance of $d = 1 \,\text{au}$. Each partition of the earth's surface $A_\text{e} = 4 \pi r_\text{e}^2$ emits radiation due to its own temperature $T_\text{e}$. Simultaneously the earth absorbs radiation emitted by the sun. At the distance of the earth, the rate of incident energy amounts to $S \cdot A_\text{e}^\perp$ where $A_\text{e}^\perp$ is the cross section area of the earth perpendicular to the incident radiation. Incident Radiation The total rate of incident energy $P_\text{in}$ [W] is the product of the energy flux density $J_\text{sun}$ [W m$^{-2}$] of the solar irradiance at the mean radius of the earth's orbit $d = 1 \,\text{au}$ (astronomical unit) and the projection surface of the earth ${ A_\text{e}^\perp = \pi r_\text{e}^2 }$. The Solar Constant The former quantity is in general referred to as the solar constant and its experimental value is \begin{align} S &:= J_\text{sun} \left( d \right) \\ &= 1370 \,\text{W}\,\text{m}^{-2}. \end{align} This number is a measure of how much energy we can receive from the sun and you can illustrate it as follows: If you had a perfect $1\text{m}^2$ solar cell that is capable of converting solar radiation into electricity without loss, it could serve as a power supply for up to 14 100W light bulbs, 3 washing machines or one hair dryer. Thereby the total rate of incident energy can be expressed as \begin{align} P_\text{in} = S \cdot \pi r_\text{e}^2 . \end{align} [8] The Role of Albedo At this point one important modification is necessary: Originally we were interested in the heat transferred to the earth. But until now we ignored the fact that a significant amount ($\approx 30\%$) of the incident radiation is reflected (e.g. by clouds or ice) immediately without transferring heat at all. This percentage is called albedo and amounts to $a = 0.3$ for the earth. Thus, only the proportion of ${1-a}$ of the incident radiation transfers heat to the earth's surface and one obtains:[9] \begin{align} P_\text{in} = (1-a) \cdot S \cdot \pi r_\text{e}^2 . \label{pIn} \end{align} Emitted Radiation To determine the rate of outgoing energy $P_\text{out}$ we approximate the earth as a black body with surface temperature $T_\text{e}$. Then the total rate of energy emitted by the earth's surface $P_\text{out}$ is the product of the energy flux density radiated per unit area (according to the Stefan-Boltzmann law) and the earth's surface area $A_\text{e} = 4 \pi r_\text{e}^2$: \begin{align} P_\text{out} = \sigma T_{e}^4 \cdot 4 \pi r_\text{e}^2 \label{pOut} \end{align} Maybe you wonder why we first used the projection area of the earth and now the total spherical area. The explanation is that in the first case we considered parallel radiation and in that case the perpendicular surface is plane. On the contrary now we considered radially emitted rays and in this case the perpendicular surface has a curved, spherical shape. Now one can insert eqs. \eqref{pIn} and \eqref{pOut} into eq. \eqref{equilibriumCondition} and one obtains: \begin{align} P_\text{in} &= P_\text{out} \\ (1-a) \cdot S \cdot \pi r_\text{e}^2 &= \sigma T_{e}^4 \cdot 4 \pi r_\text{e}^2 \\[2ex] \Leftrightarrow \quad T_\text{e} &= \sqrt[4]{\frac{(1-a) \cdot S}{4\sigma}} \end{align} [10] [11] When inserting the values as given in the previous sections, this calculation yields a surface temperature of about $T_\text{e} = 255\,\text{K}$. Even though this value is not quite bad for a very simple model, it still deviates significantly from the actual value of $T_\text{e} = 288\,\text{K}$. What are the main flaws of this model? It was assumed that the earth is a closed system with a sharp surface surrounded by a vacuum. However, this is not valid for the earth's surface since there is additionally the atmosphere which influences the radiation balance considerably. In the next article a more sophisticated model will comprise the atmosphere's impact. [1] David G. Andrews An Introduction to Atmoshperic Physics Cambridge University Press 2000 (p. 5) [2] J. I. Lunine Earth - Evolution of a habitable world Cambridge University Press 2013 (p. 238) [3] E. Boeker, R. van Grondelle Environmental Physics Wiley 2011 (ch. 1.2) [4] D. Randall Atmosphere, Clouds and Climate Princeton University Press 2012 (p. 28) [5] M. de Oliveira Equilibrium Thermodynamics Springer 2013 (ch. 18.1.1) [7] E. Boeker, R. van Grondelle Environmental Physics Wiley 2011 (ch 2.1.1) [8] D. Randall Atmosphere, Clouds and Climate Princeton University Press 2012 (pp. 28, 31) [10] E. Boeker, R. van Grondelle Environmental Physics Wiley 2011 (ch. 1.2) [11] David G. Andrews An Introduction to Atmoshperic Physics Cambridge University Press 2000 (ch. 1.3.1) About / Terms of use tw-web.solutions Your browser does not support all features of this website! more	CommonCrawl
Oxylipins in triglyceride-rich lipoproteins of dyslipidemic subjects promote endothelial inflammation following a high fat meal Anita Rajamani1, Kamil Borkowski2, Samir Akre ORCID: orcid.org/0000-0003-1495-63991, Andrea Fernandez1, John W. Newman ORCID: orcid.org/0000-0001-9632-65712,3,4, Scott I. Simon1 & Anthony G. Passerini1 Scientific Reports volume 9, Article number: 8655 (2019) Cite this article Lipoproteins Mechanisms of disease Elevated triglyceride-rich lipoproteins (TGRL) in circulation is a risk factor for atherosclerosis. TGRL from subjects consuming a high saturated fat test meal elicited a variable inflammatory response in TNFα-stimulated endothelial cells (EC) that correlated strongly with the polyunsaturated fatty acid (PUFA) content. This study investigates how the relative abundance of oxygenated metabolites of PUFA, oxylipins, is altered in TGRL postprandially, and how these changes promote endothelial inflammation. Human aortic EC were stimulated with TNFα and treated with TGRL, isolated from subjects' plasma at fasting and 3.5 hrs postprandial to a test meal high in saturated fat. Endothelial VCAM-1 surface expression stimulated by TNFα provided a readout for atherogenic inflammation. Concentrations of esterified and non-esterified fatty acids and oxylipins in TGRL were quantified by mass spectrometry. Dyslipidemic subjects produced TGRL that increased endothelial VCAM-1 expression by ≥35%, and exhibited impaired fasting lipogenesis activity and a shift in soluble epoxide hydrolase and lipoxygenase activity. Pro-atherogenic TGRL were enriched in eicosapentaenoic acid metabolites and depleted in esterified C18-PUFA-derived diols. Abundance of these metabolites was strongly predictive of VCAM-1 expression. We conclude the altered metabolism in dyslipidemic subjects produces TGRL with a unique oxylipin signature that promotes a pro-atherogenic endothelial phenotype. Disorders of metabolism are accompanied by an elevated systemic inflammatory state that accelerates the progression of atherosclerotic cardiovascular disease (CVD), the leading cause of death and disability in our society1,2. One-third of Americans are clinically categorized at higher risk for CVD in that they show a cluster of risk factors associated with high fat western diet and sedentary lifestyle defined as metabolic syndrome3. Among these risk factors is hypertriglyceridemia, which is characterized by elevated levels of serum triglycerides, circulating in ApoB-containing triglyceride-rich lipoprotein (TGRL). Elevated TGRL is a strong predictor for CVD4,5,6,7, and its uptake by the endothelium leads to inflammation that precedes and promotes atherosclerosis. Here we investigate how changes in the relative abundance of fatty acid metabolites circulating in TGRL link to its ability to prime an inflammatory response to a high fat meal. Atherosclerosis has been proposed to be a postprandial process that is tied to transient elevation of postprandial triglycerides8,9. A high fat meal induces a postprandial inflammatory state, characterized by elevated levels of circulating TGRL and other inflammatory mediators, including cytokines (e.g. TNFα) that promote endothelial dysfunction accompanying atherosclerosis. A high-fat test meal challenge has been established as an effective tool to elicit inflammation of arterial endothelium and subsequent monocyte capture, changes not typically observed under fasting conditions. Therefore, a postprandial response may provide greater insight into the effects of metabolism on insipient inflammation than the fasting state10,11. We have investigated the capacity of TGRL produced by human subjects after consuming a high saturated fat meal, typical of western diet, to elicit a pro-atherogenic endothelial response. Among the earliest inflammatory responses by the endothelium that precedes atherosclerosis is the acute upregulation of cell adhesion molecules (CAMs), notably vascular cell adhesion molecule (VCAM-1), the predominant receptor for monocyte recruitment from the circulation4,12. VCAM-1 expression correlates with coronary lesion severity in atherosclerotic patients, and is causative in mouse models of atherosclerosis13,14. We showed that TGRL isolated from subjects after a standardized fast food breakfast meal high in saturated fat, can prime either a pro- or anti-atherogenic state, defined as a net up- or down-regulation of VCAM-1 expression in TNFα-stimulated human aortic endothelial cells (HAEC)15,16,17,18. In these studies, the level of VCAM-1 expression was strongly associated with the relative capacity to induce recruitment of monocytes from healthy subjects assessed in TNFα-inflamed HAEC16. The capacity for TGRL to alter VCAM-1 expression correlated closely with an individual's metabolic status as reflected by their level of postprandial triglycerides and abdominal obesity. However, TGRL obtained from the same individuals when fasting did not prime inflamed HAEC for increased VCAM-1 expression15. Unknown is how the composition of TGRL is altered postprandially following the high fat meal to elicit a distinct endothelial inflammatory response by individuals exhibiting metabolic risk factors. Dietary fatty acids of all saturation levels are packaged into and circulated by TGRL, which consist of chylomicrons and VLDL, synthesized in the gut and liver respectively8. The fatty acid composition of circulating TGRL thus reflects both the content of a meal and an individual's overall metabolic state, including the presence of diabetes or dyslipidemia19. We previously reported that pro- and anti-atherogenic TGRL were composed of distinct fatty acid profiles, and that pro-atherogenic TGRL in particular was enriched in non-esterified polyunsaturated fatty acids (PUFA)16. Though overall less abundant in TGRL, PUFA and their oxygenated metabolites, oxylipins, are potent mediators of inflammation20. PUFA are metabolized by three main enzymatic pathways to produce an array of oxylipins: cyclooxygenases (COX), lipoxygenases (LOX), and cytochrome P450s (CYP). COX-derived prostaglandins are generally considered pro-inflammatory. While LOX-derived mid-chain alcohols, leukotrienes, and ketones can be either pro- or anti-inflammatory (depending on the LOX isoform, the parent fatty acid and hydroxylation position), CYP-derived epoxides are considered anti-inflammatory. However, epoxides are readily hydrolyzed by soluble epoxide hydrolase (sEH) to form diols, which are reported to promote or have a neutral effect on inflammation20. Fatty acids and oxylipins are found in both esterified (bound) and non-esterified (free) pools in TGRL, and can be released by the action of lipases or upon receptor-mediated uptake by endothelial cells (EC)21. Although 90% of oxylipins are esterified in lipoproteins, their exact origins are unclear, and the significance of their transport within lipoproteins is largely unknown. It has been demonstrated that diet and metabolism affect oxylipin abundance and distribution among lipoprotein pools. For example, plasma oxylipins were enriched in proportion to the fatty acid composition of a high-fat meal10,11. Metabolic syndrome is associated with a disruption of the oxylipin pattern across all lipoprotein classes that was putatively pro-inflammatory, and was mitigated by prescription omega-3 PUFA intervention22. Moreover, there is evidence that oxylipin metabolism can directly affect atherosclerosis. For example, inhibition of sEH reduced atherosclerosis in ApoE null mice on a high fat diet, which correlated with enrichment in the plasma epoxide to diol ratio23. We address whether oxylipins circulating in TGRL act to exacerbate endothelial inflammatory responses promoting atherosclerosis. The objective of the current study was to identify, using an unbiased metabolomics approach, novel fatty acid constituents that can significantly predict the pro- or anti-atherogenic postprandial response after uptake and processing of an individual's TGRL by endothelium. We hypothesized that metabolic dysregulation in dyslipidemic individuals is associated with a unique fasting and postprandial TGRL oxylipin signature in response to a high fat test meal. We report that the composition of sEH-derived diols and LOX-derived alcohols in postprandial TGRL strongly discriminated the response to the meal between pro- and anti-atherogenic subjects. A postprandial oxylipin signature was identified in pro-atherogenic subjects, characterized by enrichment in eicosapentaenoic acid (EPA) metabolites and depletion of esterified sEH-derived diols. Moreover, fasting indices of lipogenesis activity and the abundance of sEH-derived diols in TGRL were the best predictors of enhanced VCAM-1 expression. Plasma markers of dyslipidemia correlate with TGRL priming of HAEC inflammation To evaluate the capacity of TGRL to mediate an inflammatory response in arterial endothelium, we examined a cohort of 39 subjects, characterizing each individual by their anthropometric, lipid, and metabolic characteristics, fasting and 3.5 hrs after the high fat test meal (Supplementary Table S1). Subjects exhibited a broad range in BMI (19.81–30.25 kg/m2) and waist circumference (24.5–43.0 in), and ranged from normal to hypertriglyceridemic (fasting triglycerides 30–279 mg/dl). We measured the capacity of postprandial TGRL isolated from each subject's plasma to alter TNFα-stimulated CAM surface expression in HAEC in an ex vivo assay. An increase in VCAM-1 surface expression from stimulation with a low dose of TNFα (EC50 = 0.3 ng/ml) correlated most strongly with the postprandial spike in triglyceride levels (ΔTG = postprandial - fasting) (Fig. 1). Notably, postprandial TGRL isolated from subjects whose triglycerides increased >120 mg/dl in response to the meal routinely increased TNFα-stimulated VCAM-1 surface expression. In contrast, postprandial TGRL from those subjects in whom triglycerides increased <120 mg/dl reduced or had a neutral effect on TNFα-stimulated VCAM-1 surface expression in HAEC (Fig. 1a, inset). Each increase in plasma triglycerides of 100 mg/dl corresponded to a 12.5% increase in TNFα-stimulated VCAM-1 expression. In contrast, the change in triglycerides did not alter the inflammatory upregulation of ICAM-1 expression across subjects, which tended to be less variable (Fig. 1b, inset). Postprandial TGRL modulates VCAM-1 surface expression in endothelial cells in proportion to an individual's postprandial change in triglycerides. Human aortic endothelial cells (HAEC) were treated with TNFɑ (0.3 ng/ml) in the presence or absence of postprandial TGRL (10 mg/dl ApoB) for 4 hr, and CAM surface expression quantified by flow cytometry and reported relative to TNFα stimulation alone. (a) VCAM-1 expression across all subjects (black, N = 39) significantly correlated with the change in serum triglycerides in response to the meal. A threshold for change in plasma triglycerides (ΔTG, postprandial - fasting) >120 mg/dl distinguished the capacity of a subject's TGRL to elicit an increase in VCAM-1 surface expression (inset). (b) There was no such correlation with ICAM-1 expression. (c) Postprandial TGRL from pro-atherogenic subjects (red, n = 5) significantly enhanced VCAM-1 expression compared to anti-atherogenic subjects (blue, n = 5), but did not differ with respect to their effect on ICAM-1 expression. In order to evaluate changes in the composition of TGRL that correlate most strongly with a characteristic postprandial inflammatory response in HAEC, five individuals were selected for metabolomics analysis from this cohort whose TGRL produced a pro-atherogenic (pro-) response, eliciting the greatest increase in VCAM-1 surface expression (47.4 ± 22.5%) and expressing a ΔTG > 120 mg/dl. Another five subjects were selected whose TGRL produced an anti-atherogenic (anti-) response, eliciting the greatest decrease in VCAM-1 surface expression (40.2 ± 19.5%) and expressing a ΔTG < 120 mg/dl (Fig. 1c). The subject characteristics and response to the meal are reported in aggregate (Supplementary Table S2) for comparison to the larger study cohort (Supplementary Table S1). On average triglycerides more than doubled in these subjects after the meal. Consistent with the larger study cohort all subjects exhibited substantially increased TG:HDL ratio and TC:HDL ratio 3.5 hrs postprandial compared to fasting. Postprandial TGRL did not significantly alter the TNFα-stimulated increase in ICAM-1 surface expression, which is consistent with previous studies16,17 (Fig. 1c). We next compared anthropometric characteristics, lipid, and metabolic profiles between these same 10 subjects stratified into pro- and anti-atherogenic responders (Table 1). Consistent with our previous studies, pro-atherogenic subjects were distinguished by markers of abdominal obesity (higher BMI and waist circumference), and enrichment in ApoB containing lipoproteins (notably triglycerides), accompanied by lower HDL, both fasting and postprandial. Subjects did not meet the clinical criteria for metabolic syndrome24 since waist circumference, fasting glucose, and blood pressure were within normal ranges. The modulation of TNFα-stimulated VCAM-1 surface expression by postprandial TGRL in these subjects correlated directly with fasting triglycerides, TG:HDL, ApoB, TC:HDL, and inversely correlated with HDL (Table 2). This is noteworthy since TGRL isolated from subjects after an overnight fast did not itself alter endothelial inflammation. Therefore, in the following investigation into the effect of the meal on pro- and anti-atherogenic TGRL composition, oxylipin and fatty acid abundance are expressed as a ratio of postprandial/ fasting. In contrast to VCAM-1, changes in ICAM-1 surface expression did not correlate with any clinical indicators of dyslipidemia. Table 1 Anthropometric, lipid, and metabolic characteristics of subjects selected for metabolomics profiling stratified into anti- and pro- atherogenic responders. Table 2 Correlation of endothelial VCAM-1 surface expression in response to postprandial TGRL with subject (n = 10) anthropometric characteristics and fasting measures of lipid metabolism. A high fat meal challenge exacerbates a postprandial inflammatory state in pro-atherogenic subjects To further characterize the differences in postprandial inflammatory state in dyslipidemic subjects eliciting a pro-atherogenic response, a panel consisting of cytokines and other markers of systemic inflammation was measured in plasma. Postprandial plasma levels of CRP, IFNγ, s-VCAM-1, and IL-6 were all significantly elevated in concert with a subject's TGRL-dependent rise in endothelial VCAM-1 surface expression (Table 3). Among the inflammatory markers, an increase in CRP levels in response to the meal was most indicative of the pro-atherogenic endothelial response (Spearman's ρ = 0.7, p = 0.03). Elevated CRP levels in both the fasting and postprandial states strongly discriminated between pro- and anti-atherogenic subjects (Supplementary Fig. S1), consistent with its utility as a general biomarker of inflammation and cardiovascular risk25. Moreover, elevated CRP levels of this magnitude in metabolic syndrome were associated with increased liver enzyme activity26,27. Also notable was that the postprandial change in s-VCAM-1, but not s-ICAM-1, in plasma correlated directly with a subject's TGRL-mediated atherogenic phenotype in the ex vivo assay, in which pro-TGRL enhanced endothelial VCAM-1, but not ICAM-1 surface expression (Fig. 1c). Together these data confirm that VCAM-1 upregulation in HAEC and in the circulating plasma are faithful inflammatory biomarkers that correlate with clinical measurers of dyslipidemia and metabolic dysregulation in the subject cohort. Table 3 Correlation of endothelial VCAM-1 surface expression in response to postprandial TGRL with fold increase in abundance (postprandial/ fasting) of subject (n = 10) cytokine and inflammatory biomarker levels. A meal in high saturated fat increases n-6 oxylipins in circulating TGRL We next investigated whether the high fat test meal elicited an effect on specific classes of fatty acids or oxylipins across all subjects. The complete composition of the test meal is found in the supplement (Supplementary Table S3). A previous analysis of the postprandial composition of TGRL that was limited to parent fatty acids, revealed enrichment in PUFA in pro-TGRL16. Since PUFA are readily metabolized, this motivated a more detailed unbiased mass spectrometry analysis of the abundance of PUFA-derived oxylipins in addition to PUFA, SFA, and MUFA in fasting and postprandial TGRL. We identified 22 of 153 metabolites that changed significantly in response to the meal. The results are reported separately for the esterified and non-esterified pools, superimposed on network maps displaying the enzymes and parent fatty acids specific to each metabolite (Figs 2 and 3). Among the SFA, the test meal substantially increased the ratio of stearic (C18:0) to palmitic acid (C16:0) (Fig. 2), reflecting the higher abundance of stearic acid in the meal. An overload of SFA in response to the test meal was also evident as a decrease in the ratio of MUFA to SFA, particularly MUFAs derived from palmitic and stearic acids. In the esterified PUFA categories, there was significant postprandial reduction in abundance of several n-3 PUFAs, particularly docosahexaenoic acid (DHA), docosapentaenoic acid (DPA), and eicosapentaenoic acid (EPA), and in n-6 PUFA, dihomo-γ-linolenic acid (DGLA). There was a remarkable increase in abundance of sEH-derived oxylipins in response to the meal in both the esterified and non-esterified pools (Figs 2 and 3). Specifically, 12,13-dihydroxy octadecaenoic acid (12,13-DiHOME), 9,10-dihydroxy octadecaenoic acid (9,10-DiHOME), and 15,16-dihydroxy octadecadienoic acid (15,16-DiHODE) postprandially increased in abundance in both the esterified and non-esterified pools. The changes in the composition of the non-esterified pool were predominantly associated with sEH- and LOX-derived oxylipins originating from the C18 PUFA, linoleic (LA) and alpha-linolenic acid (ALA) (Fig. 3). These results highlight characteristic changes in the composition of TGRL after a meal high in saturated fat, including enrichment in diols and mid-chain alcohols that may mediate inflammation. Network map depicting the fatty acids and metabolites measured in the esterified pool of TGRL. Superimposed in color are the fold change for fatty acids and oxylipins identified as significantly enriched (red) or depleted (blue) postprandially over all subjects. Colored arrows depict significant changes in the ratio of the abundance of the metabolites connected. Black boxes indicate metabolites that were detected but unchanged by the meal. White boxes are metabolites that were part of the analysis but not detected. Significance was determined using a two factor ANOVA comparing fasting to postprandial in the same subjects, and accounting for pro- and anti-atherogenic response. A Benjamini-Hochberg FDR multiple test correction was applied. Network map depicting the fatty acids and metabolites measured in the non-esterified pool of TGRL. Superimposed in color are the fold change for fatty acids and oxylipins identified as significantly enriched (red) or depleted (blue) postprandially over all subjects. Colored arrows depict significant changes in the ratio of the abundance of the metabolites connected. Black boxes indicate metabolites that were detected but unchanged by the meal. White boxes are metabolites that were part of the analysis but not detected. Significance was determined using a two factor ANOVA comparing fasting to postprandial in the same subjects, and accounting for pro- and anti-atherogenic response. A Benjamini-Hochberg FDR multiple test correction was applied. sEH-derived diols and LOX-derived alcohols discriminate the response to the meal between pro- and anti-atherogenic subjects To identify the differences in relative abundance of fatty acids and oxylipins in pro- vs. anti-atherogenic TGRL in response to the meal, a multivariate partial least squares discriminant analysis (PLS-DA) was performed. Postprandial levels of each constituent were normalized by fasting levels for each individual (postprandial/ fasting) to adjust for baseline metabolic variation. Oxylipins yielded the strongest discrimination between the pro- and anti-TGRL samples (Q2 = 0.32) (Fig. 4). In contrast, discrimination was not achieved with the addition of the abundance of parent fatty acids to the analysis. The scores plot displays the separation between the pro- and anti-atherogenic TGRL groups, and the loadings plot superimposes the 22 oxylipins that contributed significantly to the separation. Differential enrichment in the free to bound oxylipin pools (indicated by solid versus open circles) emerged as a major trend distinguishing pro-TGRL. Mapping to the enzymatic pathways, distinguished by distinct color coding, revealed that postprandial change in non-esterified sEH-derived diols was a strong differentiator between pro- and anti-atherogenic subjects. Specifically, 9,10-DiHOME, 14,15-dihydroxy eicosatrienoic acid (14,15-DiHETrE), and 19,20-DiHDoPE contributed maximally to the discrimination of pro- and anti-atherogenic subjects in the analysis (Fig. 4). LOX-derived metabolites also contributed to the discrimination of subjects, including non-esterified alcohols such as 5-hydroxy eicosapentaenoic acid (5-HEPE), 9-HOTE, and 9-HODE. This analysis revealed an interdependence of the effect of the meal and the atherogenic phenotype of the subject on TGRL composition. sEH-derived diols and LOX-derived alcohols strongly discriminate the response to the meal between pro- and anti-atherogenic subjects. PLS-DA analysis of postprandial TGRL samples characterized as pro- and anti-atherogenic (5 per group), using postprandial oxylipin abundance adjusted by fasting levels in the same individuals (postprandial/ fasting). The scores plot reveals the separation between pro- and anti-atherogenic-TGRL (Q2 = 0.32). The PLS-DA model was constructed using all detected oxylipins. The loadings plot displays the 22 oxylipins that contributed significantly to discrimination between the groups (VIP > 1). The size of the circles corresponds to the VIP score. Each oxylipin is color coded by the enzyme from which it is derived. To further evaluate the involvement of these 22 oxylipins in mediating a differential response to the meal, they were grouped into 4 clusters based on similar patterns of expression. Non-esterified diols and alcohols of C18-PUFA LA and ALA in cluster 1, represented by 9-HODE, were enriched to a greater extent postprandially in pro-atherogenic TGRL. Oxylipins in clusters 2 and 3, represented by non-esterified 5,6-DiHETrE and 19,20-DiHDoPE, respectively, consisted primarily of diols and alcohols of C20- and C22-PUFA, which were highly expressed in the fasting state in anti-atherogenic TGRL. These same metabolites were depleted in the postprandial state of anti-atherogenic subjects (Supplementary Fig. 2). In contrast, esterified diols derived from C18-PUFA LA and ALA (cluster 4), represented by 15,16-DiHODE, were enriched to a greater extent postprandially in anti-atherogenic TGRL. Consistent with the lack of inflammatory response induced by fasting TGRL in altering VCAM-1 expression, PLS-DA analysis did not discriminate pro-atherogenic from anti-atherogenic subjects when using the abundance of fatty acids and oxylipins in fasting TGRL alone. Together these data implicate heterogeneity of LOX and sEH activity in subjects producing TGRL that elicit differential inflammatory responses to the same high fat meal. VCAM-1 surface expression correlates strongly with the postprandial change in estimated sEH activity A striking finding was that dyslipidemic subjects displayed a unique shift in abundance of sEH-derived diols and LOX-derived alcohols in response to the meal compared to normolipidemic subjects. We next investigated whether these subjects exhibited differences in the activity of enzymes including LOX, COX, CYP, and sEH that could account for the observed range in ex vivo VCAM-1 response. Indices of enzyme activity were calculated as a ratio of abundance of metabolite to parent fatty acid or precursor metabolite, independently for the esterified and non-esterified pools in TGRL. Overall the postprandial change (postprandial/ fasting) in non-esterified sEH activity index, i.e. the diol to epoxide ratio summed over all parent fatty acids measured in the non-esterified pool, strongly correlated with induction in VCAM-1 surface expression (Spearman ρ = 0.66 p = 0.04, Table 4). This enhanced activity is consistent with the PLS-DA analysis that identified the change in diols as a key feature in discriminating pro- and anti-atherogenic TGRL. Also noteworthy in the non-esterified pool was sEH activity associated with ALA, and LOX activity specific to EPA metabolism, both of which correlated strongly with pro-atherogenic activity. In the esterified pool, LOX activity linked to ALA, and CYP activity associated with EPA metabolism, were positively correlated with a pro-TGRL response. Together these results implicate a possible role for altered sEH and LOX activity in subjects whose TGRL elicits the greatest upregulation in VCAM-1 expression in inflamed HAEC. Table 4 Correlation of VCAM-1 surface expression with indices of enzyme activity (n = 10). EPA-derived metabolites and esterified diols are predictive of changes in VCAM-1 surface expression We next investigated the extent to which the abundance of specific oxylipins and fatty acids composing postprandial TGRL were predictive of the elicited changes in ex vivo VCAM-1 surface expression. A multiple linear regression model was generated iteratively using 19 clusters that were comprised of all the oxylipins and fatty acids and exhibited similar pattern of expression in postprandial TGRL across all subjects. The model with the lowest Akaike's information criteria (AIC) score was selected as the best predictor of the postprandial VCAM-1 surface expression (Supplementary Table S4). The strongest model (R2 = 0.76, p-value = 0.0067) was composed of the two cluster components depicted by the equation in Fig. 5. The strongest positive correlator with the measured ex vivo VCAM-1 surface expression was Cluster 1, which consisted predominantly of EPA and esterified EPA-derived metabolites (17,18-EpETE, 9-HEPE, 15-HEPE, 5-HEPE). This is in accordance with the elevated EPA-specific LOX activity observed postprandially in pro-atherogenic subjects. Cluster 2, comprised predominantly of C18-derived esterified diols (12,13-DiHOME, 9,10- DiHOME, 15,16-DiHODE, 9,10-DiHODE), negatively correlated with VCAM-1 expression. These findings are consistent with the observations from the PLS-DA model. Postprandial fatty acid and oxylipin abundance in TGRL predicts changes in VCAM-1 surface expression by HAEC. The model was generated iteratively from all oxylipin, fatty acids, and calculated lipogenesis indices, clustered based on postprandial expression levels in TGRL across all subjects. Cluster scores were calculated as a linear combination of abundance of each cluster constituent, and used to fit the measured Johnson normalized VCAM-1 expression. The strongest model for predicting VCAM-1 expression consisted of a combination of two clusters as depicted. Dashed lines = 95% confidence interval, =Pro-atherogenic TGRL and =Anti-atherogenic TGRL; n = 5. Reduction in lipogenesis indices and non-esterified diols in fasting subjects are strong predictors of postprandial VCAM-1 expression Given the strong correlation of postprandial VCAM-1 expression with fasting markers of dyslipidemia, we evaluated whether the composition of TGRL in fasting subjects was also predictive of the postprandial VCAM-1 response. A second multiple linear regression model was generated iteratively using 18 clusters of oxylipins and fatty acids, identified by their similar pattern of expression in TGRL from fasting subjects. As above the AIC score was used to select a model that best fit measured VCAM-1 expression data (Supplementary Table S5). Three clusters emerged as strong negative correlators with VCAM-1 surface expression, which provided the best fit for the data (Fig. 6, R2 = 0.95, p = 0.0002). Cluster 1 consisted predominantly of non-esterified diols suggesting elevated levels of sEH activity that contributes to the non-esterified oxylipin pool in fasting anti-atherogenic subjects. Clusters 2 and 3 were comprised of various lipogenesis indices and Σn-6/Σn-3, which are emerging biomarkers for dyslipidemia28. Both the fasting esterified and non-esterified ELOVL-6 indices (stearic acid to palmitic acid ratio) inversely correlated with VCAM-1 expression. Esterified Δ5D and SCD-1 and non-esterified levels of Σn-6/Σn-3 and ELOVL-2 were also inversely correlated to the measured VCAM-1 expression. Together, these data support the utility of these lipogenesis indices as potential biomarkers that predict a subject's susceptibility to CVD. Moreover, the regression analysis reveals that fasting indicators of metabolism reflected in the composition of TGRL strongly predict the inflammatory postprandial response. Fasting fatty acid and oxylipin abundance in TGRL predicts changes in VCAM-1 surface expression by HAEC. The model was generated iteratively from all oxylipin, fatty acids, and calculated lipogenesis indices, clustered based on fasting expression levels in TGRL across all subjects. Cluster scores were calculated as a linear combination of abundance of each cluster constituent, and used to fit the measured Johnson normalized VCAM-1 expression. The strongest model for predicting VCAM-1 expression consisted of a combination of three clusters as depicted. Dashed lines = 95% confidence interval, =Pro-atherogenic TGRL and =Anti-atherogenic TGRL; n = 5. Linoleic acid-derived oxylipins are regulators of TNFα-stimulated VCAM-1 expression in HAEC To demonstrate that the differences in concentrations of individual oxylipins identified by our analysis of TGRL could directly affect endothelial inflammation, we delivered several of them as methyl esters to stimulated HAEC and measured the dose response in VCAM-1 expression. The methyl ester form is readily taken up by cells and is analogous to the non-esterified or free form measured in TGRL. In this analysis we selected representative non-esterified oxylipins that contributed significantly to distinguishing pro- from anti-atherogenic TGRL (Fig. 4), and strongly influenced the modeled VCAM-1 expression (Figs 5 and 6). The linoleic acid metabolites 9-HODE and 12,13-DiHOME, which were present in relatively high abundance in TGRL and had a strong negative influence on the modeled VCAM-1 expression, exhibited a dose-dependent effect in reducing HAEC VCAM-1 expression (Fig. 7). On the other hand, the DHA metabolite19,20-DiHDoPE, which was present at lower abundance but was also a strong contributor to distinguishing pro-atherogenic TGRL (Fig. 4), did not itself elicit a detectable effect on VCAM-1. These data provide evidence that changes in individual oxylipins transported in TGRL could skew the nature of an inflammatory response defining an atherogenic phenotype. 9-HODE and 12,13-DiHOME reduce VCAM-1 surface expression in TNFα-stimulated HAEC. Representative methyl esters of oxylipins identified as contributing significantly to distinguishing a pro- atherogenic phenotype were delivered to TNFα-stimulated HAEC (0.3 ng/ml, 4hrs). 9-HODE and 12,13-DiHOME reduced VCAM-1 surface expression, whereas 19, 20-DiHDoPE did not alter VCAM-1 surface expression relative to TNFα. p < 0.05, p < 0.005, p < 0.0005, **p < 0.0001 by one-way ANOVA followed by Dunnet's post-test. n = 3–4. Herein we quantify the effects of metabolic stress associated with the progression of atherosclerosis, through direct measurement of relevant inflammatory outcomes in a monolayer of arterial endothelial cells under conditions that mimic a postprandial response15. Our platform provided a means to directly link changes in the fatty acid and oxylipin composition of an individual's TGRL with its inflammatory capacity measured ex-vivo for the first time. By characterizing TGRL before and after administration of a test meal representative of a high fat western diet, we have established a gauge for an individual's acute postprandial inflammatory response15,16,17,18. Postprandial TGRL mediated a pro- or anti-inflammatory response unique to each subject that was marked by an increase or decrease in endothelial VCAM-1 receptors. The magnitude of this inflammatory response was strongly influenced by an individual's fasting metabolic status, in that it correlated strongly with plasma markers of metabolic dysregulation and dyslipidemia. In response to the high fat meal challenge, TGRL characterized as pro- or anti-atherogenic revealed different oxylipin signatures, a key distinguishing feature of which was a shift in the relative abundance of non-esterified (free) diols. Pro-inflammatory TGRL was enriched in EPA and EPA-derived oxylipins and depleted in esterified (bound) C18-PUFA-derived diols in response to the meal, reflecting a putative shift in sEH and LOX activity in those subjects. A remarkable finding was the ability to model the measured postprandial VCAM-1 response using the relative abundance of free diols and lipogenesis indices measured in the fasting state. We conclude that specific changes in the composition of TGRL elicited by a high fat meal reflect an individual's in vivo lipoprotein metabolism and mitigate the TGRL-induced ex vivo inflammatory response. In quantifying the abundance of fatty acids of various saturation levels in TGRL we found that pro-atherogenic TGRL was enriched in long chain PUFA, consistent with our previous report16. Though PUFA are not the most abundant type of fatty acid in TGRL, by virtue of their double bonds they are readily metabolized to form oxylipins, which are known to elicit potent effects on inflammation, depending upon the metabolizing enzyme, the position of the double bond, and the parent fatty acid. Moreover, the fatty acids and oxylipins are present in both esterified and non-esterified forms in lipoproteins, which could lead to differences in their processing and effects on inflammation10,21. Esterified fatty acids and oxylipins in lipoproteins are released into tissues after their uptake and lipolysis21. We previously reported that the inflammatory response to postprandial TGRL was largely dependent upon their receptor-mediated uptake by HAEC18. In contrast, the non-esterified fatty acids constitute only a small proportion of the total fatty acid constituents in TGRL. However, free oxylipins and fatty acids in TGRL are more readily available upon uptake and thus may be significant mediators of endothelial inflammation. Our data points to the enrichment of non-esterified PUFA and oxylipins in pro-atherogenic TGRL, consistent with our previous analysis16. We proposed that the oxylipin metabolites constituting TGRL are bioactive and can in part account for the relative atherogenicity of these particles by altering endothelial inflammation. We found that an increase in the abundance of certain vicinal (1,2)-diols, including non-esterified 9,10-DiHOME, 12,13-DiHOME, and esterified 19,20-DiHDoPE, were among the strongest discriminators of a pro-inflammatory response to postprandial TGRL. Diols are downstream products of epoxides enzymatically converted by sEH10. sEH inhibition is associated with anti-inflammatory and vascular relaxation properties mediated by elevated epoxides such as EETs and EpOMEs29. For example, 11,12-EpETrE reduced TNFɑ-induced VCAM-1 surface expression in HAEC30. Treatment with sEH inhibitors has also been reported to lower systemic inflammation in a mouse model of septic shock31, and to reduce atherosclerosis in ApoE null mice32. Another study observed an increase in sEH-derived diols and LOX-derived HODEs in the atherosclerotic plaques of rabbits33. In the current study, we observed enhanced sEH and LOX activity in dyslipidemic subjects after the high fat meal challenge. However, a higher relative abundance of non-esterified diols of C20- and C22-PUFA which were depleted postprandially was a defining feature of fasting anti-atherogenic TGRL. Our analysis consistently revealed a shift in the levels of diols in TGRL as markers of an endothelial inflammatory response. However, additional studies are needed to determine whether these diols themselves mediate the inflammatory response in HAEC, or rather reflect a shift in metabolism away from an overall anti-inflammatory state that involves cross-talk with other pathways. This study further revealed enrichment in C18-derived midchain alcohols in the TGRL of dyslipidemic subjects. Moreover, a significant positive correlation was observed between the calculated postprandial shift in EPA-specific LOX activity and VCAM-1 expression. Although midchain alcohols are products of LOX metabolism, they can also be derived from auto-oxidation processes associated with oxidative stress20. As we did not directly measure the hepatic activity, the elevated LOX index may also reflect auto-oxidation within subjects. However, in monitoring markers of auto-oxidation within our samples, we found them to be expressed at low levels, and to not differ significantly between the pro- and anti-atherogenic groups. The postprandial abundance of esterified EPA and EPA-derived oxylipins (including midchain alcohols) were positive predictors for TGRL-induced VCAM-1 expression by HAEC. Additional studies are needed to determine whether the observed postprandial enrichment of EPA-derived oxylipins that mark pro-TGRL are themselves proactive mediators of inflammation. Establishing causation for a single oxylipin in the context of a shifting oxylipin profile in the lipoprotein pools of any individual is not trivial. Indeed, an advantage of our approach is the demonstration of the functional importance of an individual's mixture of oxylipins and their potential capacity to skew the inflammatory response. Since cells in vivo are exposed to a suite of fatty acids, oxylipins, and other metabolites present in lipoproteins which could have opposing effects on inflammation, characterizing the integrated response to the native oxylipin profile and evaluating correlative trends with inflammatory response ex-vivo provides physiological relevance. Nevertheless, herein we demonstrate that individual oxylipins identified by our analysis of TGRL could directly affect endothelial inflammation (Fig. 7). Specifically, we found that several oxylipins derived from LA that contributed negatively to the modeled VCAM-1 expression in fact dose-dependently decreased VCAM-1 expression alone. On the other hand, representative oxylipins that contributed positively in distinguishing pro-atherogenic TGRL had a neutral effect when tested alone. These data support the assertion that although isolated oxylipins may elicit an effect on inflammation, their combined effects and variable abundance in TGRL are more representative of the in vivo influence on inflammation. One possibility is that the expression patterns we see comparing the different responses to TGRL represent more of a shift away from an overall anti-inflammatory state to a more neutral or permissive one, rather than being overtly inflammatory. Alternatively, they may reflect a feedback response to the high fat meal in hypertriglyceridemic subjects to mitigate inflammation. Since subjects consumed an identical high fat meal, differences in the postprandial response, especially as they correlate with markers of fasting metabolism, are likely attributable to altered liver metabolism. Though we did not directly measure hepatic enzyme activities as markers of liver function, elevated CRP levels have been associated with higher alkaline phosphatase and alanine aminotransferase activities in metabolic syndrome subjects. The magnitude of elevated CRP observed at fasting and postprandial in pro-atherogenic subjects could therefore indirectly point to elevated liver function26,27. A finding of the current study was that fasting measures of lipogenesis activity (SCD-1, ELOVL-6, and Δ5D) inversely correlated with the measured endothelial VCAM-1 expression. The liver metabolizes LA and ALA derived from the diet to produce downstream long chain PUFAs using ELOVL and desaturase enzymes. These enzymes also catalyze the production of MUFAs from SFA, and elongate SFAs. Impairment or hyperactivity of these enzymes is associated with altered lipid metabolism and disease states such as type 2 diabetes and hypertriglyceridemia, both risk factors for CVD34,35,36. We observed an overall reduction in ELOVL-6 activity in response to the high fat meal challenge, particularly in pro-atherogenic subjects, which implies impairment of elongation or saturation of palmitic acid. Elevated levels of palmitic acid are strongly associated with a sedentary lifestyle and hypertriglyceridemia37. In a recent ten-year longitudinal study, ELOVL-6 and Δ5D activity were significantly reduced in obese subjects who developed dyslipidemia28. Other reports correlated impaired Δ5D activity to high BMI, CRP, and elevated triglyceride and CVD related mortality34,38. Consistent with these studies, we report that Δ5D and ELOVL-6 activity inversely correlated with endothelial VCAM-1 surface expression, a direct measure of inflammation and marker of atherosclerosis14. These findings support the utility of measures of lipogenesis and lipid metabolism (sEH, LOX, Δ5D, ELOVL-6) in identifying subjects in a state of metabolic dysregulation associated with greater cardiovascular risk. We have previously reported that the upregulation of TGRL-induced VCAM-1 is in part signaled via the endoplasmic reticulum (ER) stress response that promotes increased activity of the transcription factor interferon regulatory factor 1 (IRF-1)16,18. This mechanism distinguishes VCAM-1 expression from ICAM-1, which is independent of both IRF-1 and ER stress. By extension the varying oxylipin constituents transported in TGRL may directly induce downstream activation or inhibition of ER stress and subsequent increase or decrease in VCAM-1 surface expression. It is established that an excessive load of fatty acids can induce a state of ER stress in hepatocytes, macrophages and adipocytes39. ER is the hub of lipogenesis activity in cells and ER stress correlates strongly with obesity and CVD. Inhibition or knock-down of sEH can reduce high fat diet or pharmacologically induced ER stress in mouse models40,41. Individual oxylipins including 9-HODEs and 12/15-HETEs induced ER stress in macrophages and adipocytes, respectively42,43. 12(S)-HETE was shown to directly enhance the monocyte adhesion in HAEC44. Further, pharmacological inhibition of LOX, COX, CYP, and sEH enzymes have consistently demonstrated that perturbing oxylipin metabolism can alter systemic inflammation. For example, knock-out of 12/15 LOX in Apo-E null mice resulted in a 95% reduction in monocyte arrest within the aorta45. These observations motivate a need for further investigation to establish a causative role for enrichment in individual oxylipins within TGRL in modulating ER stress responses that promote endothelial inflammation. Herein we develop regression models to predict the relative atherogenicity of an individual's postprandial TGRL based on the oxylipins present in these lipoproteins. The models were generated from a cohort of subjects whose TGRL elicited the greatest modulation in VCAM-1 expression. Validation of these models in a larger independent cohort of subjects that includes patients with established metabolic syndrome and coronary artery disease, would ultimately demonstrate the predictive potential for gauging cardiovascular risk and disease progression over time. This approach could also prove useful in quantifying the effects of dietary intervention, statins, or other lipid-lowering therapies, by providing direct real-time measures of an individual's lipid metabolism and inflammatory status. The aim of this study was to shed light on the variable inflammatory response in individuals elicited by an identical meal rather than to compare the response to meals of different fatty acid composition. Thus, we chose a test meal that we have used to elicit a variable postprandial endothelial inflammatory response in multiple studies15,16,18. The meal is high in saturated fat and calories, and the cholesterol levels are above recommended intake for healthy diets. However, the meal is representative of a fast food breakfast meal not uncommon in the Western diet. Since this study identifies differences in the fatty acid composition of TGRL (particularly PUFA and PUFA-derived oxylipins) that correlate strongly with the nature of an inflammatory response, we envision future controlled dietary intervention studies that vary the fatty acid composition of the meal (e.g. replacing SFA with PUFA), or supplement with n-3 PUFA, and quantify the postprandial response. Previous experience shows that the differential VCAM-1 expression observed in the postprandial state is not induced by fasting TGRL15. One possible explanation is that the fasting state is primarily anti-inflammatory among most subjects, though it can reflect a different baseline on which a postprandial insult superposes to amplify differences in the inflammatory response of an individual. Consistent with this is the observation that the postprandial increase in subjects' plasma markers CRP and sVCAM predicted the magnitude of the inflammatory response to TGRL measured ex vivo. Thus, this technique has the potential to be employed to discriminate patient populations who do not present elevated levels of conventional clinical risk factors but are still at high risk for atherosclerotic CVD. It is also noteworthy that hypertriglyceridemic subjects exhibit lower clearance of TGRL particles and higher accumulation of lipoproteins. In this regard, the differences we have observed in the oxylipin composition between pro- and anti-TGRL may exacerbate an inflammatory response in endothelium exposed to these particles for a prolonged period. In conclusion, we have identified a characteristic oxylipin signature in postprandial TGRL that distinguishes an acute pro- versus anti-inflammatory endothelial response in subjects consuming an identical high fat meal. Dyslipidemic subjects were uniquely characterized by a shift in sEH and LOX activity, and produced pro-atherogenic TGRL, postprandially enriched in EPA-metabolites and depleted in C18-PUFA-derived esterified diols. In contrast, normolipidemic subjects exhibited higher indices of fasting lipogenesis activity, and produced anti-atherogenic TGRL enriched in esterified C18-PUFA-derived diols, but depleted in non-esterified diols after the high fat meal challenge. These results implicate a role for the fatty acids and oxylipins constituting TGRL in mediating a pro- or anti-atherogenic endothelial phenotype. We further conclude that lipoproteins such as TGRL can serve as a depot for fatty acids and oxylipins, and propose that these bioactive fatty acid metabolites transported in TGRL could act remotely to directly modulate an endothelial inflammatory response. Human subject recruitment and characterization Human subjects (N = 39) were recruited according to an Institutional Review Board approved protocol at the University of California, Davis in accordance with the Helsinki principles and institutional guidelines and regulations. Informed consent was obtained from all research participants. The subjects varied from normal lipidemic to dyslipidemic, but did not have hypertension or elevated fasting glucose, and thus were not overtly characterized as having metabolic syndrome (Supplementary Table S1). Subjects were recruited without restriction to age, gender, race, or socioeconomic status. Exclusion criteria include subjects with kidney disease, liver disease, untreated thyroid dysfunction and ongoing hormone replacement therapy, omega-3 (n-3) fatty acid (i.e. fish oil) supplementation, lipid lowering medications (i.e. statins), excessive or prolonged use of alcohol, and consumption of nonsteroidal anti-inflammatory drugs or caffeinated beverages within 12 hrs of blood collection. Venipuncture was performed to collect 20 ml of blood after an overnight fast and again 3.5 hrs after consuming a test meal. The meal was a standardized fast food breakfast high in saturated fatty acid content and calories (Supplementary Table S3). Blood samples were immediately centrifuged (1932g, 10 min, 25 C) to obtain plasma. Plasma aliquots (both fasting and postprandial) were sent to the UC Davis Medical Center clinical laboratory for standard lipid panels, including triglycerides (TG), total cholesterol (TC), and high-density lipoprotein cholesterol (HDL). Low density lipoprotein cholesterol (LDL) was calculated using Friedewald equation. In some cases, LDL was not reported as the postprandial spike in triglycerides rendered the equation unusable. Plasma glucose and ApoB levels were also measured. A panel of cytokines and other inflammatory markers in plasma, including interleukin (IL)-1α, IL-1β, IL-6, IL-10, TNFα, IL-17A, soluble VCAM-1 (sVCAM-1), soluble ICAM-1 (sICAM-1), c-reactive protein (CRP) and interferon γ (IFNγ), were measured using a multiplex immune assay (V-Plex Meso Scale Discovery). TGRL isolation Plasma was spiked with 1x antibiotic/ antimycotic (GIBCO). Density-based ultracentrifugation (40,000 RPM, 18hrs, 4 C) was used to isolate TGRL particles from plasma (ρ < 1.0063 g/ml) using a Beckman XL-90 ultracentrifuge. ApoB content was quantified using an ELISA kit (ALERCHEK) that measures both ApoB48 and ApoB100. In the fasting state TGRL consists of mainly ApoB100-containing very low-density lipoproteins (VLDL), whereas the postprandial TGRL is a mixture of VLDL and ApoB48-containing chylomicrons. ApoB concentrations were used to normalize the TGRL quantity for inflammatory characterization of HAEC and for mass spectrometry analysis. TGRL samples were characterized within 7 days of isolation or were aliquoted and frozen for future use. All samples were immediately flushed with nitrogen gas upon collection to prevent auto-oxidation. The extent of oxidation in the samples was monitored by tracking concentrations of 9-HETE, total TriHOME and F2-isoprostanes, which did not differ significantly between experimental groups. TGRL characterization Human aortic endothelial cells (HAEC), passage 6–8 (Genlantis, lot #2228 derived from a 21-yr-old female, and lot #7F4409 from a 34-yr-old male), were cultured with endothelial cell growth media-2 (EGM-2, Lonza) supplemented with 1x antibiotic/ antimycotic (GIBCO). Cells were seeded on tissue culture plates (Falcon) and proliferated to form a 85–95% confluent monolayer. HAEC were then incubated with postprandial TGRL (10 mg/dl ApoB) for 4 hrs in the presence of TNFα (0.3 ng/ml, the calibrated EC50 for CAM upregulation, R&D). HAEC were dissociated from the tissue culture plate using 1 mM EDTA (10 mins, Sigma), and stained with phycoerythrin-conjugated VCAM-1 antibody (BD Biosciences no. 555647, 10 µl per 100,000 cells) and Alexafluor 488-conjugated intercellular adhesion molecule-1 (ICAM-1) antibody (Biolegend no. 322714, 5 µl per 100,000 cells). VCAM-1 and ICAM-1 surface expression were measured using flow cytometry (Attune NXT). To account for day-to-day cell variability in HAEC response, CAM expression is reported as % change from TNFα stimulation alone. Based on previous studies, TGRL eliciting an increase in VCAM-1 expression ≥10% are characterized as pro-atherogenic, and those that decrease VCAM-1 by ≥10% as anti-atherogenic, as these changes correlate strongly with mononuclear cell recruitment. Quantification of fatty acids and oxylipins in fasting and postprandial TGRL Non-esterified fatty acids (NEFA) and oxylipins, along with total fatty acids (TFA) and alkaline stable oxylipins were quantified using mass spectrometry: ultra-performance liquid chromatography mass spectrometry (UPLC-MS/MS) for oxylipins, and gas chromatography-mass spectrometry (GC-MS) for FA. For simplicity, since the non-esterified pools account for a minor component of the total pool, the total fatty acid and total alkaline stable oxylipin pools are referred to as esterified lipids for the remainder of this manuscript. The abundance of oxylipins was reported as pmol/mg ApoB and fatty acids as nmol/mg ApoB. Lipids were isolated using a liquid-liquid extraction with cyclohexane/isopropanol/ammonium acetate after enrichment with a suite of deuterated and rare compounds used as analytical surrogates46,47. Fatty acids were transformed into methyl esters, using either TMS-diazomethane or methanolic transesterification for NEFA and TFA, respectively. The fatty acid methyl esters were separated on 30 m x 0.25 mm, 0.25 µm DB-225ms (Agilent Technologies), detected with a 5973 A mass selective detector (Agilent Technologies) using electron impact ionization and selected ion monitoring, and quantified against authentic standards. Results were corrected for the recoveries of analytical surrogates as previously described47. Non-esterified oxylipins were isolated directly from TGRL solution in PBS using solid phase extraction. Esterified oxylipins were transformed into oxylipin free acids by base hydrolysis, and isolated by subsequent hydrophilic/lipophilic interaction solid phase extraction. Non-esterified and alkaline stable esterified oxylipins were analyzed by UPLC-MS/MS using negative mode electrospray ionization and multi-reaction monitoring on a Sciex 4000 QTRAP47. Concentrations were calibrated using analytical standards as previously reported47,48. Characterization of oxylipin response Oxylipins were synthesized in methyl esterified forms dissolved in ethanol by the Hammock Lab (UC Davis). They were delivered to TNFα (0.3 ng/ml) stimulated HAEC at doses ranging from 0.1–100 nM for 4hrs. Change in VCAM-1 surface expression in HAEC from vehicle control was quantified by flow cytometry. Enzyme activity index estimation Fatty acid and oxylipin data were categorized into non-esterified and esterified fatty acid pools and metabolites were grouped by their parent fatty acids. An activity index for each parent fatty acid was calculated based on product to substrate abundance ratios according to the following equations: $${{\rm{L}}{\rm{O}}{\rm{X}}}_{{\rm{F}}{\rm{A}}}{\rm{i}}{\rm{n}}{\rm{d}}{\rm{e}}{\rm{x}}={{\rm{\Sigma }}}_{{\rm{F}}{\rm{A}}}({\rm{m}}{\rm{i}}{\rm{d}}{\rm{c}}{\rm{h}}{\rm{a}}{\rm{i}}{\rm{n}}\,{\rm{a}}{\rm{l}}{\rm{c}}{\rm{o}}{\rm{h}}{\rm{o}}{\rm{l}}{\rm{s}}\,+\,{\rm{k}}{\rm{e}}{\rm{t}}{\rm{o}}{\rm{n}}{\rm{e}}{\rm{s}})/{\rm{p}}{\rm{a}}{\rm{r}}{\rm{e}}{\rm{n}}{\rm{t}}\,{\rm{f}}{\rm{a}}{\rm{t}}{\rm{t}}{\rm{y}}\,{\rm{a}}{\rm{c}}{\rm{i}}{\rm{d}}$$ $${{\rm{CYP}}}_{{\rm{FA}}}{\rm{index}}={{\rm{\Sigma }}}_{{\rm{FA}}}({\rm{epoxides}})/{\rm{parent}}\,{\rm{fatty}}\,{\rm{acid}}$$ $${{\rm{s}}{\rm{E}}{\rm{H}}}_{{\rm{F}}{\rm{A}}}{\rm{i}}{\rm{n}}{\rm{d}}{\rm{e}}{\rm{x}}={{\rm{\Sigma }}}_{{\rm{F}}{\rm{A}}}({\rm{d}}{\rm{i}}{\rm{o}}{\rm{l}}{\rm{s}})/{{\rm{\Sigma }}}_{{\rm{F}}{\rm{A}}}({\rm{e}}{\rm{p}}{\rm{o}}{\rm{x}}{\rm{i}}{\rm{d}}{\rm{e}}{\rm{s}})$$ An overall activity index was also estimated using the overall product to substrate ratio summed across all measured fatty acids. Analyses were then conducted on the postprandial to fasting ratio of these calculated enzyme activities. Indices of lipogenesis were calculated as a ratio of the measured abundance of fatty acid product to precursor in the fasting and postprandial plasma or TGRL: stearoyl-CoA desaturase-1 (SCD-1, C16:1n-7/C16:0), SCD-C18 (C18:1n-9/C18:0), SCD-C16 (C16:1n-9/C16:0), delta 5 desaturase (Δ5D, C20:4n-6/C20:3n-6), elongation of very long fatty acid-6 (ELOVL-6, C18:0/C16:0), ELOVL-2 (C22:5n-3/C20:5n-3). The ratio of n-6 to n-3 PUFA was calculated across all measured PUFA (Σn-6/Σn-3). All statistical analyses were performed using JMP Pro 13 software. Fatty acid and oxylipin abundance data were Johnson transformed and checked for normality using the Shapiro-Wilk test. Outliers were assessed based on an interquartile range calculated per metabolite. Outliers were excluded if interquartile range <Q1 – (3 x interquartile range) or >Q3 + (3 x interquartile range). Missing values were imputed via a multivariate approach. Undetectable levels of metabolites were substituted by 0.5 x the lowest detected expression level measured for that specific metabolite. To assess differences in FA and oxylipin composition in TGRL as a function of the meal, TGRL-induced inflammatory phenotype, and interaction between the two factors, a two-factor analysis of variance (ANOVA) was performed. P-values were corrected for multiple testing using a Benjamini-Hochberg false discovery rate (FDR) approach. Likewise, a two-factor ANOVA and FDR was used to assess differences in plasma inflammatory markers. The Pearson method was used to correlate the change in VCAM-1 surface expression with clinical biomarkers in the full subject cohort (N = 39), since normality was satisfied. In the smaller set of subjects chosen for metabolomics analysis (n = 10), a non-parametric Spearman rank method was used to assess correlations between CAM expression and subject characteristics, plasma lipid, plasma cytokines or TGRL constituents. To identify the shift in fatty acid and oxylipin constituents differentiating pro- and anti-atherogenic TGRL, a partial least squares discriminant analysis (PLS-DA) was performed. Postprandial abundance data were adjusted to the fasting values (postprandial/ fasting) for each subject (n = 10). This approach reduced the multivariate data for projection on the 2 coordinate axes resulting in the maximum separation between the pro- and anti-TGRL groups based on fatty acid and oxylipin abundance. The scores plot depicts the separation between the samples characterized for their relative atherogenicity, while the loadings plot superposes the metabolites that contribute to the maximum variability between the two groups. Variable importance in projection (VIP) scores the relative contribution of each metabolite to the separation visualized in the scores plot. Metabolites with a VIP score of over 1.1 were considered to contribute significantly to the separation. The leave-one-out method was used to cross-validate the model. Next, the 22 oxylipins identified as the significant contributors to the PLS-DA model were clustered by expression pattern using the JMP 13 Pro cluster variable algorithm. The fasting to postprandial change of the most representative metabolite in the cluster, i.e. that for which the expression correlated most strongly to the principal component of the cluster, was graphed in order to visualize the change in abundance of the metabolite in response to the meal. Regression modeling A stepwise linear regression modeling approach was used to fit metabolite expression data to measured change in VCAM-1 surface expression in HAEC. Metabolites were first clustered to account for collinearity and maintain independence between the input variables in the model. A cluster variable algorithm in JMP Pro 13 was used to group metabolites based on the correlation of their expression levels across all subjects, using the fatty acids, oxylipins, and lipogenesis indices as inputs. This resulted in 19 clusters for the postprandial case, and 18 clusters for the fasting case. Each cluster was reduced to a representative cluster component49 that was an input into the iterative regression model. To satisfy the linearity requirement, VCAM-1 expression was Johnson transformed50. The regression model fit the response variable (y, VCAM-1 expression), using stepwise addition of cluster components as independent input variables (xi). The Akaike's information criteria (AIC) and Beyesian information criteria (BIC) scores were used in model optimization with a goal of minimizing the number of independent terms. The addition of a term was considered to result in a stronger model if it improved the p-value and R2 while also reducing the AIC score by 2 or more51,52. The resulting model can be expressed by the following equation, where b0, b1, b2, b3, b4, …, bx are coefficients corresponding to each input variable (cluster): $${\rm{y}}={{\rm{b}}}_{0}+{{\rm{b}}}_{1}{{\rm{x}}}_{1}+{{\rm{b}}}_{2}{{\rm{x}}}_{2}+{{\rm{b}}}_{3}{{\rm{x}}}_{3+}{{\rm{b}}}_{4}{{\rm{x}}}_{4}\ldots .+{{\rm{b}}}_{{\rm{i}}}{{\rm{x}}}_{{\rm{i}}}$$ The raw data generated by the study are provided as a supplementary data file to accompany the manuscript submission. Grundy, S. M., Hansen, B., Smith, S. C. Jr., Cleeman, J. I. & Kahn, R. A. Clinical management of metabolic syndrome: report of the American Heart Association/National Heart, Lung, and Blood Institute/American Diabetes Association conference on scientific issues related to management. Arter. Thromb. Vasc. Biol. 24, 19–24 (2004). Miller, M. et al. 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Endocrinol. Metab. 302, 654–665 (2012). Kim, P. M. et al. Lipoxygenase Products Increase Monocyte Adhesion to Human Aortic Endothelial. Cells. Arter. Thromb. Vasc. Biol. 19, 2615–2622 (1999). T., B. D., Suseela, S., Angela, W., C., F. L. & C., H. C. 12/15 Lipoxygenase Mediates Monocyte Adhesion to Aortic Endothelium in Apolipoprotein E–Deficient Mice Through Activation of RhoA and NF-κB. Arter. Thromb. Vasc. Biol. 26, 1260–1266 (2006). Smedes, F. Determination of total lipid using non-chlorinated solvents. Analyst 124, 1711–1718 (1999). Grapov, D., Adams, S. H., Pedersen, T. L., Garvey, W. T. & Newman, J. W. Type 2 diabetes associated changes in the plasma non-esterified fatty acids, oxylipins and endocannabinoids. PLoS One 7, 48852, https://doi.org/10.1371/journal.pone.0048852 (2012). Agrawal, K. et al. Sweat lipid mediator profiling: a noninvasive approach for cutaneous research. J. Lipid Res. 58, 188–195 (2017). Cortez, P., Cerdeira, A., Almeida, F., Matos, T. & Reis, J. Modeling wine preferences by data mining from physicochemical properties. Decis. Support Syst. 47, 547–553 (2009). Cook, R. D. & Weisberg, S. Transforming a Response Variable for Linearity. Biometrika 81, 731–737 (1994). Ward, E. J. A review and comparison of four commonly used Bayesian and maximum likelihood model selection tools. Ecol. Modell. 211, 1–10 (2008). Chatterjee, S. & Hadi, A. S. Variable selection procedures. in Regres. Anal. by Ex. (eds Chatterjee, S. & Hadi, A. S.) 299–334 (John Wiley & Sons 2015). This study was supported by National Institute of Health (NIH) grant HL082689 (S.I.S. and A.G.P.). Partial support of metabolomics profiling was provided by NIH DK097154, and USDA Intramural Project 2032-51530-022-00D (J.W.N). The USDA is an equal opportunity employer and provider. Methyl esters of representative oxylipins were synthesized by Bogdan Barnych of the Bruce Hammock Lab (UC Davis). The content is solely the responsibility of the authors and does not necessarily represent the official views of the National Institutes of Health. Department of Biomedical Engineering, University of California, Davis, 451 Health Sciences Dr., Davis, CA, 95616, USA Anita Rajamani, Samir Akre, Andrea Fernandez, Scott I. Simon & Anthony G. Passerini West Coast Metabolomics Center, Genome Center, University of California, Davis, 451 Health Sciences Dr., Davis, CA, 95616, USA Kamil Borkowski & John W. Newman Department of Nutrition, University of California, Davis, 3135 Meyer Hall, One Shields Avenue, Davis, CA, 95616, USA John W. Newman Western Human Nutrition Research Center, Obesity and Metabolism Research Unit, Agricultural Research Service, United States Department of Agriculture, 430 West Health Sciences Dr., Davis, CA, 95616, USA Anita Rajamani Kamil Borkowski Samir Akre Andrea Fernandez Scott I. Simon Anthony G. Passerini A.R. and A.G.P. wrote the manuscript. A.R., A.G.P, and S.I.S. conceived and designed the study. A.F. aided in subject recruitment, sample collection, and characterization. K.B. processed the samples for metabolomics. A.R., K.B., J.W.N., and S.A. prepared the figures and performed data analysis. All authors reviewed and edited the manuscript. Correspondence to Anthony G. Passerini. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Raw data used for the analysis Rajamani, A., Borkowski, K., Akre, S. et al. Oxylipins in triglyceride-rich lipoproteins of dyslipidemic subjects promote endothelial inflammation following a high fat meal. Sci Rep 9, 8655 (2019). https://doi.org/10.1038/s41598-019-45005-5 Serum metabolomic biomarkers of perceptual speed in cognitively normal and mildly impaired subjects with fasting state stratification Ameer Y. Taha The oxylipin profile is associated with development of type 1 diabetes: the Diabetes Autoimmunity Study in the Young (DAISY) Teresa Buckner Lauren A. Vanderlinden Jill M. Norris Diabetologia (2021)	CommonCrawl
Quantum Computing Meta Quantum Computing Stack Exchange is a question and answer site for engineers, scientists, programmers, and computing professionals interested in quantum computing. It only takes a minute to sign up. Are correlations stronger than those allowed by quantum mechanics possible? We know how a quantum correlation setup can help us with a better probability of winning games like the CHSH. But what is the upper bound that physics can allow? Is it the quantum correlation setup? Or can we exceed them in general sense to get much stronger correlations? non-locality games correlations foundations Adam Zalcman Siddhant SinghSiddhant Singh $\begingroup$ you are asking about theories beyond quantum mechanics. that's a tricky issue, since we don't know if there are any, and how they look like. but if you believe in relativity, there are non-trivial bounds on stronger correlations beyond quantum mechanics, see PR boxes: en.wikipedia.org/wiki/… $\endgroup$ – Norbert Schuch $\begingroup$ If that's not what you are after, it would help if you could make your question more precise. $\endgroup$ $\begingroup$ Here's a specific proposal for a supra-quantum correlation set which satisfies a lot of nice physical constraints: arxiv.org/abs/1403.4621 $\endgroup$ – Jalex Stark $\begingroup$ This question is precisely answered within the following work: arxiv.org/abs/1511.08144 $\endgroup$ Yes, it is possible to conceive theories with "stronger correlations" than those given by quantum mechanics. One way to make this statement precise is to consider some kind of "measurement apparatus" (you can think of it as simply a black a box with some buttons that you can push and different LEDs that correspond to different possible outputs), and analyse the set of correlations between inputs and outputs that different physical theories allow for. For example, if you have two possible inputs and two possible outputs than the set of possible theories, often referred to as behaviours in this context, is the set of probabilities $\{p(a\|x)\}_{x,a\in\{0,1\}}$. This is the set of vectors $\boldsymbol p$ in $[0,1]^{2^2}\subset\mathbb R^{2^2}$ normalised to one. In other words, it's a section of the $2^2$-dimensional hyperplane. More generally, in this context, it is common to consider a Bell-like scenario in which two parties are involved and each has its own box. Then, if each party can choose between $m$ possible inputs and can get $\Delta$ possible outputs, the set of possible behaviours is a hyperplane $\mathcal P\subset\mathbb R^{\Delta^2 m^2}$, which therefore has dimension $(\Delta^2-1)m^2$. The behaviours allowed by quantum mechanics are a strict subset of $\mathcal P$. One can consider different restrictions imposed on a physical theory and study the corresponding set of possible behaviours. A first natural assumption is to require a theory to be no-signalling, which means that it doesn't allow for faster-than-light communication. The set $\mathcal{NS}$ of no-signalling behaviours is strictly larger than the set $\mathcal Q$ of quantum behaviours. In the context of CHSH inequalities, the boundary between $\mathcal{NS}$ and $\mathcal Q$ would be observable via Tsirelson's bound, which tells us that quantum mechanics cannot produce correlations such that $S>2\sqrt 2$ (where $S$ is the usual operator defined for CHSH inequalities). Similarly, the boundary between $\mathcal Q$ and the set $\mathcal L$ of local (classical) behaviours can be witnessed via the standard Bell's inequalities. See the following picture representing the relations between these different sets (taken from Brunner et al., pag. 7): Have a look at Brunner et al.'s review (1303.2849) for more details. If you don't want to assume anything about a theory, then no, there are no restrictions at all. Any correlation between present and future is in principle possible. glS♦glS This question is precisely answered within the following work: https://arxiv.org/abs/1511.08144 They basically present that quantum bounds are basically the upper bounds for classical probabilistic correlations. And this is not unique to quantum mechanics, the upper bounds can be attained by other frameworks which are deterministic. They present a simple experiment based on tree network via which this can be realized. Sanchayan Dutta $\begingroup$ Very cool and useful reference! $\endgroup$ Thanks for contributing an answer to Quantum Computing Stack Exchange! Not the answer you're looking for? Browse other questions tagged non-locality games correlations foundations or ask your own question. What is the no-signaling set and how can it be related to other types of correlations? In Bell nonlocality, why does $P(ab\|xy)\neq P(a\|x)P(b\|y)$ mean the variables are not statistically independent? What is the relationship between the Toffoli gate and the Popescu-Rohrlich box? Is there a relation between the factorisation of the joint conditional probability distribution and Bell inequality? Determining whether $P(ab\|xy)$ factorizes in Bell experiments Proof of optimality for CHSH game classical strategy Why is $P(1,2)_{\text{same}} = \frac{1}{4}$ and not $\frac{1}{2}$ in Preskill's Bell experiment? How is Bell's Inequality converted to the CHSH inequality? Is Connes' Embedding Problem akin the word problem for finitely presented groups? How does a classical computer simulate nonclassical correlations? $\rho_{SE}(0)=\rho_S(0)\otimes\rho_E(0)$: No coupling or no entanglement?	CommonCrawl
Genetic effects of FASN, PPARGC1A, ABCG2 and IGF1 revealing the association with milk fatty acids in a Chinese Holstein cattle population based on a post genome-wide association study Cong Li1, Dongxiao Sun1, Shengli Zhang1, Shaohua Yang1, M. A. Alim1, Qin Zhang1, Yanhua Li2 & Lin Liu2 BMC Genetics volume 17, Article number: 110 (2016) Cite this article A previous genome-wide association study deduced that one (ARS-BFGL-NGS-39328), two (Hapmap26001-BTC-038813 and Hapmap31284-BTC-039204), two (Hapmap26001-BTC-038813 and BTB-00246150), and one (Hapmap50366-BTA-46960) genome-wide significant single nucleotide polymorphisms (SNPs) associated with milk fatty acids were close to or within the fatty acid synthase (FASN), peroxisome proliferator-activated receptor gamma, coactivator 1 alpha (PPARGC1A), ATP-binding cassette, sub-family G, member 2 (ABCG2) and insulin-like growth factor 1 (IGF1) genes. To further confirm the linkage and reveal the genetic effects of these four candidate genes on milk fatty acid composition, genetic polymorphisms were identified and genotype-phenotype associations were performed in a Chinese Holstein cattle population. Nine SNPs were identified in FASN, among which SNP rs41919985 was predicted to result in an amino acid substitution from threonine (ACC) to alanine (GCC), five SNPs (rs136947640, rs134340637, rs41919992, rs41919984 and rs41919986) were synonymous mutations, and the remaining three (rs41919999, rs132865003 and rs133498277) were found in FASN introns. Only one SNP each was identified for PPARGC1A, ABCG2 and IGF1. Association studies revealed that FASN, PPARGC1A, ABCG2 and IGF1 were mainly associated with medium-chain saturated fatty acids and long-chain unsaturated fatty acids, especially FASN for C10:0, C12:0 and C14:0. Strong linkage disequilibrium was observed among ARS-BFGL-NGS-39328 and rs132865003 and rs134340637 in FASN (D´ > 0.9), and among Hapmap26001-BTC-038813 and Hapmap31284-BTC-039204 and rs109579682 in PPARGC1A (D´ > 0.9). Subsequently, haplotype-based analysis revealed significant associations of the haplotypes encompassing eight FASN SNPs (rs41919999, rs132865003, rs134340637, rs41919992, rs133498277, rs41919984, rs41919985 and rs41919986) with C10:0, C12:0, C14:0, C18:1n9c, saturated fatty acids (SFA) and unsaturated fatty acids (UFA) (P = 0.0204 to P < 0.0001). Our study confirmed the linkage between the significant SNPs in our previous genome-wide association study and variants in FASN and PPARGC1A. SNPs within FASN, PPARGC1A, ABCG2 and IGF1 showed significant genetic effects on milk fatty acid composition in dairy cattle, indicating their potential functions in milk fatty acids synthesis and metabolism. The findings presented here provide evidence for the selection of dairy cows with healthier milk fatty acid composition by marker-assisted breeding or genomic selection schemes, as well as furthering our understanding of technological processing aspects of cows' milk. Recently, an increasing number of genes have been reported as associated with milk production for dairy cattle breeding, and great improvements have been obtained. Many quantitative trait locus (QTL) analysis and association studies revealed the DGAT1, GHR, FASN and PPARGC1A genes as promising candidate genes for milk production traits [1–12]. Nevertheless, there have been few reports [13–22] of association studies involving milk fatty acid traits, which should be considered because of their close relation with milk flavor and nutritional properties. High concentrations of saturated fatty acids (SFAs) such as C12:0, C14:0 and C16:0 increase the risks of coronary artery disease (CAD) by promoting the concentrations of blood low density lipoprotein (LDL) cholesterol [23], while polyunsaturated fatty acids (PUFAs) have the ability to reduce blood fat and cholesterol levels by inhibiting fat formation and enzyme activities acting on fat [24, 25]. Thus, increasing the ratio of PUFAs to SFAs would be beneficial to human health. A previous genome-wide association study (GWAS) revealed that several significant single nucleotide polymorphisms (SNPs) close to or within the FASN, PPARGC1A, ABCG2 and IGF1 genes were associated with milk fatty acids in Chinese Holstein dairy cattle [26]. In addition, the FASN, PPARGC1A, ABCG2 and IGF1 genes were observed to be associated significantly with milk production traits in our previous candidate genes analysis in Chinese Holstein cattle [27–30]. Therefore, we deduced that the significant SNPs might be linked with the causative mutations in these four genes. The purpose of the present study was to identify the genetic effects of the FASN, PPARGC1A, ABCG2 and IGF1 genes on traits of milk fatty acids in a Chinese Holstein cattle population. In addition, linkage disequilibrium (LD) analyses were conducted among the SNPs identified in our previous GWAS and in this study. Phenotypic data and traits Complete details of the milk sample collection and the detection method for milk fatty acids have been reported previously [26]. Briefly, fat was extracted from 2 mL of milk and then methyl esterification of fats was performed. One milliliter of methyl esters of fatty acids were prepared and determined by gas chromatography using a gas chromatograph (6890 N, Agilent) equipped with a flame-ionization detector and a high polar fused silica capillary column (SPTM-2560, 100 m × 0.25 mm ID, 0.20 μm film; Cat. No. 24056). About 1 μL of the sample was injected under the specific gas chromatography conditions. Finally, individual fatty acids were identified and quantified by comparing the methyl ester chromatograms of the milk fat samples with the chromatograms of pure fatty acids methyl ester standards (SupelcoTM 37 Component FAME Mix), and were measured as the weight proportion of total fat weight (wt/wt%). Phenotypic values of 10 main milk fatty acids were tested directly using gas chromatography, which included SFAs of C10:0, C12:0, C14:0, C16:0, C18:0, monounsaturated fatty acids (MUFAs) of C14:1, C16:1, C18:1n9c, and PUFAs of CLA (cis-9, trans-11 C18:2), C18:2n6c. Based on the phenotypes of 10 tested milk fatty acids, six additional traits were obtained including SFA, UFA, SFA/UFA (the ratio of SFA to UFA), C14 index, C16 index and C18 index. The three indices were calculated as $ \frac{\mathrm{cis}\hbox{-} 9\kern0.5em \mathrm{unsaturated}}{\mathrm{cis}\hbox{-} 9\kern0.5em \mathrm{unsaturated}+\mathrm{saturated}}\ast 100 $, [31]. The population in this study comprised 346 Chinese Holstein cows, which were the daughters of 13 sire families from 13 farms of the Beijing Sanyuan Dairy Farm Center. Sixteen main milk fatty acid traits were considered in this association study. Genomic DNA extraction The whole blood samples corresponding to the 346 Chinese Holstein cows with phenotypic values were collected. Genomic DNA was extracted from blood samples of the cows using a TIANamp Genomic DNA kit (TianGen, Beijing, China) according to the manufacturer's instructions and frozen semen of the sires using a standard phenol-chloroform procedure. The quantity and quality of the extracted DNA were measured using a NanoDrop™ ND-2000c Spectrophotometer (Thermo Scientific, Inc.) and by gel electrophoresis. SNP identification and genotyping A DNA pool was constructed from aforementioned 13 Holstein bulls (50 ng/μL for each individual) whose daughters were used for the association analysis to identify potential SNPs in the FASN, PPARGC1A, ABCG2 and IGF1 genes. For FASN, a total of 30 pairs of PCR primers (Additional file 1, Table S1) were designed to amplify all the exons and their partial flanking intronic sequences based on the reference sequence of the bovine FASN referring to Bos_taurus_UMD_3.1 assembly (NCBI Reference Sequence: AC_000176.1) using Primer3 web program (v.0.4.0) [32]. Following with the same method, a pair of specific primers was designed for selective amplification based on the exon 9 and partial intron 9 sequence of PPARGC1A (NCBI Reference Sequence: AC_000163.1): forward 5′- GCC GGT TTA TGT TAA GAC AG-3′ and reverse 5′- GGT ATT CTT CCC TCT TGA GC-3′. Primers were also designed from exon 7 and partial flanking intronic sequences of the ABCG2 gene (NCBI Reference Sequence: AC_000163.1): forward 5′- TAA AGG CAG GAG TAA TAA AG-3′ and reverse 5′- TAA CAC CAA ACT AAC CGA AG-3′, and the 5′-flanking region of the IGF1 gene (NCBI Reference Sequence: AC_000162.1): forward 5′- ATT ACA AAG CTG CCT GCC CC-3′ and reverse 5′- CAC ATC TGC TAA TAC ACC TTA CCC G-3′. Polymerase chain reaction (PCR) amplifications for the pooled DNA from the 13 sires were performed in a final reaction volume of 25 μL comprising of 50 ng of genomic DNA, 0.5 μL of each primer (10 mM), 2.5 μL of 10 × PCR buffer, 2.5 mM each of dNTPs, and 1 U of Taq DNA polymerase (Takara, Dalian, China). The PCR protocol was 5 min at 94 °C for initial denaturation followed by 34 cycles at 94 °C for 30 s; 56 ~ 60 °C for 30 s; 72 °C for 30 s; and a final extension at 72 °C for 7 min for all primers. The PCR products were purified to remove residual primers, dNTPs and reagents from the amplification reaction. A gel purification kit (DNA Gel Extraction Kit, TransGen Biotech, China) was used to extract the target DNA band. Then, 15 μL of each purified PCR product with 1 μL of each forward and reverse primer, was bi-directionally sequenced using an ABI3730XL sequencer (Applied Biosystems, Foster City, CA, USA). Matrix-assisted laser desorption/ionization time of flight mass spectrometry (MALDI-TOF MS, Sequenom MassARRAY, Bioyong Technologies Inc. HK) was used for subsequent genotyping of the 346 Chinese Holstein cows. Linkage disequilibrium (LD) analysis and haplotype construction Pair-wise LD was measured between the genotyped SNPs of each gene and the corresponding adjacent SNPs that were significantly associated with target traits identified in our previous GWAS based on the criterion of D' using the software Haploview [33]. Accordingly, haplotype blocks where SNPs are in high LD (D' > 0.90) were also determined based on confidence interval methods [34]. A haplotype with a frequency >5 % was treated as a distinguishable haplotype, and those haplotypes each with relative frequency <5 % were pooled into a single group. Association analyses Hardy-Weinberg equilibrium tests were performed on each identified SNP. A goodness-of-fit test (Chi-square) was used to compare the number of expected and observed genotypes, using 0.05 as significant threshold value. The mixed procedure of SAS 9.3 software (SAS Institute Inc., Cary, NC) with the following animal model was performed to estimate the genetic effects of each candidate SNP or haplotype on the milk fatty acid traits. $$ {y}_{i\mathrm{jklmn}}=\mu +{\mathrm{F}}_{\mathrm{i}}+{\mathrm{P}}_{\mathrm{j}}+{L}_k+{G}_l+{\alpha}_m+{e}_{ijklmn} $$ where, yijklmn was the phenotypic value of each trait of the cows; μ was the overall mean; Fi was the fixed effect of the farm; Pj was the fixed effect of parity; Lk was the fixed effect of the stage of lactation; Gl was the fixed effect corresponding to the genotype of polymorphisms or haplotype; αm was the random polygenic effect, distributed as N (0, Aσa 2), with the additive genetic relationship matrix A and the additive genetic variance σ 2a ; and eijklmn was the random residual, distributed as N (0, Iσe 2), with identity matrix I and residual error variance σ 2e . Bonferroni correction was adopted to correct for multiple testing. The significance level of the multiple tests was equal to the raw P value divided by number of tests. In the present study, three genotypes were compared for each trait mean that three multiple comparisons needed to be performed, therefore, Bonferroni corrected significance levels of 0.05/3 = 0.0167 and 0.01/3 = 0.0033 were used. For the haplotype, the Bonferroni corrected significance levels were presented as 0.05/N, where N refers to the number of formed haplotypes. The additive (a), dominance (d) and allele substitution (α) effects were estimated according to the equation proposed by Falconer & Mackay [35], i.e. $ \mathrm{a}=\raisebox{1ex}{$\left(\mathrm{AA}-\mathrm{B}\mathrm{B}\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right. $, $ \mathrm{d}=\mathrm{AB}-\raisebox{1ex}{$\left(\mathrm{AA}+\mathrm{B}\mathrm{B}\right)$}\!\left/ \!\raisebox{-1ex}{$2$}\right. $ and α = a + d(q − p), where AA and BB represent the two homozygous genotypes, AB is the heterozygous genotype, and p and q are the allele frequencies of the corresponding alleles. SNPs identification After sequencing the PCR products directly using the pooled genomic DNA, a total of nine SNPs were identified for the FASN gene. Of these, three were located in the intronic region and six were in exons. The SNP in exon 39 (rs41919985) was predicted to result in an amino acid replacement (A2266T) from threonine (ACC) to alanine (GCC) in the FASN protein, and the other five SNPs in the coding region (rs136947640, rs134340637, rs41919992, rs41919984 and rs41919986) were synonymous mutations. Regarding PPARGC1A, ABCG2 and IGF1, only one SNP was detected in each gene (rs109579682, rs137757790 and rs109763947, respectively), of which rs109763947 is located in the 5′-untranslated region (UTR) and the other two SNPs are in intronic regions. The detailed SNP information is shown in Table 1, and the five significant SNPs for milk fatty acids that are close to FASN, PPARGC1A, ABCG2 and IGF1 identified in our previous GWAS [26] are listed as well. All the identified SNPs in this study were found to be in Hardy-Weinberg equilibrium (P > 0.01, Tables 2 and 3). Table 1 SNPs information identified in this study and in a previous GWA study Table 2 Genotypic and allelic frequencies and Hardy-Weinberg equilibrium test of nine SNPs of the FASN gene in Chinese Holstein cattle Table 3 Genotypic and allelic frequencies and Hardy-Weinberg equilibrium test of SNPs of the PPARGC1A, ABCG2 and IGF1 genes in Chinese Holstein cattle Associations between the four candidate genes and milk fatty acid traits Associations between the nine SNPs of FASN and 16 milk fatty acid composition traits are presented in Table 4. We found that all nine SNPs showed significant associations with at least one milk fatty acid trait. Of these, three SNPs (rs136947640, rs132865003 and rs134340637) were only significantly associated with C18:2n6c (P < 0.0001, P = 0.0128, P = 0.0128), two SNPs (rs41919992 and rs133498277) showed strong associations with seven traits of C10:0, C12:0, C14:0, C18:1n9c, C16 index, SFA and UFA (P = 0.0190 to < 0.0001), three SNPs (rs41919984, rs41919985 and rs41919986) were strongly associated with the above seven traits plus SFA/UFA (P = 0.045 to P <0.0001), and one SNP (rs41919999) showed significant association with C10:0 (P = 0.0012), C12:0 (P = 0.0041) and C14:0 (P = 0.0071). Meanwhile, for C14:1, C16:0, C16:1, C18:0, CLA, C14 index and C18 index, no significant SNPs in FASN were detected. Furthermore, the results showed that heterozygous genotypes of these SNPs were the dominant type for saturated fatty acids (C10:0, C12:0, C14:0, SFA and SFA/UFA), and the homozygotic genotypes of these SNPs were dominant for unsaturated fatty acids (C18:1n9c, C16 index and UFA). Table 4 Associations of nine SNPs of the FASN gene with milk medium-chain fatty acids (MCFAs) in Chinese Holstein cattle (LSM ± SE) The effects of the three genotyped polymorphisms in PPARGC1A, ABCG2 and IGF1 on 16 milk fatty acid compositions are shown in Table 5. SNP rs109579682 in PPARGC1A was significantly associated with eight milk fatty acid traits, such as C10:0 (P = 0.0251), C12:0 (P = 0.0340), C14:0 (P = 0.0188), C16:1 (P = 0.0401), C18:1n9c (P = 0.0015), C16 index (P = 0.0010), SFA (P = 0.0065) and UFA (P = 0.0038). Correspondingly, the CC genotype was the dominant type for saturated fatty acids (C10:0, C12:0, C14:0 and SFA), and the TT genotype was dominant for unsaturated fatty acids (C16:1, C18:1n9c, C16 index and UFA). Table 5 Associations of SNPs of PPARGC1A, ABCG2 and IGF1 genes with milk medium-chain fatty acids (MCFAs) in Chinese Holstein cattle (LSM ± SE) For ABCG2, SNP rs137757790 was significantly associated with C14:0 (P = 0.0026), C18:1n9c (P = 0.0048), SFA (P = 0.0343) and UFA (P = 0.0266). The AA genotype was dominant for saturated fatty acids (C14:0 and SFA), and the CC genotype was dominant for unsaturated fatty acids (C18:1n9c and UFA). For IGF1, SNP rs109763947 was significantly associated with C10:0 (P = 0.0342), C18:1n9c (P = 0.0024), C18:2n6c (P < 0.0001), C16 index (P = 0.0239), SFA (P = 0.0090) and UFA (P = 0.0023). The homozygous genotype of TT was the dominant type for saturated fatty acids (C10:0 and SFA), and the heterozygous genotype of CT was the dominant type for unsaturated fatty acids (C18:1n9c, C16 index, C18:2n6c and UFA). Additionally, the significant dominant, additive and allele substitution effects of the significant SNPs on the target milk fatty acid traits were observed (Tables 6 and 7). Table 6 Additive, dominant and allele substitution effects of the nine SNPs on milk fatty acids traits of FASN in Chinese Holstein cattle Table 7 Additive, dominant and allele substitution effects of the SNPs on milk fatty acids traits of PPARGC1A, ABCG2 and IGF1 in Chinese Holstein cattle LD between the SNPs identified in the four candidate genes and our previous GWAS Pair-wise D' measures showed that all nine SNPs in FASN were highly linked (D' > 0.9), and one haplotype block comprising eight SNPs was inferred (Fig. 1) in which three haplotypes were formed. The common haplotypes TCGCCTGC, CCGTTCAT and CTACCTGC occurred at a frequency of 54.2 %, 27.8 % and 17.2 %, respectively (Table 8). Most importantly, the significant SNP (rs41921177) identified in our previous GWAS [26] showed strong linkage with the three FASN SNPs (rs136947640, rs132865003 and rs134340637). Subsequently, haplotype-based analysis showed significant associations of the haplotypes encompassing the eight FASN SNPs (rs41919999, rs132865003, rs134340637, rs41919992, rs133498277, rs41919984, rs41919985 and rs41919986) with C10:0, C12:0, C14:0, C18:1n9c, SFA and UFA (P = 0.0204 to P < 0.0001; Table 9). Linkage disequilibrium (LD) plot for 10 SNPs close to or within FASN. The values in boxes are pair-wise SNP correlations (D'), bright red boxes without numbers indicate complete LD (D' = 1). The blocks indicate haplotype blocks and the texts above the horizontal numbers are the SNP names Table 8 Main haplotypes and their frequencies observed in the FASN gene Table 9 Haplotype associations of the eight SNPs in FASN with milk production traits in Chinese Holstein cattle (LSM ± SE) Strong linkage among the two significant SNPs (rs110131167 and rs108967640) detected in our previous GWAS [26] and the SNP (rs109579682) in PPARGC1A was also observed (D' > 0.9, Fig. 2). However, no LD was observed between the SNPs located in the ABCG2 and IGF1 genes. Linkage disequilibrium (LD) plot for three SNPs in PPARGC1A. The values in boxes are pair-wise SNP correlations (D'), the brighter shade of red indicates higher linkage disequilibrium Information on the effects of DNA polymorphisms on milk fatty acid composition is scarce, because milk fatty acid composition data, unlike those of milk fat percentage and fat yield, are not collected routinely in milk recording schemes. Therefore, we attempted to explore the genetic variants of candidate genes identified by our previous GWAS on milk fatty acid composition [26]. In this study, we first investigated the associations between the tested SNPs of FASN, PPARGC1A, ABCG2 and IGF1 and milk fatty acid traits in Chinese Holstein cows. In our previous GWAS, the SNP rs41921177, at a distance of 58,172 bp away from FASN, showed significant association with C10:0 (P = 8.54E-06), C12:0 (P = 1.16E-07) and C14:0 (P = 6.01E-06) [26]. As expected, we found that this SNP was also strongly linked with the three SNPs in FASN (rs136947640, rs132865003 and rs134340637) that were significantly associated with C18:2n6c. Furthermore, if the haplotype block was defined based on the solid spine of the LD method, one haplotype block was constructed by the above three SNPs plus two SNPs, rs41921177 and rs41919999, that were associated with C10:0, C12:0 and C14:0. Similarly, strong linkages between the two significant SNPs (rs110131167 and rs108967640) for the C18 index, UFA and SFA/UFA identified in our previous GWAS and the SNP (rs109579682) in PPARGC1A for UFA and SFA identified in this study were observed. Probably as a result of the limited number of SNPs identified for ABCG2 and IGF1, and the farther distance between SNPs in the previous GWAS and their adjacent SNPs identified for ABCG2 and IGF1 in this study, no linkages with the significant SNPs identified in GWAS were observed. Six out of nine SNPs in FASN (rs41919999, rs41919992, rs133498277, rs41919984, rs41919985 and rs41919986) were markedly associated with C10:0, C12:0 and C14:0, and five of these six SNPs (rs41919992, rs133498277, rs41919984, rs41919985 and rs41919986) also showed significant associations with SFA, which suggested that the FASN gene mainly affects the medium-long chain saturated fatty acid traits. FASN is a complex, multifunctional enzyme that catalyzes de novo biosynthesis of long-chain saturated fatty acids [36] and plays an essential role in the determination of fatty acid synthesis and release of newly synthesized SFAs [37, 38]. In addition, several previous linkage studies [8, 39, 40] and GWA studies [13–15] have reported that the FASN gene is located in a quite large region associated with the medium-chain saturated milk fatty acids on BTA19, which is in agreement with our results that the SNPs in FASN mainly showed significant associations with C10:0, C12:0 and C14:0. Moreover, the five SNPs (rs41919992, rs133498277, rs41919984, rs41919985 and rs41919986) also showed associations with the C18:1n9c, C16 index and UFA, and three SNPs (rs136947640, rs132865003 and rs134340637) showed associations with C18:2n6, revealing that the FASN gene affects the long-chain unsaturated fatty acid traits. The haplotype-based association analysis showed their significant associations with C10:0, C12:0, C14:0, C18:1n9c, SFA and UFA, also confirming the genetic effects of the FASN gene on the medium-chain saturated and long-chain unsaturated milk fatty acids. Kim & Ntambi [41] reported that FASN is a key gene involved in the pathway for MUFAs synthesis and incorporation into triacylglycerols and phospholipids, which is consistent with our results. However, the effect of FASN on PUFAs has not been reported elsewhere. It was reported that the SNPs in different exons of the FASN gene were associated with milk-fat percentage [9] and with the medium- and long-chain fatty acid content of milk [8] and beef [42]. Morris et al. [8] identified five SNPs in FASN, including the non-synonymous SNP, rs41919985, observed in this study, which had been reported in different studies. The allele frequency of rs41919985 A (0.29) in our population is lower than that reported in Friesian and Jersey cattle (0.31 and 0.13, respectively) [8], 0.53 in Dutch Holstein–Friesian population [43] and 0.62 in Angus beef cattle [42]. Morris et al. [8] also reported that rs41919985 affected the C18:1cis9 and the total index, while other SNPs in FASN affected C14:0 and C18:2, which were consistent with our findings. Associations of the rs41919985 G allele with higher C14:0 and lower C18:1cis9 were also found in beef cattle [42]. Abe et al. [44] revealed that the FASN gene had a significant effect on the fatty acid composition of backfat, intramuscular and intermuscular fat in an F2 population from Japanese Black and Limousin cattle. For all nine significant SNPs in FASN, the heterozygous genotypes were associated with a higher proportion of milk SFAs, while the homozygous genotypes were associated with much higher levels of long-chain MUFAs and PUFAs. Thus, decreasing the number of individuals with heterozygous genotypes for these target SNPs in FASN will be beneficial to produce high-quality milk with a high proportion of unsaturated fatty acids (UFAs). PPARGC1A is involved in mammary gland metabolism, and the expression of PPARGC1A correlates with milk fat content [45]. Moreover, it is a key factor in energy metabolism and plays a central role in thermogenesis, gluconeogenesis, glucose transport and β-oxidation of fatty acids [46]. The finding that PPAR agonists are able to increase stearoyl-CoA desaturase (SCD) mRNA levels in humans, mice and rats suggested that PPARs are able to regulate SCD [47]. As the SCD enzyme is involved in the desaturation of saturated fatty acids into cis9-unsaturated fatty acids, PPARs might have an effect on unsaturation indices via their regulation of SCD [43]. Our findings supported the above research that PPARGC1A was significantly associated with the C16 index. In our study, PPARGC1A mainly affected medium-chain saturated fatty acids and long-chain unsaturated fatty acids. Only a few studies have reported associations between PPARGC1A and milk fatty acid composition [13, 43]. Schennink et al. [43] found that one SNP in PPARGC1A, c.1790 + 514G > A, was associated with the C16:1 and C16 index, and Bouwman et al. [13] reported another significant SNP associated with C16:1, which are in agreement with the results in this study that rs109579682 in PPARGC1A is associated significantly with the C16:1 and C16 index. The significant associations between PPARGC1A c.1790 + 514G > A and the C14:1, C14 index, and C18 index [43] were not found in this study. The conflicting findings could be explained by the two different genetic backgrounds of the studied populations or by the different number of individuals included in each study. Phenotypic data were available from 1,905 cows in the study reported by Schennink et al. [43], while 346 cows were available in our study. The bovine ABCG2 gene is located in the narrow region of chromosome 6 (BTA6), harboring a QTL with a large impact on milk production traits [48, 49]. The ABCG2 protein is responsible for the secretion of xenobiotics and some quantitatively minor nutrients, such as vitamin K3 or cholesterol, into milk [50, 51]. The insulin-like growth factor (IGF) signaling pathway plays a crucial role in the regulation of growth and development of mammals. Liang et al. [52] reported that IGF1 stimulates de novo fatty acid biosynthesis by Schwann cells during myelination. For ABCG2 and IGF1, most studies focused on investigating the association between the identified SNPs in these two genes and milk fat traits [4, 53–58], while limited studies on their association with milk fatty acid composition have been reported [13]. Bouwman et al. [13] reported that one QTL region underlying the ABCG2 gene showed significant effects on C12:1, C14:1 and C16:1. No association between IGF1 and milk fatty acids composition has been reported. Further studies will be necessary to confirm our results in different cattle population and to elucidate the mechanisms underlying the association found in this study. In this study, we not only confirmed the deduction that the significant SNPs close to the FASN and PPARGC1A genes identified in our previous GWAS were strongly linked with the key mutations in these two candidate genes, but also presented a link of several variants of FASN, PPARGC1A, ABCG2 and IGF1 with milk fatty acid traits. In particular, FASN and PPARGC1A mainly affected medium-chain saturated fatty acids and long-chain unsaturated fatty acids. Our findings regarding genes and polymorphisms responsible for the variation of milk fatty acids composition provide useful information that can be combined with breeding programs to tailor the fatty acid content in cow's milk. 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Olsen HG, Nilsen H, Hayes B, Berg PR, Svendsen M, Lien S, Meuwissen T. Genetic support for a quantitative trait nucleotide in the ABCG2 gene affecting milk composition of dairy cattle. BMC Genet. 2007;8:32. Ron M, Cohen-Zinder M, Peter C, Weller JI, Erhardt G. Short communication: a polymorphism in ABCG2 in Bos indicus and Bos taurus cattle breeds. J Dairy Sci. 2006;89(12):4921–3. Siadkowska E, Zwierzchowski L, Oprzadek J, Strzalkowska N, Bagnicka E, Krzyzewski J. Effect of polymorphism in IGF-1 gene on production traits in Polish Holstein-Friesian cattle. Anim Sci Pap Rep. 2006;24(3):225–37. Szewczuk M, Zych S, Czerniawska-Piatkowska E, Wojcik J. Association between IGF1R/i16/TaqI and IGF1/SnaBI polymorphisms and milk production traits in Polish Holstein-Friesian cows. Anim Sci Pap Rep. 2012;30(1):13–24. We appreciate the Dairy Data Center of China and Beijing Dairy Cattle Center for providing pedigree and milk samples for the Chinese Holstein cows. This work was supported by the National Science and Technology Programs of China (2013AA102504, 2011BAD28B02, 2014ZX08009-053B), Beijing Natural Science Foundation (6152013), Beijing Dairy Industry Innovation Team, earmarked fund for Modern Agro-industry Technology Research System (CARS-37), and Program for Changjiang Scholar and Innovation Research Team in University (IRT1191). All relevant data are available within the manuscript and its Supporting Information files. CL conducted the association analysis and wrote the manuscript. DS and SZ designed the study and revised the manuscript. SY and MA prepared the DNA samples for SNP identification and genotyping. QZ participated in the data analysis and provided suggestions for the manuscript. YL and LL provided milk samples and participated in the result interpretation. All authors read and approved the final manuscript. Animal handling and sample collection procedures were performed in accordance with protocols approved by the Institutional Animal Care and Use Committee (IACUC) at China Agricultural University. Department of Animal Genetics and Breeding, College of Animal Science and Technology, Key Laboratory of Animal Genetics and Breeding of Ministry of Agriculture, National Engineering Laboratory for Animal Breeding, China Agricultural University, 2 Yuanmingyuan West Road, Beijing, 100193, China Cong Li, Dongxiao Sun, Shengli Zhang, Shaohua Yang, M. A. Alim & Qin Zhang Beijing Dairy Cattle Center, Beijing, 100085, China Yanhua Li & Lin Liu Cong Li Dongxiao Sun Shengli Zhang Shaohua Yang M. A. Alim Qin Zhang Yanhua Li Lin Liu Correspondence to Dongxiao Sun or Shengli Zhang. Additional file 1: Table S1. Primers used to identify SNPs in the FASN gene. (PDF 104 kb) Li, C., Sun, D., Zhang, S. et al. Genetic effects of FASN, PPARGC1A, ABCG2 and IGF1 revealing the association with milk fatty acids in a Chinese Holstein cattle population based on a post genome-wide association study. BMC Genet 17, 110 (2016). https://doi.org/10.1186/s12863-016-0418-x DOI: https://doi.org/10.1186/s12863-016-0418-x Milk fatty acids Single nucleotide polymorphism	CommonCrawl
Tagged: nonsingular matrix True or False Problems on Midterm Exam 1 at OSU Spring 2018 The following problems are True or False. Let $A$ and $B$ be $n\times n$ matrices. (a) If $AB=B$, then $B$ is the identity matrix. (b) If the coefficient matrix $A$ of the system $A\mathbf{x}=\mathbf{b}$ is invertible, then the system has infinitely many solutions. (c) If $A$ is invertible, then $ABA^{-1}=B$. (d) If $A$ is an idempotent nonsingular matrix, then $A$ must be the identity matrix. (e) If $x_1=0, x_2=0, x_3=1$ is a solution to a homogeneous system of linear equation, then the system has infinitely many solutions. If $\mathbf{v}, \mathbf{w}$ are Linearly Independent Vectors and $A$ is Nonsingular, then $A\mathbf{v}, A\mathbf{w}$ are Linearly Independent Let $A$ be an $n\times n$ nonsingular matrix. Let $\mathbf{v}, \mathbf{w}$ be linearly independent vectors in $\R^n$. Prove that the vectors $A\mathbf{v}$ and $A\mathbf{w}$ are linearly independent. Find a Nonsingular Matrix $A$ satisfying $3A=A^2+AB$ (a) Find a $3\times 3$ nonsingular matrix $A$ satisfying $3A=A^2+AB$, where \[B=\begin{bmatrix} 2 & 0 & -1 \\ 0 &2 &-1 \\ -1 & 0 & 1 (b) Find the inverse matrix of $A$. Determine whether the Matrix is Nonsingular from the Given Relation Let $A$ and $B$ be $3\times 3$ matrices and let $C=A-2B$. \[A\begin{bmatrix} \end{bmatrix}=B\begin{bmatrix} \end{bmatrix},\] then is the matrix $C$ nonsingular? If so, prove it. Otherwise, explain why not. Determine whether the Given 3 by 3 Matrices are Nonsingular Determine whether the following matrices are nonsingular or not. (a) $A=\begin{bmatrix} 2 &1 &2 \\ 1 & 0 & -1 (b) $B=\begin{bmatrix} 4 & 1 & 4 For What Values of $a$, Is the Matrix Nonsingular? Determine the values of a real number $a$ such that the matrix 3 & 0 & a \\ 0 & 18a & a+1 \end{bmatrix}\] is nonsingular. Are Coefficient Matrices of the Systems of Linear Equations Nonsingular? (a) Suppose that a $3\times 3$ system of linear equations is inconsistent. Is the coefficient matrix of the system nonsingular? (b) Suppose that a $3\times 3$ homogeneous system of linear equations has a solution $x_1=0, x_2=-3, x_3=5$. Is the coefficient matrix of the system nonsingular? (c) Let $A$ be a $4\times 4$ matrix and let \[\mathbf{v}=\begin{bmatrix} \end{bmatrix} \text{ and } \mathbf{w}=\begin{bmatrix} \end{bmatrix}.\] Suppose that we have $A\mathbf{v}=A\mathbf{w}$. Is the matrix $A$ nonsingular? If $M, P$ are Nonsingular, then Exists a Matrix $N$ such that $MN=P$ Suppose that $M, P$ are two $n \times n$ non-singular matrix. Prove that there is a matrix $N$ such that $MN = P$. The Vector $S^{-1}\mathbf{v}$ is the Coordinate Vector of $\mathbf{v}$ Suppose that $B=\{\mathbf{v}_1, \mathbf{v}_2\}$ is a basis for $\R^2$. Let $S:=[\mathbf{v}_1, \mathbf{v}_2]$. Note that as the column vectors of $S$ are linearly independent, the matrix $S$ is invertible. Prove that for each vector $\mathbf{v} \in V$, the vector $S^{-1}\mathbf{v}$ is the coordinate vector of $\mathbf{v}$ with respect to the basis $B$. Click here if solved 5 Diagonalize a 2 by 2 Symmetric Matrix Diagonalize the $2\times 2$ matrix $A=\begin{bmatrix} 2 & -1\\ \end{bmatrix}$ by finding a nonsingular matrix $S$ and a diagonal matrix $D$ such that $S^{-1}AS=D$. If the Sum of Entries in Each Row of a Matrix is Zero, then the Matrix is Singular Let $A$ be an $n\times n$ matrix. Suppose that the sum of elements in each row of $A$ is zero. Then prove that the matrix $A$ is singular. Two Matrices are Nonsingular if and only if the Product is Nonsingular An $n\times n$ matrix $A$ is called nonsingular if the only vector $\mathbf{x}\in \R^n$ satisfying the equation $A\mathbf{x}=\mathbf{0}$ is $\mathbf{x}=\mathbf{0}$. Using the definition of a nonsingular matrix, prove the following statements. (a) If $A$ and $B$ are $n\times n$ nonsingular matrix, then the product $AB$ is also nonsingular. (b) Let $A$ and $B$ be $n\times n$ matrices and suppose that the product $AB$ is nonsingular. Then: The matrix $B$ is nonsingular. The matrix $A$ is nonsingular. (You may use the fact that a nonsingular matrix is invertible.) A Singular Matrix and Matrix Equations $A\mathbf{x}=\mathbf{e}_i$ With Unit Vectors Let $A$ be a singular $n\times n$ matrix. \[\mathbf{e}_1=\begin{bmatrix} \vdots \\ \end{bmatrix}, \mathbf{e}_2=\begin{bmatrix} \end{bmatrix}, \dots, \mathbf{e}_n=\begin{bmatrix} \end{bmatrix}\] be unit vectors in $\R^n$. Prove that at least one of the following matrix equations \[A\mathbf{x}=\mathbf{e}_i\] for $i=1,2,\dots, n$, must have no solution $\mathbf{x}\in \R^n$. The Matrix $[A_1, \dots, A_{n-1}, A\mathbf{b}]$ is Always Singular, Where $A=[A_1,\dots, A_{n-1}]$ and $\mathbf{b}\in \R^{n-1}$. Let $A$ be an $n\times (n-1)$ matrix and let $\mathbf{b}$ be an $(n-1)$-dimensional vector. Then the product $A\mathbf{b}$ is an $n$-dimensional vector. Set the $n\times n$ matrix $B=[A_1, A_2, \dots, A_{n-1}, A\mathbf{b}]$, where $A_i$ is the $i$-th column vector of $A$. Prove that $B$ is a singular matrix for any choice of $\mathbf{b}$. The Transpose of a Nonsingular Matrix is Nonsingular Let $A$ be an $n\times n$ nonsingular matrix. Prove that the transpose matrix $A^{\trans}$ is also nonsingular. Find the Inverse Matrices if Matrices are Invertible by Elementary Row Operations For each of the following $3\times 3$ matrices $A$, determine whether $A$ is invertible and find the inverse $A^{-1}$ if exists by computing the augmented matrix $[A\|I]$, where $I$ is the $3\times 3$ identity matrix. (b) $A=\begin{bmatrix} -1 &-3 &2 \\ If a Group $G$ Satisfies $abc=cba$ then $G$ is an Abelian Group Determine Whether the Following Matrix Invertible. If So Find Its Inverse Matrix. Give a Formula for a Linear Transformation if the Values on Basis Vectors are Known Vector Space of Polynomials and a Basis of Its Subspace	CommonCrawl
A convergent finite difference method for computing minimal Lagrangian graphs February 2022, 21(2): 355-392. doi: 10.3934/cpaa.2021181 A second-order accurate structure-preserving scheme for the Cahn-Hilliard equation with a dynamic boundary condition Makoto Okumura 1,, , Takeshi Fukao 2, , Daisuke Furihata 3, and Shuji Yoshikawa 4, Research Institute for Electronic Science, Hokkaido University, N12W7, Kita-Ward, Sapporo, Hokkaido, 060-0812, Japan Department of Mathematics, Faculty of Education, Kyoto University of Education, 1 Fujinomori, Fukakusa, Fushimi-ku, Kyoto, 612-8522, Japan Cybermedia Center, Osaka University, 1-32 Machikaneyama, Toyonaka, Osaka, 560-0043, Japan Division of Mathematical Sciences, Faculty of Science and Technology, Oita University, 700 Dannoharu, Oita, 870-1192, Japan * Corresponding author Received February 2021 Revised September 2021 Published February 2022 Early access November 2021 Fund Project: This work was partially supported by JSPS KAKENHI, Grant No. JP20KK0308, JP20K03687, JP20K20883, JP21K03309, JP21K20314, and The Sumitomo Foundation, Grant No. 190367 Figure(22) / Table(4) We propose a structure-preserving finite difference scheme for the Cahn–Hilliard equation with a dynamic boundary condition using the discrete variational derivative method (DVDM) proposed by Furihata and Matsuo [14]. In this approach, it is important and essential how to discretize the energy which characterizes the equation. By modifying the conventional manner and using an appropriate summation-by-parts formula, we can use a standard central difference operator as an approximation of an outward normal derivative on the discrete boundary condition of the scheme. We show that our proposed scheme is second-order accurate in space, although the previous structure-preserving scheme proposed by Fukao–Yoshikawa–Wada [13] is first-order accurate in space. Also, we show the stability, the existence, and the uniqueness of the solution for our proposed scheme. Computation examples demonstrate the effectiveness of our proposed scheme. Especially through computation examples, we confirm that numerical solutions can be stably obtained by our proposed scheme. Keywords: Finite difference method, structure-preserving scheme, Cahn–Hilliard equation, dynamic boundary condition, and error estimate. 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Google Scholar Figure 1. Numerical solution by our scheme with $ \Delta x = 1/2 $ Figure 2. Numerical solution by Fukao-Yoshikawa-Wada scheme with $ \Delta x = 1/2 $ Figure 3. Numerical solution by our scheme with $ \Delta x = 1/40 $ Figure 4. Numerical solution by Fukao-Yoshikawa-Wada scheme with $ \Delta x = 1/40 $ Figure 5. Time development of $ M_{\rm d}(\boldsymbol{U}^{(n)}) $ obtained by our scheme with $ \Delta x = 1/40 $: $ M_{\rm d}(\boldsymbol{U}^{(n)}) $ is preserved to accuracy $ 10^{-11} $ Figure 6. Time development of $ E_{\rm d}^{(n)} - J_{\rm d}(\boldsymbol{U}^{(0)}) $ obtained by our scheme with $ \Delta x = 1/40 $: $ E_{\rm d}^{(n)} $ is preserved to accuracy $ 10^{-6} $ Figure 7. The discrete $ L^{\infty} $-norm error $ \\|\boldsymbol{e}_{\Delta x} \\|_{L_{\rm d}^{\infty}} $ versus the space mesh size $ \Delta x $ at time $ T = 400 $: our scheme is second-order accurate in space Figure 8. The discrete $ L^{\infty} $-norm error $ \\|\boldsymbol{e}_{\Delta t} \\|_{L_{\rm d}^{\infty}} $ versus the time mesh size $ \Delta t $ at time $ T = 400 $: our scheme is second-order accurate in time Figure 10. Numerical solution by Fukao-Yoshikawa-Wada scheme with $ \Delta x = 1/25 $ Figure 11. Numerical solution by our scheme with $ \Delta x = 1/50 $ Figure 13. Time development of $ {M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}}) $ obtained by our scheme with $ \Delta x = 1/50$: ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ is preserved to accuracy 10−14 Figure 14. Time development of $ E_{\rm{d}}^{(n)} - {J_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(0)}}) $ obtained by our scheme with $ \Delta x = 1/50$: $E_{\rm{d}}^{(n)}$ is preserved to accuracy 10−11 Figure 15. The discrete L∞-norm error ${\left\\| {{\mathit{\boldsymbol{e}}_{\Delta x}}} \right\\|_{L_{\rm{d}}^\infty }}$ versus the space mesh size Δx at time T = 1000: our scheme is second-order accurate in space Figure 16. The discrete L∞-norm error ${\left\\| {{\mathit{\boldsymbol{e}}_{\Delta t}}} \right\\|_{L_{\rm{d}}^\infty }}$ versus the time mesh size Δt at time T = 1000: the convergence rates of our scheme approach three as Δt decreases Figure 17. Numerical solution to (1.1)–(1.2) with (1.5) and (6.1) obtained by our scheme Figure 18. Time development of ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ obtained by our scheme: ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ is preserved to accuracy 10−11 Figure 19. Time development of $E_{_{\rm{d}}}^{(n)} - {J_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(0)}})$ obtained by our scheme: $E_{_{\rm{d}}}^{(n)}$ is preserved to accuracy 10−10 Figure 20. Numerical solution to (1.1)–(1.2) with (7.16) obtained by the discrete variational derivative scheme Figure 21. Time development of ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ obtained by the discrete variational derivative scheme: ${M_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(n)}})$ is preserved to accuracy 10−14 Figure 22. Time development of $A_{_{\rm{d}}}^{(n)} - {{\bar J}_{\rm{d}}}({\mathit{\boldsymbol{U}}^{(0)}})$ obtained by the discrete variational derivative scheme: $A_{_{\rm{d}}}^{(n)}$ is preserved to accuracy 10−9 Table 1. The discrete $ L^{\infty} $-norm error $ \\|\mathit{\boldsymbol{e}}_{\Delta x} \\|_{L_{\rm d}^{\infty}} $ and the convergence rates $ \log_{2}(\\|\mathit{\boldsymbol{e}}_{2\Delta x} \\|_{L_{\rm d}^{\infty}}/\\|\mathit{\boldsymbol{e}}_{\Delta x} \\|_{L_{\rm d}^{\infty}}) $ at time $ T = 400 $ $ \Delta x $ $ 2^{-1} $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $ $ \\|\mathit{\boldsymbol{e}}_{\Delta x} \\|_{L_{\rm d}^{\infty}} $ 3.5272e-3 8.6474e-4 2.1507e-4 5.1156e-5 Rate / 2.0282 2.0075 2.0718 Table 2. The discrete $ L^{\infty} $-norm error $ \\|\mathit{\boldsymbol{e}}_{\Delta t} \\|_{L_{\rm d}^{\infty}} $ and the convergence rates $ \log_{2}(\\|\mathit{\boldsymbol{e}}_{2\Delta t} \\|_{L_{\rm d}^{\infty}}/\\|\mathit{\boldsymbol{e}}_{\Delta t} \\|_{L_{\rm d}^{\infty}}) $ at time $ T = 400 $ $ \Delta t $ $ 2^{-1} $ $ 2^{-2} $ $ 2^{-3} $ $ 2^{-4} $ $ \\|\mathit{\boldsymbol{e}}_{\Delta t} \\|_{L_{\rm d}^{\infty}} $ 2.2345e-6 5.6404e-7 1.4274e-7 3.4246e-8 Table 3. The discrete $ L^{\infty} $-norm error $ \\|\mathit{\boldsymbol{e}}_{\Delta x} \\|_{L_{\rm d}^{\infty}} $ and the convergence rates $ \log_{2}(\\|\mathit{\boldsymbol{e}}_{2\Delta x} \\|_{L_{\rm d}^{\infty}}/\\|\mathit{\boldsymbol{e}}_{\Delta x} \\|_{L_{\rm d}^{\infty}}) $ at time $ T = 1000 $ Table 4. The discrete $ L^{\infty} $-norm error $ \\|\mathit{\boldsymbol{e}}_{\Delta t} \\|_{L_{\rm d}^{\infty}} $ and the convergence rates $ \log_{2}(\\|\mathit{\boldsymbol{e}}_{2\Delta t} \\|_{L_{\rm d}^{\infty}}/\\|\mathit{\boldsymbol{e}}_{\Delta t} \\|_{L_{\rm d}^{\infty}}) $ at time $ T = 1000 $ $ \Delta t $ $ 1/10 $ $ 1/20 $ $ 1/40 $ $ 1/80 $ Makoto Okumura, Daisuke Furihata. A structure-preserving scheme for the Allen–Cahn equation with a dynamic boundary condition. Discrete & Continuous Dynamical Systems, 2020, 40 (8) : 4927-4960. doi: 10.3934/dcds.2020206 Takeshi Fukao, Shuji Yoshikawa, Saori Wada. Structure-preserving finite difference schemes for the Cahn-Hilliard equation with dynamic boundary conditions in the one-dimensional case. Communications on Pure & Applied Analysis, 2017, 16 (5) : 1915-1938. doi: 10.3934/cpaa.2017093 Tetsuya Ishiwata, Kota Kumazaki. Structure preserving finite difference scheme for the Landau-Lifshitz equation with applied magnetic field. Conference Publications, 2015, 2015 (special) : 644-651. doi: 10.3934/proc.2015.0644 Qi Hong, Jialing Wang, Yuezheng Gong. Second-order linear structure-preserving modified finite volume schemes for the regularized long wave equation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6445-6464. doi: 10.3934/dcdsb.2019146 Sergey P. Degtyarev. On Fourier multipliers in function spaces with partial Hölder condition and their application to the linearized Cahn-Hilliard equation with dynamic boundary conditions. Evolution Equations & Control Theory, 2015, 4 (4) : 391-429. doi: 10.3934/eect.2015.4.391 Yuto Miyatake, Tai Nakagawa, Tomohiro Sogabe, Shao-Liang Zhang. A structure-preserving Fourier pseudo-spectral linearly implicit scheme for the space-fractional nonlinear Schrödinger equation. Journal of Computational Dynamics, 2019, 6 (2) : 361-383. doi: 10.3934/jcd.2019018 Alain Miranville, Sergey Zelik. 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Numerical investigation of space fractional order diffusion equation by the Chebyshev collocation method of the fourth kind and compact finite difference scheme. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2025-2039. doi: 10.3934/dcdss.2020402 Gisèle Ruiz Goldstein, Alain Miranville. A Cahn-Hilliard-Gurtin model with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (2) : 387-400. doi: 10.3934/dcdss.2013.6.387 Laurence Cherfils, Madalina Petcu. On the viscous Cahn-Hilliard-Navier-Stokes equations with dynamic boundary conditions. Communications on Pure & Applied Analysis, 2016, 15 (4) : 1419-1449. doi: 10.3934/cpaa.2016.15.1419 Vladislav Balashov, Alexander Zlotnik. An energy dissipative semi-discrete finite-difference method on staggered meshes for the 3D compressible isothermal Navier–Stokes–Cahn–Hilliard equations. 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Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1737-1747. doi: 10.3934/dcdsb.2014.19.1737 Andreas C. Aristotelous, Ohannes Karakashian, Steven M. Wise. A mixed discontinuous Galerkin, convex splitting scheme for a modified Cahn-Hilliard equation and an efficient nonlinear multigrid solver. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2211-2238. doi: 10.3934/dcdsb.2013.18.2211 Ciprian G. Gal, Alain Miranville. Robust exponential attractors and convergence to equilibria for non-isothermal Cahn-Hilliard equations with dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 113-147. doi: 10.3934/dcdss.2009.2.113 Makoto Okumura Takeshi Fukao Daisuke Furihata Shuji Yoshikawa	CommonCrawl
Kinect and wearable inertial sensors for motor rehabilitation programs at home: state of the art and an experimental comparison Bojan Milosevic ORCID: orcid.org/0000-0002-8921-59931, Alberto Leardini ORCID: orcid.org/0000-0003-3547-73702 & Elisabetta Farella ORCID: orcid.org/0000-0001-9047-98681 Emerging sensing and communication technologies are contributing to the development of many motor rehabilitation programs outside the standard healthcare facilities. Nowadays, motor rehabilitation exercises can be easily performed and monitored even at home by a variety of motion-tracking systems. These are cheap, reliable, easy-to-use, and allow also remote configuration and control of the rehabilitation programs. The two most promising technologies for home-based motor rehabilitation programs are inertial wearable sensors and video-based motion capture systems. In this paper, after a thorough review of the relevant literature, an original experimental analysis is reported for two corresponding commercially available solutions, a wearable inertial measurement unit and the Kinect, respectively. For the former, a number of different algorithms for rigid body pose estimation from sensor data were also tested. Both systems were compared with the measurements obtained with state-of-the-art marker-based stereophotogrammetric motion analysis, taken as a gold-standard, and also evaluated outside the lab in a home environment. The results in the laboratory setting showed similarly good performance for the elementary large motion exercises, with both systems having errors in the 3–8 degree range. Usability and other possible limitations were also assessed during utilization at home, which revealed additional advantages and drawbacks for the two systems. The two evaluated systems use different technology and algorithms, but have similar performance in terms of human motion tracking. Therefore, both can be adopted for monitoring home-based rehabilitation programs, taking adequate precautions however for operation, user instructions and interpretation of the results. Emerging sensing and communication technologies are driving the innovation of a vast number of application fields, including fitness, healthcare and rehabilitation therapy [1]. Major drivers of healthcare innovation include the priority changes from treatment to prevention, and the search to provide personalized and patient-centric solutions. Both trends are enabled by unobtrusive sensing technologies, allowing for continuous monitoring and increased engagement with the patient outside the clinic [2]. Movement analysis and its use for motor rehabilitation is one of the many application fields where innovative technical solutions for unconstrained and autonomous monitoring of the patients are being adopted [3]. Standard practices for motor rehabilitation include the clinician's supervision and evaluation of the patient's movements, when performed during therapy sessions in clinic, and no supervision or any feedback when the exercises are executed at home. Computer vision and stereophotogrammetry-based technologies have been widely proven as accurate and reliable tools for objective measurement of human motion [4, 5]. However, the costs and difficulties of operation of such systems have limited their use to research rather than in everyday clinical and rehabilitation practice. The development of miniaturized inertial sensors paved the way for the development of wearable Inertial Measurement Units (IMUs) and their use for motion capture [6, 7]. Such technologies have also been validated in lab environments for medical applications and motor rehabilitation analyses [8, 9]; however, the available solutions involve cost and complexity-related limitations. Nowadays, both research and commercial applications are experiencing a push in ubiquitous computing and the use of wearable and interconnected sensing devices for a wide range of applications, from entertainment to fitness and wellbeing [10]. The adoption of the use of fitness and activity trackers is driven by their low cost and ease of use, but these have usually limited accuracy in the reported data [11]. For a successful adoption of these new technologies in rehabilitation, there is a need to evaluate their accuracy and reliability and to provide insights on their proper use in order to define best practices and standardized protocols [12]. The recent innovative low-cost sensing solutions and relevant algorithms for data analysis, once validated, can be effectively introduced in rehabilitation protocols both in specialized centers and at home, and truly enable a patient-centric, preventive and smart healthcare revolution [13]. In the field of human motion analysis, both video and inertial-based solutions have now low-cost options, suitable for wide adoption and everyday use; examples include the Kinect camera [14] and various activity tracking and wearable inertial sensors [15]. Their integration into bio-feedback-based systems and combination with exergames and appropriate back-end infrastructure allows for the development of innovative solutions for real-time monitoring of home-based rehabilitation therapies and for a continuous remote supervision by the clinician [16]. The first platforms providing such functionalities include DoctorKinetic (DoctorKinetic, Netherlands), SilverFit (SilverFit, Netherlands) and Riablo (Corehab, Italy). This paper reports an overview of these major systems, analyzing in the literature the state-of-the-art of the Kinect and of wearable motion sensing in rehabilitation, but mainly focuses on a validation work for the quantitative assessment of these systems. Since there is a lack of direct comparison and discussion on the differences of the two technologies, an original experimental study was performed and here reported to evaluate and directly compare the Kinect v2 and a commercially available wearable IMU (EXLs3 by Exel srl, Italy). Their technological characteristics and state-of-the-art algorithms for IMU-based analysis were assessed, using a marker-based stereophotogrammetry motion analysis system as the gold standard. In addition to laboratory tests, the two systems were also assessed in a typical home environment, to evaluate and fully compare their final usability and robustness. The two systems are here compared based on exactly the same human motion exercises, both in a laboratory setting for a thorough comparison with state-of-the-art motion analysis, and at home, for simulation of final users' conditions. Finally, drawing from the current state-of-the-art and from the present experimental comparisons, the main advantages and disadvantages of the two systems are discussed, analyzing their strengths and weaknesses, and highlighting the challenges for their successful future adoption in the rehabilitation context. Review of sensing technologies for motion analysis in rehabilitation This section will analyze current systems in human motion analysis, starting from the well-established video motion capture used in gait laboratories, and then focusing on innovative and low-cost alternatives, suitable for autonomous use at home. The reported references are summarized and compared in Table 1. Table 1 Summary of the main validation studies analyzing motion capture accuracy of the Kinect (v1 and v2) and of wearable IMUs Video-based motion capture The use of cameras and computer vision algorithms for the analysis of human motion is a well-established application field, and has notable contributions from both research and industry [17]. Video-based motion capture and Marker-Based Stereophotogrammetry systems (MBS) are now the de-facto standard for high-precision applications, including biomechanics research and clinical gait analysis [4]. In MBS systems, multiple cameras employ Infra-Red (IR) illuminators and triangulation algorithms to track the 3D position of reflective markers moving within a calibrated field of view. When used for human motion capture, the subject is instrumented with a set of reflective markers to identify and track relevant anatomical landmarks, and the system uses their positions to reconstruct and track subject's body segments and joints [18]. These systems have been proven to offer accurate and reliable motion tracking and are being widely used in human motion research and clinical studies. The established accuracy is less than 1 mm error for the position of single markers, which translates in the errors in the range of 1–4 degrees for the estimation of joint angles, according to the specific marker cluster configuration [18, 19]. There are a number of commercially available systems, equipped with high-performance cameras and different software solutions for out of the box motion analysis, including Vicon Nexus (Vicon Motion Systems, UK), Elite (BTSengineering, Milan) and Optitrack Motive (NaturalPoint, USA). The main downside of the MBS is the high cost and the complexity of its setup and use. To address these issues, research solutions explore the use of marker-less motion capture systems and their integration with depth sensors [20, 21]. Despite promising results, the accuracy and reliability implied with these new techniques do not yet meet the needs of healthcare applications, due to cumbersome hardware and extensive data processing requirements [22, 23]. Low-cost video sensing: the Kinect Microsoft first introduced the Kinect sensor in November 2010 to be used as a motion capture input device, as an add-on for the Xbox game console. It featured a standard digital video camera, a depth sensor based on structured IR illumination, and a directional microphone. The integration of the Kinect with dedicated algorithms allowed marker-less tracking of the user's segments pose and movements, creating a natural user interface based on gestures [24]. Although it was developed and sold as a game controller, its offer of RGB video and IR-based depth sensing (RGB+D), at a very low price, made it appealing for a wide range of users, also in biomechanical and clinical research [25, 26]. With the availability of drivers and of a Software Development Kit (SDK) for a more general use beyond gaming, the Kinect has been applied to a vast range of academic and industrial projects, including the fields of interaction, robotics and, in fact, biomechanics [27]. The first version of the Kinect (Kinect v1) was followed by a re-designed sensor presented in 2013 (Kinect v2), which introduced an improved RGB camera and a new IR time-of-flight depth sensor [28]. Kinect v2 and its new SDK improved the sensor's tracking capabilities and enhanced its use in applications based on human motion tracking [29]. The Kinect sensors have been extensively evaluated in relation to several application fields. The accuracy of the sensors and their depth estimation capabilities have been analyzed carefully [30], as well as the differences between the two versions [31,32,33]. Focusing on human motion capture applications, the use of the Kinect v1 in such scenario was triggered by the release of reverse-engineered open-source drivers and tracking software [34] and then propelled by the release of the Microsoft's SDK [35]. The second-generation device and its updated algorithm have been validated further within the context of clinical motion analysis, with applications such as posture and balance evaluation [36, 37], fall detection [38], rehabilitation exercises [39,40,41], and gait assessment [42,43,44]. Moreover, the usability of Kinect-based home rehabilitation systems has been investigated, providing insights on the user acceptance with good results and indications for future improvements [45, 46]. The two generations of Kinect sensors have been compared in validation studies: when applied to posture or movement evaluations, these showed similar results, with the Kinect v2 just slightly outperforming its predecessor [47, 48]. The new sensor achieved good overall performance in the tracking of human pose and elementary movements, but showed obvious limits when dealing with more complex exercises or when the movements were not performed with the subject standing facing the sensor. These results necessarily reduce the use of the Kinect as an accurate tool for possible exploitations in the clinical context, but open the door for a possible use in somehow qualitative evaluations of posture and exercise, and also show the potential of such system for at-home monitoring of rehabilitation therapies. Inertial-based motion capture The availability of Microelectromechanical Systems (MEMS) and their development for miniaturized sensors, combined with integrated processing and communication technologies, enabled the development of wearable sensing devices for human body monitoring [1, 15]. To obtain information regarding specific human locomotion parameters, one or more sensing devices are worn directly on relevant body parts and connected to a central processing hub for data collection and processing, forming a so-called Body Sensor Network [69]. However, this multi-sensor setup presents a number of technological requirements in terms of sensing capabilities, signal bandwidth, throughput and other general challenges such as device wearability, system usability, and data reliability [70]. There are several commercial examples, starting with high-end solutions for body motion capture [71], which are mainly used for animation and clinical movement analysis, all the way to ubiquitous motion trackers and sensors embedded in smartphones [72]. Notable examples include MVN Biomech (Xsens Technologies, Netherlands) and Opal (APDM Technologies, USA). The research and academic community is also very active on this topic, with several proposed platforms [73,74,75,76,77,78]. A wearable IMU provides unobtrusive methods to collect motion data relative to the body segment where it is worn; by combining a network of sensors, to form a whole-body model, joint motion can also be deduced. The integration of multiple sensors within the same device (accelerometer, gyroscope and magnetometer) allows to deploy robust sensor fusion algorithms in order to provide reliable and detailed information in a wide range of dynamic conditions and application contexts. In biomechanics, the most used application is the estimation of the device's orientation from the embedded sensors and its use for the estimation of joint angles [79, 80]. Algorithms derived from navigation applications are adapted to infer the orientation of the body segment of interest and include the Kalman Filter (KF), its extended and unscented variations, and also several implementations of Complementary Filters (CF) [81,82,83,84]. Moreover, IMU sensor data can be exploited to analyze various features of human motion and dedicated algorithms have been developed for tasks such as activity recognition [85], exercise recognition and evaluation [86, 87], gait analysis [63, 88, 89] and jump analysis [90, 91]. Research and clinical studies have validated the use of wearable IMUs also in various conditions and applications [80, 92]. Notable examples include balance and postural evaluation [93, 94], fall monitoring and prediction [95], gait analysis [96] and rehabilitation [64]. Laboratory evaluations and comparisons with high-precision MBS systems have shown high accuracy and reliability of wearable motion sensors. Hence, these can be used in clinical practice for the evaluation of human motion and can provide a valuable and portable tool for standardized motor tests [97]. Usability aspects of the employment of such systems for home rehabilitation evaluation have also been investigated providing encouraging results [98,99,100]. However, the scientific community still faces the challenges implied in the development of accurate, reliable and easy to use wearable solutions for motion analysis, and in their extensive validation in real-life contexts. Another emerging approach is to combine the outputs of the two systems [66, 67, 101]. In [67] the authors propose a sensor fusion algorithm to combine the Kinect and a set of wearable IMUs showing how the combined result achieves higher accuracy than any of the two systems. Additionally, in [66] an integration of IMUs and Kinect for the tracking of upper limb motions is proposed, showing again improved results when compared to the two separate systems. The work in [68] compares the use of IMUs and a Kinect-based system (Reha@Home) for gait analysis. This comparison, however, was limited to only one subject performing a short walk in front of the camera. In addition, the typical major problem of Kinect, i.e., the occlusions of body segments during the motion exercise, was promoted by the experimental setup adopted and the exercise performed. Separately, the different systems have been extensively evaluated, but direct comparison and the discussion of their tradeoffs are still very limited. Moreover, all the reported studies focus on laboratory-based validation, despite huge potential of such systems lies in at-home use and therefore these systems should be evaluated also in this context. This work provides a detailed analysis of the literature on the two approaches (see Table 1) and an original comparison performed both in laboratory with a state-of-the-art motion tracking reference and in different home settings. Review of segment and joint kinematics estimation algorithms The estimation and tracking of human segment and joint kinematics using video or wearable sensing is a well-documented research field, with several available solutions. This section will outline the main methods used in this work with the Kinect and with inertial sensors, whose estimates will be directly compared. Microsoft provides a comprehensive SDK for the Kinect, which includes a ready-to-use algorithm for the estimation and tracking of the user's complete body pose. The latest update provides real-time tracking for up to six people and it provides estimated 3D position for a complete skeletal model formed of 21 body joints and quaternion-based rotations of the relevant segments. The algorithm is based on the identification of the different body segments from the RGB+D video stream and uses a Random Forest recognition approach, which was trained with a wide dataset composed by real and synthetic data [24]. The research community has proposed some alternatives and there is still on-going work on pose estimation from RGB+D streams [102]. However, Microsoft's solution is the de-facto standard thanks to its robustness and ease of use. For these reasons, it was used in several validation and exploitation studies [36, 39, 42, 43] and is also used in the present work. The provided data are only low-pass filtered to eliminate noise. Further offline smoothing or any other processing was avoided because, in the present study, real-time tracking of the exercise was targeted. Inertial sensing Most of the previous validation studies used commercial solutions to obtain the orientation of the wearable sensors, which are used to estimate the orientation of the body segment they are attached to. Their outputs are then combined to form a partial or complete body pose estimation, based on the number of sensors in use [9, 59,60,61,62,63,64]. While there are several proposals for algorithms for the estimation of orientation from inertial sensors' data, the present work analyzes the most used ones, to provide a comparative analysis targeting robust and well-established solutions. Moreover, to evaluate the standard use at-home of these systems, robust but ready-to-use approaches, without the need for system calibration or additional operations, were considered. In particular, although all the proposed methods provide the full orientation of the device, its horizontal component is not frequently considered, since it is heavily affected by the environmental ferro-magnetic disturbances. Although inertial and magnetic sensors may be influenced by the environment (e.g., temperature [103, 104]), environment-aware calibration and rejection techniques are out of the scope of the present work. The EXLs3 sensors used in the present study are calibrated in factory, and all the experiments had a limited duration with standard and stationary environmental conditions, therefore effects from the environment are assumed to be null. All the orientation estimation algorithms are based on a combination of triaxial sensor inputs composed by accelerometer readings ${\varvec{a}} = \{a_x,a_y,a_z\}$, gyroscope readings ${\varvec{\omega }} = \{ \omega _x, \omega _y, \omega _z\}$ and magnetometer readings ${\varvec{m}} = \{m_x,m_y,m_z\}$, providing as output the orientation of the sensor, expressed either in quaternions (${\varvec{q}} = \{q_0,q_1,q_2,q_3\}$), Euler angles (${\varvec{E}} = \{E_x,E_y,E_z\}$) or rotation matrix (${\varvec{R}}$). Each of these three orientation representation methods has its advantages, but usually the quaternions are preferred for the computation efficiency and the results are converted to Euler angles because of their better clarity [105]. Orientation estimation from accelerometer (ACC) Using accelerometer (or accelerometer and magnetometer) outputs, the sensor's orientation is estimated by applying trigonometric functions. This approach assumes that the accelerometer is measuring only the gravity acceleration, and hence it is reliable only in static conditions. Accelerometer readings are used to estimate a partial orientation of the device, ${\varvec{E}}^a$ as $$\begin{aligned} E^a_x\,=\, & {} atan2(a_y,a_z) \end{aligned}$$ $$\begin{aligned} E^a_y\,= \,& {} atan2(a_x, \sqrt{a_y^2 + a_z^2}). \end{aligned}$$ From the magnetometer measures, the missing horizontal heading is estimated as $$\begin{aligned} E_z\,=\, & {} atan2(-m^a_y,m^a_x) \end{aligned}$$ where ${\varvec{m^a}}$ is the magnetometer reading projected to the accelerometer-estimated orientation plane identified by ${\varvec{E}}^a$. Gyroscope integration (GYR) Orientation of the sensor can also be estimated by integration of the angular velocity provided by the gyroscope. This estimate is reliable in dynamic situations, but suffers from drifts due to numerical integration errors. According to the chosen orientation representation, there are several implementations of its derivative; here, the quaternion one is adopted resulting in ${\varvec{{\dot{q}}}} = {\varvec{\Omega }}{\varvec{q}}$, which is integrated as $$\begin{aligned} {\varvec{q}}(t)= & {} \left({\varvec{I}} + \frac{1}{2}{\varvec{\Omega }} dt\right) {\varvec{q}}(t-dt); \end{aligned}$$ $$\begin{aligned} {\varvec{\Omega }}= & {} \begin{bmatrix} 0 &{} -\omega _x &{} -\omega _y &{} -\omega _z \\ \omega _x &{} 0 &{} \omega _z &{} -\omega _y \\ \omega _y &{} -\omega _z &{} 0 &{} \omega _x \\ \omega _z &{} \omega _y &{} -\omega _x &{} 0 \\ \end{bmatrix} \end{aligned}$$ where ${\varvec{I}}$ is a $(4 \times 4)$ identity matrix and ${\varvec{q}}(0)$ computed using the ACC estimation during a short static initialization. Kalman filter (KF) The Kalman filter is a widely used approach for optimal fusion of accelerometer and gyroscope orientation estimates [79, 81]. Several variations have been proposed and here a straightforward application of a quaternion-based KF is applied, using the GYR derivate as the state equation, which is then corrected by the ACC measurement. The KF state and measurement equations are implemented as $$\begin{aligned} {\left\{ \begin{array}{ll} {\varvec{{\dot{q}}}}(t) = {\varvec{\Omega }}{\varvec{q}}(t) + {\varvec{w}}(t) \\ {\varvec{q}}_a(t) = {\varvec{q}}(t) + {\varvec{v}}(t), \end{array}\right. } \end{aligned}$$ where ${\varvec{q}}(t)$ is the state estimate, ${\varvec{w}}\sim {\mathcal {N}}(0,{\varvec{R}})$ the zero-mean gaussian process noise with covariance matrix ${\varvec{R}}$, ${\varvec{q}}_a$ the accelerometer-based orientation estimate and ${\varvec{v}}\sim {\mathcal {N}}(0,{\varvec{Q}})$ the measurement noise with covariance matrix ${\varvec{Q}}$. The two covariance matrices were set to be diagonal with constant coefficients: 0.0001 for Q and 0.1 for R. Madgwick filter (MAD) Another approach for a quaternion-based iterative fusion of ACC and GYR estimates has been proposed by Madgwick [84] and it has been well received because of its high-quality estimate and limited computational and memory requirements. It is based on a gradient descent algorithm, which iteratively finds the optimal orientation given the input signals and it is governed by the following differential equations: $$\begin{aligned} {\varvec{q}}(t)\,=\, & {} {\varvec{q}}(t-1) + {\varvec{{\dot{q}}}}_{est}(t) dt \end{aligned}$$ $$\begin{aligned} {\varvec{{\dot{q}}}}_{est}(t)\,=\, & {} {\varvec{{\dot{q}}}}_{\omega }(t) -\beta {\varvec{{\dot{q}}}}_{a}(t) \end{aligned}$$ $$\begin{aligned} {\varvec{{\dot{q}}}}_{a}(t)\,=\, & {} \frac{\nabla f}{\Vert \nabla f\Vert }. \end{aligned}$$ The filter calculates the orientation ${\varvec{q}}$ by numerically integrating the estimated orientation rate ${\varvec{{\dot{q}}}}_{est}$, which is computed as the rate of change of orientation measured by the gyroscopes, ${\varvec{{\dot{q}}}}_{\omega }$, with the magnitude of the gyroscope measurement error, $\beta$, removed in the direction of the estimated error, ${\varvec{{\dot{q}}}}_{a}$, computed from accelerometer and magnetometer measurements. ${\varvec{{\dot{q}}}}_{a}$ is computed with the gradient descent method and f represents the function that provides the orientation from accelerometer and magnetometer readings. The implementation of the algorithm makes use of established matrix and quaternion operations, and the correction parameter $\beta$ was empirically set to 0.01 [84]. Complementary filter (CF) Another class of orientation estimation algorithms was developed by Mahony et al. using non-linear complementary filters [83]. Such approach is also becoming popular for its accuracy and reduced computational complexity when compared to KF. In this case, a rotation matrix representation is used and, contrary to the KF, the filter combines accelerometer and gyroscope estimates with a constant correction factor, following the equation: $$\begin{aligned} {\varvec{{\dot{R}}}}\,=\, & {} [{\varvec{R}} [\omega ]_{\text {x}} + k_p {\varvec{R}} \gamma ]_{\text {x}} {\varvec{R}}; \end{aligned}$$ $$\begin{aligned} \quad [{\varvec{\omega }}]_{\text {x}}= & {} \begin{bmatrix} 0 &{} -\omega _z &{} \omega _y \\ \omega _z &{} 0 &{} -\omega _x \\ -\omega _y &{} \omega _x &{} 0 \end{bmatrix} \end{aligned}$$ where ${\varvec{R}}$ is the rotation matrix, $k_p$ the filter gain empirically set to 0.5 and $\gamma$ the correction term given by the difference of the previous estimate and the current one from the accelerometer [83]. The experimental part of this work directly compared the two systems in a laboratory setting, using a high-precision MBS tracking system together with an internationally established technique as the gold standard. In addition, the use of these two systems out of the laboratory was compared, performing a pilot evaluation in unconstrained environments such as patient's house. Detailed description of experimental methodology and protocols is provided in "Methods" section. Laboratory evaluation Table 2 reports the mean differences, i.e., root mean square errors (RMSE), and the standard deviation for the considered techniques for IMU orientation estimation and for the estimates provided by the Kinect, when compared to the corresponding results established through MBS. The different IMU approaches have a similar performance, with the KF and MAD algorithms outperforming the others. It is interesting to note that the single-sensor algorithms have only limited degradation compared to the sensor fusion ones and in some cases even outperform the CF. This is mainly due to a combination of the type of performed exercises (large motions with a relatively low dynamic) and their short duration, which allow also ACC or GYR estimates to have limited RMSE. The IMU estimates employing sensor fusion algorithms outperform the Kinect's output, though by a limited margin, revealing that both approaches (by sensors or cameras) have a good overall performance, with errors in the range of 3 to 8 degree for all the joint angles analyzed, which is consistent with existing literature [39, 55, 59,60,61, 68]. Table 2 Mean errors for various IMU orientation estimation algorithms and for the Kinect v2 when compared to the MBS output As expected, the exercises performed while wearing clothes show a slightly higher RMSE and deviations; however, they are consistent with the standard GA case. The comparison between the different approaches confirms again that the sensors fusion algorithms outperform the single-sensor estimates and are aligned with the Kinect ones. Given the limited sample size, a thorough analysis on the performance degradation cannot be provided; however, additional errors in this case can be attributed to the motion artifacts caused by clothing, which allows for relative motion also between the markers and the underlying anatomical landmarks and also between IMUs and markers. Figure 1 shows resulting angles for the frontal lunge exercise in standard gait analysis while undressed (left), and the corresponding with clothing (right) from the same subject. Patterns from both IMU and Kinect match generally well with the gold standard from gait analysis, with consistent results for all the executed exercises and across all users. Timing of the waveforms is exactly the same, whereas peak values show differences, though consistent over repetitions. These can be accounted for the different technique applied for calculation of orientation, i.e., IMU tracking a single limited area of the body segment, while the Kinect is searching for the overall orientation of this segment, this being affected also by its deformation during motion. Sample data for a lunge exercise as performed in the lab without clothes (left), and with clothes (right), as monitored by wearable IMUs (IMU KF), Kinect v2 and gait analysis. The plots show in series over time 5 repetitions of normal execution, followed by 5 repetitions with larger forward inclination of the trunk At-Home evaluation Without the availability of a gold standard, it is not possible to calculate errors for the two systems for a quantitative evaluation. However, it is possible to qualitatively evaluate the outcomes and compare them to the data collected in the lab. In particular, it is possible to compute the average difference and standard deviation between the two estimates and use such parameters to compare the lab and home sessions. Table 3 collects the root mean square difference, its standard deviation and the maximum differences between IMU and Kinect estimates. For the three subjects who performed the exercises both in the lab and at home, there is a direct comparison of the two environments, while the average results for all sessions performed at home provide a qualitative insight of the performance outside the lab. A sample of the joint angles resulting from at-home acquisitions is plotted in Fig. 2, where the frontal lunge exercise is shown to facilitate the comparison with Fig. 1 showing the same exercise from the same subject performed in the lab. Table 3 Differences between the IMU KF and Kinect estimates during lab and at-home tests for each of the three subjects analyzed Sample data for a lunge exercise as performed at home and monitored by wearable IMUs (IMU KF) and the Kinect v2. The plots show in series over time 5 repetitions of normal execution, followed by 5 repetitions with larger forward inclination of the trunk All the performed exercises were correctly acquired by both systems in all the four test environments. A comparison of the outputs shows that they reported expected outcomes and the two systems show again similar performance. For the subjects monitored both in lab and at home, the difference between the two systems is consistent in both cases, with the exception of the knee flexion angle, which exhibits higher deviations in the un-controlled environments. The same outcomes are observed also for the average over all the subjects who performed exercises in the home environments. Considering both Figs. 1 and 2, it emerges that the systems show a difference in the estimated range of motion, with the Kinect underestimating it when compared to both the MBS reference and the IMU output. Similar results were observed in all acquired sessions, though such behavior should be analyzed further to establish systematic evaluations of the outcomes from the two systems. Moreover, Kinect-based estimates show also considerable peak discontinuities, as depicted in Fig. 2, right column. This result was observed throughout the dataset and it can be caused by image glitches disturbing the vision-based tracking algorithm. Of course, this influences the reported measurements of estimate differences. To mitigate this effect, ad-hoc filtering or smoothing techniques may be applied in the future. In the last decade, the consumer market opened the way for a broader acceptance and use of wearable sensing devices. Activity trackers are now widely employed in everyday life, but with limited reliability and validation of results [11, 106]. More accurate wearable inertial sensors have been adopted for a wide range of clinical applications [2, 107], with a huge potential to innovate and improve nearly every aspect of healthcare applications. But for a successful exploitation of these systems in healthcare and in particular in rehabilitation, there is definitely the need for their careful quantitative validation. In addition to these, unobtrusive sensing systems based on video and depth cameras are available at a low price and high performance, such as the Kinect v2 here assessed. It was originally developed as an interaction controller for home video games, but it has gained attention also for general research and clinical applications for its capability to track human subjects' movements in real time [27, 108]. With respect to inertial sensors, video-based tracking is even less invasive, as the body of the tracked subject is free of any instrument. Several studies have analyzed the performance and validated these two systems for the tracking of human motion in clinical applications, including postural and balance control, rehabilitation exercises, gait, or specific conditions such as Parkinson's or post stroke rehabilitation. Laboratory tests showed the limits of the low-cost tracking technologies when compared to state-of-the-art MBS systems; however, these also highlighted their overall applicability to ubiquitous patient monitoring (see Table 1 and the references therein). The development and adoption of innovative monitoring systems for effective patient monitoring in unconstrained environments open new research challenges in these systems' reliability, sensitivity to environmental and operational factors, usability and acceptability by the clinicians and the end users, i.e., the patients. In the present study, a thorough experimental analysis was performed to assess the accuracy of two instruments for human motion tracking in the context of rehabilitation. The experiments are established however as preliminary measurements on a limited sample size. Nevertheless, the state-of-the-art gait analysis was arranged as gold standard, and a large number of exercises were analyzed. These were a limited specific set within all possible rehabilitation exercises, particularly used to recover from a large number of orthopedic disorders and treatments. The scope in fact was to test the two instruments in a number of general yet well representative motor tasks; in the future these two instruments shall be tested also in other possible exercises. It is important to note, however, that among those analyzed here, the squat position is definitely very physically demanding for the extreme joint positions implied, and as such particularly suitable to reveal large measurement differences. As additional limitation, the two clothing conditions were tested in a single subject only, but this was thought just to reveal the additional artifact introduced by the clothes used routinely in these exercises, knowing that the gold standard for these measurements is represented by the motion at the skeletal system. Validation against state-of-the-art gait analysis was performed in two different conditions, though in a very small number of subjects. The standard procedure always requires the subject to be undressed, with all the markers attached to the skin in correspondence of relevant anatomical landmarks. This is recommended for a repeatable application of the marker set (for intra- and inter-subject comparisons) and to avoid the disturbances of the clothing, which adds considerable artifactual measurements. However, this is not the typical condition for the users of these systems; therefore, the validation was repeated, in one subject, also imitating a more realistic dressing condition, with the user wearing comfortable fitness clothing, typical of physical exercises in the gym or at home. In the latter case, the measures were less accurate, but they are more representative of a real scenario. The preliminary results here reported for the two systems highlight the importance of instructing the users to perform the exercises with limited and appropriate clothing and to tightly wear the sensors to limit occlusions and motion artifacts. Although all the present sensing technologies are likely to be affected by environmental factors (e.g., temperature, humidity, etc.) and by their status (duration of use, etc.), a detailed analysis of such influences is out of the scopes of this work. The present experimental protocol was rather designed to minimize the impact of any such external factors and environmental conditions. Moreover, the aim was to limit the differences between the acquired sessions and with respect to the corresponding conditions in the relevant literature (Table 1). The two systems showed similar performance in terms of final angle estimations when considering simple large-motion exercises. The measurements from this experimental work on both the laboratory and at-home sessions show good repeatability and consistency, therefore providing reliable evaluation of the performance of relevant rehabilitation exercises. However, the results also showed differences in the body segment orientations and therefore joint rotations, but these are consistent and small with respect to the corresponding overall range of motion. These findings are aligned with the reported literature, which generally reports errors below 10 degrees [40, 109]. Today, there is no consensus on the necessary accuracy that these motion tracking systems should provide for these to be appropriate in physical rehabilitation. However, based on the existing literature [23, 110], reports from therapists and physicians, as well as practical experience, errors in human segment or joint rotations smaller than 3 degrees would be tolerable for most rehabilitation programs in orthopedics; errors between 3 and 6 degrees can still be acceptable, depending on the joint, the pathology and treatment, and the status of the patient. For example, after the replacement of shoulder, hip and knee joints, the range of motion usually restored is far larger than 100 degrees, and this error therefore would be only a very small percentage. In this context, the two analyzed technologies perform well, and the errors here revealed can be well acceptable in most major human diarthrodial joints, compatible with the status of the patient and the rehabilitation exercises under observation. Direct or indirect, i.e., for at-home sessions, careful supervision and evaluation should be guaranteed in any case by trained therapists. This is in any case a step forward with respect to qualitative observations, which is biased by therapist experience. Nevertheless, the different basic technology of these two systems introduces additional considerations on their effective use. The Kinect is a well-supported commercial platform and benefits from its very simple operational requirements. To track movements, it just needs to be placed at 3–4 m in front of the subject and connected to a personal computer, without the need for additional instrumentation or further requirements. However, its vision-based approach imposes a limit on the tracked area, particularly a frontal view, and no object interposition; also, its low sampling frequency limits the range of movements correctly tracked. In particular, fast and complex movements as well as those with large components out-of-the-frontal plane of the sensor are not tracked by the system [44], thus precluding its use in applications such as real-life monitoring of patients and rehabilitation exercises performed while lying or with support devices. In addition, its limited field of view precludes its use for unconstrained gait monitoring. Wearable IMUs are now a mature and widely adopted technology, with several commercial solutions ranging from whole-body motion tracking suites to sensor kits and stand-alone units. The use of IMUs attached to a target body segment and the adoption of relevant sensor fusion algorithms is nowadays commonly employed to analyze human motion within a large spectrum of motor tasks and exercises, from up-right posture to complex sports activities [109, 111, 112]. IMU use for clinical motion analysis has been extensively evaluated regarding accuracy and reliability, but evaluation studies are mostly confined to laboratories [64, 93, 96]. Considering at-home uses, wearable IMUs have an additional requirement when compared to the Kinect, since the user has to wear the sensors. Such operation usually consists in mounting a simple elastic band, which can be considered simple enough for autonomous use at home even for children and elderly, but it can be, in theory, a source of uncertainty (i.e., sensor misplacement) or it can be problematic for severely impaired users. On the other hand, wearing the sensors on the user's body allows for a less-constrained tracking and for the development of a mobile solution capable of acquiring movements in a truly unconstrained and pervasive manner. The vast range of available sensors, paired with state-of-the-art processing algorithms, allows for the development of diversified solutions covering a wide spectrum of human motions, including static and postural analysis, rehabilitation exercises, jump analysis, gait analysis, fall detection, etc. This work addresses two of the most promising technologies for at-home rehabilitation monitoring based on real-time motion analysis, i.e., wearable IMUs and Kinect. The Kinect incorporates video and depth sensors and provides easy to use, real-time, full-body tracking at a low price. Wearable inertial sensors are now emerging as another reliable tool for movement analysis, providing an additional instrument for patient monitoring also in clinical and research settings. In the first part of this study, a detailed critical analysis of the literature on these technologies was performed (see Table 1), and in the second part original comparisons between the two are reported, after thorough experiments performed both in a state-of-the-art motion capture laboratory and in direct home settings. From the literature it emerged that the two different technologies have been assessed extensively, though mostly separately, with very limited direct experimental comparisons. In addition, only a few studies have addressed the final real conditions of use, i.e., at-home. Therefore, an original experimental analysis was performed, in both environments. The two systems showed similar performance in tracking elementary exercises with large range of motion, and provided comparable results both in the laboratory setting and in-home tests. In the former, IMUs combined with different sensor fusion algorithms showed an average RMSE of $5.5^\circ (\pm 2.3)$ over the performed exercises, which matches well with those from the Kinect, $5.6^\circ (\pm 2.0)$. These exercises were replicated with the same experimental protocol and with the same users in home environments, showing results much in support of those obtained in the laboratory. The Kinect has the advantage of very simple operational requirements, but it lacks the capabilities to track complex and highly dynamic movements, especially when the user does not move in front of the sensor. On the other hand, IMUs must be worn, but work well in a large variety of human movements, also at high speed. Both technologies, however, can be adopted for home-based rehabilitation monitoring, after taking adequate precautions about user instructions and about correct interpretation of the results. With further developments and large-scale real-life evaluations, these technologies will allow careful and pervasive patient monitoring and relevant clinical studies in the near future. This section describes the methodology and the comparison protocols employed for the experimental analysis. Our institution's Review Board (Comitato Etico dell'Istituto Ortopedico Rizzoli) approved the study conducted in the present work. All participants received detailed information about the study and provided written consent for the use of acquired data. All acquired data were anonymous and only age, gender, weight and height were stored along with the exercise data here reported. The subjects were recruited among graduate students at our institution. The direct instrumental comparison of the two systems was performed at the Movement Analysis Laboratory of the Rizzoli Orthopaedic Institute (Bologna, Italy) as shown in Fig. 3. Subjects' motion was concurrently monitored by a Kinect v2 (Microsoft, Seattle, USA), a set of three EXLs3 wearable IMUs (Exel srl, Bologna, Italy) and a high-precision 8-camera MBS motion tracking system (Vicon 612, Vicon Motion Systems Ltd, Oxford, UK) sampling at 100 Hz. Data collection sessions in the gait analysis laboratory: the same overall setup with the instrumentation mounted on a subject for a standard gait analysis (left, undressed) and for more realistic final user condition (right, dressed). Instrumentation includes IMUs on relevant body segments and reflective markers on relevant anatomical landmarks according to the gait analysis protocol [113] During the acquisitions, the Kinect was placed in front of the subject, at a distance of approximately 3.50 m, and at 1 m from the ground (Fig. 3). It was checked whether the subject was at the center of the field of view of the sensor, as recommended from the product guidelines. The Kinect4Windows 2.0 SDK was used for data acquisition and processing. It provides the reconstruction of the full body segments, formed by the position and angles of 21 joints [24]. These data were saved for offline analysis by means of a custom application. The SDK does not allow control over data acquisition and it provides an approximate sampling rate of 30 Hz. For IMU tracking, a 3-sensor kit of EXLs3 wireless IMUs was used. This study focused on the evaluation of lower limbs movements, hence the three sensors were placed on the frontal aspects of the subject's thorax and of left thigh and shank. The devices are self-worn using elastic bands with a dedicated pocket for the IMU. Each EXLs3 device is calibrated in factory and provides an on-board estimation of its orientation, in addition to triaxial sensor data for accelerometer ($\pm 2~{\text{g}}$ full scale), gyroscope ($\pm 500~{\text{dps}}$ full scale) and magnetometer ($\pm 1200~\mu T$ full scale). These are equipped with a Bluetooth transceiver for data streaming to a host device. In the performed tests, sensor data were sampled at 100 Hz and streamed to a personal computer for offline analysis. Given the placement of the IMUs and combining the orientation of the three sensors, it is possible to estimate the thorax sagittal and frontal orientation, the hip joint sagittal and frontal angles and the knee joint flexion/extension. As a gold standard reference, a state-of-the-art MBS motion capture system and an established gait analysis protocol were used. Before starting the data collection, 33 spherical 15-mm reflective markers were located on the lower limbs, pelvis and thorax in correspondence of known anatomical landmarks according to a validated protocol [113]. From these markers, anatomical-based reference frames were defined for each segment, and three-dimensional joint rotation angles were calculated according to international recommendations and conventions [114]. Thorax sagittal and frontal plane inclinations, hip joint sagittal and frontal angles and knee sagittal angle, i.e. flexion/extension, from these measurements and calculations were used as the gold standard for the comparison of the corresponding Kinect and IMU-based estimates. These gait analysis results were stored for offline comparative analysis. The study involved three healthy subjects (female 1.75 m 26 years, female 1.65 m 31 years, male 1.83 m 34 years) who performed physical exercises typical of rehabilitation programs after replacement of lower limb joints. For all three, standard gait analysis was performed which implies instrumenting the subjects without clothing (Fig. 3 left). This is considered the optimal experimental setting, with the best possible accuracy of the measurements because of the direct attachment of the markers to the skin without interposition. For one of the three subjects gait analysis was repeated days later while wearing comfortable fitness clothing (Fig. 3 right). It is worth noting however, that when collecting data while wearing clothes, MBS measurements are likely to be affected by noise, since the markers are attached to the clothing and some tissue motion artifacts are inevitable. The subjects were wearing adherent fitness clothing, which can limit this motion artifacts. The three subjects were first instructed about the functioning of the acquisition systems and how to wear the inertial sensors. In addition to squat (SQ), the following six exercises were performed by the left leg only: frontal lunge (FL), lateral lunge (LL), hip abduction (HA), hip flexion (HF), and hip extension (HE). These motion exercises include both basic and more complex movements and are typical of many rehabilitation programs targeting lower limbs functional recovery [115, 116]. For each exercise, the subjects were instructed to perform five repetitions as for standard correct execution first, i.e., with the trunk up-right, and then five more repetitions with the trunk in a large inclination forward, to mimic a common mistake in performing these rehabilitation exercises [115, 116]. The overall quality of the exercises was assessed by analyzing thorax orientation and hip and knee joint rotations; among these measurements, target parameters, i.e., those to determine the biofeedback, and control parameters, i.e., those to be checked for a correct performance of the exercise, are specified in Table 4. Table 4 Collected exercises and corresponding target and control parameters Spatial and temporal alignment of the reference frames from the three systems was performed offline. A short static up-right double-leg posture of the subject was acquired at the beginning of each data collection session and used to align the body segment orientations provided by the three systems. Moreover, a sharp right leg movement was performed at the beginning of a session to facilitate offline time alignment of the data streams. All data were stored for offline analysis, which were performed in Matlab. For a direct comparison, the joint rotations streams from the three systems were all re-sampled at 30 Hz. One of the main advantages of these two innovative approaches for human motion tracking is their low cost, which together with their small dimensions offer the possibility for ubiquitous adoption in rehabilitation centers, gyms and even at home. In addition to lab comparison, therefore, a pilot study was conducted to evaluate their use in the latter uncontrolled environment. To test the variability associated to different environmental conditions in real-life scenarios, the two systems were used to collect data in five additional locations. In particular, two homes and three different office spaces were used, where a total of 10 subjects were asked to perform the same set of exercises as during the laboratory evaluation. The same three subjects who performed the exercises in the laboratory were also among the home test group, to allow for a direct comparison of their performance. The spaces were different in dimensions and lighting conditions, going from a small office with artificial light to a large living room under direct sunlight. All sessions followed the same protocol as for the lab evaluation, except for the MBS-based gait analysis and the reference tracking, which was not available outside of the lab. 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The reliability of three-dimensional kinematic gait measurements: a systematic review. Gait Posture. 2009;29(3):360–9. Ricci L, Taffoni F, Formica D. On the orientation error of IMU: investigating static and dynamic accuracy targeting human motion. PLoS ONE. 2016;11(9):e0161940. Roell M, Roecker K, Gehring D, Mahler H, Gollhofer A. Player monitoring in indoor team sports: concurrent validity of inertial measurement units to quantify average and peak acceleration values. Front Physiol. 2018;9:141. Leardini A, Sawacha Z, Paolini G, Ingrosso S, Nativo R, Benedetti MG. A new anatomically based protocol for gait analysis in children. Gait Posture. 2007;26(4):560–71. Wu G, Siegler S, Allard P, Kirtley C, Leardini A, Rosenbaum D, et al. ISB recommendation on definitions of joint coordinate system of various joints for the reporting of human joint motion-part I: ankle, hip, and spine. J Biomech. 2002;35(4):543–8. Moran RW, Schneiders AG, Major KM, Sullivan SJ. How reliable are functional movement screening scores? A systematic review of rater reliability. Br J Sports Med. 2016;50(9):527–36. Nae J, Creaby MW, Nilsson G, Crossley KM, Ageberg E. Measurement properties of a test battery to assess postural orientation during functional tasks in patients undergoing anterior cruciate ligament injury rehabilitation. J Orthopaedic Sports Phys Ther. 2017;47(11):863–73. The authors wish to thank all subjects for participating in the study. This work was founded by the Autonomous Province of Trento (PAT) under the L6/99 project Riablo@Home, by the Italian Ministry of Economy and Finance within the program "5 per mille" and Fondazione Bruno Kessler (Trento, IT). E3DA, Fondazione Bruno Kessler (FBK), Trento, Italy Bojan Milosevic & Elisabetta Farella Movement Analysis Laboratory, IRCCS Istituto Ortopedico Rizzoli, Bologna, Italy Alberto Leardini Bojan Milosevic Elisabetta Farella BM performed the state-of-the-art analysis; BM, AL and EF conceived and designed the experiments and data collection. BM performed the data analysis analyzed and all authors participated in the writing of the manuscript the final draft. All authors read and approved the final manuscript. Correspondence to Bojan Milosevic. Our institution's Review Board (Comitato Etico IRCCS Istituto Ortopedico Rizzoli) approved the study conducted in the present work. All participants received detailed information about the study and provided written consent for the use of acquired data. All acquired data were anonymous and only age, gender, weight and height were stored along with the exercise data here reported. Milosevic, B., Leardini, A. & Farella, E. Kinect and wearable inertial sensors for motor rehabilitation programs at home: state of the art and an experimental comparison. BioMed Eng OnLine 19, 25 (2020). https://doi.org/10.1186/s12938-020-00762-7 Motor rehabilitation Home rehabilitation wearable inertial sensors	CommonCrawl
Only show content I have access to (13) Last 3 years (11) Language and Linguistics (2) Journal of Tropical Ecology (22) Journal of Applied Probability (7) Annales de Limnologie - International Journal of Limnology (6) AI EDAM (3) Canadian Journal of Mathematics (3) Microscopy and Microanalysis (3) The New Phytologist (3) ASTIN Bulletin: The Journal of the IAA (2) Bilingualism: Language and Cognition (2) Bulletin of Entomological Research (2) Invasive Plant Science and Management (2) Journal of Biosocial Science (2) RAIRO - Operations Research (2) Weed Science (2) Australian and New Zealand Journal of Family Therapy (1) International Journal of Astrobiology (1) Mycologist (1) Proceedings of the London Mathematical Society (1) The ANZIAM Journal (1) Applied Probability Trust (7) Weed Science Society of America (4) Canadian Mathematical Society (3) International Actuarial Association (IAA) (2) AMA Mexican Society of Microscopy MMS (1) Australian Association of Family Therapy Inc (1) Australian Mathematical Society Inc (1) European Association of Archaeologists (1) Institute and Faculty of Actuaries (1) International Society for Biosafety Research (1) MSC - Microscopical Society of Canada (1) MiMi / EMAS - European Microbeam Analysis Society (1) National Institute of Economic and Social Research (1) Nestle Foundation - enLINK (1) Nutrition Society (1) Society for Economic Measurement (SEM) (1) Cambridge Series in Chemical Engineering (2) The Archimedes Code: a dialogue between science, practice, design theory and systems engineering Yoram Reich Journal: Design Science / Volume 9 / 2023 Published online by Cambridge University Press: 27 January 2023, e2 Archimedes, the founder of statics and hydrostatics, in his mathematics and physics studies, created methods related to his inventions of new machines, for example, the method of mechanical theorems based on his lever invention. He also used the principles of decomposition and replication underlying his heat ray invention, and these two principles permeate his work. Analysis of Archimedes' work shows how he was perhaps the first to use methodically a strategy for solving diverse complex problems. In this article, we use the term Archimedes Code to encompass the way Archimedes approached problems including those two principles. Archimedes was perhaps the first design theorist and the first to think systematically about how to address design challenges. Furthermore, his work demonstrates the fundamental role of engineering practice in advancing science. The new insights regarding the Archimedes Code and its value in design practice may inspire both design researchers and practitioners. A decomposition for Lévy processes inspected at Poisson moments Onno Boxma, Michel Mandjes Journal: Journal of Applied Probability , First View Published online by Cambridge University Press: 15 November 2022, pp. 1-13 We consider a Lévy process Y(t) that is not continuously observed, but rather inspected at Poisson( $\omega$ ) moments only, over an exponentially distributed time $T_\beta$ with parameter $\beta$ . The focus lies on the analysis of the distribution of the running maximum at such inspection moments up to $T_\beta$ , denoted by $Y_{\beta,\omega}$ . Our main result is a decomposition: we derive a remarkable distributional equality that contains $Y_{\beta,\omega}$ as well as the running maximum process $\bar Y(t)$ at the exponentially distributed times $T_\beta$ and $T_{\beta+\omega}$ . Concretely, $\overline{Y}(T_\beta)$ can be written as the sum of two independent random variables that are distributed as $Y_{\beta,\omega}$ and $\overline{Y}(T_{\beta+\omega})$ . The distribution of $Y_{\beta,\omega}$ can be identified more explicitly in the two special cases of a spectrally positive and a spectrally negative Lévy process. As an illustrative example of the potential of our results, we show how to determine the asymptotic behavior of the bankruptcy probability in the Cramér–Lundberg insurance risk model. What factors drive gender differences in the body mass index? Evidence from Turkish adults Ebru Caglayan-Akay, Merve Ertok-Onurlu, Fulden Komuryakan Journal: Journal of Biosocial Science , First View Published online by Cambridge University Press: 05 May 2022, pp. 1-26 In recent years, studies show that obesity has become an important health condition, especially among adults. The first aim of this study is to examine socio-demographic and behavioural factors on body mass index distribution of male and female adults over 20 years old in Turkey. The second aim is to determine the body mass index disparity by gender and the socio-demographic and behavioural factors that might wider or narrow it. This study adopts unconditional quantile regression and decomposition methods, and the data set covers the Turkish Health Surveys for 2014, 2016, and 2019. The findings document that high level of body mass index are associated with being married, aging, and physical inactivity. Interestingly, employment status has different contributions on the body mass index of males and females. The results also claim a body mass index gap among males and females as a result of differences in some potential socio-demographic and behavioural factors, and the gap gets higher at the upper and lower quantiles of BMI distribution. This study may provide a clear understanding for policymakers on how to design efficacious obesity policies considering the differences in the effect of socio-demographic and behavioural factors on the distribution of body mass index across females and males. The results suggest that the Ministry of Health should specifically target different groups for males and females and should reduce the differences in socio-demographic and behavioural determinants between females and males to prevent and reduce obesity prevalence in Turkey. 4 - Measuring Market Efficiency Carlo Altomonte, Università Commerciale Luigi Bocconi, Milan, Filippo di Mauro Book: The Economics of Firm Productivity Published online: 14 April 2022 Print publication: 21 April 2022, pp 54-63 We present in this chapter the empirical techniques used to measure market efficiency and its implications, starting with the so-called OP gap and its dynamic version. We will then discuss the Foster productivity decomposition method and the Hsieh and Klenow techniques to measure allocative inefficiency. Market inefficiency is also related to concentration and market power. When competition in a market is reduced, aggregate productivity growth may decrease, reducing consumer welfare. Design without representation Sotirios D. Kotsopoulos Journal: AI EDAM / Volume 36 / 2022 Published online by Cambridge University Press: 09 February 2022, e12 Shapes are perceived unanalyzed, without rigid representation of their parts. They do not comply with standard symbolic knowledge representation criteria; they are treated and judged by appearance. Resolving the relationship of parts to parts and parts to wholes has a constructive role in perception and design. This paper presents a computational account of part–whole figuration in design. To this end, shape rules are used to show how a shape is seen, and shape decompositions having structures of topologies and Boolean algebras reveal alternative structures for parts. Four examples of shape computation are presented. Topologies demonstrate the relationships of wholes, parts, and subparts, in the computations enabling the comparison and relativization of structures, and lattice diagrams are used to present their order. Retrospectively, the topologies help to recall the generative history and establish computational continuity. When the parts are modified to recognize emergent squares locally, other emergent shapes are highlighted globally as the topology is re-adjusted. Two types of emergence are identified: local and global. Seeing the local parts modifies how we analyze the global whole, and thus, a local observation yields a global order. 2 - Mixed-Integer Programming from Part I - Background Christos T. Maravelias, Princeton University, New Jersey Book: Chemical Production Scheduling Print publication: 06 May 2021, pp 32-64 This chapter provides an overview of mixed-integer programming (MIP) modeling and solution methods. In Section 2.1, we present some preliminary concepts on optimization and mixed-integer programming. In Section 2.2, we discuss how binary variables can be used to model features commonly found in optimization problems. In Section 2.3, we present some basic MIP problems and models. Finally, in Section 2.4, we overview the basic approaches to solving MIP models and present some concepts regarding formulation tightness and decomposition methods. Finally, we discuss software tools for modeling and solving MIP models in Section 2.5. 12 - Solution Methods: Sequential Environments from Part IV - Special Topics Print publication: 06 May 2021, pp 289-317 The goal of the present chapter, as well as Chapter 13, is to illustrate how problem features can be exploited to develop more efficient models and/or specialized algorithms. We start, in the present chapter, with solution methods for problems in sequential environments. Specifically, we discuss four methods: (1) a decomposition approach, in Section 12.1; (2) preprocessing algorithms and tightening constraints, in Section 12.2; (3) a reformulation and tightening constraints based on time windows, in Section 12.3; and (4) a two-step algorithm, combining the advantages of discrete and continuous time models, in Section 12.4. While all presented methods can be applied to a wide range of problems, we present them for a subset of problems for the sake of brevity. Also, all methods can be applied to problems under different processing features, but to keep the presentation simple, we discuss problems with no shared utilities and no storage constraints. 13 - Climatic Impacts on Salt Marsh Vegetation from Part III - Marsh Response to Stress By Katrina L. Poppe, John M. Rybczyk Edited by Duncan M. FitzGerald, Boston University, Zoe J. Hughes, Boston University Book: Salt Marshes The salt marsh response to a changing climate may be more complex than that of either terrestrial or marine ecosystems because salt marshes exist at the interface of land and sea and both bring changes to the marsh. Climate change may exacerbate anthropogenic-related stresses that salt marsh plants are already experiencing, limiting their resilience (Keddy 2011). In this chapter we discuss major climate change impacts likely to affect salt marshes including temperature, sea level rise (SLR), salinity, CO2, freshwater flow, sediment, and nutrients, and consider how salt marsh plants respond to these impacts and potential interactions of these impacts. Specifically, we explore changes in plant productivity and decomposition rates, aboveground and belowground biomass, and stem density as they are central to understanding marsh responses on a larger scale, with implications for species composition, elevation change, nutrient cycling, carbon sequestration, food webs, and ultimately marsh survival. Although this chapter is focused on salt marshes, examples from tidal fresh and brackish marshes are also included to a limited extent where relevant. Synchronized Lévy queues Operations research and management science Offer Kella, Onno Boxma Journal: Journal of Applied Probability / Volume 57 / Issue 4 / December 2020 We consider a multivariate Lévy process where the first coordinate is a Lévy process with no negative jumps which is not a subordinator and the others are non-decreasing. We determine the Laplace–Stieltjes transform of the steady-state buffer content vector of an associated system of parallel queues. The special structure of this transform allows us to rewrite it as a product of joint Laplace–Stieltjes transforms. We are thus able to interpret the buffer content vector as a sum of independent random vectors. Decomposition of degenerate Gromov–Witten invariants Projective and enumerative geometry Families, fibrations Dan Abramovich, Qile Chen, Mark Gross, Bernd Siebert Journal: Compositio Mathematica / Volume 156 / Issue 10 / October 2020 We prove a decomposition formula of logarithmic Gromov–Witten invariants in a degeneration setting. A one-parameter log smooth family $X \longrightarrow B$ with singular fibre over $b_0\in B$ yields a family $\mathscr {M}(X/B,\beta ) \longrightarrow B$ of moduli stacks of stable logarithmic maps. We give a virtual decomposition of the fibre of this family over $b_0$ in terms of rigid tropical maps to the tropicalization of $X/B$. This generalizes one aspect of known results in the case that the fibre $X_{b_0}$ is a normal crossings union of two divisors. We exhibit our formulas in explicit examples. Thermal degradation kinetics of sepiolite Yüksel Sarıkaya, Müşerref Önal, Abdullah Devrim Pekdemir Journal: Clay Minerals / Volume 55 / Issue 1 / March 2020 Published online by Cambridge University Press: 13 March 2020, pp. 96-100 The kinetic parameters of the thermal degradation of sepiolite were evaluated with a new method based on thermal analysis data. Thermogravimetric/differential thermal analysis curves were recorded for the natural and preheated sepiolite samples in the temperature range 25–800°C for 4 h. The temperature-dependent height of the exothermic heat flow peak for the thermal decomposition of sepiolite located at ~850°C on the differential thermal analysis curve was taken as a kinetic variable for the thermal degradation. A thermal change coefficient was defined depending on this variable because this coefficient fit to the Arrhenius equation was assumed as a rate constant for the thermal degradation. The Arrhenius plot showed that the degradation occurs in three steps. Two of these are due to stepwise dehydration and the third originated from dehydroxylation of sepiolite. Three activation energies were obtained that increase with the increasing temperature interval of the steps. Products of Involutions of an Infinite-dimensional Vector Space Basic linear algebra Clément de Seguins Pazzis Journal: Canadian Journal of Mathematics / Volume 73 / Issue 1 / February 2021 Print publication: February 2021 We prove that every automorphism of an infinite-dimensional vector space over a field is the product of four involutions, a result that is optimal in the general case. We also characterize the automorphisms that are the product of three involutions. More generally, we study decompositions of automorphisms into three or four factors with prescribed split annihilating polynomials of degree $2$. Socio-economic inequality in unhealthy snacks consumption among adolescent students in Iran: a concentration index decomposition analysis Vahid Yazdi-Feyzabadi, Arash Rashidian, Mostafa Amini Rarani Journal: Public Health Nutrition / Volume 22 / Issue 12 / August 2019 Published online by Cambridge University Press: 14 June 2019, pp. 2179-2188 The present study aimed to assess and decompose the socio-economic inequality in unhealthy snacks consumption among adolescent students in Kerman, Iran. The data were obtained from a cross-sectional study. Principal component analysis was done to measure the socio-economic status (SES) of the adolescents' families and the normalized concentration index (NCI) was used to measure the inequality in unhealthy snacks consumption among adolescent students of different SES. The contributions of environmental and individual explanatory variables to inequality were assessed by decomposing the concentration index. Forty secondary schools of Kerman Province in Iran in 2015. Eighth-grade adolescent students (n 1320). The data of 1242 adolescent students were completed for the current study. Unhealthy snacks consumption was unequally distributed among adolescent students and was concentrated mainly among the high-SES adolescents (NCI = 0·179; 95 % CI 0·056, 0·119). The decomposition showed that higher SES (62 %) and receiving pocket money allowance (31 %), as environmental variables, had the highest positive contributions to the measured inequality in unhealthy snacks consumption. Taste and sensory perception (7 %) as well as cost sensitivity (5 %), as individual variables, followed them in terms of their contribution importance. It is highly suggested that both environmental and individual factors should be addressed at different settings including schools, families and suppliers of unhealthy snacks. These findings can help future health promotion strategies in Iran to tackle the observed inequality in unhealthy snacks consumption. The decline in China's fertility level: a decomposition analysis Quanbao Jiang, Shucai Yang, Shuzhuo Li, Marcus W. Feldman Journal: Journal of Biosocial Science / Volume 51 / Issue 6 / November 2019 Many factors have contributed to the decline in China's fertility level. Using China's population census data from 1990, 2000 and 2010, the present study investigates the factors causing the decline in China's fertility rate by decomposing changes in two fertility indices: the total fertility rate (TFR) and the net reproduction rate (NRR). The change in the TFR is decomposed into the change in the marital fertility rate (MFR) and the change in the proportion of married women (PMW). Four factors contribute to the change in the NRR. The following are the main findings. A drop in the MFR caused a decrease in the TFR and the NRR between 1989 and 2000. However, the change in MFR increased TFR and NRR between 2000 and 2010. Marriage postponement caused a decline in the fertility level between 1989 and 2000 as well as between 2000 and 2010. The effect of the MFR and marriage postponement varied with age and region and also between urban and rural areas. Quantifying Uncertainty from Mass-Peak Overlaps in Atom Probe Microscopy Andrew J. London Journal: Microscopy and Microanalysis / Volume 25 / Issue 2 / April 2019 Print publication: April 2019 There are many sources of random and systematic error in composition quantification by atom probe microscopy, often, however, only statistical error is reported. Significantly larger errors can occur from the misidentification of ions and overlaps or interferences of peaks in the mass spectrum. These overlaps can be solved using maximum likelihood estimation (MLE), improving the accuracy of the result, but with an unknown effect on the precision. An analytical expression for the uncertainty of the MLE solution is presented and it is demonstrated to be much more accurate than the existing methods. In one example, the commonly used error estimate was five times too small. Literature results containing overlaps most likely underestimate composition uncertainty because of the complexity of correctly dealing with stochastic effects and error propagation. The uncertainty depends on the amount of overlapped intensity, for example being ten times worse for the CO/Fe overlap than the Cr/Fe overlap. Using the methods described here, accurate estimation of error, and the minimization of this could be achieved, providing a key milestone in quantitative atom probe. Accurate estimation of the composition uncertainty in the presence of overlaps is crucial for planning experiments and scientific interpretation of the measurements. The Decline of British Manufacturing, 1973–2012: The Role of Total Factor Productivity Richard Harris, John Moffat Journal: National Institute Economic Review / Volume 247 / February 2019 Published online by Cambridge University Press: 01 January 2020, pp. R19-R31 This paper uses plant-level estimates of total factor productivity covering 40 years to examine what role, if any, productivity has played in the decline of output share and employment in British manufacturing. The results show that TFP growth in British manufacturing was negative between 1973 and 1982, marginally positive between 1982 and 1994 and strongly positive between 1994 and 2012. Poor TFP performance therefore does not appear to be the main cause of the decline of UK manufacturing. Productivity growth decompositions show that, in the latter period, the largest contributions to TFP growth come from foreign-owned plants, industries that are heavily involved in trade, and industries with high levels of intangible assets. Effects of soil temperature and tidal condition on variation in carbon dioxide flux from soil sediment in a subtropical mangrove forest Mitsutoshi Tomotsune, Shinpei Yoshitake, Yasuo Iimura, Morimaru Kida, Nobuhide Fujitake, Hiroshi Koizumi, Toshiyuki Ohtsuka Journal: Journal of Tropical Ecology / Volume 34 / Issue 4 / July 2018 Published online by Cambridge University Press: 26 July 2018, pp. 268-275 The variation in CO2 flux from the forest floor is important in understanding the role of mangrove forests as a carbon sink. To clarify the effects of soil temperature and tidal conditions on variation in CO2 flux, sediment–atmosphere CO2 fluxes were measured between June 2012 and May 2013. We used the closed chamber method for two plots, with a 0.5 m difference in elevation (B, high elevation; R-B, low elevation), in a mangrove forest in south-western Japan. CO2 fluxes were highest in the warm season and showed a weak positive correlation with soil temperature in both forests. Estimated monthly CO2 flux showed moderate seasonal variation in accordance with the exposure duration of the soil surface under tidal fluctuation. Additionally, measured CO2 flux and soil temperature were slightly higher in the R-B plot than the B plot, although estimated annual CO2 flux was higher in the B plot than the R-B plot due to different exposure durations. These results suggest that variation in the exposure duration of the forest floor, which changes seasonally and microgeographically, is important in evaluating the annual CO2 flux at a local scale and understanding the role of mangrove ecosystems as regulators of atmospheric CO2. Fourier transform infrared spectroscopy study of acid birnessites before and after Pb2+ adsorption Wei Zhao, Fan Liu, Xionghan Feng, Wenfeng Tan, Guohong Qiu, Xiuhua Chen Journal: Clay Minerals / Volume 47 / Issue 2 / June 2012 To provide fundamental knowledge for studying the relative content of vacant sites and exploring the mechanism of interaction between Pb2+ and birnessite, Fourier transform infrared spectroscopy (FTIR) of birnessites with different Mn average oxidation states (AOS) before and after Pb2+ adsorption were investigated. The number of absorption bands of FTIR spectra was determined by using the second derivatives of the original spectra. The band at 899–920 cm–1 was assigned to the bending vibration of -OH located at vacancies. The bands at 1059–1070, 1115–1124 and 1165–1171 cm–1 could be attributed to the vibrations of Mn(III)-OH in MnO6 layers, and the intensities of these bands increased with decreasing Mn AOS. The bands at 990 and 1023–1027 cm–1 were ascribed to the vibrations of Mn(III)-OH in the interlayers. Mn(III) in MnO6 layers partially migrated to interlayers during Pb2+ adsorption, which led to an increased intensity of the band at 990 cm–1. The band at 564–567cm–1 was assigned to the vibration of Mn-O located at vacancies. This band could split by coupling of vibrations due to Pb2+ and/or Mn2+ adsorbed at vacant sites. The large distance between the band at 610–626 cm–1 and that at 638–659 cm–1 might reflect small Mn(III) ions located in Mn(III)-rich rows. Composition variation of illite-vermiculitesmectite mixed-layer minerals in a bentonite bed from Charente (France) A. Meunier, B. Lanson, B. Velde Journal: Clay Minerals / Volume 39 / Issue 3 / September 2004 Mineralogical and chemical variations were studied in the upper half of a 1 m thick discontinuous bentonite bed interlaminated in the Lower Cenomanian sedimentary formations of the northern Aquitaine basin (France). X-ray diffraction patterns obtained from the <2 mm fraction in the Ca and K-saturated states were decomposed and compared to those calculated from decomposition parameters. They revealed the presence of two highly expandable illite-expandable (I-Exp) mixedlayer minerals (MLMs). The relative proportions of the two MLMs evolve steadily with depth leading to the decrease of the cation exchange capacity and of the (Na + Ca) content towards the centre of the bentonite bed. However, the system is essentially isochemical and Mg, Al, Si, K and Fe are roughly constant in the bulk samples. It is thought that the mineralogical zonation results from the initial stages of the smectite formation in an ash layer. In the Ca-saturated state, the expandable component of these MLMs was for the most part homogeneous with the presence of 2 sheets of ethylene glycol molecules in the interlayer. However, the heterogeneous hydration behaviour of these expandable layers was enhanced by the K-saturation test. From this test, the presence of three layer types with contrasting layer charge was evidenced from their contrasting swelling abilities. The C12-alkylammonium saturation test applied to samples in which the octahedral charge had previously been neutralized (Hofmann-Klemen treatment) showed that the tetrahedral charge is located on specific layers. These layers are responsible for the heterogeneous hydration behaviour. Low-charge smectite layers are mostly octahedrally substituted, whereas for intermediate- and high-charge layers this montmorillonitic charge is complemented by additional tetrahedral substitutions (0.30 and 0.35–0.40 charge per O10(OH)2, respectively). Mösbauer spectroscopic study of the decomposition mechanism of ankerite in CO2 atmosphere A. E. Milodowski, B. A. Goodman, D. J. Morgan Journal: Mineralogical Magazine / Volume 53 / Issue 372 / September 1989 Mössbauer spectroscopy has been used to identify the iron-containing products that are formed during the thermal decomposition of ankerite in a CO2 atmosphere. The decomposition takes place in three stages and evidence is produced to show that the first stage involves decomposition of ankerite to yield a periclase-wustite solid solution, (Mg,Fe)O, along with calcite and CO2, the periclase-wustite then reacting with CO2 to produce magnesioferrite (MgFe2O4) and CO. In the second stage the magnesioferrite and calcite react to produce periclase and dicalcium ferrite. The third stage does not involve reaction of Fe-containing phases and corresponds to the decomposition of calcite to CaO.	CommonCrawl
Trigonometric Fourier Series – Definition and Explanation Signals and SystemsElectronics & ElectricalDigital Electronics A periodic signal can be represented over a certain interval of time in terms of the linear combination of orthogonal functions, if these orthogonal functions are trigonometric functions, then the Fourier series representation is known as trigonometric Fourier series. Consider a sinusoidal signal $x(t)=A\:sin\:\omega_{0}t$ which is periodic with time period $T$ such that $T=2\pi/ \omega_{0}$. If the frequencies of two sinusoids are integral multiples of a fundamental frequency $(\omega_{0})$, then the sum of these two sinusoids is also periodic. We can prove that a signal $x(t)$ that is a sum of sine and cosine functions whose frequencies are integral multiples of the fundamental frequency $(\omega_{0})$, is a periodic signal. Let the signal $x(t)$ is given by, $$\mathrm{x(t)=a_{0}+a_{1}\:cos\:\omega_{0}t+a_{2}\:cos\:2\omega_{0}t+a_{3}\:cos\:3\omega_{0}t+....+a_{k}\:cos\:k\omega_{0}t}$$ $$\mathrm{\:\:\:\:\:\:\:\:\:\:+b_{1}\:sin\:\omega_{0}t+b_{2}\:sin\:2\omega_{0}t+b_{3}\:sin\:3\omega_{0}t+...+b_{k}\:sin\:k\omega_{0}t}$$ $$\mathrm{\Rightarrow\:x(t)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos\:n\omega_{0}t+b_{n}\:sin\:n\omega_{0}t… (1)}$$ Where, $a_{0},a_{1},a_{2}....a_{k}$ and $b_{0},b_{1},b_{2}....b_{k}$ are the constants and $\omega_{0}$ is the fundamental frequency. Again, if a signal $x(t)$ is a periodic signal, then it must satisfy the following condition − $$\mathrm{x(t)=x(t+T);\:\:for\:all\:t}$$ $$\mathrm{\Rightarrow\:x(t+T)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos\:n\omega_{0}(t+T)+b_{n}\:sin\:n\omega_{0}(t+T)}$$ $$\mathrm{\because\:Time\:period,T=\left ( \frac{2\pi}{\omega_{0}}\right )}$$ $$\mathrm{\Rightarrow\:x(t+T)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos\:n\omega_{0}(t+\frac{2\pi}{\omega_{0}})+b_{n}\:sin\:n\omega_{0}(t+\frac{2\pi}{\omega_{0}})}$$ $$\mathrm{\Rightarrow\:x(t+T)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos(n\omega_{0} t+2n\pi)+b_{n}\:sin(n\omega_{0} t+2n\pi)}$$ $$\mathrm{\because\:cos(2n\pi+\theta )=cos\:\theta \:\:and\:\:sin(2n\pi+\theta )=sin\:\theta}$$ Using these trigonometric identities, we get, $$\mathrm{\Rightarrow\:x(t+T)=a_{0}+\sum_{n=1}^{k}a_{n}\:cos(n\omega_{0} t)+b_{n}\:sin(n\omega_{0} t)=x(t)… (2)}$$ From equation (2) it is clear that the signal $x(t)$ which is a sum of sine and cosine functions of frequencies 0,$\omega_{0},2\omega_{0},...k\omega_{0}$ is a periodic signal with a time period T. If in the expression of $x(t),k\rightarrow \infty$ then we can obtain the Fourier series representation of any periodic signal $x(t)$. Therefore, any periodic signal can be represented as an infinite sum of sine and cosine functions which themselves are periodic signals of angular frequencies 0,$\omega_{0},2\omega_{0},...k\omega_{0}$ . This set of harmonically related sine and cosine functions form a complete set of orthogonal functions over the time interval $t$ to $(t+T)$ Hence, the trigonometric form of Fourier series can be defined as under − The infinite series of sine and cosine terms of frequencies 0,$\omega_{0},2\omega_{0},...k\omega_{0}$ is called the trigonometric form of Fourier series and can be represented as, $$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\: n\omega_{0} t+b_{n}\:sin\:n\omega_{0} t… (3)}$$ Where, $a_{0},a_{n}$ and $b_{n}$ are called trigonometric Fourier series confidents. $$\mathrm{a_{0}=\frac{1}{T} \int_{t_{0}}^{(t_{0}+T)}x(t)\:dt… (4)}$$ $$\mathrm{a_{n}=\frac{2}{T} \int_{t_{0}}^{(t_{0}+T)}x(t)\:cos\:n\omega_{0}t\:dt… (5)}$$ $$\mathrm{b_{n}=\frac{2}{T} \int_{t_{0}}^{(t_{0}+T)}x(t)\:sin\:n\omega_{0}t\:dt… (6)}$$ The coefficient $a_{0}$ is known as the DC component. $(a_{1}\:cos\:\omega_{0}t+b_{1}\:sin\:\omega_{0}t)$ is called the first harmonic term. $(a_{2}\:cos\:\omega_{0}t+b_{2}\:sin\:2\omega_{0}t)$is called the second harmonic term. Similarly, $(a_{n}\:cos\:n\omega_{0}t+b_{n}\:sin\:n\omega_{0}t)$ is called the nth harmonic term. Numerical Example Find the trigonometric Fourier series for the waveform shown below. As we can see the given waveform is periodic with a time period $T= 2\pi$. Mathematically, the given waveform can be described as, $$\mathrm{x(t)=\begin{cases}(\frac{A}{\pi})t & for\:0 ≤ t ≤\:\pi\0 & for\:\pi≤ t ≤2\pi\end{cases}}$$ $$\mathrm{t_{0}=0\:\:and\:\:(t_{0}+T)= 2\pi}$$ Then, the fundamental frequency of the given function is, $$\mathrm{\omega_{0}=\frac{2\pi}{T}=\frac{2\pi}{2\pi}=1}$$ Thus, the coefficient $a_{0}$ is given by, $$\mathrm{a_{0}=\frac{1}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)dt}$$ $$\mathrm{\Rightarrow\:a_{0}=\frac{1}{2\pi}\int_{0}^{2\pi}x(t)\:dt=\frac{1}{2\pi}\int_{0}^{\pi}(\frac{A}{\pi})t\:dt+\frac{1}{2\pi}\int_{0}^{2\pi}0\:dt=\frac{A}{2\pi^{2}}\left [ \frac{t^{2}}{2}\right ]_{0}^{\pi}=\frac{A}{4}}$$ The coefficient $a_{n}$ is given by, $$\mathrm{a_{n}=\frac{2}{T} \int_{t_{0}}^{(t_{0}+T)}x(t)cos\:n\omega_{0}t\:\:dt}$$ $$\mathrm{\Rightarrow\:a_{n}=\frac{2}{2\pi} \int_{0}^{\pi}(\frac{A}{\pi})t\:cos\:nt\:dt=\frac{A}{\pi^{2}}\int_{0}^{\pi}t\:cos\:nt\:dt}$$ By solving the above integration, we get, $$\mathrm{\Rightarrow\:a_{n}=\frac{A}{\pi^{2}n^{2}}[cos\:n\pi]}$$ $$\mathrm{\therefore\:a_{n}=\begin{cases}-(\frac{2A}{\pi^{2}n^{2}}) & for\:odd\:n\0 & for\:even \:n\end{cases}}$$ Similarly, the coefficient $b_{n}$ is given by, $$\mathrm{b_{n}=\frac{2}{T}\int_{t_{0}}^{(t_{0}+T)}x(t)sin\:n\omega_{0}t\:dt}$$ $$\mathrm{\Rightarrow\:b_{n}=\frac{2}{2\pi}\int_{0}^{\pi}(\frac{A}{\pi})t\:sin\:nt\:dt=\frac{A}{\pi^{2}}\int_{0}^{\pi}t\:sin\:nt\:dt}$$ On solving this integration, we have, $$\mathrm{b_{n}=\frac{A}{\pi^{2}}\left [-\frac{\pi\:cos\:n\pi}{n} +\left (\frac{sin\:nt}{n^{2}} \right )_{0}^{\pi} \right ]}$$ $$\mathrm{\Rightarrow\:b_{n}=-\frac{A}{n\pi}cos\:n\pi=\frac{A}{n\pi}(-1)^{n+1}}$$ $$\mathrm{\therefore\:b_{n}=\begin{cases}(\frac{A}{n\pi}) & for\:odd\:n\(-\frac{A}{n\pi}) & for\:even\:n\end{cases}}$$ Therefore, the trigonometric Fourier series is, $$\mathrm{x(t)=a_{0}+\sum_{n=1}^{\infty}a_{n}\:cos\:n\:\omega_{0}t+b_{n}\:sin\:n\omega_{0}t}$$ $$\mathrm{\Rightarrow\:x(t)=\frac{A}{4}-\frac{2A}{\pi^{2}}\sum_{n=odd}^{\infty}\frac{cos\:nt}{n^{2}}+\frac{A}{\pi}\sum_{n=1}^{\infty}(-1)^{n+1}\cdot \frac{sin\:nt}{n}}$$ Manish Kumar Saini Fourier Cosine Series – Explanation and Examples Relation between Trigonometric & Exponential Fourier Series Expressions for the Trigonometric Fourier Series Coefficients Difference between Fourier Series and Fourier Transform Fourier Series – Representation and Properties Derivation of Fourier Transform from Fourier Series Time Series Analysis: Definition and Components Series-Parallel Circuit: Definition and Examples GIBBS Phenomenon for Fourier Series Linearity and Conjugation Property of Continuous-Time Fourier Series Signals & Systems – Complex Exponential Fourier Series Expression for Exponential Fourier Series Coefficients Fourier Series Representation of Periodic Signals Time Differentiation and Integration Properties of Continuous-Time Fourier Series Convolution Property of Continuous-Time Fourier Series	CommonCrawl
Franks' lemma for $\mathbf{C}^2$-Mañé perturbations of Riemannian metrics and applications to persistence JMD Home This Volume On small gaps in the length spectrum 2016, 10: 353-377. doi: 10.3934/jmd.2016.10.353 Typical dynamics of plane rational maps with equal degrees Jeffrey Diller 1, , Han Liu 1, and Roland K. W. Roeder 2, Department of Mathematics, University of Notre Dame, Notre Dame, IN 46556, United States, United States IUPUI Department of Mathematical Sciences, LD Building, Room 270, 402 North Blackford Street, Indianapolis, Indiana 46202, United States Received January 2016 Revised April 2016 Published August 2016 Let $f:\mathbb{CP}^2⇢\mathbb{CP}^2$ be a rational map with algebraic and topological degrees both equal to $d\geq 2$. Little is known in general about the ergodic properties of such maps. We show here, however, that for an open set of automorphisms $T:\mathbb{CP}^2\to\mathbb{CP}^2$, the perturbed map $T\circ f$ admits exactly two ergodic measures of maximal entropy $\log d$, one of saddle type and one of repelling type. Neither measure is supported in an algebraic curve, and $f_T$ is 'fully two dimensional' in the sense that it does not preserve any singular holomorphic foliation of $\mathbb{C}\mathbb{P}^2$. In fact, absence of an invariant foliation extends to all $T$ outside a countable union of algebraic subsets of $Aut(\mathbb{P}^2)$. Finally, we illustrate all of our results in a more concrete particular instance connected with a two dimensional version of the well-known quadratic Chebyshev map. Keywords: Rational maps, dynamical degrees., ergodic properties. Mathematics Subject Classification: Primary: 37F10; Secondary: 32H5. Citation: Jeffrey Diller, Han Liu, Roland K. W. Roeder. Typical dynamics of plane rational maps with equal degrees. 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Fourier (Grenoble), 64 (2014), 645-680. doi: 10.5802/aif.2861. Google Scholar Almut Burchard, Gregory R. Chambers, Anne Dranovski. Ergodic properties of folding maps on spheres. Discrete & Continuous Dynamical Systems, 2017, 37 (3) : 1183-1200. doi: 10.3934/dcds.2017049 Mariko Arisawa, Hitoshi Ishii. Some properties of ergodic attractors for controlled dynamical systems. Discrete & Continuous Dynamical Systems, 1998, 4 (1) : 43-54. doi: 10.3934/dcds.1998.4.43 Guizhen Cui, Yan Gao. Wandering continua for rational maps. Discrete & Continuous Dynamical Systems, 2016, 36 (3) : 1321-1329. doi: 10.3934/dcds.2016.36.1321 Cezar Joiţa, William O. Nowell, Pantelimon Stănică. Chaotic dynamics of some rational maps. Discrete & Continuous Dynamical Systems, 2005, 12 (2) : 363-375. doi: 10.3934/dcds.2005.12.363 Eriko Hironaka, Sarah Koch. A disconnected deformation space of rational maps. Journal of Modern Dynamics, 2017, 11: 409-423. doi: 10.3934/jmd.2017016 Richard Sharp, Anastasios Stylianou. Statistics of multipliers for hyperbolic rational maps. Discrete & Continuous Dynamical Systems, 2021 doi: 10.3934/dcds.2021153 Karma Dajani, Cor Kraaikamp, Pierre Liardet. Ergodic properties of signed binary expansions. Discrete & Continuous Dynamical Systems, 2006, 15 (1) : 87-119. doi: 10.3934/dcds.2006.15.87 Gerhard Knieper, Norbert Peyerimhoff. Ergodic properties of isoperimetric domains in spheres. Journal of Modern Dynamics, 2008, 2 (2) : 339-358. doi: 10.3934/jmd.2008.2.339 Russell Lodge, Sabyasachi Mukherjee. Invisible tricorns in real slices of rational maps. Discrete & Continuous Dynamical Systems, 2021, 41 (4) : 1755-1797. doi: 10.3934/dcds.2020340 Yan Gao, Jinsong Zeng, Suo Zhao. A characterization of Sierpiński carpet rational maps. Discrete & Continuous Dynamical Systems, 2017, 37 (9) : 5049-5063. doi: 10.3934/dcds.2017218 Huaibin Li. An equivalent characterization of the summability condition for rational maps. Discrete & Continuous Dynamical Systems, 2013, 33 (10) : 4567-4578. doi: 10.3934/dcds.2013.33.4567 Yan Gao, Luxian Yang, Jinsong Zeng. Subhyperbolic rational maps on boundaries of hyperbolic components. Discrete & Continuous Dynamical Systems, 2022, 42 (1) : 319-326. doi: 10.3934/dcds.2021118 Cristina Lizana, Vilton Pinheiro, Paulo Varandas. Contribution to the ergodic theory of robustly transitive maps. Discrete & Continuous Dynamical Systems, 2015, 35 (1) : 353-365. doi: 10.3934/dcds.2015.35.353 Rafael Alcaraz Barrera. Topological and ergodic properties of symmetric sub-shifts. Discrete & Continuous Dynamical Systems, 2014, 34 (11) : 4459-4486. doi: 10.3934/dcds.2014.34.4459 Francesco Cellarosi, Ilya Vinogradov. Ergodic properties of $k$-free integers in number fields. Journal of Modern Dynamics, 2013, 7 (3) : 461-488. doi: 10.3934/jmd.2013.7.461 Vadim S. Anishchenko, Tatjana E. Vadivasova, Galina I. Strelkova, George A. Okrokvertskhov. Statistical properties of dynamical chaos. 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About AEM The Alternative Sigma Factor σB and the Virulence Gene Regulator PrfA Both Regulate Transcription of Listeria monocytogenes Internalins Patrick McGann, Martin Wiedmann, Kathryn J. Boor Patrick McGann Department of Food Science, Cornell University, Ithaca, New York 14853 Martin Wiedmann Kathryn J. Boor For correspondence: [email protected] Some Listeria monocytogenes internalins are recognized as contributing to invasion of mammalian tissue culture cells. While PrfA is well established as a positive regulator of L. monocytogenes virulence gene expression, the stress-responsive σB has been recognized only recently as contributing to expression of virulence genes, including some that encode internalins. To measure the relative contributions of PrfA and σB to internalin gene transcription, we used reverse transcription-PCR to quantify transcript levels for eight internalin genes (inlA, inlB, inlC, inlC2, inlD, inlE, inlF, and inlG) in L. monocytogenes 10403S and in isogenic ΔprfA, ΔsigB, and ΔsigB ΔprfA strains. Strains were grown under defined conditions to produce (i) active PrfA, (ii) active σB and active PrfA, (iii) inactive PrfA, and (iv) active σB and inactive PrfA. Under the conditions tested, σB and PrfA contributed differentially to the expression of the various internalins such that (i) both σB and PrfA contributed to inlA and inlB transcription, (ii) only PrfA contributed to inlC transcription, (iii) only σB contributed to inlC2 and inlD transcription, and (iv) neither σB nor PrfA contributed to inlF and inlG transcription. inlE transcript levels were undetectable. The important role for σB in regulating expression of L. monocytogenes internalins suggests that exposure of this organism to environmental stress conditions, such as those encountered in the gastrointestinal tract, may activate internalin transcription. Interplay between σB and PrfA also appears to be critical for regulating transcription of some virulence genes, including inlA, inlB, and prfA. The gram-positive bacterium Listeria monocytogenes is recognized as an important human food-borne pathogen, causing manifestations ranging from mild febrile gastroenteritis to severe invasive infections. High mortality rates are associated with systemic infections, which occur predominantly among pregnant women, the immunocompromised, and the elderly (19, 24). In animals, L. monocytogenes infections can result in a variety of manifestations, including abortion, encephalitis, septicemia, and, less commonly, keratoconjunctivitis and mastitis (43). A variety of L. monocytogenes surface proteins, including several members of the internalin family (11), are important for facilitating interactions between this pathogen and mammalian host cells. For example, internalin A (InlA) promotes invasion of human nonphagocytic cells that express the host cell receptor E-cadherin, such as the Caco-2 epithelial cell line (45). InlB mediates entry into several host cell types, including hepatocytes and several endothelial and epithelial cell lines of various human and animal origins, including HepG-2 (human hepatocyte), TIB73 (mouse hepatocyte), HUVEC (human endothelial), and Vero (African green monkey epithelial) cells (15, 29, 40, 50). InlB facilitates invasion of mammalian cells by interacting with cellular receptors, including hepatocyte growth factor receptor (Met) and other host cell components, including gC1q-R (9, 30, 31). Interactions between host-cell receptors and InlA or InlB are species specific. To illustrate, InlA recognizes human and guinea pig E-cadherin, but not mouse or rat intestinal E-cadherin, due to a single amino acid substitution in the binding site of mouse and rat E-cadherin (38). Studies with human trophoblastic cell lines suggest that InlA may contribute to L. monocytogenes targeting and crossing of the human maternofetal barrier (39); however, no clear role has been established for InlA in the crossing of the guinea pig maternofetal barrier (1). InlB also displays species specificity. Although both guinea pig and rabbit cell lines express Met and gC1q-R, neither responds to InlB. However, following transfection with the human Met gene, both guinea pig and rabbit cells support InlB-dependent entry (34). Transcription of inlC, which encodes the secreted protein InlC, is strongly induced when L. monocytogenes is in the host cell cytoplasm (18). In the absence of InlB, InlC and InlGHE appear to be required for InlA-dependent invasion of Caco-2 cells (5). The specific functions of the proteins encoded by other members of the internalin gene family (inlC2, inlD, inlE, inlF, inlG, and inlH) remain unclear. Transcription of many confirmed and putative L. monocytogenes virulence genes is activated, at least in part, by positive regulatory factor A (PrfA) (for a recent review, see reference 37). Among the internalin genes, inlA, inlB, and inlC are at least partially regulated by PrfA (17, 18, 44). Previous reports have shown that PrfA can be present in two functional states, weakly active and highly active, which are influenced by environmental stimuli, including temperature, growth in charcoal, and the availability of readily metabolized sugars, such as cellobiose (4, 25, 28, 41, 46, 53). PrfA is highly active in intracellular L. monocytogenes when mammalian host cells are grown at 37°C (22), as well as when cells are grown in media supplemented with activated charcoal (54). On the other hand, PrfA is present in its inactive form during L. monocytogenes growth in the presence of cellobiose and other easily fermentable sugars (3, 4, 46, 53). While PrfA is well recognized as an important activator of L. monocytogenes virulence gene expression, emerging evidence indicates that the stress-responsive alternative sigma factor, σB, encoded by sigB, also contributes to transcription of at least some L. monocytogenes virulence genes, in addition to regulating expression of a stress regulon of at least 50 genes (2, 32, 35, 59). Specifically, microarray studies using an L. monocytogenes sigB null mutant showed that transcription of the internalin genes inlA, inlB, inlC2, inlD, and inlE is at least partially σB dependent (32); σB-dependent transcription of inlA and inlB has also been confirmed by quantitative reverse transcription-PCR (qRT-PCR) (36). The objective of this study was to measure the relative contributions of PrfA and σB to internalin gene transcription under conditions that yield (i) the active form of PrfA, (ii) the active σB and the active form of PrfA, (iii) the inactive form of PrfA, and (iv) the active σB and the inactive form of PrfA. We used qRT-PCR assays to measure mRNA transcript levels for eight internalin genes (inlA, inlB, inlC, inlC2, inlD, inlE, inlF, and inlG), two PrfA-dependent genes (prfA and plcA), two σB-dependent genes (opuCA and gadA), and two housekeeping genes (rpoB and gap) in ΔprfA and ΔsigB strains as well as in a ΔprfA ΔsigB double mutant strain grown under each of the four defined conditions. Our results, which demonstrate that internalin genes group into regulons with distinct expression patterns, provide novel insight into the regulation of L. monocytogenes internalin gene expression. Bacterial strains.L. monocytogenes serotype 1/2a strain 10403S (6) and three isogenic in-frame null mutants, including a ΔsigB mutant (FSL A1-254 [63]), a ΔprfA strain (FSL B2-046) with a 339-bp in-frame deletion in prfA previously described by Wong and Freitag (64), and a ΔsigB ΔprfA mutant (FSL B2-068 [this study]) were used. The ΔsigB ΔprfA double mutant was generated by cloning a PCR-amplified allele with an in-frame prfA deletion from L. monocytogenes strain FSL B2-046 into the Escherichia coli-L. monocytogenes shuttle vector pKSV7. The prfA mutant allele was then introduced into the L. monocytogenes ΔsigB strain by allelic-exchange mutagenesis as previously described (12). The final double mutant was confirmed by PCR, followed by sequencing of the mutant prfA allele. Growth conditions and cell collection for RNA isolation.Bacterial cells for RNA collection were grown at 37°C with shaking (200 rpm) in brain heart infusion broth (BHI; Difco Laboratories, MD). For exposure to different environmental conditions, L. monocytogenes cells that had been grown in BHI to early log phase (defined as an optical density at 600 nm of 0.4) were subsequently incubated for 120 min at 37°C in (i) BHI with 0.2% charcoal (BHI-charcoal), (ii) BHI-charcoal with NaCl added (0.3 M final concentration) for the final 10 min (BHI-charcoal-NaCl), (iii) BHI with 25 mM cellobiose (BHI-cellobiose), and (iv) BHI-cellobiose with NaCl added (0.3 M final concentration) for the final 10 min (BHI-cellobiose-NaCl). Following incubation, 2 volumes of RNAprotect (QIAGEN Inc., Valencia, CA) were added to each culture. After 5 min, cells were collected for subsequent RNA isolation by centrifugation at 5,000 × g for 5 min. Optimization of growth conditions for active or inactive PrfA.To determine specific growth conditions that yield maximum PrfA activity in broth-grown L. monocytogenes, qRT-PCR was used to measure transcript levels for the PrfA-dependent gene, plcA, using mRNA obtained from L. monocytogenes 10403S grown in BHI as well as 10403S under conditions reported to induce (growth in 0.2% charcoal) or repress (growth in 25 mM cellobiose) PrfA activity. Specifically, early-log-phase L. monocytogenes 10403S cells were inoculated into BHI, BHI-charcoal, or BHI-cellobiose and subsequently incubated at 37°C with shaking. Normalized plcA transcript levels (averages of two independent experiments; see below) for bacteria grown in BHI-charcoal for 5 min, 30 min, 120 min, and overnight were 0.11, 0.22, 1.14, and 0.72, respectively; these plcA transcript levels were over 1 log higher than the corresponding transcript levels (averages of two independent experiments) for cells grown in BHI alone. Normalized plcA transcript levels for cells grown in BHI-cellobiose for 120 and 240 min were −2.25 and −1.37, respectively. While plcA transcript levels after growth in BHI-cellobiose for 120 min were over one-half log lower than corresponding plcA transcript levels for cells grown in BHI alone, plcA transcript levels after 240 min growth in BHI-cellobiose were similar to corresponding plcA transcript levels for cells grown in BHI alone. Based on these data, growth for 120 min at 37°C in BHI-charcoal or in BHI-cellobiose was selected as a condition that yields L. monocytogenes cells with high or low levels of PrfA activity, respectively. Optimization of growth conditions for increased σB activity.To define conditions that yield high σB activity in the presence of either the active or the inactive form of PrfA, qRT-PCR was used to measure transcript levels for the σB-dependent genes opuCA and gadA and the PrfA-dependent gene plcA in the presence of 0.3 M NaCl, as exposure of log-phase L. monocytogenes cells to 0.3 M NaCl in BHI for 10 min has been shown to produce high levels of σB activity (59). qRT-PCR was performed on RNA extracted from early-log-phase L. monocytogenes 10403S cells that were grown for (i) an additional 120 min in BHI with 0.3 M NaCl, (ii) an additional 120 min in BHI with 0.3 M NaCl added for the final 10 min, (iii) an additional 120 min in BHI with 0.2% charcoal and 0.3 M NaCl, (iv) an additional 120 min in BHI with 0.2% charcoal and 0.3 M NaCl added for the final 10 min, (v) an additional 120 min in BHI with 25 mM cellobiose and 0.3 M NaCl, and (vi) an additional 120 min in BHI with 25 mM cellobiose and 0.3 M NaCl added for the final 10 min. All experiments were performed on two independent RNA preparations. Transcript levels for the σB-dependent genes opuCA and gadA were similar under all six conditions, indicating that exposure to NaCl for 10 min and exposure for 120 min provided similar levels of σB activity (see Fig. S1 in the supplemental material). plcA transcript levels for cells grown in BHI-charcoal with 10 min or 120 min of exposure to NaCl were up to 1 log lower than plcA transcript levels for cells grown in BHI-charcoal without exposure to NaCl. On the other hand, plcA transcript levels for cells grown in BHI-cellobiose without exposure to NaCl were similar to transcript levels for cells grown in BHI-cellobiose with NaCl exposure for either 10 or 120 min (see Fig S2b in the supplemental material). Based on these data, we chose growth of log-phase L. monocytogenes for 120 min in BHI with 25 mM cellobiose with exposure to 0.3 M NaCl for the final 10 min as a condition that provides a high level of σB activity and minimal PrfA activity. Growth of log-phase L. monocytogenes cells for 120 min in BHI with 0.2% charcoal with exposure to 0.3 M NaCl for the final 10 min was chosen as a condition that provides high levels of σB activity and intermediate PrfA activity (as salt exposure appears to repress plcA transcription [see Fig. S2b in the supplemental material]). No in vitro conditions tested allowed maximal activity of both σB and PrfA. All subsequent experiments were performed with cells grown in the presence of charcoal or cellobiose with or without a final 10 min of exposure to 0.3 M NaCl to provide mRNA transcripts under conditions generating either active or inactive PrfA in the presence or absence of active σB and salt. Total RNA isolation.Total RNA was purified from bacterial cells using the RNeasy Midi kit (QIAGEN Inc.) as described previously (60). For cultures exposed to charcoal, an additional RNA clean-up step using the protocol recommended by QIAGEN was performed prior to DNase treatment. The final RNA pellet was resuspended in RNase-free water (QIAGEN). Total nucleic acid concentration and purity were estimated using absorbance readings (260/280 nm) on an ND-1000 spectrophotometer (NanoDrop Technologies, Wilmington, DE). qRT-PCR.Absolute quantification of transcript levels for eight internalin genes (inlA, inlB, inlC, inlC2, inlD, inlE, inlF, and inlG), prfA, the PrfA-dependent gene plcA, and two σB-dependent genes (gadA and opuCA) as well as two housekeeping genes (rpoB and gap) was performed using TaqMan primer and probes (see Table S1 in the supplemental material) and the ABI Prism 7000 sequence detection system (Applied Biosystems, Foster City, CA) essentially as previously described (14, 33). While probes and primers for prfA, plcA, gadA, opuCA, rpoB, gap, inlA, and inlB have been reported previously (32, 33, 35, 59, 60), for this study, TaqMan primer and probes for six internalin genes (inlC, inlC2, inlD, inlE, inlF, and inlG) were designed with Primer Select (Applied Biosystems) based on internalin gene sequence data for L. monocytogenes 10403S and 18 other lineage II L. monocytogenes strains sequenced in our laboratory (61; see Table S1 in the supplemental material). Primers and probes were synthesized by IDT Technologies (Coralville, IA) and Applied Biosystems, respectively. In preliminary experiments, all qRT-PCR primer/probe sets were able to reproducibly detect as few as 60 DNA copies. Immunogold electron microscopy.Polyclonal anti-InlB antibodies were kindly provided by Paul Leonard, Dublin City University, Ireland. Goat anti-mouse immunoglobulin G labeled with 10-nm gold particles, bovine serum albumin (BSA) background-suppressing reagent, goat normal serum, and cold-water-fish-skin gelatin manufactured by Aurion (Wageningen, The Netherlands) were supplied by Electron Microscopy Services (Hatfield, PA). All other reagents were supplied by Sigma-Aldrich Inc. (St. Louis, MO). To prepare bacterial cells for immunogold labeling, the L. monocytogenes 10403S parent strain as well as the isogenic ΔsigB, ΔprfA, and ΔinlB strains were initially grown to early logarithmic phase (optical density at 600 nm of 0.4) in 5 ml BHI at 37°C with shaking. A 2-ml aliquot of each culture was then added to 2 ml of fresh BHI followed by incubation at 37°C for a further 2 h with addition of NaCl (at a final concentration of 0.3 M) for the final 10 min. Cells were then harvested by centrifugation and resuspended in 500 μl of phosphate-buffered saline (PBS). Immunogold labeling of whole bacterial cells was performed as described by the manufacturer (Aurion) with minor modifications. Briefly, 30 μl of suspended cells was allowed to adhere to Formvar-coated nickel grids for 45 min. The grids were washed in blocking buffer (PBS, 5% BSA, 5% normal goat serum, 0.1% cold-water-fish-skin gelatin) for 15 min, followed by two 5-min washes in incubation buffer (PBS, 0.2% BSA, 15 nM NaN3 [pH 7.4]). Grids were then transferred onto drops of a 1:10 dilution of the primary antibody, followed by incubation for 1 h and six subsequent washes with incubation buffer. Grids were then transferred to drops of a 1:20 dilution of gold conjugate reagent, followed by incubation for 1 h and six subsequent washes with incubation buffer as well as two washes with PBS. The grids were fixed for 5 min with 2% paraformaldehyde. Finally, grids were washed twice with distilled water, negatively stained with 0.5% aqueous potassium phosphotungstate (pH 6.5), and examined using a transmission electron microscope (FEI Philips TECNAI 12 BioTwin) at 120 kV. The number of gold particles per μm2 was quantified for each strain using five separate immunogold electron microscopy images for each strain per experiment in four independent experiments conducted over four separate days. The number of gold particles in each image was recorded independently by two researchers, and particles per μm2 were calculated based on the average number of gold particles counted for each image. L. monocytogenes 10403S incubated with only the secondary gold conjugate was used as a negative control in all experiments, and no labeling was observed (data not shown). Statistical analyses.Absolute mRNA transcript levels for target genes were normalized prior to final statistical analyses as initially described by Vandesompele et al. (62), who demonstrated that measurement of transcript levels from a single housekeeping gene can be inadequate for normalizing quantitative transcript levels obtained from target genes under widely varying physiological conditions. Typically, housekeeping genes that are expected to maintain stable levels of expression under the test conditions are selected as targets for control reactions in quantitative gene expression studies (48); however, housekeeping gene expression can also vary under different conditions (14, 33, 60, 62). Therefore, we normalized mRNA transcript levels for the target genes to the geometric mean of the transcript levels for two housekeeping genes (gap and rpoB) as previously described (14, 33). Average log-transformed (log10) gap and rpoB transcript levels for different strains (see Fig. S3a in the supplemental material) and growth conditions (see Fig. S3b in the supplemental material) were plotted to assess whether transcript levels for the two housekeeping genes followed similar trends. While transcript levels were considerably higher for gap than for rpoB, the two genes showed similar trends in transcript levels. Thus, unless otherwise indicated, all statistical analyses were performed on the log-transformed (log10) mRNA transcript levels normalized to the geometric mean of rpoB and gap transcript levels calculated using the following formula: $$mathtex$$\[\mathrm{Log}_{10}(\frac{\mathrm{mRNA}\mathrm{transcript}(\mathrm{gene})}{\mathrm{mRNA}\mathrm{transcript}(rpoB{+}gap)/2})\]$$mathtex$$ Three factors, including strain (parent strain 10403S and ΔsigB, ΔprfA, and ΔsigB ΔprfA strains), salt stress (presence or absence of NaCl), and presence of cellobiose or charcoal, were included in analysis of variance (ANOVA) models of the absolute mRNA transcript levels to determine if these factors affected transcript levels for the eight internalin genes and prfA, plcA, opuCA, and gadA (Table 1). If a significant effect of the factor "strain" was observed, a second ANOVA was conducted to determine whether sigB deletion, prfA deletion, or statistical interactions between data generated for strains with these deletions had a significant effect on transcript levels for a given gene under each of the four growth conditions (see Table S2 in the supplemental material). In cases in which mRNA transcript levels were below the qRT-PCR detection limit, a value representing the detection limit cutoff was used for statistical analyses. Initial analysis showed the mRNA transcript level data to be heteroscedastic and strongly skewed; therefore, logarithmic transformation (log10 [mRNA transcript level]) was used to correct the skewness and stabilize the variance to approximate normality. Tukey's multiple-comparison procedure was used to determine whether transcript levels for a gene differed among strains within a given condition (see Fig. 1↴↴ to 4) or among conditions for a given strain (see Fig. S2 and S4 through S7 in the supplemental material). All statistical analyses were performed with Splus 6.2 (Insightful Corp, Seattle, WA). Standard regression diagnostics were computed for all models. Unless indicated otherwise, P values are reported as not significant (P > 0.05) or significant at P values of <0.05, <0.01, <0.001, or <0.0001. Normalized, log-transformed transcript levels of opuCA and gadA (A) and of prfA and plcA (B) for L. monocytogenes parent strain 10403S (wild type) as well as isogenic ΔsigB, ΔprfA, and ΔsigB ΔprfA strains cultured under four different conditions, including (i) growth in BHI with 0.2% charcoal for 120 min, (ii) growth in BHI with 25 mM cellobiose for 120 min, (iii) growth in BHI with 0.2% charcoal for 120 min with exposure to 0.3 M NaCl for the last 10 min, and (iv) growth in BHI with 25 mM cellobiose for 120 min with exposure to 0.3 M NaCl for the last 10 min. Data represent average normalized and log-transformed transcript levels for three independent replicates (i.e., three RNA isolations performed on different days); error bars represent one standard deviation each. Tukey's multiple-comparison procedure was used to determine whether transcript levels for a gene and a given condition differ among the four different strains; bars labeled with different letters indicate that the transcript levels differ significantly (P < 0.05), while bars labeled with the same letter indicate that the transcript levels do not differ significantly. Normalized, log-transformed transcript levels of inlA and inlB for L. monocytogenes parent strain 10403S (wild type) as well as isogenic ΔsigB, ΔprfA, and ΔsigB ΔprfA strains cultured under the four different conditions outlined in Materials and Methods and in the legend for Fig. 1. Data represent average normalized and log-transformed transcript levels for three independent replicates (i.e., three RNA isolations performed on different days); error bars represent one standard deviation each. Tukey's multiple-comparison procedure was used to determine whether transcript levels for a gene and a given condition differ among the four different strains; bars labeled with different letters indicate that the transcript levels differ significantly (P < 0.05), while bars labeled with the same letter indicate that the transcript levels do not differ significantly. Normalized, log-transformed transcript levels for inlC for L. monocytogenes 10403S parent strain (wild type) as well as isogenic ΔsigB, ΔprfA, and ΔsigB ΔprfA strains cultured under the four different conditions outlined in Materials and Methods and in the legend for Fig. 1. Data represent average normalized and log-transformed transcript levels for three independent replicates (i.e., three RNA isolations performed on different days); error bars represent one standard deviation each. Tukey's multiple-comparison procedure was used to determine whether transcript levels for a gene and a given condition differ among the four different strains; bars labeled with different letters indicate that the transcript levels differ significantly (P < 0.05), while bars labeled with the same letter indicate that the transcript levels do not differ significantly. NS indicates that no individual transcript levels were significantly different by Tukey's multiple-comparison procedure. Normalized, log-transformed transcript levels of inlC2 and inlD for L. monocytogenes parent strain 10403S (wild type) as well as isogenic ΔsigB, ΔprfA, and ΔsigB ΔprfA strains cultured under the four different conditions outlined in Materials and Methods and the legend for Fig. 1. Data represent average normalized and log-transformed transcript levels for three independent replicates (i.e., three RNA isolations performed on different days); error bars represent one standard deviation each. Tukey's multiple-comparison procedure was used to determine whether transcript levels for a gene and a given condition differ among the four different strains; bars labeled with different letters indicate that the transcript levels differ significantly (P < 0.05), while bars labeled with the same letter indicate that the transcript levels do not differ significantly. Effects of different factors on transcript levels of target internalin genes and opuCA, gadA, plcA, and prfA Statistical analysis of quantitative data for immunogold labeling used one-way ANOVA and Tukey's multiple-comparison procedure. qRT-PCR was used to measure mRNA transcript levels for eight internalin genes, four control genes, and two housekeeping genes in L. monocytogenes 10403S and in otherwise isogenic ΔsigB, ΔprfA, and ΔsigB ΔprfA strains grown under defined conditions to ensure (i) the active form of PrfA, (ii) active σB and the active form of PrfA, (iii) the inactive form of PrfA, and (iv) active σB and the inactive form of PrfA. ANOVA models with the factors "strain," "salt stress," and "presence of cellobiose or charcoal" were initially fitted to identify which, if any, of these factors had significant effects on transcript levels for seven internalin genes as well as for opuCA, gadA, prfA, and plcA. No detectable transcripts were found for inlE; therefore, no analyses were conducted for this gene. The factor "strain" had a significant effect on transcript levels for opuCA, gadA, prfA, and plcA as well as for five internalin genes (inlA, inlB, inlC, inlC2, and inlD) (Table 1), indicating that transcript levels for inlF and inlG were independent of both σB and PrfA. A second ANOVA model was fitted to the normalized mRNA transcript levels for the genes that were affected by the factor "strain" to investigate the main effects and interactions of the sigB and prfA deletions on transcript levels for cells grown under the four different conditions (see Table S2 in the supplemental material). The factor "salt stress" had a significant effect on transcript levels for six internalin genes (inlA, inlC, inlC2, inlD, inlF, and inlG) as well as for opuCA, gadA, and plcA. The factor "presence of cellobiose or charcoal" had a significant effect on transcript levels of all genes, including the seven internalin genes (inlA, inlB, inlC, inlC2, inlD, inlF, and inlG), indicating that the presence of charcoal or cellobiose also affects transcript levels for internalin genes that are not PrfA dependent (i.e., inlC2, inlD, inlF, and inlG). σB-dependent transcription of opuCA and gadA and PrfA-dependent transcription of plcA confirm that these genes are effective reporters for monitoring σB and PrfA activity.Transcription of the compatible solute transporter protein encoded by opuCA and transcription of the glutamate dehydrogenase protein encoded by gadA have been shown to be σB dependent (21, 32, 59, 60). PrfA-dependent transcription of plcA, which encodes the phosphatidylinositol-specific phospholipase C, also has been established (8, 44). To ensure that the opuCA and gadA genes and plcA are appropriate reporter genes for σB and PrfA, respectively, the relative contributions of σB and PrfA to opuCA, gadA, and plcA transcript levels were measured under all four growth conditions. Further, as prfA has been shown to have one σB-dependent promoter (33, 52, 55), contributions of σB to prfA transcript levels were also measured. opuCA and gadA transcript levels in the ΔsigB strain were significantly lower than those in both the isogenic parent strain and the ΔprfA strain under all four growth conditions (Fig. 1A). The sigB deletion highly significantly affected gadA and opuCA transcript levels (P < 0.0001) (see Table S2 in the supplemental material). opuCA and gadA transcript levels were similar in the ΔprfA strain and the isogenic parent strain (Fig. 1A); thus, the prfA deletion did not significantly affect opuCA and gadA transcript levels (see Table S2 in the supplemental material). Salt stress significantly affected opuCA and gadA transcript levels (Table 1); opuCA and gadA transcript levels in the ΔprfA strain and the isogenic parent strain exposed to NaCl were significantly higher than those in bacteria not exposed to NaCl (Fig. 1A) (see Fig. S2a in the supplemental material; mRNA transcript levels Fig. S2 and S4 through S6 in the supplemental material are identical to those in Fig. 1 to 4, but the data are rearranged to enable direct comparisons of transcript levels in the same strain under different growth conditions). The presence of cellobiose or charcoal significantly affected opuCA and gadA transcript levels (Table 1). While no clear trends of the effects of cellobiose or charcoal on gadA transcript levels are apparent (Fig. 1A; see Fig. S2a in the supplemental material), opuCA transcript levels in all four strains were generally lower in cells cultured with cellobiose than in cells cultured with charcoal. This effect appears to be independent of PrfA activity, as opuCA transcript levels for bacteria grown in BHI-cellobiose were significantly lower than those for bacteria grown in BHI-charcoal, even for the ΔprfA strain. The plcA transcript levels in the ΔprfA strain were significantly lower than these levels in either 10403S or the ΔsigB mutant background, regardless of growth condition (Fig. 1B), consistent with the previously reported PrfA dependence of this gene (44). The presence of cellobiose or charcoal significantly affected plcA transcript levels (Table 1); plcA transcript levels in the isogenic parent and the ΔsigB strain grown in the presence of cellobiose were significantly lower than those of bacteria grown in the presence of charcoal (Fig. 1B; see Fig. S2b in the supplemental material). Based on data collected for the 10403S and ΔsigB strains (Table 1), prfA transcript levels were determined by ANOVA to be significantly affected by the sigB null mutation and by the presence of cellobiose or charcoal. Specifically, prfA transcript levels in the parent strain grown in the presence of cellobiose were higher than those of the strain grown in the presence of charcoal (Fig. 1B; see Fig. S2b in the supplemental material). prfA transcript levels were only significantly lower in the ΔsigB strain than in the parent strain during growth in BHI-cellobiose with a final 10-min exposure to NaCl (Fig. 1B), indicating a limited effect of σB on prfA transcription, which may be detectable only under selected growth conditions. prfA transcription could not be characterized in the ΔprfA or ΔsigB ΔprfA strain, as the prfA qRT-PCR primer and probe binding sites were deleted from these strains to create the prfA null mutations. σB and PrfA regulate inlA and inlB transcription.The factors "strain" and "presence of cellobiose or charcoal" affected inlA and inlB transcript levels and the factor "salt stress" affected inlA transcript levels (Table 1). inlA and inlB transcript levels were consistently and considerably lower in the ΔsigB strain than in the isogenic parent (in both the wild-type and ΔprfA backgrounds) (Fig. 2), indicating that σB plays an important role in inlA and inlB transcription. For example, when cells were cultured in BHI-charcoal and BHI-charcoal-NaCl, inlA and inlB transcript levels in the ΔsigB strain were up to 30-fold lower than those in the isogenic parent strain. inlA and inlB transcript levels were not significantly different between 10403S and the ΔprfA strain under any of the four growth conditions (Fig. 2; see Fig. S4 in the supplemental material), suggesting that PrfA has a limited role in regulating transcription of inlA and inlB when σB is also present. The ΔsigB ΔprfA strain, however, showed consistently lower inlA transcript levels than the ΔsigB strain showed; this difference was statistically significant for cells grown in charcoal with and without NaCl (Fig. 2). The sigB and prfA deletions were determined to interact significantly to affect inlA transcript levels for cells grown in the presence of charcoal (see Table S2 in the supplemental material). When grown in BHI-charcoal, the ΔsigB ΔprfA strain also showed inlB transcript levels that were lower than those of the ΔsigB strain; a significant effect of prfA deletion on inlB transcript levels for cells grown in BHI-charcoal (see Table S2 in the supplemental material) was confirmed by ANOVA. Taken together, our data indicate that transcriptional regulation of inlA and inlB is affected by interplay between active PrfA and σB. The absence of a significant multiplicative effect of the sigB and prfA null mutations on inlA transcript levels for cells grown in the presence of cellobiose (Fig. 2; see Table S2 in the supplemental material) supports the observation that interactions between the two regulatory proteins specifically require active PrfA. PrfA, but not σB, regulates inlC transcription.inlC transcript levels were affected by the factors "strain," "presence of cellobiose or charcoal," and "salt stress" (Table 1). A significant effect of the prfA deletion on inlC transcript levels was shown by ANOVA for both cells grown in BHI-charcoal and cells grown in BHI-charcoal-NaCl (see Table S2 in the supplemental material). In both the parent strain and the ΔsigB strain, inlC transcript levels for cells grown in BHI-charcoal-NaCl were lower than those for cells grown in BHI-charcoal (see Fig. S5 in the supplemental material), indicating σB-independent down-regulation of inlC in the presence of 0.3 M NaCl. Both the ΔprfA and ΔsigB ΔprfA strains showed inlC transcript levels in BHI-charcoal that were lower than those in the 10403S and the ΔsigB strains, indicating that PrfA activates transcription of inlC under these conditions. Overall, inlC transcript levels in the wild-type strain grown in BHI-charcoal (Fig. 3) were low compared to transcript levels observed for inlA, inlB, inlC2, and inlD (Fig. 2). For cells cultured in BHI-cellobiose, inlC transcript levels were consistently below the qRT-PCR detection limit (Fig. 3). The very low absolute transcript levels measured for inlC likely contributed to the conclusion that differences in inlC transcript levels were not significant by analysis of individual comparisons by Tukey's multiple-comparison procedure (Fig. 3). σB, but not PrfA, regulates transcription of inlC2 and inlD.inlC2 and inlD transcript levels were affected by the factors "strain," "salt stress," and "presence of cellobiose or charcoal" (Table 1). Under all growth conditions, inlC2 and inlD transcript levels in the wild-type strain and in the ΔprfA strain were similar, but those in the ΔsigB and ΔsigB ΔprfA strains were significantly lower and were below the qRT-PCR detection limit (Fig. 4; see Fig. S6 in the supplemental material). A highly significant effect of the sigB deletion on inlC2 and inlD transcript levels was confirmed by ANOVA, with no effect of the prfA deletion on transcript levels (see Table S2 in the supplemental material). These data clearly indicate that, under the conditions tested, inlC2 and inlD transcription is σB dependent and PrfA independent and that the two regulators do not interact to control transcription of these two genes. inlF and inlG transcription are not dependent on either σB or PrfA under the conditions tested.The inlF and inlG transcript levels for 10403S and the ΔsigB, ΔprfA, and ΔsigB ΔprfA strains were similar, irrespective of growth conditions (see Fig. S7 in the supplemental material). The factor "strain" did not affect inlF or inlG transcript levels (Table 1). Exposure to salt induces expression of five internalin genes and reduces expression of the PrfA-dependent inlC.Among the six internalin genes significantly affected by exposure to NaCl, five (inlA, inlC2, inlD, inlF, and inlG) showed generally higher transcript levels for 10403S cells exposed to salt, while one (inlC) showed generally lower mRNA transcript levels after salt exposure (Fig. 2 to 4; see Fig. S4 through S7 in the supplemental material). inlA transcript levels in both 10403S and the ΔprfA strain grown with cellobiose with a 10-min final exposure to NaCl were higher than those in cells grown in cellobiose without exposure to NaCl (Fig. 2; see Fig. S4 in the supplemental material). However, in ΔsigB and ΔsigB ΔprfA cells, inlA transcript levels for cells exposed to NaCl were not higher than those of cells not exposed to NaCl; rather, in the presence of charcoal, inlA transcript levels were generally lower for cells exposed to NaCl. These data indicate that increased inlA mRNA transcript levels in cells grown with NaCl reflect σB-dependent induction of inlA transcription. No clear pattern of higher transcript levels in cells exposed to NaCl was observed for inlB, which is downstream of inlA and shows lower transcript levels than inlA under our experimental conditions (Fig. 2; see Fig. S4 in the supplemental material). Lower inlB transcript levels may indicate limited NaCl-based induction of inlB transcription, possibly due to low levels of read-through transcription from inlA. It is also possible that more than 10 min of post-NaCl exposure is needed to yield a detectable increase in inlB transcription. Trends for inlC2 and inlD transcript levels were similar to those of inlA; for cells with an intact sigB, transcript levels for both inlC2 and inlD were consistently higher for cells exposed to NaCl than for cells not exposed to NaCl, particularly in the presence of cellobiose (Fig. 4; see Fig. S6 in the supplemental material). As inlC2 and inlD transcript levels in the ΔsigB background were below the qRT-PCR detection limit, it is not possible to determine whether higher inlC2 and inlD transcript levels in cells exposed to salt are solely σB dependent or also include a σB-independent component. inlF and inlG transcript levels in cells exposed to NaCl were also generally higher than the transcript levels in cells not exposed to NaCl, with a relative increase for inlG transcripts larger than that for inlF transcripts. The presence of higher inlF and inlG transcript levels in cells exposed to NaCl was independent of strain (see Fig. S7 in the supplemental material), indicating that induction of inlF and inlG transcription in the presence of 0.3 M NaCl is independent of both σB and PrfA. inlC was the only internalin gene that showed transcript levels in cells exposed to NaCl that were significantly lower than the transcript levels in cells not exposed to NaCl (Fig. 3; see Fig. S5 in the supplemental material). These observations parallel the finding that transcript levels for the PrfA-dependent plcA in strains with an intact prfA (i.e., 10403S and the ΔsigB strains) that were cultured in BHI-charcoal were also higher than those in such strains in BHI-charcoal-NaCl (Fig. 1B; see Fig. S2b in the supplemental material). As inlC transcript levels were below the qRT-PCR detection limit for cells grown in the presence of cellobiose (Fig. 3; see Fig. S5 in the supplemental material), comparisons of inlC transcript levels were possible only between cells grown in BHI-charcoal or BHI-charcoal-NaCl. We hypothesize that (i) NaCl exposure down-regulates PrfA activity (rather than prfA transcription), as prfA transcript levels appear unaffected by NaCl exposure (Fig. 1B; see Fig. S2b in the supplemental material) and (ii) reduced PrfA activity in NaCl-exposed cells results in reduced transcription of PrfA-dependent genes such as plcA and inlC. The hypothesis of NaCl-mediated down-regulation of PrfA activity is supported by the observation that transcript levels for the PrfA-dependent plcA were not affected by NaCl exposure in the ΔprfA and ΔprfA ΔsigB strains (Fig. 1B; see Fig. S2b in the supplemental material). inlA, inlB, inlC, inlC2, inlD, inlF, and inlG show lower transcript levels when grown in BHI-cellobiose than when grown in BHI-charcoal.All seven internalin genes with detectable transcript levels (i.e., inlA, inlB, inlC, inlC2, inlD, inlF, and inlG) were significantly affected by the presence of cellobiose or charcoal (Table 1). The transcript levels for all of these genes in cells grown in BHI-cellobiose were lower than those in cells grown in BHI-charcoal (Fig. 2 to 4; see Fig. S4 through S7 in the supplemental material). Similarly, plcA transcript levels in cells of the 10403S and ΔsigB strain cells grown in BHI-cellobiose were also lower than those in cells grown in BHI-charcoal (Fig. 1B; see Fig. S2b in the supplemental material). In 10403S, prfA transcript levels in cells grown in BHI-cellobiose were significantly higher than those in cells grown in BHI-charcoal (Fig. 1B; see Fig. S2b in the supplemental material); this effect was not apparent in the ΔsigB strain. This may indicate that active PrfA (produced in the presence of charcoal) down-regulates its own transcription, possibly in a σB-dependent manner. Overall, these data also indicate that cellobiose and charcoal differentially affect transcript levels of different virulence and stress response genes. When cells were cultured in BHI-cellobiose (with or without NaCl), inlA and inlB transcript levels were lower than the levels in cells grown in BHI-charcoal (with or without NaCl); these differences were often but not always statistically significant (Fig. 2; see Fig. S4 in the supplemental material). While lower transcript levels in the presence of cellobiose were observed for both inlA and inlB, differences in inlB transcript levels were smaller and generally not statistically significant. For inlA, the transcript levels in the presence of cellobiose that were lower than those in the presence of charcoal were generally less pronounced when growth conditions included a final 10-min exposure to NaCl. Interestingly, the effect of cellobiose on inlA and inlB transcript levels appears to be mediated through mechanisms that are both PrfA dependent and PrfA independent. Specifically, even in the ΔprfA strain, inlA and inlB transcript levels of cells grown in the presence of cellobiose were lower than those of cells grown in the presence of charcoal (Fig. 2; see Fig. S4 in the supplemental material), suggesting that a cellobiose-dependent mechanism that is at least partially PrfA independent down-regulates inlA and inlB expression. inlC transcript levels, while detectable for all four strains grown in BHI-charcoal (with or without NaCl), were below the qRT-PCR detection limit for all four strains grown in the presence of cellobiose (Fig. 3; see Fig. S5 in the supplemental material), clearly indicating that inlC transcription is down-regulated during growth in the presence of cellobiose. Lower inlC transcript levels in the presence of cellobiose were also observed for ΔprfA cells, providing evidence that down-regulation of inlC in the presence of cellobiose is at least partially PrfA independent. inlC2, inlD, inlF, and inlG transcript levels of cells grown in the presence of cellobiose generally were lower than those of cells grown in the presence of charcoal (Fig. 4; see Fig. S6 and S7 in the supplemental material). Lower transcript levels for these genes in the presence of cellobiose were observed in both the wild-type and ΔprfA backgrounds, consistent with the observation that transcription of these genes is PrfA independent. These data suggest that a PrfA-independent mechanism is responsible for differential transcript levels between cells grown in the presence of either charcoal or cellobiose for inlC2, inlD, inlF, and inlG. InlB immunogold microscopy confirms that a sigB deletion has a greater effect on inlB expression than a prfA deletion has.Surface localization of L. monocytogenes InlB by immunoelectron microscopy on intact cells of the 10403S parent strain as well as the ΔprfA, ΔsigB, and ΔinlB strains clearly showed reduced InlB on the surface of the ΔsigB strain (Fig. 5) relative to the wild-type and ΔprfA strains. The ΔinlB mutant, which was used as a negative control, showed virtually no labeling. Average counts of gold particles per μm2 were 126 ± 38 for the wild type, 91 ± 16 for the ΔprfA strain, 32 ± 12 for the ΔsigB strain, and 2 ± 5 for the ΔinlB strain. The ΔsigB and ΔinlB strains both showed significantly (P < 0.05; Tukey's multiple-comparison procedure) lower gold particle counts than the 10403S and ΔprfA strains. The lower gold particle counts associated with the ΔsigB strain relative to the ΔprfA strain confirm that σB contributes more than PrfA to InlB expression under the conditions examined in these experiments. Surface localization of L. monocytogenes InlB by immunoelectron microscopy using whole cells of L. monocytogenes 10403S (A) and the ΔprfA (B), ΔsigB (C), and ΔinlB (D) strains. Arrows indicate the bacterial cell surface and point towards the cytoplasm. The average numbers of gold particles per μm2 for each strain were estimated to be 126 ± 38 (10403S); 91 ± 16 (ΔprfA strain); 32 ± 12 (ΔsigB strain); and 2 ± 5 (ΔinlB strain). Bacterial cells sense changes in their external environment and respond rapidly by altering protein expression via complex interactions among signal transduction pathways and global gene regulators. In L. monocytogenes, induction of gene expression under conditions of environmental stress is controlled, at least in part, by the stress-responsive alternative sigma factor σB (13, 20, 32, 60, 63). Regulation of many genes involved in host-pathogen interactions is coordinated by PrfA (8, 48, 56, 58, 64). In this report, we show that L. monocytogenes internalin genes can be grouped into distinct regulons controlled by σB, PrfA, both regulators, or neither. Our data also show that transcript levels for opuCA, gadA, inlA, inlC, inlC2, inlD, inlF, and inlG are significantly affected by the presence of NaCl. Transcript levels were lower for inlC, but higher for all others, in the presence of NaCl. Transcript levels for opuCA, gadA, plcA, inlA, inlB, inlC, inlC2, inlD, inlF, and inlG were also significantly affected by the presence of cellobiose or charcoal; for all of these genes, transcript levels were higher in the presence of charcoal. Internalin genes can be grouped into distinct regulons controlled by σB, PrfA, both regulators, or neither.σB and PrfA contribute differentially to the transcriptional regulation of different internalins under the conditions tested in these experiments such that (i) both σB and PrfA contribute to transcription of inlA and inlB, (ii) only σB contributes to transcription of inlC2 and inlD; (iii) only PrfA contributes to transcription of inlC, and (iv) neither σB nor PrfA contributes to transcription of inlF and inlG. L. monocytogenes internalin genes thus group into four distinct regulons, which suggests that specific internalins are expressed under different environmental conditions. PrfA is considered to be a key regulator of L. monocytogenes virulence gene expression and has been shown to directly regulate 12 genes (48). The observation that σB and PrfA contribute to transcription of at least four and three internalin genes, respectively, illustrates that both proteins are important for regulation of known and putative L. monocytogenes virulence genes. This hypothesis is further supported by the fact that other L. monocytogenes genes that contribute to intrahost survival and infection are also regulated by σB or by σB and PrfA (e.g., bsh, opuCA, hfq [32, 33]). While previous studies have shown that the prfAp2 promoter region includes a functional σB-dependent promoter (49, 52, 55), our data indicate that transcription of inlA and inlB is predominantly and directly dependent on σB, as σB-dependent transcription of these genes was also observed in a ΔprfA background. Further, our observations of the relative importance of σB contributions to inlA transcription provide mechanistic support for the previous observation that σB is critical for L. monocytogenes gastrointestinal invasion in guinea pigs, which requires InlA (23). Combined with previous reports, our data support an emerging model in which σB is critical for expression of virulence and stress response genes important in the environment external to the host and for intestinal stages of infection, while PrfA predominantly up-regulates genes important for intracellular stages of infection. Our data clearly show that inlC2 and inlD expression is σB dependent and independent of PrfA under the conditions studied. These findings extend results from previous microarray experiments with a ΔsigB strain, which provided initial evidence of σB-dependent expression of inlC2 and inlD, along with identification of a σB consensus promoter sequence upstream of the inlC2D operon (32). The PrfA-dependent expression of inlC reported here, which is also consistent with previous findings (18, 42, 48), has been further confirmed through a comparison of inlC transcript levels in isogenic prfA* (which expresses a constitutively active PrfA [57]) and ΔprfA strains (P. McGann, M. Wiedmann, and K. Boor, unpublished data). One previous study (5) reported that InlC plays a supportive role for InlA-mediated invasion in Caco-2 cells. In addition, inlC is strongly transcribed in the cytoplasm of phagocytic J774 cells (18), suggesting that it also may play a postinvasion role in L. monocytogenes infection. Although intracellular replication of a ΔinlC strain in Caco-2 and J774 cells appeared comparable to that of the wild-type strain, the strain showed reduced virulence in an intravenous mouse model (18). Our expression data, which indicate that inlC groups into a regulon with plcA and other PrfA-dependent genes (with transcription patterns that are both PrfA dependent and repressed by salt), while being distinct from inlA in its transcriptional regulation, support a possible role for inlC in systemic spread, which also requires plcA and other PrfA-dependent genes. The low inlE transcript levels reported here (i.e., transcript levels below the qRT-PCR detection limit) are in agreement with the observations of Dramsi et al. (16), who showed, using Western blot analysis, that InlE was not expressed during growth in bacterial medium. Exposure to 0.3 M NaCl affected transcript levels of six internalin genes, with all but one (inlC) showing higher transcript levels under salt stress.Exposure of L. monocytogenes to 0.3 M NaCl for 10 min induces high levels of σB activity (60); therefore, these parameters were selected for conditions requiring σB activity. Interestingly, in addition to inducing σB activity and, thus, higher transcript levels for most σB-dependent internalin genes, exposure to 0.3 M NaCl also induced higher transcript levels for the σB-independent internalin genes, inlF and inlG. These results suggest that salt or osmotic stress may be an important environmental stimulus for up-regulation of at least some L. monocytogenes virulence genes. Since 0.3 M NaCl mimics osmotic stress conditions that could be encountered by L. monocytogenes in the lumen of the human intestine (60), it is tempting to speculate that salt or osmotic stress conditions might represent a stimulus that signals to L. monocytogenes that it is present in an intestinal tract environment. If this hypothesis is true, then the specific internalin genes that are induced by salt stress might be important during gastrointestinal infection. In support of the importance of salt and osmotic stress as a stimulus that induces virulence gene expression, transcriptome analysis of Yersinia pestis identified a number of virulence genes that are up-regulated in response to salt and/or hyperosmotic stress conditions (26). Further, Heusipp et al. reported that transcription of Yersinia enterocolitica rpoE, which encodes the alternative sigma factor σE, is induced in vivo in infected mice and by osmotic stress, further supporting a link between salt stress and virulence, particularly as σE has been shown to contribute to virulence for a number of bacteria (27). Interestingly, transcript levels for inlC, the only internalin gene that showed only PrfA-dependent transcription, were lower for cells exposed to NaCl than for unexposed cells. It is thus tempting to speculate that InlC contributes to L. monocytogenes virulence and/or survival during systemic spread in a host or during intracellular infection, with limited contributions outside of the host cell. Transcript levels for the PrfA-dependent gene plcA were also reduced in the presence of NaCl. These data suggest that osmotic or salt stress down-regulates all or some PrfA-dependent genes. Since plcA and inlC transcript levels, but not prfA transcript levels, were lower in the presence of NaCl, exposure to salt stress appears to mediate a decrease in PrfA activity, by acting directly either on the PrfA protein itself or on some other PrfA-regulating element (e.g., PrfA-activating factor [7]). Both PrfA-dependent and -independent mechanisms mediate lower transcript levels for the seven internalin genes with detectable transcripts when cells are grown in BHI-cellobiose than when cells are grown in BHI-charcoal.A wider range of L. monocytogenes genes beyond those that are PrfA dependent, including the σB-dependent opuCA, as well as all seven internalins with detectable transcripts examined in this study, showed reduced transcript levels when cells were grown in cellobiose. Specifically, transcription of the PrfA-dependent plcA and inlC for cells grown in BHI-cellobiose was lower than that for cells grown in BHI-charcoal, consistent with previous data, which showed that cellobiose mediates repression of PrfA activity (46). Down-regulation of PrfA-independent internalin gene transcription (i.e., inlC2, inlD, inlF, and inlG) in BHI-cellobiose, as well as that of PrfA-dependent genes in a ΔprfA null mutant background, indicates that differential regulation of internalin gene transcription in the presence of cellobiose or charcoal occurs through both PrfA-dependent and -independent mechanisms. These findings indicate that cellobiose and/or other easily metabolized sugars may serve as important environmental signals for L. monocytogenes to down-regulate transcription of virulence genes, including a number of internalin genes. In support of this hypothesis, previous studies have shown that repression of L. monocytogenes virulence gene expression by the presence of sugars is not confined to the presence of the β-glucoside cellobiose but is also associated with other sugars, including arbutin (10, 25, 46, 51). PrfA-dependent gene repression by sugars appears to be regulated by at least two separate mechanisms, one involving the catabolite repression pathway and another that specifically responds to β-glucosides (10). Although Milenbachs Lukowiak et al. (47) proposed that both repression pathways may act through PrfA, our data suggest that at least one pathway also acts through a PrfA-independent mechanism. Further studies will be needed to better understand the global effects of cellobiose, as well as other easily metabolized sugars, on gene expression in L. monocytogenes, particularly since initial transcriptome analyses by Milohanic et al. (48) also indicated broad and complex effects of cellobiose on L. monocytogenes gene expression. To illustrate, while a prfA deletion in strain 10403S did not affect opuCA transcript levels in our study, opuCA transcript levels were consistently lower in bacteria grown in the presence of cellobiose. In contrast, for a different L. monocytogenes strain (EGD), Milohanic and coworkers (48) classified opuCA as a class III gene, which was found to be up-regulated by PrfA, even in the presence of cellobiose. PrfA-dependent and -independent mechanisms of gene regulation under different environmental conditions, including the presence of easily fermentable sugars, appear complex. A better understanding of transcriptional regulation under these conditions is likely to provide insight into how different L. monocytogenes strains respond to different environments encountered during transmission. Transcription of L. monocytogenes internalin genes is regulated by a diversity of mechanisms that may provide clues into the roles of specific internalin proteins in different environments and host-associated niches.Taking into consideration the ubiquitous nature of L. monocytogenes and its remarkable ability to invade a wide variety of cell types, the existence of multiple and complex regulatory systems for controlling expression of cell surface molecules is not surprising. Our data demonstrate that expression of the L. monocytogenes internalin gene family is subject to a diverse set of regulatory mechanisms. Specifically, our results suggest that σB and PrfA function in an interactive network, as illustrated by the fact that L. monocytogenes grown in the presence of charcoal, a condition that activates PrfA (54), also down-regulates prfA transcription in 10403S but not in the ΔsigB strain, indicating a σB-dependent negative-feedback loop for prfA transcription and possibly providing a mechanistic explanation for early reports of the ability of PrfA to negatively regulate its own transcription (22). In summary, both PrfA and σB contribute to transcription of internalin genes and other genes involved in virulence. Our results indicate that interactions between PrfA and σB are complex, suggesting the existence of an intricate network that allows L. monocytogenes to regulate gene expression in response to changing environmental cues. We thank B. 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Regular Article - Theoretical Physics Traversable wormholes in $R+\alpha R^n$ gravity Nisha Godani1 & Gauranga C. Samanta2 The European Physical Journal C volume 80, Article number: 30 (2020) Cite this article In this work, the study of traversable wormholes in f(R) gravity with the function $f(R)=R+\alpha R^n$, where $\alpha $ and n are arbitrary constants, is taken into account. The shape function $b(r)=\frac{r}{\exp (r-r_0)}$, proposed by Samanta et al. (arXiv:1811.06834v1 [gr-qc], 2018), is considered. The energy conditions with respect to both constant and variable redshift functions are discussed and the existence of wormhole solutions without presence of exotic matter is investigated. Literature survey Traversable wormholes can be interpreted as hypothetical structures that allow the observers to traverse freely through the throat. These structures appear as a tool connecting two different space-times or two different locations of the same space-time. The study of wormholes was initiated by Flamm [1]. After that, Einstein and Rosen [2] developed Einstein–Rosen bridge connecting two asymptotically flat space-times. Morris and Thorne [3] proposed traversable wormholes for the fast interstellar travel of an observer through the space-time. Wormholes are obtained as classical solutions of Einstein's gravitational field equations which exist in the presence of a matter with negative energy called exotic matter. The exotic matter includes a stress energy tensor that does not satisfy the null energy condition (NEC). Hochberg and Visser [4] also proved the violation of NEC as a common feature for a static wormhole in general relativity. Later on, they extended this result for dynamic wormhole [5]. Several cosmologists have tried to explore the stability of wormholes and find the ways to avoid or minimize the violation of NEC. Shinkai and Hayward [6] studied the stability of traversable wormholes. Bergliaffa and Hibberd [7] examined the stress-energy tensor and obtained that the rotating wormhole can be explained neither by a perfect fluid nor by a fluid with anisotropic stress. Kuhfittig [8] using time dependent angular velocity studied rotating axially symmetric wormholes as a generalization of static and spherically symmetric traversable wormhole. They also analyzed the effect of angular velocity on weak energy condition in case of both axially and spherically symmetric wormholes. Aygün et al. [9] studied rigidly rotating wormhole in the background of Einstein's general theory of relativity. Böhmer et al. [10] used a linear dependence between energy density and pressure and studied wormhole solutions. Bronnikov and Galiakhmetov [11] using the framework of Einstein–Cartan theory studied the existence of static traversable wormholes without exotic matter. Wang and Meng [12] obtained wormhole solutions in the framework of bulk viscosity using three classes of viscous models. Moradpour [13] investigated traversable wormholes in both Einstein's general relativity and Lyra geometry. Barros and Lobo [14] obtained wormhole solutions using three form fields and analyzed the validation of weak and null energy conditions. Tsukamoto and Kokubu [15] studied stability of thin shell wormholes in the presence of barotropic fluid. The latest astronomical observations suggest that the expansion of the universe is in accelerating way [16, 17]. This acceleration is driven by a gravitationally repulsive energy called dark energy. For the explanation of this phenomenon, several proposals have been proposed during last two decades. One of them is the modification of the Einstein's general relativity by modifying the Lagrangian gravitational action R, where R is the Ricci scalar curvature. Several theories are introduced in literature that modifies Einstein's action. A significant theory that explains the cosmic acceleration and other cosmological issues is f(R) theory of gravity. In this theory, the Einstein's gravitational action is replaced by a general function of R, f(R). For the particular case $f(R)=R$, the f(R) theory is equivalent to general relativity. The idea behind this generalization is that the results may vary with the variation of choice of function f(R). Earlier the exploration of inflation scenario created interest in f(R) theory and Starobinsky [18] provided an inflation model using this theory. Nojiri and Odintsov [19] defined an f(R) model with positive and negative powers of R supporting early inflation and late time acceleration. Carroll et al. [20] showed the early and late time accelerations by doing some tiny modifications in action of general relativity. Lin et al. [21] examined the local and cosmological tests for f(R) theory by several observations. Various other f(R) models are also developed and studied [22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56]. The study of wormholes is also extended using f(R) theory. Lobo and Oliveira [57] investigated traversable wormholes using the framework of f(R) gravity and obtained the factors responsible for the violation of null energy condition and supporting the existence of wormholes. They also obtained wormhole solutions for different shape functions. Bronnikov et al. [58] discussed the existence of wormholes in scalar tensor theory and f(R) gravity. Saiedi and Esfahani [59] studied null and weak energy conditions for wormholes with constant shape and redshift functions in f(R) gravity. Bahamonde et al. [60] developed dynamical wormholes in f(R) gravity. Peter [61] computed wormhole solutions using different types of shape functions in f(R) gravity and studied traversable wormholes. Kim [62] studied FRW model with traversable wormhole. They proved that the violation of energy conditions is not necessary by the total matter in the cosmological model. Hochberg et al. [63] solved semi classical field equations representing wormholes. Lemos et al. [64] studied static and spherically symmetric traversable wormholes in the presence of cosmological constant. They explored the properties of traversable wormholes due to the presence of cosmological term. H. Maeda and M. Nozawa [65] studied the properties of n-dimensional static wormhole solutions, where $n\ge 5$, using Einstein–Gauss–Bonnet gravity. Celis et al. [66] studied thin shell wormholes using theories beyond general relativity in greater than 4-dimensional space-time. Rahaman et al. [67] obtained various wormhole solutions using Finslerian structure of space-time. Zubair et al. [68] assumed fluids of three types and explored energy conditions for static and spherically symmetric wormholes in $f(R,\phi )$ gravity. Godani and Samanta [69] studied traversable wormhole in f(R) gravity and explored energy conditions using two shape functions. Samanta et al. [70] defined a new shape function and studied energy conditions in both f(R) and f(R, T) theories. Recently, Godani and Samanta [71] and Samanta and Godani [72] investigated energy conditions for traversable wormholes in f(R) gravity. The aim of this paper is to study the wormhole geometry equipped with minimum amount of exotic matter near the throat in $f(R) = R+\alpha R^n$ gravity with shape function $b(r)=\frac{r}{\exp (r-r_0)}$. This shape function was introduced by Samanta et al. [70] to compare the wormhole solutions in f(R) and f(R, T) gravity theories. They found the satisfaction of energy conditions for small range of r in f(R) gravity with $f(R)=R-\mu R_c\tanh \frac{R}{R_c}$, where $\mu $ and $R_c$ are constants, and for a wide range of r in f(R, T) theory of gravity with function $f(R,T)=R+2\lambda T$, where $\lambda $ is an arbitrary constant. However, there may be possibility of getting the validation of energy conditions for wide range of r or there may be possibility of wormholes with some significant geometric configuration, in the framework of f(R) gravity with some different choice of f(R) function. To explore these possibilities, we have considered the same shape function with different f(R) function defined as $f(R)= R+\alpha R^n$, where $\alpha $ and n are constants. Further, Samanta et al. [70] considered constant redshift function to investigate wormhole solutions. Anchordoqui et al. [73] considered variable redshift function $\Phi (r)=-\frac{\alpha }{r}$, $\alpha >0$ and obtained analytical solutions explaining wormhole geometries. Sarkar et al. [74] assumed $\Phi (r)=\frac{\alpha }{r}$, where $\alpha $ is a constant, to explore wormhole solutions in $\kappa (R,T)$ gravity and obtained wormhole solutions filled with exotic type matter everywhere. Rahaman et al. [75] used two forms of redshift function (i) $\Phi (r)=\frac{\alpha }{r}$ and (ii) $\Phi (r)=\ln (\frac{\sqrt{\gamma ^2++r^2}}{r})$, where $\alpha $ and $\gamma $ are constant. Using these forms of redshift functions with specific choice of shape function, they determined generating functions comprising the wormhole like geometry. Further using $\Phi (r)=\frac{\alpha }{r}$ with particular form of generating function, they derived shape function of wormhole solutions. Further, Pavlovic and Sossich [76] investigated possible wormhole solutions in the context of four viable f(R) models, namely the MJWQ model [77], the exponential gravity model [78, 79], the Tsujikawa model [80, 81] and the Starobinsky model [80, 82,83,84,85]. They proposed the redshift function $\Phi (r)=\ln (\frac{r_0}{r}+1)$ and for first three models, they obtained wormhole solutions without need of exotic matter which is a very significant result. It could be possible because of the suitable choice of redshift function. This work of Pavlovic and Sossich [76] inspires to check the validity of energy conditions in other f(R) models and hence to examine the type of matter required to sustain the wormhole solutions. Therefore, we have taken the same redshift function $\Phi (r)=\ln (\frac{r_0}{r}+1)$, in the present work, to analyze the energy conditions and investigate the wormhole solutions. The section-wise description is as follows: In Sect. 2, field equations and wormhole geometry is presented. In Sect. 3, solutions of the wormhole geometry are presented. In Sect. 4, various energy conditions are discussed. Results obtained are presented in Sect. 5. Finally, conclusions are provided in Sect. 6. Field equations and wormhole geometry The four dimensional Einstein Hilbert action with matter content of the universe can be written as $$\begin{aligned} S=\int \left( \sqrt{-g}\frac{R}{2\kappa ^2}+L_m\right) d^4x, \end{aligned}$$ where $c=1$ and $\kappa ^2=8\pi G$, R is the Ricci scalar, g is the determinant of the metric tensor and $L_m$ is the lagrangian for the matter part of the universe. The Einstein field equations can be obtained by varying the action (1) with respect to the metric tensor $g_{\mu \nu }$. The Einstein Hilbert action plays an important role for modifying geometry. Hence, the modified gravity can be obtained by modifying the Einstein Hilbert action. Therefore, to obtain f(R) gravity, we have to replace f(R) in place of R in Einstein Hilbert action. Hence, the four dimensional modified Einstein Hilbert action with matter content for f(R) gravity can be written as $$\begin{aligned} S=\int \left( \sqrt{-g}\frac{f(R)}{2\kappa ^2}+L_m\right) d^4x. \end{aligned}$$ Now, varying the action (2) with respect to $g_{\mu \nu }$, we can have $$\begin{aligned} f^{'}(R)R_{\mu \nu }-\frac{1}{2}f(R)g_{\mu \nu }= & {} \bigtriangledown _{\mu } \bigtriangledown _{\nu }f^{'}(R)\nonumber \\&-g_{\mu \nu }\bigtriangledown _{\sigma } \bigtriangledown ^{\sigma }f^{'}(R) +\kappa ^2T_{\mu \nu },\nonumber \\ \end{aligned}$$ where prime denotes the derivative with respect to the scalar curvature R, i.e. $f^{'}=\frac{df}{dR}$. The trace of Eq. (3) gives $$\begin{aligned} 3\square f^{'}(R)+Rf^{'}(R)-2f(R)=\kappa ^2T, \end{aligned}$$ where $\square \equiv g^{\mu \nu }\bigtriangledown _{\mu }\bigtriangledown _{\nu }$, $T=g^{\mu \nu }T_{\mu \nu }$ is the trace of the energy momentum tensor $T_{\mu \nu }$ and the Ricci scalar $R=g^{\mu \nu }R_{\mu \nu }$. The field Eq. (3) can be written in the following form $$\begin{aligned} G_{\mu \nu }=\chi _{eff}\left( T_{\mu \nu }+T_{\mu \nu }^{f(R)}\right) , \end{aligned}$$ where $G_{\mu \nu }=R_{\mu \nu }-\frac{1}{2}g_{\mu \nu }$, $\chi _{eff}=\frac{\kappa ^2}{f^{'}(R)}$ and $T_{\mu \nu }^{f(R)}$ could be observed as effective energy momentum tensor in modified f(R) gravity, which is expressed as $$\begin{aligned} T_{\mu \nu }^{f(R)}= & {} (\bigtriangledown _{\mu }\bigtriangledown _{\nu }-g_{\mu \nu } \bigtriangledown _{\sigma }\bigtriangledown ^{\sigma })f^{'}(R)\nonumber \\&+\frac{1}{2}(f(R)-Rf^{'}(R))g_{\mu \nu }. \end{aligned}$$ For $f(R)=R$, the aforesaid modified theory reduces to general relativity. In this paper, we have considered a spherically symmetric and static wormhole metric [3], which is defined as $$\begin{aligned} ds^2=-e^{2\Phi (r)}dt^2+\frac{dr^2}{1-\frac{b(r)}{r}} + r^2(d\theta ^2+\sin ^2\theta d\phi ^2). \end{aligned}$$ The metric function $\Phi (r)$ is related to gravitational redshift and b(r) determines the shape of the wormholes [3, 86]. Hence b(r) and $\Phi (r)$ are, respectively, called the shape and redshift functions of radial coordinate r that varies from $r_0$ to $\infty $, where $r_0$ is known as radius of the throat. At the throat of the wormhole, the shape function must satisfy $b(r_0)=r_0$. The metric coefficient $g_{rr}$ becomes infinity at the throat, which is gestured by the coordinate singularity. The proper radial distance $l(r)=\pm \int _{r_0}^{r}\left( 1-\frac{b(r)}{r}\right) ^{-\frac{1}{2}}dr$ is required to be finite everywhere. The absence of horizons is necessary for traversable wormhole. This implies that $e^{2\Phi (r)}\ne 0$, so $\Phi (r)$ must be finite everywhere. Another interesting feature of the redshift function is: the derivative of the redshift function with respect to radial coordinate determines the attractive or repulsive nature of the wormhole geometry. Since our metric is spherically symmetric, so without loss of generality, one may consider an equatorial slice $\theta =\frac{\pi }{2}$ and for a fixed moment of time i.e. $t=$ constant, the metric (7) becomes $$\begin{aligned} ds^2=\frac{dr^2}{1-b(r)/r}+r^2d\phi ^2. \end{aligned}$$ The Eq. (8) can be written in cylindrical coordinates, $(r, \phi , z)$ as $$\begin{aligned} ds^2=dz^2+dr^2+r^2d\phi ^2. \end{aligned}$$ In the three-dimensional Euclidean space the embedded surface has equation $z=z(r)$, so that the metric (9) of the surface can be written as $$\begin{aligned} ds^2=\bigg [1+\left( \frac{dz}{dr}\right) ^2\bigg ]dr^2+r^2d\phi ^2. \end{aligned}$$ Comparing the Eqs. (8) and (10), we can have $$\begin{aligned} \frac{dz}{dr}=\pm \left( \frac{r}{b(r)}-1\right) ^{-\frac{1}{2}}. \end{aligned}$$ The geometry of the wormhole solution has least radius at the throat, i.e. $r=b(r)=r_0$, where $r_0$ denotes the radius of the throat of the wormhole. The embedded surface is vertical at the throat, i. e. $\frac{dz}{dr}\rightarrow \infty $ at $r=r_0$, and the space is asymptotically flat as $r\rightarrow \infty $, i. e. $\frac{dz}{dr}\rightarrow 0$ as $r\rightarrow \infty $. One also needs to impose the flaring-out condition. The flaring-out condition demands that the inverse of the embedding function r(z) must satisfy $\frac{d^2r}{dz^2}>0$ near or at the throat $r_0$. Now, differentiating $\frac{dr}{dz}=\pm \left( \frac{r}{b(r)}-1\right) ^{\frac{1}{2}}$ with respect to z, we obtain $$\begin{aligned} \frac{d^2r}{dz^2}=\frac{b(r)-b^{'}(r)r}{2b(r)^2}>0. \end{aligned}$$ This flaring-out condition is an important constituent of wormhole physics and plays a major role in the analysis of the violation of the energy conditions. In the light of the above discussion, for a traversable wormhole, the shape function should satisfy the following properties: (i) $\frac{b(r)}{r}<1$ for $r>r_0$, (ii) $b(r_0)=r_0$ at $r=r_0$, (iii) $\frac{b(r)}{r}\rightarrow 0$ as $r\rightarrow \infty $, (iv) $\frac{b(r)-b'(r)r}{b(r)^2}>0$ for $r>r_0$ and (v) $b'(r)-1< 0$ at $r=r_0$. The energy momentum tensor for the matter source of the wormhole is defined as $$\begin{aligned} T_{\mu \nu }= & {} \frac{\partial L_m}{\partial g^{\mu \nu }},\nonumber \\= & {} (\rho + p_t)u_\mu u_\nu - p_tg_{\mu \nu }+(p_r-p_t)X_\mu X_\nu , \end{aligned}$$ where $\rho $ is the energy density, $p_t$ and $p_r$ are tangential and radial pressures respectively and $u_{\mu }$ & $X_\mu $ denote the four velocity and radial vectors respectively such that $$\begin{aligned} u^{\mu }u_\mu =-1 \quad \text{ and } \quad X^{\mu }X_\mu =1. \end{aligned}$$ The effective field equations for the metric (7) can be expressed as follows: $$\begin{aligned} \rho= & {} \frac{Fb'(r)}{r^2}-\Bigg (1-\frac{b(r)}{r}\Bigg )F'\Phi ^{'}(r)-H \end{aligned}$$ $$\begin{aligned} p_r= & {} -\Bigg [\frac{b(r)}{r^3}-2\Bigg (1-\frac{b(r)}{r}\Bigg ) \frac{\Phi ^{'}(r)}{r}\Bigg ]F\nonumber \\&-\Bigg (1-\frac{b(r)}{r} \Bigg )\Bigg [F''+\frac{F'(rb'(r)-b(r))}{2r^2\Big (1-\frac{b(r)}{r}\Big )}\Bigg ]+H \end{aligned}$$ $$\begin{aligned} p_t= & {} F\Bigg (1-\frac{b(r)}{r}\Bigg )\Bigg [\Phi ^{''}(r) -\frac{(b'(r)r-b(r))}{2r(r-b(r))}\Phi '(r)+\Phi ^{'2}\nonumber \\&+\frac{\Phi ^{'}(r)}{r}- \frac{(b'(r)r-b(r))}{2r^2(r-b(r))}\Bigg ]-\frac{F'}{r}\Bigg (1-\frac{b(r)}{r}\Bigg )+H,\nonumber \\ \end{aligned}$$ where $F\equiv \frac{df}{dR}$, $R=\Bigg [\frac{4\Phi ^{'}(r)}{r}+2\Phi ^{''}(r)+2\Phi ^{'2}(r)\Bigg ] \Bigg (1-\frac{b(r)}{r}\Bigg ) -\frac{(b^{'}(r)r-b(r))}{r^2}\Phi ^{'}(r)- \frac{2b^{'}(r)}{r^2}, H(r)=\frac{1}{4}(FR+\square F+T)$, $\square F=\left( 1-\frac{b(r)}{r}\right) \bigg [F^{''}+\frac{rb^{'}(r)-b(r)}{2r^2\left( 1-\frac{b(r)}{r}\right) }F^{'}+2\frac{F^{'}}{r}+F^{'}\Phi ^{'}(r)\bigg ]$ and $T=-\rho +p_r+2p_t$. These are the standard terminologies of the matter threading the wormhole, as a function of the shape function b(r), redshift function $\Phi (r)$ and function F(r). One can comprehend the matter content of the wormhole by specifying the above functions. Thus, one can consider a specific choice of shape function to obtain a wormhole solution. Therefore, in this paper a specific form of the shape function $b(r)=\frac{r}{\exp (r-r_0)}$ is considered. Morris and Thorne [3] defined the dimensionless function $\xi =\frac{\tau -\rho }{\|\rho \|}$. However, in this paper, we define the dimensionless function $$\begin{aligned} \xi ^{eff}=\frac{-p_r^{eff}-\rho ^{eff}}{\|\rho ^{eff}\|}. \end{aligned}$$ where $\tau = -p_r^{eff}$ and $\rho = \rho ^{eff}$. Now, the Eqs. (15) and (16) yields $$\begin{aligned} \xi ^{eff}=\frac{b(r)-rb^{'}(r)}{r\|b^{'}(r)\|} \end{aligned}$$ Let us combine the flaring-out condition, given in Eq. (12), with the Eq. (19), the effective exotic function takes the form $$\begin{aligned} \xi ^{eff}=\frac{2b(r)}{r\|b^{'}(r)\|}\frac{d^2r}{dz^2} \end{aligned}$$ At the throat, we have the following condition $$\begin{aligned} \xi (r_0)=\frac{-p_r^{eff}(r_0)-\rho ^{eff}(r_0)}{\|\rho ^{eff}(r_0)\|}>0 \end{aligned}$$ Thus, from the above condition it is observed that the radial tension should exceed the total density of mass energy i.e. $-p_r^{eff}(r_0)>\rho ^{eff}(r_0)$. We shall call matter with this property, $-p_r^{eff}(r_0)>\rho ^{eff}(r_0)>0$, exotic [3]. The presence of exotic matter at the throat of the wormhole indicates that the observer who moves through the throat with a radial velocity close to the speed of light will see a negative energy density. Overall, we can say that the field equations plus the absence of a horizon at throat indicates $-p_r^{eff}(r_0)>\rho ^{eff}(r_0)$ at the throat. This implies traveler moving through the throat with speed very close to the speed of light can see the negative energy density. This implies the violation of the null, weak, strong and dominant energy conditions at the throat. Wormhole solutions In literature, the cosmologists have studied several models with quantum corrections of Einstein's field equations [87, 88]. Starobinsky followed the same idea and studied the cosmology of a model which in its simplified form is known as Starobinsky model [18]. He replaced Einstein's action R with $R+\alpha R^2$ and presented a first compatible model of inflation. In this action, term $R^2$ is a responsible factor for acceleration at high energies in the early stage of the universe. The results obtained from the Planck satellite [89] are also consistent with the Starobinsky model. This model has been studied in several aspects [90,91,92,93,94]. In this paper, a general form of this model $f(R)=R+\alpha R^n$, where $\alpha $ and n are arbitrary constants, is considered. Further, the shape function $b(r)=\frac{r}{\exp (r-r_0)}$ defined by Samanta et al. [70] and the redshift function introduced by Pavlovic and Sossich [76] $\Phi (r)=\ln (\frac{r_0}{r}+1)$ is taken into account. The aim is to investigate the wormhole geometry equipped with less amount of exotic matter near the throat in the setting of f(R) gravity with two redshift functions (i) $\Phi (r) = $ constant and (ii) $\Phi (r)=\ln (\frac{r_0}{r}+1)$. In Sect. 2, the field equations are derived for arbitrary redshift, shape and f(R) functions. Using forms (i) and (ii) of $\Phi (r)$, the expressions for the energy density ($\rho $), radial pressure ($p_r$), tangential pressure $p_t$ and different combinations of $\rho $, $p_r$ and $p_t$ are determined which are follows: Constant redshift function I. $\Phi (r) = $ constant $$\begin{aligned} \rho= & {} \frac{\alpha 2^{n-2} e^{r-2 {r_0}} \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n+1}}{(r-1)^4}\nonumber \\&\times \Bigg [ \left( e^{{r_0}} \left( 2 n^3 \left( -r^2+r{+}1\right) ^2+n^2 (r (r (r (6{-}5 r)+12)\right. \right. \nonumber \\&- \left. \left. 13)-6)+n \left( (5 (r-2) r+2) r^2+r+6\right) \right. \right. \nonumber \\&\left. \left. -2 (r-1)^3 r\right) -2 (n-1) n e^r \left( n \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&-\left. \left. r \left( r^3-5 r+4\right) -2\right) \right) \Bigg ]-\frac{(r-1) e^{{r_0}-r}}{r^2} \end{aligned}$$ $$\begin{aligned} p_r= & {} \frac{1}{2 r^2}\Bigg [\frac{1}{(r-1)^2}\Bigg (\alpha 2^n r \left( n^2 \left( -2 r^2+2 r+2\right) \right. \nonumber \\&\left. +n \left( r^3-3\right) -(r-1)^2 r\right) \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n\Bigg )\nonumber \\&-\frac{1}{r-1}\Bigg (\alpha 2^{n+1} (n-1) n \left( r^2-r-1\right) \nonumber \\&\times \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n-1}\Bigg )-2 e^{{r_0}-r}\Bigg ] \end{aligned}$$ $$\begin{aligned} p_t= & {} \frac{1}{4 (r{-}1)^3 r}\Bigg [e^{-r-{r_0}} \left( {-}\alpha 2^{n+1} n e^{2 r} \left( n^2 \left( -r^2{+}r{+}1\right) ^2\right. \right. \nonumber \\&\left. \left. +n \left( -2 r^4+3 r^3+4 r^2-6 r-2\right) \right. \right. \nonumber \\&+\left. \left. r^4-r^3-3 r^2+4 r+1\right) \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n\right. \nonumber \\&\left. +\alpha 2^n \left( 2 n^3 \left( -r^2+r+1\right) ^2+n^2 \left( -5 r^4\right. \right. \right. \nonumber \\&+\left. \left. \left. 8 r^3+8 r^2-13 r-4\right) \right. \right. \nonumber \\&\left. \left. +n \left( 5 r^4-11 r^3+2 r^2+6 r+2\right) -2 (r-1)^3 r\right) e^{r+{r_0}}\right. \nonumber \\&\times \left. \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n+2 (r-1)^3 e^{2 {r_0}}\right) \Bigg ] \end{aligned}$$ From Eqs. (22), (23) and (24), the null and dominated energy condition terms are obtained as $$\begin{aligned} \rho +p_r= & {} \frac{\alpha 2^{n-2} e^{r-2 {r_0}} \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n+1}}{(r-1)^4}\nonumber \\&\times \Bigg [ \left( e^{{r_0}} \left( 2 n^3 \left( -r^2+r+1\right) ^2+n^2 (r (r (r (6-5 r)+12)\right. \right. \nonumber \\&- \left. \left. 13)-6)+n \left( (5 (r-2) r+2) r^2+r+6\right) \right. \right. \nonumber \\&\left. \left. -2 (r-1)^3 r\right) -2 (n-1) n e^r \left( n \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&-\left. \left. r \left( r^3-5 r+4\right) -2\right) \right) \Bigg ]-\frac{(r-1) e^{{r_0}-r}}{r^2}\nonumber \\&+\frac{1}{2 r^2}\Bigg [\frac{1}{(r-1)^2}\Bigg (\alpha 2^n r \left( n^2 \left( -2 r^2+2 r+2\right) \right. \nonumber \\&+\left. n \left( r^3-3\right) -(r-1)^2 r\right) \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n\Bigg )\nonumber \\&-\frac{1}{r-1}\Bigg (\alpha 2^{n+1} (n-1) n\nonumber \\&\times \left( r^2-r-1\right) \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n-1}\Bigg )-2 e^{{r_0}-r}\Bigg ]\nonumber \\ \end{aligned}$$ $$\begin{aligned} \rho +p_t= & {} \frac{\alpha 2^{n-2} e^{r-2 {r_0}} \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n+1}}{(r-1)^4}\nonumber \\&\Bigg [ \left( e^{{r_0}} \left( 2 n^3 \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&\left. \left. +n^2 (r (r (r (6-5 r)+12)\right. \right. \nonumber \\&- \left. \left. 13)-6)+n \left( (5 (r-2) r+2) r^2+r+6\right) \right. \right. \nonumber \\&\left. \left. -2 (r-1)^3 r\right) -2 (n-1) n e^r \left( n \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&-\left. \left. r \left( r^3-5 r+4\right) -2\right) \right) \Bigg ]-\frac{(r-1) e^{{r_0}-r}}{r^2}\nonumber \\&+\frac{1}{4 (r-1)^3 r}\Bigg [e^{-r-{r_0}} \left( -\alpha 2^{n+1} n e^{2 r}\right. \nonumber \\&\left. \left( n^2\times \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&\left. \left. \times +n \left( -2 r^4+3 r^3+4 r^2-6 r-2\right) \right. \right. \nonumber \\&\left. \left. +r^4-r^3-3 r^2+4 r+1\right) \right. \nonumber \\&\times \left. \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n+\alpha 2^n \left( 2 n^3 \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&\left. \left. +n^2 \left( -5 r^4+8 r^3+8 r^2-13 r-4\right) \right. \right. \nonumber \\&+\left. \left. n \left( 5 r^4-11 r^3+2 r^2+6 r+2\right) -2 (r-1)^3 r\right) \right. \nonumber \\&\left. e^{r+{r_0}}\left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n +2 (r-1)^3 e^{2 {r_0}}\right) \Bigg ] \end{aligned}$$ $$\begin{aligned} \rho - \|p_r\|= & {} \frac{\alpha 2^{n-2} e^{r-2 {r_0}} \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n+1}}{(r-1)^4}\nonumber \\&\times \Bigg [ \left( e^{{r_0}} \left( 2 n^3 \left( -r^2+r+1\right) ^2+n^2 (r (r (r (6-5 r)+12)\right. \right. \nonumber \\&- \left. \left. 13)-6)+n \left( (5 (r-2) r+2) r^2+r+6\right) \right. \right. \nonumber \\&\left. \left. -2 (r-1)^3 r\right) -2 (n-1) n e^r \left( n \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&-\left. \left. r \left( r^3-5 r+4\right) -2\right) \right) \Bigg ]-\frac{(r-1) e^{{r_0}-r}}{r^2}\nonumber \\&-\Bigg \|\frac{1}{2 r^2}\Bigg [\frac{1}{(r-1)^2}\Bigg (\alpha 2^n r \left( n^2 \left( -2 r^2+2 r+2\right) \right. \nonumber \\&+\left. n \left( r^3-3\right) -(r-1)^2 r\right) \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n\Bigg )\nonumber \\&-\frac{1}{r-1}\Bigg (\alpha 2^{n+1} (n-1) n\nonumber \\&\times \left( r^2-r-1\right) \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n-1}\Bigg )-2 e^{{r_0}-r}\Bigg ]\Bigg \|\nonumber \\ \end{aligned}$$ $$\begin{aligned} \rho -\|p_t\|= & {} \frac{\alpha 2^{n-2} e^{r-2 {r_0}} \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n+1}}{(r-1)^4}\nonumber \\&\times \Bigg [ \left( e^{{r_0}} \left( 2 n^3 \left( -r^2+r+1\right) ^2+n^2 (r (r (r (6-5 r)+12)\right. \right. \nonumber \\&- \left. \left. 13)-6)+n \left( (5 (r-2) r+2) r^2+r+6\right) \right. \right. \nonumber \\&\left. \left. -2 (r-1)^3 r\right) -2 (n-1) n e^r \left( n \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&-\left. \left. r \left( r^3-5 r+4\right) -2\right) \right) \Bigg ]-\frac{(r-1) e^{{r_0}-r}}{r^2}\nonumber \\&-\Bigg \|\frac{1}{4 (r-1)^3 r}\Bigg [e^{-r-{r_0}} \left( -\alpha 2^{n+1} n e^{2 r} \left( n^2\right. \right. \nonumber \\&\left. \left. \times \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&\left. \left. +n \left( -2 r^4+3 r^3+4 r^2-6 r-2\right) \right. \right. \nonumber \\&\left. \left. +r^4-r^3-3 r^2+4 r+1\right) \right. \nonumber \\&\times \left. \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n+\alpha 2^n \left( 2 n^3 \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&\left. \left. +n^2 \left( -5 r^4+8 r^3+8 r^2-13 r-4\right) \right. \right. \nonumber \\&+\left. \left. n \left( 5 r^4-11 r^3+2 r^2+6 r+2\right) \right. \right. \nonumber \\&\left. \left. -2 (r-1)^3 r\right) e^{r+{r_0}}\left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n \right. \nonumber \\&+\left. 2 (r-1)^3 e^{2 {r_0}}\right) \Bigg ]\Bigg \| \end{aligned}$$ $$\begin{aligned} p_t-p_r= & {} \frac{1}{4 (r-1)^3 r}\Bigg [e^{-r-{r_0}} \left( -\alpha 2^{n+1} n e^{2 r} \left( n^2 \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&\left. \left. +\,n \left( -2 r^4+3 r^3+4 r^2-6 r-2\right) \right. \right. \nonumber \\&\left. \left. +\,r^4-r^3-3 r^2+4 r+1\right) \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n\right. \nonumber \\&\left. +\,\alpha 2^n \left( 2 n^3 \left( -r^2+r+1\right) ^2+n^2 \left( -5 r^4+8 r^3\right. \right. \right. \nonumber \\&\left. \left. \left. +\,8 r^2-13 r-4\right) +n \left( 5 r^4-11 r^3+2 r^2+6 r+2\right) \right. \right. \nonumber \\&\left. \left. -\,2 (r-1)^3 r\right) e^{r+{r_0}}\times \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n\right. \nonumber \\&\left. +\,2 (r-1)^3 e^{2 {r_0}}\right) \Bigg ]-\frac{1}{2 r^2}\Bigg [\frac{1}{(r-1)^2}\Bigg (\alpha 2^n r \left( n^2\right. \nonumber \\&\left. \left( -2 r^2+2 r+2\right) +n \left( r^3-3\right) -(r-1)^2 r\right) \nonumber \\&\left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n\Bigg )-\frac{1}{r-1}\Bigg (\alpha 2^{n+1} (n-1)\nonumber \\&n \left( r^2-r-1\right) \times \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n-1}\Bigg )-2 e^{{r_0}-r}\Bigg ]\nonumber \\ \end{aligned}$$ $$\begin{aligned} \frac{p_r}{\rho }= & {} \frac{1}{2 r^2}\Bigg [\frac{1}{(r-1)^2}\Bigg (\alpha 2^n r \left( n^2 \left( -2 r^2+2 r+2\right) \right. \nonumber \\&\left. +n \left( r^3-3\right) -(r-1)^2 r\right) \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^n\Bigg )\nonumber \\&-\frac{1}{r-1}\Bigg (\alpha 2^{n+1} (n-1) n \left( r^2-r-1\right) \nonumber \\&\times \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n-1}\Bigg )-2 e^{{r_0}-r}\Bigg ]\nonumber \\&\div \Bigg [\frac{\alpha 2^{n-2} e^{r-2 {r_0}} \left( -\frac{(r-1) e^{{r_0}-r}}{r}\right) ^{n+1}}{(r-1)^4}\nonumber \\&\times \Bigg [ \left( e^{{r_0}} \left( 2 n^3 \left( -r^2+r+1\right) ^2+n^2 (r (r (r (6-5 r)+12)\right. \right. \nonumber \\&- \left. \left. 13)-6)+n \left( (5 (r-2) r+2) r^2+r+6\right) \right. \right. \nonumber \\&\left. \left. -2 (r-1)^3 r\right) -2 (n-1) n e^r \left( n \left( -r^2+r+1\right) ^2\right. \right. \nonumber \\&\left. \left. - r \left( r^3-5 r+4\right) -2\right) \right) \Bigg ]-\frac{(r-1) e^{{r_0}-r}}{r^2}\Bigg ] \end{aligned}$$ Table 1 Summary of results for $\phi (r) = constant$ with $\alpha =0$ Variable redshift function II. $\Phi (r)=\ln (\frac{r_0}{r}+1)$ $$\begin{aligned} \rho= & {} \frac{1}{\left( 2 r^2+2 (r_0-1) r-3 r_0\right) ^3}\\&+\frac{1}{2}\left[ \alpha e^{r-2 r_0} \left( -e^r+e^{r_0}\right) (n-1) \right. \\&n r_0 \left( 2 r^4+(4 r_0-2) r^3+\left( 2 r_0^2 -5 r_0-2\right) r^2\right. \\&\left. -3 r_0 (r_0+2) r-3 r_0^2\right) \\&\left. \times \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^{n+1}\right] \\&\times \left( \frac{e^{r_0-r} \left( -2 r^2-2 (r_0-1) r+3 r_0\right) }{r (r+r_0)}\right. \\&\left. -\alpha n \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right) +\frac{1}{2} \left( \alpha \right. \\&\times \left. \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \\&\left. +\frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) \\&+\alpha e^{r-2 r_0} (n-1) n \times \frac{1}{2 \left( 2 r^2+2 (r_0-1) r-3 r_0\right) ^4}\\&\times \left[ \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^{n+1}\right. \\&\times \left( e^{r_0} \left( 4 (2 n-3) r^8 +8 (-6 r_0+n (4 r_0-2)+1) r^7\right. \right. \\&\left. \left. +\left( -72 r_0^2+44 r_0+8 n \left( 6 r_0^2-9 r_0-1\right) +40\right) r^6\right. \right. \\&\left. \left. +4 \left( -12 r_0^3+21 r_0^2+ 46 r_0\right. \right. \right. \\&\left. \left. \left. +2 n \left( 4 r_0^3-15 r_0^2-5 r_0+2\right) -9\right) r^5\right. \right. \\&\left. \left. +\left( -12 r_0^4+68 r_0^3+299 r_0^2-198 r_0+n \left( 8 r_0^4-88 r_0^3\right. \right. \right. \right. \\&\left. \left. \left. \left. - 62 r_0^2+88 r_0+8\right) -16\right) r^4+2 r_0 \left( 10 r_0^3+103 r_0^2\right. \right. \right. \\&\left. \left. \left. -183 r_0-6 n \left( 2 r_0^3+3 r_0^2-14 r_0-4\right) -44\right) r^3\right. \right. \\&\left. \left. -3 r_0^2 \left( -17 r_0^2+95 r_0+2 n \left( r_0^2-22 r_0-16\right) +60\right) r^2\right. \right. \\&\left. \left. +9 r_0^3 (-9 r_0+4 n (r_0+2)-16) r+18 (n-2) \right. \right. \\&\left. \left. \times r_0^4\right) -2 e^r \left( 4 (n-1) r^8+8 (n (2 r_0-1)-2 r_0) r^7\right. \right. \\&\left. \left. +4 \left( -6 r_0^2+r_0+n \left( 6 r_0^2-9 r_0-1\right) +5\right) r^6 \right. \right. \\ \end{aligned}$$ $$\begin{aligned}&\left. \left. +4 \left( -4 r_0^3+3 r_0^2+23 r_0+n \left( 4 r_0^3-15 r_0^2-5 r_0+2\right) \right. \right. \right. \nonumber \\&\left. \left. \left. -4\right) r^5+\left( -4 r_0^4+12 r_0^3+151 r_0^2-88 r_0\right. \right. \right. \nonumber \\&\left. \left. \left. + n \left( 4 r_0^4-44 r_0^3-31 r_0^2+44 r_0+4\right) -8\right) r^4\right. \right. \nonumber \\&\left. \left. -2 r_0 \left( -2 r_0^3-53 r_0^2+81 r_0+3 n \left( 2 r_0^3+3 r_0^2-14 r_0\right. \right. \right. \right. \nonumber \\&\left. \left. \left. \left. -4\right) +22\right) r^3-3 r_0^2 \left( -9 r_0^2+42 r_0+n \left( r_0^2-22 r_0-16\right) \right. \right. \right. \nonumber \\&\left. \left. \left. +30\right) r^2+18 (n-2) r_0^3 (r_0+2) r\right. \right. \nonumber \\&\left. \left. \left. +9 (n-2) r_0^4\right) \right) \right] -\frac{1e^{r_0-r} (r-1)}{r^2} \nonumber \\&\times \left( \alpha n \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^{n-1}+1\right) \end{aligned}$$ $$\begin{aligned} p_r= & {} \frac{1}{2 r^2 \left( 2 r^2+2 (r_0-1) r-3 r_0\right) ^2 (r+r_0)}\\&\times \left[ 4 \alpha (n-1) r^7 \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \\&+4 \alpha (n-1) r^6 (2 n+3 r_0-2) \\&\times \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n+4 \alpha r^5 \left( (6 r_0-2) n^2\right. \\&+\left. \left( 3 r_0^2-13 r_0+2\right) n-3 r_0^2+7 r_0-1\right) \\&\times \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n-4 \alpha e^{r-r_0} n r \\&\times (r+r_0) \left( 2 (n-1) r^4+2 (n-1) (2 r_0-1) r^3\right. \\&\left. +\left( -2 r_0^2+7 r_0+n \left( 2 r_0^2-5 r_0-2\right) +2\right) r^2\right. \\&+\left. r_0 (5 r_0-3 n (r_0+2)+4) r-3 n r_0^2\right) \\&\times \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n-36 r_0^3-2 e^{r_0-r} \\&\times (r-r_0) \left( 2 r^2+2 (r_0-1) r-3 r_0\right) ^2\\&-6 r r_0^2 \left( r_0 \left( \alpha n (2 n-1) \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \right. \\&-\left. \left. 8\right) +8\right) +4 r^4\\&\times \left( \alpha (n-1) r_0^3 \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \\&\left. +2 \alpha \left( 3 n^2-7 n+4\right) r_0^2\right. \end{aligned}$$ $$\begin{aligned}&\qquad \times \left. \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \nonumber \\&\qquad \left. -\alpha n (2 n-3) \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \nonumber \\&\qquad -\left. r_0 \left( \alpha \left( 7 n^2-10 n+4\right) \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \right. \nonumber \\&\qquad \left. \left. +4\right) \right) -r^2 r_0 \left( \left( \alpha \left( 12 n^2-25 n+9\right) \right. \right. \nonumber \\&\qquad \times \left. \left. \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n+16\right) r_0^2\right. \nonumber \\&\qquad \left. +4 \left( \alpha n (9 n-8) \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \right. \nonumber \\&\qquad \left. \left. -20\right) r_0+16\right) +r^3 r_0 \left( 4 \alpha \left( 2 n^2-5 n+3\right) \right. \nonumber \\&\qquad \left. r_0^2 \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n-2 \alpha n (16 n-19)\right. \nonumber \\&\qquad \times \left. \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \nonumber \\&\qquad \left. -r_0 \left( \alpha \left( 32 n^2-57 n+21\right) \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \right. \nonumber \\&\qquad \left. +\left. \left. 32\right) 32\right) \right] \end{aligned}$$ $$\begin{aligned} p_t= & {} \frac{1}{2 r^2 (r+r_0)}\Bigg [\frac{1}{r (r+r_0)}\Bigg (2 \alpha e^{r_0-2 r} \left( e^r-e^{r_0}\right) \\&(n-1) n \left( 2 r^4+(4 r_0-2) r^3+\left( 2 r_0^2-5 r_0-2\right) r^2\right. \\&-\left. 3 r_0 (r_0+2) r-3 r_0^2\right) \left( \frac{e^{r_0-r}}{r (r+r_0)} \right. \\&\left. \times \left( 2 r^2+2 (r_0-1) r-3 r_0\right) \right) ^{n-2}\Bigg )\\&+r \left( \alpha n r (r+r_0) \left( \frac{e^{r_0-r}}{r (r+r_0)}\right. \right. \\&\times \left. \left. \left( 2 r^2+2 (r_0-1) r-3 r_0\right) \right) ^n\right. \\&\left. +e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) \right) \\&+\frac{1}{2 r^2+2 (r_0-1) r-3 r_0}\Bigg [e^{-r-r_0}\\&\times \left( 2 e^r r_0+e^{r_0} \left( r^2-2 r_0\right) \right) \left( \alpha e^r n r (r+r_0) \right. \\&\left. \times \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \\&\left. +e^{r_0} \left( 2 r^2+2 (r_0 -1) r-3 r_0\right) \right) \Bigg ]\\&+r \left( e^{r_0-r} \left( -2 r^2-2 (r_0-1) r+3 r_0\right) \right. \\&\left. -\alpha r \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\right. \\&\times \left. (r+r_0)\right) +\frac{1}{\left( 2 r^2+2 (r_0-1) r-3 r_0\right) ^3}\\&\times \Bigg (\alpha e^{-r_0} (n-1) n r \left( \frac{e^{r_0-r} \left( 2 r^2+2 (r_0-1) r-3 r_0\right) }{r (r+r_0)}\right) ^n\\&\times \left( 2 e^r \left( 4 (n-1) r^8+8 (n (2 r_0-1)-2 r_0) r^7\right. \right. \\&\left. \left. +4 \left( -6 r_0^2+r_0+n \left( 6 r_0^2-9 r_0-1\right) +5\right) r^6\right. \right. \\&+\left. \left. 4 \left( -4 r_0^3+3 r_0^2+23 r_0+n \left( 4 r_0^3-15 r_0^2-5 r_0+2\right) \right. \right. \right. \\&\left. \left. \left. -4\right) r^5+\left( -4 r_0^4+12 r_0^3+151 r_0^2-88 r_0\right. \right. \right. \\&+\left. \left. \left. n \left( 4 r_0^4-44 r_0^3-31 r_0^2+44 r_0+4\right) -8\right) r^4\right. \right. \\&\left. \left. -\,2 r_0 \left( -2 r_0^3-53 r_0^2+81 r_0+3 n \left( 2 r_0^3+3 r_0^2-14 r_0\right. \right. \right. \right. \\&\left. \left. \left. \left. -\,4\right) +22\right) r^3-3 r_0^2 \left( -9 r_0^2+42 r_0\right. \right. \right. \\&\left. \left. \left. +\,n \left( r_0^2-22 r_0-16\right) +30\right) r^2+18 (n-2) r_0^3 (r_0+2) r\right. \right. \\ \end{aligned}$$ $$\begin{aligned}&\qquad \left. \left. +\,9 (n-2) r_0^4\right) +e^{r_0} \left( (12-8 n) r^8-8 (-6 r_0\right. \right. \nonumber \\&\qquad \left. \left. +\,n (4 r_0-2)+1) r^7+\left( 72 r_0^2-44 r_0+n \left( -48 r_0^2\right. \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. \left. +\,72 r_0+8\right) -40\right) r^6-4 \left( -12 r_0^3+21 r_0^2+46 r_0\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. +\,2 n \left( 4 r_0^3-15 r_0^2-5 r_0+2\right) -9\right) r^5+\left( 12 r_0^4\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. -\,68 r_0^3-299 r_0^2+198 r_0+n \left( -8 r_0^4+88 r_0^3+62 r_0^2\right. \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. \left. -\,88 r_0-8\right) +16\right) r^4+2 r_0 \left( -10 r_0^3-103 r_0^2\right. \right. \right. \nonumber \\&\qquad \left. \left. \left. +\,183 r_0+6 n \left( 2 r_0^3+3 r_0^2-14 r_0-4\right) +44\right) r^3\right. \right. \nonumber \\&\qquad \left. \left. +\,3 r_0^2 \left( -17 r_0^2+95 r_0+2 n \left( r_0^2-22 r_0-16\right) \right. \right. \right. \nonumber \\&\qquad \left. \left. \left. +\,60\right) r^2-9 r_0^3 (-9 r_0+4 n (r_0+2)-16) r\right. \right. \nonumber \\&\qquad \left. \left. -\,18 (n-2) r_0^4\right) \right) \Bigg )\Bigg ] \end{aligned}$$ Similar to $\phi (r) = $ constant, the expressions for $\rho +p_r$, $\rho +p_t$, $\rho +p_r+2p_t$, $\rho -\|p_r\|$, $\rho -\|p_t\|$, $p_t-p_r$ and $\frac{p_r}{\rho }$ can be determined. Since the expressions for $\rho $, $p_r$ and $p_t$ are too large, therefore we are not mentioning these combinations here. Table 2 Summary of results for $\phi (r) = constant$ with $n=0$ Table 6 Summary of results for $\phi (r) = constant$ with $n=-0.1$ Table 7 Summary of results for $\phi (r) = constant$ with $n=-1$ The geometric nature of wormholes can be determined using the anisotropy parameter which is defined as $\triangle =p_t-p_r$. For $\triangle >0$, the geometry is said to be repulsive; for $\triangle <0$, the geometry is said to be attractive and for $\triangle =0$, the geometry is called isotropic. The equation of state parameter is defined as $\omega =\frac{p_r}{\rho }$. Its value determines the type of the fluid present in the wormhole structure. Energy conditions In the literature [3], it is pointed out that not only a throat of the spherically static wormhole threaded by exotic matter, but this is also true for any traversable, non-static and non-spherical wormhole. The main reason is that the bundles of null geodesics that enter the wormhole at one end (mouth) and arise from the other must have cross-sectional areas that initially decrease and then increase. The translation from shrinking to growing can only be formed by gravitational repulsion of matter through which the light rays pass. So, negative energy density is required for this repulsion [3, 86]. In the 1960s and an early 1970s, most physicists claim that no observer should ever be able to measure a negative energy density. This claim brings the name weak energy condition, and when this improved by some additional limitations, it is called the dominant energy condition or the strong energy condition. These energy conditions are allowed to violate, for the matter with the property $-p_r>\rho $. And, these are key foundations for a number of important theorems, for example: the positive mass theorem, which says that objects made of matter can never repel other bodies gravitationally, provided it satisfies the dominant energy condition [95,96,97,98,99,100]. A variety of theorems that forecast that if one or more of the energy conditions are satisfied, then space time singularities will be formed in cosmological situations and in gravitational collapse [101], and the second law of black hole mechanics, which says that if stress energy near a black hole horizon satisfies the strong energy condition, then the horizons surface area can never decrease [102]. The energy momentum tensor at every point $x\in {\mathbb {M}}$ must follow the inequality $T_{\mu \nu }W^{\mu }W^{\nu }\ge 0$ for any timelike vector $W\in {\mathbb {T}}_x$, where ${\mathbb {M}}$ is the 4-dimensional space-time and ${\mathbb {T}}_x$ is the tangent space at $x\in {\mathbb {M}}$. And, this will be true for any null vector $W\in {\mathbb {T}}_x$. An observer whose world line at x has unit tangent vector U, the local energy density seems to be $T_{\mu \nu }U^{\mu }U^{\nu }$. Thus, this supposition is corresponding to saying that the energy density as measured by any observer is non-negative. That is $\text{ NEC }\Leftrightarrow ~~ \rho +p_i\ge 0, ~\forall i$. For our model, the null energy condition (NEC) is said to be satisfied, if $\rho + p_r\ge 0$ and $\rho +p_t\ge 0$ for all $r>0$. The Week Energy Condition (WEC) is defined as $\text{ WEC } \Leftrightarrow T_{\mu \nu }W^{\mu }W^{\nu }\ge 0$, where $W^{\mu }$ is any time like vector. As it is true for any timelike vector, so it will also suggest the Null Energy Condition (NEC). The physical significance of this condition is that it claims the local energy density must be positive as measured by any timelike observer. That is $\text{ WEC }\Leftrightarrow \rho \ge 0, ~~ \text{ and } ~~ \rho +p_i\ge 0, ~\forall i$. For our model, the week energy condition (WEC) is said to be satisfied, if $\rho \ge 0$, $\rho + p_r\ge 0$ and $\rho +p_t\ge 0$ for all $r>0$. For any timelike vector $W^{\mu }$, the Strong Energy Condition (SEC) is defined as $\text{ SEC }\Leftrightarrow \left( T_{\mu \nu }-\frac{T}{2}g_{\mu \nu }\right) W^{\mu }W^{\nu }\ge 0$, where T is the trace of the stress energy tensor, $T=T_{\mu \nu }g^{\mu \nu }$. The SEC also suggest the NEC, but it does not imply, in general, the WEC. Precisely, $\text{ SEC }\Leftrightarrow \rho +p_i\ge 0, \text{ and } \rho +\sum p_i\ge 0, \forall i$. For our model, the strong energy condition (SEC) is said to be satisfied, if $\rho + p_r\ge 0$, $\rho +p_t\ge 0$ and $\rho +p_r+2p_t\ge 0$ for all $r>0$. For any timelike vector $W^{\mu }$, the Dominant Energy Condition (DEC) is defined as $\text{ DEC }\Leftrightarrow T_{\mu \nu }W^{\mu }W^{\nu }\ge 0$, and $T_{\mu \nu }W^{\mu }$ is not spacelike. The physical significance of this energy condition says that the energy density will be always positive locally, and that the energy flux is timelike or null. The dominant energy condition (DEC) is said to be satisfied if $\rho -\|p_r\|\ge 0$ and $\rho -\|p_t\|\ge 0$ for all $r>0$. The DEC implies the WEC, and thus also the NEC, however, it does not necessarily imply the SEC. Precisely, we can write $\text{ DEC }\Leftrightarrow \rho \ge 0, \text{ and } p_i\in [-\rho , +\rho ], \forall i$. In modified gravity the gravitational field equations can be rewritten as an effective Einstein equation, given by $G_{\mu \nu }=\kappa ^2T_{\mu \nu }^{eff}$, where $T_{\mu \nu }^{eff}$ is an stress energy tensor containing the stress energy tensor and curvature, arising from the specific modified gravity considered [103]. Hence the generalized NEC for the modified gravity is defined as $T_{\mu \nu }^{eff}W^{\mu }W^{\nu }\ge 0$. The violation of the NEC is necessary, for the existence of wormhole solution. Therefore, in modified gravity, the violation of the generalized NEC is necessary, for the existence of wormhole solution. Hence, $T_{\mu \nu }^{eff}W^{\mu }W^{\nu }< 0$ is required. This may reduce the violation of the NEC in classical general relativity, i. e. $T_{\mu \nu }W^{\mu }W^{\nu }<0$. In order to ensure the flaring-out condition, the generalized NEC is required to be violated, i. e. $T_{\mu \nu }^{eff}W^{\mu }W^{\nu }< 0$ . Moreover, in modified gravity one may impose some constraints, such that the matter stress energy tensor satisfies the standard NEC, i. e. $T_{\mu \nu }W^{\mu }W^{\nu }\ge 0$, while the respective generalized NEC will be violated. From Eqs. (15) and (16), it is observed that $\rho ^{eff}(r_0)+p_r^{eff}(r_0)=-\frac{1}{r_0}$, this indicates that the generalized NEC does not satisfy at the throat $r_0$ of the wormhole, which supports the existence of traversable wormhole in modified gravity. In the exploration of wormhole geometries, the modified theories have contributed significantly. The modified f(R) theory is one which has been used by several cosmologists to study the wormhole geometry. The wormhole metric is defined in terms of shape and redshift functions. We have considered variable redshift function $\phi (r) = \log (\frac{r_0}{r}+1)$, proposed by P. Pavlovic and M. Sossich [76], as well as constant redshift function $\phi (r)= $ constant with the shape function $b(r)=\frac{r}{\exp (r-r_0)}$, introduced by Samanta et al. [70]. The energy conditions are investigated to obtain the wormhole geometries in the framework of $f(R)=R+\alpha R^n$ gravity, where $\alpha $ and n are arbitrary constants. Since R is a function of r, f(R) depends on r, $\alpha $ and n. For different possible values of these parameters, the validity of energy conditions and nature of anisotropy and equation of state parameters, calculated in Sect. 3, are analyzed. Further, the spherical regions obeying the energy conditions are also determined. I: $\phi (r)= constant$ Case I(a): $\alpha =0$ In this case, $f(R)=R$. The results are summarized in Table 1. The energy density is positive only for $0<r<1$. The first NEC term $\rho + p_r$ is negative for $r\in (0,\infty )-{1}$ and indeterminate for $r=1$. This shows that NEC and hence WEC are not satisfied everywhere. The first DEC term $\rho -\|p_r\|< 0$ for all $r\in (0,\infty )-\{1\}$. This shows the violation of DEC throughout. Thus, this case is not of interest. Case I(b): $n=0$ In this case, $f(R)=R+\alpha $. The results are summarized in Table 2. It is clear from Table 2 that all the energy conditions are also violated here like case I(a). So, this case does not give favourable results. Case I(c): $n>0$ If n is not an integer, then the energy density and all energy condition terms have either negative, imaginary or indeterminate values for $r\in (0,\infty )$. If n is a positive integer, then we have different results for $n=1,2$ and $>2$. For $n>2$, we have taken $n=6$. In Tables 3, 4, 5, the results are summarized for $n=1,2$ and six respectively with the variation of $\alpha $ and r. When $n=1$ and $\alpha >0$, the energy density is positive only for $r\in (0,1)$ and all energy conditions are violated. For When $n=1$ and $\alpha <0$, NEC, WEC and DEC are satisfied for $r\in (2,\infty )$. In this region, the matter filled is ordinary with attractive geometry. Thus, energy conditions are violated near the throat of wormhole and satisfied away from it. When $n=2$, $f(R)=R+\alpha R^2$. The results are specified in Table 4. For $\alpha >0$, there is no spherical region obeying the energy conditions. For $\alpha <0$, the energy density is positive for $r\in [0.4,2.7]$ and NEC is satisfied for $r\in (1.5,2.1)$. Consequently, WEC is valid for $r\in (1.5,2.1)$. Further, SEC is satisfied for $r\in (1.5,2.1)$ and DEC is satisfied for $r\in (1.5,2.1]$. Thus, all energy conditions are valid for $r\in (1.5,2.1)$. In this range, the geometry is filled with non-phantom fluid having attractive geometry. Thus, we have desired results but in a very small region. When $n=6$, $f(R)=R+\alpha R^6$, the results obtained are summarized in Table 5. For $\alpha >0$, all the energy conditions are found to be violated. For $\alpha <0$, the energy density is positive for $r\in (0.1,1)$ and NEC is satisfied for $r\in (0.8,1)$. Thus, WEC is satisfied for $r\in (0.8,1)$. In this region, the geometry is repulsive and matter content is ordinary. Both SEC and DEC are violated throughout. So we do not have good results for $n>2$. Case I(d): $n<0$ We have found different results for $n=-2, n\ne -2$ but a negative integer and n not an integer. For $n\ne -2$ but a negative integer, we have taken $n=-1$. Thus, we have summarized the results, in particular, for $n=-0.1, -1$ and -2 in Tables 6, 7, 8 respectively. For $n=-0.1$, $f(R)=R+\alpha R^{-0.1}$. The results obtained are mentioned in Table 6. If $\alpha >0$, the energy density is positive for $r\in (0,1)$ and NEC is satisfied for $r\in [0.4,1)$. Consequently, WEC is valid for $r\in [0.4,1)$. For this range of r, the matter content is non-phantom with attractive geometry. SEC and DEC are violated everywhere. If $\alpha <0$, the energy conditions are dissatisfied everywhere. For $n=-1$, $f(R)=R+ \frac{\alpha }{R}$. In Table 7, the results are declared for $\alpha >0$ and $\alpha <0$. If $\alpha >0$, we have $\rho >0$ for $r\in (0,\infty )-\{1\}$ and NEC validates for $r\in (0.4,1)\cup (1.6,\infty )$. Thus, WEC is also satisfied for $r\in (0.4,1)\cup (1.6,\infty )$. SEC is satisfied nowhere and DEC is satisfied for $r\in (0.3,1)\cup (1.6,\infty )$. Hence, NEC, WEC and DEC are satisfied for $r\in (0.4,1)\cup (1.6,\infty )$ with non-phantom fluid having attractive geometry. If $\alpha <0$, energy conditions are violated everywhere. This depicts the presence of exotic matter with repulsive geometric configuration near the throat for $r\in (0,0.4)$. So, we have desirable results in this subcase. These results are plotted in Fig. 1a–g. For $n=-2$, $f(R)=R+ \frac{\alpha }{R^2}$. In Table 8, the results are specified for $\alpha >0$ and $\alpha <0$. When $\alpha >0$, $\rho >0$ for $r\in (0,1)$ and NEC, WEC, SEC and DEC are satisfied for $r\in (0.3,1)$. In this range of r, the geometry is filled with non-phantom fluid having attractive geometry. Thus, the results are favourable but in a small spherical region. When $\alpha <0, \rho >0$ for $r\in (1,\infty )$. The first NEC term is positive for $r>1$ and second NEC term is positive for $r\in (0,0.4)\cup (1.6,\infty )$. Thus, NEC and WEC are obeyed for $r>1.6$. The SEC term $\rho +p_r+2p_t$ is positive for $r\in (0,1)$ only, therefore SEC is disobeyed everywhere. Futhere, the first and second DEC terms are positive for $r>1$ and $r>1.6$ respectively. Thus, DEC is satisfied for $r>1.6$. The geometric structure is filled with ordinary or non-phantom fluid with repulsive geometry near the throat for $r<1$ and attractive geometry for $r>1$. II: $\phi (r)=\ln (\frac{r_0}{r}+1)$ Case II(a): $\alpha =0$ In this case, the energy density is negative and all energy conditions are invalid everywhere. Case II(b): $n=0$ The results are similar to Case II(a) for every value of r and $\alpha $. Case II(c): $n>0$ The parameter $\alpha $ can be positive or negative. When $\alpha >0$, the results are similar to Case II(a) everywhere. When $\alpha <0$, the energy density is observed to be positive for $r>1$ with $\alpha \le -1$ and $n=1$ and negative otherwise. For $\alpha \le -1$ and $n=1$, the results are summarized in Table 8. For these values of parameters $\alpha $ and n, the first and second NEC terms are positive for $r>0$ and $r\in (3,3.5)$. Thus, NEC and WEC are valid for $r\in (3,3.5)$. SEC term $\rho +p_r+2p_t>0$ for all $r>0$. This shows the satisfaction of SEC for $r\in (3,3.5)$. Both DEC terms are found to be negative everywhere which means that DEC is violated everywhere. The anisotropy parameter $\triangle <0$ and the equation of state parameter $\omega >0$ for every $r>0$ which indicates the presence of attractive geometry filled with ordinary fluid. For this case, the results are summarized in Table 9. Plots for Density, NEC, SEC, DEC, $\triangle $ & $\omega $ using $\phi (r)$ = constant with $n=-1$ and $\alpha >0$ Table 9 Summary of results for $\phi (r) = \log (\frac{r_0}{r}+1)$ with $n=1$ Table 10 Summary of results for $\phi (r) = \log (\frac{r_0}{r}+1)$ with $n<0$ Case II(d): $n<0$ In this case, either $\alpha >0$ or $\alpha <0$. When $\alpha >0$, $\rho >0$ for $r>1$. The first and second NEC terms are positive for $r>1$ and $r>1.6$ respectively. Thus, NEC as well as WEC are satisfied for $r>1.6$. The SEC term $\rho +p_r+2p_t$ is positive for $r\in (0,1]$. Thus, SEC is dissatisfied everywhere. Like NEC, the first and second DEC terms are positive for $r>1$ and $r>1.6$ respectively. Therefore, DEC is also valid for $r>1.6$. All NEC, WEC and DEC hold for $r>1.6$. The anisotropy parameter $\triangle <0$ for $r>1$ and $\triangle >0$ for $r\le 1$. This shows that the geometry is repulsive near the throat and attractive away from it. The equation of state parameter $\omega >0$ for $r\in (1,1.7)$ and $-1<\omega <0$ for $r\in (0,1.04]\cup [1.67,\infty )$. This shows the presence of non-phantom or ordinary fluid inside the wormhole. Hence, we have good results for $\alpha >0$. At last, when $\alpha <0$, then the energy density is negative and all energy conditions are violated. The results are also summarized in Table 10 and for $n<0$, $\alpha >0$, the results are plotted in Fig. 2a, g. Morris and Thorne [3] constructed traversable wormhole solutions with constant redshift function in general relativity and claimed that exotic matter is required at least at throat unless the throat may be closed. However, the study on traversable wormhole without exotic matter is a really interesting topic. Therefore, recently, several authors have tried to investigate wormhole solutions in f(R, T) gravity some of them are listed here: Zubair et al. [104] investigated static spherically symmetric wormhole in f(R, T) gravity by considering isotropic, anisotropic and barotropic fluids and obtained solutions are supported by non-exotic matter in few regions of the space time. Moraes et al. [105] constructed static traversable wormhole with constant throat radius in f(R, T) gravity and they obtained energy conditions are satisfied for wide range of r. Subsequently, several authors have studied wormhole solutions in f(R, T) gravity [70, 106,107,108,109,110,111,112]. It is natural to compare the results in the setting of f(R, T) gravity with the results in general relativity (GR). In this paper, for $\alpha =0$, the model reduces to GR. In Cases I(a) and II(a), the results are mentioned for constant and variable redshift functions respectively. In each case, all energy conditions are violated not only at throat, but also outside of the throat. It indicates that the presence of exotic type matter is necessary to support the existence of wormhole geometries using the concept of general relativity. Furthermore, in modified gravity, the results are obtained for $n=0$, $n>0$ and $n<0$. In case of constant redshift function, the results are favorable for (a) $n=1$, $\alpha <0$ and (b) $n<0$, an integer, $\alpha >0$. For variable redshift function, we have desirable results for (a) $n=1$, $\alpha \le -1$ and (b) $n<0$, $\alpha >0$. Thus, the suitable choices of f(R) function, shape function b(r) and redshift function have led to the favourable results confirming the existence of wormhole geometries without support of exotic matter. Plots for Density, NEC, SEC, DEC, $\triangle $ & $\omega $ using $\phi (r) = \log (\frac{r_0}{r}+1)$ with $n<0$ and $\alpha >0$ Starobinsky [18] replaced Einstein's action R with the function $f(R) = R+\alpha R^2$ and proposed a consistent inflationary cosmological model that well explains the acceleration at early epoch of the universe and has been studied extensively in literature. In this work, we have assumed its general form $f(R)=R+\alpha R^n$, where $\alpha $ and n are arbitrary constants and studied it in the context of wormhole metric in f(R) gravity. Since the wormhole metric is dependent on two arbitrary functions, namely redshift function and shape function. The redshift function can be constant or variable. In this work, we have used the variable redshift function $\phi (r)=\ln (\frac{r_0}{r}+1)$ [76] as well as the constant redshift function. Further, the shape function is taken as $b(r)=\frac{r}{\exp (r-r_0)}$. The goal of the present work is to find the existence of wormhole structures containing minimum amount of exotic matter at or near the throat and large amount of matter satisfying the energy conditions outside the throat. For each redshift function, the energy density, null, weak, strong and dominated energy condition terms, anisotropy and equation of state parameters are determined and analyzed for different possible values of parameters. For I. $\phi (r)= $ constant, NEC, WEC and DEC are observed to be validated for (i) $r>2$ with $\alpha <0$, $n=1$; (ii) $r\in (0.4,1)\cup (1.6,\infty )$ with $\alpha >0$, $n<0$, an integer except $n=-2$; (iii) $r\in (0.3,1)$ with $\alpha >0$, $n=-2$ and (iv) $r>1.6$ with $\alpha <0$, $n=-2$. Further, for II. $\phi (r)=\ln (\frac{r_0}{r}+1)$, NEC, WEC and SEC are obeyed for (i) $r\in (3,3.5)$ with $\alpha \le -1$, $n=1$ and NEC, WEC and DEC are satisfied for (ii) $r>1.6$ with $\alpha >0$, $n<0$. For these ranges of parameters, the wormhole geometry contains exotic matter only at the throat that to a very small portion of the geometry i.e. near the throat energy conditions are violated, matter content is phantom and geometric configuration is repulsive, however, outside thebibliography throat, the energy conditions, namely NEC, WEC and DEC, are satisfied and the matter content is non-phantom or ordinary having attractive geometric configuration. Hence, it could be concluded that the wormhole with variable redshift function is more appropriate than constant redshift function. Because, in variable redshift function case, the presence of exotic matter could be avoided by assuming the size of the throat is more than 1.6 (i. e. $r>1.6$) for any $n<0$, however, in case of constant redshift function, the presence of exotic matter could be avoided by assuming the size of the throat is more than two (i. e. r>2) for particular choice of $n=1$. 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Godani, G.C. Samanta, Chin. J. Phys. 62, 161 (2019) The authors are very much thankful to the anonymous reviewer and editor for the constructive comments for improvement of the work. The work of second author GCS is supported by CSIR Grant no. 25(0260)/17/EMR-II. Department of Mathematics, Institute of Applied Sciences and Humanities, GLA University, Mathura, Uttar Pradesh, India Nisha Godani Department of Mathematics, BITS Pilani K K Birla Goa Campus, Sancoale, Goa, India Gauranga C. Samanta Search for Nisha Godani in: Search for Gauranga C. Samanta in: Correspondence to Gauranga C. Samanta. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Funded by SCOAP3 Godani, N., Samanta, G.C. Traversable wormholes in $R+\alpha R^n$ gravity. Eur. Phys. J. C 80, 30 (2020) doi:10.1140/epjc/s10052-019-7587-5 DOI: https://doi.org/10.1140/epjc/s10052-019-7587-5	CommonCrawl
A unified finite difference Chebyshev wavelet method for numerically solving time fractional Burgers' equation On the existence and uniqueness of the solution of an optimal control problem for Schrödinger equation June 2019, 12(3): 513-531. doi: 10.3934/dcdss.2019034 Weak Galerkin mixed finite element methods for parabolic equations with memory Xiaomeng Li , Qiang Xu , and Ailing Zhu , School of Mathematical and Statistics, Shandong Normal University, Jinan 250014, China * Corresponding author: Qiang Xu, Ailing Zhu. Received June 2017 Revised September 2017 Published September 2018 Fund Project: Project supported by the Natural Science Foundation of Shandong Province (No. ZR2014AM033). Figure(1) / Table(2) We develop a semidiscrete and a backward Euler fully discrete weak Galerkin mixed finite element method for a parabolic differential equation with memory. The optimal order error estimates in both $ \|\\|·\|\\| $ and $ L^2 $ norms are established based on a generalized elliptic projection. In the numerical experiments, the equation is solved by the weak Galerkin schemes with spaces $ \{[P_{k}(T)]^2, P_{k}(e), P_{k+1}(T)\} $ for $ k = 0 $ and the numerical convergence rates confirm the theoretical results. Keywords: Weak Galerkin, mixed finite element, discrete weak divergence, integro-differential equations, error estimates. Mathematics Subject Classification: Primary: 65N15, 65N30; Secondary: 65R20. Citation: Xiaomeng Li, Qiang Xu, Ailing Zhu. Weak Galerkin mixed finite element methods for parabolic equations with memory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (3) : 513-531. doi: 10.3934/dcdss.2019034 H. Che, Z. Zhou, Z. Jiang and Y. Wang, $ H^1 $-Galerkin expanded mixed finite element methods for nonlinear pseudo-parabolic integro-differential equations, Numer. Methods Partial Differential Equations, 29 (2013), 799-817. doi: 10.1002/num.21731. Google Scholar Z. Jiang, $ L^∞(L^2) $ and $ L^∞(L^∞) $ error estimates for mixed methods for integro-differential equations of parabolic type, ESAIM Math. Model. Numer. Anal., 33 (1999), 531-546. doi: 10.1051/m2an:1999151. Google Scholar L. Mu, J. Wang, Y. Wang and X. Ye, A weak Galerkin mixed finite element method for biharmonic equations, in Numerical Solution of Partial Differential Equations: Theory, Algorithms, and Their Applications, Springer Press, 45 (2013), 247-277. doi: 10.1007/978-1-4614-7172-1_13. Google Scholar L. Mu, J. Wang and X. Ye, A hybridized formulation for the weak Galerkin mixed finite element method, J. Comput. Appl. Math., 307 (2016), 335-345. doi: 10.1016/j.cam.2016.01.004. Google Scholar A. K. Pani and G. Fairweather, $ H^1 $-Galerkin mixed finite element methods for parabolic partial integro-differential equations, IMA J. Numer. Anal., 22 (2002), 231-252. doi: 10.1093/imanum/22.2.231. Google Scholar R. K. Sinha, R. E. Ewing and R. D. Lazarov, Mixed finite element approximations of parabolic integro-differential equations with nonsmooth initial data, SIAM J. Numer. Anal., 47 (2009), 3269-3292. doi: 10.1137/080740490. Google Scholar J. Wang and X. Ye, A weak Galerkin finite element methods for elliptic problems, J. Comput. Appl. Math., 241 (2013), 103-115. doi: 10.1016/j.cam.2012.10.003. Google Scholar J. Wang and X. Ye, A weak Galerkin mixed finite element method for second-order elliptic problems, Math. Comp., 83 (2014), 2101-2126. doi: 10.1090/S0025-5718-2014-02852-4. Google Scholar J. Wang and X. Ye, A weak Galerkin finite element method for the Stokes equations, Adv. Comput. Math., 42 (2016), 155-174. doi: 10.1007/s10444-015-9415-2. Google Scholar E. G. Yanik and G. Fairweather, Finite element methods for parabolic and hyperbolic partial integro-differential equations, Nonlinear Anal., 12 (1988), 785-809. doi: 10.1016/0362-546X(88)90039-9. Google Scholar Q. Zhang and R. Zhang, A weak Galerkin mixed finite element method for second-order elliptic equations with Robin boundary conditions, J. Comput. Math., 34 (2016), 532-548. doi: 10.4208/jcm.1604-m2015-0413. Google Scholar C. Zhou, Y. Zou, S. Chai, Q. Zhang and H. Zhu, Weak Galerkin mixed finite element method for heat equation, Appl. Numer. Math., 123 (2018), 180-199. doi: 10.1016/j.apnum.2017.08.009. Google Scholar Z. Zhou, An H1-Galerkin mixed finite element method for a class of heat transport equations, Appl. Math. Model., 34 (2010), 2414-2425. doi: 10.1016/j.apm.2009.11.007. Google Scholar A. Zhu, Discontinuous mixed covolume methods for linear parabolic integrodifferential problems, J. Appl. Math. , 2014 (2014), Art. ID 649468, 8 pp. doi: 10.1155/2014/649468. Google Scholar A. Zhu, T. Xu and Q. Xu, Weak Galerkin finite element methods for linear parabolic integro-differential equations, Numer. Methods Partial Differential Equations, 32 (2016), 1357-1377. doi: 10.1002/num.22053. Google Scholar Figure 1. A typical uniform mesh on $(0, 1)\times(0, 1)$ with $h = 1/8$ Table 1. Error behaviors of FWG-MFEM for the first example with $\Delta t = 4h^2$ $h$ $\|\|\|e_{h}\|\|\|$ $\mbox{order}\approx$ $\\|\varepsilon_{h}\\|$ $\mbox{order}\approx$ $2^{-3}$ 4.8132e-002 - 1.7834e-003 - $2^{-4}$ 2.3657e-002 1.0247 4.3564e-004 2.0334 Table 2. Error behaviors of FWG-MFEM for the second example with $\Delta t = 4h^2$ Jiwei Jia, Young-Ju Lee, Yue Feng, Zichan Wang, Zhongshu Zhao. Hybridized weak Galerkin finite element methods for Brinkman equations. Electronic Research Archive, 2021, 29 (3) : 2489-2516. doi: 10.3934/era.2020126 Changling Xu, Tianliang Hou. Superclose analysis of a two-grid finite element scheme for semilinear parabolic integro-differential equations. Electronic Research Archive, 2020, 28 (2) : 897-910. doi: 10.3934/era.2020047 Xiu Ye, Shangyou Zhang, Peng Zhu. A weak Galerkin finite element method for nonlinear conservation laws. Electronic Research Archive, 2021, 29 (1) : 1897-1923. doi: 10.3934/era.2020097 Bin Wang, Lin Mu. Viscosity robust weak Galerkin finite element methods for Stokes problems. Electronic Research Archive, 2021, 29 (1) : 1881-1895. doi: 10.3934/era.2020096 Patricio Felmer, Ying Wang. Qualitative properties of positive solutions for mixed integro-differential equations. Discrete & Continuous Dynamical Systems, 2019, 39 (1) : 369-393. doi: 10.3934/dcds.2019015 Tianling Jin, Jingang Xiong. Schauder estimates for solutions of linear parabolic integro-differential equations. Discrete & Continuous Dynamical Systems, 2015, 35 (12) : 5977-5998. doi: 10.3934/dcds.2015.35.5977 Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5217-5226. doi: 10.3934/dcdsb.2020340 Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114 Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 Guanrong Li, Yanping Chen, Yunqing Huang. A hybridized weak Galerkin finite element scheme for general second-order elliptic problems. Electronic Research Archive, 2020, 28 (2) : 821-836. doi: 10.3934/era.2020042 Olivier Bonnefon, Jérôme Coville, Jimmy Garnier, Lionel Roques. Inside dynamics of solutions of integro-differential equations. Discrete & Continuous Dynamical Systems - B, 2014, 19 (10) : 3057-3085. doi: 10.3934/dcdsb.2014.19.3057 Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Singular integro-differential equations with applications. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021051 Ruishu Wang, Lin Mu, Xiu Ye. A locking free Reissner-Mindlin element with weak Galerkin rotations. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 351-361. doi: 10.3934/dcdsb.2018086 Samir K. Bhowmik, Dugald B. Duncan, Michael Grinfeld, Gabriel J. Lord. Finite to infinite steady state solutions, bifurcations of an integro-differential equation. Discrete & Continuous Dynamical Systems - B, 2011, 16 (1) : 57-71. doi: 10.3934/dcdsb.2011.16.57 Lijuan Wang, Jun Zou. Error estimates of finite element methods for parameter identifications in elliptic and parabolic systems. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1641-1670. doi: 10.3934/dcdsb.2010.14.1641 Jie Shen, Xiaofeng Yang. Error estimates for finite element approximations of consistent splitting schemes for incompressible flows. Discrete & Continuous Dynamical Systems - B, 2007, 8 (3) : 663-676. doi: 10.3934/dcdsb.2007.8.663 Hui Peng, Qilong Zhai. Weak Galerkin method for the Stokes equations with damping. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021112 Martin Burger, José A. Carrillo, Marie-Therese Wolfram. A mixed finite element method for nonlinear diffusion equations. Kinetic & Related Models, 2010, 3 (1) : 59-83. doi: 10.3934/krm.2010.3.59 Wolf-Jüergen Beyn, Janosch Rieger. Galerkin finite element methods for semilinear elliptic differential inclusions. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 295-312. doi: 10.3934/dcdsb.2013.18.295 Tomás Caraballo, P.E. Kloeden. Non-autonomous attractors for integro-differential evolution equations. Discrete & Continuous Dynamical Systems - S, 2009, 2 (1) : 17-36. doi: 10.3934/dcdss.2009.2.17 Xiaomeng Li Qiang Xu Ailing Zhu	CommonCrawl
CO/VL Angry by Choice Doc Madhattan Games with Words Genomics, Medicine, and Pseudoscience History of Geology Moss Plants and More Pleiotropy Plektix RRResearch Skeptic Wonder The Culture of Chemistry The Phytophactor The View from a Microbiologist Variety of Life Multiscale gigapixel photography By Gianluigi Filippelli on Wednesday, June 27, 2012 posted by @ulaulaman #photography #computer #science #technology Sometimes, using the Android application Wondershare Panorama, I try to shot some panoramical photos. The result is good, but it could be better, like everything: indeed the actual camera (also in our smartphones) work near the fundamental limit of 1–10 megapixels for millimetre-scale apertures(1). In theory(2) we are able to upgrade this limit. If we define $SW$ like the upper limit for the number of data channels which can be handled in parallel(2), after some calculations, Adolf W. Lohmann found that without aberration, it would increase quadratically with the scaling factor $M$(2). This result is different from the purely geometry result that states $SW$ indipendent from scaling. That result cannot be realistic, otherwise, very long focal length lens systems would be fairly useless. In practice, the apertures of these long systems are reduced, more or less according to the empirical rule(2) Now a team of researchers develop a new photographical device that can resolve at 50 gigapixels: AWARE-2 camera(1, ). And below there are some examples of its capabilities(1): Labels: applied physics, computer science, engineering, optics, photography, technology By Gianluigi Filippelli on Tuesday, June 26, 2012 #solarpower #wolframalpha In the image the solar power generated by country (via wolframalpha) Labels: solar power, wolfram alpha A quick tour in LHC By Gianluigi Filippelli on Sunday, June 24, 2012 (via cosmiclog) Labels: animation, cern, lhc, video, youtube Turing patterns in coats and sounds By Gianluigi Filippelli on Saturday, June 23, 2012 posted by @ulaulaman #AlanTuring #mathematics #biology #genetics #ecology One hundred years ago was born Alan Turing. He was known essentially for his role during the Second World War: he encrypted the Enigma machine. But He is also a brillant mathematichian and today I would try to describe one of his better model, that today biologists are applying to their research field. A vibrating object tends to vibrate at certain preferred frequencies, called natural frequencies. These frequencies depend on properties such as the density and tension of the vibrating object. Mathematicians and physicists can determine the natural frequencies of an object when they know the values for these other properties. This article describes new work being done to solve the reverse problem - calculation of properties such as density when the natural frequencies of the object are known. Elizabeth Veomett about Good Vibrations by Joyce R. McLaughlin. American Scientist, July - August 1998 Cymatics was the study of the waves' patterns. The first interested in this subject was Galileo Galilei: As I was scraping a brass plate with a sharp iron chisel in order to remove some spots from it and was running the chisel rather rapidly over it, I once or twice, during many strokes, heard the plate emit a rather strong and clear whistling sound: on looking at the plate more carefully, I noticed a long row of fine streaks parallel and equidistant from one another. Scraping with the chisel over and over again, I noticed that it was only when the plate emitted this hissing noise that any marks were left upon it; when the scraping was not accompanied by this sibilant note there was not the least trace of such marks.(1) Some years after Galilei (1680), Robert Hooke was able to see the nodal patterns associated with the modes of vibration of glass plates. In 1787 Ernst Chladni repeated Hooke's experiments and published his results in Entdeckungen über die Theorie des Klanges (Discoveries in the Theory of Sound). Finally in 1967 Hans Jenny published Kymatik (Cymatics), a book based on Chladni's work, and cymatics became an interesting science, in particular for artists! For example, Jeff Volk, poet, writes an interesting article about Jenny and the pattern of sound: From Vibration to Manifestation (pdf). In particular he presents an image from Alexander Lauterwasser's Water Sound Images Pay attention: following Lauterwasser and Volk we could explain the pattern of leopard's coat, but the first explanation come from one of the Alan Turing's paper The Chemical Basis of Morphogenesis(2). In this paper Turing is interested in the formation and development of path in biology (the so called phenomenon of morphogenesis). Any pattern or shape observed in nature, even though governed by genetics, is most likely produced by an unknown mechanism. Thus, determining these mechanisms that generate pattern and shape in organisms is an important goal of theoretical biologists.(3) The most used model for this type of systems is the reaction-diffusion system, described by the following formula: \[u_t = d \Delta u + f (\gamma, u)\] where $u$is the position of the gene, $u_t$ the diffusion speed, $d$, $\gamma$ real constants. We can write two similar formulas for every morphogene in the system. Reaction-diffusion models are particularly compelling with regard to their ability to capture complex evolving patterns.(3) Similar equations are really complex to analyze, due to their local and general phenomena. In an intuitively way, we can see the pattern formation like a challenge between reaction mode and diffusion mode. In his paper, Turing suggested that a system of reacting and diffusing chemicals (morphogens) can interact to produce stable patterns in concentration (Turing patterns).(3) Or in a more simple way: we can imagine the presence of an activator molecule of the morphogenesis. This molecule will be produced more and more thanks an autocatalysis process, but the activator will produce also an inhibitor, that will limit the production of the activator. The dynamics between activator and inhibitor will generate the pattern observed in nature (for example tigers' stripes). Tipically the two diffusion velocities are different, so we can explain the great variety of patterns. Labels: alan turing, biology, evolution, genetics, mathematics The shield of Captain America posted by @ulaulaman #physics #chemistry #superhero #CaptainAmerica After the releasing of the movie Captain America: The first Avenger in 2011 by Paramount Picture, Suveen N. Mathaudhu, the Program Manager responsible for Synthesis and Processing of Materials at the U.S. Army Research Office in Durham, NC, written a brief article, The Making of Captain America's Shield (pdf), where he try to understand if today we have the ability to construct the Captain America's shield. Thanks to Lynne Robinson(1) and the Avengers movie, Mathaudhu and his little review returned to the attention of people. First of all we try to resume the story of the Cap's shield. The first comics shield was a triangular shield but starting from Captain America Comics #2 (april 1941), Cap was equipped by the most famous circular shield: A concavo-convex metal disc approximately 0.76 m in diameter, it is virtually indestructible and has remained his most constant shield over the decades. Following Captain America #255 (march 1981), the shield was presented to Rogers by president Franklin Roosevelt(6). It is created by the scientist Myron MacLain during some experiments with vibranium, an extraterrestrial metal introduced in Fantastic Four #53 with the ability to absorb vibrations(6). A useful utilization of the vibranium was made by Thor in Avengers #68 in order to contain the explosion of Ultron-6(5). During the same saga (started on Avengers #66), MacLain presented for the first time the adamantium(6). This metal was created (or founded) by MacLain some years after the creation of Cap's shield: this last was made by an alloy of vibranium and steel with an unkown catalyst; so MacLain try to reproduce that experiment and he accidentaly created adamantium, like the same metallurgist telled in Captain America #303. Labels: captain america, chemistry, marvel comics, material science, physics, superheroes The Graham Bell's tetrahedronal shed By Gianluigi Filippelli on Monday, June 18, 2012 posted by @ulaulaman Thanks to @fadesingh #geometry #chemistry #math Tony Smith realized an interesting shed that it seems inspired by the tetrahedron, a particular polyhedron, but following Tropolism, this idea was just used by Alexander Graham Bell: The tetrahedron is in general a polyhedron constituted by four triangular sides. Now, if we describe every vertices of every sides with the vector $(x_i, y_i, z_i)$, where $i = 1, \cdots, 4$, then the volume of the tetrahedron is given by: \[V = \frac{1}{3!} \begin{vmatrix} x_1 & y_1 & z_1 & 1\\ x_2 & y_2 & z_2 & 1\\ x_3 & y_3 & z_3 & 1\\ x_4 & y_4 & z_4 & 1 \end{vmatrix}\] And if the tetrahedron is regular, we can relate in one beutiful formula, the volume $V$, the area $\Delta$ of the triangles and the radius $R$ of the sphere outside the tetrahedron (or the circumsphere)(1, ) \[6RV = \Delta^2\] The regulartetrahedron is also the platonic solid $P_5$ (...) with four polyhedron vertices, six polyhedron edges, and four equivalent equilateral triangular faces. His symmetries are a bit complex, with 12 rotational symmetries, and the tetrahedral group is isomorphic with the symmetric group $S_4$, i.e. the group of all permutations of 4 elements. The tetrahedron is also the basic idea to the Four corner project, developed by the artist David Barr in 1976, with the idea to realize a Erth-size tetrahedron in order to span our planet. Finally we can find tetrahedron also in chemistry: methane, xenon tetroxide, perchlorate ion, sulfate ion, phosphate ion and others. It's also interesting observe that also water presents a structure like a tetrahedron, but in this case isn't a regular polyhedron. In particular: The most common arrangement of liquid water molecules is tetrahedral with two hydrogen atoms covalently attached to oxygen and two attached by hydrogen bonds. Since the hydrogen bonds vary in length many of these water molecules are not symmetrical and form transient irregular tetrahedra between their four associated hydrogen atoms.(2, 3) (1) MathWorld: Weisstein, Eric W. Tetrahedron; Jackson, Frank and Weisstein, Eric W. Regular Tetrahedron (2) Wikipedia: Tetrahedral molecular geometry (3) P. E. Mason and J. W. Brady (2007). "Tetrahedrality" and the Relationship between Collective Structure and Radial Distribution Functions in Liquid Water. J. Phys. Chem. B 111 (20): 5669–5679 Labels: alexander bell, chemistry, geometry, mathematics posted by @ulaulaman thanks to @archivioDFW This comics watercolor is drawned by Davide Osenda, a computer engineer and cartoonists. His first comics is about mathematics, the italian graphic novel L'ultima lezione a Gottinga (The last lesson in Gottinga), about Cantor and the mathematics of transfinite numbers. Now it seems that he's working to a comics inspired by This is water, a speech by David Foster Wallace. So I propose you the audio of that speech: Labels: comics, david fosetr wallace, davide osenda, video, water, youtube The science behind tears By Gianluigi Filippelli on Friday, June 15, 2012 posted by @ulaulaman about #chemistry #comics #tears From the collaboration between the National Cartoonists' Society and the Center for Cartoon Studies, it is born the first issue of the first volume of the on-line magazine The Cartoon Crier. The tabloid is a collection of the saddest strips and cartoons from a lot of great cartoonists. In particular there is also a science comics, Lacrimal studies 101 by Jon Chad, from the Fizzmont institute of rad science(1). The structure of the comic is like the series of stories named A Goofy Look At... and Goofy as a famous hystoric persons drawned in particular by Hector Adolfo de Urtiága, one of the cartoonists of the Jaime Diaz's studios. But stop to write about comics and start with science, in particular about the three type of tears that our eyes can produce: basal, reflex and emotional. Basal tears are produced by our eyes constantly to keep them moist. These tears contain glucose, mucin, lysozyme, lactoferrin, lipocaln, potassium and sodium Reflex tears are produced when an irritant either physical (a poke in the eye) or chemical (onion fumes) agitates an eye!! About the emotional tears(2) I find, instead, an interesting paper published last year on Science(3). First of all there is the composition: Tears are drops of liquid produced by the lacrimal, accessory lacrimal, and Meibomian glands, which contain proteins, enzymes, lipids, metabolites, electrolytes, and traces of drugs. In mice, tears contain a chemosignal or pheromone. Because the chemical makeup of human emotional tears differs from that of reflexive eye-protective tears, we hypothesized that human tears may similarly convey a chemosignal. The research team, in order to test their hypothesis, choses a group of women between 30 and 31 years and has occurred the effect of their tears on various groups of men. For various types of emotions and tears were used different groups of donors (each group had an average age 30 years old) and after the samples are submitted to the attention of different groups of men (mean ages of groups were between 28 and 29 years old). First of all, we must note that the first test has been necessary to understand if the tears had some odor able to distinguish them than, for example, a saline solution. After determining that the tears do not have characteristic odors, they went ahead quietly with the actual experiment that aimed to test one of the two hypotheses under consideration, i.e. either that tears contain chemical signals related to the context of sadness in which they were produced, or that human tears, such as those of the mice, are capable of signaling information related to the behavior sociosexual. We can summarize the results with the following paragraph from the abstract: We found that merely sniffing negative-emotion–related odorless tears obtained from women donors induced reductions in sexual appeal attributed by men to pictures of women's faces. Moreover, after sniffing such tears, men experienced reduced self-rated sexual arousal, reduced physiological measures of arousal, and reduced levels of testosterone. Finally, functional magnetic resonance imaging revealed that sniffing women's tears selectively reduced activity in brain substrates of sexual arousal in men. The results, of course, carry with them a series of questions, such as which are the substances inside the tears responsible for this type of response, or if such signals is restricted to emotional tears, or if we can still find the same effect even in the tears of men than women. It's also interesting to note what is not said in the paper, namely that this kind of research can provide the best information to more effectively convey a certain kind of commercial messages. Discover something about ourselves, like in this case, presents a downside: it can trivially be used against us. However I think that the beauty of the world around us is equal to similar risks, especially if certain findings are not closed to the rooms of the researchers and donors. (1) A fake institute where we talk about REAL science! (2) About emotional tears, I translate you a breaf quote from an italian pdf about tears: This last type of tears [the emotional tears] contains very high percentages of manganese and some hormones including prolactin (3) Gelstein, S., Yeshurun, Y., Rozenkrantz, L., Shushan, S., Frumin, I., Roth, Y., & Sobel, N. (2011). Human Tears Contain a Chemosignal Science, 331 (6014), 226-230 DOI: 10.1126/science.1198331 (4) Other links about research: Christine Dell'Amore for National Geographic and Janelle Weaver for Scientific American Labels: chemistry, comics, tears Master dregree in Physics in scattering theory. PhD in Physics in group theory (ray representation in quantum mechanics). After a master in e-learning I'm instructional designer for Italian Astronomical Olympiads at Brera's Astronomycal Observatory (Italy). Maths in Europe: John Conway alan turing (10) albert einstein (10) arxiv (22) higgs boson (18) lhc (29) neutrinos (14) nobel prize (10) quantum mechanics (21) towel day (7) A is for Aspirin C6-H12-O6 Epiphenom Inkfish Memoirs of a Defective Brain Protein Evolution and Other Musings Rule of 6ix The Allotrope The Astronomist The Biology Files	CommonCrawl
The position of the joint of shape memory alloy and bias springs DCDS-S Home Long-time behaviour of a thermomechanical model for adhesive contact April 2011, 4(2): 247-271. doi: 10.3934/dcdss.2011.4.247 Global and exponential attractors for a Ginzburg-Landau model of superfluidity Alessia Berti 1, , Valeria Berti 2, and Ivana Bochicchio 3, Facoltà di Ingegneria, Università e-Campus, 22060 Novedrate (CO), Italy Dipartimento di Matematica, Università di Bologna, 40126 Bologna, Italy Dipartimento di Matematica e Informatica, Università di Salerno, 84084 Fisciano (SA), Italy Received October 2008 Revised June 2009 Published November 2010 The long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid 4He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. 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Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934/dcdsb.2018145 Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934/dcds.1998.4.205 Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934/cpaa.2008.7.1345 Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934/dcds.2010.28.1713 Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. 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Stadium Paradox 1 Paradox 2 Resolution 3 Historical Note Consider three rows of bodies: $\begin{array} {ccccc} \text {(A)} & 0 & 0 & 0 & 0 \\ \text {(B)} & 0 & 0 & 0 & 0 \\ \text {(C)} & 0 & 0 & 0 & 0 \\ \end{array}$ Let row $\text {(A)}$ be at rest, while row $\text {(B)}$ and row $\text {(C)}$ are travelling at the same speed in opposite directions. $\begin{array} {ccccccc} \text {(A)} & & 0 & 0 & 0 & 0 & \\ \text {(B)} & 0 & 0 & 0 & 0 & & \\ \text {(C)} & & & 0 & 0 & 0 & 0 \\ \end{array}$ By the time they are all in the same part of the course, $\text {(B)}$ will have passed twice as many of the bodies in $\text {(C)}$ as $\text {(A)}$ has. Therefore the time it takes to pass $\text {(A)}$ is twice as long as it takes to pass $\text {(C)}$. But the time which $\text {(B)}$ and $\text {(C)}$ take to reach the position of $\text {(A)}$ is the same. Therefore double the time is equal to half the time. You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by expanding it. When this page/section has been completed, {{Stub}} should be removed from the code. If you would welcome a second opinion as to whether your work is correct, add a call to {{Proofread}} the page (see the proofread template for usage). Historical Note The Stadium Paradox is one of Zeno's Paradoxes, as famously raised by Zeno of Elea. 1937: Eric Temple Bell: Men of Mathematics ... (previous) ... (next): Chapter $\text{II}$: Modern Minds in Ancient Bodies Retrieved from "https://proofwiki.org/w/index.php?title=Stadium_Paradox&oldid=258015" Zeno's Paradoxes Antinomies This page was last modified on 6 June 2016, at 15:05 and is 1,368 bytes	CommonCrawl
How much heating does Earth inner core provide to the surface? Compared to the energy that the Earth's surface receives from the Sun, how much power comes from the inner melted core? How important is this contribution to the surface temperature? thermodynamics geophysics thermal-conductivity weather agemOagemO $\begingroup$ Related to the aspect of the core & Earth's surface temperature: physics.stackexchange.com/questions/80159, physics.stackexchange.com/questions/66169, physics.stackexchange.com/questions/137229 $\endgroup$ – Kyle Kanos May 10 '15 at 17:30 $\begingroup$ The heat from outer core (the lava) is isolated from the surface. If lave wasn't hot you wouldn't notice it here. The inner core is much much deeper than lava, so we can't even measure it's temperature (we can guess it though). $\endgroup$ – zoran404 May 10 '15 at 17:32 $\begingroup$ There are estimation of inner core temperature, but I was more asking about inner earth contribution to surface temperature $\endgroup$ – agemO May 10 '15 at 17:43 $\begingroup$ @zoran404 - The outer core is not "lava". Lava is partially molten rock, and is generally a near-surface feature. (The mantle is essentially solid.) The outer core is mostly molten iron/nickel, plus some lighter elements (most likely silicon or sulfur), plus trace amounts of heavier elements. $\endgroup$ – David Hammen May 10 '15 at 18:50 $\begingroup$ To clarify, do you mean a global average? The answer is regionally different, as in some concentrated spots, the answer is different from the rest of the Earth. $\endgroup$ – gerrit Jan 4 '18 at 17:59 Very little. The Earth's surface emits about 503 watts per square meter (398.2 W/m2 as infrared radiation, 86.4 W/m2 as latent heat, and 18.4 W/m2 via conduction/convection), or about 260,000 terawatts over all of the Earth's surface (Trenberth 2009). The ultimate source of almost all of this energy is the Sun. Estimates vary on how much heat crosses the core/mantle boundary, from 4 TW to 17 TW. Even the larger value is much, much smaller than the heat emitted by the Earth's surface. Estimates of the total heat flow from the interior of the Earth (core, mantle, crust) are much tighter, 46 TW ± 3 TW (Jaupart 2007) (cf 47 TW ± 2 TW (Davis 2010)). This is considerably more than the heat coming from the core, but it's still small compared to the Earth's total heat budget: $$\frac{\text{heat from interior of Earth}}{\text{total}}\ = \frac{46\ \text{TW}}{260,000\ \text{TW}}\ =\ 0.02\% $$ Davies, J. H., and D.R. Davies (2010), "Earth's surface heat flux," Solid Earth 1.1:5-24. Jaupart, C., and J. C. Mareschal. "Heat flow and thermal structure of the lithosphere." Treatise on Geophysics 6 (2007): 217-252. Trenberth, Kevin E., John T. Fasullo, and Jeffrey Kiehl (2009), "Earth's global energy budget," Bulletin of the American Meteorological Society 90.3:311-323. David HammenDavid Hammen $\begingroup$ Thanks ! I must add I missed this wikipedia page en.wikipedia.org/wiki/Geothermal_gradient, which gives an inner/solar ratio of 0.03% $\endgroup$ – agemO May 10 '15 at 18:29 $\begingroup$ @agemO - That number is a bit unclear, but roughly correct. It's the ratio of 44.2 TW to the 340 W/m^2 received by the Earth at the top of the atmosphere, rounded up to 0.03%. (Aside: Where did wikipedia get that precise 44.2 TW? One of the things I don't like about wikipedia.) The Earth's surface receives only about 163.3 W/m^2 directly from the Sun, but also receives about 340.3 W/m^2 from the atmosphere thanks to the greenhouse effect. $\endgroup$ – David Hammen May 10 '15 at 18:48 $\begingroup$ @DavidHammen - just to clarify, that $340 \mathrm{\;W/m^2}$ number is $\frac14$ of the "direct" number because the earth presents a disk of $\pi R^2$ to the sun, but has a surface area of $4\pi R^2$, right? $\endgroup$ – Floris May 10 '15 at 19:19 $\begingroup$ @Floris - That's correct. That $340 W/m^2$ is averaged over the Earth's surface (actually, top of the atmosphere) over a period of 10+ years. After accounting for day and night, and for the equator to pole, you get a factor of 1/4. $\endgroup$ – David Hammen May 10 '15 at 19:41 $\begingroup$ @DavidHammen - thanks. I always learn a lot from your answers; please keep it up! $\endgroup$ – Floris May 10 '15 at 20:34 Not the answer you're looking for? Browse other questions tagged thermodynamics geophysics thermal-conductivity weather or ask your own question. For how much energy does the heat generated in earth's interior account in earth's total energy budget? Why has Earth's core not become solid? Why is Earth's climate so stable? Why isn't the Earth's core temperature the average of its surface temperatures? How does the inner core relieve stress as the Earth's rotation slows? Amount of thermal energy in the Earth? What is the evidence for the super-rotation of Earth's inner core? Does one square centimenter of the sun core really radiate this amount of energy? How does the sun's surface conduct thermal energy from the convective zone to the corona? How much of water's surface tension is entropic? Why doesn't the heat of the Earth's core diffuse to the surface? Why does the Earth even have a magnetic field? How does the Earth's center produce heat?	CommonCrawl
Search E-alert Submit My Account Login Article \| Open \| Published: 02 October 2018 A High-Speed SSVEP-Based BCI Using Dry EEG Electrodes Xiao Xing1,2, Yijun Wang1,2,3, Weihua Pei ORCID: orcid.org/0000-0002-1631-01991,2,3, Xuhong Guo1,2, Zhiduo Liu1,2, Fei Wang1,2, Gege Ming1,2, Hongze Zhao1,2, Qiang Gui1 & Hongda Chen1,2 Scientific Reportsvolume 8, Article number: 14708 (2018) \| Download Citation Sensors and probes A high-speed steady-state visual evoked potentials (SSVEP)-based brain-computer interface (BCI) system using dry EEG electrodes was demonstrated in this study. The dry electrode was fabricated in our laboratory. It was designed as claw-like structure with a diameter of 14 mm, featuring 8 small fingers of 6 mm length and 2 mm diameter. The structure and elasticity can help the fingers pass through the hair and contact the scalp when the electrode is placed on head. The electrode was capable of recording spontaneous EEG and evoked brain activities such as SSVEP with high signal-to-noise ratio. This study implemented a twelve-class SSVEP-based BCI system with eight electrodes embedded in a headband. Subjects also completed a comfort level questionnaire with the dry electrodes. Using a preprocessing algorithm of filter bank analysis (FBA) and a classification algorithm based on task-related component analysis (TRCA), the average classification accuracy of eleven participants was 93.2% using 1-second-long SSVEPs, leading to an average information transfer rate (ITR) of 92.35 bits/min. All subjects did not report obvious discomfort with the dry electrodes. This result represented the highest communication speed in the dry-electrode based BCI systems. The proposed system could provide a comfortable user experience and a stable control method for developing practical BCIs. The brain-computer interface (BCI) technique provides a direct communication pathway between the brain and the external world by translating signals from brain activities into machine codes or commands1. It has a wide application potential in our daily life. For example, electroencephalogram (EEG) has been used to control different types of external devices, such as a computer cursor, cellphone, home equipment or a wheelchair2,3,4,5,6. These EEG-based BCIs have provided new communication methods for either disabled or healthy people. BCI has a variety of paradigms including P300, motor imagery and steady-state visual evoked potentials (SSVEPs). SSVEP has been widely used in BCI due to high information transfer rate (ITR), little training and high reliability7,8,9,10. For example, in ref.11, Chen et al. obtained an ITR of 267 bits/min, the highest ITR to date, in a 40-target SSVEP BCI system. However, this result was achieved by gel-based wet electrodes. Gel needs to be injected to the electrode before system use. During recording, the gel tends to dry out over time. After recording, the user needs to wash out the gel in the user's hair and the EEG cap also needs to be cleaned. These procedures are time consuming and result in an uncomfortable user experience. Despite the high ITR of an SSVEP BCI, there is a certain distance between the BCI equipment and the requirement for practical applications. Major challenges include electrodes and devices for convenient EEG acquisition, and highly efficient artifact removal techniques. In order to simplify the preparation and process of wet electrodes, many types of dry-contact electrodes have been developed. They can be classified as micro-needle12,13, tips14,15, spring pin16, and soft conductive polymer17,18,19 electrodes. Those types of electrodes reach the advantage of low contact impedance, high signal quality, and comfortable user experience. However, there are few reports about dry electrodes used in BCI systems and the results of ITRs still showed large room for improvement. Table 1 summarizes the literatures of dry electrode used in BCI in recent years. Many groups have made efforts to improve the performance of the dry-electrode based BCI system. For example, Mihajlovic et al.20 used 8 metal pin-based dry electrodes to acquire SSVEPs, which could identify 4 targets with accuracy of 63% and ITR of 23 bits/min. Chi et al.14 designed a mobile BCI system, in which the data acquisition module was a wireless portable box and the signal processing was accomplished on a cellphone. The system could identify 12 targets from 3 spring-loaded-pin dry electrodes. Unlike traditional Fast Fourier Transform (FFT) algorithm to extract the features of SSVEPs, they used canonical correlation analysis (CCA) method to match the templates of sin/cos waveform. The average accuracy and ITR were 89% and 26.5 bits/min. Luo et al.21 proposed a novel stimulus-locked inter-trace correlation method for 4-target SSVEP classification using EEG time-locked to stimulus onsets, which only needed one spring-loaded-pin dry electrode. The accuracy and peak ITR reached 75.8% and 34.3 bits/min. In ref.22, Lo et al. designed a wireless control system using a non-contact metal plate electrode and an FFT-based algorithm. The accuracy and the best ITR of the system were 91.1% and 38.28 bits/min in the classification of 12 targets. In addition, Martin Spuler et al.23 used 15 g.Sahara electrodes (g.tec, Graz) to acquire code-modulated VEP, which could identify 32 targets with average accuracy of 76% and average ITR of 46 bits/min, suggesting peak ITRs over 100 bit/min are possible using dry EEG electrodes. Table 1 The performance of dry-electrode based BCI in recent years. Although dry electrodes show advantages of easy preparation and no need for cleaning after use, the signal quality acquired by dry electrodes is generally lower than that of wet electrodes. The quality of EEG is the key to the communication speed of BCI. Data with lower signal-to-noise ratio (SNR) require longer lengths for accurate target identification, leading to decreased ITRs. To compensate for the SNR decrease in dry electrodes, more efficient target identification algorithms can be employed to improve the speed and accuracy in SSVEP detection. Currently, for EEG recorded with dry electrodes, the efficacy of the individual template based SSVEP detection method still remains unknown. This study aimed to explore the speed limit of an SSVEP BCI using dry electrodes. Dry electrodes and the state-of-the-art algorithm were combined to develop a high-speed SSVEP BCI with improved feasibility and practicality. Firstly, we designed a claw-like flexible dry electrode to replace the traditional wet electrode or rigid dry electrodes. It can improve user comfort and acquire good signal quality. Secondly, we adopted an efficient SSVEP detection algorithm based on task-related component analysis (TRCA). The system only required 1-second-long SSVEP data to distinguish 12 targets with high accuracy. By combining these two factors, the proposed BCI system achieved high accuracy (93.2%) and ITR (92.35 bits/min).We hope the speed improvement of the dry-electrode based SSVEP BCI can pave the road for applications in both patients with motor disabilities and healthy people. For example, patients with Amyotrophic Lateral Sclerosis (ALS) can communicate with others using the SSVEP based BCI system. Dry Electrodes The electrode used in this study was shown in Fig. 1(a). It was designed as a claw-like structure, which was similar to the design of flexible dry electrodes in24,25, with a diameter of 14 mm, consisting of 8 small fingers, each showing 6 mm in length and 2 mm in diameter. Thermoplastic polyurethanes (TPU), a polymer elastomer was chosen to fabricate the electrode. The shape of the electrode was manufactured by molding. The surface of the electrode was coated with conductive ink of Ag for conductivity, and the tips of electrode were coated with conductive ink of Ag/AgCl mixture to improve electrochemical performance. The structure and material characteristic made it light and flexible to wear. The electrode is capable to go through the hair and make good contact to the scalp. As shown in Fig. 1(b), considering the effect of contact area on contact impedance, the tips of claw-like dry electrode were designed to be hemispherical. The hemispherical shape helps maintain contact area at a certain range of pressure. The shape and the elastic TPU improve comfort level compared with dry electrodes made of stiff material. As shown in Fig. 1(c), the material of soft headband was elasticized fabric. Soft and elastic headband can help to maintain a stable connection between the electrodes and the scalp. (a)The dry claw electrode, (b) the illustration of electrodes on the scalp, and (c) the soft headband with dry claw electrodes. This study designed an SSVEP-based BCI experiment to test the dry-electrode based BCI system. During the experiment, eight dry claw electrodes were placed at a soft headband (Fig. 1(c)). The size of headband was 18 cm × 9 cm. It could acquire the EEG signals from occipital and parietal areas (PO5, PO3, POz, PO4, PO6, O1, Oz and O2), where the SSVEPs show maximal amplitudes and SNRs7,11, referring to the international 10–10 system. The reference and ground electrodes were placed at the forehead using commercial hydrogel skin electrodes, which can ensure good and stable contact. The signals were recorded by a Neuroscan Synamps2 system at a sampling rate of 1000 Hz. After the headband was put on and the dry electrodes were connected to the Neuroscan System, impedance of the electrodes were tested and displayed. The impedance of each the electrode can be adjusted in two minutes to ensure the largest impedance lower than 50 KΩ. If the impedances were still lower than this value after the experiment, we believed the contact was stable. Since the frequencies of the SSVEP components are generally below 90 Hz7, in online analysis, data were down-sampled to 250 Hz to reduce computational cost. For the comparison purpose, after subjects completed the dry electrodes-based BCI experiment, they were also asked to use the Ag/AgCl wet electrodes to complete the experiment in the same way. The test order of dry and wet electrodes was randomized. To avoid the interference of visual fatigue, the test interval between dry and wet electrodes was 4 hours. The subjects cleaned hair after the test of wet electrodes. The visual stimulator of the BCI consisted of 12 flickering stimuli rendered on a PC monitor with a 60 Hz refresh rate. This study used the sampled sinusoidal stimulation method9 to present the frequency-phase coded stimuli, which were proved accurately in previous studies11,26. As shown in Fig. 2(a), the stimuli were arranged in a 3 × 4 matrix as a virtual keypad of a phone26, and tagged with different frequencies and phases. The frequency range was selected from 9.25 Hz to 14.75 Hz with an interval of 0.5 Hz. The phase values started from 0 and the phase interval was 0.5π. The stimulus program was developed under MATLAB (MathWorks, Inc.) using the Psychphysics Toolbox Version 327. (a) Visual stimulus layout together with frequency and phase values for encoding the stimuli. (b) The timeline of a trial in the experiment. Eleven subjects (4 females, mean age: 25 years old) were recruited from Chinese Academy of Sciences (CAS) and took part in the experiment. All of them had normal or corrected to normal vision and had no history of central nervous system abnormalities. This study was approved by CAS's Institutional Review Board. All experimental protocols were conducted in accordance with CAS ethical guidelines and informed consent was obtained from all participants. During the experiment, subjects seated in a comfortable chair 60 cm in front of the screen in a normally lit room. The experiment was divided into a training stage and a testing stage. The training data were used to design spatial filters and EEG templates, which were used for target detection in the testing stage. The training stage consisted of 10 blocks. Each block contained 12 trials corresponding to 12 targets presented in a random order. As shown in Fig. 2(b), each trial started with a visual cue indicating a target stimulus. The cue appeared for 0.3 s on the screen. Subjects were asked to shift their gaze to the target as soon as possible within the cue duration. Then the stimuli started to flicker for 1 s. After stimulus offset, the screen was blank for 0.7 s before the next trial began. Therefore, each trial lasted for 2 s. There was a rest to avoid visual fatigue between two consecutive blocks. The testing stage consisted of 5 blocks, each including 12 trials. The cue time, stimulus time and rest time were as the same as the training stage. A short beep was sounded after a target was correctly identified by the online analysis program. At the same time, the target character was typed in the text input field on the top of the screen. Identification Algorithm Figure 3 showed the flowchart of the TRCA-based target identification method28. It was mainly divided into four steps: preprocessing, construction of spatial filters, feature extraction and identification. Individual calibration data and single-trial test data for the n-th stimulus are denoted by ${{\rm{\chi }}}_{{\rm{njkh}}}\in {{\rm{R}}}^{{\rm{Nf}}\times {\rm{Nc}}\times {\rm{Ns}}\times {\rm{Nt}}}\,\,$and ${\rm{X}}\in {{\rm{R}}}^{{\rm{Nc}}\times {\rm{Ns}}}$ respectively. Here, n indicates the stimulus index, Nf is the number of stimuli, j indicates the channel index, Nc is the number of channels, k indicates the index of sample points, Ns is the number of sampling points in each trial, h indicates the index of training trials, and Nt is the number of training trials. The flowchart of the TRCA-based identification method. First, the training data χ are processed by filter bank analysis (FBA), where the SSVEPs are decomposed into m sub-band components. In filter bank analysis, the lower and upper cut-off frequencies of the m-th sub-band were set to m × 8 Hz and 90 Hz, respectively. According to7, we chose m = 5. After applying m zero-phase Chebyshev Type I Infinite impulse response (IIR) filters, the training data and test data are denoted as χ(m) and X(m). Zero-phase forward and reverse filtering was implemented using the filtfilt () function in MATLAB. Second, spatial filters for the n-th stimuli ${{\rm{W}}}_{{\rm{n}}}^{({\rm{m}})}$ are obtained through TRCA from individual calibration data X(m) as Equation (1): $$\hat{w}={{\rm{\arg }}}_{w}\,{\rm{\max }}\,\frac{{w}^{T}Sw}{{w}^{T}Qw}$$ The optimal coefficient vector is the first eigenvector of the matrix Q−1S29, where Q is the covariance of concatenated matrix of all training trials across the stimuli, and S is the correlation coefficient matrix of the n-th stimulus between all training trials. By integrating all Nf coefficient vectors, we can obtain ensemble spatial filters W(m). Meanwhile, data for the multiple training trials are averaged (MTA) to obtain the individual template ${\overline{{{\rm{\chi }}}_{{\rm{n}}}}}^{({\rm{m}})}$. Third, individual template ${\overline{{{\rm{\chi }}}_{{\rm{n}}}}}^{({\rm{m}})}$ and test data X(m)are multiplied with spatial filters W(m)respectively, and then the Pearson's correlation coefficient ${{\rm{\gamma }}}_{{\rm{n}}}^{({\rm{m}})}$ between them can be calculated. Last, the final features ρn can be obtained by merging the correlation coefficients ${{\rm{\gamma }}}_{{\rm{n}}}^{({\rm{m}})}$ as Equation (2), and the target class τ can be identified as Equation (3): $${{\rm{\rho }}}_{{\rm{n}}}=\sum _{m=1}^{{N}_{m}}a(m)\cdot {({\gamma }_{n}^{(m)})}^{2}$$ $${\rm{\tau }}={{\rm{\arg }}}_{n}{\max {\rm{\rho }}}_{n}\,n=1,2\ldots {N}_{f}$$ where, Nm is the total number of sub-bands, and ${\rm{a}}(m)={m}^{-1.25}+0.25$ according to7. Impedance of the electrode The impedance property of dry electrode was evaluated using electrochemical impedance spectroscopy (EIS). Firstly, in order to make a controlled and consistent testing environment, the data were measured by an electrochemical workstation (CHI 660D, China) and tested in 0.9% NaCl solution. The proposed dry electrode was set as the measurement electrode and standard Pt electrodes were set as the reference and ground electrodes. Sinusoidal AC signals with voltage amplitude of 10 mV, frequencies from 0.1 Hz to 1000 Hz, were applied to measure the EIS. For comparison purposes, commercial Ag/AgCl wet electrodes were measured using the same method. The experiment was repeated three times. Secondly, the impedance of the dry electrode was tested on volunteers' head30. Two dry electrodes, one as the working electrode and the other as the reference electrode, were placed 3 cm away on the occipital region. An elastic headband was used to compress them on the head of subjects. EIS of the dry electrode was tested with the same condition as that in the wet environment. The value would be divided by 2 to get the average impedance of one dry electrode. For comparison, commercial Ag/AgCl wet electrodes were measured with commercial gel (Compumedics Neuromedical SuppliesTM) on head using the same method. The test was performed on five subjects. Signal quality of the electrode The signal quality of the dry electrode was evaluated by calculating the correlation coefficient and comparing the SNR of SSVEP components between the dry and wet electrodes. The signal was collected from the dry electrode and Ag/AgCl wet electrode at the same time. They were placed at the area near the Oz position, and the distance was 2 cm apart. The reference and ground were wet electrodes on the ear lobes (A1 and A2 position, respectively). The signals were recorded by a Neruoscan Synamps2 system at a sampling rate of 1000 Hz. The test consisted of 5 trials. In one trial, there were 4 flickers rendered on a monitor in turn and the frequencies were 10 Hz, 12 Hz, 15 Hz, and 20 Hz. Each stimulus lasted for 10 s followed by a rest for 5 s. Before analyzing the data, data epochs of 10 s data length were extracted according to the event triggers generated by the stimulus program. All data epochs were first down-sampled to 250 Hz and then filtered with a 50 Hz notch filter to remove the power line noise. The correlation coefficient was calculated as follows: $${\rm{R}}={\rm{cov}}(x1,x2)/\sqrt{{\rm{cov}}(x1,x1)\,{\rm{cov}}(x2,x2)}\,$$ where R symbolized the correlation coefficient, cov(.) represented covariance, and x1 and x2 represented filtered signals recorded by the dry and wet electrodes respectively. The SNR was calculated as follows: $${\rm{SNR}}=20\ast {\mathrm{log}}_{10}\frac{y(f)}{y(f-1)+y(f+1)}\,$$ The amplitude spectrum y(f) was calculated by FFT. SNR in decibels (dB) was defined as the ratio of y(f) to the mean value of the 2 neighboring frequencies (i.e. one frequency on each side). Performance of the system The performance of the BCI system was evaluated using the classification accuracy and ITR. The ITR was calculated as follows: $${\rm{ITR}}=\frac{60}{T}[{\mathrm{log}}_{2}N+P{\mathrm{log}}_{2}P+(1-P)\,{\mathrm{log}}_{2}\frac{1-P}{N-1}))]$$ where P was the accuracy, T was the average time for a command (T = 2 s), and N was the number of commands (N = 12). Comfort level assessment The comfort level of the electrodes was assessed in the form of a survey. All subjects were asked to report the comfort level of the proposed electrode and a commercial dry electrode (purchased from Wearable Sensing Company, its pin is fabricated from a stiff metal material) after wearing the electrodes longer than 1 h. Comfort levels are divided into four grades: (1) Comfort with gentle sense of pressure but no pain, (2) Slight pain, (3) Pain but acceptable, and (4) Obvious pain with discomfort19,31,32. Dry Electrode Performance Results of EIS in wet condition were presented in Fig. 4(a). The impedance will decrease as the frequency increase. At the frequency around 10 Hz, the average impedance of dry electrode and wet electrode were 1.52 ± 1.54 KΩ and 1.72 ± 1.53 KΩ respectively. There was no clear difference of the mean impedance. It means that the proposed dry electrodes will have similar impedance property to that of the commercial wet electrodes if they can contact with the scalp with large enough area. However, without the help of gel, the effective contact area of the dry electrode is very small. As shown in Fig. 4(b), the dry-contact test result shows that the average impedance is 38.6 KΩ ± 9.5 KΩ (@10 Hz) on the scalp. Compare with the impedance of wet electrodes, 8.3 KΩ ± 1.67 KΩ (@10 Hz), the value is 4–5 fold higher. (a) The amplitude of EIS from dry electrode and wet electrode in 0.9% NaCl solution. (b) The amplitude of EIS from dry electrode and wet electrode on the scalp. EEG signal quality Figure 5 showed the averaged temporal waveform and amplitude spectrum of SSVEP at 10 Hz collected from the dry electrode and wet electrode from all subject. Both electrodes could acquire clear SSVEPs that exhibited peak frequencies at 10 Hz and 20 Hz. The temporal waveforms for the two electrodes were highly correlated to each other (R = 0.83). (a) The averaged temporal waveform and (b) amplitude spectrum of 10 Hz SSVEP from all subjects corresponding to dry electrode (black) and wet electrode (red). Table 2 showed the SNR and correlation coefficients from the two electrodes at 4 stimuli for all subjects. The average SNR across frequencies for the dry electrode and the wet electrode were 12.83 ± 2.85 dB and 14.1 ± 3.24 dB respectively. The average correlation coefficient between dry electrode and wet electrode was 0.66 ± 0.02. Table 2 The quality of SSVEPs from dry electrode and wet electrode. BCI Performance The results of the online BCI experiment were shown in Table 3. For the wet electrode, the average accuracy and ITR of all 11 subjects was 97.35 ± 4.33% and 101.28 ± 9.46 bits/min. Compared to the wet electrode, the average accuracy of dry electrode was 93.2 ± 5.74% and ITR was 92.35 ± 12.08 bits/min. Paired t-tests indicated significant difference of classification accuracy (p < 0.05) and ITR (p < 0.05) between the two type of electrodes. Four (S3, S5, S6, and S8) and two subjects (S3, S5) obtained 100% accuracy using the wet electrode and the dry electrode respectively. Compared with the TRCA algorithm, offline analysis using the CCA algorithm obtained decreased accuracy and ITR. The difference of accuracy between CCA and TRCA was significant for both wet (89.9% vs 97.35%, p < 0.01) and dry (74.82% vs 93.2%, p < 0.01) electrodes. These results indicate that the TRCA algorithm plays a key role in achieving high ITR with the dry-electrode based SSVEP BCI. Table 3 The performance of BCI from dry and wet electrodes. Comfort Level of Electrode Most participants in this study had previous experience of traditional gel-based wet electrodes and different types of dry electrodes. All of them thought the pin-type dry electrode caused stronger feeling of oppression than the claw dry electrode. For the claw dry electrode, four subjects chose level 1, six subjects chose level 2, and one subject chose level 3. In contrast, for the pin-type electrode, one subject chose level 1, four subjects chose level 2, and others chose level 3. Compared with the pin-type electrode, the deformation of the claw electrode relieves the skin pressure to a certain extent. This study obtained the highest ITR (92.35 bits/min) reported in the dry-electrode based BCIs (see Table 1). Specifically, the ITRs of the calibration-free SSVEP BCI systems ranged from 14.5–38.28 bits/min. The 32-target code-modulated VEP system using the individual template-based CCA algorithm23 obtained an ITR of 46 bits/min. The proposed system benefited from the higher classification accuracy and shorter stimulation duration, which were contributed by the self-fabricated dry electrodes and the adopted TRCA-based detection algorithm. By optimizing the structure and materials, the claw-like dry electrode acquired good signal quality and provided reliable data for subsequent signal processing. The TRCA algorithm fully considered the individual differences and took individual calibration data as template, so it achieved high classification accuracy with a short data length. The TRCA algorithm requires system calibration to obtain individual templates and spatial filters. In this study, the training procedure included 10 blocks, which lasted 4 minutes in total. The systems that do not require system calibration are more user friendly. However, as shown in Table 3, the unsupervised CCA algorithm showed a large drop (TRCA: 93.2%, CCA: 74.82%) of accuracy with the dry electrodes. This finding suggests that a much longer stimulation duration is required to obtain high accuracy using the CCA method. Alternatively, zero-training methods such as the session-to-session transfer algorithm33 could be an effective way to facilitate system implementation using the TRCA algorithm. The online BCI system used 1 s stimulation duration towards high classification accuracy. However, the ITR can be further improved by optimizing the stimulus duration. Figure 6 showed the results of average classification accuracy and ITR across all subjects with different training data lengths from dry electrode and wet electrode. The accuracy and the ITR were estimated by a leave-one out cross validation, in which 9 blocks were used as training data and 1 block was used as test data. For the wet electrode, the highest ITR was 129.92 ± 16.56 bits/min with the data length of 400 ms and the accuracy was 93.34 ± 5.58%. For the dry electrode, the highest ITR was 102.37 ± 26.92 bits/min with the data length of 500 ms and the accuracy was 85.45 ± 11.06%. This result suggested we could obtain higher ITR by optimizing stimulus duration (e.g., 500 ms). Results of average classification accuracy (a) and ITR (b) with different data lengths from dry electrode (black) and wet electrode (red). (p < 0.05, *p < 0.01). Paired t-tests showed the accuracy and ITR of two kinds of electrodes had significant difference for all data lengths. It meant the signal quality of short data was still different between dry and wet electrodes. Figure 7(a) showed the original 10 Hz SSVEP of 1 s stimulation duration without filtering. From the amplitude spectrum in Fig. 7(b), in the signal frequency band of SSVEP (8–30 Hz), there was no large difference. But in the low frequency band (0–5 Hz) and at the power line frequency (50 Hz), the dry electrode acquired more interference than wet electrode obviously. The low frequency noise mainly comes from the unstable contact between electrode and skin. This instability will cause the change of electrode-skin electron double layer and therefore produce the noise. There are two ways to produce the power line noise. First, when the contact impedances of work electrode and reference electrode do not match, the power line interference will be amplified by acquisition circuit in the form of differential-mode signal and then mixed in the EEG signals. Second, the power line interference can directly couple into the EEG signals through the unshielded electrode. Therefore, in the future, the anti-interference ability of the dry electrode need to be improved from the following two aspects. First, the mechanical properties such as elasticity and hardness should be adjusted to make the dry electrode contact with the scalp in a stable and comfortable way. It is helpful to avoid the relative sliding of the electrode and thus reduce the low frequency noise. Second, the power line noise can be reduced by designing a shielding layer for dry electrode. The original temporal waveform (a) and spectrum (b) of 1s-long data from dry electrode (black) and wet electrode (red). This study demonstrates a high performance SSVEP-based BCI system using dry claw-like electrodes. The proposed dry electrode reduces the system preparation time. Its flexible increases the wearing comfort and improves the user experience. The adopted TRCA algorithm improves the classification accuracy and ITR of system. Both ergonomic factors and system performance factors are optimized towards practical BCI applications. The material as well as structure of the dry electrode have been designed and adjusted elaborately. It can be wore comfortably on hair covered head area and the pins of the electrode can go easily through the hair and contact the scalp. The impedance of the electrode is stable and therefore the electrode is capable to record reliable and high quality signals for subsequent signal processing. The adopted TRCA-based identification algorithm can achieve high accuracy with short data by fully considering the individual differences of SSVEPs. 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The authors acknowledge the support from the National Natural Science Foundation of China (61634006, 61335010, and 61671424); National Key Technologies R&D Program (2017YFA0205903, 2017YFJC020030, 2016YFB0401303, 2016YFB0402405); The Basic Research Project of Shanghai Science and Technology Commission (16JC1400101); The Key project of Chinese Academy of Science (QYZDY-SSW-JSC004, KJZD-EW-L11-01);The Recruitment Program of Young Professionals; Key research program of frontier science, CAS (QYZDY-SSW-JSC004); National Military Science Foundation of China (AWS16J028) and Beijing S&T planning task (Z161100002616019). The State Key Laboratory on Integrated Optoelectronics, Institute of Semiconductors, Chinese Academy of Sciences, Beijing, 100083, China Xiao Xing , Yijun Wang , Weihua Pei , Xuhong Guo , Zhiduo Liu , Fei Wang , Gege Ming , Hongze Zhao , Qiang Gui & Hongda Chen The University of Chinese Academy of Sciences, Beijing, 100049, China CAS Center for Excellence in Brain Science and Intelligence Technology, Shanghai, China Yijun Wang & Weihua Pei Search for Xiao Xing in: Search for Yijun Wang in: Search for Weihua Pei in: Search for Xuhong Guo in: Search for Zhiduo Liu in: Search for Fei Wang in: Search for Gege Ming in: Search for Hongze Zhao in: Search for Qiang Gui in: Search for Hongda Chen in: Xiao Xing designed and manufactured the sensor, performed experiments and processed data. Yijun Wang gave some advice and guidance about algorithm. Weihua Pei offered help in the sensor fabrication. Xuhong Guo, Zhiduo Liu, Fei Wang, Qiang Gui and Hongda Chen contributed to writing the manuscript and assisting experiment. Gege Ming and Hongze Zhao assisted in processing data. Yijun Wang and Weihua Pei gave constructive advice in whole process. All the authors discussed the results. Corresponding authors Correspondence to Yijun Wang or Weihua Pei. The authors declare no competing interests. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 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Costa , Filippo Spina , Pasindu Lugoda , Leonardo Garcia-Garcia , Daniel Roggen & Niko Münzenrieder Technologies (2019) Portable brain-computer interface based on novel convolutional neural network Yu Zhang , Xiong Zhang , Han Sun , Zhaowen Fan & Xuefei Zhong Computers in Biology and Medicine (2019) Scientific Reports menu Scientific Reports Top 100 2017 Scientific Reports Top 10 2018 Guest Edited Collections Editorial Board Highlights Author Highlights Scientific Reports rigorous editorial process Open Access Funding Support	CommonCrawl
Many presentations of axiomatic set theory contain an error Another test for divisibility by 7 [ Content warning: highly technical mathematics ] I realized recently that there's a small but significant error in many presentations of the Zermelo-Frankel set theory: Many authors omit the axiom of the empty set, claiming that it is omittable. But it is not. The overarching issue is as follows. Most of the ZF axioms are of this type: If !!\mathcal A!! is some family of sets, then [something derived from !!\mathcal A!!] is also a set. The axiom of union is a typical example. It states that if !!\mathcal A!! is some family of sets, then there is also a set !!\bigcup \mathcal A!!, which is the union of the members of !!\mathcal A!!. The other axioms of this type are the axioms of pairing, specification, power set, replacement, and choice. There is a minor technical problem with this approach: where do you get the elements of !!\mathcal A!! to begin with? If the axioms only tell you how to make new sets out of old ones, how do you get started? The theory is a potentially vacuous one in which there aren't any sets! You can prove that if there were any sets they would have certain properties, but not that there actually are any such things. This isn't an entirely silly quibble. Prior to the development of axiomatic set theory, mathematicians had been using a model called naïve set theory, and after about thirty years it transpired that the theory was inconsistent. Thirty years of work about a theory of sets, and then it turned out that there was no possible universe of sets that satisfied the requirements of the theory! This precipitated an upheaval in mathematics a bit similar to the quantum revolution in physics: the top-down view is okay, but the most basic underlying theory is just wrong. If we can't prove that our new theory is consistent, we would at least like to be sure it isn't trivial, so we would like to be sure there are actually some sets. To ensure this, the very least we can get away with is this axiom: !!A_S!!: There exists a set !!S!!. This is enough! From !!A_S!! and specification, we can prove that there is an empty subset of !!S!!. Then from extension, we can prove that this empty subset is the unique empty set. This justifies assigning a symbol to it, usually !!\varnothing!! or just !!0!!. Once we have the empty set, pairing gives us !!\{0,0\} = \{0\} = 1!!, then !!\{0, 1\} = 2!! , and so on. Once we have these, the axioms of union and infinity show that !!\omega!! is a set, then from that the axiom of power sets gets us uncountable sets, and the sky is the limit. But we need something like !!A_S!! to get started. In place of !!A_S!! one can have: !!A_\varnothing!!: There exists a set !!\varnothing!! with the property that for all !!x!!, !!x\notin\varnothing!!. Presentations of ZF sometimes include this version of the axiom. It is easily seen to be equivalent to !!A_S!!, in the sense that from either one you can prove the other. I wanted to see how this was handled in Thomas Jech's Set Theory, which is a standard reference text for axiomatic set theory. Jech includes a different version of !!A_S!!, initially given (page 3) as: !!A_∞!!: There exists an infinite set. This is also equivalent to !!A_S!! and !!A_\varnothing!!, if you are willing to tolerate the use of the undefined term "infinite". Jech of course is perfectly aware that while this is an acceptable intuitive introduction to the axiom of infinity, it's not formally meaningful without a definition of "infinite". When he's ready to give the formal version of the axiom, he states it like this: $$\exists S (\varnothing \in S\land (\forall x\in S) x\cup\{x\}\in S).$$ ("There is a set !!S!! that includes !!\varnothing!! and, whenever it includes some !!x!!, also includes !!x\cup\{x\}!!." (3rd edition, p. 12)) Except, oh no, "!!\varnothing!!" has not yet been defined, and it can't be, because the thing we want it to refer to cannot, at this point, be proved to actually exist. Maybe you want to ask why we can't use it without proving that it exists. That is exactly what went wrong with naïve set theory, and we don't want to repeat that mistake. I brought this up on math Stack Exchange and Asaf Karagila, the resident axiomatic set theory expert, seemed to wonder why I complained about !!\varnothing!! but not about !!\{x\}!! and !!\cup!!. But the issue doesn't come up with !!\{x\}!! and !!\cup!!, which can be independently defined using the axioms of pairing and union, and then used to state the axiom of infinity. In contrast, if we're depending on the axiom of infinity to prove the existence of !!\varnothing!!, it's circular for us to assume it exists while writing the statement of the axiom. We can't depend on !!A_∞!! to define !!\varnothing!! if the very meaning of !!A_∞!! depends on !!\varnothing!! itself. That's the error: the axioms, as stated by Jech, are ill-founded. This is a little hard to see because of the way he prevaricates the actual statement of the axiom of infinity. On page 8 he states !!A_\varnothing!!, which would work if it were included, but he says "we have not included [!!A_\varnothing!!] among the axioms, because it follows from the axiom of infinity." But this is wrong. You really do need an explicit axiom like !!A_\varnothing!! or !!A_S!!. As far as I can tell, you cannot get away without it. This isn't specifically a criticism of Jech or the book; a great many presentations of axiomatic set theory make the same mistake. I used Jech as an example because his book is a well-known authority. (Otherwise people will say "well perhaps, but a more careful writer would have…". Jech is a careful writer.) This is also not a criticism of axiomatic set theory, which does not collapse just because we forgot to include the axiom of the empty set.	CommonCrawl
DOI:10.3847/2041-8213/ab042a Identifying Interstellar Objects Trapped in the Solar System through Their Orbital Parameters @article{Siraj2019IdentifyingIO, title={Identifying Interstellar Objects Trapped in the Solar System through Their Orbital Parameters}, author={Amir Siraj and Abraham Loeb}, journal={The Astrophysical Journal}, A. Siraj, A. Loeb The first interstellar object, `Oumuamua, was discovered in the Solar System by Pan-STARRS in 2017, allowing for a calibration of the abundance of interstellar objects of its size and an estimation of the subset of objects trapped by the Jupiter-Sun system. Photographing or visiting these trapped objects would allow for learning about the conditions in other planetary systems, saving the need to send interstellar probes. Here, we explore the orbital properties of captured interstellar objects… The New Astronomical Frontier of Interstellar Objects The upcoming commencement of the Vera C. Rubin Observatory's Legacy Survey of Space of Time (LSST) will greatly enhance the discovery rate of interstellar objects (ISOs). 'Oumuamua and Borisov were… No evidence for interstellar planetesimals trapped in the Solar system A. Morbidelli, K. Batygin, R. Brasser, S. Raymond In two recent papers published in MNRAS, Namouni and Morais claimed evidence for the interstellar origin of some small Solar system bodies, including: (i) objects in retrograde co-orbital motion with… Exobodies in Our Back Yard: Science from Missions to Nearby Interstellar Objects T. Eubanks, J. Schneider, A. Hein, A. Hibberd, R. Kennedy Engineering, Physics The recent discovery of the first confirmed Interstellar Objects (ISOs) passing through the Solar System on clearly hyperbolic objects opens the potential for near term ISO missions, either to the… On the Capture of Interstellar Objects by Our Solar System K. Napier, F. Adams, K. Batygin Motivated by recent visits from interstellar comets, along with continuing discoveries of minor bodies in orbit of the Sun, this paper studies the capture of objects on initially hyperbolic orbits by… Capture of interstellar objects: a source of long-period comets T. Hands, W. Dehnen We simulate the passage through the Sun-Jupiter system of interstellar objects (ISOs) similar to 1I/`Oumuamua or 2I/Borisov. Capture of such objects is rare and overwhelmingly from low incoming… View 4 excerpts, cites results High-drag Interstellar Objects and Galactic Dynamical Streams T. Eubanks The nature of 1I/'Oumuamua (henceforth, 1I), the first interstellar object known to pass through the solar system, remains mysterious. Feng \& Jones noted that the incoming 1I velocity vector "at… Halo Meteors Amir Siraj, A. Loeb The stellar halo contains some of the oldest stars in the Milky Way galaxy and in the universe. The detections of `Oumuamua, CNEOS 2014-01-08, and interstellar dust serve to calibrate the production… Galactic tide and local stellar perturbations on the Oort cloud: creation of interstellar comets S. Torres, M. Cai, A. Brown, S. Zwart Comets in the Oort cloud evolve under the influence of internal and external perturbations, such as giant planets, stellar passages, and the Galactic gravitational tidal field. We aim to study the… Detecting Interstellar Objects through Stellar Occultations Stellar occultations have been used to search for Kuiper Belt and Oort Cloud objects. We propose a search for interstellar objects based on the characteristic durations ($\sim 0.1 \mathrm{s}$) of… Exploring Long-Period Comets from Multiple Staging Orbits Gabriel Prescinotti Vivan, J. Hudson The Journal of the Astronautical Sciences This paper explores multiple heliocentric staging orbits around Lagrange points to serve as departure positions for future missions, prior to objects' detections, by utilizing more than one staging orbit concurrently. Spectroscopy and thermal modelling of the first interstellar object 1I/2017 U1 'Oumuamua A. Fitzsimmons, C. Snodgrass, +7 authors P. Lacerda During the formation and evolution of the Solar System, significant numbers of cometary and asteroidal bodies were ejected into interstellar space1,2. It is reasonable to expect that the same… A brief visit from a red and extremely elongated interstellar asteroid K. Meech, R. Weryk, +15 authors S. Chastel Observations and analysis of the object 1I/2017 U1 ('Oumuamua) that demonstrate its extrasolar trajectory, and that enable comparisons to be made between material from another planetary system and from the authors' own, reveal it to be asteroidal with no hint of cometary activity despite an approach within 0.25 astronomical units of the Sun. An interstellar origin for Jupiter's retrograde co-orbital asteroid F. Namouni, M. Morais Asteroid (514107) 2015 BZ509 was discovered recently in Jupiter's co-orbital region with a retrograde motion around the Sun. The known chaotic dynamics of the outer Solar System have so far precluded… Will the Large Synoptic Survey Telescope Detect Extra-Solar Planetesimals Entering the Solar System? A. Moro-martin, E. Turner, A. Loeb Planetesimal formation is a common by-product of the star formation process. Taking the dynamical history of the solar system as a guideline—in which the planetesimal belts were heavily depleted due… Interstellar Interlopers: Number Density and Origin of 'Oumuamua-like Objects A. Do, M. Tucker, J. Tonry We provide a calculation of Pan-STARRS' ability to detect objects similar to the interstellar object 1I/2017 U1 (hereafter 'Oumuamua), including the most detectable approach vectors and the effect of… An Observational Upper Limit on the Interstellar Number Density of Asteroids and Comets T. Engelhardt, R. Jedicke, +4 authors B. Meinke We derived 90% confidence limits (CL) on the interstellar number density ($\rho_{IS}^{CL}$) of interstellar objects (ISO; comets and asteroids) as a function of the slope of their size-frequency… Col-OSSOS: Colors of the Interstellar Planetesimal 1I/'Oumuamua M. Bannister, M. Schwamb, +10 authors M. Lehner The recent discovery by Pan-STARRS1 of 1I/2017 U1 (`Oumuamua), on an unbound and hyperbolic orbit, offers a rare opportunity to explore the planetary formation processes of other stars, and the… Observational Constraints on the Centaur Population R. Jedicke, J. D. Herron Abstract Spacewatch was the first large-scale astronomical survey program employing automated and real-time software for moving object detection. The search has resulted in the discovery of three new… The Feasibility and Benefits of In Situ Exploration of 'Oumuamua-like Objects Darryl Z. Seligman, G. Laughlin A rapid accumulation of observations and interpretation have followed in the wake of 1I `Oumuamua's passage through the inner Solar System. We briefly outline the consequences that this first… Kinematics of the Interstellar Vagabond 1I/'Oumuamua (A/2017 U1) E. 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Pattern Recognition and Machine Intelligence International Conference on Pattern Recognition and Machine Intelligence PReMI 2015: Pattern Recognition and Machine Intelligence pp 33-43 \| Cite as Towards a Robust Scale Invariant Feature Correspondence Shady Y. El-Mashad Amin Shoukry In this paper, we introduce an improved scale invariant feature correspondence algorithm which depends on the Similarity-Topology Matching algorithm. It pays attention not only to the similarity between features but also to the spatial layout of every matched feature and its neighbours. The features are represented as an undirected graph where every node represents a local feature and every edge represents adjacency between them. The topology of the resulting graph can be considered as a robust global feature of the represented object. The matching process is modeled as a graph matching problem; which in turn is formulated as a variation of the quadratic assignment problem. The Similarity-Topology Matching algorithm achieves superior performance in almost all the experiments except when the image has been exposed to scaling deformations. An amendment has been done to the algorithm in order to cope with this limitation. In this work, we depend not only on the distance between the two interest points but also on the scale at which the interest points are detected to decide the neighbourhood relations between every pair of features. A set of challenging experiments conducted using 50 images (contain repeated structure) representing 5 objects from COIL-100 data-set with extra synthetic deformations reveal that the modified version of the Similarity-Topology Matching algorithm has better performance. It is considered more robust especially under the scale deformations. Features matching Features extraction Topological Relations Graph matching Performance evaluation Download conference paper PDF Image matching or in other words, comparing images in order to obtain a measure of their similarity, is an important computer vision task. It is involved in many different applications, such as object detection and recognition, image classification, content based image retrieval, video data mining, image stitching, stereo vision, and 3D object modeling. A general solution for identifying similarities between objects and scenes within a database of images is still a faraway goal. There are a lot of challenges to overcome such as viewpoint or lighting variations, deformations, and partial occlusions that may exist across different examples. Furthermore, image matching as well as many other vision applications rely on representing images with sparse number of distinct keypoints. A real challenge is to efficiently detect and describe keypoints, with robust representations invariant against scale, rotation, view point change, noise, as well as combinations of them [1]. Keypoint detection and matching pipeline has three distinct stages which are feature detection, feature description and feature matching. In the feature detection stage, every pixel in the image is checked to see if there is a unique feature at this pixel or not. Subsequently, during the feature description stage, each region (patch) around the selected keypoints is described with a robust and invariant descriptor which can be used to match against other descriptors. Finally, at the feature matching stage, an efficient search for prospective matching descriptors in other images is made [2]. In the context of matching, a lot of studies have been used to evaluate interest point detectors as in [3, 4]. On the other hand, little efficient work has been done on the evaluation of local descriptors. K. Mikolajczyk and C. Schmid [5], proposed and compared different feature detectors and descriptors as well as different matching approaches in their study. Although this work proposed an exhaustive evaluation of feature descriptors, it is still unclear which descriptors are more appropriate in general and how their performance depends on the interest point detector. D.G. Lowe [6], proposed a new matching technique using distinctive invariant features for object recognition. Interest points are matched independently via a fast nearest-neighbour algorithm to the whole set of interest points extracted from the database images. Therefore, a Hough transform to identify clusters belonging to a single object has been applied. Finally, least-squares solution for consistent pose parameters has been used for the verification. Another technique to find correspondences is RANSAC. The most beneficial side of RANSAC is the ability of jointly estimating the largest set of mutual compatible correspondences between two views. Zhang and Kosecka [7] demonstrate the shortcomings of RANSAC when dealing with images containing repetitive structures. The failure of RANSAC in these cases is due to the fact that similarity measure is used to find matching based only on feature descriptor and, with repetitive structures, the chosen descriptors can change dramatically. Therefore, the nearest neighbor strategy is not an appropriate solution. There are two levels to measure the images similarity which are patch and image levels. In the patch level, the distance between any two patches is measured based on their descriptors. In the image level, the overall similarity between any two images is calculated which in most cases contain many patches. The Minkowski-type metric has been used to measure the distance between patches in most of researches. The Minkowski metric is defined as in (1): $$\begin{aligned} D(X,Y) = (\sum _{i=1}^P \|X_i- Y_i \|^r )^{1/r} \end{aligned}$$ when r = 2, it is the Euclidean distance (L2 distance), and it is the Manhattan distance (L1 distance) when r = 1 [8]. In the approach proposed in the present paper both local and global features are considered simultaneously. We try to retain the locality of the features advantages in addition to preserving the overall layout of the objects. The similarity between the local features has been used in conjunction with the topological relations between them as a global feature of the object. In this paper, the approach presented in [9, 10] is modified to be scale invariant. In addition, intensive experiments are conducted mainly focused on images with different resolutions as the objective of the modified algorithm to be more scale invariant. The images contain a duplication of the same object which reflects the scope of work (dealing with repeated structure). This paper is organized as follows: The proposed scale invariant feature correspondence algorithm is introduced in Sect. 2. Section 3, presents the conducted experiments to evaluate the performance of the modified matching approach. Finally, the conclusions of this work and the recommendations for future work are presented in Sects. 4 and 5, respectively. 2 Proposed Matching Approach Conventional matching approaches reduce the matching problem to a metric problem. Therefore, the choice of a metric is substantial for the matching of local features. Most approaches depend mainly on finding the minimum distance between features (descriptors) in feature space as shown in (2), where $D_{ij}$ is the distance measure between feature i from the first image and feature j from the second image. $X_{ij}$ is a matching indicator between feature i and feature j, i.e. $X_{ij}$ = 1 if feature i in the 1st image is mapped to feature j in the 2nd image and $X_{ij}$ = 0 otherwise. Note that $ X_{ij} \in \left\{ 0,1\right\} $. $$\begin{aligned} Min \,\; F = \sum _{\forall _{i,j}} D_{ij}\; X_{ij} \end{aligned}$$ Limitations: The similarity measure between features deals with each feature individually rather than a group of features. Consequently, the minimum distance between features can be misleading in some cases and as a result the performance of the algorithm deteriorates. In other words, the minimum distance criterion has no objection for a feature to be wrongly matched as long as it successfully achieves the minimum distance objective. 2.1 Similarity-Topology Matching Algorithm In [9], a new matching algorithm called "Similarity-Topology Matching" has been proposed. This algorithm pays attention not only to the similarity between features but also to the spatial layout of every matched feature and its neighbors. A new term, describing the neighbourhood/ topological relations between every pair of features has been added $\alpha \sum _{\forall _{i,j,k,l}} X_{ij} \; X_{kl}\; P_{ij,kl}$. In addition, another term has been added to relax the constraints $\beta \;(Min(m,n) \; - \sum _{\forall _{i,j}} X_{ij})$ as shown below in (3). $$\begin{aligned} Min \,\; F&= \sum _{\forall _{i,j}} D_{ij}\; X_{ij} + \alpha \sum _{\forall _{i,j,k,l}} X_{ij} \; X_{kl}\; P_{ij,kl} + \beta \;(Min(m,n) \; - \sum _{\forall _{i,j}} X_{ij}) \end{aligned}$$ Subject to: $$\begin{aligned}&\sum _{j=1}^{n}\; X_{ij} \; \le 1\qquad \qquad \qquad \qquad (a)\\&\sum _{i=1}^{m}\; X_{ij}\; \le 1\qquad \qquad \qquad \qquad (b) \end{aligned}$$ The second term in (3) represents a penalty term over all pairs of features. $P_{ij,kl}$ is called a penalty matrix. It is used to penalize matching pairs of features i and k in one image with corresponding pairs j and l in the other image if they have different topologies. It is binary and of $(m \times n, m \times n)$ dimension; where m, n are the number of features in the first and the second images respectively. $P_{ij,kl} = 1$ if the features j, l in the second image have different topology when compared to features i, k in the first image. Accordingly, the penalty matrix is calculated by applying the XOR logical operation to the adjacency matrices(AM1, AM2) of the two images as in (4). In XOR, the output is true whenever both inputs are different from each other. For example, if one input is true and the other is false. The output is false whenever both inputs are similar to each other, i.e., both inputs are true or false. $$\begin{aligned} P(i,j,k,l) = XOR \; ( AM1(i,k), \; AM2(j,l)) \end{aligned}$$ ($\alpha $) is called a topology coefficient. It indicates how much the matching algorithm depends on the topology between images. In the experiments, ($\alpha $) is chosen in a range from 0 to 0.1. The topology coefficient is effective and has a great impact when the interest points are similar to each other. On the contrary, it has almost no impact when the difference of similarities between the interest points is high. ($\beta $) is called a threshold coefficient. It indicates how much the matching algorithm depends on the features matching threshold. In the experiments, ($\beta $) is chosen in a range from 0 to 0.5. These parameters are determined by cross validations. Constraints Interpretation: Constraint (a): There exists at most one $'1'$ in every column of x. Constraint (b): There exists at most one $'1'$ in every row of x. The two constraints ensure that every feature in the first image should match to at most one feature in the second image. 2.2 Scale Invariant Similarity-Topology Matching Algorithm Analysis and Modification. An analysis is done to determine why the algorithm isn't accurate enough in case of the scaling deformation. It is noticed that the adjacency matrix (AM) of an image is constructed using the neighbourhood idea. In other words, if the distance between any two interest points in the same image is less than a threshold then they are called neighbours to each other. Consequently, the neighbourhood relation between each two interest points depends only on the distance between them, which is not valid specially when dealing with different scales. The two interest points in Fig. 1 are the same. The algorithm considers them as neighbours to each other in the left image but not in the right image which is counter intuitive, as they are neighbours in both cases. Scaling problem example An amendment is done to the algorithm in order to cope with this limitation. The modification makes the Neighbourhood Relation (NR) depend not only on the distance between the two interest points as in the Similarity-Topology but also on the scales at which the two interest points are detected. Hence, the Neighbourhood Relation (NR) between two interest points i and k in an image is defined as shown in (5). $$\begin{aligned} NR = \frac{Distance \; between \; two \; interest \; points}{Average \;scale \;of \;the \; two \; interest \; points } \end{aligned}$$ $$\begin{aligned} = \frac{d_{ik}}{Avg (\sigma _{i}, \sigma _{k} ) } \end{aligned}$$ Accordingly, the adjacency matrix is modified and calculated as in (6): $$\begin{aligned} AM(i,k) = \begin{Bmatrix} 1&\,\, if \,\,\, NR < Threshold \\ 0&otherwise \end{Bmatrix} \end{aligned}$$ where $d_{ik}$ is the Euclidean distance between interest points i and k in the same image spatial domain. $\sigma _{i}$ and $\sigma _{k}$ are the scales at which the interest points i and k are detected respectively. Scale Invariant Similarity-Topology Matching Algorithm. Algorithm (1) gives a summary of the modified version of the "Similarity-Topology Matching" approach. This new algorithm achieves superior performance in almost all the experiments specially when the images are exposed to scaling deformations. This investigated problem has a quadratic-objective function which is subject to linear constraints. It is called a binary (0-1) Quadratic Programming problem. Consequently, the objective function formulated in (3) can be rewritten as in (7): $$\begin{aligned} Min \,\; F&= \sum _{\forall _{i,j}} X_{ij} (D_{ij}-\beta ) + \alpha \sum _{\forall _{i,j,k,l}} X_{ij} \; X_{kl}\; P_{ij,kl} \end{aligned}$$ This optimization problem is solved using IBM ILOG CPLEX Optimization Studio (usually called just CPLEX for simplicity) which is an optimization software package. 3 Experimental Results 3.1 Data-Set Columbia Object Image Library (COIL-100) has been used in the experiments [11]. COIL-100 is a database of color images which has 7200 images of 100 different objects (72 images per object). These collections of objects have a wide diversity of complex geometric and reflectance characteristics. Consequently, it is the most suitable data-set which can be helpful in the proof of concept of the proposed feature correspondence approach. Figure 2, depicts 10 objects from the Coil-100 data-set. Examples of objects from the COIL-100 data-set used for the evaluation The Challenge. Fifty images representing five objects of the aforementioned data-set are chosen to perform the experiments. These objects with extra synthetic deformations such as rotation, scaling, partial occlusion and heavy noise are used for this purpose. In addition, a duplication of the same object is put in the same image with deformations, but one as a whole and one as parts to make the matching more challenging and to test the principle goal of the new matching strategy. In this case, a feature in the first image has almost two similar features in the second image. Figure 3, shows an example to illustrate the idea. The feature in the first image (left) has two similar features in the second image (right). This raises a question, which one should be matched. This challenge demonstrates the idea of the proposed approach, that rely on the similarity as well as the topological relations between the features as shown in the experiments in the next subsection. An illustrative example of the duplication of the same feature 3.2 Experiments Three different experiments are conducted to test the modification introduced in the "Similarity-Topology Matching" algorithm to make it scale invariant. All of these tests are done on images having different resolutions. The first test is done between a pair of images with different scales only. The second test is done between a pair of images with different scales as well as a duplication of the same object as parts in the second image. The last test is done like the second experiment but with extra deformations such as rotation and view point changes. These tests are ranged in difficulty from easiest to hardest as shown Table 2. Features Detection and extraction: the interest points are detected and extracted using SURF (Speeded Up Robust Features) [12]. We demonstrate in [10] that SURF algorithm can be used prior to the proposed matching approach to get more robust feature correspondence. Evaluation criterion: For each pair of images, every interest point in image 1 is compared to all interest points in image 2 according to their descriptors. The detection Rate and the False Positive Rate (FPR) are calculated in order to evaluate the performance. The detection rate R is defined as the ratio between the number of correct matches and the number of all possible matches (number of correspondence). The target is to maximize the detection rate and to minimize the false positive rate. $$\begin{aligned} \text{ R } = \frac{\text {Number of correct matches} }{\text {Number of possible matches within the full instance} } \end{aligned}$$ The experiments have been done using three state-of-the-art strategies which are Threshold, Nearest Neighbour (NN) and Nearest Neighbour Distance Ratio (NNDR) in addition to the Similarity-Topology Matching as well as its modified version. In the proposed algorithm and its modified version as well, the values of the topology penalty coefficient and the threshold penalty coefficient are 0.05 and 0.3 respectively. The modified version of the "Similarity-Topology Matching" algorithm has better performance. It is considered more robust specially under the scale deformations. As shown in Table 1, the Modified-Version of the algorithm not only has higher detection rate (0.65), but also it almost eliminates the false matches (0.01) which is more important specially in the localization problem. The experimental results summary Matching strategy Detection rate NNDR Similarity-topology Modified-version Scale invariant feature correspondence examples In this paper, an improved scale invariant feature correspondence algorithm which depends on the "Similarity-Topology Matching" algorithm has been introduced. In this approach, both local and global features are considered simultaneously and a set of control parameters is employed to tune the performance by adjusting the significance of global vs. local features. The major contribution of this research is depending not only on the distance between the two interest points but also on the scale at which the interest points are detected to decide the neighbourhood relations between every pair of features. Three different tests focusing on scaling deformations have been conducted. From the experimental results, it is noticed that the number of correctly matched features is increased. In conclusion, the modified version of the "Similarity-Topology Matching" algorithm has superior performance specially when the images have been exposed to scaling deformations. 5 Future Work After the proof of concept of the aforementioned approach has been verified, a lot of work remains to be done in order to generalize the local features matching approach and achieve high degree of robustness and computational efficiency. First, a preprocessing step is required to automatically evaluate the parameters values (alpha, beta). Second, an optimization of the algorithm to be more computationally efficient should be made without any loss in the algorithm accuracy as this algorithm may be used in real-time applications. Finally, applying this approach in a particular robot application such as mobile robot localization. The proposed approach can be used in conjunction with other approach [13] which depends on wifi-signals to determine the location of a mobile robot (such as KheperaIII) in indoor limited areas. This research has been supported by the Ministry of Higher Education (MoHE) of Egypt through a Ph.D. fellowship. Our sincere thanks to Egypt-Japan University for Science and Technology (E-JUST) for guidance and support. I wish to express an extended appreciation to Prof. Mohamed Hussein for his fruitful discussions and helpful suggestions. Zitova, B., Flusser, J.: Image registration methods: a survey. Image Vision Comput. 21(11), 977–1000 (2003)CrossRefGoogle Scholar Szeliski, R.: Computer Vision: Algorithms and Applications. Springer, London (2011)CrossRefGoogle Scholar Mikolajczyk, K., Schmid, C.: An affine invariant interest point detector. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002, Part I. LNCS, vol. 2350, pp. 128–142. Springer, Heidelberg (2002) CrossRefGoogle Scholar Mikolajczyk, K., Tuytelaars, T., Schmid, C., Zisserman, A., Matas, J., Schaffalitzky, F., Kadir, T., Van Gool, L.: A comparison of affine region detectors. Int. J. Comput. Vision 65(1–2), 43–72 (2005)CrossRefGoogle Scholar Mikolajczyk, K., Schmid, C.: A performance evaluation of local descriptors. IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1615–1630 (2005)CrossRefGoogle Scholar Lowe, D.G.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vision 60(2), 91–110 (2004)CrossRefGoogle Scholar Zhang, W., Kosecka, J.: Generalized RANSAC framework for relaxed correspondence problems. In: Third International Symposium on 3D Data Processing, Visualization, and Transmission, pp. 854–860. IEEE (2006)Google Scholar Liu, Y., Zhang, D., Lu, G., Ma, W.Y.: A survey of content-based image retrieval with high-level semantics. 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In: 2014 13th International Conference on Machine Learning and Applications (ICMLA), pp. 225–230, December 2014Google Scholar 1.Computer Science and Engineering DepartmentEgypt-Japan University for Science and Technology (E-JUST)AlexandriaEgypt El-Mashad S.Y., Shoukry A. (2015) Towards a Robust Scale Invariant Feature Correspondence. In: Kryszkiewicz M., Bandyopadhyay S., Rybinski H., Pal S. (eds) Pattern Recognition and Machine Intelligence. PReMI 2015. Lecture Notes in Computer Science, vol 9124. Springer, Cham The International Association for Pattern Recognition	CommonCrawl
Volume 19 Supplement 14 Selected articles from the 5th International Work-Conference on Bioinformatics and Biomedical Engineering: bioinformatics The poly-omics of ageing through individual-based metabolic modelling Elisabeth Yaneske1 & Claudio Angione1 Ageing can be classified in two different ways, chronological ageing and biological ageing. While chronological age is a measure of the time that has passed since birth, biological (also known as transcriptomic) ageing is defined by how time and the environment affect an individual in comparison to other individuals of the same chronological age. Recent research studies have shown that transcriptomic age is associated with certain genes, and that each of those genes has an effect size. Using these effect sizes we can calculate the transcriptomic age of an individual from their age-associated gene expression levels. The limitation of this approach is that it does not consider how these changes in gene expression affect the metabolism of individuals and hence their observable cellular phenotype. We propose a method based on poly-omic constraint-based models and machine learning in order to further the understanding of transcriptomic ageing. We use normalised CD4 T-cell gene expression data from peripheral blood mononuclear cells in 499 healthy individuals to create individual metabolic models. These models are then combined with a transcriptomic age predictor and chronological age to provide new insights into the differences between transcriptomic and chronological ageing. As a result, we propose a novel metabolic age predictor. We show that our poly-omic predictors provide a more detailed analysis of transcriptomic ageing compared to gene-based approaches, and represent a basis for furthering our knowledge of the ageing mechanisms in human cells. Ageing is a complex process characterised by phenotypes such as greying hair and wrinkles, as well as age-associated diseases such as cancer, osteoarthritis and cardiovascular disease. Phenotypes of ageing and age-associated diseases can be linked to age-associated changes in metabolic subsystems [1–3]. Identifying these metabolic links has recently led to the discovery of age-associated biomarkers [4, 5]. There are many different theories of the underlying mechanisms of ageing, including the mitochondrial theory of ageing, accumulation of metabolic by-products and dysregulation of regulatory pathways. The mitochondrion is the primary organelle responsible for metabolic cellular respiration; it takes in oxygen and nutrients and converts them into energy in the form of adenosine triphosphate (ATP). The mitochondrial theory of ageing states that oxidative damage caused by reactive oxygen species (ROS) produced by the mitochondria contributes to ageing by causing damage to mitochondrial DNA, lipids and proteins, which ultimately leads to cell death [6, 7]. Mitochondrial dysfunction and oxidative damage have been linked to age-associated neurodegenerative disorders such as Alzheimer's disease, Parkinson's disease and Huntington's disease [8, 9], as well as to the pathogenesis of cancer [10, 11]. Metabolism is increasingly being considered as a driver, rather than a marker, of the ageing process [12]. Three examples of metabolic by-products linked with ageing are amyloid proteins, advanced glycation end-products (AGEs) and lipofuscin. Accumulation of amyloid proteins in the central nervous system is associated with neurodegenerative disease in ageing [13, 14]; for instance, β-amyloid plaques in brain tissue are linked with the pathogenesis of Alzheimer's disease [15, 16]. AGEs can be ingested in foods or formed in the body by non-enzymatic glycation of lipids, nucleic acids and proteins, and their accumulation is thought to contribute to the ageing process [17, 18]. AGEs are formed when foods are processed at high temperatures such as deep-frying, grilling and roasting. They can increase oxidative stress, upregulate inflammation, and form cross-links with proteins [19], which cause impaired elasticity to blood vessels, therefore leading to poor heart health [20, 21]. Upregulation of inflammation caused by AGEs has been also linked to cancer [22–24]. Eating raw foods or foods cooked at lower temperatures can help to reduce dietary intake of AGEs [17]. Lipofuscin is a non-degradable metabolic by-product that builds up in lysosomes with time, and has been associated with age-related cellular degeneration [25], particularly macular degeneration [26]. One important example of the dysregulation of regulatory pathways as we age is chronic inflammation. The term 'inflammaging' was proposed by Franceschi et al. [27] to describe the imbalance between pro- and anti-inflammatory networks, which contributes to the chronic diseases of ageing. The function of the immune system declines as we age, leading to increased susceptibility to infectious diseases such as influenza [28], as well as decreased response to vaccinations against them [29]. This decline in function has been reported in CD4 T-cells [30, 31], which are used in this study, along with changes in the ageing transcriptome [32]. Age can be defined as chronological or transcriptomic/biological. Chronological age is a measure of the time that has passed since our birth, whereas transcriptomic age represents the difference in how time and the environment have affected the cells and organs of our body as compared to others of the same chronological age. Our transcriptomic age can therefore be older or younger than our chronological age. Until now, transcriptomic age has been calculated for individuals using transcriptomic-only data [33]. The limitation of this approach is that it does not take into account how age-associated gene expression affects the metabolism within cells and thus their observable cellular phenotype. This paper aims to improve on the current understanding of ageing (based on transcriptomics data alone) by modelling how age-associated gene expression changes metabolic processes, therefore enabling the identification of metabolic age predictors, selected using machine learning techniques. Metabolic models have proven to be valuable computational tools to study metabolism, as they allow predicting phenotypes from genotypes. By modelling most of the known biochemistry of a cell, they allow achieving a mechanistic understanding of the genotype-phenotype relationship. Coupled with tools for integration of omics data, metabolic models have been successfully exploited in a wide range of applications in health and disease, including personalised, condition- and tissue-specific cancer modelling [34–37]. Gene expression data and other types of omics-derived data can be used to constrain metabolic models for phenotype prediction [38]. The process of linking metabolic networks to phenotypes enables a better prediction of cellular phenotype compared to predictions from gene expression alone [39]. Exploiting this idea to generate a poly-omic model of ageing, we first generate individual-based genome-scale metabolic models and the associated fluxomic profiles. Specifically, we use CD4 T-cell transcriptomics data to modify a constraint-based metabolic model and achieve the predicted flux distributions (fluxomic profiles) for each individual in the cohort. Then, we adapt machine learning techniques in order to investigate metabolic changes linked to the chronological age of the individuals. We compare transcriptomic- and fluxomic-based clustering with chronological age and find that metabolic models are a better predictor of chronological age [40]. Our poly-omic pipeline also enables us to identify metabolic biomarkers of ageing, which are validated by recent literature, and to obtain metabolic age predictors. As a result, we build a metabolic age predictor capable of calculating the metabolic ages of individuals. Although a small number of metabolomics biomarkers have been proposed [41], to our knowledge this is the first time genome-scale predictors have been identified. We conclude that moving towards a poly-omic understanding of biological ageing can help provide a more accurate prediction of biological age, therefore leading to more targeted therapies for ageing individuals in a variety of environmental and physiological conditions. The poly-omic ageing pipeline Our pipeline starts from a meta-analysis of CD4 T-cell data containing the gene expression levels from human peripheral blood mononuclear cells and the chronological ages of 499 healthy individuals in the Boston area, comprised of 294 females and 205 males [42]. As the CD4 T-cell expression data was profiled on Affymetrix Human Gene 1.0 ST microarrays, it was first normalised using RMA (Robust Multi-array Average) [43]. In absence of a control profile, the gene expression values were then divided by the mean value for their associated probe. Using this normalised gene expression data, the transcriptomic ages for all 499 individuals were calculated as described in the next subsection (formulas (1) and (2)). Having obtained the transcriptomic ages, we then used individuals' transcriptomic data to generate their personalised CD4 T-cell metabolic models. These were created using constraint based modelling of the CD4 T-cell [44] augmented with transcriptomics through GEMsplice [45], by setting individual constraints on the CD4 model (see the following subsections on constraint-based modelling for details on how the mapping was achieved). On the personalised models, we finally adapted a set of statistical and machine learning methods based on clustering, PCA analysis and elastic net regression to identify metabolic predictors of ageing. Our pipeline thus enabled us to progress to a poly-omic understanding of ageing in human cells (Fig. 1; see also the following subsections for details on the modelling approach adopted in this manuscript). Poly-omic ageing pipeline. We start with the transcriptomic data and chronological ages from the CD4 T-cells of 499 individuals. We use the chronological data and corresponding age-associated transcriptomic predictors to obtain the effect of both chronological and transcriptomic ageing on the transcriptomic layer. We then combine with the functional biological network data determined by the metabolism and poly-omic model to obtain individual-based metabolic models and their fluxomic profiles. Finally, we adapt machine learning techniques to show that fluxomic data clusters better with chronological age than transcriptomic data, and to identify metabolic predictors of ageing (the poly-omic ageing map). Definition of key terms. Transcriptomic – gene expression data represented by a measurement of the mRNA transcripts within a cell. Fluxomic – fluxomic data refers to reaction flux rates, namely the value for the rate of metabolite conversion, measured in millimoles per hour per grams of dry weight, for each reaction or collection of related reactions (subsystems) within a cell. Poly-omic – the integration of more than one type of 'omic' data e.g. transcriptomic and fluxomic data Transcriptomic age predictor The transcriptomic age of an individual within a sample can be calculated by first obtaining their transcriptomic predictor, Z, using the gene expression levels of 1497 age-associated genes (i.e. those found to be differentially expressed with chronological age). This is achieved through a linear combination of the expression levels, where coefficients are their associated effect sizes [33]: $$ Z = \sum\limits_{i} b_{i} x_{i}, $$ where xi is the gene expression level of the ith probe, and bi is the effect size for the ith probe. Effect sizes were associated with individual genes, whereas the original data contained gene expression data associated with probes. Therefore, where a probe was mapped to more than one age-associated gene, the effect sizes for those genes were averaged to give an overall average effect size bi for that probe. The transcriptomic predictor for each individual is then scaled using the mean and standard deviation of the chronological ages, and the mean and standard deviation of the transcriptomic predictors from all the individuals in the sample [33]. This allows defining the transcriptomic age of an individual: $$ \ SZ = \mu_{age} + \left(Z-\mu_{Z}\right) \frac{\sigma_{age}}{\sigma_{Z}}, $$ where μage and σage are the mean and the standard deviation of the chronological age across all the individuals within the sample, while μZ and σZ are the mean and the standard deviation of the predictor Z across all the individuals in the sample. Constraint-based modelling to generate individual-based metabolic models Metabolic models can be analysed using constraint-based modelling and flux balance analysis (FBA), the most widely-used technique to simulate metabolic models at steady state [46]), to enable predictions of the distribution of reaction flux rates in the cell. Given the matrix S of all known metabolic biochemical reactions and their stoichiometry, and given the vector v of reaction flux rates in a given growth or physiological condition, the steady-state condition is set by the constraint Sv=0. Additional constraints are added on lower and upper bounds of v (vmin and vmax). Constraints are included according to the growth or physiological condition that is simulated; these can also be set taking into account multiple omics data (e.g. transcriptomics data as used in our pipeline) [47]. Further constraints can include codon usage [48], splice-isoforms [45, 49], and can be analysed using pathway-oriented approaches [50, 51]. The metabolic network is then solved by maximising one or more cellular objectives (usually the biomass and energy-related or application-specific production of metabolites). For a comprehensive introduction to constraint-based metabolic modelling and its poly-omic extensions, the reader is referred to the reviews by Palsson and Vijayakumar et al. [52, 53]. As omics data to constrain the model, here we use transcriptomic data from each individual to generate personalised metabolic models. Through GEMsplice [45], we modify the upper- and lower- limits of reactions as a function of the expression levels of the genes involved in the reaction. More specifically, for each individual, to predict the cellular flux distribution (fluxomic profile) when multiple objectives have to be taken into account, we use the following bilevel linear program: $$ \begin{aligned} & \text{max}\; g^{\intercal} v \\ & {\text{such that}} & \text{max}\;\; f^{\intercal} v, \quad Sv = 0,\\ & & v^{\text{min}} \varphi(\Theta) \leq v \leq v^{\text{max}} \varphi(\Theta). \end{aligned} $$ The vectors f and g are weights to select (or combine) the objectives to be maximised from the vector v. The vector Θ represents the expression of a biochemical reaction, defined from the individual-based expression levels of its genes with a rule involving the max and min operators, depending on the type of enzyme (single gene, isozyme, or enzymatic complex). The function φ, which acts on Θ, converts the reaction expression values into coefficients for the bounds of reactions activated by those genes [54]. Here we set the primary objective f as biomass and the secondary objective g as ATP maintenance. Simulations were performed in Matlab. Cluster analysis was used in order to group individual response according to both their transcriptomic and fluxomic profiles, and visualise them with chronological age. We compared both agglomerative hierarchical clustering (AHC) and k-means clustering using a novel application of the silhouette method. The silhouette method calculates a value which is a measure of the similarity of the values within a cluster (cohesion) and the dissimilarity of the values within that cluster to other clusters (separation). The silhouette calculation gives a value between −1 and 1. Silhouette values close to 1 are desirable as they indicate a cluster has high cohesion and high separation; if most values are close to 1 then the number of clusters is a good representation of the data. Here we use the silhouette value to measure the cohesion and separation of the clustering of individuals by chronological age [55]. We define the silhouette value of an individual within a cluster as: $$ \ s(c) = \frac{l(c)-a(c)}{max(a(c),l(c))}, $$ where s is the silhouette value (−1≤s(c)≤1), c is the chronological age of the individual, a is the average dissimilarity of c to the other ages in the same cluster and l is the lowest average dissimilarity of c to any other age in a different cluster. Our motivation for using the silhouette method was twofold. Firstly, we wanted to statistically compare the silhouette values for AHC and k-means to see which method performed better at clustering the data with chronological age. Secondly, we wanted to statistically compare whether transcriptomic-based or fluxomic-based clusters of individuals were consistent with chronological age. Multidimensional data such as fluxomic datasets can be visualised using Principal Component Analysis (PCA). PCA can reduce multidimensional datasets to as few as two or three latent dimensions (components), which allows inference of variables causing the largest variations in the data. Here we use PCA to identify the fluxes accounting for the greatest variation between individuals in different age groups according to chronological age. In our PCA analysis, the fluxomic data was split according to three chronological age groups: 21 and under (112 individuals: 64 female and 48 male), 22 to 49 (360 individuals: 219 female and 141 male), and 50 and over (27 individuals: 11 female and 16 male). The analysis was performed in R and visualised using FactoMineR [56]. The CD4 T-cell model contains 4229 flux variables (metabolic reactions). PCA analysis gives the contribution of each variable (reaction) to the variability of each component. The proportion of variability accounted for by a component is defined numerically by its eigenvalue. The total contribution Tv of a given variable across the components can be calculated by determining its overall weighted sum as follows: $$ T_{v} = \sum\limits_{i=1}^{n} E_{i} V_{i}, $$ where Ei is the eigenvalue for the principal component i, Vi is the variable (reaction) contribution to the principal component i, and n is the number of components chosen to represent the data. Variables (reactions) can be mapped to a subsystem (metabolic pathway), which contains a number of reactions that are interlinked to perform a cellular metabolic function. The CD4 T-cell model contains 95 pathways, each of which corresponds to a number of reactions and their flux values. A flux value for each pathway was calculated as the mean of its reaction flux values, and PCA was also performed on the pathway flux rates obtained for each individual. Similarly, the total contribution Ts of a given pathway across the components can be found using: $$ T_{s} = \sum\limits_{i=1}^{n} E_{i} S_{i}, $$ where Ei is the eigenvalue for the principal component i, Si is the subsystem (pathway) contribution to the principal component i, and n is the number of components. Elastic net regression We use elastic net regression to identify metabolic predictors of chronological age and their effect sizes. Elastic net regression is a linear hybrid of the L2 penalty of ridge regression [57] and the L1 penalty of lasso regression [58]. For α between 0 and 1, where 0 is ridge regression and 1 is lasso regression, and a strictly non-negative λ, elastic net is defined as [59]: $$ \left(\hat{\beta},\hat{\beta_{0}}\right) \,=\, \underset{\beta,\beta_{0}}{\text{argmin}} \left(\frac{1}{2N} \sum\limits_{i=1}^{N}\left(y_{i} - \beta_{0} - x^{T}_{i}\beta\right)^{2} \,+\, \lambda P_{\alpha}(\beta)\right), $$ $$ \begin{aligned} P_{\alpha}(\beta) &= \frac{(1 - \alpha)}{2}\left\\|\beta\right\\|_{2}^{2} + \alpha \left\\|\beta\right\\|_{1} \\ &= {\sum\nolimits}_{j=1}^{p} \left(\frac{(1 - \alpha)}{2}\beta_{j}^{2} + \alpha \left\|\beta_{j}\right\|\right), \end{aligned} $$ N is the number of individuals, yi is the chronological age of individual i, xi is a p×1 vector of p metabolic pathway fluxes at individual i, α is set to 0.5 to achieve a balance between L1 and L2 norms, λ is a positive regularization parameter, β0 is a scalar parameter, and β is a p×1 vector of effect sizes on chronological age (regression coefficients), where p is the number of metabolic pathways. We also performed an equivalent analysis directly on reaction fluxes (Additional file 1). Elastic net overcomes some of the limitations of using the lasso method alone [59]. When analysing high dimensional data, such as the individual by reaction data (499×4229), where the number of predictors is greater than the number of observations, the lasso method can only select at most the same number of variables as observations. However, omic data tends by its nature to often be highly correlated, which means there is high correlation between regression predictors. Where there is a group of highly correlated predictors, the lasso method will only select one variable from the group. The advantage of the elastic net method in our context is that while the L1 part of the penalty generates a sparse model, the quadratic part of the penalty (L2 regularisation ∥β∥2), taken from ridge regression, allows the number of selected variables to be greater than the number of observations, and allows groups of strongly correlated variables to be selected. Clustering shows best dataset for prediction of chronological age From the plot of average age-based silhouette values (Fig. 2a), optimal cluster numbers were chosen from the point closest to 1 at which there is an 'elbow bend' in the curve, indicating a drop in the amount of variance explained by the clusters after this point [60]. For the transcriptomic data seven clusters were chosen, while for fluxomic data six clusters were chosen. The pairwise distance of the chosen cluster numbers and the results of clustering for both transcriptomic and fluxomic data with chronological and transcriptomic age are shown in Figs. 2e, 2c, 2d, and 2b respectively. We selected k-means as a clustering algorithm because it performed consistently better than hierarchical clustering both in the transcriptomic (Kolmogorov-Smirnov test statistic =0.66, p-value =2.91·10−6) and in the fluxomic-based clustering (Kolmogorov-Smirnov test statistic = 0.8, p-value =5.59·10−9). k-means clustering with age. We propose and investigate an average age-based silhouette value a. This is calculated using the chronological age, and clustering data from both hierarchical and k-means clustering. The silhouette values are calculated by averaging all the individuals' silhouette scores, for a number of clusters ranging from 2 to 30. k-means clustering performs consistently better than hierarchical clustering. In both types of clustering, fluxomic data clusters better with chronological age than transcriptomic data. The pairwise distance of clusters with chronological age is visualised in scatter plots for both transcriptomic e and fluxomic c data. Clusters are annotated with different shapes, while age is shown with colour. Individual clusters are plotted against transcriptomic and chronological age for both transcriptomic d and fluxomic b data. Note that since transcriptomic age was calculated from transcriptomic data, we would expect to see more distinction in transcriptomic age between clusters for the transcriptomic data Remarkably, with both methods, and considering a variable number of clusters between 2 and 30 (Fig. 2a and Additional file 2), fluxomic-based clustering consistently outperformed transcriptomic-based clustering in terms of age-based average silhouette values (Kolmogorov-Smirnov test statistic =0.38, p-value =0.022 for k-means; Kolmogorov-Smirnov test statistic =0.62, p-value =1.15·10−5 for hierarchical). We also analysed the pattern of silhouette values based on deviations from linearity. In general, we found that drops in silhouette values corresponded to smaller clusters being merged into much larger, less distinct clusters, or cluster boundaries changing such that there was a loss of intra-cluster cohesion and inter-cluster separation (more details can be found in Additional file 3). Our results therefore suggest that, compared to gene expression values, individual-based poly-omic models and their predicted flux rates are a better predictor of chronological age. Principal component analysis identifies predictors of ageing The number of components to retain in the analysis was determined by their eigenvalues and the total contribution to variance explained by the components. Three criteria were applied to the data: (i) according to Kaiser's criterion [61] only those components with eigenvalus greater than 1 should be retained; (ii) the overall contribution to variance of the retained components should be 50% or greater; and (iii) for an n×m matrix, if the data were randomly distributed the expected contribution to variance of the eigenvalue for each axis would be 100/(n−1) % in terms of rows [62]. Therefore, any axis with a contribution to variance larger than this proportion should be retained as "significant". The threshold percentage variable contributions to variance for each group used in the analysis is shown in Table 1. The resulting number of components retained for analysis for each group is shown in Table 2. Table 1 Eigenvalue threshold % variance values for PCA Table 2 Number of components retained in PCA Using only the significant components, the overall contribution of each reaction to component variability was calculated using (5) for each of the three age groups, 21- (21 and under), 22-49 and 50 + (50 and over). The contributions of each of the reactions to the different age groups were then compared by calculating the difference between them. The differences between the reaction contributions for (i) 21- and 22-49, (ii) 21- and 50 +, and (iii) 22-49 and 50 + were obtained in order to determine the reactions that vary the most with age (see Additional file 4). In the results of the analysis of the differences in overall contribution of the 95 pathways, four pathways appeared in the top 20 of all the age group comparisons: CoA synthesis, vitamin D metabolism, hyaluronan metabolism and pyruvate metabolism. All four of these pathways also decreased in their contribution with age. Pyruvate and CoA are both key components of the citric acid cycle, which is essential in energy production in the mitochondria. Reduced stamina observed in the ageing population is thought to be related to impairment of mitochondrial energy production [63, 64]. Hypovitaminosis D in the ageing population is a major cause of impaired bone formation and mineralisation (osteoporosis) [65, 66]. Hyaluronan or Hyaluronic acid (HA) has a high capacity to bind and retain water molecules and is found in high levels in the extracellular matrix of skin where it regulates skin moisture. Reduced levels of HA are associated with the loss of moisture in ageing skin [67, 68]. Changes in HA size contribute to age-related impairment of wound healing in skin [69, 70] and to the viscoelasticity of synovial fluid, which can contribute to osteoarthritis, a common disease of ageing [71–73]. In the 21- to 22-49 group, the overall difference in contribution values for all but one of the top 20 pathways increased from the 21- group to the 22-49 group. Interestingly, CoA synthesis increased but CoA catabolism decreased, suggesting higher CoA levels due to increased synthesis and decreased degradation. Conversely, for the 21- group and 50 + group, the overall contribution values for all but one of the top 20 pathways decreased. Only Vitamin B2 metabolism increased. This is consistent with recent studies showing that the activity of vitamin B2 metabolism does not decrease until after the age of 50 [74]. All of the top 20 pathways contribution values decreased for the 50 + group compared to the 22-49 group. Squalene and cholesterol synthesis, vitamin A metabolism and glycine, serine, alanine and threonine metabolism overall contributions all decrease with age. Sebum, produced by the sebaceous glands, contains both cholesterol and squalene. Squalene is correlated with α-tocopherol (vitamin E) levels on the surface of the skin. α-tocopherol is the main antioxidant on the skin [75] and its decrease with ageing may contribute towards the signs of ageing skin [76, 77]. The reactions that make up the squalene and cholesterol synthesis pathway are part of the mevalonate pathway, which produces the precursors of all the steroid hormones, heme cholesterol, coenzyme Q10, and vitamin K [78]. The decrease in plasma levels of high density lipoprotein (HDL) cholesterol is correlated with a higher risk of atherosclerosis [79]. Low vitamin K has been associated with osteoarthritis and impaired cognitive function in older adults [80, 81]. The synthesis of mitochondrial coenzyme Q10 can decrease with age [82]; this constitutes an important factor for health, as coenzyme Q10 is an antioxidant that protects against diseases that involve oxidative stress, such as cardiovascular and neurogenerative diseases [83–88]. Furthermore, the synthesis of heme, the major functional form of iron in the body, decreases with age [89, 90] and has been linked with neurodegenerative disorders such as Alzheimer's disease [91]. Steroid hormones are also known to decline with age [92, 93]; decreasing sex steroid hormone deficiency in oestrogens and androgens contributes to ageing skin [94, 95] and increased risk of cardiovascular disease in women [93, 96]. Glycine, serine, alanine and threonine are all non-essential amino acids. Decreasing glycine levels have been linked with age-associated oxidative stress [97], while age-associated decrease in serine metabolism has been linked with impaired memory function in the brain [98, 99]. Alanine metabolism is associated with the liver, and alanine transaminase has been suggested as a biomarker for ageing [100]. Vitamin A cannot be produced by the body and is therefore obtained through diet; a decrease in the active form of vitamin A in the body, retinol, is thought to be linked with age-associated slowing in visual dark-adaption [101, 102]. Figure 3a-c show the factor maps for the top 20 pathways contributing to variance for each age group. For the 21- group, the overall contribution of components 1 and 2 to variation is 15.2%. For the 22-49 group, it is 12.4% and for the 50 + group it is 24%. If the variance was evenly distributed across all pathways, then the expected average variation would be 100/95 %=1.05%. Interestingly, the variance explained by the first two components in the 50+ group is higher compared to the other two groups, suggesting that less latent pathways, but with an increasingly strong role, characterise the ageing phenotype. Principal component analysis factor maps. Factor maps of the top 20 contributing pathways for components 1 and 2 in each age group, 21-, 22-49 and 50 + (a, b, c). The quality of the contribution of each pathway is shown by its colour As a result, the biplot (Fig. 4d) of individuals grouped by age and the top 10 pathways contributing to overall variance of components 1 and 2 shows differentiation between the three age groups, with the biggest differentiation shown in the axis of the 50 + group compared to the other two. The 50 + age group appears to lie along the glyoxylate and dicarboxylate metabolism axis. To identify the amount of intercorrelation in the pathway flux data, correlation plots were created for the pathways with the top 20 overall difference in contribution in each group, 21- and 22-49, 21- and 50 +, and 22-49 and 50 + (Fig. 4a-c). The plots show the large amount of intercorrelation between pathways. Principal component analysis applied to pathways. The correlation plots of fluxes are shown for the pathways with the top-20 overall difference in each group, 21- and 22-49, 21- and 50 +, and 22-49 and 50 + (a, b, c). The plots show the large amount of intercorrelation between pathways. A biplot of individuals grouped by age and the top 10 pathways contributing to overall variance of components 1 and 2 is also shown d. We note some differentiation between the three age groups, with the biggest differentiation shown in the axis of the 50 + group compared to the other two. The 50 + group appears to lie along the glyoxylate and dicarboxylate metabolism axis Exchange/demand reaction and oxidative phosphorylation remain the high contributors to variance for all three age groups. The most notable rise in contribution to variation with age is vitamin C metabolism, which moves from rank 18 in the 21- to rank 2 in the 50 + group. Changes in oxidative stress are known to affect the levels of vitamin C in the body [103]. Furthermore, oxidative damage to the genome caused by ROS (reactive oxygen species) is thought to be one of the causes of ageing [104]. Interestingly, ROS detoxification is one of the top 10 contributors in the 21- and 22-49 groups but does not appear in the top 20 contributors in the 50 + group, suggesting a possible link with ageing, oxidative stress and the changing variation contributed by vitamin C metabolism. Glyoxylate and dicarboxylate metabolism also has a large rise in contribution to variance with age. Glyoxylate and dicarboxylate metabolism is strictly linked with glycine, serine and threonine metabolism, pyruvate metabolism, ascorbate metabolism (a mineral salt of vitamin C), all of which our results have identified as linked to ageing. Elastic net regression identifies metabolic-age predictors The analysis was performed using tenfold cross validation for both metabolic reactions and metabolic pathways. The results of the metabolic pathways analysis are reported here, while the results for the metabolic reactions can be found in Additional file 1. Elastic net regression of the 95 metabolic pathways returned 100 possible λ values with their associated effect sizes. The value λ=1.57 was chosen as this had the lowest mean squared error value. This gave metabolic effect sizes for three pathways: butanoate metabolism, pyrimidine synthesis and beta-alanine metabolism (Table 3). Table 3 Metabolic effect sizes Both butanoate and beta-alanine metabolism have been linked to sarcopenia, deterioration of skeletal muscle, with age [105]. Butanoate and beta-alanine metabolism are facilitated by the enzyme aldehyde dehydrogenase. One of the substrates of this enzyme is nicotinamide adenine dinucleotide (NAD). Reversal of NAD loss as we age is currently undergoing human trials following successful age reversal of mitochondrial function in the skeletal muscle cells of mice [106]. Mitochondrial dysfunction, which is linked to ageing, also leads to a reduction of pyrimidine synthesis [107, 108]. In order to calculate the biological/metabolic ages of the individuals, equations (1) and (2) were modified to use the metabolic effect sizes. We therefore define the following metabolic predictor: $$ M = \sum\limits_{i} b_{i} f_{i}, $$ where M is the metabolic predictor, fi is the flux value of the ith pathway, and bi is the effect size for the ith pathway. The metabolic age was then defined as: $$ \ SM = \mu_{age} + \left(M-\mu_{M}\right) \frac{\sigma_{age}}{\sigma_{M}}, $$ where μage and σage are the mean and the standard deviation of the chronological age, while μM and σM are the mean and the standard deviation of the metabolic predictor M. The metabolic ages showed correlation (p-value =4.7·10−4) with chronological age. Both the metabolic ages and the chronological ages of all 499 individuals can be found in Additional file 1, which also includes the results of the regression performed directly on reactions. This is a promising starting point for a metabolic age predictor, which can be further refined using more samples to improve predictor accuracy. While chronological age gives an accurate measurement of the time since an individual's birth, biological age – also called transcriptomic age – gives a more accurate representation of the relative health of an individual compared to others of the same age. Where an individual has a biological age greater than their chronological age, they are ageing more quickly than their peers, and therefore have decreased life expectancy. Finding predictors of biological ageing and measuring their presentation in individuals can allow targeted and personalised interventions, both medication and lifestyle-based, to improve health and life expectancy. A number of age-associated genes along with their effect sizes have previously been identified in the literature and used to calculate the biological age of individuals [33]. The limitation of this approach is that it does not take into account the metabolic effects of those gene expression values on the cellular phenotype. For example, a gene that has been found to be differentially expressed with age may have little or no effect on cellular metabolism. Here we have achieved the first steps towards a metabolic age predictor to overcome the limitations of previous transcriptomic-only approaches. Using a genome-scale metabolic model of CD4 T-cells combined with transcriptomic data, we were able to obtain individual-specific metabolic models and generate fluxomic data (Additional file 5). The subsequent stage of our analysis used a novel application of the silhouette method and clustering techniques, from which we identified that fluxomic data clusters better with chronological age, therefore suggesting that metabolic models are a better predictor of the chronological age of an individual. Applying PCA analysis and elastic net regression enabled us to identify potential metabolic predictors of ageing. Many of these predictors have also been identified in the literature as linked to the ageing process, validating the reliability of the method. Finally, elastic net regression produced metabolic age predictors and their effect sizes, from which we calculated the metabolic age of each individual. Our next step will be to further refine the metabolic age predictors obtained from elastic net regression with more data from individuals across the different age ranges. 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All flux data and results are available as additional files. Source code and normalised transcriptomic data is available on request for academic use. About this supplement This article has been published as part of BMC Bioinformatics Volume 19 Supplement 14, 2018: Selected articles from the 5th International Work-Conference on Bioinformatics and Biomedical Engineering: bioinformatics. The full contents of the supplement are available online at https://bmcbioinformatics.biomedcentral.com/articles/supplements/volume-19-supplement-14. Department of Computer Science and Information Systems, Teesside University, Borough Road, Middlesbrough, UK Elisabeth Yaneske & Claudio Angione Elisabeth Yaneske Claudio Angione CA conceived and coordinated the study. EY and CA developed the methods. EY wrote the code and performed the analysis. EY and CA wrote the manuscript. Both authors read and approved the final manuscript. Correspondence to Elisabeth Yaneske. Metabolic ages (XLSX 33 kb) Silhouette values raw data (XLSX 11 kb) Plots of k-means clusters (PDF 283 kb) Supplementary PCA analysis raw data (XLSX 412 kb) Fluxomic data (XLSX 9174 kb) Yaneske, E., Angione, C. The poly-omics of ageing through individual-based metabolic modelling. BMC Bioinformatics 19, 415 (2018). https://doi.org/10.1186/s12859-018-2383-z Biological age Metabolic age Metabolic modelling Flux balance analysis Poly-omics CD4 T-cells	CommonCrawl
Semi-group A set with one binary operation satisfying the law of associativity. A semi-group is a generalization of the concept of a group: only one of the group axioms is retained — associativity; this is the explanation of the term "semi-group" . Semi-groups are called monoids if they have, in addition, an identity element. The theory of semi-groups is one of the relatively young branches of algebra. The earliest studies of semi-groups date back to the 1920's and are associated with the name of A.K. Sushkevich, who, in particular, determined the structure of the kernel (the minimal ideal) of a finite semi-group and thus, in particular, the structure of any finite semi-group without proper ideals (cf Kernel of a semi-group). This result was later generalized by D. Rees to arbitrary completely-simple semi-groups (cf. Completely-simple semi-group) and refined by the introduction of the concept of matrices over a group (see Rees semi-group of matrix type). Rees' theorem, which may be regarded as a sort of analogue of Wedderburn's theorem for simple algebras, is one of the fundamental propositions of the theory of semi-groups. Other early research on semi-groups was done by A. Clifford; one of his first significant achievements was the introduction and investigation of semi-groups which are unions of groups; these semi-groups are now known as completely regular or Clifford semi-groups (cf. Clifford semi-group). By the end of the 1950's, the theory of semi-groups had become a self-contained branch of modern algebra with a rich store of problems, a broad range of methods and strong links with many fields of mathematics, both properly algebraic (primarily the theory of groups and rings) and others, such as functional analysis (semi-groups of operators on Banach spaces), differential geometry (semi-groups of partial transformations), and the algebraic theory of automata (semi-groups of automata). There are an extraordinary number of examples of semi-groups. Among these are various sets of numbers which are closed under addition or multiplication; semi-groups of matrices with respect to multiplication; semi-groups of functions with respect to "pointwise" multiplication $$, defined by $(fg)(x) = f(x) g(x)$; semi-groups of sets with respect to intersection or union; etc. The following example is important in the general theory and in some applications. Let $X$ be an arbitrary set, and let an operation on the set $F_X$ of all finite sequences of elements of $X$ be defined by the formula $$ (x_1,\ldots,x_n) * (y_1,\ldots,y_m) = (x_1,\ldots,x_n,y_1,\ldots,y_m) $$ Then $F_X$ is a semi-group relative to $*$; it is known as the free semi-group on $X$. Every semi-group is a homomorphic image of some free semi-group. Any set of mappings of an arbitrary set $M$ into itself, which is closed under composition (i.e. the operation of successive application, also known as superposition), is a semi-group with respect to that operation; this is the case, in particular, for the set of all self-mappings of a set $M$, known as the symmetric semi-group on $M$. Many important sets of transformations turn out to be semi-groups, and they often fail to be groups. On the other hand, any semi-group is isomorphic to some semi-group of mappings. Thus, the concept of a semi-group proves to be most suitable for the investigation of mappings in a quite general context, and it is largely through the consideration of mappings that the links between the theory of semi-groups and other fields of mathematics have been realized. In this framework, semi-groups appear very frequently as semi-groups of endomorphisms (see Endomorphism semi-group) of the system being studied: spaces, algebras, graphs, etc. Semi-groups also appear in the theory of partial transformations and binary relations with respect to multiplication. As in other algebraic theories, one of the main problems of the theory of semi-groups is the classification of all semi-groups and a description of their structure. This is achieved by imposing various restrictions on the semi-groups under consideration and thereby specifying various types of semi-groups. Such restrictions may be of different kinds. A semi-group may satisfy a fixed system of identities (typical examples are commutative semi-groups and semi-groups of idempotents) or other conditions, expressed by a formula of first-order predicate calculus (examples are semi-groups with a cancellation law and regular semi-groups). The cancellation law and regularity are examples of restrictions which in a sense constitute weak versions of group properties; the introduction of such conditions was particularly popular in the early days of the theory of semi-groups (among the most "group-like" types thus defined are right groups (cf. Right group)). In many cases, however, the classes of semi-groups obtained in this way include semi-groups with properties not at all similar to those of groups (typical examples are semi-groups of idempotents). The concept of a regular semi-group arose in analogy with that of a regular ring (in the sense of von Neumann). Regular semi-groups are among the most intensively investigated in the theory of semi-groups. They include the following important classes: the multiplicative semi-groups of regular rings (in particular, the semi-group of all matrices of a given order over a division ring), symmetric semi-groups, the semi-groups of all partial transformations of sets, inverse semi-groups, Clifford semi-groups, in particular, semi-groups of idempotents and completely-simple semi-groups, completely $0$-simple semi-groups, etc. Another type of commonly imposed restriction are restrictions on the system of all or some of the sub-semi-groups, in particular the ideals, and also on certain relations on semi-groups, in particular congruences. Such restrictions give rise, for example, to various types of simple semi-groups (cf. Simple semi-group) and various finiteness conditions (see Semi-group with a finiteness condition; Periodic semi-group; Locally finite semi-group; Residually-finite semi-group; Minimal ideal), semi-groups with different types of ideal series and ideal systems (see Ideal series; Nil semi-group); an essential role in the investigation of many problems of the theory of semi-groups is played by the Green equivalence relations. The restrictions may relate to generating sets, delineating various types according to the nature of the generating elements (e.g. idempotents; any semi-group can be imbedded in an idempotently-generated semi-group), the number of such elements (finitely-generated semi-groups play an essential part in many investigations), or the interaction of the generators — semi-groups given by defining relations and, in particular, finitely-presented semi-groups (see Algorithmic problem; Semi-group with a finiteness condition); finally, the types of generating sets may be specified in both of the above respects (see, e.g., Bicyclic semi-group). When the structure of semi-groups is considered, much importance is attached to various constructions that reduce the description of the semi-groups in question to that of "better" types. Quite frequently, the latter are groups, and the principle of description "modulo groups" is common in semi-group-theoretical contexts; in fact, it already appeared in the above-mentioned classical theorem of Rees, according to which any completely $0$-simple (completely-simple) semi-group is isomorphic to a regular Rees matrix semi-group over a group with a zero (over a group). Groups take part in the constructions that describe inverse semi-groups, and in those that describe commutative Archimedean semi-groups (cf. Archimedean semi-group) with a cancellation law and without idempotents. The description of semi-groups with several finiteness conditions reduces to that of groups with the corresponding conditions. Among the constructions figuring in the description of semi-groups one has both general algebraic ones, such as direct and subdirect products, and specific semi-group-theoretical ones. The latter include, besides the above-mentioned Rees semi-groups, various other constructions, such as that of a band — a partition into sub-semi-groups such that the corresponding equivalence relation is a congruence. Particularly important bands are the commutative bands (or semi-lattices) and matrix (rectangular) bands (see Band of semi-groups). Many types of semi-groups can be described in terms of bands. Thus, Clifford's theorem for completely-regular semi-groups means, essentially, that these semi-groups are semi-lattices of completely-simple semi-groups; the completely-simple semi-groups are precisely the rectangular bands of groups; the Tamura–Kimura theorem states that any commutative semi-group admits a unique decomposition as a band of Archimedean semi-groups (see [3]). As always in algebra, the concept of a homomorphism plays an essential role also in the theory of semi-groups, and hence so does the concept of a congruence. Semi-groups belong to the class of universal algebras whose congruences are not uniquely determined by any canonical coset ( "kernel" ), as is the case, say, for groups and rings. This more complicated situation has led to the development of a fairly extensive branch of the theory of semi-groups, devoted to the investigation of semi-group congruences from various points of view. The problems involved fall mainly into two categories: 1) to investigate some special type of congruence on arbitrary semi-groups; 2) to describe all congruences on some special semi-groups, belonging to some class of importance. The first category includes, in particular, the study of principal congruences (see [3]), and also of ideal, or Rees, congruences, associated with the two-sided ideals of a semi-group (if $I$ is an ideal in a semi-group $S$, then the corresponding Rees congruence classes are $I$ itself and the singletons $\{x\}$, $x \in S \setminus I$). Rees congruences are frequently used in various problems and explain the importance of studying ideals; the quotient semi-group modulo a Rees congruence is known as the Rees quotient semi-group modulo the corresponding ideal. Worthy of mentioning among the solved problems in the second category are the description of the congruences on symmetric semi-groups and on completely $0$-simple semi-groups, and the far-reaching investigation of congruences on inverse semi-groups; the theory of radicals of semi-groups (cf. Radical in a class of semi-groups) has been developed, not without the influence of the analogous branch of ring theory. Thanks to the investigation of homomorphisms of semi-groups into semi-groups with specified "good" properties, it has been possible to formulate a branch of the theory dealing with approximations (see Separable semi-group; Residually-finite semi-group). In connection with the theory of sub-semi-groups, one finds a self-contained branch of the theory dealing with the study of lattice properties of semi-groups, i.e. the relationship between the properties of semi-groups and the properties of their semi-group lattice (see Lattice). Another extensive branch of the theory of semi-groups is connected with various imbeddings of semi-groups. The roots of this branch go back to the classical problem of imbedding of semi-groups in groups. For some problems and results in this branch of the theory, see Extension of a semi-group. Intensive attention has been devoted to varieties of semi-groups; concerning this field see Variety of semi-groups. First steps have been taken towards a theory of quasi-varieties of semi-groups (see Algebraic systems, quasi-variety of) and certain other classes of semi-groups similar in some sense to varieties. The general theory of semi-groups links up with specific semi-groups in many ways. Among the problems solved are the abstract characterization of various important concrete semi-groups (such as transformation semi-groups; in particular, there are several characterizations of symmetric semi-groups), and several of their abstract properties have been described. For some fundamental results concerning transformation semi-groups see Transformation semi-group. Consideration has been given to isomorphisms and homomorphisms of abstract semi-groups into various specific semi-groups, first and foremost — transformation semi-groups and matrix semi-groups (see Representation of a semi-group). The investigation of homomorphisms of semi-groups into certain semi-groups of numbers, mainly into the multiplicative semi-group of complex numbers, forms the subject of the theory of semi-group characters (cf. Character of a semi-group). One of the specific fields in the theory of semi-groups is the study of semi-groups with an additional structure compatible with the multiplication operation. In this context one should mention, above all, the structure of a topological space (see Topological semi-group) and the structure of a partial or total order (see Ordered semi-group). Theories have also been developed for certain types of generalized semi-groups. These are primarily algebras with one $n$-ary operation satisfying a generalized associative law (known as $n$-associative or $n$-semi-group operations). Another variant is that of algebras with one partial associative binary operation (a natural situation of this type arises in the theory of categories). [1] A.K. Sushkevich, "The theory of generalized groups" , Khar'kov-Kiev (1937) (In Russian) [2] E.S. Lyapin, "Semigroups" , Amer. Math. Soc. (1974) (Translated from Russian) [3] A.H. Clifford, G.B. Preston, "Algebraic theory of semi-groups" , 1–2 , Amer. Math. Soc. 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Dynamics of a ratio-dependent predator-prey system with a strong Allee effect On the dynamics of two-consumers-one-resource competing systems with Beddington-DeAngelis functional response November 2013, 18(9): 2315-2329. doi: 10.3934/dcdsb.2013.18.2315 Blow-up results for semilinear wave equations in the superconformal case Mohamed-Ali Hamza 1, and Hatem Zaag 2, Département de Mathématiques, Faculté des Sciences de Tunis, Université de Tunis El-Manar, Campus Universitaire 1060, Tunis, Tunisia Université Paris 13, Sorbonne Paris Cit, LAGA, CNRS (UMR 7539), F-93430, Villetaneuse, France Received March 2013 Revised June 2013 Published September 2013 We consider the semilinear wave equation in higher dimensions with power nonlinearity in the superconformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up solutions previously obtained by Killip, Stovall and Vişan [22]. 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Kerr black hole Revision as of 17:38, 8 December 2013 by Igor (Talk \| contribs) (→‎Problem 4: some more simple algebra) Kerr solution$^{}$ is the solution of Einstein's equations in vacuum that describes a rotating black hole (or the metric outside of a rotating axially symmetric body) . In the Boyer-Lindquist coordinates$^{}$ it takes the form \begin{align}\label{Kerr} &&ds^2=\bigg(1-\frac{2\mu r}{\rho^2}\bigg)dt^2 +\frac{4\mu a \,r\;\sin^{2}\theta}{\rho^2} \;dt\,d\varphi -\frac{\rho^2}{\Delta}\;dr^2-\rho^2\, d\theta^2 +\qquad\nonumber\\ &&-\bigg( r^2+a^2+\frac{2\mu r\,a^2 \,\sin^{2}\theta}{\rho^2} \bigg) \sin^2 \theta\;d\varphi^2;\\ \label{Kerr-RhoDelta} &&\text{where}\quad \rho^2=r^2+a^2 \cos^2 \theta,\qquad \Delta=r^2-2\mu r+a^2. \end{align} Here $\mu$ is the black hole's mass, $J$ its angular momentum, $a=J/\mu$; $t$ and $\varphi$ are time and usual azimuth angle, while $r$ and $\theta$ are some coordinates that become the other two coordinates of the spherical coordinate system at $r\to\infty$. $^{}$ R.P. Kerr, Gravitational field of a spinning mass as an example of algebraically special metrics. Phys. Rev. Lett. 11 (5), 237 (1963). $^{*}$ R.H. Boyer, R.W. Lindquist. Maximal Analytic Extension of the Kerr Metric. J. Math. Phys 8, 265–281 (1967). 1 General axially symmetric metric 1.1 Problem 1: preliminary algebra 1.2 Problem 2: integrals of motion 1.3 Problem 3: Zero Angular Momentum Observer/particle 1.4 Problem 4: some more simple algebra 2 Limiting cases 2.1 Problem 5: Schwarzshild limit 2.2 Problem 6: Minkowski limit 2.3 Problem 7: weak field rotation effect 3 Horizons and singularity 3.1 Problem 8: on null surfaces 3.2 Problem 9: null surfaces in Kerr metric 3.3 Problem 10: horizon area 3.4 Problem 11: black holes and naked singularities 3.5 Problem 12: $r=0$ is not a point. 3.6 Problem 13: circular singularity 4 Stationary limit 4.1 Problem 14: geometry of the stationary limit surfaces in Kerr 4.2 Problem 15: natural angular velocities 4.3 Problem 16: angular velocities for massive particles and rigidity of horizon's rotation 4.4 Problem 17: redshift 5 Ergosphere and the Penrose process 5.1 Problem 18: bounds on particle's energy 5.2 Problem 19: negative energy 5.3 Problem 20: unambiguity of negativeness 5.4 Problem 21: profit! 6 Integrals of motion 6.1 Problem 22: massless particles on circular orbits 6.2 Problem 23: massive particles on circular orbits 6.3 Problem 24: general case 7 The laws of mechanics of black holes 7.1 Problem 25: Killing horizons 7.2 Problem 26: surface gravity 7.3 Problem 27: horizon's area evolution 7.4 Problem 28: irreducible mass 7.5 Problem 29: extremal limit 8 Particles' motion in the equatorial plane 8.1 Problem 30: preparatory algebra 8.2 Problem 31: zero energy particles 8.3 Problem 32: geodesics and effective potential 8.4 Problem 33: principal null geodesics 8.5 Problem 34: innermost stable circular orbits, massless case 8.6 Problem 35: circular orbits for massive particles 8.7 Problem 36: innermost stable circular orbits, massive case General axially symmetric metric A number of properties of the Kerr solution can be understood qualitatively without use of its specific form. In this problem we consider the axially symmetric metric of quite general kind \begin{equation}\label{AxiSimmMetric} ds^2=A dt^2-B(d\varphi-\omega dt)^{2}- C\,dr^2-D\,d\theta^{2},\end{equation} where functions $A,B,C,D,\omega$ depend only on $r$ and $\theta$. Problem 1: preliminary algebra Find the components of metric tensor $g_{\mu\nu}$ and its inverse $g^{\mu\nu}$. The metric is: \begin{equation}\label{AxiSimmMetricmatrix} g_{\mu\nu}=\begin{pmatrix} A-\omega^2 B&0&0&\omega B\\ 0&-C&0&0\\ 0&0&-D&0\\ \omega B&0&0&-B \end{pmatrix}. \end{equation} Taking into account the structure of $g_{\mu\nu}$, for the inverse matrix we get \begin{align} &g^{rr}=\frac{1}{g_{rr}};\quad g^{\theta\theta}=\frac{1}{g_{\theta\theta}};\\ &g^{tt}= \frac{g_{\varphi\varphi}}{\|G\|};\quad g^{\varphi\varphi}=\frac{g^{tt}}{\|G\|}; \quad g^{t\varphi} =-\frac{g_{t\varphi}}{\|G\|};\\ &\text{where}\quad G=g_{tt}g_{\varphi\varphi}-g_{t\varphi}^{2}. \end{align} Using the explicit expression for $g_{\mu\nu}$, we see that $G=-AB$ and thus finally \begin{equation}\label{AxiSimmMetricInvmatrix} g^{\mu\nu}=\begin{pmatrix} 1/A&0&0&\omega/A\\ 0&-1/C&0&0\\ 0&0&-1/D&0\\ \omega/A&0&0&\frac{\omega^2 B-A}{AB} \end{pmatrix} \end{equation} Problem 2: integrals of motion Write down the integrals of motion corresponding to Killing vectors $\boldsymbol{\xi}_{t}=\partial_t$ and $\boldsymbol{\xi}_{\varphi}=\partial_\varphi$. A particle's integrals of motion are \begin{equation}\label{AxiSimm-Integrals} \boldsymbol{u}\cdot\boldsymbol{\xi}_t=u_{t}; \quad \boldsymbol{u}\cdot\boldsymbol{\xi}_{\varphi}=u_{\varphi}.\end{equation} Energy and angular momentum are defined the same way as in the Schwarzshild case \[E=mc^{2}u_{t};\qquad L=-m u_\varphi.\] Problem 3: Zero Angular Momentum Observer/particle Find the coordinate angular velocity $\Omega=\tfrac{d\varphi}{dt}$ of a particle with zero angular momentum $u_{\mu}(\partial_{\varphi})^{\mu}=0$. For a particle moving in the axially symmetric field \begin{align} u^{t}=&g^{t\mu}u_{\mu}= g^{tt}u_{t}+g^{t\varphi}u_{\varphi};\\ u^{\varphi}=&g^{\varphi\mu}u_{\mu}= g^{\varphi t}u_{t}+g^{\varphi\varphi}u_{\varphi}. \end{align} Then for a particle with zero angular momentum (ZAMO) $u_{\varphi}=0$ we get \[u^{t}=g^{tt}u_{t}; \quad u^{\varphi}=g^{t\varphi}u_{t},\] and therefore its angular velocity is \[\frac{d\varphi}{dt} =\frac{d\varphi/ds}{dt/ds}= \frac{u^{\varphi}}{u^t}= \frac{g^{t\varphi}u_t}{g_{tt}u_t}= \frac{\omega/A}{1/A}=\omega(r,\theta).\] Now we see the physical meaning of the quantity $\omega(r,\theta)$. Problem 4: some more simple algebra Calculate $A,B,C,D,\omega$ for the Kerr metric. Let us introduce notation \begin{align} \Sigma^{2}=&(r^2+a^2)^{2}-a^{2}\Delta \sin^{2}\theta,\\ &\Delta=r^2 -2\mu r +a^2 , \end{align} so that for the Kerr metric $g_{\varphi\varphi}=-\tfrac{\Sigma^2}{\rho^2}\sin^{2}\theta$. After some straightforward calculations then we obtain \begin{equation}\label{Kerr-ABCD} A=\frac{\Delta \rho^2}{\Sigma^2},\quad B=\frac{\Sigma^2}{\rho^2}\sin^2 \theta,\quad C=\frac{\rho^2}{\Delta},\quad D=\rho^2,\quad \omega=\frac{2\mu ra}{\Sigma^2}. \end{equation} Limiting cases Problem 5: Schwarzshild limit Show that in the limit $a\to 0$ the Kerr metric turns into Schwarzschild with $r_{g}=2\mu$. For $a=0$ we'll have $\rho^2=r^2$ and $\Delta=r^{2}h(r)$. On substituting this into the Kerr metric, we see it is reduced to the Schwarzshild one. Problem 6: Minkowski limit Show that in the limit $\mu\to 0$ the Kerr metric describes Minkowski space with the spatial part in coordinates that are related to Cartesian as \begin{align} &x=\sqrt{r^2+a^2}\;\sin\theta\cos\varphi, \nonumber\\ &y=\sqrt{r^2+a^2}\;\sin\theta\sin\varphi,\\ &z=r\;\cos\theta\nonumber,\\ &\text{where}\quad r\in[0,\infty),\quad \theta\in[0,\pi],\quad \varphi\in[0,2\pi).\nonumber \end{align} Find equations of surfaces $r=const$ and $\theta=const$ in coordinates $(x,y,z)$. What is the surface $r=0$? In the limit $\mu\to0$ we get $\Delta=a^2+r^2$ and \begin{align}\label{KerrM=0} ds^2&=dt^2-\frac{\rho^2}{\Delta}\;dr^2- \rho^2\, d\theta^2- \big(r^2+a^2\big) \sin^2 \theta\;d\varphi^2= \nonumber\\ &=dt^2-\frac{\rho^2}{r^2+a^2}\;dr^2- \rho^2\, d\theta^2- \big(r^2+a^2\big) \sin^2 \theta\;d\varphi^2,\\ &\text{where}\quad \rho^2=r^2+a^2 \cos^2 \theta. \nonumber \end{align} On the other hand, the Euclidean line element $ds^{2}_{eu}=dx^{2}+dy^{2}+dz^{2}$ takes the same form in spherical coordinates $(x,y,z)\to(r,\theta,\varphi)$. Surfaces $r=const$ are ellipsoids of revolution around the z axis \[\frac{x^2+y^2}{r^2+a^2}+\frac{z^2}{r^2}=1,\] and the special case $r=0$ corresponds to a disk of radius $a$ in the equatorial plane. Surfaces $\theta=const$ are hyperboloids of revolution around the same axis \[\frac{x^2+y^2}{\sin^{2}\theta}- \frac{z^2}{\cos^2\theta}=a^2.\] In the limit $a/r\to 0$ this coordinate system turns into the spherical one. Problem 7: weak field rotation effect Write the Kerr metric in the limit $a/r \to 0$ up to linear terms. In the zeroth order it is the Schwarzshild metric, and terms linear by $a$ are only present in $g_{t\varphi}$, thus \begin{align}\label{KerrAsymp} ds^2=\Big(1-\frac{2\mu}{r}\Big)dt^2- \Big(1-\frac{2\mu}{r}\Big)^{-1}dr^2-r^2d\Omega^2 +\frac{4a\mu}{r}\sin^{2}\theta\,dt\,d\varphi +O(a^2)=\nonumber\\ =ds^{2}_{Schw}+2r\,dt\,\omega_{\infty}\! d\varphi, \quad\text{where}\quad \omega_{\infty}=\frac{2a\mu}{r^3}\sin^{2}\theta. \end{align} Horizons and singularity Event horizon is a closed null surface. A null surface is a surface with null normal vector $n^\mu$: \[n^{\mu}n_{\mu}=0.\] This same notation means that $n^\mu$ belongs to the considered surface (which is not to be wondered at, as a null vector is always orthogonal to self). It can be shown further, that a null surface can be divided into a set of null geodesics. Thus the light cone touches it in each point: the future light cone turns out to be on one side of the surface and the past cone on the other side. This means that world lines of particles, directed in the future, can only cross the null surface in one direction, and the latter works as a one-way valve, -- "event horizon" Problem 8: on null surfaces Show that if a surface is defined by equation $f(r)=0$, and on it $g^{rr}=0$, it is a null surface. The normal vector $n_\mu$ to a surface $f(x)=0$ is directed along $\partial_{\mu}f$. It is null if \[g^{\mu\nu}(\partial_{\mu}f)(\partial_{\nu}f)=0.\] The normal to the surface $f(r)=0$ is directed along $\partial_{r}f$, i.e. $\partial_{\mu}f\sim \delta_{\mu}^{r}$ and the null condition takes the form \[0=g^{\mu\nu}\delta^{r}_{\mu}\delta^{r}_{\nu} =g^{rr}.\] Problem 9: null surfaces in Kerr metric Find the surfaces $g^{rr}=0$ for the Kerr metric. Are they spheres? Equations of surfaces, on which $g^{rr}$ terms to zero and $g_{rr}$ to infinity, are \begin{align} \Delta=0\quad\Leftrightarrow\quad r^2-2\mu r+a^2=0\quad\Leftrightarrow\quad r=r_{\pm},\quad\text{where}\nonumber\\ \label{Kerr-Rhor+-} r_{\pm}\equiv\mu\pm\sqrt{\mu^2-a^2}. \end{align} Although those are surfaces of constant $r$, their intrinsic metric is not spherical. Plugging $r=r_{\pm}$ into the spatial section $dt=0$ of the Kerr metric (\ref{Kerr}) ans using the relation $r_{\pm}^{2}+a^2=2\mu r_{\pm}$, which holds on the surfaces $r=r_\pm$, we obtain \[dl^{2}_{r=r_\pm}= \rho^{2}_{\pm}d\theta^{2}+ \Big(\frac{2\mu r_\pm}{\rho_\pm}\Big)^2 \sin^{2}\theta \,d\varphi^2 =\rho_{\pm}^{2} \big(d\theta^2 +\sin^{2}\theta d\varphi^{2} \big) +2a^{2}(r^2 +a^2 \cos^{2}\tfrac{\theta}{2}) \sin^{4}\theta\,d\varphi^{2}.\] The first terms is the metric of a sphere, while the second gives additional positive contribution to the distance measured along $\varphi$. Thus if we embedded such a surface into a three-dimensional Euclidean space, we'd get something similar to an oblate ellipsoid of rotation. Problem 10: horizon area Calculate surface areas of the outer and inner horizons. The horizon's area is \begin{equation}\label{Kerr-HorizonSurface} S_{\pm}=\int\rho_\pm d\theta\cdot \frac{2\mu r_\pm}{\rho_\pm}\sin\theta d\varphi= 2\mu r_{\pm} \int\limits_{0}^{\pi}\sin\theta d\theta \int\limits_{0}^{2\pi}d\varphi= 8\pi \mu r_{\pm}=4\pi (r_{\pm}^{2}+a^2). \end{equation} Problem 11: black holes and naked singularities What values of $a$ lead to existence of horizons? Solutions of $\Delta=0$ exist when \[a<m.\] On calculating curvature invariants, one can see they are regular on the horizons and diverge only at $\rho^2 \to 0$. Thus only the latter surface is a genuine singularity. Problem 12: $r=0$ is not a point. Derive the internal metric of the surface $r=0$ in Kerr solution. Let us consider the set of points $r=0$. Assuming $r=0$ and $dr=0$ in (\ref{Kerr}), from (\ref{Kerr-RhoDelta}) we obtain \[ \rho^2=a^2 \cos^2 \theta;\quad \Delta=a^2;\quad\Rightarrow\quad g_{\theta\theta}=-a^2,\quad g_{\varphi\varphi}=-a^{2}\sin^{2}\theta, \] so metric takes the form \begin{equation}\label{KerrR=0} ds^{2}_{r=0}=dt^2-a^{2}\cos^{2}\theta d\theta^2 -a^{2}\sin^{2}\theta d\varphi^2= \Big\\| a\sin\theta=\eta\Big\\|= dt^{2}-\Big(d\eta^2+\eta^2 d\varphi^2 \Big). \end{equation} This is a flat disk of radius $a$, center $\eta=\theta=0$, with distance to the center measured by $\eta=a\sin\theta$. Problem 13: circular singularity Show that the set of points $\rho=0$ is a circle. How it it situated relative to the inner horizon? The boundary of the disk is a circle $\eta=a$, or in original variables \[\{r=0,\;\theta=\pi/2,\;\varphi\in[0,2\pi)\},\] which lies beyond the inner horizon. If $a=\mu$ (extremal Kerr black hole), then $r_{-}=0$ and it lies on the horizon. Stationary limit Stationary limit is a surface that delimits areas in which particles can be stationary and those in which they cannot. An infinite redshift surface is a surface such that a phonon emitted on it escapes to infinity with frequency tending to zero (and thus its redshift tends to infinity). The event horizon of the Schwarzschild solution is both a stationary limit and an infinite redshift surface (see the problems on blackness of Schwarzshild black hole). In the general case the two do not necessarily have to coincide. Problem 14: geometry of the stationary limit surfaces in Kerr Find the equations of surfaces $g_{tt}=0$ for the Kerr metric. How are they situated relative to the horizons? Are they spheres? Equations of surfaces $g_{tt}=0$ are \begin{align} 2\mu r=\rho^2,\quad\Leftrightarrow\quad r^2-2\mu r+a^{2}\cos^{2}\theta=0, \quad\Leftrightarrow\quad r=r_{S\pm},\quad\text{where} \nonumber\\ \label{Kerr-RS+-} r_{S\pm}\equiv \mu\pm\sqrt{\mu^2-a^{2}\cos^{2}\theta}. \end{align} Those are two axially symmetric surfaces, which in the limit $a\to 0$ meld into the horizons and tend to $r=0$ and $r=2\mu$. Their intrinsic metric is obtained by plugging $r=r_{S\pm}$ into (\ref{Kerr}) and using that $r_{S\pm}^{2}+a^{2}=2\mu r_{S\pm}+a^{2}\sin^{2}\theta$: \[ dl^{2}_{r=r_{S\pm}} =2\mu r_{S\pm} ( d\theta^{2}+\sin^{2}\theta\,d\varphi^2 )+ 2a^{2}\sin^{4}\theta\,d\varphi^2.\] They are not spheres either, but surfaces similar to oblate ellipsoids, if embedded into a three-dimensional Euclidean space. Comparing (\ref{Kerr-Rhor+-}) and (\ref{Kerr-RS+-}), we see, that the inner ergosurface $r=r_{S-}$ lies entirely inside the inner horizon $r=r_-$, touching it at the poles $\theta=0,\pi$. Its intersection with the equatorial plane is the circular singularity. The outer ergosurface $r=r_{S+}$, on the contrary, encloses the outer horizon $r=r_+$ while touching it at the poles. This is a schematic view of the Kerr solution in section $t,\theta=const$. Solid lines denote ergosurfaces, the dashed ones are horizons. The thin circle shows the value of $a$, equal to its radius. a=0.5 (slowly rotating) a=0.97 (generic astrophysical) a=1.0 (extremal) a=1.1 (naked singularity) Problem 15: natural angular velocities Calculate the coordinate angular velocity of a massless particle moving along $\varphi$ in the general axially symmetric metric (\ref{AxiSimmMetric}). There should be two solutions, corresponding to light traveling in two opposite directions. Show that both solutions have the same sign on the surface $g_{tt}=0$. What does it mean? Show that on the horizon $g^{rr}=0$ the two solutions merge into one. Which one? Let us consider motion of a light ray along the angular coordinate $\varphi$, such that only $u_{t}$ and $u_{\varphi}$ are different from zero. Equation of its worldline is \[0=ds^2=g_{tt}dt^2+2g_{t\varphi}\,dt\,d\varphi +g_{\varphi\varphi}d\varphi^2,\] and plugging in the metric $g_{\mu\nu}$ in terms of parameters $A,B,C,D,\omega$ from (\ref{AxiSimmMetricmatrix}), we get \begin{equation}\label{Kerr-OmegaPM} \Omega_{\pm}\equiv \frac{d\varphi_{\pm}}{dt}= -\frac{g_{t\varphi}}{g_{\varphi\varphi}}\pm \sqrt{\Big( \frac{g_{t\varphi}}{g_{\varphi\varphi}} \Big)^{2}- \frac{g_{tt}}{g_{\varphi\varphi}}}= \omega\pm \sqrt{\omega^{2}- \frac{g_{tt}}{g_{\varphi\varphi}}}= \omega\pm\sqrt{A/B}.\end{equation} The two signs correspond to light rays emitted in two opposite directions, the prograde and the retrograde one. If \[g_{tt}=0\] the retrograde angular velocity turns to zero \[\frac{d\varphi_+}{dt}=2\omega,\quad \frac{d\varphi_-}{dt}=0,\] and at $g_{tt}<0$ both solutions are of the same sign. This means the dragging is so strong that light cannon propagate in the direction opposite to that of black hole's rotation. In other words, the locally inertial frame on the surface $g_{tt}=0$ has linear coordinate velocity along $\varphi$ equal to the speed of light. The root is double in case $A=0$ or $B\to\infty$. From (\ref{Kerr-ABCD}) we see that the first condition is realized on the horizon, therefore on the outer horizon $\Omega\pm=\omega$. Problem 16: angular velocities for massive particles and rigidity of horizon's rotation What values of angular velocity can be realized for a massive particle? In what region angular velocity cannot be zero? What can it be equal to near the horizon? As worldlines of massive particles lie in the light cone, the interval of possible values of angular velocities for them is \[ \Omega\in(\Omega_{-},\Omega_{+}).\] Thus, angular velocity cannot be equal to zero if the value $\Omega=0$ does not belong to this interval, which holds beyond the stationary limit, in the region where$g_{tt}<0$. This is the reason for the term "stationary limit". In the vicinity of the horizon all particles rotate with one angular velocity $\Omega_{H}=\omega\big\|_{r=r_+}$, which is often called the angular velocity of the horizon: \begin{equation}\label{Kerr-OmegaHorizon} \Omega_{H}\equiv \omega\Big\|_{r=r_+} =\frac{2\mu ra}{\Sigma^{2}}\Big\|_{r=r_+} =\frac{a}{2\mu r_{+}}.\end{equation} Problem 17: redshift A stationary source radiates light of frequency $\omega_{em}$. What frequency will a stationary detector register? What happens if the source is close to the surface $g_{tt}=0$? What happens if the detector is close to this surface? Let us consider stationary observers, with $4$-velocities directed along $\boldsymbol{\xi}_{t}$. Due to normalizing condition $\boldsymbol{u}\cdot \boldsymbol{u}=1$ we have $\boldsymbol{\xi}_{t}=\alpha \boldsymbol{u}$, where $\alpha^2=\boldsymbol{\xi}_{t}\cdot\boldsymbol{\xi}_{t}=g_{tt}$. Then the frequency of a photon with $4$-wavevector $k^\mu$ as measured by the stationary observer is \begin{equation}\label{StaticOmega} \omega_{stat}=\boldsymbol{k}\cdot \boldsymbol{u}= \frac{1}{\alpha}\boldsymbol{k}\cdot\boldsymbol{\xi}_{t}= \frac{1}{\alpha}\omega_{0}= \frac{\omega_{0}}{\sqrt{g_{tt}}},\end{equation} where $\omega_{0}=\boldsymbol{k}\cdot \boldsymbol{\xi}_t$ is its frequency in the world time, which is the integral of motion. In case both emitter and detector are stationary, \[\omega_{em}\sqrt{g_{tt}^{(em)}}= \omega_{det}\sqrt{g_{tt}^{(det)}}.\] Thus, the frequency of a photon emitted in the outer region (where $g_{tt}>0$), measured by a stationary observer near to the stationary limit, on which $g_{tt}\to 0$, tends to infinity. Conversely, if a stationary emitter close to the stationary limit emits a photon, its frequency measured by a stationary observer at finite (or infinite) distance, will tend to zero, and its redshift to infinity. This is why those surfaces are also called the surfaces of infinite redshift. Ergosphere and the Penrose process Ergosphere is the area between the outer stationary limit and the outer horizon. As it lies before the horizon, a particle can enter it and escape back to infinity, but $g_{tt}<0$ there. This leads to the possibility of a particle's energy in ergosphere to be also negative, which leads in turn to interesting effects. All we need to know of the Kerr solution in this problem is that it \emph{has an ergosphere}, i.e. the outer horizon lies beyond the outer static limit, and that on the external side of the horizon all the parameters $A,B,C,D,\omega$ are positive (you can check!). Otherwise, it is enough to consider the axially symmetric metric of general form. Problem 18: bounds on particle's energy Let a massive particle move along the azimuth angle $\varphi$, with fixed $r$ and $\theta$. Express the first integral of motion $u_t$ through the second one $u_{\varphi}$ (tip: use the normalizing condition $u^\mu u_{\mu}=1$). ${}^{}$ Note: relations ((7)) and ((8)) do not hold, as they were derived in assumption that $g_{00}>0$. From normalization condition \[1=u^{\mu}u_{\mu}=g^{\mu\nu}u_{\mu}u_{\nu}\] we obtain \[g^{tt}(u_{t})^{2}+2g^{t\varphi}u_{t}u_{\varphi}+ \big(g^{\varphi\varphi}(u_{\varphi})^{2}-1\big)=0,\] therefore plugging in $g_{\mu\nu}$, we have \begin{align} u_{t}=&-\frac{g^{t\varphi}}{g^{tt}}u_{\varphi}\pm \sqrt{\Big( \frac{g^{t\varphi}}{g^{tt}}u_{\varphi}\Big)^{2}- \frac{1}{g^{tt}}\big[ g^{\varphi\varphi}(u_{\varphi})^{2}-1\big]}=\\ &=-\omega u_{\varphi}\pm \sqrt{\omega^{2} (u_{\varphi})^{2}- A\Big[\frac{\omega^{2}B-A}{AB}(u_\varphi)^{2} -1\Big]}=\\ &=-\omega u_{\varphi}+ \sqrt{A+\frac{A}{B}(u_{\varphi})^{2}} \end{align} We chose the sign "$+$" here because far from the black hole, where $C,D,B,A-\omega^2 B \to 1$ and $\omega\to0$, and thus $A\to 1$, $u^t$ should tend to $+1$ when velocity turns to zero. Problem 19: negative energy Under what condition a particle can have $u_{t}<0$? In what area can it be fulfilled? Can such a particle escape to infinity? For $u_{t}$ to be negative we need the condition $u_{\varphi}>0$ to hold (with $a>0$, so that $\omega>0$) and for the square root to be less than $\omega u_{\varphi}$. As all the coefficients $A,B,C,D$ are positive, the condition of negativity of $u_t$ can be written as \begin{equation}\label{PenroseUlimit} \omega u_{\varphi}> \sqrt{A+\frac{A}{B}(u_{\varphi})^{2}}\quad \Rightarrow\quad (u_{\varphi})^{2} (\omega^{2}-A/B)>A \quad\Leftrightarrow\quad (u_{\varphi})^{2}(-g_{tt})>AB.\end{equation} This can only be true if $g_{tt}<0$, i.e. in the ergosphere. Problem 20: unambiguity of negativeness What is the meaning of negative energy? Why in this case (and in GR in general) energy is not defined up to an additive constant? For the Schwarzschild black hole the energy of a particle at rest near the horizon tends to zero. This means the work needed to pull away to infinity, where energy is $mc^2$, equals to its rest mass. If a particle' energy in the ergosphere is negative, it means the work needed to pull it away to infinity is greater than its rest mass. Note that in GTR the starting point for energy is not arbitrarily chosen, as any mass serves as a source of gravitational field. Problem 21: profit! Let a particle $A$ fall into the ergosphere, decay into two particles $B$ and $C$ there, and particle $C$ escape to infinity. Suppose $C$'s energy turns out to be greater than $A$'s. Find the bounds on energy and angular momentum carried by the particle $B$. The answer is for the most part already obtained above. This can happen if particle $B$'s energy is negative. The restriction on its angular momentum $l_{m}=-mu_{\varphi}$ is obtained from condition (\ref{PenroseUlimit}): \begin{equation}\label{PenroseMomLimit} u_{\varphi}>0,\quad (u_{\varphi})^{2}>\frac{A}{\Omega_{+}\Omega_{-}} \quad\Leftrightarrow\quad l_{m}<-m \sqrt{\frac{A}{\Omega_{+}\Omega_{-}}}.\end{equation} Thus it should be directed in the \emph{opposite} direction to the momentum of the black hole, and negative energy is achieved on retrograde orbits. Integrals of motion Problem 22: massless particles on circular orbits Find the integrals of motion for a massless particle moving along the azimuth angle $\varphi$ (i.e. $dr=d\theta=0$). What signs of energy $E$ and angular momentum $L$ are possible for particles in the exterior region and in ergosphere? The 4-velocity of a particle moving along the azimuth angle is \[u=u^{t}(1,0,0,\Omega).\] For massless ones \[0=u^{\mu}u_{\mu}=g_{tt}(u^{t})^{2} +2g_{t\varphi}u^{t}u^{\varphi} +g_{\varphi\varphi}(u^\varphi)^{2}= (u^{t})^{2}\big(g_{\varphi\varphi}\Omega^{2} +2g_{t\varphi}\Omega+g_{tt}\big).\] In parenthesis we have the equation for $\Omega_{\pm}$ (\ref{Kerr-OmegaPM}) \[\Omega_{\pm}=\omega\pm\sqrt{A/B};\quad \Omega_{+}+\Omega_{-}=2\omega,\quad \Omega_{+}\Omega_{-} =\frac{g_{tt}}{g_{\varphi\varphi}}= \omega^{2}-A/B,\] so \begin{equation}\label{MasslessCircleU} \boldsymbol{u}=u^{t}(\partial_{t}+ \Omega_{\pm}\partial_{\varphi}).\end{equation} Then the integrals of motion are \begin{align} u_{t}=g_{tt}u^{t}+g_{t\varphi}u^{\varphi}& =u^{t}(g_{tt}+g_{t\varphi}\Omega_{\pm}) =u^{t}\big[A-\omega^{2}B+ \omega B(\omega\pm\sqrt{A/B})\big]=\nonumber\\ &=u^{t}(A\pm\omega\sqrt{AB}) =\pm \Omega_{\pm}\sqrt{AB}\cdot u^{t};\nonumber\\ \label{MasslessCircleIntegrals} u_{\varphi}=g_{t\varphi}u^{t} +g_{\varphi\varphi}u^{\varphi}& =u^{t}(g_{t\varphi} +g_{\varphi\varphi}\Omega_{\pm}) =u^{t}\big[\omega B- B(\omega\pm\sqrt{A/B})\big] =\mp\sqrt{AB}\cdot u^{t}\\ \Rightarrow\quad&\quad \frac{E}{L}=-\frac{u_t}{u_\varphi}=\Omega_{\pm}. \end{align} Note that $u^{t}$ is always positive. On one hand, it cannot turn to zero in any point of the worldline, as this would violate the normalizing condition. So in the outer region, where $g_{tt}>0$, it is obvious that $u^{t}>0$, and any particle in the ergosphere can be pulled there along some trajectory from the outside (this is obvious for massive particles, but can also be imagined for the massless ones). As $u^t$ does not turn into zero, it will remain positive. Then in the outer region, where $\Omega_{+}\Omega_{-}<0$, a photon's energy is always positive, while its angular momentum can be of any sign. In the ergosphere, where $\Omega_{+}\Omega_{-}>0$, energy is positive and angular momentum (remember it differs in sign from $u_\varphi$) is positive on prograde orbits $\Omega=\Omega_+$; on retrograde orbits $\Omega=\Omega_-$ both energy and are negative. Problem 23: massive particles on circular orbits Calculate the same integrals for massive particles. Derive the condition for negativity of energy in terms of its angular velocity $d\varphi/dt$. In what region can it be fulfilled? Show that it is equivalent to the condition on angular momentum found in the problem on negative energy. For massive particles we have the same quadric equation in the normalizing condition and plugging in $g_{\varphi\varphi}=-B$, we obtain \begin{align} 1=&g_{\mu\nu}u^{\mu}u^{\nu}=(u^{t})^{2} \big[g_{\varphi\varphi}\Omega^{2} +2g_{t\varphi}\Omega+g_{tt}\big]=\\ &= (u^{t})^{2}\cdot (-B) (\Omega-\Omega_{+})(\Omega-\Omega_{-})= (u^{t})^{2} \big(A-B(\Omega-\omega)^{2}\big). \end{align} Then \begin{equation}\label{MassiveCircleU} \boldsymbol{u}= \frac{\partial_{t}+\Omega\partial_{\varphi}} {\sqrt{-B(\Omega-\Omega_+)(\Omega-\Omega_-)}}= \frac{\partial_{t}+\Omega\partial_{\varphi}} {\sqrt{A-B(\Omega-\omega)^{2}}};\quad \Omega\in(\Omega_-,\Omega_+).\end{equation} and the integrals of motion are \begin{align} u_{t}=&u^{t}(g_{tt}+g_{t\varphi}\Omega)= u^{t}(A+B\omega (\Omega-\omega))= \frac{A+B\omega (\Omega-\omega)} {\sqrt{A-B(\Omega-\omega)^{2}}};\\ u_{\varphi}=&u^{t} (g_{t\varphi}+g_{\varphi\varphi}\Omega)= u^{t}(\omega B-\Omega B)= \frac{B(\omega-\Omega)} {\sqrt{A-B(\Omega-\omega)^{2}}}. \end{align} We see that angular momentum of a particle with $\Omega=\omega$ is zero: \[L=-u_{\varphi}=0;\quad E=u_{0}=\sqrt{A}.\] Energy is negative if \[u_{t}<0\quad\Leftrightarrow\quad \omega-\Omega>\frac{A/B}{\omega} \quad\Leftrightarrow\quad \Omega<\frac{\omega^{2}-A/B}{\omega}= \frac{\Omega_{+}\Omega_{-}}{\omega} =-\frac{g_{tt}}{g_{t\varphi}},\] thus angular velocity should be \emph{less} than the critical value \[\label{PenroseAngVelocity} \Omega_{P}\equiv -\frac{g_{tt}}{g_{t\varphi}}\equiv \frac{\Omega_{+}\Omega_{-}}{\omega} \in(\Omega_{-},\omega);\] Clearly $\Omega_{P}>0$ only in the ergosphere, where $\Omega_{\pm}$ are of the same sign. Taking into account that $\Omega_{\pm}=\omega\pm\sqrt{A/B}$, we can put down the following chain of inequalities for the characteristic angular velocities in the ergosphere \begin{equation}\label{ErgospereOmegas} 0<\Omega_{P}= \frac{\Omega_{+}\Omega_{-}}{\omega} <\omega<\Omega_{+}<2\omega.\end{equation} It follows that the window for the Penrose process always exists. $\Omega_{\pm}(\xi)$ and $\omega(\xi)$ in the equatorial plane on the Kerr solution for $\alpha=a/\mu=0.9375$. A particle's energy in the ergosphere is negative if $\Omega\in(\Omega_{-},\Omega_{P})$; $\Omega_{P}(\xi)$ in the equatorial plane is a straight line. In terms of angular momentum $l_{m}=-mu_{\varphi}$ energy is negative when \[u_{\varphi}> \frac{B\cdot \frac{A/B}{\omega}} {\sqrt{A-B\cdot \frac{A^2 /B^{2}}{\omega^{2}}}}= \frac{A/\omega} {\sqrt{A-\frac{A^2 }{B\omega^{2}}}}= \sqrt{\frac{A}{\omega^{2}-A/B}}= \sqrt{\frac{A}{\Omega_{+}\Omega_{-}}}= \sqrt{A\frac{\omega}{\Omega_{P}}}\] and finally we derive the same condition as the one obtained above: \[l_{m}<-m\sqrt{\frac{A}{\Omega_{+}\Omega_{-}}}= -m\sqrt{A\frac{\omega}{\Omega_P}}.\] Problem 24: general case Derive the integrals of motion for particles with arbitrary $4$-velocity $u^{\mu}$. What is the allowed interval of angular velocities $\Omega=d\varphi/dt$? Show that for any particle $(E-\tilde{\Omega} L )>0$ for any $\tilde{\Omega}\in(\Omega_{-},\Omega_{+})$. For massive particles moving in arbitrary direction the normalizing condition takes the form \[1=(u_{t})^{2}\big( A-B(\Omega-\omega)^{2}-C(\tfrac{dr}{dt})^{2} -D(\tfrac{d\theta}{dt})^{2}\big).\] Then the interval of possible angular velocities is (\ref{MassiveCircleU}): \[\Omega\in(\Omega_{-},\Omega_{+}), \quad\text{where}\quad \Omega_{\pm}=\omega\pm\sqrt{A/B},\] which should be expected. The limiting values are realized for motion along $\varphi$ in the ultrarelativistic limit $E\to\infty$. The denominators in (\ref{MassiveCircleU}), which ensure the normalizing condition for $u^\mu$, are different now, but in the same way otherwise for the integrals of motion we obtai \begin{align} &u_{t}=u^{t}\cdot \big(A+B\omega(\Omega-\omega)\big),\\ &u_{\varphi}=-u^{t}\cdot B(\Omega-\omega), \end{align} so \[\frac{E}{L}=-\frac{u_{t}}{u_{\varphi}} =\omega+\frac{A/B}{\Omega-\omega}.\] Taking into account that $\|\Omega-\omega\|\leq \sqrt{A/B}$, we can extract from this the restrictions on $E/L$, but it is easier to achieve in a different way for all possible signs of $E$ and $L$ simultaneously. Let there be a geodesic particle with $u_{t}$ and $u_\varphi$ (those are conserving quantities, other components are arbitrary), and let there be an observer moving on a circular orbit with angular velocity $\Omega\in(\Omega_{-},\Omega_{+})$, so that his $4$-velocity is $\tilde{u}=\tilde{u}^{t}(\partial_{t}+\tilde{\Omega}\partial_{\varphi})$. The energy of the first particle as measured by this observer is the invariant \[\tilde{E}=\tilde{u}^{\mu}u_{\mu} =\tilde{u}^{t} \big(u_{t}+\tilde{\Omega}u_{\varphi}\big) >0.\] Then \[E-\tilde{\Omega} L>0\quad\Rightarrow\quad L<\frac{E}{\tilde{\Omega}} \quad\text{for}\quad\forall \tilde{\Omega}\in(\Omega_{-},\Omega_{+}),\] which is what we wanted to prove. Observers near the event horizon can only have $\tilde{\Omega}=\Omega_{H}$, so for a particle crossing the horizon \[L<\frac{E}{\Omega_{H}}.\] The laws of mechanics of black holes If a Killing vector is null on some null hypersurface $\Sigma$, $\Sigma$ is called a Killing horizon. Problem 25: Killing horizons Show that vector $K=\partial_{t}+\Omega_{H}\partial_{\varphi}$ is a Killing vector for the Kerr solution, and it is null on the outer horizon $r=r_{+}$. Here $\Omega_{H}=\omega\big\|_{r=r_+}$ is the angular velocity of the horizon. First note, that due to linearity of the Killing equation $\xi_{\mu;\nu}+\xi_{\nu;\mu}=0$, a linear combination of two Killing vector fields with constant coefficients is also a Killing vector field. As $\Omega_H$ is a constant, this holds for $K^\mu$. Next, \begin{align} g_{\mu\nu}K^{\mu}K^{\nu}=&g_{\mu\nu} (\delta_{t}^{\mu}+\Omega_{H}\delta_{\varphi}^{\mu}) (\delta_{t}^{\nu}+\Omega_{H}\delta_{\varphi}^{\nu}) =g_{tt}+2g_{t\varphi}\Omega_{H} +g_{\varphi\varphi}\Omega_{H}^2=\\ &=A-\omega^{2}B+2\omega\Omega_{H}B -\Omega_{H}^{2}B=A-B(\omega-\Omega_H)^2. \end{align} In the vicinity of the horizon $\omega\to\Omega_{H}$, and also $A\sim\Delta\to 0$ (\ref{Kerr-ABCD}), so $K^{\mu}K_{\mu}\to 0$. Problem 26: surface gravity Let us define the surface gravity for the Kerr black hole as the limit \[\kappa=\lim\limits_{r\to r_{+}} \frac{\sqrt{a^{\mu}a_{\mu}}}{u^0}\] for a particle near the horizon with $4$-velocity $\boldsymbol{u}=u^{t}(\partial_{t}+\omega\partial_{\varphi})$. In the particular case of Schwarzschild metric this definition reduces to the one given in the corresponding problem. Calculate $\kappa$ for particles with zero angular momentum in the Kerr metric. What is it for the critical black hole, with $a=\mu$? Recall the normalizing condition \[1=u^{\mu}u_{\mu}=(u^{t})^{2} (g_{tt}+2\Omega g_{t\varphi} +\Omega^{2}g_{\varphi\varphi}).\] As all the metric components depend only on $r$ and $\theta$, acceleration can be reduced to \begin{align} a^{\mu}=&{\Gamma^{\mu}}_{\nu\lambda} u^{\nu}u^{\lambda} =(u^t)^{2}\big({\Gamma^{\mu}}_{tt} +2\Omega{\Gamma^{\mu}}_{t\varphi} +\Omega^{2}{\Gamma^{\mu}}_{\varphi\varphi}\big)=\\ &=(u^t)^{2}\frac{g^{\mu\nu}}{2} \big(\!-\partial_{\nu}g_{tt} -2\Omega\partial_{\nu}g_{t\varphi} -\Omega^2\partial_{\nu}g_{\varphi\varphi}\big)= -(u^t)^{2}\frac{g^{\mu\nu}}{2}\partial_{\nu} \big(g_{tt}+2\Omega g_{t\varphi} +\Omega^2 g_{\varphi\varphi}\big)=\\ &=-g^{\mu\nu}\tfrac{1}{2}(u^t)^{2}\partial_{\nu} \frac{1}{(u^t)^2}= -g^{\mu\nu}\partial_{\nu}\ln u^t. \end{align} Again taking into account the symmetries, we get \[a^2\equiv a^{\mu}a_{\mu} =\|g^{rr}\|(\partial_{r}\ln u^t)^2 +\|g^{\theta\theta}\|(\partial_{\theta}\ln u^t)^2.\] Let us now consider a particle with zero angular momentum $\Omega=\omega$, for which (see the previous problem and the one on integrals of motion of massless particles) \[(u^{t})^2=\frac{1}{A} =\Big(\frac{\rho^2 \Delta}{\Sigma^2}\Big)^{-1},\quad \text{thus}\quad \partial_{\mu}\ln u^{t}=-\tfrac{1}{2} \frac{\partial_{\mu} A}{A}.\] we are interested only in the part of $a^2$ that is divergent on the horizon, so differentiate only $\Delta$: \[\partial_{\mu}\ln u^t \sim -\frac{1}{2} \frac{\partial_{\mu}\Delta}{\Delta} =-\frac{1}{2}\frac{\partial_{\mu}(r^2-2\mu r+a^2)} {\Delta} =-\frac{r-\mu}{\Delta}\delta_{\mu}^{1}.\] Plugging $g^{rr}=\Delta/\rho^2$, we get \[a^2\approx\frac{(r-\mu)^2}{\rho^2 \Delta^2},\] and on substitution of $(u^t)^2$, we obtain the surface gravity: \begin{align} \kappa^2 =&\lim\limits_{r\to r_{+}} \frac{a^2}{(u^t)^2} =\frac{(r-\mu)^2}{\Sigma^2}\Big\|_{r=r_+} \quad\Rightarrow\nonumber\\ \label{SurfaceGravity} \kappa=&\frac{r_{+}-\mu}{r_{+}^{2}+a^2} =\frac{r_{+}-\mu}{2\mu r_{+}} =\frac{1}{2\mu} \frac{\sqrt{1-\alpha^2}}{1+\sqrt{1-\alpha^2}} =\Omega_{H}\frac{\sqrt{1-\alpha^2}}{\alpha}, \quad \alpha=\frac{a}{\mu}. \end{align} For the extremal Kerr solution, in the limit $\alpha\to 1$, it tends to zero. Problem 27: horizon's area evolution Find the change of (outer) horizon area of a black hole when a particle with energy $E$ and angular momentum $L$ falls into it. Show that it is always positive. In case a particle of mass $m$ crosses the event horizon, the black hole's mass increases by $\delta \mu=E$, and its angular momentum by $\delta J=L$. Then, using the result of problem on particle's integrals of motion, for any continuous process of accretion on a black hole the following holds \begin{equation}\label{Kerr-JM} \delta J<\frac{\delta \mu}{\Omega_H}.\end{equation} Note that this relation works both for positive and negative $E$ and $L$. As $\alpha=\tfrac{a}{\mu}=\tfrac{J}{\mu^2}$, the area of the horizon is expressed through $\mu$ and $J$ this way \[A_{+}=4\pi(r_{+}^{2}+a^{2})=8\pi\mu r_{+}= 8\pi\mu^{2}(1+\sqrt{1-\alpha^2})= 8\pi(\mu^2+\sqrt{\mu^4-J^2}),\] and $\Omega_{H}$ can be rewritten in terms of the same quantities as \[\Omega_{H}\equiv\omega(r_{+}) =\frac{a}{r_{+}^{2}+a^2} =\frac{a}{2\mu r_{+}} =\frac{J/\mu}{\mu^2+\sqrt{\mu^4-J^2}}.\] On differentiating $A_{+}$, we obtain then \[\delta A_{+}=\frac{2\mu A_{+}}{\sqrt{\mu^4-J^2}} \Big\{\delta\mu-\Omega_{H}\delta J\Big\}.\] Expressing the factor by the braces though $\alpha$, we obtain surface gravity (\ref{SurfaceGravity}): \begin{equation}\label{BHThermodynamics} \delta A_{+}=\frac{8\pi}{\kappa} \Big\{\delta\mu-\Omega_{H}\delta J\Big\}. \end{equation} Due to condition (\ref{Kerr-JM}) the surface area of the horizon always increases: \begin{equation}\label{AreaTheorem} \delta A_{+}>0\end{equation} Problem 28: irreducible mass Let us define the irreducible mass $M_{irr}$ of Kerr black hole as the mass of Schwarzschild black hole with the same horizon area. Find $M_{irr}(\mu,J)$ and $\mu(M_{irr},J)$. Which part of the total mass of a black hole can be extracted from it with the help of Penrose process? As defined, \[A_{+}=4\pi (2M_{irr})^{2}=16\pi M_{irr}^2.\] Then \[M_{irr}^{2}=\tfrac{1}{2} \Big(\mu^2 + \sqrt{\mu^4-J^2}\Big) \quad\Leftrightarrow\quad \mu^2=M_{irr}^{2}+\Big(\frac{J}{2M_{irr}}\Big)^{2}.\] This relation provides interesting interpretation: the full mass of a black hole $\mu$ consists of the irreducible mass $M_{irr}$ and the rotational energy $J/2M_{irr}$, which add up squared. The second term can in principle be extracted through the Penrose process. Problem 29: extremal limit Show that an underextremal Kerr black hole (with $a<\mu$) cannot be turned into the extremal one in any continuous accretion process. \[\delta\alpha=\delta\Big(\frac{J}{\mu^2}\Big) =\frac{1}{\mu^3} \Big[\mu \delta J-2J \delta\mu\Big],\] and using the condition (\ref{Kerr-JM}), we get \[\delta\alpha<\frac{2\delta\mu}{\mu} \frac{1+\sqrt{1-\alpha^2}}{\alpha} \cdot\sqrt{1-\alpha^2}.\] When $\alpha\to1$ the last factor tends to zero, so $\alpha$ cannot become equal or greater than unity in any continuous accretion process. This problem's results can be presented in the form that provides far-reaching analogy with the laws of thermodynamics. 0: Surface gravity $\kappa$ is constant on the horizon of a stationary black hole. The zeroth law of thermodynamics: a system in thermodynamic equilibrium has constant temperature $T$. 1: The relation \[\delta\mu=\frac{\kappa}{8\pi}\delta A_{+} +\Omega_{H}\delta J\] gives an analogy of the first law of thermodynamics, energy conservation. 2: Horizon area $A_+$ is nondecreasing. This analogy with the second law of thermodynamics hints at a correspondence between the horizon area and entropy. 3: There exists no such continuous process, which can lead as a result to zero surface gravity. This is an analogy to the third law of thermodynamics: absolute zero is unreachable. Particles' motion in the equatorial plane The following questions refer to a particle's motion in the equatorial plane $\theta=\pi/2$ of the Kerr metric. Problem 30: preparatory algebra Put down explicit expressions for the metric components and parameters $A,B,C,D,\omega$ \[g_{\mu\nu}=\begin{pmatrix} 1-\frac{2\mu}{r}&0&0&\frac{2\mu a}{r}\\ 0&-\frac{r^2}{\Delta}&0&0\\ 9&0&-r^2&0\\ \frac{2\mu a}{r}&0&0& -\frac{\Sigma^2}{r^2} \end{pmatrix},\quad g^{\mu\nu}=\begin{pmatrix} \frac{\Sigma^2}{r^2\Delta} &0&0&\frac{2\mu a}{\Delta r}\\ 0&-\frac{\Delta}{r^2}&0&0\\ 9&0&-\frac{1}{r^2}&0\\ \frac{2\mu a}{\Delta r}&0&0& -\frac{1}{\Delta}\Big(1-\frac{2\mu}{r}\Big) \end{pmatrix}\] where \[\Delta=r^{2}+a^{2}-2\mu r;\quad \Sigma^{2}=r^{2}(r^2+a^2)+2\mu r a^2;\] the other expressions, for $A,B,\ldots,\omega,\Omega_\pm$ etc. are not simplified essentially. Problem 31: zero energy particles What is the angular velocity of a particle with zero energy? For $\theta=\pi/2$ and $\rho^2=r^2$ we get \[\label{PenroseAngVelocity2} \Omega_{P}=\frac{\frac{2\mu r}{\rho^2}-1} {\frac{2\mu r}{\rho^2}a\sin^2 \theta}= \frac{1-\frac{\rho^2}{2\mu r}}{a\sin^2 \theta}= \Big\\| \rho^2=r^2,\; \sin\theta=1\Big\\| =\frac{1-\frac{r}{2\mu}}{a}.\] It s clear also that $\Omega_P$ turns to zero at the ergosurface, where $2\mu r=\rho^2$. On the other hand, due to the system of inequalities (\ref{ErgospereOmegas}) on the horizon it should be equal to the horizon's angular velocity $\Omega_H$ (\ref{Kerr-OmegaHorizon}). Indeed, the latter can be written using $r_{+}r_{-}=a^2$ as \[\Omega_{H}\equiv\omega\Big\|_{r=r_+} =\frac{2\mu ra}{\Sigma^2}\big\|_{r=r_+} =\frac{a}{2\mu r_+}=\frac{r_-}{2\mu a}.\] and then we affirm that \[\Omega_{P}\|_{r_+} =\frac{1}{a}\Big(1-\frac{r_+}{2\mu}\Big) =\frac{1}{a}\frac{2\mu-r_+}{2\mu} =\frac{r_-}{2\mu a}=\Omega_H.\] Problem 32: geodesics and effective potential Use the normalizing conditions for the $4$-velocity $u^{\mu}u_{\mu}=\epsilon^2$ and two conservation laws to derive geodesic equations for particles, determine the effective potential for radial motion. The integrals of motion are \[\left\{\begin{array}{l} E\equiv u_{t}=g_{tt}u^{t}+g_{t\varphi}u^\varphi,\\ -L\equiv u_{\varphi}=g_{t\varphi}u^{t}+ g_{\varphi\varphi}u^{\varphi} \end{array}\right.\;\Rightarrow\; \left\{\begin{array}{l} u^{t}=\frac{1}{G}(g_{\varphi\varphi}E+ g_{t\varphi}L)\\ u^{\varphi}=-\frac{1}{G}(g_{tt}L-g_{t\varphi}E) \end{array}\right.,\;\text{where}\quad G\equiv \begin{vmatrix} g_{tt}&g_{t\varphi}\\ g_{t\varphi}&g_{\varphi\varphi}\end{vmatrix}.\] Plugging in the metric, we get $G=-\Delta$ and \[ u^{t}=\frac{1}{\Delta}\Big[ \big(r^2+a^2+\frac{2\mu a^2}{r}\big)E- \frac{2\mu a}{r} L\Big];\quad u^\varphi =\frac{1}{\Delta}\Big[ \frac{2\mu a}{r}E +\big(1-\frac{2\mu}{r}\big)L \Big].\] Let us write the normalizing condition as \[u^{\mu}u_{\mu}=\epsilon^2,\] so that $\epsilon^2=1$ for a massive particle, and $\epsilon^2=0$ to a massless one. Then \[\epsilon^2=g^{tt}u_{t}^{2} +2g^{t\varphi}u_{t}u_{\varphi} +g^{\varphi\varphi}u_{\varphi}^{2}+g^{rr}u_{r}^{2},\] and taking into account that $g_{rr}=1/g_{rr}$, \[\Big(\frac{dr}{ds}\Big)^{2}\equiv (u^{r})^{2} =g^{rr}(g_{rr}u_{r})^{2}= g^{rr}\Big(\epsilon^2-g^{tt}E^2+2g^{t\varphi}EL -g^{\varphi\varphi}L^2\Big).\] This can be transformed to an equation for $r(s)$ in the form \begin{equation}\label{KerrEqPotential} \frac{1}{2}\Big(\frac{dr}{ds}\Big)^{2}+U_{eff}=0, \quad\text{where}\quad U_{eff}=\frac{\epsilon^2-E^2}{2} -\frac{\epsilon^2 \mu}{r} +\frac{L^2-a^{2}(E^2-\epsilon^2)}{2r^2} -\frac{\mu(L-aE)^{2}}{r^3}\end{equation} is effective gravitational energy, with both $E$ and $L$ acting as parameters. As both of them are present in the left hand side, we can as well leave just zero on the right. In terms of dimensionless variables \[\xi=\frac{r}{\mu},\quad \alpha=\frac{a}{\mu}, \quad \lambda=\frac{L}{\mu}\] the full system of equations is \begin{align} &\frac{1}{2}(u^r)^{2}+U_{eff}=0;\quad U_{eff}= \frac{\epsilon^{2}-E^2}{2} -\frac{\epsilon^{2}}{\xi} +\frac{\lambda^2 +\alpha^{2}(\epsilon^2-E^2)} {2\xi^2} -\frac{(\lambda-\alpha E)^{2}}{\xi^3}.\\ &u^{t}=\frac{\mu^2 E}{\Delta} \Big[\xi^2+\alpha^2+\frac{2\alpha^2}{\xi} -\frac{2\alpha}{\xi}\frac{\lambda}{E}\Big];\\ &u^\varphi =\frac{\mu E}{\Delta} \Big[\frac{2\alpha}{\xi}+ \big(1-\frac{2}{\xi}\big)\frac{\lambda}{E}\Big]. \end{align} Problem 33: principal null geodesics Integrate the equations of motion for null geodesics with $L=aE$, investigate the asymptotes close to the horizons, limits $a\to 0$ and $a\to \mu$. When $L=Ea$, which corresponds to the impact parameter at infinity equal to $a$, the equations for null geodesics are essentially simplified: \begin{align} &(u^r)^{2}=E^2;\\ &u^{t}\equiv\frac{dt}{ds} =\frac{E}{\Delta}\big[r^2+a^2\big];\\ &u^\varphi\equiv\frac{d\varphi}{ds} =\frac{aE}{\Delta}. \end{align} Note that this relation, $L=Ea$, singles out quite peculiar (critical) particles, for which the asymptote of the effective potential at small $\xi$ is $\sim \xi^{-2}$, as opposed to any other particle, for whichh $U_{eff}\sim \xi^{-3}$. Then $ds=\pm dr/E$, where the plus sign corresponds to a photon falling to the center and minus to the one moving from the center, and \begin{align} &\mp\varphi(r)=\mp\int d\varphi=\int\frac{a dr}{\Delta} =\int\frac{a\, dr}{(r-r_{-})(r-r_{+})} =\ldots=\frac{\alpha}{2\sqrt{1-\alpha^2}} \ln \Big\|\frac{r-r_-}{r-r_+}\Big\|;\\ &\mp t(r)=\mp\int dt=\int \frac{dr(r^2+a^2)}{\Delta} =\ldots =r\mu \ln\Big\|\frac{r-r_-}{r-r_+}\Big\|+ \frac{\mu}{\sqrt{1-\alpha^2}}\ln\|\Delta\|. \end{align} In the limit $r\to r_{+}$ both $\varphi(r)$ and $t(r)$ diverge as logarithms, but it is not hard to show that \[\frac{\varphi}{t}\approx \frac{\alpha/2\mu}{1+\sqrt{1-\alpha^2}} =\frac{a}{2\mu r_{+}}\equiv \Omega_{H}.\] This should be expected, as we know that at the horizon all particles should rotate with the angular velocity of the horizon. Assuming $\alpha\to0$ we get the usual Schwarzshild solutions $\varphi=0$, $\mp t=r+\mu.$ In the opposite limit $\alpha\to 1$ the quantity $\Delta$ has a double root and recalculating the integrals, we see that $t$ and $\varphi$ now diverge not logarithmically but as $(r-r_{+})^{-1}$. Problem 34: innermost stable circular orbits, massless case Find the minimal radii of circular geodesics for massless particles, the corresponding values of integrals of motion and angular velocities. Show that of the three solutions one lies beyond the horizon, one describes motion in positive direction and one in negative direction. Explore the limiting cases of Schwarzschild $a\to0$ and extreme Kerr $a\to\mu$. For a massless particle $\epsilon=0$ the equation for $r(t)$ is \[\frac{1}{2}\Big(\frac{dr}{d\tilde{s}}\Big)^2 +U_{eff}=0,\quad\text{where}\quad U_{eff}=-\frac{1}{2} +\frac{\lambda^2-\alpha^2 E^2}{2\xi^2} -\frac{(\lambda-\alpha E)^2}{\xi^3}.\] It is convenient, as usual for massless particles, to introduce the parameter $p=L/E$, which at $r\to\infty$ has the meaning of the impact parameter. For a circular orbit \[U_{eff}=0,\quad \frac{dU_{eff}}{d\xi}=0.\] The second condition gives \[\xi_{0}=3\frac{\lambda-\alpha E}{\lambda+\alpha E}= 3\frac{\sigma-1}{\sigma+1},\quad\text{where}\quad \sigma\equiv\frac{\lambda}{\alpha E}= \frac{p}{a}=\frac{p}{\alpha\mu}.\] The first one leads to a cubic equation for $\sigma$: \[(\sigma+1)^{3}=\frac{27}{\alpha^2}(\sigma-1).\] When $\alpha\ll 1$ (the near-Schwarzschild case) the three roots are $\sigma\approx 1+\frac{8}{27}\alpha^2,\pm \frac{3\sqrt{3}}{\alpha}$. The first one gives $\xi_0\approx\alpha^2$, i.e. this orbit lies beyond the horizons (we do not consider this region), and the other two \[\alpha\ll1:\qquad \sigma\approx3\sqrt{3}/\alpha; \quad\Rightarrow\quad p\approx 3\sqrt{3}\mu,\quad\xi_{0}\approx 3.\] Those are the familiar parameters of a Schwarzschild black hole's photon sphere of , at $r=3\mu=\frac{3}{2}r_{g}$. It is natural that in this limit the radius does not depend on the sign of $L$. In the limit $\alpha\to1$ (extremal Kerr black hole) the equation is reduced to $(\sigma+1)^{3}=27(\sigma-1)$, the roots of which are, as can be checked, $\sigma=2,2,-7$. Substituting this in $\xi_0$ and $p$, we get \[p_{+}^{(extr)}\approx2\mu,\quad \xi_{+}^{(extr)}\approx 1;\qquad p_{-}^{(extr)}\approx-7\mu,\quad \xi_{-}^{(extr)}\approx 4.\] The first root corresponds to prograde photons, the second one to the retrograde ones. It can be shown that in the limit $a\to 0$ one of the two close roots, which merge at $a=1$, lies before the horizon, and the other one is beyond it. In the general case it is convenient to rewrite the equation for $\sigma$ as \[\nu^3=\nu-2\beta,\quad\text{where}\quad \beta=\frac{\alpha}{3^{3/2}},\quad \nu=\beta(\sigma+1);\qquad \xi_{0}=3\nu^2.\] This is a reduced equation, solved by the change of variables $\nu=a+b$ with additional constraint $3ab=1$. The resulting system for $a^{3},b^{3}$ is \[a^{6}-2\beta a^{3}+\frac{1}{27}=0.\] Its solution is \[a^{3}=-\frac{\alpha\pm\sqrt{\alpha^2-1}}{3^{3/2}}= \pm 3^{-3/2}\exp\{\pm i\omega\},\quad\text{where} \quad \omega=\arccos \alpha \in\Big(0,\frac{\pi}{2}\Big).\] Extracting the root and selecting the pairs $(a,b)$ which obey the imposed condition $3ab=1$, we obtain the three roots \[\nu_1=-\frac{2}{3}\cos\frac{\omega}{3};\quad \nu_2=\frac{2}{3}\cos\frac{\pi-\omega}{3};\quad \nu_3=\frac{2}{3}\cos\frac{\pi+\omega}{3}.\] The "radii" of corresponding orbits are \[\xi_{1}=4\cos^{2}\frac{\omega}{3};\quad \xi_{2}=4\cos^{2}\frac{\pi-\omega}{3}\quad \xi_{3}=4\cos^{2}\frac{\pi+\omega}{3}.\] As $\omega\in(0,\pi/2)$, it is not hard to show that a sequence of inequalities hold \[0<\xi_3 <1<\xi_2 <3<\xi_1 <4;\] it also turns out that $\xi_2<\xi_{+}<\xi_3$ (in terms of $\omega$ the "radius" of the outer horizon is $\xi_{+}=1+\sin\omega$, so the problem is reduced to trigonometric inequalities). Thus, the second root always lies on the outside of the horizon, while the third one is beyond it and is unphysical. The two remaining solutions correspond to positive and negative $p$. Taking into account that $\pi-\omega=\arccos(-\alpha)$, they both can be expressed in the form \begin{equation}\label{KerrPhotonOrbits} \xi_{1,2}=4\cos^{2}\Big[\frac{1}{3} \arccos(\pm\alpha)\Big];\end{equation} The corresponding angular momenta are \[p_{i}=\mu\alpha\sigma_{i} =\mu\alpha(\nu_{i}/\beta-1) =\mu(3\sqrt{3}\nu_{i}-\alpha)\quad\Rightarrow\quad p_{1,2}=\mp 6\mu\cos\Big[\frac{1}{3} \arccos(\pm\alpha)\Big]-a.\] The angular velocities for photons on circular orbits are \[\Omega_{1,2}=\Omega_{\mp}(\xi=\xi_{1,2}),\] where the upper sign corresponds to retrograde orbits ($\xi_1$) and the lower sign to prograde ones ($\xi_2$). Problem 35: circular orbits for massive particles Find $L^2$ and $E^2$ as functions of radii for circular geodesics of the massive particles. Let us write the effective potential for massive particles in terms of $u=1/\xi$: \[U_{eff}=\frac{1-E^2}{2}-u+\tfrac{\beta}{2}u^2 -\nu^2 u^3,\quad\text{where}\quad \nu=\lambda-\alpha E,\quad \beta=\lambda^{2}+\alpha^2 (1-E^2).\] For a circular orbit (the full effective energy with the chosen potential is zero) \[U_{eff}(u)=0,\quad {U_{eff}}'(u)=0,\] that is \begin{align} &\frac{1-E^2}{2}-u +\frac{\beta}{2}u^2-\nu^2 u^{3}=0;\\ &-1+\beta u-3\nu^2 u^2=0. \end{align} The orbit $u=u_{i}(E,\lambda)$ is stable if for the given $E$ and $\lambda$ in the neighborhood of $u_{i}$ holds $U_{eff}<0$, so that $(u^r)^{2}<0$ and there are no other solutions. Thus a stable orbit corresponds to a point in which $U_{eff}$ touches zero from above, and unstable orbits to a point in which is touches zero from below. The condition of stability is $d^{2}U_{eff}/dr^{2}>0$. Taking into account $0=U_{eff}=U_{eff}'$, it is equivalent to $d^{2}U_{eff}/du^{2}>0$, and thus \[6\nu^{2}u<\beta.\] An equality would mean that the two touching points merge into an inflection point, which gives us the minimal radius of the stable orbit and the maximal radius of the unstable one. Subtracting the second equation multiplied by $u/2$ from the first one, we get \[\nu^2 u^3 -u=E^2-1.\] Using $U_{eff}'=0$, we can exclude $\beta$ and $E^2$ \[\beta=3\nu^{2}u+u^{-1};\quad E=\frac{u\nu^2 (3u-1)+1-\alpha^2 u}{2\alpha u\nu},\] and obtain a quadratic equation for $\nu^2$: \begin{equation}\label{KerrEq-M-Nu2Eq} u^{2}\big[(3u-1)^{2}-4\alpha^{2}u^{3}\big]\nu^4 -2u\big[\alpha^{2}u(u+1)-(3u-1)\big]\nu^2 -(1-\alpha^2 u)^{2}=0. \end{equation} After straightforward calculation the discriminant can be brought to the form \[4\alpha^2 u^3 (\alpha^2 u^2 -2u+1)^{2}.\] When $r>r_{+}$, the expression in braces can be shown to always be positive. The two solutions for $\nu^2$ and $E^2=(\nu^2 u^3 +1-u)$ then take the form \begin{equation}\label{KerrEq-M-Nu2Sol} \nu^{2}=\frac{(u^{-1/2}\pm\alpha)^2} {1-3u\mp 2\alpha u^{3/2}},;\qquad E^{2}=\frac{(1-2u\mp\alpha u^{3/2})^{2}} {1-3u\mp 2\alpha u^{3/2}}. \end{equation} Restoring relative signs of $E$ and $\nu$ from the condition $U_{eff}'=0$, we finally obtain \begin{equation}\label{KerrEq-M-La2Sol} \lambda^{2} =\frac{(u^{-1/2}\pm 2\alpha u+\alpha^2 u^{3/2})^{2}} {1-3u\mp 2\alpha u^{3/2}}. \end{equation} Problem 36: innermost stable circular orbits, massive case Derive equation for the minimal radius of a stable circular orbit; find the energy and angular momentum of a particle on it, the minimal radius in the limiting cases $a/\mu\to 0,1$. An orbit's stability condition is $3\nu^2 u^2 <1$. Substituting $\beta$, it is brought to \[3\nu^2 u^2 <1,\] and substituting $\nu^2$, to \[3\alpha^2 u^2 \pm 8\alpha u^{3/2}+6u-1<0.\] In terms of $\xi$ \[\xi^2-6\xi\pm 8\alpha \sqrt{\xi}-3\alpha^2>0.\] In the limit $\alpha\to 0$ we get $\xi>6$, i.e. $r>3\mu=\frac{3}{2}r_{g}$, which is the familiar result for Schwarzschild. " Radius of innermost stable circular orbit as function of $\alpha=a/\mu$: the upper curve for $L<0$, the lower one for $L>0$. The horizon is shown by the dashed line." In the limit $a\to 1$ different signs lead to different inequalities, which are easier to solve numerically. From the curves we see, that for the "$-$" sign (retrograde or counter-rotating orbits) the stability condition holds when $\xi>9$, and for the "$+$" sign (prograde or co-rotating orbits) when $\xi>1$. In the general case then \begin{align} &''-'':\qquad \alpha\in(0,1)\;\Rightarrow\; \xi_{min}\in(6,9);\\ &''+'':\qquad \alpha\in(0,1)\;\Rightarrow\; \xi_{min}\in(6,1). \end{align} The binding energy on the minimal stable (prograde) orbit is \[E^2=\nu^2 u^3+1-u=u/3+1-u=1-\frac{2}{3\xi},\] and for $\alpha\approx 1$ for the most strongly bound particle with $\xi\approx 1$ \[E_{min} \approx \sqrt{1-\frac{2}{3}} =\frac{1}{\sqrt{3}}, \] so the binding energy in the units of rest mass is \[E_{acc}\approx 1-\frac{1}{\sqrt{3}}\approx 0.42.\] In the model of $\alpha$-disk accretion on a compact object this number gives the upper limit to the accretion effectiveness, i.e. the part of a particle's rest mass that can be radiated into outer space due to dissipation in the disk caused by slow slipping of particles into the gravitational well along almost circular orbits. Retrieved from "http://universeinproblems.com/index.php?title=Kerr_black_hole&oldid=1795"	CommonCrawl
If I leave a glass of water out, why do only the surface molecules vaporize? If I leave a glass of water out on the counter, some of the water turns into vapor. I've read that this is because the water molecules crash into each other like billiard balls and eventually some of the molecules at the surface acquire enough kinetic energy that they no longer stay a liquid. They become vapor. Why is it only the molecules on the surface that become vapor? Why not the molecules in the middle of the glass of water? After all, they too are crashing into each other. If I put a heating element under the container and increase the average kinetic energy in the water molecules to the point that my thermometer reads ~100°C, the molecules in the middle of the glass do turn into vapor. Why doesn't this happen even without applying the heat, like it does to the surface molecules? thermodynamics statistical-mechanics temperature everyday-life evaporation edited Jan 30 at 9:43 lyndonlyndon $\begingroup$ Read the answer because they are answers indeed. But it is " similar" to say "why I can exit the room through the door only,and not through the middle?" Not mechanistically but quite the same. $\endgroup$ – Alchimista Jan 30 at 8:24 $\begingroup$ This "room model" explains the role of the surface but it does not explain why, above a certain temperature, the liquid can come out in the middle of the room. The model can be refined by adding the famous Star Trek transporter which would be efficient only above the ebullition temperature. But it loose a little its original simplicity.... $\endgroup$ – Vincent Fraticelli Jan 31 at 6:11 From a thermodynamic point of view, at fixed pressure, the vaporization takes place when the temperature exceeds the temperature of change of state $ Tc (P ) $ Within the liquid, the pressure that is to be taken into account is the hydrostatic pressure. This pressure is a little greater than 1 bar and the associated vaporization temperature is 100 ° C. On the surface (thickness of some mean free path), the environment of the molecules is different. the pressure to be taken into account is the partial pressure of water vapor, which is related to the moisture content of the air. If the humidity is less than 100%, this pressure is well below 1 bar and evaporation takes place at a much lower temperature. Vincent FraticelliVincent Fraticelli 1,83111 gold badge22 silver badges55 bronze badges $\begingroup$ I am aware of the possibilities of delaying the ebullition related to the nucleation phenomenon and which depend on the state of purity of the water. But it is known that water in a vacuum chamber boils. For exemple, on Youtube :youtube.com/watch?v=glLPMXq6yc0 Since English is not my native language, I hope to understand your comments properly. (and sorry for my poor english) $\endgroup$ – Vincent Fraticelli Jan 30 at 19:16 There's a fundamental difference between a liquid changing to a gas at the surface vs. in the bulk: the formation of new surface area, which costs energy. Net evaporation from the surface is spontaneous whenever the relative humidity is less than 100% because energy fluctuations enable surface molecules to detach into the gas phase, as you describe. Here, the total surface area doesn't change. In contrast, gas formation within the bulk (i.e., the formation of an evaporative bubble) requires the formation of an additional liquid-gas interface, which carries an energy cost because bonds tend to be unsatisfied at interfaces. In fact, the energy penalty from having to form the surface of a nucleating vapor bubble is so important that we generally can only achieve boiling by (1) providing an existing surface or (2) waiting a large amount of time for a large gas cluster to randomly assemble or (3) superheating the liquid to increase the driving force for boiling (or a combination of these). ChemomechanicsChemomechanics The water molecules in the liquid attract each other. Their thermal velocity distribution allows some molecules to be fast enough to overcome this attraction. If it happens to a molecule at the surface to be kicked by such a fast molecule, it may be kicked with an impulse stronger than the attractive forces, and therefore leave the liquid. The same kick inside the liquid would be passed on to other molecules very efficiently. If a gas bubble forms inside the liquid, the reduced attraction between the molecules in the gas is part of the energy penalty to be paid for the bubble formation in the form of heat from an external heat source. flaudemusflaudemus Not the answer you're looking for? Browse other questions tagged thermodynamics statistical-mechanics temperature everyday-life evaporation or ask your own question. Smell the Coffee Why doesn't a glass french press shatter when boiling water is poured in? Relation between Free Energy barrier and relaxation time? Why doesn't the gravitational energy in this system of evaporating and condensing water violate the second law of thermodynamics? Temperature of steam rising off boiling water Does $dS = \frac{dQ}{T}$ explain why evaporation increases total entropy? Crash simulation on Mythbusters Dependence of relative humidity on temperature and pressure Is water vapor in a metastable state at 1 atm and at room temperature? Help me in understanding the general principles behind evaporation and condensation of a liquid	CommonCrawl
Hostname: page-component-7ccbd9845f-dzwm5 Total loading time: 1.487 Render date: 2023-01-29T08:57:53.692Z Has data issue: true Feature Flags: { "useRatesEcommerce": false } hasContentIssue true >Political Analysis >Generalized Synthetic Control Method: Causal Inference... Estimation Strategy Monte Carlo Evidence Empirical Example Generalized Synthetic Control Method: Causal Inference with Interactive Fixed Effects Models Part of: PA editors' choice articles Published online by Cambridge University Press: 21 February 2017 Yiqing Xu Yiqing Xu* Department of Political Science, University of California, San Diego. 9500 Gilman Drive #0521, La Jolla, CA 92093, USA. Email: [email protected] *Email: [email protected] Save PDF (2 mb) View PDF[Opens in a new window] Difference-in-differences (DID) is commonly used for causal inference in time-series cross-sectional data. It requires the assumption that the average outcomes of treated and control units would have followed parallel paths in the absence of treatment. In this paper, we propose a method that not only relaxes this often-violated assumption, but also unifies the synthetic control method (Abadie, Diamond, and Hainmueller 2010) with linear fixed effects models under a simple framework, of which DID is a special case. It imputes counterfactuals for each treated unit using control group information based on a linear interactive fixed effects model that incorporates unit-specific intercepts interacted with time-varying coefficients. This method has several advantages. First, it allows the treatment to be correlated with unobserved unit and time heterogeneities under reasonable modeling assumptions. Second, it generalizes the synthetic control method to the case of multiple treated units and variable treatment periods, and improves efficiency and interpretability. Third, with a built-in cross-validation procedure, it avoids specification searches and thus is easy to implement. An empirical example of Election Day Registration and voter turnout in the United States is provided. Political Analysis , Volume 25 , Issue 1 , January 2017 , pp. 57 - 76 DOI: https://doi.org/10.1017/pan.2016.2[Opens in a new window] Copyright © The Author(s) 2017. Published by Cambridge University Press on behalf of the Society for Political Methodology. Difference-in-differences (DID) is one of the most commonly used empirical designs in today's social sciences. The identifying assumptions for DID include the "parallel trends" assumption, which states that in the absence of the treatment the average outcomes of treated and control units would have followed parallel paths. This assumption is not directly testable, but researchers have more confidence in its validity when they find that the average outcomes of the treated and control units follow parallel paths in pretreatment periods. In many cases, however, parallel pretreatment trends are not supported by data, a clear sign that the "parallel trends" assumption is likely to fail in the posttreatment period as well. This paper attempts to deal with this problem systematically. It proposes a method that estimates the average treatment effect on the treated using time-series cross-sectional (TSCS) data when the "parallel trends" assumption is not likely to hold. The presence of unobserved time-varying confounders causes the failure of this assumption. There are broadly two approaches in the literature to deal with this problem. The first one is to condition on pretreatment observables using matching methods, which may help balance the influence of potential time-varying confounders between treatment and control groups. For example, Abadie (Reference Abadie2005) proposes matching before DID estimations. Although this method is easy to implement, it does not guarantee parallel pretreatment trends. The synthetic control method proposed by Abadie, Diamond, and Hainmueller (Reference Abadie, Diamond and Hainmueller2010, Reference Abadie, Diamond and Hainmueller2015) goes one step further. It matches both pretreatment covariates and outcomes between a treated unit and a set of control units and uses pretreatment periods as criteria for good matches.Footnote 1 Specifically, it constructs a "synthetic control unit" as the counterfactual for the treated unit by reweighting the control units. It provides explicit weights for the control units, thus making the comparison between the treated and synthetic control units transparent. However, it only applies to the case of one treated unit and the uncertainty estimates it offers are not easily interpretable.Footnote 2 The second approach is to model the unobserved time-varying heterogeneities explicitly. A widely used strategy is to add in unit-specific linear or quadratic time trends to conventional two-way fixed effects models. By doing so, researchers essentially rely upon a set of alternative identification assumptions that treatment assignment is ignorable conditional on both the fixed effects and the imposed trends (Mora and Reggio Reference Mora and Reggio2012). Controlling for these trends, however, often consumes a large number of degrees of freedom and may not necessarily solve the problem if the underlying confounders are not in forms of the specified trends. An alternative way is to model unobserved time-varying confounders semiparametrically. For example, Bai (Reference Bai2009) proposes an interactive fixed effects (IFE) model, which incorporates unit-specific intercepts interacted with time-varying coefficients. The time-varying coefficients are also referred to as (latent) factors while the unit-specific intercepts are labeled as factor loadings. This approach builds upon an earlier literature on factor models in quantitative finance.Footnote 3 The model is estimated by iteratively conducting a factor analysis of the residuals from a linear model and estimating the linear model that takes into account the influences of a fixed number of most influential factors. Pang (Reference Pang2010, Reference Pang2014) explores nonlinear IFE models with exogenous covariates in a Bayesian multi-level framework. Stewart (Reference Stewart2014) provides a general framework of estimating IFE models based on a Bayesian variational inference algorithm. Gobillon and Magnac (Reference Gobillon and Magnac2016) show that IFE models outperform the synthetic control method in DID settings when factor loadings of the treatment and control groups do not share common support.Footnote 4 This paper proposes a generalized synthetic control (GSC) method that links the two approaches and unifies the synthetic control method with linear fixed effects models under a simple framework, of which DID is a special case. It first estimates an IFE model using only the control group data, obtaining a fixed number of latent factors. It then estimates factor loadings for each treated unit by linearly projecting pretreatment treated outcomes onto the space spanned by these factors. Finally, it imputes treated counterfactuals based on the estimated factors and factor loadings. The main contribution of this paper, hence, is to employ a latent factor approach to address a causal inference problem and provide valid, simulation-based uncertainty estimates under reasonable assumptions. This method is in the spirit of the synthetic control method in the sense that by essence it is a reweighting scheme that takes pretreatment treated outcomes as benchmarks when choosing weights for control units and uses cross-sectional correlations between treated and control units to predict treated counterfactuals. Unlike the synthetic matching method, however, it conducts dimension reduction prior to reweighting such that vectors to be reweighted on are smoothed across control units. The method can also be understood as a bias correction procedure for IFE models when the treatment effect is heterogeneous across units.Footnote 5 It treats counterfactuals of treated units as missing data and makes out-of-sample predictions for posttreatment treated outcomes based on an IFE model. This method has several advantages. First, it generalizes the synthetic control method to cases of multiple treated units and/or variable treatment periods. Since the IFE model is estimated only once, treated counterfactuals are obtained in a single run. Users therefore no longer need to find matches of control units for each treated unit one by one.Footnote 6 This makes the algorithm fast and less sensitive to the idiosyncrasies of a small number of observations. Second, the GSC method produces frequentist uncertainty estimates, such as standard errors and confidence intervals, and improves efficiency under correct model specifications. A parametric bootstrap procedure based on simulated data can provide valid inference under reasonable assumptions. Since no observations are discarded from the control group, this method uses more information from the control group and thus is more efficient than the synthetic matching method when the model is correctly specified. Third, it embeds a cross-validation scheme that selects the number of factors of the IFE model automatically, and thus is easy to implement. One advantage of the DID data structure is that treated observations in pretreatment periods can naturally serve as a validation dataset for model selection. We show that with sufficient data, the cross-validation procedure can pick up the correct number of factors with high probability, therefore reducing the risks of overfitting. The GSC method has two main limitations. First, it requires more pretreatment data than fixed effects estimators. When the number of pretreatment periods is small, "incidental parameters" can lead to biased estimates of the treatment effects. Second, and perhaps more importantly, modeling assumptions play a heavier role with the GSC method than the original synthetic matching method. For example, if the treated and control units do not share common support in factor loadings, the synthetic matching method may simply fail to construct a synthetic control unit. Since such a problem is obvious to users, the chances that users misuse the method are small. The GSC method, however, will still impute treated counterfactuals based on model extrapolation, which may lead to erroneous conclusions. To safeguard against this risk, it is crucial to conduct various diagnostic checks, such as plotting the raw data, fitted values, and predicted counterfactuals. The rest of the paper is organized as follows. Section 2 sets up the model and defines the quantities of interest. Section 3 introduces the GSC estimator, describes how it is implemented, and discuss the parametric bootstrap procedure. Section 4 reports simulation results that explores the finite sample properties of the GSC estimator and compares it with several existing methods. Section 5 illustrates the method with an empirical example that investigates the effect of Election Day Registration (EDR) laws on voter turnout in the United States. The last section concludes. 2 Framework Suppose $Y_{it}$ is the outcome of interest of unit $i$ at time $t$ . Let ${\mathcal{T}}$ and ${\mathcal{C}}$ denote the sets of units in treatment and control groups, respectively. The total number of units is $N=N_{tr}+N_{co}$ , where $N_{tr}$ and $N_{co}$ are the numbers of treated and control units, respectively. All units are observed for $T$ periods (from time $1$ to time $T$ ). Let $T_{0,i}$ be the number of pretreatment periods for unit $i$ , which is first exposed to the treatment at time $(T_{0,i}+1)$ and subsequently observed for $q_{i}=T-T_{0,i}$ periods. Units in the control group are never exposed to the treatment in the observed time span. For notational convenience, we assume that all treated units are first exposed to the treatment at the same time, i.e., $T_{0,i}=T_{0}$ and $q_{i}=q$ ; variable treatment periods can be easily accommodated. First, we assume that $Y_{it}$ is given by a linear factor model. Assumption 1. Functional form: $$\begin{eqnarray}Y_{it}=\unicode[STIX]{x1D6FF}_{it}D_{it}+x_{it}^{\prime }\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D706}_{i}^{\prime }f_{t}+\unicode[STIX]{x1D700}_{it},\end{eqnarray}$$ where the treatment indicator $D_{it}$ equals 1 if unit $i$ has been exposed to the treatment prior to time $t$ and equals 0 otherwise (i.e., $D_{it}=1$ when $i\in {\mathcal{T}}$ and $t>T_{0}$ and $D_{it}=0$ otherwise).Footnote 7 $\unicode[STIX]{x1D6FF}_{it}$ is the heterogeneous treatment effect on unit $i$ at time $t$ ; $x_{it}$ is a $(k\times 1)$ vector of observed covariates, $\unicode[STIX]{x1D6FD}=[\unicode[STIX]{x1D6FD}_{1},\ldots ,\unicode[STIX]{x1D6FD}_{k}]^{\prime }$ is a $(k\times 1)$ vector of unknown parameters,Footnote 8 $f_{t}=[f_{1t},\ldots ,f_{rt}]^{\prime }$ is an $(r\times 1)$ vector of unobserved common factors, $\unicode[STIX]{x1D706}_{i}=[\unicode[STIX]{x1D706}_{i1},\ldots ,\unicode[STIX]{x1D706}_{ir}]^{\prime }$ is an $(r\times 1)$ vector of unknown factor loadings, and $\unicode[STIX]{x1D700}_{it}$ represents unobserved idiosyncratic shocks for unit $i$ at time $t$ and has zero mean. Assumption 1 requires that the treated and control units are affected by the same set of factors and the number of factors is fixed during the observed time periods, i.e., no structural breaks are allowed. The factor component of the model, $\unicode[STIX]{x1D706}_{i}^{\prime }f_{t}=\unicode[STIX]{x1D706}_{i1}f_{1t}+\unicode[STIX]{x1D706}_{i2}f_{2t}+\cdots +\unicode[STIX]{x1D706}_{ir}f_{rt}$ , takes a linear, additive form by assumption. In spite of the seemingly restrictive form, it covers a wide range of unobserved heterogeneities. First and foremost, conventional additive unit and time fixed effects are special cases. To see this, if we set $f_{1t}=1$ and $\unicode[STIX]{x1D706}_{i2}=1$ and rewrite $\unicode[STIX]{x1D706}_{i1}=\unicode[STIX]{x1D6FC}_{i}$ and $f_{2t}=\unicode[STIX]{x1D709}_{t}$ , then $\unicode[STIX]{x1D706}_{i1}f_{1t}+\unicode[STIX]{x1D706}_{i2}f_{2t}=\unicode[STIX]{x1D6FC}_{i}+\unicode[STIX]{x1D709}_{t}$ .Footnote 9 Moreover, the term also incorporates cases ranging from unit-specific linear or quadratic time trends to autoregressive components that researchers often control for when analyzing TSCS data.Footnote 10 In general, as long as an unobserved random variable can be decomposed into a multiplicative form, i.e., $U_{it}=a_{i}\times b_{t}$ , it can be absorbed by $\unicode[STIX]{x1D706}_{i}^{\prime }f_{t}$ while it cannot capture unobserved confounders that are independent across units. To formalize the notion of causality, we also use the notation from the potential outcomes framework for causal inference (Neyman Reference Neyman1923; Rubin Reference Rubin1974; Holland Reference Holland1986). Let $Y_{it}(1)$ and $Y_{it}(0)$ be the potential outcomes for individual $i$ at time $t$ when $D_{it}=1$ or $D_{it}=0$ , respectively. We thus have $Y_{it}(0)=x_{it}^{\prime }\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D706}_{i}^{\prime }f_{t}+\unicode[STIX]{x1D700}_{it}$ and $Y_{it}(1)=\unicode[STIX]{x1D6FF}_{it}+x_{it}^{\prime }\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D706}_{i}^{\prime }f_{t}+\unicode[STIX]{x1D700}_{it}$ . The individual treatment effect on treated unit $i$ at time $t$ is therefore $\unicode[STIX]{x1D6FF}_{it}=Y_{it}(1)-Y_{it}(0)$ for any $i\in {\mathcal{T}}$ , $t>T_{0}$ . We can rewrite the DGP of each unit as: $$\begin{eqnarray}Y_{i}=D_{i}\circ \unicode[STIX]{x1D6FF}_{i}+X_{i}\unicode[STIX]{x1D6FD}+F\unicode[STIX]{x1D706}_{i}+\unicode[STIX]{x1D700}_{i},\quad i\in 1,2,\ldots N_{co},N_{co}+1,\ldots ,N,\end{eqnarray}$$ where $Y_{i}=[Y_{i1},Y_{i2},\ldots ,Y_{iT}]^{\prime }$ ; $D_{i}=[D_{i1},D_{i2},\ldots ,D_{iT}]^{\prime }$ and $\unicode[STIX]{x1D6FF}_{i}=[\unicode[STIX]{x1D6FF}_{i1},\unicode[STIX]{x1D6FF}_{i2},\ldots ,\unicode[STIX]{x1D6FF}_{iT}]^{\prime }$ (symbol " $\circ$ " stands for point-wise product); $\unicode[STIX]{x1D700}_{i}=[\unicode[STIX]{x1D700}_{i1},\unicode[STIX]{x1D700}_{i2},\ldots ,\unicode[STIX]{x1D700}_{iT}]^{\prime }$ are $(T\times 1)$ vectors; $X_{i}=[x_{i1},x_{i2},\ldots ,x_{iT}]^{\prime }$ is a $(T\times k)$ matrix; and $F=[f_{1},f_{2},\ldots ,f_{T}]^{\prime }$ is a $(T\times r)$ matrix. The control and treated units are subscripted from 1 to $N_{co}$ and from $N_{co}+1$ to $N$ , respectively. The DGP of a control unit can be expressed as: $Y_{i}=X_{i}\unicode[STIX]{x1D6FD}+F\unicode[STIX]{x1D706}_{i}+\unicode[STIX]{x1D700}_{i}$ , $i\in 1,2,\ldots N_{co}$ . Stacking all control units together, we have: (1) $$\begin{eqnarray}Y_{co}=X_{co}\unicode[STIX]{x1D6FD}+F\unicode[STIX]{x1D6EC}_{co}^{\prime }+\unicode[STIX]{x1D700}_{co},\end{eqnarray}$$ in which $Y_{co}=[Y_{1},Y_{2},\ldots ,Y_{N_{co}}]$ and $\unicode[STIX]{x1D700}_{co}=[\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2},\ldots ,\unicode[STIX]{x1D700}_{N_{co}}]$ are $(T\times N_{co})$ matrices; $X_{co}$ is a three-dimensional $(T\times N_{co}\times p)$ matrix; and $\unicode[STIX]{x1D6EC}_{co}=[\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2},\ldots ,\unicode[STIX]{x1D706}_{N_{co}}]^{\prime }$ is a $(N_{co}\times r)$ matrix, hence, the products $X_{co}\unicode[STIX]{x1D6FD}$ and $F\unicode[STIX]{x1D6EC}_{co}^{\prime }$ are also $(T\times N_{co})$ matrices. To identify $\unicode[STIX]{x1D6FD}$ , $F$ and $\unicode[STIX]{x1D6EC}_{co}$ in Equation (1), more constraints are needed. Following Bai (Reference Bai2003, Reference Bai2009), I add two sets of constraints on the factors and factor loadings: (1) all factor are normalized, and (2) they are orthogonal to each other, i.e.: $F^{\prime }F/T=I_{r}$ and $\unicode[STIX]{x1D6EC}_{co}^{\prime }\unicode[STIX]{x1D6EC}_{co}=\text{diagonal}$ .Footnote 11 For the moment, the number of factors $r$ is assumed to be known. In the next section, we propose a cross-validation procedure that automates the choice of $r$ . The main quantity of interest of this paper is the average treatment effect on the treated (ATT) at time $t$ (when $t>T_{0}$ ): $$\begin{eqnarray}\mathit{ATT}_{t,t>T_{0}}=\frac{1}{N_{tr}}\mathop{\sum }_{i\in {\mathcal{T}}}[Y_{it}(1)-Y_{it}(0)]=\frac{1}{N_{tr}}\mathop{\sum }_{i\in {\mathcal{T}}}\unicode[STIX]{x1D6FF}_{it}.^{12}\end{eqnarray}$$ Note that in this paper, as in Abadie, Diamond, and Hainmueller (Reference Abadie, Diamond and Hainmueller2010), we treat the treatment effects $\unicode[STIX]{x1D6FF}_{it}$ as given once the sample is drawn.Footnote 13 Because $Y_{it}(1)$ is observed for treated units in posttreatment periods, the main objective of this paper is to construct counterfactuals for each treated unit in posttreatment periods, i.e., $Y_{it}(0)$ for $i\in {\mathcal{T}}$ and $t>T_{0}$ . The problem of causal inference indeed turns into a problem of forecasting missing data.Footnote 14 2.1 Assumptions for causal identification In addition to the functional form assumption (Assumption 1), three assumptions are required for the identification of the quantities of interest. Among them, the assumption of strict exogeneity is the most important. Assumption 2. Strict exogeneity. $$\begin{eqnarray}\unicode[STIX]{x1D700}_{it}\bot \!\!\!\bot D_{js},x_{js},\unicode[STIX]{x1D706}_{j},f_{s}\quad \forall i,j,t,s.\end{eqnarray}$$ Assumption 2 means that the error term of any unit at any time period is independent of treatment assignment, observed covariates, and unobserved cross-sectional and temporal heterogeneities of all units (including itself) at all periods. We call it a strict exogeneity assumption, which implies conditional mean independence, i.e., $\mathbb{E}[\unicode[STIX]{x1D700}_{it}\|D_{it},x_{it},\unicode[STIX]{x1D706}_{i},f_{t}]=\mathbb{E}[\unicode[STIX]{x1D700}_{it}\|x_{it},\unicode[STIX]{x1D706}_{i},f_{t}]=0$ .Footnote 15 Assumption 2 is arguably weaker than the strict exogeneity assumption required by fixed effects models when decomposable time-varying confounders are at present. These confounders are decomposable if they can take forms of heterogeneous impacts of a common trend or a series of common shocks. For instance, suppose a law is passed in a state because the public opinion in that state becomes more liberal. Because changing ideologies are often cross-sectionally correlated across states, a latent factor may be able to capture shifting ideology at the national level; the national shifts may have a larger impact on a state that has a tradition of mass liberalism or has a higher proportion of manufacturing workers than a state that is historically conservative. Controlling for this unobserved confounder, therefore, can alleviate the concern that the passage of the law is endogenous to changing ideology of a state's constituents to a great extent. When such a confounder exists, with two-ways fixed effects models we need to assume that $(\unicode[STIX]{x1D700}_{it}+\unicode[STIX]{x1D706}_{i}f_{t})\bot \!\!\!\bot D_{js},x_{js},\unicode[STIX]{x1D6FC}_{j},\unicode[STIX]{x1D709}_{s},\forall i,j,t,s$ (with $\unicode[STIX]{x1D706}_{i}f_{t}$ , $\unicode[STIX]{x1D6FC}_{j}$ and $\unicode[STIX]{x1D709}_{s}$ representing the time-varying confounder for unit $i$ at time $t$ , fixed effect for unit $j$ , and fixed effect for time $s$ , respectively) for the identification of the constant treatment effect. This is implausible because $\unicode[STIX]{x1D706}_{i}f_{t}$ is likely to be correlated with $D_{it}$ , $x_{it}$ , and $\unicode[STIX]{x1D6FC}_{i}$ , not to mention other terms. In contrast, Assumption 2 allows the treatment indicator to be correlated with both $x_{js}$ and $\unicode[STIX]{x1D706}_{j}^{\prime }f_{s}$ for any unit $j$ at any time periods $s$ (including $i$ and $t$ themselves). Identifying the treatment effects also requires the following assumptions: Assumption 3. Weak serial dependence of the error terms. Assumption 4. Regularity conditions. Assumptions 3 and 4 (see the Online Appendix in Supplementary Materials for details) are needed for the consistent estimation of $\unicode[STIX]{x1D6FD}$ and the space spanned by $F$ (or $F^{\prime }F/T$ ). Similar, though slightly weaker, assumptions are made in Bai (Reference Bai2009) and Moon and Weidner (Reference Moon and Weidner2015). Assumption 3 allows weak serial correlations but rules out strong serial dependence, such as unit root processes; errors of different units are uncorrelated. A sufficient condition for Assumption 3 to hold is that the error terms are not only independent of covariates, factors, and factor loadings, but also independent both across units and over time, which is assumed in Abadie, Diamond, and Hainmueller (Reference Abadie, Diamond and Hainmueller2010). Assumption 4 specifies moment conditions that ensure the convergence of the estimator. For valid inference based on a block bootstrap procedure discussed in the next section, we also need Assumption 5 (see Online Appendix for details). Heteroscedasticity across time, however, is allowed. Assumption 5. The error terms are cross-sectionally independent and homoscedastic. Remark 1. Assumptions 3 and 5 suggest that the error terms $\unicode[STIX]{x1D700}_{it}$ can be serially correlated. Assumption 2 rules out dynamic models with lagged dependent variables; however, this is mainly for the purpose of simplifying proofs (Bai Reference Bai2009, p. 1243). The proposed method can accommodate dynamic models as long as the error terms are not serially correlated. 3 Estimation Strategy In this section, we first propose a GSC estimator for treatment effect of each treated unit. It is essentially an out-of-sample prediction method based on Bai (Reference Bai2009)'s factor augmented model. The GSC estimator for the treatment effect on treated unit $i$ at time $t$ is given by the difference between the actual outcome and its estimated counterfactual: $\hat{\unicode[STIX]{x1D6FF}}_{it}=Y_{it}(1)-{\hat{Y}}_{it}(0)$ , in which ${\hat{Y}}_{it}(0)$ is imputed with three steps. In the first step, we estimate an IFE model using only the control group data and obtain $\hat{\unicode[STIX]{x1D6FD}},\hat{F},\hat{\unicode[STIX]{x1D6EC}}_{co}$ : $$\begin{eqnarray}\displaystyle & \displaystyle \mathbf{Step1.}\qquad (\hat{\unicode[STIX]{x1D6FD}},\hat{F},\hat{\unicode[STIX]{x1D6EC}}_{co})=\underset{\tilde{\unicode[STIX]{x1D6FD}},\tilde{F},\tilde{\unicode[STIX]{x1D6EC}}_{co}}{\operatorname{argmin}}\mathop{\sum }_{i\in {\mathcal{C}}}(Y_{i}-X_{i}\tilde{\unicode[STIX]{x1D6FD}}-\tilde{F}\tilde{\unicode[STIX]{x1D706}}_{i})^{\prime }(Y_{i}-X_{i}\tilde{\unicode[STIX]{x1D6FD}}-\tilde{F}\tilde{\unicode[STIX]{x1D706}}_{i}) & \displaystyle \nonumber\\ \displaystyle & \mathit{s.t.}\qquad \tilde{F}^{\prime }\tilde{F}/T=I_{r}\quad \text{and}\quad \tilde{\unicode[STIX]{x1D6EC}}_{co}^{\prime }\tilde{\unicode[STIX]{x1D6EC}}_{co}=\text{diagonal}. & \displaystyle \nonumber\end{eqnarray}$$ We explain in detail how to estimate this model in the Online Appendix. The second step estimates factor loadings for each treated unit by minimizing the mean squared error of the predicted treated outcome in pretreatment periods: $$\begin{eqnarray}\displaystyle \mathbf{Step2.}\qquad \hat{\unicode[STIX]{x1D706}}_{i} & = & \displaystyle \underset{\tilde{\unicode[STIX]{x1D706}}_{i}}{\operatorname{argmin}}(Y_{i}^{0}-X_{i}^{0}\hat{\unicode[STIX]{x1D6FD}}-\hat{F^{0}}\tilde{\unicode[STIX]{x1D706}}_{i})^{\prime }(Y_{i}^{0}-X_{i}^{0}\hat{\unicode[STIX]{x1D6FD}}-\hat{F^{0}}\tilde{\unicode[STIX]{x1D706}}_{i})\nonumber\\ \displaystyle & = & \displaystyle (\hat{F}^{0\prime }\hat{F}^{0})^{-1}\hat{F}^{0\prime }(Y_{i}^{0}-X_{i}^{0}\hat{\unicode[STIX]{x1D6FD}}),\quad i\in {\mathcal{T}},\nonumber\end{eqnarray}$$ in which $\hat{\unicode[STIX]{x1D6FD}}$ and $\hat{F}^{0}$ are from the first-step estimation and the superscripts "0"s denote the pretreatment periods. In the third step, we calculate treated counterfactuals based on $\hat{\unicode[STIX]{x1D6FD}}$ , $\hat{F}$ , and $\hat{\unicode[STIX]{x1D706}}_{i}$ : $$\begin{eqnarray}\mathbf{Step3.}\qquad {\hat{Y}}_{it}(0)=x_{it}^{\prime }\hat{\unicode[STIX]{x1D6FD}}+\hat{\unicode[STIX]{x1D706}}_{i}^{\prime }\hat{f}_{t}\quad i\in {\mathcal{T}},~t>T_{0}.\end{eqnarray}$$ An estimator for $ATT_{t}$ therefore is: $\widehat{ATT}_{t}=(1/N_{tr})\sum _{i\in {\mathcal{T}}}[Y_{it}(1)-{\hat{Y}}_{it}(0)]$ for $t>T_{0}$ . Remark 2. In the Online Appendix, we show that, under Assumptions 1–4, the bias of the GSC shrinks to zero as the sample size grows, i.e., $\mathbb{E}_{\unicode[STIX]{x1D700}}(\widehat{ATT}_{t}\|D,X,\unicode[STIX]{x1D6EC},F)\rightarrow ATT_{t}$ as $N_{co},T_{0}\rightarrow 0$ ( $N_{tr}$ is taken as given), in which $D=[D_{1},D_{2},\ldots ,D_{N}]$ is a $(T\times N)$ matrix, $X$ is a three-dimensional $(T\times N\times p)$ matrix; and $\unicode[STIX]{x1D6EC}=[\unicode[STIX]{x1D706}_{1},\unicode[STIX]{x1D706}_{2},\ldots ,\unicode[STIX]{x1D706}_{N}]^{\prime }$ is a $(N\times r)$ matrix. Intuitively, both large $N_{co}$ and large $T_{0}$ are necessary for the convergences of $\hat{\unicode[STIX]{x1D6FD}}$ and the estimated factor space. When $T_{0}$ is small, imprecise estimation of the factor loadings, or the "incidental parameters" problem, will lead to bias in the estimated treatment effects. This is a crucial difference from the conventional linear fixed effects models. 3.1 Model selection In practice, researchers may have limited knowledge of the exact number of factors to be included in the model. Therefore, we develop a cross-validation procedure to select models before estimating the causal effect. It relies on the control group information as well as information from the treatment group in pretreatment periods. Algorithm 1 describes the details of this procedure. Algorithm 1 (Cross-validating the number of factors). A leave-one-out-cross-validation procedure that selects the number of factors takes the following steps: The basic idea of the above procedure is to hold back a small amount of data (e.g., one pretreatment period of the treatment group) and use the rest of data to predict the held-back information. The algorithm then chooses the model that on average makes the most accurate predictions. A TSCS dataset with a DID data structure allows us to do so because (1) there exists a set of control units that are never exposed to the treatment and therefore can serve as the basis for estimating time-varying factors and (2) the pretreatment periods of treated units constitute a natural validation set for candidate models. This procedure is computationally inexpensive because with each $r$ , the IFE model is estimated only once (Step 1). Other steps involves merely simple calculations. In the Online Appendix, we conduct Monte Carlo exercises and show that the above procedure performs well in term of choosing the correct number of factors even with relatively small datasets. Remark 3. Our framework can also accommodate DGPs that directly incorporate additive fixed effects, known time trends, and exogenous time-invariant covariates, such as: (2) $$\begin{eqnarray}Y_{it}=\unicode[STIX]{x1D6FF}_{it}D_{it}+x_{it}^{\prime }\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D6FE}_{i}^{\prime }l_{t}+z_{i}^{\prime }\unicode[STIX]{x1D703}_{t}+\unicode[STIX]{x1D706}_{i}^{\prime }f_{t}+\unicode[STIX]{x1D6FC}_{i}+\unicode[STIX]{x1D709}_{t}+\unicode[STIX]{x1D700}_{it},\end{eqnarray}$$ in which $l_{t}$ is a $(q\times 1)$ vector of known time trends that may affect each unit differently; $\unicode[STIX]{x1D6FE}_{i}$ is $(q\times 1)$ unit-specific unknown parameters; $z_{i}$ is a $(m\times 1)$ vector of observed time-invariant covariates; $\unicode[STIX]{x1D703}_{t}$ is a $(m\times 1)$ vector of unknown parameters; $\unicode[STIX]{x1D6FC}_{i}$ and $\unicode[STIX]{x1D709}_{t}$ are additive individual and time fixed effects, respectively. We describe the estimation procedure of this extended model in the Online Appendix. 3.2 Inference We rely on a parametric bootstrap procedure to obtain the uncertainty estimates of the GSC estimator (deriving the analytical asymptotic distribution of the GSC estimator is a necessary step for future research). When the sample size is large, when $N_{tr}$ is large in particular, a simple nonparametric bootstrap procedure can provide valid uncertainty estimates. When the sample size is small, especially when $N_{tr}$ is small, we are unable to approximate the DGP of the treatment group by resampling the data nonparametrically. In this case, we simply lack the information of the joint distribution of $(X_{i},\unicode[STIX]{x1D706}_{i},\unicode[STIX]{x1D6FF}_{i})$ for the treatment group. However, we can obtain uncertainty estimates conditional on observed covariates and unobserved factors and factor loadings using a parametric bootstrap procedure via resampling the residuals. By resampling entire time series of residuals, we preserve the serial correlation within the units, thus avoiding underestimating the standard errors due to serial correlations (Beck and Katz Reference Beck and Katz1995). Our goal is to estimate the conditional variance of ATT estimator, i.e., $\text{Var}_{\unicode[STIX]{x1D700}}(\widehat{ATT}_{t}\|D,X,\unicode[STIX]{x1D6EC},F)$ . Notice that the only random variable that is not being conditioned on is $\unicode[STIX]{x1D700}_{i}$ , which are assumed to be independent of treatment assignment, observed covariates, factors and factor loadings (Assumption 2). We can interpret $\unicode[STIX]{x1D700}_{i}$ as measurement errors or variations in the outcome that we cannot explain but are unrelated to treatment assignment.Footnote 16 In the parametric bootstrap procedure, we simulate treated counterfactuals and control units based on the following resampling scheme: $$\begin{eqnarray}\displaystyle & \displaystyle {\tilde{Y}}_{i}(0)=X_{i}\hat{\unicode[STIX]{x1D6FD}}+\hat{F}\hat{\unicode[STIX]{x1D706}}_{i}+\tilde{\unicode[STIX]{x1D700}}_{i},\quad \forall i\in {\mathcal{C}}; & \displaystyle \nonumber\\ \displaystyle & \displaystyle {\tilde{Y}}_{i}(0)=X_{i}\hat{\unicode[STIX]{x1D6FD}}+\hat{F}\hat{\unicode[STIX]{x1D706}}_{i}+\tilde{\unicode[STIX]{x1D700}}_{i}^{p},\quad \forall i\in {\mathcal{T}}, & \displaystyle \nonumber\end{eqnarray}$$ in which ${\tilde{Y}}_{i}(0)$ is a vector of simulated outcomes in the absence of the treatment; $X_{i}\hat{\unicode[STIX]{x1D6FD}}+\hat{F}\hat{\unicode[STIX]{x1D706}}_{i}$ is the estimated conditional mean; and $\tilde{\unicode[STIX]{x1D700}}_{i}$ and $\tilde{\unicode[STIX]{x1D700}}_{i}^{p}$ are resampled residuals for unit $i$ , depending on whether it belongs to the treatment or control group. Because $\hat{\unicode[STIX]{x1D6FD}}$ are $\hat{F}$ are estimated using only the control group information, $X_{i}\hat{\unicode[STIX]{x1D6FD}}+\hat{F}\hat{\unicode[STIX]{x1D706}}_{i}$ fits $X_{i}\unicode[STIX]{x1D6FD}+F\unicode[STIX]{x1D706}_{i}$ better for a control unit than for a treated unit (as a result, the variance of $\tilde{\unicode[STIX]{x1D700}}_{i}^{p}$ is usually bigger than that of $\tilde{\unicode[STIX]{x1D700}}_{i}$ ). Hence, $\tilde{\unicode[STIX]{x1D700}}_{i}$ and $\tilde{\unicode[STIX]{x1D700}}_{i}^{p}$ are drawn from different empirical distributions: $\tilde{\unicode[STIX]{x1D700}}_{i}$ is the in-sample error of the IFE model fitted to the control group data, and therefore is drawn from the empirical distribution of the residuals of the IFE model, while $\tilde{\unicode[STIX]{x1D700}}_{i}^{p}$ can be seen as the prediction error of the IFE model for treated counterfactuals.Footnote 17 Although we cannot observe treated counterfactuals, $Y_{it}(0)$ is observed for all control units. With the assumptions that treated and control units follow the same factor model (Assumption 1) and the error terms are independent and homoscedastic across space (Assumption 5), we can use a cross-validation method to simulate $\unicode[STIX]{x1D700}_{i}^{p}$ based on the control group data (Efron Reference Efron2012). Specifically, each time we leave one control unit out (to be taken as a "fake" treat unit) and use the rest of the control units to predict the outcome of left-out unit. The difference between the predicted and observed outcomes is a prediction error of the IFE model. $\unicode[STIX]{x1D700}_{i}^{p}$ is drawn from the empirical distributions of the prediction errors. Under Assumptions 1–5, this procedure provides valid uncertainty estimates for the proposed method without making particular distributional assumptions of the error terms. Algorithm 2 describes the entire procedure in detail. Algorithm 2 (Inference). A parametric bootstrap procedure that gives the uncertainty estimates of the ATT is described as follows: 4 Monte Carlo Evidence In this section, we conduct Monte Carlo exercises to explore the finite sample properties of the GSC estimator and compare it with several existing methods, including the DID estimator, the IFE estimator, and the original synthetic matching method. We also investigate the extent to which the proposed cross-validation scheme can choose the number of factors correctly in relatively small samples. We start with the following data generating process (DGP) that includes two observed time-varying covariates, two unobserved factors, and additive two-way fixed effects: (3) $$\begin{eqnarray}Y_{it}=\unicode[STIX]{x1D6FF}_{it}D_{it}+x_{it,1}\cdot 1+x_{it,2}\cdot 3+\unicode[STIX]{x1D706}_{i}^{\prime }f_{t}+\unicode[STIX]{x1D6FC}_{i}+\unicode[STIX]{x1D709}_{t}+5+\unicode[STIX]{x1D700}_{it}\end{eqnarray}$$ where $f_{t}=(f_{1t},f_{2t})^{\prime }$ and $\unicode[STIX]{x1D706}_{i}=(\unicode[STIX]{x1D706}_{i1},\unicode[STIX]{x1D706}_{i2})^{\prime }$ are time-varying factors and unit-specific factor loadings. The covariates are (positively) correlated with both the factors and factor loadings: $x_{it,k}=1+\unicode[STIX]{x1D706}_{i}^{\prime }f_{t}+\unicode[STIX]{x1D706}_{i1}+\unicode[STIX]{x1D706}_{i2}+f_{1t}+f_{2t}+\unicode[STIX]{x1D702}_{it,k},k=1,2$ . The error term $\unicode[STIX]{x1D700}_{it}$ and disturbances in covariates $\unicode[STIX]{x1D702}_{it,1}$ and $\unicode[STIX]{x1D702}_{it,2}$ are i.i.d. $N(0,1)$ . Factors $f_{1t}$ and $f_{2t}$ , as well as time fixed effects $\unicode[STIX]{x1D709}_{t}$ , are also i.i.d. $N(0,1)$ . The treatment and control groups consist of $N_{tr}$ and $N_{co}$ units. The treatment starts to affect the treated units at time $T_{0}+1$ and since then 10 periods are observed ( $q=10$ ). The treatment indicator is defined as in Section 2, i.e., $D_{it}=1$ when $i\in {\mathcal{T}}$ and $t>T_{0}$ and $D_{it}=0$ otherwise. The heterogeneous treatment effect is generated by $\unicode[STIX]{x1D6FF}_{it,t>T_{0}}=\bar{\unicode[STIX]{x1D6FF}}_{t}+e_{it}$ , in which $e_{it}$ is i.i.d. N(0,1). $\bar{\unicode[STIX]{x1D6FF}}_{t}$ is given by: $[\bar{\unicode[STIX]{x1D6FF}}_{T_{0}+1},\bar{\unicode[STIX]{x1D6FF}}_{T_{0}+1},\ldots ,\bar{\unicode[STIX]{x1D6FF}}_{T_{0}+10}]=[1,2,\ldots ,10]$ . Factor loadings $\unicode[STIX]{x1D706}_{i1}$ and $\unicode[STIX]{x1D706}_{i2}$ , as well as unit fixed effects $\unicode[STIX]{x1D6FC}_{i}$ , are drawn from uniform distributions $U[-\sqrt{3},\sqrt{3}]$ for control units and $U[\sqrt{3}-2w\sqrt{3},3\sqrt{3}-2w\sqrt{3}]$ for treated units ( $w\in [0,1]$ ). This means that when $0\leqslant w<1$ , (1) the random variables have variance 1; (2) the supports of factor loadings of treated and control units are not perfectly overlapped; and (3) the treatment indicator and factor loadings are positively correlated.Footnote 18 4.1 A simulated example We first illustrate the proposed method, as well as the DGP described above, with a simulated sample of $N_{tr}=5$ , $N_{co}=45$ , and $T_{0}=20$ (hence, $N=50$ , $T=30$ ). $w$ is set to be $0.8$ , which means that the treated units are more likely to have larger factor loadings than the control units. Figure 1 visualizes the raw data and estimation results. In the upper panel, the dark and light gray lines are time series of the treated and control units, respectively. The bold solid line is the average outcome of the five treated units while the bold dashed line is the average predicted outcome of the five units in the absence of the treatment. The latter is imputed using the proposed method. Figure 1. Estimated ATT for a simulated sample $N_{tr}=5$ , $N_{co}=45$ , $T=30$ , $T_{0}=10$ . The lower panel of Figure 1 shows the estimated ATT (solid line) and the true ATT (dashed line). The 95 percent confidence intervals for the ATT are based on bootstraps of 2,000 times. It shows that the estimated average treated outcome fits the data well in pretreatment periods and the estimated ATT is very close to the actual ATT. The estimated factors and factor loadings, as well as imputed counterfactual and individual treatment effect for each treat unit, are shown in the Online Appendix. 4.2 Finite sample properties We present the Monte Carlo evidence on the finite sample properties of the GSC estimator in Table 1 (additional results are shown in the Online Appendix). As in the previous example, the treatment group is set to have five units. The estimand is the ATT at time $T_{0}+5$ , whose expected value equals 5. Observables, factors, and factor loadings are drawn only once while the error term is drawn repeatedly; $w$ is set to be 0.8 such that treatment assignment is positively correlated with factor loadings. Table 1 reports the bias, standard deviation (SD), and root mean squared error (RMSE) of $\widehat{ATT}_{T_{0}+5}$ from 5,000 simulations for each pair of $T_{0}$ and $N_{co}$ .Footnote 19 It shows that the GSC estimator has limited bias even when $T_{0}$ and $N_{co}$ are relatively small and the bias goes away as $T_{0}$ and $N_{co}$ grow. As expected, both the SD and RMSE shrink when $T_{0}$ and $N_{co}$ become larger. Table 1 also reports the coverage probabilities of 95 percent confidence intervals for $\widehat{ATT}_{i,T_{0}+5}$ constructed by the parametric bootstrap procedure (Algorithm 2). For each pair of $T_{0}$ and $N_{co}$ , the coverage probability is calculated based on 5,000 simulated samples, each of which is bootstrapped for 1,000 times. These numbers show that the proposed procedure can achieve the correct coverage rate even when the sample size is relatively small (e.g., $T_{0}=15$ , $N_{tr}=5$ , $N_{co}=80$ ). Table 1. Finite sample properties and coverage rates In the Online Appendix, we run additional simulations and compare the proposed method with several existing methods, including the DID estimator, the IFE estimator, and the synthetic matching method. We find that (1) the GSC estimator has less bias than the DID estimator in the presence of unobserved, decomposable time-varying confounders; (2) it has less bias than the IFE estimator when the treatment effect is heterogeneous; and (3) it is usually more efficient than the original synthetic matching estimator. It is worth emphasizing that these results are under the premise of correct model specifications. To address the concern that the GSC method relies on correct model specifications, we conduct additional tests and show that the cross-validation scheme described in Algorithm 1 is able to choose the number of factors correctly most of the time when the sample is large enough. 5 Empirical Example In this section, we illustrate the GSC method with an empirical example that investigates the effect of EDR laws on voter turnout in the United States. Voting in the United States usually takes two steps. Except in North Dakota, where no registration is needed, eligible voters throughout the country must register prior to casting their ballots. Registration, which often requires a separate trip from voting, is widely regarded as a substantial cost of voting and a culprit of low turnout rates before the 1993 National Voter Registration Act (NVRA) was enacted (e.g., Highton Reference Highton2004). Against this backdrop, EDR is a reform that allows eligible voters to register on Election Day when they arrive at polling stations. In the mid-1970s, Maine, Minnesota, and Wisconsin were the first adopters of this reform in the hopes of increasing voter turnout; while Idaho, New Hampshire, and Wyoming established EDR in the 1990s as a strategy to opt out the NVRA (Hanmer Reference Hanmer2009). Before the 2012 presidential election, three other states, Montana, Iowa, and Connecticut, passed laws to enact EDR, adding the number of states having EDR laws to nine.Footnote 20 Most existing studies based on individual-level cross-sectional data, such as the Current Population Surveys and the National Election Surveys, suggest that EDR laws increase turnout (the estimated effect varies from 5 to 14 percentage points).Footnote 21 These studies do not provide compelling evidence of a causal effect of EDR laws because the research designs they use are insufficient to address the problem that states self-select their systems of registration laws. "Registration requirements did not descend from the skies," as Dean Burnham puts it (Reference Burnham and Rose1980, p. 69). A few studies employ time-series or TSCS analysis to address the identification problem.Footnote 22 However, Keele and Minozzi (Reference Keele and Minozzi2013) cast doubts on these studies and suggest that the "parallel trends" assumption may not hold, as we will also demonstrate below. In the following analysis, we use state-level voter turnout data for presidential elections from 1920 to 2012.Footnote 23 The turnout rates are calculated with total ballots counted in a presidential election in a state as the numerator and the state's voting-age population (VAP) as the denominator.Footnote 24 Alaska and Hawaii are not included in the sample since they were not states until 1959. North Dakota is also dropped since no registration is required. As mentioned above, up to the 2012 presidential election, nine states had adopted EDR laws (hereafter referred to as treated) and the rest thirty-eight states had not (referred to as controls). The raw turnout data for all forty-seven states are shown in the Online Appendix.Footnote 25 First, we use a standard two-way fixed effects mode, which is often referred to as a DID model in the literature. The results are shown in Table 2 columns (1) and (2). Standard errors are produced by nonparametric bootstraps (blocked at the state level) of 2,000 times. In column (1), only the EDR indicator is included, while in column (2), we additionally control for indicators of universal mail-in registration and motor voter registration. The estimated coefficients of EDR laws are 0.87 and 0.78 percent using the two specifications, respectively, with standard errors around 3 percent. Table 2. The effect of EDR on voter turnout Note: Standard errors in columns (1) and (2) are based on nonparametric bootstraps (blocked at the state level) of 2,000 times. Standard errors in columns (3) and (4) are based on parametric bootstraps (blocked at the state level) of 2,000 times. The two-way fixed effects model presented in Table 2 assumes a constant treatment effect both across states and over time. Next we relax this assumption by literally employing a DID approach. In other words, we estimate the effect of EDR laws on voter turnout in the posttreatment period by subtracting the time intercepts estimated from the control group and the unit intercepts based on the pretreatment data. The predict turnout for state $i$ in year $t$ , therefore, is the summation of unit intercept $i$ and time intercept $t$ , plus the impact of the time-varying covariates. The result is visualized in the upper panel of Figure 2. Figure 2a shows the average actual turnout (solid line) and average predicted turnout in the absence of EDR laws (dashed line); both averages are taken based on the number of terms since (or before) EDR laws first took effect. Figure 2b shows the gap between the two lines, or the estimated ATT. The confidence intervals are produced by block bootstraps of 2,000 times. It is clear from both figures that the "parallel trends" assumption is not likely to hold since the average predicted turnout deviates from the average actual turnout in the pretreatment periods. Figure 2. The effect of EDR on turnout: Main results. Next, we apply the GSC method to the same dataset. Table 2 columns (3) and (4) summarize the result.Footnote 26 Again, both specifications impose additive state and year fixed effects. In column (3), no covariates are included, while in column (4), mail-in and motor voter registration are controlled for (assuming that they have constant effects on turnout). With both specifications, the cross-validation scheme finds two unobserved factors to be important and after conditioning on both the factors and additive fixed effects, the estimated ATT based on the GSC method is around 5 percent with a standard error of 2.3 percent.Footnote 27 This means that EDR laws are associated with a statistically significant increase in voter turnout, consistent with previous OLS results based on individual-level data. The lower panel of Figure 2 shows the dynamics of the estimated ATT. Again, in the left figure, averages are taken after the actual and predicted turnout rates are realigned to the timing of the reform. With the GSC method, the average actual turnout and average predicted turnout match well in pretreatment periods and diverge after EDR laws took effect. The right figure shows that the gaps between the two lines are virtually flat in pretreatment periods and the effect takes off right after the adoption of EDR.Footnote 28 Figure 3. The Effect of EDR on turnout: Factors and loadings. Figure 3 presents the estimated factors and factor loadings produced by the GSC method.Footnote 29 Figure 3a depicts the two estimated factors. The $x$ -axis is year and the $y$ -axis is the magnitude of factors (rescaled by the square root of their corresponding eigenvalues to demonstrate their relative importance). Figure 3b shows the estimated factors loadings for each treated (black, bold) and control (gray) units, with $x$ - and $y$ -axes indicating the magnitude of the loadings for the first and second factors, respectively. Bearing in mind the caveat that estimated factors may not be directly interpretable because they are, at best, linear transformations of the true factors, we find that the estimated factors shown in this figure are meaningful. The first factor captures the sharp increase in turnout in the southern states because of the 1965 Voting Rights Act that removed Jim Crow laws, such as poll taxes or literacy tests, that suppressed turnout. As shown in the right figure, the top eleven states that have the largest loadings on the first factor are exactly the eleven southern states (which were previously in the confederacy).Footnote 30 The labels of these states are underlined in Figure 3b. The second factor, which is set to be orthogonal to the first one, is less interpretable. However, its nonnegligible magnitude indicates a strong downward trend in voter turnout in many states in recent years. Another reassuring finding shown by Figure 3b is that the estimated factor loadings of the nine treated units mostly lie in the convex hull of those of the control units, which indicates that the treated counterfactuals are produced mostly by more reliable interpolations instead of extrapolations. Finally, we investigate the heterogeneous treatment effects of EDR laws. Previous studies have suggested that the motivations behind enacting these laws are vastly different between the early adopters and later ones. For example, Maine, Minnesota, and Wisconsin, which established the EDR in mid-1970s, did so because officials in these states sincerely wanted the turnout rates to be higher, while the "reluctant adopters," including Idaho, New Hampshire, and Wyoming, introduced the EDR as a means to avoid the NVRA because officials viewed the NVRA as "a more costly and potentially chaotic system" (Hanmer Reference Hanmer2009). Because of the different motivations and other reasons, we may expect the treatment effect of EDR laws to be different in states that adopted them in different times. Table 3. The effect of EDR on voter turnout: Three waves Note: Standard errors are based on parametric bootstraps (blocked at the state level) of 1,000 times. The estimation of heterogeneous treatment effects is embedded in the GSC method since it gives individual treatment effects for all treated units in a single run. Table 3 summarizes the ATTs of EDR on voter turnout among the three waves of EDR adopters. Again, additive state and year fixed effects, as well as indicators of two other registration systems, are controlled for. Table 3 shows that EDR laws have a large and positive effect on the early adopters (the estimate is about 7 percent with a standard error of 3 percent) while EDR laws were found to have no statistically significant impact on the other six states.Footnote 31 Such differential outcomes can be due to two reasons. First, the NVRA of 1993 substantially reduced the cost of registration: since almost everyone who has some intention to vote is a registrant after the NVRA was enacted, "there is now little room for enhancing turnout further by making registration easier" (Highton Reference Highton2004). Second, because states having a strong "participatory culture" is more likely to be selected into an EDR system in earlier years, costly registration, as a binding constraint in these states, may not be a first-order issue in a state where many eligible voters have low incentives to vote in the first place. It is also possible that voters in early adopting states formed a habit to vote in the days when the demand for participation was high (Hanmer Reference Hanmer2009). In short, using the GSC method, we find that EDR laws increased turnout in early adopting states, including Maine, Minnesota, and Wisconsin, but not in states that introduced EDR as a strategy to opt out the NVRA or enacted EDR laws in recent years. These results are broadly consistent with evidence provided by a large literature based on individual-level cross-sectional data (see, for example, Leighley and Nagler Reference Leighley and Nagler2013 for a summary). They are also more credible than results from conventional fixed effects models when the "parallel trends" assumption appears to fail.Footnote 32 In this paper, we propose the GSC method for causal inference with TSCS data. It attempts to address the challenge that the "parallel trends" assumption often fails when researchers apply fixed effects models to estimate the causal effect of a certain treatment. The GSC method estimates the individual treatment effect on each treated unit semiparametrically. Specifically, it imputes treated counterfactuals based on a linear interactive fixed effects model that incorporates time-varying coefficients (factors) interacted with unit-specific intercepts (factor loadings). A built-in cross-validation scheme automatically selects the model, reducing the risks of overfitting. This method is in spirit of the original synthetic control method in that it uses data from pretreatment periods as benchmarks to customize a reweighting scheme of control units in order to make the best possible predictions for treated counterfactuals. It generalizes the synthetic control method in two aspects. First, it allows multiple treated units and differential treatment timing. Second, it offers uncertainty estimates, such as standard errors and confidence intervals, that are easy to interpret. Monte Carlo exercises suggest that the proposed method performs well even with relatively small $T_{0}$ and $N_{co}$ and show that it has advantages over several existing methods: (1) it has less bias than the two-way fixed effects or DID estimators in the presence of decomposable time-varying confounders, (2) it corrects bias of the IFE estimator when the treatment effect is heterogeneous across units; and (3) it is more efficient than the synthetic control method. To illustrate the applicability of this method in political science, we estimate the effect of EDR laws on voter turnout in the United States. We show that EDR laws increased turnout in early adopting states but not in states that introduced them more recently. Two caveats are worth emphasizing. First, insufficient data (with either a small $T_{0}$ or a small $N_{co}$ ) cause bias in the estimated treatment effect. In general, users should be cautious when $T_{0}<10$ or $N_{co}<40$ . Second, excessive extrapolations based on imprecisely estimated factors and factor loading can lead to erroneous results. To avoid this problem, we recommend the following diagnostics upon using this method: (1) plot raw data of treated and control outcomes as well as imputed counterfactuals and check whether the imputed values are within reasonable intervals; (2) plot estimated factor loadings of both treated and control units and check the overlap (as in Fig. 3). We provide software routines gsynth in R to implement the estimation procedure as well as these diagnostic tests. When excessive extrapolations appear to happen, we recommend users to include a smaller number of factors or switch back to the conventional DID framework. We also recommend users to benchmark the results with estimates from the IFE model (Bai Reference Bai2009) as well as Bayesian multi-level factor models (e.g., Pang Reference Pang2014) whenever it is possible. Another limitation of the proposed method is that it cannot accommodate complex DGPs that often appear in TSCS data (when $T$ is much bigger than panel data), such as (1) dynamic relationships between the treatment, covariates, and outcome (e.g., Pang Reference Pang2010, Reference Pang2014, Blackwell and Glynn Reference Blackwell and Glynn2015), (2) structural breaks (e.g., Park Reference Park2010, Reference Park2012), and (3) multiple times of treatment and variable treatment intensity. Nor does it allow random coefficients for the observed time-varying covariates, as such modeling setups become increasing popular with Bayesian multi-level analysis. Future research is needed to accommodate these scenarios. For supplementary material accompanying this paper, please visit https://doi.org/10.1017/pan.2016.2. Author's note: The author is indebted to Matt Blackwell, Devin Caughey, Justin Grimmer, Jens Hainmueller, Danny Hidalgo, Simon Jackman, Jonathan Katz, Luke Keele, Eric Min, Molly Roberts, Jim Snyder, Brandon Stewart, Teppei Yamamoto, as well as seminar participants at the 2015 MPSA Annual Meeting and 2015 APSA Annual Meeting for helpful comments and suggestions. The author thanks the editor, Mike Alvarez, and two anonymous reviewers for their extremely helpful suggestions. He also thanks Jushan Bai for generously sharing the Matlab codes used in Bai (2009) and Melanie Springer for kindly providing the state-level voter turnout data (1920–2000). The source code and data used in the paper can be downloaded from the Political Analysis Dataverse at dx.doi.org/10.7910/DVN/8AKACJ (Xu 2016) as well as the author's website. Contributing Editor: R. Michael Alvarez 1 See Hsiao, Ching, and Wan (Reference Hsiao, Ching and Wan2012) and Angrist, Jord, and Kuersteiner (Reference Angrist, Jord and Kuersteiner2013) for alternative matching methods along this line of thought. 2 To gauge the uncertainty of the estimated treatment effect, the synthetic control method compares the estimated treatment effect with the "effects" estimated from placebo tests in which the treatment is randomly assigned to a control unit. 3 See Campbell, Lo, and MacKinlay (Reference Campbell, Lo and Craig MacKinlay1997) for applications of factor models in finance. 4 For more empirical applications of the IFE estimator, see Kim and Oka (Reference Kim and Oka2014) and Gaibulloev, Sandler, and Sul (Reference Gaibulloev, Sandler and Sul2014). 5 When the treatment effect is heterogeneous (as it is almost always the case), an IFE model that imposes a constant treatment effect assumption gives biased estimates of the average treatment effect because the estimation of the factor space is affected by the heterogeneity in the treatment effect. 6 For example, Acemoglu et al. (Reference Acemoglu, Johnson, Kermani, Kwak and Mitton2016), who estimate the effect of Tim Geithner connections on stock market returns, conduct the synthetic control method repeatedly for each connected (treated) firm; Dube and Zipperer (Reference Dube and Zipperer2015) estimate the effect of minimum wage policies on wage and employment by conducting the method for each of the 29 policy changes. The latter also extend Abadie, Diamond, and Hainmueller (Reference Abadie, Diamond and Hainmueller2010)'s original inferential method to the case of multiple treated units using the mean percentile ranks of the estimated effects. 7 Cases in which the treatment switches on and off (or "multiple-treatment-time") can be easily incorporated in this framework as long as we impose assumptions on how the treatment affects current and future outcomes. For example, one can assume that the treatment only affect the current outcome but not future outcomes (no carryover effect), as fixed effects models often do. In this paper, we do not impose such assumptions. See Imai and Kim (Reference Imai and Kim2016) for a thorough discussion. 8 $\unicode[STIX]{x1D6FD}$ is assumed to be constant across space and time mainly for the purpose of fast computation in the frequentist framework. It is a limitation compared with more flexible and increasingly popular random coefficient models in Bayesian multi-level analysis. 9 For this reason, additive unit and time fixed effects are not explicitly assumed in the model. An extended model that directly imposes additive two-way fixed effects is discussed in the next section. 10 In the former case, we can set $f_{1t}=t$ and $f_{2t}=t^{2}$ ; in the latter case, for example, we can rewrite $Y_{it}=\unicode[STIX]{x1D70C}Y_{i,t-1}+x_{it}^{\prime }\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D700}_{it}$ as $Y_{it}=Y_{i0}\cdot \unicode[STIX]{x1D70C}^{t}+x_{it}^{\prime }\unicode[STIX]{x1D6FD}+\unicode[STIX]{x1D708}_{it}$ , in which $\unicode[STIX]{x1D708}_{it}$ is an AR(1) process and $\unicode[STIX]{x1D70C}^{t}$ and $Y_{i0}$ are the unknown factor and factor loadings, respectively. See Gobillon and Magnac (Reference Gobillon and Magnac2016) for more examples. 11 These constraints do not lead to loss of generality because for an arbitrary pair of matrices $F$ and $\unicode[STIX]{x1D6EC}_{co}$ , we can find an $(r\times r)$ invertible matrix $A$ such that $(FA)^{\prime }(FA)/T=I_{r}$ and $(A^{-1}\unicode[STIX]{x1D6EC}_{co})^{\prime }A^{-1}\unicode[STIX]{x1D6EC}_{co}$ is a diagonal matrix. To see this, we can then rewrite $\unicode[STIX]{x1D706}_{i}^{\prime }F$ as $\tilde{\unicode[STIX]{x1D706}}_{i}^{\prime }\tilde{F}$ , in which $\tilde{F}=FA$ and $\tilde{\unicode[STIX]{x1D706}}_{i}=A^{-1}\unicode[STIX]{x1D706}_{i}$ for units in both the treatment and control groups such that $\tilde{F}$ and $\tilde{\unicode[STIX]{x1D6EC}}_{co}$ satisfy the above constraints. The total number of constraints is $r^{2}$ , the dimension of the matrix space where $A$ belongs. It is worth noting that although the original factors $F$ may not be identifiable, the space spanned by $F$ , a $r$ -dimensional subspace of in the $T$ -dimensional space, is identified under the above constraints because for any vector in the subspace spanned by $\tilde{F}$ , it is also in the subspace spanned by the original factors $F$ . 12 For a clear and detailed explanation of quantities of interest in TSCS analysis, see Blackwell and Glynn (Reference Blackwell and Glynn2015). Using their terminology, this paper intends to estimate the Average Treatment History Effect on the Treated given two specific treatment histories: $\mathbb{E}[Y_{it}(\text{}\underline{a}_{t}^{1})-Y_{it}(\text{}\underline{a}_{t}^{0})\|\text{}\underline{D}_{i,t-1}=\text{}\underline{a}_{t-1}^{1}]$ in which $\text{}\underline{a}_{t}^{0}=(0,\ldots ,0)$ , $\text{}\underline{a}_{t}^{1}=(0,\ldots ,0,1,\ldots ,1)$ with $T_{0}$ zeros and $(t-T_{0})$ ones indicate the histories of treatment statuses. We keep the current notation for simplicity. 13 We attempt to make inference about the ATT in the sample we draw, not the ATT of the population. In other words, we do not incorporate uncertainty of the treatment effects $\unicode[STIX]{x1D6FF}_{it}$ . 14 The idea of predicting treated counterfactuals in a DID setup is also explored by Brodersen et al. (Reference Brodersen, Gallusser, Koehler, Remy and Scott2014) using a structural Bayesian time-series approach. 15 Note that because $\unicode[STIX]{x1D700}_{it}$ is independent of $D_{is}$ and $x_{is}$ for all $(t,s)$ , Assumption 2 rules out the possibility that past outcomes may affect future treatments, which is allowed by the so called "sequential exogeneity" assumption. A directed acyclic graph (DAG) representation is provided in the Online Appendix. See Blackwell and Glynn (Reference Blackwell and Glynn2015) and Imai and Kim (Reference Imai and Kim2016) for discussions on the difference between the strict ignorability and sequential ignorability assumptions. What is unique here is that we conditional on unobserved factors and factor loadings. 16 $\unicode[STIX]{x1D700}_{it}$ may be correlated with $\hat{\unicode[STIX]{x1D706}}_{i}$ when the errors are serially correlated because $\hat{\unicode[STIX]{x1D706}}_{i}$ is estimated using the pretreatment data. 17 The treated outcome for unit $i$ , thus can be drawn from ${\tilde{Y}}_{i}(1)={\tilde{Y}}_{i}(0)+\unicode[STIX]{x1D6FF}_{i}$ . We do not directly observe $\unicode[STIX]{x1D6FF}_{i}$ , but since it is taken as given, its presence will not affect the uncertainty estimates of $\widehat{ATT}_{t}$ . Hence, in the bootstrap procedure, we use ${\tilde{Y}}_{i}(0)$ for both the treatment and control groups to form bootstrapped samples (set $\unicode[STIX]{x1D6FF}_{i}=\mathbf{0}$ , for all $i\in {\mathcal{T}}$ ). We will add back $\widehat{ATT}_{t}$ when constructing confidence intervals. 18 The DGP specified here is modified based on Bai (Reference Bai2009) and Gobillon and Magnac (Reference Gobillon and Magnac2016). 19 Standard deviation is defined as: $SD(\widehat{ATT}_{t})=\sqrt{\mathbb{E}[\widehat{ATT}_{t}^{(k)}-\mathbb{ E}(\widehat{ATT}_{t}^{(k)})]^{2}}$ , while root mean squared error is defined as: $RMSE(\widehat{ATT}_{t})=\sqrt{\mathbb{E}(\widehat{ATT}_{t}^{(k)}-ATT_{t}^{(k)})^{2}}$ . The superscript $(k)$ denotes the $k$ th sample. We see that they are very close because the bias of the GSC estimator shrinks to zero as the sample size grows. 20 In the Online Appendix, we list the years during which EDR laws were enacted and first took effect in presidential elections. 21 See Wolfinger and Rosenstone (Reference Wolfinger and Rosenstone1980), Mitchell and Wlezien (Reference Mitchell and Wlezien1995), Rhine (Reference Rhine1992), Highton (Reference Highton1997), Timpone (Reference Timpone1998), Timpone (Reference Timpone2002), Huang and Shields (Reference Huang and Shields2000), Alvarez, Ansolabehere, and Wilson (Reference Alvarez, Ansolabehere and Wilson2002), Brians and Grofman (Reference Brians and Grofman2001), Hanmer (Reference Hanmer2009), Burden et al. (Reference Burden, Canon, Mayer and Moynihan2009), Cain, Donovan, and Tolbert (Reference Cain, Donovan and Tolbert2011), Teixeira (Reference Teixeira2011) for examples. The results are especially consistent for the three early adopters, Maine, Minnesota, and Wisconsin. 22 See, for example, Fenster (Reference Fenster1994), King and Wambeam (Reference King and Wambeam1995), Knack and White (Reference Knack and White2000), Knack (Reference Knack2001), Neiheisel and Burden (Reference Neiheisel and Burden2012), Springer (Reference Springer2014). 23 The data from 1920 to 2000 are from Springer (Reference Springer2014). The data from 2004 to 2012 are from The United States Election Project, http://www.electproject.org/. Indicators of other registration laws, including universal mail-in registration and motor voter registration, also come from Springer (Reference Springer2014), with a few supplements. Replication files can be found in Xu (Reference Xu2016). 24 We do not use the voting-eligible population (VEP) as the denominator because they are not available in early years. 25 As is shown in the figure and has been pointed out by many, turnout rates are in general higher in states that have EDR laws than states that have not, but this does not necessarily imply a causal relationship between EDR laws and voter turnout. 26 Note that although the estimated ATT of EDR on voter turnout is presented in the same row as the coefficient of EDR using the FE model, the GSC method does not assume the treatment effect to be constant. In fact, it allows the treatment effect to be different both across states and over time. Predicted counterfactuals and individual treatment effect for each of the nine treated states are shown in the Online Appendix. 27 The results are similar if additive state and year fixed effects are not directly imposed, though not surprisingly, the algorithm includes an additional factor. 28 Although it is not guaranteed, this is not surprising since the GSC method uses information of all past outcomes and minimizes gaps between actual and predicted turnout rates in pretreatment periods. 29 The results are essentially the same with or without controlling for the other two registration reforms. 30 Although we can control for indicators of Jim Crow laws in the model, such indicators may not be able to capture the heterogeneous impacts of these laws on voter turnout in each state. 31 In the Online Appendix, we show that the treatment effects are positive (and relatively large) for all three early adopting states, Maine, Minnesota, and Wisconsin. Using a fuzzy regression discontinuity design, Keele and Minozzi (Reference Keele and Minozzi2013) show that EDR has almost no effect on the turnout in Wisconsin. The discrepancy with this paper could be mainly due to the difference in the estimands. Two biggest cities in Wisconsin, Milwaukee and Madison constitute a major part of Wisconsin's constituency but have neglectable influence to their local estimates. One advantage of Keele and Minozzi (Reference Keele and Minozzi2013)'s approach over ours is the use of fine-grained municipal level data. 32 Glynn and Quinn (Reference Glynn and Quinn2011) argue that traditional cross-sectional methods in general overestimate the effect of EDR laws on voter turnout and suggest that EDR laws are likely to have minimum effect on turnout in non-EDR states (the ATC). In this paper, we focus on the effect of EDR in EDR states (the ATT) instead. Abadie, Alberto. 2005. Semiparametric difference-in-differences estimators. 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Help us improve our products. Sign up to take part. A Nature Research Journal Search E-alert Submit My Account Login Highly selective detection of methanol over ethanol by a handheld gas sensor J. van den Broek ORCID: orcid.org/0000-0003-1432-99931, S. Abegg ORCID: orcid.org/0000-0002-7442-05801, S. E. Pratsinis ORCID: orcid.org/0000-0003-2042-249X1 & A. T. Güntner ORCID: orcid.org/0000-0002-4127-752X1,2 Nature Communications volume 10, Article number: 4220 (2019) Cite this article 123 Altmetric Medical and clinical diagnostics Methanol poisoning causes blindness, organ failure or even death when recognized too late. Currently, there is no methanol detector for quick diagnosis by breath analysis or for screening of laced beverages. Typically, chemical sensors cannot distinguish methanol from the much higher ethanol background. Here, we present an inexpensive and handheld sensor for highly selective methanol detection. It consists of a separation column (Tenax) separating methanol from interferants like ethanol, acetone or hydrogen, as in gas chromatography, and a chemoresistive gas sensor (Pd-doped SnO2 nanoparticles) to quantify the methanol concentration. This way, methanol is measured within 2 min from 1 to 1000 ppm without interference of much higher ethanol levels (up to 62,000 ppm). As a proof-of-concept, we reliably measure methanol concentrations in spiked breath samples and liquor. This could enable the realization of highly selective sensors in emerging applications such as breath analysis or air quality monitoring. Ingestion, inhalation, or skin absorption of methanol leads to irreversible tissue damage, especially to eyes and nervous system, or even death1. This is attributed to metabolization of methanol to toxic formic acid and formaldehyde2, if not immediately treated3. Especially in developing countries, methanol poisoning outbreaks occur frequently with hundreds of victims due to adulterated alcohol as shown recently in Iran (Oct. 2018, 959 cases)4, Cambodia (May 2018, 237 cases)5, and India (Feb. 2019, > 95 cases)6. Furthermore, methanol is often used as solvent or chemical feedstock7 in laboratories and chemical plants, posing a potential hazard of intoxication. The gold standard for detection of methanol intoxication is blood analysis by gas–liquid chromatography, but more frequent in hospitals is the indirect diagnosis through blood gas analysis8. However, both require trained personnel, are expensive and rarely available in developing countries where most outbreaks occur9. Blood methanol levels can also be determined non-invasively in exhaled breath10, analogous to ethanol as widely applied by law enforcement11. The challenge is thereby the selective detection of methanol in the presence of much higher ethanol background typically present after consumption of tainted alcoholic beverages and during therapy where ethanol is used as an antidote12. Even more interesting might be simple methods for screening of alcoholic beverages to prevent methanol poisoning. But here too, the same challenge is met. Thus, inexpensive and portable devices are needed for rapid screening of methanol poisoning in breath and liquor by paramedics or even laymen. Chemical gas sensors are promising due to their low cost13, high miniaturization potential14, and simple use15. In particular, metal-oxide sensors show high sensitivity when nanostructured, capable to detect analytes down to 5 ppb within seconds16. But, such sensors are typically non-selective17, especially for chemically similar molecules (like methanol and ethanol), representing a long-standing challenge in the field. Therefore, current chemoresistive18 and electrochemical19 methanol gas sensors show cross-interferences to ethanol and other alcohols, hindering them for the targeted applications. Filters can drastically improve the selectivity of chemical sensors by exploiting additional molecular properties of the target analyte. For instance, highly selective (>100–1,000) formaldehyde detection was possible even with a non-specific SnO2-based sensor by placing ahead a microporous zeolite membrane to filter molecules by size20. This way, formaldehyde was detected down to 30 ppb in 90% relative humidity (RH) without interference of 1 ppm ammonia, acetone, isoprene, and ethanol20. Also a sorption packed bed separation column of polar, nanostructured alumina enabled separation of hydrophilic from hydrophobic compounds21, analogous to a gas chromatographic (GC) column22. This has led to highly selective (>100) sensing of isoprene down to 5 ppb at 90% RH despite the presence of much higher (4–8 times) methanol, ammonia and acetone concentrations21. Non-polar adsorbents, such as Tenax TA, on the other hand, can separate molecules by their molecular weight and chemical functional groups23. They are widely used in air sampling, whereby heavy molecules are retained longer than lighter ones due to stronger adsorption by van-der-Waals forces23. Thus, they are also promising to separate methanol from ethanol as done already in GC for the analysis of liquor (e.g., detected by olfactometry with humans24) and human breath (e.g., by mass spectrometry25). Here, we present a handheld and inexpensive methanol detector (Fig. 1a) capable to quantify methanol selectively in the presence of ethanol and other analytes (e.g., acetone, H2). It consists of a small packed bed of Tenax (Fig. 1b) to separate the analytes and a highly sensitive, but non-specific microsensor (Fig. 1d) consisting of flame-made Pd-doped SnO2 nanoparticles on interdigitated sensing electrodes. In comparison to typical GC instruments22, our device is much smaller and less expensive. It is benchmarked by detection of methanol in the relevant concentration range in the presence of much higher ethanol levels at high RH. Ultimately, the methanol detector is tested to sense toxic methanol levels in tainted rum and even in spiked human breath. Images of the handheld methanol detector. a It consists of a microsensor (in Teflon housing) connected to a separation column (Tenax TA particles in Teflon tube). b Close-up of the separation column with particles inside a glass tube for better visibility. c Magnified images of a particle's surface. d The sensor chip carrier with a mounted microsensor. e, f Top-view images of the sensing films consisting of a fine network of agglomerated and aggregated Pd-doped SnO2 nanoparticles Detector design Figure 1a shows the handheld methanol detector. It consists of a separation column upstream of a micromachined metal-oxide gas sensor housed inside a Teflon chamber. Breath or the headspace of a beverage can be drawn by a pump through the separation column to the sensor. The separation column is a miniaturized GC column with Tenax TA as the stationary phase (shown in Fig. 1b) featuring lower adsorption strength to methanol over ethanol23. Figure 1c shows a scanning electron microscopy (SEM) image of a Tenax particle's surface revealing its high specific surface area (35 m2 g−1) and porosity (average pore size 200 nm). Compared to typical GC columns22, the separation column used here is much shorter (4.5 cm) and thicker (4 mm inner diameter). Together with the small amount of adsorbent used (150 mg) and its large particle size (~200 µm), this results in a sufficiently small pressure drop (<20 mbar) to provide the required flow rate (25 mL min−1) to the sensor. Figure 1d shows the sensor bonded on a chip carrier. It is micromachined, offering small size and minimal power requirement (76 mW at 350 °C) readily suitable for integration into a handheld device. Figure 1e, f show top-view SEM images of the sensing film made of chemoresistive Pd-doped SnO2 nanoparticles26 offering high porosity and specific surface area (~80 m2 g−1 for similarly prepared Pt-doped SnO2)27. The open film structure enables fast diffusion of analytes and interaction with the large surface area, important for rapid and highly sensitive methanol sensing. Such sensors have been used, for instance, for detection of only 3 ppb formaldehyde with fast response (140 s) and recovery (190 s) times, and good reproducibility (<10% response variation)28. Selective methanol detection Figure 2a shows responses of the Pd-doped SnO2 sensor without separation column to 10 s exposures of 5 ppm hydrogen (purple line), methanol (red line), acetone (green line), and ethanol (blue line) at 50% RH. The sensor quickly reacts to all these analytes with responses between 10 and 25. However, it cannot differentiate between them. This becomes even more evident when exposing the sensor to a mixture of these analytes (Fig. 2b). The sensor gives now a much higher response, slightly lower than the sum of the individual ones, as typically observed for chemoresistive sensors at such ppm concentrations29. As a consequence, this sensor cannot measure methanol selectively in the presence of such interferents. Single and mixed analyte responses without and with separation column. Response of the methanol detector a, b without and c, d with separation column to 10 s exposures of 5 ppm hydrogen (purple line), methanol (red line), ethanol (blue line), and acetone (green line) as well as their mixture (black line in b and d) at 50% RH and 25 mL min−1 total flow rate. Also given are the corresponding retention times (tR, dashed lines, for hydrogen tR = 0 min as it is not retained). Note the different ordinate scale between a, b and c, d. The presence of the polymer sorbent packed bed upstream of the Pd-doped SnO2 sensor facilitates the separation of the mixture components When combined with the separation column, the sensor responses are separated as in a chromatograph. The response to hydrogen (purple line) remains the same (Fig. 2c). This is expected as hydrogen features low molecular weight and is not retained by Tenax30. For the other analytes, however, a different behavior is observed. In fact, methanol (red line) is now detected after >1 min with a maximum sensor response (i.e., retention time tR, dashed lines) after 1.7 min. Note that the maximum methanol response is lower than without separation column, as the column dissipates it over a longer time period, in line with theory31. Most importantly, ethanol (blue line, tR = 8.7 min) and acetone (green line, tR = 33 min) are retained for much longer, in agreement with literature (tR = 2.2, 10.8, and 36 min for methanol, ethanol, and acetone, respectively, at 20 °C)30. As a result, the separation column enables selective methanol detection. Interestingly also, the ethanol and acetone responses decrease with increasing tR. In fact, the response to 5 ppm acetone is barely picked up by the sensor (while it was twice that of methanol without separation column, compare Fig. 2a, c). When exposing the sensor with separation column to a mixture of the same analytes and concentrations (Fig. 2d), the analytes can be detected individually at their specific retention time with very high selectivity, identical to the single analyte exposures (Fig. 2c). Most remarkably, for the targeted applications, methanol is detected without ethanol interference, superior to state-of-the-art methanol sensors where the highest selectivity to ethanol (>30) has been reported for imprinted Ag-doped LaFeO3 core-shell particles32. As shown in Fig. 2c, d, the detector fully regenerates from each analyte or mixture exposure by flushing with air. The recovery time depends on the analyte and is about two to three times its retention time, in agreement with literature33. The recovery time can be decreased considerably by simply increasing the flow rate or by slight heating of the separation column (e.g., acetone from ~60 min to 20 s30 when increasing column temperature and flow rate briefly to 80 °C and 100 mL min−1)23. Methanol concentrations in the targeted applications may occur from several ppm in breath10 up to several hundred ppm in the headspace of beverages34. Figure 3 shows the Pd-doped SnO2 sensor response with separation column when exposed for 10 s to 1–918 ppm of methanol at 50% RH. The median methanol concentration in healthy breath35 (green line), and the range of exogeneous36 (orange line) and toxic breath methanol concentrations1 (red area) are also indicated. The response curve is non-linear, in line with diffusion-reaction theory29 for such semiconductive metal-oxide films at high analyte concentrations. Nevertheless, as a result, this separation column–sensor system can discriminate clearly toxic from nontoxic levels and even detect low concentrations of 1 ppm with a signal-to-noise ratio >100. Lower concentrations are not relevant for the liquor headspace and breath analyses, but such Pd-doped SnO2 gas sensors can detect volatile organic compounds down to single ppb levels (e.g., 3 ppb formaldehyde28). In contrast, other benchtop methanol detectors (e.g., PTR-TOF-MS) can detect such low concentrations as well, but they have a much smaller dynamic range and require dilution to measure the high ppm concentrations present in breath or the headspace of beverages. Dynamic range of the detector. Response of the detector (Tenax separation column + Pd-doped SnO2 sensor) to 1–918 ppm methanol concentrations (black squares and dashed line). The median methanol concentration in healthy breath35 (green dashed line), exogeneous36 (orange dashed line), and toxic1 breath levels (red dashed line and shaded area) are indicated. Measurements were performed with 10 s exposure of all methanol concentrations at 50% RH and a flow rate of 25 mL min−1 through the detector Please note that the response curve in Fig. 3 is valid for a separation column temperature of 22 °C at 50% RH. With increasing column temperature, the sensor responses become higher as tR decreases (Supplementary Fig. 1a). Most importantly, however, methanol is clearly separated and detected individually from ethanol even at 40 °C. Such temperature effects could be accounted for by a temperature sensor. At higher humidity, tR does not change, in line with literature23, but the responses decrease (Supplementary Fig. 1b). This is typical for such doped SnO2 sensors20 and can be addressed by using a sensor material less sensitive to humidity (e.g., Sb-doped SnO237) or by correction with a humidity sensor as done with sensor arrays to monitor volatile emission from human breath and skin38. High ethanol background To analyze methanol in the headspace of alcoholic beverages or in intoxicated breath, the detector must remain accurate in the presence of very high ethanol concentrations. Figure 4a shows the response of the detector when exposed to 1 ppm methanol with interfering ethanol concentrations of 5 (green line), 650 (1% relative saturation, blue line), and 32,500 ppm (50% relative saturation, red line). Despite the significantly higher ethanol concentration, methanol is detected first (tR = 1.5–1.7 min) giving comparable responses to the single gas calibration (Fig. 3). Ethanol is detected later with breakthrough times (tB, dashed lines) that decrease with increasing concentration (5.7 min at 5 ppm to 2.2 min at 50% saturation) but are always higher than the tR of methanol. In GC, the same phenomenon is observed when overloading the column with analyte39. Methanol detection with high ethanol interference. a Responses of the methanol detector upon exposure to 1 ppm methanol in the presence of 5 (green line), 650 (1% relative saturation, blue line), and 32,500 ppm (50% relative saturation, red line) ethanol. Corresponding ethanol breakthrough times (tB, dashed lines) are indicated. b The ethanol tB (squares and dashed line) and methanol retention time (tR, circles and solid line) as a function of interfering ethanol concentration of 5 (green), 650 (1% relative saturation, blue), 6500 (10% relative saturation, orange), 32,500 ppm (50% relative saturation, red), and 62,000 ppm (95% relative saturation, purple) Interestingly, at 50% ethanol saturation concentration, methanol is detected slightly earlier with higher peak maximum and narrower peak width. Probably, this is due to competitive adsorption on Tenax where methanol is displaced by ethanol that adsorbs more strongly40. Nevertheless, the resulting error of 17% is sufficiently small for the targeted applications as the difference between normal and toxic methanol concentrations in liquor and breath are much larger (e.g., human breath median 0.46 ppm35 vs. intoxicated >133 ppm1). If higher accuracy is required, alternatively, the area below the methanol response could be evaluated, as commonly done in gas chromatography22. In fact, the peak areas below the methanol responses are basically identical (within 2%), irrespective of the ethanol concentration (Supplementary Fig. 2). Most importantly, the methanol response is clearly separated from that of ethanol even at very high concentrations. This is shown in Fig. 4b where the tR of methanol (solid line) and tB of ethanol (dashed line) are plotted for ethanol concentrations in the range of 5–62,000 ppm (95% saturation). The tB decreases exponentially with increasing concentration, in line with literature at lower concentrations31. Even at the most extreme conditions of 95% saturated ethanol atmosphere, methanol is detected independently of ethanol as its response is clearly separated from the breakthrough of ethanol. These results are astonishing considering the simplicity of this device and outperform other methanol detectors. Methanol-spiked liquor and breath Drinking as little as 6 mL methanol can be fatal41. Thus, a methanol detector for screening of alcoholic beverages could help to prevent methanol poisoning outbreaks. The safety threshold for naturally occurring methanol in liquor (40 vol% ethanol) is 0.4 vol% (US34 and EU36), as such low levels are a byproduct of fermentation34. The detector must therefore be able to distinguish "safe" alcoholic beverages from tainted ones with typically much higher methanol content. Figure 5a shows the responses to pure (green line) and laced Arrack (common liquor in Southeast Asia) with 0.3 (blue line), 0.4 (orange line), 0.5 (purple line), and 1 vol% (red line) methanol. The detector clearly recognizes the added methanol at the expected tR = 1.7 min, matching the retention time of methanol in laboratory gas mixtures (Figs. 2c and 4). Detection of methanol in laced liquor (Arrack). a Response of the detector to pure liquor (green line) containing 40 vol% ethanol and laced with methanol of 0.3 (blue line), 0.4 (orange line), 0.5 (purple line), and 1 vol% (red line). b Sensor responses as a function of methanol content (0–10 vol%) in the liquor (black circles). The red dashed line indicates the legally allowed (US34 and EU36) naturally occurring methanol content in liquor (40 vol% ethanol). Error bars indicate the standard deviation of at least three measurements (i.e., repeatability) with <15% variation Most importantly, the response increases with increasing methanol concentration and even small differences between 0.3, 0.4 and 0.5 vol% (i.e., close to the allowed limit) can be clearly resolved by the sensor with high signal-to-noise ratio >100. In all cases, the response steeply increases after 3 min, corresponding to the high concentration of ethanol. Also at higher methanol contents of 5 and 10 vol% the sensor response continues to increase (Fig. 5b). As a result, the methanol detector can clearly distinguish pure Arrack from that laced with toxic levels of methanol with good repeatability (<15% variation, error bars in Fig. 5b). Owing to the high signal-to-noise ratio, also lower concentrations of methanol should be detectable, which may be interesting for the production monitoring and quality control of alcoholic beverages (e.g., naturally occurring methanol in wine42). The methanol detector features also good stability with a sensor baseline drift of 0.7% per day during 18 days of testing (Supplementary Fig. 4). Such drifts could be corrected by an additional processing algorithm43. By purging with ambient air, it fully regenerates within 15 min (Supplementary Fig. 3a), enabling rapid screening and multiple uses with no observed saturation or degradation effects over, at least, 2 weeks of repeated testing (Supplementary Fig. 4). As a proof-of-concept for breath analysis, we evaluated the methanol detector on the original and methanol-spiked breath of an intoxicated (after ingestion of ethanol) volunteer (blood alcohol level 0.54‰ as measured with a Dräger Alcotest). Poisoning volunteers with methanol is unacceptable. However, spiking the analyte to the sample (i.e., standard addition method44) is a standard approach in analytical chemistry as the complexity of the gas matrix (i.e., intoxicated breath) is preserved. Figure 6a shows the detector response for the normal (blue dashed line) and methanol-spiked breath (135 ppm methanol, red solid line). Note that a methanol concentration of 135 ppm was chosen as it is just above the threshold of serious methanol intoxication (>133 ppm1). In both cases, the detector shows identical responses to hydrogen at t = 0–30 s (not retained) and ethanol (tR = 8.3 min) with full recovery thereafter (Supplementary Fig. 3b). A clear peak associated with methanol is visible at tR = 1.7 min with high signal-to-noise ratio (>1000), identical to laboratory gas mixtures (Fig. 2c). To verify the methanol (Fig. 6b) and ethanol (Fig. 6c) peaks, the same breath samples were analyzed by benchtop PTR-TOF-MS equipped with the same separation column. Note that these high concentrations were only measurable by PTR-TOF-MS by additional dilution (please see Methods). Methanol and ethanol were detected at tR identical to the sensor, confirming the sensor results. Methanol detection in spiked breath. a Response of the methanol detector to breath of an intoxicated volunteer (0.54‰ blood alcohol level) sampled from Tedlar bags. The blue dashed line shows the measurement from the normal breath sample and the red solid line from the spiked sample with 135 ppm methanol, indicating methanol intoxication. The PTR-TOF-MS measurements (with separation column and dilution) of the same samples for b methanol and c ethanol. The instrument shown in c is a PTR-TOF-MS 1000 (Ionicon, Austria) used for sensor validation. Hydrogen (H2) is not retained by the separation column and does not interfere with the methanol detector As a result, this detector can clearly differentiate between normal and methanol-spiked breath. Therefore, this it is promising for fast and non-invasive sensing of methanol poisoning. Given the high signal-to-noise ratio at 135 ppm methanol, it also shows promise for monitoring methanol elimination during treatment10. Of course, the results are rather preliminary (only one subject tested) and further validation with extended cohorts is required as done recently with breath acetone and a similar sensor (Si-doped WO3) for body fat burn monitoring during exercise45 and dieting46. Interestingly, in liquor (Fig. 5) and human breath (Fig. 6), only methanol, ethanol and hydrogen (breath) are clearly detected by the sensor, although both liquor47 and breath48 are complex mixtures with more than 100 and 800 analytes, respectively. This is probably due to the higher molecular weight and different functional groups (e.g., diols or glycols) of most interferants, resulting in longer retention in the separation column than methanol (e.g., ethylene glycol 100 times longer than methanol30). The most likely reason, however, is the much lower concentration of most confounders (e.g., 0.003 ppm trimethylamine in breath49 compared to >133 ppm of methanol in case of intoxication1). To the best of our knowledge, this is the first methanol sensor for the detection of relevant concentrations in the presence of ethanol in realistic samples of liquor and breath. Other sensors are either liquid sensors that cannot be used for breath (e.g., photoluminescent Tb3+-based metal-organic framework sensor50), do not offer the required detection limit (e.g., Quartz tuning fork-based sensor51) or were not tested in gas mixtures (e.g., optical fiber sensor52). We created an inexpensive, handheld and reliable methanol detector based on a separation column–sensor concept. The separation column is a small packed bed of polymer adsorbent (Tenax TA) that separates methanol from ethanol and other interferants including hydrogen and acetone analogous to a column in gas chromatography. So, methanol is detected within 2 min by a non-specific but highly sensitive nanostructured Pd-doped SnO2 gas sensor in a wide concentration range from 1 to 918 ppm without interference of much higher ethanol concentrations (up to 62,000 ppm). The detector successfully quantified methanol concentrations in laced rum (Arrack) down to 0.3 vol% by analyzing its headspace and distinguished it from pure liquor. As first proof-of-concept, the detector was also tested on breath of an intoxicated volunteer, where it could clearly identify the sample spiked with toxic methanol concentrations. Thus, it shows promise for quick and non-invasive screening of methanol poisoning from breath and laced alcoholic beverages and could be used by first responders in developing countries, where most outbreaks occur. In a broader sense, the present detector demonstrates how to possibly address a long-standing challenge of chemical sensors: the discrimination between analytes from the same chemical family. Giving comparable performance to a gas chromatographic column, such separation columns are much simpler in design, modular, and can be combined flexibly with other sensor technologies that often lack selectivity, such as optical sensors (e.g., plasmonic53, fluorescent54), gas ionization detectors55, electrochemical cells56, and carbon-nanotube57- or graphene-based sensors58. Based on their small size and low price, such separation columns could enable highly selective, compact, and portable gas detectors for emerging applications including medical breath analysis, food spoilage, and air quality monitoring. Sensor fabrication Palladium-doped SnO2 nanoparticles were produced by flame spray pyrolysis (FSP). So, Pd-acetylacetonate (Sigma-Aldrich, 99%) was dissolved in tin(-II-)ethylhexanoate (Strem Chemicals, ~90% in 2-ethylhexanoic acid) and xylene (Sigma-Aldrich, ≥98.5%) to obtain a total metal molarity (Pd and Sn) of 0.5 M and nominal Pd content of 1 mol%28. This precursor solution was fed through a capillary at 5 mL min−1, dispersed into a fine spray by 5 L min−1 oxygen (pressure drop of 1.6 bar) and ignited by a surrounding premixed methane/oxygen flame (1.25/3.2 L min−1). The FSP reactor design is described in more detail elsewhere26. Nanoparticles were directly deposited26 for 4 min onto micromachined free-standing membrane-type sensor substrates (1.9 × 1.7 mm2, MSGS 5000i, Microsens SA, Switzerland) attached to a water-cooled holder at 20 cm height above the burner (HAB). The microsensor membranes feature an integrated heater layer underneath the interdigitated sensing electrodes. Subsequent in-situ annealing with a particle-free flame for 30 s at a HAB of 14.5 cm improved adhesion and cohesion of the highly porous sensing film59. Therefore, xylene was fed at 11 mL min−1 through the nozzle with identical dispersion flow used during nanoparticle production. Finally, the sensors were annealed at 500 °C for 5 h in an oven (CWF13/23, Carbolite, United Kingdom) and wire-bonded onto leadless chip carriers (Chelsea Technology Inc., Massachusetts, US). Separation column fabrication The separation column is a packed bed of 150 mg Tenax TA (poly(2,6-diphenyl-p-phenylene oxide), 60–80 mesh, ~35 m2 g−1, Sigma Aldrich) packed inside a Teflon tube (4 mm inner diameter) and secured on both ends with silanized glass wool plugs and tension springs. Freshly prepared columns were flushed overnight with 100 mL min−1 synthetic air (PanGas, CnHm and NOx ≤ 0.1 ppm, Switzerland) at 50% RH to desorb impurities that might be adsorbed on the Tenax. Scanning electron microscopy (SEM) images of the sensing film and the Tenax TA particle surface were made with a Hitachi S-4800 operated at 3 kV. Gas evaluation The methanol detector (Fig. 1a) consists of the separation column followed by the Pd-doped SnO2 sensor. A miniature rotary vane pump of only 12 g (135 FZ 3 VDC Schwarz Precision, Germany) downstream of the sensor draws the sample through the separation column at 25 mL min−1. The flow was validated by a calibrated bubble flow meter connected to the pump outlet. The sensor was heated by providing DC current (R&S HMC8043, Germany) through the heater of the micromachined sensor substrate. The sensing film temperature was set to 350 °C requiring only 76 mW. The ohmic resistance of the sensing film between the interdigitated electrodes was monitored with a multimeter (Keithley, 2700, USA). Sensor responses were evaluated as $$S = \frac{{R_{\rm{A}}}}{{R_{\rm{S}}}} - 1$$ where RA and RS denote the sensor film resistances measured in background air (synthetic air or ambient air in case of breath and liquor headspace analysis) and during sample measurement, respectively. The retention time tR of an analyte was defined as the time from the start of analyte exposure to the sensor's maximum response, analogous to gas chromatography60. The breakthrough time tB of an analyte was defined as the time from the start of analyte exposure to an analyte response equal to 5% of the response to 1 ppm methanol. For characterization of the sensor with synthetic gas mixtures, the methanol detector was connected to a gas delivery system illustrated schematically in Fig. 7. In specific, synthetic air was guided through a glass bubbler (Drechsel bottle, 125 mL, sintered glass frit, Sigma-Aldrich) containing ultrapure water (Milli-Q A10, Merck, Switzerland) and mixed with another stream of (dry) synthetic air to achieve 50% RH. All flows were accurately controlled by calibrated mass flow controllers (MFC, Bronkhorst, Netherlands) and the RH was verified by a humidity sensor (SHT2x, Sensirion AG, Switzerland). For generation of low analyte concentrations (5 ppm H2, 1–5 ppm methanol, 5 ppm ethanol, and 5 ppm acetone), analytes were admixed from calibrated gas standards (PanGas, in synthetic air) and added to the synthetic gas stream through a septum via a capillary. Thereby, the capillary was quickly inserted into the septum for 10 s to generate well defined analyte exposures. The Teflon gas lines were heated to ~50 °C to avoid condensation and adsorption of water or analytes. The flow rates of the synthetic air and the analyte streams were varied in the range of 300–1000 and 1–300 mL min−1, respectively, while the flow rate to the sensor was always kept constant by the pump at 25 mL min−1. Schematic of the synthetic gas mixing setup and the methanol detector. The methanol detector (orange box) consisting of the packed bed separation column of polymer (Tenax TA, red) particles, followed by the chemoresistive (Pd-doped SnO2, green) sensor and the vane pump that draws 25 mL min−1 of gas sample. For characterization with synthetic gas mixtures, the detector is connected to a gas delivery system. It supplies the detector with a constant flow of humidified air by mixing dry and humidified synthetic air (syn. air). Analyte exposures are generated by admixing analytes from calibrated gas standards or by bubbling syn. air through analyte/water mixtures to the humidified syn. air stream with a capillary through a septum. Flows are accurately controlled by calibrated mass flow controllers (MFCs) For higher methanol concentrations (15–918 ppm), dry synthetic air was guided through a glass bubbler filled with ultrapure water and 1 vol% methanol (>99.9%, Sigma-Aldrich) and dilution with synthetic air. The generated methanol concentration from the bubbler was measured with a proton-transfer-reaction time-of-flight mass spectrometer (PTR-TOF-MS 1000, Ionicon, Austria) after further inlet dilution (1:200–1000) with synthetic air to avoid device saturation. The ionization conditions were 600 V drift voltage, 60 °C drift temperature, and 2.3 mbar drift pressure. Methanol concentrations were determined in the H3O+ mode by measuring the counts per second at a mass-to-charge ratio61 (m/z) of 33.0335 and comparison to a calibration curve obtained from the methanol gas standard. Higher ethanol concentrations (250–64,000 ppm) were generated similarly by bubbling air through pure ethanol (absolute, >99.8%, Fisher Chemical) and dilution with synthetic air. Generated concentrations were calculated from the weight loss of the bubbler after bubbling with air for 0, 2, 4, 6, and 8 h, while room temperature was kept constant at 22 °C. Evaluation of the headspace of drinks and human breath For testing of methanol-spiked drinks and breath, sensors were stabilized in ambient air with analyte background concentrations of methanol <50 ppb, ethanol <500 ppb, and acetone <100 ppb as determined by PTR-TOF-MS. Liquid samples were prepared in 25 mL glass bottles by mixing 5 mL of rum (40 vol% ethanol, Boven's echter Arrak, Indonesia) with 0, 0.3, 0.4, 0.5, 1, 5, and 10 vol% of methanol. Concentrations <0.3 vol% are not relevant for the liquor screening as the legal limit is 0.4 vol% in the US34 and EU36. To guarantee equilibrium headspace concentrations, the samples were vigorously shaken manually for 30 s before sampling62. Headspace was sampled for 10 s by injecting a capillary attached to the methanol detector through a septum into the glass bottle caps. During sampling, a second capillary was inserted to keep the vial at ambient pressure. For breath sampling, a volunteer consumed an alcoholic beverage (40 vol% ethanol, Bacardi Rum Carta Blanca) containing an equivalent of 50 mL pure ethanol. After 1 h, blood alcohol concentration was estimated with a breathalyzer (Alcotest 3820, Dräger, Germany). Another two breath samples were collected in Tedlar bags (3 L, SKC Inc., USA) by direct and complete exhalation through a Teflon tube. One of the bags was spiked with 300 mL of 918 ppm methanol in synthetic air (100% RH), giving a final concentration of 135 ppm, as verified by PTR-TOF-MS (inlet dilution 1:40 with synthetic air). Also to the second bag, 300 mL of synthetic air (100% RH) without methanol was added to keep dilution of the breath samples similar. Breath samples were stored no longer than 1 h in the Tedlar bags to avoid analyte losses63. The detector was exposed to breath samples for 10 s by injecting a capillary through a septum at the cap of the Tedlar bags. 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Particle Technology Laboratory, Department of Mechanical and Process Engineering, ETH Zurich, 8092, Zurich, Switzerland , S. Abegg , S. E. Pratsinis & A. T. Güntner Department of Endocrinology, Diabetes, and Clinical Nutrition, University Hospital Zurich, 8091, Zurich, Switzerland A. T. Güntner Search for J. van den Broek in: Search for S. Abegg in: Search for S. E. Pratsinis in: Search for A. T. Güntner in: J.v.d.B. and A.T.G. conceived the concept and experiments. J.v.d.B. performed the experiments and the data evaluation. S.A. designed and provided the microsensors and contributed to the experimental design. S.E.P and A.T.G were in charge and advised on all parts of the project. J.v.d.B., S.A., S.E.P., and A.T.G co-wrote the paper. All authors gave final approval to the manuscript. Correspondence to A. T. Güntner. A patent application based on this manuscript has been submitted by J.v.d.B., S.A, S.E.P, and A.T.G. Peer Review Information Nature Communications thanks Patrik Španěl and other, anonymous, reviewer(s) for their contribution to the peer review of this work. Peer reviewer reports are available. Publisher's note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Peer Review File Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. van den Broek, J., Abegg, S., Pratsinis, S.E. et al. Highly selective detection of methanol over ethanol by a handheld gas sensor. Nat Commun 10, 4220 (2019) doi:10.1038/s41467-019-12223-4 DOI: https://doi.org/10.1038/s41467-019-12223-4 Pt/WN based fuel cell type methanol sensor Da Meng , Shendan Zhang , Tiju Thomas , Chaozhu Huang , Jingwei Zhao , Ruiyang Zhao , Ying Shi , Fengdong Qu & Minghui Yang Sensors and Actuators B: Chemical (2020) By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate. 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$\newcommand{\dollar}{\$} \DeclareMathOperator{\erf}{erf} \DeclareMathOperator{\arctanh}{arctanh} \newcommand{\lt}{<} \newcommand{\gt}{>} \newcommand{\amp}{&} $ Coordinated Calculus Nathan Wakefield, Christine Kelley, Marla Williams, Michelle Haver, Lawrence Seminario-Romero, Robert Huben, Aurora Marks, Stephanie Prahl, Based upon Active Calculus by Matthew Boelkins IndexPrevUpNext PrevUpNext Exponential and Logarithmic Functions 1Understanding the Derivative Introduction to Continuity Introduction to Limits How do we Measure Velocity? The Derivative of a Function at a Point The Derivative Function Interpreting, Estimating, and Using the Derivative The Second Derivative Differentiability 2Computing Derivatives Elementary Derivative Rules The Sine and Cosine Functions The Product and Quotient Rules Derivatives of Other Trigonometric Functions The Chain Rule Derivatives of Functions Given Implicitly The Tangent Line Approximation The Mean Value Theorem 3Using Derivatives Using Derivatives to Identify Extreme Values Global Optimization Using Derivatives to Describe Families of Functions Using Derivatives to Evaluate Limits 4The Definite Integral Determining Distance Traveled from Velocity The Definite Integral 5Evaluating Integrals Constructing Accurate Graphs of Antiderivatives Antiderivatives from Formulas The Second Fundamental Theorem of Calculus Integration by Substitution The Method of Partial Fractions Trigonometric Substitutions Numerical Integration Comparison of Improper Integrals Using Technology and Tables to Evaluate Integrals 6Using Definite Integrals Using Definite Integrals to Find Area and Volume Using Definite Integrals to Find Volume by Rotation and Arc Length Area and Arc Length in Polar Coordinates Density, Mass, and Center of Mass Physics Applications: Work, Force, and Pressure 7Sequences and Series Convergence of Series Ratio Test and Alternating Series Absolute Convergence and Error Bounds Taylor Polynomials Taylor Series Applications of Taylor Series 8Differential Equations An Introduction to Differential Equations Qualitative Behavior of Solutions to DEs Euler's Method Separable differential equations Population Growth and the Logistic Equation A Short Table of Integrals Supplemental Videos FeedbackAuthored in PreTeXt Section3.4Using Derivatives to Describe Families of Functions Motivating Questions Given a family of functions that depends on one or more parameters, how does the shape of the graph of a typical function in the family depend on the value of the parameters? How can we construct first and second derivative sign charts of functions that depend on one or more parameters while allowing those parameters to remain arbitrary constants? Mathematicians are often interested in making general observations, say by describing patterns that hold in a large number of cases. Think about the Pythagorean Theorem: it doesn't tell us something about a single right triangle, but rather a fact about every right triangle. In the next part of our studies, we use calculus to make general observations about families of functions that depend on one or more parameters. People who use applied mathematics, such as engineers and economists, often encounter the same types of functions where only small changes to certain constants occur. These constants are called parameters. Figure3.38The graph of $y=f(t)\text{,}$ where $f(t) = a \sin(b(t-c)) + d\text{,}$ based on parameters $a\text{,}$ $b\text{,}$ $c\text{,}$ and $d\text{.}$ You are already familiar with certain families of functions. For example, $f(t) = a \sin(b(t-c)) + d$ is a stretched and shifted version of the sine function with amplitude $a\text{,}$ period $\frac{2\pi}{b}\text{,}$ phase shift $c\text{,}$ and vertical shift $d\text{.}$ We know that $a$ affects the size of the oscillation, $b$ the rapidity of oscillation, and $c$ where the oscillation starts, as shown above in Figure3.38, while $d$ affects the vertical positioning of the graph. Here is another example: every function of the form $y = mx + b$ is a line with slope $m$ and $y$-intercept $(0,b)\text{.}$ The value of $m$ affects the line's steepness, and the value of $b$ situates the line vertically on the coordinate axes. These two parameters describe all possible non-vertical lines. For other less familiar families of functions, we can use calculus to discover where key behavior occurs: e.g. where members of the family are increasing or decreasing, concave up or concave down, where relative extrema occur, and more, all in terms of the parameters involved. To get started, we revisit a common collection of functions to see how calculus confirms things we already know. Example3.39 Let $a\text{,}$ $h\text{,}$ and $k$ be arbitrary real numbers with $a \ne 0\text{,}$ and let $f$ be the function given by the rule $f(x) = a(x-h)^2 + k\text{.}$ What familiar type of function is $f\text{?}$ What information do you know about $f$ just by looking at its form? (Think about the roles of $a\text{,}$ $h\text{,}$ and $k\text{.}$) Next we use some calculus to develop familiar ideas from a different perspective. To start, treat $a\text{,}$ $h\text{,}$ and $k$ as constants and compute $f'(x)\text{.}$ Find all critical numbers of $f\text{.}$ (These will depend on at least one of $a\text{,}$ $h\text{,}$ and $k\text{.}$) Assume that $a \lt 0\text{.}$ Construct a first derivative sign chart for $f\text{.}$ Based on the information you've found above, classify the critical values of $f$ as maxima or minima. $f$ is written in vertex form for this type of function; what does that tell you? Try using the chain rule to differentiate $f$ as written (instead of expanding $f$ before taking the derivative). If $c$ is a critical number of $f\text{,}$ what must be true about $f'(c)\text{?}$ Remember, there are two possibilities in general. What does the graph of $y=f(x)$ look like when $a\lt 0\text{?}$ Use what you know prior about the shape of $y=f(x)$ to support the answer you get with calculus and the first derivative sign chart you made in (d). $f$ is quadratic. $f'(x)=2a(x-h)\text{.}$ $h$ is the only critical number of $f\text{.}$ For $x\lt h\text{,}$ we have $f'(x)\gt0$ and $f(x)$ increasing. For $x\gt h\text{,}$ we have $f'(x)\lt0$ and $f(x)$ decreasing. $k$ is a maximum of $f\text{.}$ Since $f$ is a second-degree polynomial in $x\text{,}$ $f$ is a quadratic function, and the graph of $y=f(x)$ is a parabola. The vertex of this parabola is at the point $(h,k)\text{,}$ and the sign of $a$ determines whether the parabola opens upward (when $a\gt0$) or downward (when $a\lt0$). Because $h$ and $k$ are constants, their derivatives are both $0$ and this simplifies the application of the sum rule in taking the derivative. Furthermore, the product rule is unnecessary because $a$ is a constant as well, so we instead make use of the constant multiple rule. Doing so while also applying the chain rule, we find that \begin{equation} f'(x)=2a(x-h)\text{.} \end{equation} Since $f'$ is linear with slope $2a\neq0\text{,}$ it is continuous (hence, defined everywhere) and has exactly one root (where $f'(x)=0$). Thus the function $f$ will have exactly one critical number. Indeed, we see that $f'(x)=0$ only at the point $x=h\text{.}$ Therefore, the only critical number of $f$ is $h\text{.}$ We note that this agrees with our previous assertion that the vertex of $y=f(x)$ is at the point $(h,k)\text{.}$ If $a\lt0\text{,}$ then our algebra knowledge tells us that the graph of $y=f(x)$ is a parabola that opens down, so we should find that $f$ is increasing for $x\lt h$ and decreasing for $x\gt h\text{.}$ Testing this intuition, we note that When $x\lt h\text{,}$ then $x-h\lt0$ and hence $2a(x-h)\gt0$ since $a$ also is known to be negative. Therefore, on the interval $(-\infty,h)$ we have $f'(x)\gt0$ and $f$ increasing. When $x\gt h\text{,}$ then $x-h\gt0$ and hence $2a(x-h)\lt0$ since $a$ is known to be negative. Therefore, on the interval $(h,\infty)$ we have $f'(x)\lt0$ and $f$ decreasing. Applying the first derivative test, we see that at the point $x=h\text{,}$ the value of $f'(x)$ switches from positive to negative, and thus $k\text{,}$ the critical value corresponding to the critical number $h\text{,}$ is a maximum of $f\text{.}$ This agrees with our existing knowledge of parabolas, as we had established that this parabola opens downward. SubsectionDescribing Families of Functions in Terms of Parameters Our goal is to describe the key characteristics of the overall behavior of each member of a family of functions in terms of its parameters. By finding the first and second derivatives and constructing sign charts (each of which may depend on one or more of the parameters), we can often make broad conclusions about how each member of the family will appear. Consider the two-parameter family of functions given by $g(x) = axe^{-bx}\text{,}$ where $a$ and $b$ are positive real numbers. Fully describe the behavior of a typical member of the family in terms of $a$ and $b\text{,}$ including: the location of all critical numbers; where $g$ is increasing, decreasing, concave up, and concave down; and the long-term behavior of $g\text{.}$ Start by finding the first and second derivatives of $g\text{.}$ Be careful with your use of the product rule and chain rule. $g$ has one critical number, at $x=\frac1b\text{.}$ $g$ is increasing on $\big(-\infty,\frac1b\big)\text{,}$ decreasing on $\big(\frac1b,\infty\big)\text{,}$ concave down on $\big(-\infty,\frac2b\big)\text{,}$ and concave up on $\big(\frac2b,\infty\big)\text{.}$ $\lim_{x\to-\infty}g(x)=-\infty$ and $\lim_{x\to\infty}g(x)=0\text{.}$ We begin by computing $g'(x)\text{.}$ By the product rule, \begin{equation} g'(x) = \frac{d}{dx}[ax]e^{-bx} +ax \frac{d}{dx}\left[e^{-bx}\right]\text{.} \end{equation} By applying the chain rule and constant multiple rule, we find that \begin{equation} g'(x) = ae^{-bx}+axe^{-bx}(-b)\text{.} \end{equation} To find the critical numbers of $g\text{,}$ we note that $g'$ is defined everywhere and proceed to solve the equation $g'(x) = 0\text{.}$ By factoring $g'(x)\text{,}$ we find \begin{equation} 0 = ae^{-bx}(1-bx)\text{.} \end{equation} Since we are given that $a \ne 0$ and we know that $e^{-bx} \ne 0$ for all values of $x\text{,}$ the only way this equation can hold is when $1-bx = 0\text{.}$ Solving for $x\text{,}$ we find $x = \frac{1}{b}\text{,}$ and this is therefore the only critical number of $g\text{.}$ Because the factor $ae^{-bx}$ is always positive, the sign of $g'$ depends on the linear factor $(1-bx)\text{,}$ which is positive for $x \lt \frac{1}{b}$ and negative for $x \gt \frac{1}{b}\text{.}$ Hence we can not only conclude that $g$ is always increasing for $x \lt \frac{1}{b}$ and decreasing for $x \gt \frac{1}{b}\text{,}$ but also that $g$ has a global maximum at $\big(\frac{1}{b}, g\big(\frac{1}{b}\big)\big)$ and no local minimum. The first derivative sign chart for $g$ is shown below in Figure3.41. Figure3.41The first derivative sign chart for $g(x) = axe^{-bx}\text{.}$ We turn next to analyzing the concavity of $g\text{.}$ With $g'(x) = ae^{-bx}-abxe^{-bx}\text{,}$ we differentiate to find that \begin{equation} g''(x) = ae^{-bx}(-b)-ab\left(e^{-bx}+xe^{-bx}(-b)\right)\text{.} \end{equation} Combining like terms and factoring, we now have \begin{equation} g''(x) = ab^2xe^{-bx} - 2abe^{-bx} = abe^{-bx}(bx - 2)\text{.} \end{equation} We observe that $abe^{-bx}$ is always positive, and thus the sign of $g''$ depends on the sign of $(bx-2)\text{,}$ which is zero when $x = \frac{2}{b}\text{.}$ Since $b$ is positive, the value of $(bx-2)$ is negative for $x \lt \frac{2}{b}$ and positive for $x \gt \frac{2}{b}\text{.}$ The sign chart for $g''$ is shown below in Figure3.42. Thus, $g$ is concave down for all $x \lt \frac{2}{b}$ and concave up for all $x \gt \frac{2}{b}\text{.}$ Figure3.42The second derivative sign chart for $g(x) = axe^{-bx}\text{.}$ Finally, we analyze the long-term behavior of $g$ by considering two limits. First, we note that \begin{equation} \lim_{x \to \infty} g(x) = \lim_{x \to \infty} axe^{-bx} = \lim_{x \to \infty} \frac{ax}{e^{bx}}\text{.} \end{equation} This limit has indeterminate form $\frac{\infty}{\infty}\text{,}$ so with our current tools, we can not evaluate the limit algebraically. We can, however, reason that $ax$ is linear while $e^{bx}$ is exponential, and conclude that because exponential growth is so much more rapid than linear growth, we expect the limiting value to be zero. A more rigorous answer will be possible following Section3.6, but for now we can also use graphical evidence10A good tool to use here is desmos.com. to support the claim that $\lim_{x\to\infty}g(x)=0\text{.}$ We do so by examining the graphs of several different functions in this family, noting that each has a horizontal asymptote of $y=0$ as $x\to\infty\text{.}$ In the other direction, \begin{equation} \lim_{x \to -\infty} g(x) = \lim_{x \to -\infty} axe^{-bx} = -\infty\text{,} \end{equation} because $ax \to -\infty$ and $e^{-bx} \to \infty$ as $x \to -\infty\text{.}$ Hence, as we move left on its graph, $g$ decreases without bound, while as we move to the right, $g(x) \to 0\text{.}$ All of this information now helps us produce the graph of a typical member of this family of functions without using a graphing utility (and without choosing particular values for $a$ and $b$), as shown in Figure3.43 below. Figure3.43The graph of $g(x) = axe^{-bx}\text{.}$ Note that the value of $b$ controls the horizontal location of the global maximum and the inflection point, as neither depends on $a\text{.}$ The value of $a$ affects the vertical stretch of the graph. For example, the global maximum occurs at the point $\big(\frac{1}{b}, g\big(\frac{1}{b}\big)\big) = \big(\frac{1}{b}, \frac{a}{b}e^{-1}\big)\text{,}$ so the larger the value of $a\text{,}$ the greater the value of the global maximum. The work we've completed in Example3.40 can often be replicated for other families of functions that depend on parameters. Normally we are most interested in determining all critical numbers, a first derivative sign chart, a second derivative sign chart, and the limit of the function as $x \to \infty\text{.}$ Throughout, we prefer to work with the parameters as arbitrary constants. In addition, we can experiment with some particular values of the parameters present to reduce the algebraic complexity of our work. What follows are several key examples where we see that the values of the parameters substantially affect the behavior of individual functions within a given family. Consider the family of functions defined by $p(x) = x^3 - ax\text{,}$ where $a \ne 0$ is an arbitrary constant. Find $p'(x)$ and determine the critical numbers of $p\text{.}$ How many critical numbers does $p$ have? Construct a first derivative sign chart for $p\text{.}$ What can you say about the overall behavior of $p$ if the constant $a$ is positive? Why? What if the constant $a$ is negative? In each case, describe the relative extrema of $p\text{.}$ Find $p''(x)$ and construct a second derivative sign chart for $p\text{.}$ What does this tell you about the concavity of $p\text{?}$ What role does $a$ play in determining the concavity of $p\text{?}$ Without using a graphing utility, sketch and label typical graphs of $p(x)$ for the cases where $a\gt 0$ and $a \lt 0\text{.}$ Label all inflection points and local extrema. Finally, use a graphing utility to test your observations above by entering and plotting the function $p(x) = x^3 - ax$ for at least four different values of $a\text{.}$ Write several sentences to describe your overall conclusions about how the behavior of $p$ depends on $a\text{.}$ When solving $p'(x) = 0\text{,}$ think about two possible cases: when $a\gt 0$ and when $a \lt 0\text{.}$ Remember that any quadratic function can be zero at most two times. How does the graph of $y = 3x^2 - a$ look? Don't forget that $\frac{d}{dx}[a] = 0\text{.}$ Think about how the graph of a typical cubic polynomial behaves. $p$ has two critical numbers $\left(x = \pm \sqrt{\frac{a}{3}}\right)$ whenever $a \gt 0$ and no critical numbers when $a \lt 0\text{.}$ When $a \lt 0\text{,}$ $p$ is always increasing and has no relative extreme values. When $a\gt 0\text{,}$ $p$ has a relative maximum at $x = -\sqrt{\frac{a}{3}}$ and a relative minimum at $x = +\sqrt{\frac{a}{3}}\text{.}$ $p$ is concave down for $x \lt 0$ and $p$ is concave up for $x\gt 0\text{,}$ making $x = 0$ an inflection point. On the left, $a\lt0\text{;}$ on the right, $a\gt0\text{.}$ See desmos.com for additional graph samples. As $a$ decreases towards $-\infty\text{,}$ the graph of $y=p(x)$ looks more and more linear. As $a$ increases towards $\infty\text{,}$ the two bumps on the graph get more pronounced. We first note that $p'(x) = 3x^2 - a\text{,}$ so to find critical numbers we set $p'(x) = 0$ and solve for $x\text{.}$ This leads to the equation $3x^2 - a = 0\text{,}$ which implies \begin{equation} x^2 = \frac{a}{3}\text{.} \end{equation} If $a \gt 0\text{,}$ then the solutions to this equation are $x = \pm \sqrt{\frac{a}{3}}\text{;}$ if $a \lt 0\text{,}$ then the equation has no solution. Hence, $p$ has two critical numbers $\left(x = \pm \sqrt{\frac{a}{3}}\right)$ whenever $a \gt 0$ and no critical numbers when $a \lt 0\text{.}$ For the case when $a \lt 0\text{,}$ we observe that $p'(x) = 3x^2 - a$ is positive for every value of $x\text{,}$ and thus $p$ is always increasing and has no relative extreme values. (There are no critical numbers to place on the first derivative sign chart, and $p'$ is always positive.) For the case when $a\gt 0\text{,}$ we observe that $p'(x) = 3x^2 - a$ is a concave up parabola with zeros at $x = -\sqrt{\frac{a}{3}}$ and $x = +\sqrt{\frac{a}{3}}\text{.}$ It follows that for $x \lt -\sqrt{\frac{a}{3}}\text{,}$ $p'(x)\gt 0$ (so $p$ is increasing); for $-\sqrt{\frac{a}{3}} \lt x \lt \sqrt{\frac{a}{3}}\text{,}$ $p'(x)\gt 0$ (so $p$ is decreasing); and for $x\gt \sqrt{\frac{a}{3}}\text{,}$ $p'(x)\gt 0$ (so $p$ is again increasing). In this situation, we see that $p$ has a relative maximum at $x = -\sqrt{\frac{a}{3}}$ and a relative minimum at $x = +\sqrt{\frac{a}{3}}\text{.}$ Since $p'(x) = 3x^2 - a$ and $a$ is constant, it follows that $p''(x) = 6x\text{.}$ Note that $p''(x) = 0$ when $x = 0$ and that $p''(x) \lt 0$ for $x \lt 0$ and $p''(x)\gt 0$ for $x\gt 0\text{.}$ Hence $p$ is concave down for $x \lt 0$ and $p$ is concave up for $x\gt 0\text{,}$ making $x = 0$ an inflection point. Below, we show the two possible situations. At left, for the case when $a \lt 0$ and $p$ is always increasing with an inflection point at $x = 0\text{,}$ and at right for when $a\gt 0$ and $p$ has a relative maximum at $x = -\sqrt{\frac{a}{3}}$ and a relative minimum at $x = +\sqrt{\frac{a}{3}}\text{,}$ again with an inflection point at $x = 0\text{.}$ Note, too, that $p$ has its $x$-intercepts at $x = \pm \sqrt{a}\text{.}$ See desmos.com for additional graph samples. We note that as $a$ gets increasingly negative, the graph of $y=p(x)$ gets steeper near the origin, tending more and more toward a linear shape (which it never achieves). Conversely, as $a$ gets increasingly positive, the two bumps on the graph get more and more pronounced. Consider the two-parameter family of functions of the form $h(x) = a(1-e^{-bx})\text{,}$ where $a$ and $b$ are positive real numbers. Find the first derivative and the critical numbers of $h\text{.}$ Use these to construct a first derivative sign chart and determine for which values of $x$ the function $h$ is increasing and decreasing. Find the second derivative and build a second derivative sign chart. For which values of $x$ is a function in this family concave up? Concave down? What is the value of $\lim_{x \to \infty} a(1-e^{-bx})\text{?}$ What about $\lim_{x \to -\infty} a(1-e^{-bx})\text{?}$ How does changing the value of $b$ affect the shape of the curve? Without using a graphing utility, sketch the graph of a typical member of this family. Write several sentences to describe the overall behavior of a typical function $h$ and how this behavior depends on $a$ and $b\text{.}$ Expand to write $h(x) = a - ae^{-bx}$ before differentiating. Remember that $e^{-bx}$ is always positive (hence, never zero), regardless of the value of $x\text{.}$ Recall that $e^{-x} \to 0$ as $x \to \infty$ and $e^{-x} \to \infty$ as $x \to -\infty\text{.}$ Consider how $b$ affects the value of $h'(x)\text{.}$ Use your work in (a)-(d). $h$ is an always increasing function. $h$ is always concave down. $\lim_{x \to \infty} a(1-e^{-bx}) = a\text{,}$ and $\lim_{x \to -\infty} a(1-e^{-bx}) = -\infty\text{.}$ If $b$ is large and $x$ is close to zero, $h'(x)$ is relatively large near $x = 0\text{,}$ and the curve's slope will quickly approach zero as $x$ increases. If $b$ is small, the graph is less steep near $x = 0$ and its slope goes to zero less quickly as $x$ increases. Since $h(x) = a - ae^{-bx}\text{,}$ we have by the constant multiple and chain rules that \begin{equation} h'(x) = -ae^{-bx}(-b) = abe^{-bx}\text{.} \end{equation} Since $a$ and $b$ are positive constants and $e^{-bx} \gt 0$ for all $x\text{,}$ we see that $h'(x)$ is never zero (nor undefined), and indeed $h'(x) \gt 0$ for all $x\text{.}$ Hence $h$ is an always increasing function. Because $h'(x) = abe^{-bx}\text{,}$ we have that $h''(x) = abe^{-bx}(-b) = -ab^2e^{-bx}\text{.}$ As with $h'\text{,}$ we recognize that $a\text{,}$ $b^2\text{,}$ and $e^{-bx}$ are always positive, and thus $h''(x) = -ab^2e^{-bx} \lt 0$ for all values of $x\text{,}$ making $h$ always concave down. As $x \to \infty\text{,}$ $e^{-bx} \to 0\text{.}$ Thus, \begin{equation} \lim_{x \to \infty} a(1-e^{-bx}) = \lim_{x \to \infty} (a - ae^{-bx}) = a - 0 = a\text{.} \end{equation} This shows that $h$ has a horizontal asymptote at $y = a$ as we move rightward on its graph. As $x \to -\infty\text{,}$ $e^{-bx} \to \infty\text{.}$ Thus, \begin{equation} \lim_{x \to -\infty} a(1-e^{-bx}) = \lim_{x \to -\infty} (a - ae^{-bx}) = -\infty\text{.} \end{equation} Noting that $h'(x) = abe^{-bx}\text{,}$ we see that if we consider different values of $b\text{,}$ the slope of the graph changes. If $b$ is large and $x$ is close to zero, $h'(x) \approx ab$ (since $e^0 = 1$), so $h'(x)$ is relatively large near $x = 0\text{.}$ At the same time, for large $b\text{,}$ $e^{-bx}$ approaches zero quickly as $x$ increases, so the curve's slope will quickly approach zero as $x$ increases. If $b$ is small, the graph is less steep near $x = 0$ and its slope goes to zero less quickly as $x$ increases. Observing that $h(0) = 0$ and $\lim_{x \to \infty} h(x) = a\text{,}$ along with the facts that $h$ is always increasing and always concave down, we see that a typical member of this family looks like the following graph. Let $L(t) = \frac{A}{1+ce^{-kt}}\text{,}$ where $A\text{,}$ $c\text{,}$ and $k$ are all positive real numbers. Observe that we can equivalently write $L(t) = A(1+ce^{-kt})^{-1}\text{.}$ Find $L'(t)$ and explain why $L$ has no critical numbers. Is $L$ always increasing or always decreasing? Why? Given the fact that \begin{equation} L''(t) = Ack^2e^{-kt} \frac{ce^{-kt}-1}{(1+ce^{-kt})^3}\text{,} \end{equation} find all values of $t$ such that $L''(t) = 0$ and then construct a second derivative sign chart. For which values of $t$ is a function in this family concave up? Concave down? What is the value of $\lim_{t \to \infty} \frac{A}{1+ce^{-kt}}\text{?}$ $\lim_{t \to -\infty} \frac{A}{1+ce^{-kt}}\text{?}$ Find the value of $L(t)$ at the inflection point found in (b). Without using a graphing utility, sketch the graph of a typical member of this family. Write several sentences to describe the overall behavior of a typical function $L$ and how this behavior depends on the values of the parameters $A\text{,}$ $c\text{,}$ and $k\text{.}$ Explain why it is reasonable to think that the function $L(t)$ models the growth of a population over time, in a setting where the surrounding environment cannot support a population larger than $A\text{.}$ Use the chain rule, treating $A\text{,}$ $c\text{,}$ and $k$ as constants. Note that the only way $L''(t) = 0$ is if $ce^{-kt}-1 = 0\text{.}$ Remember that $e^{-t} \to 0$ as $t \to \infty$ and $e^{-t} \to \infty$ as $t \to -\infty\text{.}$ Note that at the inflection point $t_0\text{,}$ we have $ce^{-kt_0}=1\text{.}$ Think about horizontal asymptotes, where $L$ is increasing and decreasing, and concavity. Intuitively, if an environment can only support a population of a certain size, how should the population be growing when it is well below the limit? When it is approaching the limit? Does this intuition describe what you found to be true for $L\text{?}$ $L$ is an always increasing function. $L$ is concave up for all $t \lt -\frac{1}{k} \ln \big(\frac{1}{c}\big)$ and concave down for all $t\gt-\frac1k\ln\big(\frac1c\big)\text{.}$ $\lim_{t \to \infty} \frac{A}{1+ce^{-kt}} = A\text{,}$ and \begin{equation} \lim_{t \to -\infty} \frac{A}{1+ce^{-kt}} = 0\text{.} \end{equation} The inflection point on the graph of $L$ is $\big( -\frac{1}{k} \ln \big(\frac{1}{c}\big), \frac{A}{2}\big)\text{.}$ The population grows rapidly at first, but its growth rate decreases to near zero as the population approaches the limiting size of $A\text{.}$ By the chain rule and treating $A\text{,}$ $c\text{,}$ and $k$ as constants, we find that \begin{equation} L'(t) = A(-1)(1+ce^{-kt})^{-2} ce^{-kt}(-k) = Acke^{-kt}(1+ce^{-kt})^{-2}\text{.} \end{equation} Since $A\text{,}$ $c\text{,}$ and $k$ are all positive and $e^{-kt} \gt 0$ for all values of $t\text{,}$ it is apparent that $L'(t)$ is never zero (nor undefined), and indeed is positive for every value of $t\text{.}$ Thus, $L$ is an always increasing function. Given that the only way $L''(t) = 0$ is if $ce^{-kt}-1 = 0\text{.}$ Solving $ce^{-kt}-1 = 0$ for $t\text{,}$ we first write $e^{-kt} = \frac{1}{c}\text{.}$ Taking the natural logarithm of both sides yields $-kt = \ln\big(\frac{1}{c}\big)\text{,}$ so that \begin{equation} t = -\frac{1}{k} \ln \left(\frac{1}{c}\right) \end{equation} is the only value of $t$ for which $L''(t) = 0\text{.}$ Now, observe that since $ce^{-kt} \to 0$ as $t \to \infty$ and $ce^{-kt}\to\infty$ as $t\to-\infty\text{,}$ it follows that the quantity $ce^{-kt} - 1$ will be positive to the left of where it is zero and negative to the right of where it is zero. Since this is the only term in $L''(t)$ that can change sign, it follows that $L''(t) \gt 0$ for $t \lt -\frac{1}{k} \ln \big(\frac{1}{c}\big)$ and $L''(t) \lt 0$ for $t \gt -\frac{1}{k} \ln \big(\frac{1}{c}\big)\text{,}$ making $L$ concave up to the left of the noted inflection point and concave down thereafter. Recalling that $e^{-kt} \to 0$ as $t \to \infty\text{,}$ we observe that \begin{equation} \lim_{t \to \infty} \frac{A}{1+ce^{-kt}} = \frac{A}{1+0} = A\text{,} \end{equation} so $L$ has a horizontal asymptote of $y = A$ as $t \to \infty\text{.}$ On the other hand, since $e^{-kt} \to \infty$ as $t \to -\infty\text{,}$ this causes the denominator of $L$ to grow without bound (while the numerator remains constant), and therefore \begin{equation} \lim_{t \to -\infty} \frac{A}{1+ce^{-kt}} = 0\text{,} \end{equation} which means $L$ has a horizontal asymptote of $y = 0$ as $t \to -\infty\text{.}$ From (b), we know that $t = -\frac{1}{k} \ln \big(\frac{1}{c}\big)$ is the location of the inflection point of $L\text{.}$ We evaluate $L\big( -\frac{1}{k} \ln \big(\frac{1}{c}\big) \big)\text{,}$11If you prefer to evaluate the function without this apparent shortcut , be careful with parentheses as you plug in and evaluate. In particular, notice that we do indeed have $ce^{-k\left[-\frac{\ln(\frac{1}{c})}{k}\right]}=ce^{k\frac{\ln(\frac{1}{c})}{k}}=ce^{\ln(\frac{1}{c})}=c(\frac{1}{c})=1\text{.}$ The remainder of the calculation should match what is shown below. recalling that at this $t$-value we have the equality $ce^{-kt}=1\text{,}$ thus \begin{equation} L\left( -\frac{1}{k} \ln \left(\frac{1}{c}\right) \right) = \frac{A}{1+1} = \frac{A}{2}\text{.} \end{equation} Thus, the inflection point on the graph of $L$ is located at $\big( -\frac{1}{k} \ln \big(\frac{1}{c}\big), \frac{A}{2}\big)\text{.}$ We have shown that $L$ is an always increasing function that has horizontal asymptotes at $y =0$ and $y = A\text{,}$ as well as an inflection point at $\big( -\frac{1}{k} \ln \big(\frac{1}{c}\big), \frac{A}{2}\big)\text{,}$ which we note lies vertically halfway between the asymptotes. In addition, we see that $L(0) = \frac{A}{1+c}\text{.}$ The combination of all of this information shows us that a typical graph in this family of functions is given by the following figure. The population grows rapidly at first, but its growth rate decreases to near zero as the population approaches the limiting size of $A\text{.}$ This makes sense when environmental factors would affect the population to keep it at a sustainable size. SubsectionSummary Given a family of functions that depends on one or more parameters, we can often accurately describe the shape of the function in terms of the parameters by investigating how critical numbers and locations where the second derivative is zero depend on the values of these parameters. In particular, just as we can creat first and second derivative sign charts for a single function, we can often do so for entire families of functions where critical numbers and possible inflection points depend on arbitrary constants. These sign charts then reveal where members of the family are increasing or decreasing, concave up or concave down, and help us to identify relative extrema and inflection points. SubsectionSupplemental Videos Families of Functions SubsectionExercises 1Drug dosage with a parameter 2Using the graph of $g'$ 3Using the graph of $f$ 4Sign Change 5Critical and inflection points of a function with parameters 6Behavior of a function with parameters 7Analyzing and curve sketching 8Analyzing families of functions Consider the one-parameter family of functions given by $p(x) = x^3-ax^2\text{,}$ where $a \gt 0\text{.}$ Sketch a plot of a typical member of the family, using the fact that each is a cubic polynomial with a repeated zero at $x = 0$ and another zero at $x = a\text{.}$ Find all critical numbers of $p\text{.}$ Compute $p''$ and find all values for which $p''(x) = 0\text{.}$ Hence construct a second derivative sign chart for $p\text{.}$ Describe how the location of the critical numbers and the inflection point of $p$ change as $a$ changes. That is, if the value of $a$ is increased, what happens to the critical numbers and inflection point? Let $q(x) = \frac{e^{-x}}{x-c}$ be a one-parameter family of functions where $c \gt 0\text{.}$ Explain why $q$ has a vertical asymptote at $x = c\text{.}$ Determine $\lim_{x \to \infty} q(x)$ and $\lim_{x \to -\infty} q(x)\text{.}$ Compute $q'(x)$ and find all critical numbers of $q\text{.}$ Construct a first derivative sign chart for $q$ and determine whether each critical number leads to a local minimum, local maximum, or neither for the function $q\text{.}$ Sketch a typical member of this family of functions with important behaviors clearly labeled. 10Analyzing families of functions Let $E(x) = e^{-\frac{(x-m)^2}{2s^2}}\text{,}$ where $m$ is any real number and $s$ is a positive real number. Compute $E'(x)$ and hence find all critical numbers of $E\text{.}$ Construct a first derivative sign chart for $E$ and classify each critical number of the function as a local minimum, local maximum, or neither. It can be shown that $E''(x)$ is given by the formula \begin{equation} E''(x) = e^{-\frac{(x-m)^2}{2s^2}} \left(\frac{(x-m)^2 - s^2}{s^4} \right)\text{.} \end{equation} Find all values of $x$ for which $E''(x) = 0\text{.}$ Determine $\lim_{x \to \infty} E(x)$ and $\lim_{x \to -\infty} E(x)\text{.}$ Construct a labeled graph of a typical function $E$ that clearly shows how important points on the graph of $y = E(x)$ depend on $m$ and $s\text{.}$	CommonCrawl
What is the most secure/reliable CSPRNG currently available? [closed] Closed. This question is opinion-based. It is not currently accepting answers. Want to improve this question? Update the question so it can be answered with facts and citations by editing this post. Closed 1 year ago. In terms of period length, uniformity/distribution, what is the most secure/reliable CSPRNG currently available? random-number-generator randomness pseudo-random-generator pseudo-random-function FzpX FzpXFzpX $\begingroup$ Try the Yarrow suite of Algorithms by Bruce Schneier $\endgroup$ – cryptoknight Sep 1 '19 at 13:50 The most secure CSPRNG would be one which produces a stream indistinguishable from random data without knowledge of the key, and which takes a key large enough that an exhaustive search of the key space is impossible. It turns out that there are many such CSPRNGs, so there's more than one answer: ChaCha and Salsa20 are 256-bit stream ciphers which can output $2^{64} \times 512$ bits of data per key. AES in CTR mode is secure, but becomes distinguishable after generating about $2^{64} \times 128$ bits. HMAC_DRBG and Hash_DRBG are NIST standards that use a secure hash function, but are slow. There are many more which I could list, but not all are popular. ISAAC for example is a secure CSPRNG with no known weaknesses, but it hasn't been the subject of a large amount of research for this reason. If you are asking which CSPRNG is least likely to be the subject of major cryptanalytic breakthroughs in the future (i.e. which have the highest security margin), then the answer would be complex enough to fill entire cryptography journals, and indeed, they do. In general, ChaCha20 (ChaCha with 20 rounds) is currently thought by many to have one of the highest security margins. But they're all unbreakable. forestforest Not the answer you're looking for? Browse other questions tagged random-number-generator randomness pseudo-random-generator pseudo-random-function or ask your own question. What stops the Multiply-With-Carry RNG from being a Cryptographically Secure PRNG? Definition of a CSPRNG What is the difference between CSPRNG and PRNG? How do you interpret the p-values from the Dieharder testsuite to evaluate an RNG? How distinct are the meanings of the terms "CSPRNG," "DRBG" and "stream cipher"? Encryption using deterministic CSPRNG as secure as CSPRNG What is a "set synchronized CSPRNG"?	CommonCrawl
Chebyshev approximation by projection vs interpolation Suppose we want to approximate a function $f: [a, b] \rightarrow \Re$ with a Chebyshev series: $$ f(x) \approx \sum_{k=0}^n c_k \, T_k\left( \frac{2x-b-a}{b-a} \right) $$ where $T_k(x) = \cos(k\, \cos^{-1}x)$ are Chebyshev polynomials of the first kind. There are two methods of finding the coefficients $c_k$ that make this approximation good. Projection / Truncation method This uses the fact that Chebyshev polynomials are orthogonal. Let [A] = 1 if statement A is true and [A] = 0 otherwise: $$ \begin{align} \frac{\pi}{1 + [k \neq 0]} \, c_k &= \int_0^\pi f\left( \frac{b-a}{2} \cos\theta + \frac{b+a}{2} \right) \cos(k\theta) \, d\theta \\[1ex] &= \int_{-1}^1 f\left( \frac{b-a}{2}x + \frac{b+a}{2} \right) T_k(x) \, \frac{1}{\sqrt{1-x^2}} \, dx \\[1ex] &= \int_a^b f(x) \; T_k\left( \frac{2x-b-a}{b-a} \right) \frac{2}{\sqrt{(b-a)^2 - (2x-b-a)^2}} \, dx \end{align} $$ Interpolation / Collocation method The Chebyshev nodes in [–1, 1] are $x_k = \cos \displaystyle{\frac{k\pi}{n}}$ where k = 0, 1, ..., n. We can force the Chebyshev series to cut $f(x)$ at those nodes by solving the linear system $$ f\left( \frac{b-a}{2} \cos \left( \frac{i\pi}{n} \right) + \frac{b+a}{2} \right) = \sum_{j=0}^n c_j \, \cos \left( \frac{ij\pi}{n} \right) \qquad \forall \, i = 0, \dotsm, n $$ Which method is better? In terms of accuracy, speed or any relevant criteria. Which method is implemented in NumPy and Chebfun? numerical-analysis interpolation numpy approximation visitorvisitor $\begingroup$ Similar earlier question: scicomp.stackexchange.com/questions/23389 Also, your interpolation method doesn't require solving a linear system in the usual sense: it is, in fact, a discrete cosine transform. $\endgroup$ – Kirill Jan 8 '17 at 18:25 $\begingroup$ Which methods is more accurate depends on how you measure accuracy. Clearly, the $L_2$ project has the least $L_2$ error, but it may or may not have the smaller maximum norm, for example. $\endgroup$ – Wolfgang Bangerth Jan 9 '17 at 0:34 $\begingroup$ @WolfgangBangerth Since the series is truncated, the error can only be roughly as small as the truncation error, which is independent of the method. $\endgroup$ – Kirill Jan 9 '17 at 0:53 $\begingroup$ Well, maybe indeed the right thing is to increase $n$ -- but that wasn't part of the question :-) $\endgroup$ – Wolfgang Bangerth Jan 11 '17 at 2:46 $\begingroup$ @visitor: Why not just look at the source code for Numpy and Chebfun to find out? $\endgroup$ – Paul♦ Jan 13 '17 at 16:53 Please don't downvote this answer just because it's incomplete. My intention is to let whoever answering my question build on it, rather than write from scratch. If your answer is more comprehensive than mine, then I'll mark yours as the answer. Before answering your question, be aware of two kinds of Chebyshev nodes: Roots of $T_{n+1}(x)$ are $\displaystyle{x_k = \cos \left( \frac{k+1/2}{n+1}\pi \right)}$ where k = 0, 1, ..., n Extrema of $T_n(x)$ in [–1, 1] are $\displaystyle{x_k = \cos \left( \frac{k\pi}{n} \right)}$ where k = 0, 1, ..., n The former may be necessary when the endpoints are problematic, such as when integrating a function that goes to infinity. 1. Which method is better? The projection method can be made faster by using the fact that roots of $T_{n+1}(x)$ satisfy discrete orthogonality relations: $$ \sum_{k=0}^n T_r(x_k) T_s(x_k) = \left\{ \begin{array}{ll} 0 & \text{if } \, r \neq s \\ (n+1)/2 & \text{if } \, r = s \neq 0 \\ n+1 & \text{if } \, r = s = 0 \end{array} \right. $$ which can be proven by considering $$ \begin{align} S_n(\theta) &= \sum_{k=0}^n \cos (k+1/2) \theta = \frac{1}{2} \csc \frac{\theta}{2} \sin (n+1) \theta \\ S_n(0) &= n+1 \\ S_n \left( \frac{k\pi}{n+1} \right) &= 0 \qquad \qquad \forall \, k = \pm 1, \dotsm, \pm (2n+1) \\ S_n(\pm 2\pi) &= -(n+1) \end{align} $$ $$ \begin{align} \sum_{k=0}^n T_r(x_k) T_s(x_k) &= \sum_{k=0}^n \cos \left( \frac{k+1/2}{n+1} r\pi \right) \cos \left( \frac{k+1/2}{n+1} s\pi \right) \\ &= \frac{1}{2} \sum_{k=0}^n \left[ \cos \left( \frac{k+1/2}{n+1} (r+s) \pi \right) + \cos \left( \frac{k+1/2}{n+1} (r-s) \pi \right) \right] \\ &= \frac{1}{2} \left[ S_n \left( \frac{r+s}{n+1} \pi \right) + S_n \left( \frac{r-s}{n+1} \pi \right) \right] \end{align} $$ So to answer your question directly, computing the coefficients $c_r$ as follows will be faster than solving a linear system as in the interpolation method: $$ \frac{n+1}{1 + [r\neq 0]} \, c_r = \sum_{k=0}^n f\left( \frac{b-a}{2}x_k + \frac{b+a}{2} \right) T_r(x_k) $$ 2. Which method is implemented in NumPy and Chebfun? $\begingroup$ The bit about "computing $c_r$ will be faster" I think is wrong: both the formula for $c_r$ and the interpolation formula in the question are examples of discrete cosine transform (DCT), and are evaluated using FFT in practice (e.g., in ApproxFun.jl). Evaluating the sums for $c_r$ directly would be unnecessarily slow, $O(n^2)$ instead of $O(n\log n)$, as would solving the linear system with LU/GE in time $O(n^3)$. This is also why, if you compute $c_r$ the way you've described, the methods produce identical results. $\endgroup$ – Kirill Jan 14 '17 at 2:47 $\begingroup$ Also, I think the $c_r$ in this answer are not quite the same as the integrals in the question, you seem to be ignoring higher-order terms in the Chebyshev series for $f$, which makes the different $c$'s not exactly equal. I think $f=T_{n+1}$ is a counterexample. $\endgroup$ – Kirill Jan 14 '17 at 3:05 Not the answer you're looking for? Browse other questions tagged numerical-analysis interpolation numpy approximation or ask your own question. Interpolation with the roots of orthogonal polynomials & Spectral expansion How do I do Chebyshev interpolation in multi-dimentional space? What spline functions are used in Section 13.9 of "Numerical Recipes in C"? roots of polynomials with small coefficients Mathematical error when attempting to represent step function using fourier series Chebyshev and Legendre expansions Numerical evaluation of the Exponential Integral Ei by rational Chebyshev approximations fails Clenshaw-type recurrence for derivative of Chebyshev series	CommonCrawl
Journal of the American Mathematical Society Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics. The 2020 MCQ for Journal of the American Mathematical Society is 4.79. Journals Home eContent Search About JAMS Editorial Board Author and Submission Information Journal Policies Subscription Information A restriction estimate using polynomial partitioning by Larry Guth PDF J. Amer. Math. Soc. 29 (2016), 371-413 Request permission If $S$ is a smooth compact surface in $\mathbb {R}^3$ with strictly positive second fundamental form, and $E_S$ is the corresponding extension operator, then we prove that for all $p > 3.25$, $\\| E_S f\\|_{L^p(\mathbb {R}^3)} \le C(p,S) \\| f \\|_{L^\infty (S)}$. The proof uses polynomial partitioning arguments from incidence geometry. J. 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Math. 119 (1997), no. 5, 985–1026. MR 1473067, DOI 10.1353/ajm.1997.0034 R. Zhang, Polynomials with dense zero sets and discrete models of the Kakeya conjecture and the Furstenberg set problem, available at arXiv:1403.1352. Retrieve articles in Journal of the American Mathematical Society with MSC (2010): 42B20 Retrieve articles in all journals with MSC (2010): 42B20 Larry Guth Affiliation: Department of Mathematics, MIT, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139 MR Author ID: 786046 Email: [email protected] Received by editor(s): July 14, 2014 Received by editor(s) in revised form: January 23, 2015 Published electronically: May 11, 2015 Additional Notes: The author is supported by a Simons Investigator Award. Journal: J. Amer. Math. Soc. 29 (2016), 371-413 MSC (2010): Primary 42B20 DOI: https://doi.org/10.1090/jams827	CommonCrawl
Geometric interpretation of $\frac {\partial^2} {\partial x \partial y} f(x,y)$ Are there any geometric interpretation for the second partial derivative? i.e. $$f_{xy} = \frac {\partial^2 f} {\partial x \partial y}$$ In particular, I'm trying to understand the determinant from second partial derivative test for determining whether a critical point is a minima/maxima/saddle points: $$D(a, b) = f_{xx}(a,b) f_{yy}(a,b) - f_{xy}(a,b)^2$$ I have no trouble understanding $f_{xx}(x,y)$ and $f_{yy}(x,y)$ as the of measure of concavity/convexity of f in the direction of x and y axis. But what does $f_{xy}(x,y)$ means? multivariable-calculus hmakholm left over Monica Lie RyanLie Ryan $\begingroup$ there is a taylor theorem (polynomial approximation) for functions of several variables, here $\mathbb{R}^2\to\mathbb{R}$. the second derivative test looks at what kind of 2nd degree polynomial (in two variable) approximates the function. if it is a paraboloid or hyperboloid you can infer max/min/saddle properties but if it is flat in some direction it is inconclusive. $\endgroup$ – yoyo Mar 28 '11 at 18:23 $\begingroup$ $\frac{\partial}{\partial y}f$ tells you how $f$ is changing "in the $y$-direction", but that change will generally depend on the $x$-position (think of a grid tiling the plane; the partial tells you how things are changing in the vertical direction, but the change depends on which "column" you are in). If you think about it as the slope of the "tangent in the $y$-direction", then as you move the point in the $x$-direction this slope changes as well; the change in that slope is given by $\frac{\partial^2}{\partial x\partial y}$. $\endgroup$ – Arturo Magidin Mar 28 '11 at 18:30 $\begingroup$ Seems you're asking a subquestion of what you want--- not how to interpret mixed partials, but why the sign of $D(a,b)$ can give the nature of a saddle point. For this, do elementary analytic geometry on the graph of a function $Ax^2 + By^2 + Cxy$ at $(0,0)$ (add $Ex + Fy + G$, and at $(a,b)$, if you don't see how to reduce to this case). What conditions on $A,B,C$ do you get bowl that opens up, bowl that opens down, or saddle? Once you "get" this, you "get" all $f$, by the Taylor theorem. (Personally, I understand this via the algebra, not "geometric understanding" of $f_{xy}$.) $\endgroup$ – anon Mar 28 '11 at 21:14 $\begingroup$ @anon: yes, it's a subquestion of what I want to ultimately understand; I'm trying to understand why/how the determinant works from a geometric perspective, and to do that, I believe I need to understand the mixed derivative first. $\endgroup$ – Lie Ryan Mar 29 '11 at 10:24 $\begingroup$ math.harvard.edu/archive/21a_fall_08/exhibits/fxy/index.html $\endgroup$ – user26439 Mar 7 '12 at 2:45 The object that truly has geometric meaning is the Hessian, i.e. the matrix consisting of the second order partial derivatives: $$ H(x,y) := \begin{pmatrix} f_{xx} & f_{xy} \\ f_{xy} & f_{yy} \end{pmatrix}. $$ (In the following, I will denote the dot/scalar product by $\langle(u_1, u_2), (v_1, v_2)\rangle = u_1 v_1 + u_2 v_2$.) Write $\mathbf x = (x, y)$. Taylor's theorem says that the best second order approximation to the (smooth) function $f$ is given by $$ f(\mathbf x) = f(0) + \langle \nabla f(0), \mathbf x \rangle + \langle H(0) \mathbf x, \mathbf x\rangle + O( \\| \mathbf x \\|^3 ).$$ If you are at a critical point, the relevant term is the quadratic term $\langle H(0) \mathbf x, \mathbf x\rangle$. Level sets of a quadratic expression like this are conic sections. The determinant of $H$ (which is given by the formula you wrote above) allows you to determine what the level sets are, whether this quadratic function is positive definite, negative definite or indefinite. If you think of the graph of the function as a mountain range, the eigenvalues of $H$ tell you how spiky the mountain is, and the eigenvectors tell you the directions of steepest ascent/gentlest ascent (or descent, as the case may be). Maximilian Janisch Sam LisiSam Lisi Think of the mixed partial as an abstract tendency of the curve to twist like DNA, (but just because it's positive in absolute value doesn't mean the surface actually twists since other tendencies may overwhelm it), and the straight second derivatives $f_{xx}$ and $f_{yy}$ as an abstract tendency of the curve to bulge up or down in the $xz$- and $yz$- cross-sections. To tease out the individual roles of the partials, let's assume for the moment that the mixed partials are identically zero, the better to isolate their effect later. If $f_{xx}$ and $f_{yy}$ are of opposite signs, then the curve has a saddle tendency (think of two curves, one in the $yz$-plane and one in the $xz$-plane, intersecting at right angles, one opening up and the other opening down: this will produce a negative intrinsic curvature or saddle shape like a hyperbolic paraboloid). Now think of the same parabolas but both opening say down, with $f_{xx}$ and $f_{yy}$ tending to have the same signs: that will tend to produce an intrinsic positive curvature (bulging) like an ellipsoid. I'm sure you knew this. Now let's add in the mixed partials to show what their effect is. Think of the mixed partials as a pure twisting factor, also tending to produce a negative or saddle curvature, but rotated $45$ degrees! Yes, a twist is the same as a saddle tendency, but we think of it differently. Imagining your hand riding down along the wall of a saddle shape, like an airplane that dives while rotating, may help you see this. Now: ever seen a diagram of a saddle showing a kind of "X" shape over it, diagonal lines coming out of the saddle-point, where the tendency to go up in say $y$ is canceled by the tendency to go down in $x$, and the thing just holds steady along the diagonal line, with $f_{xx}$ and $f_{yy}$ both zero here? Well, if you had a properly rotated saddle shape it might not go up or down in the $x$- or $y$- directions at all--and yet it would be negatively curved or twisting nonetheless as would be plainly visible from the other directions. It is THIS twist that the mixed partials measure (just as an $xy$-term rotates a conic, by the way). If the curvature from the two ordinary second partials is negative, forget it--the mixed partials will make it even more negative. In other words, if $f_{xx}f_{yy}$ is negative because these differ in sign, the intrinsic curvature is already negative, and subtracting $f_{xy}^2$ will make it even more so. But if the curvature from the straight second partials is positive (bulging up or down), because $f_{xx}$ and $f_{yy}$ agree in sign, then possibly this positive tendency can still be overwhelmed by the independent negative-curvature twisting action of the mixed partials. That is why $f_{xy}$ is squared and subtracted: its sign doesn't matter to the saddle-ness and it is an inherently negatively-curved factor. The contest between these is the discriminant test you mention. Henry T. Horton Harvard ManHarvard Man $\begingroup$ WOW!this is BY FAR the best geometrical and intuitive explanation of this subject! $\endgroup$ – TheQuantumMan Aug 6 '15 at 20:06 $\begingroup$ This is a great answer! $\endgroup$ – Arrow Nov 9 '20 at 11:14 Try thinking about the Hessian as expressing the curvature of the surface at a particular point. Negative curvature corresponds to a saddle point, positive with a relative extrema, etc. If you want a real geometric interpretation you should just pick up a differential geometry book (Do Carmo's is a good one) and look up sectional curvature. I tend to think of the $f_{xy}$ as expressing how much the curves obtained by intersecting the graphs with planes parallel to the xz,y-planes deviate from having maximum acceleration with respect to curves tangent to the surface intersecting your point of interest. Also the determinant is independant of your choice of coordinates of the tangent space-hence coordinate free. Jeremy MannJeremy Mann Roughly, the mixed partial represents how fast (and in what direction) a tangent line "spins" as you "drag" the tangent point across a surface. At least this is how I think of it. Consider a surface such as $z = xy$, which is a fairly simple case. Its mixed partial is identically $1$, so the discriminant is identically $-1$ and the critical point at $(0, 0)$ is a saddle point (as expected). If you draw a tangent line at $(-1, 0)$ parallel to the y-z plane, you get the line $(-1, 0, 0) + t(0, 1, -1)$. Now drag this line toward the origin, and it spins around to meet the y axis, then farther until you reach $(1, 0, 0) + t(0, 1, 1)$. In fact this tangent coincides with the surface at all points, but in general this will not be the case; try $z = x^2 + 3xy + y^2 = (x+y)^2 + xy$, which has an extra confounding term but still has the same basic behavior (and the same saddle point since its discriminant is identically $-5$). Paul ZPaul Z Not the answer you're looking for? Browse other questions tagged multivariable-calculus or ask your own question. What can/do mixed second-order partial derivatives represent? Intuition explaining $\frac{\partial^2 f}{\partial x \partial y}$ geometric meaning of $\partial_{xy}f(x,y)?$ Geometric interpretation of mixed partial derivatives? Geometric intuition for mixed partial derivatives Intuition behind one second partial derivative test case Monotonicity of a function in two variables Partial Derivatives and Physics meaning Second partial derivative test question Finding Critical Points and Local Maxima/Minima or Saddle Point Characterizing the critical point of a two-variable function when the Hessian determinant is zero Interpretation of eigenvectors of Hessian in context of local min/max/saddle? Second derivative test failure? Why use Bordered Hessian than "simple" Hessian as second derivative test? Is $(0,0)$ a saddle point for the given function?	CommonCrawl
Saddle Phase Portrait Using index theory to rule out limit cycles. tion, bifurcation diagrams, maximum Lyapunov exponents diagrams, phase portraits and Poincaré maps. Show that there is a trajectory connection two saddle points. The stability analysis for this example is verified by the following direction field and phase portrait of the nonlinear system: The phase portrait confirms the presence of the two saddle points and fixed point attractor suggested by the direction field diagram. Anyone who has taken music lessons enough to know what the interval of a fifth is, knows there is no such thing as a minor fifth. We draw the vector field given at each point (x,y) by the vector f(x,y) g(x,y). What do the foxes say? Style 21 Disgustingly Hot Silver Foxes That'll Make You Fall In Love With Gray Hair. ) b) With the aid of computer if necessary, sketch the phase portrait for a < 0 and a > 0. This will create homoclinic orbits on each saddle. Women, Handbags at thebay. Introduction 1. portrait do, as in the simpler cases, change at their bifurcation point. Such phase portrait is called saddle. One problem with approaching a saddle point is that the initial condition, as well as the subsequent integration, is approximate. org Center manifold; Использование Saddle-node phase portrait with central manifold. The phase portrait thus has a distinct star-burst shape. Add and label nullclines and any real eigenspaces in all phase portraits. Hence, the phase portrait is that of the saddle. "Slowly getting back in the saddle, I had the honor to contribute a piece for Rumble. Now let us analyse the phase portrait of the system. Stability in case B cannot be decided with the information available from the picture alone. Phase plane analysis is a technique of the qualitative theory of dynamic systems. Why does this happen? Does it have any relevance to differential equations or other areas of math?. with negative real parts. In this context, the Cartesian plane where the phase portrait resides is called the phase plane. Find definitions, meanings, synonyms, pronunciations, translations, origin and examples. Shop online the latest FW19 collection of designer for Women on SSENSE and find the perfect clothing & accessories for you among a great selection. Mathematica provides us at least two options: either StreamPlot or VectorPlot (for the latter we give two versions without normalization and with it). examine Ww2 Harrogate Portrait Authentic audits and profound jump for more Ww2 Harrogate Portrait Authentic here from Ebay. 18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015 View the complete course: http://ocw. The global phase portraits of system (2. † Look at particular solution with Y(0) = (1;1). She was said to have be making a full recovery after suffering from a brain aneurysm two years ago. There are three fixed points: a stable node, a saddle point, and an unstable focus or node. The picture depicts a number of the identi ed folded and cusp-shaped saddle-node bifurcation. On this page I explain how to use Matlab to draw phase portraits for the the two linear systems. William Link was born on December 15, 1933 in Philadelphia, Pennsylvania, USA as William Theodore Link Jr. middle branch correspond to a linear saddle (with 1 <0 < 2). An Interactive Applet powered by Sage and MathJax. A fixed point of a linear system eigenvalues having different signs is called a saddle. I Examples include changes in number or stability of fixed points, closed orbits, or saddle connections. The two dimensional case is specially relevant, because it is simple enough to give us lots of information just by plotting it. Lab Report Math 302, Fall 2019 Junping Shi In this lab, we study the dynamical behavior of a one-parameter family of nonlinear, rst order systems consisting of predator-prey equations. Gates of Vienna News Feed 1/17/2013 Tonight's news feed is unusually fat, due to the inclusion of last night's items, which were never used because of the Blogger outage. A saddle-node bifurcation model of magnetic reconnection onset P. This page plots a system of differential equations of the form dx/dt = f(x,y), dy/dt = g(x,y). Sketch the graph of the potential function V(x. 03, Spring, 1999 It is convenient to represent the solutions to an autonomous system ~x0= f~(~x)(where ~x= x y ) by means of a phase portrait. We show by treating a concrete example how you can use Matlabto plot the phase portrait of a linear system in the plane. Orbits,phase portraits,and invariant sets appear before any differential equations,which are treated as one of the ways to define a dynamical system. The original non-linear system will also have a saddle point at the origin, oriented in. Driver's licenses for residents age 21 and older have also been redesigned. point taking the same valule. (There are a few other special cases that we did. We use cookies to provide and improve our services. Add and label nullclines and any real eigenspaces in all phase portraits. (a, b) (c, d) 9. Phase Plane Analysis - demo simulations. Draw the nullclines and equilibrium points for the following phase portrait. We are knowledge lovers and seekers. orbit collides with a saddle point. 4) by applying linear transformation x= Py(see Figure 1b). † Look at what happens as t ! 1 and as t ! ¡1. Phase portraits in two dimensions 18. 6 with that of Fig. Google has many special features to help you find exactly what you're looking for. Phase Portraits: A phase portrait is a plot of the phase plane showing multiple solutions to a given differential equation. portrait for (3. favorite this post Nov 10 Veggie Bullet Electric Spiralizer & Food Processor, Silver $85 (Saddle Brook) pic hide this posting restore restore this posting. • (d) The phase plane is divided into two parts by a curve that con-sists of the stable manifold of the saddle point — which includes the heteroclinic orbit from the spiral (0,0) to the saddle (1,1) — and the. Free for commercial use No attribution required High quality images. Associated with each pair of numbers is a pair of time-derivatives, (\dot x, \dot y), which depend on x and y. The eigenvalues are of opposite sign so the phase portrait is a saddle. Qualtitative analysis of the homogeneous ODE system. Introduction to the Phase Plane The following figure shows a typical phase portrait of this case. The latest Tweets and replies from Caring 4 You. Existence, uniqueness, consequences. Transitions from the oscillatory to the excitable states go via saddle-node infinite period bifurcations. Shop online adidas YEEZY adidas x Yeezy Powerphase £161 as well as new season, new arrivals daily. (b) Find the xed points in R2, and analyse the linearisation of the system at each of these xed points. Does not saddle businesses with a burdensome mandate. The sketch should show all special trajectories and a few generic trajectories. The eigenvalues are of opposite signs. A phase portrait shows the 2 stable states separated by a basin boundary on which the saddle lies (Fig. Since we are only interested in positive populations, we obtain the following phase-portrait for the linearized system. Whilst in the two dimensional case, additionally programs are available to plot nullclines and stable/unstable manifolds of saddle points. Using Matlab to get Phase Portraits Once upon a time if you wanted to use the computer to study continuous dynamical systems you had to learn a lot about numerical methods. For the phase-plane III, the origin is a sink. Therefore, if there are no saddle points the phase portrait is fairly simple. 3B provided that the slope of the u-nullcline is sufficiently positive. Case of a Saddle Point A = [1 3; 1 1] eig(A) A = 1 3 1 1 ans = 2. Include several phase curves. The rocky mole stands for strength and perseverance. Attractors and repellers for 1-di m vector fields. The global phase portraits of system (2. uk rigorous standards as "new", with all the benefits of Amazon. The phase portraits near these positive equilibria are studied. The theoretical principles of phase plane analysis were developed by H. STABILITY Phase portraits and local stability (0,0) is an unstable saddle for the linear approximation Phase portrait is therefore symmetric about x- and y. Enphase IQ Cable Portrait SINGLE - Wholesale Solar. By using our site, you consent to cookies. In this video lesson we will look at Phase Plane Portraits. ``same'' means that type and stability for the nonlinear problem are the same as for the corresponding linear problem. 18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015 View the complete course: http://ocw. We convey Ww2 Harrogate Portrait Authentic at discount costs. Poincaré–Bendixson Theorem. One problem with approaching a saddle point is that the initial condition, as well as the subsequent integration, is approximate. Below the window the name of the phase portrait is displayed, along with the matrix A and the eigenvalues of A. Phase Portraits and Equilibrium As discussed many times in the previous sections, there are three general solution types, Numerical, Analytic, and Qualitative. Hence, the two eigenvalues are negative. The x, y plane is called the phase y plane (because a point in it represents the state or phase of a system). The -plane is called phase plane. Phase portrait of the uncontrolled system We consider this dynamics under the application of a chemo-therapeutic agent. where E is a constant. Phase-portrait of Malthusian and Verhulst models, discussion of equilibria's role. The phase portrait varies depending on the signs of the eigenvalues. Enphase IQ Cable Portrait SINGLE - Wholesale Solar. He is a writer and producer, known for Columbo (1971), Rehearsal for Murder (1982) and Murder, She Wrote (1984). The family of open curves which approaches on the left of the singular straight. See the complete profile on LinkedIn and discover B. 3B provided that the slope of the u-nullcline is sufficiently positive. The x;yplane is called the phase plane (because a point in it represents the state or phase of a system). Using pplane5, draw the phase portrait of the saddle. edu/RES-18-009F. Moreover, Vasari describes the portrait of Mona Lisa as "unfinished" in 1568, that is, the drawing was never turned into a complete oil painting. Introduction 1. • In this context, the Cartesian plane where the phase portrait resides is called the phase plane. org Programowanie w systemie UNIX/c grafika. Polking of Rice University. Wayne dropped out, and that. Note that if the real parts of the eigenvalues of Awere positive, the phase portrait would look the same except that the orbits would spiral outward from the origin. in towards the origin as shown in the phase portrait below. And Joni Mitchell, 73, looked in good spirits as she attended Elton John's 70th birthday party. Saddle point 4. by horatia k. There are two homoclinic orbits that approach x = 0 as t !1 in which the left or right part of the. Shop online the latest FW19 collection of designer for Women on SSENSE and find the perfect clothing & accessories for you among a great selection. Phase Portrait Jiwen He, University of Houston Math 3331 Di↵erential Equations Summer, 2014 5 / 24 9. Portrait on the Phase Plane 4. If the eigenvalue are real you need to compute the eigenvectors and indicate them clearly on the phase portrait. portrait; the thin line depicts the perturbed trajectory. 2 MATH 134: PRACTICE FINAL SOLUTIONS The fixed points occur at (−1,0), (0,0, and (1,0) and they are a stable node, a saddle point, and a stable node respectively. First let us consider the under damped (0 < a < 1) case. Sims was honored for his contributions to the industry as he received the Saddle and Sirloin Portrait Award in 2010 during a special ceremony in conjunction with the North American International Livestock Exposition (NAILE) in Louisville, Ky. By a not-so-out-of-season pun, a post on both a saddle tree and its skirts could conceivably be called Tree Skirts. Equation Parameter Distribution of Poles Phase Portrait Singular Point Phase Trajectory Equation 23. 630 zenith phase 4 hardcover Who is the Photograph Of Per Morberg Portrait for? 1974 Press Photo Miss Skeenes Side Saddle Dfpb85735. Please try again later. Awesome prices of Bottle 1890s Cardui and comparable items. The biggest advantage of the Arca-Swiss quick release system is this ability to slide the plate without having to worry about mounting or dismounting anything. Simple Non-Linear Example. We use cookies to provide and improve our services. Know the "square root scaling law" for the period near a saddle node bifurcation. Phase portraits and eigenvectors. Dodge Long Port 1500 3in W System Ram Square Arh 1-34in 2006-2008 Cats For X 2006-2008 Dodge Square Ram Port System Long Cats X W For 1-34in 1500 3in Arh. An efficient methodfor solving any linear system of ordinary differential equations is presentedin Chapter 1. 2 Consider the system ˜x = x¡x2: a) Find and classify all equilibrium points. The real parts of the eigenvalues. 1 B (left). 24 Phase portrait for Example 2. The important feature of saddles is that there are special trajectories (the eigendirections) that limit on the origin in either forward or backward time. , sketch the phase portrait. I'm quite new to this so any help would be appreciated. For alpha=0, the critical point is a saddle. In this chapter, we consider methods for sketching graphs of the solutions. The state of a coupled system is represented by a pair of numbers, (x,y), which in turn can be represented as a point in a 2D plane that we call phase space. General case: If two eigenvalues of A are λ 1 < 0 and λ 2 > 0, with two corresponding eigenvectors vV 1 ,vV 2. From various parts, they made whips, saddle pads, glues, toys, drums, belts, stirrups, shields, knife cases, boats, thread, and of course - FOOD. The applications of such studies are applicable to virtually all Science or Phenomenon that we model using numerical data. Phase portrait for x' = -x, y' = -2y. For a much more sophisticated phase plane plotter, see the MATLAB plotter written by John C. Finally, if the determinant is positive, the eigenvalues (or their real part, if they are complex) is the same sign as the trace of A. Keywords Duffing Equation, Melnikov Method, Numerical Simulations 1. ``same'' means that type and stability for the nonlinear problem are the same as for the corresponding linear problem. The phase-portrait of the reduced HH model is shown in Fig. Phase portrait(a) is correct, but here's how the given trajectories can be extended. Find the fixed points. Phase Space Portrait of The Damped Simple Pendulum. (10) Find a solution of the DE, subjected to the initial condition: (x + 2y)dx + (2x + y) = 0 3) (15) Solve the linear differential equation with the IC y(0) - 2. 001 Mechanics 1 Phase portraits 1. It is convenient to rep resent the solutions of an autonomous system x˙ = f(x) (where x = x. Dynamical Systems. This phase portrait is impossible. Now the dynamics become clear. Locate on the diagram any bifurcations. The phase portrait for different values of p is shown in Figure 2. point taking the same valule. Glow is the debut album from Japanese Wallpaper about navigating the simultaneously beautiful and incredibly awkward phase that is one's late teens, growing up and trying to figure out your place in the world. (c) A few phase portraits are seen below. Asymptotic stability A fixed point is asymptotically stable if it is stable and nearby initial conditions tend to the fixed point in positive time. Teresa Maras Weishoff is on Facebook. In such a situation this is also called a (complex) phase portrait. , a class of van der Pol-Duffing oscillators. Phase Portraits of Nonlinear Systems Consider a , possibly nonlinear, autonomous system , (autonomous means that the independent variable , thought of as representing time, does not occur on the right sides of the equations). The vertical nullclines occur at x = −1, x = 0, and x = 1 and the horizontal nullcline occurs at y = 0. data related to themes of Angel Exhaust magazine. Glossary of Dynamical Systems Terms. Since 1856, Orvis has offered our customers distinctive clothing, the world's finest fly fishing rods and tackle, upland hunting gear, dog beds, luggage, and unique gifts. Our friend, saddlemaker Mike Brennan of Meeker, Colorado sent us this picture of these old saddle irons (also called "running irons. In this lesson, we will learn how to classify 2D systems of Differential Equations using a qualitative approach known as Phase Portraits. The critical point (0,0) is an improper nodal sink and so asymptotically stable. This Demonstration plots an extended phase portrait for a system of two first-order homogeneous coupled equations and shows the eigenvalues and eigenvectors for the resulting system. Please try again later. He is a writer and producer, known for Columbo (1971), Rehearsal for Murder (1982) and Murder, She Wrote (1984). PHASE PLANE PORTRAITS Phase Plane Portraits: plots in the phase plane for typical solutions to y0= Ay, for n= 2. Local codim 2 bifurcations cusp Bogdanov-Takens Bautin. For further details can be found in Grayling (2014) 0 so it is a centre. Lyapunov functions: know how and why they work, and how to derive them for simple examples. 4 Phase portrait. The solution is on the ray in the opposite direction. 18-009 Learn Differential Equations: Up Close with Gilbert Strang and Cleve Moler, Fall 2015 View the complete course: http://ocw. Our friend, saddlemaker Mike Brennan of Meeker, Colorado sent us this picture of these old saddle irons (also called "running irons. Read Details. 1 in textbook). The whole picture should be very symmetrical. ) from the eigenvalues and have found the $\infty$-isocline and $0$-isocline, how do I determine the direction of the arrows on the portrait?. 5: Homoclinic bifurcation For small parameter values, there is a saddle point at the origin and a limit cycle in the first quadrant. Maple Phase Portrait Modeling. A phase portrait is given on the next slide. Lonesome Dove was originally a much shorter screenplay. points, and they have the names saddle, center. Phase portrait of the uncontrolled system We consider this dynamics under the application of a chemo-therapeutic agent. Chaos Looking You in the Face. One problem with approaching a saddle point is that the initial condition, as well as the subsequent integration, is approximate. section (actual study of the phase portrait for the parameter values corresponding to the F-point is yet to be done for the R osler model). changes (for example, for <2 the portrait might be a saddle-point, for = 2 it might be something else, and for >2 it might be something else), and (c) draw a phase portrait for a value of less than the critical value, and draw a phase portrait for a value of greater than the critical value. 4 Phase portrait. Moreover, Vasari describes the portrait of Mona Lisa as "unfinished" in 1568, that is, the drawing was never turned into a complete oil painting. Ready to upgrade your camera support systems? Really Right Stuff provides the world's toughest, lightest, and most reliable camera support systems. 4 for the phase portrait. 03SC (Alternatively, make the change of variables x 1 = x − x 0, y 1 = y − y 0, and drop all terms having order higher than one; then A is the matrix of coefficients for the linear. The narrator says two musicians were playing in "minor fifths". ) Solutions tend to one of the equilibria on the unit circle. He competed in saddle bronc and showed promise. \) Its phase portrait is a geometric representation of the trajectories of a dynamical system in the phase plane. Thus phase portrait 3 could be correct. This bifurcation diagram yields 27 phase portraits for systems in QTS¯ counting phase portraits with and without limit cycles. For a one-dimensional autonomous ODE, it plots the phase portrait, i. 10 Draw a phase portrait that has exactly three closed orbits and one fixed point. But at a. Direction Fields. The rest of this paper is arranged as follows: In section 2, the nonlinear phase portrait model will be described. The phase portrait for different values of p is shown in Figure 2. Use Sage to graph the direction field for the system linear systems $d\mathbf x/dt = A \mathbf x$ in Exercise Group 3. It specialises in offering great deals on returned, warehouse-damaged, used, or refurbished products that are in good condition but do not meet Amazon. Let's go hang out, get to know each other, and make some gorgeous images. Join our Mailing List. For system (1), especially assuming that detA6= 0, this problem. 2 MATH 134: PRACTICE FINAL SOLUTIONS The fixed points occur at (−1,0), (0,0, and (1,0) and they are a stable node, a saddle point, and a stable node respectively. In our previous lessons we learned how to solve Systems of Linear Differential Equations, where we had to analyze Eigenvalues and Eigenvectors. Similar to a directionfield, a phase portrait is a graphical tool to visualize how the solutions of agiven system of differential equations would behave in the long run. On this page I explain how to use Matlab to draw phase portraits for the the two linear systems. However, phase portraits do not account for the effects of applied inputs;. A quick guide to sketching phase planes Our text discusses equilibrium points and analysis of the phase plane. The term 'saddle-node bifurcation' is most often used in reference to continuous dynamical systems. When quantitive aspects are needed, the Runge-Kutta method is used. In the one-dimensional case, a program is also available to plot the phase portrait. Whilst in the two-dimensional case, programs are additionally available to plot nullclines and stable/unstable manifolds of saddle points. If you don't see that email in your inbox shortly, fill out the. That said, an understanding of the meaning of. More important, good memories to link with them. From the phase portrait, and from what we know about the linear systems that approximate the non-linear system near the critical points, we can see that there are only three long-term possibilities for these two populations. Vector fields: motivation, definition Vector fields: why, where? A vector field arises in a situation where, for some reason, there is a direction and a magnitude assigned to each point of the space or of a surface, typically examples are fluid dynamics, wheather prediction, A classical example would be to represent. point taking the same valule. as in Figure ??. favorite this post Nov 10 Veggie Bullet Electric Spiralizer & Food Processor, Silver $85 (Saddle Brook) pic hide this posting restore restore this posting. The phase plane. Both saddle tree customization and tooled leather skirts are featured in this first post on the second Clyde Goehring saddle, numbered Timaru Star II #457. Be certain to display all of its characteristic features as in the example above. Phase portraits and eigenvectors. 03SC (Alternatively, make the change of variables x 1 = x − x 0, y 1 = y − y 0, and drop all terms having order higher than one; then A is the matrix of coefficients for the linear. Dynamical Systems. backwards time [Peixoto 1962]. Our most popular A4, A5, both Portrait and Landscape booklets. Handron Exam #3 Review 1. View PNG (It opens in a new tab, so you may need to allow popups). The fact that the phase portraits can be fully described in terms of the two one-variable real functions F and G allows, as well, a complete study of the bifurcation diagrams in complete families of vector ¯elds. This text discusses the qualitative properties of dynamical systems including both differential equations and maps. We work every day to bring you discounts on new products across our entire store. and I am asked to draw a phase portrait, once I have found the type of portrait (saddle point, node, spiral, etc. By using our site, you consent to cookies. This is a first. 3A, the middle one is a saddle point as in Fig. A direction field for a two-dimensional system of first-order ODEs, drawn in the phase plane for the system, is similar to the direction field for a single first-order ODE (see Lesson 1, Lesson 3, or Lesson 11). Click enough different initial values to get a good idea of what the phase portrait looks like. Sketch the phase plane portrait of a 2D system of first order differential equations. Being prepared for an upcoming storm is vital and can save a life. Phase Portrait 55/168 www. ) b) With the aid of computer if necessary, sketch the phase portrait for a < 0 and a > 0. For details on Poincar e-Lyapunov constants and weak foci we refer to [24], [18]. in this context as the phase plane. This might be of the form y x The arrows indicate the direction of increasing t. It stands for kindness and love. The week-long. Find definitions, meanings, synonyms, pronunciations, translations, origin and examples. 2, DynPac 10. And Joni Mitchell, 73, looked in good spirits as she attended Elton John's 70th birthday party. And as before if we find solutions, we draw the trajectories by plotting all points x(t),y(t) for a certain range of t. Next we discuss the phase portraits of linear saddles. It is convenient to rep resen⎩⎪t the solutions of an autonomous system x˙ = f(x) (where x = ) by means of a phase portrait. By hand, sketch typical solution curves in the regions in the xy - plane determined by the graphs of the equilibrium solutions, dy/dx = (y-s)^4. Solutions to assignment 5 1. For second order systems, it is convenient to do this graphically using the phase portrait, which is a set of phase trajectories in the coordinate plane. Now we can draw the phase portrait easily, noting that for large xthe curves essen- Find the phase. You've just been sent an email that contains a confirm link. Classify each critical point as asymptotically stable, unstable, or semi-stable. Note that the isocline for is the curve (shown in blue), which has positive slope and goes through the two equilibrium points. The parametric curves traced by the solutions are. But I received permission from my fantastic client to show you this painting before her husband even sees it on Christmas morning. com and find the best online deals on everything for your home. orbit collides with a saddle point. Closed-loop phase portraits demonstrate the potential for augmenting a vehicle's open-loop dynamics through steering and braking. the picture of the orbits of the system (which are level sets H(x,y)=constants), must be either like a center or like a saddle depending on the determinant being positive or, respectively, negative. 3 Distinct Eigenvalues Complex Eigenvalues Borderline Cases. By using our site, you consent to cookies. Miracle Wonderland In Alice 36 X 48 48 Alice Miracle With 36 Painting Oil On In X Wonderland Canvas Cats. He competed in saddle bronc and showed promise. Below are the possible behaviors for linear systems with nonzero eigenvalues, sketch a plausible phase portrait for each behavior. Convert a second-order differential equation to a system of two first-order equations. A graph of solution curves in the space of the dependent variables is a phase portrait for the system. Much of the functionality of P5 is inherited by P4: Functionality of P4/P5. Know how to find the period of a periodic orbit. 03 Recitation Problems 23 May 4, 2004 Linear Phase Portraits 1. By using our site, you consent to cookies. edu/RES-18-009F. 2D phase portraits 1. The theoretical principles of phase plane analysis were developed by H. 03, Spring, 1999 It is convenient to represent the solutions to an autonomous system ~x0= f~(~x)(where ~x= x y ) by means of a phase portrait. The eigenvalues are p 4 16 = 8i, so the angular frequency is != 8. They consist of a plot of typical trajectories in the state space. There are two homoclinic orbits that approach x = 0 as t !1 in which the left or right part of the. The phase space diagram is given by figure 4. 5 , the critical point is still a saddle, but when alpha =1 the critical point is a nodal sink. For the contour, which is positively oriented in the counter-clockwise direction, we find that I f (D) = 1 2π I J dθ f (x) = 1 (5) Inside the contour, there are 8 equilibria of which 4 are saddles. Dynamical Systems. Important skills: Know how to nondimensionalize systems involving parameters Know how to use nullclines and the principle of linearized stability to sketch the phase portrait of general, two{dimensional, autonomous systems. favorite this post Nov 10 Veggie Bullet Electric Spiralizer & Food Processor, Silver $85 (Saddle Brook) pic hide this posting restore restore this posting. He received a B. and I am asked to draw a phase portrait, once I have found the type of portrait (saddle point, node, spiral, etc. First let us consider the under damped (0 < a < 1) case. Distribute the values of the parameter a below among teams and have them record the answers on the board. 2-D Autonomous System. the system of equations, set the minimum and maximum axis limits for plotting the phase plane, andset parameters of the vector field. 2D phase portraits 1. Typically, we like to look at how position, velocity, acceleration, and time in high school physics. Limit cycles: ruling out their existence. Polking of Rice University. Next we discuss the phase portraits of linear saddles. 226 Hybrid equipment makes use of both technology and magic; though not necessarily in equal measure, both forces are crucial to the item's function.	CommonCrawl
Analysis of the lipid extraction performance in a cascade process for Scenedesmus almeriensis biorefinery I. Papachristou ORCID: orcid.org/0000-0003-1735-14081, S. Akaberi1, A. Silve1, E. Navarro-López2, R. Wüstner1, K. Leber1, N. Nazarova1, G. Müller1 & W. Frey1 Biotechnology for Biofuels volume 14, Article number: 20 (2021) Cite this article Microalgae have attracted considerable interest due to their ability to produce a wide range of valuable compounds. Pulsed Electric Fields (PEF) has been demonstrated to effectively disrupt the microalgae cells and facilitate intracellular extraction. To increase the commercial viability of microalgae, the entire biomass should be exploited with different products extracted and valorized according to the biorefinery scheme. However, demonstrations of multiple component extraction in series are very limited in literature. This study aimed to develop an effective lipid extraction protocol from wet Scenedesmus almeriensis after PEF-treatment with 1.5 MJ·kgDW−1. A cascade process, i.e., the valorization of several products in row, was tested with firstly the collection of the released carbohydrates in the water fraction, then protein enzymatic hydrolysis and finally lipid extraction. Biomass processed with high pressure homogenization (HPH) on parallel, served as benchmark. Lipid extraction with ethanol:hexane (1:0.41 vol/vol) offered the highest yields from the different protocols tested. PEF-treatment promoted extraction with almost 70% of total lipids extracted against 43% from untreated biomass. An incubation step after PEF-treatment, further improved the yields, up to 83% of total lipids. Increasing the solvent volume by factor 2 offered no improvement. In comparison, extraction with two other systems utilizing only ethanol at room temperature or elevated at 60 °C were ineffective with less than 30% of total lipids extracted. Regarding cascade extraction, carbohydrate release after PEF was detected albeit in low concentrations. PEF-treated samples displayed slightly better kinetics during the enzymatic protein hydrolysis compared to untreated or HPH-treated biomass. The yields from a subsequent lipid extraction were not affected after PEF but were significantly increased for untreated samples (66% of total lipids), while HPH displayed the lowest yields (~ 49% of total lipids). PEF-treatment successfully promoted lipid extraction from S. almeriensis but only in combination with a polar:neutral co-solvent (ethanol:hexane). After enzymatic protein hydrolysis in cascade processing; however, untreated biomass displayed equal lipid yields due to the disruptive effect of the proteolytic enzymes. Therefore, the positive impact of PEF in this scheme is limited on the improved reaction kinetics exhibited during the enzymatic hydrolysis step. Microalgae have traditionally been part of the diet of various cultures across the globe [1]. After the 1970 oil crisis, they were considered as a potential source for biodiesel production due to their high lipid content, which can reach up to 50% of dry weight [2]. Since then, microalgae have captivated research interest due to their flexible outputs. These microbial factories, depending on their cultivation conditions, can accumulate significant amounts of protein or lipids along with other high value compounds, such as carbohydrates, carotenoids etc. [3]. In relatively simple terms, when the microalgae are cultivated in nitrogen-replete conditions, they are producing proteins. However, upon entering into nitrogen-deplete conditions, they switch to lipid production (additional ways to boost lipid production do exist, however [4]). Microalgae lipids are composed mainly of triglycerides, which are three long chain fatty acids attached to a glycerol backbone, although other types of lipids can be encountered such as glycolipids or phospholipids [5]. They function either as energy storage or membrane structural components [6]. The length of the carbon chain along with the degree of saturation of the fatty acids directly affects their commercial application. Most microalgae produce saturated or monounsaturated fatty acids, which make an excellent source for biodiesel production [5]. However, certain species can generate significant amounts of polyunsaturated fatty acids (PUFAs) such as docosahexaenoic acid (DHA) and eicosapentaenoic acid (EPA) [7], which are poor feedstock for biodiesel [8] but are of high value for human nutrition and animal feed [9]. Proteins are large biomolecules composed of amino acids and are crucial for the metabolism's proper function. Microalgae are capable of producing all essential amino acids and in significant portions [10]. This, coupled with high biomass productivity rates and proteins with similar quality to conventional plant-derived ones [11], makes microalgae a potential answer to worldwide growing food demands. Moreover, protein-rich microalgae are considered as a nutrient feedstock by another growing sector, that of biofertilizers [12], due to their potential to enhance crop yields and increased sustainability [13]. Production of high value intracellular components is not enough though, since an efficient and economical extraction technique must also be developed for the commercial deployment. Microalgae proteins are typically extracted through physical or biochemical processes accompanied by recovery of the product through a separation method such as centrifugation and ultrafiltration [14]. Often, to improve the digestibility of the final product, the proteins are hydrolyzed chemically or enzymatically to free amino acids [15]. This strategy is especially effective for biofertilizer production, since it improves the plant's ability to absorb directly the amino acids by avoiding the protein hydrolysis process, which requires energy [16]. Lipids in contrast to proteins are not water soluble. Their extraction, therefore, requires the use of an organic solvent [17]. An extra challenge is added here since microalgae, unlike land plants, are producing both polar and neutral lipids which in turn requires the usage of a mixture of polar and neutral solvents [18] although successful extractions utilizing only polar solvents (alcohols) have been reported [19]. The traditional solvents used for analytics, chloroform and methanol, are unsuitable for beyond laboratory-scale applications due to their toxicity. Conventional solvents evaluated are ethanol, hexane, acetone and isopropanol among others, with the first two already accepted in the food processing industry [20]. Microalgae cells present a distinctive resistance to intracellular extraction, usually attributed to their rigid cell walls [21]. To overcome this barrier, a pre-treatment method is applied. There are various disruption techniques of different nature and approach (physical, mechanical, chemical) currently evaluated by several research groups as attested by various reviews [22,23,24]. In principle, the pre-treatment method should be energy efficient, applicable in industrial scales and not harmful to the target compounds. Pulsed Electric Fields (PEF) is a non-thermal technology, which guarantees mild and, therefore, non-damaging operating conditions and has been successfully demonstrated to facilitate extraction of various microalgae intracellular components [25,26,27]. Even if pilot-scale demonstrations of microalgae treatment are scarce, a number of proven PEF-related applications in the food industry support the applicability of this technology [28,29,30]. During PEF-treatment, repetitive high voltage pulses of short duration are applied to the microalgal biomass located between two electrodes. The resulting increase of the transmembrane potential leads to the reorganization of the cell membrane and its eventual permeabilization or 'electroporation' [31]. Despite the high potential of microalgae, their commercial exploitation remains limited and mostly focused on the production of high value commodities, such as cosmetics or food supplements [32]. While currently being the only profitable options, these applications serve a niche market, easily saturated [33]. A single output process is also gradually phased out in favor of a biorefinery approach. This concept, much like the conventional crude oil refinery, aims towards the complete utilization of the biomass through selective and cascade extraction of different components. A typical strategy would involve the disruption of the microalgae cells, preferably on wet basis to minimize drying costs, followed by the extraction of the water soluble- fraction. Once the aqueous phase is removed, the introduction of an appropriate organic solvent initiates the lipid extraction. The spent biomass can then be further exploited either through gasification for production of energy or directly for animal feed. The majority of already conducted studies, focused on a single product extraction although some work has been reported on cascade processes as well. Imbimbo et al. performed cascade extraction on the red microalgae Galdieria phlegrea with French press as pretreatment method [34]. Proteins along phycocyanin were recovered in the supernatant, whereas carotenoids and lipids were afterwards extracted utilizing pressurized liquid extraction and supercritical fluid extraction, respectively. Francavilla et al. performed lipid extraction from freeze-dried Dunaliella tertiolecta with chloroform:methanol followed by fast pyrolysis of the residue. They reported that the resulting bio-oil was in need of upgrading for fuel usage but the produced char could have potential as biofertilizer [35]. Lupatini et al. studied the extraction of proteins and carbohydrates after prior defatting Arthrospira platensis with soxhlet extraction [36]. Ansari et al. tested different cascade extraction pathways of different components from dried Scenedesmus obliquus concluding that the protein-lipid-carbohydrate route resulted in the optimum recovery of each individual fraction [37]. Interestingly, the authors observed product loss after each step. Another cascade process for S. obliquus, was developed by Gilbert-López [38]. In this work, biomass was subjected to High Pressure Homogenization (HPH), followed by freeze-drying. Firstly, triglycerides were extracted through supercritical CO2, then various carotenoids were removed with gas expanded liquids and finally pressurized liquid extraction with water was performed for proteins and sugars. In a recent study, Zhang et al. performed a two-stage aqueous and organic extraction from Nannochloropsis oculata using High Voltage Electrical Discharges (HVED) and Vacuum Drying (VD) [39]. It was reported that HVED combined with pre-washing had a positive impact on the water-soluble extraction. Moreover, the VD of the spent biomass was accelerated and the subsequent organic solvent of lipids and pigments was improved. The same group also compared HVED and HPH in a multi-step extraction from Phaeodactylum tricornutum [40]. HPH proved to be more effective in water-soluble extraction. However, HVED allowed for a more selective aqueous extract and for improved subsequent non-aqueous extraction with chloroform/methanol. Easy separability of the different fractions is important for reduction of contaminations. PEF offers the advantage of no debris generation [41] since the cells retain their original shape after treatment. Therefore, the integration of PEF in a cascade scheme is a prospect worth examining. A few studies regarding the extraction of multiple products from microalgae with PEF (usually, proteins and carbohydrates) do exist [42, 43] although in these cases, the compounds are extracted together and not in a true cascade format. Guo et al. also evaluated the valorization of residual biomass through hydrothermal liquefaction after the extraction of one product (lipids, proteins or amino acids) [44]. Αn example of cascade extraction of different components from microalgae utilizing PEF, was a previous work of our group [45], where carbohydrate and lipid extractions were performed in two different steps after PEF-treatment of wet Auxenochlorella protothecoides. In a recent study from our group, enzymatic hydrolysis of proteins from wet Scenedesmus almeriensis (S. almeriensis) was performed after PEF-treatment and compared to HPH treatment as a benchmark. In both cases, the disrupted biomass displayed similar hydrolysis yields and kinetics [15]. S. almeriensis is a good candidate for commercial applications since it exhibits significant protein content (50–55% of dry weight) with satisfactory growth rates in a broad range of environmental conditions [46] and is already studied in medium scale production [47]. This microalga has been mainly studied for lutein production and extraction [48, 49]. Bauer et al. performed lipid extraction from S. almeriensis among other microalgae with liquefied dimethyl ether although the authors utilized freeze-dried biomass in their study. A proposed scheme for the serial valorization of several components of S. almeriensis is shown in Fig. 1. In this cascade process, the initial step consists of submitting the wet, concentrated biomass to PEF-treatment and evaluating the spontaneous release of intracellular carbohydrates in the water fraction Enzymatic hydrolysis was then performed to cleave and release the intracellular proteins in the form of amino acids in the surrounding aqueous medium. Finally, after removing the water fraction, lipid extraction with organic solvents was performed on the rest biomass. Cascade extraction scheme utilizing Pulsed Electric Fields The goal of this work was to study more specifically the lipid extraction segment of the process described above. Lipid extraction of freshly harvested, wet S. almeriensis after PEF-treatment was performed evaluating three different extraction solvents, pure ethanol at room temperature, an ethanol:hexane blend (1:0.41 vol/vol) and pure ethanol at elevated temperature (60 °C). Given the dynamic nature of the PEF effect on cells [45], experiments were conducted immediately after PEF-treatment and after a 24 h incubation step in inert conditions. Once the lipid extraction methodology was established, cascade extraction after 3 h enzymatic hydrolysis was carried out. Thus, a cascade extraction of independent products from wet microalgae utilizing a combination of PEF, enzymatic hydrolysis and lipid extraction could be demonstrated and evaluated. S. almeriensis composition All S. almeriensis cultivations were conducted in photobioreactors (PBR) for 5 days. The protein, lipid, carbohydrate and ash content of the biomass was determined for each harvest. Total protein content was evaluated through a modified Lowry method (DC™, Protein Assay, BioRad) [50], total carbohydrates with sulfuric hydrolysis with Anthrone reagent [45], total lipid content with chloroform:methanol (2:1 vol/vol) in a modified Kochert protocol [51] while inorganics by overnight ashing in a high temperature furnace. Results are presented in Table 1 and indicate that more than half of the biomass, i.e., 55.9% consisted of proteins which made it very suitable for enzymatic hydrolysis. Table 1 Biomass composition of S. almeriensis Each component is displayed in percentage of dry weight. Values are the average ± std of three independent cultivations. Fatty acid content of S. almeriensis The fatty acid content of the produced biomass was evaluated through gas chromatography after direct transesterification of lyophilized biomass that was priorly molturated with alumine. Results from three independent cultivations are presented in Table 2 and show that S. almeriensis is capable of producing noteworthy amounts of polyunsaturated fatty acids. Table 2 Fatty acid content of S. almeriensis Pulsed electric fields treatment of S. almeriensis An indirect but rapid way to determine the efficiency of PEF-treatment on any microalgae suspension is through the conductivity increase measurement due to the release of ions and small charged molecules. After PEF-treatment with 1.5 MJ·kgDW−1, the conductivity values of the suspension increased by a factor 2, from 1.02 ± 0.1 mS cm−1 for untreated biomass, to 2.4 ± 0.1 mS·cm−1 for PEF-treated, both values normalized to 20 °C according to Eq. (1). The temperature rise of the suspension due to the Joule effect was equal to 282 K after treatment, with maximum recorded temperature being 33 °C. The conductivity increase after PEF-treatment was in agreement with previous experience performed on this microalgae [15] and confirmed the efficiency of the PEF-treatment. Lipid extraction from S. almeriensis after PEF-treatment Three different systems were compared for lipid extraction from wet S. almeriensis, namely 24 h extraction with ethanol:hexane (1:0.41 vol/vol), 24 h extraction with pure ethanol and 0.5 h extraction with pure ethanol at 60 °C. As mentioned in Sect. 5.4., the protocols were adapted from literature and were proven effective for lipid extraction from other microalgae strains. The effect of solvent volume was additionally examined by increasing it by a factor 2. For the first two extraction systems, experiments were also conducted on biomass that was incubated for 24 h after PEF-treatment. In Fig. 2, the results from two independent cultivations are presented along with standard deviation. Comparison of lipid extraction from wet S. almeriensis after PEF-treatment using three different extraction systems, 24 h extraction with ethanol:hexane at ratios 1:0.41 vol/vol (a), 24 h extraction with pure ethanol (b) and 0.5 h extraction with ethanol at 60 °C (c). In the same graph, the effect of increasing the solvent volume by a factor 2 along with the effect of incubating the biomass for 24 h after PEF-treatment, are shown for systems A and B. In blocks, the average values of lipid yields from two independent experiments are presented along with the standard deviation as error bars. The straight grey line indicates the total lipid content as evaluated from extraction with the reference method, chloroform:methanol 2:1 vol/vol As it can be seen for system A, direct extraction from untreated biomass offered 10% yields on dry weight, which would correspond to 43% of the total lipid content as estimated by the reference method. PEF-treatment with 1.5 MJ·kgDW−1 followed by immediate lipid extraction offered an increase of the lipid yields, up to 16.3% in dry weight or 70% of the total lipids. PEF-treatment thus increased the yields by almost 60% compared to control biomass. Increasing the solvent amount by a factor 2 did not increase the yields from untreated biomass and offered a slight increase of 3% dry weight for PEF-treatment. Incubating the biomass for 24 h after PEF-treatment, results in a very similar yield increase by 3% dry weight, with Control samples remaining unaffected. For pure ethanolic extraction (system B), lipid yields were relatively low for all conditions tested. More specifically, untreated biomass never exceeded 5% of lipid yields in dry biomass. PEF-treatment offered similar yields with incubation after PEF-treatment, offering a borderline increase by 3%. Interestingly enough, both solvent volumes offered almost identical results. System C, ethanolic extraction at 60 °C painted a similar picture with the previously discussed system. Only 7% lipid yields were achieved with PEF-treatment, less than 30% of the overall lipid content. Solvent volume did not affect the yields. For this experiment, no incubated biomass was tested. Even though yields never reached the ones obtained from freeze-dried and bead milled biomass, the extraction system utilizing ethanol:hexane was selected to be applied after the enzymatic hydrolysis since based on the results from Fig. 2, it offered the highest yields. Cascade processing of S. almeriensis The selected lipid extraction protocol was then tested within the context of a biorefinery scheme. Biomass treated with PEF at 1.5 MJ·kgDW−1 without any incubation was tested along with untreated microalgae. HPH served as benchmark for comparison. Water fraction The potential valorization of the water fraction was examined by evaluating the intracellular release of carbohydrates after PEF-treatment as observed on a previous study on A. protothecoides [45]. The microalgae suspension was centrifuged immediately after, 2 h or 24 h after PEF-treatment. The spontaneous carbohydrate release in the supernatant over these time points was determined with the Anthrone method as descripted in session 5.2.3. The results from two independent cultivations are shown in Fig. 3. As it can be seen, the carbohydrate concentration in the supernatant of untreated samples remained stable at 0.2 g L−1. Treatment with PEF offered a slight release immediately to ~ 0.3 g L−1 which almost doubled after 2 h incubation and reached up to ~ 0.8 g L−1 after 24 h. Carbohydrate release from S. almeriensis in the supernatant after PEF- treatment, with no incubation, 2 h incubation and 24 h incubation. Results of two independent cultivations are displayed in average with the error bars indicating the standard deviation. On left y-axis, results in g carbohydrates per L while on the right y-axis mg carbohydrates per g dry weight Enzymatic hydrolysis of wet S. almeriensis took place for 3 h at 50 °C, at controlled temperature and pH. As benchmark for comparison to PEF, HPH was also tested. At the end of the reaction, the samples were centrifuged to separate the aqueous phase along with the free amino acids and the degree of hydrolysis was determined. The kinetics of the enzymatic hydrolysis over 3 h are presented in Fig. 4. Kinetics of the enzymatic hydrolysis of wet S. almeriensis. Biomass was either untreated (Control), fed into high pressure homogenizer (HPH) or treated with pulsed electric fields, using 3% enzymes (vol/w). The average of three independent cultivations are presented with standard deviations as error bars At 0 h right before the addition of the enzymes, HPH-treatment released higher amounts of free amino acids with a degree of hydrolysis equal to ~ 7% against PEF-treatment, which released 1.7% into the suspension. It has to be mentioned that in untreated biomass no amino acids have been detected which confirms that the washing step had no effect on the cells. After 1 h of hydrolysis, PEF-treated samples displayed a similar degree of hydrolysis as HPH at 44% and 40%, respectively. Untreated samples were still lagging behind with a degree of hydrolysis of 30%. After 2 h of reaction, the hydrolysis was slowly reaching equilibrium. At this time point, PEF-, HPH- and untreated samples had a degree of hydrolysis of 53%, 49% and 46%, respectively. With a prolongation of the reaction, up to 3 h, the degree of hydrolysis for PEF-treated biomass increased up to 57%, Control reached 51%, whereas HPH samples remained unaffected. Lipid extraction after enzymatic hydrolysis The residual biomass was then subjected to lipid extraction to evaluate the feasibility of cascade processing. The results from three independent cultivations are shown in Fig. 5. Lipid extraction from wet S. almeriensis following enzymatic hydrolysis within a cascade processing. Biomass was either untreated, fed into high pressure homogenizer or PEF-treated. Lipid extraction was performed either directly after pre-treatment or after enzymatic hydrolysis. In blocks, the average values of lipid yields from three independent experiments are presented with standard deviations as error bars. The straight grey line displays the total lipid content as evaluated from chloroform:methanol (2:1 vol/vol) Lipid extraction from fresh S. almeriensis without enzymatic hydrolysis behaved in a similar manner with the results described before in Fig. 2. Untreated biomass had on average 10% lipid yields on dry weight, whereas PEF-treated samples exhibited increased yields, up to 17%. HPH was also highly efficient, displaying similar yields, equal to 17% dry weight. When extraction was performed in cascade after enzymatic hydrolysis, the lipid yields of PEF-treated samples were not affected. However, a sharp increase in the yields of the untreated sample before the enzymatic hydrolysis was observed. Indeed, as shown in Fig. 5, the yields had risen up to ~ 17% of biomass dry weight, i.e., very similar to samples initially pre-treated with PEF. In contrast, if the biomass is treated with HPH before the enzymatic hydrolysis, the lipid yields after extraction of the rest biomass are lower compared to previous conditions, namely a little higher than 10% of dry weight. The produced biomass was rich in protein content (more than 50% of dry weight), with lipids being the second higher component with approximately 24% of dry weight. Only 12% of carbohydrates were produced, while the non-volatile inorganics made up 5–6%. The overall mass balance was close to 100% to a satisfactory degree (98.4%). The composition was slightly different compared to S. almeriensis as reported by Romero García et al. with values 41.8%, 11.2%, 38.7% and 8.3% for proteins, lipids, carbohydrates and ashes, respectively [52]. In that study, however, the biomass was cultivated in continuous mode with daily harvests and in Mann and Myers medium instead of Arnon medium used in the current study, which could explain the differences. Approximately 40% of the fatty acids of the produced biomass were saponifiable, an important parameter regarding their potential for biodiesel. Moreover, the lipids were rich in polyunsaturated fatty acids, namely alpha-Linolenic acid C18:3n3 (30% of total fatty acids) and linoleic acid C18:2n6 (15.8% of total fatty acids). This might make such lipids unsuitable for biodiesel production since polyunsaturated fatty acids are considered bad feedstock [53, 54], but are highly valued in the nutrient sector. As reported by Ruiz et al., microalgae for food has three times higher potential value compared to other biomass usages [33]. Kumar et al. also state that lipid extraction from even microalgae strains with lipid content as little as 10% will be feasible given a large enough production [55]. Therefore, the composition of S. almeriensis, reported in this work with high protein and modest polyunsaturated lipid amounts, was deemed appropriate for the cascade process examined here. As shown in Fig. 2, PEF-treatment promoted lipid extraction but only in combination with the appropriate solvent. The ethanol:hexane blend (1:0.41 vol/vol) that was successfully applied for lipid extraction from A. protothecoides in a previous work [56], offered the highest yields in this study. On the other hand, utilizing only ethanol, the yields were significantly lower, both for untreated and PEF-treated samples. These results are in agreement with a study from Sean Lai et al. who performed lipid extraction using chloroform:methanol, pure ethanol and pure hexane after subjecting Scenedesmus spp. in a double pass through PEF with an overnight storage and subsequent freeze-drying. The authors reported that ethanol offered the lowest yields from the systems examined, while the co-solvent approach was favored by PEF-treatment [57]. Pure ethanolic extraction is promoted as a viable alternative for lipid extraction as seen in Yang et al. who achieved high lipid yields from the wet microalga Picochlorum sp [19]. Navarro López et al. using only ethanol were able to successfully extract lipids from wet Nannochloropsis gaditana [58], whereas in this study using a similar methodology (extraction system C) only 30% of the total lipids were extracted. Yao et al. also reported an effective usage of isopropanol as lipid solvent from Nannochloropsis sp. at 80 °C [59]. The same authors state that elevated temperatures favor extractions with alcohol. In this study, a comparison of systems B and C, i.e., ethanol at room temperature versus 60 °C, resulted in slightly increased lipid yields for the latter. Elevated temperatures generally favor the extraction of thermally stable products, which could explain this difference [60] although this might lead to damage of the unsaponifiable fraction. From the above it can be concluded, that PEF can serve as a pre-treatment method from S. almeriensis, however, a mixture of polar and neutral solvents, in relatively large and potentially unsustainable volumes, is necessary for effective lipid extraction, in this case ethanol:hexane. Whether the necessity to use a co-solvent is due to the microalgae structure, e.g., the cell wall composition, or whether it is imposed by the lipid type, was not investigated in this study. Regarding the long-term effect of PEF-treatment, the lipid yield was higher when microalgae were incubated for 24 h before submitted to lipid extraction. Indeed, 83% of total lipids were extracted with 24 h incubation compared to 70% without incubation. A similar tendency was observed in previous experiments with A. protothecoides [45]. It is peculiar though that independently of the state of the biomass (fresh or 24 h incubated), 100% total lipid extraction was never achieved compared to the reference method. A significant increase of the solvent volume by a factor 2 had practically no effect on the lipid yields, indicating that the solvent volume was not the limiting factor. The reference lipid extraction was performed on freeze-dried biomass after bead milling which completely shatters the microalgae. As shown in Fig. 5, the lipid yields of HPH-treatment samples were very similar with PEF-treated ones. This fact implies that this inability to reach total lipid extraction was not because of the pre-treatment itself but instead due to the limitation of the solvation ability of ethanol:hexane, potentially further reduced by the presence of water. A second possible explanation would be a slight overestimation of lipid yields with the reference method. In a chloroform:methanol extraction process, once the system is turned biphasic, the lower phase containing the lipids is separated from the upper phase by pipetting. It has to be considered that this pipetting step carries an increased risk of contaminations from the upper phase [61]. In ethanol:hexane systems though, the hexane along the lipids form the upper phase, resulting in a more clean separation without any contaminants in the gravimetric determination. Regarding the spontaneous release of other compounds into the water fraction after PEF-treatment during the first step of a cascade process, only small amounts of carbohydrates were detected in the supernatant, up to ~ 0.8 g·L−1. For comparison, in a previous study with A. protothecoides up to 8.0 g·L−1 of carbohydrate were released in the supernatant after PEF treatment of a 100 g·L−1 suspension [45]. One explanation for this different behavior could be due to the very different composition of the two microalgae. Indeed, S. Almeriensis had a relatively low total carbohydrate content, i.e., ~ 12.5% dry biomass, while A. protothecoides ranged between 20 and 30% dry biomass (unpublished observations). Another possible scenario could be that S. Almeriensis responds less efficiently to the PEF-treatment regarding the release of intracellular soluble molecules. The cell wall of Scenedesmus strains contains an additional pectin layer compared to Chlorella species [62] so it is possible that this additional barrier hampers any intracellular component diffusion. Nonetheless, the fact that only little amount of carbohydrates were detected, does not render the water fraction necessarily without value. The biostimulant activity of the supernatant after PEF-treatment could be evaluated in a similar study like Navarro-López et al. [63]. Concerning the enzymatic hydrolysis, the results were in agreement with data previously reported from our group with samples that underwent HPH exhibiting slightly lower yields [15]. This discrepancy was attributed to the fact that HPH treatment in the previous work was performed at concentrations of 50–80 g·L−1, whereas in this study it was conducted at 100 g·L−1 which resulted in a reduced efficiency using our apparatus. PEF-treated samples though were not affected by this increase of concentration. The lipid extraction that was performed after enzymatic hydrolysis gave interesting insights on the possibility of combining the two processes in a cascade. From Fig. 5, it can be derived that for PEF-treated samples, the enzymatic hydrolysis has a minimum impact on the lipid content and the lipid yields. However, while PEF-treatment was beneficial for lipid extraction, it appears that at the end of the enzymatic hydrolysis reaction, the untreated biomass displayed similar lipid yields as the PEF-treated one. It is apparent, that enzymatic hydrolysis acted as a pre-treatment with the enzymes damaging the cell wall [64], resulting in an increased subsequent lipid extraction. The HPH-treated lipid yields after enzymatic hydrolysis were less compared to both untreated and PEF-treated samples (11% versus ~ 16% dry weight). The destruction of the cells and the resulting emulsification of the various cell components could have led in lipid losses during the removal of the aqueous phase at the end of the enzymatic hydrolysis something, which might explain this observation. While the above could be interpreted as proof that PEF is applicable in a cascade process, the increased lipid yields displayed by untreated samples after enzymatic hydrolysis, render the effect of PEF-treatment moot as far as the lipid extraction is concerned. Any potential benefits from PEF-treatment in this scheme, therefore, have to be detected in the possible valorization of the water fraction or in the improved kinetics displayed during the enzymatic hydrolysis. In this study, three different lipid extraction systems were carried out on wet S. almeriensis biomass that was pre-treated with PEF at 1.5 MJ·kgDW−1. Among the three extraction systems (ethanol:hexane, pure ethanol at room temperature and pure ethanol at 60 °C) that were tested, ethanol:hexane clearly displayed the best performance, with 70% of total lipid content extracted (increased up to 82% of total lipids if a 24 h incubation step is introduced after PEF-treatment) with a clearly positive effect of PEF observed. The utilization of PEF in a biorefinery processing of S. almeriensis composed of a water fraction extraction, an enzymatic hydrolysis and a lipid extraction was examined and compared to HPH as benchmark. Very little amounts of spontaneously released carbohydrates were detected in the water fraction after PEF-treatment. During the enzymatic hydrolysis, PEF and HPH accelerated the reaction kinetics in an equal manner. In the subsequent lipid extraction, however, PEF-treated samples retained their high lipid yields in contrast to HPH-treated biomass, which displayed diminished results. The most likely explanation for this observation is the complete cell fractionation after HPH treatment and the lipid losses in the formed agglomerates. However, completely untreated samples displayed equal lipid yields with PEF after hydrolysis, limiting thus any positive impact of PEF to the eventual valorization of the water fraction or to the improved reaction kinetics exhibited during the enzymatic hydrolysis step. Cultivation and harvest of biomass The cultivation conditions of S. almeriensis were identical to the description provided in [15]. In brief, the biomass was cultivated in Arnon medium in a 25 L bubble column annular bioreactor illuminated 24 h at 250 μ·mol−2·s−1 with temperature maintenance at 25 °C. Supply of air and CO2 was provided at a rate of 5000 cm3·min−1 and 25 cm3·min−1, respectively. The pH was fixed at 8. The cultivation lasted for 5 days. Harvest was carried out using a separator (STC 3–06-170, GEA Westphalia, Germany). The resulting biomass paste was re-suspended with deionized water with a twofold purpose. First, to adjust the biomass concentration at ~ 100 g·L−1, i.e., as high as possible to reduce the energy input of the PEF-treatment. It has to be considered that the biomass needs to be still liquid enough to be pumped through the PEF treatment chamber. Second, to reduce the conductivity of the microalgae suspension from the initial 4.2 mS·cm−1 down to 1–1.2 mS·cm−1. The obtained conductivity corresponds to the design parameters of the treatment chamber for matched conditions and, therefore, ensures square electric pulses. As shown in the previous study, S. almeriensis is resistant to any osmotic shock resulting from this washing step [15]. The exact final concentration was determined by overnight drying of known amounts of the final suspension and supernatant in a drying oven (Universalshrank model U, Memmert, Germany) [40]. After each harvest, part of the biomass would be freeze-dried and stored in vacuum-sealed bags at -20 °C for composition determination of the biomass. Freeze-drying was conducted in a laboratory freeze-drier (Alpha 1–4 LDplus, Christ) for at least 24 h and stored afterwards in vacuum-sealed bags. Biomass composition characterization After each harvest, the composition of S. almeriensis was determined and more specifically, the total protein, carbohydrate, lipid and inorganic (ashes) content. Protein determination took place in fresh microalgae, while all the other analyses were performed on freeze-dried biomass. Total protein determination To determine total protein content, a chemical extraction was performed using sodium hydroxide. From concentrated suspension, a volume that contained 5 mg of microalgae biomass was resuspended in 2 mL sodium hydroxide (1 M), followed by 1 h of incubation at 95 °C. After this step and upon reaching room temperature, the sample was centrifuged at 10,000×g for 10 min and the supernatant was processed for protein determination with a modified Lowry method (DC™ Protein Assay, BioRAd) using bovine serum albumin as standard [50]. Total lipid determination Chloroform:methanol extraction was performed on freeze-dried S. almeriensis utilizing a modified Kochert protocol to determine the total lipid content [51]. Freeze-dried biomass was bead-milled at 30 Hz, 5 times for 15 s (Mixer mill, MM400, Retsch, Haan, Germany) and approximately 100 mg were recovered and measured in a precision balance. 2 mL of chloroform:methanol (2:1 vol/vol) were mixed with the biomass, vortexed and immediately centrifuged at 1800×g for 4 min. After the centrifugation, the supernatant was removed and collected into a separate glass tube. 2 mL of fresh solvent were added in the biomass and the above process was repeated. Overall, 7 mL of solvent were used, in four separate extraction steps (3 × 2 mL and 1 × 1 mL for the last step). In the glass tube with the collected solvent, 3 mL of HCl 0.1 N and 0.3 mL MgCl2 0.5% were added to facilitate phase separation. The lipid-containing lower phase was removed with a Pasteur pipette into pre-weighted glass tubes and evaporated under N2. The lipid yield was determined gravimetrically. All samples were performed in duplicates. Total carbohydrate determination The determination of carbohydrate release was conducted using the Anthrone sulfuric acid assay [45]. Freeze-dried biomass was resuspended in deionized water in concentrations ~ 0.1–0.2 g·L−1. On parallel, fresh starch aqueous solutions with concentrations ranging from 0.02 g·L−1 to 0.4 g·L− 1 were prepared from starch powder (Merck 1.01257) to be used as standards and processed in a similar manner with the samples. The anthrone reagent was prepared on the day of the experiment by dissolving anthrone (Merk 1.01468) in 95% sulfuric acid (AnalaR NORMAPUR: VWR Chemicals 20,700) at a final concentration of 0.1% w/v. 400 μL of diluted sample or standard along 800 μL of anthrone reagent were mixed in 1.5 mL Eppendorf Safe Lock tube. After 5 min incubation in ice, the sample was placed into a thermo-incubator pre-heated at 95 °C and shaken at 300 rpm for 16 min followed by cooling down on ice for again 5 min. Optical density of the cooled samples was measured at 625 nm in a spectrophotometer (Genesys 10S UV–Vis, Thermo Scientific) and carbohydrate concentration was calculated using the standard curve and considering the dilution factors. All measurements were performed in duplicate. Inorganic components measurement (ashes) Approximately 200 mg of freeze-dried biomass were measured in a precision balance in alumina crucibles and placed in a high temperature furnace (Hochtemperaturofen Supertherm HT04/17, Nabertherm, Germany) for overnight ashing. After removal from the furnace, the samples were let to cool down to room temperature, whereupon they were measured again in the precision balance. The ash content was determined gravimetrically and in duplicate. PEF-treatment and incubation of biomass PEF-treatment of freshly harvested biomass took place in a custom-made continuous flow treatment chamber with a 4 mm distance between the electrodes. The apparatus was identical with previous works. The generator was described in [56] and photos of the chamber and electrodes are available in [65]. In brief, the pulse parameters were set to duration of Δt = 1 μs, electric field intensity of 4 MV·m−1 and repetition rate of 3 Hz with a constant flow microalgae rate in the treatment chamber equal to 0.1 mL·s−1. These parameters correspond to an energy input of 150 kJ·L−1 or 1.5 MJ·kg−1DW, treatment, conditions that are demonstrably effective to S. almeriensis and to other microalgae based on our previous works [15, 56]. Further details on the estimation of the energy input can be found in [45]. The conductivity value of the microalgae suspension before and after PEF-treatment was measured with a conductivity meter (WTW, cond 3310), without automatic temperature compensation. The conductivity values were normalized to 20 °C using the Eq. (1), where σ stands for the conductivity, T for the measured temperature and α20 the temperature compensation coefficient at 20 °C which is equal to 2.38% per degree of centigrade [15]. $${\sigma }_{20}={\sigma }_{T}\frac{1}{1+{\alpha }_{20}(T-20)}$$ Lipid extraction from wet S. almeriensis The lapsed time after PEF-treatment has been demonstrated to be an important parameter that can directly affect the lipid extraction yields [45]. Therefore, lipid extraction experiments were performed on biomass immediately after PEF-treatment and after an incubation step. During this incubation, both PEF-treated and untreated biomass were stored in inert conditions (flushed with N2, in dark, without any agitation) for 24 h prior to further handling. Three different protocols were tested for lipid extraction from wet S. almeriensis. The first one was a co-solvent ethanol: hexane (1:041 vol/vol) system (system A), adapted from Grima et al. [66]. The second system was pure ethanol (system B) adapted from Eing et al.[67]. Both these protocols were priorly proven to be very robust for lipid extraction from A. protothecoides [56, 67]. The third one, pure ethanol in elevated temperature (system C) was adapted from Navarro-López et al.[58] who demonstrated its effectiveness for lipid extraction from Nannochloropsis gaditana. The protocols are summarized in Table 3 and described in detail below. All reagents were of analytical grade. Two independent cultivations were studied, with each sample processed in duplicate. Table 3 Composition of the different extraction systems Ethanol:hexane extraction (system A) For each sample, approximately 3 mL of suspension were measured in Teflon tubes (Nalgene® Oak Ridge Centrifuge Tubes, Teflon® FEP, 50 mL Thermo Scientific), which at 100 g·L−1 concentration corresponds to 0.3 g biomass. The probes were then centrifuged (Heraeustrade; Megafugetrade 8R, ThermoFischer Scientific, Germany) at 10,000 × g for 10 min and the supernatant was removed and measured to evaluate the remaining water in the system which was equal to approx. 1.5 mL. The biomass pellet was then re-suspended by adding 16.1 mL ethanol and 6.6 mL hexane. The composition of the system at the beginning of the extraction was thus 1:0.41:0.09 ethanol:hexane:water with a ratio of 81 mL of solvent per 1 g dry weight. It should be noted, that the water present is the leftover in the biomass pellet from the above centrifugation step, without any extra water addition at this stage. After rigorous vortexing, the samples were left to agitate on an agitator for 24 h, in the dark and at room temperature. Once extraction was completed, the probes were centrifuged at 10,000 × g for 10 min, to separate the solvent from the residual biomass. From the supernatant, 6.1 mL were removed into a separate tube, where an additional 18.2 mL hexane and 2.9 mL distilled water were added. From the two distinct phases formed, 15 mL from the upper, hexane lipid-rich phase was removed into pre-weighted glass tubes and evaporated under N2. The lipid yields were then calculated gravimetrically. Incubated samples were treated in an identical manner. Ethanol extraction (system B) The samples were prepared in a similar manner with the previous protocol up to the addition of the solvent, where instead of the ethanol:hexane blend, 22.7 mL of pure ethanol were added resulting in a ~ 96% ethanol extraction system at a ratio of 81 mL of solvent per 1 g dry biomass. Extraction took place in the dark, on an agitator for 24 h. The extraction solvent was separated by the spent biomass by a 10 min centrifugation at 10,000 × g. From the supernatant, 11.4 mL were removed into a separate tube, where 11.4 mL of hexane and 5.7 mL 10% NaCl deionized water were added. From the two phases formed, 9 mL of the upper phase were removed into pre-weighted glass tubes and evaporated under N2. The lipid yields were then calculated gravimetrically. Incubated samples were treated in an identical manner. Ethanol extraction at elevated temperature (system C) Like the previous systems, 3 mL of concentrated microalgae suspension were measured in a precision balance in teflon tubes and further de-watered after centrifugation. The biomass pellet was re-suspended by adding 22.7 mL pure ethanol resulting in a ~ 96% ethanol extraction system at a ratio of 81 mL of solvent per 1 g dry biomass. Extraction took place in the dark, for 30 min at 60 °C in a water bath (SONOREX SUPER RK 510 H, Bandelin, Germany). Following this, the samples were centrifuged at 10,000 × g for 10 min. The supernatant was completely removed into a separate tube, where 6.81 mL hexane and 13.6 mL deionized water was added. After the formation of two distinct phases, 5 mL from the upper phase was removed into pre-weighted glass tubes and evaporated under N2. The above step was repeated with the addition of 5 mL fresh hexane. Gravimetric yields were measured gravimetrically. Increase of the extraction solvents by factor 2 Experiments were conducted, where the extraction solvent was increased by a factor 2. The protocols followed were identical to the description above, the main difference being that instead of 3 mL sample, 1.5 mL were measured instead, corresponding to 162 mL solvent per 1 g of dry biomass. Lipid transesterification and gas chromatography analysis To evaluate the saponifiable content of S. almeriensis, the biomass underwent a direct transesterification reaction, followed by gas chromatography (GC) analysis, as described by Jiménez Callejón et al. [68]. In brief, 10 mg of freeze-dried milled biomass were molturated with 10 mg of alumine for 5 min and stored at -21 °C until use. The molturated biomass was directly transesterified using 1 mL of acetyl chloride:methanol solution 1:20 vol/vol and 1 mL hexane. The reaction took place for 20 min at 105 °C and agitation. Afterwards, the mixture was left to reach room temperature, followed by the addition of 1 mL distil. water. The samples were then agitated and centrifuged. Two phases were formed, the upper one containing hexane and the produced FAMEs was removed and analyzed by GC. This was conducted in a Agilent Technologies 6890 N (Santa Clara, USA) GC, equipped with a capillary column of fused silica OmegaWax™ (0.25 mm x 30 m, 0.25 μm standard film, Supelco, Bellefonte, PA) and a flame ionization detector (FID). As carrier gas, nitrogen was used. Further technical details can be found in [68]. Carbohydrate analysis in water fraction The water fraction containing the released carbohydrates was removed by centrifugation in a similar manner as described in 5.4.1. The samples were then stored in -20 °C until processing. After thawing, the carbohydrate content was determined following the same procedure with 5.2.3. Enzymatic protein hydrolysis of S. almeriensis Enzymatic hydrolysis was performed on wet biomass at concentration of 100 g·L−1, either directly after the PEF-treatment or, after centrifugation, removal of the supernatant and replacement by an equivalent volume of water (in case the water fraction was previously extracted, as in this case). The protocol of enzymatic hydrolysis itself was described in full detail in [15]. In brief, enzymatic hydrolysis took place in 50 mL glass tubes with screw caps (Roth, Germany). Temperature was fixed at 50 °C in a water bath, placed atop a magnetic stirrer with heating function (neoLab, Germany) which also provided constant agitation. The pH was adjusted at 8 using sodium hydroxide (1 M). As enzymes, Alcalase (subtilisin) 2.5 L (Novozyme, Denmark) and Flavourzyme 1000 L (Novozyme, Denmark) were added at 3% (vol/w) each with regard to cell dry weight of the biomass. The hydrolysis reaction lasted for 3 h and samples were removed every 1 h with immediate deactivation of enzymes by heating at 80 °C for 10 min. Centrifugation at 10,000 × g for 10 min separated the water fraction along the free amino acids from the biomass and the amino acid content was determined spectrophotometrically using ortho-phtaldialdehyde (OPA) assay, with serine as standard. The ratio of the number of cleaved peptide bonds over the total number of peptide bonds in the sample, also defined as Degree of Hydrolysis (DH) offers an indication of the reaction rate. High pressure homogenization (HPH) was used as a benchmark. HPH took place in an EmulsiFlex-C3 homogenizer (Avestin Europe GmbH, Germany) at 2 MPa for 5 passes. Lipid extraction from S. almeriensis following enzymatic hydrolysis After 3 h of hydrolysis, approximately 3 mL microalgae suspension per sample for all conditions, were measured precisely into Teflon tubes. As mentioned in Sect. 5.5.2, the free amino acids were separated from the rest of the biomass through centrifugation. For lipid extraction, the following step of the cascade, the residual biomass pellet was re-suspended in 22.7 mL ethanol: hexane co-solvent 1: 0.41 vol/vol. Lipid extraction then took place as described in Sect. 5.4.1 (system A). All data generated or analyzed during this study are included in this published article. PEF: Pulsed electric fields HPH: High pressure homogenization Sathasivam R, Radhakrishnan R, Hashem A, Abd-Allah EF. Microalgae metabolites: a rich source for food and medicine. 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Comparison between extraction of lipids and fatty acids from microalgal biomass. J Am Oil Chemists' Society. 1994;71:955–9. Eing C, Goettel M, Straessner R, Gusbeth C, Frey W. Pulsed Electric Field Treatment of Microalgae—Benefits for Microalgae Biomass Processing. IEEE Trans Plasma Sci. 2013;41:2901–7. Jiménez Callejón MJ, Robles Medina A, Macías Sánchez MD, Hita Peña E, Esteban Cerdán L, González Moreno PA, et al. Extraction of saponifiable lipids from wet microalgal biomass for biodiesel production. Biores Technol. 2014;169:198–205. Non applicable. Open Access funding enabled and organized by Projekt DEAL. This work was conducted in the framework and financed by the Helmholtz Research Program on Renewable Energies [Topic 3: Bioenergy] and by the European Union's Horizon 2020 Research and Innovation program [Grant Agreement No. 727874]. We acknowledge support by the KIT-Publication Fund of the Karlsruhe Institute of Technology. Institute for Pulsed Power and Microwave Technology (IHM), Karlsruhe Institute of Technology (KIT), Hermann-von-Helmholtz-Platz 1, Bldg 630, 76344, Eggenstein-Leopoldshafen, Germany I. Papachristou, S. Akaberi, A. Silve, R. Wüstner, K. Leber, N. Nazarova, G. Müller & W. Frey Department of Chemical Engineering, University of Almería, 04120, Almería, Spain E. Navarro-López I. Papachristou S. Akaberi A. Silve R. Wüstner K. Leber N. Nazarova G. Müller W. Frey IP contributed to the acquisition, the analysis and interpretation of data and drafted the manuscript. SA made contributions to the acquisition of the data and to the manuscript writing. ENL contributed to the acquisition, analysis and interpretation of the data. RW, NN, KL contributed to the acquisition of the data. GM contributed to the conception and revision of the manuscript. AS and WF organized the study, contributed to interpretation of data and editing and revising the manuscript. Correspondence to I. Papachristou. All authors agree to publish this article. The authors declare no competing interest. Papachristou, I., Akaberi, S., Silve, A. et al. Analysis of the lipid extraction performance in a cascade process for Scenedesmus almeriensis biorefinery. Biotechnol Biofuels 14, 20 (2021). https://doi.org/10.1186/s13068-020-01870-1 Lipid extraction Cascade processing	CommonCrawl
Levan from Leuconostoc citreum BD1707: production optimization and changes in molecular weight distribution during cultivation Jin Han1, Huafeng Feng1, Xiaohua Wang1, Zhenmin Liu1 & Zhengjun Wu ORCID: orcid.org/0000-0001-5200-06251 Levan is a well-known homopolymer of fructose composed predominantly of β-(2, 6) fructofuranosyl linkages in the backbone with occasional β-(2, 1) linkages in the branch chains with varied applications. However, high production cost due to low yield of microbial levan has become a bottleneck for its practical applications. Furthermore, factors affecting the molecular mass of the synthesized levan by Leuconostoc spp. during prolonged cultivation is not fully elucidated. The cultivation condition for Leuconostoc citreum BD1707 to synthesize levan was optimized by single-factor experiments and subsequently with response surface methodology (RSM). The average molecular weight (Mw) of levan synthesized by the strain L.citreum BD1707 under the optimized cultivation conditions was monitored by high-performance size exclusion chromatography (HPSEC). Finally, the enzyme with levan-degrading activity was determined by sodium dodecyl sulfate polyacrylamide gel electrophoresis (SDS-PAGE). The levan yield of BD1707 reached 34.86 g/L with a corresponding productivity of 7.47 g/L/d under the optimal cultivation conditions deduced by RSM, i.e., cultivation at 26 °C and 200 rpm for 112 h in tomato juice supplemented with 172 g/L sucrose with an initial pH value of 6.12. The Mw of levan reached a peak value of 2.320 × 107 Da at 6 h of cultivation under the optimized cultivation conditions and then gradually decreased to 8.809 × 106 Da after 120 h of cultivation. The levan yield of the strain L.citreum BD1707 could be sufficiently enhanced via cultivation condition optimization. The decrease in molecular mass of the synthesized levan was attributed predominantly to the hydrolytic activity of levansucrase secreted by L.citreum BD1707 during cultivation, with an estimated Mw of 130 KD by SDS-PAGE, while the effect of acid hydrolysis could be nearly neglected. Levan is a well-known homopolymer of fructose composed predominantly of β-(2, 6) fructofuranosyl linkages in the backbone with occasional β-(2, 1) linkages in the branch chains and is widespread in both of plants and microorganisms. Levan from different sources differs in molecular weight and degree of branching. The molecular weight of bacterial levan (ranging from 2.0 × 106 to 1.0 × 108 Da) is generally much higher than that of levan from plants (ranging from 2000 to 3,733,000 Da) [1]. Microbial levan is synthesized by the transglycosylation activity of levansucrase (sucrose: 2–6-β-D-fructan 6-β-D-fructosyltransferase, E.C.2.4.1.10) in the presence of sucrose. Levansucrase, belonging to the glycoside hydrolase family 68 (GH68) [2] has been reported in a variety of microorganisms, such as Bacillus and Pseudomonas species and a few lactic acid bacteria (LAB) [3]. Due to its well water solubility, high molecular mass and low viscosity, microbial levan can be used as emulsifing, stabilizing, thickening and encapsulating agent in the food industry [3] and as blood plasma volume extender [4], antiobesity agent or hypocholesterolemic agent, etc., in the pharmaceutical industry [5]. Furthermore, owing to its moisturizing properties, low cell cytotoxicity, promoting mammalian cell proliferation and anti-inflammatory effect, levan can also be utilized in cosmetic products [6]. However, low yield of microbial levan has become a bottleneck for its practical application [7]. In recent years, researchers have exploited various agricultural raw materials such as molasses [8, 9] and syrup [10, 11] to replace chemically defined medium (CDM) to reduce the production costs of levan. The weight-average molecular weight (Mw) of microbial levan, an important index rendering this biopolymer with varied physiochemical and functional properties, is affected by the type of producer [12] and cultivation conditions, such as sucrose concentration [13], temperature [14], ionic strength [15] and initial pH value of the medium [16]. Calazans GTM et al. found that levan with Mw of 456,900 Da showed a superior antitumor activity than those with lower or higher molecular mass (Mw = 353,500 Da, 720,200 Da, 769,500 Da and 1,073,500 Da) and concluded that antitumor activity of levan was restricted to the biopolymer with molecular weight in specific ranges [17]. Because the Mw of levan might greatly affect its application, factors influencing the molecular mass of levan synthesized by individual microbial producer should be carefully checked. Tomato (Lycopersicon esculentum) is one of the most widely cultivated vegetables of the family Solanaceae. Tomato juice is not only well recognized as a healthy beverage but also as a natural medium suitable for the growth and enrichment of beneficial metabolites of lactic acid bacteria (LAB). Tomato juice fermented by four LAB species (Lactobacillus acidophilus, L. plantarum, L. casei and L. delbrueckii) was suggested to be a probiotic beverage suitable for vegetarians as well as subjects allergic to dairy-based products [18]. In our previous study, Leuconostoc citreum BD1707 (=CGMCC 6431) was reported to synthesize levan in sucrose-supplemented tomato juice (tomato juice-sucrose medium, TJSM) with a yield of 28 g/L [19]. However, no study regarding the optimization of cultivation conditions on levan production as well as its molecular weight distribution by Leuconostoc spp. in tomato juice has been reported. In the present study, the effects of cultivation period (h), temperature (°C), initial pH value of the medium and sucrose concentration (g/L) in TJSM on levan production by L. citreum BD1707 were investigated by single-factor experiments and response surface methodology (RSM). Furthermore, the growth characteristics of L. citreum BD1707 under optimized cultivation conditions, including changes in the levels of sucrose, fructose, glucose and levan production, were also assayed. Finally, the variation of the average molecular weight (Mw) of the levan synthesized in the broth was determined, which was gradually decreased during the prolonged cultivation of the strain L.citreum BD1707 and caused predominantly by enzyme(s) secreted by the producer cells with levan degrading activity. Bacterial strain, propagation and storage L. citreum BD1707 (=CGMCC 6431) was provided by the State Key Laboratory of Dairy Biotechnology, Bright Dairy & Food Co., Ltd., Shanghai, China. The bacterial strain was routinely streaked on M17 agar (Merck, Germany) supplemented with 50 g/L sucrose and incubated at 30 °C aerobically. The strain was stored in M17 broth (Merck, Germany) supplemented with 10% glycerol at − 80 °C. Preparation of TJSM TJSM was prepared according to the method described previously [20]. Briefly, tomatoes (Solanum lycopersicum) purchased from local market were cut into small cubes, ground in a pulper and filtered through cotton gauze to remove the majority of the peel and seeds. The filtrated tomato juice was further centrifuged to remove the fruit debris. The major biochemical components of the tomato juice were determined in triplicate and listed in Table 1. No obvious difference in the components was determined among different variety of tomato (see Supplementary material, Figure S1, Table S1). The tomato juice was supplemented with sucrose at 50, 100, 150, 200 or 250 g/L, and the pH value of the mixture was adjusted to 4.5, 5.5, 6.5, 7.5, or 8.5 individually by addition of 5.0 M NaOH. The TJSM was sterilized at 121 °C for 20 min. Table 1 Major biochemical components and parameters of tomato juice (values are the average ± range of triplicate (3) analyses) Single-factor experiment To quantify the relationship between individual factor and the response variable (levan yield), single-factor experiments were conducted. In the present study, four factors, i.e., cultivation period, cultivation temperature, initial pH and sucrose concentration in TJSM, were investigated; these factors were presumed to significantly affect levan synthesis by the strain L.citreum BD1707. A loop of fresh culture of the strain L.citreum BD1707 grown on M17 agar was inoculated into 100-mL Erlenmeyer flask containing 25 mL of sterile M17 broth and cultivated at 30 °C on a rotary shaker (Model I2400, New Brunswick Scientific Inc., Edison, NJ, U.S.) at 200 rpm for 24 h. The cells were collected by centrifugation at 15,000×g for 5 min at 4 °C and washed twice with sterile buffered physical saline. Subsequently, the washed cells were suspended in initial volume of buffered physical saline (9.2 log10 cfu/mL) and inoculated at a 2% (v/v) ratio into 250-mL Erlenmeyer flasks containing 100 mL of sterilized TJSM supplemented with varying levels of sucrose (50, 100, 150, 200 and 250 g/L) at different initial pH values (4.5, 5.5, 6.5, 7.5 or 8.5). The inoculated TJSM was then incubated at 15, 20, 25, 30 or 35 °C on a shaker at 200 rpm. Samples were withdrawn at 0, 24, 48, 72, 96 and 120 h to assay to the yield of levan as described below. RSM experimental design RSM based on a Box-Behnken design (BBD), which was generated using Design Expert 8.0.5b software (Stat-Ease Corporation, USA), was utilized to investigate the interactions among the individual factors tested in the aforementioned single-factor experiment on levan production by BD1707. The variables and their coded levels were listed in Table 2. The three-level four-factor factorial BBD developed a total of 27 runs containing 3 replications of the central points (to check if there was a nonlinear relationship between the variables and the responses) and 24 trials organized in a fractional factorial design (Table 3). The experimental data were analyzed by the response surface regression procedure to fit the following second-order polynomial equation: $$ \mathrm{Y}={\upbeta}_0+{\upbeta}_1\mathrm{A}+{\upbeta}_2\mathrm{B}+{\upbeta}_3\mathrm{C}+{\upbeta}_4\mathrm{D}+{\upbeta}_{11}{\mathrm{A}}^2+{\upbeta}_{22}{\mathrm{B}}^2+{\upbeta}_{33}{\mathrm{C}}^2+{\upbeta}_{44}{\mathrm{D}}^2+{\upbeta}_{12}\mathrm{AB}+{\upbeta}_{13}\mathrm{AC}+{\upbeta}_{14}\mathrm{AD}+{\upbeta}_{23}\mathrm{BC}+{\upbeta}_{24}\mathrm{BD}+{\upbeta}_{34}\mathrm{CD} $$ where Y was the predicted response corresponding to levan production; A, B, C and D were the coded independent variables; β0 is an offset term; β1, β2, β3, and β4 are linear effects; β11, β22, β33, and β44 are quadratic coefficients; and β12, β13, β14, β23, β24, and β34 are interaction terms. Table 2 Independent variables and their coded levels chosen for BBD Table 3 Box-Behnken design (matrix and responses) for levan production by BD1707 Preparation and quantification of Levan Levan in the cultivated broth was prepared according to the procedure described previously [19]. In brief, the cultivated broth was centrifuged at 15,000×g for 5 min at 4 °C, and 4 volumes of prechilled ethanol were added to the supernatant. The mixture was stored at 4 °C overnight. The precipitate was collected by centrifugation at 15,000×g at 4 °C for 20 min and washed twice with prechilled ethanol. The pellet was redissolved in deionized water, dialyzed (MWCO 14,000 Da) against deionized water at 4 °C for 72 h, freeze-dried using FreeZone12 (Labconco corporation, Kansas, USA) and weighed. As reported previously, the lyophilized powder was composed solely of levan [19]. Change of the molecular weights of Levan during prolonged bacterial cultivation The change of the molecular weights of levan in the cultivated TJSM under optimized conditions, except for the cultivation period, was investigated. The resuspended BD1707 cells were inoculated at a 2% (v/v) ratio into 250-mL Erlenmeyer flasks containing 100 mL of sterile TJSM supplemented with 172 g/L sucrose, with the initial pH value adjusted to 6.12. The inoculated broth was cultivated at 26 °C on a shaker at 200 rpm. Samples at 3, 6, 12, 24 and 120 h of cultivation were withdrawn and treated as mentioned above to prepare levan for further Mw assays. Influence of Levan-degrading enzymes and organic acids on the mw of Levan The suspended BD1707 cells were inoculated at a 2% (v/v) ratio into 250-mL Erlenmeyer flasks containing 100 mL of sterile TJSM supplemented with 172 g/L sucrose, with the initial pH value adjusted to 6.12. The inoculated broth was first cultivated at 26 °C on a shaker at 200 rpm for 72 h. Then, the cultivated broth was divided into 4 groups undertaking different treatments: (A) the cultivated broth was centrifuged, precipitated by absolute alcohol, dialyzed against deionized water and lyophilized to prepare levan as aforementioned; (B) the cultivated broth was centrifuged at 15,000×g for 5 min at 4 °C to remove bacterial cells, and the supernatant (with a pH value of 4.00) was filtrating sterilized with a 0.22-μm membrane (Sartorius AG, Goettingen, GER); (C) The pH value of the cell free supernatant as obtained in group B was firstly neutralized to 7.0 with 1 M NaOH and then boiled for 5 min to inactivate enzymes capable of degrading levan. Then, the pH of the heat treated supernatant was readjusted to 4.00 (the predicted pH of 120-h fermented medium) after cooling to room temperature; (D) the control group without any treatment. Group B to D were further incubated at 26 °C on a shaker at 200 rpm for an additional 48 h. After incubation, levan in groups B to D was extracted as described above and used to determine the change in the distribution of molecular weights. Enzyme involved in Levan synthesis and degradation secreted by L.citreum BD1707 To reduce the interference of levan on the precipitation of enzymes secreted by L. citreum BD1707 in the broth by ammonium sulfate, the resuspended BD1707 cells were inoculated into tomato juice supplemented with 2% (v/w) sucrose and cultivated at 26 °C for 72 h. The cultivated broth was centrifuged at 15,000×g for 15 min at 4 °C to remove bacterial cells. The proteins in the supernatant was precipitated by ammonium sulfate at 60% saturation and collected by centrifugation at 20,000×g for 30 min at 4 °C. The pellet was dissolved in sodium acetate buffer (20 mM, pH 5.6) containing 2 mM CaCl2 and dialyzed (MWCO 14,000 Da) against the same buffer at 4 °C overnight with 2 changes of buffer to remove the residual ammonium sulfate. The lyophilized protein was dissolved with 1x Laemmli Sample Buffer (Bio-Rad, USA) and incubated at 37 °C for 1 h. For SDS-PAGE, 6% acrylamide containing 0.1% SDS was used. Proteins in the gel were stained with Coomassie blue G-250 and de-stained with 10% acetic acid solution. The standard protein markers were purchased from Bio-Rad (USA). For in situ detection of the levansucrase activity, the SDS-PAGE gel was incubated in 20 mM sodium acetate buffer containing 50 g/L sucrose buffered at pH 5.6, as described by Dols, Remaud-Simeon, Willemot, Vignon, and Monsan [21]. Briefly, the gel was washed three times with 20 mM sodium acetate buffer (pH 5.6) containing 2 mM CaCl2 and 0.1% (vol/vol) Triton X − 100 at room temperature to eliminate the SDS and then soaked in 20 mM sodium acetate buffer (pH 5.6) containing 2 mM CaCl2 and 50 g/L of sucrose at 30 °C for 48 h. The protein bands capable of EPS synthesis were detected by the appearance of opaque polymers in the gel [22]. The polysaccharide formed on the gel was extracted by warm water, and structure characterized as described in previous study [19]. The band in a parallel SDS-PAGE gel with corresponding polymerization activity in situ was cut off and processed with peptide mass fingerprinting to determine the sequence of the enzyme [23]. For levan-degrading activity assay, single protein band in the SDS-PAGE gel was cut and individually incubated in one milliliter of 20 mM sodium acetate buffer containing 10 g/L levan (prepared previously) and 2 mM CaCl2 (pH 5.6) at 30 °C for 72 h. The released fructose was determined by HPLC method described below [24]. The levels of sucrose, glucose and fructose in the cultivated TSJM were determined by a high-performance liquid chromatography (HPLC)-based method described previously [24]. The Mw of the levan obtained was determined by high-performance size exclusion chromatography (HPSEC) using a Perkin-Elmer series 200 liquid chromatography (PerkinElmer, Waltham, MA) equipped with a Perkin-Elmer series 200 refractive index detector. Two TSK-GEL columns (G6000PWXL and G4000PWXL, Tosoh Bioscience Co., Japan) were maintained in series, utilizing 0.1 M NaNO3 as the eluent at a flow rate of 0.6 ml/min. The columns were maintained at 25 °C, and 5 mg/ml of BD1707 levan dissolved in the 0.1 M NaNO3 was filtered through a 0.22-μm filter before injection. Commercial pullulans with molecular mass ranging from 6000 to 2,560,000 Da (6000 Da, 12,000 Da, 110,000 Da, 800,000 Da, 2,560,000 Da) (Sigma, St. Louis, MO, USA) were used as standards (see Supplementary material, Table S2). The viable cell counts of BD1707 in the cultivation broth were enumerated by plating the serially 10-fold diluted sample on M17 agar (Merck, USA) and incubating at 30 °C aerobically. The pH was measured by using a pH meter (PB − 10, Sartorius AG, Goettingen, GER) [19]. All experiments and analyses at every time point for each experiment were performed in triplicate. The means, standard errors, and standard deviations were calculated from replicate experiments and analyzed using Design Expert 8.0.5b and OriginPro8.0. Effect of cultivation period, temperature, initial pH and sucrose concentration of TJSM on Levan production Figure 1a illustrated the effect of cultivation temperatures from 15 °C to 35 °C on levan production by the strain L.citreum BD1707. In TJSM with sucrose at 150 g/L and an initial pH of 6.5, high yields of levan (exceeding 28 g/L) was produced by the strain L. citreum BD1707 at 30 °C and 25 °C after 96 h of cultivation, while low yields of levan (below 20 g/L) were observed at 15, 20 and 35 °C. A maximal yield of levan (33.40 g/L) was observed at 25 °C, which was chosen as the center point with 5 °C step changes in subsequent tests. Effects of individual cultivation factors on levan production by BD1707. a Cultivation temperature, b sucrose concentration and c initial pH Levan production by the strain L.citreum BD1707 in TJSM at pH 6.5 with different levels of sucrose (50, 100, 150, 200 or 250 g/L) at 25 °C for 120 h is shown in Fig. 1b. Overall, high levels of sucrose are beneficial for levan synthesis in TJSM by the strain L.citreum BD1707. The levan yield at 96 h was enhanced from 15.8 to 33.4 g/L by increasing the sucrose concentration from 50 to 150 g/L. Therefore, sucrose concentration of 150 g/L was selected as the center point with a step change of 50 g/L. The effect of initial pH value (4.5–8.5) on levan production by BD1707 was examined in TJSM with 150 g/L sucrose at 25 °C (Fig. 1c). The highest levan yield (33.40 g/L) was observed in TJSM with an initial pH of 6.5. Therefore, the center point of the initial pH was fixed at 6.5 with step changes set at 1.0. In the single-factor experiments (Fig. 1a, b, c), high levan yields were observed in all batches after 72–120 h of cultivation, and A peak yield of levan (33.40 g/L) occurred at 96 h. Therefore, 96 h was chosen as the center point of cultivation time with step changes set at 24 h. Response surface methodology (RSM) A three-level four-factor Box-Behnken experimental design for RSM with 27 runs was employed to determine the optimal cultivation variables for levan production by L. citreum BD1707 in TJSM. The results derived from the experimental data and simulated values predicted by the constructed model employing levan yield as the response variable were listed in Table 3. By applying multiple regression analysis on the experimental data, a second-order polynomial equation describing the relationship of levan production (Y) in TJSM with cultivation time (A), cultivation temperature (B), initial pH (C) and sucrose concentration (D) is established in terms of coded factors as eq. 2. $$ \mathrm{Y}=33.36+0.68\mathrm{A}+4.95\mathrm{B}+0.16\mathrm{C}+3.92\mathrm{D}-0.13\mathrm{AB}-1.54\mathrm{AC}+0.66\mathrm{AD}-0.29\mathrm{BC}+0.18\mathrm{BD}-0.28\mathrm{CD}-1.13{\mathrm{A}}^2-9.26{\mathrm{B}}^2-1.42{\mathrm{C}}^2-5.11{\mathrm{D}}^2 $$ As shown in Table 4, based on the analysis of p-value and F-value, the four factors were ranked in order of positive effect on levan production as follow: B (Cultivation temperature, F-value of 299.24, p-value<0.0001)>D (sucrose concentration, F-value of 187.95, p-value<0.0001)>A (cultivation time, F-value of 5.71, p-value of 0.0342)>C (initial pH value, F-value of 0.3, p-value of 0.594). Linear effect of cultivation temperature and sucrose concentration was highly remarkable, indicating that they might act as limiting factors on levan yield. According to the coefficients of interactions, AD (0.66) and BD (0.18) had positive effect on levan yield, while negative effect could be seen in the AB (− 0.13), AC (− 1.54), BC (− 0.29) and CD (− 0.28), but all the interactions had no significant effect on responses. Table 4 Model coefficient estimated by multiple linear regression By means of analysis of variance (ANOVA), the quadratic regression model with a low p-value (p < 0.0001) and insignificant result from the lack-of-fit test (p = 0.1295) was proven to be suitable and had good prediction ability. The coefficient of determination (R2) measuring the model's goodness of fit was 0.9887, which indicated that the model was capable of explaining 98.87% of the variation in the response and that only 1.13% of the total variations could not be accounted for by the model. The "adjusted R2" and the "predicted R2" were 0.9756 and 0.9363, respectively, which indicated that the model was highly reliable according to the principle of "the nearer to 1.0 the R2 value was, the more fit the model was deemed to be". The "adequate precision" value of the present model, reflecting the signal-to-noise ratio, was 31.217, which was much greater than the desirable value of 4, suggesting that the model could be used to navigate the design space. The standard deviation, mean, and predicted residual sum of squares (PRESS) values were 0.99, 25.84, and 66.74, respectively, and the low variation coefficient value (C.V. =3.84) provided further evidence for the high preciseness and reliability of the model. Six response surface graphs were obtained from this model, two of which were chosen to illustrate the combined effects of individual variables on levan production. Figure 2a shows the effect of sucrose concentration and cultivation temperature on the response at the fixed center values of initial pH and cultivation time. The yield of levan was correlated with increasing sucrose concentrations and cultivation temperatures up to approximately 150 g/L and 25 °C, respectively, while the levan yields decreased with further increase in the levels of these two variables. The interaction effect of initial pH value and cultivation time on levan production was also explored while keeping the sucrose concentration and cultivation temperature constant at the center values (Fig. 2b). Increasing the initial pH value and cultivation time led to an increase in levan production but at a modest rate compared with the increase in sucrose concentration and cultivation temperature. A maximum levan yield of 35.10 g/L by the strain L.citreum BD1707 was predicted by the point prediction tool of Design Expert software under the optimized cultivation conditions, i.e., cultivation time of 112 h, cultivation temperature of 26 °C, initial pH value of 6.12 and sucrose level of 172 g/L. 3D response surface plots of the combined effects of independent factors on levan production. a Sucrose concentration and cultivation temperature, b initial pH value of the medium and cultivation period To validate the second-order model, three independent replications were conducted under optimal conditions. Meanwhile, cultivation in triplicate under the non-optimized conditions described previously [19] was also carried out as a control. Figure 3 provides a comparison of the changes in the levels of sucrose, glucose and levan between optimized and non-optimized fermentation. Changes in sucrose, glucose, fructose and levan production by L. citreum BD1707 under optimal and nonoptimal cultivation conditions (——, optimal; ———, nonoptimal; ■□, levan production; ●○, sucrose; ◆◇, glucose; ▲△, fructose) The highest yield of levan from the strain L.citreum BD1707 under optimized cultivation conditions was 34.86 g/L, approximately the maximum value (35.10 g/L) predicted by the second-order model, which was reached at a cultivation time of 112 h and was much higher (P < 0.05) than the yield obtained under non-optimized conditions. Meanwhile, the sucrose concentration decreased by 9.22% (w/v) from the initial level of 17.20% (w/v) to 7.98% (w/v) at 120 h under the optimized conditions, while only a decrease of 6.92% (w/v) in sucrose concentration was observed in non-optimized cultivation conditions, which could be resulted from the high expression and/or high enzyme activity of levansucrase in BD1707 under optimal conditions. Levansucrase, a member of the glycoside hydrolase family 68, is crucial in the formation of levan with two catalytic functions: hydrolysis of sucrose and transglycosylation of the fructose moiety of sucrose to the elongated fructan chain. This result indicated that process optimization is an efficient way to increase levan production. Additionally, glucose, one of the hydrolysis products of levansucrase, accumulated gradually during cultivation, and the increment of 4.03% (w/v) under optimal conditions was markedly higher (p < 0.05) than that (2.71% (w/v)) under non-optimized conditions, which provided further evidence that the activity of levansucrase expressed by the strain L.citreum BD1707 is stronger under optimized conditions than that under non-optimized conditions. Unlike glucose, no significant change in the concentration of fructose was observed under these two cultivation conditions, which fluctuated around 1% (w/v). Degradation of the Levan during prolonged cultivation of the strain L.citreum BD1707 To investigate the changes in Mw of levan synthesized under optimal conditions, BD1707 was cultivated in TJSM (pH 6.12) containing 172 g/L sucrose at 26 °C and 200 rpm for 120 h. Samples at different intervals (3, 6, 12, 24 and 120 h) were obtained, and the molecular weight distribution of levan was analyzed. As shown in Fig. 4, the calculated Mw of levan obtained at different cultivation periods were 2.245 × 107 Da (3 h), 2.320 × 107 Da (6 h), 2.053 × 107 Da (12 h), 1.554 × 107 Da (24 h) and 8.809 × 106 Da (120 h) , indicating the molecular mass of levan synthesized by the strain L.citrem BD1707 reached the highest value at 6 h of cultivation and then decreased with prolonged cultivation. Changes in the Mw of BD1707 levan, pH and viable cell counts during fermentation under optimal conditions. a Retention volume of levan in different cultivation periods; b calculated Mw, pH and viable cell counts To explore the factor responsible for the degradation of levan, cultivated TJSM by the strain L.citreum BD1707 at 72 h was chosen for further investigation. At this stage, cells of the strain L.citreum BD1707 were in the declining phase (shown in Fig. 4b), inclining to secret some degrading enzymes including the presumed levan-degrading enzymes (levansucrase and/or levanase, induced by sucrose and/or levan, respectively) to salvage the residual nutrients in the environment to maintain the viability of the bacterial cells. On the other hand, at this stage, the broth was with a rather low pH value of approximately 4.0, an ideal acidic environment to explore the impact of organic acid on the change of the Mw of existed levan in the broth during elongated cultivation. As shown in Fig. 5, the Mw of levan decreased markedly from 1.305 × 107 Da at 72 h of cultivation (levan from group A) to 8.824 × 106 Da at 120 h of cultivation (levan from group D), which might be attributed to the synergetic hydrolysis by both secreted enzymes (levanase/levansucrase) and organic acids. The Mw of levan from group C from the filter sterilized, enzyme inactivated and pH-readjusted medium was 1.197 × 107 Da, which was slightly lower than that of levan from group A, indicating that organic acids were not responsible for the degradation. Compared with the levan from group A, a significantly higher degree of levan degradation than that from group B could be observed (9.268 × 106 Da, obtained in the existence of pH value of 4.0 and degrading enzyme for an additional 48 h). As organic acid existed in both Group B and Group C, except for the presumed degrading enzyme(s) which was heat inactivated in the latter, the Mw of the levan in group B was much lower than that of levan in group C strongly suggested that the existence of levan degrading enzyme(s) (levanase/levansucrase) secreted into the medium during the first 72 h of cultivation. This result was further supported by the fact that the Mw of levan in group B was almost consistent with that in sample D. Molecular weight distribution of levan with different treatments. a Levan from 72-h cultivation sample; b 72-h cultivation sample centrifuged, filtered through a 0.22-μm membrane, and incubated for 120 h; c 72-h cultivation sample centrifuged, filtered through a 0.22-μm membrane, pH-adjusted to 7.0, boiled, pH-adjusted to 4.0, and incubated for 120 h; d control Enzymes involved in Levan synthesis and degradation by L.citreum BD1707 The protein expressed by L.citreum BD1707 in tomato juice supplemented with 2% (w/v) sucrose could be more efficiently precipitated by ammonium sulfate at 60% saturation (see supplementary material, Fig. S3). As shown in Fig. 6, although seven visible protein bands (Band A to G) were observed on the stained SDS-PAGE gel, only one obvious white and opaque band occurred at the position of 130 KD in the unstained SDS-PAGE gel soaked in a 50 g/L sucrose solution for 48 h at 30 °C. This result indicated the protein band at this position possessed polymerization activity. The polymers in the gel were extracted and characterized as levan (data not shown). After cutting off and subjecting to peptide mass fingerprinting, the corresponding SDS-PAGE protein band (Band B) was identified as levansucrase (tr\|A0A192S224\|A0A192S224_LEUME) of L.mesenteroides with a sequence coverage of 25%, which indicated the gene encoding the levansucrase in L. citreum BD1707 is phylogenetically diversified from its alleles in other levan producing bacteria. The levansucrase showed an unusual Mw of 130 KD, which was greatly differed from the reported Mw of levansucrase, usually in the range of 50-90KD, secreted by other levan producing bacterial species [25,26,27]. The individual protein band (Band A to G) on the stained SDS-PAGE gel was cut off and assayed for their fructose releasing activity from levan. Among the seven protein bands, only band B showed detectable levan-degrading activity, which released about 0.04% (w/v) fructose from levan during the incubation of the band with 10 g/L levan at 30 °C for 48 h. In situ gel polysaccharides synthesis To enhance the levan yield by the strain L. citreum BD1707 in tomato juice, 4 cultivation factors i.e. cultivation period, cultivation temperature, initial pH and sucrose concentration were selected from a variety of nutrition and cultivation parameters to investigate their impacts on the synthesis of levan. In the single factor test, cultivation temperature and the level of sucrose exert more significant impact on the levan yield of the strain L.citreum BD1707. In the tested cultivation temperature of 15-35 °C, 25-30 °C is more suitable for L. citreum BD1707 to synthesize levan in the presence of sucrose. As temperature is one of the most important environmental factors affecting the growth and development of bacterial cells by influencing intracellular bioactivities [28], alike other Leuconostoc species, L. citreum is a mesophilic LAB, preferring to grow, proliferate and accumulate secondary metabolites such as dextran [29], inulin [30], mannitol [31] and bacteriocin [32] at mild temperatures. The low yield of levan of L. citreum BD1707 at cultivation temperature outside this scope might be due to 1) decreased levansucrase expression caused by slow bacterial growth at low temperature [33], 2) inactivation of extracellular levansucrase at elevated temperatures, 3) and irreversible denaturation of levansucrase to synthesize levan at higher cultivation temperature [34]. Sucrose was determined to be the sole carbon source for the strain L.citreum BD1707 to synthesize levan in our previous study [19], which was in consistent with the results of other researchers [2]. Consequently, sucrose was selected as a crucial variable to optimize levan production in this study. The low yield of levan by the strain L.citreum BD1707 at sucrose levels lower than 150 g/L might be attributed either to the insufficient substrates for the synthesis of the fructan-type polymer or the rapid hydrolysis of the synthesized polymer by enzymes with levan degrading activity, i.e. levanase [35] and levansucrase [36] during cultivation. In our study, the degradation of the synthesized levan seemed less obvious in TJSM with a high concentration of sucrose (≥ 150 g/L), which might be caused by the repressed hydrolytic activity of the enzyme at the presence of high levels of reducing sugar generated during the cultivation process [37] (Fig. 1b). A further increase in sucrose concentration (250 g/L) would result in a high osmotic pressure unsuitable to the growth and metabolism of the strain L.citreum BD1707, and thus led to a lower yield of levan after 96 h of cultivation, as also observed in cultivation of other microbes such as Zymomonas spp. in the presence of high osmotic pressure [38, 39]. A similar result was obtained by SHIH et al., who observed that the yield of levan by Bacillus subtilis (natto) decreased at both extremely high and low sucrose concentrations [40]. The initial pH value of the culture medium might also affect the yield and size of the soluble levan synthesized by microorganisms [41] by altering the morphologic features of the producer cells [39]. At all tested pH values, levan production was higher than 29 g/L after 96 h except in TJSM with an initial pH of 4.5 (17.48 g/L), which might be due to either poor growth of BD1707 cells and /or the levan polymerization activity of levansucrase being lower than the hydrolytic activity [37]. Nevertheless, the levan yield by the strain L.citreum BD1707 at pH 4.5 was still higher than those of other microbial producers [37]. A plausible explanation to this phenomenon might be the strong acidity tolerance ability of L. citreum, which endows the bacteria in this species to grow well in low-pH niche and subsequently inhibit the growth of other spoilage microbes and thus prolong the shelf life of kimchi [42]. The fluctuation in levan production by the strain L.citreum BD1707 in TJSM with initial pH values of 5.5 to 8.5 during the cultivation exhibited a similar tendency, with levan yield peaking at 96 h and decreasing thereafter, indicating that the strain L.citreum BD1707 could synthesize levan in a wide range of pH values. Levan degradation might be related to two biochemical process: (1) hydrolysis by levan-degrading enzymes secreted by BD1707, alike those expressed by other levan producers via a carbohydrate regulation mechanism [35, 36]; (2) acid hydrolysis by organic acids [15]. In the present work, while the Mw of levan peaked at 6 h of cultivation (6 h), the strain L.citreum BD1707 underwent dramatically rapid growth (P < 0.05), with the viable cell counts increasing from an initial value of 7.6 log10 to 9.25 log10 cfu/mL. At this point of cultivation, the growth of the strain L.citreum BD1707 was at the late exponential phase (Fig. 4b), with a levan yield of 5.8 g/L (Fig. 3), indicating the existence of levansucrase in the cultivation medium. Meanwhile, the pH of the medium decreased sharply from 6.13 to 5.16, which might be low enough to shift the catalytic activity partially from polymerization to degradation upon further cultivation [15]. As the Mw of levan of 1.305 × 107 Da synthesized by the strain L.citreum BD1707 at 72 h (levan from group A) was decreased to 8.824 × 106 Da at 120 h of cultivation (levan from group D) in the presence of both organic acids and continued secretion of presumed levan degrading enzyme, whilst the Mw of the levan sample shifted slightly to 1.197 × 107 Da (levan from group C) in the absence of the presumed levan degrading enzyme, compared to a sharply decrease of Mw to 9.268 × 106 Da in levan levan from group B in the presence of presumed degrading enzyme(s) at the same acidic environment. Therefore, the degradation of the existed levan could in reasonably linked to the activity of some enzyme secreted into the broth by the levan producer. Our result was in disagreement with those of Bekers et al. and Runyon et al., who concluded that acid hydrolysis was the main levan-degrading factor [43] and that levan degradation was more likely to occur at pH values lower than 5.5 and faster at even lower pH values [15]. The discrepancy of our result and the published literature can be plausibly explained by the fact that L. citreum is acid tolerant while producers such as Zymomonas mobilis produced little organic acid into its growth medium [43]. By means of SDS-PAGE and in situ polymerization, an active protein band responsible for levan synthesis was observed in the cultivated TJSM with 20 g/L sucrose by at 30 °C for 72 h by the strain L. citreum BD1707 and identified as levansucrase, which also showed detectable fructose releasing activity from levan. The identified levansucrase was with an unusual Mw of 130 KD, much higher than the reported Mw of levansucrase secreted by other levan producing bacterial species [25,26,27], which was usually in the range of 50–90 KD and also with low amino acid sequence coverage with that reported in L. mesenteroides. Our result is in agreement with previously published literature in that levansucrase could display hydrolytic activity just as that of levanase. Mendez-Lorenzo L. et al. found that while SacB (Bacillus subtilis 168 levansucrase) released 1 μmol of fructose per min from 100 mg/mL of sucrose, the enzyme released 1/18 μmol of fructose from 100 mg/mL of levan. The authors hypothesized that the levan hydrolytic activity of levansucrase was minor at the first stage of reaction with enough sucrose as substrate for the polymerization, but it might become obvious with the decrease of sucrose and the increase of levan in the middle or late stages of the reaction [44]. In blasting the full genome sequence of L. citreum BD1707, a fragment of DNA in the genome was inferred to encode a protein related to levanase with protein ID of WP_040177126.1. Unfortunately, in our study, no levanase was precipitated from the tomato juice with 20 g/L sucrose cultivated with L. citreum BD1707 for 72 h. A plausible explanation might be levan synthesized by L. citreum BD1707 at the presence of 20 g/L of sucrose is insufficient to induce the bacteria cells to express detectable levanase. The functions of exopolysaccharides, especially those homopolymer of EPS were molecular weight depended. Generally, levan with high molecular weight was commonly used as encapsulating agent, emulsifier, stabilizer and thickener for its specific rheological and physical-chemical properties [45], whilst levan with relatively low molecular weight behaved more efficiently in healthy promotion, which is usually employed as dietary fiber or prebiotics with extremely lower degree of polymerization. Esawy M.A. et al. proved that levan (9.53 KD) from Bacillus subtilis M was promising inhibitor of cytochrome for its inhibitory effect on carcinogens induced-DNA fragmentation [46]. The antitumor activity of levan (456,900 Da) was much stronger than that of other levan samples with higher molecular weights (720,200 Da, 769,500 Da and 1,073,500 Da) [17]. Levan of 2.25 × 106 Da molecular weight exhibited a moisturizing effect as well as a similar cell proliferation effect on human fibroblast and keratinocyte cells [6]. Therefore, factors affecting the Mw of levan in the large scale preparation of this polymer should be noticed as molecular weight might influence its bioactivity and functionality. TJSM was previously proven to be a low-cost medium suitable for L. citreum BD1707 growth for levan synthesis. In the present study, RSM based on a 27-factorial BBD was successfully employed to optimize the cultivation conditions to further enhance levan production by BD1707. The optimal cultivation conditions were predicted to be as follows: cultivation time, 112 h; cultivation temperature, 26 °C; initial pH, 6.12; and sucrose concentration, 172 g/L. Under the optimized cultivation conditions, a maximum levan yield of 34.86 g/L was attained. During the cultivation of BD1707 in TJSM, the Mw of the levan produced by BD1707 reached a maximum value of 2.320 × 107 Da at 6 h of cultivation and then gradually decreased to 8.809 × 106 Da at 120 h of cultivation. Levansucrase with Mw of 130 KD secreted by the strain L.citreum BD1707 into the medium during cultivation was presumed for the degradation of levan, leading the levan with a lower molecular weight, while the hydrolysis at low pH caused by organic acid accumulation could be neglected. The 16sRNA sequence of L. citreum BD1707 has been deposited at DDBJ/ENA/GenBank under the accession KT626384.The whole genome information of L. citreum BD1707 has been deposited at DDBJ/ENA/GenBank under the accession JACDIP000000000. All supporting data are included in the manuscript and supplementary materials. 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Process Biochem. 2005;40(5):1535–9. Mndez-Lorenzo L, Porras-Dom Nguez JR, Raga-Carbajal E, et al. Intrinsic levanase activity of Bacillus subtilis 168 levansucrase (SacB). PLoS ONE. 2015. https://doi.org/10.1371/journal.pone.0143394. Goncalves BCM, Baldo C, Celligoi MAPC. Levan and Levansucrase-a mini review. Int J Scie Technol Res. 2005;4(5):100–4. Esawy MA, Amer H, Gamal-Eldeen AM, et al. Scaling up, characterization of Levan and its inhibitory role in carcinogenesis initiation stage. Carbohydr Polym. 2013;95(1):578–87. The authors are thankful to Dr. Jingyu Liu, Research Institute of Bright Dairy & Food, Co. Ltd., for her assistance in the blasting of genes encoding levansucrase and levanase in the genome sequence of L.citreum BD1707, and Dr. Zhenyi Qiao from the same institute for his help in the statistic of the RSM experiment respectively. The authors also gratefully acknowledge Instrumental Analysis Center of Shanghai Jiao Tong University for their technical support in MALTI-TOF-MS of the protein band in the SDS-PAGE gel. This work was supported by Shanghai Engineering Center of Dairy Biotechnology (16DZ2280600) and Shanghai Committee of Science and Technology (17391901100). The fundings made no decision on the collection and final interpretion of the experimental data or publication. State Key Laboratory of Dairy Biotechnology, Shanghai Engineering Center of Dairy Biotechnology, Research Institute of Bright Dairy & Food Co., Ltd., Shanghai, 200436, China Jin Han, Huafeng Feng, Xiaohua Wang, Zhenmin Liu & Zhengjun Wu Jin Han Huafeng Feng Xiaohua Wang Zhenmin Liu Zhengjun Wu JH carried out the main body experiments and the writing of the manuscript; HF took part in preparation and assay of Mw of Levan; XW carried out sampling at different cultivation intervals and determination of the mono-, and di- sugars in the cultivated broth; ZL undertook the exploring of factors capable of degrading levan during elongated cultivation; ZW carried out the designing of experimental protocol and the writing of the manuscript. The author(s) read and approved the final manuscript. Correspondence to Zhengjun Wu. All the authors declare no competing interests with the others. Additional file 1: Table S1. Major Biochemical components and parameters of tomato juices prepared from different variety of Lycopersicon esculentum (values are the average ± range of triplicate analyses). Figure S1. Different variety of Lycopersicon esculentum and the prepared tomato juice (I: Lycopersicon esculentum var. vulgare, II: Lycopersicon esculentum var. grandifolium, III:Lycopersicon esculentum var. valiudmbaily). Figure S2. GPC-HPLC profiles of pullulan with molecular mass ranging from 6,000 to 2,560,000 Da. Figure S3. SDS-PAGE profiles of proteins expressed by the strain L.citreum BD1707 in juices supplemented with 2% (w/v) sucrose. The proteins were precipitated from the supernatant of the cultivated TJSM either by ammonium sulfate at 40% (40%ASP) or 60% saturation (60%ASP). Han, J., Feng, H., Wang, X. et al. Levan from Leuconostoc citreum BD1707: production optimization and changes in molecular weight distribution during cultivation. BMC Biotechnol 21, 14 (2021). https://doi.org/10.1186/s12896-021-00673-y Leuconostoc citreum BD1707 Distribution of molecular weight Levansucrase	CommonCrawl
All issues Volume 35 / No Suppl. 1 (2003) Genet. Sel. Evol., 35 Suppl. 1 (2003) S19-S34 Abstract Genet. Sel. Evol. Second International Symposium on Candidate Genes for Animal Health https://doi.org/10.1051/gse:2003014 Genet. Sel. Evol. 35 (2003) S19-S34 DOI: 10.1051/gse:2003014 Application of disease-associated differentially expressed genes - Mining for functional candidate genes for mastitis resistance in cattle Manfred Schwerina, Diana Czernek-Schäfera, Tom Goldammera, Srinivas R. Katab, James E. Womackb, Ravi Pareekc, Chandra Pareekc, Krzysztof Walawskic and Ronald M. Brunnera a Research Unit for Molecular Biology, Research Institute for the Biology of Farm Animals, Dummerstorf, Germany b Department of Veterinary Pathobiology, Texas A&M University, College Station, TX 77843, USA c Department of Animal Genetics, University of Warmia and Mazury, Olsztyn, Poland (Accepted 4 February 2003) In this study the mRNA differential display method was applied to identify mastitis-associated expressed DNA sequences based on different expression patterns in mammary gland samples of non-infected and infected udder quarters of a cow. In total, 704 different cDNA bands were displayed in both udder samples. Five hundred-and-thirty two bands, (75.6%) were differentially displayed. Ninety prominent cDNA bands were isolated, re-amplified, cloned and sequenced resulting in 87 different sequences. Amongst the 19 expressed sequence tags showing a similarity with previously described genes, the majority of these sequences exhibited homology to protein kinase encoding genes (26.3%), to genes involved in the regulation of gene expression (26.3%), to growth and differentiation factor encoding genes (21.0%) and to immune response or inflammation marker encoding genes (21.0%). These sequences were shown to have mastitis-associated expression in the udder samples of animals with and without clinical mastitis by quantitative RT-PCR. They were mapped physically using a bovine-hamster somatic cell hybrid panel and a 5000 rad bovine whole genome radiation hybrid panel. According to their localization in QTL regions based on an established integrated marker/gene-map and their disease-associated expression, four genes (AHCY, PRKDC, HNRPU, OSTF1) were suggested as potentially involved in mastitis defense. Key words: mastitis / expressed sequence tag / gene expression / cattle / RH mapping Correspondence and reprints: Manfred Schwerin e-mail: [email protected] Assessment of the immune capacity of mammary epithelial cells: comparison with mammary tissue after challenge with Escherichia coli Vet. Res. 40, 1-14 (2009) Real-time RT-PCR and cDNA macroarray to study the impact of the genetic polymorphism at the $\alpha_{s1}$-casein locus on the expression of genes in the goat mammary gland during lactation Identification of the bovine $\alpha$1-acid glycoprotein in colostrum and milk Vet. Res. 36, 735-746 (2005) Escherichia coli, but not Staphylococcus aureus triggers an early increased expression of factors contributing to the innate immune defense in the udder of the cow Bacterial lipopolysaccharide induces increased expression of toll-like receptor (TLR) 4 and downstream TLR signaling molecules in bovine mammary epithelial cells Genetics Selection Evolution ISSN: 0999-193X - eISSN: 1297-9686	CommonCrawl
Assessing the between-country genetic correlation in maize yield using German and Polish official variety trials Waqas Ahmed Malik ORCID: orcid.org/0000-0001-6455-53531, Harimurti Buntaran1, Marcin Przystalski2, Tomasz Lenartowicz2 & Hans-Peter Piepho1 Theoretical and Applied Genetics volume 135, pages 3025–3038 (2022)Cite this article Key message We assess the genetic gain and genetic correlation in maize yield using German and Polish official variety trials. The random coefficient models were fitted to assess the genetic correlation. Official variety testing is performed in many countries by statutory agencies in order to identify the best candidates and make decisions on the addition to the national list. Neighbouring countries can have similarities in agroecological conditions, so it is worthwhile to consider a joint analysis of data from national list trials to assess the similarity in performance of those varieties tested in both countries. Here, maize yield data from official German and Poland variety trials for cultivation and use (VCU) were analysed for the period from 1987 to 2017. Several statistical models that incorporate environmental covariates were fitted. The best fitting model was used to compute estimates of genotype main effects for each country. It is demonstrated that a model with random genotype-by-country effects can be used to borrow strength across countries. The genetic correlation between cultivars from the two countries equalled 0.89. The analysis based on agroecological zones showed high correlation between zones in the two countries. The results also showed that 22 agroecological zones in Germany can be merged into five zones, whereas the six zones in Poland had very high correlation and can be considered as a single zone for maize. The 43 common varieties which were tested in both countries performed equally in both countries. The mean performances of these common varieties in both countries were highly correlated. Variety examination offices evaluate the performance of newly bred varieties for their value of cultivation and use (VCU) before their addition to the national list and admission for commercial use in a country. The main objective of testing is to assess the relative phenotypic performance of the new varieties and only release the best varieties for commercial use. For example, every year on average 20 new maize varieties are registered in Germany. Before registration, they are tested for 2 to 3 years at up to 25 locations in the main growing area of maize in Germany. Similarly, in Poland, on average 30 new maize varieties are registered, after testing in official trials at up to 32 locations over 2 to 3 years. According to Laidig et al. (2008), 50% of the candidate varieties are usually withdrawn by the breeder after the first testing year, and only 25–30% reach the last year and are able to get registration. Maize is grown across a wide range of environments, and the growth of plants and the harvested product depend on the climatic and environmental conditions. An important objective of plant breeders is to develop broadly adapted varieties for a wider target region. On the basis of similarity in the agroclimatic conditions, a larger region can be subdivided into zones. These zones can extend beyond the political or national boundaries and can be regarded as mega-environments in which genotypes perform relatively homogeneously (Gauch and Zobel 1997). Kleinknecht et al. (2013) and Piepho and Möhring (2005) presented an approach showing how best linear unbiased prediction (BLUP) can be used for selection of genotypes for zones or mega-environments. The BLUP method allows the borrowing of information across zones or mega-environments to exploit genetic correlation between zones (Buntaran et al. 2019, 2020), thereby providing more accurate estimates of genotype performance as compared to best linear unbiased estimation (BLUE), which cannot exploit such correlation. The predictive ability of models can be improved by incorporating soil or environmental covariates (van Eeuwijk et al. 2016). Usually, in the analysis of multi-environment trial (MET) data, the variety-specific regression terms for covariates are taken as fixed effects (Denis 1988). However, when the target region is divided into zones and variety effects are modelled as random to borrow strength across zones, then variety-specific regression coefficients must be modelled as random, giving rise to random coefficient models (Longford 1993; Buntaran et al. 2021). This paper aims to assess the genetic correlation between the maize agroecological zones of Germany and Poland using official variety trials of maize from 1987 to 2017. The 43 common varieties allow the assessment of the genetic correlation between the two countries and among agroecological zones in these countries. Environmental covariates are also incorporated into the statistical models to achieve better fit and predictions. The rest of the paper is structured as follows. First, we describe the datasets from German and Poland official registration trials for maize. The models are then outlined in detail, followed by an illustration of results for maize trials from Germany and Poland. Finally, a discussion on the results is presented, focusing on the correlation between agroecological zones of Germany and Poland. In this study, datasets from the official grain maize variety trials of the Bundessortenamt (Hannover) in Germany from 1987 to 2016 and the Research Centre for Cultivar Testing (COBORU) in Poland from 1994 to 2017 were used. The German trials were conducted for assessing the value for cultivation and use (VCU), whereas the Polish trials were both VCU and post-registration trials (PDO). The trait yield in dt/ha is considered for the analysis. All trials from Germany were laid out as randomised complete block designs with three replications, while trials from Poland were laid out in a 1-resolvable design with three replicates. The crop management in the trials included standard fertilisation adapted to conditions in each location. The applied levels of nitrogen fertilisation are given in Fig. 1. Application of nitrogen in field trials in Germany and Poland from 1987 to 2018 The datasets were non-orthogonal because new varieties were tested for two or three years, and after each year, some varieties were withdrawn, whereas new varieties entered the trials. In any particular year, the same set of varieties was tested at each location in that year. Therefore, data within years are balanced for varieties and location; however, data are unbalanced for varieties by years. Some basic information about the datasets is given in Table 1, whereas the year-wise yields for both countries are plotted in Fig. 2. The lines in Fig. 2 represent year mean yield for each country. An increasing trend can be seen in both countries. Between 1987 and 2016, there were 43 varieties tested in both countries. A list of common varieties along with testing year is given in Table 2. Table 1 Basic information on the yield trial data of grain maize from Germany and Poland Year-wise yield (dt/ha) of maize field trials in Germany (left) and Poland (right) from 1987 to 2017. The lines represent yearly mean yield Table 2 List of 43 common varieties that were tested in Germany and Poland from 1987 to 2017 Based on the agroclimatic condition and soil type, the maize-growing area in Germany has been classified into 22 agroecological zones (Graf et al. 2009; Maiskomitee 2022). Similarly, the maize-growing area in Poland is classified into six zones (Fig. 3). Each location used for official maize trials was assigned to one of these zones. Zones with fewer locations were merged with the neighbouring zone because some zones were represented by only a few locations, as depicted in Fig. 3. Also, the merging of the zones was performed after analysing the data and looking into the genetic correlations between zones (results not shown). For example, Zone 1 and Zone 3 in Poland have a higher correlation as compared to the correlation between Zone 1 and Zone 2. Therefore, Zone 1 and Zone 3 were merged to form one zone. This zone merging resulted in five zones in Germany and four zones in Poland, which were subsequently used in the analysis. Maize agroecological zones of Germany and Poland. a Maize-growing area in Germany is classified into 22 agroecological zones, while maize-growing area in Poland is classified into 6 zones. b merged agroecological zones Models for the analysis of data from Germany and Poland maize variety trials The basic model for analysis of multi-environment trial data is expressed as: $${yield}_{ijk}=\mu +{g}_{i}+{s}_{j}+{y}_{k}+{(sy)}_{jk}+{(gs)}_{ij}+{(gy)}_{ik}+ {(gsy)}_{ijk}^{^{\prime}}$$ where ${yield}_{ijk}$ is the mean yield of the ith genotype (or variety) in the jth location (or site) and the kth year $(i = 1, \ldots ,n_{g} ;\;j = 1, \ldots ,n_{s} ;\;k = 1, \ldots ,n_{y} )$, μ is the overall mean, ${g}_{i}$ is the main effect of the ith genotype, ${s}_{j}$ is the main effect of the jth location, ${y}_{k}$ is the main effect of the kth year, ${(sy)}_{jk}$ is the interaction effect of the jth location and kth year, ${(gs)}_{ij}$ is the interaction effect of the ith genotype and jth location, ${(gy)}_{ik}$ is the interaction effect of the ith genotype and kth year, and ${(gsy)}_{ijk}^{^{\prime}}$ is the sum of the interaction effect of the ith genotype, jth location and kth year, and the residual for the mean yield ${yield}_{ijk}$. Country and agroecological zone levels analyses Since locations within countries are taken to be a representative sample within countries, Model 1 can be extended as follows, including a country main effect and its interaction with other main effects. $${yield}_{ijlk}=\mu +{c}_{l}+{g}_{i}+{(gc)}_{il}+{y}_{k}+{(yc)}_{kl}+{(gy)}_{ik}+{(gyc)}_{ikl}+{s}_{jl}+{(sy)}_{jlk}+{(gs)}_{ijl}+{(gsy)}_{ijlk}^{^{\prime}}$$ where ${c}_{l}$ is the effect of the lth country $(l=1,\dots ,{n}_{l})$, ${s}_{jl}$ is the effect of the jth location nested within the lth country, ${(gc)}_{il}$ is the interaction effect of the ith genotype and the lth country, $({yc)}_{kl}$ is the interaction effect of the kth year and the lth country, ${(gyc)}_{ikl}$ is the interaction effect of the ith genotype, kth year and the lth country, and ${(gsy)}_{ijlk}^{^{\prime}}$ is the sum of the interaction effect of the ith genotype, jth location within the lth country and the kth year, and the residual. In general, to get the genotype-by-country or genotype-by-zone means, we can model the main effects for genotype and country, and their interaction as fixed effects, and the rest of the effects as random, i.e. location, year and any interaction effects with year and location. Stacking effects of the same type into vectors (e.g. all location main effects, etc.), we may assume that these are independently normally distributed with zero mean and variance–covariance structures, ${\mathbf{G}}_{s}$, ${\mathbf{G}}_{y}$, ${\mathbf{G}}_{sy}$, ${\mathbf{G}}_{yc}$, ${\mathbf{G}}_{gs}$, ${\mathbf{G}}_{gy}$, ${\mathbf{G}}_{gyc}$ and ${\mathbf{R}}_{gsy}$, respectively. In the case of exploiting the genetic correlation between countries, the vector g of genotype main effect is modelled as random with the assumption $g\sim N(0,{\mathbf{G}}_{g})$. In this case, the variance of the main effect is equivalent to the genetic covariance between the two countries. For the country level analysis, the fixed genotype effect (FG) model and the random genotype effect (RG) model were fitted. The different agroecological zones for maize in Germany and Poland can be used for the agroecological zone-level analysis. In this case, the country effect $({c}_{l})$ in Model 2 is replaced by the zone effect $({z}_{l})$, and all other effects stay the same. Thus, Model 2 becomes $${yield}_{ijlk}=\mu +{z}_{l}+{g}_{i}+{(gz)}_{il}+{y}_{k}+{(yz)}_{kl}+{(gy)}_{ik}+{(gyz)}_{ikl}+{s}_{jl}+{(sy)}_{jlk}+{(gs)}_{ijl} +{(gsy)}_{ijlk}^{^{\prime}}$$ where ${z}_{l}$ is the effect of the lth zone. As some agroecological zones were represented by only a few locations, we grouped a few of the neighbouring zones into mega-environments, which resulted in five zones in Germany and four zones in Poland (see Fig. 3). Another extension of Model 3 can be made by considering location nested within zones and zones nested in countries. However, we are not considering this model because our objective here is to determine the correlation between the zones in the countries without considering the effect of the country. For zone-level analyses, only the RG model was fitted since the aim is to exploit the genetic correlation between agroecological zones. Furthermore, in the RG model, different variance–covariance structures for genotype and genotype × country $(gc)$ or genotype × zone $(gz)$ effects were fitted. The models with fixed genotype and random genotype effects are given in Table 3, while Table 4 contains an overview of variance–covariance structures for each term in the models. Table 3 Two fixed genotype effect (FG) and six random genotype effect models Table 4 Variance–covariance structures for each random term in the models Genetic and non-genetic trend analysis and incorporation of nitrogen application as a covariate in the country level analysis Model 2 can be augmented for conducting genetic and non-genetic trend analysis as demonstrated by Piepho et al. (2014), Laidig et al. (2014) and Hadasch et al. (2020). The inclusion of covariates for nitrogen application and year of trial allows a correction for non-genetic trend. Equally importantly, fitting a covariate for the first trial year of a variety allows the estimation of genetic correlation not overly driven by large variation across years generated by genetic gain in the trial period, but more driven by the genetic correlation of common varieties released in a single year. The extended model is $${yield}_{ijlk}=\mu +{c}_{l}+{\beta }_{1}{x}_{1jl}+{\eta }_{l}{x}_{1jl}+{\beta }_{2}{x}_{2i}+{\beta }_{3}{x}_{3i}+{g}_{i}+{(gc)}_{il}+{y}_{k}+{(yc)}_{kl}+{(gy)}_{ik}+ {(gyc)}_{ikl}+{s}_{jl}+ {(sy)}_{jlk}+{(gs)}_{ijl}+{(gsy)}_{ijlk}^{^{\prime}}$$ where ${x}_{1jl}$ is location-specific nitrogen application, ${x}_{2i}$ is the first trial year of a variety, ${x}_{3i}$ is the calendar year of testing of a variety. The notation for fixed regression terms involving the covariate is ${\beta }_{1}{x}_{1jl}+{\eta }_{l}{x}_{1jl}+{\beta }_{2}{x}_{2i}+{\beta }_{3}{x}_{3i}$, where ${\beta }_{1}$, ${\beta }_{2}$ and ${\beta }_{3}$ are the fixed effects for the slopes of nitrogen application, first trial year of a variety and calendar year of testing of a variety, whereas ${\eta }_{l}$ is the country-specific slope of the nitrogen application of the lth country. The nitrogen covariate was mean-centred and then divided by 500, since this scaling resulted in non-negative variance estimates for the random coefficient models. The first year of testing and calendar year were mean-centred. The fixed and random effects genotype models are used in Model 4. The FGC is Model 4 with fixed genotype effects, and the RGC is Model 4 with random genotype effects. For the random genotype effects, the model can be extended to a random coefficient model, which includes the interaction of genotype × nitrogen application and genotype × country × nitrogen application. In this paper, four random coefficient models are fitted. RC1 is a model with random coefficients. A random coefficients model was fitted for the genotype term, ${g}_{i}$, and the genotype × country term, $({gc})_{il}$. Hence, this model has genotype and genotype × country-specific coefficients for the intercepts and slopes. RC2 is a reduced model of RC1, where the random regression coefficients in the genotype main effect are dropped. RC3 is another modification of RC1, in which the random regression coefficients in the genotype × country effect are dropped. RC4 is a modification of RC1 with random regression coefficients for the genotype and genotype-by-country terms with ${\Sigma }_{reg}\otimes {\mathbf{G}}_{{c}_{l}}$ variance–covariance structure (where symbol $\otimes$ represents the Kroneker product of matrices). Thus, the variance structure of the intercepts and slopes is country-specific and allows for covariances between the slopes and intercepts for each of the countries. The details of variance–covariance structures for random coefficient models are explained in the next section and summarised in Table 4. All models were fitted in R (R Core Team, 2022) using ASReml-R package version 4.1.0.130 (Butler et al. 2017). Variance–covariance structures in relation to the random effects of genotype Genotypes in trials can be regarded as a random sample from a population of genotypes and to exploit the genetic correlation between countries and to extend the variance–covariance structure implied by the Model 2, we merged the genetic main effect ${g}_{i}$ and the genotype × country interaction ${(gc)}_{il}$ into a composite genetic effect ${(g\left(c\right))}_{il}$ nested within countries, that is, $$\left( {g\left( c \right)} \right)_{il} = g_{i} + \left( {gc} \right)_{il}$$ If we consider agroecological zones to exploit the genetic correlation between zones, the factor ($c$) is replaced by the factor ($z$). The variance–covariance structure in Eq. 5 is based on the common genotypes in both countries, i.e. $({g}_{i\left(ger\right)}, {g}_{i\left(pol\right)})$, and can have different structures. The compound symmetry (CS) structure in Eq. 6 is implied by Model 2 if both ${g}_{i}$ and ${(gc)}_{il}$ have constant variance. The term ${\sigma }_{g}^{2}+{\sigma }_{gc}^{2}$ on the diagonal of the matrix in Eq. 6 is the variance of genotypes within one country, and the covariance of the same genotype between countries is ${\sigma }_{g}^{2}$, which is on the off-diagonal of the matrix. $$\left[\begin{array}{cc}{\sigma }_{g}^{2}+{\sigma }_{gc}^{2}& {\sigma }_{g}^{2}\\ {\sigma }_{g}^{2}& {\sigma }_{g}^{2}+{\sigma }_{gc}^{2}\end{array}\right]$$ The CS structure assumes a common variance within countries and a common covariance between countries, which is very restrictive and, in many cases, is an unrealistic assumption. The unstructured (UN) variance–covariance given in Eq. 7 is more flexible than the CS structure in the sense that it allows different variances and a separate parameter ${\rho }_{l{l}^{^{\prime}}}$ that specifies the correlation between countries l and ${l}^{^{\prime}}$. $$\left[\begin{array}{cc}{\sigma }_{g(l)}^{2}& {\sigma }_{g(l)}{\sigma }_{g({l}^{^{\prime}})}{\rho }_{{ll}^{^{\prime}}}\\ {\sigma }_{g(l)}{\sigma }_{g({l}^{^{\prime}})}{\rho }_{{ll}^{^{\prime}}}& {\sigma }_{g({l}^{^{\prime}})}^{2}\end{array}\right]$$ In the case of two countries, the UN structure is easy to fit. However, this variance–covariance structure becomes very complex if there are several countries or agroecological zones. For example, with five zones, 15 variance–covariance component estimates need to be estimated in the unstructured model. A less computationally expensive structure allowing for heterogeneity of variances and covariances is the factor analytic (FA) model. The first-order FA (FA1) model is composed of covariance terms that are defined as the product ${\lambda }_{l}{\lambda }_{{l}^{^{\prime}}}$, where ${\lambda }_{l}$ is the loading for the lth country/zone. The variance for the lth country/zones is represented by the term ${\lambda }_{l}^{2}$ and an additional variance component ${\psi }_{l}$. It is also possible to have an FA structure by omitting the term ${\psi }_{l}$, which here is called reduced rank FA or FA-0. The FA1 structure for five zones needs only five ${\lambda }_{l}$ terms and five ${\psi }_{l}$ terms as shown in Eq. 8, which is less expensive than the UN structure. When Eq. 8 uses the reduced rank FA order 1 structure (FA-01), then it needs only five ${\lambda }_{l}$ terms. $${\varvec{\Lambda}}{{\varvec{\Lambda}}}^{\mathrm{T}}+{\varvec{\Psi}}=\left[\begin{array}{ccccc}{\lambda }_{1}^{2}+{\psi }_{1}& {\lambda }_{1}{\lambda }_{2}& {\lambda }_{1}{\lambda }_{3}& {\lambda }_{1}{\lambda }_{4}& {\lambda }_{1}{\lambda }_{5}\\ {\lambda }_{2}{\lambda }_{1}& {\lambda }_{2}^{2}+{\psi }_{2}& {\lambda }_{2}{\lambda }_{3}& {\lambda }_{2}{\lambda }_{4}& {\lambda }_{2}{\lambda }_{5}\\ {\lambda }_{3}{\lambda }_{1}& {\lambda }_{3}{\lambda }_{2}& {\lambda }_{3}^{2}+{\psi }_{3}& {\lambda }_{3}{\lambda }_{4}& {\lambda }_{3}{\lambda }_{5}\\ {\lambda }_{4}{\lambda }_{1}& {\lambda }_{4}{\lambda }_{2}& {\lambda }_{4}{\lambda }_{3}& {\lambda }_{4}^{2}+{\psi }_{4}& {\lambda }_{4}{\lambda }_{5}\\ {\lambda }_{5}{\lambda }_{1}& {\lambda }_{5}{\lambda }_{2}& {\lambda }_{5}{\lambda }_{3}& {\lambda }_{5}{\lambda }_{4}& {\lambda }_{5}^{2}+{\psi }_{5}\end{array}\right]$$ In Model 4, we include the nitrogen application as a covariate. Thus, when the genotype effect is random, the genotype-specific regression must be modelled as random effects as well, as demonstrated by Buntaran et al. (2021). Therefore, the term ${g}_{i}$ can be expanded as a random coefficient of genotype × nitrogen application, ${g}_{i}={a}_{i}+{b}_{i}{x}_{1jl}$, where ${a}_{i}$ is the random intercept for the $i$ th genotype and ${b}_{i}$ is the random slope for the $i$ th genotype. In this case, the variance–covariance structure of ${\mathbf{G}}_{g}$ has variance estimates for the random intercept and random slopes with a covariance between random intercept and random slope. Furthermore, in the random coefficient model, it is important to have the UN structure for ${\mathbf{G}}_{g}$ to ensure invariance with respect to translation and scale transformation of the covariate (Longford, 1993; Wolfinger, 1996; Piepho and Ogutu, 2002). The random coefficient can also be applied for the genotype × country × nitrogen application term, which is expanded as ${(gc)}_{il}={p}_{il}+{q}_{il}{x}_{1jl}$, where ${p}_{il}$ is the random intercept for the $i$ th genotype in the $l$ th country and ${q}_{il}$ is the random slope for the $i$ th genotype in the $l$ th country. Country level analysis The fit statistics for all models listed in Table 3 are reported in Table 5. The Akaike information criterion (AIC) based on the full maximum likelihood method is used to compare different fixed effects terms in the models, while the AIC based on the restricted maximum likelihood (REML) was used to compare models with different variance–covariance structures for the random effects. For all models, the AIC was calculated (the smaller AIC is a better fit) using the infoCriteria function from the asremlPlus library. Table 5 Fit statistics for model selection of eight models fitted with (restricted) maximum likelihood method from country-based analysis The infoCriteria function could not compute the full likelihood of FG and FGC models (as the iterations did not converge), so we could not compare these two models. Furthermore, the main purpose of the analysis is to borrow strength between countries, so the comparison of the RG and RGC models is more essential. The complex variance–covariance structure in the RG model did not improve the fit statistics since the AIC based on REML was slightly higher for FA-01 and UN structures than for CS. Therefore, the simpler CS structure was sufficient to explain the variations of $gc$ interaction effects. For the RGC model, the UN structure had the smallest REML-based AIC, although the value was only slightly smaller than that for the CS structure. On the other hand, the FA-01 structure had a far larger AIC compared to the UN and CS structures. Compared to the RG model, the RGC model had a smaller AIC-full maximum likelihood based on the same variance–covariance structure for $gc$. Thus, the covariates improved the fit statistics. Among the random coefficient models, the RC4 model had the smallest AIC based on both REML and full maximum likelihood. The RC4 model fitted slightly differently from the other random coefficient models because it had the ${\Sigma }_{reg}\otimes {\mathbf{G}}_{{c}_{l}}$ structure as shown in Table 4, which combined the random coefficient regression for the $g$ and $gc$ terms in a single term $g\left(c\right)$. The AIC between RC1 and RC2 were the same, which showed that dropping the random coefficient term in the genotype did not change the fit statistics. However, when the random coefficient was only retained for the genotype term, the AIC increased. Table 6 presents the variance component estimates for model RGC-UN and RC4, with the smallest AIC (estimates of variance components from all models are given in the supplementary Table S1). In these two models, the genetic correlations between the two countries, ${\rho }_{g({c}_{\mathrm{1,2}})}$, were very similar, i.e. 0.890 and 0.884, for the RGC-UN and the RC4 models, respectively. This suggests that the performance and the rank of the overlapping genotypes between the two countries were quite similar. Moreover, the genetic and non-genetic trends and the nitrogen application effects were similar for the two models. Both the regression coefficient of the nitrogen application and the genetic trend were positive. The regression coefficient for non-genetic trend was negative but non-significant. However, the genetic trend and nitrogen application coefficients were positive and larger than the non-genetic trend, so overall the yield was still increasing. Table 6 Estimates of covariates and variance components of two best model (RGC and RC4) and associated standard errors (s.e.) Figure 4 depicts the country pair-wise scatterplots of genotype estimates of genotye × country interaction effects from the FG, FGC, RG-UN, RGC-UN and RC4 models. There is a clear distinction between the fixed genotype effect models (FG and FGC) and the random genotype effect models (RGs and RC models). The estimates in the random genotype effect models were more similar between the two countries compared to the fixed genotype effect models, as a result of borrowing information, i.e. the genetic correlation exploited between the two countries. Moreover, we can see that the correlations between mean yield from the two countries from random genotype effect models were close to 1. The high correlation implies that the performance and ranking of the common genotypes were very similar between Germany and Poland. Yield prediction of 43 common varieties in Germany and Poland from a FG, b FGC, c RG-UN, d RGC-UN and e RC4 models Agroecological zone level analysis The zone-based analysis per country was performed prior to the joint analysis of agroecological zones between countries which examines the correlations between zones within a country. The analysis was performed using the RGC model and replacing country effect (${c}_{l}$) with zone effect (${z}_{l}$). The estimates of variance components from the analysis are given in Table 7. The variance estimate of the genotype × zone effect in Poland was zero, which means that there is no necessity for agroecological zonation or division for maize. However, a small variation in genotype × zone effect can be seen in the German data. The estimates of genotype variance in each zone and zone correlations for German data are given in supplementary Table S2. The small genotype × zone variance estimate in Table 7 for both countries can be explained by the fact that the genotype variance estimates were very similar across zones and there were strong correlations between zones. Table 7 Estimates of variance components and their standard errors of using RGC model for agroecological zones in Poland and Germany As there were not enough genotypes tested for several years across zones in the German data to fit more complex models, the heterogeneous diagonal zone-specific variance structure for genotype × year × zone effect was used. By comparison, in the Polish data, the unstructured variance could be fitted for the genotype × year × zone effect. However, to be consistent for both countries, we also fitted the heterogeneous diagonal zone-specific structure for the genotype × year × zone effect in the Polish data. Since the variance estimate of genotype × zone effects in the Polish data was zero, in the joint analysis of German and Polish agroecological zones, we merged all zones of Poland into one zone. Thus, the dataset for the joint analysis consisted of five zones in Germany and Poland as one zone. The analysis based on agroecological zones was performed using the RGC model. The estimates of variance components from zone-based analysis using the RGC model are given in supplementary Table S3. The genetic correlations between German zones and Poland are given in Fig. 5. In most of the cases, the genetic correlations are high between the German zones; however, only German zone D3 has a high correlation with Poland. The genetic correlation between zones was based on the performance of the common genotypes. Although the zones are far from each other on the map, the performances of the common genotypes between these zones were very similar, making the genetic correlation high. Genetic correlations between German zones and Poland using RGC model A high genetic correlation was observed between Germany and Poland (Table 6), and this is reflected in the similarity of genetic ranking based on the mean yield of common varieties from Germany and Poland reported in Fig. 4. The fitting of heteroscedastic variance–covariance structures to the genotype × country classification provides a view of the similarities between countries. The fit in terms of AIC was not improved for the heterogeneous UN model; however, higher-order models should be preferred to avoid underfitting and the resulting bias (Piepho, 2008). The incorporation of covariates allows for improving models as given in Table 5. The random coefficient models enable the user to fit genotype and genotype × country regressions. However, the results show that random coefficient models could not improve the fit because of the high similarity between Germany and Poland, and genotypes performing similarly in both countries. Because of the high genetic correlation between Germany and Poland, one country can benefit from using maize trial data from the other country and vice versa. The stratification of locations into zones and the use of random effects models for the genotype × zone classification allows for the borrowing of strength across zones when estimating mean yield per zone. However, to attain reliable estimates of genotype effects, it is necessary for the zone to be represented by a suitable number of locations (Kleinknecht et al. 2013). Many zones in Germany and Poland were not represented by enough locations (Fig. 3); therefore, these zones were merged with the neighbouring zones after assessing the genetic correlations between zones. This resulted in five zones in Germany and four zones in Poland, subsequently used for zone-specific analysis. The heteroscedastic variance–covariance structures used for the genotype × zone classification provided estimates of similarities between zones. The variance estimate of the genotype × zone effect in Poland was zero (Table 7), which means that there is no necessity for agroecological zonation or division for maize. However, a small variation in the genotype × zone effect can be seen in German data. The estimates of genetic correlation between Germany and Poland zones (Fig. 5) indicate that the German zone D3 is highly correlated with Poland. The German zones are also very similar to one another in terms of the mean yield comparisons among genotypes. The high genetic correlation between German and Polish zones does not necessarily mean that zones are agroecologically similar, although climatic and soil conditions in these zones are distinct from one another (Graf et al. 2009). It is well possible that the mean yield of genotypes in the zones are not significantly affected by agroecological differences among the zones, so a high genetic correlation can occur despite such differences, especially when the genetic variance is not very large compared to variance components for genotype × environment interaction effects. Data were provided by the Federal Plant Variety Office Germany and Research Centre for Cultivar Testing Poland for exclusive use in this study and are in general not publicly available. Reasonable requests may be addressed to Federal Plant Variety Office, Hannover, Germany, and Research Centre for Cultivar Testing, Słupia Wielka, Poland. Buntaran H, Piepho HP, Hagman J, Forkman J (2019) A cross-validation of statistical models for zoned-based prediction in cultivar testing. Crop Sci 59:1544–1553. https://doi.org/10.2135/cropsci2018.10.0642 Buntaran H, Piepho HP, Schmidt P, Rydén J, Halling M, Forkman J (2020) Cross-validation of stage-wise mixed-model analysis of Swedish variety trials with winter wheat and spring barley. Crop Sci 60:2221–2240. https://doi.org/10.1002/csc2.20177 Buntaran H, Forkman J, Piepho HP (2021) Projecting results of zoned multi-environment trials to new locations using environmental covariates with random coefficient models: accuracy and precision. Theor Appl Genet 134:1513–1530. https://doi.org/10.1007/s00122-021-03786-2 Butler DG, Cullis B, Gilmour A, Gogel BJ, Thompson R (2017) ASReml-R reference manual, version 4. University of Wollongong, Wollongong Denis JB (1988) Two-way analysis using covariates. Statistics 19:123–132. https://doi.org/10.1080/02331888808802080 Gauch HG, Zobel RW (1997) Identifying mega-environments and targeting genotypes. Crop Sc 37:311–326 Graf R, Michel V, Roßberg D, Neukampf R (2009) Definition pflanzenartspezifischer Anbaugebiete für ein regionalisiertes Versuchswesen im Pflanzenbau. J Für Kulturpflanzen 61(7):247–253. https://doi.org/10.5073/JfK.2009.07.02 Hadasch S, Laidig F, Macholdt J, Bönecke E, Piepho HP (2020) Trends in mean performance and stability of winter wheat and winter rye yields in a long-term series of variety trials. Field Crops Res 252:107792. https://doi.org/10.1016/j.fcr.2020.107792 Kleinknecht K, Möhring J, Singh KP, Zaidi PH, Atlin GN, Piepho HP (2013) Comparison of the performance of best linear unbiased estimation and best linear unbiased prediction of genotype effects from zoned Indian maize data. Crop Sci 53:1384–1391. https://doi.org/10.2135/cropsci2013.02.0073 Laidig F, Drobek T, Meyer U (2008) Genotypic and environmental variability of yield for cultivars from 30 different crops in German official variety trials. Plant Breed 127:541–547. https://doi.org/10.1111/j.1439-0523.2008.01564.x Laidig F, Piepho HP, Drobek T, Meyer U (2014) Genetic and non-genetic long-term trends of 12 different crops in German official variety performance trials and on-farm yield trends. Theor Appl Genet 127:2599–2617. https://doi.org/10.1007/s00122-014-2402-z Longford NT (1993) Random coefficient models. Oxford University Press, New York Maiskomitee D (2022) Deutsches Maiskomitee e.V. (DMK). www.maiskomitee.de Piepho HP, Möhring J (2005) Best linear unbiased prediction of cultivar effects for subdivided target regions. Crop Sci 45:1151–1159. https://doi.org/10.2135/cropsci2004.0398 Piepho HP, Ogutu JO (2002) A simple mixed model for trend analysis in wildlife populations. J Agric Biol Environ Stat 7:350. https://doi.org/10.1198/108571102366 Piepho HP, Möhring J, Melchinger AE, Büchse A (2008) BLUP for phenotypic selection in plant breeding and variety selection. Euphytica 161:209–228. https://doi.org/10.1007/s10681-007-9449-8 Piepho HP, Laidig F, Drobek T, Meyer U (2014) Dissecting genetic and non-genetic sources of long-term yield trend in German official variety trials. Theor Appl Genet 127:1009–1018. https://doi.org/10.1007/s00122-014-2275-1 R Core Team (2022) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. https://www.R-project.org van Eeuwijk FA, Bustos-Korts DV, Malosetti M (2016) What should students in plant breeding know about the statistical aspects of genotype × environment interactions? Crop Sci 56:2119–2140. https://doi.org/10.2135/cropsci2015.06.0375 Wolfinger RD (1996) Heterogeneous variance: covariance structures for repeated measures. J Agric Biol Environ Stat 1:205–230. https://doi.org/10.2307/1400366 This work was supported by German Research Foundation (DFG), Grant No. PI 377/20-2. We are thankful to Dr. Friedrich Laidig (Institute of Crop Science, University of Hohenheim, Germany) for his suggestion for the data analysis and valuable comments for improving the paper. Biostatistics Unit, Institute of Crop Science, University of Hohenheim, Fruwirthstrasse 23, 70599, Stuttgart, Germany Waqas Ahmed Malik, Harimurti Buntaran & Hans-Peter Piepho Research Centre for Cultivar Testing, Słupia Wielka 34, 63-022, Słupia Wielka, Poland Marcin Przystalski & Tomasz Lenartowicz Waqas Ahmed Malik Harimurti Buntaran Marcin Przystalski Tomasz Lenartowicz Hans-Peter Piepho WAM and HPP perceived the idea of investigating the topic. WAM and HB performed the analysis with the subsequent input by MP and TL. WAM wrote the early version of the manuscript. HB and HPP amended it. All authors read and reviewed the manuscript. Correspondence to Waqas Ahmed Malik. The authors declare that the experiments comply with the current laws of the countries in which the experiments were performed. Communicated by Daniela Bustos-Korts. Supplementary file1 (DOCX 46 kb) Malik, W.A., Buntaran, H., Przystalski, M. et al. Assessing the between-country genetic correlation in maize yield using German and Polish official variety trials. Theor Appl Genet 135, 3025–3038 (2022). https://doi.org/10.1007/s00122-022-04164-2 Issue Date: September 2022	CommonCrawl
$L_1$-norm of combinations of products of independent random variables [PDF] Rafa? Lata?a Mathematics , 2013, DOI: 10.1007/s11856-014-1076-1 Abstract: We show that $L_1$-norm of linear combinations (with scalar or vector coefficients) of products of i.i.d. nonnegative mean one random variables is comparable to $l_1$-norm of coefficients. On the number of zeros of linear combinations of independent characteristic polynomials of random unitary matrices [PDF] Yacine Barhoumi,Chris Hughes,Joseph Najnudel,Ashkan Nikeghbali Mathematics , 2013, Abstract: We show that almost all the zeros of any finite linear combination of independent characteristic polynomials of random unitary matrices lie on the unit circle. This result is the random matrix analogue of an earlier result by Bombieri and Hejhal on the distribution of zeros of linear combinations of $L$-functions, thus providing further evidence for the conjectured links between the value distribution of the characteristic polynomial of random unitary matrices and the value distribution of $L$-functions on the critical line. Distribution functions of linear combinations of lattice polynomials from the uniform distribution [PDF] Jean-Luc Marichal,Ivan Kojadinovic Abstract: We give the distribution functions, the expected values, and the moments of linear combinations of lattice polynomials from the uniform distribution. Linear combinations of lattice polynomials, which include weighted sums, linear combinations of order statistics, and lattice polynomials, are actually those continuous functions that reduce to linear functions on each simplex of the standard triangulation of the unit cube. They are mainly used in aggregation theory, combinatorial optimization, and game theory, where they are known as discrete Choquet integrals and Lovasz extensions. Linear programming problems for l_1- optimal frontier estimation [PDF] Stéphane Girard,Anatoli Iouditski,Alexander Nazin Statistics , 2011, Abstract: We propose new optimal estimators for the Lipschitz frontier of a set of points. They are defined as kernel estimators being sufficiently regular, covering all the points and whose associated support is of smallest surface. The estimators are written as linear combinations of kernel functions applied to the points of the sample. The coefficients of the linear combination are then computed by solving related linear programming problem. The L_1 error between the estimated and the true frontier function with a known Lipschitz constant is shown to be almost surely converging to zero, and the rate of convergence is proved to be optimal. Sign changes in linear combinations of derivatives and convolutions of Polya frequency functions [cached] Steven Nahmias,Frank Proschan International Journal of Mathematics and Mathematical Sciences , 1979, DOI: 10.1155/s0161171279000090 Abstract: We obtain upper bounds on the number of sign changes of linear combinations of derivatives and convolutions of Polya frequency functions using the variation diminishing properties of totally positive functions. These constitute extensions of earlier results of Karlin and Proschan. Linear combinations of sections and tails of Mittag-Leffler functions and their zeros [PDF] N. A. Zheltukhina Abstract: The zero distribution of sections of Mittag-Leffler functions of order >1 was studied in 1983 by A. Edrei, E.B. Saff and R.S. Varga. In the present paper, we study the zero distribution of linear combinations of sections and tails of Mittag-Leffler functions of order >1. Expected number of real zeros for random linear combinations of orthogonal polynomials [PDF] D. S. Lubinsky,I. E. Pritsker,X. Xie Abstract: We study the expected number of real zeros for random linear combinations of orthogonal polynomials. It is well known that Kac polynomials, spanned by monomials with i.i.d. Gaussian coefficients, have only $(2/\pi + o(1))\log{n}$ expected real zeros in terms of the degree $n$. On the other hand, if the basis is given by Legendre (or more generally by Jacobi) polynomials, then random linear combinations have $n/\sqrt{3} + o(n)$ expected real zeros. We prove that the latter asymptotic relation holds universally for a large class of random orthogonal polynomials on the real line, and also give more general local results on the expected number of real zeros. Some inequalities of linear combinations of independent random variables: II [PDF] Xiaoqing Pan,Maochao Xu,Taizhong Hu Statistics , 2013, DOI: 10.3150/12-BEJ429 Abstract: Linear combinations of independent random variables have been extensively studied in the literature. However, most of the work is based on some specific distribution assumptions. In this paper, a companion of (J. Appl. Probab. 48 (2011) 1179-1188), we unify the study of linear combinations of independent nonnegative random variables under the general setup by using some monotone transforms. The results are further generalized to the case of independent but not necessarily identically distributed nonnegative random variables. The main results complement and generalize the results in the literature including (In Studies in Econometrics, Time Series, and Multivariate Statistics (1983) 465-489 Academic Press; Sankhy\={a} Ser. A 60 (1998) 171-175; Sankhy\={a} Ser. A 63 (2001) 128-132; J. Statist. Plann. Inference 92 (2001) 1-5; Bernoulli 17 (2011) 1044-1053). Multi-Target Regression via Random Linear Target Combinations [PDF] Grigorios Tsoumakas,Eleftherios Spyromitros-Xioufis,Aikaterini Vrekou,Ioannis Vlahavas Computer Science , 2014, DOI: 10.1007/978-3-662-44845-8_15 Abstract: Multi-target regression is concerned with the simultaneous prediction of multiple continuous target variables based on the same set of input variables. It arises in several interesting industrial and environmental application domains, such as ecological modelling and energy forecasting. This paper presents an ensemble method for multi-target regression that constructs new target variables via random linear combinations of existing targets. We discuss the connection of our approach with multi-label classification algorithms, in particular RA$k$EL, which originally inspired this work, and a family of recent multi-label classification algorithms that involve output coding. Experimental results on 12 multi-target datasets show that it performs significantly better than a strong baseline that learns a single model for each target using gradient boosting and compares favourably to multi-objective random forest approach, which is a state-of-the-art approach. The experiments further show that our approach improves more when stronger unconditional dependencies exist among the targets. Delay on broadcast erasure channels under random linear combinations [PDF] Nan Xie,Steven Weber Computer Science , 2013, Abstract: We consider a transmitter broadcasting random linear combinations (over a field of size $d$) formed from a block of $c$ packets to a collection of $n$ receivers, where the channels between the transmitter and each receiver are independent erasure channels with reception probabilities $\mathbf{q} = (q_1,\ldots,q_n)$. We establish several properties of the random delay until all $n$ receivers have recovered all $c$ packets, denoted $Y_{n:n}^{(c)}$. First, we provide upper and lower bounds, exact expressions, and a recurrence for the moments of $Y_{n:n}^{(c)}$. Second, we study the delay per packet $Y_{n:n}^{(c)}/c$ as a function of $c$, including the asymptotic delay (as $c \to \infty$), and monotonicity properties of the delay per packet (in $c$). Third, we employ extreme value theory to investigate $Y_{n:n}^{(c)}$ as a function of $n$ (as $n \to \infty$). Several results are new, some results are extensions of existing results, and some results are proofs of known results using new (probabilistic) proof techniques.	CommonCrawl
Algorithms for Molecular Biology On a greedy approach for genome scaffolding Tom Davot ORCID: orcid.org/0000-0003-4203-51401, Annie Chateau1,2, Rohan Fossé3, Rodolphe Giroudeau1 & Mathias Weller4 Algorithms for Molecular Biology volume 17, Article number: 16 (2022) Cite this article Scaffolding is a bioinformatics problem aimed at completing the contig assembly process by determining the relative position and orientation of these contigs. It can be seen as a paths and cycles cover problem of a particular graph called the "scaffold graph". We provide some NP-hardness and inapproximability results on this problem. We also adapt a greedy approximation algorithm on complete graphs so that it works on a special class aiming to be close to real instances. The described algorithm is the first polynomial-time approximation algorithm designed for this problem on non-complete graphs. Tests on a set of simulated instances show that our algorithm provides better results than the version on complete graphs. In this paper, we focus on a bioinformatic problem occurring in the production of genomes. Genomes are usually obtained by sequencing. Sequencing produces an important amount of small sequences of nucleotides called reads. Herein, the lengths range from hundreds to tens of thousands of characters, depending on the sequencing technology. As a rule of thumb, shorter reads, produced for example by second generation sequencing (Illumina) have a higher quality (contain less read-errors) than long reads produced by third generation sequencing technologies (PacBio or Oxford Nanopore) [1]. The assembly process exploits overlaps between reads to reconstruct the targeted sequence. However, this is complicated by repeated parts in real-world genomes. Assembly algorithms cannot uniquely infer the original genome if it contains such repeated regions (the longer the repeated region with respect to the read length, the harder it is to infer the original genome). To avoid misassembly, such algorithms reconstruct only parts of the genome which is then returned as set of "contiguous regions" (or contigs). A thus fragmented genome is not ideal for further processing, and one would like to have as few contigs as possible while avoiding misassembly. A way to approach this are hybrid strategies using both long and short reads [2]. However, many genomes comprising current databases have been assembled before the development of third generation sequencing, preventing such hybrid strategies. One way to reduce the fragmentation of genomes in these databases while avoiding costly re-sequencing, is the exploitation of "meta-information" about the available reads. Genome scaffolding In second generation sequencing, short reads come in pairs, indicating that a fragment of the DNA molecule exists whose ends correspond to the two reads of a pair. In particular, the total length of said fragment is known approximately. This pairing information can be used to infer the order (and orientation) of the given contigs on the chromosome, thus completing the genome (modulo possible gaps between the contigs). The mathematical problem modeling this inference, called scaffolding, is made complicated by possible inconsistencies in the pairing information. See [3] for a recent overview of models, variants, and methods in this context. The problem we study here is an optimization problem in a special graph called scaffold graph. The present formulation use both pairing information and some genomic structural constraints, like a fixed number of linear and circular chromosomes. In [4], we presented preliminary results about the complexity of this problem and a first polynomial-time approximation on complete graphs. Those results were extended and completed by another polynomial-time algorithm [5] and by a randomized approach [6]. We also explored exact algorithms [7], and studied some sparse special cases of scaffold graphs [8]. The contribution of the present paper is a continuation of published works [9, 10], where special classes of graphs have been studied (from sparse to very dense). Since real instances are usually sparse but contain some dense regions, due to abundance of repeats [11], we are interested in graphs built from cliques that are separated by bridges (i.e. edges whose removal disconnects the graph). The main contribution is the extension of the approximation algorithm on complete graphs of Chateau and Giroudeau [5] to a particular class called "connected cluster graph ". Ultimately, the objective is to adapt the algorithm to sparse classes of graphs. To keep the approximation algorithm in polynomial time, one condition is that the decision problem of the scaffolding must be solvable in polynomial time. We propose a negative result, (i.e. it is $\mathcal{NP}\mathcal{}$-complete) for a particular sparse graph class. Finally, since the presented approximation has a polynomial approximation ratio in some particular cases, we show that the scaffolding problem can not be approximed with a ratio better than a polynomial function in such cases. Organization of the paper The next section is devoted to notations and the description of the scaffold problem. In "Computational hardness" section, we show a $\mathcal{NP}\mathcal{}$-hardness results for sparse scaffolding graphs. In "Non-approximability" section, we address inapproximability. "Feasibility function for connected cluster graphs" section is devoted to a greedy algorithm for a special class of graph called connected cluster graph. Finally, we provide experimental results for the greedy algorithm. Notation and problem description Graph definitions For a graph G, we denote by V(G) and E(G) the set of vertices and edges of G, respectively. Let u be a vertex of G, the degree d(u) of u is the number of edges incident with u. The girth g(G) of G is the length of the smallest cycle of G. A graph is bipartite if its vertices can be partitioned into two sets of non-adjacent vertices. A graph is planar if it can be drawn in the two-dimensional plane without crossing edges. A matching $M^\subseteq E(G)$ of G is a set of non-adjacent edges. $M^$ is called perfect if it touches all vertices of G. For a vertex u, we let $M^(u)$ denote the unique vertex v (if it exists) such that $uv \in M^$. In a scaffold graph, vertices represent extremities of contigs. Given a matching $M^$, the matching edges represent contigs and edges outside the matching represent possible contiguity relationship between contigs. The confidence that two contigs (more precisely, contig-extremities) occur consecutively in the genomic sequence is represented by a weight on edges outside the matching. An alternating path (resp. alternating cycle) is a path (resp. cycle) such that its edges alternatingly belong to $M^$ or not. The extremal edges of an alternating path must be in $M^$. A clique of G is a set of vertices such that all vertices are adjacent. A bridge (resp. cut vertex) of G is an edge (resp. vertex) such that its deletion increases by one the number of connected components of G. In "Feasibility function for connected cluster graphs" section, we study a particular class of graph called connected cluster graph, defined as follows. Definition 1 A connected cluster graph G is a graph that admits a decomposition of its edges $E(G)=E'\cup B$ such that the subgraph induced by $E'$ is a disjoint union of cliques and each edge $e \in B$ is a bridge of G. An example of a connected cluster graph is given in Fig. 1. Example of a connected cluster graph. The bridge edges are bold Scaffolding problem A scaffold graph $(G^,M^,\omega )$ is a simple, loopless graph $G^$ with a perfect matching $M^$ and a weight function $\omega$ on the non-matching edges. The matching $M^$ represents the set of contigs and for an edge uv, $\omega (uv)$ indicates the confidence that the contig extremity v follows the contig extremity u in the genomic sequence.Footnote 1 The alternating girth of a scaffold graph denoted by $g^(G^)$ is the number of matching edges in the smallest alternating cycle of $(G^,M^,\omega )$. In this paper, we study a decision and optimization version of scaffolding, defined as follows. The two integers $\sigma _p$ and $\sigma _c$ are used to model restrictions on the sought genomic structure by representing the number of linear and circular chromosomes, respectively. Let S be a collection of p alternating paths and c alternating cycles. We call the number $p+c$ the cardinality of S and, we let $\sigma _p(S):=p$ and $\sigma _c(S):=c$. The main contribution of this paper is an extension of a known polynomial-time 3-approximation [5] to connected cluster graphs. Whereas the original algorithm was developed to work in complete graphs, it can be adapted for the general case, as shown in Algorithm 1. The idea of this greedy algorithm is to consider each non-matching edge in decreasing order of weight and add it into a partial solution, if possible. The key instruction is the feasibility function: given a partial solution S and an edge e, this function indicates whether $S\cup e$ can still be extended into a collection of $\sigma _c$ alternating cycles and $\sigma _p$ alternating paths in $(G^,M^)$. Proposition 1 Let f be a feasibility function with time complexity $\mathcal {O}(t)$. Algorithm 1 gives an approximate solution for max scaffolding (if it exists) in $\mathcal {O}(\|E(G^)\| \cdot (t+\log \|E(G^)\|))$. The solution S given in the input of the feasibility function is called initiating solution. In general, since scaffolding is $\mathcal{NP}\mathcal{}$-complete, feasibility cannot be decided in polynomial-time, even if $S=\varnothing$ (unless $\mathcal {P}$ =$\mathcal{NP}\mathcal{}$). Thus, we focus on restricted classes of graphs. In [5], a constant-time feasibility function was developed for complete graphs, leading to the following result. Theorem 1 ([5]) In complete graphs, Algorithm 1 gives a solution with an approximation factor of 3. In "Feasibility function for connected cluster graphs" section, we develop a feasibility function for connected cluster graphs and show that Algorithm 1 gives a 5-approximate solution in this case. Notice that, on graph classes containing the $2\times k$ grids, the worst-case approximation factor of the greedy algorithm cannot be better than polynomial, even if a polynomial-time feasibility function exists (see Fig. 2). Unbounded ratio of Algorithm 1 in the general case. Let $(G^,M^,\omega )$ be a $2 \times k$ grid where the perfect matching (bold edges) corresponds to the edges between the two rows. Let $(x_1,\dots ,x_k)$ and $(y_1,\dots ,y_k)$ be the vertices of the first and second row, respectively. We are looking for a solution of max scaffolding with $\sigma _c =0$ and $\sigma _p=1$. If the algorithm chooses first the edge $x_1x_2$, then the only feasible solution is $S = \{x_{\ell }x_{\ell } \mid \ell \mod 2 = 1 \} \cup \{ y_{\ell }y_{\ell +1} \mid \ell \mod 2 = 0 \}$ (dashed edges). Suppose that an optimal solution is $S_{opt}=E(G^)\setminus (M^\cup S)$ (solid edges). If all edges of $S_{opt}$ and $x_1x_2$ are valued by one and all edges of $S \setminus \{x_1x_2\}$ are valued by zero, then we have $(k-1)\cdot \omega (S) = \omega (S_{opt})$ which leads to an unbounded ratio We conclude this section with a note on real-world instances, which are too sparse to fall into our considered class. However, we can transform them by adding some non-matching edges with weight zero. This technique was used to run the feasibility function for complete graphs on simulated instances [5] and the computed solution was close to the optimal. One of the reasons we develop a feasibility function for connected cluster graphs is that we conjecture that using a feasibility function for a graph class that is closer to the original instance (edge-deletion distance from the class) provides better approximation in practice, even though the theoretical approximation factor of the algorithm becomes worse. We test this hypothesis in "Experimental results" section. Computational hardness Like said in the previous section, when using the greedy algorithm on a real instance, we must complete the original instance by adding non-matching edges with weight zero. To minimize the number of added edges, the solution is to adapt the greedy algorithm to a sparse class of graphs. In order to do that, scaffolding must be solvable in polynomial time in this particular class since otherwise, the feasibility function can not be run in polynomial time. In this section, we show that scaffolding is $\mathcal{NP}\mathcal{}$-hard for the particular class of graphs where $\|M^\|=2\sigma _c + \sigma _p$. That is, we show that the greedy algorithm can not be executed in polynomial time in this special case. In such instance, any feasible solution S contains only alternating paths of length one and alternating cycles of length four (i.e. the smallest possible elements). While scaffolding is polynomial in this case [5], a natural extension would be to consider slightly longer alternating paths and alternating cycles. Unfortunately however, it turns out that deciding whether $(G^,M^)$ contains a collection with alternating paths of length one and alternating cycles of length six is already $\mathcal{NP}\mathcal{}$-complete. In order to show this, we focus on the value of the alternating girth of the scaffold graph. Indeed, in a solution of scaffolding with $g^(G^) \cdot \sigma _c + \sigma _p$ edges, each alternating path consists of exactly one matching edge and each alternating cycle is an alternating girth. We show that finding such a solution is $\mathcal{NP}\mathcal{}$-complete, even if $g^(G^)=3$, by reducing independent set to it. IS is $\mathcal{NP}\mathcal{}$-complete in general graphs. In order to build our reduction, we need G to be subcubic and triangle-free (i.e. $\Delta (G)\le 3$ and $g(G) > 3$). Note that Lozin et Milanič [12] showed that independent set remains $\mathcal{NP}\mathcal{}$-complete in $\mathcal {F}$-free planar subcubic graphs if $\mathcal {F}$ does not contain a tree with exactly three leaves. By choosing $\mathcal {F}:=\{C_3\}$ (where $C_3$ is the cycle on three vertices), we obtain the desired $\mathcal{NP}\mathcal{}$-completeness. Our reduction uses the following construction. (see Fig. 3) Given a subcubic, triangle-free graph G, construct a scaffold graph $(G^,M^,\omega )$ as follows: for each edge $e_i \in E(G)$, construct a matching edge $u_i\overline{u}_i$, and for each vertex $v_t \in V(G)$, introduce the matching edges $\{u^j_t\overline{u}^j_t \mid j \le 3-deg(v_t)\}=:E_t$ and construct an alternating 6-cycle $C_t$ on the vertices $E_t \cup \{u_i\overline{u}_i \mid v_t \in e_i\}$ such that no two u (or $\overline{u}$) vertices are adjacent. The alternating cycles $C_i$ are called vertex-cycles. A bipartition is given by the u- and $\overline{u}$-vertices. Note that, if G is planar, it is also possible to construct a planar graph (which may no longer be bipartite). To show hardness of scaffolding when $\|M^\|= g^(G^)\cdot \sigma _c + \sigma _p$, we use the following properties of graphs resulting from Construction 1. Example of a scaffold graph produced by Construction 1. Left: input graph with an independent set of size two given by the black vertices. Right: output graph with a collection of two alternating cycles and one alternating path in black. A bipartition is given by gray and white vertices. An example of a vertex-cycle is $C_{v_1}=\{\overline{v}^1_1,v^1_1,\overline{u}_1,u_1,\overline{u}_2,u_2\}$. It is possible to turn this graph into a planar graph by replacing the edges $\{u_3\overline{u}_2,u_2\overline{u_3},u_5\overline{u}_3\}$ with $\{u_3u_2,u_5\overline{u_2},\overline{u}_5\overline{u}_3\}$ Lemma 1 Let G be a subcubic triangle-free graph and let $(G^,M^,\omega )$ be its scaffold graph produced by Construction 1. Let S be a collection of $\sigma _c = k$ alternating cycles and $\sigma _p= \|M^\|-3k$ alternating paths. Then, $g^(G^)=3$, every alternating cycle in S is a vertex-cycle, and let $C_t$ and $C_{t'}$ be vertex-cycles in S, the vertices $v_t$ and $v_{t'}$ are not adjacent in G. By construction, each vertex-cycle contains exactly three matching edges and, thus, $g^(G^) \le 3$. Suppose there is an alternating cycle containing exactly two matching edges e and $e'$. Let $C_t$ be a vertex-cycle containing e. Since $C_t$ has length six, there is another vertex-cycle $C_{t'}\ne C_t$ that contains $e'$. Indeed, e and $e'$ are both in $C_t$ and $C_{t'}$ since, otherwise, their extremities cannot be adjacent. By construction, there are two edges $e_i$ and $e_j$ in G that are incident to both $v_t$ and $v_{t'}$, contradicting G being simple. Hence, there is no alternating cycle with two matching edges and $g^(G^)=3$. Let C be an alternating cycle in S. By Lemma 1(a), $\|M^\| = g^(G^)\cdot \sigma _c + \sigma _p$, implying that C has length six. Let $u_i\overline{u}_i$ be a matching edge of C. If there is a matching edge $v^1_t\overline{v}^1_t \in C$ then, by construction, the third matching edge of C is either $v_t^2\overline{v}_t^2$ (if $deg(v_t)=1$) or $u_j\overline{u}_j$ (where $v_t \in e_j$ in G). Thus, C is the vertex-cycle $C_t$. Suppose there is no matching edge $v^1_t\overline{v}^1_t$ in C. For any pair of matching edges $(u_k\overline{u}_k,u_{k'}\overline{u}_{k'})$ of C, $e_k$ and $e_{k'}$ are incident to a same vertex in G. Let $u_i\overline{u_i},u_j\overline{u_j}$ and $u_k\overline{u_k}$ be the three matching edges of C. Since G is triangle-free, $e_i, e_j$ and $e_k$ are adjacent in G, hence, C is a vertex-cycle. Let $e_i = v_tv_{t'}\in E(G)$. The matching edge $u_i\overline{u}_i$ is in $C_t$ and $C_{t'}$ and, thus, S cannot contain both $C_t$ and $C_{t'}$. In the proof of correctness, we simulate vertices of the independent set with vertex-cycles. If a solution S contains two vertex cycles $C_i$ and $C_j$, then $v_i$ and $v_j$ are not adjacent in G. Hence, if a solution S contains k vertex-cycles, then there is an independent set of k vertices in G. scaffolding is $\mathcal{NP}\mathcal{}$-complete, even in bipartite (or planar) subcubic scaffold graphs $(G^,M^,\omega )$ were $\|M^\| = g^(G^) \cdot \sigma _c + \sigma _p$ and $g^=3$. Since, clearly, scaffolding is in $\mathcal{NP}\mathcal{}$, it remains to show that Construction 1 is a reduction, that is, G has an independent set of size k if and only if there is a collection of k alternating cycles and $\|M^\|-3k$ alternating paths in $(G^,M^)$. "$\Rightarrow$": Let I be an independent set of size k in G. We build a solution of scaffolding as follows. For each vertex $v_t \in I$, we construct the vertex-cycle $C_t$ in S. For each remaining matching edge in $M^\setminus \bigcup _{v_t\in I}C_t$, we construct an alternating path of length one. We obtain a solution S as thought. "$\Leftarrow$": Let S be a solution in $(G^,M^)$ containing k alternating cycles and $\|E(G)-k$ alternating paths and let $I := \{v_t \| C_t \in S\}$. By Lemma 1(b), any alternating cycle of S is a vertex-cycle in $(G^,M^)$ and, thus, $\|I\|=k$. Moreover, by Lemma 1(c), I is independent in G.□ Note that Theorem 2 can be generalized to $g^(G^)>3$ by modifying Construction 1 as follows. First, we build our construction from a graph G with $g(G)>\ell \ge 3$. IS remains $\mathcal{NP}\mathcal{}$-complete in such graphs by the result of Lozin and Milanič: it suffices to take $F=\{C_{i} \mid i \le \ell \}$, where $C_i$ is the cycle of order i. Then, we increase the length of every vertex-cycle by taking $E_t = \{u^j_t\overline{u}^j_t \mid j\le 3+\ell - deg(v_t)\}$ for each $v_t \in V(G)$. By making these modifications, we construct a scaffold graph with $g^(G^) = \ell$ and we preserve properties Lemma 1(b) and Lemma 1(c). This leads to the following result. Corollary 1 scaffolding is $\mathcal{NP}\mathcal{}$-complete even in bipartite (or planar) subcubic scaffold graphs $(G^,M^)$ were $\|M^\| = g^(G^) \cdot \sigma _c + \sigma _p$, for all $g^(G^)\ge 3$. Non-approximability In this section, we discuss the hardness of approximating max scaffolding. Notice that, since scaffolding is $\mathcal{NP}\mathcal{}$-complete, there is no polynomial-time approximation algorithm for max scaffolding (unless $\mathcal {P} =\mathcal{NP}\mathcal{}$). However, this argument does not hold for graph classes where scaffolding is in $\mathcal {P}$ (i.e. classes for which the feasibility function (and, thus, the greedy algorithm) runs in polynomial time). We show that, in this case, max scaffolding is still Poly-APX-hard, that is, it is not possible to approximate max scaffolding within a factor better than a polynomial function in $\|V(G^)\|+\|E(G^)\|$ (unless $\mathcal {P} =\mathcal{NP}\mathcal{}$). Recall that Fig. 2 already shows that the greedy algorithm can not approximate max scaffolding with a ratio better than a polynomial function. The inapproximability result presented in this section shows that it is the case for any polynomial-time algorithm. In the following, we construct an S-reduction (see [13]) from the optimization version of independent set. (see Fig. 4) Let G be a graph. Then, construct the following scaffold graph $(G^,M^,\omega )$: For each $e_i= v_tv_q\in E(G)$, construct a clique $\{u_i^t,\overline{u}_i^t,u^q_i,\overline{u}_i^q,e_i,\overline{e}_i\}$ with $u_i^t\overline{u}_i^t,u_i^q\overline{u}_i^q,e_i\overline{e}_i \in M^$. For each $v_t\in V(G)$, construct a cycle $(v^t_1,\overline{v}^t_1,\overline{v}^t_2,v^t_2)$ with $v^t_1\overline{v}^t_1,v^t_2\overline{v}^t_2 \in M^$. Let $v_t \in V(G)$ and let $\mathcal {A}_t$ be a list of all edges incident with $v_t$ in G. Construct an alternating cycle containing all vertices in $\{v^t_1,\overline{v}^t_1,v^t_2,\overline{v}^t_2\} \cup \{ u^t_i,\overline{u}^t_i \mid \forall e_i \in \mathcal {A}_t\}$ as follows: For all $k < d(v_t)$, let $e_i$ and $e_j$ be the $k^\text {th}$ and $k+1^\text {st}$ edges of $\mathcal {A}_t$, respectively, and add a non-matching edge between $\overline{u}^t_i$ and $u^t_j$. Let $e_i$ and $e_j$ be the first and last edges of $\mathcal {A}_t$, respectively, and add the non-matching edges $v^t_1u^t_i$ and $v^t_2\overline{u}^t_j$. Each non-matching edge has weight zero, except the edges $v^t_2\overline{e}_j$ which have weight one. Example of a scaffold graph produced by Construction 2. The input graph is composed by the edges $e_1 = v_1v_2$, $e_2 = v_2v_3$, $e_3= v_1v_3$, $e_4= v_1v_4$ and $e_5 = v_3v_4$. Gray vertices in the figure belong to an edge gadget and white vertices belong to a vertex gadget. Matching edges are bold. Solid edges have weight zero and dashed edges have weight one. The long vertex-cycle $C(v_2)$ corresponds to the vertices $\{\overline{v}^2_1,v^2_1,u^2_2,\overline{u}^2_2,u^2_1,\overline{u}^2_1,v^2_2,\overline{v}^2_2\}$ Let $v_t\in V(G)$. The cycle on $\{v^t_1,\overline{v}^t_1,v^t_2,\overline{v}^t_2\} \cup \{u^t_i,\overline{u}^t_i \mid \exists q\; e_i = v_tv_q \in E(G)\}$ is called the long vertex-cycle of $v_t$ and is denoted by $C(v_t)$. Note that a long vertex-cycle has weight one. Now consider the following properties. Let G be a graph and let $(G^,M^,\omega )$ be the scaffold graph produced by Construction 2. Let S be a collection of $\|V(G)\|+\|E(G)\|$ alternating cycles in $(G^,M^,\omega )$. Every non-zero-weight alternating cycle C of S is a long vertex-cycle. Let $C(v_t)$ and $C(v_q)$ be two long vertex-cycles of S. Then, $v_tv_q\notin E(G)$. Note that it is always possible to build a collection of $\|V(G)\|+\|E(G)\|$ (weight-0) alternating cycles in $(G^,M^,\omega )$ by constructing the alternating cycle $\{u^t_i, \overline{u}^t_i,u^q_i,\overline{u}^q_i,e_i,\overline{e}_i\}$ for each edge $e_i=v_tv_q$ of G and the alternating cycle $\{v^t_1,\overline{v}^t_{1},v^t_2,\overline{v}^t_2\}$ for each vertex $v_t\in V(G)$.□ Let $v_t\in V(G)$ and $e_i\in E(G)$. Then, no alternating cycle of S contains both $e_i\overline{e}_i$ and $v^t_1\overline{v}^t_1$. Towards a contradiction, assume that there is such an alternating cycle C. By pidgeonhole principle, one of the $\|V(G)+E(G)\|$ alternating cycles in S, say $C'$, avoids both $e_i\overline{e}_i$ and $v^t_1\overline{v}^t_1$ for all $i,t\in \mathbb {N}$. Let $u^t_i\overline{u}^t_i$ be a matching edge of $C'$ for some $e_i=v_tv_q$. Then, $C'$ cannot contain $u^q_i\overline{u}^q_i$ as, otherwise, $e_i\overline{e}_i$ cannot be part of an alteranting cycle in S, implying that S is not a solution. Thus, each matching edge of $C'$ is on the long vertex-cycle $C(v_t)$. Since the graph induced by the vertices of $C(v_t) \setminus v^t_1\overline{v}^t_1$ is a path, it is not possible to construct $C'$. Hence, we conclude that C does not exist.□ (a): Let C be a non-zero-weight alternating cycle of S and assume towards a contradiction that C is not a long vertex-cycle. Since C contains a non-zero-weight edge $v^t_2\overline{u}^1_i$, the matching edge $v^t_2\overline{v}^t_2$ is in C. As C is not a long vertex-cycle, there is some $e_i=v_tv_q$ such that C contains both $u^t_i\overline{u}^t_i$ and $u^q_i\overline{u}^q_i$. Thus, either the matching edge $e_i\overline{e}_i$ is in C, contradicting Claim 1, or $e_i\overline{e}_i$ consists of a single-edge alternating path of S, contradicting our choice of S. (b): Towards a contradiction, assume that S contains $C(v_t)$ and $C(v_q)$ such that $e_i=v_tv_q\in E(G)$. Then, the matching edge $e_i\overline{e}_i$ is a single-edge alternating path of S, contradicting our choice of S. We now show the Poly-APX-hardness of max scaffolding, even for graph classes for which $\textsc {Scaffolding} \in \mathcal {P}$. Reusing the same idea of Theorem 2, we simulate the vertices of the independent set with long vertex-cycles. If a solution S of max scaffolding has weight k, then S contains k long vertex-cycles and, since their related vertices cannot be adjacent, we can construct an independent set with k vertices in G. max scaffolding is Poly-APX-hard, even for graph classes for which $\textsc {Scaffolding} \in \mathcal {P}$. Let G be an instance of independent set and let $(G^,M^,\omega )$ be the scaffold graph produced by Construction 2. Let $\mathcal {S}$ be the set of all collections of $\sigma _p=0$ alternating paths and $\sigma _c = \|V(G)\|+\|E(G)\|$ alternating cycles in $(G^,M^,\omega )$. Recall that independent set is Poly-APX-complete for general graphs [14]. We show that G has a size-k independent set if and only if $\mathcal {S}$ contains a solution S of score k. "$\Rightarrow$": Let I be an independent set of size k in G. We construct a solution $S\in \mathcal {S}$ as follows. First, for each $v_t\in I$, construct the alternating cycle $C(v_t)$ in S. Second, for each $v_t\in V(G)\setminus I$, construct the alternating cycle $(v^t_1,\overline{v}^t_{1},\overline{v}^t_2,v^t_2)$ in S. Third, for each edge $e_i=v_tv_q$ not incident with a vertex in I, construct the alternating cycle $(u^t_i, \overline{u}^t_i,\overline{u}^q_i,u^q_i,e_i,\overline{e}_i)$ in S. Fourth, for each edge $e_i=v_tv_q$ with $v_t\in I$, (the matching edge $u^t_i\overline{u}^t_i$ is in $C(v_t)$ which is already in S), construct the alternating cycle $(u^q_i,\overline{u}^{q}_i,e_i,\overline{e}_i)$. Since each long vertex-cycle has weight one, we obtain a solution S with $\omega (S)= k$. "$\Leftarrow$": Let $S\in \mathcal {S}$ with $\omega (S) = k$. We construct an independent set I by taking all vertices whose long vertex-cycle is in S, that is, $I := \{ v_t \mid C(v_t) \in S\}$. Since each long vertex-cycle has weight one, Lemma 2a implies that S contains k long vertex-cycles. Thus, $\|I\| = k$. Further, by Lemma 2b, I is independent. Let f be the function corresponding to Construction 2 and let g be a function that computes an independent set in G from a solution in f(G), as described above. Suppose that there is a polynomial-time algorithm A with approximation factor $\rho$ for max scaffolding. The approximation factor of $g \circ A \circ f$ is equal to $\rho$, thus Construction 2 constitutes an S-reduction. Non-approximability results of independent set transfer to max scaffolding.□ Feasibility function for connected cluster graphs In this section, we present a feasibility function for connected cluster graphs using dynamic programming. For simplicity, we consider in the following scaffold graphs $(G^,M^,\omega )$ such that $G^$ is a connected cluster graph and no matching edge is a bridge. The case were a bridge can be a matching edge is included in the feasibility function for block graph that (see "Experimental results" section). Notice that the structure of a connected cluster graph is close to a tree (that is, collapsing each clique of $G^$ into a single vertex leads to a tree), so we will use a similar vocabulary: a rooted connected cluster graph is a connected cluster graph where a clique r is designated as a root. Then, the following notation applies: the parent of a clique x is the clique connected to x on the unique x-r-path. A child of a clique c is clique of which c is the parent. Any clique without children is called a leaf. A vertex v of a clique c is a door of c if v is adjacent to a vertex u in a child of c. In that case, for simplicity, we say that the clique containing u is a child of v. The upper door of a clique $c\ne r$ is the unique vertex v that is adjacent to a vertex of the parent of c. Let c be a clique of $G^$ and let S be a partial solution in $G^$. Let $S'$ be the intersection of S and c, an alternating element of c is either an alternating cycle of $S'$ or an alternating path of $S'$. Notice that an alternating path of S can be decomposed into several alternating elements if it belongs to several cliques. Let e be the alternating element containing the upper door of c. The subclique $c'$ of c is the subgraph containing every vertex of c that does not belong to e. Formally, $c'=G^[V(c)\setminus V(e)]$. We use the tree-structure to develop a bottom-up algorithm, that is, we construct and assemble some partial solutions from the leaves to the root. We define some operations to combine this partial solutions. Let $G_1$ and $G_2$ be two edge-disjoint subgraphs. We can build a solution in the graph induced by $V(G_1) \cup V(G_2)$ from a solution in $G_1$ and a solution in $G_2$, using four operations. Let $G_1$ and $G_2$ be edge-disjoint subgraphs of $G^$. Let $S_1$ and $S_2$ be solutions of $G_1$ and $G_2$, respectively. Let S be a solution of $G^[V(G_1) \cup V(G_2)]$. S is a composition of $S_1$ and $S_2$ if S can be obtained from $S_1\cup S_2$ by at most one of the following operations: Merger:: merge an alternating path $(u_1,u_2,\ldots ,u_{2t})$ of $S_1$ with an alternating path $(v_1,v_2,\ldots ,v_{2q})$ of $S_2$ by adding the non-matching edge $u_{2t}v_1$. Closing:: close an alternating path $(u_1,u_2,\ldots ,u_{2t})$ of $S_1$ and an alternating path $(v_1,v_2,\ldots , v_{2q})$ of $S_2$ into an alternating cycle by adding the non-matching edges $u_{2t}v_1$ and $v_{2q}u_1$. Absorption:: replace a non-matching edge $v_{2i}v_{2i+1}$ of an alteranting path in $S_2$ with an alternating path $(u_1,u_2,\ldots ,u_{2t}$ of $S_1$ by removing $v_{2i}v_{2i+1}$ and adding the non-matching edges $v_{2i}u_1$ and $u_{2t}v_{2i+1}$. We call $v_{2i}v_{2i+1}$ absorbent. Finally, if no operation is necessary to obtain S from $S_1\cup S_2$, we say that S is obtained by juxtaposition. Note that all presented operations add only edges of $E(G^) \setminus (E(G_1) \cup E(G_2))$. Note further that not all compositions of two solutions are guaranteed to exist for a pair $S_1$ and $S_2$. In the algorithm, we manipulate sets of solutions: we can create a new set of solutions from two sets of solution if all pairs of solutions of the two input sets are used in the resulting set. Let $G_1$ and $G_2$ be two edge-disjoint subgraphs of $G^$ and let $\mathcal {S}_1$ and $\mathcal {S}_2$ be sets of solutions of subgraphs $G_1$ and $G_2$, respectively. Let op be one the operation described in Definition 2. Then, we call the set $\mathcal {S}=\{op(S_1,S_2)\mid \forall S_1\in \mathcal {S}_1 \wedge \forall S_2\in \mathcal {S}_2\;\}$ the complete composition of $\mathcal {S}_1$ and $\mathcal {S}_2$. To ensure the possibility of building a complete composition from two sets of solutions, it is useful to characterize a solution according to the operations we can perform on it. Let G and $G'$ be two edge-disjoint subgraphs of $G^$ and let S be a feasible solution of scaffolding for $(G,M^,\omega )$. S is closeable if S contains an alternating path $(u_1,u_2\ldots ,u_{2t})$ and $G'$ contains an alternating path $(v_1,v_2,\ldots ,v_{2q})$ such that $u_{2t}v_1$ and $v_{2q}u_1$ are edges of $E(G^\setminus M^$. S is extensible by $G'$ if S contains a vertex v such that v is an extremity of an alternating path and v has a neighbor in $G'$ . S is frozen to $G'$ if S is not extensible. S is absorbent to $G'$ if S contains an alternating path $(u_1,u_2,\ldots ,u_{2t})$ and $G'$ contains an alternating path with extremities v and w such that $vu_{2i}, wu_{2i+1} \in E(G^)\setminus M^$ for some $i<t$. Note that an absorbent solution can also be closeable, alternating or frozen. When omitted, $G'$ defaults to $G^- V(G)$. Note that all closeable solutions are also extensible. If a solution S is closeable by a subgraph $G'$, then we can close an alternating path of S into an alternating cycle by adding some edges of $G'$. If a solution S is extensible by a subgraph $G'$, then we can add some edges of $G'$ in an extremity of an alternating path of S without changing the cardinality of the solution. Finally, if a solution S is absorbent to a subgraph $G'$, then we can replace an absorbent edge of S by a path of length three without changing the cardinality of S. An example of the different operations of Definition 4 is given in Fig. 5. The solution S is composed of a single alternating path $\{v_1,\dots ,v_6\}$. S is closeable by subgraph $G_3= \{x_3,y_3\}$: we can close the alternating path of S into an alternating cycle by adding the edges $v_1x_3$, $x_3y_3$ and $y_3v_6$. S is extensible by subgraph $G_2 = \{x_2,y_2\}$: we can extend the alternating path of S by adding the edges $v_6y_2$ and $y_2x_2$ without changing the number of paths in S. S is absorbent to $G_4 = \{x_4,y_4\}$: we can replace the edge $v_2v_3$ of S by the edges $v_2y_4,y_4x_4$ and $x_4v_3$ without changing the number of paths in S. S is frozen to $G_1= \{x_1,y_1\}$ Since the number of possible solutions can be exponential, we just store the possible cardinalities in the table entries, which is sufficient to answer the question of feasibility. Recall that, if $X, Y \subseteq \mathbb {N}$ are two sets of integers, then the sum of X and Y is defined as $X+Y = \{ x+y\mid x \in X, y \in Y \}$. Note that $X + \varnothing = \varnothing$. In the following, we call an integer j eligible with respect to a set $\mathcal {S}$ of solutions and an integer i if there is a solution $S \in \mathcal {S}$ containing i alternating cycles and j alternating paths. Then, our dynamic programming table has the following semantics. (Semantics) Let $\mathcal {S}$ be a set of solutions and let $i \in \mathbb {N}$. A table entry $[\mathcal {S},i]$ is the set of all integers eligible with respect to $\mathcal {S}$ and i. More formally, letting $X_i = \{S \mid S \in \mathcal {S} \wedge \sigma _c(S) = i\}$, we define $[\mathcal {S},i] = \{\sigma _p(S) \mid S \in X_i\}$. Let us highlight three particular values of $[\mathcal {S},i]$. For $\mathcal {S}=\{\varnothing \}$, we have $[\{\varnothing \},0] = \{0\}$ and, for each $i>0$, we have $[\{\varnothing \},i]=\varnothing$. For an alternating path p, we have $[\{p\},0] = \{1\}$ and $[\{p\},i]=\varnothing$ for each $i>0$. Finally, for an alternating cycle c, we have $[\{c\},1] = \{0\}$ and $[\{c\},i]=\varnothing$ for each $i\ne 1$. For brevity, we let $[\mathcal {S}]$ denote the vector $([\mathcal {S},0],\dots ,[\mathcal {S},\sigma _c])$ and, for any operator $\diamond$ and any sets $\mathcal {S}_1$ and $\mathcal {S}_2$ of solutions, we define $[\mathcal {S}_1] \diamond [\mathcal {S}_2]$ as component-wise $\diamond$, that is, $[\mathcal {S}_1,i] \diamond [\mathcal {S}_2,i]$ for each $i \in [0,\sigma _c]$. Let $G_1$ and $G_2$ be two vertex-disjoint subgraphs of $G^$ and let $\mathcal {S}_1$ and $\mathcal {S}_2$ be sets of solutions of $G_1$ and $G_2$, respectively. Let $\mathcal {S}$ be a set of solutions of $G^[V(G_1) \cup V(G_2)]$ such that $\mathcal {S}$ is a complete composition of $\mathcal {S}_1$ and $\mathcal {S}_2$. If $\mathcal {S}$ is the set of solutions composed with a merger operation, then $[\mathcal {S},k] = \bigcup _{i+j=k}([\mathcal {S}_1,i] + [\mathcal {S}_2,j] + \{-1\})$. If $\mathcal {S}$ is the set of solutions composed with a closing operation, then $[\mathcal {S},k] = \bigcup _{i+j+1=k}([\mathcal {S}_1,i] + [\mathcal {S}_2,j] + \{-2\})$. If $\mathcal {S}$ is the set of solutions composed with an absorption operation, then $[\mathcal {S},k] = \bigcup _{i+j=k}([\mathcal {S}_1,i] + [\mathcal {S}_2,j] + \{-1\})$. If $\mathcal {S}$ is the set of solutions composed with a juxtaposition operation, then $[\mathcal {S},k] = \bigcup _{i+j=k}([\mathcal {S}_1,i] + [\mathcal {S}_2,j])$. Let $S \in \mathcal {S}$ and let $S_1$ and $S_2$ denote the solutions of $\mathcal {S}_1$ and $\mathcal {S}_2$, respectively, such that S is composed by $S_1$ and $S_2$. Then, since $S_1$ and $S_2$ have a common alternating path in S, we have $\sigma _p(S) = \sigma _p(S_1) + \sigma _p(S_2) -1$ and since no cycle is formed, $\sigma _c(S) = \sigma _c(S_1) + \sigma _c(S_2)$. Thus, since $\mathcal {S}$ is a complete composition of $\mathcal {S}_1$ and $\mathcal {S}_2$, we have $[\mathcal {S},k] = \bigcup _{i+j=k}([\mathcal {S}_1,i] + [\mathcal {S}_2,j] + \{-1\})$. since one path of $S_1$ and one path of $S_2$ are closed into a single alternating cycle, we have $\sigma _p(S) = \sigma _p(S_1) + \sigma _p(S_2) -2$ and $\sigma _c(S) = \sigma _c(S_1) + \sigma _c(S_2) + 1$. Thus, since $\mathcal {S}$ is a complete composition of $\mathcal {S}_1$ and $\mathcal {S_2}$, we have $[\mathcal {S},k] = \bigcup _{i+j=k}([\mathcal {S}_1,i] + [\mathcal {S}_2,j] + \{-2\})$. since $S_1$ has an alternating path that is "absorbed" into a connected component of $S_2$, we have $\sigma _p(S) = \sigma _p(S_1) + \sigma _p(S_2) -1$ and since no cycle is formed, $\sigma _c(S) = \sigma _c(S_1) + \sigma _c(S_2)$. Thus, since $\mathcal {S}$ is a complete composition of $\mathcal {S}_1$ and $\mathcal {S_2}$, we have $[\mathcal {S},k] = \bigcup _{i+j=k}([\mathcal {S}_1,i] + [\mathcal {S}_2,j] + \{-1\})$. since all paths and cycles of $S_1$ and $S_2$ are present in S, we have $\sigma _p(S) = \sigma _p(S_1) + \sigma _p(S_2) -1$ and since no cycle is formed, $\sigma _c(S) = \sigma _c(S_1) + \sigma _c(S_2)$. Thus, since $\mathcal {S}$ is a complete composition of $\mathcal {S}_1$ and $\mathcal {S_2}$, we have $[\mathcal {S},k] = \bigcup _{i+j=k}([\mathcal {S}_1,i] + [\mathcal {S}_2,j])$. We use Lemma 3 to define the four functions juxtapose, $merge_t$, absorb, and $close_t$, which provide table entries for complete compositions "composed" with a juxtaposition, merge, absorption or closing operation, respectively. Although Lemma 3 is defined for two sets, we use a generalized version which can take as parameters more than two sets. The functions $merge_t$ and $close_t$ have a parameter t that indicates the number of paths merged or closed during the operation. For example, if we have three sets $S_1$, $S_2$, and $S_3$ and if it is possible to construct a single alternating path in the resulting composition by taking one alternating path in each set, then we use the function $merge_3(\{S_1\},\{S_2\},\{S_3\})$. Note that the parameter t can be different from the number of sets. In addition, it is sometimes possible to close a single alternating path into an alternating cycle and, in that case, the function $close_1$ is used. The four functions are defined in Algorithm 2, Algorithm 3 and Algorithm 4. However, we must ensure that the associated operation is feasible before using one these functions. In the algorithm, we traverse four different types of subgraphs defined as follows. Let $v \in V(G^)$, let child(v) be the set of children of v in $G^$ (possibly empty). Then, $G^(v)$ denotes the subgrah of $G^$ that is induced by v and every branch linked to v. Formally, $G^(v) := G^[\{v\} \cup \bigcup \limits _{x \in child(v)} V(G^(x))]$. Let e be an alternating element. Then, $G^(e)$ denotes the subgraph of $G^$ that is induced by e and all children of its vertices. Formally, $G^(e) = G^[\bigcup \nolimits _{v \in e} V(G^(v))]$. Let c be a clique of $G^$ and let $c'$ be the subclique of c. For all $x \in \{c,c'\}$, the subgraph $G^(x)$ is the union of x and all children of x. Formally, $G^(x)=G^[\bigcup \nolimits _{e \in M^\cap ( {\begin{array}{c}x\\ 2\end{array}}) } V(G^(e))]$. For each traversed subgraph, we use four different sets of solutions distinguishing solutions according to their properties. Let S be a partial solution of $G^$. Let x be a vertex, a partial path, a subclique or clique of $G^$ and let $S'$ be a solution of the subgraph $G^(x)$. Then, $S \in \mathcal {C}(x) \Leftrightarrow$ $S'$ is closeable and $S \cap E(G^(x)) \subseteq S'$. $S \in \mathcal {E}(x) \Leftrightarrow$ $S \notin \mathcal {C}(x)$ and S is extensible and $S \cap E(G^(x)) \subseteq S'$. $S \in \mathcal {A}(x) \Leftrightarrow$ S is frozen and absorbent and $S \cap E(G^(x)) \subseteq S'$. $S \in \mathcal {F}(x) \Leftrightarrow$ $S \notin \mathcal {A}(x)$ and S is frozen and $S \cap E(G^(x)) \subseteq S'$. We now present a method to provide the feasibility function needed by Algorithm 1. In the next paragraphs, we describe the algorithms that calculate the table entries for the four types of subgraphs described above. Let $v\in V(G^)$. We show in this part how to compute the table entries for the sets $\mathcal {F}(v)$ and $\mathcal {E}(v)$. Note that, since the edge between $G^(v)$ and its parent is a bridge, any solution $S'$ for $G^(v)$ can have at most one edge incident to v. Thus, the sets $\mathcal {C}(v)$ and $\mathcal {A}(v)$ are empty. If v is not incident to an edge of $S \cap E(G^(v))$, then we construct the table entries by successively merging the table entries of the children adjacent to v. For that, we use at each step an intermediate graph $G_i$. Let $V_i$ be the set of the first i children of v. $G_i$ is the subgraph of $G^$ induced by v and all vertices in $V_i$. If v is incident with an edge $S \cap E(G^(v))$, then any solution containing S is in $\mathcal {E}(v)$. An example of solutions computed by Algorithm 5 is depicted in Fig. 6. Example of two solutions $S_1$ (left, frozen) and $S_2$ (right, extensible) in $G^(v)$. The cliques $c_1$ and $c_2$ are children of c(v). Each of $S_1$ and $S_2$ contains two alternating elements (solid black lines). The frozen solution is obtained with the juxtaposition of two frozen solutions in $c_1$ and $c_2$. The extensible solution is obtained with the juxtaposition of a frozen solution in $c_1$ and with the merge between v and an extensible solution in $c_2$ For any vertex v, the values of the table entries provided by Algorithm 5 are correct for the set $\mathcal {F}(v)$ and $\mathcal {E}(v)$. First, if there is no child linked to v, then $G^(v)$ consists of the single vertex v. In that case, the only solution for $G^(v)$ consists of zero alternating cycles and paths and this solution is frozen. Thus, the initial values given to $[\mathcal {F}(v)]$ and $[\mathcal {E}(v)]$ in the initialization step (i.e. lines 1 to 2) are correct. Assume that table entries returned by $compute\_clique$ are correct. Let $S'$ be a solution of $G^(v)$ such that $S\cap E(G^(v)) \subseteq S'$. We distinguish two cases. Case 1:: there is an edge $uv \in S\cap E(G^(v))$. Thus, $S'$ is extensible and is composed by the merge of an extensible solution in $G^(c_u)$ with uv and the juxtaposition of any solution for each child $c_{u'}\ne c_u$. Hence, lines 9 and 11 are correct. there is no edge $uv \in S\cap E(G^(v))$. Then, $S'$ is frozen if and only if it does not contain an edge incident to v. As there is no edge uv in any child $c_t$, $S'$ is composed by juxtaposition of any solution for each child $c_t$ and the assignment in line 13 is correct. If $S'$ is extensible, then there is a unique child $c_t$ of v such that an alternating path from $S' \cap E(G^(c_t))$ has been expanded to v and, therefore, the solution $S' \cap E(G^(c_t))$ is extensible. Thus, $S'$ is composed by a merge of a extensible solution of a unique child and the juxtaposition of any solution in other children. Hence, line 14 is correct. Alternating element Let c be a clique of $G^$ and let e be an alternating element of c such that e does not contain the upper door of c. We show in this part how to compute the table entries for the sets $\mathcal {C}(e)$, $\mathcal {F}(e)$ and $\mathcal {E}(e)$. If e is a u-v-path, then the idea is to merge the computed table entries of u and v and juxtapose the frozen solutions of the inner vertices. If e is an alternating cycle, then there is no choice and the only solution containing S is frozen. An example of solutions computed by Algorithm 6 is depicted in Fig. 7. Example of solutions (black edges) in $G^(e)$ where e is a u-v-path. The left solution is closeable, the center solution is extensible and the right solution is frozen. The closeable solution is obtained by the juxtaposition of e, any solution in $G^(u)$ and any solution in $G^(v)$. The extensible solution is obtained by the merge of e with an extensible solution in $G^(u)$ and the juxtaposition of any solution in $G^(v)$. The frozen solution is obtained by the merge of e, an extensible solution of $G^(u)$ and an extensible solution in $G^(v)$ For any alternating element e, the values of the table entries provided by Algorithm 6 are correct for the sets $\mathcal {C}(e)$, $\mathcal {F}(e)$ and $\mathcal {E}(e)$. Note that the only possibility to obtain an absorbent solution of $G^(e)$ is when e is a path that is closed into an alternating cycle. However, if an absorption operation is done in the function $compute\_subclique$, then the resulting solution can also be obtained by a closing operation with a solution in $\mathcal {C}(e)$. Thus, our dynamic programming will not compute the value of $[\mathcal {A}(e)]$. Suppose that the values of the table entries provided by the function $compute\_vertex$ are correct. First note that, for each inner vertex $v_t$ of e, the subsolutions of $G^(v_t)$ are necessarily frozen, then a solution of $G^(e)$ contains a juxtaposition of frozen solutions of the inner vertices of e. If e is an alternating cycle, then the only possible solution is obtained by the juxtaposition of frozen solutions of the inner vertices and the alternating cycle e. Thus, the assignment line 5 is correct. Suppose that e is a partial path. All possible values of the juxtaposition of the frozen solutions of the inner vertices are assigned in the table entry $[\mathcal {I}_e]$. A solution $S'$ of $G^(e)$ is closeable if the degree of the extremities of e are equal to one. Then, the subsolutions of $S'$ in $G^(v_0)$ and $G^(v_k)$ are frozen. Thus, the assignment line 7 is correct. A solution $S'$ of $G^(e)$ is frozen if the degree of the extremities of e are equal to two. It is the case if (1) the subsolutions of $S'$ in $G^(v_0)$ and $G^(v_k)$ are extensible or (2) the subsolutions of $S'$ in $G^(v_0)$ and $G^(v_k)$ are frozen and e is closed into an alternating cycle. Thus, the assignment line 8 is correct. A solution $S'$ of $G^(e)$ is extensible and not closeable if and only if exactly one vertex in $\{v_0,v_k\}$ has degree one. Then, exactly one subsolution of S in $G^(v_0)$ or $G^(v_k)$ is extensible. Thus, the assignment line 9 is correct. Subclique Let $c'$ be a subclique of $G^$ containing k alternating elements. We show in this part how to compute the table entries for the sets $\mathcal {C}, \mathcal {F}, \mathcal {A}$ and $\mathcal {E}$. The idea is to construct the table entry by merging successively each table entry of the alternating elements of $c'$. For that, we use at each step an intermediate graph $G_t$ and two intermediate sets $\mathcal {A}_+$ and $\mathcal {E}_+$, defined as follows. Let $L(c') = \{e_1,\dots ,e_k\}$ be a list of alternating elements of $c'$, let $t\le k$, let $E_t=\{e_1,\dots ,e_t\}$, and let $V_t =\bigcup _{e \in E_t} V(G^(e))$. Let $G_t$ be the subgraph of $G^$ induced by $V_t$. At step t, a solution $S'$ is in $\mathcal {A}_+$ (resp. $\mathcal {E}_+$) if and only if (1) $S'$ is a solution of $G_t$, (2) $S'$ contains a set $C \ne \varnothing$ of closeable paths and (3) $S \setminus C$ is not extensible (resp. extensible). For any subclique $c'$, the value of the table entries provided by Algorithm 7 are correct for the sets $\mathcal {C}(c'), \mathcal {F}(c'), \mathcal {A}(c')$ and $\mathcal {E}(c')$. Assume table entries returned by $compute\_alternating\_element$ are correct. We show by induction that the values calculated in each step t are correct for the graph $G_t$. First, $G_0$ is the empty graph and the unique solution is that containing zero alternating cycles and paths and this solution is frozen. Thus, lines 1 to 3 are correct. Now, consider the alternating element $e_t$ and suppose the previously computed values are correct (i.e. values stored in $\mathcal {F}',\mathcal {A}',\mathcal {E}',\mathcal {A}_+'$ and $\mathcal {E}_+'$). Let $S_1$ be a solution in $G_{t-1}$, let $S_2$ be a solution in $G^(e_t)$ and let $S'$ be a composition of $S_1$ and $S_2$. We have the following properties: if $S'$ is obtained by a juxtaposition, then $S_1$ is in $\mathcal {F}',\mathcal {A}',\mathcal {E}',\mathcal {A}_+'$ or $\mathcal {E}_+'$ and $S_2$ is in $\mathcal {C}(e_t),\mathcal {F}(e_t)$ or $\mathcal {E}(e_t)$, if $S'$ is obtained by a merge, then $S_1$ is in $\mathcal {E}',\mathcal {A}_+'$ or $\mathcal {E}_+'$ and $S_2$ is in $\mathcal {C}(e_t)$ or $\mathcal {E}(e_t)$, if $S'$ is obtained with an absorption, then $S_1$ is in $\mathcal {A}'$ or $\mathcal {A}_+'$ and $S_2$ is in $\mathcal {C}(e_t)$, and if $S'$ is obtained by a closing, then $S_1$ is in $\mathcal {A}_+'$ or $\mathcal {E}_+'$ and $S_2$ is in $\mathcal {C}(e_t)$. Thus, there are 25 complete compositions to consider. If $S_2 \in \mathcal {C}(e_t)$ (resp. $\mathcal {E}(e_t)$) and $S'$ is obtained by a closing (resp. merge), then $S'$ is closeable (resp. extensible) if $S_1$ contains more than one closeable (resp. extensible) alternating path or absorbent, otherwise. Hence, a complete composition obtained with a closing or a merge is not included in a unique set among those defined. This problem can be solved by ignoring certain solutions: $S'$ can be ignored if there is another solution in $G_t$ with the same cardinality. Suppose $S'$ is obtained with a closing (resp. merge) and $S_1$ contains more than one closeable (resp. extensible) alternating path. Let $p_1$ and $p_2$ be closeable (resp. extensible) alternating paths of $S_1$. There is a solution $S'_1$ similar to $S_1$ except that $p_1$ and $p_2$ have been closed into a cycle (resp. merged into a unique alternating path) during a previous step. We can obtain a solution in $G_t$ with the same cardinality as $S'$ by juxtaposing $S'_1$ and $S_2$. Thus, $S'$ can be ignored, and we suppose that a solution obtained with a closing does not contain a closeable alternating path (i.e. is not in $\mathcal {A}_+$ or $\mathcal {E}_+$). Likewise, we can suppose a solution obtained with a merge between a solution of $\mathcal {E}'\cup \mathcal {E}_+'$ and a solution of $\mathcal {E}(e_t)$ does not contain an extensible alternating path (i.e. is not in $\mathcal {E}(c')$ or $\mathcal {E}_+$). Assume that one of the following conditions is true. (1) $S_1 \in \mathcal {A}_+', S_2 \in \mathcal {E}(e_t)$ and $S'$ is obtained by a merge, (2) $S_2 \in \mathcal {E}_+', S_2 \in \mathcal {F}(e_t)$ and $S'$ is obtained by a merge, (3) $S_1 \in \mathcal {A}_+', S_2 \in \mathcal {F}(e_t)$ and $S'$ is obtained by an absorption. Let p be a closeable alternating path of $S_1$ that is absorbed or merged in $S'$. There is a solution $S'_1$ similar to $S_1$ except that all non-matching edges of p have been merged or absorbed during previous steps. We can obtain a solution in $G_t$ with the same cardinality as $S'$ by juxtaposing $S'_1$ and $S_2$. Thus, $S'$ can be ignored. Fig. 8 shows an example of case (3). Example of a case ignored by the algorithm. At the top, the solution is obtained after juxtaposing a closeable alternating path $p_1$ and absorbing a closeable alternating path $p_2$. The intermediate solution is in the set $\mathcal {A}'_+$ during the second step. Below, a solution of the same cardinality is obtained after absorbing $p_1$ and juxtaposing $p_2$ and in this case, the intermediate solution is in $\mathcal {A}'$. The upper solution is not considered by the algorithm because the bottom solution has the same cardinality The second item allows us to ignore three complete compositions: there are 22 still to be considered. Each of these complete compositions is in only one of the six sets of solutions among $\mathcal {F}(c')$, $\mathcal {A}(c')$, $\mathcal {E}(c')$, $\mathcal {A}'_+$ and $\mathcal {E}'_+$. Suppose $S'$ is frozen. The only feasible operation to obtain $S'$ is juxtaposition because an addition of an edge of $E(c')\setminus S$ creates an absorbent solution. $S_1$ and $S_2$ are frozen as, otherwise, their juxtaposition is not frozen. Thus, line 9 is correct. Suppose $S'$ is absorbent. Thus, $S'$ contains at least one edge in $\mathcal {E}(c')\setminus S$. If $S_2$ is frozen, then the only feasible operation is juxtaposition and $S_1$ is absorbent. If $S_2$ is extensible, then its extensible alternating path is merged with an extensible alternating path of $S_1$ that is not closeable. Thus, $S_2$ is in $\mathcal {E}'$. If $S'$ results from an absorption, then $S_1$ is absorbent and $S_2$ is closeable. If $S'$ results from a closing, then $S_1$ and $S_2$ are closeable. Since the resulting solution is absorbent, $S_1$ is in $\mathcal {A}_+'$. Hence, line 10 is correct. Suppose $S'\in \mathcal {A}_+$. Then, $S'$ is extensible and does not contain any extensible alternating paths. If $S'$ results from a juxtaposition, then $S_1$ does not contain an extensible alternating path and $S_2$ is either frozen or closeable. In the first case, $S_1$ must be closeable and therefore $S_1\in \mathcal {A}_+'$. In the second case, $S_1$ is in $\mathcal {F}', \mathcal {A}'$ or $\mathcal {A}_+'$. If $S'$ results from a merge, then $S_1$ is closeable and $S_2$ is either extensible or closeable. In the first case, the extensible alternating path of $S_1$ is merged with an extensible alternating path of $S_2$ so that the resulting solution is not extensible. Thus, $S_1$ is in $\mathcal {E}_+'$. In the second case, $S_1$ does not contain an extensible alternating path since otherwise $S'$ is extensible. Thus, $S_1$ is in $\mathcal {A}_+$. Suppose $S'$ is extensible. Then, either $S_1$ contains an extensible alternating path or $S_2$ is extensible. If $S_1$ is extensible and $S'$ results from a juxtaposition, then $S_2$ is not closeable since otherwise the resulting solution is also closeable. Thus, $S_2$ is frozen or extensible. If $S_1$ is extensible and $S'$ results from a merge. Then, since we only consider solutions of $\mathcal {E}'$ with a unique extensible alternating path, $S_2$ cannot be extensible since otherwise the resulting solution is absorbent. Thus, $S_2$ is closeable. If $S_1$ is in $\mathcal {E}_+'$, then since $S'$ is not closeable, the extensible alternating path of $S_1$ is either merged with an alternating path or closed into a cycle with a closeable alternating path. Thus, $S_2$ is extensible and $S'$ results from a merge or $S_2$ is closeable and $S'$ results from a closing. If $S_2$ is extensible and $S_1$ does not contain any extensible or closeable alternating path, then $S'$ results from a juxtaposition and $S_1$ is frozen or absorbent. Suppose $S'$ is in $\mathcal {E}_+$. Then, $S'$ is closeable and contains one extensible alternating path. Recall that we ignore solutions resulting from merge between a solution of $\mathcal {E}_+$ and a closeable solution. Thus, $S'$ results from a juxtaposition and either $S_1$ or $S_2$ contains an extensible alternating path. If $S_1$ is in $\mathcal {E}_+'$, then $S_2$ can be any solution. If $S_1$ is in $\mathcal {A}_+'$, then for $S'$ to contain an extensible alternating path, $S_2$ must be extensible. If $S_1$ is extensible, then for $S'$ to contain a closeable alternating path, $S_2$ must be closeable. As after these assignments, each of the solutions of $G_t$ is in a unique set and is a composition of a solution of $G_{t-1}$ and $G^(e_t)$, computed values for the table entries are correct for $G_t$. Finally, after the execution of the loop, computed values for sets $\mathcal {F}(c'), \mathcal {A}(c')$ and $\mathcal {E}(c')$ are correct for $G_k = G^(c')$. It remains to compute the value of the table entry for $\mathcal {C}(c')$. Sets containing closeable alternating paths are exactly the sets $\mathcal {A}_+$ and $\mathcal {E}_+$, thus $\mathcal {A}_+\cup \mathcal {E}_+= \mathcal {C}(c')$. Hence, the assignment line 15 is correct.□ Let c be a clique of $G^$ and let d be the upper door of c. We show in this part how to compute the table entries for the sets $\mathcal {F}(c)$ and $\mathcal {E}(c)$. Note that, since the edge between $G^(c)$ and its parent is a bridge, the sets $\mathcal {C}(c)$ and $\mathcal {A}(c)$ are empty. Let e be the alternating element of c containing the upper door d of c. The idea is to first compute the table entries for the graph $G^(e)$ and then merge the obtained table entries to the table entries of the subclique. If e is an alternating path and d is an extremity of e, we replace $\mathcal {E}(e)$ by two intermediate sets $\mathcal {\mathcal {E}}_{d}$ and $\mathcal {\mathcal {E}}_{d'}$. Let $S'$ be a solution of $G^(e)$. Then, $S' \in \mathcal {\mathcal {E}}_{d}$ if and only if $S' \in \mathcal {E}(e)$ and d is an extremity of an alternating path of $S'$. Likewise, $S' \in \mathcal {\mathcal {E}}_{d'}$ if and only if $S' \in \mathcal {E}(e)$ and d is not an extremity of an alternating path of $S'$. Note that $\mathcal {E}(e)= \mathcal {\mathcal {E}}_{d}\cup \mathcal {\mathcal {E}}_{d'}$. In order to compute these two sets, we reuse the value of $\mathcal {I}_e$, computed in $compute\_alternating\_element$. For any clique c, the values of the table entries provided by Algorithm 8 are correct for the sets $\mathcal {F}(c)$ and $\mathcal {E}(d)$. Suppose e is an alternating path and the upper door d of c is an extremity of e. Let $d'$ be the other extremity of e. First, we compute the table entries for the sets $\mathcal {C}(e),\mathcal {F}(e),\mathcal {\mathcal {E}}_{d}$ and $\mathcal {\mathcal {E}}_{d'}$. Suppose that the values of the table entries provided by $compute\_alternating\_element(p)$ are correct for the sets $\mathcal {C}(e)$ and $\mathcal {F}(e)$. It remains to compute the table entries for the sets $\mathcal {\mathcal {E}}_{d}$ and $\mathcal {\mathcal {E}}_{d'}$. We recall that $\mathcal {I}_e$ is the juxtaposition of all frozen solutions of the inner vertices of e. A solution $S'$ of $G^(e)$ is in $\mathcal {\mathcal {E}}_{d}$ if and only if $S'$ is in $\mathcal {E}(e)$ and no non-matching edge is incident to d in $G^(e)$. Thus, $\mathcal {\mathcal {E}}_{d}$ is the juxtaposition of e, $\mathcal {I}_e$, $\mathcal {F}(d)$ and $\mathcal {E}(d')$, implying that line 5 is correct. Similarly, a solution $S'$ of $G^(e)$ is in $\mathcal {\mathcal {E}}_{d'}$ if and only if $S'$ is in $\mathcal {E}(e)$ and no non-matching edge is incident to $d'$ in $G^(e)$. Thus, $\mathcal {\mathcal {E}}_{d}$ is the juxtaposition of e, $\mathcal {I}_e$, $\mathcal {E}(d)$ and $\mathcal {F}(d')$, implying that line 6 is correct. Further, we show that the table entries computed for the set $\mathcal {F}(c)$ and $\mathcal {E}(c)$ are correct. A solution $S'$ of $G^(c)$ is frozen if and only if $S'$ contains an edge incident to d. This is the case if the subsolution of $S'$ in $G^(e)$ is in $\mathcal {F}(e)$ or $\mathcal {\mathcal {E}}_{d'}$ or if S is obtained by a merger operation, an absorption operation or a closing operation. Thus, line 8 is correct. A solution $S'$ of $G^(c)$ is extensible if and only if S does not contain an edge incident to d. This is the case if the subsolution of S in $G^(e)$ is in $\mathcal {C}(e)$ or $\mathcal {\mathcal {E}}_{d}$ or if $S'$ is obtained by a merger operation and $d'$ is an extremity of an alternating path in the subsolution of S in $G^(e)$. Thus, line 10 is correct. Now, suppose that the upper door d of c is an inner vertex of e. In that case, a subsolution $S'$ of $G^(c)$ is necessarily frozen. Then any feasible composition of a solution of $G^(c')$ and a solution of $G^(e)$ is a frozen solution and thus, line 13 is correct. Similarly, since no extensible solution of $G^(c)$ exist, line 14 is correct.□ Feasibility function We can now provide an answer to the feasibility of finding a solution for $\textsc {Scaffolding}$ by using Algorithm 9. Let r be the root of $G^$. Notice than since r does not have an upper door then the subclique of r corresponds to r. Thus, it is not possible to call $compute\_clique$ on r. That is why the first recursive call of the algorithm is made with the function $compute\_subclique$. Given a partial solution S, Algorithm 9 returns true if and only if $(G^, M^)$ can be decomposed into $\sigma _p$ alternating paths and $\sigma _c$ alternating cycles. The time complexity of the algorithm is $\mathcal {O}(\|V(G^)\| \cdot \sigma _c^2)$. Since $G^(root) = G^$, there is a solution S with $\sigma _p(S)=\sigma _p$ and $\sigma _c(S)=\sigma _c$, if and only if S is in $\mathcal {C}(root)$, $\mathcal {F}(root)$, $\mathcal {A}(root)$, or $\mathcal {E}(root)$. Thus, the return of the function indicates if such a solution exists and then the algorithm is correct. Concerning the time complexity, the composition operations are executable in $\mathcal {O}(\sigma _c^2)$ time. Thus, without taking into account the recursive calls, the time complexity of Algorithm 5, Algorithm 6, Algorithm 7 and Algorithm 8 in one iteration of a loop is $\mathcal {O}(\sigma _c^2)$. Let C denote the number of cliques in GG. In Algorithm 5, the number of iterations made by all calls of this function depends on C and then the time complexity of all these iterations is $\mathcal {O}(C \cdot \sigma _c^2)$. Similarly, we can show that the time complexities of the iterations made by all calls of Algorithm 6, Algorithm 7 and Algorithm 8 are $\mathcal {O}(\|V\| \cdot \sigma _c^2)$, $\mathcal {O}(\|M^\| \cdot \sigma _c^2)$ and $\mathcal {O}(C \cdot \sigma _c^2)$. Then, the time complexity of all iterations in all functions is $\mathcal {O}((\|V)\| + \|M^\| + C) \cdot \sigma _c^2)$ and since the number of matching edges and the number of cliques is bounded by the number of vertices of $G^$, we have a time complexity $\mathcal {O}(\|V(G^)\| \cdot \sigma _c^2)$.□ A running example is depicted in Fig. 9 and Example 1 (Tables 1, 2, 3, 4 and 5). Left: The connected cluster graph $G^$ used for the pratical example. The graph contains the following cliques: $c_1 =\{a,b,c,d\}$, $c_2 = \{f,e\}$, $c_3=\{h,g\}$, $c_4=\{i,j,k,l\}$, $c_5=\{m,n,o,p,q,r,s,t\}$ and $c_6=\{u,v,w,x\}$. Right: Tree structure of $G^$ used in the algorithm. The root of these structure is the clique $c_6$ Table 1 Compute_vertex Table 2 Compute_alternating_element Table 3 Compute_subclique Table 4 Compute_clique Table 5 Detailled computation for subclique $c'_5$ Running example on the graph depicted in Fig. 9. Tables 1, 2, 3 and 4 depicte the table entries resulting from Algorithms 5 to 8, respectively. Table 5 display the values of the table entries after each iteration of alternating element for the subclique $c'_5$. Let c be the value given by the column "#cycles" and x be the item considered in the first column. For each X in $\mathcal {F} ,\mathcal {A}, \mathcal {A}_+, \mathcal {E}, \mathcal {E}_+$ and $\mathcal {C}$, the interval given by the column X corresponds to [X(x), c]. Approximation result We now prove the following approximation result. Algorithm 1 provides a solution for $(\sigma _p,\sigma _c)$-scaffolding in connected cluster graphs with an approximation ratio of at most five and a time complexity $\mathcal {O}(\|V\|\cdot \|E(G^)\| \cdot \sigma _c^2)$. The approximation ratio is tight. Proof We suppose that the input of the algorithm is a scaffold graph $(G^,M^,\omega )$ with non-negative weights and such that $G^$ is a path connected cluster graph. We first show that the algorithm is correct. Note that, since each time we add an edge e to S, we remove from E all incident non matching edges to e, the set S induces only paths and cycles. If it is not possible to build a solution from the graph, then the feasibility condition is not verified and then the algorithm returns an error. Otherwise, since we ensure that the feasibility condition is verified at each step, when the algorithm terminates, then it builds $\sigma _p$ paths and $\sigma _c$ cycles. Now, we prove the approximation ratio. Since they always appear in any solution, we do not consider the edges of $M^$ in what follows. Notice that, since there is, for each path, one chosen edge less than the number of involved matching edges, and for a cycle, the same number of chosen edge as the number of involved matching edges, then the number of non-matching edges in every solution is exactly $n - \sigma _p$. We denote by $e_1,\dots ,e_m$ the edges of the graph $G^$, sorted in non-increasing order by their weights. We denote by $e^A_1,\dots ,e^A_{n-\sigma _p}$ the edges of the solution $S_A$ given by Algorithm 1, sorted in non-increasing order by their weights. In the same way, we denote by $e^{opt}_1,\dots ,e^{opt}_{n-\sigma _p}$ the edges of an optimal solution $S_{opt}$ for the problem, also sorted in non-increasing order. Both sequences $e^A_1,\dots ,e^A_{n-\sigma _p}$ and $e^{opt}_1,\dots ,e^{opt}_{n-\sigma _p}$ are clearly subsequences of $e_1,\dots ,e_m$. Let $\varphi : S_{opt} \rightarrow S_A$ be a mapping such that $$\begin{aligned} \forall e \in S_{opt}, \omega (e) \le \omega (\varphi (e)) \end{aligned}$$ $$\begin{aligned} \forall e \in S_A, \|\varphi ^{-1}(\{e\})\| \le 5 \end{aligned}$$ Inequality (1) indicates that for each $e\in E$ in an optimal solution, there is an edge $\varphi (e) \in S_A$ such that the weight of this latter edge is at least the weight of e. Whereas (2) states that for each $e \in S_A$, we may associate e to at most four edges of the optimal solution. In the following, we prove that it is possible to define a mapping $\varphi$ satisfying these inequalities. A greedily chosen edge can eliminate up to two optimal edges by the update_edge function The algorithm may decide not to choose an edge $e^{opt}_i$ for four main reasons: $e^{opt}_i$ is eliminated because it is in R, when an edge $e^A_j$ is chosen. In this case, we have $\omega (e^A_j) \ge \omega (e_i^{opt})$ because only edges appearing after $e^A_j$ in the ordered list can be in R. When an edge $e^A_j$ is chosen, it can eliminate at most two edges of optimal solution by updating of the list of edges (see Fig. 10). We assign $\varphi (e^{opt}_i) = e^A_j$ in this case. (1) is satisfied by construction, and (2) holds when considering only the optimal edges which are eliminated by this way. $e^{opt}_i$ is eliminated because its addition disconnects the graph and the number of alternating cycles and alternating paths required to cover the graph becomes too big. This happens in one of the following two cases. $e^{opt}_i$ closes a cycle. In that case, there is at least one edge $e^A_{j}$ in this cycle, and since it has been chosen before the algorithm considers $e^{opt}_i$, we necessarily have $\omega (e^{A}_j) \ge \omega (e^{opt}_i)$. Thus, we assign $\varphi (e^{opt}_i)= e^A_j$. Then, (1) is satisfied by construction. The edge $e^A_j$ has been already chosen, may have eliminated at most two optimal edges, but (2) is still satisfied. $e^{opt}_i$ closes a door d and one bridge dx incident to d is necessary to construct a solution with the remaining edges. There is a door y which has been closed by an edge $e^A_j$ in a previous step and this forces dx to be in $S_A$. Since closing a door increases by at most one the minimum number of alternating paths required to cover the graph, the closing of y forces at most one bridge of $G^$ to be in $S_A$. Thus, the closing of y prevents d and x from closing, that is, at most two edges of $S^{opt}$, incident to d and x respectively, can be associated to $e^A_j$ Then, (1) is satisfied by construction. The edge $e^A_j$ may have eliminated at most two optimal edges in R and may prevent the closing of a cycle, but (2) is still satisfied. $e^{opt}_i$ is eliminated because its inclusion would merge two paths $p_1$ and $p_2$. If $e^{opt}_i$ is not a bridge and $p_1$ and $p_2$ are a single-edge paths, then the number of alternating cycles and paths are reached in S, that is $\sigma _c = c$, $\sigma _p = p$ and $S = S_A$. Then, we can find an edge $e^A_j$ such that $\|\varphi ^-1(e^A_j)\|=0$ and we assign $\varphi (e^{opt}_i) = e^A_j$. Then, (1) and (2) are satisfied by construction. Otherwise, the algorithm eliminates $e^{opt}_i$ because one of the merged paths must be closed into a cycle to reach the correct number of alternating cycles. Otherwise, there is an edge $e^A_j$ in $S_A$ considered before $e^{opt}_i$ in the algorithm such that $\|\varphi ^-1(e^A_j)\| \le 3$ (since otherwise the path would be already closed into a cycle) and then we assign $\varphi (e^{opt}_i) = e^A_j$. Again, (1) and (2) are satisfied by construction. From the previous discussion and by (1) and (2), clearly we have: $$\begin{aligned} \omega (S_{opt}) \le \omega (\varphi (S_{opt})) \le 5 \omega (S_A). \end{aligned}$$ The ratio is tight, as shown by the example depicted in Fig. 11. Concerning the complexity, the edges can be sorted in $\mathcal {O}(\|V(G^)\| \log \|E(G^)\|)$ time. The feasibility function is called $\|E(G^)\|$ times. Thus, the time complexity of the algorithm is $\mathcal {O}(\|E(G^)\| \cdot \|V(G^)\| \cdot \sigma _c^2)$. The approximation ratio of five for the greedy algorithm is tight. Matching edges are bold, dashed edges are in the approximate solution and solid edges are in the optimal solution. $G^$ is composed by the cliques ${C_1 = \{a,b,c,d,e,f\}}$, ${C_2=\{g,h\}}$, ${C_3 = \{i,j,k,l\}}$ and $C_4 = \{m,n,o,p\}$. All edges have weight zero except ac and the edges of $S_{opt}$. We suppose that $\sigma _p = 3$ and $\sigma _c=0$, and the greedy algorithm chooses "the wrong edge" ac first. Consequently, the solution $S_A$ given by the greedy algorithm is of weight 1, whereas an optimal solution would be of weight 5 In this section, we compare the performance of Algorithm 1 with three different feasibility functions and an integer linear programming formulation [15] implemented with ILOG CPLEX [16]. We reuse the dataset already used in [9], which was obtained with the following pipeline: Table 6 Real dataset Choice of a reference genome, for instance on the nucleotide database from NCBIFootnote 2. Table 6 presents selected genomes used for our experiments. We chose a panel of genomes of various origins and sizes. Simulation of paired-end reads, using wgsim [17]. The chosen parameters are an insert size of 500bp and a read length L of 100bp. Assembly using the de novo assembly tool, based on a De Bruijn graph efficient representation: minia [18] with k-mer size $k=30$. Mapping of reads on contigs, using bwa [19]. This mapping tool was chosen according to results obtained by Hunt [20], a survey on scaffolding tools. Generation of scaffold graph from the mapping file. Statistics on the numbers of vertices and edges in produced scaffold graphs can be viewed in Table 7. Feasibility functions There is no polynomial-time computable feasibility function in the general case. Thus, to use the greedy algorithm with a specific feasibility function on a real instance, we must transform it. For this, we construct a supergraph by adding edges of weight zero. We compare three feasibility functions, defined on complete graphs, connected cluster graphs and block graphsFootnote 3, respectively. Note that the construction of a complete supergraph requires the largest amount of edge additions whereas the least amount of edge additions is required for the construction of a block supergraph. We already showed in [9] that the computed ratio is close to one on real instances, that is, relatively far from the theoretical ratio of 3. The aim of these experiments is to answer the two following questions: Can greedy algorithms on connected cluster graphs and block graphs be used on large scaffold graphs, and what is its associated computation time? Do we get a better practical ratio if the amount of additional edges is smaller (e.g. the completion rate, see Table 7, is smaller)? In other words, do we obtain better results on block graphs and connected cluster graphs than in complete graphs? Table 7 Statistics on scaffold graphs Experiments were run on a personal computer with four i7 processors at 1.9GHz and 16GB RAM. Memory usage was very light, even on the biggest instance anopheles. Table 8 shows scores and computation times for every instance. We can see that greedy computation times are less than few seconds except for anopheles, where the connected cluster graph version and the block graph version need a few minutes. As expected, the greedy algorithms are much faster than the ILP formulation in every case. These results let us answer to our first question: connected cluster graph and block graph versions of the greedy algorithm are capable of treating big instances, however the computation time is significantly bigger than the complete version. Concerning the scores, we can see that the three greedy algorithms have the same score for most of the data. The connected cluster graph and block graph versions have a slightly better score in four instances: anopheles, anthrax, sacchr3 and sacchr12. Moreover, connected cluster graph and block graph versions have the same score in all instances except in anopheles, where the block graph version improves the score of the connected cluster graph version by three (which is not really significant compared to the absolute values). These results indicate that the answer to the second question is positive. However, the differences between scores are not significant enough to be completely affirmative. We can think that using the greedy algorithm with feasibility function defined on a sparser class of graphs may lead to better results. Table 8 Results statistics Conclusion and future work We presented in this paper the first polynomial-time algorithm approximating the scaffolding problem on non-complete graphs. Using a dynamic programming approach, we exploited the tree-like nature of connected cluster graphs to extend the feasibility function and the analysis of the approximation ratio. We also showed that this new algorithm provides slightly better results on real data than the greedy algorithm on complete graphs, although its theoretical ratio is worse. This leads us to the hypothesis that using a feasibility function defined on a graph class close to the original instance produces better results. This is surprising since, intuitively, algorithms on superclasses can choose from a larger set of edges to build solutions (any solution on the more restricted class is also a solution in the more general class). A natural extension of this work is to consider sparser graphs: for example, one could replace cliques in connected cluster graphs by co-bipartite graphs as the feasibility function is polynomial-time computable in this case [8]. One may also explore the possibility of exploiting randomized algorithms to improve the ratio [6]. The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request. 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The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data. Davot, T., Chateau, A., Fossé, R. et al. On a greedy approach for genome scaffolding. Algorithms Mol Biol 17, 16 (2022). https://doi.org/10.1186/s13015-022-00223-x Accepted: 20 September 2022 DOI: https://doi.org/10.1186/s13015-022-00223-x Poly-APX-hardness Submission enquiries: [email protected]	CommonCrawl
Antagonistic potential of an Egyptian entomopathogenic nematode, compost and two native endophytic bacteria isolates against the root-knot nematode (Meloidogyne incognita) infecting potato under field conditions Hamida A. I. Osman1, Hoda H. Ameen1, Mostafa M. A. Hammam1, Ghada M. El-Sayed2, Usama Samy Elkelany1 & Mahfouz M. M. Abd-Elgawad1 Egyptian Journal of Biological Pest Control volume 32, Article number: 137 (2022) Cite this article The root-knot nematode, Meloidogyne spp., are one of the most dominant and dangerous group of pests. The deformations and discolorations make tubers unmarketable and/or of less quality. Therefore, management of Meloidogyne spp. becomes an obligatory challenge that warrants intervention. Biological control agents are the best alternative tools for controlling plant-parasitic nematodes that comply with the requirements of the development of the green agriculture and that reduce the reliance on these harmful chemicals. Therefore, this study aimed to evaluate the effectiveness of compost singly, and in combinations with the bio-agents Heterorhabditis bacteriophora, and two bacterial isolates Nem 212 and Nem 213 against the root-knot nematode Meloidogyne incognita infecting potato plants under field conditions. Among 15 bacterial isolates (Nem205-Nem219) obtained from the rhizosphere of tomato and eggplant from Giza, Egypt, the two isolates (Nem 212 and Nem 213) were molecularly characterized based on the partial 16S rDNA sequencing analysis. These two bacterial isolates were deposited in the GenBank as Bacillus cereus Nem 212 and B. cereus Nem 213 and were tested against M. incognita J2s in vitro. Results showed that the cell filtrates of B. cereus Nem 212 and B. cereus Nem 213 gave the highest percentage of M. incognita J2s mortality (100%), after 48 h of the in vitro application. Moreover, all the applied treatments significantly suppressed the reproductive of M. incognita on potato plants and enhanced the potato crop yield under the field conditions. Compost enriched with B. cereus Nem 212 cell suspension was the most effective treatment. The combination between the bacterial cell suspension and the compost offered an increase in the disease curing and the potato plant growth and yield production, compared to the treatment with compost alone. The entomopathogenic nematode, Heterorhabditis bacteriophora, was relatively less effective in controlling M. incognita on potato, compared to B. cereus Nem 212 and/or B. cereus Nem 213 treatments. However, when compost was enriched with H. bacteriophora, it increased its capability to control the nematodes. This study provides insights into the practical usage of EPNs H. bacteriophora, and the endophytic bacteria (B. cereus Nem 212 or B. cereus Nem 213) as biocontrol agents against M. incognita on potato plants. The application of compost enriched with the bacterial cell suspensions of either B. cereus Nem 212 or B. cereus Nem 213 and H. bacteriophora within Galleria mellonella cadaver proved efficient control of M. incognita infecting potato plants and improved the growth and yield of potato plants under field conditions. Potato (Solanum tuberosum L.) is a carbohydrate-rich vegetable crop that has a high nutritional value and is highly popular worldwide. Potato production is negatively affected by plenty of various pests and diseases, including plant-parasitic nematodes that adversely affect the quantity and quality of the tuber yields (Mugniery and Phillips 2007). The root-knot nematodes (RKNs) Meloidogyne spp. are considered as the most damaging plant-parasitic nematode (Abd-Elgawad 2020), especially on potato plants. Moreover, RKNs cause deformations and discolorations of potato tubers which make tubers unmarketable (Vovlas et al. 2005). Contrary to chemical nematicides, biological control agents (BCAs) are safe tools to control plant-parasitic nematodes (PPNs). They also fulfill the requirements of the development of the green agriculture and reduce the reliance on the harmful chemicals. Among these alternative control strategies, various compost treatments are gaining a lot of interest because of their low cost and positive agronomical effect on plant growth, physical, chemical and biological properties of the soil and in its effectiveness in controlling plant pathogens, which eventually leads to an increase in crop production (Al-Hendy et al. 2021). Large numbers of soil amendments could be used as plant protectants or organic fertilizers (Ntalli et al. 2020). Furthermore, these organic wastes can act in integration with the microorganisms leading to an increase in the nematode-antagonistic populations (Hernandes et al. 2020). Management of PPNs using BCAs is a promising alternative to the chemical nematicides (Viljoen et al. 2019). Bacillus spp. are a group of bacterial agents that have been recognized as one of the most promising groups of nematode antagonists, e.g., B. cereus that has been found to be effective in managing RKN and enhancing crop production (El-Wakeel et al. 2020). The action of the genus Bacillus on the plant-parasitic nematodes has been widely confirmed (Yin et al. 2021). Entomopathogenic nematodes (EPNs) of the families: Steinernematidae and Heterorhabditidae, are obligate pathogens of soil insects and are employed as biocontrol agents for a number of insect pests (Kaya and Gaugler 1993) that provide environmentally safe management tools for PPNs without affecting the free living nematodes, which play an important role in nutrient cycling (Somasekhar et al. 2002). These nematodes are mutualistically associated with certain bacterial species of the genus Xenorhabdus that frequently associated with Steinernema and the genus Photorhabdus with Heterorhabditis. Third-stage infective juveniles (IJs) of EPNs of the families Steinernematidae and Heterorhabditis are the only stage of the nematode that can survive outside the host in the soil. The IJs retain cells of the bacterial symbiont in their intestines when they leave the host. When the IJs find a susceptible insect host, they enter the insect through the natural openings (anus or mouth) and penetrate into the hemocoel, where the nematodes produce toxins and release their mutualistic bacteria. The bacterial cells proliferate rapidly in the hemocoel, and the host is consequently killed by toxemia (Kusakabe et al. 2022). The endosymbionts bacteria: Xenorhabdus and Photorhabdus, produce a wide range of secondary metabolites. These metabolites are different kinds of antibiotics, proteases, adhesions, lipases and hemolysins, and they are used as biocontrol negotiators for management of virus, fungi, nematodes and insects (Lulamba et al. 2021). Only two secondary metabolites molecules, a stilbene derivative (3,5-dihydroxy-4-isopropylstilbene) and indole, had been found to have nematicidal activity. These metabolites are toxic to many species of nematodes including those of the genus Meloidogyne (Tomar et al. 2022). It is well known that the efficiency of bio-agents varies upon both environmental conditions and nematode species. Bacterial strains collected from one set of environmental conditions may be less effective under other environmental conditions (Schisler et al. 1997). Therefore, one of the means to increase the potency of these biocontrol agents is to use the native ones that are well adapted to the local environmental conditions. This study was conducted to assess the efficacy of compost singly or in combination with the Egyptian EPN, Heterorhabditis bacteriophora, and the two bacterial isolates (B. cereus Nem 212 and Nem 213) against M. incognita, infecting potato plants under field conditions. Source of seeds Potato tuber seeds (Solanum tuberosum L. cv. Sponta) weighing 40–50 g per tuber imported from Denmark by Chance Company for Import and Export, Egypt, were used for the field experiment. Isolation of bacterial strains Soil samples were collected from rhizosphere region of tomato and eggplant farms from Kafr-Hakim village, Giza, Governorate, Egypt, at 10–15 cm depth. About one gram of soil was dissolved in 100 ml of sterile saline solution. Afterward, serial dilution up to 10–7 using sterile saline solution (0.85%, NaCl w/v) was made. After thorough mixing, 50 µl aliquot from suitable dilutions was pour plates, in triplicate, on the nutrient agar (NA) plates (Biobasic, India) and incubated for 24 h at 30 ± 0.5 °C. To obtain axenic bacterial cultures, single colonies were then suspended in sterile saline solution and distributed on NA plates as the same in soil samples. Standard microbiological methods were used to isolate bacteria from the rhizosphere (Flores-Vargas and O'Hara 2006). Fifteen pure bacterial cultures were isolated and maintained on NA slants at 4 °C for further studies and coded as Nem205 to Nem219. Preparation of second-stage juveniles of Meloidogyne incognita M. incognita population was maintained on susceptible tomato plants in the greenhouse. The nematode species was previously identified based on the morphological features of the perineal pattern (Taylor and Sasser 1978). The nematode eggs were extracted from the roots of tomato plants using a 0.5% NaOCl according to the method described by Hussey and Barker (1973). The eggs were incubated in egg hatching plastic cups at the room temperature for 72 h to obtain second-stage juveniles (J2). The J2s were surface sterilized by 0.01% streptomycin sulfate solution for 1 h before use. Nematicidal activity of bacterial isolates in vitro To determine the nematicidal activity of the bacterial isolates, axenic bacterial cultures were prepared in nutrient agar plates as described earlier. Bacterial cultures were centrifuged at 500 rpm for 15 min. The supernatant solution was passed through a 0.22 µm in diameter nitrocellulose filter, and the flow through was used as a cell-free filtrate for the bioassay test. Petri dishes (5 cm in diameter) were supplied separately with 1 ml of cell-free filtrate from each bacterial isolate plus 3 ml of nematode suspension in distilled water containing 50 ± 5 freshly hatched M. incognita second-stage juveniles J2. A volume of 4 ml of distilled water containing 50 ± 5 freshly hatched M. incognita juveniles served as a control. All treatments and control were replicated five times. All dishes were kept in incubator at 35 °C. Dishes were partially covered to permit aeration and less evaporation. The mortality rates of juveniles were recorded after 24 and 48 h under a light microscope. After the exposure periods, the nematodes in each treatment were transferred to distilled water and left for 24 h before microscopic examination whether immobile juveniles resumed activity or not. The corrected percentages of nematode mortality were calculated according to the following equation: Mortality % = (m-n)/(100-n) × 100, where m and n indicate the percentages mortality in treatment and control, respectively (Abbott 1925). Extraction of genomic DNA from bacterial isolates Two bacterial isolates (B. cereus Nem 212 and Nem 213) were selected for amplification of the 16S rDNA. Single colonies of bacterial isolates were cultured in conical flasks containing 20 ml of nutrient broth medium for 18 h at 120 rpm and 30C°. The cultures were centrifuged at 12,000 rpm for three minutes at 4 °C. The pellets were subjected to genomic DNA extraction using the (QIAamp DNA Mini Kit, QIAGEN, Germany). The extracted DNA was used as a template for PCR to amplify the 16S rDNA gene using the universal primers: forward primer sequence (5′AGAGTTTGATCCTGGCTCAG3′) and reverse primer sequence (5′CTACGGCTACCTTGTTACGA3′), thereby producing an amplicon of ~ 1500 bp (El-Sayed et al. 2018). The amplicon obtained by PCR was purified using the QIA quick PCR purification kit (Qiagen, Germany), following resolving by electrophoresis on 1% agarose gels and compared to a 100 bp DNA ladder (Thermo scientific, USA). Purified PCR products were subjected to sequencing by sanger sequencing method using sequencer 3500 genetic analyzer, big dye X terminator kit (thermo fisher, USA) for forward and reverse directions in biomedical laboratory of colors in Clinilab, Egypt. The sequences were edited using Bioedit 7.1.10 (Hall 1999) software, and BLAST was used to detect the homology with other relatives (Altschul et al. 1997). Assay of protease activity A colony from each bacterial isolate (B. cereus Nem 212 and Nem 213) was cultured in Luria Bertani (LB) broth media pH 7 for 24 h at 30 °C and 200 rpm in an orbital shaker to develop bacterial growth. The liquid medium used for the production of alkaline protease had the following composition (% w/v): 2.0% glucose, 2.0% hydrolysate casein, 0.04% CaCl2 and 0.02% MgCl2. The pH of the medium was adjusted to 8.5. The production medium was inoculated with 5% inoculum. The flasks were incubated for 72 h in a 30 °C shaking incubator (200 rpm). The contents were then centrifuged (12,000 g, 4 °C, 20 min), and the cell-free supernatant was used for determining extracellular protease activity. Casein dissolved in pH buffer 9 was used as the substrate for the assay. The reaction mixture containing casein and the enzyme solution was incubated for 10 min at 37 °C. The amount of enzyme required to liberate 1 μg tyrosine per ml per minute under the standard conditions, defined one unit of protease activity (Asha and Palaniswamy 2018). Source, culturing and treatments of entomopathogenic nematodes The IJs of an Egyptian EPN Heterorhabditis bacteriophora routinely cultured in last-instar larvae of the greater wax moth, Galleria mellonella L. (Woodring and Kaya 1988), were used in two delivery forms. Five days after adding 10 G. mellonella larvae to a 10-cm-diameter Petri dish lined with # 1 Whatman filter paper and wetted by one ml distilled water having 200 IJs, the consequently infected insect cadavers were used for application. The cadavers were buried directly about 3 cm below the soil surface, beneath the potato seedling stem at a rate of 5 insects/seedling (cadaver application). The other delivery form used the IJs in water suspension. The IJs were collected from White traps and applied within 3 days of nematode emergence at a rate of 100 ml of IJ suspension/potato seedling (125 active juveniles/ml). A one-liter hand sprayer was used to deliver the suspension on the soil surface around the potato seedling (spray application). The two IJ treatments were applied early in the morning as soil drench; the soil was wetted by water. Source of compost Organic compost was purchased from Shafei Compost Company, Giza, Egypt. The chemical analysis was provided to the authors by the company (Table 1). Table 1 The chemical composition of the organic compost used in this study Nematode identification and field experiment The field experiment was carried out during the period of January–June 2021, at Kafr-Hakim village, Giza, Governorate, Egypt. The experimental area was naturally infested with M. incognita. The roots of potato plants previously planted in the experimental field were collected, and the adult females were picked out from the infected roots to identify the nematode species using the morphological characteristics of their perineal pattern according to female perineal pattern (Taylor and Sasser 1978). This area was divided into 3 plots, each comprising 8 rows of 8 m. length and 50 cm. width, and the distance between plants was 30 cm. The experiment was set up in a completely randomized block design with 8 treatments; each treatment was replicated 3 times. The treatments were: 1—M. incognita-infested non-treated soil (control) (T1), 2—M. incognita-infested soil + Compost C (T2), 3—M. incognita-infested soil + C + Heterorhabditis bacteriophora-infective juveniles within infected G. mellonella larvae (T3) (cadaver application), 4—M. incognita-infested soil + C + H. bacteriophora-infective juveniles in water suspension (T4) (spray application), 5—M. incognita-infested soil + C + Bacillus cereus Nem 212 cell suspension (T5), 6—M. incognita-infested soil + C + B. cereus Nem 212 cell-free culture filtrate (T6), 7—M. incognita-infested soil + C + B. cereus Nem 213 cell suspension (T7), and 8—M. incognita-infested soil + C + B. cereus Nem 213 cell-free culture filtrate (T8). In each treatment with the bacterial isolates B. cereus Nem 212 suspension and B. cereus Nem 213 suspension (T5 and T7), soil was treated with the bacterial suspension at the rate of 50 ml/hill, and concentration was (2 × 107 CFU/ml) at planting time. In treatments (T 6 and T8), the bacterial culture filtrates were added at the rate of 50 ml/hill at planting time. Compost was applied into soil, 10 days before planting for proper decompositions of the materials at the rate of 50 g/hill. Initial population densities of M. incognita juveniles (J2) were determined at planting time according to Barker (1985) from 250 g subsamples of well mixed soil from each row (5 subsamples per 15 sample per treatment). Five months later (at harvest), five potato plants were randomly selected from every row and carefully uprooted. Potato tubers were hand harvested for yield estimation and recorded in terms of their average weights. Other plant growth parameters including: root length, plant height, number and weight of tubers/plant, were recorded. For evaluation of nematode reproductive parameters (NRPs), the numbers of root galls and egg masses/5 g roots as well as number of M. incognita juveniles/one g roots and total number of eggs/5 g roots were recorded. Assessment of root infestation: The roots from each plant were gently washed and cut into 3.0 cm length pieces. One-gram subsample was taken from each plant. The roots were stained with acid fuchsin–lactophenol (Bybd et al 1983) and M. incognita (J2s) were counted under light microscope and recorded. Assessment of eggs per 5 g roots: one-gram subsample from a 5 g potato roots was taken, cut into pieces of 2-cm long. M. incognita eggs were extracted from the roots using a 0.5% NaOCl solution for 3 min and then obtained by rinsing the egg suspensions with sterile water in a sieve with 25-µm openings according to the method of Hussey and Barker (1973). Eggs were counted under a light microscope, and their average numbers were recorded. Final nematode soil population was extracted, and densities of M. incognita were determined and expressed as the number of juveniles/250 g soil (Barker 1985). Percentage nematode reduction was determined according to Henderson and Tilton formula (Puntener 1981) as follows: $${\text{Nematode}}\,{\text{reduction}}\,\% = \left\{ {1 - ({\text{PTA}}/{\text{PTB}} \times {\text{PCB}}/{\text{PCA}})} \right\} \times 100$$ where PTA = population in treated plot after application, PTB = population in treated plot before application, PCB is population in check plot before application, and PCA = population in check plot after application. Data were subjected to the analysis of variance (ANOVA) according to Snedecor and Cochran (1980), using Assistant program version 7.6 beta. The means were compared using New Multiple Range Test (DNMRT) at P ≤ 0.05 (Duncan 1955). Screening endophytic isolates with nematicidal activity in vitro Fifteen endophytic bacteria strains were isolated from the rhizosphere of tomato and eggplants. Based on the nematicidal activity and different appearances of dead nematodes in vitro after 24 and 48 h of treatment with the isolate's cell-free filtrates, the mortality rate of the 15 isolates was above 90%. The highly percentage mortality (100%) was recorded by two bacterial isolates (Nem 212 and Nem 213) (Table 2). Table 2 Nematicidal activity of the cell-free filtrates of 15 bacterial isolates against Meloidogyne incognita juveniles (J2s), in vitro Molecular identification of Bacillus cereus Nem 212 and 213 isolates The universal primers of 16 s rDNA gene amplified a fragment of approximately ~ 1550 bp (Fig. 1) in both bacterial isolates. The amplified gene was sequenced in both directions. After trimming and sequence editing, partial DNA sequences were subjected to BLAST search on https://blast.ncbi.nlm.nih.gov/Blast against the available sequences deposited in NCBI database. Partial 16S rDNA gene sequence of bacterial isolate Nem 212 matched 100% to Bacillus cereus ATCC 14,579 16S ribosomal RNA accession no. NR_074540. In regard to Nem 213, partial 16S rDNA gene sequence matched 99.85% to Bacillus cereus strain 1H4 16S ribosomal RNA gene accession no. OK178870. Bacterial isolates Nem 212 and Nem 213 were submitted at GenBank database as B. cereus Nem 212 and B. cereus Nem 213 under accession numbers OK284601 and OK284743, respectively. Agarose gel electrophoresis for the PCR product of 16 s rDNA gene (1500 bp) in Bacillus cereus Nem 212 (1) and Bacillus cereus Nem 213 (2); M: 100 bp DNA ladder (Thermo Scientific, USA) Protease activity of the bacterial strains The results showed that B. cereus Nem 212 protease activity was 2.01 µg/min/ml; however, Bacillus cereus Nem 213 recorded 1.99 µg/min/ml. Effects of compost enriched with endophytic bacteria and entomopathogenic nematodes on the reproduction of Meloidogyne incognita All the tested treatments (Table 3) significantly suppressed the reproduction of M. incognita on potato plants at various time intervals, compared to the non-treated control. After two months of planting, application, T7 (C + B. cereus Nem 213 cell suspension) produced the highest percentage of reduction (79.4%) in the numbers of juveniles (J2s) in the soil compared to the non-treated control, followed by T8 (B. cereus Nem 213 supernatant (cell-free culture filtrate) that produced 75.6%, reduction in (J2s) in the soil. Other treatments with compost (T2, T5 and T3) exhibited 66.3%, 57.1% and 54.8% reduction in the number of J2s in the soil, respectively, than the non-treated control. However, the lowest percent reduction in the numbers of J2s in the soil was found in T6 (C + B. cereus Nem 212 supernatant (cell-free culture filtrate) that produced reduction of 21.3% in the number of J2s in the soil compared to the non-treated control. Table 3 Effect of compost and biocontrol agents on potato plant cv. Sponta infected with root-knot nematode Meloidogyne incognita under field conditions after two months of planting and at harvest The treatment: T7 and T5, resulted in the greatest percentages reduction in the number of nematode stages in the roots 79.1% and 77.3%, respectively, compared to the non-treated control. However, application of T4 (C + H. bacteriophora juveniles in water suspension) showed the least percentage reduction in the number of juveniles in the roots (15.5%). Application of T5 and T6 exhibited the greatest percentage reduction, in number of galls (72% and 70.1%), respectively, compared to non-treated control. At harvest, all treatments suppressed the nematode reproduction on potato under field conditions (Table 3). The treatment T5 (C + B. cereus Nem 212 cell suspension) provided the highest reduction in the number of galls and egg masses/plant, number of nematode stages/roots, nematode final population (Pf) and number of nematode eggs/5 g roots, followed by (T7 cell suspension (Table 3)). Application of compost alone (T2) resulted in 66.4, 38.5, 65.8, 47.1 and 46.8% decrease in the number of M. incognita (J2) in soil, number of galls and egg masses/plant, number of nematode stages in the roots and total number of eggs/5 g roots, respectively, compared to the control (Table 3). Concerning the application of entomopathogenic nematode, the treatment T4 (C + H. bacteriophora IJs in water suspension) exhibited percentage reduction of 46.6, 17.2, 19.4, 38.8 and 29.1% in the number of M. incognita J2 in soil, number of egg masses and galls/plant, number of nematode stages in the roots and the total number of eggs/5 g roots, respectively, compared to the non-treated control. Finally, the application of T3 (C + H. bacteriophora IJs within G. mellonella cadavers) resulted in a reduction of 54.9, 59.7, 64.8% in the number of M. incognita (J2s) in the soil, number of egg masses/plant and total number of eggs/5 g roots, respectively. Effects of compost enriched with endophytic bacteria and entomopathogenic nematodes on potato growth parameters In addition to the nematicidal activity of the studied treatments, especially T5, T6, T7 and T8, the treatments generally enhanced the potato growth parameters including: root length, plant height, dry weight of plant, number and weight of potato tubers and number of leaves in the various levels of potato growth, compared to the non-treated control (Table 4). However, compost and H. bacteriophora IJs in water suspension treatments were the least effective in this respect (Table 4). Table 4 Growth parameters at harvest of potato cv. Sponta infected with Meloidogyne incognita and treated with compost and bio-agents under field conditions Treatment effects on potato yield All treatments significantly (P ≤ 0.05) increased potato yield production (Table 4). T7 treatment resulted in the greatest increase (221.2%) in potato yield compared to the non-treated control, followed by T8 and T5 (209% and 206.8% increase in potato yield, respectively), compared to the non-treated control. However, the application of T3 compost and H. bacteriophora (cadaver application) and T4 containing the compost and H. bacteriophora (spray application) produced an increase of 163.6% and 90.9%, respectively, in the yield of potato, compared to the non-treated control (Table 4). The results of the present field experiments showed that all the applied treatments significantly reduced the reproductive of M. incognita on potato plants and enhanced the potato growth and tuber yield. The most nematode-suppressive treatments were those of the endophytic bacteria Bacillus cereus Nem 212 and B. cereus Nem 213. Antagonistic bacteria like B. cereus were previously found to be effective microorganisms in controlling root-knot nematodes and enhancing the plant growth (Osman et al. 2021). The present results clearly showed that applications of B. cereus (Nem212 and 213 isolates) as cell suspension were more significantly potent than when used as culture-cell-free filtrate in suppressing nematode reproductive ability. Application of different delivery formulations has been reported to affect the potentiality of bacterial preparations on the host plant (Nagachandrabose 2020). Gao et al. (2016) previously found that the supernatant of B. cereus resulted in 90.96% mortality of M. incognita J2s indicating that its capability to produce some extracellular substances can kill the nematodes. In a pot experiment, the control efficiency against M. incognita reached 81.36% for B. cereus S2 supernatant due to the nematicidal substance of sphingosine in the supernatant (Gao et al 2016). This can be explained in the light of B. cereus culture containing not only the sphingosine but also live cells or spores, which can colonize the rhizosphere of the plant to exert nematicidal activity for a long term. From another point of view, Zhou et al. (2021) stated that the endophytic bacteria B. cereus can produce some extracellular substances like protease that kill 100% of M. incognita within 72 h by degrading the cuticle and egg shell. Similarly, Yin et al. (2021) reported that B. cereus strain Bc-cm 103 caused 100% mortality of M. incognita J2 but within only 12 h. They also identified volatile organic compounds caused 97.2% in M. incognita J2 after 48 h. Siddiqui (2000) indicated that aqueous cell suspension and the cell-free culture filtrate considerably reduced nematode population in the root and soil and subsequently reduced M. javanica population in tomato plants. Aqueous cell suspension was found more effective than culture filtrate of the bacterium, indicating that the respective activity was due to cellular metabolic components. It was also speculated that compounds required for growth and suppression of nematode's reproduction might not be produced in sufficient quantity in culture media as produced by the bacterial cells in the rhizosphere. Furthermore, the success of bio-agents with respect to their biocontrol efficacy and consistency relies upon appropriate delivery mechanisms at field conditions. Earlier reports revealed that incorporation of bio-agents with organic amendments such as compost will change the soil environment in favor of the bio-agents and provided readily available nutrients to fungal and bacterial antagonists for their survival and development (Timper 2014). Moreover, Al-Hendy et al. (2021) proposed multiple mechanisms to explain the beneficial effects of organic amendments on PPNs and plants by releasing nematode-toxic compounds from decomposing materials, stimulating the nematodes' natural enemies, improving tolerance to nematodes and altering chemicals, biological and physical properties. In addition, these amendments enhance fertility of the soils which in turn improve plant growth. Walker (2004) reported that the activity of the bio-agents was directly correlated with organic amendments. Afterward, many researchers demonstrated that using bio-agents enriched with organic amendments exhibited greater antagonistic activity against plant pathogens such as root-knot nematodes (Gowda et al. 2018). In the present study, organic compost with the endophytic bacteria was found to be more effective against M. incognita than the compost alone treatment. The endophytic bacteria B. cereus Nem 212 and B. cereus Nem 213, at various delivery methods, whether cell suspension or supernatant enriched with compost were the highly effective treatments in improving plant growth and reducing nematode multiplications. These results are in harmony with findings of Park et al. (2014) who estimated the potential of B. cereus C1-7 against root-knot nematode M. hapla infecting carrot and tomato plants in pot conditions. They reported a complete inhibition of root galls and egg mass formation in treated plants, and subsequently reducing root-knot nematode damage and suppressing nematode population. From the biochemical point of view, B. cereus S2 could induce systemic resistance in tomato plant and enhance the activity of some defense-related enzymes for the biocontrol of M. incognita. These findings might be supported by several workers (Osman et al. 2021) who demonstrated that B. cereus is a strong producer of hydrolytic enzymes such as protease which may be partially involved in the suppression of root-knot nematodes reproduction. This enzyme can potentially cause harm to the external structures of the nematodes and their eggshells, accompanied by inhibition of egg hatching and increased juvenile's mortality (Hong et al. 2013). The suppression of M. incognita reproduction on potato plants as a result of the treatment with B. cereus strains has led to the enhancement of plant growth and tuber yield. The efficacy of EPNs was relatively lower when compared with B. cereus Nem 212 or B. cereus Nem 213 treatments in this study. These results are in agreement with those reported by many authors under greenhouse and field conditions (Hammam et al. 2019). Our application using EPN-IJs within G. mellonella cadavers exhibited more inhibition in NRPs than using IJs in water suspension. Similar results were obtained under laboratory and greenhouse conditions (El Aimani et al. 2022). It has also been reported that applying the IJs within cadavers has a higher dispersal capacity and prolonged longevity compared to IJs in water suspension. Moreover, the cadavers themselves offer protection against harmful environmental conditions such as freezing and desiccation (Dolinski et al. 2015). Application of IJs in aqueous solution has some disadvantages such as decreased infectivity, survival during storage and the need for adequate irrigation equipment (Grewal 2002). Eventually, cadaver or spray application significantly increased plant growth parameters and potato yield. These results are in harmony with El Aimani et al. (2022). This increase was at least partly due to competition at the root surface, reduction in root-knot infection rate, production of allelochemicals or could be a fertilizer effect of the treatment combined with relief from M. incognita infection (Sayedian et al. 2020). The application of compost enriched with bacterial cell suspensions of either B. cereus Nem 212 or B. cereus Nem 213 strains or H. bacteriophora IJs (cadaver application) gave more effective results in controlling M. incognita infecting potato plants and improved the growth parameters and production of potato tubers under field conditions. Further research experiments are still needed to determine the optimum dose–response as well as the time of application and economic value in each plant species. 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Front Microbiol 12: article 684888 This study was supported in part by the NRC In-house project No. 12050105 entitled "Pesticide alternatives against soil-borne pathogens and pests attacking economically important solanaceous crops." The facilities offered by The National Research Centre are appreciated. Financial support was partially made by National Research Centre, Egypt (In-house project No. 12050105), to develop and analyze the data. Plant Pathology Department, National Research Centre, El-Tahrir Street, Dokki, Giza, 12622, Egypt Hamida A. I. Osman, Hoda H. Ameen, Mostafa M. A. Hammam, Usama Samy Elkelany & Mahfouz M. M. Abd-Elgawad Microbial Genetic Department, National Research Centre, Giza, Egypt Ghada M. El-Sayed Hamida A. I. Osman Hoda H. Ameen Mostafa M. A. Hammam Usama Samy Elkelany Mahfouz M. M. Abd-Elgawad All authors participated in the development and implementation of the reviewing plan and subsequently written it. The first author HO discussed the different parts of the article with HA, MH, GE, UE, and MA to conduct and finalize the experimentation and write the manuscript. All authors have read and approved the final manuscript. Correspondence to Hoda H. Ameen. Osman, H.A.I., Ameen, H.H., Hammam, M.M.A. et al. Antagonistic potential of an Egyptian entomopathogenic nematode, compost and two native endophytic bacteria isolates against the root-knot nematode (Meloidogyne incognita) infecting potato under field conditions. Egypt J Biol Pest Control 32, 137 (2022). https://doi.org/10.1186/s41938-022-00635-2 Entomopathogenic nematode Endophytic bacteria isolates	CommonCrawl
Hydrocortisone enhances the barrier properties of HBMEC/ciβ, a brain microvascular endothelial cell line, through mesenchymal-to-endothelial transition-like effects Tomomi Furihata1, Shinya Kawamatsu1, Ryo Ito1, Kosuke Saito2, Shota Suzuki1, Satoshi Kishida1, Yoshiro Saito2, Atsuko Kamiichi1 & Kan Chiba1 Fluids and Barriers of the CNS volume 12, Article number: 7 (2015) Cite this article Because in vitro blood–brain barrier (BBB) models are important tools for studying brain diseases and drug development, we recently established a new line of conditionally immortalized human brain microvascular endothelial cells (HBMEC/ciβ) for use in such models. Since one of the most important functional features of the BBB is its strong intercellular adhesion, in this study, we aimed at improving HBMEC/ciβ barrier properties by means of culture media modifications, thus enhancing their use for future BBB studies. In addition, we simultaneously attempted to obtain insights on related mechanistic properties. Several types of culture media were prepared in an effort to identify the medium most suitable for culturing HBMEC/ciβ. The barrier properties of HBMEC/ciβ were examined by determining Na+-fluorescein permeability and transendothelial electric resistance (TEER). Endothelial marker mRNA expression levels were determined by quantitative real-time polymerase chain reaction. Adherens junction (AJ) formation was examined by immunocytochemistry. Cell migration ability was analyzed by scratch assay. Furthermore, cellular lipid composition was examined by liquid chromatography-time-of-flight mass spectrometry. Our initial screening tests showed that addition of hydrocortisone (HC) to the basal medium significantly reduced the Na+-fluorescein permeability and increased the TEER of HBMEC/ciβ monolayers. It was also found that, while AJ proteins were diffused in the cytoplasm of HBMEC/ciβ cultured without HC, those expressed in cells cultured with HC were primarily localized at the cell border. Furthermore, this facilitation of AJ formation by HC was in concert with increased endothelial marker mRNA levels and increased ether-type phosphatidylethanolamine levels, while cell migration was retarded in the presence of HC. Our results show that HC supplementation to the basal medium significantly enhances the barrier properties of HBMEC/ciβ. This was associated with a marked phenotypic alteration in HBMEC/ciβ through orchestration of various signaling pathways. Taken together, it appears that overall effects of HC on HBMEC/ciβ could be summarized as facilitating endothelial differentiation characteristics while concurrently retarding mesenchymal characteristics. The blood–brain barrier (BBB), which is formed primarily by brain microvascular endothelial cells (BMECs), is an interface between the central nervous system (CNS) and the systemic circulation [1]. Several cell types located adjacent to BMECs (including astrocytes and pericytes) are also known to contribute to BBB function. One of the most important features of the BBB is its extremely strong intercellular adhesion, which is established by adherens junctions (AJs) and tight junctions (TJs) between the endothelial cells [1]. This forceful adhesion seals the paracellular route and prevents entry of a variety of substances, both small and large, into brain from blood, while simultaneously creating a foundation that allows BMEC transporters to take up or expel molecules indispensable for, or harmful to, the physiological functions of the brain. Based on this functional importance in maintaining brain homeostasis, it has become increasingly evident that impairment of BBB function is associated with various CNS diseases, such as multiple sclerosis, amyotrophic lateral sclerosis and Alzheimer's disease, even though it remains currently inconclusive whether BBB impairment is a cause or consequence of those diseases [2,3]. On the other hand, because the BBB appears to prevent passage of more than 98% of all small therapeutic molecules [4], the barrier is considered a primary obstacle that prevents drugs from exercising their pharmacological actions in brain. This is a critical reason why the development of CNS drugs is difficult and time consuming. Collectively, the BBB is a pivotal research target for various brain diseases and CNS drug development studies. In vitro BBB models are among the most important tools used in BBB studies [5,6]. While it is probably inevitable that in vitro BBB models will never reflect the full range of in vivo BBB functionalities due to their differing environments, such models offer multiple experimental benefits in terms of simplicity, scalability, and versatility. In in vitro BBB models, BMECs are cultured on a porous membrane in a transwell culture system, thereby resulting in a cell layer that creates separate blood-side and brain-side compartments. To date, human and animal primary BMECs, as well as immortalized human and animal BMECs, have been extensively utilized in such models [6]. The primary cells show excellent functionality, but they suffer from several experimental limitations, such as scarcity, low cell proliferation potential and sample to sample variations. On the other hand, even though the functional levels of immortalized cells are not regarded as being as high as that of the freshly isolated primary cells, they show infinite proliferation ability and stable phenotypes, which makes them useful to researchers conducting a variety of experiments. Therefore, it would be ideal if, through the refinement and elaboration of their culture methods, the BBB functions of immortalized cells could be improved to levels that are comparable to primary cells. Recently, we reported establishment of a new line of human immortalized BMECs, HBMEC/ciβ [7]. In addition to excellent proliferative ability, HBMEC/ciβ express a series of endothelial marker genes (e.g., von Willebrand factor (vWF) and vascular endothelial-cadherin (VE-cadherin)) along with BBB-related genes (e.g., claudin-5, glucose transporter 1, P-glycoprotein, and transferrin receptor), and limited sucrose and Na+-fluorescein (Na-F) penetration across cell monolayers. Based on these characteristics, it is logical to conjecture that HBMEC/ciβ have significant potential for providing the foundation of uniquely effective in vitro BBB models. However, to achieve this goal, further improvements to the intercellular junctional property of the cells are absolutely necessary. Since it is well known that the barrier function of in vitro BBB models can be reinforced or degraded by various biological and chemical factors [6], it is also likely that the culture media composition plays a crucial role in determining the strength of the HBMEC/ciβ barrier function. In this regard, it was noted that the culture medium we used initially was designed for pan-primary cells, which caused us to consider the possibility that media optimization for HBMEC/ciβ culture might enhance barrier properties. Accordingly, the primary purpose of the present study was to clarify the effects of culture media modifications on HBMEC/ciβ barrier functions. In addition, we provide results that show mechanistic insights into the effects of those modifications. Culture medium CSC medium (Complete Medium Kit containing 10% fetal bovine serum, 4Z0-500-R, Cell Systems, Kirkland, WA, USA) or EBM2 medium (Lonza, Walkersville, MD, USA) was used as a basal medium. CultureBoost-R (cbR, 2% v/v, Cell Systems) and SingleQuots (SQs, Lonza) were used as culture supplements. Although the components of cbR are not published, according to the manufacturer's information it contains several growth factors. SQs consist of a series of vials, each of which contains 2% (v/v) fetal bovine serum, 0.1% (v/v) vascular endothelial growth factor, 0.1% (v/v) long R3 insulin-like growth factor, 0.1% (v/v) recombinant human epidermal growth factor, 0.4% (v/v) recombinant human basic fibroblast growth factor, 0.1% (v/v) hydrocortisone (HC), 0.1% (v/v) heparin or 0.1% (v/v) ascorbate. The actual HC concentration is 180 nM based on our determination using enzyme-linked immunosorbent assay. Additionally, all culture media were supplemented with blastcidin S (4 μg/mL) and penicillin-streptomycin. Medium information is also provided in Additional file 1: Figure S1. HBMEC/ciβ were routinely grown on type-I collagen-coated dishes in CSC-cbR at 33°C with 5% CO2/95% air. They were seeded (day 0) at 1.0 × 105 or 4.0 × 105 cells/mL onto a dish or a membrane filter of an insert culture system (polyethylene terephthalate, 0.4 μm high-density pores, and 0.3 cm2, BD Falcon, Franklin Lakes, NJ, USA), respectively. At day three, the medium was changed to either the same or a differently supplemented medium, depending on the experiments (for example, the medium was changed from CSC-cbR to CSC-HC at day three). Then, the cells were continuously cultured for 12 days, during which a medium change was conducted every other day. All functional or gene expression analyses were performed on day 12. The culture schedule is also provided in Additional file 1: Figure S1. Permeability assay The apparent permeability (Papp, cm/min) and the permeability coefficients (Pe, cm/min) of Na-F (Sigma, St. Louis, MO, USA) and [14C] sucrose (GE Healthcare, Giles, UK) were determined essentially as described previously [7]. Briefly, the medium was replaced with serum-free CSC medium 30 min before the assay (4Z3-500-R, Cell Systems). The assay was initiated by adding [14C] sucrose (0.1 μCi/mL) or Na-F (500 ng/mL) to the insert at 37°C. The incubation time was 40 min. The Pe values were calculated using the permeability-surface area product (PS, μL/min) described in our previous report [7], and the Papp values were calculated using the following equation: $$ {P}_{app}\left( cm/ min\right)=\frac{V_{baso}\ \left(c{m}^3\right)}{\mathrm{A}\ \left(c{m}^2\right)\times {\left[{C}_0\right]}_{api}\ \left( ng/\mu L\right)}\times \frac{\varDelta {\left[C\right]}_{baso}\ \left( ng/\mu L\right)}{\varDelta T\ (min)} $$ where V baso is medium volume at the basolateral side, A is membrane surface area (0.3 cm2), [C 0 ] api is Na-F concentration at the apical side at T = 0, [C 0 ] baso is Na-F concentration at the basolateral side at T = 40, and ΔT is time of experiment. Determination of transendothelial electric resistance (TEER) TEER was examined using the Millicell ERS-2 (Millipore, Billerica, MA, USA) before performing permeability analysis. After subtracting the ohms of a blank insert membrane from the ohms of cell monolayer, the value was multiplied by 0.33 cm2 (Ω × cm2). Total RNA isolation, cDNA synthesis, and quantitative real-time polymerase chain reaction (qPCR) Total RNA extraction and cDNA synthesis of the HBMEC/ciβ (cultured as described above) were conducted using methods described previously [7]. qPCR was performed using the previously-described SYBR green-based method [7] to determine the following mRNA expression levels: glucocorticoid-induced leucine zipper (GILZ), nuclear factor-Kappa B inhibitor alpha (NFκBIA), annexin A1 (ANXA1), matrix metalloproteinase 1 (MMP-1), MMP-2, MMP-16, vWF, Duffy antigen/receptor for chemokines (DARC), angiopoietin 2 (ANGPT2), early growth response 1 (EGR-1), histone deacetylase 7 (HDAC7), inhibitor of differentiation or DNA binding-1 (Id-1), Ras-proximate-1 or Ras-related protein 1 (RAP1), exchange protein directly activated by cAMP (EPAC), VE-cadherin, claudin-5, occludin, glyceronephosphate O-acyltransferase (GNPAT), alkylglyceronephosphate synthase (AGPS), fatty acyl-CoA reductase 1 (FAR1), and glyceraldehyde 3-phosphate dehydrogenase (GAPDH). The primers used for qPCR are described in (see Additional file 2: Table S1), and the amplification efficiency of each PCR was confirmed to be close to one. Data was calculated using the delta-delta-CT method, where GAPDH was used as a control. Western blotting analysis HBMEC/ciβ cells were cultured on dishes as described above. Homogenates were prepared using methods described previously [7]. Proteins were separated by sodium dodecyl sulfate-polyacrylamide gel electrophoresis, and then transferred onto a polyvinyldene difluoride membrane. The membrane was blocked with 5% skim milk. The primary antibodies used were rabbit anti-glucocorticoid receptor (GR) polyclonal IgG (1,000-fold dilution, sc-1002, Santa Cruz Biotechnology, Santa Cruz, CA, USA), rabbit anti-VE-cadherin polyclonal IgG (1,000-fold dilution, sc-28644, Santa Cruz Biotechnology), rabbit anti-claudin-5 polyclonal IgG (1,000-fold dilution, ab53765, Abcam, Cambridge, UK), or rabbit anti-occludin polyclonal IgG (1,000-fold dilution, 71–1500, Zymed Laboratories, San Francisco, CA, USA). The secondary antibody used was goat anti-rabbit IgG–peroxidase antibody (10,000-fold dilution, A9196, Sigma). Fluorescence detection of adherens junction-related proteins The insert membrane filter on which HBMEC/ciβ were cultured was taken out, followed by incubation with BD Cytofix/Cytoperm Fixation and Permeabilization Solution (BD Biosciences) for 40 min at 4°C (fixation). The membrane was then incubated with BD Perm/Wash Buffer (BD Biosciences) for 15 min at room temperature (permeation), and blocked with BLOCKACE (DS Pharma Biomedical, Osaka, Japan) for 30 min. For immunocytochemistry, the primary antibodies used were anti-VE-cadherin rabbit polyclonal IgG (100-fold dilution, sc-28644, Santa Cruz Biotechnology), anti-β-catenin rabbit monoclonal IgG (100-fold dilution, #8480, Cell Signaling Technology, Danvers, MA, USA), and anti-zonula occludens-1 (ZO-1) rabbit polyclonal IgG (100-fold dilution, #5406S, Cell Signaling Technology). The secondary antibodies used were Rhodamine (TRITC)-AffiniPure F(ab')2 Fragment goat anti-rabbit IgG (100-fold dilution, Jackson Immuno Research Laboratories, West Grove, PA, USA) or Alexa Fluor 488 donkey anti-rabbit IgG (100-fold dilution, Life Technologies). The above antibodies were diluted to the indicated concentrations with CanGetSignal immunostain solution A (TOYOBO, Osaka, Japan). It was confirmed that the secondary antibodies did not bind to cellular proteins in a non-specific manner. For F-actin detection, the membrane was incubated with Acti-stain 488 Fluorescent Phalloidin (150-fold dilution, Cytoskeleton, Denver, CO, USA) for 30 min at room temperature. Fluorescence was detected using the OLYMPUS LSM (Olympus, Tokyo, Japan). The above immunocytochemical analyses were also performed in the presence of GGTI298 (5 μM, Sigma), a RAP inhibitor. The inhibitor or its vehicle (DMSO) was added during every medium change (Days 3, 5, 7, 9 and 11). Scratch assay HBMEC/ciβ cells were cultured on dishes as described above. A scratch on the cell monolayer was created using a tip, immediately after which the medium was changed to wash away the floating cells (time = 0). Cell migration status was observed at time = 0, 6, and 12 hrs. The width of the scratch was calculated using Motic Image Plus 2.2S (Shimadzu, Tokyo, Japan). Lipidomics analysis HBMEC/ciβ cells were cultured on dishes with CSC-free or CSC-HC. The cells were also cultured with CSC-HC containing RU486. After twelve days of culturing (at the confluent status), cells were washed twice with phosphate buffered saline (PBS), and collected. Lipids, corresponding to half of one dish, were extracted from the cells with 200 μL of methanol with internal standards (2 μM of 12:0/12:0 phosphatidylcholine (PC) [Avanti Polar Lipids, Alabaster, AL, USA] for PC, 2 μM of 12:0/12:0 phosphatidylethanolamine (PE) [Avanti Polar Lipids] for PE and 0.5 μM of d18:1/17:0 sphingomyelin (SM) [Avanti Polar Lipids] for SM). Next, filtering was performed for the following non-targeted measurement of PC, ether-type PC (ePC), PE, ether-type PE (ePE), and SM by liquid chromatography-time-of-flight mass spectrometry (LC-TOFMS; ACQUITY UPLC System [Waters, Milford]-LCT Premier XE [Waters, Milford]), as described previously [8]. The relative standard deviation of the internal standards (PC, PE and SM), which monitor experimental quality throughout extraction and measurement, were 5.6%, 7.9% and 9.7%, respectively. Raw data obtained by LC-TOFMS were processed using 2DICAL software (Mitsui Knowledge Industry, Tokyo, Japan), which allows detection and alignment of the ion peaks of each ionized biomolecule obtained at the specific m/z and column retention time (RT). The main 2DICAL parameter was set as described previously with a few modifications [8]. The RT range was set from 2.0 to 38.0 min in the negative ion mode in order to extract the ion peaks. Ion peaks with the top 300 signal intensities were set as the cut-off and were used in the following data analyses. Extracted ion peaks were subjected to identification of lipid molecules by comparison of ion features, including RT, m/z, preferred adducts, and in-source fragments, of the experimental samples with those of our reference library of lipid molecule entries, as described previously [8]. Processing of extracted ion peaks yielded 104 lipid molecules, including 40 PC, 7 ePC, 26 PE, 21 ePE and 10 SM (see Additional file 3: Table S2). To determine the amount of each lipid molecule, the intensities of each extracted ion peak were normalized to those of the internal standards. One-way analysis of variance was first performed to determine whether there was a significant difference among values, after which a Student's t-test was performed to determine statistical significance of difference between values. A statistical software package (Statcell, OMS, Saitama, Japan) was used for these analyses. Culture media composition differentially affected junctional properties of HBMEC/ciβ Several media that have been optimized for endothelial cell culture are currently commercially available. Taking advantage of the successful use of these media in endothelial cell culture, we conducted preliminary screenings to identify the medium most suitable for HBMEC/ciβ culture. Although the results are not shown here, it was determined that, among the media, the EBM basal 2 medium supplemented with SingleQuots (EBM2-SQs) provided differential effects on the morphology and gene expression profile of HBMEC/ciβ when compared to those cultured with our initially-used medium, which was a CSC-complete recombinant serum-containing medium supplemented with cbR (CSC-cbR). (Please note that medium is called "basal medium-supplement" throughout the manuscript.) We then proceeded to examine the effects of different combinations of basal media and culture supplements (CSC-cbR, CSC-SQs, EBM2-cbR and EBM2-SQs) on Na-F permeability to determine whether either EBM-2 and SQs or both would affect HBMEC/ciβ barrier properties (Figure 1). The results showed that neither EBM2-cbR nor EBM2-SQs improved the junctional properties of HBMEC/ciβ over the levels obtained using CSC-cbR. However, CSC-SQs significantly decreased the Na-F permeability of HBMEC/ciβ. Therefore, it appeared that SQs was effective in strengthening the barrier property of HBMEC/ciβ, while the CSC basal medium was also essential. Effects of different basal medium and supplement combinations on Na-F permeability in HBMEC/ciβ-based blood–brain barrier model. Four media types were prepared, where either CSC or EBM2 basal medium (indicated by "Basal") was supplemented by either cbR or SQs (indicated by "Suppl"). Three days after cell seeding, the initial medium was changed to one of the above-mentioned media. The cells were cultured continuously for 12 days, after which permeability assay was performed. Na-F permeability (Papp, × 10−3 cm/min) was expressed as mean ± S.D. of three independent experiments. The asterisk indicates p < 0.05 compared to the value of CSC-cbR. Hydrocortisone was identified as a component of SQs responsible for improving HBMEC/ciβ barrier properties It was expected that improvements in HBMEC/ciβ barrier properties by CSC-SQs culture conditions would be due to either the withdrawal of cbR from or the addition of SQs to the CSC basal medium. To clarify which was responsible, we first compared barrier properties of cells cultured with CSC-cbR to those cultured with CSC basal medium alone (CSC-free). The results showed the Na-F permeability levels were comparable to each other (Figure 2A), thus suggesting that SQs were likely to play the primary improvement role in the HBMEC/ciβ barrier property. Identification and characterization of barrier strengthening effects of hydrocortisone on HBMEC/ciβ-based blood–brain barrier model. (A) The effect of cbR withdrawal from the culture medium on HBMEC/ciβ barrier properties was examined. The "-" symbol indicates the absence of cbR. (B) SQs components were screened for identification of the factors responsible for barrier property improvement. Here, the "-" symbol indicates the absence of growth factors. (C) The effect of cbR on the HC-mediated decrease in Na-F permeability of HBMEC/ciβ was examined. Here, the "-" symbol indicates the absence of HC. (D) The effect of HC on barrier function of HBMEC/ciβ was confirmed by determining the relative Pe values of Na-F (left) and sucrose (right) permeability. The value obtained from the cells cultured with CSC-HC was calculated relative to the value obtained from CSC-cbR cells (set as basal level =1) in each assay. (E) TEER was examined and compared between HBMEC/ciβ cultured with CSC-cbR and CSC-HC. The values of Na-F permeability (Papp or Pe) (×10−3 cm/min) and TEER (Ω × cm2) were expressed as mean ± S.D. of three independent experiments. The single, double, and triple asterisks indicate p < 0.05, p < 0.01, and p < 0.005, respectively, compared to the value of CSC-SQs in (B) or CSC-cbR in (D) and (E). SQs consists of several culture support agents. To identify the primary component(s) of SQs that was responsible for favorable effects on HBMEC/ciβ barrier properties, individual factors were examined separately (Figure 2B). Among those factors, HC alone (180 nM) was found to be effective in strengthening HBMEC/ciβ barrier function. The Na-F Papp value of the cells cultured with CSC-HC was 0.71 ± 0.10 (×10−3 cm/min), which was significantly lower than that of CSC-cbR (1.46 ± 0.15 [×10−3 cm/min]). Next, the cooperative actions of cbR or other SQ factor(s) with HC on barrier tightening were tested. However, no combination showed effects that were superior to that of HC alone (Figure 2C and see Additional file 4: Figure S2). To further verify the barrier-strengthening effect of HC, Na-F and sucrose Pe values, along with TEER, were determined. Results showed that Na-F and sucrose Pe values of cells cultured with CSC-HC were less than 40% of those cultured with CSC-cbR (Figure 2D), and that TEER values were more than 10-fold higher in cells cultured with CSC-HC compared with that of CSC-cbR (Figure 2E). Taken together, the results showed that HC was a primary component in the SQs that could enhance the HBMEC/ciβ barrier function. Glucocorticoid receptor was involved in hydrocortisone-mediated functional improvements in HBMEC/ciβ barrier properties It is well known that HC is a ligand of the nuclear receptor, GR, and that binding of HC to GR stimulates its translocation to the nucleus, where GR modulates gene expression [9]. This is considered to be the primary pathway via which HC elicits pleiotropic effects on cellular function. To clarify the involvement of GR in HC-mediated improvements in HBMEC/ciβ barrier properties, the functional expression of GR was first characterized. Western blotting analysis showed that GR protein expression was clearly detected in HBMEC/ciβ (Figure 3A). In addition, qPCR showed that mRNA levels of known GR-target genes, GILZ [10], NFκBIA [11,12], and ANXA1 [13,14], were significantly increased in HC-containing medium (Figure 3B). Furthermore, increased mRNA expression of these genes was prevented by addition of RU486 (250 nM), a GR-antagonist. Given this clarification of functional GR expression in HBMEC/ciβ, the effects of RU486 on the Na-F permeability and TEER of these cells were examined. RU486 treatment nullified the barrier-tightening effects of HC (Figure 3C). These results indicate that GR plays a crucial role in HC-mediated improvements in HBMEC/ciβ barrier properties. Involvement of glucocorticoid receptor in the effects of hydrocortisone on HBMEC/ciβ barrier function. (A) GR protein expression was examined by Western blot analysis. The arrowhead indicates 100 kDa. (B) mRNA expression level of representative GR-target genes (ANXA1, GILZ, and NFκBIA) in cells cultured with CSC-cbR, CSC-free, or CSC-HC were examined by qPCR. Effect of a GR antagonist, RU486 (RU), on HC-mediated induction of mRNA expression was also investigated. mRNA expression levels were calculated relative to the value obtained from CSC-cbR cells (set as the basal level =1). (C) Effects of RU486 on HC-mediated barrier reinforcement was examined by determining Na-F permeability (left) and TEER (right). Each value in the above experiments is expressed as mean ± S.D. obtained from three independent experiments. In (B) and (C), the "-" symbol indicates DMSO (0.1%) was added to the medium as a control. The single and double asterisks indicate p < 0.05 and p < 0.01, respectively, compared to the value of CSC-free in (B) or CSC-cbR in (C). Adherens junction formation was facilitated by hydrocortisone To further clarify the mechanisms underlying the barrier-strengthening effect of HC, the status of intercellular junction formation in HBMEC/ciβ cultured with each medium was examined by immunocytochemistry for VE-cadherin and β-catenin, which are representative components of AJ [15], and ZO-1, which is an important molecule for both AJ and TJ [16,17] (Figure 4). The results showed that VE-cadherin, β-catenin and ZO-1 were localized at the cell-to-cell border membrane in cells cultured with CSC-HC, while they primarily diffused throughout the cytoplasm in cells cultured with CSC-cbR and CSC-free. The plasma membrane localization in cells cultured with CSC-HC was significantly impaired by RU486 treatment. Immunocytochemical analysis of adherens junction proteins and the effect of hydrocortisone. Immunocytochemistry was performed to identify the cellular localization of VE-cadherin, β-catenin, and ZO-1 in cells cultured with CSC-cbR, CSC-free, or CSC-HC. Phalloidin staining was performed to analyze F-actin distribution. The effect of RU486 on the distribution of the junction-related proteins was also investigated in cells cultured with CSC-HC. The experiments were repeated six times, and representative results are shown: "+" or "-" symbols indicate the addition of RU486 or DMSO (0.1%) to the medium, respectively. In addition, the intracellular organization of F-actin was determined using fluorescently labeled phalloidin, because it has been known that F-actin undergoes structural remodeling to localize to the peri-plasma membrane region, where it can stabilize AJ [18]. The results of phalloidin-staining showed that F-actin appeared concentrated in the vicinity of the plasma membrane in cells cultured with CSC-HC, but not in those cultured with other media. Similar to the junctional proteins, this circumferential organization was severely disrupted by RU486 treatment. Because it was also possible that HC enhanced VE-cadherin expression in HBMEC/ciβ, qPCR and Western blotting analyses were performed. However, no increase in VE-cadherin mRNA or protein was observed with HC (see Additional file 5: Figure S3). Tight junctions play a crucial role in BBB function [1,19], therefore, qPCR and Western blot analyses were performed to examine the key TJ proteins, claudin-5 and occludin, [19,20]. However, no changes in claudin-5 or occludin protein expression levels were observed in the presence of HC, even though occludin mRNA levels were increased (see Additional file 6: Figure S4). Next, immunocytochemical analyses of claudin-5 and occludin were performed. However, those analyses showed that they were not clearly identified at the cell border, thus indicating tight junction immaturity (data not shown). These results suggested that facilitation of AJ formation in HBMEC/ciβ was likely related to barrier function improvements by HC. The EPAC-RAP1 pathway is involved in hydrocortisone-mediated facilitation of adherens junction formation in HBMEC/ciβ The HC-mediated facilitation of AJ formation described above is reminiscent of findings that show the EPAC-RAP1 pathway plays a pivotal role in AJ formation by recruiting AJ proteins to the plasma membrane [21]. Thus, to explore the relationship between these two events, the effect of GGTI298 (a RAP1 inhibitor) on the actions of HC in HBMEC/ciβ was examined. Results showed that in GGTI298-treated cells, VE-cadherin, β-catenin, and ZO-1 were dispersed intracellularly, which clearly differs from their cellular localization profiles in cells with CSC-HC (Figure 5A). Consistently, the lowered Na-F permeability with HC was completely reversed in the presence of GGTI298 (results not shown). Therefore, inhibition of RAP1 activity appears to abrogate the actions of HC. Impairment of hydrocortisone-mediated facilitation of adherens junction formation by inhibiting RAP1 activity. (A) Immunocytochemistry was performed to examine the cellular localization of VE-cadherin, β-catenin, and ZO-1 in cells cultured with CSC-HC in the presence or absence of a RAP1 inhibitor, GGTI298 (5 μM). Phalloidin staining was performed to analyze F-actin distribution. The experiments were repeated three times and representative results are shown. The "+" or "-" symbols indicate the addition of GGTI298 or DMSO (0.1%) to the medium, respectively. (B) EPAC and RAP1 mRNA expression levels in cells cultured with CSC-free or CSC-HC were examined by qPCR. Effect of RU486 on HC-mediated induction of mRNA expression was also investigated. mRNA expression levels were calculated relative to the value obtained from CSC-free cells (set as the basal level =1). Each value is expressed as mean ± S.D. obtained from three independent experiments. The single and double asterisks indicate represent p < 0.05 and p < 0.01 respectively, relative to CSC-free. An interplay between the EPAC-RAP1 and HC signaling pathways was also identified by real-time PCR analyses (Figure 5B). EPAC and RAP1 mRNA levels were, respectively, 3.7-fold and 2.5-fold higher in HBMEC/ciβ cultured with CSC-HC than in CSC-free cultures. In addition, enhanced EPAC and RAP1 mRNA levels were not observed in CSC-HC cells when RU486 was present. These results suggest that the EPAC-RAP1 pathway plays a critical role in HC-mediated facilitation of AJ formation in HBMEC/ciβ. The differentiation status of HBMEC/ciβ was enhanced by hydrocortisone Proper AJ formation is generally associated with endothelial cell differentiation [15,22]. When cell morphology was examined during the above experiments, it was found that cells cultured with CSC-HC showed more spindle-like shapes with overall streamline contours, compared with cells cultured with other media (Figure 6A). However, this was not observed in the presence of RU486. Subsequently, the mRNA expression levels of endothelial differentiation marker genes vWF and DARC were analyzed (Figure 6B). The mRNA levels for both genes were significantly higher in CSC-HC cells than in those with other media. Thus, it appears that the endothelial cell differentiation of HBMEC/ciβ was promoted by HC through a GR pathway. Promotion of endothelial cell differentiation of HBMEC/ciβ by hydrocortisone. (A) Cell morphology of HBMEC/ciβ cultured with CSC-cbR, CSC-free, or CSC-HC was analyzed by microscopy. The effect of RU486 on cell morphology was also tested using cells cultured with CSC-HC. The experiments were repeated four times and representative results are shown. (B) mRNA expression levels of representative endothelial cell differentiation marker genes (vWF and DARC) were determined by qPCR and calculated relative to the value obtained from CSC-cbR cells (set as basal level = 1). The "+" or "-" symbols indicate addition of RU486 or DMSO (0.1%) to the medium, respectively. Each value is expressed as mean ± S.D. obtained from three independent experiments. The single and double asterisks indicate p < 0.05 and p < 0.01, respectively, compared to CSC-free. HBMEC/ciβ migration ability was retarded by hydrocortisone It is known that endothelial cells reduce their growth activity and motility upon establishment of intercellular junctions [15]. Therefore, the migration ability of HBMEC/ciβ was examined by the scratch assay (Figure 7). The results showed that, as expected, the migration ability of the cells was retarded in the absence of cbR. Furthermore, it was found that addition of HC to the medium resulted in a further significant reduction in cell migration. The effects of HC were completely negated by RU486 treatment. Therefore, these results showed that cell migration was inhibited by HC via the GR function. Retardation of HBMEC/ciβ migration by hydrocortisone. Migration abilities of HBMEC/ciβ cultured with CSC-cbR, CSC-free, or CSC-HC were examined by scratch assay. The effect of RU486 on cell morphology was also tested using cells cultured with CSC-HC. Wound width was determined under microscopic observation at 0, 6 and 12 hours after the scratch had been made. The experiments were repeated three times and the representative results are shown. In the right side, the relative value of wound width was calculated as the value obtained at time 0 (immediately after scratch) was set to 1, and each value is expressed as mean ± S.D. obtained from three independent experiments. The asterisk indicates p < 0.05 compared to the value of CSC-free. A key contributor to cell motility is the family of MMPs that is capable of degrading the extracellular matrix in order to facilitate cell migration [23,24]. Thus, we hypothesized that the differential cell migration shown in Figure 7 would be associated with altered MMP expression. Since our preliminary microarray analysis had shown that, among the family members, MMP-1, MMP-2, and MMP-16 expressions were detected in HBMEC/ciβ (data not shown), the mRNA expression levels of these MMPs were examined in cells cultured with each medium (Figure 8). The results showed that cbR withdrawal from the medium caused more than 350-fold and 2-fold decrease in MMP-1 and MMP-16 mRNA levels, respectively, while it also led to more than 16-fold increase in MMP-2 mRNA levels. This might, at least partially, explain less significant reduction of migration ability observed in cells cultured with CSC-free conditions compared with those cultured in CSC-cbR. In addition, the results showed that HC treatment significantly reversed MMP-2 mRNA levels and further reduced MMP-16 mRNA levels (by 3.2-fold and 2.8-fold, respectively), while maintaining the repressive status of MMP-1 expression (Figure 8). Again, the effects of HC were prevented by RU486 treatment. These results were apparently consistent with the minimum migration ability of HBMEC/ciβ cultured with CSC-HC among the conditions, suggesting that altered MMP expression levels were involved in the effects of HC on HBMEC/ciβ mortality. Matrix metalloproteinase mRNA expression profile in HBMEC/ciβ cultured with each medium. MMP-1, -2, and -16 mRNA expression levels in cells cultured with CSC-cbR, CSC-free, or CSC-HC were examined by qPCR. The effect of a GR antagonist, RU486, on HC-mediated modulation of mRNA expression was also investigated. The inset of the figure of MMP-1 shows magnified results obtained from cells cultured with CSC-free and CSC-HC (in the absence or presence of RU486) (left, middle, and right bar, respectively). mRNA expression levels were calculated relative to the value obtained from cells of CSC-cbR (set as basal level =1). Each value in the above experiments is expressed as mean ± S.D. obtained from three independent experiments. The "-" symbol indicates that DMSO (0.1%) was added to the medium as a control. The asterisk indicates p < 0.05 compared to the value of CSC-free. Ether-type phosphatidylethanolamine levels were affected by hydrocortisone Since the morphology and mobility of HBMEC/ciβ differed significantly depending on the presence and absence of HC, it was speculated that HC may actively influence the membrane lipid composition necessary to adapt to global cellular phenotypic alterations. Thus, a lipidomics approach, primarily focusing on major membrane lipid molecules (glycerophospholipids and sphingomyelin), was utilized in order to determine whether lipid composition differed among cells cultured with CSC-free, CSC-HC, or CSC-HC with RU486. While PC, ePC, PE, and SM levels were not significantly different among the conditions, ePE level in cells with CSC-HC was increased to a slight, but significant, degree compared to CSC-free (Figure 9A). Furthermore, this increase was mostly reversed by RU486 treatment. Identification of unique ether-phosphatidylethanolamine increases in HBMEC/ciβ cultured with CSC-HC. (A) Total lipids were extracted from cells cultured with CSC-free, CSC-HC or CSC-HC + RU486, and then subjected to LC-TOFMS analysis with a primary focus on the major lipid molecules that make up cell membranes. Each lipid concentration was normalized with that of the internal standard, and each bar represents a mean ± S.D. obtained from three independent experiments. (B) ePE concentrations were further determined using the same method. Among the twenty-one ePE molecules showing effective signal intensity, the results of five molecules (38:5ePE, 36:5ePE, 38:7ePE, 37:5ePE, and 36:3ePE) are shown. The inset shows magnified results of three minor species. Each lipid concentration was normalized with that of internal standard, and each bar represents a mean ± S.D. obtained from three independent experiments. (C) FAR1, GNPAT and AGPS mRNA expression levels in cells cultured with CSC-free or CSC-HC were examined by qPCR. The effect of RU486 on the HC-mediated induction of mRNA expression was also investigated. mRNA expression levels were calculated relative to the values obtained from CSC-free cells (set as the basal level =1). Values are expressed as means ± S.D. obtained from three independent experiments. In all the above, single and double asterisks indicate p < 0.05 and p < 0.01, compared to the CSC-free value. Based on those results, detailed ePE molecule determination was conducted in order to identify the ePE species primarily affected by HC. The results showed that, among twenty-one molecules showing effective signal intensities, five molecules (38:5ePE, 36:5ePE, 38:7ePE, 37:5ePE, and 36:3ePE) were significantly increased in cells cultured with CSC-HC compared to CSC-free (Figure 9B). Furthermore, 38:5ePE and 36:5ePE were found to be the primary ePE species among the five (Figure 9B). It has previously been known that these lipid molecules have the 1-O-(1Z-alkenyl)-2-acyl-sn-glycerophosphatidylethanolamine structure. When the two acyl moieties of these molecules were searched for by referring to the reference data, 38:5ePE and 36:5ePE were found to be primarily derived from 18:0p/20:4 PE and 16:0p/20:4 PE, respectively. Next, the effects of HC on mRNA levels of three key ePE biosynthesis enzymes were examined. While FAR1 and AGPS mRNAs were not affected by HC, GNPAT mRNA levels in cells cultured with CSC-HC were significantly higher (over 3-fold) than cells cultured with CSC-free (Figure 9C). This enhancement was not detected in cells cultured with CSC-HC + RU486. Taken together, these results showed that HC significantly modulated cellular ePE levels, specifically 38:5ePE and 36:5ePE, partially via enhancing GNPAT mRNA levels. Our results clearly show that culture media composition indeed plays an important role in determining HBMEC/ciβ functionality. Among the factors examined in this study, the supplementation of HC to a medium at a physiological concentration provides the highest beneficial impact on the barrier functionality of the HBMEC/ciβ-based in vitro BBB model. In addition to HC, our results showed that both the CSC basal medium and cbR likely affect HBMEC/ciβ biology. However, since details of cbR and CSC basal media composition have not been published and since neither was capable of inducing a notable functional improvement to the barrier property of HBMEC/ciβ, the remainder of this discussion will focus solely on the effects of HC. HC-mediated enhancement of the HBMEC/ciβ barrier function is consistent with previous results obtained from porcine, mouse, rat, and human BMECs [25-28]. The Na-F Papp of cells cultured with CSC-HC (0.71 [×10−3 cm/min]) is comparable to that reported in rat or bovine primary cell-based BBB models (0.92 and 0.66 [×10−3 cm/min]) [29,30]. The Na-F Papp is also close to those obtained from other human-derived immortalized cell lines (hBMEC, hCMEC/D3, and TY08), which are 0.30 to 0.75 (×10−3 cm/min) [31]. Therefore, even though the different experimental conditions employed among those studies should be considered when interpreting their data, it is believed that improvement of the HBMEC/ciβ barrier function by HC significantly enhances their usability in various BBB studies. It has been acknowledged that AJs play an important role in the control of vascular permeability [15,22], and it is thus considered likely that the barrier tightening effect of HC on HBMEC/ciβ is primarily mediated by facilitating AJ formation. On the other hand, our results also indicate immature TJ formation in the same culture condition. Because both AJs and TJs are essential for ensuring mature barrier integrity between cells through their reciprocal interaction, it seems that a critical point required for the achievement of further strengthening barrier tightness of HBMEC/ciβ lies in facilitating TJ formation. Since our present results are in accord with the notion that AJ formation precedes TJ assembly [22,32], the CSC-HC culture is presumed to provide HBMEC/ciβ with a cellular foundation for TJ formation. This indicates that additional stimuli that can be used to build up firm TJs in the cells should be sought in future experiments. These, if identified, can be expected to not only further enhance HBMEC/ciβ functionality, but also provide key clues towards understanding the molecular process of TJ formation. It has been shown that several soluble factors, such as retinoic acid, cAMP, and hepatocyte growth factor, along with co-culture with astrocytes and/or pericytes are capable of enhancing junctional property of in vitro BBB models in an independent or collaborative manner [28,33-35]. Therefore, it would be worthwhile to test these culture modifications in order to identify ways whereby TJ formation could be promoted. In such experiments, it may be important to consider crosstalk among media components in the optimization of HBMEC/ciβ culture method, because our results show that cbR apparently counteracts HC's effects, even though the mechanism behind this observation is currently unclear. Enhanced barrier tightness, AJ formation, spindle-like cell morphology, and abundant endothelial marker gene expression, are all signs of endothelial differentiation. In particular, AJ formation is critically important in not only stabilizing intercellular adhesion but also in promoting endothelial differentiation [15,22]. On the other hand, our results also show that cell migration is significantly disturbed by HC when it is associated with a reduction of MMP-2 and MMP-16 mRNA levels. In line with the reversed characteristics observed in HBMEC/ciβ cultured without HC (CSC-cbR or CSC-free), we suggest taking a comprehensive view of the phenotype transition; specifically, the mesenchymal-to-endothelial transition (MEndT), into consideration in order to gain a global picture of HC effects on HBMEC/ciβ. Recently, it has become evident that the plasticity of endothelial cells allows them to change their phenotype from endothelial to mesenchymal, and vice versa, which are regarded as the endothelial-to-mesenchymal transition (EndMT) and the MEndT, respectively [36]. EndMT molecular events are apparently similar to those of epithelial-to-mesenchymal transition (EMT) in epithelial cells, which includes abnormal intercellular junction formation, extensive cell migration, and remarkable expression of matrix remodeling genes [37]. Similarly, the reverse processes, the MEndT and the mesenchymal-to-epithelial transition (MET), appear to share numerous hallmarks, such as cell-to-cell junction stabilization, actin filament reorganization to the peri-plasma membrane region, cell migration ability retardation, and an increase in marker gene expression [37]. From this viewpoint, the overall effects of HC on HBMEC/ciβ appear to be closely related to MEndT. This is also consistent with previous results showing that glucocorticoid induces the MET or prevents the EMT in several cell types [38-40]. However, alterations in mRNA expression of some representative mesenchymal genes (such as α-smooth muscle actin) were not associated with phenotype transition in our cells (data not shown). Taken together, it appears that the effects of HC on HBMEC/ciβ can be regarded as "MEndT-like". When this view is applied to the interpretations of previous literature, it appears that MEndT-like glucocorticoid effects have also been found in rat and mouse BMEC lines [41-43]. Because the MEndT/MET is a drastic phenotypic transition, it is not surprising that re-organization of various cellular processes are involved during the MEndT-like transition of HBMEC/ciβ. The present study, for the first time, verified that increased ePE levels are one of those processes. Although there have been no reports associating altered ePE levels with the MEndT/EndMT process, it has been reported that primary bovine aortic endothelial cells at the confluent state contain higher ePE levels than at the sub-confluent state [44], and that increased ePE levels were observed during the MET in MDCK cells [45]. Thus, our results may share similar findings with those reports. The ePE species identified here, 18:0p/20:4 PE and 16:0p/20:4 PE are classified as plasmalogens, and recent reports have shown that plasmalogens have diverse biological activities [46]. Among them, effects reducing membrane fluidity might be, at least partially, involved in lowering Na-F permeability and/or retarding mobility in HBMEC/ciβ cultured in CSC-HC. Nevertheless, it remains mostly unclear whether increased plasmalogen levels are adapted responses to (and/or necessary for) the HC-mediated MEndT-like process of HBMEC/ciβ. Therefore, progress in this new area of study has the potential to allow us to advance towards a greater understanding of a previously unacknowledged aspect of BBB biology and/or its barrier property. Finally, the mechanisms by which an HC-GR-pathway facilitates AJ formation should be discussed. Based on our results, it appears that the EPAC-RAP1 pathway is a pivotal player involved in HC-mediated AJ protein translocation to the action site. Although it has been shown that the pathway is critically involved in AJ formation [21], our results are (to our knowledge) the first showing that the EPAC-RAP1 pathway can be located downstream of the HC-GR signaling pathway. Considering that GR is a transcription factor, it is reasonable to assume that GR activation enhances EPAC and RAP1 mRNA levels in order to expand their potential activity upon stimulation, but it is difficult to believe that the GR function directly activates the EPAC-RAP1 activity. Rather, activated GR is likely to enhance or repress target gene expression, which in turn elicits various signals necessary to increase intracellular cAMP levels leading to EPAC-RAP1 pathway activation. Although it remains unknown what signaling pathways are activated by HC in order to stimulate an increase in cellular cAMP levels, we have observed that the expression levels of various transcription and autocrine/paracrine factors were targets of the HC-GR axis, as can be seen in the representative results (see Additional file 7: Figure S5). Therefore, it can be speculated that at least one of the targets might be involved in the intermediate signaling pathway(s) that bridge the gap between the HC-GR and EPAC-RAP1 pathways. Identifying the target(s) involved and the related signaling pathway(s) will be an interesting and important challenge in future studies, not only of HBMEC/ciβ but also of other BBB cells/cell lines. Our results show that HC clearly improves HBMEC/ciβ barrier functionality through GR activation, thus leading to our recommendation that HC be regarded as an essential media component for HBMEC/ciβ-based in vitro BBB models. It is also considered likely that these HC effects result from the orchestration of diverse cellular signaling networks, including those involved in modulating plasmalogen levels and accelerating AJ formation, which appears to be adapted to (and/or lead to) MEndT-like phenotype transitions. Nevertheless, since HBMEC/ciβ differentiation status, including TJ formation, is likely to be further promoted by optimization of culture conditions, we hope that such research efforts will result in the development of unique and useful HBMEC/ciβ-based in vitro BBB models, while simultaneously providing new opportunities to explore molecular events related to MEndT/EndMT in BMEC. 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Laboratory of Pharmacology and Toxicology, Graduate School of Pharmaceutical Sciences, Chiba University, 1-8-1 Inohana, Chuo-ku, Chiba-shi, Chiba, 260-8675, Japan Tomomi Furihata , Shinya Kawamatsu , Ryo Ito , Shota Suzuki , Satoshi Kishida , Atsuko Kamiichi & Kan Chiba Division of Medical Safety Science, National Institute of Health Sciences, 1-18-1 Kamiyoga, Setagaya, Tokyo, Japan Kosuke Saito & Yoshiro Saito Search for Tomomi Furihata in: Search for Shinya Kawamatsu in: Search for Ryo Ito in: Search for Kosuke Saito in: Search for Shota Suzuki in: Search for Satoshi Kishida in: Search for Yoshiro Saito in: Search for Atsuko Kamiichi in: Search for Kan Chiba in: Correspondence to Tomomi Furihata. TF and KC participated in the design of the study. SK, RI, SS, SK, KS and AK performed the experiments. TF, SK, YS and KC analyzed the results and wrote the manuscript. All authors read and approved the final manuscript. Illustration of experimental procedure and medium information. Culture schedules are shown at the top. At Day 3, CSC-cbR was changed to fresh CSC-cbR, CSC-HC, or CSC-free medium, as indicated at the center of the illustration. All functional analyses, including immunocytochemistry, gene expression analyses, and Na-F permeability assays, were performed at Day 12. Medium composition is shown at the bottom. Primers used for qPCR in this study. Summary of identification of ether-phosphatidylethanolamine molecules (PC, ePC, PE, ePE and SM) in HBMEC/ciβ cells. Exploration of cooperative effects of other SQs component(s) with hydrocortisone on barrier property of HBMEC/ciβ. CSC-HC supplemented with ascorbate and heparin, which are components of SQs, was used as a basic medium. Different SQs growth factor combinations were tested to determine if they had the potential to further enhance HBMEC/ciβ barrier properties. The "-" and "+" symbols indicate absence and presence of respective growth factors. Three days after cell seeding, the medium was changed to one of the two above-mentioned media. The cells were continuously cultured for 12 days, after which Na-F permeability assay was performed. The Papp value obtained from the cells cultured with CSC-HC in the absence of any growth factors was set to the basal level (=1) in each assay. Each bar represents the mean ± S.D. of the relative Na-F permeability values, which were obtained from three independent experiments. VE-cadherin expression profile in HBMEC/ciβ. A, VE-cadherin mRNA expression in HBMEC/ciβ cultured with CSC-cbR or CSC-HC were determined by real-time PCR. Each value represents mean ± S.D. of three independent assays, each performed in duplicate. The mean value obtained from HBMEC/ciβ cultured with CSC-cbR was set to 1. B, VE-cadherin protein expressions in HBMEC/ciβ cultured with CSC-cbR or CSC-HC was determined by Western blot. β-actin protein expression was used as a loading control. The representative results of three independent assays are shown. Claudin-5 and occludin expression profiles in HBMEC/ciβ. A, claudin-5 and occludin mRNA expressions in HBMEC/ciβ cultured with CSC-cbR or CSC-HC were determined by real-time PCR. Each value represents mean ± S.D. of three independent assays, each performed in duplicate. The mean value obtained from HBMEC/ciβ cultured with CSC-cbR was set as 1. B, claudin-5 and occludin protein expression in HBMEC/ciβ cultured with CSC-cbR or CSC-HC was determined by Western blot. β-actin protein expression was used as a loading control. The representative results of three independent assays are shown. ANGPT2, Id-1, Egr-1 and HDAC7 mRNA expression in HBMEC/ciβ cultured with CSC-cbR, CSC-free, or CSC-HC. ANGPT2, Id-1, Egr-1 and HDAC7 mRNA expression levels in cells cultured with CSC-cbR, CSC-free, or CSC-HC, were examined by qPCR. The effect of a GR antagonist, RU486, on HC-mediated modulation of mRNA expression was also investigated. mRNA expression levels were calculated relative to the value obtained from CSC-cbR cells (set as basal level =1). Each value in the above experiments is expressed as mean ± S.D. obtained from three independent experiments. The "-" symbol indicates that DMSO (0.1%) was added to the medium as a control. The single and double asterisks indicate p < 0.05 and p < 0.01, respectively, compared with the value of CSC-free. ANGPT2 is a secreted glycoprotein member of the angiopoietin family of growth factors. It has been shown to be involved in functional impairment of the BBB [47]. Id-1 is a basic helix-loop-helix transcription factor family member, while lacking a basic DNA binding domain. It has been shown that Id-1 plays a facilitative role in EMT [48]. EGR-1 is a C2H2-class zinc finger transcription factor and its knockdown has been shown to be associated with retardation of cell migration ability [49]. HDAC7 is a member of the HDACs that contribute to gene transcription control via modification of acetylation level of histones, along with other proteins. It has been reported that HDAC7 mRNA knockdown causes reduction of endothelial migration [50]. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Furihata, T., Kawamatsu, S., Ito, R. et al. Hydrocortisone enhances the barrier properties of HBMEC/ciβ, a brain microvascular endothelial cell line, through mesenchymal-to-endothelial transition-like effects. Fluids Barriers CNS 12, 7 (2015). https://doi.org/10.1186/s12987-015-0003-0 In vitro BBB model Mesenchymal-to-endothelial transition Plasmalogen	CommonCrawl
Pt. 36, App. A A36.1 Introduction. A36.2 Noise Certification Test and Measurement Conditions. A36.3 Measurement of Airplane Noise Received on the Ground. A36.4 Calculations of Effective Perceived Noise Level From Measured Data. A36.5 Reporting of Data to the FAA. A36.6 Nomenclature: Symbols and Units. A36.7 Sound Attenuation in Air. A36.8 [Reserved] A36.9 Adjustment of Airplane Flight Test Results. Section A36.1 Introduction A36.1.1 This appendix prescribes the conditions under which airplane noise certification tests must be conducted and states the measurement procedures that must be used to measure airplane noise. The procedures that must be used to determine the noise evaluation quantity designated as effective perceived noise level, EPNL, under §§ 36.101 and 36.803 are also stated. A36.1.2 The instructions and procedures given are intended to ensure uniformity during compliance tests and to permit comparison between tests of various types of airplanes conducted in various geographical locations. A36.1.3 A complete list of symbols and units, the mathematical formulation of perceived noisiness, a procedure for determining atmospheric attenuation of sound, and detailed procedures for correcting noise levels from non-reference to reference conditions are included in this appendix. A36.1.4 For Stage 4 airplanes, an acceptable alternative for noise measurement and evaluation is Appendix 2 to ICAO Annex 16, Volume I, Amendment 7 (incorporated by reference, see § 36.6). A36.1.5 For Stage 5 airplanes, an acceptable alternative for noise measurement and evaluation is Appendix 2 to ICAO Annex 16, Volume 1, Amendment 11-B (incorporated by reference, see § 36.6). Section A36.2 Noise Certification Test and Measurement Conditions A36.2.1 General. A36.2.1.1 This section prescribes the conditions under which noise certification must be conducted and the measurement procedures that must be used. Many noise certifications involve only minor changes to the airplane type design. The resulting changes in noise can often be established reliably without resorting to a complete test as outlined in this appendix. For this reason, the FAA permits the use of approved equivalent procedures. There are also equivalent procedures that may be used in full certification tests, in the interest of reducing costs and providing reliable results. Guidance material on the use of equivalent procedures in the noise certification of subsonic jet and propeller-driven large airplanes is provided in the current advisory circular for this part. A36.2.2 Test environment. A36.2.2.1 Locations for measuring noise from an airplane in flight must be surrounded by relatively flat terrain having no excessive sound absorption characteristics such as might be caused by thick, matted, or tall grass, shrubs, or wooded areas. No obstructions that significantly influence the sound field from the airplane must exist within a conical space above the point on the ground vertically below the microphone, the cone being defined by an axis normal to the ground and by a half-angle 80° from this axis. Those people carrying out the measurements could themselves constitute such obstruction. A36.2.2.2 The tests must be carried out under the following atmospheric conditions. (a) No precipitation; (b) Ambient air temperature not above 95 °F (35 °C) and not below 14 °F (−10 °C), and relative humidity not above 95% and not below 20% over the whole noise path between a point 33 ft (10 m) above the ground and the airplane; Care should be taken to ensure that the noise measuring, airplane flight path tracking, and meteorological instrumentation are also operated within their specific environmental limitations. (c) Relative humidity and ambient temperature over the whole noise path between a point 33 ft (10 m) above the ground and the airplane such that the sound attenuation in the one-third octave band centered on 8 kHz will not be more than 12 dB/100 m unless: (1) The dew point and dry bulb temperatures are measured with a device which is accurate to ±0.9 °F (±0.5 °C) and used to obtain relative humidity; in addition layered sections of the atmosphere are used as described in section A36.2.2.3 to compute equivalent weighted sound attenuations in each one-third octave band; or (2) The peak noy values at the time of PNLT, after adjustment to reference conditions, occur at frequencies less than or equal to 400 Hz.; (d) If the atmospheric absorption coefficients vary over the PNLTM sound propagation path by more than ±1.6 dB/1000 ft (±0.5 dB/100m) in the 3150Hz one-third octave band from the value of the absorption coefficient derived from the meteorological measurement obtained at 33 ft (10 m) above the surface, "layered" sections of the atmosphere must be used as described in section A36.2.2.3 to compute equivalent weighted sound attenuations in each one-third octave band; the FAA will determine whether a sufficient number of layered sections have been used. For each measurement, where multiple layering is not required, equivalent sound attenuations in each one-third octave band must be determined by averaging the atmospheric absorption coefficients for each such band at 33 ft (10 m) above ground level, and at the flight level of the airplane at the time of PNLTM, for each measurement; (e) Average wind velocity 33 ft (10 m) above ground may not exceed 12 knots and the crosswind velocity for the airplane may not exceed 7 knots. The average wind velocity must be determined using a 30-second averaging period spanning the 10 dB-down time interval. Maximum wind velocity 33 ft (10 m) above ground is not to exceed 15 knots and the crosswind velocity is not to exceed 10 knots during the 10 dB-down time interval; (f) No anomalous meteorological or wind conditions that would significantly affect the measured noise levels when the noise is recorded at the measuring points specified by the FAA; and (g) Meteorological measurements must be obtained within 30 minutes of each noise test measurement; meteorological data must be interpolated to actual times of each noise measurement. A36.2.2.3 When a multiple layering calculation is required by section A36.2.2.2(c) or A36.2.2.2(d) the atmosphere between the airplane and 33 ft (10 m) above the ground must be divided into layers of equal depth. The depth of the layers must be set to not more than the depth of the narrowest layer across which the variation in the atmospheric absorption coefficient of the 3150 Hz one-third octave band is not greater than ±1.6 dB/1000 ft (±0.5 dB/100m), with a minimum layer depth of 100 ft (30 m). This requirement must be met for the propagation path at PNLTM. The mean of the values of the atmospheric absorption coefficients at the top and bottom of each layer may be used to characterize the absorption properties of each layer. A36.2.2.4 The airport control tower or another facility must be aproved by the FAA for use as the central location at which measurements of atmospheric parameters are representative of those conditions existing over the geographical area in which noise measurements are made. A36.2.3 Flight path measurement. A36.2.3.1 The airplane height and lateral position relative to the flight track must be determined by a method independent of normal flight instrumentation such as radar tracking, theodolite triangulation, or photographic scaling techniques, to be approved by the FAA. A36.2.3.2 The airplane position along the flight path must be related to the noise recorded at the noise measurement locations by means of synchronizing signals over a distance sufficient to assure adequate data during the period that the noise is within 10 dB of the maximum value of PNLT. A36.2.3.3 Position and performance data required to make the adjustments referred to in section A36.9 of this appendix must be automatically recorded at an approved sampling rate. Measuring equipment must be approved by the FAA. Section A36.3 Measurement of Airplane Noise Received on the Ground A36.3.1 Definitions. For the purposes of section A36.3 the following definitions apply: A36.3.1.1 Measurement system means the combination of instruments used for the measurement of sound pressure levels, including a sound calibrator, windscreen, microphone system, signal recording and conditioning devices, and one-third octave band analysis system. Practical installations may include a number of microphone systems, the outputs from which are recorded simultaneously by a multi-channel recording/analysis device via signal conditioners, as appropriate. For the purpose of this section, each complete measurement channel is considered to be a measurement system to which the requirements apply accordingly. A36.3.1.2 Microphone system means the components of the measurement system which produce an electrical output signal in response to a sound pressure input signal, and which generally include a microphone, a preamplifier, extension cables, and other devices as necessary. A36.3.1.3 Sound incidence angle means in degrees, an angle between the principal axis of the microphone, as defined in IEC 61094-3 and IEC 61094-4, as amended and a line from the sound source to the center of the diaphragm of the microphone (incorporated by reference, see § 36.6). When the sound incidence angle is 0°, the sound is said to be received at the microphone at "normal (perpendicular) incidence;" when the sound incidence angle is 90°, the sound is said to be received at "grazing incidence." A36.3.1.4 Reference direction means, in degrees, the direction of sound incidence specified by the manufacturer of the microphone, relative to a sound incidence angle of 0°, for which the free-field sensitivity level of the microphone system is within specified tolerance limits. A36.3.1.5 Free-field sensitivity of a microphone system means, in volts per Pascal, for a sinusoidal plane progressive sound wave of specified frequency, at a specified sound incidence angle, the quotient of the root mean square voltage at the output of a microphone system and the root mean square sound pressure that would exist at the position of the microphone in its absence. A36.3.1.6 Free-field sensitivity level of a microphone system means, in decibels, twenty times the logarithm to the base ten of the ratio of the free-field sensitivity of a microphone system and the reference sensitivity of one volt per Pascal. The free-field sensitivity level of a microphone system may be determined by subtracting the sound pressure level (in decibels re 20 µPa) of the sound incident on the microphone from the voltage level (in decibels re 1 V) at the output of the microphone system, and adding 93.98 dB to the result. A36.3.1.7 Time-average band sound pressure level means in decibels, ten times the logarithm to the base ten, of the ratio of the time mean square of the instantaneous sound pressure during a stated time interval and in a specified one-third octave band, to the square of the reference sound pressure of 20 µPa. A36.3.1.8 Level range means, in decibels, an operating range determined by the setting of the controls that are provided in a measurement system for the recording and one-third octave band analysis of a sound pressure signal. The upper boundary associated with any particular level range must be rounded to the nearest decibel. A36.3.1.9 Calibration sound pressure level means, in decibels, the sound pressure level produced, under reference environmental conditions, in the cavity of the coupler of the sound calibrator that is used to determine the overall acoustical sensitivity of a measurement system. A36.3.1.10 Reference level range means, in decibels, the level range for determining the acoustical sensitivity of the measurement system and containing the calibration sound pressure level. A36.3.1.11 Calibration check frequency means, in hertz, the nominal frequency of the sinusoidal sound pressure signal produced by the sound calibrator. A36.3.1.12 Level difference means, in decibels, for any nominal one-third octave midband frequency, the output signal level measured on any level range minus the level of the corresponding electrical input signal. A36.3.1.13 Reference level difference means, in decibels, for a stated frequency, the level difference measured on a level range for an electrical input signal corresponding to the calibration sound pressure level, adjusted as appropriate, for the level range. A36.3.1.14 Level non-linearity means, in decibels, the level difference measured on any level range, at a stated one-third octave nominal midband frequency, minus the corresponding reference level difference, all input and output signals being relative to the same reference quantity. A36.3.1.15 Linear operating range means, in decibels, for a stated level range and frequency, the range of levels of steady sinusoidal electrical signals applied to the input of the entire measurement system, exclusive of the microphone but including the microphone preamplifier and any other signal-conditioning elements that are considered to be part of the microphone system, extending from a lower to an upper boundary, over which the level non-linearity is within specified tolerance limits. Microphone extension cables as configured in the field need not be included for the linear operating range determination. A36.3.1.16 Windscreen insertion loss means, in decibels, at a stated nominal one-third octave midband frequency, and for a stated sound incidence angle on the inserted microphone, the indicated sound pressure level without the windscreen installed around the microphone minus the sound pressure level with the windscreen installed. A36.3.2 Reference environmental conditions. A36.3.2.1 The reference environmental conditions for specifying the performance of a measurement system are: (a) Air temperature 73.4 °F (23 °C); (b) Static air pressure 101.325 kPa; and (c) Relative humidity 50%. A36.3.3. General. Measurements of aircraft noise that are made using instruments that conform to the specifications of this section will yield one-third octave band sound pressure levels as a function of time. These one-third octave band levels are to be used for the calculation of effective perceived noise level as described in section A36.4. A36.3.3.1 The measurement system must consist of equipment approved by the FAA and equivalent to the following: (a) A windscreen (See A36.3.4.); (b) A microphone system (See A36.3.5): (c) A recording and reproducing system to store the measured aircraft noise signals for subsequent analysis (see A36.3.6); (d) A one-third octave band analysis system (see A36.3.7); and (e) Calibration systems to maintain the acoustical sensitivity of the above systems within specified tolerance limits (see A36.3.8). A36.3.3.2. For any component of the measurement system that converts an analog signal to digital form, such conversion must be performed so that the levels of any possible aliases or artifacts of the digitization process will be less than the upper boundary of the linear operating range by at least 50 dB at any frequency less than 12.5 kHz. The sampling rate must be at least 28 kHz. An anti-aliasing filter must be included before the digitization process. A36.3.4 Windscreen. A36.3.4.1 In the absence of wind and for sinusoidal sounds at grazing incidence, the insertion loss caused by the windscreen of a stated type installed around the microphone must not exceed ±1.5 dB at nominal one-third octave midband frequencies from 50 Hz to 10 kHz inclusive. A36.3.5 Microphone system. A36.3.5.1 The microphone system must meet the specifications in sections A36.3.5.2 to A36.3.5.4. Various microphone systems may be approved by the FAA on the basis of demonstrated equivalent overall electroacoustical performance. Where two or more microphone systems of the same type are used, demonstration that at least one system conforms to the specifications in full is sufficient to demonstrate conformance. An applicant must still calibrate and check each system as required in section A36.3.9. A36.3.5.2 The microphone must be mounted with the sensing element 4 ft (1.2 m) above the local ground surface and must be oriented for grazing incidence, i.e., with the sensing element substantially in the plane defined by the predicted reference flight path of the aircraft and the measuring station. The microphone mounting arrangement must minimize the interference of the supports with the sound to be measured. Figure A36-1 illustrates sound incidence angles on a microphone. A36.3.5.3 The free-field sensitivity level of the microphone and preamplifier in the reference direction, at frequencies over at least the range of one-third-octave nominal midband frequencies from 50 Hz to 5 kHz inclusive, must be within ±1.0 dB of that at the calibration check frequency, and within ±2.0 dB for nominal midband frequencies of 6.3 kHz, 8 kHz and 10 kHz. A36.3.5.4 For sinusoidal sound waves at each one-third octave nominal midband frequency over the range from 50 Hz to 10 kHz inclusive, the free-field sensitivity levels of the microphone system at sound incidence angles of 30°, 60°, 90°, 120° and 150°, must not differ from the free-field sensitivity level at a sound incidence angle of 0° ("normal incidence") by more than the values shown in Table A36-1. The free-field sensitivity level differences at sound incidence angles between any two adjacent sound incidence angles in Table A36-1 must not exceed the tolerance limit for the greater angle. A36.3.6 Recording and reproducing systems. A36.3.6.1 A recording and reproducing system, such as a digital or analog magnetic tape recorder, a computer-based system or other permanent data storage device, must be used to store sound pressure signals for subsequent analysis. The sound produced by the aircraft must be recorded in such a way that a record of the complete acoustical signal is retained. The recording and reproducing systems must meet the specifications in sections A36.3.6.2 to A36.3.6.9 at the recording speeds and/or data sampling rates used for the noise certification tests. Conformance must be demonstrated for the frequency bandwidths and recording channels selected for the tests. A36.3.6.2 The recording and reproducing systems must be calibrated as described in section A36.3.9. (a) For aircraft noise signals for which the high frequency spectral levels decrease rapidly with increasing frequency, appropriate pre-emphasis and complementary de-emphasis networks may be included in the measurement system. If pre-emphasis is included, over the range of nominal one-third octave midband frequencies from 800 Hz to 10 kHz inclusive, the electrical gain provided by the pre-emphasis network must not exceed 20 dB relative to the gain at 800 Hz. A36.3.6.3 For steady sinusoidal electrical signals applied to the input of the entire measurement system including all parts of the microphone system except the microphone at a selected signal level within 5 dB of that corresponding to the calibration sound pressure level on the reference level range, the time-average signal level indicated by the readout device at any one-third octave nominal midband frequency from 50 Hz to 10 kHz inclusive must be within ±1.5 dB of that at the calibration check frequency. The frequency response of a measurement system, which includes components that convert analog signals to digital form, must be within ±0.3 dB of the response at 10 kHz over the frequency range from 10 kHz to 11.2 kHz. Microphone extension cables as configured in the field need not be included for the frequency response determination. This allowance does not eliminate the requirement of including microphone extension cables when performing the pink noise recording in section A36.3.9.5. A36.3.6.4 For analog tape recordings, the amplitude fluctuations of a 1 kHz sinusoidal signal recorded within 5 dB of the level corresponding to the calibration sound pressure level must not vary by more than ±0.5 dB throughout any reel of the type of magnetic tape used. Conformance to this requirement must be demonstrated using a device that has time-averaging properties equivalent to those of the spectrum analyzer. A36.3.6.5 For all appropriate level ranges and for steady sinusoidal electrical signals applied to the input of the measurement system, including all parts of the microphone system except the microphone, at one-third-octave nominal midband frequencies of 50 Hz, 1 kHz and 10 kHz, and the calibration check frequency, if it is not one of these frequencies, the level non-linearity must not exceed ±0.5 dB for a linear operating range of at least 50 dB below the upper boundary of the level range. Note 1: Level linearity of measurement system components may be tested according to the methods described in IEC 61265 as amended. Microphone extension cables configured in the field need not be included for the level linearity determination. A36.3.6.6 On the reference level range, the level corresonding to the calibration sound pressure level must be at least 5 dB, but no more than 30 dB less than the upper boundary of the level range. A36.3.6.7 The linear operating ranges on adjacent level ranges must overlap by at least 50 dB minus the change in attenuation introduced by a change in the level range controls. It is possible for a measurement system to have level range controls that permit attenuation changes of either 10 dB or 1 dB, for example. With 10 dB steps, the minimum overlap required would be 40 dB, and with 1 dB steps the minimum overlap would be 49 dB. A36.3.6.8 An overload indicator must be included in the recording and reproducing systems so that an overload indication will occur during an overload condition on any relevant level range. A36.3.6.9 Attenuators included in the measurement system to permit range changes must operate in known intervals of decibel steps. A36.3.7 Analysis systems. A36.3.7.1 The analysis system must conform to the specifications in sections A36.3.7.2 to A36.3.7.7 for the frequency bandwidths, channel configurations and gain settings used for analysis. A36.3.7.2 The output of the analysis system must consist of one-third octave band sound pressure levels as a function of time, obtained by processing the noise signals (preferably recorded) through an analysis system with the following characteristics: (a) A set of 24 one-third octave band filters, or their equivalent, having nominal midband frequencies from 50 Hz to 10 kHz inclusive; (b) Response and averaging properties in which, in principle, the output from any one-third octave filter band is squared, averaged and displayed or stored as time-averaged sound pressure levels; (c) The interval between successive sound pressure level samples must be 500 ms ±5 milliseconds(ms) for spectral analysis with or without slow time-weighting, as defined in section A36.3.7.4; (d) For those analysis systems that do not process the sound pressure signals during the period of time required for readout and/or resetting of the analyzer, the loss of data must not exceed a duration of 5 ms; and (e) The analysis system must operate in real time from 50 Hz through at least 12 kHz inclusive. This requirement applies to all operating channels of a multi-channel spectral analysis system. A36.3.7.3 The minimum standard for the one-third octave band analysis system is the class 2 electrical performance requirements of IEC 61260 as amended, over the range of one-third octave nominal midband frequencies from 50 Hz through 10 kHz inclusive (incorporated by reference, see § 36.6). IEC 61260 specifies procedures for testing of one-third octave band analysis systems for relative attenuation, anti-aliasing filters, real time operation, level linearity, and filter integrated response (effective bandwidth). A36.3.7.4 When slow time averaging is performed in the analyzer, the response of the one-third octave band analysis system to a sudden onset or interruption of a constant sinusoidal signal at the respective one-third octave nominal midband frequency, must be measured at sampling instants 0.5, 1, 1.5 and 2 seconds(s) after the onset and 0.5 and 1s after interruption. The rising response must be −4 ±1 dB at 0.5s, −1.75 ±0.75 dB at 1s, −1 ±0.5 dB at 1.5s and −0.5 ±0.5 dB at 2s relative to the steady-state level. The falling response must be such that the sum of the output signal levels, relative to the initial steady-state level, and the corresponding rising response reading is −6.5 ±1 dB, at both 0.5 and 1s. At subsequent times the sum of the rising and falling responses must be −7.5 dB or less. This equates to an exponential averaging process (slow time-weighting) with a nominal 1s time constant (i.e., 2s averaging time). A36.3.7.5 When the one-third octave band sound pressure levels are determined from the output of the analyzer without slow time-weighting, slow time-weighting must be simulated in the subsequent processing. Simulated slow time-weighted sound pressure levels can be obtained using a continuous exponential averaging process by the following equation: Ls (i,k) = 10 log [(0.60653) 100.1 Ls[i, (k−1)] + (0.39347) 100.1 L (i, k)] where Ls(i,k) is the simulated slow time-weighted sound pressure level and L(i,k) is the as-measured 0.5s time average sound pressure level determined from the output of the analyzer for the k-th instant of time and i-th one-third octave band. For k = 1, the slow time-weighted sound pressure Ls[i, (k − 1 = 0)] on the right hand side should be set to 0 dB. An approximation of the continuous exponential averaging is represented by the following equation for a four sample averaging process for k ≥4: Ls (i,k) = 10 log [(0.13) 100.1 L[i,(k−3)] + (0.21) 100.1 L[i, (k−2)] + (0.27) 100.1 L[i, (k−1)] + (0.39) 100.1 L[i, k]] where Ls (i, k) is the simulated slow time-weighted sound pressure level and L (i, k) is the as measured 0.5s time average sound pressure level determined from the output of the analyzer for the k-th instant of time and the i-th one-third octave band. The sum of the weighting factors is 1.0 in the two equations. Sound pressure levels calculated by means of either equation are valid for the sixth and subsequent 0.5s data samples, or for times greater than 2.5s after initiation of data analysis. The coefficients in the two equations were calculated for use in determining equivalent slow time-weighted sound pressure levels from samples of 0.5s time average sound pressure levels. The equations do not work with data samples where the averaging time differs from 0.5s. A36.3.7.6 The instant in time by which a slow time-weighted sound pressure level is characterized must be 0.75s earlier than the actual readout time. The definition of this instant in time is needed to correlate the recorded noise with the aircraft position when the noise was emitted and takes into account the averaging period of the slow time-weighting. For each 0.5 second data record this instant in time may also be identified as 1.25 seconds after the start of the associated 2 second averaging period. A36.3.7.7 The resolution of the sound pressure levels, both displayed and stored, must be 0.1 dB or finer. A36.3.8 Calibration systems. A36.3.8.1 The acoustical sensitivity of the measurement system must be determined using a sound calibrator generating a known sound pressure level at a known frequency. The minimum standard for the sound calibrator is the class 1L requirements of IEC 60942 as amended (incorporated by reference, see § 36.6). A36.3.9 Calibration and checking of system. A36.3.9.1 Calibration and checking of the measurement system and its constituent components must be carried out to the satisfaction of the FAA by the methods specified in sections A36.3.9.2 through A36.3.9.10. The calibration adjustments, including those for environmental effects on sound calibrator output level, must be reported to the FAA and applied to the measured one-third-octave sound pressure levels determined from the output of the analyzer. Data collected during an overload indication are invalid and may not be used. If the overload condition occurred during recording, the associated test data are invalid, whereas if the overload occurred during analysis, the analysis must be repeated with reduced sensitivity to eliminate the overload. A36.3.9.2 The free-field frequency response of the microphone system may be determined by use of an electrostatic actuator in combination with manufacturer's data or by tests in an anechoic free-field facility. The correction for frequency response must be determined within 90 days of each test series. The correction for non-uniform frequency response of the microphone system must be reported to the FAA and applied to the measured one-third octave band sound pressure levels determined from the output of the analyzer. A36.3.9.3 When the angles of incidence of sound emitted from the aircraft are within ±30° of grazing incidence at the microphone (see Figure A36-1), a single set of free-field corrections based on grazing incidence is considered sufficient for correction of directional response effects. For other cases, the angle of incidence for each 0.5 second sample must be determined and applied for the correction of incidence effects. A36.3.9.4 For analog magnetic tape recorders, each reel of magnetic tape must carry at least 30 seconds of pink random or pseudo-random noise at its beginning and end. Data obtained from analog tape-recorded signals will be accepted as reliable only if level differences in the 10 kHz one-third-octave-band are not more than 0.75 dB for the signals recorded at the beginning and end. A36.3.9.5 The frequency response of the entire measurement system while deployed in the field during the test series, exclusive of the microphone, must be determined at a level within 5 dB of the level corresponding to the calibration sound pressure level on the level range used during the tests for each one-third octave nominal midband frequency from 50 Hz to 10 kHz inclusive, utilizing pink random or pseudo-random noise. Within six months of each test series the output of the noise generator must be determined by a method traceable to the U.S. National Institute of Standards and Technology or to an equivalent national standards laboratory as determined by the FAA. Changes in the relative output from the previous calibration at each one-third octave band may not exceed 0.2 dB. The correction for frequency response must be reported to the FAA and applied to the measured one-third octave sound pressure levels determined from the output of the analyzer. A36.3.9.6 The performance of switched attenuators in the equipment used during noise certification measurements and calibration must be checked within six months of each test series to ensure that the maximum error does not exceed 0.1 dB. A36.3.9.7 The sound pressure level produced in the cavity of the coupler of the sound calibrator must be calculated for the test environmental conditions using the manufacturer's supplied information on the influence of atmospheric air pressure and temperature. This sound pressure level is used to establish the acoustical sensitivity of the measurement system. Within six months of each test series the output of the sound calibrator must be determined by a method traceable to the U.S. National Institute of Standards and Technology or to an equivalent national standards laboratory as determined by the FAA. Changes in output from the previous calibration must not exceed 0.2 dB. A36.3.9.8 Sufficient sound pressure level calibrations must be made during each test day to ensure that the acoustical sensitivity of the measurement system is known at the prevailing environmental conditions corresponding with each test series. The difference between the acoustical sensitivity levels recorded immediately before and immediately after each test series on each day may not exceed 0.5 dB. The 0.5 dB limit applies after any atmospheric pressure corrections have been determined for the calibrator output level. The arithmetic mean of the before and after measurements must be used to represent the acoustical sensitivity level of the measurement system for that test series. The calibration corrections must be reported to the FAA and applied to the measured one-third octave band sound pressure levels determined from the output of the analyzer. A36.3.9.9 Each recording medium, such as a reel, cartridge, cassette, or diskette, must carry a sound pressure level calibration of at least 10 seconds duration at its beginning and end. A36.3.9.10 The free-field insertion loss of the windscreen for each one-third octave nominal midband frequency from 50 Hz to 10 kHz inclusive must be determined with sinusoidal sound signals at the incidence angles determined to be applicable for correction of directional response effects per section A36.3.9.3. The interval between angles tested must not exceed 30 degrees. For a windscreen that is undamaged and uncontaminated, the insertion loss may be taken from manufacturer's data. Alternatively, within six months of each test series the insertion loss of the windscreen may be determined by a method traceable to the U.S. National Institute of Standards and Technology or an equivalent national standards laboratory as determined by the FAA. Changes in the insertion loss from the previous calibration at each one-third-octave frequency band must not exceed 0.4 dB. The correction for the free-field insertion loss of the windscreen must be reported to the FAA and applied to the measured one-third octave sound pressure levels determined from the output of the analyzer. A36.3.10 Adjustments for ambient noise. A36.3.10.1 Ambient noise, including both an acoustical background and electrical noise of the measurement system, must be recorded for at least 10 seconds at the measurement points with the system gain set at the levels used for the aircraft noise measurements. Ambient noise must be representative of the acoustical background that exists during the flyover test run. The recorded aircraft noise data is acceptable only if the ambient noise levels, when analyzed in the same way, and quoted in PNL (see A36.4.1.3 (a)), are at least 20 dB below the maximum PNL of the aircraft. A36.3.10.2 Aircraft sound pressure levels within the 10 dB-down points (see A36.4.5.1) must exceed the mean ambient noise levels determined in section A36.3.10.1 by at least 3 dB in each one-third octave band, or must be adjusted using a method approved by the FAA; one method is described in the current advisory circular for this part. Section A36.4 Calculation of Effective Perceived Noise Level From Measured Data A36.4.1.1 The basic element for noise certification criteria is the noise evaluation measure known as effective perceived noise level, EPNL, in units of EPNdB, which is a single number evaluator of the subjective effects of airplane noise on human beings. EPNL consists of instantaneous perceived noise level, PNL, corrected for spectral irregularities, and for duration. The spectral irregularity correction, called "tone correction factor", is made at each time increment for only the maximum tone. A36.4.1.2 Three basic physical properties of sound pressure must be measured: level, frequency distribution, and time variation. To determine EPNL, the instantaneous sound pressure level in each of the 24 one-third octave bands is required for each 0.5 second increment of time during the airplane noise measurement. A36.4.1.3 The calculation procedure that uses physical measurements of noise to derive the EPNL evaluation measure of subjective response consists of the following five steps: (a) The 24 one-third octave bands of sound pressure level are converted to perceived noisiness (noy) using the method described in section A36.4.2.1 (a). The noy values are combined and then converted to instantaneous perceived noise levels, PNL(k). (b) A tone correction factor C(k) is calculated for each spectrum to account for the subjective response to the presence of spectral irregularities. (c) The tone correction factor is added to the perceived noise level to obtain tone-corrected perceived noise levels PNLT(k), at each one-half second increment: PNLT(k) = PNL(k) + C(k) The instantaneous values of tone-corrected perceived noise level are derived and the maximum value, PNLTM, is determined. (d) A duration correction factor, D, is computed by integration under the curve of tone-corrected perceived noise level versus time. (e) Effective perceived noise level, EPNL, is determined by the algebraic sum of the maximum tone-corrected perceived noise level and the duration correction factor: EPNL = PNLTM + D A36.4.2 Perceived noise level. A36.4.2.1 Instantaneous perceived noise levels, PNL(k), must be calculated from instantaneous one-third octave band sound pressure levels, SPL(i, k) as follows: (a) Step 1: For each one-third octave band from 50 through 10,000 Hz, convert SPL(i, k) to perceived noisiness n(i, k), by using the mathematical formulation of the noy table given in section A36.4.7. (b) Step 2: Combine the perceived noisiness values, n(i, k), determined in step 1 by using the following formula: $$\begin{aligned} \mathrm{N}(\mathrm{k}) &=\mathrm{n}(\mathrm{k})+0.15\left\{\left[\sum_{\mathrm{i}=1}^{24} \mathrm{n}(\mathrm{i}, \mathrm{k})\right]-\mathrm{n}(\mathrm{k})\right\} \\ &=0.85 \mathrm{n}(\mathrm{k})+0.15 \sum_{\mathrm{i}=1}^{24} \mathrm{n}(\mathrm{i}, \mathrm{k}) \end{aligned}$$ where n(k) is the largest of the 24 values of n(i, k) and N(k) is the total perceived noisiness. (c) Step 3: Convert the total perceived noisiness, N(k), determined in Step 2 into perceived noise level, PNL(k), using the following formula: \[ \text{PNL (k)} = 40.0 + \frac{10}{\text{log}\ 2} \text{log}\ N (k) \] PNL(k) is plotted in the current advisory circular for this part. A36.4.3 Correction for spectral irregularities. A36.4.3.1 Noise having pronounced spectral irregularities (for example, the maximum discrete frequency components or tones) must be adjusted by the correction factor C(k) calculated as follows: (a) Step 1: After applying the corrections specified under section A36.3.9, start with the sound pressure level in the 80 Hz one-third octave band (band number 3), calculate the changes in sound pressure level (or "slopes") in the remainder of the one-third octave bands as follows: s(3,k) = no value s(4,k) = SPL(4,k)−SPL(3,k) s(i,k) = SPL(i,k)−SPL(i−1,k) s(24,k) = SPL(24,k)−SPL(23,k) (b) Step 2: Encircle the value of the slope, s(i, k), where the absolute value of the change in slope is greater than five; that is where: \|Δs(i,k)\| = \|s(i,k)−s(i−1,k)\|>5 (c) Step 3: (1) If the encircled value of the slope s(i, k) is positive and algebraically greater than the slope s(i−1, k) encircle SPL(i, k). (2) If the encircled value of the slope s(i, k) is zero or negative and the slope s(i−1, k) is positive, encircle SPL(i−1, k). (3) For all other cases, no sound pressure level value is to be encircled. (d) Step 4: Compute new adjusted sound pressure levels SPL′(i, k) as follows: (1) For non-encircled sound pressure levels, set the new sound pressure levels equal to the original sound pressure levels, SPL′(i, k) = SPL(i, k). (2) For encircled sound pressure levels in bands 1 through 23 inclusive, set the new sound pressure level equal to the arithmetic average of the preceding and following sound pressure levels as shown below: SPL′(i,k) = 1⁄2[SPL(i−1,k) + SPL(i + 1,k)] (3) If the sound pressure level in the highest frequency band (i = 24) is encircled, set the new sound pressure level in that band equal to: SPL′(24,k) = SPL(23,k) + s(23,k) (e) Step 5: Recompute new slope s′(i, k), including one for an imaginary 25th band, as follows: s′(3,k) = s′(4,k) s′(4,k) = SPL′(4,k)−SPL′(3,k) s′(i,k) = SPL′(i,k)−SPL′(i−1,k) s′(24,k) = SPL′(24,k)−SPL′(23,k) s′(25,k) = s′(24,k) (f) Step 6: For i, from 3 through 23, compute the arithmetic average of the three adjacent slopes as follows: s(i,k) = 1⁄3[s′(i,k) + s′(i + 1,k) + s′(i + 2,k)] (g) Step 7: Compute final one-third octave-band sound pressure levels, SPL′ (i,k), by beginning with band number 3 and proceeding to band number 24 as follows: SPL′(3,k) = SPL(3,k) SPL′(4,k) = SPL′(3,k) + s(3,k) SPL′(i,k) = SPL′(i−1,k) + s(i−1,k) SPL′(24,k) = SPL′(23,k) + s(23,k) (h) Setp 8: Calculate the differences, F (i,k), between the original sound pressure level and the final background sound pressure level as follows: F(i,k) = SPL(i,k)-SPL′(i,k) and note only values equal to or greater than 1.5. (i) Step 9: For each of the relevant one-third octave bands (3 through 24), determine tone correction factors from the sound pressure level differences F (i, k) and Table A36-2. (j) Step 10: Designate the largest of the tone correction factors, determined in Step 9, as C(k). (An example of the tone correction procedure is given in the current advisory circular for this part). Tone-corrected perceived noise levels PNLT(k) must be determined by adding the C(k) values to corresponding PNL(k) values, that is: For any i-th one-third octave band, at any k-th increment of time, for which the tone correction factor is suspected to result from something other than (or in addition to) an actual tone (or any spectral irregularity other than airplane noise), an additional analysis may be made using a filter with a bandwidth narrower than one-third of an octave. If the narrow band analysis corroborates these suspicions, then a revised value for the background sound pressure level SPL′(i,k), may be determined from the narrow band analysis and used to compute a revised tone correction factor for that particular one-third octave band. Other methods of rejecting spurious tone corrections may be approved. A36.4.3.2 The tone correction procedure will underestimate EPNL if an important tone is of a frequency such that it is recorded in two adjacent one-third octave bands. An applicant must demonstrate that either: (a) No important tones are recorded in two adjacent one-third octave bands; or (b) That if an important tone has occurred, the tone correction has been adjusted to the value it would have had if the tone had been recorded fully in a single one-third octave band. A36.4.4 Maximum tone-corrected perceived noise level A36.4.4.1 The maximum tone-corrected perceived noise level, PNLTM, must be the maximum calculated value of the tone-corrected perceived noise level PNLT(k). It must be calculated using the procedure of section A36.4.3. To obtain a satisfactory noise time history, measurements must be made at 0.5 second time intervals. Figure A36-2 is an example of a flyover noise time history where the maximum value is clearly indicated. In the absence of a tone correction factor, PNLTM would equal PNLM. A36.4.4.2 After the value of PNLTM is obtained, the frequency band for the largest tone correction factor is identified for the two preceding and two succeeding 500 ms data samples. This is performed in order to identity the possibility of tone suppression at PNLTM by one-third octave band sharing of that tone. If the value of the tone correction factor C(k) for PNLTM is less than the average value of C(k) for the five consecutive time intervals, the average value of C(k) must be used to compute a new value for PNLTM. A36.4.5 Duration correction. A36.4.5.1 The duration correction factor D determined by the integration technique is defined by the expression: where T is a normalizing time constant, PNLTM is the maximum value of PNLT, t(1) is the first point of time after which PNLT becomes greater than PNLTM-10, and t(2) is the point of time after which PNLT remains constantly less than PNLTM-10. A36.4.5.2 Since PNLT is calculated from measured values of sound pressure level (SPL), there is no obvious equation for PNLT as a function of time. Consequently, the equation is to be rewritten with a summation sign instead of an integral sign as follows: where Δt is the length of the equal increments of time for which PNLT(k) is calculated and d is the time interval to the nearest 0.5s during which PNLT(k) remains greater or equal to PNLTM-10. A36.4.5.3 To obtain a satisfactory history of the perceived noise level use one of the following: (a) Half-Second time intervals for Δt; or (b) A shorter time interval with approved limits and constants. A36.4.5.4 The following values for T and Δt must be used in calculating D in the equation given in section A36.4.5.2: T = 10 s, and Δt = 0.5s (or the approved sampling time interval). Using these values, the equation for D becomes: \[ \text{D} = 10\ \text{log} \left[ \sum^{2d}_{k=0} \text{antilog} \frac{\text{PNLT(k)}}{10} \right] - \text{PNLTM} - 13 \] where d is the duration time defined by the points corresponding to the values PNLTM-10. A36.4.5.5 If in using the procedures given in section A36.4.5.2, the limits of PNLTM-10 fall between the calculated PNLT(k) values (the usual case), the PNLT(k) values defining the limits of the duration interval must be chosen from the PNLT(k) values closest to PNLTM-10. For those cases with more than one peak value of PNLT(k), the applicable limits must be chosen to yield the largest possible value for the duration time. A36.4.6 Effective perceived noise level. The total subjective effect of an airplane noise event, designated effective perceived noise level, EPNL, is equal to the algebraic sum of the maximum value of the tone-corrected perceived noise level, PNLTM, and the duration correction D. That is: where PNLTM and D are calculated using the procedures given in sections A36.4.2, A36.4.3, A36.4.4. and A36.4.5. A36.4.7 Mathematical formulation of noy tables. A36.4.7.1 The relationship between sound pressure level (SPL) and the logarithm of perceived noisiness is illustrated in Figure A36-3 and Table A36-3. A36.4.7.2 The bases of the mathematical formulation are: (a) The slopes (M(b), M(c), M(d) and M(e)) of the straight lines; (b) The intercepts (SPL(b) and SPL(c)) of the lines on the SPL axis; and (c) The coordinates of the discontinuities, SPL(a) and log n(a); SPL(d) and log n = −1.0; and SPL(e) and log n = log (0.3). A36.4.7.3 Calculate noy values using the following equations: SPL ≥SPL (a) n = antilog {(c)[SPL−SPL(c)]} SPL(b) ≤SPL n = antilog {M(b)[SPL−SPL(b)]} SPL(e) ≤SPL n = 0.3 antilog {M(e)[SPL−SPL(e)]} SPL(d) ≤SPL n = 0.1 antilog {M(d)[SPL−SPL(d)]} A36.4.7.4 Table A36-3 lists the values of the constants necessary to calculate perceived noisiness as a function of sound pressure level. Section A36.5 Reporting of Data to the FAA A36.5.1.1 Data representing physical measurements and data used to make corrections to physical measurements must be recorded in an approved permanent form and appended to the record. A36.5.1.2 All corrections must be reported to and approved by the FAA, including corrections to measurements for equipment response deviations. A36.5.1.3 Applicants may be required to submit estimates of the individual errors inherent in each of the operations employed in obtaining the final data. A36.5.2 Data reporting. An applicant is required to submit a noise certification compliance report that includes the following. A36.5.2.1 The applicant must present measured and corrected sound pressure levels in one-third octave band levels that are obtained with equipment conforming to the standards described in section A36.3 of this appendix. A36.5.2.2 The applicant must report the make and model of equipment used for measurement and analysis of all acoustic performance and meteorological data. A36.5.2.3 The applicant must report the following atmospheric environmental data, as measured immediately before, after, or during each test at the observation points prescribed in section A36.2 of this appendix. (a) Air temperature and relative humidity; (b) Maximum, minimum and average wind velocities; and (c) Atmospheric pressure. A36.5.2.4 The applicant must report conditions of local topography, ground cover, and events that might interfere with sound recordings. A36.5.2.5 The applicant must report the following: (a) Type, model and serial numbers (if any) of airplane, engine(s), or propeller(s) (as applicable); (b) Gross dimensions of airplane and location of engines; (c) Airplane gross weight for each test run and center of gravity range for each series of test runs; (d) Airplane configuration such as flap, airbrakes and landing gear positions for each test run; (e) Whether auxiliary power units (APU), when fitted, are operating for each test run; (f) Status of pneumatic engine bleeds and engine power take-offs for each test run; (g) Indicated airspeed in knots or kilometers per hour for each test run; (h) Engine performance data: (1) For jet airplanes: engine performance in terms of net thrust, engine pressure ratios, jet exhaust temperatures and fan or compressor shaft rotational speeds as determined from airplane instruments and manufacturer's data for each test run; (2) For propeller-driven airplanes: engine performance in terms of brake horsepower and residual thrust; or equivalent shaft horsepower; or engine torque and propeller rotational speed; as determined from airplane instruments and manufacturer's data for each test run; (i) Airplane flight path and ground speed during each test run; and (j) The applicant must report whether the airplane has any modifications or non-standard equipment likely to affect the noise characteristics of the airplane. The FAA must approve any such modifications or non-standard equipment. A36.5.3 Reporting of noise certification reference conditions. A36.5.3.1 Airplane position and performance data and the noise measurements must be corrected to the noise certification reference conditions specified in the relevant sections of appendix B of this part. The applicant must report these conditions, including reference parameters, procedures and configurations. A36.5.4 Validity of results. A36.5.4.1 Three average reference EPNL values and their 90 percent confidence limits must be produced from the test results and reported, each such value being the arithmetical average of the adjusted acoustical measurements for all valid test runs at each measurement point (flyover, lateral, or approach). If more than one acoustic measurement system is used at any single measurement location, the resulting data for each test run must be averaged as a single measurement. The calculation must be performed by: (a) Computing the arithmetic average for each flight phase using the values from each microphone point; and (b) Computing the overall arithmetic average for each reference condition (flyover, lateral or approach) using the values in paragraph (a) of this section and the related 90 percent confidence limits. A36.5.4.2 For each of the three certification measuring points, the minimum sample size is six. The sample size must be large enough to establish statistically for each of the three average noise certification levels a 90 percent confidence limit not exceeding ±1.5 EPNdB. No test result may be omitted from the averaging process unless approved by the FAA. Permitted methods for calculating the 90 percent confidence interval are shown in the current advisory circular for this part. A36.5.4.3 The average EPNL figures obtained by the process described in section A36.5.4.1 must be those by which the noise performance of the airplane is assessed against the noise certification criteria. Section A36.6 Nomenclature: Symbols and Units antilog Antilogarithm to the base 10. C(k) Tone correction factor. The factor to be added to PNL(k) to account for the presence of spectral irregularities such as tones at the k-th increment of time. Duration time. The time interval between the limits of t(1) and t(2) to the nearest 0.5 second. Duration correction. The factor to be added to PNLTM to account for the duration of the noise. EPNL EPNdB Effective perceived noise level. The value of PNL adjusted for both spectral irregularities and duration of the noise. (The unit EPNdB is used instead of the unit dB). EPNLr Effective perceived noise level adjusted for reference conditions. f(i) Frequency. The geometrical mean frequency for the i-th one-third octave band. F (i, k) Delta-dB. The difference between the original sound pressure level and the final background sound pressure level in the i-th one-third octave band at the k-th interval of time. In this case, background sound pressure level means the broadband noise level that would be present in the one-third octave band in the absence of the tone. dB-down. The value to be subtracted from PNLTM that defines the duration of the noise. Relative humidity. The ambient atmospheric relative humidity. Frequency band index. The numerical indicator that denotes any one of the 24 one-third octave bands with geometrical mean frequencies from 50 to 10,000 Hz. Time increment index. The numerical indicator that denotes the number of equal time increments that have elapsed from a reference zero. Logarithm to the base 10. log n(a) Noy discontinuity coordinate. The log n value of the intersection point of the straight lines representing the variation of SPL with log n. M(b), M(c), etc Noy inverse slope. The reciprocals of the slopes of straight lines representing the variation of SPL with log n. noy The perceived noisiness at any instant of time that occurs in a specified frequency range. n(i,k) The perceived noisiness at the k-th instant of time that occurs in the i-th one-third octave band. n(k) Maximum perceived noisiness. The maximum value of all of the 24 values of n(i) that occurs at the k-th instant of time. Total perceived noisiness. The total perceived noisiness at the k-th instant of time calculated from the 24-instantaneous values of n (i, k). p(b), p(c), etc Noy slope. The slopes of straight lines representing the variation of SPL with log n. PNdB The perceived noise level at any instant of time. (The unit PNdB is used instead of the unit dB). PNL(k) The perceived noise level calculated from the 24 values of SPL (i, k), at the k-th increment of time. (The unit PNdB is used instead of the unit dB). PNLM Maximum perceived noise level. The maximum value of PNL(k). (The unit PNdB is used instead of the unit dB). PNLT TPNdB Tone-corrected perceived noise level. The value of PNL adjusted for the spectral irregularities that occur at any instant of time. (The unit TPNdB is used instead of the unit dB). PNLT(k) The tone-corrected perceived noise level that occurs at the k-th increment of time. PNLT(k) is obtained by adjusting the value of PNL(k) for the spectral irregularities that occur at the k-th increment of time. (The unit TPNdB is used instead of the unit dB). PNLTM Maximum tone-corrected perceived noise level. The maximum value of PNLT(k). (The unit TPNdB is used instead of the unit dB). PNLTr Tone-corrected perceived noise level adjusted for reference conditions. s (i, k) Slope of sound pressure level. The change in level between adjacent one-third octave band sound pressure levels at the i-th band for the k-th instant of time. Δs (i, k) Change in slope of sound pressure level. s′ (i, k) Adjusted slope of sound pressure level. The change in level between adjacent adjusted one-third octave band sound pressure levels at the i-th band for the k-th instant of time. Average slope of sound pressure level. dB re20 µPa Sound pressure level. The sound pressure level that occurs in a specified frequency range at any instant of time. SPL(a) Noy discontinuity coordinate. The SPL value of the intersection point of the straight lines representing the variation of SPL with log n. SPL(b)SPL (c) Noy intercept. The intercepts on the SPL-axis of the straight lines representing the variation of SPL with log n. SPL (i, k) The sound pressure level at the k-th instant of time that occurs in the i-th one-third octave band. SPL′ (i, k) Adjusted sound pressure level. The first approximation to background sound pressure level in the i-th one-third octave band for the k-th instant of time. SPL(i) Maximum sound pressure level. The sound pressure level that occurs in the i-th one-third octave band of the spectrum for PNLTM. SPL(i)r Corrected maximum sound pressure level. The sound pressure level that occurs in the i-th one-third octave band of the spectrum for PNLTM corrected for atmospheric sound absorption. Final background sound pressure level. The second and final approximation to background sound pressure level in the i-th one-third octave band for the k-th instant of time. Elapsed time. The length of time measured from a reference zero. t(1), t(2) Time limit. The beginning and end, respectively, of the noise time history defined by h. Δt Time increment. The equal increments of time for which PNL(k) and PNLT(k) are calculated. Normalizing time constant. The length of time used as a reference in the integration method for computing duration corrections, where T = 10s. t(°F) (°C) °F, °C Temperature. The ambient air temperature. α(i) dB/1000ft db/100m Test atmospheric absorption. The atmospheric attenuation of sound that occurs in the i-th one-third octave band at the measured air temperature and relative humidity. α(i)o Reference atmospheric absorption. The atmospheric attenuation of sound that occurs in the i-th one-third octave band at a reference air temperature and relative humidity. First constant climb angle (Gear up, speed of at least V2 + 10 kt (V2 + 19 km/h), takeoff thrust). Second constant climb angle (Gear up, speed of at least V2 + 10 kt (V2 + 19 km/h), after cut-back). δε Thrust cutback angles. The angles defining the points on the takeoff flight path at which thrust reduction is started and ended respectively. Approach angle. ηr Reference approach angle. Noise angle (relative to flight path). The angle between the flight path and noise path. It is identical for both measured and corrected flight paths. Noise angle (relative to ground). The angle between the noise path and the ground. It is identical for both measured and corrected flight paths. Engine noise emission parameter. μr Reference engine noise emission parameter. Δ1 PNLT correction. The correction to be added to the EPNL calculated from measured data to account for noise level changes due to differences in atmospheric absorption and noise path length between reference and test conditions. Adjustment to duration correction. The adjustment to be made to the EPNL calculated from measured data to account for noise level changes due to the noise duration between reference and test conditions. Source noise adjustment. The adjustment to be made to the EPNL calculated from measured data to account for noise level changes due to differences between reference and test engine operating conditions. Section A36.7 Sound Attenuation in Air A36.7.1 The atmospheric attenuation of sound must be determined in accordance with the procedure presented in section A36.7.2. A36.7.2 The relationship between sound attenuation, frequency, temperature, and humidity is expressed by the following equations. A36.7.2(a) For calculations using the English System of Units: $$\begin{aligned} \alpha(\mathrm{i}) &=10^{\left[2.05 \log \left(f_{0} / 1000\right)+6.33 \times 10^{-4} \theta-1.45325\right]} \\ &+\eta(\delta) \times 10^{\left[\log \left(f_{0}\right)+4.6833 \times 10^{-3} \theta-2.4215\right]} \end{aligned}$$ $$\begin{aligned} \delta &=\sqrt{\frac{1010}{f(0)}} 10^{\left(\log \mathrm{H}-1.97274664+2.288074 \times 10^{-2} \theta\right)} \\ & \times 10^{\left(-9.589 \times 10^{-5} \mathrm{\theta}^{2}+3.0 \times 10^{-7} \theta^{3}\right)} \end{aligned}$$ η(δ) is listed in Table A36-4 and f0 in Table A36-5; α(i) is the attenuation coefficient in dB/1000 ft; θ is the temperature in °F; and H is the relative humidity, expressed as a percentage. A36.7.2(b) For calculations using the International System of Units (SI): $$\begin{aligned} \alpha(\mathrm{i}) &=10^{\left[2.05 \log \left(f_{0} / 1000\right)+1.1394\times 10^{-3} \theta-1.916984\right]} \\ &+\eta(\delta) \times 10^{\left[\log \left(f_{0}\right)+8.42994 \times 10^{-3} \theta-2.755624\right]} \end{aligned}$$ $$\begin{aligned} \delta &=\sqrt{\frac{1010}{f_{0}}} 10^{\left(\log \mathrm{H}-1.328924+3.179768 \times 10^{-2} \theta\right)} \\ & \times 10^{\left(-2.173716 \times 10^{-4} \mathrm{\theta}^{2}+1.7496 \times 10^{-6} \theta^{3}\right)} \end{aligned}$$ α(i) is the attenuation coefficient in dB/100 m; θ is the temperature in °C; and A36.7.3 The values listed in table A36-4 are to be used when calculating the equations listed in section A36.7.2. A term of quadratic interpolation is to be used where necessary. Section A36.8 [Reserved] Section A36.9 Adjustment of Airplane Flight Test Results. A36.9.1 When certification test conditions are not identical to reference conditions, appropriate adjustments must be made to the measured noise data using the methods described in this section. A36.9.1.1 Adjustments to the measured noise values must be made using one of the methods described in sections A36.9.3 and A36.9.4 for differences in the following: (a) Attenuation of the noise along its path as affected by "inverse square" and atmospheric attenuation (b) Duration of the noise as affected by the distance and the speed of the airplane relative to the measuring point (c) Source noise emitted by the engine as affected by the differences between test and reference engine operating conditions (d) Airplane/engine source noise as affected by differences between test and reference airspeeds. In addition to the effect on duration, the effects of airspeed on component noise sources must be accounted for as follows: for conventional airplane configurations, when differences between test and reference airspeeds exceed 15 knots (28 km/h) true airspeed, test data and/or analysis approved by the FAA must be used to quantify the effects of the airspeed adjustment on resulting certification noise levels. A36.9.1.2 The "integrated" method of adjustment, described in section A36.9.4, must be used on takeoff or approach under the following conditions: (a) When the amount of the adjustment (using the "simplified" method) is greater than 8 dB on flyover, or 4 dB on approach; or (b) When the resulting final EPNL value on flyover or approach (using the simplified method) is within 1 dB of the limiting noise levels as prescribed in section B36.5 of this part. A36.9.2 Flight profiles. As described below, flight profiles for both test and reference conditions are defined by their geometry relative to the ground, together with the associated airplane speed relative to the ground, and the associated engine control parameter(s) used for determining the noise emission of the airplane. A36.9.2.1 Takeoff Profile. Figure A36-4 illustrates a typical takeoff profile. (a) The airplane begins the takeoff roll at point A, lifts off at point B and begins its first climb at a constant angle at point C. Where thrust or power (as appropriate) cut-back is used, it is started at point D and completed at point E. From here, the airplane begins a second climb at a constant angle up to point F, the end of the noise certification takeoff flight path. (b) Position K1 is the takeoff noise measuring station and AK1 is the distance from start of roll to the flyover measuring point. Position K2 is the lateral noise measuring station, which is located on a line parallel to, and the specified distance from, the runway center line where the noise level during takeoff is greatest. (c) The distance AF is the distance over which the airplane position is measured and synchronized with the noise measurements, as required by section A36.2.3.2 of this part. A36.9.2.2 Approach Profile. Figure A36-5 illustrates a typical approach profile. (a) The airplane begins its noise certification approach flight path at point G and touches down on the runway at point J, at a distance OJ from the runway threshold. (b) Position K3 is the approach noise measuring station and K3O is the distance from the approach noise measurement point to the runway threshold. (c) The distance GI is the distance over which the airplane position is measured and synchronized with the noise measurements, as required by section A36.2.3.2 of this part. The airplane reference point for approach measurements is the instrument landing system (ILS) antenna. If no ILS antenna is installed an alternative reference point must be approved by the FAA. A36.9.3 Simplified method of adjustment. A36.9.3.1 General. As described below, the simplified adjustment method consists of applying adjustments (to the EPNL, which is calculated from the measured data) for the differences between measured and reference conditions at the moment of PNLTM. A36.9.3.2 Adjustments to PNL and PNLT. (a) The portions of the test flight path and the reference flight path described below, and illustrated in Figure A36-6, include the noise time history that is relevant to the calculation of flyover and approach EPNL. In figure A36-6: (1) XY represents the portion of the measured flight path that includes the noise time history relevant to the calculation of flyover and approach EPNL; XrYr represents the corresponding portion of the reference flight path. (2) Q represents the airplane's position on the measured flight path at which the noise was emitted and observed as PNLTM at the noise measuring station K. Qr is the corresponding position on the reference flight path, and Kr the reference measuring station. QK and QrKr are, respectively, the measured and reference noise propagation paths, Qr being determined from the assumption that QK and QrKr form the same angle θ with their respective flight paths. (b) The portions of the test flight path and the reference flight path described in paragraph (b)(1) and (2), and illustrated in Figure A36-7(a) and (b), include the noise time history that is relevant to the calculation of lateral EPNL. (1) In figure A36-7(a), XY represents the portion of the measured flight path that includes the noise time history that is relevant to the calculation of lateral EPNL; in figure A36-7(b), XrYr represents the corresponding portion of the reference flight path. (2) Q represents the airplane position on the measured flight path at which the noise was emitted and observed as PNLTM at the noise measuring station K. Qr is the corresponding position on the reference flight path, and Kr the reference measuring station. QK and QrKr are, respectively, the measured and reference noise propagation paths. In this case Kr is only specified as being on a particular Lateral line; Kr and Qr are therefore determined from the assumptions that QK and QrKr: (i) Form the same angle θ with their respective flight paths; and (ii) Form the same angle ψ with the ground. For the lateral noise measurement, sound propagation is affected not only by inverse square and atmospheric attenuation, but also by ground absorption and reflection effects which depend mainly on the angle ψ. A36.9.3.2.1 The one-third octave band levels SPL(i) comprising PNL (the PNL at the moment of PNLTM observed at K) must be adjusted to reference levels SPL(i)r as follows: A36.9.3.2.1(a) For calculations using the English System of Units: SPL(i)r = SPL(i) + 0.001[α(i)−α(i)0]QK + 0.001α(i)0(QK−QrKr) + 20log(QK/QrKr) In this expression, (1) The term 0.001[α(i)−α(i)0]QK is the adjustment for the effect of the change in sound attenuation coefficient, and α(i) and α(i)0 are the coefficients for the test and reference atmospheric conditions respectively, determined under section A36.7 of this appendix; (2) The term 0.001α(i)0(QK − QrKr) is the adjustment for the effect of the change in the noise path length on the sound attenuation; (3) The term 20 log(QK/QrKr) is the adjustment for the effect of the change in the noise path length due to the "inverse square" law; (4) QK and QrKr are measured in feet and α(i) and α(i)0 are expressed in dB/1000 ft. A36.9.3.2.1(b) For calculations using the International System of Units: SPL(i)r = SPL(i) + 0.01[α(i)−α(i)0]QK + 0.01α(i)0 (QK − QrKr) + 20 log(QK/QrKr) (1) The term 0.01[α(i) − α(i)0]QK is the adjustment for the effect of the change in sound attenuation coefficient, and α(i) and α(i)0 are the coefficients for the test and reference atmospheric conditions respectively, determined under section A36.7 of this appendix; (2) The term 0.01α(i)0(QK − QrKr) is the adjustment for the effect of the change in the noise path length on the sound attenuation; (4) QK and QrKr are measured in meters and α(i) and α(i)0 are expressed in dB/100 m. A36.9.3.2.1.1 PNLT Correction. (a) Convert the corrected values, SPL(i)r, to PNLTr; (b) Calculate the correction term Δ1 using the following equation: Δ1 = PNLTr − PNLTM A36.9.3.2.1.2 Add Δ1 arithmetically to the EPNL calculated from the measured data. A36.9.3.2.2 If, during a test flight, several peak values of PNLT that are within 2 dB of PNLTM are observed, the procedure defined in section A36.9.3.2.1 must be applied at each peak, and the adjustment term, calculated according to section A36.9.3.2.1, must be added to each peak to give corresponding adjusted peak values of PNLT. If these peak values exceed the value at the moment of PNLTM, the maximum value of such exceedance must be added as a further adjustment to the EPNL calculated from the measured data. A36.9.3.3 Adjustments to duration correction. A36.9.3.3.1 Whenever the measured flight paths and/or the ground velocities of the test conditions differ from the reference flight paths and/or the ground velocities of the reference conditions, duration adjustments must be applied to the EPNL values calculated from the measured data. The adjustments must be calculated as described below. A36.9.3.3.2 For the flight path shown in Figure A36-6, the adjustment term is calculated as follows: Δ2 = −7.5 log(QK/QrKr) + 10 log(V/Vr) (a) Add Δ2 arithmetically to the EPNL calculated from the measured data. A36.9.3.4 Source noise adjustments. A36.9.3.4.1 To account for differences between the parameters affecting engine noise as measured in the certification flight tests, and those calculated or specified in the reference conditions, the source noise adjustment must be calculated and applied. The adjustment is determined from the manufacturer's data approved by the FAA. Typical data used for this adjustment are illustrated in Figure A36-8 that shows a curve of EPNL versus the engine control parameter μ, with the EPNL data being corrected to all the other relevant reference conditions (airplane mass, speed and altitude, air temperature) and for the difference in noise between the test engine and the average engine (as defined in section B36.7(b)(7)). A sufficient number of data points over a range of values of μr is required to calculate the source noise adjustments for lateral, flyover and approach noise measurements. A36.9.3.4.2 Calculate adjustment term Δ3 by subtracting the EPNL value corresponding to the parameter μ from the EPNL value corresponding to the parameter μr. Add Δ3 arithmetically to the EPNL value calculated from the measured data. A36.9.3.5 Symmetry adjustments. A36.9.3.5.1 A symmetry adjustment to each lateral noise value (determined at the section B36.4(b) measurement points), is to be made as follows: (a) If the symmetrical measurement point is opposite the point where the highest noise level is obtained on the main lateral measurement line, the certification noise level is the arithmetic mean of the noise levels measured at these two points (see Figure A36-9(a)); (b) If the condition described in paragraph (a) of this section is not met, then it is assumed that the variation of noise with the altitude of the airplane is the same on both sides; there is a constant difference between the lines of noise versus altitude on both sides (see figure A36-9(b)). The certification noise level is the maximum value of the mean between these lines. A36.9.4 Integrated method of adjustment A36.9.4.1 General. As described in this section, the integrated adjustment method consists of recomputing under reference conditions points on the PNLT time history corresponding to measured points obtained during the tests, and computing EPNL directly for the new time history obtained in this way. The main principles are described in sections A36.9.4.2 through A36.9.4.4.1. A36.9.4.2 PNLT computations. (a) The portions of the test flight path and the reference flight path described in paragraph (a)(1) and (2), and illustrated in Figure A36-10, include the noise time history that is relevant to the calculation of flyover and approach EPNL. In figure A36-10: (1) XY represents the portion of the measured flight path that includes the noise time history relevant to the calculation of flyover and approach EPNL; XrYr represents the corresponding reference flight path. (2) The points Q0, Q1, Qn represent airplane positions on the measured flight path at time t0, t1 and tn respectively. Point Q1 is the point at which the noise was emitted and observed as one-third octave values SPL(i)1 at the noise measuring station K at time t1. Point Qr1 represents the corresponding position on the reference flight path for noise observed as SPL(i)r1 at the reference measuring station Kr at time tr1. Q1K and Qr1Kr are respectively the measured and reference noise propagation paths, which in each case form the angle θ1 with their respective flight paths. Qr0 and Qrn are similarly the points on the reference flight path corresponding to Q0 and Qn on the measured flight path. Q0 and Qn are chosen so that between Qr0 and Qrn all values of PNLTr (computed as described in paragraphs A36.9.4.2.2 and A36.9.4.2.3) within 10 dB of the peak value are included. (b) The portions of the test flight path and the reference flight path described in paragraph (b)(1) and (2), and illustrated in Figure A36-11(a) and (b), include the noise time history that is relevant to the calculation of lateral EPNL. (1) In figure A36-11(a) XY represents the portion of the measured flight path that includes the noise time history that is relevant to the calculation of lateral EPNL; in figure A36-11(b), XrYr represents the corresponding portion of the reference flight path. (2) The points Q0, Q1 and Qn represent airplane positions on the measured flight path at time t0, t1 and tn respectively. Point Q1 is the point at which the noise was emitted and observed as one-third octave values SPL(i)1 at the noise measuring station K at time t1. The point Qr1 represents the corresponding position on the reference flight path for noise observed as SPL(i)r1 at the measuring station Kr at time tr1. Q1K and Qr1Kr are respectively the measured and reference noise propagation paths. Qr0 and Qrn are similarly the points on the reference flight path corresponding to Q0 and Qn on the measured flight path. Q0 and Qn are chosen to that between Qro and Qrn all values of PNLTr (computed as described in paragraphs A36.9.4.2.2 and A36.9.4.2.3) within 10 dB of the peak value are included. In this case Kr is only specified as being on a particular lateral line. The position of Kr and Qr1 are determined from the following requirements. (i) Q1K and Qr1Kr form the same angle θ1 with their respective flight paths; and (ii) The differences between the angles 1 and r1 must be minimized using a method, approved by the FAA. The differences between the angles are minimized since, for geometrical reasons, it is generally not possible to choose Kr so that the condition described in paragraph A36.9.4.2(b)(2)(i) is met while at the same time keeping 1 and r1 equal. For the lateral noise measurement, sound propagation is affected not only by "inverse square" and atmospheric attenuation, but also by ground absorption and reflection effects which depend mainly on the angle. A36.9.4.2.1 In paragraphs A36.9.4.2(a)(2) and (b)(2) the time tr1 is later (for Qr1Kr >Q1K) than t1 by two separate amounts: (1) The time taken for the airplane to travel the distance Qr1Qr0 at a speed Vr less the time taken for it to travel Q1Q0 at V; (2) The time taken for sound to travel the distance Qr1Kr-Q1K. For the flight paths described in paragraphs A36.9.4.2(a) and (b), the use of thrust or power cut-back will result in test and reference flight paths at full thrust or power and at cut-back thrust or power. Where the transient region between these thrust or power levels affects the final result, an interpolation must be made between them by an approved method such as that given in the current advisory circular for this part. A36.9.4.2.2 The measured values of SPL(i)1 must be adjusted to the reference values SPL(i)r1 to account for the differences between measured and reference noise path lengths and between measured and reference atmospheric conditions, using the methods of section A36.9.3.2.1 of this appendix. A corresponding value of PNLr1 must be computed according to the method in section A36.4.2. Values of PNLr must be computed for times t0 through tn. A36.9.4.2.3 For each value of PNLr1, a tone correction factor C1 must be determined by analyzing the reference values SPL(i)r using the methods of section A36.4.3 of this appendix, and added to PNLr1 to yield PNLTr1. Using the process described in this paragraph, values of PNLTr must be computed for times t0 through tn. A36.9.4.3 Duration correction. A36.9.4.3.1 The values of PNLTr corresponding to those of PNLT at each one-half second interval must be plotted against time (PNLTr1 at time tr1). The duration correction must then be determined using the method of section A36.4.5.1 of this appendix, to yield EPNLr. A36.9.4.4 Source Noise Adjustment. A36.9.4.4.1 A source noise adjustment, Δ3, must be determined using the methods of section A36.9.3.4 of this appendix. A36.9.5 Flight Path Identification Positions Start of Takeoff roll. Lift-off. Start of first constant climb. Start of thrust reduction. Start of second constant climb. End of noise certification Takeoff flight path. Start of noise certification Approach flight path. Position on Approach path directly above noise measuring station. Start of level-off. Touchdown. Noise measurement point. Reference measurement point. Flyover noise measurement point. Lateral noise measurement point. Approach noise measurement point. End of noise certification Takeoff flight track. Threshold of Approach end of runway. Start of noise certification Approach flight track. Position on measured Takeoff flight path corresponding to apparent PNLTM at station K See section A36.9.3.2. Position on corrected Takeoff flight path corresponding to PNLTM at station K. See section A36.9.3.2. Airplane test speed. Airplane reference speed. A36.9.6 Flight Path Distances Feet (meters) Length of takeoff roll. The distance along the runway between the start of takeoff roll and lift off. Takeoff measurement distance. The distance from the start of roll to the takeoff noise measurement station along the extended center line of the runway. Takeoff flight track distance. The distance from the start of roll to the takeoff flight track position along the extended center line of the runway after which the position of the airplane need no longer be recorded. Measured noise path. The distance from the measured airplane position Q to station K. QrKr Reference noise path. The distance from the reference airplane position Qr to station Kr. K3H Airplane approach height. The height of the airplane above the approach measuring station. Approach measurement distance. The distance from the runway threshold to the approach measurement station along the extended center line of the runway. Approach flight track distance. The distance from the runway threshold to the approach flight track position along the extended center line of the runway after which the position of the airplane need no longer be recorded. [Amdt. 36-54, 67 FR 45212, July 8, 2002; Amdt. 36-24, 67 FR 63195, 63196, Oct. 10, 2002; 68 FR 1512, Jan. 10, 2003; Amdt. 36-26, 70 FR 38749, July 5, 2005; FAA Doc. No. FAA-2015-3782, Amdt. No. 36-31, 82 FR 46131, Oct. 4, 2017]] § 36.1583 Noncomplying agricultural and fire fighting airplanes Appendix B to Part 36—Noise Levels for Transport Category and Jet Airplanes Under § 36.103	CommonCrawl
Eliminating any subset of quantifiers by Skolemization: no constants, please! Can you use Skolemization to reduce a formula to the variables you want it to be about? I was trying to think of a nice algorithmic way to do it but only ended up having problems. Say you have a formula $G$ in prenex normal form, i.e. something looking like this: $$G \equiv Q_1x_1...Q_nx_n\;F$$ where $Q_i$ are quantifiers binding variables $x_i$ and occuring exclusively on the left of formula $F$ (which, consequently does not contain quantifiers itself). It is known that $G$ can be transformed into Skolem normal form, i. e. all existential quantifiers can be eliminated by substituting the variables they are binding by functions. But can any variable be substituted so that in the end the formula expresses something about the variables you wish to keep? The possibly cheapest way to do it involves bracketing $F$ with its nearest quantifier. [EDIT: as you can read in the comments, this manoeuvre involves a major flaw, which shall stay undeleted for the record.] If the quantifier nearest to $F$ is $\exists$, it would go like this: $$Q_1x_1...Q_{n-1}x_{n-1}\;\exists x_n\;F$$ $$Q_1x_1...Q_{n-1}x_{n-1}\;(\exists x_n\;F)$$ $$Q_1x_1...Q_{n-1}x_{n-1}\;(F[x_n/c_1])$$ ...where $F[x_n/c_1]$ expresses that $x_n$ has been substituted by constant $c_1$ in $F$. If the quantifier nearest to $F$ is $\forall$, the procedure takes some extra legwork and two negations: $$Q_1x_1...Q_{n-1}x_{n-1}\;(\forall x_n F)$$ $$Q_1x_1...Q_{n-1}x_{n-1}\;(\neg \exists x_n\;\neg F)$$ $$Q_1x_1...Q_{n-1}x_{n-1}\;\neg (\exists x_n\;\neg F)$$ $$Q_1x_1...Q_{n-1}x_{n-1}\;(\neg F[x_n/c_1])$$ Iterating the process for all bound variables would leave you with a formula with no quantifiers but many constants you cannot assume much about. To sustain "expessive power" regarding the variables designated to stay, there ought to be a way to substitute the rest of the variables by functions with arities higher than 0. I was thinking of (if required, repeatedly) putting the formula in the following form: $$Q_1x_1...Q_{n-1}x_{n-1}\;(\forall y_{k} ... \forall y_n\;\exists x_n F)$$ $$Q_1x_1...Q_{n-1}x_{n-1}\;(\forall y_{k} ... \forall y_n\;F[x_n/f(y_{k},..., y_n)])$$ ...where $y_i$ shall be the variables preserved in the final form and $x_i$ the variables to be eliminated. Two problems arise: 1) As any quantifiers can bind $y_i$ in the original prenex normal form, the existential quantifiers binding $y_i$ must first be rephrased into universal ones for the Skolemization to yield functions of $y_i$. This does not always lead to a chain of universal quantifiers with no negations in between. It seems like it's not always possible to get rid of the negation in $\forall y_i \; \neg \forall y_{i-1}$. 2) Consequently, opportunities to rephrase $Q_ix_i\; Q_{i+1}x_{i+1}$ as $Q_{i+1}x_{i+1}\;Q_ix_i$ are limited. So far, I managed to fabricate formulas containing only the variables I picked as survivors in the initial stage. Anyway, I could not avoid constants in every case. Can you think of a procedure with rules that would prevent constants appearing in any resulting formula? logic predicate-logic first-order-logic quantifiers RauteRaute Your version of Skolemization is incorrect, and we can see this at the very first step. "$\forall xL(x)$" is not equivalent to "$\neg L(x/c)$:" while the former implies the latter, the latter doesn't imply the former (how do we know $c$ wasn't "chosen badly"?) Similarly, "$\exists x L(x)$" is not equivalent to "$L(x/c)$:" while the latter implies the former,the former doesn't imply the latter (for the same reason as above. So the way you've introduced constants is broken. Skolemization doesn't introduce new functions - rather, it introduces new existential quantifiers over functions. E.g. the Skolemized version of "$\forall x\exists yL(x, y)$" is "$\exists F^1\forall x(L(x, F(x)))$." (The "$^1$" indicates that $F$ is a unary function symbol.) Things get clearer once we have more than one existential quantifier. The Skolemized version of "$\exists x\forall y\exists z L(x, y, z)$" is "$\exists F^0\exists G^1\forall y(L(F^0, y, G(y)))$." Note two things here: All the existential function quantifiers are together, at the front. We've proceeded from the outside in: the $F^0$ corresponds to $x$, and the $G^1$ corresponds to $y$. Basically, the Skolemized version of a formula "$\overline{Q}\overline{x}L(\overline{x})$" is "There are functions picking witnesses for the existential quantifiers in $\overline{Q}$ appropriately." By the way, we can also perform "partial" Skolemization: we can Skolemize away any subset of the first-order existential quantifiers while leaving the rest. For instance, $$\forall x\exists y\forall z\exists w\forall a\exists b(L(x, y, z,w, a, b))$$ is equivalent to $$\exists F^1\exists G^2\exists H^3\forall x \forall z\forall a(L(x, F(x), z, G(x, z), a, H(x, z, a)))$$ (this is the fully Skolemized version), but it's also equivalent to the partially Skolemized sentence $$\exists F^1\exists H^3\forall x\forall z\exists w\forall a(L(x, F(x), z, w, a, H(x, z, a))).$$ Note that when we leave an existential quantifier alone it stays where it was - it doesn't move to the front. Noah SchweberNoah Schweber Unless I misunderstand you, this doesn't work. Say you have $\forall x \exists y Loves(x,y)$ : Everyone loves someone. By your method you get (again, if I understand you correctly): $\forall x Loves(x,c_1)$ But that can't be right, for that would imply that there is something $c_1$ that everyone loves. But if I have $a$ and $b$ (and not other objects in my domain) and $a$ loves $b$, $b$ loves $a$, but $a$ and $b$ do not love themselves, then $\forall x \exists y Loves(x,y)$ is true but $\forall x Loves(x,c_1)$ is false: $c_1$ cannot be $a$, and also not be $b$, and there are no other objects. This is exactly why you can only eliminate existentials by using a function that shows that the 'something' is dependent on the choice of objects for the universal quantifiers in front of it: if I pick $a$, I can point to $b$ as the something that $a$ loves, but if I pick $b$, I point to $a$. So you have to think about this process 'outside-in', rather than 'inside-out'. Bram28Bram28 $\begingroup$ Nice, +1! While the one-quantifier step is already broken (see my answer), this drives the point home, and might be clearer to the OP. $\endgroup$ – Noah Schweber Jul 3 '17 at 15:24 Not the answer you're looking for? Browse other questions tagged logic predicate-logic first-order-logic quantifiers or ask your own question. What does it mean when $x$ is 'free'? When is de-Skolemizing statements appropriate? How is quantifier elimination accomplished in second and higher order logic? How to define new constants in Tableaux Method of predicate logic Distributive Property of Quantifiers How to move quantifiers to the front of a formula? Should $\to$-elimination always have precedence over $\lnot$-elimination when transforming formulas to Prenex CNF? Transform a formula via prenex normal form to to Skolem normal form Prove formula using skolemization and resolution	CommonCrawl
Extraction of Computer Image Analysis Information by Desk Top Computer from Beef Carcass Cross Sections Karnuah, A.B.;Moriya, K.;Sasaki, Y. 1171 https://doi.org/10.5713/ajas.1999.1171 PDF The precision and reliability of the Computer Image Analysis technique using a desk top computer for extracting information from carcass cross section scans was evaluated by the repeatability (R) and coefficient of variation (CV) for error variance. The 6th and 7th ribs cross section of carcasses from 55 fattened Japanese Black steers were used. The image analysis was conducted using a desk top computer (Macintosh-Apple Vision 1710 Display) connected to a scanner and an image capture camera. Two software applications, Adobe Photoshop and Mac Scope were used interchangeably. The information extracted and measured were individual muscle area, circumference length, long and short axes lengths, muscle direction; distance between any two muscle centers of gravity; cross section total area, lean, fat, and bone. The information was extracted after the processes of scanning, digitization, masking, muscle separation, and binarization. When using the Computer Image Analysis technique by desk top computer, proper digitization and selection of scanning resolution are very important in order to obtain accurate information. The R-values for muscle area, circumference, long and axes lengths, and direction ranged from 0.95 to 0.99, whereas those of the distance between any two muscle centers of gravity ranged from 0.96 to 0.99, respectively. For the cross section total area, lean, fat, and bone it ranged from 0.83 to 0.99. Excellent repeatability measurements were observed for muscle direction and distance between any two muscle centers of gravity. The results indicate that the Computer Image Analysis technique using a desk top computer for extracting information from carcass cross section is reliable and has high precision. Growth Data of Broiler Chickens Fitted to Gompertz Function Duan-yai, S.;Young, B.A.;Lisle, A.;Coutts, J.A.;Gaughan, J.B. 1177 This study describes the growth of broiler chickens to the two forms of Gompertz function for application in broiler production models. The first form is based on the estimated mature weight ($W_A$), while the second is based on the estimated hatch weight ($W_O$). Both equations gave identical estimation because they are mathematically identical. To fit the growth curve of commercial broilers that marketed at 35-42 days, it is unnecessary to keep broilers to near maturity (> day 140) to obtain growth data for deriving the Gompertz function. This date does not improve the curve fitting of the early growing period. Additionally, a high mortality and health problem occurred to this type of chicken after day 105. Development of the Gonads Derived from Hetero-Sexually Transferred Primordial Germ Cells (PGCs) between Embryos in the Chicken Furuta, H.;Yamaguchi, H.;Fujihara, N. 1188 Primordial germ cells (PGCs) of White Leghorn chicken embryos as a donor were transferred to Rhode Island Red chicken embryos as a recipient. At 48-50 h (stage 13-15) of incubation of fertilized eggs, donor PGCs, which were taken out from blood vessels of donor embryos, were injected into blood vessels of recipient embryos. Sex of the treated embryos was determined after the transfer of PGCs using remaining blood samples. In the present experiments, survival rate of the treated embryos was 33.3% for homo-sexual and 35.4% for hetero-sexual transfers of PGCs, respectively, when determined at 17 days of incubation. In this study, most of the treated embryos could not survive more than 18 days of incubation, though the reason for that was not clarified in the present work. The gonalds removed from embryos that died after 18 days of incubation and the organs from newly hatched chicks were examined for morphological and histological features. The gonads removed from the embryos with homo-sexual transfer of PGCs showed normal development in appearance. On the contrary, some (35.3%) of the embryos with hetero-sexual transfer of PGCs possessed abnormal gonads similar to ovotestis by histological observation. In cases where the gonads developed to be normal organs (64.7%) the sex of embryos was the same as recipient ones. The present results suggest that hetero-sexual transfer of the PGCs may bring about the possibility of development of the embryos bearing sexually different gametes, spermatogonia or oogonia. Sward Characteristics and Nutritive Value of Two Cultivars of Subterranean Clover Ru, Y.J.;Fortune, J.A. 1192 Two cultivars of subterranean clover (Trifolium subterraneum L.), "Dinninup" and "Seaton Park" were sown at Shenton Park Field Station, Western Australia, in May 1992 and 1993. The characteristics of Dinninup related to animal production were compared with Seaton Park under grazing conditions with herbage utilization efficiencies of 60% in 1992 and 65% in 1993. The results showed that Dinninup and Seaton Park had similar dry matter digestibility (77-78%) and dry matter production (1,290 kg/ha in 1992; 930 kg/ha in 1993) before flowering initiation even though Dinninup had more (p<0.05) branches, leaves and petioles per plant. After flowering, the herbage on offer of Dinninup was higher (p<0.05) and dry matter digestibility was lower (p<0.05) than that of Seaton Park while the sward structure was similar for both cultivars. The variation in nutritive value among plant parts increased with maturation. Leaf was more digestible than stem and petiole with a higher nitrogen content, and stem had the lowest dry matter digesitibility and nitrogen content in late of the season. Sheep did not show any preference for Seaton Park over Dinninup. The predicted bodyweight gain of sheep grazing pure Seaton Park and Dinninup swards using Grazfed software indicated that sheep grazing Dinninup were predicted to have a similar bodyweight gain in early growing stage and a significantly lower gain after flowering compared with those grazing Seaton Park. Effect of Feeding Urea Treated Rice and Wheat Straw on Intake and Milk Yield of Lactating Buffaloes under Farmers Conditions Khanal, R.C.;Gurung, D.B.;Kadariya, R.K. 1200 Two experiments were conducted to study the effect of urea treatment of rice and wheat straw on feed intake, dry matter (DM) digestibility and milk yield of lactation buffaloes in their late lactation under farmers' management conditions in the western hills of Nepal during 1995 and 1997. Dry matter intake (DMI) from urea treated rice and wheat straw was not improved significantly (p<0.05) nor the total DMI of the lactating buffaloes was improved significantly. However, feeding urea treated rice straw increased straw DMI by 14.2% and total DMI by 10.63% units over the untreated rice straw. Similarly, the increase in straw and total DMI were 20.18 and 17.40% units over the untreated wheat straw fed animals. Although there was no significant effect of urea treatment of both straw on DM digestibility, it was higher for treated than untreated straw at all locations. An overall increment of 18.1% units for rice straw and 13.3% units for wheat straw was observed. There was a significant effect (p<0.01) of feeding urea treated rice and wheat straw on the milk yield of lactating buffaloes during late lactation under farmers conditions. Post experiment milk yield was also significantly (p<0.05) higher for the animals fed treated straw in both the experiments. Buffalo milk yield was also significantly affected by breed (p<0.01), location (p<0.01) and parity (p<0.01) of the animals. General response of the farmers about the technology and their observed effect on animal performance was also very positive. Determination of Optimal Conditions of Pressure Toasting on Legume Seeds for Dairy Deed Industry : I. Effects of Pressure Toasting on Nutritive Values of Lupinus albus in Lactating Dairy Cows Yu, P.;Goelema, J.O.;Tamminga, S. 1205 Whole lupinus albus seeds were pressure toasted at temperatures of 100, 118 and $136^{\circ}C$ for 3, 7, 15 and 30 min to study rumen degradation and post-rumen digestion and to determine optimal heating conditions for the Dutch dairy feed industry. In sacco nylon bag and mobile bag techniques were employed for rumen and intestine incubations to determine ruminal degradation characteristics and intestinal digestion of crude protein (CP) in 4 lactation rumen cannulated and 4 lactating intestinal cannulated Dutch dairy cows fed 47% hay and 53% concentrate according to Dutch dairy requirements. Measured rumen degradation characteristics were soluble fraction (S), undegradable fraction (U), potentially degradable fraction (D), lag time (T0) and rate of degradation (Kd) of insoluble but degradable fraction. Percentage bypass feed protein (BCP), ruminal microbial protein synthesized based on available nitrogen (N_MP) and that based on available energy (E_MP), true protein supplied to the small intestine (TPSI), truly absorbed BCP (ABCP), absorbed microbial protein (AVP) in the small intestine, endogenous protein losses in the digestion (ENDP), true digested protein in the small intestine (TAP or DVE in Dutch) and degraded protein balance (PDB or OEB in Dutch) were totally evaluated using the new Dutch DVE/OEB System. Pressure toasting decreased (p<0.001) rumen degradability of CP. It reduced S (p<0.05) and Kd (p=0.06), increased D (p<0.05) and U (p<0.01) but did not alter T0 (p>0.05), thus resulting in dramatically increased BCP (p<0.001) with increasing time and temperature from 73.7 (raw) up to 182.5 g/kg DM ($136^{\circ}C/15min$). Although rumen microbial protein synthesized based on available energy (E_MP) was reduced, true protein (microbial and bypass feed protein) supplied to the small intestine (TPSI) was increased (p<0.001) from 153.1 (raw) to 247.6 g/kg DM ($136^{\circ}C/15min$). Due to digestibility of BCP in the intestine not changing (p>0.05) average 87.8%, the absorbed BCP increased (p<0.001) from 62.3 (raw) to 153.7 g/kg DM ($136^{\circ}C/15min$). Therefore DVE value of true digested protein in the small intestine was significantly increased (p<0.001) from 118.9 (raw) to 197.0 g/kg DM ($136^{\circ}C/15min$) and OEB value of degraded protein balance was significantly reduced (p<0.001) from 147.2 (raw) to 63.1 g/kg DM ($136^{\circ}C/15min$). It was concluded that pressure toasting was effective in shifting degradation of CP of lupinus albus from the rumen to small intestine without changing intestinal digestion. Further studies are required on the degradation and digestion of individual amino acids and on the damaging effects of processing on amino acids, especially the first limiting amino acids. Effect of Different Levels of Rumensin in Diet on Rumen Fermentation, Nutrient Digestibility and Methane Production in Cattle Singh, G.P.;Mohini, M. 1215 Twelve rumen fistulated cross-bred calves were divided into three groups and fed wheat straw and concentrate mixture according to their maintenance requirement. Animals in group II and III were fed 50 and 100mg rumensin per day, in addition to basal feed. Supplementation of rumensin in the diet decreased the dry matter intake significantly (p<0.05) along with a significant decrease in the straw intake. Digestibility coefficients of all the nutrients were not affected significantly except that of CF digestibility which was lower (p<0.05) in groups II and III as compared to group I. Among N-parameters in the rumen fluid, mean $NH_3-N$ was significantly lower in groups II and III (19.13 and 18.63 mg N/100 ml respectively) than in group I (22.68); total-N and TCA-ppt-N did not differ among the three groups. Total VFA concentration did also not differ among the three groups, however, propionate increased from 24.33 molar % to 32.73 while acetate and butyrate decreased respectively from 65.85 to 58.81% and 9.79 to 8.46%. Total VFA, bacteria and protozoa production rates were not affected significantly due to rumensin in diet. Methane production per kg DDM as well as % of methane in total gas were reduced at both the levels of rumensin on different concentrate ratios with wheat straw as roughage. Similar trend was also observed with rice straw and concentrate mixture as substrate with rumensin addition. Effect of Season and Fertilizer on Species Composition and Nutritive Value of Native Grasses Khan, R.I.;Alam, M.R.;Amin, M.R. 1222 Effect of three major cropping seasons and five fertilizer treatments on botanical composition, nutritional composition and in sacco digestibility of native grasses grown in 30 experimental plots of a medium fertile land was determined. It was observed that all the major grass species were grown in all seasons but their predominancy of growth was different. During the study the predominant grass species were Panicum repens (Angta), Fimvristylis miliacea (Joina), Cyanolis axillaries (Kanainala), Cynodon dactylon (Durba) and Cyperus iria (Phulchaise) which contributed about 27, 20, 13, 11 and 9% of the total grass yield, respectively. Dry matter (DM) contents was higher in dry followed by monsoon and summer seasons (p<0.05). Crude protein (CP) content in the summer and monsoon appeared to be higher (p<0.05) than that of dry season. Organic matter (OM) and neutral detergent fibre (NDF) were higher (p<0.05) in dry and monsoon than in summer season. Application of urea fertilizer and cowdung increased 28.2% of CP content of the grasses, but decreased 19.5 and 9.8% of DM and NDF contents, respectively. The potential degradation of DM and CP of the grasses grown in summer were 4.1 and 8.4% and 3.9 and 5.8% higher than those of monsoon and dry seasons, respectively, and both of these increased (11.3 and 5.9%, respectively) with the application of cowdung and urea fertilizer. Effect of Stage of Maturity and Cultivars on the Digestibility of Whole Maize Plant and its Morphological Fractions Firdous, R.;Gilani, A.H. 1228 A study was conducted on four maize cultivars to determine the dry matter and fibre digestibility as influenced by advancing plant age. Samples of maize cultivars Akbar, Neelum, UM-81 and IZ-31 were harvested at weekly intervals/ growth stages. The samples of morphological fractions such as leaf and stem were also collected at various growth stages. Whole mixed fodder and different fractions of maize plant were analysed for their chemical composition and in vitro digestibility. The results showed that in vitro dry matter digestibility (IVDMD) of whole maize plant, leaf and stem decreased significantly with advancing stage of maturity. Digestibility of NDF, ADF, hemicellulose and cellulose decreased significantly in all plant parts with advancing plant age/growth stages. Maximum values for the digestibility of dry matter and various cell wall constituents were observed in leaf, followed by whole plant and stem fractions. Cultivars were observed to have significant effect of IVDMD and digestibility of NDF, ADF and cellulose in all plant fractions. The results indicated that digestibility of maize fodder was affected by stage of maturity and cultivars. However, maturity had a greater effect on digestibility in all plant fractions than did cultivars. Dry matter contents were found to be significantly and negatively correlated with IVDMD of whole plant and its leaf and stem fractions. Based on correlations, regression equations were computed to predict IVDMD. The Utilization of Rumen Content-Barley Meal in Diets of Growing Lambs Abouhief, M.A.;Kraidees, M.S.;Al-Selbood, B.A. 1234 The nutritive value of rumen contents and barley mixture (4:1 w/w; RCB) was evaluated and the effect of their feeding on growth performances in Najdi lambs was studied. A metabolism trial was conducted with 16 rams divided into four dietary groups. The diets were: a whole-mixed control diet and three diets where RCB was incorporated at the rates of 25, 50 and 100%, replacing an equal amount of control diet. The results showed that there was a depression (p<0.05) in DM digestibility for the 100% RCB diet in comparison with other diets. The digestibility of CP was higher(p<0.05) for the 25% RCB diet as compared to the control diet; there was a trend for a small (p>0.05) decrease in digestibility as level of RCB increased. Lambs in all studied diets were in positive nitrogen balance; the differences between diets were not significant. A total of 45 lambs were allotted into three groups and used to evaluate the effect of dietary inclusion of RCB (0, 25 and 50%) on growth performance and carcass traits. Daily DM intake, final body weight, carcass weight and dressing percentage were not different among treatments. Average weight gain and ether extract (EE) in 9-11th control joint were higher (p<0.05) in lambs fed control diet than those fed RCB diets. The substitution of RCB for 50% of control diet exhibited 11.8% reduction in feeding cost for each kg of body weight gain. Influence of Controlling Protozoa on the Degradation and Utilization of Dietary Fibre and Protein in the Rumen and Nitrogenous Flow Entering the Duodenum of Sheep Han, C.Y.;Lu, D.X.;Hu, M.;Tan, Z.L. 1241 Nine two-year old sheep fitted with rumen and duodenum cannulas were used to study the effect of controlling protozoa flora on the degradation and utilization of dietary fibre and protein in the rumen and on nitrogenous flow to the duodenum. There were three groups in this experiment: defaunation (DF); partial defaunation (PDF); faunation (F) as control. Results showed that: 1,There were no differences between treatments in dietary DM degradation in the rumen, but defaunation and partial defaunation increased the quantity of nitrogenous material in the rumen and the flow of N to duodenum. 2, partial defaunation and defaunation improved the degradabilities of dietary NDF, ADF and HC, but there were no differences between the defaunated and partially defaunated groups. 3, Partial defaunation decreased the degradability of dietary protein in the rumen. There was no difference between defaunated and faunated groups. 4, Defaunation and partial defaunation increased the quantity of total N (TN) and microbial N (MCN) in the rumen and the amounts entering the duodenum. The protozoa N (PN) flow in the faunated group was higher than that in the partially defaunated group, and the amino acid pattern in the digesta at the proximal duodenum in the defaunated group was closer to the ideal amino acid pattern. 5, There were differences in the mole percent of acetic, propionic, total-VFAs and the non-glucogenic to glucogenic VFAs ratio (NGR) value in the rumen fluids. The order was as follows: mole percent of acetate: F>PDF>DF; mole percent of propionate: DF>PDF>F; total-VFAs: PDF>F>DF; NGR: F>PDF>DF. Effect of Graded Dietary Levels of Neem (Azadirachta indica) Seed Kernel Cake on Carcass Characteristics of Broiler Rabbits Vasanthakumar, P.;Sharma, K.;Sastry, V.R.B.;Kumar, S. 1246 Rabbits (48) of Soviet chinchilla (24) and White giant (24) were fed from 6 weeks to 12 weeks of age intensively on either of four isonitrogenous - isocaloric diets containing 0 ($D_1$), 5($D_2$), 10($D_3$) and 20($D_4$) percent raw neem seed kernel cake (NSKC), respectively as per NRC (1977) requirements in a Randomized block design and slaughtered at the end to find out differences in their carcass traits due to NSKC feeding. Dietary treatment had no significant effect on weight of edibles and inedibles and their percentages and dressing percentage in terms of carcass, carcass with pluck and carcass with pluck and head. Similarly, the meat-bone ratio of various primal cuts and overall carcass, yield of edibles per unit of inedibles and eye muscle area were not influenced due to the dietary variations. Chemical composition of fresh meat, and organoleptic evaluation of cooked meat with and without salt did not vary significantly due to incorporation of NSKC in the diets. The rabbits fed 20% NSKC ($D_4$) though consumed more (p<0.05) DM and DE per kg meat production, the intake of crude protein and total digestible nutrients was similar with other dietary treatments. Feed cost per unit meat production was, however, lower on 5 and 10% NSKC containing diets by 7.75 and 12.56%, respectively, as compared to deoiled ground nut cake containing control diet. It appears that NSKC could be used as a wholesome vegetable protein supplement upto 10% in diet of rabbits without any adverse effect on commercial carcass traits. Extrusion Processing of Low-Inhibitor Soybeans Improves Growth Performance of Early-Weaned Pigs Kim, I.H.;Hancock, J.D.;Jones, D.B.;Reddy, P.G. 1251 Two experiments were conducted to determine the effects of roasting and extrusion on nutritional value of conventional and low-inhibitor soy beans for nurser-age pigs. In Exp. 1, 100 weaning pigs (7.5 kg average initial BW) were used in a 35-d growth assay to determine the effects of processing method (roasting in a Rast-A-Tron$^{TM}$ raster vs extrusion in an Insta-Pro$^{TM}$ extruder) on the nutritional value of Williams 82 soybeans with (+K) and without (-K) gene expression for the Kunitz trypsin inhibitor. Treatments were 48% soybean meal with added soybean oil, +K roasted, +K extruded, -K roasted and -K extruded. All diets were formulated to contain 3.5 Mcal DE/kg, with 0.92% lysine for d 0 to 14 and 0.76% lysine for d 14 to 35 of the experiment. The lysine concentrations were 80% of NRC (1988) recommendations to accentuate difference in response to protein quality and lysine availability. For d 0 to 14, pigs fed extruded soybeans (+K and -K) had greater ADG (p<0.001), ADFI (p<0.09) and gain/feed (p<0.01) than pigs fed roasted soybeans. For d 14 to 35 and overall, the same effects were noted, i.e., pigs fed extruded soybeans had greater ADG, ADFI and gain/feed than pigs fed roasted soybeans (p<0.03). Also, pigs fed -K soybeans were more efficient (p<0.008) than pigs fed +K soybeans. In Exp. 2, 150 weanling pigs (7.0 kg average initial BW) were used in a 35-d growth assay. All diets were formulated to contain 3.5 Mcal DE/kg, with 1.25% lysine for d 0 to 14 and 1.10% lysine for d 14 to 35 of the experiment. The lysine concentrations were formulated to be in excess of NRC recommendation to determine if differences in nutritional value of the soybean preparations could be detected in protein-adequate diets. For d 0 to 14 (p<0.06), 14 to 35 (p<0.03) and 0 to 35 (p<0.02), pigs fed extruded soybeans had greater ADG and gain/feed than pigs fed roasted soybeans. Apparent digestibilities of DM, N and GE were greater for diets with extruded soybeans than diets with roasted soybeans and diets with soybean meal and soybean oil were intermediate. The response to extrusion processing was greater with -K than +K soybeans, with pigs fed extruded -K soybeans having the greatest growth performance and nutrient digestibilities and lowest skin-fold thickness of any treatment. In conclusion, extrusion yielded a full-fat soy product of greater nutritional value than roasting. Also, selection against genetic expression of the Kunitz trypsin inhibitor improved nutritional value of the resulting soybean preparations. The Lipogenic Capacity of Hepatocytes and Lipolytic Rate of Adipocytes in Tsaiya Ducks during Growing and Laying Periods Lien, T.F.;Jan, D.F. 1258 With an attempt to elucidating the lipid metabolism of Tsaiya ducks, thirty ducks at growing (8 weeks of age) and laying periods (10 weeks after the onset of laying) were examined, respectively. The ducks were randomly allocated into ad libitum feeding and 3-day fasting groups, to investigate their in vitro hepatocytes lipogenesis capacity and adipocytes lipolysis rate. Results indicate that (1) the capacity of hepatocytes incorporation of glucose and acetate into total lipid and metabolite of $^{14}CO_2$ production during the laying period was greater than during the growing period. Approximately 50% of the glucose or acetate converted into triacylglycerol (TG) by the hepatocytes were recovered as fatty acid during the growing period, while it was 65-70% during the laying period. (2) Acetate used for lipogenesis ability was superior to glucose in both periods. (3) The adipocytes lipolysis rate was increased significantly (p<0.05) by fasting. In contrast, the capacity of incorporated glucose or acetate into total lipid, triacylglycerol, fatty acid and glycerol by hepatocytes was reduced significantly (p<0.05) by fasting. Effect of L-Carnitine and Source of Dietary Fat on Growth Performance and Serum Biochemical Parameters of Piglets Weaned at 35 Days of Age Li, Defa;Qiao, Q.;Johnson, E.W.;Jiang, J.;Wang, F.;Blum, R.;Allee, G. 1263 The effects of carnitine in diets with or without added fat (5% lard or soybean oil) were evaluated in 72 Large White ${\times}$ Landrace ${\times}$ Duroc pigs weaned at 35 days of age. Pigs were fed a 1.30% lysine corn-soybean basal diet+15% dried whey+4% fish meal with carnitine at 0 or 50 mg/kg and either 0% added fat, 5% soybean oil or 5% lard for 6 weeks in a $2{\times}3$ factorial trial (6 treatments, 3 pens per treatment, 4 pigs per pen). Addition of carnitine increased average daily gain (ADG) and average daily feed intake (ADFI) in the second two weeks of the six-week trial and overall, but had no significant effect on feed per gain (F/G). Lard alone depressed ADG (p<0.05) in the last two weeks of the trial and overall, but the ADG for pigs fed lard+carnitine was similar to the control. Lard reduced feed intake in the first two weeks of the trial (p<0.05). Carnitine reduced the percentage of pigs with poor (ADG<375 g/d) growth (15 vs 40%; p<0.05). The greater uniformity of growth was most evident in low-weaning-weight pigs in the second period (16 vs 62%, p<0.005). Addition of fat did not produce any positive effect on uniformity and had no interaction with carnitine on uniformity. Carnitine addition increased serum total carnitione and short-chain acyl-carnitine levels (p<0.05), but did not modify free carnitine levels. Serum carnitine levels were lower at weaning than at 14, 28, or 39 days after weaning (p<0.05). Carnitine increased serum protein levels on day 14 (p<0.05). Addition of fat in the form of soybean oil or lard did not improve piglet growth performance. Addition of 50 mg/kg of carnitine to the diet of weanling pigs enhanced postweaning performance. Thermoregulatory Responses of Swamp Buffaloes and Friesian Cows to Diurnal Changes in Temperature Koga, A.;Kurata, K.;Furukawa, R.;Nakajima, M.;Kanai, Y.;Chikamune, T. 1273 Several reports have indicated that a rectal temperature of buffaloes is easily influenced by their surroundings. To clarify an effect of changing environmental temperature on thermoregulatory responses of buffaloes, an environment with diurnal temperature changes of $25^{\circ}C$ to $35^{\circ}C$ was created using an artificial climate laboratory. Three swamp buffaloes and three Friesian cows were exposed to three different experimental periods as follows: Period 1 (constant temperature of $30^{\circ}C$, Period 2 (diurnally changing temperature) and Period 3 (diurnally changing temperature and fasting). Heat production, rectal temperature, respiration rate, heart rate and respiration volume were measured during each period. Rectal temperature of the buffaloes fluctuated diurnally with the changing temperature (Periods 2 and 3), but remained constant in cows. Mean heat production was significantly lower in buffaloes than in cows in Period 2 and 3. However, the maximum rectal temperature and the increment of heat production were not always lower in buffaloes than in cows during Period 2. These results show that a rectal temperature and heat production in buffaloes are markedly influenced by the diurnal changes in temperature. Compared with Bos Taurus cows, the differences may be attributed to the physiological features of buffaloes including a high heat conductivity of their bodies and an lower heat production. Effect of Plant Fibre on the Solubility of Mineral Elements Ibrahim, M.N.M.;Zemmelink, G. 1277 Eight feeds and their residues left after washing with tap water (water residue) or incubation in the rumen (rumen residues) were treated with hydrochloric acid, neutral detergent solution without EDTA (NDS) or both, and the release or sorption of minerals (Ca, Mg, P, Na, K, Cu and Zn) assessed. Six of the feeds were from Sri Lanka (Panicum maximum ecotype Guinea A, Glyricidia maculate, Artocarpus heterophyllus (jak leaves), untreated and urea-treated rice straw, and rice bran) and two from the Netherlands (maize silage and wheat straw). The initial concentration of mineral elements, the concentration of neutral detergent fibre (NDF) and the type of feed significantly influenced (p<0.01). The proportion of the mineral elements released or sorbed. In general, feeds with high NDF content (straws and guinea grass) sorbed Ca from tap water, or released less in the rumen, and within these feeds the extent of sorption varied with source of fibre. Acid or NDS treatment removed little of the sorbed Ca, but they removed much of the Mg from both water and rumen residues. Fibres of wheat straw and jak leaves showed an affinity for Mg in the rumen. All feeds and their water and rumen residues sorbed P and Na from NDS, and the extent of sorption varied with the initial concentrations of these elements and with the type of fibre. Acid treatment removed part of the sorbed Na, but not the P. The solubility of K was not affected by the content of NDF, the type of fibre or the initial concentration of K. All feeds and their residues, except for the rumen residues of rice bran sorbed Cu from tap water and in the rumen. The recovery of Cu in rumen residues declined from 353% to 147% after NDS treatment, and with some feeds (glyricidia and jak leaves) the recovery was below 100%. Acid treatment removed part of the Zn sorbed by the water and rumen residues, but the capacity of residues to retain Zn varied with the type of feed. Nutrient Utilization and Compensatory Growth in Crossbred (Bos indicus×Bos taurus) Calves Santra, A.;Pathak, N.N. 1285 A feeding trial was carried out over 238 days to determine the effect of compensatory growth in crossbred calves having 166 kg body weight. Fifteen crossbred calves were divided into two groups of five calves (G1 group) and ten calves (G2 group) as per randomized block design. Growth study was conducted on the feeding of wheat straw based diet containing 60 and 30 percent concentrate supplying equal amount of protein in group G1 and G2 respectively for 119 days (phase - I). At the end of phase-I, calves of G2 group were subdivided in to two groups (G3 and G4). One sub group (G4) received 60% concentrate in their diet (during 120 to 238 days of experiment) while other subgroup G3 received 30% concentrate in their diet (phase-II). The calves of G1 group continued to receive the same diet as during phase-I experiment. Mean DM intake was significantly higher in calves fed high level of concentrate (in G1 and G4 groups), which resulted in significantly higher digestibility of all nutrients except NDF. Nitrogen balance was positive in all the groups and showed significant differences in phase-II (higher nitrogen retention in G4 group than G1 group). ME intake was significantly affected by the level of dietary concentrate, being higher in high concentrate fed group (G1 and G4 than G2 and G3 group). Higher daily body weight gain in the calves of G4 group during phase-II than in G1 and G3 groups was due to compensatory growth on shifting animals from low concentrate to high concentrate based ration. Average daily body weight gain was higher in phase-I than in the phase-II. Protein and energy intake per unit body weight gain were significantly lower in calves fed high concentrate diet. Preference Test on Feed and Nutrient Intakes in Male and Female Lesser Mouse Deer (Tragulus Javanicus) in Captivity Darlis, N. Abdullah;Liang, J.B.;Jalaludin, S.;Ho, Y.W. 1292 A preference test on feed and nutrient intakes were conducted on four male ($1.25{\pm}0.08kg$) and four female ($1.21{\pm}0.15kg$) lesser mouse deer (Tragulus javanicus) in captivity. Each animal was kept in individual cages placed in a well-ventilated animal house. The experiment was conducted in two weeks, where the first week was for adaptation to the feeds and the second week for measurements of nutrient intake, nutrient digestibility and nitrogen balance. The feeds offered were kangkong (Ipomoea aquatica), long bean (Vigna sinensis) and french bean (Phaseolus vulgaris) as roughages and proteinaceous feeds; sweet potato (Ipomoea batatas) and carrot (Daucus carota) as carbohydrate-rich feeds; and commercial rabbit pellet (0.3 cm diameter and 0.5 cm long) as a complete feed. The dry matter (DM) content of each feed in the order mentioned above was 7.1, 6.1, 3.9, 18.5, 6.2 and 87.6%, respectively. Long bean had the highest protein (CP) content (29.7%), while sweet potato had the lowest (6.2%). The CP contents of other feeds were within the range of 14.2 - 25.1%. Among the feeds, carrot had the lowest energy content (3.83 kcal/g) and long bean the highest (4.67 kcal/g). When fresh weight of the feed was considered, the male mouse deer consumed sweet potato the most ($86.3{\pm}12.90g/d$), but the female had a high preference for carrot ($79.2{\pm}9.76g/d$). The other feeds were consumed in lesser amounts. However, in terms of DM of the feed, the amount of commercial pellet consumed was the highest for both male ($45.0{\pm}5.10%$) and female ($44.7{\pm}7.38%$) mouse deer, followed by sweet potato ($33.1{\pm}4.43%$ and $22.4{\pm}7.73%$ for male and female, respectively). Significant (p<0.05) differences in DM, organic matter (OM) and gross energy (GE) intakes were observed between male and female mouse deer. The male consumed higher amount of DM, OM and GE than the female. The total DM intake was $40.7{\pm}2.24g/d/kg$ $W^{0.75}$ for male and $35.9{\pm}1.72g/d/kg$ $W^{0.75}$ for female mouse deer. Percentage digestibilities of DM, OM, CP and GE were within 72.7~80.8% and were not significantly different between male and female mouse deer. However, male mouse deer had significantly (p<0.05) higher digestible DM, OM and GE intakes than the female. Both male and female mouse deer were in positive nitrogen balance (0.6 g N/d/kg $W^{0.75}$). The male mouse deer gained $7.6{\pm}3.45g/d$, while the female gained $4.3{\pm}2.40g/d$. Recent Advances in Amino Acid Nutrition for Efficient Poultry Production - Review - Ishibashi, T.;Ohta, Y. 1298 The nutritional value of protein varies between feedstuffs. It is possible to feed animals using crystalline amino acids as a sole nitrogen source, but in practice only some limiting amino acids are added to the diet. In order to use feedstuffs efficiently, it is important to determine exact amino acid requirements. Reported values differ widely because the requirements are affected by various factors. In this report, therefore, the factors affecting amino acid requirements are reviewed as follows: 1) availability of dietary amino acids, conversion factors of nitrogen to protein, interaction of amino acids, and strain, sex and age of animals; 2) amino acid requirements for maximum performance and maintenance, usefulness of non-essential amino acids; 3) plasma amino acid concentration as a parameter to determine amino acid requirements; and 4) nitrogen excretion to reduce environmental pollution. These factors should be considered, it is to improve the dietary efficiency, which is to reduce excess nitrogen excretion for environmental pollution. Better Housing for Effective Pig Production - Review - Choi, H.L.;Song, J.I.;An, H.K. 1310 Air quality in confinement pig houses is important to production and health. Mechanical ventilation and confinement is known to be the most practical tool for maintaining adequate air quality in pig houses through extensive researches since Millier (1950) invented the 'slotted inlet' ventilation system. A variety of mechanical ventilation systems have been applied to confined nursery pig houses in Korea without scientific verification of their ventilation effectiveness. Ventilation systems with three feasible combinations (NA, NB, and NC) of inlets and outlets in a confined nursery pig house were tested to evaluate their ventilation efficiency, of which the one with the performance was supposed to be taken as a standard ventilation system for nursery pig houses in Korea. Field data of air velocity and temperature fields, and ammonia concentration with three ventilation systems were taken and compared to determine the best system. The air velocity and temperature fields predicted by the PHOENICS computer program were also validated against the available experimental data to investigate the feasibility of computer simulation of air and temperature distribution with an acceptable accuracy in a confined house. NC system with duct-induced in-coming air, performed best among the three different ventilation systems, which created higher velocity field and evener distribution ($2.5m/s{\pm}0.3m/s$) over the space with a Reynolds number of $10^4$. The experimental data obtained also fitted well with the simulated values using the modified PHOENICS, which suggested a viable tool for the prediction of air and temperature field with given calculation geometries. Dairy Cows of High Genetic Merit for Yields of Milk, Fat and Protein - Review - Norman, H.D.;Powell, R.L. 1316 Extensive emphasis on milk and milk fat yields with no diversion for beef performance increased the yield efficiency of North American dairy cattle. Heavy demand for North American genetics followed national strain comparison trials in Poland, and US and Canadian dairy cattle and germplasm still are an important source of genetics for many countries. Genetic improvement has accelerated in many countries because of the implementation of sampling programs for young bulls and improved evaluation procedures. Rapid access to information and more frequent calculation of genetic information also are having a positive impact on genetic improvement. Traits other than yield should be considered in a breeding program, but those traits mist have a reasonable opportunity for improvement and sufficient economic worth. Because of ever increasing efficiency, the world's milk supply comes from fewer cows each year. However, no decline in the rate of genetic improvement is apparent under current genetic practices; estimates of heritability are increasing, and a decline in yield efficiency is unlikely in the near future. As management improves, especially for subtropical conditions, many of the selection principles used in temperate climates will be adopted for more adverse environmental conditions. Use of Water Buffalo for Environmental Conservation of Waterland - Review - Georgoudis, A.G.;Papanastasis, V.P.;Boyazoglu, J.G. 1324 The aim of this paper is to propose the preservation of buffaloes not only as productive livestock, but also as a part of the biodiversity of wetlands and especially of the Greek wetlands. The water buffalo used to be an integral part of the biodiversity of many Greek wetland ecosystems, enriched their landscape, and provided invaluable services and products to the rural people and to the economy in general. Its total population before the 1950s was over 100,000 animals. Presently, it is found only in four wetland sites in Macedonia and Thrace and in the estuaries of Rivers Gallikos and Axios, with a total population of a few hundred animals. These wetlands are Ramsar Sites. Even this small population is threatened with immediate extinction because of the rapidly changing rural socio-economic conditions and the expansion of cultivated fields into wet meadows. Farmers and consumers are rapidly losing contact with this mammal and its products. This species possesses minimum requirements for treatment and is characterized by the ability of utilizing roughage of variable nutritional value. These factors are promising to render buffalo breeding a valuable branch of the Greek livestock sector, which can also contribute to the maintenance of the wetlands.	CommonCrawl
Topics in Catalysis September 2016 , Volume 59, Issue 15–16, pp 1438–1457 \| Cite as H2 Production via Ammonia Decomposition Using Non-Noble Metal Catalysts: A Review T. E. Bell L. Torrente-Murciano The wide-spread implementation of the so-called hydrogen economy is currently partially limited by an economically feasible way of storing hydrogen. In this context, ammonia has been commonly presented as a viable option for chemical storage due its high hydrogen content (17.6 wt%). However, its use as an energy carrier requires the development of catalytic systems capable of releasing hydrogen at adequate rates and conditions. At the moment, the most active catalytic systems for the decomposition of ammonia are based on ruthenium, however its cost and scarcity inhibit the wide scale use of these catalysts. This issue has triggered research on the development of alternative catalysts based on more sustainable systems using more readily available, non-noble metals mainly iron, cobalt and nickel as well as a series of transition metal carbides and nitrides and bimetallic systems, which are reviewed herein. There have been some promising cobalt- and nickel-based catalysts reported for the decomposition of ammonia but metal dispersion needs to be increased in order to become more attractive candidates. Conversely, there seems to be less scope for improvement of iron-based catalysts and metal carbides and nitrides. The area with the most potential for improvement is with bimetallic catalysts, particularly those consisting of cobalt and molybdenum. Hydrogen storage Sustainable catalysts Ammonia decomposition Cobalt Iron Nickel Bimetallic It is generally accepted, not only by the scientific community but also by politicians, environmentalists and the general public that our current energy system based on fossil fuels cannot sustain the predicted trends in the growth of the population and the increase of energy demand per capita without damaging our environment and contributing to global warming [1]. These complex and interrelated global challenges are continuously motivating the search for new energy sources and vectors. In this framework, hydrogen is often presented as an attractive alternative. Hydrogen is a clean energy vector for portable applications using fuel cells, with water as the only by-product of combustion [2]. It can be produced sustainably via water splitting using surplus renewable energy to balance the grid demands. However, the potential of the called "hydrogen economy" is currently limited by the inability to store hydrogen in a safe and economical manner with a sufficiently high density without using controversial high pressures [3]. In 2015, the US Department of Energy (DoE) released targets for the physical storage of hydrogen. These include: high storage capacity (9 wt% hydrogen content, 81 g L−1 volumetric capacity as shown in Fig. 1), low cost, operational temperatures below 60 °C, rapid system filling and inert and non-toxic materials [4, 5]. Hydrogen can be stored in high pressure cylinders, although there is an inherent cost associated to this technique as well as the risk of explosion with poor public acceptance. Indeed, the safety of the storage and transportation methods of hydrogen is a concern which is currently slowing the widespread uptake of the existing hydrogen fuel cell technologies. Furthermore, as shown in Fig. 1 pressurised hydrogen at 350 bar and 700 bar does not meet the 2015 US DoE targets in terms of volumetric and gravimetric capacity. Volumetric versus gravimetric hydrogen density for various hydrogen storage compounds. The US Department of Energy targets for hydrogen storage are shown by dashed lines [5, 6, 7, 8, 9, 10]. Circle hydrogen under different conditions, triangle hydrocarbons, pentagon materials for H2 physisorption, right angle triangle metal hydrides, diamond water, square ammonia and related compounds A more attractive way of physically storing hydrogen is by its adsorption in porous materials such as zeolites, porous carbons, microporous polymers and metal organic frameworks (MOFs) [1, 5, 6]. In the last decade, there has been a great progress in this area with the development of MOFs, achieving the gravimetric targets of the US DoE. However, these materials are currently unable to uptake and release hydrogen at the demanded rates and in most cases, the adsorption processes require very low operational temperatures (e.g. −196 °C for 7.5 wt% H2 adsorption on MOF-177) [1, 7]. Aside from physical methods, hydrogen can also be chemically stored in molecules such as methanol, methane, metal amine salts (e.g. Mg(NH3)6Cl2), ammonia and related compounds (e.g. NH3BH3) or in hydrides (interstitial H as in LaNi5H6 or complex hydrides such as NaAlH4) [7, 10]. As shown in Fig. 1, most of these hydrogen-rich molecules meet the 2015 US DoE targets for hydrogen storage. Out of these, ammonia has highly attractive chemical and physical properties as a carbon-free hydrogen vector, containing a significantly higher amount of hydrogen than liquefied hydrogen on a volumetric and gravimetric basis [8]. Additionally, from a safety point of view, ammonia has a relatively narrow combustion range of 16–25 % in air, compared with 4–75 % for H2 [11, 12]. Although the toxicity of ammonia may be a concern for specific uses, its strong smell is useful for identifying leaks or alternatively this issue can be completely overcome by the use of metal ammines such as Mg(NH3)6Cl2 or Ca(NH3)8Cl2 [7, 11]. Furthermore, ammonia can be liquefied at low pressure of 10 bar at 298 K, facilitating its transport and storage [13]. 1.1 Hydrogen Production via Ammonia Decomposition The production of COx-free hydrogen via ammonia decomposition (Reaction 1) for its use in a proton exchange membrane fuel cell (PEMFC) was first proposed by Green [14] in 1982. It is important to note that this is a reversible reaction, thermodynamically limited at low temperatures. Any NH3 remaining in the inlet stream to the PEMFC can potentially damage the Nafion™ polymer membrane so a robust separation method is required for the pre-purification of hydrogen [7]. Alternatively, an attractive process is the use of membrane reactor as reported [15, 16]. $$2 {\text{NH}}_{ 3} \left( {\text{g}} \right) \, \rightleftarrows {\text{ 3H}}_{ 2} \left( {\text{g}} \right) \, + {\text{ N}}_{ 2} \left( {\text{g}} \right)$$ Catalytic cracking or decomposition of ammonia is the reverse reaction of the Haber–Bosch synthesis of ammonia, one of the most extensively researched processes over the past 150 years [17]. Ammonia is commonly used in the production of fertilisers and household cleaning products and therefore has well-established protocols for its handling and usage and a safe existing transportation and distribution network [12]. In order to benefit from the absence of COx emissions associated to hydrogen as a fuel in PEMFC, the whole process, from production to consumption needs to be COx free [10]. At the moment, ammonia is produced globally at a large scale of over 100 million tonnes annually, mainly from fossil fuels but new research is demonstrating its sustainable production from heat waste and renewable electricity sources such as solar, wind, hydro or geothermal energy in combination with air and water or biomass or organic waste [10, 12]. A similar important challenge, which is the subject of this review, is the delivery of hydrogen from ammonia. The US Department of Energy has clearly indicated that the feasibility of ammonia as hydrogen storage molecule relies on its decomposition at temperatures aligned with those of the PEM fuel cell, in the range of 150–180 °C making necessary the development of catalysts active under these conditions [18]. To date, the most effective catalyst for ammonia decomposition consists of ruthenium particles supported on carbon nanotubes (CNT) due to their high conductivity (6353 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ru}}^{-1}}{\text{h}}^{-1}$ at 430 °C) [2, 17]. The low temperature activity can be further improved by the addition of an electron donating promoter such as cesium (7870 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ru}}^{-1}}{\text{h}}^{-1}$ at 370 °C) [3, 17, 19, 20]. Furthermore, the synergetic effect of cesium and the graphitisation of the carbon nanotubes support on the ruthenium nanoparticles have recently enabled the decomposition of ammonia at temperatures as low as 180 °C, representing a breakthrough in the field [17]. Despite the excitement of these results, the cost and scarcity of ruthenium and cesium as shown in Fig. 2 is likely to limit the economic feasibility of the in situ production of H2 from ammonia in PEMFC using Ru-based catalysts. Thus, the scientific community has been working on the development of alternative catalysts based on more readily available and more sustainable metals. Progress in this field to date is the focus of this review, including both non-noble monometallic (Fe, Co, Ni) and bimetallic systems (based on Fe, Co, Ni) based on elements with a high annual production (Fig. 2). Global production of elements in 2010 as a function of atomic number. Elements shown in blue are not obtained directly. Reproduced from Ref. [21] with permission from The Royal Society of Chemistry 1.2 Mechanism of Ammonia Decomposition Over Heterogeneous Catalysts In order to understand the superior activity of ruthenium-based catalysts, it is important to consider the reaction mechanism of ammonia decomposition, which is initiated by the adsorption of ammonia onto the active site surface. The adsorbed ammonia molecule undergoes successive N–H bond cleavage, releasing hydrogen atoms than can combine to form molecular hydrogen. The final step involves the recombinative desorption of nitrogen adatoms to yield molecular nitrogen [22]. Interestingly, even though the decomposition of ammonia is the reverse of the synthesis, catalyst do not necessarily exhibit the same activity in both directions due to a difference in rate limiting step as discussed in the work by Boisen et al. [23]. A study by Chellappa et al. [24], demonstrates that the kinetics of ammonia decomposition vary depending on the temperature and the concentration of ammonia. Across the temperature range of 520–690 °C with ammonia pressure from 7 to 104 MPa and high ammonia concentration, the reaction is said to be first order with respect to the ammonia concentration. Work by Ganley et al. [25] highlighted that at 580 °C, the rate limiting step depends on the metal component of the catalyst with the nitrogen desorption step rate limiting on iron and cobalt systems whereas the N–H bond scission step limits the kinetic rate on rhodium, iridium, palladium, platinum and copper. Based on their results, no distinction was made for the rate limiting step when using ruthenium and nickel based catalysts. However, irrespective of the metal, at low temperatures nitrogen desorption is rate limiting as demonstrated by Wang et al. [26] using NH3 tracking experiments to confirm that strongly bound N adatoms limit the rate of NH3 decomposition. Consequently the metal-N binding energy is a key parameter in the design of catalysts for the low temperature ammonia decomposition. While the ammonia molecule needs to be adsorbed on the active metal to be activated, strongly adsorbed N adatoms would, on the other hand, poison the metal active sites. Therefore, there exists an optimum nitrogen binding energy for ammonia decomposition catalysts within the range of 544–586 kJ mol−1, with the optimum activity observed at 561 kJ mol−1, lower than that for ammonia synthesis catalysts [27, 28, 29]. As shown in Fig. 3, the ammonia decomposition rate of different metals and their nitrogen binding enthalpy, presents a volcano-type relationship, where ruthenium has the optimum value. This relationship constitutes one of the fundamental guidelines used in the development of alternative catalysts to the ruthenium-based ones. It is proposed that the nitrogen binding enthalpy of the ruthenium could be mimicked with stable bimetallic systems or suitable combinations of metal, promoters and support. Top Relationship between ammonia synthesis (0.02 % NH3) and decomposition (99 % NH3) TOF and nitrogen desorption energy. The bold line shows ammonia decomposition TOF with 20 % NH3 inlet against N binding energy. The straight line shows the optimal value for nitrogen binding energy for 20 % ammonia gas composition. Bottom Experimental ammonia decomposition rate over various catalysts at 500 °C, 1 bar, 3:1 H2:N2 and 20 % NH3. Reproduced directly with permission from Boisen et al. [23] The two dashed lines in the top of Fig. 3 shows the difference of the reaction rate for ammonia synthesis (0.02 % NH3) and ammonia decomposition (99 % NH3) with respect to the metal binding energy. By comparing top and bottom of Fig. 3, it can be seen that the optimal decomposition curve is closer to Co whereas the optimal ammonia synthesis catalyst is closer to Fe in terms of nitrogen binding energy, explaining the difference in decomposition activity between Co and Fe [23]. 2 Monometallic Systems A wide range of monometallic catalytic systems have been tested for the hydrogen production via ammonia decomposition. The catalytic activity is highly dependent on the choice of metal component, the catalytic support and the potential use of promoters as well as the ammonia decomposition conditions used. Taking this into consideration, the general activity trend of monometallic systems supported on activated alumina is Ru > Ni > Rh > Co > Ir > Fe ≫ Pt > Cr > Pd > Cu ≫ Te, Se, Pb [25]. It is important to notice that the resulting activity also depends on the catalyst structure and the active site configuration in order to anchor the ammonia molecule as well as the presence of vacant sites for the release of N and H atoms [20]. Out of these metals, this review is only going to focus on those metals widely available with a focus on iron, cobalt and nickel. Some other ammonia decomposition catalysts such as transition metal carbide and nitride systems will also be considered in this section. It is important to mention that transition metal catalysts are reported to be deactivated by sulphur impurities, resulting in the need for a pre-desulphurisation step if the ammonia feed contains sulphur. Alternatively, research has also focused on the development of sustainable catalysts that are not prone to sulphur poisoning such as the red mud catalyst reported by Uemiya et al. [30], which was believed to be resistant to sulphur poisoning due to the presence of FeCx. Recent work by Wang et al. [31] studied the synergy between plasma and Fe, Co and Ni catalysts for ammonia decomposition, reporting at least a five-fold improvement in catalytic activity when plasma was combined with the catalyst. A DFT theoretical study by Duan et al. [32] calculated the activation energy and adsorption energies of reaction intermediates on Fe (110), Co (111) and Ni (111) close-packed surfaces during ammonia decomposition. The results showed that the adsorption energy of NH3 onto Co and Ni is lower than on Fe and Fe has the highest activation energy, agreeing with experimental observations of iron-based catalysts generally having the lowest activity compared to cobalt- and nickel-based catalysts. 2.1 Iron-Based Catalysts Iron-based catalysts are an extensively studied monometallic system for ammonia decomposition, due to its industrial use in the ammonia synthesis reaction. In this context, it is important to note that the most popular commercial catalysts for the large-scale production of ammonia are based on iron promoted with K2O, CaO, SiO2, and Al2O3, active at temperatures above 400 °C [29]. While most of the fundamental research regarding catalyst development for ammonia decomposition is on ruthenium-based systems, the bulk price of iron is 50,000 times lower than that of ruthenium, giving rise to an obvious significant cost benefit associated to the development of effective iron-based catalysts with similar activity than ruthenium ones [18]. The relatively lower activity of iron with respect to ruthenium-based catalysts can be explained by the stronger bond enthalpy of Fe–N compared to Ru–N which can lead to the formation of surface nitrides which slows down the reaction rate and eventually deactivates the catalysts by poisoning. It is known that iron forms stable nitrides and although industrial nitridation usually takes place in the presence of ammonia at 600 °C, it has also been reported from temperatures as low as 300 °C [13]. This deactivation process is reversible at high reaction temperatures where desorption of the nitride species takes place but it is usually accompanied by sintering of iron, which deactivates the catalyst irreversibly [33]. The simultaneous occurrence of ammonia decomposition and nitridation of the iron active species was addressed by Arabczyk and Pelka [34, 35] by comparing the decomposition activity of Fe and Fe4N. The activation energy of the latter was approximately double compared to the former highlighting that surface nitrides formation is undesirable during ammonia decomposition with a consequent reduction of the reaction rates. Detection of FeNx as the predominant active phase on the surface was identified by a combination of high resolution TEM, elemental mapping and electron energy loss spectroscopy (EELS). There are numerous reports focused on the kinetics of ammonia decomposition using iron-based catalysts. Generally, it is accepted that the nitrogen associative desorption is the rate limiting step [36, 37, 38]. However, several studies highlight that the limiting step varies depending on the reaction conditions (mainly temperature) due to the aforementioned formation of nitrides on the iron surface. In this context, Takezawa and Toyoshima [39] reported that while the rate limiting step at low temperatures (<479 °C) is the nitrogen desorption, at higher temperatures (>479 °C), N–H scission dictates the rate of reaction. A density functional theory (DFT) study by Lanzani and Laasonen [40] considered the mechanism of ammonia decomposition over a Fe55 cluster. They computed stable geometries of ammonia, N and H adsorbed to the cluster, mapping the energy landscape of the reaction mechanism. Interestingly, their results suggested that the first N–H scission step is rate limiting, contrary to the experimental kinetic studies at low temperature. They propose that the rate of decomposition of ammonia is faster with nano-sized Fe, in agreement with experimental observations as discussed in the follow subsection. The rate of ammonia decomposition over iron-based catalysts depends on the partial pressure of ammonia and hydrogen as described by the Temkin-Pyzhev Eq. (2) [41]. $$r = k\left( {\frac{{p_{{NH_{3} }}^{2} }}{{p_{{H_{2} }}^{3} }}} \right)^{0.25}$$ The purity of the inlet ammonia stream and the presence of other gases and/or impurities can also have a beneficial or detrimental effect on the catalyst, some of them capable of altering the iron surface [41]. For example, the presence of CO2 and H2O was found to maintain the metallic active state of iron, leading to an enhanced activity [13]. 2.1.1 Effect of the Support on Iron-Based Systems A range of supported and unsupported iron based catalysts have been reported for the decomposition of ammonia, as summarised in Table 1. Unsupported catalysts typically have low surface areas and large metal particle sizes, reducing the number of exposed active sites and thus the catalytic activity. However, the unsupported systems can pose an economic advantage if they are cheap to manufacture or obtain in spite of the lower activity. Physical properties and ammonia decomposition catalytic activity of reported iron catalysts Fe Content (wt%) Support (Promoter) NP size (nm) Support S BET a (m2 g−1) Catalyst S BET a (m2 g−1) T (°C) GHSVb (cm3 g cat −1 h−1) Conversion (%) Ratec (${\text{mol}}_{\text{H}_2}$ g cat −1 h−1) Ratec (${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Fe}}^{-1}} \, {\text{h}}^{-1}$) Red mud (Fe2O3, SiO2, TiO2, and Al2O3) 100 (h−1) Fe containing CNT αFe2O3 – SiO2 (Core–shell) 50 (Fe) 20 (Si) Nano Fe2O3 – SiO2 (core–shell) 83 (2Si:10Fe) Fe-CeO2 composite 83 (5Fe:1Ce) 21 (Fe) 13 (Ce) Limonite (α-FeOOH rich ore) Goethite (FeOOH ore) Fe fused with Al2O3, CaO, K2O (3.3 % Al, 3.2 % Ca, 0.8 % K) Fe in Al2O3 matrix (Al2O3) Fe/CNT Fe/CMK-5 CMK-5 Fe/C-SBA-15 Carbonised SBA-15 Fe/CNF,Mica CNF on mica K-Fe/C Graphitised C (K) Fe/coal char Coal char Fe/Al2O3 Dash indicates data not published or insufficient data to enable calculation aSurface area calculated by the BET method. Corresponds to metallic surface area when value followed by (M) bGas hourly space velocity (GHSV) cRate quoted with respect to number of moles of H2 produced Amongst the unsupported ones, a few inexpensive iron containing materials such as ores [42] and waste products e.g. red mud [30] have been tested for ammonia decomposition. Unfortunately insufficient data was provided to enable to calculation of rate for comparison sake but nevertheless, considering the high iron content and high temperatures of the studies, it is clear that they are not exceptionally active catalysts. Nevertheless, several characteristics of the red mud present it as a viable, disposable catalyst for ammonia decomposition. Notably, since red mud is a waste product from the extraction of aluminium in the Bayer process and currently its disposal represents a problem for the mining industry, any potential uses of red mud represent not only a highly attractive economic advantage for the industry but also a sustainable solution. Additionally, red mud was reported to be resistant to poisoning by sulphur and exhibited good stability over 200 h of operation [30]. However, the composition of red mud varies depending on the bauxite source. Additionally, red mud contains a complex mixture of Fe2O3, SiO2, TiO2 and Al2O3, making it difficult to identify the component(s) acting as co-catalyst or promoter in the mixture. A more systematic study would be necessary to understand the role of each component on the resulting activity. The activity of a bulk iron catalyst [43] fused with small amounts (<3.3 wt%) of metal oxides (Al2O3, CaO and K2O) is poor (1.14 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Fe}}^{-1}}{\text{h}}^{-1}$, 500 °C), which may be due to the low catalyst surface area of 11 m2/g. In fact, iron incorporated within an Al2O3 matrix with a higher surface area (77 m2/g) exhibited a higher rate of hydrogen formation (125 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Fe}}^{-1}}{\text{h}}^{-1}$, 500 °C), making the case instead for the development of supported catalysts, which typically possess high surface area [44]. An alternative stabilisation strategy is the use of core–shell systems such as α-Fe2O3 nanoparticles surrounded by a shell of porous SiO2 [18, 45]. The silica shell provides high thermal stability versus sintering. However, as a result of the additionally stability, higher temperatures are needed to facilitate the mass transfer diffusion of ammonia through the silica shell. As such these catalysts are only active at temperatures above 400 °C, with a promising rate of hydrogen formation of 127 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Fe}}^{-1}}{\text{h}}^{-1}$ [18] at 500 °C when only 12 wt% Fe is used. The rate per mole of iron for the core–shell catalyst containing 83 % iron is significantly lower. Nevertheless, the α-Fe2O3–SiO2 core–shell catalysts are the most promising unsupported iron catalyst with respect to rate at 500 °C. Iron nanoparticles have been supported on a range of materials including coal char, carbon nanotubes (CNT), carbon nanofibers (CNF) and structured porous carbons such as CMK-5. Research has been mainly focused on carbon materials with metal oxides being used primarily as promoters rather than supports. Carbon materials have high thermal stability and electrical conductivity, making them attractive catalyst supports [51]. Depending on the properties of the carbon support, especially its surface area and porosity, and the iron impregnation method, iron nanoparticles of different sizes have been achieved. Iron nanoparticles with average sizes of ~6 nm in diameter were stabilised on CMK-5 support (a mesoporous carbon with a dual pore network, which is formed by the templating of SBA-15 porous silica structure and subsequent removal of the template) and carbonised SBA-15 (similar to the previous support but with the SBA-15 template present) [51]. Fe/CMK-5 was found to be a more active catalyst (515 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Fe}}^{-1}}{\text{h}}^{-1}$, 500 °C) than iron supported on a carbon-SBA-15 composite (152 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Fe}}^{-1}}{\text{h}}^{-1}$, 500 °C). However, the carbonised SBA-15 support resulted in a more stable catalyst in the long term as the iron nanoparticles diffused through the carbon wall and anchored to the silica walls inhibiting their sintering. Jedynak et al. [53] reported high TOF (0.016 s−1 at 400 °C, 20 % NH3) with an iron catalyst supported on graphitised carbon and linked the high activity to small particles of Fe of 13 nm (compared to 24 nm in the less active catalyst). Iron nanoparticles in the size range of 20–50 nm supported on coal chars [54] presented low activity (56 %) at high temperatures of 750 °C, even though the rate cannot be calculated, a strong link between particle size and activity can nevertheless be inferred. From the data presented in Table 1 for supported systems, the rate of hydrogen formation (at 500 °C) is highest for the 6 nm Fe nanoparticles supported on CMK-5 [51], suggesting that small iron nanoparticles are more active for ammonia decomposition although the extent of validity of this conclusion needs to be corroborated by further work in this area. Although, this conclusion is in agreement with computational simulations of iron clusters which suggest that nano-sized particles of iron less than 10 nm in diameter increase catalytic activity [40]. By contrast, iron nanoparticles of significantly larger average sizes of 85 nm and a wide particle size distribution (40 and 160 nm) [52], achieved a superior rate (119 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Fe}}^{-1}}{\text{h}}^{-1}$ at 500 °C) to some of the catalysts with particles smaller than 10 nm. It is worth noting that with the Fe/coal char catalyst, the formation of Fe4N under reaction conditions was inferred by a decrease in N2 formation rate, the presence of which could have also played a role in the lower reactivity in combination with the effect of particle size [54]. Regardless, it is clear that control of particle size by use of well-defined porous supports provides a degree of control over catalyst activity and presents an opportunity for improvement of iron based catalysts. 2.1.2 Effect of Promoters on Iron-Based Systems Itoh et al. [47] synthesised Fe powders containing metal oxide components (CeO2, Al2O3, SiO2, SrO and ZrO2). Out of them, CeO2 and Al2O3 were the most effective at enhancing the catalytic activity of iron. The improvement in activity was believed to be due to the enhanced surface area and the role of the oxide as an acidic adsorbent of ammonia. Additionally, CeO2 was also found to inhibit sintering of iron, with a 5:1 ratio of Fe:Ce showing the highest activity amongst the studied materials. The addition of alkali metals as promoters have been reported to be effective at preventing sintering of Fe nanoparticles [55]. As an example, the addition of K2O on iron fused with Al2O3 and CaO resulted in a six fold increase in ammonia decomposition rate (400 °C, 30 % NH3) compared to the same catalysts without K2O [49]. In the fresh promoted catalyst, KxOy was present both on the iron surface and on the Al2O3 islands found on the iron surface. Interestingly, the promoter effect of potassium varied as a function of the inlet NH3 concentration. This was attributed to chemisorption competition on the acidic alumina sites between the more basic KxOy promoter and the less basic NH3 reactant. As the inlet NH3 concentration increases, some KxOy is displaced by surface diffusion onto bare iron surface and thus its effect as promoter becomes more prevalent. No notable improvement of activity was observed with the inclusion of potassium when these catalysts were tested for ammonia synthesis [49]. An alternative explanation of the promoting effect of potassium is its effect on the resulting iron particle size. Fe/graphitised carbon catalyst promoted with a K:Fe molar ratio of 4.4:1 resulted in a fivefold increase in rate compared with the catalyst with a 1:1 ratio, which may be linked to the presence of smaller iron nanoparticles (13 nm compared with 25 nm) in the catalyst with more potassium [53]. Since both acidic (e.g. Al2O3) and basic (e.g. K) promoters have been reported to increase the catalytic activity of iron, the role of the promoter may lie fundamentally in the control of nanoparticle size as opposed to mediating the basicity/acidity. From the experimental works in this area, it is not clear yet if the higher activity of smaller iron particles is due to stronger promoter effects or an increase in concentration of C7 active sites, which are well known to be highly active in ammonia synthesis [53]. 2.2 Cobalt-Based Catalysts Due to the relative low cost and its nitrogen adsorption energy, cobalt has also been explored as an alternative to ruthenium for hydrogen production via the ammonia decomposition reaction. A kinetic study by Lendzion-Bielun et al. [27]. Calculated the activation energy for NH3 decomposition to be 27 kJ mol−1 lower on cobalt than on iron-based catalysts. This finding can be attributed to the weaker nitrogen binding of the former compared with the latter, resulting in a superior cobalt catalyst activity, especially at low reaction temperatures. Beyond these general aspects, the effect of the physical and chemical properties of the support (e.g. basicity and electron conductivity) [56] as well as the presence of promoters have been explored in order to understand their role and facilitate the development of active cobalt-based catalysts. Based on the current literature, the active cobalt phase for ammonia decomposition has not been clearly identified, although some reports suggest that it is metallic cobalt [27]. Consideration of the preparation and pre-treatment conditions, such as the choice of cobalt salt [57] and the calcination conditions [58], is critical as it can alter the catalysts properties, affecting the resulting ammonia decomposition activity of the catalyst. For example, work by Varisli and Kaykac [57] reported superior catalysts when synthesised from cobalt acetate compared to cobalt acetyl acetonate and cobalt nitrate. The activity and physical properties of cobalt based systems for ammonia decomposition reported in the literature are summarised in Table 2. Physical properties and ammonia decomposition catalytic activity of reported cobalt catalysts Co Content (wt%) Ratec (${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{co}}^{-1}} \, {\text{h}}^{-1}$) Co incorporated in Silicate (Na silicate) Co3O4 promoted with mixed oxides (2.9 Al, 1.6 Ca, 0.5 K, 0.23 Cr) Co3O4 promoted with Ca/Al/K oxide (2.8 Al, 2.4 Ca, 0.55 K) Co-SiO2 core–shell Co in Al2O3 matrix Co/mixed oxide 2MgAl 2MgCe 2MgLa Co/MgO-La2O3 MgO-La2O3 Co containing CNT Co/MWCNT MWCNT 6000 (h−1) 0.2 (M) Co/Al2O3 bGas hourly space velocity (GHSV). If units differ and could not be converted, they are shown in brackets 2.2.1 Effect of the Support on Cobalt-Based Catalysts A series of materials have been studied as cobalt supports for the decomposition of ammonia reaction. In general, the studies reveal that the role of the support and its characteristics can be related to the cobalt particle size (nature of anchoring points), stability (depending on the metal-support interaction) and activity (e.g. electron donating properties of the support, conductivity) amongst others. The current literature shows a range of techniques used to stabilise cobalt nanoparticles, including core–shell structures [45], incorporation within silica [55, 57] or alumina [44] matrices and ceramic [56] or carbon [20, 61, 62] supports. Carbon materials are the most studied supports due to their mechanical stability and in most cases, a good metal-support interaction with cobalt, resulting in an improved electron transfer and consequently a reduction in the nitrogen desorption energy [20]. The activity of cobalt supported on multi-walled carbon nanotubes (MWCNT) was superior to the equivalent iron and nickel catalysts with 60 % conversion at 500 °C for cobalt compared with (14.8 and 25.4 % for iron and nickel respectively) [20, 61]. The effect of the metal-support interaction on cobalt/MWCNT was studied by varying the pre-treatment temperature (230–700 °C) and gas (nitrogen and hydrogen) [62]. It was reported that pre-treatment in nitrogen resulted in higher catalytic activity compared with hydrogen and lower pre-treatment temperatures resulted in smaller nanoparticles (~6 nm) with higher dispersion, in agreement with activity trends reported by Podila et al. [58]. However, contrary to expectations, the activity of the Co/MWCNT catalyst pre-treated at 600 °C under pure nitrogen with an average Co particle size of 57.4 nm was comparable to the activity of the catalyst pre-treated at 500 °C, even though the average diameter of the cobalt nanoparticles is 9.3 nm [62]. This result is surprising as larger particles are usually less active owing to a lower surface area and a lower concentration of active sites at the surface but may be due to the effect of the nitrogen pre-treatment. Pre-treatment of the carbon support with acid (e.g. CMK-3 treated with nitric acid [63]) has also been shown to have an effect on the control of the cobalt particle size (4–20 nm) and improve the dispersion of the cobalt nanoparticles on carbon materials [63], likely due to the creation of anchoring points however, the activity of these materials for the ammonia decomposition reaction have not yet been reported. An alternative way of achieving cobalt size control is by the incorporation in commercial cobalt nanoparticles with sizes of 4–20 nm on CNTs, which proved to be highly active with a rate of formation of 542 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Co}}^{-1}}{\text{h}}^{-1}$ at 600 °C, although a drop in rate at lower temperatures was observed (11 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Co}}^{-1}}{\text{h}}^{-1}$ at 500 °C) [46]. The resulting cobalt particle size can alternatively be controlled by carefully selecting a support material with appropriate and beneficial physical properties. In general, the higher the surface area of the support, the higher the metal dispersion that can be achieved. However, other characteristics of the support can override this general rule. In this context, cobalt supported on active carbons showed a lower catalytic activity than those supported on MWCNT despite of having a higher specific surface area. Nitrogen temperature programmed desorption studies show that nitrogen desorbs at lower temperatures from the Co/MWCNT catalyst than from the Co/AC, suggesting that the nitrogen binding energy is lower in the former due to superior electron conductivity of the MWCNTs respect to AC, resulting in a higher catalytic activity. A similar trend of support dependence has been reported for ruthenium catalysts which show highest activity (6353 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ru}}^{-1}}{\text{h}}^{-1}$ at 430 °C) when supported on graphitic carbon and CNTs due to the high conductivity [17]. Due to the limited number of studies of cobalt catalysts for ammonia decomposition, it is difficult to draw concrete conclusions about the most active cobalt particle size, although in general the highest rates of conversion (at 500 °C) have been reported within the size range 10–20 nm [44, 56]. 2.2.2 Effect of Promoters on Cobalt-Based Systems A series of promoters have been reported in the literature to enhance the activity of cobalt active sites, especially alkali and alkaline earth metals. Co-impregnation of unsupported cobalt with calcium, aluminium and potassium oxides promoters was found to enhance the catalytic activity [59, 60]. Specifically, the promoter effect of potassium on Co/silicate catalysts increased as the quantities of KOH increased, which may be due to an increase in surface area and reduced pore diameter [55]. On the other hand, the presence of chromium and manganese on Co/mixed oxide catalyst resulted in lower activity with respect to the cobalt-only catalyst, yet no explanations were provided following this reporting [59]. Podila and co-workers [56] incorporated Mg oxide supports with Al, Ce and La oxides with a Mg:M ratio of 2:1 for use as a support of cobalt catalysts. Out of the three, the addition of LaO provided the highest activity enhancement with a rate of reaction of 385 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Co}}^{-1}}{\text{h}}^{-1}$ at 500 °C. Further studies on the Mg:La ratio, revealed an optimum 5:1 Mg:La ratio due to the stabilization of a cobalt average particle size of 15.6 nm and a high basicity of the support. There is a direct link between particle size and activity in this case, with the inactive MgAl supported catalyst possessing large cobalt nanoparticles of 170 nm whereas significantly smaller (<20 nm) cobalt nanoparticles are stabilised in the presence of lanthanum and cerium oxides, both of which yield highly active catalysts. Pre-treatment of the MgO-La2O3 support with nitrogen has been shown to yield the most active catalyst due to a modification of catalyst basicity and morphology [58] As shown in Table 2, for 5 wt% Co supported on MgO-La2O3, the rate of hydrogen formation at 500 °C increases from 385 to 602 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Co}}^{-1}}{\text{h}}^{-1}$ due to calcination of the support in air [56] and nitrogen [58] respectively. However, it is worth noting that the ratio of Mg:La is 2 in the former [56] and 3 in the latter [58] which may also contribute to the improvement in activity. Ceria and barium have been studied as promoters for unsupported cobalt catalysts for ammonia synthesis. It was observed that addition of ceria inhibited the sintering of the cobalt particles due to the stabilisation of the hexagonal close-packed (HCP) phase of Co3O4 under the reaction conditions, while the addition of barium led to heat resistivity at 600 °C over 160 h [64], phenomena that can be pertinent to the development of ammonia decomposition catalysts based on cobalt. A range of oxides materials such as Al, Ce, La, K, Mn and Cr have been studied as promoters [44, 56, 59] for cobalt based catalysts for ammonia decomposition, with the most notable enhancement in activity by lanthanum. The limited literature within this domain suggests that there is scope for further improvement of cobalt systems using novel promoter elements or methods. Although it is worth noting that the use of ceramics as promoters (or equally as supports) does present a possibility of the formation of inactive, irreducible mixed oxides such as cobalt silicate in the Co/SiO2 catalyst [20], resulting in lower activities, identifying an area for further development of these catalysts. 2.3 Nickel-Based Catalysts Another attractive alternative from the economic and availability points of view to substitute ruthenium on ammonia decomposition catalysts is nickel. However, nickel's high structure sensitivity, formation of irreducible Ni compounds and strong binding of hydrogen to the nickel active sites are some of the issues where further research is required for the development of an active catalyst containing highly disperse, homogeneous Ni particles [65, 66]. Computational studies of the ammonia decomposition reaction on nickel-based catalysts confirmed that similarly to the iron and cobalt catalysts, the energy of the associative desorption of nitrogen is higher than the N–H scission step [67]. This is in agreement with the experimental kinetic studies that suggest that nitrogen recombination is the rate limiting step. Indeed, Ertl et al. [68] calculated the activation energy of ammonia decomposition over a clean Ni surface to be 197 kJ mol−1. This value is similar to the energy of nitrogen desorption, confirming that the associative nitrogen desorption is the rate limiting step for nickel-based catalysts. Temperature programmed reduction and in situ XRD have been used to identify the active phase of nickel to be the metallic form, Ni0 [70]. It is generally accepted that the key effect of the nickel particle size is on its activity for the ammonia decomposition reaction. Figure 4 shows the relationship between nanosized nickel particles and their turn-over frequency (TOF). Only Ni0 particles with average sizes below 2.9 nm showed considerable activity, with 2.3 nm its optimum value. Interestingly, this particle size falls into the range where the presence of B5 sites, a particular configuration of 5 atoms, is maximised as speculated by several authors [66, 69] although without experimental verification. It is important to mention here that this specific B5 sites are well known to be related to the high activity of ruthenium nanoparticles with sizes between 3–5 nm [71]. Relationship between activity measured as forward TOF and average Ni0 particle size where solid and hollow squares are Ni/Al2O3 and Ni/–La–Al2O3 respectively. The data highlighted in the grey shaded area is expanded in the inset graph. Directly reproduced with permission from Zhang et al. [69] A study by Zhang et al. [69] revealed the strong effect of not only the size but also the structure on the catalytic activity of the nickel-based catalysts. Indeed, this relationship between activity and structure has been confirmed by the difference of ammonia adsorption energy between Ni (110) and Ni (111) surfaces [13]. In addition, the nitrogen desorption energy on the stepped (211) nickel surface is higher than on a close packed (111) terrace surface and in the former case, strongly adsorbed nitrogen can block up to two-thirds of the active stepped sites. Simulations studies of the ammonia decomposition reaction on nickel-based catalysts show that surfaces with too many or too few stepped sites are likely to show low activity. The nature and concentration of these stepped active sites can be controlled by varying the particle size as discussed above. The nickel catalyst preparation method is also known to influence the catalytic activity, with co-precipitation and adsorption methods reportedly yielding more active catalysts than impregnation techniques, due to the difference in resulting nickel particle size and dispersion [65, 66, 69]. Not only the choice of method is important, but also the conditions used during the chosen procedure. For example, in the deposition–precipitation method with silica as a support, the type of Ni2+ phase deposited on the surface depends on the synthesis time as well as the surface area of the silica. Longer synthesis times results in increased formation of phyllosilicate which is linked to a decrease in surface and pore volume [72]. Li et al. [66] found that by using the template ion exchange method, the nanoparticles formed were predominantly on the internal walls of the support and were too small to contain a high concentration of active sites, although the exact size of these particles was not reported. Another parameter affecting the reactivity of nickel is its loading as it can affect not only its particle size but also the nickel phase formed. In this context, Fig. 5 shows the effect of nickel loading supported on alumina via impregnation. At low loading, a high coverage of Ni atoms is formed but mainly as α-NiO. This phase is easily reduced but it also has a weak interaction with the support, making it susceptible to sintering. Higher nickel loadings resulted in the formation of γ-Ni aluminate in the spinel phase which requires reduction temperatures in excess of 800 °C but this high temperature would ultimately lead to a reduction of surface area and consequently activity [65]. Schematic of interaction of Ni with alumina support with increased Ni loading synthesised by impregnation. Directly reproduced with permission from Zhang et al. [65] 2.3.1 Effect of the Support on Nickel-Based Catalysts The effect of the physical and chemical properties of the support are key in determining the reactivity of nickel catalysts on the ammonia decomposition reaction, mainly due their effect on the particle size and metal-support interaction discussed above. As shown in Table 3, the majority of nickel based catalysts tested for ammonia decomposition are based on ceramic supports, predominantly SiO2 and Al2O3, with significantly fewer studies using carbon based supports. Physical properties and ammonia decomposition catalytic activity of reported nickel catalysts Ni Content (wt%) Ratec (${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ni}}^{-1}} \,{\text{h}}^{-1}$) Ni–SiO2 core–shell Ni in Al2O3 matrix Ni/monolith Al2O3 covered monolith Ni/microfibre Al2O3 (CeO2) Ni/Al2O3 Al2O3 (La) Ni/SiO2 Ni/SiO2/Al2O3 SiO2 + Al2O3 Ni/MgO 15,000 (h−1) Ni–Ce–Al–O microsphere (Ni:Ce:Al:O 0.5:0.1:0.4:x) Ni/SBA-15 SBA-15 Ni/MCM-41 MCM-41 Ni/CNT MWCNT-COOH SWCNT ~0 Ni/Graphene aSurface area calculated by the BET method Based on the current literature, carbon supports appear to be ineffective for nickel based catalysts as shown by the very low or negligible conversion at 500 °C [50, 78]. However, functionalising the multi-walled CNT (MWCNT) support with –COOH [78], resulted in an improvement in activity. The authors suggest that the increased activity may be due to nickel anchoring points created by the –COOH groups, but their presence had negligible effect on the nickel particle size, phase composition and nickel reducibility [78]. On the other hand, the use of SiO2 and Al2O3 as supports has resulted in highly active of nickel catalysts. In particular, a high rate of 578 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ni}}^{-1}}{\text{h}}^{-1}$ at 500 °C was achieved with mesoporous SBA-15 support for nickel [72] closely rivalled by the use of Al2O3 (496 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ni}}^{-1}}{\text{h}}^{-1}$, 500 °C) [75]. Two highly active Ni/Al2O3 catalysts [65, 75] possess similar sized average nickel particles of 3.5–3.9 nm, however this size is larger than the optimal, average 2.3 nm size reported by Zhang et al. [69]. Regardless, the ability of ceramic supports to stabilise small nickel nanoparticles is promising and may play a crucial in the development of these active Ni/ceramic systems in future. Alumina has also been shown to be effective not only as a support but also as encapsulation material of high surface area nickel microfibers [74]. The large void volume and open structure of the alumina facilitated a good heat and mass transfer with a high permeability and good heat resistance, leading to a high rate of reaction of 703 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ni}}^{-1}}{\text{h}}^{-1}$ at 600 °C with high stability over 100 h [74]. Interestingly, complete ammonia decomposition conversion was achieved with an Al2O3 coated monolithic nickel catalyst at temperatures 100 °C lower compared to a packed bed of the same Ni/Al2O3 catalyst [73]. This is likely due to the increase in exposed nickel, although this catalyst does not outperform other Ni/Al2O3 presented in Table 3 Nickel nanoparticles were found to be anchored to the alumina surface as opposed to blocking the mesopores of the monolith. Thus, catalysts supported on monoliths are promising in the development of cheap, efficient and robust catalysts with a lower pressure drop, making them suitable for potential mobile applications [73]. 2.3.2 Effect of Promoters on Nickel-Based Catalysts Contrary to the observations on ruthenium-, iron- and cobalt-based catalysts, the addition of alkali metals such as potassium (by using KOH precursor) does not seem to have an effect on the catalytic activity of nickel catalysts supported on silica [66]. On the other hand, the addition of transition metals clearly shows a beneficial effect on the nickel-based catalysts. In this context, the use of lanthanum as promoter of Ni/Al2O3 catalysts (423 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ni}}^{-1}}{\text{h}}^{-1}$, 500 °C) [65] results not only in morphological modifications of the nickel active sites but also in electronic effects. The presence of lanthanum can alter the local arrangement of the nickel atoms to maximise the number of stepped nickel active sites. Additionally, the surface reaction between lanthanum oxide and nickel promotes an electron transfer towards the nickel active sites which facilitates the nitrogen recombinative desorption and thus increases the rate of decomposition [69]. Alternative explanations of the beneficial effect of lanthanum suggest the promotion of a more open mesoporous structure of the Al2O3 support and consequently an increased nickel dispersion [65]. The addition of ceria to Ni/Al2O3 catalysts results in a beneficial electron transfer to the active sites with high rates reported (496 ${\text{mol}}_{\text{H}_2}$ ${{\text{mol}}_{\text{Ni}}^{-1}}{\text{h}}^{-1}$, 500 °C) [69, 75]. Indeed, the addition of 10 wt% ceria to Ni/Al2O3 catalysts reduces by 100 °C the temperature at which catalysts show ammonia decomposition activity [74]. The optimum Ce:Ni molar ratio is reported to be 0.1 with Ni/Al2O3 catalysts [74]. Increasing the Ce loading resulted in a decrease of the Ni0 (111) diameter and an increase of the CeO2 (111) sites, suggesting that the optimal Ce loading could inhibit the growth of Ni, allowing a degree of control on Ni particle size [75]. Remarkable catalytic activity was reported for a triple metal microsphere catalyst containing Ni, Ce and Al [70]. The precise rate of hydrogen formation cannot be deduced due to insufficient information, however based on the small mass of catalyst tested (0.05 g) and high hydrogen formation rate in terms of ${\text{mol}}_{\text{H}_2}$ g cat −1 h−1, it seems that the rate is exceedingly high on a per mole of metal basis. The activity of the reported Ni–Ce–Al microsphere catalyst was higher than for the analogous bimetallic Ni–Ce and Ni–Al catalysts, suggesting a synergistic effect between Ce and Al for promoting the activity of Ni. Thus, there is scope for enhancement of Ni/Al2O3–Ce catalysts as well as exploring other potential promoter metals. To our knowledge, there are no reports of promoted Ni/SiO2 catalysts, which is surprising given the high activity of these systems, presenting an attractive opportunity for further improvement. 2.4 Other Monometallic Systems A range of other monometallic systems have been studied for the production of hydrogen from ammonia using non-noble metals including transition metal carbides (MoCx, VCx, WCx and FeCx) and nitrides (MoNx, VNx and WNx), as well as zirconium oxynitride. Out of these, molybdenum nitride and tungsten carbide are the most studied in the ammonia decomposition reaction. However, it is worth noting that these catalysts are typically tested under conditions comparable to the clean-up of gasification mixtures as well as for the production of hydrogen [13]. As shown in Table 4, the conversion of these catalysts by the rates of hydrogen formation at 500 °C per mole of metal are in general disappointingly low due to the high metal content. Physical properties and ammonia decomposition catalytic activity of reported miscellaneous bulk monometallic catalysts Ratec (${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) Li2NH Meso WC MoO3 (MoNx active) Mo2C Molybdenum nitride (MoNx) is considered the most active catalyst amongst the studied transition metal carbides and nitrides, with a catalytic activity comparable to that of platinum, however the rate of reaction at 500 °C is inferior (10 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) [82] to mesoporous WC per mole of metal (111 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) [13, 81]. Note the increase in rate at 500 °C from 3 to 111 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1when mesoporous WC [81] is used rather than bulk WC [80], likely due to the increase in catalyst surface area from 1.5 to 138 m2 g−1. It is known that formation of metal carbides modifies the electronic structure of the tungsten atom, responsible of its activity. However, the chemical stability of WC is low, being irreversibly poisoned in the presence of CO and H2 at temperatures in excess of 500 °C, likely due to the decomposition of the WC compound, making it impractical for use as an industrial catalyst [80]. The decomposition of WC results in the loss of carbon at the surface, which is most likely needed to electronically modify and activate tungsten for ammonia decomposition [13]. Additionally, WC exhibited an induction period which is believed to be related to the restructuring of the WC surface in the presence of ammonia [80]. For example at 500 °C (in the absence of CO and H2) the initial reaction rate was 3 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1 but increased after 60 min the rate to 6 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1, after which the rate remained constant for the subsequent 300 min tested [80]. MoNx is the active compound formed during reaction in the presence of ammonia when molybdenum oxide (MoO3) is used as fresh catalyst. The activity of MoNx was considerably improved after ball milling of the MoO3 catalyst due to the increase in specific surface area from 1 to 13 m2 g−1 [82]. Complementary experimental and theoretical studies by Zheng et al. [83] demonstrated that the high rate of ammonia decomposition over molybdenum carbide and nitride can be attributed to the energetic sites comprising of twin boundaries, faults in stacking, steps and defect sites. Further development of the MoNx catalyst is needed to achieve a net cost effective system compared to the ruthenium-based catalysts, especially considering that the cost of molybdenum is half of that of ruthenium, the molybdenum-based catalyst have a very low surface area (bulk systems) compared to the highly dispersed ruthenium ones. Despite this, unless higher active surface area is achieved on molybdenum-based systems, no economic advantage would be achieved versus ruthenium systems owing to the significantly lower current rate of hydrogen formation of MoNx (10 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1, 500 °C) [82] compared to Ru/CNT (6353 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1, 430 °C) [3]. On a similar note, some metal amides, such as lithium amide, can decompose under heating, forming imides compounds or mixtures of imide-amide. These compounds are not expected to facilitate the decomposition of ammonia although in some cases, for example the lithium imide-amide system is effective in non-stoichiometric quantities at catalysing the decomposition of ammonia [79]. Chromium oxide has also been investigated however, it exhibits low hydrogen formation rate of 90 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1 at elevated temperature of 600 °C [84]. Mo2C on the other hand exhibits higher and more promising rates at this elevated temperature of 1112 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1. In general, however, the low temperature activity of these catalysts is not particularly promising with little scope for further improvement without screening and testing novel classes of compounds. 3 Bimetallic Systems It is well established that in general, bimetallic catalysts present different chemical properties to those of their individual monometallic components, achieving synergetic effects in particular cases [85]. Consequently, the specific arrangement of atoms within the bimetallic system can alter the catalytic activity [86]. There are different types of bimetallic systems such as core–shell or alloys. If the atoms are distributed evenly both on the surface and within the core, a perfect alloy system is formed. Core–shell particles contain a core formed by one metal surrounded by a monolayer of the second metal at the surface. In these cases, unusual chemical properties can be achieved due to the ligand effect created by the interaction of the two metals and the strain effect as the monolayer metal is constrained by the lattice of the core metal. Depending on the specific distribution of the metal atoms in a bimetallic system, its properties can result in a linear combination of the properties of the individual components or present a synergetic effect [86]. In general, bimetallic nitrides show higher activity for ammonia decomposition than their respective bimetallic oxides [87]. A requirement for bimetallic systems is a high stability under reaction conditions versus segregation into monometallic particles which would alter the surface properties and thus catalytic activity [22]. Additionally, in some supported bimetallic systems, the conditions of thermal treatment or high reaction temperatures can promote the formation of less reducible oxides such as Fe–Al, Co–Al, Ni–Al and Co–Si when supported on alumina or silica, resulting in considerably lower activity [88]. Figure 6 shows the volcano-type relationship between the catalytic activity (measured as TOF) of different monometallic systems for the synthesis of ammonia and their respective nitrogen binding energy. Out of the systems investigated, ruthenium-based catalysts present an optimum value as previously discussed herein [89]. The bimetallic guidelines mentioned above, can be applied to the rational design of bimetallic systems by combining two metals with lower and higher nitrogen binding energy than ruthenium to potentially mimic the chemical properties of ruthenium. Relationship between the turnover frequency (TOF) of different metals for the NH3 synthesis reaction at 400 °C with respect to their nitrogen adsorption energy. Reprinted with permission from Jacobsen et al. [89]. Copyright (2001) American Chemical Society A potential bimetallic system for ammonia decomposition identified by Fig. 6 is FeCo. FeCo confined on the internal surface of CNT has experimentally been shown to be an active system with a hydrogen formation rate of 4080 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1 at 600 °C [88]. The confinement of the FeCo nanoparticles within the internal CNT structure was crucial to inhibit their sintering. Additionally, a strong synergistic effect resulted not only in a high rate but also a high stability over 1000 h of operation. Elemental mapping in combination with TEM showed that the CoFe particles were alloyed and no change in the degree of alloying was found between fresh and spent catalysts. Perhaps slightly unexpected (based on the energies plotted in Fig. 6) is the reported high activity of a FeMo catalyst [90], the rate (6642 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) of which is superior to the aforementioned FeCo (4080 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) [88] catalyst at 600 °C, which may be due to the inclusion of a lanthanum promoter in the former case. Although the formation of the FeMo alloy in the fresh catalyst was verified by pXRD, under reaction conditions, the FeMo alloy is converted into the respective iron and molybdenum nitrides, but without deterioration of the activity, which suggests that these are the real active species. Additionally, the use of La2O3 modified Al2O3 as support of the FeMo particles proved to be a more effective support than Y promoted ZiO2 due to the increased support basicity [90]. At a lower reaction temperature of 500 °C, the rate of hydrogen formation of NiFe/Al2O3 (640 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) [91] is superior to that of FeMo (79 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) [90]. In the synthesis of NiFe/Al2O3, interestingly, both incipient wetness impregnation and co-precipitation methods resulted in alloy formation with comparable activities [91]. In addition, the support was shown to influence the activity and stability of the resulting NiFe bimetallic nanoparticles. The activity of the NiFe alloy was highest when supported on Al2O3 and Mg–Al-Spinel compared with SiO2, TiO2 and ZrO2, which may be due to a loss in surface area of the latter supports after reduction at 800 °C. The use of first principle calculations and interpolation across the periodic table has identified CoMo as an attractive bimetallic candidate to substitute ruthenium based catalysts for the production of hydrogen from ammonia [22, 89, 98]. As shown in Fig. 6, a microkinetic simulation model identifies the CoMo combination to have a similar nitrogen binding energy to Ru. As a result, several experimental studies have focused on Co–Mo catalysts [92, 95] for ammonia decomposition, in which Co3Mo3N is believed to be the active species, in agreement with recent work on bulk Co–Mo catalysts by Duan et al. [93] and Podila et al. [94]. Several studies agree with the synergetic effect of the CoMo bimetallic system supported on γ-Al2O3 to be more active than the equivalent monometallic Co and Mo catalysts [87, 92]. The optimum Co:Mo atomic ratio varies slightly amongst studies between 7:3 [95] and 8:2 [87]. The effect of the support on the CoMo systems has also been studied by several authors with support dependant activity following the trend of γ-Al2O3 > MCM-41 > SiO2 [92, 95]. In general, the activity of CoMo nitrides in ammonia decomposition increases as the surface acidity and support surface area increases. The trend in activity may also be linked to alloy particle size, with γ-Al2O3 and MCM-41 supports effectively stabilising small 1.8–4 nm sized particles. Unfortunately the CoMo particle size was not reported when supported on SiO2, preventing a more in depth deduction of correlation between activity and particle size resulting from the choice of support material. Interestingly, when the bimetallic CoMo catalyst [92] was prepared from a salt containing both metal species (i.e. Co(en)3MoO4), a higher activity and stability was achieved compared with the use of the equivalent monometallic salts. It is likely that Co and Mo have a strong interaction in the Co(en)3MoO4 salt and the higher activity can be attributed to a higher content of the Co3Mo3N active species. The stability of the bimetallic catalyst using Co(en)3MoO4 salt as metal precursor did not vary after 1200 h on stream, whereas the activity of the monometallic cobalt catalyst declined over this period, possibly due to the migration of cobalt to the tetrahedral γ-Al2O3 sites of the support forming inactive CoAl2O4. Whilst the average particle size of both the CoMo/γ-Al2O3 and Co/γ-Al2O3 catalysts was 1.8 nm, the presence of larger particles in the latter catalyst suggest that the addition of molybdenum promotes a narrower particle size distribution [92]. Further work [93] within this framework, reported significantly lower catalytic activity at 500 °C for unsupported Co(en)3MoO4 (32 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) compared with the supported precursor (4564 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) [92], likely due to the reduction in exposed active metal species. Regardless, they [93] report on the importance of calcination atmosphere and pre-nitridation temperature on the catalyst activity, with calcination in air followed by treatment at 750 °C giving the most stable and active catalyst. It is important to mention that work by Jacobsen et al. [89] demonstrated that the activity of CoMo nitrides in ammonia synthesis is highly dependent on the inlet concentration of ammonia in the system as shown in Fig. 7. Their work showed that, Co3Mo3N can present a higher activity than a ruthenium counterpart catalyst when low concentration of NH3 below 5 % is used, although the activity dramatically decreases as the ammonia concentration increases. Whilst these results are related specifically to ammonia synthesis catalysts, this work highlights a significant limitation of Co3Mo3N as an ammonia decomposition catalyst. The ammonia inlet concentration is consequently an important consideration for the testing and use of Co3Mo3N, due to the reported poisoning at NH3 concentrations above 5 %. Relationship between the ammonia synthesis activity of Co3Mo3N, Ru and Fe catalysts as a function of the ammonia concentration in the inlet stream. Reprinted with permission from Jacobsen et al. [89]. Copyright (2001) American Chemical Society Few studies have been focussed on nickel-based bimetallic systems, amongst them, theoretical calculations of nitrogen binding energy predict a nitrogen binding energy of 582 kJ mol−1 for nickel supported on WC [28], which is close to the theoretical optimal value of 561 kJ mol−1, however, no experimental verification is available. On the other hand, nitriding of Ni and NiMo catalysts greatly enhances the ammonia decomposition activity, although there was not a significant improvement in catalytic activity from the addition of molybdenum [96]. Whilst platinum itself is not active for ammonia decomposition, theoretical studies by Hansgen et al. [22, 86] show that when combined with nickel, iron or cobalt the binding energy increases from 418 kJ mol−1, to just below the Ru (0001) binding energy of 561 kJ mol−1, suggesting that platinum could be effective in enhancing the activity of nickel, iron or cobalt. However, their results show that this is only applicable for M–Pt–Pt(111) but not Pt–M–Pt(111) where M=Ni, Fe, Co and Pt are monolayers. Specifically, the stability of nickel on Pt(111) has been reported as an issue due to the migration of Ni into the first layer of Pt at 450 K [28]. Additionally, Cu, both in monometallic and bimetallic systems (in conjunction with platinum), is predicted to be inactive in this context [86]. Based on the catalytic results of bimetallic systems containing a sustainable metal substituent (iron, cobalt or nickel) it is clear from the data presented in Table 5 that CoMo is the most promising bimetallic candidate with the highest rate reported at 500 °C (e.g. CoMo/γ-Al2O3 4564 ${\text{mol}}_{\text{H}_2}$ mol M −1 h−1) [92]. Physical properties and ammonia decomposition catalytic activity of reported bimetallic catalysts containing at least one of iron, cobalt or nickel Combined M Content (wt%) FeCo in CNT 5 (Co:Fe, 1:5) FeMo/ La–Al2O3 10 (Fe:Mo, 1:1) La–Al2O3 NiFe/Al2O3 10 (Ni:Fe, 1:4) CoMo/γ-Al2O3 4.8 (Co:Mo, 1:1.6) γ-Al2O3 Co(en)3MoO4 Co+ Mo2N 93 (Co:Mo 3:90) CoMo/MCM-41 5 (Mo:Co, 1:2.3) CoMo/SiO2 NiMoN/α-Al2O3 10.8 (Ni:Mo, 1:1.6) α-Al2O3 Ni2Mo3N 97 (Ni: Mo 1:1.3) Ni-Pt/Al 1–5 Ni, <1 Pt aSurface area calculated by the BET method. Value in brackets for catalyst SA c Rate quoted with respect to number of moles of H2 produced 4 Outlook and Conclusions Chemical hydrogen storage has the potential of resolving most of the current issues associated to the physical storage of hydrogen, which is currently limiting the implementation of the hydrogen economy. In this context, ammonia is presented as a highly attractive alternative due to the existing distribution network, expertise for handling and, most importantly, high hydrogen content—almost three times higher than the current storage target. The use of ammonia for on-demand hydrogen production requires the development of a new generation of catalysts based on readily available, non-noble metals as alternatives to the highly active ruthenium-based ones, which currently lead not only the highest rates but also the lowest temperature activity. It is widely accepted that the limiting step for the ammonia decomposition, especially at low temperatures, is the associative desorption of nitrogen from the catalyst surface. Thus, the nitrogen binding energy of different monometallic and bimetallic systems is generally used as guideline for the rational design of novel catalysts. Despite the low cost of iron and its industrial use in ammonia synthesis, its activity for the decomposition of ammonia is relatively low as shown in Fig. 8 due to its high nitrogen binding energy, which consequently poisons the catalyst. Based on this, there seems to be little opportunity for further development of monometallic iron catalysts although there may be potential for further improvement using acidic and basic promoters with a view to improving particle size control. Summary of hydrogen formation rate for reported iron, cobalt, nickel, other monometallic, bimetallic and ruthenium (benchmark) catalysts calculated for each catalyst quoted both with respect to the mass of catalyst tested (Open circle, left axis) and with respect to the number of moles of metal (X, right axis) at 500 °C or lower (if temperature differs, shown by different coloured marker). Refer to Tables 1, 2, 3, 4, 5 for the reference of each study and further details about the catalyst. Ruthenium benchmark catalysts are 7 wt% Ru/gCNT (green marker) and 7 wt% Ru/gCNT with 4 wt% Cs promoter (pink marker) [17] The most active cobalt catalysts tend to possess nanoparticles in the size range 10–20 and for further work the choice of support based on electronic properties and acidity/basicity is a vital consideration as well as the addition of electronic promoters. Unlike iron and cobalt, nickel supported on carbon materials is virtually inactive but the nickel activity can be enhanced using ceramic materials as support of 2–4 nm nanoparticles. While current studies focus on the high temperature activity of these systems, they show potential for the low temperature decomposition of ammonia, probably using similar promotion strategies to the ones used in ruthenium and iron systems. As shown in Fig. 8 there are a few promising reports of cobalt- and nickel-based catalysts with a high rate of hydrogen formation per gram of metal. However when these results are reported per mole of metal, none of the catalysts exhibit high activity. In addition to these metals, only lithium imide exhibits potential as an alternative to ruthenium with the reported rate at 450 °C exceeding the rate of Ru/CNT when considering the rate per gram of catalyst. However, since lithium imide is a bulk material, when the rate is quoted per mole of metal, the rate is significantly lower. Figure 8 highlights the need for increased dispersion of metallic components in order to reduce the metal content of the catalysts, resulting in improved rate per mole of metal. Figure 8 shows that of all the catalysts reported, at 500 °C only supported Co–Mo exhibits activity close to that of Ru/CNT on a per mole of metal basis, with ruthenium based-catalysts remaining superior. Thus, we believe that the development of bimetallic catalysts, using theoretical predictions, is the most promising one. It is important to note that experimental results may disagree with theoretical predictions of proposed high activity for a particular bimetallic system but this is likely due to the difficultly of producing a pure alloy without segregation of the metallic counterparts. As such, extensive characterisation is needed to fully understand the alloy structure. 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Top Catal (2016) 59: 1438. https://doi.org/10.1007/s11244-016-0653-4	CommonCrawl
Photocatalytic degradation of tetracycline in aqueous systems under visible light irridiation using needle-like SnO2 nanoparticles anchored on exfoliated g-C3N4 Adewumi Olufemi Oluwole1 & Olatunde Stephen Olatunji ORCID: orcid.org/0000-0001-8348-13531 Environmental Sciences Europe volume 34, Article number: 5 (2022) Cite this article Pharmaceuticals is one of the groups of contaminants of emerging concern that are resistant to decomposition or removal by most of the existing water and wastewater treatment procedures, hence the need to develop techniques to facilitate the removals of this group of organic contaminants from water systems. In this study, needle-like SnO2 nanoparticles was synthesised and loaded on exfoliated g-C3N4 nanosheet through a hydrothermal method, for use as sensitive visible light induce-photocatalyst for the decomposition of tetracycline in aqueous systems. The synthesised composites was characterized and analysed for the nature of the heterojunction between the SnO2 nanoparticle and g-C3N4 nanosheet using microscopic and spectroscopic techniques. The composites were of improved surface properties and enhanced visible-light absorption. The synthesised SnO2/g-C3N4 nanocomposites with various amounts of SnO2 (10–50 mg), employed in the degradation of tetracycline under visible light irradiation, were of good degradation efficiency. The degradation efficiencies of tetracycline by 1 wt.%, 2 wt.%, 3 wt.% and 5 wt.% SnO2/g-C3N4 photocatalyst were 81.54%, 90.57%, 95.90% and 92.15% as compared to g-C3N4 and SnO2 with 40.92% and 51.32% degradation efficiencies. The synergistic interaction between the needle-like SnO2 and exfoliated g-C3N4 nanosheet promoted the separation of photogenerated electron holes pairs, which enhanced their migration rate between SnO2 and g-C3N4 heterojunction, thereby facilitating the degradation of tetracycline. The ·O2− was noted to be the major reactive species in the photocatalytic of the 3 wt.% SnO2/g-C3N4 nanocomposite. The fabricated SnO2 nanoparticles anchored on exfoliated g-C3N4 showed good performance for the decomposition of tetracycline in water, with possible application on other pharmaceuticals having same moiety (similar chemical structures). Low concentrations, ranging from ng/L to µg/L of over 3000 chemicals substances, including pharmaceutical compounds (PCs), have been detected in wastewater, groundwater, surface water, soil and even in drinking water world over, for over a decade now [1,2,3]. The occurrence of these PCs in water bodies is due to outfalls from their global usage as therapeutic and prophylactic medicines and nutrient supplements in humans, livestocks, and agricultural health management [4,5,6]. The growth in population, urbanization and lifestyle places pressure on PCs demand/supply shift dynamics, resulting in increase, in the production and consumption of tonnes of different pharmaceuticals in recent times. Small to significant proprtion of pharmaceuticals administered/consumed by humans and animals are excreted from their bodies in metabolized/un-metabolized forms, and discharged via domestic wastewater into sewers and other drainage systems, from where they are transferred/translocated into freshwater surface systems and other environmental compartments. Industrial wastewater release, as well as indiscriminate disposal of unused and expired medicinal products, contributes to the total load of chemical contaminants in different environmental media [7, 8]. The environmental stability of these pharmaceutical substances (materials) coupled with the continuous unabated and uncontrollable environmental reception, has resulted in growing concerns about the potential adverse effects these compounds will have on the ecosystem and human health [9, 10]. The removal of pharmaceuticals from wastewater and other aqueous matrices, especially drinking water, is of intense focus. Unfortunately, most of the existing conventional potable water (PW) and wastewater treatment plant (WWTP) technologies encompassing biological, chemical, and adsorption processes are not designed to resolve/degrade residues of pharmaceutical compounds such as tetracycline and other antibiotics during the treatment process, prior to the release of treated water use into the environment [11]. The inability of WWTPs to eliminate many pharmaceuticals and chemical contaminants calls for the need for the development of new technologies and process designs for the degradation of these substances during water purification/treatment. One of the possible technologies gaining traction for the resolution of pharmaceuticals and other organic contaminants in aqueous systems is the advanced oxidation procedures (AOP). AOP processes, which relies on the production of highly reactive free radicals, have proven to be a potential efficient means through which pharmaceuticals can be degraded [12, 13]. Advance oxidation procedures have been adapted in different forms ranging from ozonolysis [14], Fenton methods [15, 16], UV–Vis photo-catalysed [17], and many others. Most of these AOP adapted techniques are simple in operation; possess high catalytic activity with no selectivity for organic and inorganic compounds. They are however, limited by energy needs, non-recyclable, waste generation, incomplete decomposition and generation of solid sludge, which hinders their practical application in the degradation of pollutants within the environment [18, 19]. Oxidative degradation of pharmaceutical compounds using semiconducting photo-catalysts appears a viable option due to their simplicity, energy-saving, cheap and environmental friendliness, as well as their mild reaction condition requirements [20]. However, a major challenge arising from the use of photo-catalyst in advance oxidative photo-catalysed degradation is the narrow light response range and low quantum efficiency. It is therefore important to develop and fabricate an effective, stable and cheap visible-light-driven, efficient photo-catalyst for the degradation of organic pollutants [21]. Heterostructured semiconducting nanocomposites have a promising photocatalytic activity that is amenable to tuning for improved oxidative decomposition of hazardous pollutants within an aqueous environment [22]. This is because heterostructured nanocomposites possess heterojunction properties that improve the charge transfer mechanism and slow down the fast recombination of the charge carrier. They also possess excellent physical and chemical properties that enhance the activity of active radicals during the degradation process [23]. Despite these advantages, their photocatalytic performance is hindered by the quick recombination of photoinduced charge carriers and the low surface area that limits their quantum efficiency and photocatalytic activity. The development and use of hetero-structured semiconducting nanocomposites doped on g-C3N4 is suggested as a route to address the limited visible light adsorption, and to enhance charge separation associated with many photocatalysts [24]. However, information on the use of hetero-structured nanocomposites with improved photocatalytic potentials for the oxidative degradation of pharmaceutical compounds in aqueous systems is scanty. In this study, visible light driven needle-like SnO2/g-C3N4 photocatalyst was synthesised using a facile and economical hydrothermal method by loading various amounts of SnO2 nanosheets on exfoliated g-C3N4, characterized for phase structures, morphologies, optical properties, chemical compositions and photocatalytic performance, and tested for the oxidative photodegradation of tetracycline (TCN), an organic pollutant, under visible light irradiation. Tetracycline (TCN) is one of the commonly used broad-spectrum, cheap and readily available antibiotic pharmaceuticals. As far as we know, the degradation of tetracycline using SnO2/g-C3N4 composites has not been fully explored, and very little knowledge is available concerning its use for the treatment of other pharmaceutical compounds in aqueous media. Melamine (99.9%), hydrazine monohydrate (80%), ethanol, SnCl2.5H2O, benzoquinone, isopropyl alcohol, thiourea, potassium iodide, tetracycline, acetonitrile, methanol were purchased from Sigma- Aldrich, South Africa. All the reagents were of analytical grade and were used without further purification. Preparation of thermally exfoliated g-C3N4 Nanosheet The g-C3N4 nanosheet was synthesised by the calcination process as described by Yousaf, et al. [25] with slight modification. Briefly, 5 g of melamine was weighed into a crucible with a cover and transferred into a laboratory model muffle furnace. This was calcined at 550 °C for 4 h at a heating rate set at 10 °C/min. The resulting bulky yellow powder was allowed to cool to room temperature and thereafter crushed to a fine powder using a porcelain pestle and mortar. This resulting g-C3N4 fine was exfoliated by thermal treatment at 550 °C for another 2 h, at a heating rate of 10 °C/min to yield a thermal exfoliated g-C3N4 nanosheet. Preparation of SnO2/g-C3N4 composite photocatalysts About 1 g of the exfoliated g-C3N4 nanosheet was dispersed in 120 ml of deionized water in a 250 ml beaker and ultra-sonicated for 30 min at room temperature. The sonicated g-C3N4 suspension was stirred continuously at room temperature for 30 min, followed by the addition of different masses of tin chloride pentahydrate (10, 20, 30, 50 mg) to different aliquots to achieve, equivalent to 1, 2, 3, 5 wt.% equivalent loadings, respectively. The resulting homogenous mixtures were stirred, thereafter, 3 ml of hydrazine hydrate was added to each of them. The resulting suspensions were stirred again for another 30 min, and then transferred into 100 ml Teflon-lined stainless-steel autoclaves and kept at 180 °C in an oven for 24 h. The autoclave was removed from the oven and allowed to cool to room temperature. The resulting precipitates of the different coupled amounts of SnO2 nanoparticles were obtained via centrifugation, and afterwards, washed severally with deionized water and ethanol, and then dried overnight in a hot air oven at 80 °C to obtain needle-like SnO2/g-C3N4. The SnO2 nanoparticles were also synthesised using the same method but without the addition of g-C3N4. The X-ray diffraction pattern of the prepared nanocomposites were obtained using a Shimadzu 6100 X-ray diffractometer with a CU Kα radiation source (Bruker D6). The UV–Vis diffuse reflectance spectra (DRS) of the synthesised SnO2/g-C3N4 composites were measured with Pelkin-Elmer UV–Vis spectrophotometer, the photoluminescence (PL) emissions were obtained via Perkin Elmer LS 55 spectrofluorometer. The BET analysis (Micrometrics Tristar 3000) of the synthesised composites were carried out to determine their specific surface areas and their textural characteristics. Their surface morphologies were also determined using the Scanning Electron Microscopy instrument (Zeiss 10 kV field). Elemental analysis of the composites were also carried out by energy dispersive spectrometry (EDS, Shimadzu), while the interfacial interaction within the synthesised composites were obtained via High-resolution transmission electron microscopy (JEOL TEM), along with the selected area electron diffraction (SAED) pattern. The FTIR of the composites were measured on Perkin Elmer Series 100 Spectrum to determine the functional characteristics of the composites. Electrochemical impedance spectroscopy (EIS) plots were performed by an electrochemical workstation (CHI660C, Shanghai Chenhua Instrument Corporation, China) with a three-electrode system consisting of glassy carbon electrode (GCE) as reference electrode, Pt wire as the counter electrode and ITO glass coated with SnO2/g-C3N4, in an electrolyte solution made up of a mixture of 0.5 mol/L Na2SO4, 2.5 mmol/L potassium hexacyanoferrate (III) (K3[Fe(CN)6]), and 2.5 mmol/L potassium ferrocyanide (K4Fe(CN)6•3H2O). Photocatalytic activity evaluation of the synthesised catalyst The photocatalytic activity of the synthesised SnO2/g-C3N4 composite was evaluated for the photodegradation of tetracyclines under visible light irradiation (250 W xenon lamp with a cut-off filter of 420 nm) in a photocatalytic reactor (Lelesil Innovative System, India). About 50 mg of the SnO2/g-C3N4 composite materials were dispersed in 100 ml, 30 mg/L tetracycline solution. The resulting suspension of mixtures were stirred (using a magnetic stirrer) in the dark (to minimize the decomposition of tetracycline) for 30 min to attain the tetracycline—photocatalyst adsorption–desorption equilibrium. The resultant suspension solutions were exposed to continuous illumination under visible light irradiation (250 W, wavelength λ = 420 nm) for 120 min, with constant magnetic stirring to maintain cooling at temperatures between 25 and 30 °C. About 4 ml aliquots of the visible light irradiated suspensions were taken at 20 min intervals, centrifuged to remove the SnO2/g-C3N4 photocatalysts, and then analysed with UV–visible spectroscopy (UV-3600-Shimadzu, UV–Vis-NIR) at an absorbance of 356 nm to determine the extent of the degradation of tetracycline. The removal efficiency of tetracyclines by the photocatalyst was calculated as follows; $$\mathrm{Removal \,efficiency }= \frac{Co-C }{Co}\times 100\%$$ where Co is the initial concentration of tetracycline solution at time = 0 (t0) (initial time), degradation before treatment, while C is the concentration value at a final time after treatment. Morphology studies The morphologies of the synthesised composites were characterised using SEM, EDX mapping, TEM, SAED, and HRTEM to confirm their crystallinity, composition, and the elemental distribution of the synthesized heterostructure composites. Figure 1(i); shows the SEM images of the exfoliated g-C3N4 nanosheet, SnO2 nanoparticle and SnO2/g-C3N4 heterostructure composites. (iii): TEM images of a g-C3N4 nanosheet, b SnO2 nanoparticles, c SnO2/g-C3N4 and, d HR-TEM image SnO2/g-C3N4 composites The prepared exfoliated g-C3N4 nanosheet are made up of lamellar structure with rough pore surfaces; the SnO2 nanoparticles appear as tiny grain-like particles with near uniform sizes, while the morphology of the exfoliated g-C3N4 sheet remains unchanged. The SEM image of synthesised SnO2/g-C3N4 heterostructure composites showed that the tiny grain-like SnO2 were evenly decorated on the surface of the exfoliated g-C3N4 nanosheet, which probably partly accounts for the suppression of the recombination of photogenerated electron–hole pairs. In addition, the composites were further evaluated to confirm the presence of SnO2 in the SnO2/g-C3N4 by the EDS analysis. The EDS of 3 wt.% SnO2/g-C3N4 composites (Fig. 1(ii)), where the elements Sn, N, C and O were examined, further proved that the composites are composed of SnO2 and g-C3N4. Combined with the content distribution of each element in the surface analysis and comparison with the selected area, Sn is the most abundant, covering the surface of the composites, which is indicative that SnO2 was successfully loaded on g-C3N4. The TEM micrographs of g-C3N4 (Fig. 1(iii)) showed a lamellar shape with several rough sheets while the synthesised needle-like SnO2 nanoparticles were compactly and uniformly distributed on the surface of the synthesised g-C3N4 in the SnO2/g-C3N4 composites. The HRTEM images further confirms the deposition of SnO2 on the surface of exfoliated g-C3N4 nanosheets, and the fabrication of heterostructured SnO2/g-C3N4 nanocomposites (Fig. 1(iii)). The crystallographic nanoparticles lattice fringe pattern of the heterostructures SnO2/g-C3N4 nanocomposites were measured to be 0.24 nm, 0.27 nm and 0.34 nm. The measured lattice fringe patterns with spacings of 0.24 nm, 0.27 nm and 0.34 nm could be respectively ascribed to the interplanar distance of (200), (101) and (110) planes of SnO2 nanocrystals, which when combined with the XRD patterns indicates the formation of heterostructured SnO2/g-C3N4 composites. Moreover, the selected area electron diffraction (SAED) pattern (Fig. 1(iii)d) displays several discrete concentric rings with superimposed bright spots, which confirms that SnO2 are deposited on the surface of the heterostructured SnO2/g-C3N4 nanocomposites. This confirms the successful formation of hybrid heterostructure between SnO2 nanoparticles and exfoliated g-C3N4 nanosheet. X-ray diffraction (XRD) analysis The phase analysis and the crystal orientations of the synthesised SnO2 nanoparticles, g-C3N4 and the corresponding needle-like SnO2/g-C3N4 heterostructure composites, comprising different mass loading of SnO2 were examined by powder X-ray diffraction (XRD). The diffraction patterns of the needle-like SnO2 nanoparticles, g-C3N4 nanosheet and each of the composites were slightly variable, with those for the composites having similar 2θ diffraction (Additional file 1: Fig. S1). The major diffraction peaks in the SnO2 nanosheet diffractogram are 26.54, 34.02, 38.03, 51.72, 54.50, 65.88 and 78.54 2θ degrees. The 2θ diffraction angles correspond to the (110), (101), (200), (211), (002), (301) and (202) crystal planes, which is consistent with the formation of pure cassiterite SnO2 crystallite, devoid of impurities (JCPDS No 041-1445) [26]. Two significant peaks were observed on the diffractogram of the synthesised g-C3N4 nanosheet at 13.19 and 27.39 2θ degrees; indexed to the (002) and (100) diffraction planes assigned to the tris-s-triazine units, which is a characteristic of interlayer stacking of conjugated aromatics systems of graphitic materials (JCPDS No 87-1526) [27]. The diffractogram of the synthesized heterostructure needle-like SnO2/g-C3N4 composites had two characteristic 2θ diffraction peaks at 13.02 and 27.22 2θ, attributed to the crystalline nature of g-C3N4 with a reduced intensity devoid of distinct diffraction peaks corresponding to the diffraction peaks of SnO2 nanoparticles could be observed (Additional file 1: Fig. S2). This might be due to the small amount of SnO2 contents, in the surface of exfoliated g-C3N4 nanosheet and their high dispersion in the interior of polymeric g-C3N4 nanosheet during the preparation of the nanocomposites. This is indicative that a good synergistic interaction occurs between the exfoliated g-C3N4 nanosheet and SnO2 nanoparticles [28]. However, on increasing the amount of SnO2 loaded on g-C3N4 from 1 wt.% and 2 wt.% to 3 wt.% and 5 wt.%, two weak peaks at 33.60° and 51.55° 2θ were observed on the XRD pattern, which were assigned to the (101) and (211) planes of tetragonal SnO2. This is indicative that the as-synthesised photocatalyst demonstrates high crystallinity without any impurity [27, 29]. Additionally, the values of the crystallite sizes for the synthesized heterostructure composites were calculated using the Scherrer equation (D = $\frac{\mathrm{K}\lambda }{\beta \mathrm{Cos}\theta }$), as given [30]; where D is the crystallite size in nm, K is Scherrer's constant ≈ 0.9, λ is the wavelength of the X-ray radiation (CuKα = 0.15406 nm), β is the corrected band broadening (full width at half-maximum (FWHM)) of the diffraction peak, and θ is the diffraction angle. The particle particle size distribution of the synthesized materials generally falls within 8–23 nm (Table 1), which is indicative that they are polycrystalline in nature. Table 1 Specific surface area, pore volume, pore diameter and crystallite sizes of the synthesized SnO2, g-C3N4 and SnO2/g-C3N4 composites Optical characteristics: absorption and photoluminescence The optical properties of the synthesised composites were studied to estimate their ability to absorb visible light. The bandgap of all the synthesised composites were calculated according to the Kubelka–Munk equation, αhѴ = A(hѴ-Eg)1/2 [31]; where α represents the adsorption coefficient; h, Planck's constant; ν, for light frequency; while Eg and A are the bandgap energy, and constantly called band tailing parameter respectively. The synthesised needle-like SnO2 nanoparticles absorb in the visible light region at about 478 nm (Fig. 2). Therefore, the bandgap energy of the as-synthesised needle-like SnO2 nanoparticles were calculated to be approximately 2.64 eV. The observed absorption and calculated bandgap may be attributed to the higher oxygen concentration due to the formation of extra ionized oxygen vacancies within the synthesised SnO2 nanoparticles when compared to other synthesised SnO2 with bandgap energy of about 3.60 eV [32, 33]. The g-C3N4 nanosheet absorbs at approximately around 458 nm, with a calculated energy bandgap of about 2.70 eV; while the energy bandgap of the synthesised composites with different weights of SnO2 loaded on exfoliated g-C3N4 are close to 2.52 eV for 10% SnO2/g-C3N4, and 2.45 eV for each of 2, 3, 5 wt.% SnO2/g-C3N4 heterostructure composites, respectively. The calculated energy band gap values for various composites of SnO2/g-C3N4 composites imply that they are photosensitive, and have the ability to respond to visible light. This also signifies that the fabrication of heterostructure composites can hinder the recombination of charge carriers, thereby causing a shift in their light response-ability towards the visible region of the spectrum. Thus, the materials synthesised SnO2/g-C3N4 possess narrow bandgap, which consequently, lead to improved light harvesting capacity of the material, compared to 2.50 eV and 2.68 eV reported for the degradation of some organic pollutants [34, 35]. a UV–Vis DRS and, b Kubelka–Munk model of SnO2, g-C3N4 and its corresponding heterostructured composites The photoluminescence (PL) spectra of the synthesised composites were assessed to evaluate the migration, transfer, and recombination rate of photogenerated electron–hole pairs. This is important as it helps to understand the interaction between the catalyst involved in the formation of heterojunction structure within the composites photocatalyst and the separation of the electron-holes pair, which leads to their photocatalytic activity in photodegradation application [36]. The maximum absorption wavelength used as PL excitation wavelength, was 350 nm as obtained from the UV DSR analysis. The exfoliated g-C3N4 nanosheets showed a strong emission peak which appears at about 425 nm (Fig. 3a). This is characteristic, for fast recombination of photogenerated charge carriers in g-C3N4 [37]. The observed emission bands for the synthesized needle-like SnO2 nanoparticles were 425, 460, 484 and 532 nm (inserts in Fig. 3a). The above emission bands can be attributed to phenomenons such as crystal defects or surface defects, oxygen vacancies, tin vacancies and tin interstitials attributes in the materials [38]. Pure SnO2 nanoparticles showed a less intense peak when compared to the intensity of the peak of the synthesised exfoliated g-C3N4. This suggests that the synthesized needle-like SnO2 nanoparticles consist of a lower electron–hole recombination rate or slow recombination of photoinduced charge carriers. a PL spectra of g-C3N4, SnO2 and SnO2/g-C3N4 composites with different compositions of SnO2 and, b their corresponding EIS analysis plot Moreover, the introduction of different masses of SnO2 nanoparticles on the exfoliated g-C3N4 nanosheets significantly reduced the emission peak of the exfoliated g-C3N4 nanosheet. The decrease in the emission intensities of the different needle-like SnO2/g-C3N4 composites result from the formation of heterojunction between needle-like SnO2 nanoparticles and the g-C3N4 nanosheet, which lower electron–hole recombination or slows recombination of photoinduced charge carriers, weakened by the addition of SnO2 nanoparticles [39]. This suggests that the synergistic interaction between SnO2 nanoparticles and g-C3N4 nanosheet considerably leads to the decrease in the recombination of the photoinduced charge carrier, which is evident in its ability to degrade tetracycline under visible light. Furthermore, electrochemical impedance spectroscopy (EIS) analysis was conducted to confirm the relevant charge transfer process and the recombination rate between the photogenerated electrons and holes. The EIS plot (Fig. 3b) reveals that the steady state perturbation in the different SnO2/g-C3N4 heterostructure composites (indicated by the magnitude of the electrical impedance frequency) are much smaller than that of g-C3N4. This indicates that a higher charge transfer rate occurred in SnO2/g-C3N4 composites, resulting in less obstruction in the transfer of electrons-holes, and thus, a more efficient separation of the charge [40]. Therefore, the % mass content of SnO2 nanoparticles in the SnO2/g-C3N4 composites clearly influences the magnitude of EI, which is consistent with the PL analysis. The 3 wt.% SnO2/g-C3N4 showed lower resistance and higher separation efficiency for the photogenerated electrons and holes, thereby enhancing their potential photocatalytic efficiency for organic pollutants' degradation compared to other composites. The FTIR spectra of the synthesised needle-like SnO2 nanoparticles, g-C3N4 nanosheet and all the SnO2/g-C3N4 composites were measured (Additional file 1: Fig. S3). The spectra patterns provide information and insight into the changes in the structure of the exfoliated g-C3N4 nanosheet. The FTIR spectrum of SnO2 showed a broad peak around 527 cm−1, characteristic of Sn–O stretching vibration of Sn–O-Sn, while the peaks at 1648 and 3362 cm−1 are attributed to the molecular water bending vibration and hydroxyl groups stretching vibration, respectively [39]. FTIR spectrum of g-C3N4 shows a peak at 804 cm−1 which is considered as an out-of-plane bending vibration characteristic of a tris-s-triazine ring of the g-C3N4 main building block. Peaks from 1204 to 1626 cm−1 are related to the stretching vibration modes of CN heterocycles, while the broad peak at around 3086 cm−1 and 3158 cm−1 are attributed to the stretching mode of N–H bond from the uncondensed amino groups and O–H band from absorbed water molecules [41]. The FTIR spectrum of SnO2/g-C3N4 composites was made up of the main characteristic peaks of g-C3N4, with a slight shift in their wavenumbers, which indicates a strong interaction between exfoliated g-C3N4 nanosheet and SnO2 particles. Sorption–desorption studies The N2 adsorption–desorption isotherms of the synthesised composites are shown in Fig. 4. The four composites of SnO2/g-C3N4 composites with the different masses of SnO2 particles with SnO2 and g-C3N4 showed type IV isotherms, while its hysteresis loops are categorized as type H3. According to the IUPAC classification, heterostructured composite materials of this type are listed as mesoporous in nature [42]. Mesoporous materials are of high surface area, and when used as photocatalyst; they have been reported to enhanced visible-light harvesting capacity vis-a-viz mesostructure light trapping in their molecules, which in turn enhances their photocatalytic efficiency by the generation of more reactive species that improves reactant adsorption for better photon-reactant molecules interactions [41]. N2 adsorption of a g-C3N4, b SnO2, c 1% SnO2/g-C3N4, d 2% SnO2/g-C3N4, e 3% SnO2/g-C3N4, f 5% SnO2/g-C3N4 g and with their respective pore size distribution plot inserted The textural properties of the synthesised SnO2, g-C3N4, and the different masses of SnO2/g-C3N4 heterostructure composites such as; BET surface area, pore volume and average pore diameter are listed in Table 1. The synthesised needle-like SnO2 nanoparticles display a high surface area of 38.06 m3/g when compared to the synthesised exfoliated g-C3N4 nanosheet with surface area of 25.22 m3/g. However, the introduction of SnO2 into the g-C3N4 nanosheet results in a slight improvement in the surface area with an increase in their pore volume and pore diameter of the different composites. The 3 wt.% SnO2/g-C3N4 composites showed the best textural properties of 34.37 m3/g, 0.137 cm3/g, 28.62 nm for it's surface area, pore volume and pore diameter, respectively. The improvement in the surface area of the composites indicates that some changes occurred during the incorporation of SnO2 on g-C3N4, which could facilitate the transfer of photoinduced electron–hole pairs, hence, improving their photocatalytic efficiency for the degradation of organic pollutants. Thermogravimetric (TGA) analysis The thermal stability of the synthesised materials were analysed by thermogravimetric analysis in an atmosphere of air at a heating rate of 10.00 °C/min. An initial weight loss at the temperature of 50–200 °C was observed for all materials (Additional file 1: Fig. S4a). This can be attributed to the loss of absorbed water and other substances on the surface of the materials [43]. The gradual weight loss for g-C3N4 after 550 °C may be associated with the combustion of the carbon skeleton and the decomposition of defects and edge functional groups like uncondensed amine functional groups and the edge cyano-group that exist within g-C3N4 nanosheet [44]. Meanwhile, the weight loss of the different masses of SnO2 nanoparticle loaded on the g-C3N4 sheet between the temperature range of 400 and 550 °C may be ascribed to the combustible carbon and the decomposition of the functional groups in the g-C3N4 structure, while the SnO2 nanoparticle maintains its stability with a nearly straight line [45]. The specific decomposition temperatures of the synthesised materials are 703 °C, 658 °C, 635 °C with 50% SnO2/g-C3N4 showing the lowest stability at 595 °C, while the g-C3N4 nanosheet shows the highest stability of 720 °C (Additional file 1: Fig. S4b). The TGA revealed that the synthesised materials are thermally stable, while its composition are consistent with the original materials. Photocatalytic evaluation The photocatalytic activity of the composites with different needle-like SnO2 nanoparticles mass loads on exfoliated g-C3N4 nanosheets for the photocatalytic degradation of tetracycline under visible light irradiation the effects of SnO2 mass loads was investigated. The visible light-assisted degradation of tetracycline in reaction systems in which the different synthesised composites were dispersed and dark stabilised for 30 min to attain adsorption–desorption equilibrium prior to visible light exposure were fairly. The exposure of the tetracycline solution to visible light irradiation caused minimum degradation of about 26.90%, while a 40.92% and 51.32% degradation was observed when exfoliated g-C3N4 nanosheet and needle-like SnO2 were added to the tetracycline solution as a catalyst, respectively, after 120 min (Fig. 5a). a photocatalytic activities of the synthesized composite, b kinetics studies The concentration of tetracycline decreased significantly when the different composites SnO2/g-C3N4 were used as catalysts. The degradation efficiencies of tetracycline by the 1 wt.%, 2 wt.%, 3 wt.% and 5 wt.% SnO2/g-C3N4 photocatalyst were 81.54%, 90.57%, 95.90% and 92.15%, respectively. It is important to note that the photocatalytic ability of the different SnO2/g-C3N4 composites increased with respect to the amount of SnO2 nanoparticles added to the g-C3N4 nanosheet until 3 wt.% SnO2/g-C3N4 composite, which displayed the highest degradation of 95.90% tetracycline degradation within 120 min irradiation under visible light, after which a decrease was observed with increase in % wt. SnO2 load. This implies that an increase in the content of SnO2 in the SnO2/g-C3N4 composites plays an important role in catalysing the degradation of tetracycline molecules vis-a-viz the formation of heterojunction within the structural configuration of the SnO2/gC3N4 composites. The heterostructured characteristics of the SnO2/gC3N4 play a crucial role in improving the photocatalytic activity of the composites. Hence, the photocatalytic efficiency can be optimized by adjusting the content of SnO2 nanoparticles on exfoliated g-C3N4 nanosheet. More also, the efficiently high tetracycline degradation of 95.90% by the 30% SnO2/g-C3N4 composite could be because of their promoted charge separation, which results in increased quantum confinement leading and low recombination of photogenerated charge, as well as its larger surface area when compared to other composites (Table 1). On the other hand, increasing the SnO2 composite content to 5 wt.% in the SnO2/g-C3N4 led to a decrease in the photocatalytic activity. This may be due to the shielding of the active sites as a result of the excess addition of SnO2 on the exfoliated g-C3N4 nanosheet, which in turn hinders the light-capturing capacity of g-C3N4 under visible light irradiation during the photodegradation process [46]. The photocatalytic kinetics for the degradation of tetracycline was studied. The degradation process obeyed the pseudo-first order kinetics model ($-In(\frac{C}{Co})=kt$); where k represents the reaction rate constant while Co and C stand for the initial concentration and final concentration of tetracycline at time t, respectively. The calculated rate constant (k) and the mean regression coefficients (R2) values for the kinetics of the degradation of tetracycline with different amounts of SnO2 nanoparticles loaded on the g-C3N4 sheets are presented in Fig. 5b. The calculated k values are 0.0062, 0.00671, 0.0107, 0.0215, 0.0235 and 0.0222 min−1 for 1 wt.%, 2 wt.%, 3 wt.% and 5 wt.% SnO2QDs/g-C3N4, respectively (Table 2). The as-synthesised 3 wt.% SnO2/g-C3N4 nanocomposites exhibited a superior k value when compared to the other nanocomposites, generating a rate constant that is 6.32 and 4.33 times better than those of pure g-C3N4 nanosheet and SnO2 nanoparticle, respectively, for the degradation of tetracycline. This showed that loading different amounts of SnO2 on g-C3N4 can improve the degradation of tetracycline under visible light. This can be attributed to a decrease in the recombination of the electron–hole pairs through the formation of localized states between the conduction and valence bands. Table 2. Summary of photocatalytic degradation of tetracycline visible light irradiation using SnO2/g-C3N4 heterostructured composites Photocatalyst reusability Recyclability and stability are among the important parameters considered in the practicality of a photocatalyst. This was evaluated for the practical application of the synthesised SnO2/g-C3N4 heterostructured composites in five cycling experiments, for the degradation of tetracycline under the same reaction condition (50 mg of the catalyst dispersed in 100 ml, 30 mg/L tetracycline solution). At the end of each reaction cycle, the spent solutions were centrifuged to recover the SnO2/g-C3N4 composites used as catalysts and washed severally with water and ethanol, followed by overnight drying in the oven at 80 °C. About 86.78% of the tetracycline was degraded on the fifth run after 120 min, showing a slight reduction in the photocatalytic efficiency of 3 wt.% SnO2/g-C3N4 compared to the first run which had 95.90% degradation, under the same photocatalytic condition (Additional file 1: Fig. S5). This might be due to the loss of the photocatalyst charge during the recycling process. Furthermore, the crystal structure of the 3 wt.% SnO2/g-C3N4 composite photocatalyst was investigated prior to- and after use in five-cycle experiments of the photocatalytic process, to evaluate and confirm whether there was any distortion or changed in the crystal structure, using XRD analysis. The results showed that there was no significant change in their crystalline structure, thus suggesting that the chemical structures of the photocatalysts were not affected after use, hence their stability. It can therefore be inferred that the 3 wt.% SnO2/g-C3N4 displayed high stability under visible light for the degradation of tetracycline, probably because of the π-π stacking interaction that exists between the photocatalyst and the tested pollutants [47]. To understand the photocatalytic mechanisms and elucidate the reactive species involved during the photocatalytic degradation of tetracycline using 3 wt.% SnO2/g-C3N4 heterostructured composite under visible light irradiation, various scavengers such as potassium iodide KI for H+ scavenging, p-benzoquinone (PBQ) for ·O2−, Isopropyl alcohol (IPA) for ·OH scavenging and lastly thiourea for scavenging OH·, H+ and e− were used as suggested by Bui, et al. [48]. The same experimental procedure for the photocatalytic activity were used. Results revealed that the introduction of thiourea into the reaction medium led to a considerable decrease in tetracycline degradation (65.54%); KI addition led to 87.62% removal efficiency, PBQ caused 56.41 removals. However, the addition of IPA significantly suppresses tetracycline degradation by 41.38% when compared to degradation without scavenger (Additional file 1: Fig. S6). These results indicate that the presence of •OH and •O2− radical proved to be the major reactive species for the degradation of tetracycline using 3 wt.% SnO2/g-C3N4 as photo-catalyst. The calculated conduction band (CB) and valence band (VB) of SnO2 are 3.04 eV and 0.40 eV while that of g-C3N4 nanosheet are − 1.12 eV and 1.58 eV, respectively; showing that the compositing semiconductors have suitable band potentials that led to the formation of heterojunction structures to restrain the recombination of electron-holes pairs. Based on the characterisation and experimental data obtained in this study, a tentative mechanism for the photodegradation of tetracycline using SnO2/g-C3N4 composites under visible light irradiation is proposed (Additional file 1: Fig. S7). Upon reception of visible light irradiation on the composites, both g-C3N4 and SnO2 generated electrons and holes, leading to te migration of the photoexcited electrons from the CB of g-C3N4 nanosheet to the CB of SnO2 nanoparticles. In contrast, the photoexcited holes migrates from the VB of g-C3N4 to the VB of SnO2 resulting in the effective separation of the photoexcited charge carriers, which enhances their photocatalytic performance for the degradation of tetracycline under visible light irradiation. From the Mott-Schottky plot, as shown in Additional file 1: Fig. S8, the flat band potential for the g-C3N4, SnO2 and the 3 wt.% SnO2/g-C3N4 composites were estimated to be − 1.33, − 0.12 and − 1.15 eV. Generally, the conduction band potential of a semiconductor is − 0.1 or − 0.2 eV, more negative than the flat band potential. Therefore the estimated conductor band from the recorded flat band potentials are − 1.43, − 0.22, − 1.25 eV. Moreover, the valence band potential of g-C3N4, SnO2 and SnO2/g-C3N4 were calculated (1.27, 2.42 and 1.20 eV) using the Butler and Ginley equations [49] as given in Eqs. 1 and 2: $$ {{E}}_{{{\text{VB}}}} = \chi - {\text{E}}^{{\text{e}}} + 0.{\text{5E}}_{{\text{g}}} $$ $$ {{E}}_{{{\text{CB}}}} = {\text{E}}_{{{\text{VB}}}} - {\text{E}}_{{\text{g}}} $$ where the value of Ec is 4.5 eV, X value of SnO2 is 6.22 eV, while that of g-C3N4 is 4.73 eV which was similar to the values reported by Sun et al. [50]. The bandgap energy (Eg) of g-C3N4 and SnO2 as obtained from UV–Vis DRS are 2.70 eV and 2.64 eV. The electronic structure of the composites was influenced by the large percentage composition of g-C3N4 being the base material, while the small quantity of SnO2 in the composites led to the suppression of its electronic structure. From the above discussion, it is evident that the conduction band potential of the composites is higher than the required electron reduction potential (− 0.33) for the conversion of an oxygen molecule to superoxide anions radicals. Hence the reason superoxide anion radical is the determining oxygen species for the photocatalytic degradation of tetracycline. On the other hand, the valence band (1.20 eV) is less positive than the required oxidative potential for the conversion of H2O molecule and OH− ions to hydroxyl radicals (HO•) which explains the insufficient amount of hydroxyl radicals in the system. Moreover, the conduction band position of g-C3N4 nanosheet is −1.12 eV vs. NHE, which is more negative than that of SnO2 nanoparticle (0.04 eV vs. NHE) with a redox potential of O2/•O2− (+ 0.28 eV vs NHE). This shows that the photogenerated electrons could react with the adsorbed O2 to produce active oxygen species •O2− radicals. Series of visible light-driven SnO2/g-C3N4 heterostructure composites were synthesised via a facile hydrothermal method. The loading of different masses of SnO2 nanoparticles on the synthesised exfoliated g-C3N4 sheets produces a strong heterojunction and improved specific surface area coupled with good optical properties. As revealed by the SEM, TEM and HR-TEM the synthesised composites generated a needle-like crystal structure with the uniform distribution of the SnO2 nanoparticles on the surface of the g-C3N4 nanosheet while the PL result confirm the decrease in the recombination rate due to the introduction of SnO2 nanoparticle on to the surface of the exfoliated g-C3N4 sheet. After comparison, 3 wt.% SnO2/g-C3N4 composites exhibited a higher photocatalytic performance than other composites as a result of improved specific surface area and charge separation efficiency. From the photocatalytic analysis, it was observed that the •O2− and •OH, radicals are the main active species generated during the photocatalysis process due to the action of electron reduction. Moreover, the kinetic rate constant, recyclability and stability of the synthesised composites for the degradation of tetracycline were also analysed with obtained results showing excellent capability and robustness of the composites towards the degradation of tetracycline under visible light irradiation. The heterostructure composite mechanism pathway for the degradation of tetracycline under visible light irradiation were also proposed. Attributable to the improved performances of the synthesised composites for visible-light responsive applications, materials with such efficiency have potential environmental applications. The data sets used and/or analyzed during the current study are available from the corresponding author on reasonable request. He K, Borthwick AG, Lin Y, Li Y, Fu J, Wong Y, Liu W (2020) Sale-based estimation of pharmaceutical concentrations and associated environmental risk in the Japanese wastewater system. 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ACS Appl Mater Interfaces 6(19):16745–16754 School of Chemistry and Physics, College of Agriculture, Engineering and Science, University of KwaZulu-Natal, Westville, Durban, 4000, South Africa Adewumi Olufemi Oluwole & Olatunde Stephen Olatunji Adewumi Olufemi Oluwole Olatunde Stephen Olatunji AO and OO contributed equally to this work. Both authors read and approved the final manuscript. Correspondence to Olatunde Stephen Olatunji. The authors wish to state that there was no conflict of interest. Additional file 1: Fig. S1. XRD patterns of (a) SnO2 nanoparticles, (b) g-C3N4 nanosheet and, (c) different mass combinations of SnO2/g-C3N4 heterostructure composites. Fig. S2. Diffractograms of synthesized heterostructure needle-like SnO2/g-C3N4 composites devoid of distinct diffraction peaks of SnO2 nanoparticles. Fig. S3. FTIR Spectra of pure SnO2, g-C3N4, and SnO2/g-C3N4 composites. Fig. S4. (a) TGA thermograms and (b) the corresponding derivative thermograms of the synthesized heterostructured composites. Fig. S5. (a) recyclability test of 30% SnO2/g-C3N4 composite and (b) XRD pattern of 30%SnO2/g-C3N4 before photoreaction and used after five repeated cycles of photodegradation experiments. Fig. S6. showing the effects of various scavengers on the tetracycline degradation using 3% SnO2/g-C3N4 under visible light irradiation. Fig. S7. Reaction mechanism pathway for the degradation of tetracycline through 30% SnO2/g-C3N4. Fig. S8. Mott-Schottky plots measured at 1 kHz in 5 mM [Fe(CN)6]4−/3− solution. Oluwole, A.O., Olatunji, O.S. Photocatalytic degradation of tetracycline in aqueous systems under visible light irridiation using needle-like SnO2 nanoparticles anchored on exfoliated g-C3N4. Environ Sci Eur 34, 5 (2022). https://doi.org/10.1186/s12302-021-00588-7 Needle-like SnO2 SnO2/g-C3N4 Nanocomposite Tetracycline degradation Reactive species	CommonCrawl
Revisiting design principles of Salsa and ChaCha AMC Home Embedding cover-free families and cryptographical applications November 2019, 13(4): 645-688. doi: 10.3934/amc.2019040 Another look at success probability of linear cryptanalysis Subhabrata Samajder and Palash Sarkar , Applied Statistics Unit, Indian Statistical Institute, 203, B.T.Road, Kolkata, 700108, India * Corresponding author Received October 2018 Revised January 2019 Published June 2019 Figure(3) / Table(2) This work studies the success probability of key recovery attacks based on using a single linear approximation. Previous works had analysed success probability under different hypotheses on the distributions of correlations for the right and wrong key choices. This work puts forward a unifying framework of general key randomisation hypotheses. All previously used key randomisation hypotheses as also zero correlation attacks can be seen as special cases of the general framework. Derivations of expressions for the success probability are carried out under both the settings of the plaintexts being sampled with and without replacements. Compared to previous analysis, we uncover several new cases which have not been considered in the literature. For most of the cases which have been considered earlier, we provide complete expressions for the respective success probabilities. Finally, the full picture of the dependence of the success probability on the data complexity is revealed. Compared to the extant literature, our work provides a deeper and more thorough understanding of the success probability of single linear cryptanalysis. Keywords: Linear cryptanalysis, success probability, data complexity. Mathematics Subject Classification: 94A60, 11T71, 68P25, 62P99. Citation: Subhabrata Samajder, Palash Sarkar. Another look at success probability of linear cryptanalysis. Advances in Mathematics of Communications, 2019, 13 (4) : 645-688. doi: 10.3934/amc.2019040 T. 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WKRH) is an abbreviation for right (resp. wrong) key randomisation hypothesis; std (resp. adj) denotes whether the standard (resp. adjusted) key randomisation hypothesis is considered type samp. RKRH WKRH cond. previous $P_{S}$ new $P_{S}$ $p \neq 1/2$ wr std std - [33] Section 5.4, Eqn (41) std adj - [12] Section 5.5, Eqn (44) adj std - - Section 5.6, Eqn (46) adj adj - [8] Section 5.7, Eqn (48) wor std std - - Section 5.4, Eqn (42) std adj - [1] Section 5.5, Eqn (44) adj adj - - Section 5.7, Eqn (49) $p = 1/2$ wr std adj - - Section 7.1, Eqn (57) adj adj ${\mathtt{ELP}}>2^{-n}$ [7] Section 7.3, Eqn (62) adj adj ${\mathtt{ELP}} < 2^{-n}$ - Section 7.3, Eqn (63) wor std adj - - Section 7.1, Eqn (58) Table 2. Summary of the different cases and sub-cases showing the dependence of the success probability on the data complexity for $ p\neq 1/2 $. 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Complete characterization of the first descent point distribution for the k-error linear complexity of 2n-periodic binary sequences. Advances in Mathematics of Communications, 2017, 11 (3) : 429-444. doi: 10.3934/amc.2017036 Jianqin Zhou, Wanquan Liu, Xifeng Wang, Guanglu Zhou. On the $ k $-error linear complexity for $ p^n $-periodic binary sequences via hypercube theory. Mathematical Foundations of Computing, 2019, 2 (4) : 279-297. doi: 10.3934/mfc.2019018 G. R. Cirmi, S. Leonardi. Higher differentiability for solutions of linear elliptic systems with measure data. Discrete & Continuous Dynamical Systems - A, 2010, 26 (1) : 89-104. doi: 10.3934/dcds.2010.26.89 Rainer Steinwandt, Adriana Suárez Corona. Cryptanalysis of a 2-party key establishment based on a semigroup action problem. Advances in Mathematics of Communications, 2011, 5 (1) : 87-92. doi: 10.3934/amc.2011.5.87 Valentin Afraimovich, Lev Glebsky, Rosendo Vazquez. Measures related to metric complexity. 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physics- and probability-related approaches to integer partitioning problems (approximately chronological bibliography) F.C. Auluck, S. Chowla and G. Gupta, "On the maximum value of the number of partitions of $n$ into $k$ parts", Journal of the Indian Mathematical Society 6 (1942) 105 F.C. Auluck and D.S. Kothari, "Statistical mechanics and the partitions of numbers", Proceedings of the Cambridge Philosophical Society 42 (1946) 272 [abstract:] "The properties of partitions of numbers extensively investigated by Hardy and Ramanujan have proved to be of outstanding mathematical interest. The first physical application known to us of the Hardy–Ramanujan asymptotic expression for the number of possible ways any integer can be written as the sum of smaller positive integers is due to Bohr and Kalckar* for estimating the density of energy levels for a heavy nucleus. The present paper is concerned with the study of thermodynamical assemblies corresponding to the partition functions familiar in the theory of numbers. Such a discussion is not only of intrinsic interest, but it also leads to some properties of partition functions, which, we believe, have not been explicitly noticed before. Here we shall only consider an assembly of identical (Bose–Einstein, and Fermi–Dirac) linear simple-harmonic oscillators. The discussion will be extended to assemblies of non-interacting particles in a subsequent paper." *N. Bohr and F. Kalckar, "On the transmutation of atomic nuclei by impact of material particles. I. General theoretical remarks", Kgl. Danske Vid. Selskab. Math. Phys. Medd. 14 No. 10 (1937) [see also C. Van Lier and G.E. Uhlenbeck, "On the statistical calculation of the density of the energy levels of the nuclei", Physica 4 (1937) 531] H.N.V. Temperley, "Statistical mechanics and the partition of numbers. I. The transition of liquid helium", Proceedings of the Royal Society of London A 199 (1949) 361–375 [abstract:] "The existing theory of 'Bose–Einstein condensation' is comparared with some results obtained from the theory of partition of numbers. Two models are examined, one in which the energy levels are all equally spaced, the other being the perfect gas model. It is concluded that orthodox theory can be relied upon at very high and at very low temperatures, also that the condensation phenomenon is a real one, but that it is not correctly described by orthodox theory, the position of the transition temperature and the form of the specific heat anomaly both being given wrongly." H.N.V. Temperley, "Statistical mechanics and the partition of numbers: II. The form of crystal surfaces", Proceedings of the Cambridge Philosophical Society 48 (1952) 683–697 [abstract:] "The classical theory of partition of numbers is applied to the problem of determining the equilibrium profile of a simple cubic crystal. It is concluded that it may be thermodynamically profitable for the surface to be 'saw-toothed' rather than flat, the extra entropy associated with such an arrangement compensating for the additional surface energy. For both a two- and a three-dimensional 'saw-tooth' the extra entropy varies, to a first approxomation, in the same way as the surface energy, i.e. is proportional to $N^{1/2}$ or $N^{2/3}$ respectively, where $N$ is the number of molecules in a 'tooth'. For the simple cubic lattice, the entropy associated with the formation of a tooth containing $N$ atoms is estimated to be $3.3kN^{2/3}$. It is also possible to estimate the variation of the 'equilibrium roughness' of a crystal with temperature, if its surface energy is known." V.S. Nanda, "Partition theory and thermodynamics of multi-dimensional oscillator assemblies", Proceedings of the Cambridge Philosophical Society 47 (1951) 591–601 [abstract:] "The close similarity between the basic problems in statistical thermodynamics and the partition theory of numbers is now well recognized. In either case one is concerned with partitioning a large integer, under certain restrictions, which in effect means that the 'Zustandsumme' of a thermodynamic assembly is identical with the generating function of partitions appropriate to that assembly. The thermodynamic approach to the partition problem is of considerable interest as it has led to generalizations which so far have not yielded to the methods of the analytic theory of numbers. An interesting example is provided in a recent paper of Agarwala and Auluck where the Hardy–Ramanujan formula for partitions into integral powers of integers is shown to be valid for non-integral powers as well. The present paper is concerned with the problems in partition theory of numbers corresponding to the thermodynamic assemblies of two and three-dimensional oscillators. Asymptotic expressions are deduced which constitute a generalization of the Hardy–Ramanujan formula for $p(n)$ which corresponds to an assembly of $\emph{linear oscillators}$. Generating functions similar to those considered here were studied earlier by MacMahon in his work on combinatory analysis. It is remarkable that the Zustandsumme of an assembly of a variable number of two-dimensional oscillators is identical with the generating function of plane partitions. The problem, thus, becomes one of establishing a relationship between the two seemingly different types of partitions. Further, it is noticed that a study of two-dimensional oscillator assembly is connected with the partitions of bi-partite numbers." V.S. Nanda, "Bose–Einstein condensation and the partition theory of numbers", Proc. Nat. Inst. Sci. (India) 19 (1953) 681–690 In Section 2.3 of Bernard Julia's seminal 1990 paper "Statistical theory of numbers", the author turns briefly from multiplicative to additive number theory, in particular to generating functionals associated with integer partition problems. He relates these to the Veneziano open string model, the tachyon mode, and the phenomenon of "bosonization" which is discussed elsewhere in the paper. F.Y. Wu, G. Rollet, H.Y. Huang, J.M. Maillard, C.K. Hu and C.N. Chen, "Directed compact lattice animals, restricted partitions of an integer, and the infinite-state Potts model", Phys. Rev. Lett. 76 (1996) 173–176 [abstract:] "We consider a directed compact site lattice animal problem on the $d$-dimensional hypercubic lattice, and establish its equivalence with (i) the infinite-state Potts model and (ii) the enumeration of $(d-1)$-dimensional restricted partitions of an integer. The directed compact lattice animal problem is solved exactly in $d = 2,3$ using known solutions of the enumeration problem. The maximum number of lattice animals of size $n$ grows as $\exp(cn^{(d-1)/d})$. Also, the infinite-state Potts model solution leads to a conjectured limiting form for the generating function of restricted partitions for $d>3$, the latter an unsolved problem in number theory." F.Y. Wu, "The infinite-state Potts model and restricted multidimensional partitions of an integer", Mathematical and Computer Modelling 26 (1997) 269–274 [abstract:] "It is shown that the partition function of the $q$-state Potts model on a finite $d$-dimensional hypercubic lattice in the $q\rightarrow\infty$ limit is precisely the generating function of $(d-1)$-dimensional restricted partitions of an integer. For $d=2,3$, this equivalence leads to closed-form expressions of the $q=\infty$ Potts partition function. Our discussion also establishes symmetry and reciprocal properties for the generating function of restricted partitions in higher dimensions." A.M. Vershik, "Statistical mechanics of combinatorial partitions, and their limit shapes", Funkts. Anal. Prilozh. 30 (1996) 19–30 [English translation: Funct. Anal. Appl. 30 (1996) 90–105] A.M. Vershik, "Limit distribution of the energy of a quantum ideal gas from the viewpoint of the theory of partitions of natural numbers", Uspekhi Mat. Nauk 52 (1997) 139–146 [English translation: Russian Math. Surveys 52 (1997) 379–386] S. Grossmann and M. Holthaus, "Microcanonical fluctuations of a Bose system's ground state occupation number", Phys. Rev. E 54 (1996) 3495–3498 [abstract:] "Employing asymptotic formulas from the partition theory of numbers, we derive the microcanonical probability distribution of the ground state occupation number for a one-dimensional ideal Bose gas confined at low temperatures by a harmonic potential. We compare the grand canonical analysis to the microcanonical one, and show how the fluctuation catastrophe characteristic for the grand canonical ensemble is avoided by the proper microcanonical approach." M. Holthaus, E. Kalinowski and K. Kirsten, "Condensate fluctations in trapped Bose gases: Canonical vs. microcanonical ensemble", Annals of Physics 270 (1998) 198–230 [abstract:] "We study the fluctuation of the number of particles in ideal Bose–Einstein condensates, both within the canonical and the microcanonical ensemble. Employing the Mellin–Barnes transformation, we derive simple expressions that link the canonical number of condensate particles, its fluctuation, and the difference between canonical and microcanonical fluctuations to the poles of a Zeta function that is determined by the excited single-particle levels of the trapping potential. For the particular examples of one- and three-dimensional harmonic traps we explore the microcanonical statistics in detail, with the help of the saddle-point method. Emphasizing the close connection between the partition theory of integer numbers and the statistical mechanics of ideal Bosons in one-dimensional harmonic traps, and utilizing thermodynamical arguments, we also derive an accurate formula for the fluctuations of the number of summands that occur when a large integer is partitioned." S. Grossmann and M. Holthaus, "From number theory to statistical mechanics: Bose–Einstein condensation in isolated traps", Chaos, Solitons and Fractals 10 No. 4–5 (1999) 795–804 [abstract:] "We question the validity of the grand canonical ensemble for the description of Bose–Einstein condensation of small ideal Bose gas samples in isolated harmonic traps. While the ground state fraction and specific heat capacity can be well approximated with the help of the conventional grand canonical arguments, the calculation of the fluctuation of the number of particles contained in the condensate requires a microcanonical approach. Resorting to the theory of restricted partitions of integer numbers, we present analytical and numerical results for such fluctuations in one- and three-dimensional traps, and show that their magnitude is essentially independent of the total particle number." C. Weiss and M. Holthaus, "Asymptotics of the number partitioning distribution", Europhys. Lett. 59 (4) (2002) 486–492 [abstract:] "The number partitioning problem can be interpreted physically in terms of a thermally isolated noninteracting Bose gas trapped in a one-dimensional harmonic-oscillator potential. We exploit this analogy to characterize, by means of a detour to the Bose gas within the canonical ensemble, the probability distribution for finding a specified number of summands in a randomly chosen partition of an integer $n$. It is shown that this distribution approaches is asymptotics only for $n>10^{10}$." C. Weiss, M. Block, M. Holthaus and G. Schmieder, "Cumulants of partitions", J. Phys. A 36 (2003) 1827–1844 [abstract:] "We utilize the formal equivalence between the number-partitioning problem and a harmonically trapped ideal Bose gas within the microcanonical ensemble for characterizing the probability distribution which governs the number of addends occurring in an unrestricted partition of a natural number $n$. By deriving accurate asymptotic formulae for its coefficients of skewness and excess, it is shown that this distribution remains non-Gaussian even when $n$ is made arbitrarily large. Both skewness and excess vary substantially before settling to their constant-limiting values for $n>10^{10}$." M. Holthaus, K.T. Kapale, V.V. Kocharovsky and M.O. Scully, "Master equation vs. partition function: Canonical statistics of ideal Bose–Einstein condensates", Physica A 300 (2001) 433–467 [abstract:] "Within the canonical ensemble, a partially condensed ideal Bose gas with arbitrary single-particle energies is equivalent to a system of uncoupled harmonic oscillators. We exploit this equivalence for deriving a formula which expresses all cumulants of the canonical distribution governing the number of condensate particles in terms of the poles of a generalized Zeta function provided by the single-particle spectrum. This formula lends itself to systematitic asymptotic expansions which capture the non-Gaussian character of the condensate fluctuations with utmost precision even for relatively small, finite systems, as confirmed by comparison with exact numerical calculations. We use these results for assessing the accuracy of a recently developed master equation approach to the canonical condensate statistics; this approach turns out to be quite accurate even when the master equation is solved within a simple quasithermal approximation. As a further application of the cumulant formula we show that, and explain why, all cumulants of a homogeneous Bose–Einstein condensate "in a box" higher than the first retain a dependence on the boundary conditions in the thermodynamic limit." D.P. Bhatia, M.A. Prasad and D. Arora, "Asymptotic results for the number of multidimensional partitions of an integer and directed compact lattice animals", J. Phys. A 30 (1997) 2281–2285 [abstract:] "There is an exact one-to-one correspondence between the number of $(d-1)$-dimensional partitions of an integer and the number of directed compact lattice animals in $d$ dimensions. Using enumeration techniques, we obtain upper and lower bounds for the number of multidimensional partitions (both restricted and unrestricted). We show that asymptotically the number of unrestricted $(d-1)$-dimensional partitions of an integer $n$ goes as $\exp(Cn^{(d-1)/d})$. We also show that for restricted partitions in $(d-1)$ dimensions (with $j$ dimensions finite, $0<j<d-1$), this number goes as $\exp(Cn^{(d-j-1)/(d-j)}(\prod_{k=1}^j L_k)^{1/(d-j)})$, where $L_k$ is the extent of the lattice along the $k$th axis." F.F. Ferreira and J.F. Fontanari, "Probabilistic analysis of the number partitioning problem", J. Phys. A 31 (1998) 3417 [abstract:] "Given a sequence of $N$ positive real numbers $\{a_1,a_2,..., a_N \}$, the number partitioning problem consists of partitioning them into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. In the case that the $a_j$'s are statistically independent random variables uniformly distributed in the unit interval, this NP-complete problem is equivalent to the problem of finding the ground state of an infinite-range, random anti-ferromagnetic Ising model. We employ the annealed approximation to derive analytical lower bounds to the average value of the difference for the best constrained and unconstrained partitions in the large $N$ limit. Furthermore, we calculate analytically the fraction of metastable states, i.e. states that are stable against all single spin flips, and found that it vanishes like $N^{-3/2}$." F.F. Ferreira and J.F. Fontanari, "Statistical mechanics analysis of the continuous number partitioning problem", Physica A 269 (1999) 54–60 [abstract:] "The number partitioning problem consists of partitioning a sequence of positive numbers ${a_1,a_2,..., a_N}$ into two disjoint sets, ${\cal A}$ and ${\cal B}$, such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We use statistical mechanics tools to study analytically the Linear Programming relaxation of this NP-complete integer programming. In particular, we calculate the probability distribution of the difference between the cardinalities of ${\cal A}$ and ${\cal B}$ and show that this difference is not self-averaging." F.F. Ferreira and J.F. Fontanari, "Instance space of the number partitioning problem", J. Phys. A 33 (2000) 7265 [abstract:] "Within the replica framework we study analytically the instance space of the number partitioning problem. This classic integer programming problem consists of partitioning a sequence of N positive real numbers $\{a_1, a_2,..., a_N}$ (the instance) into two sets such that the absolute value of the difference of the sums of $a_j$ over the two sets is minimized. We show that there is an upper bound $\alpha_c N$ to the number of perfect partitions (i.e. partitions for which that difference is zero) and characterize the statistical properties of the instances for which those partitions exist. In particular, in the case that the two sets have the same cardinality (balanced partitions) we find $\alpha_c=1/2$. Moreover, we show that the disordered model resulting from the instance space approach can be viewed as a model of replicators where the random interactions are given by the Hebb rule." S. Mertens, "Phase transition in the number partitioning problem", Phys. Rev. Lett. 81 (1998) 4281–4284 [abstract:] "Number partitioning is an NP-complete problem of combinatorial optimization. A statistical mechanics analysis reveals the existence of a phase transition that separates the easy from the hard to solve instances and that reflects the pseudo-polynomiality of number partitioning. The phase diagram and the value of the typical ground state energy are calculated." S. Mertens, "The easiest hard problem: number partitioning", A.G. Percus, G. Istrate and C. Moore, eds., Computational Complexity and Statistical Physics (Oxford University Press, 2006) 125–140 [abstract:] "Number partitioning is one of the classical NP-hard problems of combinatorial optimization. It has applications in areas like public key encryption and task scheduling. The random version of number partitioning has an "easy-hard" phase transition similar to the phase transitions observed in other combinatorial problems like k-SAT. In contrast to most other problems, number partitioning is simple enough to obtain detailled and rigorous results on the "hard" and "easy" phase and the transition that separates them. We review the known results on random integer partitioning, give a very simple derivation of the phase transition and discuss the algorithmic implications of both phases." S. Mertens, "A physicist's approach to number partitioning", Theor. Comp. Science 265 (2001) 79–108 [abstract:] "The statistical physics approach to the number partioning problem, a classical NP-hard problem, is both simple and rewarding. Very basic notions and methods from statistical mechanics are enough to obtain analytical results for the phase boundary that separates the "easy-to-solve" from the "hard-to-solve" phase of the NPP as well as for the probability distributions of the optimal and sub-optimal solutions. In addition, it can be shown that solving a number partioning problem of size $N$ to some extent corresponds to locating the minimum in an unsorted list of $O(2^N)$ numbers. Considering this correspondence it is not surprising that known heuristics for the partitioning problem are not significantly better than simple random search." M. Latapy, "Generalized integer partitions, tilings of zonotopes and lattices", Formal Power Series and Algebraic Combinatorics: 12th International Conference, FPSAC'00, Moscow, Russia, June 2000, Proceedings (Springer, 2000) 256–267 [abstract:] "In this paper, we study two kinds of combinatorial objects, generalized integer partitions and tilings of two dimensional zonotopes, using dynamical systems and order theory. We show that the sets of partitions ordered with a simple dynamics, have the distributive lattice structure. Likewise, we show that the set of tilings of zonotopes, ordered with a simple and classical dynamics, is the disjoint union of distributive lattices which we describe. We also discuss the special case of linear integer partitions, for which other dynamical systems exist. These results give a better understanding of the behaviour of tilings of zonotopes with flips and dynamical systems involving partitions." G. Freiman, A. M. Vershik, and Yu. V. Yakubovich, "A local limit theorem for random strict partitions", Theory Probab. Appl. 44 (2000), 453–468 [abstract:] "We consider a set of partitions of natural number n on distinct summands with uniform distribution. We investigate the limit shape of the typical partition as $n\to\infty$, which was found in [A. M. Vershik, Funct. Anal. Appl., 30 (1996), pp. 90–105], and fluctuations of partitions near its limit shape. The geometrical language we use allows us to reformulate the problem in terms of random step functions (Young diagrams). We prove statements of local limit theorem type which imply that joint distribution of fluctuations in a number of points is locally asymptotically normal. The proof essentially uses the notion of a large canonical ensemble of partitions." C. Borgs, J. Chayes and B. Pittel, "Phase transition and finite-size scaling for the integer partitioning problem", Random Structures and Algorithms 19 (2001) 247–288 [abstract:] "We consider the problem of partitioning n randomly chosen integers between $1$ and $2m$ into two subsets such that the discrepancy, the absolute value of the difference of their sums, is minimized. A partition is called $\emph{perfect}$ if the optimum discrepancy is $0$ when the sum of all $n$ integers in the original set is even, or $1$ when the sum is odd. Parameterizing the random problem in terms of $\kappa=m/n$, we prove that the problem has a phase transition at $\kappa=1$, in the sense that for $\kappa < 1$, there are many perfect partitions with probability tending to $1$ as $n\rightarrow\infty$, whereas for $\kappa > 1$, there are no perfect partitions with probability tending to $1$. Moreover, we show that this transition is first-order in the sense the derivative of the so-called entropy is discontinuous at $\kappa=1$. We also determine the finite-size scaling window about the transition point: $\kappa_n=1-(2n)^{-1}\log_2n+\lambda_n/n$, by showing that the probability of a perfect partition tends to $1$, $0$, or some explicitly computable $p(\lambda)\in(0,1)$, depending on whether $\lambda_n$ tends to $-\infty, \infty$, or $\lambdaz\in(-\infty,\infty)$, respectively. For $\lambda_n\rightarrow -\infty$ fast enough, we show that the number of perfect partitions is Gaussian in the limit. For $\lambda_n\rightarrow\infty$, we prove that with high probability the optimum partition is unique, and that the optimum discrepancy is $\Theta(2^{\lambda_n})$. Within the window, i.e., if $\|\lambda_n\|$ is bounded, we prove that the optimum discrepancy is bounded. Both for $\lambda_n\rightarrow\infty$ and within the window, we find the limiting distribution of the (scaled) discrepancy. Finally, both for the integer partitioning problem and for the continuous partitioning problem, we find the joint distribution of the $k$ smallest discrepancies above the scaling window." M.N. Tran, M.V.N. Murthy, R.K. Bhaduri, "On the quantum density of states and partitioning an integer", Ann. Phys. 311 204–219 [abstract:] "This paper exploits the connection between the quantum many-particle density of states and the partitioning of an integer in number theory. For $N$ bosons in a one dimensional harmonic oscillator potential, it is well known that the asymptotic ($N\rightarrow\infty$) density of states is identical to the Hardy–Ramanujan formula for the partitions $p(n)$, of a number $n$ into a sum of integers. We show that the same statistical mechanics technique for the density of states of bosons in a power-law spectrum yields the partitioning formula for $p^s(n)$, the latter being the number of partitions of $n$ into a sum of $s$-th powers of a set of integers. By making an appropriate modification of the statistical technique, we are also able to obtain $d^s(n)$ for $\emph{distinct]$ partitions. We find that the distinct square partitions $d^2(n)$ show pronounced oscillations as a function of $n$ about the smooth curve derived by us. The origin of these oscillations from the quantum point of view is discussed. After deriving the Erdös–Lehner formula for restricted partitions for the $s=1$ case by our method, we generalize it to obtain a new formula for distinct restricted partitions." M.N. Tran and R.K. Bhaduri, "Number fluctuation and the fundamental theorem of arithmetic" Physical Review E 68 (2003) 026206 [abstract:] "We consider $N$ bosons occupying a discrete set of single-particle quantum states in an isolated trap. Usually, for a given excitation energy, there are many combinations of exciting different number of particles from the ground state, resulting in a fluctuation of the ground state population. As a counter example, we take the quantum spectrum to be logarithms of the prime number sequence, and using the fundamental theorem of arithmetic, find that the ground state fluctuation vanishes exactly for all excitations. The use of the standard canonical or grand canonical ensembles, on the other hand, gives substantial number fluctuation for the ground state. This difference between the microcanonical and canonical results cannot be accounted for within the framework of equilibrium statistical mechanics." M. Planat, "Quantum 1/f noise in equilibrium: from Planck to Ramanujan", Physica A 318 (2003) 371 [abstract:] "We describe a new model of massless thermal bosons which predicts an hyperbolic fluctuation spectrum at low frequencies. It is found that the partition function per mode is the Euler generating function for unrestricted partitions $p(n)$. Thermodynamical quantities carry a strong arithmetical structure: they are given by series with Fourier coefficients equal to summatory functions $\sigma_k(n)$ of the power of divisors, with $k=-1$ for the free energy, $k=0$ for the number of particles and $k = 1$ for the internal energy. Low frequency contributions are calculated using Mellin transform methods. In particular the internal energy per mode diverges as $\frac{\tilde{E}}{kT}=\frac{\pi^2}{6 x}$ with $x=\frac{h \nu}{kT}$ in contrast to the Planck energy $\tilde{E}=kT$. The theory is applied to calculate corrections in black body radiation and in the Debye solid. Fractional energy fluctuations are found to show a $1/\nu$ power spectrum in the low frequency range. A satisfactory model of frequency fluctuations in a quartz crystal resonator follows. A sketch of the whole Ramanujan–Rademacher theory of partitions is reminded as well." I. Junier and J. Kurchan, "Microscopic realizations of the trap model", J. Phys. A 37 (2004) 3945–3965 [abstract:] "Monte Carlo optimizations of Number Partitioning and of Diophantine approximations are microscopic realizations of 'Trap Model' dynamics. This offers a fresh look at the physics behind this model, and points at other situations in which it may apply. Our results strongly suggest that in any such realization of the Trap Model, the response and correlation functions of smooth observables obey the fluctuation-dissipation theorem even in the aging regime. Our discussion for the Number Partitioning problem may be relevant for the class of optimization problems whose cost function does not scale linearly with the size, and are thus awkward from the statistical mechanic point of view." N.M. Chase, "Global structure of integer partition sequences" (preprint 04/2004, submitted to The Electronic Journal of Combinatorics) [abstract:] "Integer partitions are deeply related to many phenomena in statistical physics. A question naturally arises which is of interest to physics both on "purely" theoretical and on practical, computational grounds. Is it possible to apprehend the global pattern underlying integer partition sequences and to express the global pattern compactly, in the form of a "matrix" giving all of the partitions of $N$ into exactly $M$ parts? This paper demonstrates that the global structure of integer partitions sequences (IPS) is that of a complex tree. By analyzing the structure of this tree, we derive a closed form expression for a map from $(N,M)$ to the set of all partitions of a positive integer $N$ into exactly $M$ positive integer summands without regard to order. The derivation is based on the use of modular arithmetic to solve an isomorphic combinatoric problem, that of describing the global organization of the sequence of all ordered placements of $N$ indistinguishable balls into $M$ distinguishable non-empty bins or boxes. This work has the potential to facilitate computations of important physics and to offer new insights into number theoretic problems." A. Kubasiak, J. Korbicz, J. Zakrzewski, M. Lewenstein, "Fermi–Dirac statistics and the number theory", Europhysics Letters 72 (2005) 506 [abstract:] "We relate the Fermi–Dirac statistics of an ideal Fermi gas in a harmonic trap to partitions of given integers into distinct parts, studied in number theory. Using methods of quantum statistical physics we derive analytic expressions for cumulants of the probability distribution of the number of different partitions." V.P. Maslov, V.E. Nazaikinskii, "On the distribution of integer random variables related by a certain linear inequality: I", Mat. Zametki 83 (2008), 232–263 [Math. Notes 83 (2008) 211–237] [abstract:] "We consider the mathematical problem of the allocation of indistinguishable particles to integer energy levels under the condition that the number of particles can be arbitrary and the total energy of the system is bounded above. Systems of integer as well as fractional dimension are considered. The occupation numbers can either be arbitrary nonnegative integers (the case of ``Bose particles'') or lie in a finite set $\{0,1,\dots,R\}$ (the case of so-called parastatistics; for example, $R = 1$ corresponds to the Fermi–Dirac statistics). Assuming that all allocations satisfying the given constraints are equiprobable, we study the phenomenon whereby, for large energies, most of the allocations tend to concentrate near the limit distribution corresponding to the given parastatistics." V.P. Maslov, V. E. Nazaikinskii, "On the distribution of integer random variables related by a certain linear inequality: II", Mat. Zametki 83 (2008) 381–401 [Math. Notes 83 (2008) 345–363] [abstract:] "We continue our study of the problem on the allocation of indistinguishable particles to integer energy levels under the condition that the total energy of the system is bounded above. It is shown that the $\emph{Bose condensation}$ phenomenon can occur in this model. Systems of dimension $d < 1$ (including negative dimensions) are studied." V. P. Maslov and V. E. Nazaikinskii, "On the distribution of integer random variables related by a certain linear inequality, III", Mat. Zametki 83 (2008) 880–898 [Math. Notes 83 (2008) 804–820] [abstract:] "We consider tuples $\{N_{ jk}\}, j=1,2,\dots,k=1,\dots,q_j$ , of nonnegative integers such that $$ \sum\limits_{j = 1}^\infty {\sum\limits_{k = 1}^{q_j } {jN_{jk} } } \leqslant M. $$ Assuming that $q_j\sim j^{d-1}, 1 < d < 2$, we study how the probabilities of deviations of the sums $$ \sum\nolimits_{j = j_1 }^{j_2 } {\sum\nolimits_{k = 1}^{q_j } {N_{jk} } } $$ $N_{jk}$ from the corresponding integrals of the Bose‐Einstein distribution depend on the choice of the interval $[j_1,j_2]$." M. Nardelli, "On the physical interpretation of the Riemann zeta function, the rigid surface operators in gauge theory, the adeles and ideles groups applied to various formulae regarding the Riemann zeta function and the Selberg trace formula, p-adic strings, zeta strings and p-adic cosmology and mathematical connections with some sectors of string theory and number theory" (preprint 10/2008) [abstract:] "This paper is a review of some interesting results that has been obtained in the study of the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence and some aspects of the rigid surface operators in gauge theory. Furthermore, we describe the mathematical connections with some sectors of string theory (p-adic and adelic strings, p-adic cosmology) and number theory. In the Section 1 we have described various mathematical aspects of the Riemann Hypothesis, matrix/gravity correspondence and master matrix for FZZT brane partition functions. In the Section 2, we have described some mathematical aspects of the rigid surface operators in gauge theory and some mathematical connections with various sectors of number theory, principally with the Ramanujan's modular equations (thence, prime numbers, prime natural numbers, Fibonacci's numbers, partitions of numbers, Euler's functions, etc...) and various numbers and equations related to the Lie Groups. In the Section 3, we have described some very recent mathematical results concerning the adeles and ideles groups applied to various formulae regarding the Riemann zeta function and the Selberg trace formula (connected with the Selberg zeta function), hence, we have obtained some new connections applying these results to the adelic strings and zeta strings. In the Section 4 we have described some equations concerning p-adic strings, p-adic and adelic zeta functions, zeta strings and p-adic cosmology (with regard the p-adic cosmology, some equations concerning a general class of cosmological models driven by a nonlocal scalar field inspired by string field theories). In conclusion, in the Section 5, we have showed various and interesting mathematical connections between some equations concerning the Section 1, 3 and 4." M. Nardelli, "On some equations concerning fivebranes and knots, Wilson loops in Chern–Simons theory, cusp anomaly and integrability from string theory. Mathematical connections with some sectors of number theory" (preprint 09/2011) [abstract:] "The present paper is a review, a thesis of some very important contributes of E. Witten, C. Beasley, R. Ricci, B. Basso et al. regarding various applications concerning the Jones polynomials, the Wilson loops and the cusp anomaly and integrability from string theory. In this work, in Section 1, we have described some equations concerning the knot polynomials, the Chern–Simons from four dimensions, the D3-NS5 system with a theta-angle, the Wick rotation, the comparison to topological field theory, the Wilson loops, the localization and the boundary formula. We have described also some equations concerning electric-magnetic duality to $N = 4$ super Yang-Mills theory, the gravitational coupling and the framing anomaly for knots. Furthermore, we have described some equations concerning the gauge theory description, relation to Morse theory and the action. In Section 2, we have described some equations concerning the applications of non-abelian localization to analyze the Chern–Simons path integral including Wilson loop insertions. In the Section 3, we have described some equations concerning the cusp anomaly and integrability from string theory and some equations concerning the cusp anomalous dimension in the transition regime from strong to weak coupling. In Section 4, we have described also some equations concerning the "fractal" behaviour of the partition function. Also here, we have described some mathematical connections between various equation described in the paper and (i) the Ramanujan's modular equations regarding the physical vibrations of the bosonic strings and the superstrings, thence the relationship with the Palumbo-Nardelli model, (ii) the mathematical connections with the Ramanujan's equations concerning $\pi$ and, in conclusion, (iii) the mathematical connections with the golden ratio $\phi$ and with $1.375$ that is the mean real value for the number of partitions $p(n)$." J. Roccia and P. Leboeuf, "Level density of a Fermi gas and integer partitions: a Gumbel-like finite-size correction", Phys. Rev. C 81 (2010) 044301 [abstract:] "We investigate the many-body level density of gas of non-interacting fermions. We determine its behavior as a function of the temperature and the number of particles. As the temperature increases, and beyond the usual Sommerfeld expansion that describes the degenerate gas behavior, corrections due to a finite number of particles lead to Gumbel-like contributions. We discuss connections with the partition problem in number theory, extreme value statistics as well as differences with respect to the Bose gas." V.P. Maslov, "Zeno-line, binodal, T-$\rho$ diagram and clusters as a new Bose-condensate bases on new global distributions in number theory" (preprint 07/2010) [abstract:] "We present the correspondence principle between the T-$\rho$ diagram, the Zeno line and the binodal for a given interaction potential of Lennard–Jones type. We use this correspondence further to construct a distribution of the Bose–Einstein type for a classical gas with the help of the new notion of Bose condensate, making it possible to decrease fractal dimension while simultaneously preserving the number of particles. In so doing, we use new global distributions in number theory." J.H. Bruinier and K. Ono, "An algebraic formula for the partition function" (A.I.M. preprint, 01/2011) [abstract:] "We derive a formula for the partition function $p(n)$ as a finite sum of algebraic numbers. The summands are discriminant $-24n + 1$ singular moduli for a special weak Maass form that we describe in terms of Dedekind's eta-function and Eisenstein series." A. Folsom, Z.A. Kent and Ken Ono, "$l$-adic properties of the partition function" (A.I.M. preprint, 01/2011) popularly accessible blog piece on the remarkable discoveries described in the above two papers [excerpt:] "Our work brings completely new ideas to the problems," Ono says. "We prove that partition numbers are 'fractal' for every prime. These numbers, in a way we make precise, are self-similar in a shocking way. Our 'zooming' procedure resolves several open conjectures, and it will change how mathematicians study partitions." M. Psimopoulos, "Harmonic representation of combinations and partitions" (preprint 02/2011) [abstract:] "In the present article a new method of deriving integral representations of combinations and partitions in terms of harmonic products has been established. This method may be relevant to statistical mechanics and to number theory." D. Prokhorov and A. Rovenchak, "Asymptotic formulas for integer partitions within the approach of microcanonical ensemble" (preprint 10/2012) "The problem of integer partitions is addressed using the microcanonical approach which is based on the analogy between this problem in the number theory and the calculation of microstates of a many-boson system. For ordinary (one-dimensional) partitions, the correction to the leading asymptotic is obtained. The estimate for the number of two-dimensional (plane) partitions coincides with known asymptotic results." archive tutorial mystery new search home contact	CommonCrawl
SamPler – a novel method for selecting parameters for gene functional annotation routines Fernando Cruz1, Davide Lagoa1, João Mendes1, Isabel Rocha1,2, Eugénio C. Ferreira1, Miguel Rocha1 & Oscar Dias ORCID: orcid.org/0000-0002-1765-71781 As genome sequencing projects grow rapidly, the diversity of organisms with recently assembled genome sequences peaks at an unprecedented scale, thereby highlighting the need to make gene functional annotations fast and efficient. However, the (high) quality of such annotations must be guaranteed, as this is the first indicator of the genomic potential of every organism. Automatic procedures help accelerating the annotation process, though decreasing the confidence and reliability of the outcomes. Manually curating a genome-wide annotation of genes, enzymes and transporter proteins function is a highly time-consuming, tedious and impractical task, even for the most proficient curator. Hence, a semi-automated procedure, which balances the two approaches, will increase the reliability of the annotation, while speeding up the process. In fact, a prior analysis of the annotation algorithm may leverage its performance, by manipulating its parameters, hastening the downstream processing and the manual curation of assigning functions to genes encoding proteins. Here SamPler, a novel strategy to select parameters for gene functional annotation routines is presented. This semi-automated method is based on the manual curation of a randomly selected set of genes/proteins. Then, in a multi-dimensional array, this sample is used to assess the automatic annotations for all possible combinations of the algorithm's parameters. These assessments allow creating an array of confusion matrices, for which several metrics are calculated (accuracy, precision and negative predictive value) and used to reach optimal values for the parameters. The potential of this methodology is demonstrated with four genome functional annotations performed in merlin, an in-house user-friendly computational framework for genome-scale metabolic annotation and model reconstruction. For that, SamPler was implemented as a new plugin for the merlin tool. The emergence of high-throughput sequencing techniques led to a fast increase in the number of genome sequencing projects over the years, with over 196,000 finished or ongoing projects, as of April 2018 [1]. It is clear that, nowadays, the functional annotation of these genome sequences is eased by automated methodologies and workflows. Several authors have been reporting novel computational tools [2,3,4,5,6] and workflows built essentially as meta-servers [7,8,9,10,11,12,13,14,15], to automate the annotation of genes encoding proteins. Though the effort of the Gene Ontology (GO) Consortium [16,17,18] and others [19,20,21,22,23] has been notorious, the emergence of multiple methodologies and platforms to perform genome functional annotation hindered the aim for data unification and spread redundant information across multiple repositories and databases. Surprisingly, multiple authors have reported errors in genes' functional annotations over the years, pointing out high misannotation levels (up to 80% in some cases) in single-genome annotations [24, 25], single-gene annotations [26], large-scale data repositories [27, 28], or even in the GO database [29, 30]. Concerns about this issue increase when assessing misannotations of molecular functions in the four major public protein sequence databases: Universal Protein Resource Knowledgebase (UniProtKB)/TrEMBL, UniProtKB/Swiss-Prot, GenBank NR [31] and Kyoto Encyclopedia of Genes and Genomes (KEGG) [32]. According to Schnoes and coworkers [33], the manually curated database UniProtKB/Swiss-Prot, shows levels of misannotation close to 0, whereas the three non-curated protein sequences databases (GenBank NR, UniProtKB/TrEMBL and KEGG) exhibit high levels of misannotations averaging 5–63%, and in specific cases up to 80%. Genome annotation can be divided into structural annotation and functional annotation. Whereas some of the mentioned computational tools and meta-servers can perform both [7,8,9,10,11,12,13,14], this work will focus on a novel method for leveraging genome functional annotations, namely enzyme and transporter protein functional annotation. Detailed, accurate and useful gene products' descriptions are regularly sought when retrieving high-level gene functional annotations. Although simple terms, such as dehydrogenase, can describe gene products, accurate and thoroughly refined terms as pyruvate dehydrogenase (NADP+) should be used instead. Additionally, besides systematic database accession numbers and gene products' descriptions, special identifiers such as the Enzyme Commission (EC) [34] and Transport Classification (TC) numbers [35] are vastly used nowadays to classify enzymatic and transporter proteins, respectively. These special identifiers should, whenever possible, be included in the annotations, to increase the scope and decrease the ambiguity of genome functional annotations. Most publicly available frameworks for the automation of genome functional annotations are complex pipelines made of multiple processes using several tools and algorithms, based on similarity and/or profile searches [2, 3, 9,10,11], subsystem-based functional roles [12], domain prediction and annotation [8, 11], genome annotations comparison [5, 6, 15]. These, in turn, are sensitive to parameters, cut-offs and quality assessments. Cases in which the need to obtain fast gene products' annotations justifies the use of such frameworks, should also guarantee the quality of the annotation. Interestingly, current frameworks indeed claim to provide both fast and accurate genome annotations. Nevertheless, parameters and cut-off thresholds are rarely changed when submitting data, and descriptions of the methodologies that lead to specific outcomes are seldom available [5, 7,8,9,10,11,12,13,14]. Hence, the annotations of genes encoding proteins may have been inferred from biased and redundant data without user's awareness. Genome-wide functional annotations are often performed using similarity search algorithms, such as the Basic Local Alignment Search Tool (BLAST) [36], Smith-Waterman alignments [37] or HMMER [38]. These and other similar algorithms compare sequences with other sequences (typically in sequence databases), providing clusters of genes with similar sequences, which in theory should have similar functions. Occasionally, functional annotations may be erroneously inferred due to: a biased taxonomic distribution of the homologous genes, which may be systematically reported by alignment algorithms; the presence of incorrectly annotated homologous genes; the systematic presence of homologous genes assigned with unknown functions or hypothetical proteins; and, the spread of redundancy across multiple databases. These problems enhance the need to manually curate gene functional annotations. However, manual curation is often time-consuming and requires a huge effort, even from expert curators. Cases in which the user is allowed to configure the annotation workflow may benefit from a prior meticulous analysis of the data and a fine tuning of the computational algorithm parameters, which may hasten downstream processing. Moreover, other authors highlighted the importance of pairing manual curation (namely, inferred from literature) with computational predictions or using multiple databases as information sources, to update genes' annotations [5, 6, 15, 20, 23, 39]. As far as our knowledge allows, here we provide the only method (SamPler) aimed at determining the best settings for the parameters of genome functional annotation algorithms, through the manual annotation of a random sample of entities. The approach demonstrated in this study is used to determine the best configuration of merlin's [2] enzyme annotation algorithm, setting the α parameter, which leverages two scores (namely the frequency and taxonomy scores), while proposing upper and lower thresholds to automatically accept or reject gene's metabolic annotations, respectively. Furthermore, this strategy prevents the utilization of biased and redundant data. The method was implemented as a new plugin for merlin's current version, and thus made available for the community, making the reproducibility of our results possible with minimum effort. Using the SamPler plugin, one can now choose between two gene functional annotation routines available in merlin: automatic first-hit annotation or SamPler for leveraging the enzyme's annotation algorithm. Remarkably, this method can have many other applications besides the one demonstrated herein. merlin is a user-friendly computational tool which, among other features, allows performing genome-wide metabolic (re-)annotations and the reconstruction of genome-scale metabolic models [2]. merlin performs two types of metabolic (re-) annotation, namely the enzymes and transporters annotation. The methodology proposed in this work was implemented as a new plugin that leverages the enzymes annotation algorithm, thereby being fully available for merlin's current version. In merlin, the enzymes annotation involves performing remote similarity searches (either with BLAST or HMMER) and assigning a function to each gene, taking into account the homologous genes functions. Genes encoding enzymes should have at least one homologous gene assigned with an EC number. The likelihood of a gene being assigned with an EC number is determined by an internal scoring algorithm that takes into account both the frequency (cardinality of such EC number among the orthologous genes) and the taxonomy (taxonomic proximity between the organisms for a given EC number), to calculate a score (between 0 and 1). The higher the score, the higher the chance of that gene encoding such enzyme. More information on this algorithm can be found in [2]. The trade-off between the frequency and the taxonomy scores is attained by a parameter, the α value. In previous versions of the framework, the user set this parameter and the challenge of this work is to provide an automatic method to calculate this value, found to be of a great importance in the results of the annotation. merlin also allows performing automatic gene functional annotations. In this case, a first-hit annotation pipeline is followed by using an annotation workflow such as the one described in Additional file 1. Nevertheless, manual curation and other factors influencing the outcome of the annotation such as the frequency of a given annotation is not considered. Currently, the methodology described in this work, SamPler, is implemented and used to select a value for the α parameter, together with the thresholds for accepting annotations and rejecting genes as enzymatic. The proposed method suggests the curation of a random sample of entries from an annotation project, using these to automatically select the parameters for configuring the algorithm that performs automatic enzymes annotation, being integrated in the software merlin. Alternatively, the method can be used to configure any other annotation algorithm or workflow. For this to be possible, these computational tools or meta-servers should return a score or other rank for each entry as a function of the parameters configuration. Selection and annotation of a standard of truth This process begins by setting an initial sensible value for all parameters and automatically running the enzyme annotation algorithm or workflow. Then, a sample with x1 genes (5% to 10% of the genes/proteins to be annotated) should be (randomly) selected, guaranteeing that all possible score intervals are represented and the number of entities in each interval is similar. These records should be manually curated with a defined workflow (an example of a manual curation workflow for merlin is shown in the Additional file 1). These entries will become the standard of truth for evaluating the genes functional annotations proposed by the tool. Parameters assessment This evaluation starts with the creation of a multi-dimensional array, of dimensions (x1, x2) that features the selected genes on the rows and all possible combinations of parameters (x2) on the columns, as shown in Fig. 1. Scheme of a multi-dimensional array (x1, x2). This scheme is used for evaluating randomly selected annotations (x1) automatically proposed by a tool, against annotations curated through a manual curation workflow (standard of truth). The multi-dimensional array allows evaluating all possible combinations of values for the tool's parameters (x2) This array will comprise the framework's automatic annotations for sample x1, for all possible combinations of the algorithm's parameters (x2). These combinations may lead to different automatic annotations that, when compared to the standard of truth, allow assessing the best configuration of the parameters' settings. This comparison allows creating a new multi-dimensional array of dimensions (t, x2), which helps simplifying the analysis of the results. Such array will have score thresholds (t) between the minimum and the maximum score in the rows, and all possible parameter combinations x2 in the columns. Each intersecting cell will contain a confusion matrix for each (t, x2) pair, as shown in Table S.1 of the Additional file 2. These confusion matrices evaluate the performance of the different conditions of the algorithm or workflow and report the number of incorrect annotations (IA – similar to false positives), incorrect rejections (IR – similar to false negatives), correct annotations (CA – similar to true positives) and correct rejections (CR – similar to true negatives), according to Fig. 2. Evaluation of an annotation algorithm when compared with the Standard of Truth (A or Ø), determined with a manual curation workflow. If the manual annotation is equal to the algorithm's annotation (A = A) and has a score above the threshold (≥ Threshold), it is classified as CA. When the algorithm's annotation is below the threshold (≤ Threshold), it is classified as IR. Alternatively, when the annotation is different from the manual curation (A ≠ B), the annotation will be classified either as IA or IR, when above (≥ Threshold) or below (≤ Threshold) the threshold, respectively. Finally, the manual curation may indicate that the entity should not be annotated (Ø), whereas the tool assigned an annotation (A). These cases will be classified as IA or CR, whenever the annotation's score is above or below the threshold, respectively The two former conditions represent type I and type II errors. The first case (IA) may be the outcome of two distinct situations: the algorithm annotation is incorrect (A ≠ B), or the entity should not be annotated (A ≠ Ø), though the algorithm annotation score is higher (or equal) than the provided threshold. The second case (IR) may also arise from two distinct situations: the algorithm's annotation score is below the threshold, despite being correct (A = A) or even if it is incorrect (A ≠ B). Finally, the two latter cases represent cases in which automatic annotations, above or equal to the threshold, are in agreement (A = A) with the curated annotation (CA), or entities that should not have been annotated (A ≠ Ø) have annotation scores below the threshold (CR), respectively. Nevertheless, these rules can be changed and adapted to other situations, depending on the paradigm and objective of the annotation. Different premises regarding the conditions of the algorithm and the curation workflow, will provide different outcomes that will lead to distinct classifications. Confusion matrices allow calculating several different metrics. For this work, accuracy, precision and negative predictive value ( NPV) will be considered. Accuracy reveals how often the workflow makes correct annotations and rejections, and is calculated as follows: $$ accuracy=\frac{CA+ CR}{CA+ CR+ IA+ IR} $$ On the other hand, precision assesses how often the algorithms assignments are correct, according to: $$ precision=\frac{CA}{CA+ IA} $$ Finally, the NPV evaluates how often the algorithm's rejections are correct: $$ NPV=\frac{CR}{\ CR+ IR} $$ These rates allow selecting the best values for several parameters, together with upper and lower thresholds for the annotations' scores. The average accuracy of each column allows determining which parameters' settings (x2) attain higher accuracy, i.e. which set of values for all parameters yields more correct annotations and rejections. Then, for such column, precision and NPV are used to determine the upper and lower thresholds, respectively. Every row t with a precision of 1 indicates that the algorithm's annotations are always correct above the respective threshold, hence these should not require curation. Likewise, rows with NPV of 1 indicate that the annotation algorithm correctly rejects all annotations below such t. Therefore, the upper threshold should be the lowest t with the highest precision (ideally 1), and the lower threshold the highest t with the highest NPV (ideally 1). This methodology allows accepting annotations with scores above the upper threshold, reject all below the lower threshold and encourages manual curation of entries with scores in between. However, incorrect automatic annotations of the sample retrieved for the standard of truth, with very high or very low scores, may impair this methodology, by requiring the annotation of a large number of entries. Hence, it should be allowed to relax precision and NPV thresholds to values below 1, so that the manual curation efforts are not so demanding. For instance, lowering precision will increase the number of annotations automatically accepted. Likewise, lowering NPV increases the number of annotations automatically rejected. Therefore, relaxing these metrics introduces a new trade-off that, though accepting erroneous annotations, increases the number of records automatically annotated/rejected, decreasing curation efforts. An example of the steps required to implement SamPler, is presented in Tables S.2-S.4 of the Additional file 2. As shown in Table S.2 of the Additional file 2, a sample of 50 (x1) genes, roughly 5% of the potentially metabolic genome, was manually curated to determine whether such genes encoded enzymes and which EC numbers should be assigned to them, thus becoming the standard of truth for such genes. The automatic annotations for nine (0.1– 0.9) different α values (x2) were then calculated and compared with the standard of truth in the array (x1, x2). Setting the parameter values to both edges (0.0 and 1.0) may be too extreme, as it may completely eliminate a component of the scorer, thus biasing results. Therefore, combinations with extreme values for the parameters should be carefully considered. Next, the confusion matrices for each pair (t, x2) were computed, as shown in Table S.3 of the Additional file 2. For instance, when calculating the confusion matrix for the pair (t, x2) = (0.5, 0.2), all correct annotations with scores equal or above 0.5 are considered correct (29) and inserted in the CA cell. Every correct annotation below 0.5 (12) was inserted in the IR cell. Likewise, all incorrect merlin assignments with scores above 0.5 (4) are inserted in the IA, whereas wrong annotations below the threshold (5) are included in the CR cells. This process is repeated until confusion matrices of all (t, x2) pairs are calculated. Worth mentioning is the fact that the example presented herein only depicts analysis for merlin's α value, and upper and lower thresholds, to present a clear and concise demonstration. Finally, as shown in Table S.4 of the Additional file 2, these matrices allow calculating accuracy, precision and NPV. merlin's annotation algorithm highest mean accuracy is associated with α = 0.1. For this α, all annotations with t above 0.9 have a precision of 1, which means that merlin's algorithm is correct 100% of the times. Likewise, annotations with t below 0.2 have a NPV of 1, which means that merlin correctly rejects these annotations 100% of the times. As shown in Table S.5 of the Additional file 2, the user will have to curate manually 301 genes, which represent 30% of the genes that potentially encode enzymes. Often, the α with the highest accuracy is simultaneously the one with the highest number of entities to be manually verified. Thus, a curation ratio score, which compromises accuracy with the percentage of records to be curated, is also calculated, according to Eq. 4. $$ curation\ ratio\ score=\frac{accuracy}{\% entries\ to\ be\ curated} $$ This allows a trade-off between of accuracy and curation efforts. Moreover, merlin allows users to accept lower precision and/or NPV (to at least 75%) to decrease the number of entities that will be curated, thus consenting errors in the automatic annotation. SamPler was used to calculate the parameters for several organisms. Four complete genome sequences, for Lactobacillus rhamnosus (taxonomy identifier: 568703), Streptococcus thermophilus (taxonomy identifier: 322159), Lactobacillus helveticus (taxonomy identifier: 326425) and Nitrosomonas europaea (taxonomy identifier: 228410), were retrieved from the National Center for Biotechnology Information (NCBI) database. These organisms are of interest for the host group and are being studied in different projects. S. thermophilus, L. helveticus and Lactobacillus rhamnosus belong to the lactic acid bacteria group (Lactobacillales Order), thus having a considerable number of strains and taxonomically close microorganisms, available in the UniProtKB database, whereas, N. europaea is the only sequenced strain for this organism, with a relatively low number of taxonomically close microorganisms available in UniProtKB. After integrating SamPler's plugin in merlin, an automatic genome functional annotation was performed for each genome using the BLAST algorithm against the UniProtKB. A random sample of genes was then automatically collected by SamPler. These entries were manually curated inside merlin's environment, following the manual curation scheme in the Additional file 1. Finally, the best parameters settings were calculated by SamPler, for both precision and NPV of 100%. An illustration of the SamPler workflow implemented in merlin for selecting the best parameters of the annotation algorithm is shown in Fig. 3. Scheme of the semi-automated method implemented in merlin's EC number annotation tool. Merlin proposes × 1 entries (5 to 10% of the sequences to be annotated) for manual curation which will become standard of truth. Then, for each α value the corresponding automatic annotations are retrieved and assessed against the standard of truth. After assessing each entry for each α value, merlin calculates a confusion matrix for each pair (threshold, α value). This multi-dimensional array, of confusion matrices, allows calculating the accuracy of each α, and the precision and NPV of each pair. Finally, the number of records between thresholds (taking into account the error allowed in precision and NPV) are assigned as entries to be curated and the curation ratio score helps determining the best α value and thresholds as a function of the highest accuracy divided by the ratio of entries to be curated. An error up to 25% is allowed in both precision and NPV Furthermore, besides assessing the best parameters for the annotation algorithm of the above mentioned organisms, another organism's annotation (L. rhamnosus) was analyzed in detail. In this case, the annotation was performed using initially UniProt/SwissProt as the BLAST database and later, for records without homology hits, against UniProtKB. However, in this case, all EC number annotations were reviewed with the curation workflow (described in Additional file 1), which allows assessing SamPler's calculations and predictions. SamPler allows determining the best settings for the parameters of genome functional annotation algorithms, hereby contributing to the improvement of annotations, even when based on biased and redundant data. The manual annotation of a random sample of genes together with the computation of confusion matrices, decreases the manual curation efforts, often required after submitting data to automatic procedures. SamPler was implemented as a plugin for merlin's enzymes annotation algorithm, allowing to select the best parameter values along with the thresholds that determine the number of protein coding genes to be manually annotated. The analysis performed with L. rhamnosus was used to validate SamPler's methodology. All enzymatic genes were manually annotated using the workflow shown in Additional file 1. SamPler was used to calculate parameters for two sets of genes, namely genes annotated against UniProt/SwissProt and genes annotated with UniProtKB. As shown in Figs. 4, 5, Tables S.7 and S.8 of the Additional file 2, these annotations allowed assessing precision and NPV to sample size and type of database used for annotation (curated, non-curated and merging of both). Hence, these analyses were performed for records annotated against UniProt/SwissProt (334 genes), UniProtKB (973) and both (1307), for two sample sizes in triplicate (three manually curated samples). The annotation performed with the UniProt/SwissProt provided few records with lower scores. Therefore, to balance the distribution of scores in the sample, the sizes were 42 and 75. For the same reasons, the larger sample size for UniProtKB had 98 entries, whereas, for latter assessment, the sample sizes were 50 and 100. Analysis of the number of genes automatically annotated by merlin, according to SamPler's proposals for the α parameter and upper threshold. Results shown for genes annotated against UniProt/SwissProt, using 42 genes as standard of truth, for 100, 95, 85 and 75% precision As expected, generally, lowering the acceptable precision and NPV allows annotating more records, though accepting errors in such annotations (Tables S.6, S.7 and S.8 of the Additional file 2). SwissProt results When using 42 genes to assess the parameters, lowering the acceptable overall precision to 95% will not increase the average number of automatically annotated entries, thus also not adding errors to the annotation. However, selecting the parameters to automatically accept annotations with an overall 85% precision will automatically annotate 290 ± 5 more entries, though 34 ± 6 of these are incorrectly annotated. Yet, in this case, the mean overall effective precision will be ≈88 % ± 2%. Likewise, when configuring SamPler to select parameters that provide an overall annotation precision over 75%, 313 ± 12 entries are automatically annotated, though 49 ± 5 of these are wrong, thus the overall effective precision is ≈84 % ± 1%. As shown in Table S.6 of the Additional file 2, using 75 genes to determine the standard of truth, provides roughly the same results. Accepting an overall precision of 95% automatically includes 225 ± 5 genes in the annotation, being 5 ± 4 incorrect. For an overall precision of 85 % , 253 ± 7 genes are automatically annotated, 16 ± 3 of which are incorrect. Finally, lowering the acceptable precision to 75% will automatically annotate (318 ± 2) genes, from which 43 ± 1 will be wrongly annotated. The effective precisions would be ≈98 % ± 2, ≈ 94 % ± 1 and ≈86 % ± 1 respectively. For both sample sizes, the α and the upper threshold values tend to go down, when accepting lower precisions. When analyzing annotations obtained with this database, for both sample sizes in all replicates, lowering the overall NPV does not affect results, hence no incorrect rejections are performed in the annotation. Likewise, the lower threshold and α remain stable for all acceptable NPV. UniprotKB results The analyses performed for entries annotated against UniProtKB provided overall results similar to the previous ones. Again, as shown in Table S.7 of the Additional file 2, the effective precision is always within the range accepted by SamPler. Configuring SamPler to provide an overall precision of 95%, for the 50 genes' sample, does not misannotate any genes, thus providing an effective precision of 100%. Furthermore, when the algorithm is configured to accept 15% of error in precision, only 7 ±7 incorrect entries, within 188 ± 10, are automatically accepted as correct (precision = 96 % ± 4). Finally, SamPler has a precision of 89 % ± 4, which corresponds to 23 ± 10 misannotations for 211 ± 13 automatically annotated records, when calculating the parameters for a minimum precision of 75% in the sample's annotations. Regarding the analysis of the results obtained with the larger sample (98 genes), accepting 5% of error in the precision of the annotation, provides similar results to the default 100 % precision. In this case, the mean of the automatically annotated genes is 166 ± 9, with 1 ± 2 misannotations, instead of 162 ± 4. Again, decreasing the acceptable precision, allows automatically annotating more entries, without significant errors 193 ± 3 (6 ± 2 misannotations) and 229 ± 33 (36 ± 24 misannotations), for 85% and 75 % precision respectively. For annotations in UniProtKB, though the threshold decreases along with precision, the mean of the α parameter remains fairly constant (average of 0.73 − 0.8 for the smaller sample and 0.63 − 0.77 for the larger sample). Analyzes of the automatically rejected entries in UniProtKB provided interesting results. Most annotations with low scores corresponded to incorrect annotations, thus the proposed lower threshold rejected hundreds of annotations. In fact, for a special case (gene CAR87684.1), according to the workflow, the annotation proposed by merlin was correct, though with a very small score (0.12). Hence, whenever this gene was present in the sample, the NPV would be significantly affected. Nevertheless, this effect can be softened by accepting a 5% error in NPV, as shown in Table S.7 of Additional file 2, although this was the only overall NPV for which the mean of the effective NPV 's was below the acceptable, for both sample sizes. Overall, the average α varied between (0.73 and 0.9) for both samples. As expected, the lower threshold tended to increase with the decrease of the NPV. Merged databases Performing these analyzes for all entries annotated with both homology searches, provides results similar to the entries annotated against UniProtKB, as ≈73% of the annotations were obtained from that database. As expected, the mean of the effective precision is above the minimum limits configured in SamPler, for all intervals in all samples. Regarding the NPV, results are also very good, although when configuring SamPler to provide an overall NPV of 85% or 95%, analyses show that the means of the effective NPV are slightly below the limit, for both samples. Also, the upper threshold decreases together with precision and the lower threshold increases when lowering the NPV. The α parameter values are fairly constant for both samples and metrics, though (for a precision of 100%) in sample 50.1 the calculated α is low when compared to samples 50.2 and 50.3. Other organisms' results Regarding S. thermophilus, merlin's SamPler suggested an α value of 0.4 (accuracy of 0.691) with a curation ratio score of 1.8 (Table 1), and the manual curation of 331 entries (38% of the number of potentially enzymatic genes), when setting the lower and upper thresholds to 0.2 to 0.6, respectively. As shown in Table S.9 of the Additional file 2, three α values, viz. 0.1, 0.3 and 0.7, had a higher calculated accuracy (0.696), but the number of entries to be curated would be 40%, 45% and 46%, respectively. Thus, the increase in accuracy would not justify the extra curation efforts. Table 1 Main results of implementing the semi-automated method in merlin's EC number annotation tool using three complete protein sequences as test cases, one of each single-different organism Regarding L. helveticus, merlin's SamPler proposed an α value of 0.7, with a curation ratio score of 2.11 (Table 1), and the manual curation of 264 entries (36% of the number of potentially enzymatic genes), when setting the lower and upper thresholds to 0.3 and 0.8, respectively. For this organism, as shown in Table S.10 of the Additional file 2, only one α value (0.9) had a higher accuracy (0.758), but it required the curation of 45% of the annotations, that is 67 more entries than for α = 0.7. Finally, for N. europaea, SamPler's results are substantially different from the previous. The proposed α was 0.2 and the thresholds were 0.8 and 0.1, for upper and lower thresholds, respectively. Recall that this organism has few closely related species available in the BLAST database. If α = 0.1 was selected, instead of the current 624, 693 records would have to be curated (Table S.11 of the Additional file 2). The complete calculation reports, provided by merlin's SamPler, are available in Tables S.9 to S.11 of the Additional file 2, for S. thermophilus, L. helveticus and N. europaea, respectively. Indeed, the results provided by SamPler are correlated with the organisms' taxonomic families' presence in the BLAST databases. For instance, the recommendation of αsth = 0.4 for S. thermophilus or αlhe = 0.7 for L. helveticus, can be associated with the fact that these microorganisms have multiple strains and an even higher number of closely related microorganisms (surprisingly from the same genus) with complete genome functional annotations available in UniProtKB. Hence, both the frequency of the EC number and the taxonomy of the homologous genes annotated with such EC numbers should be taken into account when calculating the annotation score. On the other hand, N. europaea is poorly described in UniProtKB. Hence, it was expected that the selected α (0.2) would enhance the taxonomy component in the final score ( αneu = 0.2 x scorefrequency + 0.8 scoretaxonomy). The absence of taxonomically close, well characterized, organisms increases the relevance of the few related records, enhancing the taxonomy score. Notice that N. europaea also presents the highest percentage of genes that should be manually curated. Here SamPler, a tool aimed at improving genome functional annotations, is showcased. The results of this work show that the parametrization of merlin's enzyme annotation is not straightforward. In fact, the selected α values for two Lactobacillales were 0.4 and 0.7, for S. thermophilus and L. helveticus, respectively. Still, for N. europaea's the α parameter value was 0.2. The automatic annotation/rejection thresholds also varied significantly among the projects. These results show that each project is unique, with several factors influencing the outcome of the annotation, such as the availability of manually curated or incorrect annotations for the organism's genes. Analyses of the complete curation of L. rhamnosus show that using non-curated databases to perform annotations provide very few correct annotations when compared with the curated ones. Also, the effective errors when allowing values both for precision and NPV below 100%, are mostly within acceptable ranges. Therefore, the results of this work demonstrate that larger projects can benefit from lower precision and NPV thresholds, as these may decrease the curation efforts, while incorrectly annotating (or rejecting) very few entries. Hence, performing high-quality semi-automatic genome functional annotations should involve systematic and reproducible methodologies, that reduce both human error and data bias, such as the SamPler presented in this work. Indeed, SamPler accepted and rejected a significant number of entries for all organisms while proposing a well-aimed number of genes that should be manually curated, clearly improving, guiding and hastening the manual curation process. merlin is freely available at http://www.merlin-sysbio.org/. SamPler is available at https://gitlab.bio.di.uminho.pt/merlin-sysbio/merlin-sampler. The data generated or analysed during this study are included in this published article and its supplementary information files. The genome functional annotation analysed during the current study is available from the corresponding author on reasonable request. BLAST: Basic Local Alignment Search Tool Correct Annotations CR: Correct Rejections EC: Enzyme Commission GO: Incorrect Annotations Incorrect Rejections Kyoto Encyclopedia of Genes and Genomes NPV: Negative Predictive Value TC: Transport Classification UniProtKB: Universal Protein Resource Knowledgebase Mukherjee S, et al. 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A guide to best practices for gene ontology (GO) manual annotation. Database. 2013:1–18. Gene Ontology Consortium. Gene ontology consortium: going forward. Nucleic Acids Res. 2015;43:D1049–56. Angiuoli SV, et al. Toward an Online repository of standard operating procedures (SOPs) for (meta) genomic annotation. Omi A J Integr Biol. 2008;12:137–41. Costanzo MC, et al. Using computational predictions to improve literature-based Gene Ontology annotations: a feasibility study. Database (Oxford). 2011, 2011; bar004. Garcia-Garcia J, et al. Biana: a software framework for compiling biological interactions and analyzing networks. BMC Bioinformatics. 2010;11:56. Mazandu GK, Mulder NJ. The use of semantic similarity measures for optimally integrating heterogeneous gene ontology data from large scale annotation pipelines. Front Genet. 2014;5:264. Park J, et al. CvManGO, a method for leveraging computational predictions to improve literature-based gene ontology annotations. Database. 2012;2012:bas001. Brenner SE. Errors in genome annotation. Trends Genet. 1999;15:132–3. Devos D, Valencia A. Intrinsic errors in genome annotation. Trends Genet. 2001;17:429–31. Naumoff DG, et al. Retrieving sequences of enzymes experimentally characterized but erroneously annotated : the case of the putrescine carbamoyltransferase. BMC Genomics. 2004;5:52. Andorf C, et al. Exploring inconsistencies in genome-wide protein function annotations: a machine learning approach. BMC Bioinformatics. 2007;8:284. Keseler IM, et al. Curation accuracy of model organism databases. Database. 2014:1–6. Jones CE, et al. Estimating the annotation error rate of curated GO database sequence annotations. BMC Bioinformatics. 2007;8:170. Škunca N, et al. Quality of computationally inferred gene ontology annotations. PLoS Comput Biol. 2012;8:e1002533. Benson DA, et al. GenBank. Nucleic Acids Res. 2012;41:D36–42. Kanehisa M, et al. KEGG: new perspectives on genomes, pathways, diseases and drugs. Nucleic Acids Res. 2017;45:D353–61. Schnoes AM, et al. Annotation error in public databases: Misannotation of molecular function in enzyme Superfamilies. PLoS Comput Biol. 2009;5:e1000605. Barrett AJ, et al. In: Enzyme Nomenclature NC-ICBMB and Webb,E.C, editor. Academic Press, San Diego. (eds) ed; 1992. Saier MH. A functional-phylogenetic classification system for transmembrane solute transporters. Microbiol Mol Biol Rev. 2000;64:354–411. Altschul SF, et al. Basic local alignment search tool. J Mol Biol. 1990;215:403–10. Smith TF, Waterman MS. Identification of common molecular subsequences. J Mol Biol. 1981;147:195–7. Eddy SR. Profile hidden Markov models. Bioinformatics. 1998;14:755–63. Eilbeck K, et al. Quantitative measures for the management and comparison of annotated genomes. BMC Bioinformatics. 2009;10:67. We would like to acknowledge José Dias and Pedro Raposo, for providing data on the Lactobacillus helveticus and the Nitrosomonas europaea genome annotations. This study was supported by the Portuguese Foundation for Science and Technology (FCT) under the scope of the strategic funding of [UID/BIO/04469] unit and COMPETE 2020 [POCI-01-0145-FEDER-006684] and BioTecNorte operation [NORTE-01-0145-FEDER-000004] funded by the European Regional Development Fund under the scope of Norte2020 - Programa Operacional Regional do Norte. The authors thank the project DD-DeCaF - Bioinformatics Services for Data-Driven Design of Cell Factories and Communities, Ref. H2020-LEIT-BIO-2015-1 686070–1, funded by the European Commission. The funding body has no involvement in the design of the study and collection, analysis, and interpretation of data and in writing the manuscript. Centre of Biological Engineering, University of Minho, 4710-057, Braga, Portugal Fernando Cruz, Davide Lagoa, João Mendes, Isabel Rocha, Eugénio C. Ferreira, Miguel Rocha & Oscar Dias Instituto de Tecnologia Química e Biológica, Universidade Nova de Lisboa, 2780-157, Oeiras, Portugal Isabel Rocha Fernando Cruz Davide Lagoa Eugénio C. Ferreira Miguel Rocha Oscar Dias OD conceived and designed the study, managed its coordination and drafted the manuscript. FC participated in the design of the study, performed the genome annotations and helped to draft the manuscript. DL participated in the design of the study, generated the data, performed the implementation of SamPler as a new plugin for merlin and helped to draft the manuscript. JM performed genome annotations. MR, ECF and IR participated in the design and coordination of the study and helped to draft the manuscript. All authors read and approved the final manuscript. Correspondence to Oscar Dias. Example of a manual curation workflow for the genome functional annotation of the microorganism Lactobacillus rhamnosus using merlin. (PDF 173 kb) SamPler demonstration. Results obtained after applying SamPler procedure to 4 genome functional annotations. (XLSX 65 kb) Cruz, F., Lagoa, D., Mendes, J. et al. SamPler – a novel method for selecting parameters for gene functional annotation routines. BMC Bioinformatics 20, 454 (2019). https://doi.org/10.1186/s12859-019-3038-4 Annotation routines Parametrization Results and data	CommonCrawl
Download article (PDF) This article is published open access. MAG120pp. 16–28 Mirrors, lenses and Monge–Ampère equations Quentin Mérigot Université Paris-Saclay, Orsay, France Boris Thibert Université Grenoble Alpes, Grenoble, France 1 Anidolic optics and Monge Ampère type equations 2 Geometric discretization of Monge–Ampère equations 3 Numerical resolution 4 Applications to anidolic optics Is it possible to shape a piece of glass so that it refracts and concentrates sunlight in order to produce a given image? The modelling of this kind of problem leads to nonlinear second-order partial differential equations, which belong to the family of Monge–Ampère equations. We will see how semi-discrete methods, that can be traced back to Minkowski's works, allow us to numerically solve such equations. Figure 1. Mirror transforming a parallel, uniform light source into the shape of a train 🅭🅯 CC BY 4.0 In anidolic optics, or non-imaging optics, one studies the design of devices that transfer light energy between a source and a target. The general problem is to design the shape of a mirror (or a lens) that reflects (or refracts) the light emitted from a given source towards a target whose geometry and intensity distributions are prescribed (see Figures ⁠1⁠ and ⁠2⁠). Applications of anidolic optics include the design of solar ovens, public lighting, car headlights, and more generally the optimization of the use of light energy and the reduction of light pollution. Figure 2. Lens transforming a parallel light source into a hikari 🅭🅯 CC BY 4.0 Near-field and far-field light sources There exists many different problems in anidolic optics, depending for instance on the geometry of the light source, the type of optical component, and the target to be illuminated. These problems are distinguished in particular by the spatial position of the target illumination. A problem is called near-field when the target is located within a finite distance, i.e., when one wishes to illuminate an area of space such as a screen. In Figures ⁠1⁠ and ⁠2⁠, the target illumination is on a wall, making the problem near-field. Most of the illustrations in this article correspond near-field targets. However, we will first consider the far-field case, which is mathematically simpler, and in which the target lies "at infinity", in the space of directions. In practice, this means that when light is reflected or refracted from a point of the optical component, we can forget the spatial position of the reflected or refracted ray, keeping only its direction, which can then be encoded by a unit vector. Note that if the near-field target illumination is far away from the optical component, each point of the target almost corresponds to a direction, so that the far-field problem is a good approximation of the near-field one in this situation. We will see in Section ⁠4.2⁠ that one can solve a problem involving a near-field target by iteratively solving problems with far-field targets. We will first present two far-field mirror problems in their continuous form, as illustrated in Figure ⁠1⁠. Then, we will explain how these continuous problems can be approached by discrete problems, following the so-called supporting quadric method introduced by Luis Caffarelli and Vladimir Oliker. This method can be traced back to work of Hermann Minkowski and Aleksandr Aleksandrov in convex geometry. Figure 3. Mirror transforming light from a point source (left) or collimated light (right) 🅭🅯 CC BY 4.0 1.1 Mirror for a point light source In this first problem, light is emitted from a point OOO, which we assume to be located at the origin of the space R3\mathbb{R}^{3}R3. The intensity of the light source is modeled by a probability density μ\muμ on the sphere of directions S2\mathbb{S}^{2}S2. Let X⊂S2X\subset\mathbb{S}^{2}X⊂S2 denote the support of the measure μ\muμ. For example, if the light is emitted in a solid cone, then XXX is a disc on the sphere. The quantity of light emanating from a measurable set of directions A⊂XA\subset XA⊂X is given by μ(A)\mu(A)μ(A). For the far-field problem, the target is described by a probability measure ν\nuν on the sphere of directions S2\mathbb{S}^{2}S2, which then represents the directions "at infinity", i.e., after reflection. Let Y⊂S2Y\subset\mathbb{S}^{2}Y⊂S2 denote the support of the measure ν\nuν. The inverse problem considered here consists in constructing the surface R\mathcal{R}R of a mirror which will transport the intensity μ\muμ of the light source to the desired light distribution ν\nuν at infinity using Snell's law of reflection. For example, if the target measure is a Dirac mass δy\delta_{y}δy, meaning that we want to reflect all the light in a single direction yyy, then the shape of the mirror is given by a paraboloid of revolution. Let ⟨⋅ ∣ ⋅⟩\langle\cdot\,\|\,\cdot\rangle⟨⋅∣⋅⟩ denote the Euclidean scalar product on R3\mathbb{R}^{3}R3. An incident ray x∈S2x\in\mathbb{S}^{2}x∈S2 is reflected by a surface R\mathcal{R}R in the direction R(x)=x−⟨x ∣ n(x)⟩n(x)R(x)=x-\langle x\,\|\,n(x)\rangle n(x)R(x)=x−⟨x∣n(x)⟩n(x), where n(x)n(x)n(x) is the unit vector normal to the surface R\mathcal{R}R at the point touched by the direction xxx and oriented so that ⟨x ∣ n(x)⟩≤0\langle x\,\|\,n(x)\rangle\leq 0⟨x∣n(x)⟩≤0. The surface R\mathcal{R}R solves the inverse mirror problem if RRR transports the source measure μ\muμ to the target measure ν\nuν, in the sense that for any measurable subset BBB of the sphere one has ∀B⊆S2,ν(B)=μ(R−1(B)).\forall B\subseteq\mathbb{S}^{2},\quad\nu(B)=\mu\bigl(R^{-1}(B)\bigr).∀B⊆S2,ν(B)=μ(R−1(B)). Note that the preservation of overall light quantity was already ensured by having chosen probability measures, i.e., μ(S2)=ν(S2)=1\mu(\mathbb{S}^{2})=\nu(\mathbb{S}^{2})=1μ(S2)=ν(S2)=1. Now assume that μ\muμ and ν\nuν are absolutely continuous measures with respect to the area measure on the sphere. Let μ(x)=ρ(x)dx\mu(x)=\rho(x)\mathrm{d}xμ(x)=ρ(x)dx and ν(x)=σ(x)dx\nu(x)=\sigma(x)\mathrm{d}xν(x)=σ(x)dx, where ρ\rhoρ and σ\sigmaσ are the densities of μ\muμ and ν\nuν respectively. The previous equation then reads ∀B⊆S2,∫Bσ(x)dx=∫R−1(B)ρ(x)dx.\forall B\subseteq\mathbb{S}^{2},\quad\int_{B}\sigma(x)\mathrm{d}x=\int_{R^{-1}(B)}\rho(x)\mathrm{d}x.∀B⊆S2,∫Bσ(x)dx=∫R−1(B)ρ(x)dx. Suppose furthermore that the densities ρ\rhoρ et σ\sigmaσ are continuous and that RRR is a diffeomorphism from XXX to YYY. By the change of variable y=R(x)y=R(x)y=R(x), the last equation is then equivalent to σ(R(x))det⁡(DR(x))=ρ(x)\sigma(R(x))\det(DR(x))=\rho(x)σ(R(x))det(DR(x))=ρ(x) for any x∈Xx\in Xx∈X. Since the mirror reflects rays emitted from the origin, we will assume that the surface R\mathcal{R}R is radially parametrized by x∈S2↦u(x)xx\in\mathbb{S}^{2}\mapsto u(x)xx∈S2↦u(x)x, where u:S2→R+u:\mathbb{S}^{2}\to\mathbb{R}^{+}u:S2→R+ is a positive function that must be determined. The unit normal to the surface R\mathcal{R}R at the point xu(x)xu(x)xu(x) and the direction of the reflected ray can both be expressed as a function of xxx and of the gradient ∇u(x)∈TxS2\nabla u(x)\in\mathrm{T}_{x}\mathbb{S}^{2}∇u(x)∈TxS2: Ru(x)=x−⟨x ∣ nu(x)⟩nu(x)andnu(x)=∇u(x)−u(x)x∥∇u(x)∥2+u(x)2.R_{u}(x)=x-\langle x\,\|\,n_{u}(x)\rangle n_{u}(x)\quad\text{and}\quad n_{u}(x)=\frac{\nabla u(x)-u(x)x}{\sqrt{\left\\|\nabla u(x)\right\\|^{2}+u(x)^{2}}}.Ru(x)=x−⟨x∣nu(x)⟩nu(x)andnu(x)=∥∇u(x)∥2+u(x)2∇u(x)−u(x)x. This allows us to formulate the problem as a system of partial differential equations, i.e., the problem of finding a positive function u:S2→R+u:\mathbb{S}^{2}\to\mathbb{R}^{+}u:S2→R+ of class C2\mathcal{C}^{2}C2 which satisfies {σ(Ru(x))det⁡(DRu(x))=ρ(x),Ru be a diffeomorphism from X to Y.\begin{cases}\sigma\bigl(R_{u}(x)\bigr)\det\bigl(DR_{u}(x)\bigr)=\rho(x),\\ \text{$R_{u}$ be a diffeomorphism from $X$ to $Y$.}\end{cases}{σ(Ru(x))det(DRu(x))=ρ(x),Ru be a diffeomorphism from X to Y. The first line of equation (⁠Mir-Ponc-C⁠) involves the determinant of a quantity which depends on the second derivatives of uuu. This equation belongs to the family of Monge–Ampère equations. Note that the requirement that RuR_{u}Ru is a diffeomorphism is non-standard and difficult to handle. In practice, it is replaced by a condition on uuu which is akin to convexity, and by the so-called second boundary condition Ru(X)=YR_{u}(X)=YRu(X)=Y. These two conditions ensure the ellipticity of the problem. Caffarelli and Oliker proved in 1994 [⁠1⁠ L. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics. Journal of Mathematical Sciences154, 39–49 (2008) ] the existence of weak solutions to this equation, i.e., the existence of a locally Lipschitz function uuu such that the application RRR defined by the last two lines of (⁠Mir-Ponc-C⁠) satisfies (⁠1⁠). The existence of regular solutions to the problem (⁠Mir-Ponc-C⁠) is due to Wang and Guan [⁠6⁠ X.-J. Wang, On the design of a reflector antenna. Inverse Problems12, 351–375 (1996) , ⁠2⁠ P. Guan and X.-J. Wang, On a monge-ampere equation arising in geometric optics. J. Differential Geom48, 205–223 (1998) ]. 1.2 Mirror for a collimated light source We now present a second inverse problem arising in anidolic optics. This time the light source is collimated, which means that all the rays of light emitted by the source are parallel. We furthermore assume that they are positively collinear to the vertical vector ez=(0,0,1)e_{z}=(0,0,1)ez=(0,0,1) and emitted from a domain of the horizontal plane X⊂R2×{0}X\subset\mathbb{R}^{2}\times\{0\}X⊂R2×{0}. For convenience, we will identify R2\mathbb{R}^{2}R2 and R2×{0}\mathbb{R}^{2}\times\{0\}R2×{0}. We assume that the surface of the optical component is smooth and parametrized by a height function u:X→Ru:X\to\mathbb{R}u:X→R. The intensity of the light source is modeled by a probability measure μ\muμ on XXX. As in the previous case, the intensity of the target illumination is modeled by a probability measure ν\nuν on the sphere of directions at infinity. At each point (x,u(x))(x,u(x))(x,u(x)) of the optical component, the gradient ∇u(x)\nabla u(x)∇u(x) encodes the direction of the normal to the surface and we denote by F(∇u(x))∈S2F(\nabla u(x))\in\mathbb{S}^{2}F(∇u(x))∈S2 the direction of the ray reflected by Snell's law. The reflector defined by uuu solves the inverse mirror problem between μ\muμ and ν\nuν if for any measurable set B⊂S2B\subset\mathbb{S}^{2}B⊂S2 one has ν(B)=μ((F∘∇u)−1(B)).\nu(B)=\mu\bigl((F\circ\nabla u)^{-1}(B)\bigr).ν(B)=μ((F∘∇u)−1(B)). Let us introduce the measure ν~\tilde{\nu}ν~ defined by ν~(B)=ν(F(B))\tilde{\nu}(B)=\nu(F(B))ν~(B)=ν(F(B)), which is supported on R2\mathbb{R}^{2}R2. We assume that μ\muμ and ν~\tilde{\nu}ν~ are absolutely continuous with respect to the Lebesgue measure, with continuous densities ρ\rhoρ and σ\sigmaσ, and that x↦∇u(x)x\mapsto\nabla u(x)x↦∇u(x) is a diffeomorphism on its image. Then, with the change of variable y=∇u(x)y=\nabla u(x)y=∇u(x), the inverse mirror problem for a collimated source can also be phrased as a partial differential equation: {σ(∇u(x))det⁡(D2u(x))=ρ(x),F∘∇u is a diffeomorphism from X to Y.\begin{cases}\sigma\bigl(\nabla u(x)\bigr)\det\bigl(D^{2}u(x)\bigr)=\rho(x),\\ \text{$F\circ\nabla u$ is a diffeomorphism from $X$ to $Y$.}\end{cases}{σ(∇u(x))det(D2u(x))=ρ(x),F∘∇u is a diffeomorphism from X to Y. We finally note that if uuu is smooth and strongly convex (or strongly concave), the application ∇u\nabla u∇u is a diffeomorphism on its image. 1.3 Lenses The construction of lenses that transform a light source into a target illumination prescribed at infinity are similar and are also described by Monge–Ampère type equations. As with mirrors, when the light is emitted from a point, the equation to be solved is on the sphere and when the light source is collimated, it is on the plane. In these problems, the light source passes through the surface of one side of the lens, either flat or spherical, and the aim is to construct the surface of the other side of the lens such that it refracts the light onto a prescribed target illumination at infinity. We do not detail the modelling of these problems here, but we will show results with lenses at the end. The two inverse problems in anidolic optics described in the previous section each involve two sets XXX and YYY, on which we have two probability measures μ\muμ and ν\nuν respectively, representing the light source and the desired target illumination. We saw that when these measures are absolutely continuous, the problems of construction of optical components correspond to partial differential equations of Monge–Ampère type. The most direct method to solve a partial differential equation numerically is to approximate the domain XXX with a discrete grid and to replace the partial derivatives with differences of the values of the function at the points of the grid divided by the grid step. In the case of Monge–Ampère equations, the application of these methods is made difficult by the non-linearity of the Monge–Ampère operator and by the diffeomorphism condition. We refer to the work of Adam Oberman, Brittany Froese, Jean-David Benamou and Jean-Marie Mirebeau for this line of research. In recent years, alternative methods, called semi-discrete methods, have been used to discretize and numerically solve Monge–Ampère type equations arising from optimal transport. In order to apply this method, one assumes that one of the two measures μ\muμ or ν\nuν is absolutely continuous, while the other is finitely supported. Here we assume that μ\muμ has a density μ(x)=ρ(x)dx\mu(x)=\rho(x)\mathrm{d}xμ(x)=ρ(x)dx on the space XXX and that ν\nuν is a discrete measure on the space Y={y1,…,yN}Y=\{y_{1},\ldots,y_{N}\}Y={y1,…,yN}, i.e., ν=∑1≤i≤Nδyiνi\nu=\sum_{1\leq i\leq N}\delta_{y_{i}}\nu_{i}ν=∑1≤i≤Nδyiνi where δyi\delta_{y_{i}}δyi is the Dirac mass in yiy_{i}yi. In this section, we describe the semi-discrete variant of the two far-field mirror problems seen in the previous section, leaving aside the problem of convergence of the solutions to the discrete problems towards those to the continuous problems. These constructions give rise to equations which can naturally be seen as discrete Monge–Ampère equations. We also propose an economic interpretation by addressing the bakeries problem. Let us go back to the problem of constructing mirrors that transform the light emitted by a point light source (see Section ⁠1.1⁠). As in the previous section, the light source is modeled by a continuous probability density ρ\rhoρ on the sphere of directions S2\mathbb{S}^{2}S2, whose support Xρ:={x∈S2, ρ(x)>0}X_{\rho}:=\{x\in\mathbb{S}^{2},\ \rho(x)>0\}Xρ:={x∈S2, ρ(x)>0} corresponds to the set of directions in which light is emitted. This time we assume that the desired target illumination is described by a probability measure ν=∑1≤i≤Nδyiνi\nu=\sum_{1\leq i\leq N}\delta_{y_{i}}\nu_{i}ν=∑1≤i≤Nδyiνi supported on a set Y={y1,…,yN}⊂S2Y=\{y_{1},\ldots,y_{N}\}\subset\mathbb{S}^{2}Y={y1,…,yN}⊂S2 of distinct directions. The problem is still to find the mirror surface R\mathcal{R}R that will reflect the measure μ\muμ onto the measure ν\nuν under Snell's law, but this time the target measure ν\nuν is discrete. Mirror composed of paraboloid pieces We use the method of supporting paraboloids proposed by Caffarelli and Oliker in 1994 [⁠1⁠ L. Caffarelli and V. Oliker, Weak solutions of one inverse problem in geometric optics. Journal of Mathematical Sciences154, 39–49 (2008) ], which was originally developed to show the existence of weak solutions in the case where both measures are absolutely continuous. Caffarelli and Oliker's idea is based on a well-known property of paraboloids of revolution: a paraboloid of revolution with focal point OOO and direction yyy reflects any ray coming from point OOO to the direction yyy. It is thus natural to seek to construct a mirror whose surface is composed of pieces of paraboloids, each paraboloid illuminating a direction yiy_{i}yi. More precisely, we take ψ=(ψ1,…,ψN)∈RN\psi=(\psi_{1},\ldots,\psi_{N})\in\mathbb{R}^{N}ψ=(ψ1,…,ψN)∈RN and denote by P(yi,ψi)P(y_{i},\psi_{i})P(yi,ψi) the solid (i.e. filled) paraboloid of direction yiy_{i}yi, with focal point at the origin OOO and focal distance ψi\psi_{i}ψi. This means that 12ψi\frac{1}{2}\psi_{i}21ψi is the distance between OOO and the paraboloid's closest point to OOO. We define by Rψ\mathcal{R}_{\psi}Rψ the surface bordering the intersection of the solid paraboloids P(yi,ψi)P(y_{i},\psi_{i})P(yi,ψi): Rψ=∂(⋂1≤i≤NP(yi,ψi)).\mathcal{R}_{\psi}=\partial\left(\bigcap_{1\leq i\leq N}P(y_{i},\psi_{i})\right).Rψ=∂(1≤i≤N⋂P(yi,ψi)). For each i∈{1,…,N}i\in\{1,\ldots,N\}i∈{1,…,N} we denote by Vi(ψ)\mathrm{V}_{i}(\psi)Vi(ψ) the set of rays x∈S2x\in\mathbb{S}^{2}x∈S2 emitted by the light source and reflected by Snell's law in the direction yiy_{i}yi. This set is called the iii-th visibility cell of the mirror Rψ\mathcal{R}_{\psi}Rψ. By construction, it corresponds to the radial projection of Rψ∩∂P(yi,ψi)\mathcal{R}_{\psi}\cap\partial P(y_{i},\psi_{i})Rψ∩∂P(yi,ψi) onto the sphere (see Figure ⁠2.1⁠). A simple calculation shows that the intersection of two confocal paraboloids ∂P(yi,ψi)\partial P(y_{i},\psi_{i})∂P(yi,ψi) and ∂P(yj,ψj)\partial P(y_{j},\psi_{j})∂P(yj,ψj) is included in a plane curve. Projecting radially onto the unit sphere, this implies that the intersection of two visibility cells Vi(ψ)∩Vj(ψ)\mathrm{V}_{i}(\psi)\cap\mathrm{V}_{j}(\psi)Vi(ψ)∩Vj(ψ) is included in a curve on the sphere. We deduce that the set of visibility cells forms a partition of the sphere S2\mathbb{S}^{2}S2, up to a set of measure zero. The paraboloid of revolution ∂P(yk,ψk)\partial P(y_{k},\psi_{k})∂P(yk,ψk) can be parametrized radially by the function x∈S2↦xρk(x)x\in\mathbb{S}^{2}\mapsto x\rho_{k}(x)x∈S2↦xρk(x), where ρk(x)=ψk/(1−⟨x ∣ yi⟩)∈R\rho_{k}(x)=\psi_{k}/(1-\langle x\,\|\,y_{i}\rangle)\in\mathbb{R}ρk(x)=ψk/(1−⟨x∣yi⟩)∈R. We deduce that xxx belongs to the visibility cell Vi(ψ)\mathrm{V}_{i}(\psi)Vi(ψ) if and only if the distance ρi(x)\rho_{i}(x)ρi(x) is smaller than the distances ρj(x)\rho_{j}(x)ρj(x) for j∈{1,…,N}j\in\{1,\ldots,N\}j∈{1,…,N}. Composing with the logarithm to linearize the expression in ψ\psiψ, we obtain an explicit expression for the visibility cells Vi(ψ)={x∈S2∣ ∀j, c(x,yi)+ln⁡(ψi)≤c(x,yj)+ln⁡(ψj)},\mathrm{V}_{i}(\psi)=\left\{x\in\mathbb{S}^{2}\mid~{}\forall j,\ c(x,y_{i})+\ln(\psi_{i})\leq c(x,y_{j})+\ln(\psi_{j})\right\},Vi(ψ)={x∈S2∣ ∀j, c(x,yi)+ln(ψi)≤c(x,yj)+ln(ψj)}, where c(x,y)=−ln⁡(1−⟨x ∣ y⟩)c(x,y)=-\ln(1-\langle x\,\|\,y\rangle)c(x,y)=−ln(1−⟨x∣y⟩). By construction, each ray emitted by the point source and belonging to the cell Vi(ψ)\mathrm{V}_{i}(\psi)Vi(ψ) hits the mirror Rψ\mathcal{R}_{\psi}Rψ at a point which belongs to the paraboloid ∂P(yi,ψi)\partial P(y_{i},\psi_{i})∂P(yi,ψi) and which is reflected in the direction yiy_{i}yi. The quantity of light received in the direction yiy_{i}yi is therefore exactly the quantity of light emanating from the visibility cell Vi(ψ)\mathrm{V}_{i}(\psi)Vi(ψ), i.e., μ(Vi(ψ))\mu(\mathrm{V}_{i}(\psi))μ(Vi(ψ)). The desired quantity of light in the direction yiy_{i}yi is νi\nu_{i}νi. The equation to be solved is therefore μ(Vi(ψ))=νi\mu(\mathrm{V}_{i}(\psi))=\nu_{i}μ(Vi(ψ))=νi for any i∈{1,…,N}i\in\{1,\ldots,N\}i∈{1,…,N}. Moreover, note that a paraboloid of revolution is only determined by its focal point, its direction and its focal distance. The free parameter remaining for each paraboloid ∂P(yi,ψi)\partial P(y_{i},\psi_{i})∂P(yi,ψi) is the focal distance ψi\psi_{i}ψi. Figure 4. Mirror composed of three pieces of paraboloides reflecting in three directions 🅭🅯 CC BY 4.0 Formulation of the problem The semi-discrete far-field mirror problem for a point source can be formulated as the problem of finding focal distances ψ=(ψ1,…,ψN)∈RN\psi=(\psi_{1},\ldots,\psi_{N})\in\mathbb{R}^{N}ψ=(ψ1,…,ψN)∈RN that satisfy ∀i, μ(Vi(ψ))=νi\forall i,~{}~{}\mu\bigl(\mathrm{V}_{i}(\psi)\bigr)=\nu_{i}∀i, μ(Vi(ψ))=νi where c(x,y)=−ln⁡(1−⟨x ∣ y⟩)c(x,y)=-\ln(1-\langle x\,\|\,y\rangle)c(x,y)=−ln(1−⟨x∣y⟩) and where Vi(ψ)={x∣ ∀j, c(x,yi)+ln⁡(ψi)≤c(x,yj)+ln⁡(ψj)}.\mathrm{V}_{i}(\psi)=\left\{x\mid~{}\forall j,\ c(x,y_{i})+\ln(\psi_{i})\leq c(x,y_{j})+\ln(\psi_{j})\right\}.Vi(ψ)={x∣ ∀j, c(x,yi)+ln(ψi)≤c(x,yj)+ln(ψj)}. We will see in Section ⁠3⁠ how to solve such systems of equations. Note that if ψ∈RN\psi\in\mathbb{R}^{N}ψ∈RN is a vector of focal distances solving the mirror problem for a point-like source, then the surface of the mirror is parametrized by Rψ: x∈S2 ↦ min⁡iψi1−⟨x ∣ yi⟩ x.\mathcal{R}_{\psi}:\ x\in\mathbb{S}^{2}\ \mapsto\ \min_{i}\frac{\psi_{i}}{1-\langle x\,\|\,y_{i}\rangle}\ x.Rψ: x∈S2 ↦ imin1−⟨x∣yi⟩ψi x. In the numerical experiments, we assume that the target illumination ν\nuν is included in the half-sphere S−2:={x∈S2, ⟨x ∣ ez⟩≤0}\mathbb{S}^{2}_{-}:=\{x\in\mathbb{S}^{2},\ \langle x\,\|\,e_{z}\rangle\leq 0\}S−2:={x∈S2, ⟨x∣ez⟩≤0}, that the support XρX_{\rho}Xρ of ρ\rhoρ is included in the half-sphere S+2:={x∈S2, ⟨x ∣ ez⟩≥0}\mathbb{S}^{2}_{+}:=\{x\in\mathbb{S}^{2},\ \langle x\,\|\,e_{z}\rangle\geq 0\}S+2:={x∈S2, ⟨x∣ez⟩≥0}, and that the mirror is parametrized above the domain XρX_{\rho}Xρ. Remark 2.1. The mirror surface is by construction the boundary of a convex set, i.e., the intersection of the solid paraboloids P(y1,ψ1),…,P(yN,ψN)P(y_{1},\psi_{1}),\ldots,P(y_{N},\psi_{N})P(y1,ψ1),…,P(yN,ψN). It is also possible to construct a mirror contained in the boundary of the union of solid paraboloids rather than an intersection. This produces mirrors that are somewhat less interesting in practice, as they are neither convex nor concave. Let us now consider the mirror problem for a collimated light source, already seen in Section ⁠1.2⁠. As before, the probability measure modelling the light source has a density ρ\rhoρ with respect to the Lebesgue measure on the plane. However the probability measure modelling the target illumination intensity is discrete ν=∑1≤i≤Nδyiνi\nu=\sum_{1\leq i\leq N}\delta_{y_{i}}\nu_{i}ν=∑1≤i≤Nδyiνi, supported on a finite set Y={y1,…,yN}⊂S2Y=\{y_{1},\ldots,y_{N}\}\subset\mathbb{S}^{2}Y={y1,…,yN}⊂S2 of distinct directions. The problem is, again, to find the surface R\mathcal{R}R of a mirror which reflects the measure μ\muμ to the measure ν\nuν. Mirror with planar faces We choose to construct the mirror surface R\mathcal{R}R as the graph of affine height functions of the form x∈R2↦max⁡i⟨x ∣ pi⟩−ψix\in\mathbb{R}^{2}\mapsto\max_{i}\langle x\,\|\,p_{i}\rangle-\psi_{i}x∈R2↦maxi⟨x∣pi⟩−ψi (see Figure ⁠2.2⁠). The vector pip_{i}pi is chosen so that the plane Pi={(x,⟨x ∣ pi⟩)∣x∈R2}P_{i}=\{(x,\langle x\,\|\,p_{i}\rangle)\mid x\in\mathbb{R}^{2}\}Pi={(x,⟨x∣pi⟩)∣x∈R2} reflects vertical rays, i.e., with direction eze_{z}ez, into the direction yi∈S2y_{i}\in\mathbb{S}^{2}yi∈S2. We need to determine the heights ψi\psi_{i}ψi of those planes. Given a family of heights ψ∈RN\psi\in\mathbb{R}^{N}ψ∈RN, we define the iii-th visibility cell as Vi(ψ)={x∈R2×{0}∣ ∀j, −⟨x ∣ pi⟩+ψi≤−⟨x ∣ pj⟩+ψj}.\mathrm{V}_{i}(\psi)=\{x\in\mathbb{R}^{2}\times\{0\}\mid~{}\forall j,\ -\langle x\,\|\,p_{i}\rangle+\psi_{i}\leq-\langle x\,\|\,p_{j}\rangle+\psi_{j}\}.Vi(ψ)={x∈R2×{0}∣ ∀j, −⟨x∣pi⟩+ψi≤−⟨x∣pj⟩+ψj}. Figure 5. Convex mirror for a collimated light source 🅭🅯 CC BY 4.0 By construction, for each i∈{1,…,N}i\in\{1,\ldots,N\}i∈{1,…,N}, any vertical ray emitted from a point x∈Vi(ψ)x\in V_{i}(\psi)x∈Vi(ψ) hits the mirror R\mathcal{R}R at a height ⟨x ∣ pi⟩−ψi\langle x\,\|\,p_{i}\rangle-\psi_{i}⟨x∣pi⟩−ψi and is reflected in the direction yiy_{i}yi. Thus, the amount of light reflected in the direction yiy_{i}yi is equal to μ(Vi(ψ))\mu(\mathrm{V}_{i}(\psi))μ(Vi(ψ)). Solving the semi-discrete far-field mirror problem for a collimated light source amounts to finding the heights ψ∈RN\psi\in\mathbb{R}^{N}ψ∈RN that satisfies where c(x,y)=−⟨x ∣ y⟩c(x,y)=-\langle x\,\|\,y\ranglec(x,y)=−⟨x∣y⟩ and Vi(ψ)={x∣∀j, c(x,yi)+ψi≤c(x,yj)+ψj}.\mathrm{V}_{i}(\psi)=\left\{x\mid\forall j,\ c(x,y_{i})+\psi_{i}\leq c(x,y_{j})+\psi_{j}\right\}.Vi(ψ)={x∣∀j, c(x,yi)+ψi≤c(x,yj)+ψj}. A solution of the equation (⁠Mir-Colli-SD⁠) induces a parametrization of the convex mirror R\mathcal{R}R that reflects μ\muμ onto ν\nuν: Rψ: x∈R2 ↦ (x,max⁡i⟨x ∣ pi⟩−ψi)∈R3.\mathcal{R}_{\psi}:\ x\in\mathbb{R}^{2}\ \mapsto\ \bigl(x,\max_{i}\langle x\,\|\,p_{i}\rangle-\psi_{i}\bigr)\in\mathbb{R}^{3}.Rψ: x∈R2 ↦ (x,imax⟨x∣pi⟩−ψi)∈R3. In practice, we only consider the part of the mirror located above the domain Xρ:={x∈R2×{0}, ρ(x)≠0}X_{\rho}:=\{x\in\mathbb{R}^{2}\times\{0\},~{}\rho(x)\neq 0\}Xρ:={x∈R2×{0}, ρ(x)=0}. Remark 2.2. The function Rψ\mathcal{R}_{\psi}Rψ being the maximum of affine functions, it is convex. The optical component which is parametrized by the graph of this application is also convex. Note that one could have the same construction by replacing the max⁡\maxmax in the formula by a min⁡\minmin. This would result in a concave function Rψ\mathcal{R}_{\psi}Rψ and a concave mirror. Remark 2.3. Problem (⁠Mir-Colli-SD⁠) is very similar (but not equivalent) to Minkowski's problem in convex geometry which is also an inverse problem. Given a set of unit vectors yiy_{i}yi and real numbers νi>0\nu_{i}>0νi>0, this problem consists in building a convex polyhedron whose iii-th facet has normal yiy_{i}yi and area νi\nu_{i}νi – which is possible only under some assumptions on the directions and areas. We also note that Oliker, who was the first to introduce semi-discrete methods for the numerical resolution of Monge–Ampère equations, was a doctoral student of the famous geometer Aleksandrov who is known (among other) for introducing and studying the "continuous" formulation of Minkowski's problem. Figure 6. Bakeries: The city XXX with its boundary drawn in blue is endowed with a probability density pictured in grayscale representing the population density. The set YYY (in red) represents the location of bakeries. Here, X,Y⊆R2X,Y\subseteq\mathbb{R}^{2}X,Y⊆R2 and c(x,y)=∣x−y∣2c(x,y)=\|x-y\|^{2}c(x,y)=∣x−y∣2. We see the Voronoi tessellation of the city (in the middle, uniform price) as well as its Laguerre tessellation (on the right, only the bread price ψ1\psi_{1}ψ1 has increased). 🅭🅯 CC BY 4.0 2.3 The bakeries problem We now present an economic analogy which leads to an equation having the same structure as in the two optical problems presented above. We assume that XXX represents a city whose population density is described by a probability density ρ\rhoρ, that the finite set Y={y1,…,yN}Y=\{y_{1},\ldots,y_{N}\}Y={y1,…,yN} represents the locations of the city's bakeries and that νi\nu_{i}νi represents the quantity of bread available in bakery yiy_{i}yi. Customers living at a location xxx in XXX naturally will look for the bakery minimizing the cost of walking from xxx to yiy_{i}yi, denoted c(x,yi)c(x,y_{i})c(x,yi). This leads to a decomposition of the city space into Voronoi cells, Vor⁡i:={x∈ΩX∣∀j, c(x,yi)≤c(x,yj)}.\operatorname{Vor}_{i}:=\bigl\{x\in\Omega_{X}\mid\forall j,\ c(x,y_{i})\leq c(x,y_{j})\bigr\}.Vori:={x∈ΩX∣∀j, c(x,yi)≤c(x,yj)}. The number of customers going to a bakery yiy_{i}yi is equal to the integral of the density ρ\rhoρ over Vor⁡i\operatorname{Vor}_{i}Vori. Suppose that a bakery yiy_{i}yi receives too many customers in comparison to its bread's production capacity νi\nu_{i}νi – this could be the case in Figure ⁠6⁠ for the downtown bakery y1y_{1}y1 where the population density is high. This means we have μ(Vor⁡1)>ν1\mu(\operatorname{Vor}_{1})>\nu_{1}μ(Vor1)>ν1, where we denote μ(x)=ρ(x)dx\mu(x)=\rho(x)\mathrm{d}xμ(x)=ρ(x)dx. The baker y1y_{1}y1 will then seek to increase the price of his bread. This will reduce the number of potential customers, but will increase the baker's profit as long as he manages to sell all his stock. We write νi≥0\nu_{i}\geq 0νi≥0 for the proportion of the population that the bakery yiy_{i}yi is able to serve, and ψi\psi_{i}ψi the price of the bread in the bakery yiy_{i}yi. If we assume that the customers living at point xxx make a compromise between walking cost and price of bread by minimizing the sum (c(x,yi)+ψic(x,y_{i})+\psi_{i}c(x,yi)+ψi), the city is then decomposed into Laguerre cells Lag⁡i(ψ)={x∈X∣ ∀j, c(x,yi)+ψi≤c(x,yj)+ψj}.\operatorname{Lag}_{i}(\psi)=\left\{x\in X\mid~{}\forall j,\ c(x,y_{i})+\psi_{i}\leq c(x,y_{j})+\psi_{j}\right\}.Lagi(ψ)={x∈X∣ ∀j, c(x,yi)+ψi≤c(x,yj)+ψj}. Note that we do not necessarily have yi∈Lag⁡i(ψ)y_{i}\in\operatorname{Lag}_{i}(\psi)yi∈Lagi(ψ), and that it is even possible to have Lag⁡i(ψ)=∅\operatorname{Lag}_{i}(\psi)=\emptysetLagi(ψ)=∅: indeed, if the bread is very expensive in a certain bakery, even people living next door may prefer going to a more distant one. Problem formulation The bakeries problem therefore boils down to finding a price vector ψ∈RN\psi\in\mathbb{R}^{N}ψ∈RN such that each bakery sells all its stock of bread νi\nu_{i}νi. This is described by the system of equations μ(Lag⁡i(ψ))=νi∀i∈{1,…,N},\mu\bigl(\operatorname{Lag}_{i}(\psi)\bigr)=\nu_{i}\quad\forall i\in\{1,\ldots,N\},μ(Lagi(ψ))=νi∀i∈{1,…,N}, This equation has exactly the same structure as (⁠Mir-Ponc-SD⁠) and (⁠Mir-Colli-SD⁠). We will see in the next section how to solve this class of equations. The discrete problems mentioned in the previous section all show the same structure; our focus will now be on their numerical resolution. We start by introducing the semi-discrete Monge–Ampère equation, and show that its solution is equivalent to finding the maximum of a concave function. Subsequently, we present a Newton method that allows us to solve these equations efficiently. 3.1 Semi-discrete Monge–Ampère equation Let XXX be a compact subset of the space R2\mathbb{R}^{2}R2 or of the sphere S2\mathbb{S}^{2}S2, let Y={y1,…,yN}Y=\{y_{1},\ldots,y_{N}\}Y={y1,…,yN}, and let c∈C1(X×Y)c\in\mathcal{C}^{1}(X\times Y)c∈C1(X×Y) be a cost function. The Laguerre cell (which corresponds to a visibility cell in optics) associated with a family of real numbers ψ=(ψ1,…,ψN)∈RN\psi=(\psi_{1},\ldots,\psi_{N})\in\mathbb{R}^{N}ψ=(ψ1,…,ψN)∈RN is given by Suppose that the cost function satisfies the Twist condition ∀x∈X,y↦∇xc(x,y) is injective,\forall x\in X,\quad y\mapsto\nabla_{x}c(x,y)\ \textrm{is injective},∀x∈X,y↦∇xc(x,y) is injective, which ensures that the Laguerre cells form a partition of the domain XXX up to a negligible set. Semi-discrete Monge–Ampère equation Let μ\muμ be a probability measure on XXX with density ρ\rhoρ with respect to the area measure, and let ν=∑iνiδyi\nu=\sum_{i}\nu_{i}\delta_{y_{i}}ν=∑iνiδyi be a probability measure on YYY. In the following equation, the discrete probability measure ν\nuν is conflated with the vector ν=(νi)1≤i≤N\nu=(\nu_{i})_{1\leq i\leq N}ν=(νi)1≤i≤N. We are seeking ψ∈RN\psi\in\mathbb{R}^{N}ψ∈RN satisfying G(ψ)=ν,G(\psi)=\nu,G(ψ)=ν, where the function G:RN→RNG:\mathbb{R}^{N}\to\mathbb{R}^{N}G:RN→RN is defined by G(ψ)=(G1(ψ),…,GN(ψ))andGi(ψ)=μ(Lag⁡i(ψ)).G(\psi)=\bigl(G_{1}(\psi),\ldots,G_{N}(\psi)\bigr)\quad\text{and}\quad G_{i}(\psi)=\mu\bigl(\operatorname{Lag}_{i}(\psi)\bigr).G(ψ)=(G1(ψ),…,GN(ψ))andGi(ψ)=μ(Lagi(ψ)). Remark 3.1. The visibility cells used in optics in (⁠Mir-Ponc-SD⁠) and (⁠Mir-Colli-SD⁠) are Laguerre cells, with c(x,y)=−log⁡(1−⟨x ∣ y⟩)andc(x,y)=−⟨x ∣ y⟩c(x,y)=-\log\bigl(1-\langle x\,\|\,y\rangle\bigr)\quad\text{and}\quad c(x,y)=-\langle x\,\|\,y\ranglec(x,y)=−log(1−⟨x∣y⟩)andc(x,y)=−⟨x∣y⟩ respectively. Equation (⁠MA⁠) is a reformulation of equations (⁠Mir-Ponc-SD⁠) and (⁠Mir-Colli-SD⁠). Note that the Laguerre cells are invariant under addition of a constant to ψ\psiψ, and that the solution of (⁠MA⁠) is therefore defined up to an additive constant. Optical problems have a similar invariance: for example, if a surface R\mathcal{R}R is a solution of the mirror problem for a point source, then so is λR\lambda\mathcal{R}λR for all λ>0\lambda>0λ>0. 3.2 Variational formulation The following theorem shows that the function GGG in the semi-discrete Monge–Ampère equation is the gradient of a concave function. Theorem 3.1. We assume that the cost function ccc satisfies (⁠Twist⁠). Then the function K:RN→R\mathcal{K}:\mathbb{R}^{N}\to\mathbb{R}K:RN→R defined by K(ψ)=∑1≤i≤N∫Lag⁡i(ψ)(c(x,y)+ψi)ρ(x)dx−∑1≤i≤Nψiνi\mathcal{K}(\psi)=\sum_{1\leq i\leq N}\int_{\operatorname{Lag}_{i}(\psi)}\bigl(c(x,y)+\psi_{i}\bigr)\rho(x)\mathrm{d}x-\sum_{1\leq i\leq N}\psi_{i}\nu_{i}K(ψ)=1≤i≤N∑∫Lagi(ψ)(c(x,y)+ψi)ρ(x)dx−1≤i≤N∑ψiνi is concave, of class C1\mathcal{C}^{1}C1 and with gradient ∇K(ψ)=G(ψ)−ν=(μ(Lag⁡i(ψ))−νi)1≤i≤N.\nabla\mathcal{K}(\psi)=G(\psi)-\nu=\bigl(\mu\bigl(\operatorname{Lag}_{i}(\psi)\bigr)-\nu_{i}\bigr)_{1\leq i\leq N}.∇K(ψ)=G(ψ)−ν=(μ(Lagi(ψ))−νi)1≤i≤N. As we will see in the next paragraph, the function K\mathcal{K}K is related to the Kantorovitch duality in optimal transport theory, and we will therefore call it the Kantorovitch functional. Moreover, since a concave function of class C1\mathcal{C}^{1}C1 reaches its maximum exactly at its critical points, we obtain the following corollary: Corollary 3.2. Under the assumptions of Theorem ⁠3.1⁠, a vector ψ∈RN\psi\in\mathbb{R}^{N}ψ∈RN is a solution to equation (⁠MA⁠) if and only if ψ\psiψ is a maximizer of K\mathcal{K}K. Since the function K\mathcal{K}K is invariant under addition of a constant, one can choose to work on the set M0\mathcal{M}_{0}M0 of vectors whose coordinates sum to zero. It can be shown that the function K\mathcal{K}K is proper on M0\mathcal{M}_{0}M0, i.e., lim⁡∥ψ∥→+∞,ψ∈M0K(ψ)=−∞\lim_{\left\\|\psi\right\\|\to+\infty,\psi\in\mathcal{M}_{0}}\mathcal{K}(\psi)=-\inftylim∥ψ∥→+∞,ψ∈M0K(ψ)=−∞, which ensures that it reaches its maximum: the problem (⁠MA⁠) thus admits a solution. 3.3 Relation to optimal transport The variational formulation of the Monge–Ampère equation, i.e., the search for a maximizer of the Kantorovitch functional, corresponds in fact to the dual of the Monge–Kantorovitch problem in optimal transport theory. We discuss this link in detail below in the semi-discrete case. The reader interested in the proofs may refer for instance to the book chapter [⁠4⁠ Q. Mérigot and B. Thibert, Optimal transport: Discretization and algorithms. In Handbook of Numerical Analysis, Geometric PDEs 22, Elsevier, 621–679 (2020) ]. Monge's problem The image of a probability measure μ\muμ on XXX under a measurable application T:X→YT:X\to YT:X→Y is the measure T#μT_{\#}\muT#μ on YYY defined by T#μ(B)=μ(T−1(B))T_{\#}\mu(B)=\mu(T^{-1}(B))T#μ(B)=μ(T−1(B)). If T#μ=νT_{\#}\mu=\nuT#μ=ν, we say that TTT transports μ\muμ to ν\nuν. Since the set YYY is finite, we have T#μ=∑1≤i≤Nμ(T−1(yi))δyiT_{\#}\mu=\sum_{1\leq i\leq N}\mu(T^{-1}(y_{i}))\delta_{y_{i}}T#μ=∑1≤i≤Nμ(T−1(yi))δyi. Monge's optimal transport problem consists in finding a transport map TTT that transports μ\muμ to ν\nuν and that minimizes the total cost ∫Xc(x,T(x))dμ(x)\int_{X}c(x,T(x))d\mu(x)∫Xc(x,T(x))dμ(x). If the cost function ccc satisfies the Twist condition, Brenier and Gangbo–McCann, relying on Kantorovich duality, proved the existence of a minimizer for this problem when the source μ\muμ is absolutely continuous. For example, one can state the following: Theorem 3.3 (Kantorovitch duality). Suppose that ccc satisfies the condition (⁠Twist⁠) and that μ\muμ is absolutely continuous. Then min⁡T:X→YT♯μ=ν∫Xc(x,T(x))dμ(x)=max⁡ψ∈RNK(ψ).\min_{{T:X\to Y}\atop{T_{\sharp}\mu=\nu}}\int_{X}c\bigl(x,T(x)\bigr)d\mu(x)=\max_{\psi\in\mathbb{R}^{N}}\mathcal{K}(\psi).T♯μ=νT:X→Ymin∫Xc(x,T(x))dμ(x)=ψ∈RNmaxK(ψ). If moreover ψ\psiψ is a maximizer of K\mathcal{K}K, then the function Tψ:X→YT_{\psi}:X\to YTψ:X→Y defined μ\muμ-a.e. by Tψ∣Lag⁡y(ψ)=yT_{\psi}\|_{\operatorname{Lag}_{y}(\psi)}=yTψ∣Lagy(ψ)=y realizes the minimum in Monge's problem. Remark 3.2. Not all Monge–Ampère equations derive from an optimal transport problem and not all of them admit a variational formulations. These two strong properties come in fact from the very particular structure of Laguerre cells, which follows from the functions ψ↦c(x,y)+ψ(y)\psi\mapsto c(x,y)+\psi(y)ψ↦c(x,y)+ψ(y) being affine. We saw that the far-field optics problems presented in Section ⁠2⁠ possess this structure. On the other hand, if we consider the mirror construction problems for a target illumination in the near-field (i.e., we are illuminating points in R3\mathbb{R}^{3}R3 and not directions at infinity), we still have semi-discrete Monge–Ampère equations to solve, but the Laguerre cells are of the form Lag⁡i(ψ)={x∈X∣∀j, G(x,yi,ψi)≤G(x,yj,ψj)},\operatorname{Lag}_{i}(\psi)=\bigl\{x\in X\mid\forall j,\ G(x,y_{i},\psi_{i})\leq G(x,y_{j},\psi_{j})\bigr\},Lagi(ψ)={x∈X∣∀j, G(x,yi,ψi)≤G(x,yj,ψj)}, where the function GGG is nonlinear in ψ\psiψ. These equations do not derive from the optimal transport problem and in fact do not admit a variational formulation. They are called prescribed Jacobian equations by Trudinger, and are the subject of recent research both in analysis and in more applied fields (optics, economics). 3.4 Laguerre cells and derivatives Before applying Newton's method to solve the equation G(ψ)=νG(\psi)=\nuG(ψ)=ν, we need to show that the function GGG is of class C1\mathcal{C}^{1}C1 (or equivalently that K\mathcal{K}K is of class C2\mathcal{C}^{2}C2), calculate its partial derivatives and study the (strict) concavity of K\mathcal{K}K. To do this, we need a genericity assumption which is somewhat technical, but which is natural and not restrictive in practice. In the optical cases mentioned in this paper, this assumption is satisfied if the intersection of three distinct Laguerre cells is finite and if the intersection of two Laguerre cells with the boundary of XXX is also finite. For more details, the reader may refer to the book chapter [⁠4⁠ Q. Mérigot and B. Thibert, Optimal transport: Discretization and algorithms. In Handbook of Numerical Analysis, Geometric PDEs 22, Elsevier, 621–679 (2020) ]. Theorem 3.4 (Differental of GGG). Suppose that the cost satisfies (⁠Twist⁠), that YYY is generic (see above), and ρ\rhoρ is continuous. Then the application G:RN→RNG:\mathbb{R}^{N}\to\mathbb{R}^{N}G:RN→RN is of class C1\mathcal{C}^{1}C1 and ∀j≠i, ∂Gi∂ψj(ψ)=∫Lag⁡ij(ψ)ρ(x)∥∇xc(x,yi)−∇xc(x,yj)∥dx,\displaystyle\forall j\neq i,~{}~{}\frac{\partial G_{i}}{\partial\psi_{j}}(\psi)=\int_{\operatorname{Lag}_{ij}(\psi)}\frac{\rho(x)}{\left\\|\nabla_{x}c(x,y_{i})-\nabla_{x}c(x,y_{j})\right\\|}\mathrm{d}x,∀j=i, ∂ψj∂Gi(ψ)=∫Lagij(ψ)∥∇xc(x,yi)−∇xc(x,yj)∥ρ(x)dx, ∀i, ∂Gi∂ψi(ψ)=−∑j≠i∂Gi∂ψj(ψ),\displaystyle\forall i,~{}~{}\frac{\partial G_{i}}{\partial\psi_{i}}(\psi)=-\sum_{j\neq i}\frac{\partial G_{i}}{\partial\psi_{j}}(\psi),∀i, ∂ψi∂Gi(ψ)=−j=i∑∂ψj∂Gi(ψ), where Lag⁡ij(ψ)=Lag⁡i(ψ)∩Lag⁡j(ψ)\operatorname{Lag}_{ij}(\psi)=\operatorname{Lag}_{i}(\psi)\cap\operatorname{Lag}_{j}(\psi)Lagij(ψ)=Lagi(ψ)∩Lagj(ψ). The formula for the partial derivatives of GGG has a geometric interpretation. In the following two figures, which are obtained for the cost c(x,y)=∥x−y∥2c(x,y)=\left\\|x-y\right\\|^{2}c(x,y)=∥x−y∥2 on R2\mathbb{R}^{2}R2, we explain why the formula for partial derivatives involves integrals over the interfaces between Laguerre cells and how the singularities of DGDGDG may occur depending on the geometry of the points yiy_{i}yi. Figure ⁠3.4⁠ illustrates that the partial derivative ∂Gi/∂ψj(ψ)\partial G_{i}/\partial\psi_{j}(\psi)∂Gi/∂ψj(ψ) is an integral over the interface Lag⁡ij(ψ)\operatorname{Lag}_{ij}(\psi)Lagij(ψ): the value Gi(ψ)G_{i}(\psi)Gi(ψ) is an integral over the Laguerre cell Lag⁡i(ψ)\operatorname{Lag}_{i}(\psi)Lagi(ψ) (in grey on the left); we increase the value ψj\psi_{j}ψj by ε>0\varepsilon>0ε>0 considering ψ+εej\psi+\varepsilon e_{j}ψ+εej; the rate of increase (Gi(ψ)−Gi(ψ+εej))/ε(G_{i}(\psi)-G_{i}(\psi+\varepsilon e_{j}))/\varepsilon(Gi(ψ)−Gi(ψ+εej))/ε is proportional to an integral over the symmetric difference between two Laguerre cells (in grey in the middle); passing to the limit we obtain an integral over the green segment Lag⁡ij(ψ)\operatorname{Lag}_{ij}(\psi)Lagij(ψ). The signs that occurs in the formula for the partial derivatives can also be interpreted with the bakeries metaphor: when the price of bread ψi\psi_{i}ψi increases, the number of customers of the bakery yiy_{i}yi decreases (i.e., the Laguerre cell Lag⁡i(ψ)\operatorname{Lag}_{i}(\psi)Lagi(ψ) shrinks) and the number of customers for the other bakeries increases, so that ∂Gi/∂ψi(ψ)≤0\partial G_{i}/\partial\psi_{i}(\psi)\leq 0∂Gi/∂ψi(ψ)≤0 and ∂Gi/∂ψj(ψ)≥0\partial G_{i}/\partial\psi_{j}(\psi)\geq 0∂Gi/∂ψj(ψ)≥0 for j≠ij\neq ij=i. In Figure ⁠3.4⁠, the genericity condition is not satisfied because y1y_{1}y1, y2y_{2}y2 and y3y_{3}y3 are aligned, and there exists ψ∈RN\psi\in\mathbb{R}^{N}ψ∈RN for which Lag⁡1(ψ)∩Lag⁡2(ψ)∩Lag⁡3(ψ)\operatorname{Lag}_{1}(\psi)\cap\operatorname{Lag}_{2}(\psi)\cap\operatorname{Lag}_{3}(\psi)Lag1(ψ)∩Lag2(ψ)∩Lag3(ψ) is a line segment. The partial derivative ∂G2/∂ψ3(ψ)\partial G_{2}/\partial\psi_{3}(\psi)∂G2/∂ψ3(ψ) is an integral on the (green) segment Lag⁡23(ψ)\operatorname{Lag}_{23}(\psi)Lag23(ψ). If we simultaneously decrease ψ1\psi_{1}ψ1 and ψ2\psi_{2}ψ2 by the same amount, we can see that the segment Lag⁡23(ψ)\operatorname{Lag}_{23}(\psi)Lag23(ψ) varies continuously and then suddenly becomes empty when the cell Lag⁡2(ψ)\operatorname{Lag}_{2}(\psi)Lag2(ψ) gets empty (bottom right of Figure ⁠3.4⁠). Thus, ∂G2/∂ψ3(ψ)\partial G_{2}/\partial\psi_{3}(\psi)∂G2/∂ψ3(ψ) is not continuous. Newton's method requires a certain regularity, and we will see below that it converges under the above genericity assumptions. Figure 7. The partial derivatives are boundary integrals 🅭🅯 CC BY 4.0 Figure 8. Non-continuous partial derivative: ∂G2/∂ψ3\partial G_{2}/\partial\psi_{3}∂G2/∂ψ3 is an integral on the green segment Lag⁡23\operatorname{Lag}_{23}Lag23 which is discontinuous. 🅭🅯 CC BY 4.0 To establish the convergence of Newton's method, we also need to study the concavity of the Kantorovitch functional K\mathcal{K}K, or equivalently the monotonicity of its gradient ∇K=G−ν\nabla\mathcal{K}=G-\nu∇K=G−ν. The functions K\mathcal{K}K and GGG are invariant by addition of a constant vector (i.e., K(ψ+C(1,…,1))=K(ψ)\mathcal{K}(\psi+C(1,\ldots,1))=\mathcal{K}(\psi)K(ψ+C(1,…,1))=K(ψ)), which can be seen in the definition of Laguerre cells. Thus, we can only hope to establish strong concavity of K\mathcal{K}K in the directions orthogonal to constant vectors, i.e., belonging to the set M0:={v∈RN∣∑1≤i≤Nvi=0}.\mathcal{M}_{0}:=\Bigl\{v\in\mathbb{R}^{N}\mid\sum_{1\leq i\leq N}v_{i}=0\Bigr\}.M0:={v∈RN∣1≤i≤N∑vi=0}. Another reason for the lack of strong concavity of K\mathcal{K}K is that if ψi\psi_{i}ψi is very large, then Lag⁡i(ψ)\operatorname{Lag}_{i}(\psi)Lagi(ψ) is empty and remains empty in a neighborhood of ψ\psiψ. In this case, Gi(ψ)G_{i}(\psi)Gi(ψ) is constant equal to zero, and the Hessian matrix D2G(ψ)=DGD^{2}G(\psi)=DGD2G(ψ)=DG has a row of zeros. We can therefore hope to establish strong concavity only if ψ\psiψ belongs to the set C+:={ψ∈RN∣∀i, Gi(ψ)>0}.\mathcal{C}_{+}:=\bigl\{\psi\in\mathbb{R}^{N}\mid\forall i,\ G_{i}(\psi)>0\bigr\}.C+:={ψ∈RN∣∀i, Gi(ψ)>0}. The next theorem shows, in a nutshell, that these are the only two obstructions to strong concavity. Theorem 3.5 (Strict concavity). We assume the hypotheses of the previous theorem hold. If the set {ρ>0}\{\rho>0\}{ρ>0} is connected, the function K\mathcal{K}K is locally strongly concave on C+\mathcal{C}_{+}C+ in the direction M0\mathcal{M}_{0}M0: ∀ψ∈C+, ∀v∈M0∖{0},⟨DG(ψ)v ∣ v⟩<0.\forall\psi\in\mathcal{C}_{+},~{}\forall v\in\mathcal{M}_{0}\setminus\{0\},\quad\langle DG(\psi)v\,\|\,v\rangle<0.∀ψ∈C+, ∀v∈M0∖{0},⟨DG(ψ)v∣v⟩<0. Remark 3.3 (Uniqueness). We saw above that the function K\mathcal{K}K has a maximum, and thus equation (⁠MA⁠) has a solution. The previous theorem implies that this maximum is unique if we impose that ψ∈M0\psi\in\mathcal{M}_{0}ψ∈M0, i.e., ψ\psiψ has zero average, since a strictly concave function admits at most one local maximum. We will see in the next paragraph how these results of regularity and monotonicity allow us to iteratively construct a sequence (ψ(k))k≥0(\psi^{(k)})_{k\geq 0}(ψ(k))k≥0 converging to the unique zero-average ψ∗\psi^{}ψ∗ satisfying G(ψ∗)=νG(\psi^{})=\nuG(ψ∗)=ν. 3.5 Newton's method Newton's method in 1D We begin by recalling Newton's method for solving the equation g(x)=0g(x)=0g(x)=0, where g:R→Rg:\mathbb{R}\to\mathbb{R}g:R→R is a real function. Newton's method starts from x0∈Rx_{0}\in\mathbb{R}x0∈R and constructs the sequence xk+1=xk−g(xk)/g′(xk)x^{k+1}=x^{k}-g(x^{k})/g^{\prime}(x^{k})xk+1=xk−g(xk)/g′(xk) by induction. If we assume that ggg is of class C1\mathcal{C}^{1}C1 and that there exists a∈Ra\in\mathbb{R}a∈R such that g(a)=0g(a)=0g(a)=0 and g′(a)≠0g^{\prime}(a)\neq 0g′(a)=0, then one can show, using Taylor–Lagrange formulas, that for x0x^{0}x0 sufficiently close to aaa, the sequence (xk)k≥0(x^{k})_{k\geq 0}(xk)k≥0 converges to aaa. The convergence is then said to be local. Thus, under a regularity hypothesis (g∈C1g\in\mathcal{C}^{1}g∈C1) and monotonicity (g′g^{\prime}g′ has constant sign in a neighborhood of aaa), Newton's method converges locally. Newton's method (local) Assume that we are given a zero-average vector ψ0∈M0\psi^{0}\in\mathcal{M}_{0}ψ0∈M0 such that the mass of all Laguerre cells is strictly positive: ε0:=12min⁡[min⁡y∈YGi(ψ0), min⁡1≤i≤Nνyi]>0.\varepsilon_{0}:=\frac{1}{2}\min\left[\min_{y\in Y}G_{i}(\psi^{0}),~{}\min_{1\leq i\leq N}\nu_{y_{i}}\right]>0.ε0:=21min[y∈YminGi(ψ0), 1≤i≤Nminνyi]>0. We define ψk+1\psi^{k+1}ψk+1 in the following way: we start by calculating the Newton direction dkd^{k}dk, i.e., the vector dkd^{k}dk satisfying DG(ψk)dk=−(G(ψk)−ν)anddik∈M0,DG(\psi^{k})d^{k}=-(G(\psi^{k})-\nu)\quad\text{and}\quad d_{i}^{k}\in\mathcal{M}_{0},DG(ψk)dk=−(G(ψk)−ν)anddik∈M0, which exists and is unique by according to Theorem ⁠3.5⁠. The second equation enables us to overcome the invariance of GGG and thus the non-invertibility of DG(ψk)DG(\psi^{k})DG(ψk). We then define ψk+1=ψk+dk\psi^{k+1}=\psi^{k}+d^{k}ψk+1=ψk+dk. As in the 1D case, it can be shown that the method converges locally: if ψ0\psi^{0}ψ0 is chosen close enough to the ψ∗\psi^{}ψ∗ solution, then the sequence (ψk)(\psi^{k})(ψk) converges to ψ∗\psi^{}ψ∗. Globally convergent Newton's method However, the condition ψ0\psi^{0}ψ0 is close to the solution ψ∗\psi^{}ψ∗ is impossible to fulfill in practice. Fortunately, a very simple modification of the method allows to ensure a global convergence, allowing us to drop this closeness assumption. To do this, one must construct ψk+1\psi^{k+1}ψk+1 in such a way that the kernel of the Jacobian DG(ψk+1)DG(\psi^{k+1})DG(ψk+1) remains equal to constant vectors, so that the system defining the direction dk+1d^{k+1}dk+1 admits a unique solution. For this purpose, we define the stepτk\tau^{k}τk as the largest real of the form 2−ℓ2^{-\ell}2−ℓ (with ℓ∈N\ell\in\mathbb{N}ℓ∈N) such that ψk,ℓ:=ψk+2−ℓdk\psi^{k,\ell}:=\psi^{k}+2^{-\ell}d^{k}ψk,ℓ:=ψk+2−ℓdk satisfies {∀i∈{1,…,N},Gi(ψk,ℓ)≥ε0,∥G(ψk,ℓ)−ν∥≤(1−2−(ℓ+1))∥G(ψk)−ν∥.\begin{cases}\forall i\in\{1,\ldots,N\},\quad G_{i}(\psi^{k,\ell})\geq\varepsilon_{0},\\ \left\\|G(\psi^{k,\ell})-\nu\right\\|\leq(1-2^{-(\ell+1)})\left\\|G(\psi^{k})-\nu\right\\|.\end{cases}{∀i∈{1,…,N},Gi(ψk,ℓ)≥ε0,∥∥G(ψk,ℓ)−ν∥∥≤(1−2−(ℓ+1))∥∥G(ψk)−ν∥∥. Finally, we define ψk+1=ψk+τkdk\psi^{k+1}=\psi^{k}+\tau^{k}d^{k}ψk+1=ψk+τkdk. By using the regularity and concavity results on K\mathcal{K}K, the step τk\tau^{k}τk can be bounded from below, thus ensuring the convergence of the sequence constructed above to a solution of the optimal transport problem [⁠4⁠ Q. Mérigot and B. Thibert, Optimal transport: Discretization and algorithms. In Handbook of Numerical Analysis, Geometric PDEs 22, Elsevier, 621–679 (2020) ]: Under the assumptions of Theorem ⁠3.5⁠, there exists τ∗>0\tau^{}>0τ∗>0 such that ∥G(ψk+1)−ν∥≤(1−τ⋆2)∥G(ψk)−ν∥.\left\\|G(\psi^{k+1})-\nu\right\\|\leq\left(1-\frac{\tau^{\star}}{2}\right)\left\\|G(\psi^{k})-\nu\right\\|.∥∥G(ψk+1)−ν∥∥≤(1−2τ⋆)∥∥G(ψk)−ν∥∥. In particular, the sequence (ψk)k≥0(\psi^{k})_{k\geq 0}(ψk)k≥0 converges to the unique solution ψ∗\psi^{}ψ∗ of (⁠MA⁠) satisfying ∑iψi∗=0\sum_{i}\psi_{i}^{}=0∑iψi∗=0. Remark 3.4 (Quadratic convergence). The above theorem shows that the convergence of Newton's method is globally exponential. This convergence is actually called linear convergence in optimization. When the cost ccc satisfies the Ma–Trudinger–Wang (MTW) condition that appears in the theory of optimal transport regularity, and the density ρ\rhoρ is Lipschitz, then the convergence is even locally quadratic [⁠3⁠ J. Kitagawa, Q. Mérigot and B. Thibert, Convergence of a Newton algorithm for semi-discrete optimal transport. Journal of the European Mathematical Society (2019) ]: for sufficiently large kkk, we have ∥G(ψk+1)−ν∥≤12∥G(ψk)−ν∥2.\left\\|G(\psi^{k+1})-\nu\right\\|\leq\frac{1}{2}\left\\|G(\psi^{k})-\nu\right\\|^{2}.∥∥G(ψk+1)−ν∥∥≤21∥∥G(ψk)−ν∥∥2. In practice, the convergence is very fast and the basin where quadratic convergence occurs seems to be quite large. This last observation is empirical, and not mathematically explained yet. In Figure ⁠9⁠, X=[0,1]2X=[0,1]^{2}X=[0,1]2 is the large white square and YYY is a set of points in the lower left corner and c(x,y)=∥x−y∥2c(x,y)=\left\\|x-y\right\\|^{2}c(x,y)=∥x−y∥2. With N=100N=100N=100 points, after three iterations the error ∥G(ψ3)−ν∥1\\|G(\psi^{3})-\nu\\|_{1}∥G(ψ3)−ν∥1 is already of order 10−910^{-9}10−9. Even difficult examples of size N=107N=10^{7}N=107 in dimension d=3d=3d=3 can be solved to high numerical precision with less than 20 iterations! In this section we present the adaptation of semi-discrete methods to the practical resolution of inverse problems in optics. These results were obtained in the PhD thesis of Jocelyn Meyron and the images are taken from the article [⁠5⁠ J. Meyron, Q. Mérigot and B. Thibert, Light in power: A general and parameter-free algorithm for caustic design. In SIGGRAPH Asia 2018 Technical Papers, ACM Transaction on Graphics, p. 224 (2018) ]. Figure 9. Convergence of the sequence (ψk)(\psi^{k})(ψk). On images 2, 3 and 4 we see the Laguerre cells Lag⁡i(ψk)\operatorname{Lag}_{i}(\psi^{k})Lagi(ψk) for k=0,1,3k{=}0{,}1{,}3k=0,1,3. 4.1 Far-field problems We saw in Section ⁠2⁠ that in several far-field problems, i.e., when the target illumination is at infinity, solving the Monge–Ampère equation (⁠MA⁠) allows us to construct an optical component. This involves modelling mirrors or lenses, with a point or collimated light source, and in each case there are two components that may be produced (one of which is convex), so that in all we have formulated eight different near-field optical problems. The main difficulty in implementing Newton's algorithm to solve (⁠MA⁠) lies in the evaluation of the function GGG and its differential DGDGDG at point ψk\psi^{k}ψk, and more precisely in the calculation of the set of Laguerre cells Lag⁡i(ψk)\operatorname{Lag}_{i}(\psi^{k})Lagi(ψk). For cells from non-imaging optics problems, also called visibility cells, it is possible to perform this calculation in almost linear time in the number NNN of Dirac masses. Take for example the mirror problem for a point source. The visibility cells are obtained by projecting radially onto the sphere an intersection of "solid" confocal paraboloids, and we have already seen that the intersection of two confocal paraboloids is included in a plane. Another simple calculation shows that the radial projection of such an intersection is also included in a (different) plane. This shows that the visibility cells are separated by hyperplanes. In fact, it can be shown that there exists a partition of R3\mathbb{R}^{3}R3 into convex polyhedra P1,…,PNP_{1},\ldots,P_{N}P1,…,PN – called a power diagram in computational geometry – such that each visibility cell is of the form Vi(ψ)=S2∩PiV_{i}(\psi)=\mathbb{S}^{2}\cap P_{i}Vi(ψ)=S2∩Pi (Figure ⁠4.1⁠). A similar property holds for each of the eight problems. The point of this reformulation is that there are powerful libraries – for example Cgal or Geogram – that allow us to compute power diagrams in dimensions 222 and 333, and thus also the Laguerre cells associated with the optics problems. It is therefore possible to implement the damped Newton algorithm, and to use it to construct – numerically and even physically – mirrors and lenses for far-field targets in anidolic optics. Figure 10. Visibility cell structure 🅭🅯 CC BY 4.0 4.2 Near-field problems It is also possible to deal with more realistic target illuminations in the near-field – i.e., when illuminating points at a finite distance rather than directions – with an iterative method that solves a far-field solution at each step [⁠5⁠ J. Meyron, Q. Mérigot and B. Thibert, Light in power: A general and parameter-free algorithm for caustic design. In SIGGRAPH Asia 2018 Technical Papers, ACM Transaction on Graphics, p. 224 (2018) ]. The convergence is very fast, requiring only a few iterations, as illustrated in Figure ⁠4.2⁠. In all the experiments presented below, the light source is assumed to be uniform, so that the light source μ\muμ has a constant density on its support. The reflection or refraction of this light on a wall is simulated in the computer by the physically realistic rendering software LuxRender. Generic method The different problems of anidolic optics having the same structure (point or collimated light sources, mirrors or lenses, convex or concave components, near-field or far-field), it is possible to solve them in a unified, precise and automatic manner with the same Figure 11. Convergence of far-field mirrors to near-field mirrors 🅭🅯 CC BY 4.0 Figure 12. Mirrors for collimated (top) and point (bottom) light; visibility cells (left), component mesh (middle) and rendering with LuxRender (right) 🅭🅯 CC BY 4.0 Figure 13. Lenses for collimated (top) and point (bottom) light; visibility cells (left), component mesh (middle) and rendering with LuxRender (right) 🅭🅯 CC BY 4.0 Figure 14. A point light (not visible) is placed in front of the mirror and the path of the light is simulated by the computer using the physically realistic renderer LuxRender 🅭🅯 CC BY 4.0 Figure 15. Concave and convex lenses 🅭🅯 CC BY 4.0 algorithm (Figures ⁠4.2⁠, ⁠4.2⁠ and ⁠4.2⁠). In Figures ⁠4.2⁠	CommonCrawl
What is the general form of a transfer function I repeatedly see two representations of the general transfer function in the literature. The first is the following which is factorization of the numerator and denominator polynomials: $$\textbf{G}(s) = k \frac{ (s - a_{1})(s - a_{2})(s - a_{3})\ldots }{ (s - b_{1})(s - b_{2})(s - b_{3})(s - b_{4})\ldots }$$ The alternative, below, eludes me and I'll checked many sources but none seem to justify it. The difference is a lowly $s$ term in the denominator raised to a power. Is this general form related the open-loop transfer function? $$\textbf{G}(s) = k \frac{ (s - a_{1})(s - a_{2})(s - a_{3})\ldots }{s^m (s - b_{1})(s - b_{2})(s - b_{3})(s - b_{4})\ldots }$$ My question is what is the difference, if any in these two forms? transfer-function rhodyrhody You may write the second equation on the first form by using $b_i = 0$ corresponding to those poles appearing at $s = 0$. Hence, the only difference is that in the second form you know that there are $m$ poles at $s=0$ while in the first form they may still be there, but one will have to check the values of $b_i$ to determine if they are there. Note that a system with poles at $s=0$ is a marginally stable system and that the poles are typically introduced by integrators, which have a transfer function $1/s$. OscarOscar The first representation of the Transfer Function can have poles at zero,this is not represented explicitly in that equation.The second form of the transfer function representation, takes into account,the poles located at z=0.The presence of a pole at zero causes instability. logamadilogamadi 8911 silver badge1010 bronze badges Not the answer you're looking for? Browse other questions tagged transfer-function or ask your own question. Z-domain transfer function question Digital filters with more zeros than poles Solving simple transfer function Transfer function sinusoidal response Basic transfer function question Transfer function and difference equations: why does $H(z)$ numerator polynomial not correspond to $Y(z)$? Identification of a transfer function What is the zero in this transfer function? Transfer Function definition	CommonCrawl
Previous (Aristotle) Next (Arius) Arithmetic or arithmetics (from the Greek word αριθμός, meaning "number") is the oldest and most fundamental branch of mathematics. It is used by almost everyone, for tasks ranging from simple daily counting to advanced science and business calculations. Some have called it the "science of numbers." Our knowledge of and skill in using arithmetic operations is part of our definition of literacy. In common usage, arithmetic refers to a branch of mathematics that records elementary properties of certain operations on numbers. Professional mathematicians sometimes use the term higher arithmetic[1] as a synonym for number theory, but this should not be confused with elementary arithmetic. The traditional arithmetic operations are addition, subtraction, multiplication, and division, although more advanced operations (such as manipulations of percentages, square root, exponentiation, and logarithmic functions) are also sometimes included in this subject. Any set of objects upon which all four operations of arithmetic can be performed (except division by zero), and wherein these four operations obey the usual laws, is called a field. 2 Decimal arithmetic 3 Addition (+) 3.1 Terminology 3.2 Notation 3.3 Properties 4 Subtraction (−) 5 Multiplication (× or ·) 6 Division (÷ or /) 7.1 Addition table 7.2 Multiplication table 8 Arithmetic in education Addition is the simplest form and combines two numbers, such as 1+1=2. This can be used for simple tasks such as adding grocery amounts or the money in one's pocket. Subtraction is the process of finding the difference between two numbered quantities, such as 5-3=2. This process can be used in tasks such as calculating the balance in a bank account after withdrawing some cash. Multiplication consists of adding a number (the multiplicand) to itself a certain number of times. For example, adding 3 to itself 5 times gives 15, which is written as 3x5=15. Division is the inverse of multiplication. It consists of dividing a number into groups of equal amounts. For example, to divide the number 20 into several groups, each containing 4 units, one would write 20/4 (or 20÷4), which would yield 5. In other words, 20 can be divided into 5 equal groups, with 4 units in each group. Our knowledge of the prehistory of arithmetic is limited by a small number of artifacts indicating a clear conception of addition and subtraction, the best-known being the Ishango Bone[2] from Africa, dating from 18,000 B.C.E. It is clear that the Babylonians had solid knowledge of almost all aspects of elementary arithmetic circa 1850 B.C.E., historians can only infer the methods utilized to generate the arithmetical results. Likewise, a definitive algorithm for multiplication and the use of unit fractions can be found in the Rhind Mathematical Papyrus dating from Ancient Egypt circa 1650 B.C.E. In the Pythagorean school, in the second half of the sixth century B.C.E., arithmetic was considered one of the four quantitative or mathematical sciences (Mathemata). These were carried over in medieval universities as the Quadrivium, which consisted of arithmetic, geometry, music, and astronomy. Together with the Trivium of grammar, rhetoric, and dialectic, they constituted the septem liberales artes (seven liberal arts). All these were thought to be fundamentally interconnected. The book Introduction to Arithmetic was written by Nicomachus of Gerasa (ca. 60? - 120 C.E.) almost 2,000 years ago and contains both philosophical prose and very basic mathematical ideas. Nichomachus, one of the first mathematicians, was schooled in Alexandria. His book covers pythagorean number theory and contains the multiplication table of Greek origin. Compared to Euclid's book, which represents numbers by lines, Nichomachus used arithmetical notation expressed in ordinary language. Nicomachus referred to Plato (429 - 347 B.C.E.) quite often, and wrote about how philosophy can be possible only if one knows enough math. This is his only complete book that has survived to our day. Nicomachus describes how natural numbers and basic mathematical ideas are eternal and unchanging, and in an incorporeal realm. Modern algorithms for arithmetic (for both hand and electronic computations) were made possible by the introduction of Arabic numerals and decimal place notation for numbers. By contrast, the ancient mathematician Archimedes (c. 287 - c. 212 B.C.E.) devoted an entire work, The Sand Reckoner, to devising a notation for a certain large integer. The flourishing of algebra in the medieval Islamic world and in Renaissance Europe was an outgrowth of the enormous simplification of computation through decimal notation. Decimal arithmetic Decimal notation is based on ten parts and constructs all real numbers from the basic digits, and the first ten non-negative integers 0,1,2,…,9. A decimal numeral consists of a sequence of these basic digits, with the "denomination" of each digit depending on its position with respect to the decimal point: for example, 507.36 denotes 5 hundreds (102), plus 0 tens (101), plus 7 units (100), plus 3 tenths (10-1) plus 6 hundredths (10-2). Decimals can also be noted in base ten, example: 0.34 = 34/100 (10-2)or 0.344 = 344/1000 (103). Algorithm comprises all of the rules of performing arithmetic computations using a decimal system for representing numbers in which numbers written using ten symbols having the values 0 through 9 are combined using a place-value system (positional notation), where each symbol has ten times the weight of the one to its right. This notation allows the addition of arbitrary numbers by adding the digits in each place, which is accomplished with a 10 x 10 addition table. (A sum of digits which exceeds 9 must have its 10-digit carried to the next place leftward.) One can make a similar algorithm for multiplying arbitrary numbers because the set of denominations {…,102,10,1,10-1,…} is closed under multiplication. Subtraction and division are achieved by similar, though more complicated algorithms. Addition (+) An example of addition: 3 + 2 = 5, using apples, a popular choice in textbooks.[3] Addition is the basic operation of arithmetic. In its simplest form, addition combines two numbers. The result of adding two quantities a and b is a + b. It is sometimes phrased as "a more than b," or "b more than a." For example, 3 + 2 = 5, since 5 is 2 more than 3. Addition is used to model many related processes, such as: joining two collections of objects, repeated incrementation, moving a point across the number line, representing two successive translations as one. The numbers or the objects to be added are generally called the "terms," the "addends," or the "summands"; this terminology carries over to the summation of multiple terms. The resultant number is called the sum. Therefore, from the above example, the terms are 3,2, and 5. The addends are 3 and 2. The sum is 5. The word terms is to be distinguished from factors, which are multiplied. Some authors call the first addend the augend. In fact, during the Renaissance, many authors did not consider the first addend an "addend" at all. Today, due to the symmetry of addition, "augend" is rarely used, and both terms are generally called addends.[4] Adding more than two numbers can be viewed as repeated addition; this procedure is known as summation and includes ways to add infinitely many numbers in an infinite series; repeated addition of the number one is the most basic form of counting. Addition is written using the plus sign "+" between the terms; that is, in infix notation. The result is expressed with an equals sign. For example, 5 + 4 + 2 = 11 (see "associativity" below) 3 + 3 + 3 + 3 = 12 (see "multiplication" below) There are also situations where addition is "understood" even though no symbol appears: A column of numbers, with the last number in the column underlined, usually indicates that the numbers in the column are to be added, with the sum written below the underlined number. A whole number followed immediately by a fraction indicates the sum of the two, called a mixed number.[5] For example, 31⁄2 = 3 + 1⁄2 = 3.5. This notation can cause confusion, since in most other contexts, juxtaposition denotes multiplication instead. Addition is said to have "commutative" and "associative" properties. The term commutative come from "commute" or "move around," and in addition it means that terms can be interchanged. For instance, "a+b = b+a." The order in which the terms are added does not matter. The associative property means to "associate" or "group," and in addition it means that terms can be added in different groups. For example, "(a+b) + c = a + (b+c)." The "identity element" of addition (or the additive identity) is 0—that is, adding zero to any number will yield that same number. Also, the "inverse element" of addition (the additive inverse) is the opposite of any number—that is, adding the opposite of any number to the number itself will yield the additive identity, 0. For example, the opposite of 7 is (-7), so 7 + (-7) = 0. Subtraction (−) An example of subtraction: "5 - 2 = 3". A subtraction problem. Subtraction is essentially the opposite of addition. It is denoted by a minus sign "−" in infix notation. Subtraction is removing objects from a group. For example, 5 - 3 = 2, which means that three objects taken away from a total of five leaves two. Subtraction is used to model several closely related processes: From a given collection, take away (subtract) a given number of objects. Combine a given measurement with an opposite measurement, such as a movement right followed by a movement left, or a deposit and a withdrawal. Compare two objects to find their difference. For example, the difference between $800 and $600 is $800 − $600 = $200. The traditional names for the parts of the formula c − b = a are minuend (c) − subtrahend (b) = difference (a). The words "minuend" and "subtrahend" are virtually absent from modern usage; Linderholm charges "This terminology is of no use whatsoever."[6] However, "difference" is very common. If the minuend is larger than the subtrahend, the difference will be positive; if the minuend is smaller than the subtrahend, the difference will be negative; and if they are equal, the difference will be zero. For example: 5 - 3 = 2; 3 - 5 = -2; 3 - 3 = 0. Imagine a line segment of length b with the left end labeled a and the right end labeled c. Starting from a, it takes b steps to the right to reach c. This movement to the right is modeled mathematically by addition: a + b = c. From c, it takes b steps to the left to get back to a. This movement to the left is modeled by subtraction: c − b = a. Now, imagine a line segment labeled with the numbers 1, 2, and 3. From position 3, it takes no steps to the left to stay at 3, so 3 − 0 = 3. It takes 2 steps to the left to get to position 1, so 3 − 2 = 1. This picture is inadequate to describe what would happen after going 3 steps to the left of position 3. To represent such an operation, the line must be extended. To subtract arbitrary natural numbers, one begins with a line containing every natural number (0, 1, 2, 3, 4, ...). From 3, it takes 3 steps to the left to get to 0, so 3 − 3 = 0. But 3 − 4 is still invalid since it again leaves the line. The natural numbers are not a useful context for subtraction. The solution is to consider the integer number line (…, −3, −2, −1, 0, 1, 2, 3, …). From 3, it takes 4 steps to the left to get to −1, so 3 − 4 = −1. Subtraction is neither commutative nor associative. For this reason, it is often helpful to look at subtraction as addition of the minuend and the opposite of the subtrahend, that is, a − b = a + (−b). When written as a sum, all the properties of addition hold. In mathematics, it is often useful to view or even define subtraction as a kind of addition, the addition of the opposite. We can view 7 − 3 = 4 as the sum of two terms: seven and negative three. This perspective allows us to apply to subtraction all of the familiar rules and nomenclature of addition. Although subtraction is not associative or commutative, the addition of signed numbers is both. Multiplication (× or ·) Multiplication is in essence repeated addition, or the sum of a list of identical numbers. a × n = a + ⋯ + a ⏟ n {\displaystyle {{a\times n=} \atop {\ }}{{\underbrace {a+\cdots +a} } \atop n}} For example, 7 × 4 is the same as 7 + 7 + 7 + 7. Fractions are multiplied by separately multiplying their denominators and numerators: a/b × c/d = (ac)/(bd). For example, 2/3 × 3/4 = (2×3)/(3×4) = 6/12 = 1/2. Multiplication is used to determine the total of the amounts in many groups of the same size. For example, if there are 6 apples in 1 bag, and you buy 4 bags, then 6+6+6+6 = 24 (repeated addition), or 6 x 4 = 24 (multiplication), giving a total of 24 apples. Multiplication is used to increase a number by a fixed amount stepwise, or to compute a product. Simple numbers are incorporated in a multiplication table ("times table") as shown below. The two numbers being multiplied are formally called the multiplicand and the multiplier, where the multiplicand is usually written first. (Some write the multiplier first, and say that 7 × 4 stands for 4 + 4 + 4 + 4 + 4 + 4 + 4, but this usage is less common.) The difference was important in Roman numerals and similar systems, where multiplication is transformation of symbols and their addition.[7] Because of the commutative property of multiplication, there is generally no need to distinguish between the two numbers so they are more commonly referred to as factors. The result of the multiplication is called the product. Multiplication can be denoted in several equivalent ways. For example, the expression "5 multiplied by 2" can be written in any of the following ways: 5·2 (5)2, 5(2), (5)(2), 5[2], [5]2, [5][2] The asterisk () is often used on computers because it is a symbol on every keyboard, but it is rarely used when writing math by hand. This usage originated in the FORTRAN programming language. Frequently, multiplication is implied by juxtaposition rather than shown in a notation. This is standard in algebra, taking forms such as 5x or xy. This notation is not used with numbers alone: 52 never means 5 × 2. Also, this notation is potentially confusing if variables are permitted to have names longer than one letter, as in computer programming languages. If the terms are not written out individually, then the product may be written with an ellipsis to mark out the missing terms, as with other series operations (like sums). Thus, the product of all the natural numbers from 1 to 100 can be written as: 1 ⋅ 2 ⋅ … ⋅ 99 ⋅ 100 {\displaystyle 1\cdot 2\cdot \ldots \cdot 99\cdot 100} or 1 ⋅ 2 ⋅ ⋯ ⋅ 99 ⋅ 100 {\displaystyle 1\cdot 2\cdot \cdots \cdot 99\cdot 100} . Multiplication is really repeated addition, is commutative and associative; further it is distributive over addition and subtraction. The multiplicative identity is 1, that is, multiplying any number by 1 will yield that same number. Also, the multiplicative inverse is the reciprocal of any number, that is, multiplying the reciprocal of any number by the number itself will yield the multiplicative identity, 1. In summary the four properties are : a0 = 0a = 0 the zero property a1 = 1a = a the identity property ab = b*a the commutative property a(b+c) = ab+ac the distributive property. For integers, fractions, real and complex numbers, multiplication has certain properties: the order in which two numbers are multiplied does not matter. This is called the commutative property, x · y = y · x. The associative property means that for any three numbers x, y, and z, (x · y)z = x(y · z). Note: the parentheses mean that the operations inside the parentheses must be done before anything outside the parentheses is done. Multiplication also has what is called a distributive property with respect to the addition, x(y + z) = xy + xz. Also of interest is that any number times 1 is equal to itself, thus, 1 · x = x. and this is called the identity property. In this regard the number 1 is known as the multiplicative identity. The sum of zero numbers is zero. This fact is directly received by means of the distributive property: m · 0 = (m · 0) + m − m = (m · 0) + (m · 1) − m = m · (0 + 1) − m = (m · 1) − m = m − m = 0. m · 0 = 0 no matter what m is (as long as it is finite). Multiplication with negative numbers also requires a little thought. First consider negative one (-1). For any positive integer m: (−1)m = (−1) + (−1) +...+ (−1) = −m This is an interesting fact that shows that any negative number is just negative one multiplied by a positive number. So multiplication with any integers can be represented by multiplication of whole numbers and (−1)'s. All that remains is to explicitly define (−1)(−1): (−1)(−1) = −(−1) = 1 Every number x, except zero, has a multiplicative inverse, 1/x, such that x × 1/x = 1. Multiplication by a positive number preserves order: if a > 0, then if b > c then ab > ac. Multiplication by a negative number reverses order: if a < 0, then if b > c then ab < ac. Division (÷ or /) Division is essentially the inverse of multiplication. Specifically, if c times b equals a, written: c × b = a {\displaystyle c\times b=a} where b is not zero, then a divided by b equals c, written: a b = c {\displaystyle {\frac {a}{b}}=c} For instance, 6 3 = 2 {\displaystyle {\frac {6}{3}}=2} 2 × 3 = 6 {\displaystyle 2\times 3=6\,} . Division is the act or process of dividing. The arithmetic process is opposite of multiplication. Division is used to find out how many times a number will go into another number. For example, two goes into nine, four and a half times. This can also be written down as 9 ÷ 2 = 4.5, or 9 / 2 = 4.5 or spoken verbally as "nine over two is four and a half." The numbers in the operation have special names: Dividend ÷ divisor = quotient. In the above expression, a is called the dividend, b the divisor and c the quotient. Division by zero (i.e., where the divisor is zero) is usually not defined. Division finds the quotient of two numbers, the dividend divided by the divisor. Any dividend divided by zero is undefined. For positive numbers, if the dividend is larger than the divisor, the quotient will be greater than one, otherwise it will be less than one (a similar rule applies for negative numbers and negative one). The quotient multiplied by the divisor always yields the dividend. Division is most often shown by placing the dividend over the divisor with a horizontal line, also called a vinculum, between them. For example, a divided by b is written a b . {\displaystyle {\frac {a}{b}}.} This can be read out loud as "a divided by b" or "a over b." A way to express division all on one line is to write the dividend, then a slash, then the divisor, like this: a / b . {\displaystyle a/b.\,} This is the usual way to specify division in most computer programming languages since it can easily be typed as a simple sequence of characters. A typographical variation which is halfway between these two forms uses a slash but elevates the dividend, and lowers the divisor: a⁄b. Any of these forms can be used to display a fraction. A fraction is a division expression where both dividend and divisor are integers (although typically called the numerator and denominator), and there is no implication that the division needs to be evaluated further. A less common way to show division is to use the obelus (division sign) in this manner: a ÷ b . {\displaystyle a\div b.} This form is infrequent except in elementary arithmetic. The obelus is also used alone to represent the division operation itself, as for instance as a label on a key of a calculator. In some non-English-speaking cultures, "a divided by b" is written a : b. However, in English usage the colon is restricted to expressing the related concept of ratios (then "a is to b"). Division is neither commutative nor associative. As it is helpful to look at subtraction as addition, it is helpful to look at division as multiplication of the dividend times the reciprocal of the divisor, that is a ÷ b = a × 1⁄b. When written as a product, it will obey all the properties of multiplication. Division also has its own simple rules: (2) All even numbers are divisible by 2. (3) Add up all the digits of a number. If the sum is divisible by 3, then so is the number. For example, consider the number 1275. In this case, 1+2+7+5=15, and 15/3=5; therefore, 1275 is divisible by 3. (4) In a number, if the group of last two digits is divisible by 4, then so is the number. For example, consider the number 1316. In this case, 16/4 = 4; therefore, 1316 is divisible by 4. (5) All numbers ending in 5 or 0 are divisible by 5. (6) If the number is divisible by 2 and 3, then it is divisible by 6. (8) In a number, if the group of last 3 digits is divisible by 8, then so is the number. For example, consider the number 57144. In this case, 144/8 = 18; therefore, 57144 is divisible by 8. (9) Add up all the digits of a number. If the sum is divisible by 9, then so is the number. (10) If the last digit of a number is 0, then the number is divisble by 10. Addition table 4 5 6 7 8 9 10 11 12 13 9 10 11 12 13 14 15 16 17 18 Multiplication table Arithmetic in education Primary education in mathematics often places a strong focus on algorithms for the arithmetic of natural numbers, integers, rational numbers (vulgar fractions), and real numbers (using the decimal place-value system). This study is sometimes known as algorism. The difficulty and unmotivated appearance of these algorithms has long led educators to question this curriculum, advocating the early teaching of more central and intuitive mathematical ideas. One notable movement in this direction was the New Math of the 1960s and 1970s, which attempted to teach arithmetic in the spirit of axiomatic development from set theory, an echo of the prevailing trend in higher mathematics [8]. Since the introduction of the electronic calculator, which can perform the algorithms far more efficiently than humans, an influential school of educators has argued that mechanical mastery of the standard arithmetic algorithms is no longer necessary. In their view, the first years of school mathematics could be more profitably spent on understanding higher-level ideas about what numbers are used for and relationships among number, quantity, measurement, and so on. However, most research mathematicians still consider mastery of the manual algorithms to be a necessary foundation for the study of algebra and computer science. This controversy was central to the "Math Wars" over California's primary school curriculum in the 1990s, and continues today [9]. Elementary arithmetic Finite field arithmetic ↑ Harold Davenport, (1999). The Higher Arithmetic: An Introduction to the Theory of Numbers, 7th ed. (Cambridge, England: Cambridge University Press. ISBN 0521634466). ↑ "Ishango Bone" Mathematics of the African Diaspora.[1]. Mathematics dept. University of Buffalo. Retrieved June 22, 2008. ↑ From Herbert Enderton. Elements of set theory. (Academic Press, 1977), 138: "…select two sets K and L with card K = 2 and card L = 3. Sets of fingers are handy; sets of apples are preferred by textbooks." ↑ Steven Schwartzman. The words of mathematics: An etymological dictionary of mathematical terms used in English. (MAA, 1994), 19 ↑ D. Devine, J. Olson, and M. Olson. Elementary mathematics for teachers. (Wiley, 1991), 263 ↑ Carl Linderholm. Mathematics made difficult. (Wolfe, 1971), 42 ↑ For example, to multiply VII by XV one changes the VII to LXX (multiplying VII by X) plus XXV (V times V) plus X (II times V), but to multiply XV by VII one changes XV into LXXV (XV times V) plus XV plus XV (each XV times I). ↑ Glossary of Terms. [2].mathematicallycorrect.com. Retrieved June 22, 2008. ↑ "Math Wars" [3].education-world. Retrieved June 22, 2008. Bunt, Jones, and Bedient. The historical roots of elementary mathematics. Prentice-Hall, 1976. ISBN 0133890155 Cunnington, Susan. The story of arithmetic, a short history of its origin and development. London: Swan Sonnenschein, 1904. Dickson, Leonard Eugene. History of the theory of numbers. Three volumes. Reprints: Carnegie Institute of Washington, Washington, 1932. Chelsea, New York, 1952, 1966. Ferreirós, José. Labyrinth of thought: A history of set theory and its role in modern mathematics. Birkhäuser, 1999. ISBN 0817657495 Fine, Henry Burchard (1858-1928). The number system of algebra treated theoretically and historically. Boston: Leach, Shewell & Sanborn, 1891. Kaplan, Robert. The nothing that is: A natural history of zero. Oxford UP, 2000. ISBN 0195128427 Karpinski, Louis Charles. The history of arithmetic. (original 1925) Reprint: New York: Russell & Russell, 1965. Ore, Øystein. Number theory and its history. New York: McGraw-Hill, 1948. Schwartzman, Steven. The words of mathematics: An etymological dictionary of mathematical terms used in English. MAA, 1994. ISBN 0883855119 Williams, Michael. A history of computing technology. Prentice-Hall, 1985. ISBN 0133899179 Weil, Andre. Number theory: an approach through history. Boston: Birkhauser, 1984. Reviewed: Math. Rev. 85c:01004. Elementary mathematics Davison, Landau, McCracken, and Thompson. Mathematics: Explorations & Applications. Prentice Hall, 1999. ISBN 0134358171 Sparks, F. and C. Rees. A survey of basic mathematics. McGraw-Hill, 1979. ISBN 0070599025 Begle, Edward. The mathematics of the elementary school. McGraw-Hill, 1975. ISBN 0070043256 California State Board of Education mathematics content standards Adopted December 1997, accessed December 2005. Devine, D., J. Olson, and M. Olson. Elementary mathematics for teachers. Wiley, 1991. ISBN 0471859478 National Research Council. Adding it up: Helping children learn mathematics. National Academy Press, 2001. ISBN 0309069955 United States National Research Council. [4]. Retrieved June 22, 2008. Van de Walle , John. Elementary and middle school mathematics: Teaching developmentally. Pearson, 2004. ISBN 020538689X Baroody and Tiilikainen. "Two perspectives on addition development." The development of arithmetic concepts and skills. 2003 ISBN 080583155X Fosnot and Dolk. Young mathematicians at work: Constructing number sense, addition, and subtraction. Heinemann, 2001 ISBN 032500353X Weaver, J. Fred. "Interpretations of number operations and symbolic representations of addition and subtraction." Addition and subtraction: A cognitive perspective. 1982 ISBN 0898591716 Wynn, Karen. "Numerical competence in infants." The development of mathematical skills. \| 1998 ISBN 086377816X Mathematical exposition Bogomolny, Alexander. 1996. "Addition" Interactive Mathematics Miscellany and Puzzles [5] cut-the-knot.org. accessdate February 3, 2006 Dunham, William. The mathematical universe. Wiley, 1994. ISBN 0471536563 Johnson, Paul. From sticks and stones: Personal adventures in mathematics. Science Research Associates, 1975 ISBN 0574191151 Linderholm , Carl. Mathematics made difficult. Wolfe, 1971 ISBN 0723404151 Smith, Frank. The glass wall: Why mathematics can seem difficult. Teachers College Press, 2002 ISBN 0807742422 Smith, Karl. The nature of modern mathematics. Wadsworth, 1980 ISBN 0818503521 Bergman, George. An invitation to general algebra and universal constructions. General Printing, 2005 ISBN 0965521141 [6] Burrill, Claude. Foundations of real numbers. McGraw-Hill, 1967 Davenport, Harold. The Higher Arithmetic: An Introduction to the Theory of Numbers, 7th ed. Cambridge, England: Cambridge University Press, 1999. ISBN 0521634466 Dummit, D. and R. Foote. Abstract algebra. =Wiley, 1999 ISBN 0471368571 Enderton, Herbert. Elements of set theory. Academic Press, 1977 ISBN 0122384407 Lee, John. Introduction to smooth manifolds. Springer, 2003 ISBN 0387954481 Martin, John. Introduction to languages and the theory of computation. McGraw-Hill 2003 ISBN 0072322004 Rudin, Walter. Principles of mathematical analysis. McGraw-Hill, 1976 ISBN 007054235X Stewart, James. Calculus: Early transcendentals. Brooks/Cole, 1999. ISBN 0534362982 Mathematical research Akian, Bapat, and Gaubert. "Min-plus methods in eigenvalue perturbation theory and generalised Lidskii-Vishik-Ljusternik theorem." INRIA reports 2005 online Baez, J. and J. Dolan. "From Finite Sets to Feynman Diagrams." Mathematics Unlimited— 2001 and Beyond. 2001 ISBN 3540669132 online Litvinov, Maslov, and Sobolevskii (1999). Idempotent mathematics and interval analysis. Reliable Computing, Kluwer. Loday, Jean-Louis. "Arithmetree." J. of Algebra 2002 online Mikhalkin, Grigory. "Tropical Geometry and its applications." Madrid ICM 2006 online Viro, Oleg (2000). Dequantization of real algebraic geometry on logarithmic paper. (HTML) Plenary talk at 3rd ECM, Barcelona. Flynn, M. and S. Oberman. Advanced computer arithmetic design. Wiley. 2001 ISBN 0471412090 Horowitz, P. and W. Hill. The art of electronics. Cambridge UP, 2001 ISBN 0521370957 Jackson, Albert. Analog computation. McGraw-Hill, 1960 Truitt, T. and A. Rogers. Basics of analog computers. John F. Rider, 1960 All links retrieved November 5, 2021. Applet for kids to practice arithmetic What is arithmetic? MathWorld article about arithmetic Arithmetic history Addition history Subtraction history Multiplication history Division_(mathematics) history History of "Arithmetic" Retrieved from https://www.newworldencyclopedia.org/p/index.php?title=Arithmetic&oldid=1059836	CommonCrawl
Topological entropy for set-valued maps DCDS-B Home Formulas for the topological entropy of multimodal maps based on min-max symbols December 2015, 20(10): 3435-3459. doi: 10.3934/dcdsb.2015.20.3435 Realizing subexponential entropy growth rates by cutting and stacking Frank Blume 1, Department of Mathematics, John Brown University, 2000 W. University St, Siloam Springs, AR 72761, United States Received October 2014 Revised March 2015 Published September 2015 We show that for any concave positive function $f$ defined on $[0,\infty)$ with $\lim_{x\rightarrow\infty}f(x)/x=0$ there exists a rank one system $(X_f,T_f)$ such that $\limsup_{n\rightarrow\infty} H(\alpha_0^{n-1})/f(n)\ge 1$ for all nontrivial partitions $\alpha$ of $X_f$ into two sets and that there is one partition $\alpha$ of $X_f$ into two sets for which the limit superior of $H(\alpha_0^{n-1})/f(n)$ is equal to one whenever the condition $\lim_{x\rightarrow\infty}\ln x/f(x)=0$ is satisfied. Furthermore, for each system $(X_f,T_f)$ we also identify the minimal entropy growth rate in the limit inferior. Keywords: Ergodic theory, entropy growth rate, cutting and stacking, zero-entropy system., rank-one system. Mathematics Subject Classification: Primary: 28D20; Secondary: 27A9. Citation: Frank Blume. Realizing subexponential entropy growth rates by cutting and stacking. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3435-3459. doi: 10.3934/dcdsb.2015.20.3435 F. Blume, An entropy estimate for infinite interval exchange transformations,, Mathematische Zeitschrift, 272 (2012), 17. doi: 10.1007/s00209-011-0919-2. Google Scholar F. Blume, Minimal rates of entropy convergence for completely ergodic systems,, Israel Journal of Mathematics, 108 (1998), 1. doi: 10.1007/BF02783038. Google Scholar F. Blume, Minimal rates of entropy convergence for rank one systems,, Discrete and Continuous Dynamical Systems, 6 (2000), 773. doi: 10.3934/dcds.2000.6.773. Google Scholar F. Blume, On the relation between entropy and the average complexity of trajectories in dynamical systems,, Computational Complexity, 9 (2000), 146. doi: 10.1007/PL00001604. Google Scholar F. Blume, On the relation between entropy convergence rates and Baire category,, Mathematische Zeitschrift, 271 (2012), 723. doi: 10.1007/s00209-011-0887-6. Google Scholar F. Blume, Possible rates of entropy convergence,, Ergodic Theory and Dynamical Systems, 17 (1997), 45. doi: 10.1017/S0143385797069733. Google Scholar F. Blume, The Rate of Entropy Convergence,, Doctoral Dissertation, (1995). Google Scholar A. Katok and J.-P. Thouvenot, Slow entropy type invariants and smooth realization of commuting measure-preserving transformations,, Annales de l'Institut Henri Poincare (B) Probability and Statistics, 33 (1997), 323. doi: 10.1016/S0246-0203(97)80094-5. Google Scholar W. Parry, Entropy and Generators in Ergodic Theory,, Benjamin, (1969). Google Scholar K. E. Petersen, Ergodic Theory,, Cambridge University Press, (1983). doi: 10.1017/CBO9780511608728. Google Scholar Paulina Grzegorek, Michal Kupsa. Exponential return times in a zero-entropy process. Communications on Pure & Applied Analysis, 2012, 11 (3) : 1339-1361. doi: 10.3934/cpaa.2012.11.1339 Manfred Einsiedler, Elon Lindenstrauss. On measures invariant under diagonalizable actions: the Rank-One case and the general Low-Entropy method. Journal of Modern Dynamics, 2008, 2 (1) : 83-128. doi: 10.3934/jmd.2008.2.83 Frank Blume. Minimal rates of entropy convergence for rank one systems. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 773-796. doi: 10.3934/dcds.2000.6.773 Wenxiang Sun, Cheng Zhang. Zero entropy versus infinite entropy. Discrete & Continuous Dynamical Systems - A, 2011, 30 (4) : 1237-1242. doi: 10.3934/dcds.2011.30.1237 Yixiao Qiao, Xiaoyao Zhou. Zero sequence entropy and entropy dimension. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 435-448. doi: 10.3934/dcds.2017018 John Kieffer and En-hui Yang. Ergodic behavior of graph entropy. Electronic Research Announcements, 1997, 3: 11-16. Xianchao Xiu, Lingchen Kong. Rank-one and sparse matrix decomposition for dynamic MRI. Numerical Algebra, Control & Optimization, 2015, 5 (2) : 127-134. doi: 10.3934/naco.2015.5.127 Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935 Karsten Keller, Sergiy Maksymenko, Inga Stolz. Entropy determination based on the ordinal structure of a dynamical system. Discrete & Continuous Dynamical Systems - B, 2015, 20 (10) : 3507-3524. doi: 10.3934/dcdsb.2015.20.3507 Denis Mercier, Virginie Régnier. Decay rate of the Timoshenko system with one boundary damping. Evolution Equations & Control Theory, 2019, 8 (2) : 423-445. doi: 10.3934/eect.2019021 Masayuki Asaoka. Local rigidity of homogeneous actions of parabolic subgroups of rank-one Lie groups. Journal of Modern Dynamics, 2015, 9: 191-201. doi: 10.3934/jmd.2015.9.191 Gabriele Link, Jean-Claude Picaud. Ergodic geometry for non-elementary rank one manifolds. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6257-6284. doi: 10.3934/dcds.2016072 Jessy Mallet, Stéphane Brull, Bruno Dubroca. General moment system for plasma physics based on minimum entropy principle. Kinetic & Related Models, 2015, 8 (3) : 533-558. doi: 10.3934/krm.2015.8.533 Radosław Kurek, Paweł Lubowiecki, Henryk Żołądek. The Hess-Appelrot system. Ⅲ. Splitting of separatrices and chaos. Discrete & Continuous Dynamical Systems - A, 2018, 38 (4) : 1955-1981. doi: 10.3934/dcds.2018079 Manfred Einsiedler, Elon Lindenstrauss. Symmetry of entropy in higher rank diagonalizable actions and measure classification. Journal of Modern Dynamics, 2018, 13: 163-185. doi: 10.3934/jmd.2018016 Katayun Barmak, Eva Eggeling, Maria Emelianenko, Yekaterina Epshteyn, David Kinderlehrer, Richard Sharp, Shlomo Ta'asan. An entropy based theory of the grain boundary character distribution. Discrete & Continuous Dynamical Systems - A, 2011, 30 (2) : 427-454. doi: 10.3934/dcds.2011.30.427 Eva Glasmachers, Gerhard Knieper, Carlos Ogouyandjou, Jan Philipp Schröder. Topological entropy of minimal geodesics and volume growth on surfaces. Journal of Modern Dynamics, 2014, 8 (1) : 75-91. doi: 10.3934/jmd.2014.8.75 César J. Niche. Topological entropy of a magnetic flow and the growth of the number of trajectories. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 577-580. doi: 10.3934/dcds.2004.11.577 Harald Garcke, Kei Fong Lam. Analysis of a Cahn--Hilliard system with non-zero Dirichlet conditions modeling tumor growth with chemotaxis. Discrete & Continuous Dynamical Systems - A, 2017, 37 (8) : 4277-4308. doi: 10.3934/dcds.2017183 Sébastien Court. Stabilization of a fluid-solid system, by the deformation of the self-propelled solid. Part II: The nonlinear system.. Evolution Equations & Control Theory, 2014, 3 (1) : 83-118. doi: 10.3934/eect.2014.3.83 Frank Blume	CommonCrawl
Chronic treatment with a smart antioxidative nanoparticle for inhibition of amyloid plaque propagation in Tg2576 mouse model of Alzheimer's disease Phetcharat Boonruamkaew1 nAff5, Pennapa Chonpathompikunlert1 nAff6, Long Binh Vong1 nAff7, Sho Sakaue1, Yasushi Tomidokoro2, Kazuhiro Ishii2, Akira Tamaoka2,3 & Yukio Nagasaki1,3,4 Scientific Reports volume 7, Article number: 3785 (2017) Cite this article The present study aimed to assess whether our newly developed redox nanoparticle (RNPN) that has antioxidant potential decreases Aβ levels or prevents Aβ aggregation associated with oxidative stress. The transgenic Tg2576 Alzheimer's disease (AD) mice were used to investigate the effect of chronic ad libitum drinking of RNPN solution for 6 months, including memory and learning functions, antioxidant activity, and amyloid plaque aggregation. The results showed that RNPN-treated mice had significantly attenuated cognitive deficits of both spatial and non-spatial memories, reduced oxidative stress of lipid peroxide, and DNA oxidation. RNPN treatment increased the percent inhibition of superoxide anion and glutathione peroxidase activity, neuronal densities in the cortex and hippocampus, decreased Aβ(1-40), Aβ(1-42) and gamma (γ)-secretase levels, and reduced Aβ plaque observed using immunohistochemistry analysis and thioflavin S staining. Our results suggest that RNPN may be a promising candidate for AD therapy because of its antioxidant properties and reduction in Aβ aggregation, thereby suppressing its adverse side effect. The number of patients affected with Alzheimer's disease (AD) is increasing1. AD is a slow progressing, neurodegenerative disorder, which exhibits the cardinal hallmarks of amyloid beta (Aβ) plaques accumulation and neurofibrillary tangles. In addition, the loss of synapses, atrophy of the cerebral cortex, neuroinflammation, and increase in free radicals in patients with AD is also reported2. In the progression of AD, the conformation and misfolding of Aβ proteins causes them to clump together into small oligomers, protofibrils, and mature fibrils3. The soluble oligomers are structurally heterogeneous, which have been determined as the main toxic forms in AD4. The oligomer-induced toxicity of the cell is related to its ability to permeabilize cellular membranes and lipid bilayers5 to form distinct pores or single channels in the membranes6 and disturb intracellular calcium homeostasis, leading to oxidative stress including overproduction of reactive oxygen species (ROS), altered signaling pathways, mitochondrial dysfunction, and neuronal death7. Currently, one of the most promising strategies for AD therapy is to decrease Aβ aggregation, especially Aβ(1-42) and/or destroy the oligomers and fibrils formed in the brain8, 9. Previous studies have proposed an effective inhibitor of Aβ, Quercetin, which blocks the formation of fibrils, including the inhibition of Aβ aggregation10, reduction in ROS production11, and alteration in the functions of immune and inflammatory cells12. Therefore, there are several targets such as inhibition of Aβ aggregation, and Aβ-induced oxidative stress, especially ROS, to improve therapeutic and preventive strategies for the development of disease-modifying drugs for AD. Antioxidants are widely used to scavenge ROS13, 14. Although versatile antioxidants such as vitamin C and E, many phytochemicals, and synthetic drugs have been developed so far, none of them work effectively. One of the serious issues in the use of antioxidants is the dysfunction of normal redox reaction in healthy cells, including the electron transport chain due to the internalization of these low-molecular-weight (LMW) antioxidants because of their size. This undesired adverse side effect limits the effective dosage of these LMW antioxidants. Recently, we designed polymer antioxidants, where the antioxidant moieties are covalently bound to the amphiphilic block copolymer backbone. Due to the high molecular weight, the internalization of these polymer antioxidants is decreased in healthy cells and their mitochondria, which significantly reduce its adverse effects15, 16. We designed the polymer poly(ethylene glycol)-b-poly[4-(2,2,6,6-tetramethylpiperidine-1-oxyl)aminomethylstyrene] (PEG-b-PMNT). The PEG segment is water-soluble part, while PMNT is the water-insoluble part. Thus, it forms a polymer micelle of several tens of nanometer size in transparent aqueous media (we abbreviate it as RNPN, as shown in Supplementary Figure 1). The 2,2,6,6-tetramethylpiperidine-1-oxyl (so called TEMPO) groups as a side chain of the PMNT segment is known to be a stable radical and does not react with each other. However, it catalytically reacts with ROS and is regarded as one of the strongest antioxidants17. Because the PMNT segment possesses repeating amino groups, it protonates in response to pH decrease and becomes hydrophilic to result in disintegration under acidic conditions15, 18, 19. After oral administration of RNPN solution, it disintegrates in the stomach, and the disintegrated antioxidant polymer is absorbed in the bloodstream from the small intestine because the total molecular weight of the polymer was limited to around 7–20 kDa19. We have previously confirmed that our redox nanoparticle exhibits an anti-apoptotic effect on Aβ-induced cell death in vitro 20. We have also confirmed that oral administration of RNPN showed an antioxidant effect on the brain of senescence-accelerated (SAMP8) mice and did not exhibit any detectable toxicity in the vital organs after continuous administration for 1 month19. The objective of this work was to confirm the effectiveness of RNPN in transgenic AD mice model (Tg2576) for an extended time by ad libitum drinking of RNPN for the prevention of Aβ accumulation in Tg2576 mice overexpressing a mutant form of amyloid precursor protein (APP). Delivery of redox polymer to the blood and brain by chronic ad libitum drinking of RNPN solution To confirm the uptake of the redox polymer in the blood by oral administration, electron spin resonance (ESR) measurements of the blood and brain were carried out. The ESR signal of LMW TEMPO derivatives showed a triplet signal due to the coupling between the unpaired electron on oxygen and 14N nuclei (Fig. 1A). After ad libitum drinking of LMW TEMPO, the triplet signal was clearly observed in both the blood and brain samples as observed in the standard solution (Fig. 1B,C). We have previously reported that the ESR pattern of an aqueous solution of RNPN does not appear as a typical triplet signal but as broadened spectra as shown in Fig. 1D. When RNPN was mixed with untreated brain, and the homogenate was analyzed by ESR measurement, the broadened spectrum was observed, similar to RNPN in saline solution (Fig. 1E). After ad libitum drinking of RNPN solution, small but definite signal was observed in the homogenized brain samples, although the signal pattern changed from broad ESR spectra (Fig. 1E) to triplet signal in both the blood and brain samples (Fig. 1F and G), indicating the disintegration of RNPN in the stomach to result in the uptake of redox polymer in the bloodstream and brain tissues. These results were consistent with the quantitative analysis of ESR spectra intensities as displayed in Supplementary Table S1. The electron spin resonance (ESR) spectra of 4-amino-TEMPO standard in saline (A), 4-amino-TEMPO in the blood after ad libitum drinking for 6 months (B) 4-amino-TEMPO in the brain after ad libitum drinking for 6 months (C) ESR spectra of RNPN in saline (D), RNPN in control brain (E), RNPN in the blood after ad libitum drinking for 6 months (F), and RNPN in brain after ad libitum drinking for 6 months (G). Effect of redox polymer on learning and memory deficit in transgenic mice expressing mutant APP We analyzed the recovery of learning and memory functions in transgenic mice expressing APP in the brain as shown in Fig. 2. In the object recognition and location memory tests, the time period was lower for AD mice aged around 10–11 months, compared with the wild type group while in the groups treated with LMW TEMPO and nRNP, which is a polymer micelle without TEMPO moiety, the exploration time and discrimination index did not increase. In contrast, the exploration time and discrimination index for the RNPN-treated group significantly increased compared with other treated groups for both the object recognition and location tests (Fig. 2A–D). Furthermore, the effect of RNPN on object recognition test after 24 h also displayed the same pattern as after 1 h of ad libitum drinking (Figure S4). The Morris water maze tests also showed a tendency similar to the object recognition tests; the RNPN-treatment significantly shortened the acquisition time compared with other groups (Fig. 2E). The aforementioned results confirmed that the RNPN-treatment significantly improved the learning and memory dysfunctions in AD transgenic mice (p < 0.05) when compare to the water-treated, LMW TEMPO-treated, and nRNP-treated groups. It is interesting to note that the LMW TEMPO-treated group did not exhibit a strong improvement effect on the learning and memory functions than those of RNPN-treated mice, although it was detected in the brain using ESR measurements. The chronic effect of oral ad libitum RNPN treatment on object recognition test at mice age 10 (A) and 11 months (B), object location test at mice age 10 (C) and 11 months (D) after 1 h of administration and Morris water maze test at mice age 12 months (E). Data were represented as mean ± SEM, # vs. wild type mice, p < 0.05; ## vs. AD mice treated with TEMPO, p < 0.05; * vs. AD mice treated with water, p < 0.05; ** vs. AD mice treated with nRNP, p < 0.05; n = 10 mice/group. Suppression of oxidative stress in the brain by ad libitum drinking of RNPN solution The lipid peroxide or malondialdehyde (MDA), 8-hydroxy-2′ -deoxyguanosine (8-OHdG) levels, the percent inhibition of superoxide anion (O2 ·−), and glutathione peroxidase (GPx) assays were used to determine the free radical scavenging capacity of RNPN in comparison with non-treated, water-treated, TEMPO-treated, and nRNP-treated groups. As shown in Fig. 3, RNPN treatment decreased MDA and 8-OHdG levels, while increasing percent inhibition of O2 ·− and GPx activity when compared to the other treatment groups (p < 0.05), indicating the effectiveness of RNPN treatment compared to LMW antioxidant treatment. The oral ad libitum treatment with RNPN improves oxidative stress parameters of MDA level (A), 8-OHdG level (B), % inhibition of O2 ·− (C) and GPx activity (D). Data were represented as mean ± SEM, # vs. wild type mice, p < 0.05; ## vs. AD mice treated with TEMPO, p < 0.05; * vs. AD mice treated with water, p < 0.05; ** vs. AD mice treated with nRNP, p < 0.05; n = 5 samples/group. Decreased Aβ(1-40), Aβ(1-42), and γ-secretase levels in the brain and plasma of transgenic mice treated with ad libitum drinking of RNPN solution As mentioned above, the formation and aggregation of Aβ(1-42) is more toxic, widespread, and abundant than Aβ(1-40)21. In addition, γ-secretase is a protease with substrates that cleave APP and catalyze Aβ aggregation via the amyloidogenic pathway22. We observed that the levels of Aβ(1-40), (1-42), and γ-secretase in both the brain and plasma samples were significantly reduced after RNPN-treatment compared to AD mice treated with water, TEMPO, and nRNP (p < 0.05) as shown in Fig. 4. The effect of ad libitum drinking of RNPN on Aβ(1-42) in the brain (A), the plasma (B), γ-secretase in the brain (C) and Aβ (1-40) in the brain (D), Aβ(1-40) in the plasma (E) of mice. Data were expressed as mean ± SEM, # vs. wild-type group, p < 0.05; * vs. AD mice treated with water, p < 0.05; ** vs. AD mice treated with nRNP, p < 0.05, n = 5 samples/group. Reduction in amyloid plaque formation by chronic ad libitum drinking of RNPN solution To confirm amyloid plaque formation and/or aggregation in mice brain, thioflavin S and immunohistochemistry staining of Aβ(1-40) and Aβ(1-42) were carried out. From the representative photomicrographs of thioflavin S staining in the cerebral cortex, as shown in Fig. 5A, no remarkable Aβ fibrils was observed in wild-type mice (Fig. 5A-1), while transgenic mice in the water-treated group (Fig. 5A-2) showed remarkable Aβ fibrils in the cortex area. Although the LMW TEMPO treatment attenuated the number Aβ fibrils to some extent, AD mice treated with RNPN showed much higher attenuation efficiency (Fig. 5A-4) (p < 0.05, Figs 5A-3,5), which was confirmed by the quantitative fluorescent intensity in Fig. 5B. The effect of oral ad libitum drinking of RNPN on Aβ-fibrils using thioflavin S staining (A), semi-quantitative analysis of Aβ-fibril (B), immunohistochemistry staining of Aβ(1-42) (C), semi-quantitative analysis of Aβ(1-42) (D), immunohistochemistry staining of Aβ(1-40) (E), and semi-quantitative analysis of Aβ(1-40) (F) of the cerebral cortex of mice brain; 1. wild type group; 2. AD mice group treated with water; 3. AD mice group treated with TEMPO; 4. AD mice group treated with RNPN; 5. AD mice group treated with nRNP; 6. negative control group. Data were expressed as mean ± SEM, # vs. wild-type group, p < 0.05; ## vs. AD mice treated with TEMPO, p < 0.05; * vs. AD mice treated with water, p < 0.05; ** vs. AD mice treated with nRNP, p < 0.05, n = 5 sample/group. White arrow indicates amyloid fibril and red arrow indicates amyloid plaque and scale bar = 100 μm. The immunostaining of Aβ(1-40) and Aβ(1-42) revealed no immunoreactive cells in the cerebral cortex of wild-type mice (Fig. 5C-1, E-1). On the contrary, in AD transgenic mice that were treated with water, numerous immunoreactive Aβ(1-40) and Aβ(1-42) positive cells were observed (Fig. 5C-2, E-2), similar to that after treatment with nRNP (Fig. 5C-5, E-5). The RNPN-treated AD mice group showed significant reduction in Aβ(1-40) and Aβ(1-42) positive cells (Fig. 5C-4, E-4). This reduction was higher than that in the LMW TEMPO group (Fig. 5C-3, E-3), which are quantitatively confirmed as shown in Fig. 5D,F. For quantitative evaluation of number of plaques as shown in Fig. 5D,F, we employed randomly captured 5 slices per group under magnification of 20X (See Figure S5). We have designed an antioxidative treatment in a transgenic AD model mice (Tg2576) using our novel amphiphilic polymer antioxidant, which self-assembles in aqueous media. Since the size of RNPN was approximately 30-40 nm, we also confirmed that the volume of ad libitum drinking of RNPN solution was the same as in water, TEMPO, and nRNP-treated mice (data not shown). The triplet signals were observed in both the blood and brain samples of LMW TEMPO-administered group (Fig. 1B and C), suggesting that the blood brain barrier of these Tg2576 mice might be weakened at tight junctions between the endothelial cells due to the significant Aβ expression23. Immediately, after preparation via the dialysis method, the ESR signal of RNPN became broader, because of the restricted mobility of the nitroxide radicals in the solid state, which proves that the confinement of nitroxide radicals in the nanoparticle core (Fig. 1D). The ESR signals of RNPN changed its pattern from broad ESR spectra to triplet in the blood of mice with ad libitum drinking (Fig. 1F) due to its disintegration in the stomach followed by absorption in the blood stream via mesentery in the small intestine. Because the redox polymer possesses cationic charge, it was previously confirmed to interact with serum proteins in the blood stream and circulate for an extended period19. By the ad libitum drinking of RNPN solution, the redox polymer continuously circulates in the bloodstream and gains access to the brain vessel walls continuously. This repeated access of the redox polymer improved the uptake in the brain as shown in Fig. 1G. The results of ESR signal intensity (a.u.) data (Supplementary Table S1) exhibited that the brain of mice treated with RNPN (3.04 × 103 a.u.) has significantly higher ESR signal intensity than that of the LMW TEMPO-treated group (1.94 × 103 a.u.), which is probably due to the rapid elimination of LMW TEMPO. Transgenic Tg2576 mice overexpressing human amyloid precursor protein (hAPP) are widely used as an AD mouse model to evaluate the treatment effects on Aβ pathology. Previous reports on the onset of cognitive deficits in Tg2576 have determined that abnormalities appear as early as 3 months and as late as 12 months24. In this study, the aged Tg2576 mice (approximate 7–12 months) were used for the evaluation of the learning and memory of AD mice; Morris water maze, novel object recognition and object location memory tests25,26,27,28. The transgenic mice we used in this study showed statistically significant reduction of the duration of both object recognition and location tests and elongation of acquisition time in Morris water maze test than that of wild-type mice, which is in accordance with the previous study where Tg2576 mice displayed impaired spontaneous alternation of Y-maze test at both 3 and 10 months of age, as well as advanced impairment in the acquisition time of Morris water maze test in 9–10 months of AD mice29. We have previously reported that oral RNPN treatment recovered both cognition and memory levels in 17-week-old SAMP8 mice19. Here, RNPN-treated group extended the time in the object recognition and location tests and decreased the acquisition time in Morris water maze test, indicating the effective attenuation of the learning and memory deficits in AD transgenic mice (Tg2576) (Fig. 2). Taking together these results and our previous data on SAMP8, it suggests that our antioxidative nanoparticle treatment is robust strategy for AD therapy. Oxidative stress has been implicated in Aβ accumulation and progression via mitochondrial dysfunction, which caused by the generation of ROS30, 31 and reduction in the level of detoxifying enzymes including superoxide dismutase (SOD), GPx and catalase (CAT) in the early stages of the AD32, 33. ROS destroys the polyunsaturated fatty acids of cellular membranes to generate lipid peroxidation products such as MDA, which may serve as an indicator of the level of oxidative damage32, 34, and induce neuronal deterioration35, 36. In addition, 8-OHdG has been determined as a pivotal biomarker of oxidative DNA damage37. Additionally, the toxic effect of Aβ peptide in neuronal cells has been proposed via the interaction between the peptide and Cu2+ and Fe3+ ions because Aβ is a metalloprotein that displays high-affinity binding to these ions, leading to amyloid plaque formation. Aβ catalyzes the reduction of Cu2+ to Cu+ and Fe3+ to Fe2+, that produce hydrogen peroxide (H2O2)38,39,40. Furthermore, in AD transgenic mouse models of mutants of APP elevated production of H2O2 and nitric oxide increases protein and lipid peroxidation. These were associated with age-related Aβ accumulation, and Aβ further enhances oxidative stress41. As expected, the observed neuroprotection by RNPN was due to its antioxidant properties through enhancement of the activity of GPx and percent inhibition of O2 ·−, while reducing MDA and 8-OHdG levels, compared to other AD groups (p < 0.05) as shown in Fig. 3. This was consistent with our previous result, where we showed that RNPN eliminates superoxide anion and hydroxyl radicals, which cause lipid peroxidation and protein and DNA oxidation19, 20. Since RNPN possesses covalently conjugated TEMPO molecules, which scavenge superoxide radicals42 it directly reacts with both carbon-centered and peroxy radicals43 preventing the reduction of hydrogen peroxide to hydroxyl radical44. Hence, RNPN might attenuate the formation of hydroxyl radical and acts similarly to SOD. The LMW TEMPO compounds easily permeate the cell membrane and functions as an intracellular scavenger of O2 ·− and other radicals45, 46, which is in sharp contrast to our redox polymer because of no internalization in healthy cells due to the high molecular weight16. Although our redox polymer, PEG-b-PMNT was not internalized in healthy cells, it was effective in APPsw/Tg2576 AD mice to attenuate oxidative damage and induced the recovery of the activities of the radical scavenging enzymes of GPx. The suppression of cell internalization tendency of the redox polymer is one of the most important factors to suppress severe adverse side effects in healthy cells and tissues unlike the LMW TEMPO16. Recently, we have been investigating the effect and potential toxicity of RNPs (pH-sensitive RNPN and pH-insensitive RNPO), TEMPOL and nRNP to several animals. For example, when zebrafish larvae were exposed to 10–30 mM of LMW TEMPOL, all were dead after 12 h, whereas no larva death was observed after exposure to RNPs at the same high concentrations. By staining mitochondria from the larvae, we found that LMW TEMPOL significantly induced mitochondrial dysfunction. In contrast, RNPs did not cause any significant reduction in the mitochondrial function of zebrafish larvae16. Oral administration of RNPO to healthy mice did not cause any damage to intestinal microflora47. In addition, long-term oral administration of RNPO to xenograft-tumor-bearing mice coupled with anti-cancer drug did not cause any adverse effect but reduced diarrhea and other undesired phenomena48,49,50. Oral administration of RNPN to transgenic non-alcoholic steatohepatitis model mice suppressed liver fibrosis but cause no adverse effect51. On the basis of our alternative investigations, we have concluded that RNPs do not cause strong toxic and adverse effects. In contrast to conventional antioxidants that can be internalized into healthy cells, which destroy its normal redox reactions (e.g., electron transport chain), the size of RNPs prevents its cellular internalization. ROS are implicated in the formation of senile plaques in the brains of patients with AD, which may result in neuronal death30. As the result displayed that scavenging ROS by orally administered RNPN significantly attenuates the neuronal loss in both the cerebral cortex and hippocampus when compared to other AD-treated groups (Supplementary Figures S2-3, p < 0.05). The development of γ-secretase inhibitors has been explored as drugs for AD52. Our in vivo result shows that RNPN-treatment significantly reduced γ-secretase levels, followed by a decrease in Aβ-level compared to AD mice treated with water, TEMPO, and nRNP respectively (p < 0.05), as shown in Fig. 4, which is in accordance with the previous in vitro and knockdown experiments. Finally, we confirmed amyloid plaque formation and/or aggregation via thioflavin S and immunohistochemistry staining of Aβ(1-40) and Aβ(1-42). The results shown in Fig. 5 display that chronic oral administration of RNPN significantly reduced Aβ plaques and/or fibril aggregation when compared to other AD treatment groups (p < 0.05). It has been reported that the accumulation of Aβ is through the interaction of soluble Aβ with metal ions, mainly Zn2+, Cu2+, and Fe3+ 38. The prevention of soluble Aβ formation by RNPN treatment inhibits Aβ aggregation. Here, we confirmed that after ad libitum drinking of our pH-sensitive redox nanoparticle, RNPN, which is a self-assembling polymer antioxidant, RNPN is internalized in the brain and eliminates the increased oxidative stress by scavenging ROS and attenuates cognitive deficits. The increased levels of Aβ and γ-secretase and radical-scavenging activity were decreased after RNPN treatments. Finally, the number of the thioflavin S-stained neurons was significantly higher in the RNPN-treated group than in other groups. On the basis of these obtained results in addition to our previous results on the senescence-accelerated mice-treated with RNPN, we suggest that RNPN may be a promising candidate for the treatment of brain disorders including AD therapy. Drugs and reagents Amino-2,2,6,6-tetramethylpiperidine-N-oxyl (TEMPO), 4-hydroxy-2,2,6,6-tetramethylpiperidine-1-oxyl (TEMPOL) were purchased from Sigma-Aldrich (MO, USA), Mouse Aβ(1-42), Aβ(1-40) and γ-secretase ELISA kits were bought from MyBioSource, Inc. (San Diego, USA), Oxidative DNA damage ELISA kit was purchased from Cell Biolab, Inc. (San Diego, USA), Liquid 3,3′-diaminobenzidine (DAB) substrate kit was bought from Life technologies Corp. (Waltham, USA) and primary anti-Aβ(1-42), Aβ(1-40) and secondary antibodies were purchased from Abcam Plc. (Tokyo, Japan). All other chemicals in this study, which are analytical-reagent grade, were purchased locally from Wako Pure Chemical Industries Ltd. Japan. Female APPSWE/hemi-rd1 Tg2576 and wild-type mice were used in this study (CLEA, Japan, Inc., Tokyo, Japan). They were housed in the experimental animal center of the University of Tsukuba under controlled temperature (23 ± 1 °C), humidity (50 ± 5%), and lighting (12 h light/dark cycles). The animals had unrestricted access to food and water. All the experiments were carried out in accordance with the guidelines for animal care and use of Faculty of Medicine, Tsukuba University and were approved by the animal ethics committee of the Institutional Animal Experiment Committee at the University of Tsukuba (Protocol number 13-407) and in accordance with the Regulation for Animal Experiments in our University and the Fundamental Guidelines for Proper Conduct of Animal Experiments and Related Activities in Academic Research Institutions under the jurisdiction of the Ministry of Education, Culture, Sports, Science, and Technology. Biodistribution of RNPN in the blood and brain after ad libitum drinking Tg2576 mice age approximately 12 months old were anesthetized via an intraperitoneal injection of sodium pentobarbital (50 mg/kg) after 6-month ad libitum drinking of RNPN solution (5 mg/mL/day) and LMW TEMPO (0.6 mg/mL/day). The blood and brain tissue were collected immediately after perfusion. Whole blood samples were subjected immediately to ESR measurement to quantify the drug levels. The brain tissue was immediately placed on ice and homogenized. The ESR signals from the blood were recorded at room temperature using a Bruker EMX-T ESR spectrometer operating at 9.8 GHz with a 100 kHz magnetic field modulation. Signals were collected with the following parameters: center field, 5000 G; sweep width, 7000 G; microwave power, 100.2 mW; receiver gain, 1 × 103; time constant, 81.92 ms; and conversion time, 160 ms and sweep time, 163.84 s. The blood samples were corrected at predetermined time points and subjected to the ESR. The total amount of drug (nitroxide radicals + hydroxyamines) in the brain was estimated from X-band ESR spectrometer (JES-TE25X; JEOL, Tokyo, Japan) at room temperature after the oxidation of hydroxylamine by K3[Fe(CN)6], which was prepared at 200 mM as a stock solution. The ESR measurements were carried out under the following conditions: frequency, 9.41 GHz; power, 8.00 mW; field, 333.8 ± 5 mT; sweep time, 1.0 min; modulation, 0.1 mT; time constant, 0.1 s. Voluntary test fluids consumption Fifty milliliters of drinking fluids were provided in 75 mL drinking bottles equipped with stainless steel spouts, placed on cage covers. During the first 5 days, mice were given water as their only drinking fluid from bottles replacement water tap system to habituate them. For the next 6 months (at 7–12 months of age), all mice were allowed drinking bottles containing RNPN (5 mg/mL), nRNP (5 mg/mL), LMW TEMPO (0.6 mg/mL), and water in each group. The consumption of the fluids and the body weights was recorded once a week, and the bottles were filled with fresh solutions. We used 7 month-old female APPSWE/hemi-rd1 Tg2576 and their non-transgenic littermates or wild-type (WT) mice (40 Tg2576 mice and 10 WT mice; weight, 25 ± 2.0 g and 30 ± 2.0 g, respectively) in this study. All Tg2576 mice were randomly assigned to various groups for RNPN (5 mg/mouse/day) or blank micelles (5 mg/mL/day) or LMW TEMPO (0.6 mg/mL/day) or vehicle (water) by unrestricted access from drinking bottles every day for a 6-month period. In the fourth and fifth month after ad libitum drinking, the animals were tested in object recognition and object location tests. In the last month of the experiment, we also tested the mice in the Morris water maze test. After 6 months of the experiment, 50 mice were sacrificed, and half of the brain tissues were used for the assays for soluble Aβ(1-40) and Aβ(1-42), γ-secretase activity, ROS production (MDA and DNA oxidation), ROS levels, scavenging enzyme activity (GPx activity), and the other halves were used for Aβ immunohistochemical, thioflavin S, and cresyl violet staining. Plasma and brain collection After their behavioral tests were finished, the mice were anesthetized with sodium pentobarbital (50 mg/kg i.p.); blood samples were kept to separate plasma, and brains were quickly isolated. The tissues were prepared for biochemical, histological and immunohistochemical examinations and stored at −80 °C until determination. Antioxidant enzyme assay The glutathione peroxidase (GPx) activity was determined by the previous method of Hussain et al.53 based on that the activity was measured indirectly by a coupled reaction with glutathione reductase. Oxidized glutathione, produced upon reduction of hydrogen peroxide by glutathione peroxidase, was recycled to its reduced state by glutathione reductase and NADPH. The oxidation of NADPH to NADP+ was accompanied by a decrease in absorbance at 340 nm. The rate of decrease in the A340 nm was directly proportional to the glutathione peroxidase activity. In the final 1 mL of the system mixture contained 48 mM sodium phosphate, 0.38 mM EDTA, 0.12 mN β-NADPH, 0.95 mM sodium azide, 3.2 units of glutathione reductase, 1 mM glutathione (GSH), 0.02 mM DL-dithiothreitol, 0.0007% H2O2, and the standard enzyme glutathione peroxidase solution or a homogenate brain sample. The glutathione peroxidase solution was used as a standard enzyme activity. The standard curve was plotted as the rate of A340 nm per minute against the GPx activity. One unit activity was defined as the amount of enzyme necessary to catalase the oxidation by H2O2 of 1 µmole of GSH to GSSG per minute at pH 7 at 25 °C. The data were reported in units of GPx per mg protein. Reactive oxygen species (ROS) products assays Lipid peroxidation (LPO) was measured by determining malonyldialdehyde (MDA) which have been used as an indicator of lipid peroxidation according to Ohkawa et al.54, were measured by using a commercial assay kit (BIOMOL International, USA). Each sample was homogenized (Potter-Elvehjem) in a 10-fold volume of ice-cold 20 mM pH 7.4 of PBS containing 0.5 mM butylated hydroxytoluene to prevent sample oxidation. The homogenized sample was centrifuged at 3,000 g at 4 °C for 10 min, and a 200 µL aliquot of the supernatant was used to measure MDA plus HAE levels according to the instructions of the manufacturer. Values were standardized to micrograms of protein. Each sample was homogenized (Potter-Elvehjem) in a 10-fold volume of ice-cold 20 mM pH 7.4 of PBS containing 1% streptomycin sulfate and incubated for 30 min at room temperature. The nucleic acid precipitates were removed by centrifuging at 6,000 g for 10 min at 4 °C to avoid erroneous contribution to a higher estimation of the carbonyl content from nucleic acid in the cells. The supernatant was used to measure protein levels according to the instructions of the manufacturer. The obtained values were standardized to milligrams of protein. Deoxyguanosine (dG) is one of the constituents of DNA and when it is oxidized, it is altered into 8-hydroxy-2′-deoxyguanosine (8-OHdG). 8-OHdG is useful as a general DNA oxidation marker in the body. A commercial assay kit (Cell Biolab, Inc., San Diego,USA) was used in this measurement. Tissue homogenate and supernatant were used to measure 8-OHdG levels according to the instructions of the manufacturer. The obtained values were standardized to milligrams of protein. Brain oxidative DNA damage determined by ELISA The remained supernatant of mice' brains were also analyzed of 8-hydroxydeoxyguanosine (8-OHdG) which is the common marker of DNA oxidative stress according to the protocol of product manual (Cell Biolab, Inc., San Diego, USA). First, prepared 8-OHdG coated plate, overnight at 4 °C. Washed, filled assay diluent to each well and incubated for 1 h at room temperature (RT). Removed the solution, added 50 µL of sample to the coated wells following with anti-8-OHdG antibody and incubated on an orbital shaker for 1 h at RT. Washed properly, filled 100 µL of secondary antibody. Added substrate solution and inhibit reaction by stop solution. Eventually, read the absorbance at 450 nm on a spectrophotometer to compare to the standard curve of 8-OHdG. Assay of superoxide anion level The reaction mixture consisted of 10 mM phosphate buffer (pH 7.4) containing 0.1 mM xanthine, 0.1 mM EDTA, 0.1 mM nitroblue tetrazolium, and 0.1 unit xanthine oxidase (XO) at a final volume of 1 mL. The formation rate of formazan produced was determined from the slope of the absorbance curve during the initial 2 min of the reaction at 560 nm. In order to analyze the anti-oxidation activity, each sample of different groups was added to the reaction mixture. The change of absorbance was compared with that of the control in the same time reaction, and anti-oxidation activity was calculated according to the following Equation (1) $$\mathrm{Anti}\mbox{-}\mathrm{oxidation}\,{\rm{activity}}\,( \% )=\mathrm{100}\times ({\rm{A}}-{\rm{B}})/{\rm{A}}$$ where A and B are the rate of formazan formation in the absence and presence of sample, respectively. Measurement soluble Aβ(1-40), soluble Aβ(1-42) and γ-secretase in brain tissue and plasma The brain tissue from one brain hemisphere of each mouse was homogenized in PBS, pH 7.4 and centrifuged at 1500 g for 15 min. The anticoagulated blood was drawn and spinned at 1000 g for 10 min. The supernatants of both types of sample were collected and protein quantification was performed using the bicinchoninic acid (BCA) assay (Bio-Rad Laboratories, Hercules, CA). Samples were analyzed for soluble Aβ(1-40), Aβ(1-42) or γ-secretase (BioSource International, Inc., Camarillo, CA) according to the manufacturer's instructions. Briefly, added 50–100 µL of samples to each well of pre-coated microtiter plate, mixed with 5–10 µL of balance solution, filled 50–100 µL of conjugate solution to each specimen and then incubated for 1 h at 37 °C on automated shaker. Washed properly, added substrate A and B respectively and then incubated for 10–15 min at 37 °C. Finally, the reaction was stopped by terminate solution and determine the optical density at 450 nm immediately to compare with the typical standard curve of mouse Aβ(1-40) or Aβ(1-42) or γ-secretase respectively. Brain was isolated for immunohistopathology. The corrected tissues were fixed in 10% formalin solution. Paraffin blocks were prepared after completing the tissue processing in different grades of alcohol and xylene. Brain sections (5 µm) were prepared from paraffin blocks using microtome, stained with 0.5% cresyl violet according to Paxions and Chorles55. Images were taken using OLYMPUS camera connected to the microscope to examine gross cellular damage and neuronal density determination using Image JTM (NIH, MD, USA) software56. Aβ plaque staining and quantification The brain sections of 5 µm were also studied of immunohistochemistry analysis for Aβ(1-40) and Aβ(1-42) respectively. Pre-heated slices with microwave in acetic solution, cool at RT, blocked with methanol and hydrogen peroxide solution for 30 min. Washed properly, prepared moist chamber, blocked with BSA and added primary anti-Aβ(1-40) and Aβ(1-42) respectively (1:100, 1:100 and 1:50) of each slices for overnight at RT. Washed again and incubated with secondary antibody conjugated-horseradish peroxidase (HRP) for 45 min. Developed color by using DAB kit and slices were represented as brown positive staining cells, compared to WT group and used without primary antibody as negative control group under light microscope (Olympus model BZ-X710 Keyence, Tokyo, Japan). Thioflavin S staining and semi-quantification This method is suitable for staining of Aβ fibrils following Schmidt et al.57. After tissue processing, sections were immersed into 0.25% potassium permanganate solution for 20 min, bleaching solution for 2 min, blocking solution for 20 min, 0.25% acetic acid for 5 s, dropped thioflavin S staining solution on slices for 3–5 min, washed by 50% ethanol and following distilled water. Mounted with glycerin and observed under fluorescent microscope (Olympus model BZ-X710 Keyence, Tokyo, Japan). The behavioural tests In this experiment using 3 behavioural tests for measuring cognition status. The Morris water maze test was selected as a method for the evaluation of the spatial learning and memory according to the Morris's method58. A circular water tank (120 cm in diameter and 50 cm in height) was filled with water to a depth of 30 cm inside the tank; an escape platform (11 cm in diameter) was placed, with the top of 1 cm below the water surface. The platform was in the middle of the target quadrant, and its position remained fixed during the experiment. Above the tank, a white floor-to-ceiling cloth curtain was drawn around the pool, and four kinds of black cardboard (circle, triangular, rhombus and square) were hung equidistantly on the interior of the curtain serving as spatial cues. Each mouse had daily sessions of one trial for 5 consecutive days. When they succeeded, mice were allowed to stay on the platform for 30 s. When the mice failed to find the platform within 60 s, they were assisted by the experimenter and allowed to stay the platform for the same time. Probe trials were performed after the last training session at 6 months after free drinking substance. The object location test or spatial novelty was conducted according to Barker et al. with some modification59. One day before object location test, all mice were exposed for 30 min to the empty test box for habituation. The object location tests consisted of 2 trials which are sample phase (T1) and test phase (T2) with a 30 min interval between the 2 trials. In sample phase, mice were exposed to object O1 and O2, which were placed in the far corner of the area. The animal was allowed to explore both objects during a sample phase for 3 min and the amount of exploration of each object was recorded. After a delay of 30 min, the test phase was started. In the test phase, object O3 was placed in the same position as object O1 in the sample test while object O4 was placed in the corner adjacent to the original position of O2, so that object O3 and O4 were in diagonal corners. Thus, both objects in the test phase were equally familiar, but one was in a new location. The position of the moved object was counterbalanced between mice. All measures experiments were made with the experimenter blind to the treatment status of each animal. The basic measure was the total time spent by mice exploring each object during T1 and T2 trials. Exploratory behavior was defined as the animal directing the nose toward the object at a distance of <2 cm. Looking around while sitting or resting against the object was not considered as exploration. On object location task the amount of time exploring each object (object in the new location versus object in familiar position) is reported as an object discrimination ratio and calculated using the following Equation (2) $$\begin{array}{c}({\rm{E}}{\rm{x}}{\rm{p}}{\rm{l}}{\rm{o}}{\rm{r}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}}\,{\rm{t}}{\rm{i}}{\rm{m}}{\rm{e}}\,{\rm{o}}{\rm{f}}\,{\rm{o}}{\rm{b}}{\rm{j}}{\rm{e}}{\rm{c}}{\rm{t}}\,{\rm{i}}{\rm{n}}\,{\rm{t}}{\rm{h}}{\rm{e}}\,{\rm{n}}{\rm{e}}{\rm{w}}\,{\rm{l}}{\rm{o}}{\rm{c}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}}-{\rm{E}}{\rm{x}}{\rm{p}}{\rm{l}}{\rm{o}}{\rm{r}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}}\,{\rm{t}}{\rm{i}}{\rm{m}}{\rm{e}}\,{\rm{o}}{\rm{f}}\,{\rm{o}}{\rm{b}}{\rm{j}}{\rm{e}}{\rm{c}}{\rm{t}}\,{\rm{i}}{\rm{n}}\\ \quad {\rm{f}}{\rm{a}}{\rm{m}}{\rm{i}}{\rm{l}}{\rm{i}}{\rm{a}}{\rm{r}}\,{\rm{l}}{\rm{o}}{\rm{c}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}})/{\rm{T}}{\rm{o}}{\rm{t}}{\rm{a}}{\rm{l}}\,{\rm{e}}{\rm{x}}{\rm{p}}{\rm{l}}{\rm{o}}{\rm{r}}{\rm{a}}{\rm{t}}{\rm{i}}{\rm{o}}{\rm{n}}\,{\rm{t}}{\rm{i}}{\rm{m}}{\rm{e}}\,{\rm{o}}{\rm{f}}\,{\rm{b}}{\rm{o}}{\rm{t}}{\rm{h}}\,{\rm{o}}{\rm{b}}{\rm{j}}{\rm{e}}{\rm{c}}{\rm{t}}{\rm{s}}\end{array}$$ The object recognition task was performed in a circle open-field apparatus (60 cm in diameter and 50 cm in height). The objects used in this task were different in shapes, colors, and textures according to Antunes and Biala60. The open field and the objects were cleaned between each trial using 70% ethanol to avoid odor trails. Before the experiment day, the animals were allowed to acclimatize to the experimental environment. During habituation, the animals were allowed to freely explore the apparatus without objects for 5 min, once a day for three consecutive days before testing. On the experimental day, animals were submitted to two trials spaced. During the first trial (T1), animals were placed in the area containing the same two identical objects for an amount of time necessary to spend 15 s exploring these two objects in a limit of 4 min. Any mice which did not explore the objects for 15 s within the 4 min period were excluded from experiments. 1 h after exposing to the first trial, the animals were exposed to the second trial (T2). 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Phetcharat Boonruamkaew Present address: School of Pharmacy, Walailak University, Thasala, Nakhon Si Thammarat, 80161, Thailand Pennapa Chonpathompikunlert Present address: College of Alternative Medicine, Chandrakasem Rajabhat University, 39/1 Ratchadaphisek Road, Khwaeng Chantharakasem, Chatuchak Districk, Bangkok, 10900, Thailand Long Binh Vong Present address: Department of Biochemistry, Faculty of Biology and Biotechnology, University of Science, Vietnam National University Ho Chi Minh City (VNU-HCM), Ho Chi Minh City, 702500, Vietnam Department of Materials Sciences, Graduate School of Pure and Applied Sciences, University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki, 305-8573, Japan Phetcharat Boonruamkaew, Pennapa Chonpathompikunlert, Long Binh Vong, Sho Sakaue & Yukio Nagasaki Institute of Clinical Medicine, Department of Neurology, University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki, 305-8575, Japan Yasushi Tomidokoro, Kazuhiro Ishii & Akira Tamaoka Master's School of Medical Sciences, Graduate School of Comprehensive Human Sciences, Tennoudai 1-1-1, Tsukuba, Ibaraki, 305-8573, Japan Akira Tamaoka & Yukio Nagasaki Satellite Laboratory, International Center for Materials Nanoarchitechtonics (WPI-MANA), National Institute for Materials Sciences (NIMS), University of Tsukuba, Tennoudai 1-1-1, Tsukuba, Ibaraki, 305-8573, Japan Yukio Nagasaki Sho Sakaue Yasushi Tomidokoro Kazuhiro Ishii Akira Tamaoka Conceived and designed the experiments: Y.N. and P.C. Performed the experiments: P.B., P.C., L.B.V., and S.S. Analyzed the data: P.B., P.C., and Y.N. Contributed reagents/materials/analysis tools: Y.T., K.I., A.T., and Y.N. Wrote the paper: P.B., P.C., L.B.V., and Y.N. Correspondence to Yukio Nagasaki. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Supplementary file Boonruamkaew, P., Chonpathompikunlert, P., Vong, L.B. et al. Chronic treatment with a smart antioxidative nanoparticle for inhibition of amyloid plaque propagation in Tg2576 mouse model of Alzheimer's disease. 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Driving steady-state visual evoked potentials at arbitrary frequencies using temporal interpolation of stimulus presentation Søren K. Andersen1 & Matthias M. Müller2 BMC Neuroscience volume 16, Article number: 95 (2015) Cite this article Steady-state visual evoked potentials have been utilized widely in basic and applied research in recent years. These oscillatory responses of the visual cortex are elicited by flickering stimuli. They have the same fundamental frequency as the driving stimulus and are highly sensitive to manipulations of attention and stimulus properties. While standard computer monitors offer great flexibility in the choice of visual stimuli for driving SSVEPs, the frequencies that can be elicited are limited to integer divisors of the monitor's refresh rate. To avoid this technical constraint, we devised an interpolation technique for stimulus presentation, with which SSVEPs can be elicited at arbitrary frequencies. We tested this technique with monitor refresh rates of 85 and 120 Hz. At a refresh rate of 85 Hz, interpolated presentation produced artifacts in the recorded spectrum in the form of additional peaks not located at the stimulated frequency or its harmonics. However, at a refresh rate of 120 Hz, these artifacts did not occur and the spectrum elicited by an interpolated flicker became indistinguishable from the spectrum obtained by non-interpolated presentation of the same frequency. Our interpolation technique eliminates frequency limitations of the common non-interpolated presentation technique and has many possible applications for future research. The steady-state visual evoked potential (SSVEP) is a continuous oscillatory response of the visual cortex which is elicited by a flickering stimulus and has the same temporal frequency as the driving stimulus [1–4]. It can be recorded by electro- or magnetoencephalography and has been widely used in basic and applied research in recent years. In particular, SSVEPs have been employed to study different aspects of selective attention such as spatial attention [5–10], feature-based attention [11–13], visual search [14], competitive stimulus interactions [15–17] and attentional function in healthy old age [18, 19]. Other studies have employed SSVEPs to assess processing of emotional stimuli [20–24] and faces [25, 26], binocular rivalry [27, 28], object processing [29, 30], and figure-ground separation [31]. SSVEPs have also been widely employed in the development of brain-computer interfaces [32, 33]. SSVEPs offer good signal-to-noise ratios, as the signal power is concentrated in a few discrete frequency bands. This allows for easy signal extraction by means of Fourier-transformation if temporally sustained effects are of interest. Dynamic changes in stimulus processing can also be studied by using time–frequency analysis techniques such as Gabor-Filters [34] or wavelets. A particularly powerful application of SSVEPs is the 'frequency tagging' technique, in which the cortical processing of multiple stimuli can be assessed simultaneously by flickering each stimulus at a specific frequency [3]. However, as stimulus presentation is necessarily synchronized to the monitor's refresh rate, the choice of frequencies is severely restricted. For example, when using a monitor with a refresh rate of 60 Hz, a 30 Hz SSVEP can be elicited by presenting the stimulus for one frame and then switching it off for one frame (two frames cycle length) while two frames on and two frames off (four frames cycle length) result in a frequency of 60/4 = 15 Hz. At 60 Hz, this allows for frequencies of 30, 20, 15, 12, 10, 8.57, 7.5 Hz, etc. Using this approach, SSVEPs cannot be elicited at frequencies that are not integer divisors of the refresh rate. Depending on the purpose of a particular study, the choice of frequencies may be restricted even further. One of the most basic such restrictions is not to use frequencies that are harmonics of other frequencies of concurrently presented stimuli. For example, if one stimulus is presented at 10 Hz, then it is not advisable to use 20 or 30 Hz for a second stimulus, as the harmonic of the 10 Hz stimulus would be superimposed on the SSVEP of the second stimulus. When perceptual differences between stimuli flickering at different rates are to be minimized [9], frequencies must be close to each other, which can only be accomplished at the lower frequencies in our 60 Hz example. Other restrictions apply when a high temporal resolution is required for the question at hand, as for example when studying cued shifts of attention [35–37]. When using Gabor-Filters or wavelets to analyze time-courses at different frequencies, the frequencies should be clearly separated in order to avoid crosstalk. If a Fourier-transform with a short window-length is used, then crosstalk can be avoided by using frequencies that have an integer number of cycles in the chosen time-window (e.g. in a 200 ms window, 10 and 15 Hz have 2 or 3 cycles, respectively). In other cases, it may be important to separate SSVEPs from frequency bands of the EEG carrying other signals, such as transient ERPs or the alpha-band, by choosing appropriate frequencies for the SSVEP. Last but not least, in some cases it might be desirable to use stimuli with equal on–off rations in order to equate stimulus luminance over time. If, for example, all stimuli are to have a 50/50 on–off ratio, then frequencies are limited to even integer divisors of the refresh rate. Whatever the particular reasons for the choice of frequencies in an SSVEP study, any additional criteria further reduce the already limited number of potential frequencies. Especially in frequency-tagging studies with multiple stimuli presented at different frequencies, the practically available set of frequencies may be far from the theoretically ideal number and choice of frequencies. This is particularly the case in the development of brain-computer interfaces (BCIs), which allow a person to convey his or her intentions by attending to a particular stimulus. Here, the information transfer rate can be improved with higher numbers of stimuli (i.e. frequencies) and certain applications, such as spelling or typing numbers, ideally require a large number of stimuli [38]. To surpass these limitations, an approximation technique for stimulus presentation has recently been proposed, which allows for driving SSVEPs at frequencies that are not limited to integer divisors of the monitor's refresh rate [39]. This technique interleaves stimulus sequences of different length to approximate a specific presentation rate. For example, a rate of 11 Hz would be approximated at a refresh rate of 60 Hz by interleaving cycles of five or six frames length (corresponding to 12 and 10 Hz, respectively). This approximation approach is successful in eliciting SSVEPs at the desired frequencies which are largely comparable to those that would be elicited by a traditional stimulation technique. However, it also elicits additional signal power at other interference frequencies [38]. Such artifacts of the stimulation technique may be irrelevant for some approaches, in which the entire analysis is limited to a few specific frequencies at which no interference occurs. However, for other applications such interference artifacts may be unacceptable. If, for example, the goal of a study is to concurrently assess SSVEPs and activity in naturally occurring frequency bands of the EEG (e.g. theta-, alpha- or gamma-band), then this frequency band should not be contaminated by such interference artifacts. We here devised and tested an interpolation technique in order to drive SSVEPs at arbitrary frequencies that can be chosen independently of the screen refresh rate. Ideally, a stimulus interpolation technique should (1) robustly produce signal power at the desired frequency (and potentially its harmonics) and (2) not produce additional signal power in other frequency bands. In order to test our technique against these criteria, we presented stimuli flickering at four different frequencies using two different monitor refresh rates. This was done in such a way, that each frequency was elicited in the standard non-interpolated way at one refresh rate, while being interpolated at the other refresh rate. If our technique satisfies the two above criteria, then the spectrum elicited when a frequency is driven non-interpolated at one refresh rate should be indistinguishable from the spectrum elicited when the same frequency is driven by interpolated stimulation at another refresh rate. Fifteen right-handed subjects (10 female, ages 19–30, average 23.8 years) with normal or corrected-to-normal visual acuity participated in the experiment after giving informed consent and received either a small financial bonus (6 € per hour) or credit points. All subjects were included in the statistical analysis. The study was conducted in accordance with the Declaration of Helsinki and the guidelines and requirements for electrophysiological studies of the ethics committee of the University of Leipzig. Interpolation technique A specific SSVEP-frequency can be elicited by presenting a stimulus for a certain number of frames and then switching it off for another number of frames. The elicited frequency is then equal to the monitor refresh rate divided by the total cycle length (number of frames 'on' plus number of frames 'off'). For example, at a refresh rate of 120 Hz, a 10.0 Hz SSVEP can be elicited by flickering a stimulus with a cycle length of 12 frames (e.g. 6 frames on, 6 frames off). However, if the monitor refresh rate is 85 Hz, it is not possible to elicit an SSVEP at 10.0 Hz using this approach, as this would require a cycle length of 85/ 10 HZ = 8.5 frames. Using a 50/50 on–off ratio, this would require the stimulus to be on and off for 4.25 frames respectively, which is technically not possible. However, this can be approximated by presenting the stimulus at full intensity for 4 frames and then presenting it at 25 % intensity for one frame (4 + 0.25 = 4.25 frames on). After this, the stimulus would be off for another 3 frames and then be presented at 50 % intensity (0.75 + 3+0.5 = 4.25 frames off). So the impossible requirement to present a stimulus for a fractional duration of frames at each on–off reversal is approximated by presenting the stimulus for a whole frame at an intermediate intensity. Using this interpolated stimulation technique the time-averaged intensity of the stimulus remains unchanged. However, the technically impossible sharp transitions from on to off at time-points that do not coincide with the monitor's refresh rate are replaced by transitions synchronous to the monitor's refresh rate at intermediate stimulus intensities (Fig. 1b). a Stimulus display. Participants performed a simple detection task at fixation while SSVEPs elicited by the flickering ring were recorded b Illustration of stimulus intensity as a function of time for the four employed frequencies at each of two monitor refresh rates. 10.0 and 15.0 Hz waveforms were non-interpolated at a refresh rate of 120 Hz and interpolated at 85 Hz. 10.625 and 14.167 Hz were non-interpolated at 85 Hz and interpolated at 120 Hz In the general case, our interpolation technique is defined as follows. With a monitor refresh rate R and a desired stimulus frequency f, the required cycle length λ is defined by: $$\lambda = R/f$$ The stimulus intensity w (1:on, 0:off) in any given frame i can then be calculated as follows: $$w = \left\{ {\begin{array}{{20}l} {1,} \hfill & {if\;1 \le i\bmod \lambda \le r_{on} \lambda } \hfill \\ {r_{on} \lambda + 1 - i\bmod \lambda ,} \hfill & {if\;r_{on} \lambda < i\bmod \lambda < r_{on} \lambda + 1} \hfill \\ {0,} \hfill & {if\;r_{on} \lambda \le i\bmod \lambda } \hfill \\ {i\bmod \lambda ,} \hfill & {if\;i\bmod \lambda < 1} \hfill \\ \end{array} } \right.$$ where ron denotes the fraction of the stimulus cycle in which the stimulus is on (ron = 0.5 in the recordings reported here). Note that i modulo λ is the position within the current flicker cycle. The first line defines the frames in which the stimulus is on at full intensity, the second line the on–off transitions, the third line the off-frames and finally the fourth line defines the off–on transitions. In order to present stimuli at intermediate intensities 0 < w <1 two further things need to be taken into account. First, the off-phase of a stimulus is not necessarily black but could be any arbitrary color Coff and thus intermediate stimulus intensities C must be a weighted average of the stimulus color Con and the background or 'off' color Coff. Second, the relationship between color values in a stimulation program Vin (e.g. RGB values) and the output of a computer monitor Vout is not linear, but defined by a power function with an exponent gamma (γ): $$V_{out} \propto V_{in}^{\gamma }$$ The exponent γ depends on the specific graphics hardware and its settings and usually lies between 1.8 and 2.2 for standard computer equipment. Taking the two points above into account, the output color C can be computed as: $$C = \left( {wC_{on}^{\gamma } + \left( {1 - w} \right)C_{off}^{\gamma } } \right)\!{^{1}\!/\!{\gamma}}$$ To test this interpolation technique, we recorded the EEG elicited by a ring flickering at four different frequencies in different conditions. The four frequencies were chosen within a frequency range commonly employed in SSVEP experiments in such a way that two of them could be presented non-interpolated at a monitor refresh rate of 85 Hz, but not at 120 Hz (10.625 and 14.167 Hz), while the two other frequencies could be presented non-interpolated at a monitor refresh rate of 120 Hz, but not at 85 Hz (10 and 15 Hz, see Fig. 1). If the differences introduced by the interpolation technique are sufficiently subtle to be imperceptible to the human brain, then the EEG elicited by interpolated and non-interpolated stimulation of the same frequency should be indistinguishable. Flicker at 60 Hz elicits robust phase locked activity in the primary visual cortex of humans and macaque monkeys [40], however the magnitude of such activity in humans is already strongly reduced at 72 Hz [41]. In light of these findings, we considered it unlikely that distortions produced by stimulus interpolation would be imperceptible at monitor refresh rates of 72 Hz or less. Therefore, we chose a somewhat higher frequency of 85 Hz as the lower refresh rate for our experiment and a rate of 120 Hz as the higher refresh rate, since this was the maximum rate available with our equipment. Stimulus material and procedure Stimulation was presented on a 19″ Belinea 10 60 75 cathode ray tube (CRT) monitor set to a resolution of 640 × 480 pixels and 32 bits per pixel color mode. At a viewing distance of 80 cm, the ring had an outer diameter corresponding to 8.1° degrees of visual angle and an inner diameter of 4.06°. The bars of the fixation cross had a length corresponding to 0.81° × 0.16°. The background had a luminance of 9.4 cd/m2 and the fixation cross and flickering ring had a luminance of 79.7 cd/m2 (Fig. 1a). To ensure correct intermediate luminance values for interpolated flicker, luminance calibration and gamma correction was performed separately for 85 and 120 Hz refresh rates. This is important, as the same RGB color values do not generally produce the same luminance values for different screen modes. We empirically estimated γ by presenting different gray values statically and measuring their luminance, resulting in γ = 2.0 for both refresh rates. Stimulation was realized using Cogent Graphics (John Romaya, LON at the Wellcome Department of Imaging Neuroscience). In each trial, the flickering ring was presented for approximately 3340 ms (284 frames at 85 Hz and 401 frames at 120 Hz). Timestamps for each frame of stimulus presentation were stored in order to verify the accuracy of stimulus timing. We did not detect any dropped frames throughout the entire dataset. To control the allocation of attention, participants performed a simple target detection task at fixation. Beginning from 300 ms after stimulation onset (after 26 frames at 85 Hz or 36 frames at 120 Hz), the length of either the horizontal or vertical bar of the fixation cross could briefly change by 0.12° of visual angle. Length decrements of either bar were defined as targets and required a detection response by pressing space bar. Responses to analogous length increments constituted false alarms. The duration of targets and distractors was about 50 ms (4 frames at 85 Hz or 6 frames at 120 HZ). Any combination of up to three targets and/or distractors could occur within a single trial and the onsets of successive targets or distractors were separated by at least 700 ms (59 frames at 85 Hz or 84 frames at 120 Hz). Responses occurring from 200 ms to 850 ms after the onset of targets or distractors were counted as hits or false alarms, respectively. Over the entire experiment, a total of 45 targets and 45 distractors were presented for each of the eight conditions (4 frequencies each presented at two different monitor refresh rates). The experiment consisted of 480 trials (60 per condition) presented in 8 blocks of 60 trials each. Trials of all conditions were presented in random order with the constraint that any single block contained only trials presented at the same monitor refresh rate. This was done to prevent frequent switching of the screen mode. Blocks at 85 and 120 Hz monitor refresh rate were presented in random order. Responding hand was switched after the first 4 Blocks. Prior to recordings, participants performed one or two blocks of task training (average 1.3 blocks). EEG recordings and analysis Brain electrical activity was recorded non-invasively at a sampling rate of 256 Hz from 64 Ag/AgCl electrodes mounted in an elastic cap using an ActiveTwo amplifier system (BioSemi, Amsterdam, The Netherlands). Lateral eye movements were monitored with a bipolar outer canthus montage (horizontal electroocculogram). Vertical eye movements and blinks were monitored with a bipolar montage positioned below and above the right eye. Processing of EEG data was performed using the EEGLab toolbox [42] in combination with custom written procedures in Matlab (The Mathworks, Natick, MA, USA). Analysis epochs were extracted from 100 ms before to 3400 ms after stimulation onset. All epochs were detrended (removal of mean and linear trends) and epochs with eye movements or blinks were rejected from further analysis. All remaining artifacts were corrected or rejected by means of an automated procedure using a combination of trial exclusion and channel approximation based on statistical parameters of the data [43]. This led to an average rejection rate of 10.6 % of all epochs, which did not differ between conditions. Subsequently, all epochs were re-referenced to average reference and averaged for each experimental condition. To quantify the spectral content in each condition, analysis windows were defined that contained an integer number of cycles of the driven SSVEP frequency (28, 30, 40 and 42 cycles for 10.0, 10.6 14.1 and 15.0 Hz, respectively). All analysis windows began 400 ms after stimulation onset in order to exclude the evoked potential to stimulation onset and to allow the SSVEP sufficient time to build up. The resulting analysis windows lasted from 400 to 3200 ms (10.0 and 15.0 Hz) or to 3225 ms (10.6 and 14.1 Hz) after stimulation onset. The amplitude spectrum of the data in these time windows was obtained by taking the absolute value of the complex Fourier coefficients for each frequency and electrode separately. Iso-contour voltage maps of amplitudes at the frequencies of the elicited SSVEPs for each condition showed a narrow peak over occipital electrodes (Fig. 2a). Accordingly, electrode Oz was chosen for statistical analysis. a Isocontour voltage maps of SSVEP amplitudes averaged over all subjects at the four stimulation frequencies for non-interpolated (top-row) and interpolated (bottom-row) stimulation; b grand-average spectrum (top) and q-values comparing amplitudes (middle) elicited by interpolated vs. non-interpolated stimulation at a refresh rate of 85 Hz reveal clear artifacts of the interpolation technique. A comparison of the variance of the spectrum elicited by interpolated vs. non-interpolated stimulation (bottom) reveals no differences, indicating that artifacts were elicited consistently across participants; c no artifacts were apparent when stimulation was interpolated at a refresh rate of 120 Hz. 0 dB corresponds to 1 µV In the following step, we compared the recorded amplitude spectrum between 0 and 48 Hz for each of the four pairs of conditions with the same SSVEP frequency (e.g. 10 Hz non-interpolated was compared to 10 Hz interpolated). In order to better encompass the large amplitude differences in this wide frequency band, all amplitudes were transformed to a decibel scale by taking the logarithm to a base of 10 and multiplying by 10 (e.g. 1 µV corresponds to 0 dB, 0.1 µV correspond to −10 dB, see Fig. 2b). The amplitudes (in dB) for each frequency above 0 Hz and below 48 Hz were compared by means of paired t-tests. Due to slightly different window lengths, this analysis comprised 134 different frequencies for 10.0 and 15.0 Hz and 135 frequencies for 10.6 and 14.1 Hz. To account for multiple comparisons, we controlled the false discovery rate (FDR) [44] by correcting p-values using the 'mafdr' function in Matlab (The Mathworks). The resulting q-values (i.e. corrected p-values) depicted in Fig. 2b are significant when they are smaller than or equal to 0.05. In order to test for possible differences in the variance between participants of the recorded spectrum, a Bartlett test was conducted analogously to the t-tests described above and using the same correction for multiple comparisons. Behavioral data The behavioral performance of participants is displayed in Table 1. Participants performed the task reliably with average hit-rates above 90 %, average false alarm rates around 10 % and average reaction times just below 500 ms. One-factorial repeated measures ANOVAs using Greenhouse-Geyser Correction for non-sphericity were conducted separately for hit rates, false alarm rates and reaction times. None of the three measures of behavioral performance differed between experimental conditions [hit rate: F(3.691, 51.571) = 1.325; false alarm rate: F(2.880, 40.320) = 0.361; reaction time: F(4.019, 56.263) = 2.022; all p > 0.1]. Table 1 Mean hit and false alarm rates and reaction times for all conditions Electrophysiological data Amplitude spectra and statistical results are depicted in Fig. 2. For each of the four SSVEP frequencies, the spectrum elicited with non-interpolated stimulation was compared against the spectrum elicited with interpolated stimulation. For 10.0 Hz, interpolated stimulation presented at a monitor refresh rate 85 Hz elicited clear additional peaks in the spectrum, that were not present when 10 Hz SSVEPs were elicited by non-interpolated flicker at a refresh rate of 120 Hz (Fig. 2b). Pairwise t-tests for each frequency in the spectrum and corrected for multiple comparisons using the false discovery rate revealed significantly higher amplitudes at 5.0 Hz [t(14) = −6.550, q = 0.000845], 15.0 Hz [t(14) = −5.089, q = 0.00632] and 44.9 Hz [t(14) = −6.471, q = 0.000845]. Although additional peaks are also visible at 25 and 35 Hz, these were not significantly different from the amplitudes elicited with non-interpolated stimulation (see Fig. 2b). Additional peaks were also visible in the spectrum when SSVEPs were elicited at 15.0 Hz using interpolated flicker at a refresh rate of 85 Hz, as compared to when 15.0 Hz SSVEPs were elicited by non-interpolated flicker at a refresh rate of 120 Hz. Significantly higher amplitudes were observed at 10.0 Hz [t(14) = −5.857, q = 0.00317], 20.0 Hz [t(14) = −4.263, q = 0.0200], 25.0 Hz [t(14) = −4.840, q = 0.00997] and 39.9 Hz [t(14) = −3.815, q = 0.0360)]. The variance across participants did not differ significantly between interpolated and non-interpolated stimulation at any point of the spectrum for any of the four SSVEP frequencies (Fig. 2b, c, bottom panels). This indicates that the pattern of presence or absence of interpolation artifacts was consistent across participants, because the presence of such artifacts in only some participants would have led to a higher variance than in the non-interpolated control condition. To provide a more detailed depiction of the recorded data, the grand mean spectrograms were computed for each condition by means of a bank of Gabor-Filters [34] located at the 134 (10.0 and 15.0 Hz) or 135 (10.6 and 14.1 Hz) frequencies of the spectrum used for the main analysis (Fig. 3). Spectrograms of the recorded data averaged over all subjects at the four stimulation frequencies for non-interpolated (top-row) and interpolated (bottom-row) stimulation. Additional signal power was elicited in specific frequency bands when stimulus presentation was interpolated at a refresh rate of 85 Hz (10 and 15 Hz), but not at a refresh rate of 120 Hz (10.63 and 14.17 Hz). 0 dB corresponds to 1 µV In summary, interpolated stimulation at a refresh rate of 85 Hz successfully elicited robust SSVEPs at the stimulated frequencies, which did not differ from those elicited by non-interpolated flicker. However, additional signal power was also elicited at other frequencies, which represents an artifact of the interpolation technique. A different picture emerged when SSVEPs were elicited at 10.6 and 14.1 Hz by interpolated stimulation with a refresh rate of 120 Hz. Here, the spectrum was indistinguishable from the one elicited by non-interpolated stimulation of the same frequencies at 85 Hz (Fig. 2c). The paired comparison yielded no significantly different amplitudes for any of the tested frequencies. We investigated the feasibility of eliciting SSVEPs by using an interpolated stimulation technique in order to overcome the restrictions of available frequencies inherent to no-interpolated stimulation. To verify the adequacy of our method, we compared the recorded spectrum to a control condition in which the same frequency was elicited at another monitor refresh rate without interpolation. An ideal interpolation technique should elicit spectra that are indistinguishable from those recorded without interpolation. Our results consistently showed that this was not the case for interpolation using a monitor refresh rate of 85 Hz: although the elicited SSVEP response in these conditions did not differ from recordings without interpolation, additional peaks at other frequencies were elicited which constitute an unwanted artifact of the interpolation. This was however not the case for interpolation at a refresh rate of 120 Hz, where amplitudes at the elicited SSVEP frequencies and all other frequencies were comparable to those elicited by non-interpolated stimulation. Thus it seems that the temporal resolution of processing in the early visual areas that have previously been identified as the sources of the SSVEP [12, 16, 45] is still high enough to detect the differences in interpolated flicker when presented at a refresh rate of 85 Hz, but not at 120 Hz. We used our interpolation technique to approximate a square wave luminance flicker at high contrast. The elicited spectrum was indistinguishable from the one elicited by non-interpolated flicker when a refresh rate of 120 Hz, but not 85 Hz, was employed. This implies that a refresh rate of 120 Hz is sufficient to interpolate stimulus presentation with our technique without eliciting unwanted artifacts in the spectrum. It should be noted that we tested our technique under 'unfavorable' conditions that were likely to reveal artifacts in the spectrum. Such artifacts may be less evident at lower stimulus contrast or if a smoother stimulus waveform than a square wave (e.g. a sine wave) was approximated. Also, luminance flicker can be perceived at higher frequencies than chromatic flicker [46]. Therefore, interpolating the flicker of chromatic stimuli of equal luminance as the background should produce less artifacts, and may thus be feasible at refresh rates lower than 120 Hz. Other authors have recently proposed a different approximation technique to elicit SSVEPs at frequencies independent of the monitor refresh rate [38, 39]. This technique elicits robust SSVEPs at the desired frequencies and is slightly simpler to implement than ours, as it does not require intermediate stimulus intensities. However, our technique is superior in that it does not produce artifacts at interference frequencies as long as a refresh rate of 120 Hz is utilized. This difference may be critical for applications that also measure activity in other frequency bands (e.g. alpha, gamma) than those at which SSVEPs are elicited or when ERPs to discrete events interleaved in the SSVEP stimulation are to be measured. We tested our interpolation technique for frequencies in the range between 10 and 15 Hz, which is often utilized in SSVEP experiments. SSVEPs can be elicited at lower frequencies too. However the frequency range below 10 Hz is of less interest for interpolated stimulation, because non-interpolated stimulation already offers much more flexibility in this range. For example, at a refresh rate of 120 Hz, the narrow frequency band between 4 and 6 Hz allows for 11 different frequencies to be driven using non-interpolated stimulation. Our interpolation technique is of high utility in cases where temporal changes in the processing of multiple stimuli are of interest. In such cases, higher frequencies are preferable because they allow for better temporal resolution. However, even at a refresh rate of 120 Hz, the wide frequency band from 10 to 60 Hz only contains 11 different frequencies that can be elicited by non-interpolated stimulation. Of these, five frequencies are higher harmonics of other frequencies, thus only leaving 6 frequencies that can be utilized together. Once other constraints are taken into consideration (e.g. equal on–off ratios, sufficient separation in frequency-space to avoid crosstalk, avoiding specific frequency bands; see introduction), the availability of frequencies is reduced even more thus rendering experiments that aim to investigate rapid dynamic changes in the processing of more than 2–3 stimuli infeasible without temporal interpolation of stimulus presentation. It should be noted that our interpolation technique is in not limited to the simple black and white stimuli employed here. By applying Eq. (4) to the red, green and blue (RGB) values of a stimulus separately, this procedure can be utilized for flickering stimuli of any arbitrary color against a background of any arbitrary color. Furthermore, stimuli do not have to be of uniform color. For example, a bitmap image flickering against a background (or two alternating bitmap images) can be implemented by applying Eq. (4) to each pixel separately. Also, the phase of the stimulation can be shifted by any desired angle φ (0 < = φ < 2π). If we define j as $$j = \left( i + \lambda \phi/ 2\pi \right)\bmod \lambda$$ and insert j into Eq. (2). Stimulus intensity w of a waveform shifted by an arbitrary phase φ is given by $$w = \left\{ {\begin{array}{{20}l} {1,} \hfill & {if\;1 \le j \le r_{on} \lambda } \hfill \\ {r_{on} \lambda + 1 - j,} \hfill & {if\;r_{on} \lambda < j < r_{on} \lambda + 1} \hfill \\ {0,} \hfill & {if\;r_{on} \lambda \le j} \hfill \\ {j,} \hfill & {if\;j < 1} \hfill \\ \end{array} } \right.$$ In conclusion, we devised and tested an interpolation technique that, using Eq. (6) allows to drive SSVEPs at any frequency up to half the monitor's refresh rate with arbitrary on/off ratio (as defined by ron) and phase φ. At refresh rates of 120 Hz, this interpolation technique elicited SSVEPs that are indistinguishable from those elicited by non-interpolated flicker without producing any detectable artifacts in other parts of the spectrum. This technique thus portends many future applications in both basic and applied research. Regan D. 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J Opt Soc Am. 1972;62:1508–15. Both authors contributed to conception and design of the study. SKA wrote stimulation programs, recorded and analysed the data. SKA and MM wrote the manuscript. Both authors read and approved the final manuscript. We thank Renate Zahn for help with data collection. This work was supported by Deutsche Forschungsgemeinschaft (AN 841/1-1, MU 972/20-1). We would like to thank A. Trujillo-Ortiz, R. Hernandez-Walls, A. Castro-Perez and K. Barba-Rojo (Universidad Autonoma de Baja California) for making Matlab code for non-sphericity corrections freely available. Copies of the data may be obtained from the corresponding author (SKA). School of Psychology, University of Aberdeen, William Guild Building, Aberdeen, AB24 3FX, UK Søren K. Andersen Institute of Psychology, University of Leipzig, Neumarkt 9-19, 04109, Leipzig, Germany Matthias M. Müller Correspondence to Matthias M. Müller. Andersen, S.K., Müller, M.M. Driving steady-state visual evoked potentials at arbitrary frequencies using temporal interpolation of stimulus presentation. BMC Neurosci 16, 95 (2015). https://0-doi-org.brum.beds.ac.uk/10.1186/s12868-015-0234-7 DOI: https://0-doi-org.brum.beds.ac.uk/10.1186/s12868-015-0234-7 SSVEP Stimulus presentation Frequency-tagging Submission enquiries: [email protected]	CommonCrawl
Article \| Open \| Published: 06 June 2019 The human glomerular endothelial cells are potent pro-inflammatory contributors in an in vitro model of lupus nephritis Paraskevi Dimou ORCID: orcid.org/0000-0001-8436-06761, Rachael D. Wright1, Kelly L. Budge1, Angela Midgley1, Simon C. Satchell3, Matthew Peak2 & Michael W. Beresford1,2,4 Juvenile-onset lupus nephritis (LN) affects up to 80% of juvenile-onset systemic lupus erythematosus patients (JSLE). As the exact role of human renal glomerular endothelial cells (GEnCs) in LN has not been fully elucidated, the aim of this study was to investigate their involvement in LN. Conditionally immortalised human GEnCs (ciGEnCs) were treated with pro-inflammatory cytokines known to be involved in LN pathogenesis and also with LPS. Secretion and surface expression of pro-inflammatory proteins was quantified via ELISA and flow cytometry. NF-κΒ and STAT-1 activation was investigated via immunofluorescence. Serum samples from JSLE patients and from healthy controls were used to treat ciGEnCs to determine via qRT-PCR potential changes in the mRNA levels of pro-inflammatory genes. Our results identified TNF-α, IL-1β, IL-13, IFN-γ and LPS as robust in vitro stimuli of ciGEnCs. Each of them led to significantly increased production of different pro-inflammatory proteins, including; IL-6, IL-10, MCP-1, sVCAM-1, MIP-1α, IP-10, GM-CSF, M-CSF, TNF-α, IFN-γ, VCAM-1, ICAM-1, PD-L1 and ICOS-L. TNF-α and IL-1β were shown to activate NF-κB, whilst IFN-γ activated STAT-1. JSLE patient serum promoted IL-6 and IL-1β mRNA expression. In conclusion, our in vitro model provides evidence that human GEnCs play a pivotal role in LN-associated inflammatory process. Lupus nephritis (LN) is one of the main complications of juvenile-onset systemic lupus erythematosus (JSLE), a rare, multi-system autoimmune disease1. LN is a chronic inflammatory renal disease characterised by periodic flares (active LN) and remissions (inactive LN)2. Each flare potentially contributes to the accumulation of damage of renal structures3 such as the glomerulus4. LN can affect up to 80% of JSLE patients5,6,7, whereas 10–15% of LN patients eventually develop end-stage renal disease8,9,10,11,12. The renal glomerulus is part of the nephron, where primary urine formation occurs13. Each glomerulus contains a network of glomerular capillaries which, in healthy individuals, effectively filter blood plasma through the glomerular filtration barrier (GFB) restricting blood cells and proteins inside the circulation13. In LN, chronic inflammation-induced GFB damage results in proteinuria and haematuria14,15. The human glomerular endothelial cells (GEnCs), a component of the GFB, line the glomerular capillary lumen13. As the GEnCs directly interact with circulating factors and immune cells in the blood and with the renal glomerular podocytes and mesangial cells, they could be central intra-renal contributors to inflammatory processes. To date, there are limited data concerning human GEnC involvement in LN inflammation and production of pro-inflammatory mediators. However, human umbilical vein endothelial cells (HUVECs)16,17 and human brain microvascular endothelial cells (HBMECs)18,19, when cultured in vitro under appropriate pro-inflammatory stimuli such as tumour necrosis factor-alpha (TNF-α), interleukin (IL)-1 beta (IL-1β) and lipopolysaccharide (LPS), have been shown to elicit inflammatory responses. These responses involve nuclear factor kappa-light-chain-enhancer of activated B-cells (NF-κB) activation and production of adhesion molecules and cytokines. Thus, the aim of this study was to investigate the potential inflammatory role of human GEnCs in an in vitro model of LN. Unravelling their potential involvement in lupus renal disease could provide further insight into LN pathogenesis. Inflammatory stimulation of ciGEnCs induces the production of urinary biomarkers and pro-inflammatory proteins The combination of all cytokines (All: 3,755 pg/mL [3,159–3,989], p = 0.008) led to statistically significantly higher levels of secreted monocyte chemoattractant protein-1 (MCP-1) compared to the untreated cells (800 pg/mL [346–1,856]) (Fig. 1a), an effect also observed for LPS (2,876 pg/mL [2,457–3,072], p = 0.04). A significant increase in secreted levels of soluble vascular cell adhesion molecule-1 (sVCAM-1) (Fig. 1b) was induced only by IL-13 (1,539 pg/mL [731–1,718], p = 0.022) compared to the untreated cells (192 pg/ml [74–661]). None of the other novel urinary biomarkers were expressed by ciGEnCs (data not shown). LN urinary biomarker, chemokine, cytokine and growth factor secretion after 24-hour ciGEnC-stimulation with individual cytokines, combination of all cytokines (All) and LPS. MCP-1 (a), sVCAM-1 (b), IL-6 (c), IL-8 (d), IL-10 (e), M-CSF (f), GM-CSF (g), MIP-1α (h), IP-10 (i), TNF-α (j) and IFN-γ (k) secretion in response to cytokine treatments. MCP-1 (l) and M-CSF secretion (m) following combined TNF-α and IL-1β treatments. N = 5–6/group, Data are presented as median concentrations (pg/ml) [range] are analysed using Kruskal-Wallis test with Dunn's post-hoc test, P < 0.05, P < 0.01, P < 0.001, *P < 0.0001 vs untreated. As the ciGEnCs were shown to mainly produce and secrete only two (MCP-1 and sVCAM-1) out of the seven identified novel urinary biomarkers for LN20, it was hypothesised that the role of GEnCs may be related to the production and local secretion of other pro-inflammatory mediators. To address this hypothesis, changes in secretion levels of pro-inflammatory cytokines (TNF-α, IFN-γ, IL-6 and IL-10), chemokines (macrophage inflammatory protein-1 alpha (MIP-1α), IFN-γ-induced protein-10 (IP-10) and IL-8) as well as the blood cell growth factors (granulocyte-macrophage and macrophage colony-stimulating factor (GM-CSF and M-CSF)) were examined via Luminex ELISA. IL-6 secretion was significantly upregulated after IL-1β (3,479 pg/mL [1,599–5,273], p = 0.0004) and LPS stimulation (3,323 pg/mL [1,083–5,228], p = 0.0007) (Fig. 1c) compared to untreated ciGEnCs (31 pg/ml [11–164]). A trend was observed for increased IL-8 secretion after combined cytokine treatment (All; 6,127 pg/mL [3,664–6,255], p = 0.066) compared to untreated cells (623 pg/mL [449–1,503]) (Fig. 1d). Combined cytokine treatment (All: 24 pg/ml [22–24], p < 0.0001) and LPS (17 pg/ml [8–21], p = 0.009) significantly increased IL-10 secretion compared to untreated cells (0.5 pg/ml [0.3–0.7]) (Fig. 1e). The combined cytokine treatment led to secretion of significantly higher amounts of M-CSF (All: 3,461 pg/ml [946–6,220], p = 0.02) compared to untreated cells (192 pg/ml [69–2,077]) (Fig. 1f). IL-1β (2,284 pg/ml [710–2,748], p = 0.0001) and TNF-α (522 pg/ml [493–722], p = 0.02) significantly increased the secretion of GM-CSF compared to untreated cells (20 pg/ml [17–40]), as did the combined cytokine treatment (All: 1,010 pg/ml [323–1,925], p = 0.002) and LPS (790 pg/ml [231–1,882], p = 0.008) (Fig. 1g). TNF-α (222.3 pg/ml [221–224], p = 0.048), IL-1β (221.5 pg/ml [214–226], p = 0.049) and the combined cytokine treatment (All: 239 pg/ml [233–243], p = 0.0002) significantly increased the secretion of MIP-1α compared to untreated cells (209 pg/ml [207–209], as did LPS (226 pg/ml [215–230], p = 0.011) (Fig. 1h). IFN-γ (3,015 pg/ml [2,530–3,201], p = 0.008) and combination of cytokines (All: 3,222 pg/ml [3,153–3,262], p = 0.0003) significantly upregulated IP-10 secretion compared to untreated cells (2 pg/ml [2–5]), as did LPS (2,870 pg/ml [847–3,218], p = 0.013) (Fig. 1i). IFN-γ (8 pg/ml [7–9], p = 0.029), IL-1β (11 pg/ml [7–13], p = 0.011) and LPS (15 pg/ml [8–21], p = 0.0005) treatments significantly upregulated TNF-α secretion compared to untreated cells (4 pg/ml [4–4]) (Fig. 1j). The single TNF-α treatment and the combined cytokine treatment were excluded as they gave false positive results due to the presence of recombinant human TNF-α. LPS treatment (28 pg/ml [2–42], p = 0.005) induced the secretion of significantly higher amounts of IFN-γ compared to untreated cells (1 pg/ml [1–1]) (Fig. 1k). The single IFN-γ treatment and the combined cytokine treatment were excluded as they gave false positive results due to the presence of recombinant IFN-γ. Due to lack of IL-6- and VEGF-induced changes in the mRNA expression levels of MCP-1, VCAM-1, IL-6 and IL-8 compared to untreated ciGEnCs (data not shown), ciGEnC pro-inflammatory protein secretion in IL-6- and VEGF-treated conditioned media was not tested in ELISA assays (the 96-well plate capacity of the Luminex assay also limited the amount of conditioned media tested to those that were expected to exhibit the most meaningful changes in protein secretion compared to the untreated ciGEnCs). However, IL-6 and VEGF were included in the combined cytokine treatments (All). The combined effect of TNF-α and IL-1β in MCP-1 and M-CSF secretion was also specifically tested. TNF-α together with IL-1β statistically significantly increased MCP-1 secretion (2,561 pg/ml [2,055–3,449], p = 0.02) compared to untreated ciGEnCs (1,300 pg/ml [916–2,401]) (Fig. 1l). TNF-α alone significantly increased M-CSF secretion (446 pg/ml [333–975], p = 0.02) compared to untreated ciGEnCs (187 pg/ml [136–304]) and combined TNF-α and IL-1β treatment further increased M-CSF secretion (629 pg/ml [446–1,354], p = 0.001) (Fig. 1m). 24-hour serum treatments induce changes in mRNA expression levels of pro-inflammatory genes Treatment of ciGEnCs for 4 h with sera from JSLE- and healthy control (HC) patients did not show any significant differences between groups (data not shown). At 24 h, JSLE serum (0.002 [0.0007–0.003], p = 0.03) significantly increased IL-6 mRNA expression compared to HC (0.001 [0.0006–0.002]) (Fig. 2a). A trend was also observed for increased JSLE sera-induced IL-1β mRNA expression (0.003 [0.0008–0.006], p = 0.07) compared to HC (0.002 [0.0008–0.004]) (Fig. 2k). Furthermore, a trend was observed for reduced IP-10 mRNA expression levels after JSLE serum treatments (0.00005 [0.000004–0.005], p = 0.055) compared to HC sera (0.0001 [0.00003–0.00080]) with the mRNA in both groups being, in general, lowly expressed (Fig. 2g). Effect of 24 h ciGEnC treatments with 5% JSLE or HC sera in pro-inflammatory gene mRNA expression. Changes in IL-6 (a), IL-8 (b), IL-10 (c), M-CSF (d), GM-CSF (e), MCP-1 (f), VCAM-1 (g), MIP-1α (h), IP-10 (i), TNF-α (j), IL-1β (k) and NF-κB (l) mRNA expression levels. Data presented as median ΔΔCt [range] are analysed with Mann-Whitney test. P </= 0.05 vs HC N = 10–20/group. When JSLE serum was categorised as either active (renal BILAG A/B) or inactive LN (renal BILAG D/E), a trend was observed for decreased IP-10 mRNA levels after inactive LN serum treatments (0.00005 [0.00002–0.0001], p = 0.06) compared to HC (0.0001 [0.00003–0.0008]) but no significant difference was demonstrated between HC and active LN serum treatments (Fig. 3g). Similarly, TNF-α mRNA expression levels were statistically significantly reduced after inactive LN serum treatments (0.0003 [0.0001–0.000901], p = 0.04) but not after active LN serum treatments when compared to HC (0.0008 [0.0004–0.002]) (Fig. 3j). A trend was observed for higher IL-10 mRNA expression after active LN serum treatments (0.0003 [0.00002–0.003]) compared to the inactive sera (0.00005 [0.00002–0.0001]) although these differences were not statistically significant (p = 0.08) (Fig. 3c). Effect of 24 h ciGEnC treatments with 5% active and inactive LN or HC sera in pro-inflammatory gene mRNA expression. Changes in IL-6 (a), IL-8 (b), IL-10 (c) M-CSF (d), GM-CSF (e), MCP-1 (f), VCAM-1 (g), MIP-1α (h), IP-10 (i), TNF-α (j), IL-1β (k) and NF-κB (l) mRNA expression levels. Data presented as median ΔΔCt [range] are analysed with Kruskal-Wallis with Dunn's post-hoc test. P </= 0.05. N = 10/group. 24-hour serum treatments induce changes in the secretion levels of IL-6 The 24 h JSLE serum treatments were able to promote an increase in the mRNA expression of IL-1β and IL-6. For this reason, we also investigated the effect of serum treatments in IL-1β and IL-6 secretion by the ciGEnCs. IL-1β and IL-6 in serum samples were also tested to ensure whether potential presence of these cytokines in the ciGEnC conditioned media could be attributed to ciGEnC production. IL-1β presence in serum samples was relatively low (frequently below the level of detection for the assay) and did not differ among HCs, active and inactive LN patients (Fig. 4a). Furthermore, when ciGEnCs were treated with 5% sera from LN patients, IL-1β levels secreted by the cells were below the level of detection for the assay (except for 1 active disease sera sample) (Fig. 4b). Effect of 24 h ciGEnC treatments with 5% active and inactive LN or HC sera in IL-1β and IL-6 secretion. IL-1β levels in active and inactive LN patient and HC sera (a) and IL-1β levels in ciGEnC conditioned media (b). IL-6 levels in active and inactive LN patient and HC sera (c) and IL-6 levels in ciGEnC conditioned media (d). Data presented as median concentrations (pg/ml) [range] are analysed using Kruskal-Wallis test with Dunn's post-hoc test. P </= 0.05, *P < 0.01 vs HC or vs untreated ciGEnC conditioned media. N = 6/group. On the contrary, a trend was observed for higher IL-6 levels in active LN sera (62 pg/ml [21–165], p = 0.054) compared to those of HC sera (13 pg/ml [13–40]) whereas inactive LN sera levels of IL-6 (52 pg/ml [1–267]) were closer to those of active LN sera (Fig. 4c). The active LN (572 pg/ml [390–741], p = 0.01]) and the inactive LN (647 pg/ml [352–792], p = 0.007) serum treatments promoted a statistically significant increase in IL-6 secretion by ciGEnCs compared to untreated ciGEnCs (263 pg/ml [22–314]); the HC serum treatments (471 pg/ml [345–698]) were not able induce as high IL-6 levels as the JSLE sera. Inflammatory protein release is not associated with increased cellular death The majority of ciGEnCs remained viable (Anx V−/PI−) after 24 h treatment with cytokines and LPS similarly to the untreated ciGEnCs (Supplementary Fig. 1). No significant changes were observed in the levels of apoptotic (Anx V+/PI−) and late apoptotic/necrotic (Anx V+/PI+) ciGEnCs among the different treatments (Supplementary Fig. 1). Inflammatory mediators induce cell adhesion and co-stimulatory molecule expression in ciGEnCs The endothelium is responsible for the recruitment and activation of leukocytes to the glomerular space by upregulating expression of cellular adhesion and co-stimulatory molecules. The role of TNF-α, IL-1β, IL-13, IFN-γ and LPS was therefore assessed in modulating VCAM-1, intercellular adhesion molecule-1 (ICAM-1), programmed death-ligand 1 (PD-L1) and inducible co-stimulator-ligand (ICOS-L) surface expression. TNF-α and IL-13, alone or combined, had no significant effect on VCAM-1 and ICAM-1 surface expression at 4 h (data not shown) in contrast to 24 h, where TNF-α (15 [7–86], p = 0.008) and more robustly the combination of IL-13 and TNF-α (54 [11–157], p = 0.0007) significantly upregulated surface VCAM-1 compared to untreated ciGEnCs (3 [0–5]) (Fig. 5a). At 24 h, TNF-α significantly upregulated surface expression of the constitutively expressed ICAM-1 (19,789 [13,348–25,897], p = 0.009) (Fig. 5c) compared to untreated cells (6,609 [3,984–9,632]) but IL-13 had no effect on ICAM-1. IL-1β and LPS treatments had an effect predominantly on ICAM-1. IL-1β and LPS significantly increased surface ICAM-1 at 4 h (data not shown) compared to untreated cells, and this effect was sustained and further enhanced after 24 h (IL-1β: (6,562 [5,953–9,673], p = 0.004), LPS: (7,997 [5,497–12,334], p = 0.002) compared to untreated cells (1,727 [1,339–3,161) (Fig. 5d). No surface VCAM-1 upregulation by IL-1β, IFN-γ and LPS was observed at 24 h (untreated ciGEnC measurements were lower than those of the isotype control and therefore have been assigned 0-values) (Fig. 5b). E-/P-selectin surface expression was not modified by cytokine treatments (Supplementary Fig. 2). IFN-γ had no prominent effect on ICAM-1 or VCAM-1 but significantly increased the surface expression of PD-L1 (IFN-γ: 3,549 [2,199–3,828], p = 0.02), a negative T-cell co-stimulatory molecule21, after 24 h compared to untreated ciGEnCs (1,763 [793–2,067]) (Fig. 5e). Finally, surface ICOS-L expression was significantly increased only at 24 h by IL-1β (380 [365–401], p = 0.04) compared to untreated ciGEnCs (55 [52–59]) (Fig. 5f). VCAM-1, ICAM-1, PD-L1 and ICOS-L surface expression following 24-hour stimulation with cytokines and LPS. (a) 24-hour VCAM-1 surface expression following IL-13 and TNF-α treatments. (b) 24-hour VCAM-1 surface expression following IL-1β, IFN-γ and LPS treatments. (c) 24-hour ICAM-1 surface expression following IL-13 and TNF-α treatments. (d) 24-hour ICAM-1 surface expression IL-1β, IFN-γ and LPS treatments. (e) 24-hour PD-L1 surface expression following TNF-α, IL-1β, IL-13, IFN-γ and LPS treatments. (f) 24-hour ICOS-L surface expression following TNF-α, IL-1β, IL-13, IFN-γ and LPS treatments. N = 4–6/group. Data presented as median [range] are analysed using Kruskal-Wallis test with Dunn's post-hoc test. P </= 0.05, P < 0.01, P < 0.001. N = 4–6/group. TNF-α promotes neutrophil adhesion to ciGEnCs Adhesion of human neutrophils on ciGEnCs was tested following 24 h incubation of ciGEnCs with TNF-α and IL-13 alone, or in combination. Following 24 h incubation, the numbers of neutrophils bound on the ciGEnCs were counted (Fig. 6a–d). TNF-α ciGEnC treatment promoted the adhesion of a statistically significantly higher number of neutrophils ([90 cells [71–142], p = 0.027) (Fig. 6b) compared to untreated ciGEnCs (52 cells [34–60]) (Fig. 6a, 6e). The effect of the combined TNF-α and IL-13 treatments did not differ from that of TNF-α alone and led to ciGEnC adhesion of a statistically significantly higher number of neutrophils ([99 cells [69–151], p = 0.03), compared to the untreated ciGEnCs (52 cells [34–60]) (Fig. 6d, 6e). Treatments of ciGEnCs with IL-13 [46 cells [23–61]) did not increase neutrophil adhesion compared to untreated ciGEnCs (52 cells [34–60]) (Fig. 6c, 6e). Neutrophil adhesion assay following 24 h stimulation of ciGEnCs with TNF-α and IL-13. (a) Neutrophil adhesion on untreated ciGEnCs. (b) Neutrophil adhesion on TNF-α-treated ciGEnCs. (c) Neutrophil adhesion on IL-13-treated ciGEnCs. (d) Neutrophil adhesion on TNF-α + IL-13-treated ciGEnCs. White arrow on (a) indicates neutrophil. (e) Graph of statistical analysis of neutrophil adhesion assay. Data presented as median cell number [range] are analysed using Kruskal-Wallis test with Dunn's post-hoc test. P </= 0.05 vs untreated. N = 6/group. Scale bars: 200 μm. Pro-inflammatory protein release is occurring through NF-κB and STAT1 induction in ciGEnCs In order to determine whether the effects observed in ciGEnCs following cytokine and LPS treatments are mediated through the NF-kB pathway, we initially used an immunoblot assay to determine nuclear factor of kappa light polypeptide gene enhancer in B-cells inhibitor, alpha (Iκ-Bα) degradation, as Iκ-Bα is the cytoplasmic inhibitor of NF-κΒ that prevents its nuclear translocation22. At 30 min, it was mainly IL-1β and, to a lesser extent, TNF-α, which were able to induce Iκ-Bα degradation (Fig. 7a). Western blotting for Iκ-Bα protein expression, 20x immunofluorescence images of ciGEnCs and diagrams of statistical analysis for NF-κB, STAT-1 and STAT-2 activation and nuclear translocation, after 30 minutes of stimulation with cytokines and LPS. (a) Representative image and graph of Iκ-Bα and beta-actin Western blots (blots have been cropped), following 30-minute stimulation. Data presented as (Mean +/− SEM) in ratio diagrams are analysed using Friedman test with Dunn's post-hoc test, P < 0.05 vs untreated. (b) Treatments of ciGEnCs with TNF-α, IL-1β and LPS for 30 minutes for NF-κB nuclear translocation. (c) Treatments of ciGEnCs with IFN-γ for 30 minutes for STAT-1 nuclear translocation. (d) Treatments of ciGEnCs with IFN-γ for 30 minutes for STAT-2 nuclear translocation. Data (r-values for nuclear co-localisation of DAPI and A488 or A568 for each treatment group) representative of three similar experimental repeats are presented as box and whisker plots and are analysed using Kruskal-Wallis test with Dunn's post-hoc test. ***P < 0.0001 vs untreated. Scale bars: 100 μm. Nuclear translocation of NF-kB and of signal transducer and activator of transcription (STAT)-1 and -2 (STAT-1 and STAT-2) was tested via immunofluorescence to determine pathway activation. TNF-α and IL-1β promoted NF-κB nuclear translocation at 30 min with IL-1β displaying a more robust effect than TNF-α, whereas LPS only induced low levels of NF-κB activation (Fig. 7b). IFN-γ induced STAT-1 nuclear translocation at 30 min (Fig. 7c) while it induced only moderate to low STAT-2 activation (Fig. 7d). LN is one of the main and most significant complications of JSLE. LN affects the majority of JSLE patients (approx. 80%)5,6,7, and can cause severe renal damage3 and GFB impairment4. This study aimed to investigate the potential role of the human glomerular endothelium in LN inflammation using a cytokine-based in vitro model of LN. The ciGEnCs were treated with cytokines previously found by our laboratory and by other studies to be elevated in plasma and/or serum samples of JSLE patients with LN (IFN-α23, IFN-γ24, TNF-α24,25, IL-1β24, IL-624, IL-1324 and VEGF24,26) or shown by other studies to be expressed in the glomerular renal tissue of JSLE or adult patients with LN (IFN-γ27), or to have certain polymorphisms associated with JSLE (IL-1β27,28). The potential effect of bacterial endotoxin inflammation was also examined via 1 μg/ml LPS treatment, which, in murine models of LN and in human endotoxemia, is known to play a critical role in LN exacerbation after a bacterial infection28,29,30,31. Our findings indicate that in line with in vitro models using HUVECs17,32,33 and HBMECs34,35,36, upon activation with TNF-α, IL-1β, IL-13, IFN-γ and with LPS, ciGEnCs significantly increase the production of key pro-inflammatory proteins. These included: cytokines, chemokines, blood cell growth factors, adhesion molecules and T-cell co-stimulatory molecules. All proteins demonstrated in this study to be upregulated in human ciGEnCs have been previously shown to be involved in human LN or other types of human renal disease and are summarised on Table 1. Table 1 Proteins produced by activated ciGEnCs and their involvement in renal disease. Single TNF-α and IL-1β treatments upregulated MIP-1α and GM-CSF and combined TNF-α and IL-1β treatment significantly increased MCP-1 secretion. LPS also had a prominent effect in MIP-1α and GM-CSF secretion. MCP-1 promotes infiltration by inflammatory dendritic cells in LN37 and MIP-1α can activate and attract human granulocytes, macrophages and monocytes38 whereas GM-CSF promotes the maturation and differentiation of macrophages, neutrophils, eosinophils and basophils39. IL-1β and LPS also upregulated IL-6 and TNF-α secretion. IL-6 promotes B- and T-cell differentiation40 whereas TNF-α is a prototypical pro-inflammatory cytokine41 affecting a broad variety of cell types, including the human endothelium. IFN-γ increased the secretion of IP-10, IL-10 and TNF-α, as did LPS. IP-10 mediates T-cell accumulation to the inflamed tissues42 whereas IL-10 is involved in autoantibody production43. The combined cytokine treatment, which could be mainly attributed to TNF-α, IL-1β and IFN-γ led to increased amounts of secreted M-CSF as the other cytokines had minimal effects individually. When the combined effect of the prototypical pro-inflammatory cytokines -TNF-α and IL-1β- was specifically tested, the ciGEnCs were found to significantly upregulate M-CSF secretion compared to the untreated ciGEnCs. M-CSF promotes the proliferation, differentiation, and survival of monocytes and macrophages44. LPS had a prominent effect on secretion of IFN-γ which is involved in anti-viral immunity45. TNF-α and IL-1β appeared to increase secretion of IL-8, a potent neutrophil chemokine46 although not to a level of statistical significance tested. These data suggest that GEnCs secrete mediators into the environment that induce the recruitment, activation and maturation of inflammatory leukocytes leading to perpetuation of the inflammatory response. Two important adhesion molecules were found to be upregulated by the ciGEnCs following pro-inflammatory stimulation; VCAM-1 and ICAM-1. VCAM-1 is involved in the binding of lymphocytes, monocytes, eosinophils, and basophils but not neutrophils to the vascular endothelium47 whereas ICAM-1 promotes all leukocytes' binding to the endothelium48. In this study, IL-13 had an effect exclusively on soluble and surface VCAM-1 expression with the latter being promoted by IL-13 in combination with TNF-α. ICAM-1 expression was significantly increased by TNF-α, IL-1β and LPS. 24 h ICAM-1 upregulation by TNF-α, IL-1β and LPS has also been confirmed in HUVECs49,50. Similar to the human ciGEnCs and in contrast to other types of human endothelial cells (carotid, coronary), lack of 24 h IL-1β- and LPS-mediated upregulation of surface VCAM-1 has been previously observed in HUVECs and human subclavian endothelial cells50. The differential cytokine regulation of surface VCAM-1 and ICAM-1 expression could imply that certain pro-inflammatory stimuli could lead to preferential intra-renal accumulation of certain immune cell populations. Indeed, our findings from the neutrophil adhesion assay indicated that combinatorial TNF-α and IL-13 treatment of ciGEnCs had an effect similar to that of TNF-α alone and led to ciGEnC adhesion of similar numbers of neutrophils, which were significantly higher than those adhering to untreated or IL-13-treated ciGEnCs. Thus, ICAM-1 that was found in this study to be induced by TNF-α but not IL-13 (IL-13 had an effect only on VCAM-1 expression), could be the adhesion molecule predominantly responsible for neutrophil binding on ciGEnCs. This assumption is further corroborated by evidence in the literature that, in contrast to VCAM-1, ICAM-1 can promote binding of neutrophils on the endothelium48. In this study an attempt was made to assess the expression of E/P-selectin by the ciGEnCs in response to our pro-inflammatory model however no expression was seen. A recent study demonstrated that GEnCs do not require E/P-selectin for neutrophil binding due to an increased dwell time by cells at this location51. This has been confirmed both in mouse models using lupus-prone mice and in human renal biopsies from 108 patients with primary renal disease and allograft recipients52,53,54,55. PD-L1 surface expression that was upregulated at 24 h in ciGEnCs by IFN-γ treatment, can prevent the activation of CD-8+ T-cells21 and inhibit the autoreactive T-cell function56. However, after 24 h, IL-1β significantly increased surface expression of the positive CD4+ T-cell co-stimulatory molecule, ICOS-L57. Therefore, T-cell-driven kidney inflammation could be promoted by CD4+ instead of CD8+ T-cells. Incubation of ciGEnCs for 4 h and 24 h reflects acute phase inflammation58 rather than chronic inflammation. The 24 h cytokine and LPS stimulation did not significantly induce apoptotic or necrotic cell death, indicating that during acute inflammation human ciGEnCs maintain their integrity and acquire a pro-inflammatory phenotype due to cell activation and not due to cell death. Downstream activation of the central pro-inflammatory transcription factor NF-κB primarily occurred following IL-1β and TNF-α stimulation and to a lesser extent by LPS and was demonstrated by nuclear NF-κB translocation as well as by Iκ-Bα degradation. In contrast, IFN-γ was found to activate downstream STAT-1 signalling but not STAT-2 which is normally activated by IFN-α. NF-κB shRNA knock-down in human microvascular endothelial cells (HMECs) has been shown to lead to strong suppression of the expression of TNF-α-induced genes such as MCP-1, M-CSF, GM-CSF, IL-6 and IL-859. Furthermore, STAT-1 siRNA knock-down in IFN-γ-treated HUVECs has been shown to downregulate secretion of IP-10 and other T-cell related chemokines60 providing supporting evidence that these pathways are activated in endothelial cells. Thus, it could be hypothesised that in human GEnCs similar mechanisms of action could be occurring. The 24 h serum treatments did not demonstrate significant changes in the mRNA expression levels of most pro-inflammatory genes tested. However, IL-6 and IL-1β mRNA levels were increased by the JSLE sera. In addition, IL-6 secretion by the ciGEnCs underwent a pronounced increase after both active and inactive LN serum treatments. On the contrary, IL-1β secretion by ciGEnCs was not affected by either of the serum treatments. This finding suggests potential lack of inflammasome and caspase-1 activation in ciGEnCs following serum treatments which could lead to a reduction in the release of active IL-1β within the microenvironment. The limited amount of blood samples due to rarity of the disease as well as the small blood sample volumes obtained from paediatric patients did not allow for further investigation and detection of the JSLE serum factors inducing the expression IL-6 and IL-1β production. A potential candidate, however, could be autoantibodies present in the sera samples. Indeed, previous research studies using affinity purified IgG antibodies from anti-endothelial cell antibody (AECA)-positive SLE patients to treat HUVECs in vitro, demonstrated increased HUVEC-derived secretion of IL-161 and IL-662. Similar findings have also been observed in scleroderma and systemic vasculitis patients62,63. The lack of strong responses induced by the JSLE sera may be due to the immunosuppressant regimens followed by patients. Increasing the serum doses used in these experiments may be required to demonstrate an effect; however, the 5% treatments (limited due to sample volumes from paediatric patients) did not suffice to induce a strong in vitro ciGEnC pro-inflammatory response. Furthermore, the serum treatments could have had a more pronounced effect on ciGEnCs in an in vitro model in which ciGEnCs would be co-cultured with human podocytes and/or mesangial cells. Lack of crosstalk between native glomerular renal cells could be affecting the results obtained from the serum treatments. The development of a co-culture model, however, was beyond the scope of this study, the aim of which was to investigate pro-inflammatory properties of the human ciGEnCs, a type of human endothelial cells that has been little studied. There are inherent limitations with using a cell line to generate a model of human disease. However, ciGEnCs retain all the morphological and functional characteristics of the primary GEnCs and express at levels comparable to those of primary GEnCs, the majority of endothelial cell-specific markers64. These characteristics render the ciGEnCs into a reliable tool for the study of human GEnCs. Future work could focus on validating these findings using primary cells or in an in vivo murine model however this was beyond the scope of this study. In conclusion, our data imply that the glomerular endothelium could have a pivotal role in LN inflammation (Fig. 8). As demonstrated by our in vitro model, the GEnCs are not just passive bystanders but actively respond to the renal pro-inflammatory environment created in kidney disease. Potential effect of stimulated GEnCs in LN based on the in vitro findings. Under the highly inflammatory environment created within the glomeruli by the presence of TNF-α, IL-1β, IL-13 and IFN-γ but also by the potential presence of LPS, the human GEnCs will be activated to produce, secrete and express a variety of pro-inflammatory cytokines, chemokines, blood cell growth factors and adhesion molecules that will further exacerbate the renal disease by either promoting the infiltration of immune cells within the glomeruli or by activating their neighbouring renal cells, the podocytes and mesangial cells. Recombinant cytokines were purchased from Peprotech, LPS was purchased from Sigma. All antibodies are listed on Supplementary Table 1. All methods reported here were carried out in accordance with relevant guidelines and regulations of the University of Liverpool. All experimental protocols were reviewed and approved by either the North West – Liverpool East Research Ethics Committee (UK JSLE Cohort Study and Repository) (REC: 06/Q1502/77) or the University of Liverpool Committee of Research Ethics (Adult Healthy Control samples) (Research Ethics Approval Number: RETH000773). Informed consent was obtained from all subjects or, if subjects are under 18, from a parent and/or legal guardian. Human conditionally immortalised glomerular cell line culture The ciGEnCs were developed and kindly donated by Professor Moin Saleem and Dr Simon Satchell (Children's Renal Unit, Bristol Royal Hospital for Children)64. The ciGEnCs were cultured at 33 °C (with 5% CO2) in endothelial growth medium 2-microvascular (EGM-2 MV Bulletkit, CC-3202, Lonza) containing 5% Fetal Bovine Serum (FBS) and growth factors as supplied (VEGF was excluded as it has been utilised for treatments). Once cells reached 50–70% confluence they were thermoswitched at 37 °C (with 5% CO2) for seven days until complete differentiation was achieved and confirmed via light microscopy. For all experiments, a minimum of three consecutive ciGEnC passages was used for reproducibility purposes. Consecutive passages between 29 and 35 were used, as the researchers who developed and donated the ciGEnCs had previously shown that ciGEnCs can retain their primary GEnC-specific morphological and functional characteristics up to passage 4164. All changes in mRNA expression or protein expression, secretion and production levels were compared to those of the untreated ciGEnCs. The fully differentiated ciGEnCs were incubated at 37 °C for 30 mins, 4 h and 24 h with 10 ng/ml (a concentration previously used to induce pro-inflammatory changes in endothelial cells in vitro65,66,67)- of seven human recombinant cytokines (IFN-α, IFN-γ, TNF-α, IL-1β, IL-6, IL-13, VEGF), individually and in combination (All) (cytokine concentration response assays were also performed and are presented on Supplementary Fig. 4) and with 1 μg/ml of LPS. Fully differentiated ciGEnCs were treated for 4 h or 24 h with 5% (5% sera concentration matching the 5% FBS concentration in standard media used for ciGEnC culture) serum samples in medium without FBS, from JSLE patients with active LN (renal BILAG: A/B, N = 10) and inactive LN (renal BILAG: D/E, N = 10) and with 5% sera from age-matched healthy controls (N = 10). Following completion of treatments, qRT-PCR was performed as above to assess changes in the mRNA expression levels between ciGEnC groups treated with HC and LN sera. Patient and HC information are presented on Table 2. To test the effect of serum samples on IL-6 and IL-1β secretion by the ciGEnCs, an additional of N = 6 of active LN, inactive LN and HC serum samples were collected and used to treat the ciGEnCs for 24 h, at a 5% concentration (patient and HC information presented on Table 3). Table 2 Serum treatments for changes in ciGEnC pro-inflammatory gene expression. Age, gender, ethnicity, renal BILAG scores and medications for JSLE patients; age, gender and ethnicity for paediatric healthy controls (N = 10/group). Table 3 Serum treatments for changes in ciGEnC IL-1β and IL-6 secretion. Age, gender, ethnicity, renal BILAG scores and medications for JSLE patients; age, gender and ethnicity for paediatric healthy controls (N = 6/group). qRT-PCR RNA was extracted using Trizol-chloroform and in-column DNase digestion (Qiagen RNeasy Mini Kit). RNA concentrations and purity were determined using Nanodrop (ND-1000 Spectrophotometer, Thermo Fisher Scientific). 200 ng RNA was transcribed into cDNA using the AffinityScript multi-temp cDNA synthesis kit (Agilent Technologies, Cheshire, UK) as per manufacturer's instructions. qRT-PCR was performed using primers described in Supplementary Table 2 with the Brilliant III Ultra-fast SYBR QPCR mastermix kit (Agilent Technologies) as per manufacturer's instructions. mRNA expression for target genes was normalised using the mean (for one housekeeping (HK) gene) or the geometric mean (for more than one HK genes) of the following HK genes: beta-actin (ACTB), TATA-binding protein (TBP), beta-tubulin (TUBB). The ΔΔCt value was calculated as follows: $${\rm{\Delta }}{\rm{\Delta }}\mathrm{Ct}={2}^{-({\bf{Mean}}{\bf{Ct}}[{\rm{target}}{\rm{gene}}]-({\bf{Geo}}){\bf{Mean}}{\bf{Ct}}[{\rm{HK}}\mathrm{gene}(s)]}.$$ The secretion of MCP-1, sVCAM-1, IL-6, IL-8, IL-10, M-CSF, GM-CSF, MIP-1α, IP-10, TNF-α and IFN-γ was tested with single ELISAs (R&D Systems DuoSets) or a Luminex multiplex ELISA assay, utilising conditioned media from ciGEnCs treated for 24 hours with cytokines (as above) and LPS. Conditioned media from IL-6 and VEGF treatments were excluded as they induced no change in the mRNA expression of the urinary biomarkers. However, IL-6 and VEGF were still included in the combined cytokine treatment (All). For VCAM-1, ICAM-1, E-/P-selectin, PD-L1 and ΙCOS-L detection, the ciGEnCs were stimulated for 24 and then were stained for 30 min on ice in the dark with fluorescence-conjugated antibodies against VCAM-1-FITC, ICAM-1-APC, E-/P-selectin-PE, PD-L1-PE and ICOS-L-PE with appropriate isotype controls. Apoptosis was assessed using the Annexin V/PI kit (Sigma) as per manufactureur's instructions. Flow cytometry was performed using a Merck Guava Flow Cytometer and data were analysed using FlowJo 10.3 software. For the WB experiments, ciGEnCs were stimulated for 30 min, proteins were extracted using RIPA buffer and protein concentrations were determined using the Pierce™ BCA Protein Assay Kit (Thermo Fisher Scientific). 15 μg of protein was diluted in NuPAGE LDS Sample buffer (4x, Invitrogen) at a final 1x concentration with 50 μM dithiothreitol (DTT, Sigma-Aldrich). The protein samples were loaded into 50 μl 12% 10-well precast polyacrylamide gels and transferred to PVDF membranes (Bio-Rad) using the Trans-Blot Turbo Transfer System (Bio-Rad). Antibodies were used for the detection of human beta-actin (β-actin) and Iκ-Bα. Following gel transfer, membranes were blocked in TBS-T with 5% milk for 1 h, were incubated with the primary antibodies overnight at 4 °C, washed in TBS-T and incubated with secondary antibodies for 1 h (room temperature). Protein detection was achieved using enhanced chemiluminescence (ECL) reagents (Li-COR). Li-COR software was used for protein band densitometry. Due to β-actin and Iκ-Bα having a similar molecular weight, the samples were loaded and were run twice onto two different gels, which were blocked for either β-actin or Iκ-Bα (full-length images are presented on Supplementary Fig. 3). Neutrophil adhesion assay ciGEnCs were stimulated with 10 ng/ml of TNF-α and IL-13, alone or in combination, for 24 h (untreated ciGEnCs were included). Whole-blood from adult healthy controls was used to isolate human neutrophils via HetaSep (StemCell)-aided separation of white blood cells from red blood cells and subsequent Histopaque (Sigma-Aldrich) centrifugation of white blood cells. Neutrophils were re-suspended in RPMI (+10% FBS) at a concentration of 3 × 106/ml. ciGEnCs were then washed in PBS and following PBS removal 1 ml of neutrophil suspension was added onto each well. Cells were then incubated for 90 min at 37 °C. Following 90 min incubation, conditioned media containing non-adherent neutrophils were collected and neutrophils were counted using a cell counter. The cells remaining on the plate were washed in PBS and images were taken at 20x magnification. 5 random images per well were collected and the number of neutrophils adherent to the ciGEnC monolayer was counted using ImageJ software. Transcription factor activation and nuclear translocation was assessed using IF. Following 30 min incubation, cells were fixed in 4% formaldehyde for 30 mins and permeabilised using 0.4% Triton-X-100 for 10 mins. Non-specific binding was blocked using PBS-1% BSA for 1 h (room temperature). Antibodies against human NF-κΒ, STAT-1 and STAT-2 were added overnight, at 4 °C. ciGEnCs were washed three times with PBS and incubated with fluorescence-conjugated secondary antibodies and 1 μg/ml of DAPI (4′,6-diamidino-2-phenylindole), for 2 h at room temperature. Cells were visualised immediately, using the EVOS FLoid Cell Imaging Station (ThermoFisher Scientific). Image analysis to determine nuclear translocation of the signal in response to cytokine or LPS treatment was performed using Fiji (ImageJ) and particularly the Coloc2 algorithm68. For every group of treatments, the Pearson's r-values for every single cell, representing the degree of nuclear colocalisation between DAPI and A488 or A568, were obtained. Statistical analysis of all data was performed using GraphPad Prism 4.0 software. Data are expressed as median values[range]. Multiple comparisons were made using Kruskal-Wallis or Friedman non-parametric test with Dunn's post-hoc test for three or more treatment groups and Mann-Whitney non-parametric test for two treatment groups. Statistical significance was set a priori at a p value < 0.05. For the IF assay, the r-values from each treatment group were used to perform non-parametric Kruskal-Wallis statistical test (with Dunn's post-hoc test) for three or more treatment groups, or Mann-Whitney tests for two treatment groups. 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The authors would like to thank: Dr I. Poursaitidis (ICR, UK) for developing an algorithm for ImageJ/Fiji to assist with multiple image analysis of immunofluorescence experiments; Professor Moin Saleem and Dr Simon C Satchell, University of Bristol, for providing the ciGEnCs; and colleagues within the UK's 'Experimental Arthritis Treatment Centre for Children' (EATC4Children.co.uk) and in particular the Lupus Research Group for their support and comments on this work. LUPUS UK provided administrative support for the UK JSLE Cohort Study and Repository. This work was supported by the funding from Alder Hey Children's NHS Foundation Trust and Alder Hey Children's Kidney Fund, the UK's Experimental Arthritis Treatment Centre for Children (supported by Arthritis Research UK, Alder Children's NHS Foundation Trust, the Alder Hey Charity, and the University of Liverpool), and the National Institute of Health Research (NIHR) Alder Hey Clinical Research Facility for Experimental Medicine. Department of Women's and Children's Health, Institute of Translational Medicine, University of Liverpool, Liverpool, UK Paraskevi Dimou , Rachael D. Wright , Kelly L. Budge , Angela Midgley & Michael W. Beresford NIHR Alder Hey Clinical Research Facility, Alder Hey Children's NHS Foundation Trust, Liverpool, UK Matthew Peak Academic Renal Unit, University of Bristol, Bristol, UK Simon C. Satchell Department of Paediatric Rheumatology, Alder Hey Children's NHS Foundation Trust, Liverpool, UK Michael W. Beresford Search for Paraskevi Dimou in: Search for Rachael D. Wright in: Search for Kelly L. Budge in: Search for Angela Midgley in: Search for Simon C. Satchell in: Search for Matthew Peak in: Search for Michael W. Beresford in: Conceptualisation, P.D., R.D.W., M.W.B. P.D. designed all figures and drawings and performed all experiments for initial manuscript submission with support from R.D.W., A.M., M.P. and M.W.B. R.D.W. designed and performed the experiments for IL-6 and IL-1β detection in serum samples/ciGEnC conditioned media and investigation of MCP-1 and M-CSF levels in ciGEnC conditioned media after combined TNF-α/IL-1β treatments. R.D.W. and K.L.B. designed and performed the neutrophil adhesion assay. S.C.S. provided the ciGEnC cell line. P.D. wrote and edited the original draft. P.D., M.P., A.M., S.C.S., R.D.W. and M.W.B. all reviewed the original draft. P.D. edited the revised draft. R.D.W. and A.M. reviewed the revised draft. Funding was secured for this study by M.W.B. M.W.B. is the Chief Investigator of the UK JSLE Cohort Study and Repository, of which this study is part. Correspondence to Michael W. Beresford. Supplementary Figures and Tables https://doi.org/10.1038/s41598-019-44868-y	CommonCrawl
www.springer.com The European Mathematical Society Pages A-Z StatProb Collection Project talk Local normal forms for dynamical systems From Encyclopedia of Mathematics 2010 Mathematics Subject Classification: Primary: 34C20,32S65,34M25 Secondary: 32Sxx37C2537C8537C20 [MSN][ZBL] $\def\l{\lambda}$ A local dynamical system is a dynamical system (flow of a vector field, cascade of iterates of a self-map, or sometimes more involved construction) defined in an unspecifiedly small neighborhood of a fixed (rest) point. Application of local invertible self-map ("change of the variables") transforms a local dynamical system to an equivalent system. The local classification problem is to describe the equivalence classes of local dynamical systems, providing, if possible, the simplest or most convenient representative in each class. The advantage of considering the local system rests in the hope that the classification will be determined by (semi)algebraic conditions imposed on the first few Taylor coefficients of the field (map). This hope is partly justified, see Algebraic decidability of local classification problems. 1 Local dynamical systems and their equivalence 1.1 Equivalence of local dynamical systems 1.1.1 (Local) topological (orbital) equivalence of vector fields 1.2 Comparison of classifications (an overview) 1.3 Other local dynamical systems 2 Analytic, formal and smooth equivalence 2.1 Resonances 3 Poincaré-Dulac formal normal form 3.1 Analytic linearization 3.2 Smooth linearization 3.3 Topological linearization and topological reduction on the center manifold 4 Nonlinear normal forms 5 Local dynamical systems with additional structure (Hamiltonian) 6 References and basic literature Local dynamical systems and their equivalence By a local dynamical system one usually understands one of the following: a (smooth, analytic, formal) vector field $v$ defined[1] on a neighborhood $(\RR^n,0)$, $v:(\RR^n,0)\owns x\mapsto T_x(\RR^n,0)$, and vanishing at the origin, $v(0)=0$, or a (smooth, analytic, formal germ of a) invertible self-map $f\in\operatorname{Diff}(\RR^n,0)=\{$invertible maps of $(\RR^n,0)$ to itself fixing the origin, $f(0)=0\}$[2]. The "dynamics" consists in the possibility to iterate the self map, producing the cyclic group $$ f^{\circ\ZZ}=\{\underbrace{f\circ \cdots\circ f}_{k\text{ times}}\,\|\,k\in\ZZ\}\subseteq\operatorname{Diff}(\RR^n,0), $$ or a one-parametric group of the flow maps[3] $$\exp \RR v=\{\exp tv\in\operatorname{Diff}(\RR^n,0)\,\|\, t\in\RR,\ \exp[(t+s)v]=(\exp tv)\circ (\exp sv),\ \tfrac{\rd}{\rd t}\|_{t=0}\exp tv=v\} $$ with $v$ as the infinitesimal generator[4]. Equivalence of local dynamical systems Two local dynamical systems of the same type are equivalent, if there exists an invertible self-map $h\in\operatorname{Diff}(\RR^n,0)$ which conjugates them: $$ f\sim f'\iff\exists h:\ f\circ h=h\circ f', \qquad\text{resp.,}\qquad v\sim v'\iff\exists h:\ \rd h\cdot v=v'\circ h. \tag{Cj} $$ Here $\rd h$ is the differential of $h$, acting on $v$ as a left multiplication by the Jacobian matrix $\bigl(\frac{\partial h}{\partial x}\bigr)$. Obviously, the equivalent systems have equivalent dynamics: if $h$ conjugates $f$ with $f'$, it also conjugates any iterate $f^{\circ k}$ with $f'^{\circ k}$, and conjugacy of vector fields implies that their flows are conjugated by $h$: $h\circ(\exp tv)=(\exp tv')\circ h$ for any $t\in\RR$. This definition (and the whole ensuing theory) depends in the most crucial way on the regularity condition imposed on the conjugacy $h$. Even if the two local systems themselves are very regular (say, real analytic), the reasonable classification may be sometimes possible only if $h$ is of lower regularity. The following classes are usually considered: For real local dynamical systems on $(\R^n,0)$: Real analytic conjugacy with $h,h^{-1}\in \operatorname{Diff}^\omega(\R^n,0)$; Smooth conjugacy by $h\in\operatorname{Diff}^k(\R^n,0)$ with $k$ continuous first derivatives, $k=1,\dots,\infty$; the $C^\infty$-case is probably the most important; Formal conjugacy defined by a tuple $h=(h_1,\dots,h_n)$ of the formal series $\R[[x_1,\dots,x_n]]$ without the free terms and with the nondegenerate Jacobian matrix $\det(\frac{\partial h_i}{\partial x_j})\ne 0$; Topological conjugacy[5] by $h\in\operatorname{Homeo}(\R^n,0)=\operatorname{Diff}^0(\R^n,0)$. For holomorphic dynamical systems on $(\C^n,0)$: Holomorphic conjugacy by a biholomorphism $h\:(\C^n,0)\to(\C^n,0)$, $\det\rd h(0)\ne 0$; Formal conjugacy by a tuple $h=(h_1,\dots,h_n)$ of the formal series $\C[[x_1,\dots,x_n]]$; Topological classification by $h\in\operatorname{Homeo}(\C^n,0)$. A singularity (or singularity type) of a local dynamical system is a subspace of germs defined by finitely many semialgebraic constraints on the initial Taylor coefficients of the germ. Hyperbolic dynamical systems: Real self-maps tangent to linear automorphisms without modulus one eigenvalues, or vector fields whose linear part has no eigenvalues on the imaginary axis; Saddle-nodes, real self-maps having only one simple egenvalue $\mu=1$, resp., vector fields, whose linearization matrix has a simple eigenvalue $\lambda=0$; Cuspidal germs of vector fields on $(\RR^2,0)$ with the nilpotent linearization matrix $\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)$; Parabolic singularity, a holomorphic self-map $(\C^1,0)\to(\C^1,0)$, tangent to the identity with finite order: $z\mapsto z+cz^p+\cdots$, $p<+\infty$, $c\ne 0$. The classification problem for a given singularity type requires to construct a list (finite or infinite, eventually involving parameters) of normal forms, such that any local dynamical system of the given type is equivalent to one of these normal forms. A particular case of classification problems is the study of linearizability. A germ of a vector field with the Taylor expansion $v(x)=Ax+O(\\|x\\|^2)$ (resp., of a self-map with the Taylor expansion $f(x)=Mx+O(\\|x\\|^2)$ is linearizable (formally, smoothly or analytically), if it is conjugated to the linear vector field $v'(x)=Ax$, (resp., to the linear automorphism $f(x)=Mx$. (Local) topological (orbital) equivalence of vector fields This is the most coarse classification, which is nevertheless widely used to designate the "same topology of phase portraits". Recall that a smooth vector field $v\in\mathscr X(\R^n,0)$ near an isolated singular point $0$ defines a foliation $\mathscr F$ of the punctured neighborhood $(\R^n,0)\smallsetminus\{0\}$ by (connected, oriented) pieces of phase trajectories of $v$, partition of $(\R^n,0)\smallsetminus\{0\}=\bigsqcup \gamma_a$ into the disjoint union of connected curves $\gamma_a$ tangent to $v$. This partition is often called the "phase portrait" of $v$. Two vector fields $v$ and $v'$, generating the foliations $\mathscr F,\mathscr F'$, are said to be (orbitally) topologically equivalent, if there exists a germ $h:(\R^n,0)\to(\R^n,0)$ of the orientation preserving homeomorphism, which sends each leaf $\gamma_a$ of $\mathscr F$ into a leaf $\gamma'_{a'}$ of $\mathscr F'$ while preserving the orientation. Example. The topological equivalence is essentially complete only for germs of analytic vector fields on the plane, where the equivalence classes are characterized by the number and relative position of sectors of different types. For vector fields with a nondegenerate linear part, $v(x)=Ax+\cdots$, $\det A\ne 0$, the topological type is mainly determined by the eigenvalues $\lambda,\mu$ of the linear part, as follows. Eigenvalues of $A$ Popular name of the singularity Representative of the topological equivalence class $\lambda\mu<0$ Saddle $\dot x=x$, $\dot y=-y$ $\lambda\mu>0$, $\lambda,\mu\in\R$ Node, unstable or stable $\dot x=x$, $\dot y=y$ or $\dot x=-x$, $\dot y=-y$ $\lambda\mu>0$, $\lambda=\bar\mu\notin i\R$ Focus, unstable or stable Topologically equivalent to the respective nodes $\lambda,\mu=\pm i\omega$, $~~\omega>0$ Center or (slow) focus $\dot x=-y$, $\dot y=x$ (center) or a node as above. In line with these four Poincare types (saddle, node, focus, center), which in fact constitute four different topological types (saddle, unstable and stable node, center), a degenerate type of a saddle node is usually mentioned, which is represented by the vector field $\dot x=x^2$, $\dot y=y$. Comparison of classifications (an overview) Unlike the left-right classification of germs of maps, the classification by conjugacy changes drastically with the regularity. Probably the easiest is the topological classification of local dynamical systems: a generic local dynamical system can be topologically linearized if it is hyperbolic. The formal classification depends on more delicate properties of eigenvalues (arithmetical identities between them). In the hyperbolic case formal and $C^\infty$-smooth classifications usually coincide, while in general the situation can be more complicated in the relatively low smoothness[6]. The analytic classification depends on more subtle arithmetic nature of the eigenvales and in the resonant cases the analytic normal form cannot be finite-parametric. A parabolic self-map $f\in\operatorname{Diff}(\CC^1,0)$, $f(z)=z+a_2z^2+a_3z^3+\cdots$ (the series converges, $a_2\ne0$) is formally equivalent to the cubic self-map $f'(z)=z+z^2+az^3$, with a formal invariant $a\in\CC$, yet the analytic classification of such self-maps has a functional invariant, the so called Ecalle-Voronin modulus, which shows that the same class of formal equivalence contains continuum of pairwise analytically non-equivalent self-maps distinguished by a certain auxiliary analytic function. The phenomenon is known today under the name of the Nonlinear Stokes phenomenon, [I93], [IY]. ↑ In the formal case instead of the germ we consider a tuple of formal Taylor series in the variables $x=(x_1,\dots,x_n)$. ↑ In the formal and analytic cases one can replace the real field $\RR$ by the field of complex numbers $\CC$. ↑ As before, the "real time" $t\in\RR$ can be replaced by the "complex time" $t\in\CC$ given the appropriate context. ↑ Note that all iterates (resp., flow maps) are defined only as germs, thus the definition of the orbit $O(a)=\{f^{\circ k}(a)\}$ of a point $a\in(\RR^n,0)$ (forward, backward or bi-infinite) requires additional work. ↑ In the definition of topological conjugacy of vector fields (Cj) cannot be applied directly, since $\rd h$ is not defined. Two (smooth) vector fields $v,v'$ are topologically conjugate if their flow maps $f_t=\exp tv$ and $f'_t=\exp tv'$ are conjugate by the same $h$ for all $t\in\R$. ↑ Yu. Ilʹyashenko, S. Yakovenko, Nonlinear Stokes phenomena in smooth classification problems. Nonlinear Stokes phenomena, 235--287, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, 1993, MR1206045. Other local dynamical systems When speaking of local dynamics, one can also consder a slightly more general situation of arbitrary finitely generated group action on $\operatorname{Diff}(\RR^n,0)$ or $\operatorname{Diff}(\CC^n,0)$. Instead of just one self-map (vector field), one can consider $r$-tuples of such objects $\{f_1,\dots,f_r\}\subseteq\operatorname{Diff}(\R^n,0)$, $r=2,3,\dots$ (eventually with $r>n$), resp., $\{v_1,\dots,v_r\}$. The "dynamics" generated by such tuples is the subgroup in $\operatorname{Diff}$, generated by these objects as generators: in the discrete time case this is the image of the free group on $r$ letters, in the continuous time case the subgroup is generated by the flow maps $\exp t v_i$, $t\in\R$, $i=1,\dots,r$. This dynamics may be very complicate in general, especially in the continuous time case, thus some additional restrictions are imposed. For instance, the vector fields are assumed to be commuting, $[v_i,v_j]=0$, $i,j=1,\dots,r$. In this case the flows also commute and the dynamics reduces to a smooth action of the multidimensional real time $\R^r$ on $\operatorname{Diff}(\R^n,0)$: $(t_1,\dots,t_r)\mapsto \exp (t_1v_1+\cdots+t_r v_r)=\exp (t_1v_1)\circ \exp (t_2v_2)\circ\cdots\circ \exp(t_r v_r)$. Classification of such "multidimensional-time" local dynamical system is formally defined by the same relation: two tuples $\mathbf f=(f_1,\dots,f_r)$ and $\mathbf '=(f_1',\dots,f_r')$ are called equivalent (with the same caveat about the regularity), if there exists a conjugacy $h\in\operatorname{Diff}(\R^n,0)$, which conjugates simultaneously all generators of the two systems: $$ \mathbf f\sim\mathbf h'\iff\exists h:\ h\circ f_i=f_i'\circ h,\qquad \forall i=1,\dots,r. $$ Finally, besides vector fields, one can also consider a problem of local classification of Pfaffian forms. A Pfaffian (differential 1-)form $\xi$ on the real plane $(\R^2,0)$ defines an integrable distribution of lines (eventually with a singularity at the origin) $\{\xi=0\}$ of null spaces which is tangent to a suitable vector field $v_\xi$. If $\xi=A(x,y)\rd x+B(x,y)\rd y$ with, say, analytic germs $A,B$ having an isolated common root at the origin, then the vector field $v_\xi$ takes the form $\dot x=-B(x,y)$, $\dot y=A(x,y)$, which is also analytic. However, the distribution of null spaces is preserved if the 1-form $\xi$ is replaced by a form $u\cdot\xi$, where $u$ is the germ of a non-vanishing function. Classification of null distributions of Pfaffian 1-forms is often referred to as the orbital classification of the respective vector fields: two vector fields are orbitally equivalent if the foliations by integral curves, generated by these fields, are conjugate by a local diffeomorphism (smooth or analytic) of $(\R^n,0)$. In higher dimension, however, the integrability of the 1-form $\xi$ has to be explicitly postulated; the classification problem for such forms (modulo a scalar multiple, as before) is equivalent to classification of codimension one foliations near a singular point. Analytic, formal and smooth equivalence Technically, the local classification problem for dynamical systems is no different from the for left-right classification problem for germs of smooth maps. In particular, one would assume looking for conjugacy $h$ in the same regularity class as the objects of classification (formal conjugacy for formal germs, smooth conjugacy for smooth germs, analytic conjugacy for analytic germs). This approach usually works for the left-right equivalence (to the extent where a meaningful classification exists). However, for local dynamical systems a completely new phenomenon of divergence arises: Resonances Linearizability of local dynamical systems very strongly depends on the arithmetical properties of eigenvalues $\l_1,\dots,\l_n$ of the operator $A=\rd v(0)$ (resp., $\mu_1,\dots,\mu_n$ of $M=\rd f(0)$). A tuple[1] $\l=(\l_1,\dots,\l_n)\in\CC^n$ is said to be in additive resonance[2][3], if there exists an integer vector $\alpha=(\alpha_1,\dots,\alpha_n)\in\ZZ_+^n$ and index $j\in\{1,\dots,n\}$ such that $$ \l_j-\left<\alpha,\l\right>=0,\quad\|\alpha\|\ge 2,\qquad\text{where }\left<\alpha,\l\right>=\sum_{i=1}^n\alpha_i\lambda_i,\ \|\alpha\|=\sum_{i=1}^n \alpha_i. $$ A tuple $\mu=(\mu_1,\dots,\mu_n)\in\CC^n_{\ne 0}$ is said to be in a multiplicative resonance, if there exists an integer vector $\alpha=(\alpha_1,\dots,\alpha_n)\in\ZZ_+^n$ and index $j\in\{1,\dots,n\}$ such that $$ \mu_j-\mu^\alpha=0,\ \|\alpha\|\ge 2,\qquad\text{where }\mu^\alpha=\mu_1^{\alpha_1}\cdots\mu_n^{\alpha_n}. $$ The corresponding resonant vector monomial is the vector function $v_{j\alpha}:\R^n\to\R^n$ whose only component which is not identically zero, is the monomial $x^\alpha$ at the position $j$, $$v_{j\alpha}(x)=(0,\dots,0,\underset{j}{x^\alpha},0,\dots,0),\qquad j=1,\dots,n,\quad \|\alpha\|\ge 2.$$ The number $\|\alpha\|\ge2$ is called the order of resonance. The characteristic property of the resonant vector monomials is their commutation with the linear part: $$ [V,v_{j,\alpha}]=0,\qquad V(x)=Ax,\quad \forall (j,\alpha)\text{ resonant}. $$ A vector field (resp., self-map) is resonant, if the eigenvalues of its linear part exhibit one or more additive (resp., multiplicative) resonances. Otherwise the local dynamical system is called non-resonant. A self-map $M:\CC^1\to\CC^1$, $x\mapsto \mu x$ is (multiplicatively) resonant if and only if $\mu$ is a root of unity, $\mu^d=1$ for some $d\in\NN$. The singleton $\{\mu\}\in\CC^1_{\ne 0}$ satisfies infinitely many resonant identities of the form $\mu=\mu^{\nu d+1}$, $\nu=1,2,\dots$, of orders $d+1,2d+1,\dots$. A tuple $(\l_1,\l_2)$ is additively resonant in two different cases. If $(\l_1:\l_2)=(1:d)$ or $(d:1)$, with $d\in\NN$, then there exists only one resonance between them, $\l_2=d\cdot\l_1$ or $\l_1=d\cdot \l_2$ respectively. The corresponding germ of vector field is usually referred to as the resonant node. If the ratio $\l_1/\l_2=-\beta_2/\beta_1$, $\gcd(\beta_1,\beta_2)=1$, is a nonpositive rational number, then the corresponding identity $\left<\beta,\l\right>=0$ implies infinitely many additive resonance identities of the form $$ \l_j=\l_j+\nu\left<\beta,\l\right>,\qquad \nu=1,2,\dots $$ of orders $\nu\|\beta\|$. In particular, if one of the numbers vanishes, say, $\l_1=0$, the resonant identities are all of the form $\l_j=\l_j+\nu\l_1$ for all $\nu$ and $j=1,2$. If $\|\beta\|>1$, the corresponding singularity is called a resonant saddle, otherwise the standard name is the saddle-node. Poincaré-Dulac formal normal form The central result on the formal classification of local dynamical systems is the Poincaré-Dulac theorem [IY, Sect. 4], [A83, Ch. V]. It claims that any vector field (resp., self-map) is formally equivalent to a formal vector field (resp., self-map) which contains only resonant monomials. $$ v\underset{\text{form.}}{\sim} v'=Ax+\sum_{(j,\alpha)\text{ res. for $A$}} c_{j\alpha}v_{j\alpha},\qquad\text{res.,}\qquad f\underset{\text{form.}}{\sim} f'=Mx+\sum_{(j,\alpha)\text{ res. for $M$}} c_{j\alpha}v_{j\alpha},\qquad c_{j\alpha}\in\CC. $$ In particular, a non-resonant vector field (self-map) is formally linearizable[4]. By definition of the resonance monomials, $v$ is in the Poincare-Dulac formal normal form if it commutes with the linear part $V(x)=Ax$, $[v,V]=0$, ditto for the maps. This allows to extend the notion of a normal form for $C^\infty$-smooth fields (self-maps). It is important to notice that if the eigenvalues satisfy a unique identity $\left<\alpha,\l\right>=0$, then the normal forms are integrable in quadratures: the equation for the (unique) resonant monomial $u(x)=x^{\alpha}$ separates, $\frac{\rd}{\rd t}u=u\,F(u)$, where $F$ is a formal series in one variable $u$; this equation can be integrated. The remaining equations all take the form $\frac{\rd x_i}{x_i\rd t}=\l_i(1+G_i(u))$ with formal series $G_i$ and separated variables. For multi-resonant tuples this is no more the case. Analytic linearization Convergence of the series bringing a local dynamical system to its Poincaré-Dulac normal form is primarily depending on the relative position of the eigenvalues and the imaginary axis (resp., the unit circle). The case where all eigenvalues $\l_1,\dots,\l_n$ of the linear part $A=\rd v(0)$ are to one side of the imaginary axis[5] (resp., all eigenvalues $\mu_i$ of $M=\rd f(0)$ are all inside the unit circle or all outside of it) is referred to as the Poincaré domain. For instance, a self-map $f:(\CC^1,0)\to(\CC^1,0)$ with the multiplicator $\mu=\rd f(0)\in\CC_{\ne 0}$ belongs to the Poincare domain if $\|\mu\|\ne 1$; a vector field on the plane is in the Poincare domain if the ratio of the eigenvalues $\frac{\l_1}{\l_2}$ is not zero or negative. The only possible additive resonance in the Poincare domain is the "nodal case" $(\l_1:\l_2)=(1:d)$. The corresponding normal form is polynomial, $$ \dot x=\l x,\quad \dot y=d \l y+cx^d,\qquad \l,c\in\R,\ \N\owns d\geqslant 2. $$ Note that this normal form is integrable in quadratures. In the Poincare domain the series bringing the local dynamical system to its Poincare-Dulac normal form, always converges. The complementary case, where eigenvalues of the linear part cannot be separated by a line from the origin (resp., by a circle from $1$), is referred to as the Siegel domain. One-dimensional self-maps are in the Siegel domain, if $\|\mu\|=1$ (resonant if $\mu$ is a root of unity, otherwise non-resonant). Two-dimensional vector fields are in the Siegel domain, if the ratio of eigenvalues $\l_1/\l_2$ is zero or negative number (resonance occurs if this number is zero or negative rational, otherwise the field is non-resonant). Convergence of the formal series linearizing analytic germs in the Siegel domain depends on certain quantitative conditions on the arithmetic nature of the (non-resonant tuples of) eigenvalues. Very roughly, if the (nonvanishing) values of the small denominators, the differences $\delta_k=\inf_{j,\ \|\alpha\|=k}\|\l_j-\left<\alpha,\l\right>\|$ (resp., $\delta_k=\inf_{j,\|\alpha\|\le k}\|\mu_j-\mu^\alpha\|$), which may decrease to zero as $k\to+\infty$, decrease not too fast (the so called Diophantine case), then the formal conjugacy is convergent. On the contrary, if the small denominators $\delta_k$ decrease anomalously fast (the so called Liouvillean case), the normalizing series in general diverge. The sufficient decay rate of the small denominators $\delta_k\to0$ was first discovered by C. L. Siegel[6] and later improved significantly by A. D. Brjuno [Br]. The sufficient Brjuno condition for self-maps $(\CC^1,0)\to(\CC^1,0)$ was shown to be sharp by J.-C. Yoccoz[7], see Diophantine conditions in dynamics. The Diophantine conditions for convergence/divergence to be imposed on the multiplicator $\rd f(0)\in\CC_{\ne 0}=\mu=\exp 2\pi i \theta$, $\theta\in\RR\smallsetminus\QQ$, are most easily formulated in terms of the expansion of rotation angle $\theta$ in the continued fraction, more precisely, in terms of the growth rate of partial denominators, $$ \theta = q_0 + \cfrac{1}{q_1 + \cfrac{1}{q_2 + \cfrac{1}{q_3 + \cdots}}}, \qquad q_0,q_1,\dots \in\NN. $$ The Siegel condition requires that the denominators' growth is bounded asymptotically by the uniform estimate $\log q_{n+1}=O(\log q_n)$ as $n\to\infty$. The Brjuno condition is equivalent to the summability of the series $$ \sum_{n=0}^\infty \frac{\log q_{n+1}}{q_n}<+\infty.\tag{Br} $$ The necessary condition for convergence, due to Cremer (1938), claims that if $$ \sup_{n\ge 0}\frac{\log q_{n+1}}{q_n}=\infty,\tag{Cr} $$ then there exists a non-linearizable analytic self-map with the multiplicator $\mu=\exp 2\pi i\theta$. For any number violating the Brjuno condition J.-C. Yoccoz constructed in 1987 an example of a quadratic self-map which is non-linearizable. ↑ We use the multi-index notation here. ↑ [A83, Chapter V], [IY, Sect. 4] ↑ Cf. with small denominators. ↑ The linear objects are equivalent to their Jordan normal forms. ↑ By a linear change of the independent variable $t\mapsto \sigma t$ one can bring to such form any vector field such that the convex hull of eigenvalues $\l_1,\dots,\l_n$ does not contain zero. ↑ C. L. Siegel, J. K. Moser, Lectures on celestial mechanics, Die Grundlehren der mathematischen Wissenschaften, Band 187. Springer-Verlag, New York-Heidelberg, 1971, MR0502448 ↑ J.-C. Yoccoz, Théorème de Siegel, nombres de Bruno et polynômes quadratiques. Petits diviseurs en dimension 1. Astérisque No. 231 (1995), 3–88, MR1367353. Smooth linearization If the local dynamical system $v(x)=Ax+\cdots$ (resp., $f(x)=Mx+\cdots$) is real and exhibits no additive (resp., multiplicative) resonances until sufficiently high order $N\le+\infty$, then this system admits a $C^n$-smooth linearization of smoothness order $n$ which grows to infinity together with $N$. The key assumption used in the proof of this theorem is the hyperbolicity: the non-resonant linear part $A$ (resp., $M$) cannot have eigenvalues on the imaginary axis, $\operatorname{Re}\l_i\ne 0$[1] for all $i=1,\dots,n$ (resp., on the unit circle, $\|\mu_i\|\ne 1$ for all $i=1,\dots,n$[2]). This result is known as the Sternberg[3]-Chen[4] theorem, see [H, Ch. IX, Sect. 12-14]. The order $N(n)$ as a function of the required smoothness $n$ grows no faster than linearly: it is sufficient to verify absence of resonances till order $N\le C\cdot n$, where the constant $C$ depends on the relative position of eigenvalues and the imaginary axis (resp., the unit circle) and can be expressed[5][6] in terms of the hyperbolicity measure, the ratio $$\frac{\max_i\|\l_i\|}{\min_i\|\operatorname{Re}\l_i\|},\qquad\text{resp.,}\qquad\frac{\max_i\|\mu_i\|}{\min_i\bigl\|\|\mu_i\|-1\bigr\|}.$$ ↑ Indeed, if $\l$ is an imaginary eigenvalue, then $\l'=\bar\l$ is also an imaginary eigenvalue, which implies that either $\l=0$, or $\l+\l'=0$, in both cases implying infinitely many resonances. ↑ Violation of this condition produces infinitely many resonances via the identity $\mu\mu'=1$, where $\mu'=\bar\mu$ is another eigenvalue. ↑ S. Sternberg, On the structure of local homeomorphisms of euclidean $n$-space, II. Amer. J. Math. 80 (1958) 623–631, MR0096854 ↑ Chen, Kuo-Tsai, Equivalence and decomposition of vector fields about an elementary critical point, Amer. J. Math. 85 (1963) 693–722, MR0160010. ↑ V. S. Samovol, Equivalence of systems of differential equations in the neighborhood of a singular point (Russian), Trudy Moskov. Mat. Obshch. 44 (1982), 213–234, MR0656287 ↑ G. R. Belitsky, Equivalence and normal forms of germs of smooth mappings, Russian Math. Surveys 33 (1978), no. 1, 107--177, MR0490708 Topological linearization and topological reduction on the center manifold The (real) topological classification of hyperbolic local dynamical systems is especially simple. If the linear part $A=\rm dv(0)$ has no eigenvalues on the imaginary axis, then $v$ is topologically equivalent to the "standard saddle" vector field $s(x)$, $$ s(x)=(s_1,\dots,s_n(x)):\quad s_i(x)=x_i,\ i=1,\dots, k,\ s_i(x)=-x_i,\ i=k+1,\dots,n,\qquad\text{for some }k, \ 0\le k\le n. $$ This statement is known as the Grobman-Hartman theorem (for vector fields). For self-maps the hyperbolicity condition requires that the linear part $M=\rd f(0)$ has no eigenvalues on the unit circle. Such a map is topologically equivalent to one of the "standard saddle maps" of the form $$ S(x)=(S_1(x),\dots,S_n(x)):\quad S_i(x)=\pm \tfrac12 x_i,\ i=1,\dots, k,\ S_i(x)=\pm 2 x_i,\ i=k+1,\dots,n, $$ with some $k$ and a certain combination of signs (some of them equivalent to each other). The number of different normal forms of "standard saddles" is finite, which implies the structural stability of hyperbolic local dynamical systems: a small perturbation of a hyperbolic system does not change its topological type (i.e., the topological equivalence class contains an open neighborhood of the hyperbolic system). The non-hyperbolic case is also partially covered by the so called Shoshitaishvili reduction principle [Ar83, Sect. 32C], [AR, App. C] [1]. For vector fields it takes the following form: any sufficiently smooth vector field is topologically equivalent to a product vector field $v'=(v'_h,v'_c)$ on $(\R^h,0)\times (\R^{n-h},0)$, where the field $v'_h$ on $(\R^h,0)$ is hyperbolic and all eigenvalues of the vector field $v'_c$ are on the imaginary axis. ↑ A. N. Šošitaĭšvili, The bifurcation of the topological type of the singular points of vector fields that depend on parameters. Trudy Sem. Petrovsk. Vyp. 1 (1975), 279–309, MR0478239, English translation in MR0685149. Nonlinear normal forms The Poincare-Dulac normal form is linear in the nonresonant case and integrable in the single-resonance case. For more degenerate cases the number of resonant monomials grows very fast, until the limit case $A=0$ (resp., $M=E$, the identity matrix) all monomials are resonant. Sometimes even in these very degenerate cases one can single out the "leading" nonlinear terms and use them to simplify the remaining part by suitable conjugacy. The first steps of this classification look rather simple [IY, Sect. 4, 5]. A (not identically zero analytic) vector field on the 1D-line $(\R^1,0)$ with vanishing linear part is formally and even analytically equivalent to the polynomial vector field $v(z)=z^{p+1}+az^{2p+1}$, $p=1,2,\dots$, or a rational vector field $v(z)=\frac{z^{p+1}}{1+bz^{p}}$. The natural number $p$ and the complex numbers $a$ (or $b$) are formal invariants (cannot be changed by the formal conjugacy). A holomorphic self-map $f(z)=z+a_{p+1}z^{p+1}+\cdots$ with $a_p\ne 0$ is formally equivalent to the polynomial self-map $z\mapsto z+z^{p+1}+az^{2p+1}$ or to the time one (flow) map of one of the two above vector fields. However, the formal series conjugating $f$ to its formal normal form, almost always diverge, see nonlinear Stokes phenomenon. A similar, although somewhat more involved but still polynomial formal normal form can be written for the self-maps tangent to rational rotations $f(z)=\mu z+\cdots$, $\mu=\exp 2\pi i \theta$, $\theta\in\Q$, with the same remark concerning divergence. A cuspidal singularity is a planar vector field with the linearization matrix $\bigl(\begin{smallmatrix}0&1\\0&0\end{smallmatrix}\bigr)$. Since this matrix is nilpotent, all both eigenvalues are zero and all monomials are resonant. The formal normal form in this case corresponds to the Liénard system of the differential equations[1][2] $$ \left\{ \begin{aligned} \dot x&=y, \\ \dot y&=\phi(x)+y\psi(x), \end{aligned}\right. \qquad \phi,\psi\in\C[[x]], \tag{Cs} $$ with the formal series $\phi,\psi$ in one variable without linear ($\phi$), resp., free ($\psi$) terms. In contrast with the previous problems, these series are not uniquely defined and can be changed by suitable conjugacies. One can show, either by careful estimations[3] or by elegant use of global classification of holomorphic bundles over $\C P^1$[4] that a cuspidal singularity can always be brought to an analytic formal form (Cs) by an analytic conjugacy (and then the series $\phi,\psi$ will automatically converge). Alas, the difficulties on the way of constructing nonlinear normal forms, mount very fast and no general theory in higher dimensions exists. ↑ Equivalently, one can consider the normal form $\dot x=y+a⁢(x),\ \dot y=b⁢(x)$ with formal series $a,b\in x^2\cdot\C[[x]]$. ↑ L. Perko, Differential Equations and Dynamical Systems, Springer, New York, 2001, MR1801796. ↑ E. Stróżyna, H. Żołądek, The analytic and formal normal form for the nilpotent singularity, J. Differential Equations 179 (2002), no. 2, 479–537 MR1885678 ↑ F. Loray, A preparation theorem for codimension-one foliations, Ann. of Math. (2) 163 (2006), no. 2, 709–722, MR2199230. Local dynamical systems with additional structure (Hamiltonian) 2010 Mathematics Subject Classification: Primary: 37J40 Secondary: 37J10 [MSN][ZBL] In parallel with the "general" dynamical systems, it is important to consider dynamical systems induced by special structure. For instance, one can assume that the local phase space $(\R^n,0)$ is equipped with a Riemannian metric, and consider the class of gradient vector fields, with two such fields being equivalent if there exists a local isometry conjugating these fields (or, what is equivalent in this case, their potentials). Another, much more important class consists of Hamiltonian systems (in continuous or discrete time) on even-dimensional space. Recall that a symplectic structure on an even-dimensional neighborhood $(\R^{2n},0)$ is the germ of a nondegenerate closed 2-form $\omega\in\varLambda^2(\R^n,0)$. By the Darboux theorem, in suitable local coordinates $(x_1,\dots,x_n,y_1,\dots,y_n)$ such a form looks as $$ \omega =\sum_{i=1}^n \rd y_i\land\rd x_i. $$ A local diffeomorphism $f\in\operatorname{Diff}(\R^{2n},0)$ is called symplectic, or canonical, if it preserves the symplectic structure, $f^\omega=\omega$. A germ of the vector field $v$ is called canonical, if all its flow maps preserve $\omega$. In this case the Lie derivative $L_v\omega$ vanishes identically, and thus by the homotopy formula the contraction $i_v\omega=\omega(v,\cdot)\in\varLambda^1$ must be a closed, hence exact 1-form: $$ 0=L_v\omega=i_v (\rd\omega)+\rd (i_v\omega)=\rd i_v(\omega)=\rd\big(\omega(v,\cdot)\big)\implies i_v\omega=\rd H. $$ The function $H:(\R^{2n},0)\to(\R,0)$ such that its differential $\rd H$ coincides with the contraction $\omega(v,\cdot)$ is called the Hamiltonian of the vector field $v$ preserving $\omega$. Two Hamiltonians $H,H'$ on the symplectic neighborhood $(\R^{2n},0)$ are called canonically equivalent, if there exists a canonical (symplectic) transformation $f$ such that $H\circ f=H'$. Such transformation necessarily conjugates also the corresponding Hamiltonian vector fields. The local classification problem for Hamiltonian systems is reduced therefore to the (right) classification of smooth functions by the action of symplectomorphisms, $$ H,H':(\R^{2n},0)\to(\R^1,0),\quad H\sim H'\iff\exists f\in\operatorname{Diff}(\R^{2n},0):\quad f^\omega=\omega,\ f^*H=H'. $$ The corresponding matrix classification problem was discussed here. For practical reasons, the most important case is that where the Hamiltonian vector field has only imaginary eigenvalues, that is, with the quadratic part linearly equivalent to $$ H_2=\sum_1^n \frac12\omega_i(x_i^2+y_i^2),\qquad \omega_1,\dots,\omega_n\in\R $$ (some of the frequencies may well be zero). For this classification problem, the notion of resonance has to be modified: of course, any of the $n$ pair of imaginary conjugate eigenvalues $\pm i\omega$ produces infinitely many Poincare-Dulac (additive) resonances, but all these resonant monomials are non-Hamiltonian and hence irrelevant. The resonances which correspond to Hamiltonian monomials, all have the form $$ \left<\omega,\alpha\right>=0,\qquad \omega=(\omega_1,\dots,\omega_n)\in\R^n,\ \alpha=(\alpha_1,\dots,\alpha_n)\in\Z^n_+,\ \|\alpha\|\ge 2.\tag{HR} $$ The formal normal form of Hamiltonian vector fields (an analog of the Poincare linearization theorem) claims [Ar74] that if the tuple of frequencies is non-resonant, then the Hamiltonian is formally symplectically equivalent to the series in the variables $I_i(x,y)=\tfrac12(x_i^2+y_i^2)$ only, $$ \begin{gathered} H(x,y)=\frac12\sum_1^n \omega_i(x_i^2+y_i^2)+O\bigl(\|x\|+\|y\|)\bigr)^3 \implies H\sim\sum_1^n\omega_i I_i+F(I_1,\dots,I_n),\\ I_i=I_i(x,y)=\tfrac12(x_i^2+y_i^2),\ F\in\R[[I_1,\dots,I_n]]. \end{gathered} $$ References and basic literature [sort] [I93] Yu. Ilyashenko, Nonlinear Stokes phenomena, Nonlinear Stokes phenomena, 1--55, Adv. Soviet Math., 14, Amer. Math. Soc., Providence, RI, 1993, MR1206041 [IY] Yu. Ilyashenko, S. Yakovenko, Lectures on analytic differential equations. Graduate Studies in Mathematics, 86. American Mathematical Society, Providence, RI, 2008 MR2363178 [A83] Arnold V. I., Geometrical methods in the theory of ordinary differential equations. Grundlehren der Mathematischen Wissenschaften, 250. Springer-Verlag, New York-Berlin, 1983, MR0695786 [Br] A. D. Brjuno, Analytic form of differential equations. I, II, Trans. Moscow Math. Soc. 25 (1971), 131--288 (1973); ibid. 26 (1972), 199--239 (1974) MR0377192. [H] P. Hartman, Ordinary differential equations, John Wiley & Sons, Inc., New York-London-Sydney 1964, MR0171038 [Ar74] V. I. Arnold, Mathematical methods of classical mechanics. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. MR1345386 [AI88] V. I. Arnold, Yu. I. Ilyashenko, Ordinary differential equations, Encyclopaedia Math. Sci., 1, Dynamical systems, I, 1--148, Springer, Berlin, 1988, MR0970794 [AAIS] V. I.Arnold, V. S. Afrajmovich, Yu. S. Ilʹyashenko, L. P. Shilnikov, Bifurcation theory and catastrophe theory, Encyclopaedia Math. Sci., 5, Dynamical systems, V, Springer, Berlin, 1994, MR1287421 [P] L. Perko, Differential equations and dynamical systems, Third edition. Texts in Applied Mathematics, 7. Springer-Verlag, New York, 2001. xiv+553 pp. ISBN: 0-387-95116-4, MR1801796. [AR] R. Abraham, J. Robbin, Transversal mappings and flows An appendix by Al Kelley W. A. Benjamin, Inc., New York-Amsterdam 1967, MR0240836. How to Cite This Entry: Local normal forms for dynamical systems. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Local_normal_forms_for_dynamical_systems&oldid=30971 Retrieved from "https://encyclopediaofmath.org/index.php?title=Local_normal_forms_for_dynamical_systems&oldid=30971" TeX done Dynamical systems and ordinary differential equations Several complex variables and analytic spaces Dynamical systems and ergodic theory About Encyclopedia of Mathematics Impressum-Legal	CommonCrawl
1 handwiki -- 3050 2022-11-28 01:35:50 Home Entry Topic Review Current: (225088) 2007 OR10 (225088) 2007 OR10 The content is sourced from: https://handwiki.org/wiki/Astronomy:(225088)_2007_OR10 (225088) 2007 OR10, proposed to be named Gonggong, is a likely dwarf planet in the Solar System beyond Neptune, and is a member of the scattered disc. It has a highly eccentric and inclined orbit during which it ranges from 33–101 astronomical units (4.9–15.1 billion kilometers) from the Sun. (As of 2019), its distance from the Sun is 88 AU (1.32×1010 km; 8.2×109 mi), and it is the sixth-farthest known Solar System object. 2007 OR10 is in a 3:10 orbital resonance with Neptune, in which it completes three orbits around the Sun for every ten orbits completed by Neptune. 2007 OR10 was discovered in July 2007 by American astronomers Megan Schwamb, Michael Brown, and David Rabinowitz at the Palomar Observatory, and the discovery was announced in January 2009. At 1,230 km (760 mi) in diameter, 2007 OR10 is approximately the size of Pluto's moon Charon, and is the fifth-largest known trans-Neptunian object in the Solar System. It is sufficiently massive to be gravitationally rounded, thereby qualifying for dwarf planet status. Its large mass also makes retention of a tenuous atmosphere of methane just possible, though such an atmosphere would slowly escape into space. 2007 OR10 is currently the largest known body in the Solar System without an official name, but in 2019, the discoverers hosted an online poll for the general public to help choose a name for the object, and the name Gonggong won. The winning name is derived from Gonggong, a Chinese water god responsible for chaos, floods and the tilt of the Earth. 2007 OR10 is red in color, likely due to the presence of organic compounds called tholins on its surface. Water ice is also present on its surface, which hints at a brief period of cryovolcanic activity in the distant past. 2007 OR10 rotates slowly compared to other trans-Neptunian objects, which typically have rotation periods less than 12 hours, which may be due to its natural satellite, provisionally designated S/2010 (225088) 1. tholins gravitationally dwarf planet Subjects: Astronomy & Astrophysics Entry Collection: HandWiki Revision: 1 time (View History) 1.1. Discovery 2007 OR10 was discovered using the Samuel Oschin telescope at Palomar Observatory. https://handwiki.org/wiki/index.php?curid=1192500 2007 OR10 was discovered by American astronomers Megan Schwamb, Michael Brown and David Rabinowitz on 17 July 2007.[1] The discovery was part of the Palomar Distant Solar System Survey, a survey conducted to find distant objects in the region of Sedna, beyond 50 astronomical unit\|AU (7.5×109 km; 4.6×109 mi) from the Sun, using the Samuel Oschin telescope at Palomar Observatory near San Diego, California .[2][3][4] The survey was designed to detect the movements of objects out to at least 1,000 AU from the Sun.[4] Schwamb identified 2007 OR10 by comparing multiple images using the blinking technique.[3] In the discovery images, 2007 OR10 appeared to move slowly, suggesting that it is a distant object.[3][5] The discovery was part of Schwamb's doctoral thesis. At that time, Schwamb was a graduate student of Michael Brown at the California Institute of Technology.[3][6] 2007 OR10 was formally announced in a Minor Planet Electronic Circular on 7 January 2009.[7] It was then given the provisional designation 2007 OR10 because it was discovered during the second half of July 2007.[7] The last letter and numbers of its designation indicate that it is the 267th object discovered during the latter half of July.[8] (As of April 2017), it has been observed 230 times over 13 oppositions, and has been identified in two precovery images, with the earliest image taken by the La Silla Observatory on 19 August 1985.[1][9] 1.2. Naming (As of 2019), 2007 OR10 is the largest known object in the Solar System without an official name.[10] Initially after the discovery of 2007 OR10, Brown nicknamed the object "Snow White" for its presumed white color based on his assumption that it may be a member of the icy Haumea collisional family.[11][12] The nickname also fitted because, by that time, Brown's team had discovered seven other large trans-Neptunian objects which were collectively referred to as the "seven dwarfs":[13] Quaoar in 2002, Sedna in 2003, Haumea, Salacia and Orcus in 2004, and Makemake and Eris in 2005. However, 2007 OR10 turned out to be very red in color, comparable to Quaoar, so the nickname was dropped.[5][11] On 2 November 2009, two years after its discovery, the Minor Planet Center assigned the minor planet number 225088 to 2007 OR10.[9] Upon the discovery and announcement of 2007 OR10, Brown did not consider naming it, as he regarded it to be an unremarkable object, despite its large size.[12][14] Brown later declared in 2011 that he now had enough information to justify giving it a name, in consideration of the discovery of water ice and the possibility of methane on its surface, which made it noteworthy enough to warrant further study.[6] Following the Kepler spacecraft's large revision of 2007 OR10's size in 2016, Schwamb justified that 2007 OR10 was eligible for naming, an acknowledgement of its large size and that the characteristics of 2007 OR10 were known with enough certainty for a name to be given to reflect them.[10] In 2019, the discoverers of 2007 OR10 hosted an online poll for the general public to choose between three possible names: Gonggong (Chinese), Holle (German), and Vili (Norse). These were selected by the discoverers in accordance with the International Astronomical Union's (IAU's) minor planet naming criteria, which state that objects with orbits like that of 2007 OR10 must be given names related to mythological figures that are associated with creation.[15][16] The three options were chosen because they were associated with water, ice, snow, and the color red—all characteristics of 2007 OR10—and because they had associated figures that could later provide a name for 2007 OR10's satellite.[17] The satellite would not be named by the hosts of the naming poll, as this privilege is reserved for its discoverers.[15][18] On 29 May 2019, the discovery team announced Gonggong as the winning name, it having gained 46 percent of the 280,000 votes casted.[18] The name has been proposed to the IAU's Committee on Small Body Nomenclature, which is responsible for naming minor planets.[18] In Chinese mythology, the water god Gonggong is depicted as having a red-haired human head and the body of a serpent. Gonggong was responsible for creating chaos, causing flooding, and tilting the Earth, and was sent into exile.[15][18] 2. Physical Characteristics 2.1. Surface and Spectra Artist's impression of 2007 OR10 depicting its red surface color. https://handwiki.org/wiki/index.php?curid=1663931 The surface of 2007 OR10 has an albedo (reflectivity) of 0.14.[19] The surface composition and spectrum of 2007 OR10 is expected to be similar to that of Quaoar, as both objects are red in color and display signs of water ice and possibly methane in their spectra.[20][21] The reflectance spectrum of 2007 OR10 was first measured in 2011 at near-infrared wavelengths, with the Folded port InfraRed Echellette (FIRE) spectrograph on the Magellan Baade Telescope at the Las Campanas Observatory in Chile .[22] 2007 OR10's spectrum exhibits a strong red spectral slope along with broad absorption bands at wavelengths of 1.5 μm and 2 μm, meaning that 2007 OR10 reflects more light at these wavelengths.[22] Additional photometric measurements from the Hubble Space Telescope's Wide Field Camera 3 instrument display similar absorption bands at 1.5 μm,[22] which are characteristic features of water ice, a substance often found on large Kuiper belt objects.[23] The presence of water ice on the surface of 2007 OR10 implies a brief period of cryovolcanism in the distant past, when water erupted from its interior, deposited onto its surface, and subsequently froze.[24] 2007 OR10 is among the reddest trans-Neptunian objects known, especially in the visible and near-infrared.[22][25] Its red color is unexpected for an object with a substantial amount of water ice on its surface,[6][24] which are typically neutral in color, hence why 2007 OR10 was initially nicknamed "Snow White".[11][12] 2007 OR10's color implies that methane is present on its surface, although it was not directly detected in the spectrum of 2007 OR10 due to the low signal-to-noise ratio of the data.[22] The presence of methane frost would account for its red color, as a result of the photolysis of methane by solar radiation and cosmic rays producing reddish organic compounds known as tholins.[22][26] Observations of 2007 OR10's near-infrared spectrum in 2015 revealed an absorption feature at 2.27 μm, indicating the presence of methanol along with its irradiation products on its surface.[27] Methanol is expected to brighten 2007 OR10's surface, although the irradiation of water ice may account for its present dark surface.[27] 2007 OR10 is large enough to be able to retain trace amounts of volatile methane on its surface,[22] even when at its closest distance to the Sun (33.5 AU),[28] where temperatures are higher than that of Quaoar.[22] In particular, the large size of 2007 OR10 means that it is likely to retain trace amounts of other volatiles, including ammonia, carbon monoxide, and possibly nitrogen, which almost all trans-Neptunian objects lose over the course of their existence.[10][20][26] Like Quaoar, 2007 OR10 is expected to be near the mass limit at which it is able to retain those volatile materials on its surface.[6][20] 2.2. Atmosphere The presence of tholins on the surface of 2007 OR10 implies the possible existence of a tenuous methane atmosphere, analogous to Quaoar.[6][24] Although 2007 OR10 occasionally comes closer to the Sun than Quaoar, where it becomes warm enough that a methane atmosphere should evaporate, its larger mass could make the retention of methane just possible.[22] During aphelion, methane along with other volatiles would condense on 2007 OR10's surface, allowing for long-term irradiation that would otherwise result in a decrease in surface albedo.[29] The lower surface albedo would contribute to the loss of highly volatile materials such as nitrogen, as a lower albedo corresponds to more light being absorbed by the surface rather than being reflected, thus resulting in greater surface heating. Hence, the nitrogen content of 2007 OR10's atmosphere is expected to be depleted to trace amounts while methane is likely retained.[29] 2007 OR10 is thought to have had cryovolcanic activity along with a more substantial atmosphere shortly after its formation.[6][24] Such cryovolcanic activity is expected to have been brief, and the resulting atmosphere gradually escaped over time.[6][24] Volatile gases, such as nitrogen and carbon monoxide, were lost, while less volatile gases such as methane are likely to remain in its present tenuous atmosphere.[24][29] 2.3. Size Size estimates 2010 1,752 km thermal [30] 2011 1,200+300 −200 km best fit albedo [22] 2012 1,280±210 km thermal [21] −467 km thermal [31] 2013 1,290 km radiometric [32] 2016 1,535+75 2018 1,230±50 km radiometric [19] 2007 OR10 compared to the Earth and the Moon. https://handwiki.org/wiki/index.php?curid=1601838 As of 2019, 2007 OR10 is estimated to have a diameter of 1,230 km (760 mi), derived from radiometric measurements, its calculated mass, and assuming a density similar to other similar bodies.[19] This would make 2007 OR10 the fifth-largest trans-Neptunian object, after Pluto, Eris, Haumea and Makemake. 2007 OR10 is approximately the size of Pluto's moon Charon, although 2007 OR10's current size estimate has an uncertainty of 50 km (31 mi).[19] The International Astronomical Union (IAU) has not addressed the possibility of officially accepting additional dwarf planets since the acceptance of Makemake and Haumea in 2008, prior to the announcement of 2007 OR10 in 2009.[33][34] Despite not satisfying the IAU's criterion of having an absolute magnitude brighter than +1,[33][35] 2007 OR10 is large enough to be considered a dwarf planet by several astronomers.[30][32][36] Brown states that 2007 OR10 "must be a dwarf planet even if predominantly rocky", based on the 2013 radiometric measurement of 1290 km.[32] Scott Sheppard and his colleagues think that it is likely to be a dwarf planet,[36] based on its minimum possible diameter—580 km under the assumption of an albedo of 1[37]—and what was at the time the expected lower size limit of around 200 km for hydrostatic equilibrium in cold icy-rocky bodies.[36] However, Iapetus is not in equilibrium despite being 1,470 km (910 mi) in diameter, so this remains just a possibility.[38] In 2010, astronomer Gonzalo Tancredi initially estimated 2007 OR10 to have a very large diameter of 1,752 km (1,089 mi), though its dwarf planet status was unclear as there was no lightcurve data or other information to ascertain its size.[30] 2007 OR10 is too distant to be resolved directly; Brown placed a rough estimate of its diameter ranging from 1,000–1,500 km (620–930 mi), based on an albedo of 0.18 which was the best fit in his model.[22] A survey led by a team of astronomers using the European Space Agency's Herschel Space Observatory in 2012 determined its diameter to be 1280±210 km, based on the thermal properties of 2007 OR10 observed in the far infrared range.[21] This measurement is consistent with Brown's estimate. Later observations in 2013 using combined thermal emission data from Herschel and the Spitzer Space Telescope suggested a smaller size of 1142+647 −467 km, though this estimate had a larger range of uncertainty.[31] In 2016, combined observations from the Kepler spacecraft and archival thermal emission data from Herschel suggested that 2007 OR10 was much larger than previously thought, giving a size estimate of 1535+75 −225 km (954 mi) based on an assumed equator-on view and a lower estimated albedo of 0.089.[26][39] This would have made 2007 OR10 the third-largest trans-Neptunian object after Eris and Pluto, larger than Makemake (1430 km).[10][39] These observations of 2007 OR10 were part of the Kepler spacecraft's K2 mission which includes studying small Solar System bodies.[10] Subsequent measurements in 2018 revised the size of 2007 OR10 to 1230±50 km, based on the mass and density of 2007 OR10 derived from the orbit of its satellite and the discovery that the viewing direction was almost pole-on.[19] With this size estimate, 2007 OR10 is again thought to be the fifth-largest trans-Neptunian object.[19] 2.4. Mass, Density and Rotation File:Kepler 2007 OR10.ogg Based on the orbit of its satellite, the mass of 2007 OR10 has been calculated to be 1.75×1021 kg (3.86×1021 lb), with a density of 1.72±0.16 g/cm3.[19] From these mass and density estimates, the size of 2007 OR10 was calculated to be about 1,230 km (760 mi), smaller than the previous 2016 size estimate of 1,535 km (954 mi).[19] Given the mass, the 2016 size estimate of 1,535 km (954 mi) would have implied an unexpectedly low (and likely erroneous) density of 0.92 g/cm3.[19] 2007 OR10 is the fifth most massive trans-Neptunian object, after Eris, Pluto, Haumea, and Makemake.[19] It is slightly more massive and denser than Charon, which has a mass of 1.586×1021 kg (3.497×1021 lb) and a density of 1.702 g/cm3.[19][40] Due to its large size, mass, and density, 2007 OR10 is expected to be in hydrostatic equilibrium, taking the shape of a MacLaurin spheroid that is slightly flattened due to its rotation.[19][26] The rotation period of 2007 OR10 was first measured in March 2016, through observations of variations in its brightness with the Kepler space telescope.[26] 2007 OR10's light curve amplitude as observed by Kepler is small, only varying in brightness by about 0.09 magnitudes.[26] The small light curve amplitude of 2007 OR10 indicates that it is being viewed at a pole-on configuration, further evidenced by the observed inclined orbit of its satellite.[19] The Kepler observations provided ambiguous values of 44.81±0.37 and 22.4±0.18 hours for the rotation period.[19][26] Based on a best-fit model for its rotation pole orientation, the value of 22.4±0.18 hours is thought to be the more plausible one.[19] 2007 OR10 rotates slowly compared to other trans-Neptunian objects, which usually have rotation periods between 6 and 12 hours.[19] Due to its slow rotation, it is expected to have a low oblateness of 0.03 or 0.007, for rotation periods of 22.4 or 44.81 hours, respectively.[19] 3. Orbit Polar view of the orbits of 2007 OR10, Eris, and Pluto. https://handwiki.org/wiki/index.php?curid=1363609 Ecliptic view of the highly inclined orbits of 2007 OR10 and Eris. https://handwiki.org/wiki/index.php?curid=1806179 A preliminary motion analysis of 2007 OR10 librating in a 3:10 resonance with Neptune. This animation consists of 16 frames covering 26,000 years.[41] Neptune (white dot) is held stationary. https://handwiki.org/wiki/index.php?curid=1773833 Apparent motion of 2007 OR10 through the constellation Aquarius (years 2000 to 2050). https://handwiki.org/wiki/index.php?curid=1462402 2007 OR10 orbits the Sun at an average distance of 67.4 AU (1.008×1010 km; 6.27×109 mi), and completes a full orbit in 553 years.[28] The orbit of 2007 OR10 is highly inclined to the ecliptic, with an orbital inclination 30.7 degrees.[28] Its orbit is also highly eccentric, with a measured orbital eccentricity of 0.503.[28] Due to its highly eccentric orbit, the distance of 2007 OR10 from the Sun varies greatly over the course of its orbit, from 101.3 AU (1.515×1010 km; 9.42×109 mi) at aphelion, its furthest point from the Sun, to around 33.5 AU (5.01×109 km; 3.11×109 mi) at perihelion, its closest point to the Sun.[1][28] 2007 OR10 had approached its perihelion in 1857, and is currently moving farther from the Sun, toward its aphelion.[42] The Minor Planet Center lists it as a scattered disc object for its eccentric and distant orbit.[43] The Deep Ecliptic Survey shows the orbit of 2007 OR10 to be in a 3:10 resonance with Neptune; 2007 OR10 completes three orbits around the Sun for every ten orbits completed by Neptune.[41] (As of 2019), 2007 OR10 is about 88 astronomical unit\|AU (1.32×1010 km) from the Sun[44] and is moving away at a speed of 1.1 kilometers per second (2,500 miles per hour).[45] It is the sixth-farthest known Solar System object from the Sun, preceding 2015 TH367 (89.5 AU), 2014 UZ224 (90.4 AU), Eris (96.1 AU), 2018 VG18 (~ 120 AU),[46] and "FarFarOut" (~ 140 AU).[44][47] 2007 OR10 is more distant than Sedna, which is located 84.8 AU from the Sun as of July 2019.[44] It has been farther from the Sun than Sedna since 2013.[45] 2007 OR10 will be farther than both Sedna and Eris by 2045,[48] and will reach its aphelion in 2130.[45] 3.1. Brightness 2007 OR10 has an absolute magnitude (H) of 2.34,[25][26] which makes it the seventh-brightest trans-Neptunian object known. It is dimmer than Orcus (H=2.31; D=917 km)[49] but brighter than Quaoar (H=2.82; D=1,110 km).[50] The Minor Planet Center and the Jet Propulsion Laboratory Small-Body Database assume a brighter absolute magnitude of 1.6 and 1.8, respectively,[1][28] which would make it the fifth brightest trans-Neptunian object.[51] Being 88 AU from the Sun, the apparent magnitude of 2007 OR10 is only 21.5,[52] and so it is too dim to be seen from Earth with the naked eye.[15][53] Although closer to the Sun than the dwarf planet Eris, 2007 OR10 appears dimmer, as Eris has a higher albedo and an apparent magnitude of 18.8.[21][54] 4. Satellite Hubble images of 2007 OR10 and its moon, taken in 2009 and 2010 with the Wide Field Camera 3. https://handwiki.org/wiki/index.php?curid=1874949 Following the March 2016 discovery that 2007 OR10 was an unusually slow rotator, the possibility was raised that a satellite may have slowed it down via tidal forces.[55] The indications of a possible satellite orbiting 2007 OR10 led Csaba Kiss and his team to analyze archival Hubble observations of 2007 OR10.[56] Their analysis of Hubble images taken on 18 September 2010 revealed a faint satellite orbiting 2007 OR10 at a distance of at least 15,000 km (9,300 mi).[57] The discovery was announced on 17 October 2016 and the satellite was subsequently provisionally designated S/2010 (225088) 1.[15] The satellite is approximately 100 km (62 mi) in diameter and has an orbital period of 25 days.[56] 5. Exploration It was calculated by planetary scientist Amanda Zangari that a flyby mission to 2007 OR10 would take a minimum of over 20 years with current rocket capabilities.[58] A flyby mission could take just under 25 years using a Jupiter gravity assist, based on a launch date of 2030 or 2031. 2007 OR10 would be approximately 95 AU from the Sun when the spacecraft arrives.[58] "225088 (2007 OR10)". Minor Planet Center. International Astronomical Union. https://www.minorplanetcenter.net/db_search/show_object?object_id=225088. Retrieved 2 November 2019. "A Search for Distant Solar System Bodies in the Region of Sedna". The Astrophysical Journal Letters 694 (1): L45–L48. March 2009. doi:10.1088/0004-637X/694/1/L45. Bibcode: 2009ApJ...694L..45S. https://dx.doi.org/10.1088%2F0004-637X%2F694%2F1%2FL45 "2007 OR10 Needs a Name!". The Planetary Society. 9 April 2019. http://www.planetary.org/blogs/guest-blogs/2019/or10-needs-a-name.html. Retrieved 24 May 2019. "Properties of the Distant Kuiper Belt: Results from the Palomar Distant Solar System Survey". The Astrophysical Journal Letters 720 (2): 1691–1707. 25 August 2010. doi:10.1088/0004-637X/720/2/1691. Bibcode: 2010ApJ...720.1691S. https://dx.doi.org/10.1088%2F0004-637X%2F720%2F2%2F1691 "There's something out there -- part 3". Mike Brown's Planets. 29 November 2010. http://www.mikebrownsplanets.com/2010/11/theres-something-out-there-part-3.html. Retrieved 10 May 2019. "Astronomers Find Ice and Possibly Methane On Snow White, a Distant Dwarf Planet". Science Daily (California Institute of Technology). 22 August 2011. https://www.sciencedaily.com/releases/2011/08/110822124955.htm. Retrieved 5 March 2018. "MPEC 2009-A42 : 2007 OR10". Minor Planet Electronic Circular. Minor Planet Center. 7 January 2009. https://minorplanetcenter.net/mpec/K09/K09A42.html. Retrieved 23 May 2019. In the convention for minor planet provisional designation, the first letter represents the half-month of the year of discovery while the second letter and numbers indicate the order of discovery within that half-month. In the case for 2007 OR10, the first letter 'O' corresponds to the second half-month of July 2007 while the last letter 'R' indicates that it is the 17th object discovered on the 10th cycle of discoveries. Each cycle consists of 25 letters representing discoveries, hence 17 + (10 cycles × 25 letters) = 267.[20] Lowe, A.. "(225088) 2007 OR10 Precovery Images". Andrew Lowe's Minor Planet Home Page. http://andrew-lowe.ca/2007or10.htm. Retrieved 6 May 2019. Dyches, P. (11 May 2016). "2007 OR10: Largest Unnamed World in the Solar System". Jet Propulsion Laboratory. http://www.jpl.nasa.gov/news/news.php?feature=6509. Retrieved 12 May 2016. "The Redemption of Snow White (Part 1)". Mike Brown's Planets. 9 August 2011. http://www.mikebrownsplanets.com/2011/08/redemption-of-snow-white-part-1.html. "Snow White needs a bailout". Mike Brown's Planets. 10 March 2009. Archived from the original on 17 May 2009. https://www.webcitation.org/5gpP1IbCr?url=http://www.mikebrownsplanets.com/2009/03/snow-white-needs-bailout.html. Retrieved 17 February 2010. Williams, M. (3 September 2015). "The (possible) dwarf planet 2007 OR10". Universe Today. https://www.universetoday.com/122154/the-possible-dwarf-planet-2007-or10/. Retrieved 2 November 2019. Plotner, T. (3 August 2011). ""Snow White" or "Rose Red" (2007 OR10)". Universe Today. https://www.universetoday.com/88377/planet-snow-white-or-rose-red/. Retrieved 8 May 2019. "Help Name 2007 OR10". https://2007or10.name. Retrieved 9 April 2019. "How Are Minor Planets Named?". Minor Planet Center. International Astronomical Union. https://www.minorplanetcenter.net/iau/info/HowNamed.html. Retrieved 8 May 2019. "Astronomers Invite the Public to Help Name Kuiper Belt Object". International Astronomical Union. 10 April 2019. https://www.iau.org/news/announcements/detail/ann19021/. Retrieved 12 May 2019. "The People Have Voted on 2007 OR10's Future Name!". The Planetary Society. 29 May 2019. http://www.planetary.org/blogs/guest-blogs/2019/or10-vote-results.html. Retrieved 29 May 2019. Kiss, C.; Marton, G.; Parker, A. H.; Grundy, W. et al. (October 2018). "The mass and density of the dwarf planet (225088) 2007 OR10". Icarus: 311.02. doi:10.1016/j.icarus.2019.03.013. 311.02. Bibcode: 2018DPS....5031102K. Initial publication at the American Astronomical Society DPS meeting #50, with the publication ID 311.02 https://dx.doi.org/10.1016%2Fj.icarus.2019.03.013 "The compositions of Kuiper belt objects". Annual Reviews 40 (1): 467–494. May 2012. doi:10.1146/annurev-earth-042711-105352. Bibcode: 2012AREPS..40..467B. http://web.gps.caltech.edu/~mbrown/papers/ps/kbcomp.pdf. Retrieved 19 May 2019. Santos-Sanz, P.; Lellouch, E.; Fornasier, S.; Kiss, C. et al. (May 2012). ""TNOs are Cool": A survey of the trans-Neptunian region. IV. Size/albedo characterization of 15 scattered disk and detached objects observed with Herschel-PACS". Astronomy & Astrophysics 541 (A92): 18. doi:10.1051/0004-6361/201118541. Bibcode: 2012A&A...541A..92S. https://dx.doi.org/10.1051%2F0004-6361%2F201118541 "The Surface Composition of Large Kuiper Belt Object 2007 OR10". The Astrophysical Journal Letters 738 (2): 4. September 2011. doi:10.1088/2041-8205/738/2/L26. Bibcode: 2011ApJ...738L..26B. http://web.gps.caltech.edu/~mbrown/papers/ps/or10.pdf. "The compositions of Kuiper belt objects". The Astrophysical Journal 143 (6). May 2012. doi:10.1088/0004-6256/143/6/146. http://web.gps.caltech.edu/~mbrown/papers/ps/kbowater.pdf. Retrieved 19 May 2019. "The Redemption of Snow White (Part 3 of 3)". Mike Brown's Planets. 20 August 2011. Archived from the original on 25 July 2014. https://web.archive.org/web/20140725023611/http://www.mikebrownsplanets.com/2011/08/redemption-of-snow-white-part-3-of-3.html. Boehnhardt, H.; Schulz, D.; Protopapa, S.; Götz, C. (November 2014). "Photometry of Transneptunian Objects for the Herschel Key Program 'TNOs are Cool'". Earth, Moon, and Planets 114 (1–2): 35–57. doi:10.1007/s11038-014-9450-x. Bibcode: 2014EM&P..114...35B. https://dx.doi.org/10.1007%2Fs11038-014-9450-x Pál, A.; Kiss, C.; Müller, T. G.; Molnár, L. et al. (May 2016). "Large Size and Slow Rotation of the Trans-Neptunian Object (225088) 2007 OR10 Discovered from Herschel and K2 Observations". The Astronomical Journal 151 (5): 8. doi:10.3847/0004-6256/151/5/117. Bibcode: 2016AJ....151..117P. https://dx.doi.org/10.3847%2F0004-6256%2F151%2F5%2F117 Holler, B. J.; Young, L. A.; Bus, S. J.; Protopapa, S. (September 2017). "Methanol ice on Kuiper Belt objects 2007 OR10 and Salacia: Implications for formation and dynamical evolution". European Planetary Science Congress 2017. 11. European Planetary Science Congress. EPSC2017-330. Bibcode: 2017EPSC...11..330H. https://meetingorganizer.copernicus.org/EPSC2017/EPSC2017-330.pdf. "JPL Small-Body Database Browser: 225088 (2007 OR10)". Jet Propulsion Laboratory. 10 April 2017. https://ssd.jpl.nasa.gov/sbdb.cgi?sstr=2225088. Retrieved 23 May 2019. Johnson, R. E.; Oza, A.; Young, L. A.; Volkov, A. N.; Schmidt, C. (August 2015). "Volatile Loss and Classification of Kuiper Belt Objects". The Astrophysical Journal 809 (1): 43. doi:10.1088/0004-637X/809/1/43. Bibcode: 2015ApJ...809...43J. https://dx.doi.org/10.1088%2F0004-637X%2F809%2F1%2F43 "Physical and dynamical characteristics of icy "dwarf planets" (plutoids)". Proceedings of the International Astronomical Union 5 (S263): 173–185. 6 April 2010. doi:10.1017/S1743921310001717. Bibcode: 2010IAUS..263..173T. https://www.cambridge.org/core/services/aop-cambridge-core/content/view/S1743921310001717. Retrieved 14 May 2019. Lellouch, E.; Santos-Sanz, P.; Lacerda, P.; Mommert, M. et al. (August 2013). ""TNOs are Cool": A survey of the trans-Neptunian region. IX. Thermal properties of Kuiper belt objects and Centaurs from combined Herschel and Spitzer observations". Astronomy & Astrophysics 557 (A60): 19. doi:10.1051/0004-6361/201322047. Bibcode: 2013A&A...557A..60L. http://www.aanda.org/articles/aa/pdf/2013/09/aa22047-13.pdf. "How many dwarf planets are there in the outer solar system?". California Institute of Technology. 20 May 2019. http://web.gps.caltech.edu/~mbrown/dps.html. Retrieved 23 May 2019. "Naming of Astronomical Objects". International Astronomical Union. https://iau.org/public/themes/naming/#dwarfplanets. Retrieved 2 November 2019. "IAU 2006 General Assembly: Result of the IAU Resolution votes" (Press release). International Astronomical Union. 24 August 2006. Retrieved 2 October 2019. http://www.iau.org/news/pressreleases/detail/iau0603/ A larger magnitude value corresponds to a dimmer brightness and vice versa. The numerical value of 2007 OR10's absolute magnitude is 2.34,[10] hence it is dimmer than the IAU's minimum absolute magnitude of 1. "A Southern Sky and Galactic Plane Survey for Bright Kuiper Belt Objects". The Astronomical Journal 142 (4): 10. October 2011. doi:10.1088/0004-6256/142/4/98. Bibcode: 2011AJ....142...98S. https://dx.doi.org/10.1088%2F0004-6256%2F142%2F4%2F98 The resulting minimum diameter of 580 km is derived from the equation [math]\displaystyle{ E=\frac{\sqrt} 10^{-0.2H} }[/math], where [math]\displaystyle{ H }[/math] is the absolute magnitude of 2007 OR10, and [math]\displaystyle{ p }[/math] is the albedo of 2007 OR10, which in this case is assumed to be 1.[43] Thomas, P. C. (July 2010). "Sizes, shapes, and derived properties of the saturnian satellites after the Cassini nominal mission". Icarus 208 (1): 395–401. doi:10.1016/j.icarus.2010.01.025. Bibcode: 2010Icar..208..395T. http://www.ciclops.org/media/sp/2011/6794_16344_0.pdf. Szabó, R. (4 November 2015). "Pushing the Limits of K2: Observing Trans-Neptunian Objects S3K2: Solar System Studies with K2". Archived from the original. Error: If you specify \|archiveurl=, you must also specify \|archivedate=. https://web.archive.org/web/20190207015945/https://lco.global/files/conferences/K2SciCon/Robert_Szabo-Szabo_TNOs_web.pdf. "The Pluto System After New Horizons". Annual Review of Astronomy and Astrophysics 56: 357–392. September 2018. doi:10.1146/annurev-astro-081817-051935. https://dx.doi.org/10.1146%2Fannurev-astro-081817-051935 "Orbit Fit and Astrometric record for 225088". Southwest Research Institute. 24 May 2019. 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Astronomy & Astrophysics 555 (A15): 22. doi:10.1051/0004-6361/201321329. Bibcode: 2013A&A...555A..15F. https://dx.doi.org/10.1051%2F0004-6361%2F201321329 Braga-Ribas, F.; Sicardy, B. (August 2013). "The Size, Shape, Albedo, Density, and Atmospheric Limit of Transneptunian Object (50000) Quaoar from Multi-chord Stellar Occultations". The Astrophysical Journal 773 (1): 13. doi:10.1088/0004-637X/773/1/26. Bibcode: 2013ApJ...773...26B. https://dx.doi.org/10.1088%2F0004-637X%2F773%2F1%2F26 "The Redemption of Snow White (Part 2 of 3)". Mike Brown's Planets. 11 August 2011. Archived from the original on 25 July 2014. https://web.archive.org/web/20140725020928/http://www.mikebrownsplanets.com/2011/08/redemption-of-snow-white-part-2.html. "(225088) 2007OR10 Observation Prediction". Asteroids Dynamic Site. Department of Mathematics, University of Pisa, Italy. Archived from the original on 24 May 2019. https://web.archive.org/web/20190524034532/https://newton.spacedys.com/astdys/index.php?n=225088&pc=1.1.4.1&oc=500&y0=2019&m0=04&d0=27&h0=00&mi0=00&s=3.0. Retrieved 24 May 2019. Under good conditions, the unaided human eye can detect objects with a visual magnitude of around +7.4 or lower.[55] "(136199) Eris Observation prediction". Asteroids Dynamic Site. Department of Mathematics, University of Pisa, Italy. Archived from the original on 24 May 2019. https://web.archive.org/web/20190524034336/https://newton.spacedys.com/astdys/index.php?n=136199&pc=1.1.4.1&oc=500&y0=2019&m0=4&d0=27&h0=00&mi0=00&s=3.0. Retrieved 2 November 2019. "DPS/EPSC update: 2007 OR10 has a moon!". The Planetary Society. 19 October 2016. http://www.planetary.org/blogs/emily-lakdawalla/2016/10190940-dpsepsc-update-2007-or10.html. Retrieved 19 October 2016. Kiss, C.; Marton, G.; Farkas-Takács, A.; Stansberry, J. et al. (March 2017). "Discovery of a Satellite of the Large Trans-Neptunian Object (225088) 2007 OR10". The Astrophysical Journal Letters 838 (1): L1. doi:10.3847/2041-8213/aa6484. Bibcode: 2017ApJ...838L...1K. https://dx.doi.org/10.3847%2F2041-8213%2Faa6484 Marton, G.; Kiss, C.; Mueller, T. G. (October 2016). "The moon of the large Kuiper-belt object 2007 OR10". Division for Planetary Sciences Abstract Book. 48. American Astronomical Society. 120.22. https://aas.org/files/dps-epsc-abstract-book-final.pdf. Retrieved 24 May 2019. Zangari, A. M.; Finley, T. J. (May 2019). "Return to the Kuiper Belt: Launch Opportunities from 2025 to 2040". Journal of Spacecraft and Rockets 56 (3): 919–930. doi:10.2514/1.A34329. https://dx.doi.org/10.2514%2F1.A34329 Handwiki, (225088) 2007 OR10. Encyclopedia. Available online: https://encyclopedia.pub/entry/37836 (accessed on 31 January 2023). Handwiki . (225088) 2007 OR10. Encyclopedia. Available at: https://encyclopedia.pub/entry/37836. Accessed January 31, 2023. 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User scheduling for multicast transmission in high throughput satellite systems Shuo Zhang1, Min Jia1, Yuming Wei1 & Qing Guo ORCID: orcid.org/0000-0003-4209-54691 EURASIP Journal on Wireless Communications and Networking volume 2020, Article number: 133 (2020) Cite this article Adopting full frequency reuse in high throughput satellite (HTS) systems is expected to cope with huge communication demands and large user populations. Moreover, a kind of multicast transmission, which embeds the data of several users in each frame, can be employed to increase the efficiency of HTS systems. Multicast precoding is usually utilized in such systems to mitigate the co-frequency interference between beams and improve the efficiency of transmission. In this context, considering that the number of users may exceed the number of available communication resources in the system, we investigate user scheduling for the multicast transmission in HTS systems with full frequency reuse and multicast precoding. We perform user scheduling according to the user channel state information and decouple the scheduling problem into two phases: intra-beam and inter-beam scheduling. Intra-beam scheduling determines the users involved in the transmission of each frame with the purpose of reducing the influence of the multicast fashion. For intra-beam scheduling, we put forth a fixed-size user grouping algorithm. In contrast to previous studies, this algorithm takes the interference among beams into consideration during the scheduling. In inter-beam scheduling, user groups belonging to different beams are scheduled to improve the performance of the multiplexed transmission. An inter-beam scheduling algorithm is proposed to improve the fairness among users. The simulation results verify the superiority of the proposed algorithms in terms of fairness and spectral efficiency. The appearance of high throughput satellite (HTS) systems has brought new opportunities to satellite communication. In addition to the advantages of satellite communication, HTS systems have low communication cost and high throughput, which compensate the deficiencies of traditional satellite systems. Thanks to the rapid development of antenna technology [1–6], a large number of high-gain beams can be created and a high order of frequency reuse can be employed in the service area of the system to achieve this large throughput [7]. A four-color frequency reuse scheme is adopted in current HTS systems to avoid co-frequency interference between adjacent beams as well as obtaining a fairly high frequency reuse factor [8]. As indicated in literature, present HTS systems have achieved a throughput over 300 Gbps [9]. Partnering terrestrial communication systems, HTS is expected to address the challenges of huge traffic amounts and large user populations. Further, with the integration of satellite and terrestrial communication, HTS systems need to support various types of services [10]. Aggressive frequency reuse emerges in HTS systems to meet future communication demands [11]. Along with augmenting the available bandwidth in each beam, aggressive frequency reuse brings about the co-frequency interference between adjacent beams. This results in the interference-limited characteristics of the system. Considering the similarity between HTS communication and multiuser Multiple Input Multiple Output (MIMO) communication, precoding techniques can be employed to handle the co-frequency interference. With channel state information (CSI), precoding can bring throughput increase through the joint processing of the signals from different feeds [12]. Relevant studies show that the adoption of precoding in HTS systems with full frequency reuse can achieve performance improvement by comparison with ones that adopt four-color frequency reuse [13, 14]. Since coding schemes with long codewords are adopted in satellite communication to cope with the large fading loss of satellite links, satellite systems may statistically multiplex the data of different users belonging to the same beam in each frame to increase the encapsulation efficiency of the long frames [15]. Precisely, the data of a certain number of users are coded as a single codeword and the codeword is transmitted as a frame. Thus, each user can only obtain his data after successfully receiving the whole frame. Although this type of transmission can increase the efficiency of the system, it leads to a multicast fashion in each beam which complicates the precoding design [16]. Besides the precoding design, user scheduling according to the locations or CSIs of users becomes an emerging research field on account of the large user populations in such systems [12, 17]. A multicast-aware user scheduling algorithm was proposed to improve the performance of multicast transmission in HTS systems, and the simulation results indicated the throughput gains brought by user scheduling [18]. Since this study only investigated the multicast transmission with a fixed number of users, the algorithm might not apply to the situation where the number of users is much larger than the fixed number. For the user scheduling problem with a large number of users, the scheduling consists of two phases, i.e., intra-beam scheduling and inter-beam scheduling. Intra-beam scheduling, aiming at reducing the influence of the multicast fashion, is employed in each beam to constitute user groups, each of which consists of users sharing the same frame for transmission. Inter-beam scheduling determines the user groups (respectively belonging to different beams) that are involved in each multiplexed transmission among the beams. The two phases should be jointly handled to obtain the best performance, but most current studies only involve one of the two phases. The multicast fashion in each beam limits the achievable rate of the users in each frame groups since users may experience different channel gains. In order to mitigate this impact, intra-beam scheduling aims at constituting user groups in which users have similar channel vectors [16]. Considering the invariable channel state of the users in fixed HTS communication, intra-beam scheduling was formulated as a clustering problem to guarantee that all the users can be served [19]. Clustering algorithms, such as spectral clustering [20] and K-means [19], were employed in intra-beam scheduling. These scheduling algorithms can obtain a fixed number of clusters with various cluster sizes. Fixed size user clustering, i.e., fixed size user grouping, has practical application for intra-beam scheduling for HTS systems because of the limited number of users in each frame. Indeed, standards with respect to satellite communication specify the length of the codewords of each coding scheme. For example, DVB-S2 standard adopts LDPC coding with fixed lengths of 16 kbit and 64 kbit [21]. The fixed length of the codeword in each frame suggests the necessity of fixed size clustering for intra-beam scheduling. A k-user grouping scheme was proposed to constitute fixed-size user groups for successive frame transmission by selecting the users based on the collinearity of channel vectors [16]. For the clustering scheduling problem, a fixed size clustering algorithm was proposed which randomly selected a user as the first user in each group and formed the group based on the similarity of the channel vectors between the first user and the others [22]. Further, the MaxDist algorithm, also a fixed size clustering algorithm, was proposed. The algorithm took a new approach to determine the first user in each group and outperformed the previous one that randomly selected each first user [19]. The aforementioned intra-beam scheduling algorithms can enhance the performance of the system. Nevertheless, these studies only focus on increasing the similarity of the channel vectors in each group through scheduling and have not taken the interference-limited characteristics of the system into consideration. The inter-beam scheduling should maximize the orthogonality between the users in any two beams to improve the performance [23]. In contrast to general scheduling problems, each unit to be scheduled in inter-beam scheduling corresponds to a group of users instead of an individual user. Considering the complexity due to this fact, present HTS systems usually adopt random scheduling as the inter-beam scheduling scheme [24]. One of the few studies of inter-beam scheduling is a geographical scheduling algorithm, which partitioned the beams into zones and scheduled the users based on their locations [24]. This algorithm is appropriate for the situation where the users in HTS systems are uniformly distributed in each beam, but it does not account for the situation where some zones have no users. Besides user locations, user CSI can also be utilized in the scheduling. To our best knowledge, however, no previous studies have investigated the utilization of user CSI for inter-beam scheduling. This paper investigates the user scheduling problem for multicast transmission in HTS systems with full frequency reuse. CSI of users is utilized for user scheduling. We divide the scheduling problem into intra-beam and inter-beam scheduling. For intra-beam scheduling, in view of the practicability of fixed size clustering, we pay attention to such a scheduling problem and seek for a new algorithm considering the existence of the co-frequency interference among the beams. For inter-beam scheduling, in addition to increasing the spectral efficiency of the system, the scheduling algorithm should ensure the fairness among users on account of the invariable channel state of users. One challenge associated with inter-beam scheduling is the handle of each unit to be scheduled, which corresponds to the CSIs of a number of users. The contributions of this paper can be concluded as follows: A fixed-size user grouping algorithm is proposed for intra-beam scheduling. In contrast to the previous studies, the interference-limited characteristics of the system is considered during the scheduling and a new similarity metric is utilized. The concept of equivalent CSIs is introduced for inter-beam scheduling to simplify the inter-beam scheduling problem (making each scheduled unit only involve one CSI). Two forms of equivalent CSIs are presented. A scheduling algorithm according to equivalent CSIs is proposed for inter-beam scheduling aiming at improving the fairness among users. The reminder of the paper is organized as follows. Section 2 models the multicast transmission of HTS systems with user scheduling, including the channel model, the signal model, and the precoding adopted in this paper. The problem statement and proposed algorithms for intra-beam and inter-beam scheduling are introduced in Sections 3 and 4, respectively. In Section 5, simulation results of different scheduling schemes, each including intra-beam and inter-beam scheduling, are presented in terms of average spectral efficiency and Jain's fairness index. Section 6 ends with the conclusion. Notation: Boldface lowercase and uppercase letters denote vectors and matrices. (∙)H and (∙)T donate the conjugate transpose operator and the transpose operator. $\mathbb {C}$ and $\mathbb {R}$ donate the complex space and the real space. ∥∙∥ correspond to the Frobenius norm. For a set, \|∙\| donates the cardinality of the set, and for a vector, the result of \|∙\| is the modulus vector of the vector. [∙]mn donates the m,nth element of the matrix. CN(0,N0)U[0,2π) and N(0,σ2) donate the circularly symmetric Gaussian distribution, the uniform distribution, and the Gaussian distribution, respectively. System model and preliminaries We consider the downlink transmission (from the satellite to the user) of a geosynchronous orbit HTS that deploys full frequency reuse to provide services for fixed users in the satellite coverage area. The multibeam antenna on the satellite adopts the single feed per beam (SFB) structure and creates N beams corresponding to N feeds with the set ${\mathcal {N}} = \left \{ {1,2, \cdots,N} \right \}$, as shown in Fig. 1. In beam $n\left ({n \in {\mathcal {N}}} \right), M_{n}$ single-antenna users are distributed in the beam area with the user set ${{\mathcal {M}}_{n}} = \left \{ {1,2, \ldots,{M_{n}}} \right \}$. The data of the users in each beam are embedded in frames. This leads to the multicast transmission. We assume that the transmission of frames in different beams are synchronized and the number of users involved in the transmission of each frame is lower than a fixed number R. Obviously, $R \ll \left \| {{{\mathcal {M}}_{n}}} \right \|$. User scheduling should sequentially determine the users participating in the transmission of each frame among all the users in these beams. In this paper, user scheduling is decoupled into intra-beam and inter-beam scheduling. The intra-beam scheduling, which is performed in each beam, divides the Mn users into Knframe groups with the group set ${{\mathcal {K}}_{n}} = \left \{ {1,2, \ldots,{K_{n}}} \right \}$ and the corresponding user set $\left \{ {{\mathcal {U}}_{1}^{\left (n \right)},{\mathcal {U}}_{2}^{\left (n \right)}, \ldots,{\mathcal {U}}_{{K_{n}}}^{\left (n \right)}} \right \}$ that satisfies ${\sum \nolimits }_{k = 1}^{{K_{n}}} {\left \| {{\mathcal {U}}_{k}^{\left (n \right)}} \right \|} = {M_{n}}$. Without loss of generality, we assume that the number of users in each beam is equal, i.e., ${M_{n}} = M,\forall n \in {\mathcal {N}}$. This assumption implies that each beam has the same number of frame groups, i.e., ${K_{n}} = K,\forall n \in {\mathcal {N}}$. The inter-beam scheduling partitions the frame groups of different beams into Kmultiplexed groups$\left \{ {{{\mathcal {G}}_{1}},{{\mathcal {G}}_{2}}, \ldots,{{\mathcal {G}}_{K}}} \right \}$ with the group set ${\mathcal {K}} = \left \{ {1,2, \ldots,K} \right \}$. Each multiplexed group ${{\mathcal {G}}_{k}}\left ({k \in {\mathcal {K}}} \right)$ consists of N frame groups, i.e., $\left \| {{{\mathcal {G}}_{k}}} \right \| = N,\forall k \in {\mathcal {K}}$, respectively, corresponding to frame group ${l_{k,n}} \in {{\mathcal {K}}_{n}}$ in beam n. After the precoding processing, the N frames groups in a multiplexed group are simultaneously transmitted with the same frequency band. It takes K successive frames to complete the transmission for all the users. The downlink transmission of the HTS system The diagram of multicast transmission with user scheduling for N=2 and K=3 is presented as an example in Fig. 2. The output of the intra-beam scheduling is the three frame groups in each beam. For each frame group, the data of the users are embedded in the same frame. Then, three multiplexed groups are obtained by the inter-beam scheduling. Each multiplexed group consists of two frames groups respectively from the two beams, and the data of the users in the two frames groups are transmitted with the same frequency band after precoding. Multicast transmission with user scheduling for N=2 and K=3 Channel model We assume that the channel is flat-fading and the influence of inter-symbol interference is ignored. The channel coefficients between each user and the feeds can be expressed as a complex vector, and the channel vector of user $j \in {{\mathcal {M}}_{n}}$ in beam n is indicated as $\mathbf {h}_{j}^{\left (n \right)} \in {\mathbb {C}^{1 \times N}}$. The ith element stands for the channel coefficient between user j and feed $i \in {\mathcal {N}}$, which can be expressed as $$ h_{j,i}^{\left(n \right)} = G_{j,i}^{\left(n \right)}{e^{j\varphi_{j,i}^{\left(n \right)}}}. $$ $G_{j,i}^{\left (n \right)}$ is the free space path loss of user j from feed i, which can be expressed as [16] $$ G_{j,i}^{\left(n \right)} = \frac{{{G_{R}}b_{j,i}^{\left(n \right)}}}{{4\pi \frac{{d_{j,i}^{\left(n \right)}}}{\lambda }\sqrt {{K_{B}}{T_{R}}{B_{W}}} }}, $$ where $d_{j,i}^{\left (n \right)}$ is the distance between user j and feed i, λ is the wavelength, and KBTRBW is the noise power, in which KB,TR, and BW are the Boltzmann constant, the noise temperature, and the user link bandwidth, respectively. $G_{R}^{2}$ is the receiver antenna gain of the user, and ${\left ({b_{j,i}^{\left (n \right)}} \right)^{2}}$ is the multibeam antenna beam gain between user j and feed i. For the SPF antenna, the beam gain can be approximated by [25] $$ {\left({b_{j,i}^{\left(n \right)}} \right)^{2}} = {b^{2}}\left({\theta_{j,i}^{\left(n \right)}} \right) = {G_{\max }}{\left({\frac{{{J_{1}}\left(u \right)}}{{2u}} + 36\frac{{{J_{3}}\left(u \right)}}{{{u^{3}}}}} \right)^{2}}, $$ where J1 and J3 are respectively the first-kind Bessel function of order 1 and 3. $u = 2.07123~{\text {sin}}~\theta _{j,i}^{(n)} \big / {\text {sin}{\theta _{3dB}}}$, where θ3dB is the 3 dB angle of each beam and $\theta _{j,i}^{(n)}$ is the angle between the location of user j and the ith beam center. The phase of the ith channel coefficient of user j is assumed as [8, 26] $$ \varphi_{j,i}^{\left(n \right)} = \theta_{j}^{\left(n \right)} + {\delta_{i}}, $$ where $\theta _{j}^{\left (n \right)} \sim U\left [ {0,2\pi } \right)$ is the same for the phases of all the channel coefficients of user j. It is the phase due to radio frequency (RF) signal propagation. Since the distance between a user and any feed is much longer than that between any two feeds, the phases caused by the RF signal propagation are almost the same for the same user. δi∼N(0,σ2) is the phase caused by the payload oscillator of feed i [16]. δi is the same for the ith channel coefficient of each user in the coverage area. The phases of the channel vectors have a great effect on the system performance [27], and the phase variation δi cannot be obtained by channel estimation [28]. Thus, we make a reasonable assumption that the CSI available for the scheduling, i.e., available CSI, is donated as $$ {\hat{\mathbf{h}}}_{j}^{\left(n \right)} = \left({\hat h_{j,1}^{\left(n \right)},\hat h_{j,2}^{\left(n \right)}, \ldots,\hat h_{j,N}^{\left(n \right)}} \right) = {e^{j\theta_{j}^{\left(n \right)}}} \cdot \left({\begin{array}{*{20}{c}} {G_{j,1}^{\left(n \right)}}&{G_{j,2}^{\left(n \right)}}& \cdots &{G_{j,N}^{\left(n \right)}} \end{array}} \right), $$ which takes the imperfection estimation of phases into consideration. Signal model In the frame transmission of multiplexed group k, the channel vector of user q in beam n is donated as $\mathbf {h}_{q,{l_{k,n}}}^{\left (n \right)} \in {\mathbb {C}^{1 \times N}}\left ({q \in {\mathcal {U}}_{{l_{k,n}}}^{\left (n \right)},{l_{k,n}} \in {{\mathcal {K}}_{n}}} \right)$, and the received signal of the user is written as $$ y_{q,{l_{k,n}}}^{\left(n \right)}{\mathrm{ = }}\mathbf{h}_{q,{l_{k,n}}}^{\left(n \right)}{\mathbf{P}_{k}}{\mathbf{s}_{k}} + n_{q,{l_{k,n}}}^{\left(n \right)}, $$ where $n_{q,{l_{k,n}}}^{\left (n \right)} \sim CN\left ({0,{N_{0}}} \right)$ is the received noise and ${\mathbf {s}_{k}} \in {\mathbb {C}^{N \times 1}}$ is the transmitted signal from the N feeds to the corresponding beams satisfying that $E\left [ {{{\left \| {{s_{k,n}}} \right \|}^{2}}} \right ] = 1,\forall n \in {\mathcal {N}}$. ${\mathbf {P}_{k}} \in {\mathbb {R}^{N \times N}}$ is the diagonal power factor matrix, and [trace(PkPk)]nn is the transmitted power of feed n. Multicast precoding is adopted to reduce the interference between adjacent beams. For the signal sk,n of beam n, with the precoding vector ${\mathbf {w}_{k,n}} \in {\mathbb {C}^{N}}$, the received signal can be expressed as $$ y_{q,{l_{k,n}}}^{\left(n \right)}{\mathrm{ = }}\mathbf{h}_{q,{l_{k,n}}}^{\left(n \right)}{\mathbf{w}_{k,n}}{s_{k,n}} + \sum\limits_{p \in {\mathcal{N}}\backslash \left\{ n \right\}} {\mathbf{h}_{q,{l_{k,n}}}^{\left(n \right)}{\mathbf{w}_{k,p}}{s_{k,p}}} + n_{q,{l_{k,n}}}^{\left(n \right)}. $$ The actual signal sent by the feeds is ${\mathbf {x}_{k}} = {\sum \nolimits }_{n \in {\mathcal {K}}} {{s_{k,n}}{\mathbf {w}_{k,n}}} $. Assume that $\left \{ {{s_{k,n}}} \right \}_{n = 1}^{N}$ are mutually uncorrelated during the transmission. The transmitted power of all the feeds is ${\sum \nolimits }_{n \in {\mathcal {N}}} {\mathbf {w}_{k,n}^{H}{\mathbf {w}_{k,n}}}$, and the transmitted power of feed n is ${\left ({{\sum \nolimits }_{n' \in {\mathcal {N}}} {{\mathbf {w}_{k,n'}}\mathbf {w}_{k,n'}^{H}}} \right)_{{nn}}}$. The signal to interference plus noise ratio (SINR) of the user after precoding is $$ SINR_{q,{l_{k,n}}}^{\left(n \right)}{\mathrm{ = }}\frac{{{{\left\| {\mathbf{h}_{q,{l_{k,n}}}^{\left(n \right)}{\mathbf{w}_{k,n}}} \right\|}^{2}}}}{{\sum\limits_{p \in {\mathcal{N}}\backslash \left\{ n \right\}} {{{\left\| {\mathbf{h}_{q,{l_{k,n}}}^{\left(n \right)}{\mathbf{w}_{k,p}}} \right\|}^{2}}} + {N_{0}}}}. $$ The actual data rate of the users in each frame group depends on the lowest SINR of the frame group members because of the multicast fashion. Thus, based on Shannon formulation, the spectral efficiency of frame group lk,n in beam n can be expressed as $$ C_{{l_{k,n}}}^{\left(n \right)} = {\log_{2}}\left({1 + \mathop {\min }\limits_{q \in {\mathcal{U}}_{{l_{k,n}}}^{\left(n \right)}} SINR_{q,{l_{k,n}}}^{\left(n \right)}} \right). $$ For multiplexed group k which consists of frame groups $\left \{ {{\mathcal {U}}_{{l_{k,n}}}^{\left (n \right)},\forall n \in {\mathcal {N}}} \right \}$, the average spectral efficiency per beam can be expressed as $$ {C_{k}} = \frac{{\sum\limits_{n \in {\mathcal{N}}} {C_{{l_{k,n}}}^{\left(n \right)}} }}{N}. $$ Multicast precoding The precoding design influences the system performance. For multicast transmission, the precoding problem is NP-hard for optimization objectives such as minimizing the transmitted power, maximizing the fairness among users [29, 30]. These algorithms are complex and involve iterative interior point methods during the calculation of precoding vectors. A low-complex precoding algorithm, which was a one-shot design, was proposed to improve the transmission efficiency by limiting the interference between adjacent beams in HTS systems [16]. The aforementioned algorithms, however, all have higher complexity than the average minimum mean squared error (MMSE) scheme, which was first proposed for multicast transmission in HTS systems [31]. Considering that this scheme can achieve satisfactory performance with low complexity, we adopt it as the multicast precoding scheme in this paper. As the available CSI shown in (5) has the same phase for each feed link, we leave out the phases of the channel coefficients. Thus, in multiplexed group k, the average available CSI of frame group lk,n used for multicast precoding is $$ {\hat{\mathbf{h}}}_{{l_{k,n}}}^{\left(n \right)} = \frac{{\sum\limits_{q \in {\mathcal{U}}_{{l_{k,n}}}^{\left(n \right)}} {\left\| {{\hat{\mathbf{h}}}_{q,{l_{k,n}}}^{\left(n \right)}} \right\|} }}{{\left\| {{\mathcal{U}}_{{l_{k,n}}}^{\left(n \right)}} \right\|}}. $$ The channel matrix used for precoding is ${\mathbf {H}_{k}} = {\left ({{\hat {\mathbf {h}}}{{{~}_{{l_{k,1}}}^{\left (1 \right)}}^{T}},{\hat {\mathbf {h}}}{{{~}_{{l_{k,2}}}^{\left (2 \right)}}^{T}}, \ldots,{\hat {\mathbf {h}}}{{{~}_{{l_{k,N}}}^{\left (N \right)}}^{T}}} \right)^{T}}$. For the sum transmitted power PT, the precoding matrix Wk=(wk,1,wk,2,…,wk,N) is $$ {\mathbf{W}_{k}} = \gamma \mathbf{H}_{k}^{H}{\left({{\mathbf{H}_{k}}\mathbf{H}_{k}^{H} + \frac{N}{{{P_{T}}}}\mathbf{I}} \right)^{- 1}}, $$ where γ is the power factor, fulfilling $trace\left ({\mathbf {W}_{k}{\mathbf {W}_{k}^{H}}} \right) \le {P_{T}}$ for the sum power constraint or ${\left [ {\mathbf {W}_{k}{\mathbf {W}_{k}^{H}}} \right ]_{nn}} \le {{P_{T}}} \bigg / {N},\forall n \in {\mathcal {N}}$ for per-antenna power constraints. Intra-beam scheduling In this section, the intra-beam scheduling problem is first introduced and a fixed-size user grouping algorithm, named Mod-MaxDist, is proposed. Performed respectively in each beam, the intra-beam scheduling divides the users into frame groups according to the CSIs. The channel condition differences in a frame group influence the data rate of the multicast transmission. An intra-beam scheduling algorithm should constitute frame groups with users having similar channel conditions to enhance the efficiency of the transmission. The date rate of the users in each frame group depends on the lowest SINR of the members. According to (8), not only the similarity of the channel vectors of the users in each frame group, but also the interference from the other beams has effects on the performance of the multicast transmission. During the algorithm design, a key issue is to select a proper metric to measure the similarity between the user channel vectors to facilitate the scheduling. For clustering scheduling algorithms involving an iterative process, since similarity metrics need to satisfy the conditions of identity of indiscernibles, symmetry, and triangle inequality [32], the Euclidean distance between the user locations or channel vectors is often utilized as a similarity metric. However, for scheduling algorithms without an iterative process, more appropriate metrics can be utilized regardless of these conditions. In this paper, the objective of the intra-beam scheduling is to design a fixed-size user grouping algorithm that allocates $\left \| {{{\mathcal {M}}_{n}}} \right \|$ users into Kn frame groups $\left \{ {{\mathcal {U}}_{1}^{\left (n \right)},{\mathcal {U}}_{2}^{\left (n \right)}, \ldots,{\mathcal {U}}_{{K_{n}}}^{\left (n \right)}} \right \}$ according to the available CSIs. Specifically, the algorithm should utilize a proper metric to measure the similarity of the channel conditions and take the interference-limited characteristics of the system into account. Mod-MaxDist The Mod-MaxDist algorithm sequentially establishes the fixed-size frames groups. For each frame group, the first user is selected employing the approach of MaxDist [19], i.e., selecting the outlier among the users as the first user, then making a certain number of users whose channel vectors have higher similarity to the channel vector of the first user than the others as the frame group members. The highlights of the proposed algorithm, also the main differences from MaxDist, lie in: The CSI used for scheduling is redefined based on the available CSI considering the interference-limited characteristics of the system. For a user in a beam, the redefined CSI only relates to the channel coefficients of the feed links that cause high interference to the beam. For beam n, the set of feed links that cause high interference is donated as ${{\mathcal {N}}_{n}}$. Notice that $\hat h_{j,n}^{\left (n \right)}$ is not considered during the scheduling, i.e., $n \notin {{\mathcal {N}}_{n}}$. The determination of ${{\mathcal {N}}_{n}}$ depends on the beam pattern of the HTS. For example, ${{\mathcal {N}}_{n}}$ can be defined as a set that comprises the six adjacent beams surrounding beam n. For user j in beam n, the refined CSI ${\tilde {\mathbf {h}}}_{j}^{\left (n \right)} = \left ({\tilde {h}_{j,1}^{\left (n \right)},\tilde {h}_{j,2}^{\left (n \right)}, \ldots,\tilde {h}_{j,N}^{\left (n \right)}} \right)$ satisfies $$ \tilde{h}_{j,i}^{\left(n \right)} = \left\{ \begin{array}{l} \hat h_{j,i}^{\left(n \right)}\ \ \ \ i \in {{\mathcal{N}}_{n}}\\ 0\ \ \ \ \ \ \ i \notin {{\mathcal{N}}_{n}} \end{array}\right.. $$ The CSI set of beam n can be denoted as ${\tilde {\mathcal {H}}^{\left (n \right)}} = \left \{ {{\tilde {\mathbf {h}}}_{1}^{\left (n \right)},{\tilde {\mathbf {h}}}_{2}^{\left (n \right)}, \ldots,{\tilde {\mathbf {h}}}_{{M_{n}}}^{\left (n \right)}} \right \}$. The redefinition of the CSI reduces the complexity of the intra-beam scheduling. Moreover, the algorithm has good scalability since only a fixed part of the CSI is necessary. A new similarity metric is utilized for the scheduling algorithm. Existing algorithms usually utilize the Euclidean distance of channel coefficients or user locations as a similarity metric to achieve clustering. Since no iterative process is involved in the fixed size clustering, the three conditions related to the metric no longer need to be satisfied. Thus, more effective metrics can be utilized. For a selected outlier (the first user in a frame group) and user j with the corresponding redefined CSIs g and ${{\tilde {\mathbf {h}}}}_{j}^{\left (n \right)}$, the metric, i.e., the norm of the inner product (NoIP), is defined as $$ P_{{kj}}^{\left(n \right)} = \left\| {\mathbf{g}{\tilde{h}}}{{{~}_{j}^{\left(n \right)}}^{H}} \right\|. $$ The geometric meaning of NoIP is the projection from ${{\tilde {\mathbf {h}}}}_{j}^{\left (n \right)}$ to g. This metric has lower computational complexity than the Euclidean distance, and its expression is similar to the desired signal, i.e., the molecular part of the SINR. The situation where Mn is a non-integer multiple of R is considered in the algorithm. In this situation, the size of one of the frame groups is smaller than R. Our proposed algorithm sets the first frame group (containing the first outlier) as the frame group with the smaller group size. The details of the algorithm are shown in Algorithm 1. ${r_{{k_{n}}}}$ stands for the size of frame group kn, i.e., the number of elements in ${\mathcal {U}}_{{k_{n}}}^{\left (n \right)}$. Algorithm 1 needs to be ran N times to complete the intra-beam scheduling process of all the beams. The number of frame groups Kn and the frame group size are first determined based on the values of R and Mn. The Mn users are allocated to the frame groups by calculating and comparing the NoIPs between the users and the first user in each frame group until all the Kn frame groups are formed. Inter-beam scheduling The inter-beam scheduling partitions the frame groups obtained from the intra-beam scheduling into multiplexed groups with the aim of enhancing the performance of multiplexed transmission. As analyzed before, the main challenge of the inter-beam scheduling is that each scheduled unit corresponds to a set of available CSIs. We handle this by introducing a concept of equivalent CSIs to make each unit correspond to only one CSI. Thus, the inter-beam scheduling contains two steps: Calculation of equivalent CSIs: Calculate the equivalent CSI of each frame group based on the available CSIs of the users in the frame group. Two forms of equivalent CSIs are presented. Frame scheduling: According to equivalent CSIs, frame groups are divided into multiplexed groups based on the relationships among the frame groups. A heuristic scheduling algorithm, named fairness scheduling, is put forth. The details of the two steps are presented in the following part. Equivalent CSI An equivalent CSI is a representative of a frame group during the scheduling. It should be determined considering the available CSIs of the frame group members. For the multicast transmission in HTS systems, we propose two forms of equivalent CSIs. For beam n, the set of equivalent CSIs used for the intra-beam scheduling is denoted as ${{\mathcal {C}}^{\left (n \right)}} = \left \{ {\mathbf {c}_{1}^{\left (n \right)},\mathbf {c}_{2}^{\left (n \right)}, \ldots,\mathbf {c}_{K}^{\left (n \right)}} \right \}$ in this paper. The direction vector of the average available CSI The direction vector of the average available CSI can serve as the equivalent CSI of a frame group. For frame group ${k_{n}} \in {{\mathcal {K}}_{n}}$, the direction vector can be expressed with Eq. (11), i.e., $\mathbf {c}_{{k_{n}}}^{\left (n \right)} = {{\hat {\mathbf {h}}}_{{k_{n}}}^{\left (n \right)}} \bigg / {\left \\| {{\hat {\mathbf {h}}}_{{k_{n}}}^{\left (n \right)}} \right \\|}$. This form implies the available CSIs for all the frame group members. Considering that average available CSIs are the inputs of the precoding, this form of equivalent CSIs is appropriate for average MMSE precoding. The center of the frame group The center of the available CSIs corresponding to the frame group can serve as the equivalent CSI. To obtain the center $\mathbf {c}_{{k_{n}}}^{\left (n \right)} \in {{\mathcal {K}}_{n}}$, we need to solve the following optimization problem for each frame group: $$ \begin{array}{l} \underset{\mathbf{c}_{{k_{n}}}^{\left(n \right)}}{\max} \underset{q \in {\mathcal{U}}_{{k_{n}}}^{\left(n \right)}}{\min}\;\; \frac{{\left\| {\mathbf{c}{{{~}_{{k_{n}}}^{\left(n \right)}}^{H}}{\hat{\mathbf{h}}}_{q,{k_{n}}}^{\left(n \right)}} \right\|}}{{\left\\| {{\hat{\mathbf{h}}}_{q,{k_{n}}}^{\left(n \right)}} \right\\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;s.t\;\;\left\\| {\mathbf{c}_{{k_{n}}}^{\left(n \right)}} \right\\| = 1 \end{array} $$ Indeed, problem (15) is equivalent to the single-group multicast precoding problem, which is NP-hard [33]. The calculation of the center has high computational complexity. This form of equivalent CSIs can reflect the feature of the frame group especially for large frame group sizes. Frame scheduling The adoption of equivalent CSIs simplifies the inter-beam scheduling. The fact that users are associated with specific beams results in constraints on the scheduling. The following part presents the problem statement and the proposed fairness scheduling algorithm for the inter-beam scheduling. Frame scheduling forms K multiplexed groups $\left \{ {{{\mathcal {G}}_{1}},{{\mathcal {G}}_{2}}, \ldots,{{\mathcal {G}}_{K}}} \right \}$ according to the equivalent CSI sets $\left \{ {{{\mathcal {C}}^{\left (1 \right)}},{{\mathcal {C}}^{\left (2 \right)}}, \ldots,{{\mathcal {C}}^{\left (N \right)}}} \right \}$. Thanks to the adoption of equivalent CSIs, the frame scheduling can learn from the multiuser MIMO scheduling. Thus, the frame scheduling can improve the performance of the multiplexed transmission with multicast precoding by ensuring that any two equivalent CSIs of the frame groups in the same multiplexed group have small similarity as far as possible. As introduced before, the objective of the inter-beam scheduling is to maximize the fairness among the users in all the frame groups. That is to say, the frame scheduling should consider the performance of not only the scheduled frame groups but also the unscheduled ones. Thus, the design objective can be expressed as $$ \underset{{{\mathcal{G}}_{1}},{{\mathcal{G}}_{2}}, \ldots,{{\mathcal{G}}_{K}}}{\max} \underset{k \in {\mathcal{K}}}{\min }{C_{k}}, $$ where Ck is the average spectral efficiency per beam for ${{\mathcal {G}}_{k}}$ and is calculated with (10). The constraints that should be satisfied during the scheduling are: $\forall k \in {\mathcal {K}}$, the N elements in ${{\mathcal {G}}_{k}}$ respectively belong to ${{\mathcal {K}}_{\mathrm {1}}},{{\mathcal {K}}_{2}}, \ldots {{\mathcal {K}}_{N}}$ (${{\mathcal {K}}_{n}}$ corresponding to the elements in ${{\mathcal {C}}^{\left (n \right)}}$); $\forall k \in {\mathcal {K}},\left \| {{{\mathcal {G}}_{k}}} \right \| = N$. The number of available scheduling results is (K!)N, which makes it impractical to perform the exhaustive search. We focus on heuristic algorithms with low complexity. Aiming at maximizing the fairness, the proposed scheduling algorithm should take the relationships between the frame groups of any two beams into consideration during the scheduling. The fairness scheduling algorithm A heuristic algorithm, named the fairness scheduling (FS) algorithm, is proposed to deal with the frame scheduling problem raised above. The algorithm sequentially allocates the K frame groups in each beam ${{\mathcal {K}}_{n}}\left ({n \in {\mathcal {N}}} \right)$ to the K multiplexed groups $\left \{ {{{\mathcal {G}}_{1}},{{\mathcal {G}}_{2}}, \ldots,{{\mathcal {G}}_{K}}} \right \}$ based on ${{\mathcal {C}}^{\left (n \right)}}\left ({n \in {\mathcal {N}}} \right)$ until $\left \| {{{\mathcal {G}}_{k}}} \right \| = N,\forall k \in {\mathcal {K}}$. The details are shown in Algorithm 2. To improve the fairness among the users in all the beams, the beams are scheduled in a certain order based on the amount of interference that each beam suffers and the beam pattern. Precisely, the beam suffering more interference should be first scheduled to avoid poor scheduling results. For the scheduling of the frame groups in the nth beam, a pairing algorithm makes a one-to-one mapping between the K frame groups in the nth beam and the K multiplexed group sets $\left \{ {{{\mathcal {G}}_{1}},{{\mathcal {G}}_{2}}, \ldots,{{\mathcal {G}}_{K}}} \right \}$ based on a relation matrix $\mathbf {Q} \in {\mathbb {R}^{K \times K}}$. Q represents the correlation between the K elements in ${{\mathcal {C}}^{n}}$ and the K incomplete multiplexed groups. The following part introduces the calculation of the relation matrix Q and our proposed pairing algorithm. Calculation of the relation matrix The $i,j{th}\left ({i,j \in {\mathcal {K}}} \right)$ element of the relation matrix Q stands for the projection of the equivalent CSI of the jth frame group in the beam to be scheduled on the space spanned by the equivalent CSIs of the frame groups in the ith incomplete multiplexed group. For the nth beam to be scheduled, the equivalent CSI of the jth frame group is cj,(n). For the group set ${{\mathcal {G}}_{i}}\left ({i \in {\mathcal {K}}} \right)$, the equivalent CSIs of the frame groups can constitute a matrix ${\mathbf {H}_{{\mathcal {G}},i}} \in {{\mathbb {C}}^{\left \| {{{\mathcal {G}}_{i}}} \right \| \times N}}$, of which each row is an equivalent CSI. Obviously, the incomplete multiplexed groups satisfy that $\left \| {{{\mathcal {G}}_{i}}} \right \| = n - 1,\forall i \in {\mathcal {K}}$. [Q]i,j, the projection, can be expressed as $$ {\left[ \mathbf{Q} \right]_{i,j}} = {\left\| {{\mathbf{c}_{j,(n)}}\mathbf{H}_{{\mathcal{G}},i}^{H}{{\left({{\mathbf{H}_{{\mathcal{G}},i}}\mathbf{H}_{{\mathcal{G}},i}^{H}} \right)}^{- 1}}{\mathbf{H}_{{\mathcal{G}},i}}} \right\|^{2}}. $$ A large value of [Q]i,j means that the jth frame group has low similarity to the multiplexed group ${{\mathcal {G}}_{i}}$, and the interference may be strong if this frame group is allocated to ${{\mathcal {G}}_{i}}$. The pairing algorithm The pairing algorithm selects K elements q1,q2,…,qK from Q, fulfilling that not only the row numbers but also the column numbers of these elements are mutually unequal. $\left \{ {{q_{k}}} \right \}_{k = 1}^{K}$ represent the one-to-one mappings between the K frame groups of the beam to be scheduled and the multiplexed group sets ${{{\mathcal {G}}_{1}},{{\mathcal {G}}_{2}}, \ldots,{{\mathcal {G}}_{K}}}$. Specifically, qk=[Q]w,v means that the vth frame group is scheduled to multiplexed group ${{\mathcal {G}}_{w}}$ in the kth one-to-one mapping. Aiming at maximizing the fairness during the scheduling, the pairing algorithm has the design purpose: $$ \underset{{{\mathcal{C}}_{\left(n \right)}} \to \left\{ {{{\mathcal{G}}_{1}},{{\mathcal{G}}_{2}}, \ldots,{{\mathcal{G}}_{K}}} \right\}}{\min} \underset{k \in {\mathcal{K}}} {\max}\ {q_{k}}. $$ The number of available pairing results is K! as there are K2 elements in Q. When K is small, the optimal pairing results are easy to achieve, but the complexity of pairing increases with the rise in K. The pairing algorithm belongs to heuristic selection algorithms and the details of it are shown in Algorithm 3. The selection of the elements in the matrix is carried out by comparing the value of [Q]i,j with a threshold α. In this way, the search space is reduced. For the selection of the K elements, only pairing results that satisfy $\forall k \in {\mathcal {K}},{q_{k}} \le \alpha $ are accepted; otherwise, the threshold is raised and a new selection starts until accepted results are achieved. The initial value of α is $$ \alpha = \max \left\{ {\underset{i \in {\mathcal{K}}}{\max} \underset{j \in {\mathcal{K}}}{\min} {{\left[ \mathbf{Q} \right]}_{i,j}}\underset{j \in {\mathcal{K}}}{\max} \underset{i \in {\mathcal{K}}}{\min} {{\left[ \mathbf{Q} \right]}_{i,j}}} \right\}. $$ Two selections are performed for each α. The first selection includes a preferred pairing process (lines 3 to 13) to achieve good pairing results as far as possible. Through the process, some frame groups and multiplexed groups are scheduled preferentially. If no available results are obtained after the first selection, the selection with the threshold α restarts without the preferred pairing process. No backtracking processing is involved in the pairing algorithm. Remarkably, the proposed pairing algorithm can be extended to other pairing situations where a relation matrix is available. Simulation result A seven-beam scenario (N=7), where a beam is surrounded by six beams, is considered in the simulation. Full frequency reuse is adopted among the seven contiguous beams, and one frequency band is involved in the data transmission of the beams. M users are randomly distributed in each beam. The simulation parameters are $f = 20~{\text {GHz}}, {B_{W}}{\mathrm { = 41}}{\mathrm {.7~MHz}}, {K_{B}}{\mathrm { = 1}}{\mathrm {.38}} \times {\mathrm {1}}{{\mathrm {0}}^{- 23}}{\mathrm {K/J}}, {G_{\max }} = 52{\mathrm {~dB}}, G_{R}^{2} = 41.7{\mathrm {~dB}}, {T_{R}} = 207{\mathrm {K}}, {\theta _{3dB}} = 0.2^{\circ }$, and σ=2∘ for phase variation [16, 34]. The height of the GEO satellite is 35786km. For different values of the user number M, the multiplexed group number K, the transmitted power PT, and the maximum frame group size R, the simulation results are obtained over Nmc Monte Carlo runs. For Algorithm 3, α is raised by 0.1 each time in the simulation. The direction vector of the average available CSI is used as the equivalent CSI for the inter-beam scheduling. The sum power constraint is considered for the average MMSE precoding. Two intra-beam scheduling algorithms, i.e., MaxDist [19] (indicated by MD) and the proposed Mod-MaxDist (indicated by Mod-MD) and two inter-beam scheduling algorithms, i.e., random scheduling (indicated by RS) and our fairness scheduling (indicated by FS) proposed in Section 4, are involved in the simulation. Thus, the performances of the four scheduling schemes, i.e., MD+RS, MD+FS, Mod-MD+RS, and Mod-MD+FS, are compared with each other. Average spectral efficiency The average spectral efficiency is calculated with (10). Figure 3 depicts the average spectral efficiency verse R under different scheduling schemes with M=90,120. It is observed that the average spectral efficiency declines with the growth in R. This is excepted as larger R may reduce the spectral efficiency of the multicast transmission. It is observed that Mod-MD significantly outperforms MD, and the scheduling schemes with FS have better performance than those with RS. With the rise in R, the performance difference between MD+FS and MD+RS becomes small. This is due to the fact that MD mainly concerns the similarity of the group members and the increased interference caused by the growth in R weakens the effectiveness of FS. In contrast to MD, Mod-MD takes account of the interference among the beams during the scheduling. This peculiarity of Mod-MD results in a consistent performance increase for Mod-MD+FS by comparison with Mod-MD+RS. Average spectral efficiency verse R, PT=10 dB Figures 4 and 5 present the spectral efficiency verse PT under different scheduling schemes with different values of M and R. The cumulative distribution function (CDF) curves of user SINR with specific parameters are depicted as supplements in Figs. 4 and 5. User SINR is the SINR corresponding to the actual transmission rate of the user. Four-color refers to the multicast transmission in the system with four-color frequency reuse where PT is uniformly distributed among the seven beams. For different values of PT, especially at low PT, the average spectral efficiency under the four-color frequency reuse scenario is inferior to that under the full frequency reuse scenario with user scheduling schemes, owing largely to the reduced available bandwidth in each beam. It is also seen that, with the rise in PT, the spectral efficiency increases under the four scheduling schemes gradually decline, while the spectral efficiency under the four-color scenario shows an almost linear growth. The results indicate that the system with four-color frequency reuse is power-limited, and system with full-frequency reuse scenario is interference-limited. Spectral efficiency verse PT,R=2. a Average spectral efficiency. b CDF of user SINR with specific parameters Spectral efficiency verse PT,R=5. a Average spectral efficiency. b CDF of user SINR with specific parameters, PT=7dBW In Fig. 6, the average spectral efficiency verse M under different scheduling schemes is plotted. It is seen that Mod-MD outperforms MD. The advantage of Mod-MD lies in the consideration of interference-limited characteristics of the system. In most cases, the average spectral efficiency of different scheduling schemes increases with the growth in M. This stems from the larger multiuser diversity gain caused by the increase in M. However, exceptions exist with regard to specific values of M, such as M=4,5 in Fig. 6a, and M=25,36 in Fig. 6b. This is due to the fact that M is a non-integer multiple of R, i.e., the size of one frame group is smaller than R. Since the average spectral efficiency may decrease with the increase in R, the users in the frame group with the smaller size may have higher spectral efficiency than the others. However, the exceptions are non-significant for Mod-MD. A possible explanation of this might be that the performance enhancement caused by Mod-MD covers the effect caused by the frame group with the smaller size. Average spectral efficiency verse M. aR=2,PT=13dBW. bPT=7dBW In Fig. 7, we provide the average spectral efficiency verse K under different scheduling schemes. For each set of K and R, it satisfies that each frame group has the same number of users. It is observed that the average spectral efficiency increases with the growth in K under all the four scheduling schemes. Since the effect caused by the unequal group sizes in each beam is eliminated, the simulation results indicate the multiuser diversity gain caused by the growth in M for user scheduling. Average spectral efficiency verse K, PT=7 dBW The above simulation results reveal that Mod-MD+FS stands out from the other scheduling schemes in terms of the average spectral efficiency. Moreover, Mod-MD and FS can be separately employed to increase the average spectral efficiency of the system, i.e., Mod-MD+RS and MD+FS both outperform MD+RS. Jain's fairness index Jain's fairness index is utilized as the indicator to evaluate the fairness among users. For a series of variables x1,x2,…,xn, Jain's fairness index is defined as $$ {\mathcal{J}}\left({{x_{1}},{x_{2}}, \ldots,{x_{n}}} \right) = \frac{{{{\left({{\sum\nolimits}_{i = 1}^{n} {{x_{i}}}} \right)}^{2}}}}{{n{\sum\nolimits}_{i = 1}^{n} {x_{i}^{2}} }}. $$ In this simulation, each Jain's fairness index is calculated based on (20) with the spectral efficiency of the NM users. Figures 8 and 9 show the Jain's fairness index verse PT under different scheduling schemes, including the average Jain's fairness index and the CDF of Jain's fairness index with respect to specific values of PT. It is seen that the schemes with FS perform better than those with RS. This indicates the superiority of FS. With the increase in PT, for all the scheduling schemes, the average Jain's fairness index shows a downward trend. This is expected as the increased spectral efficiency caused by the rise in PT leads to larger data rate differences among the users. As shown in Fig. 10, the curves with different values of PT imply that the user SINR range becomes larger with the increase in PT. For large values of PT, not the noise but the inter-beam interference has great effects on user data rates and the interference differences may be considerable. This may explain the downtrend of the average Jain's fairness index. It is also observed that the increase in user SINR declines with the increase in PT, owing largely to the interference-limited characteristics of the system. Jain's fairness index verse PT,R=5,M=25. a Average Jain's fairness index. b CDF of Jain's fairness index with specific parameters CDF of user SINR, R=4,M=70. a MD+RS. b MD+FS. c Mod-MD+RS. d Mod-MD+FS In Figs. 11 and 12, we plot the Jain's fairness index verse R under different scheduling schemes with M=90,120. It is seen that Mod-MD+FS is superior to the others and, in most cases, FS can bring the enhancement of fairness by comparison with RS. Precisely, compared with RS, FS can bring almost consistent fairness enhancement for Mod-MD+FS, while the fairness enhancement declines with the increase in R for MD+FS. The performance difference stems from the different characteristics of MD and Mod-MD. It is also seen that the Jain's fairness index with MD shows an increasing trend. This is expected due to the increase in R with fixed M. The slight reduction of fairness in Mod-MD with the increase in R is due to the ignorance of some channel coefficients. However, Mod-MD still outperforms MD in most cases. Jain's fairness index verse R, M=90,PT=13 dBW. a Average Jain's fairness index. b CDF of Jain's fairness index with specific parameters Jain's fairness index verse R, M=120,PT=7 dBW. a Average Jain's fairness index. b CDF of Jain's fairness index with specific parameters Figures 13 and 14 depict the average Jain's fairness index verse K and M, respectively. In the simulation of Fig. 13, each M is an integral multiple of the corresponding K. It is observed that the average Jain's fairness index increases with the rise in K(M), especially for the schemes with Mod-MD. Since the space of the scheduling results grows exponentially with the increase in K, the probability that RS can achieve favorable scheduling results becomes small. The simulation results show that FS can provide performance gain by comparison with RS. Additionally, Fig. 13 indicates the increased multiuser diversity gain with respect to the growth in M. When M is a non-integral multiple of R, the frame groups with smaller sizes may impact the fairness among users. For example, in Fig. 14, when R=3, the average Jain's fairness index with M=20 is higher than that with M=25 under the schemes with MD. The users in these frame groups with smaller sizes may have higher data rates than the others, which leads to unfairness among the users. However, the effect caused by the frame groups with smaller sizes is small for Mod-MD. This is because Mod-MD can achieve favorable performance for all the users. Average Jain's fairness index verse K, PT=10 dBW Average Jain's fairness index verse M, PT=13 dBW Above all, the schemes with FS can achieve better fairness performance than those with RS. In most cases, Mod-MD outperforms MD due to the fact that it emphasizes the interference among the beams. This fact can bring about apparent fairness enhancement. Moreover, the joint use of Mod-MD and FS can always achieve better performance than the other scheduling schemes. This paper addressed the user scheduling problem for the multicast transmission in HTS systems with full frequency reuse. The scheduling problem has been divided into intra-beam and inter-beam scheduling. For the intra-beam scheduling problem, a fixed-size user grouping algorithm has been proposed. For the inter-beam scheduling problem, first, the concept of equivalent CSIs has been introduced to simplify the scheduling problem by making each scheduled unit correspond to only one CSI and two forms of equivalent CSIs have been proposed. Second, according to equivalent CSIs, a scheduling algorithm aiming at improving the fairness has been proposed. The simulation results show that, in most cases, either the proposed intra-beam or inter-beam scheduling algorithm can improve the performance with respect to system spectral efficiency and user fairness. Moreover, the joint use of the two proposed algorithms can achieve a huge performance increase in the system. Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study. HTS: High throughput satellite MIMO: Multiple Input Multiple Output CSI: Channel state information Minimum mean squared error Digital video broadcasting-satellite 2 LDPC: Low density parity check SFB: Single feed per beam SINR: Signal to interference plus noise ratio NoIP: Norm of the inner product FS: fairness scheduling MaxDist Mod-MD: RS: Random scheduling D. H. Werner, S. Ganguly, An overview of fractal antenna engineering research. IEEE Antennas Propag. Mag.45(1), 38–57 (2003). E. Guariglia, Harmonic sierpinski gasket and applications. Entropy. 20(9), 1–17 (2018). K. C. Hwang, A modified Sierpinski fractal antenna for multiband application. IEEE Antennas Wirel. Propag. Lett.6:, 357–360 (2007). E. Guariglia, Entropy and fractal antennas. Entropy. 18(3), 1–12 (2016). R. G. Hohlfeld, N. Cohen, Self-similarity and the geometric requirements for frequency independence in antennae. Fractals. 07(01), 79–84 (1999). https://doi.org/10.1142/S0218348X99000098. E. 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Mignone, DVB-S2X: extending DVB-S2 flexibility for core markets and new applications. Int. J. Satell. Commun. Netw.34(3), 327–336 (2016). https://doi.org/10.1002/sat.1157. D. Christopoulos, S. Chatzinotas, B. Ottersten, Weighted fair multicast multigroup beamforming under per-antenna power constraints. IEEE Trans. Signal Process.62(19), 5132–5142 (2014). https://doi.org/10.1109/tsp.2014.2345340. D. Christopoulos, S. Chatzinotas, B. Ottersten, in Proc. IEEE International Conference on Communications (ICC2014). Multicast multigroup beamforming under per-antenna power constraints (IEEESydney, Australia, 2014), pp. 4704–4710. https://doi.org/10.1109/ICC.2014.6884064. G. Taricco, in Proc. European Wireless Conference'2014. Linear precoding methods for multi-beam broadband satellite systems (VDEBarcelona, Spain, 2014), pp. 1–6. J. Han, M. Kamber, J. Pei, in Data Mining (Third Edition), ed. by J. Han, M. Kamber, and J. Pei. Getting to know your data (Morgan KaufmannBoston, 2012), pp. 39–82. https://doi.org/10.1016/B978-0-12-381479-1.00002-2 Chap. 2. N. D. Sidiropoulos, T. N. Davidson, Z. Q. Luo, Transmit beamforming for physical-layer multicasting. IEEE Trans. Signal Process.54(6), 2239–2251 (2006). https://doi.org/10.1109/Tsp.2006.872578. A. I. Aravanis, M. R. B. Shankar, P. Arapoglou, G. Danoy, P. G. Cottis, B. Ottersten, Power allocation in multibeam satellite systems: a two-stage multi-objective optimization. IEEE Trans. Wirel. Commun.14(6), 3171–3182 (2015). https://doi.org/10.1109/twc.2015.2402682. This work was supported by the National Natural Science Foundation of China (No.61671183, 61771163). National Natural Science Foundation of China under grant numbers 61671183 and 61771163. Harbin Institute of Technology, No.2 Yikuang Street, Harbin, China Shuo Zhang, Min Jia, Yuming Wei & Qing Guo Shuo Zhang Min Jia Yuming Wei SZ and QG put forward the idea of this paper. SZ finished the design of the study and the algorithms. YW and MJ contributed to the experimental work and the data analysis. SZ and MJ drafted the manuscript. The authors read and approved the manuscript. Correspondence to Qing Guo. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. Zhang, S., Jia, M., Wei, Y. et al. User scheduling for multicast transmission in high throughput satellite systems. J Wireless Com Network 2020, 133 (2020). https://doi.org/10.1186/s13638-020-01749-7 User scheduling Radar and Sonar Networks	CommonCrawl
5. Rational Functions & Expressions Investigation: Modeling tsunami movement 5.01 Simplifying rational expressions 5.02 Adding and subtracting rational expressions 5.03 Multiplying and dividing rational expressions 5.04 Solving equations with rational expressions 5.05 Graphical and analytical solutions to rational equations United States of AmericaMO Simplifying by factoring We already learned how to simplify fractions that only involve numbers, and algebraic fractions work exactly the same way. Simply find common factors (greatest common factor [GCF] is the fastest way) between the denominator and the numerator and cancel them out until you can't find any more. For example, $\frac{49st^2}{42t}$49st242t involves numbers and the two variables $s$s and $t$t, so let's look at them all separately. Just looking at the numbers, we see that $7$7 is the $GCF$GCF between $49$49 and $42$42, so we can take it out from both to leave $7$7 and $6$6, respectively. As for $s$s terms, there are no common factors between the denominator and numerator except $1$1 so we leave them. Lastly the $t$t terms have a GCF of $t$t, so we are left with $t$t on the top and $1$1 on the bottom after canceling out. So in the end we should have $\frac{7st}{6}$7st6 after simplifying. All of the factoring methods we have learned in the past can greatly help us simplify complicated algebraic fractions. There is, however, something that you must keep in mind. An algebraic fraction will be undefined for any value of the variables that would make the denominator of the fraction equal to zero. So, these values must be restricted from use when you are working with rational expressions. When we simplify the fraction $\frac{84}{270}$84270 to $\frac{14}{45}$1445, our simplified fraction is fully equivalent. We could go backwards from $\frac{14}{45}$1445 to express the fraction as $\frac{84}{270}$84270 again. However, when we simplify, say, $\frac{3\left(x-4y\right)}{x-4y}$3(x−4y)x−4y to $3$3 by canceling a common factor of $x-4y$x−4y, our simplified answer is not fully equivalent to what we started with. This is because in $\frac{3x-12y}{x-4y}$3x−12yx−4y, the denominator cannot equal zero, so we cannot choose $x$x and $y$y such that $x-4y=0$x−4y=0. But when we cancel out the factor $x-4y$x−4y, we lose this piece of information. Worked examples Factor and simplify $\frac{x^2-6x+9}{x-3}$x2−6x+9x−3 Think: about whether the numerator can be factored using quadratic methods or perfect square methods $x^2-6x+9$x2−6x+9 can be factored using the perfect square method as $9$9 is a square number, and $6$6 is double $\sqrt{9}=3$√9=3 Therefore it becomes $\left(x-3\right)^2$(x−3)2 So our fraction can be rewritten as: $\frac{\left(x-3\right)^2}{x-3}=x-3$(x−3)2x−3=x−3 as $x-3$x−3 is a common factor of the numerator and denominator Reflect: We would have the restriction that $x\ne3$x≠3 because $x-3$x−3 was in the denominator and $x-3\ne0$x−3≠0. Factor and simplify $\frac{y+4}{y^2-3y-28}$y+4y2−3y−28 Think about which method to use for the denominator and how the negative $-28$−28 will affect it $y^2-3y-28$y2−3y−28 is a quadratic trinomial but not a perfect square as $-28$−28 is not a square number Its negativity also means the two numbers $a$a and $b$b we need to find in $\left(y+a\right)\left(y+b\right)$(y+a)(y+b) have different signs. Number pairs that give us $-28$−28 are: $1$1 & $-28$−28, $-1$−1 & $28$28, $2$2 & $-14$−14, $-2$−2 & $14$14, $4$4 & $-7$−7, $-4$−4 & $7$7 The only pair to have a sum of $-3$−3 is $4$4 & $-7$−7, which must be our $a$a and $b$b $\left(y+4\right)\left(y-7\right)$(y+4)(y−7) must then be the factored form Our fraction then becomes $\frac{y+4}{\left(y+4\right)\left(y-7\right)}=\frac{1}{y-7}$y+4(y+4)(y−7)=1y−7 as $\frac{y+4}{y+4}=1$y+4y+4=1. We have the restriction that $y\ne4$y≠4 from the original denominator. Factor $\frac{6x-16}{12}$6x−1612 and simplify. Factor and simplify $\frac{50m^2+70mn}{80m^2}$50m2+70mn80m2. Factor and simplify $\frac{a^2-81}{9-a}$a2−819−a. Here are a few more involved examples. Simplify $\frac{x^2+5x+4}{x^2-2x-3}$x2+5x+4x2−2x−3. Think: We factor the numerator and the denominator to obtain $\frac{(x+1)(x+4)}{(x+1)(x-3)}$(x+1)(x+4)(x+1)(x−3). The factor $x+1$x+1 is common. Do: So, we cancel it to see that $\frac{x^2+5x+4}{x^2-2x-3}=\frac{x+4}{x-3}$x2+5x+4x2−2x−3=x+4x−3 This equivalence is correct except when $x=-1$x=−1 for which case the original fraction was undefined, so we can say: $\frac{x^2+5x+4}{x^2-2x-3}=\frac{x+4}{x-3}$x2+5x+4x2−2x−3=x+4x−3, where $x\ne-1$x≠−1 Simplify $\frac{x^2+xy+xz+yz}{x^2+2xy+y^2}$x2+xy+xz+yzx2+2xy+y2. Factor and simplify $\frac{3a^2+24a+45}{9a^2-81}$3a2+24a+459a2−81 Further simplifying fractions with polynomial expressions Remember when factoring algebraic expressions to first try taking out the greatest common factor (GCF) from all terms. Recall that we have many different strategies for factoring polynomial expressions such as difference of squares, perfect squares or grouping. Simplify $\frac{16x^2+40x+25}{25-16x^2}$16x2+40x+2525−16x2. Think: Firstly, how do we factor the numerator $16x^4+40x^3+25x^2$16x4+40x3+25x2? First we notice, there is a common factor of $x^2$x2, so we are really trying to factor $16x^2+40x+25$16x2+40x+25. We are looking for a binomial factoring $\left(ax+b\right)\left(cx+d\right)$(ax+b)(cx+d) for some integers $a$a, $b$b, $c$c, $d$d. The product of the constant terms $bd$bd will have to equal $25$25. Since $25$25 factors to $5\times5$5×5, we should try $b=5$b=5 and $d=5$d=5 first, which would give $\left(ax+5\right)\left(cx+5\right)$(ax+5)(cx+5). Now, can we find the $x$x coefficients $a$a and $c$c such that this distributes to $16x^2+40x+25$16x2+40x+25? The product of these will be $16$16, so let's test $a=4$a=4 and $c=4$c=4 first. $\left(4x+5\right)\left(4x+5\right)=16x^2+20x+20x+25$(4x+5)(4x+5)=16x2+20x+20x+25 so does indeed equal $16x^2+40x+25$16x2+40x+25 when distributed. Hence our factoring is correct. Note that you could have also figured out this factoring by using the quadratic formula to find the zeros of the quadratic. What about the denominator? Well, notice that $25-16x^2$25−16x2 is a difference of two squares so can be factored to $\left(5-4x\right)\left(5+4x\right)$(5−4x)(5+4x). Let's now use these factorisations to simplify the fraction. $\frac{16x^4+40x^3+25x^2}{25-16x^2}$16x4+40x3+25x225−16x2 $=$= $\frac{x^2\left(16x^2+40x+25\right)}{25-16x^2}$x2(16x2+40x+25)25−16x2 Take out the common factor of $x^2$x2. $=$= $\frac{x^2\left(4x+5\right)\left(4x+5\right)}{\left(5-4x\right)\left(5+4x\right)}$x2(4x+5)(4x+5)(5−4x)(5+4x) Using our factorisations $=$= $\frac{x^2\left(4x+5\right)}{5-4x}$x2(4x+5)5−4x Cancel the common factors $4x+5$4x+5 and $5+4x$5+4x, which are equivalent Reflect: We have the restriction that $4x+5\ne0$4x+5≠0, so $x\ne\frac{-5}{4}$x≠−54. Simplify $\frac{6\left(2k^2-5\right)^4-10k\left(2k^2-5\right)^5}{8\left(2k^2-5\right)^8}$6(2k2−5)4−10k(2k2−5)58(2k2−5)8. Think: We want to think of$2k^2-5$2k2−5 as a group. $\frac{6\left(2k^2-5\right)^4-10k\left(2k^2-5\right)^5}{8\left(2k^2-5\right)^8}$6(2k2−5)4−10k(2k2−5)58(2k2−5)8 $=$= $\frac{2\left(2k^2-5\right)^4\left(3-5k\left(2k^2-5\right)\right)}{8\left(2k^2-5\right)^8}$2(2k2−5)4(3−5k(2k2−5))8(2k2−5)8 $=$= $\frac{\left(2k^2-5\right)^4\left(3-5k\left(2k^2-5\right)\right)}{4\left(2k^2-5\right)^8}$(2k2−5)4(3−5k(2k2−5))4(2k2−5)8 $=$= $\frac{3-5k\left(2k^2-5\right)}{4\left(2k^2-5\right)^4}$3−5k(2k2−5)4(2k2−5)4 Recall our exponent law that states that $b^m\div b^n=b^{m-n}$bm÷bn=bm−n to notice that dividing out $\left(2k^2-5\right)^4$(2k2−5)4 from $\left(2k^2-5\right)^8$(2k2−5)8 will leave $\left(2k^2-5\right)^4$(2k2−5)4. Finally, we distribute what remains in the numerator to get our final answer of $\frac{3+25k-10k^3}{4\left(2k^2-5\right)^4}$3+25k−10k34(2k2−5)4. A2.NQ.A.2 Create and recognize equivalent expressions involving radical and exponential forms of expressions.	CommonCrawl
EJNMMI Physics Impact of image-based motion correction on dopamine D3/D2 receptor occupancy—comparison of groupwise and frame-by-frame registration approaches Jieqing Jiao1,2, Graham E. Searle2, Julia A. Schnabel1 & Roger N. Gunn1,2,3 EJNMMI Physics volume 2, Article number: 15 (2015) Cite this article Image registration algorithms are frequently used to align the reconstructed brain PET frames to remove subject head motion. However, in occupancy studies, this is a challenging task where competitive binding of a drug can further reduce the available signal for registration. The purpose of this study is to evaluate two kinds of algorithms—a conventional frame-by-frame (FBF) registration and a recently introduced groupwise image registration (GIR), for motion correction of a dopamine D3/D2 receptor occupancy study. The FBF method co-registers all the PET frames to a common reference based on normalised mutual information as the spatial similarity. The GIR method incorporates a pharmacokinetic model and conducts motion correction by maximising a likelihood function iteratively on tracer kinetics and subject motion. Data from eight healthy volunteers scanned with [11C]-(+)-PHNO pre- and post-administration of a range of doses of the D3 antagonist GSK618334 were used to compare the motion correction performance. The groupwise registration achieved improved motion correction results, both by visual inspection of the dynamic PET data and by the reduction of the variability in the outcome measures, and required no additional steps to exclude unsuccessfully realigned PET data for occupancy modelling as compared to frame-by-frame registration. Furthermore, for the groupwise method, the resultant binding potential estimates had reduced variation and bias for individual scans and improved half maximal effective concentration (EC50) estimates were obtained for the study as a whole. These results indicate that the groupwise registration approach can provide improved motion correction of dynamic brain PET data as compared to frame-by-frame registration approaches for receptor occupancy studies. Dynamic PET brain scans are susceptible to head motion that distorts the tissue-to-voxel mapping, and this leads to degraded PET images from acquisitions that can last up to 2 h. If uncorrected, motion-induced attenuation correction mismatch, inter-frame misalignment and intra-frame blurring in the PET data will make the quantification of the tracer kinetic data unreliable [1]. Previous approaches to the motion problem have included the use of head restraints or external motion tracking systems that have been developed to try to record the motion parameters [2–5]. However, these methods have limitations either due to patient discomfort, accuracy or ease of use. Meanwhile, image-based computational methods have been developed to establish the spatial correspondences between PET data at different time frames [6–8], allowing for post-acquisition corrections to be applied. Such methods perform a rigid frame-by-frame (FBF) image registration of PET time frames to a common reference image, which is usually a PET image derived from a single frame, a weighted sum of frames or an associated magnetic resonance (MR) image for the subject. The FBF registration methods have been widely used for motion correction in recent studies [9–12] due to the ease of implementation based on existing publicly available image registration software packages, such as Statistical Parametric Mapping (SPM) (used in [13]), AIR (used in [14]), FLIRT (used in [15]) and the commercial software PMOD. FBF methods are shown to improve the integrity of PET data but have potential limitations that need consideration. For example, the FBF registration is solely based on maximising the spatial similarities between images and it can converge to inaccurate solutions in the presence of noise [16]. Recently, a groupwise image registration (GIR) framework for dynamic PET data has been introduced for motion correction [17, 18]. The GIR method enables noise modelling and accounts for tracer kinetics by incorporating a pharmacokinetic model with either an arterial input function (AIF) [18] or a reference tissue input function [17]. Improved registration results as compared to FBF methods have been demonstrated in simulation-based validations. This work aims to evaluate the motion correction performance of these image-based methods on an occupancy study where the competitive binding can impose further challenges for conducting image registration on the PET images. Data from a dopamine D3/D2 receptor occupancy study with [11C]-(+)-PHNO were used. The study was designed to measure the half maximal effective concentration (EC50) of the D3 antagonist GSK618334. For the reconstructed PET time frames, three separate approaches to motion were applied: (1) no motion correction, (2) frame-by-frame motion correction and (3) groupwise image registration motion correction with a reference tissue input. Following motion correction, kinetic analysis was applied to each dataset to derive regional binding potential estimates for each scan and then modelling of the competitive binding of the drug to derive its EC50 was performed using all scans in the study. Human [11C]-(+)-PHNO PET study The motion correction algorithms were evaluated on [11C]-(+)-PHNO PET occupancy data involving a range of doses of the antagonist GSK618334. Data from eight subjects, from a previously reported study [19], were used here. All subjects were healthy, males, drug-free, non-smoking volunteers, aged between 25 and 55 years and with body weights and BMI in the normal range. All subjects gave written informed consent, and their eligibility was confirmed via medical history, physical examinations and standard tests. Further details of the inclusion and exclusion criteria can be found on www.clinicaltrials.gov by reference to NCT00814957, and the study was approved by East of England Hatfield REC (known as NRES Committee East of England—Welwyn at the time of the study). Each subject received a baseline PET scan, then a single oral dose of 5–550 mg of GSK618334 followed by two further PET scans performed between 1.5 and 29 h post-administration of GSK618334. Venous blood was sampled for measurements of GSK618334 plasma concentration. The [11C]-(+)-PHNO PET scans were acquired using a Siemens Biograph 6 PET-CT with Truepoint gantry in 3D mode and then reconstructed using filtered back projection with corrections for dead time, random coincidences, variations in detector sensitivity, attenuation (based on a low-dose CT acquisition) and scatter. The reconstruction had the measured resolution of 9 mm (transaxial) and 7 mm (axial) in full width at half maximum at the centre of the field of view, and after reconstruction, the PET images were filtered with a Gaussian filter of 5 mm in full width at half maximum in the three orthogonal planes. Dynamic data were acquired using 26 frames (durations 8 × 15 s, 3 × 1 min, 5 × 2 min, 5 × 5 min, 5 × 10 min). Arterial blood data was also acquired and enabled plasma input-based modelling to be applied for determination of the final binding potential BPND regional outcome measures of interest. Each subject also received a high-resolution T1-weighted MR scan with a Siemens Tim Trio 3T scanner (Siemens Healthcare, Erlangen, Germany). A U-shaped head holder with foam padding designed to snugly hold the subject's head in place laterally, with a soft Velcro strap across the forehead to aid as a reminder to the subject, was used in this study. FBF motion correction The FBF method co-registers all the PET frames to a common reference based on spatial similarity. The PET frame acquired between 13 and 15 min in the scan was used as the reference frame, and normalised mutual information (NMI) was used as the cost function. The settings of the FBF method used in this work have been optimised in our previous internal investigations. Frames 13–15 min were selected based on previous (unpublished) work which optimised the method by evaluating the performance using a range of possible reference frames for [11C]-(+)-PHNO. The image of [11C]-(+)-PHNO at this time in the scan contains features that are common to both early distribution and later binding phases. Although the duration of this frame (2 min) is short, it balances the desire to select a frame with minimal motion whilst capitalising on the high imaging statistics at this time in conjunction with representation of all the key image features. The FBF algorithm was implemented in MATLAB 7.7, using functions available within the Statistical Parametric Mapping (SPM8, http://www.fil.ion.ucl.ac.uk/spm/) with the default settings for the optimisation, interpolation, image smoothing and histogram smoothing. GIR motion correction The proposed GIR algorithm conducts motion correction by solving the maximum likelihood problem. Assuming the measurement is distributed as a multivariate Gaussian, the likelihood of the measured dynamic PET data Y can be formulated as $$ L\left(\varPhi, \mathbf{T};\mathbf{Y}\right)={\displaystyle \prod_{j=1}^M{\displaystyle \prod_{k=1}^F\frac{1}{\sqrt{2\pi {\sigma}^2\left({\mathbf{x}}_j,k\right)}} \exp \left(-\frac{{\left(\mathbf{Y}\left({\mathbf{T}}_k^{-1}\left({\mathbf{x}}_j\right),{t}_k\right)-{\mathbf{Y}}_{\varPhi}\left({\mathbf{x}}_j,{t}_k\right)\right)}^2}{2\pi {\sigma}^2\left({\mathbf{x}}_j,k\right)}\right)}} $$ where Y Φ is the predicted PET data determined by the tracer kinetic parameter Φ, T k is the spatial transformation that corrupts the voxel-to-tissue mapping for the kth frame, σ 2(x j , k) is the variance term describing the measurement noise level and M and F are the numbers of voxels and time frames, respectively. The unknown T and Φ can be optimised iteratively until convergence. For brain images, T describes the rigid head motion using three translations and three rotations. Y Φ can be described by the generalised reference tissue model embedded in a basis function framework and solved by the method of basis pursuit denoising [20] as follows. Let C T (t) be the tracer concentration time course in the target tissue, then C T (t) can be expressed as an expansion on a basis as $ {C}_T(t)={\phi}_0{C}_R(t)+{\displaystyle {\sum}_{i=1}^N{\phi}_i{\psi}_i}, $ where $ {\psi}_i={e}^{-{\theta}_it}\otimes {C}_R(t) $ and C R (t) is the tracer concentration time course in the reference tissue. A discrete set of values can be selected for θ i from a physiologically plausible range spaced in a logarithmic manner to elicit a suitable coverage of the kinetic spectrum. The measured PET data Y(x, t) corresponds to C T (t) as $$ \mathbf{Y}\left({\mathbf{x}}_j,{t}_k\right)=\frac{1}{t_k^e-{t}_k^s}{\displaystyle {\int}_{t_k^s}^{t_k^e}{C}_T(t)dt=}{\mathbf{Y}}_{k,j}, $$ where $ {t}_k^s $ and $ {t}_k^e $ are the start and end times for the kth frame (k = 1 ⋯ F). Accordingly, the basis functions can be written as $$ \begin{array}{c}\hfill {\varPsi}_{0,k}=\frac{1}{t_k^e-{t}_k^s}{\displaystyle {\int}_{t_k^s}^{t_k^e}{C}_R(t)dt}\hfill \\ {}\hfill {\varPsi}_{i,k}=\frac{1}{t_k^e-{t}_k^s}{\displaystyle {\int}_{t_k^s}^{t_k^e}{e}^{-{\theta}_it}\otimes {C}_R(t)dt},\kern0.5em i=1\cdots N\hfill \end{array} $$ The unknown tracer kinetic parameter matrix of the image Φ can then be determined by solving Y ≅ ΨΦ. In practice, to account for the uncertainty of the measurements, the weighted least squares problem $$ {\mathbf{W}}^{\frac{1}{2}}\mathbf{Y}\cong {\mathbf{W}}^{\frac{1}{2}}\varPsi \varPhi $$ can be considered, where W is the inverse of the covariance matrix corresponding to the noise variance term σ 2 in Eq. 1. The noise variance for decay-corrected PET data can be modelled as $ {\sigma}^2(k)={\displaystyle \sum_j\mathbf{Y}\left({\mathbf{x}}_j,{t}_k\right)}/\left({t}_k^e-{t}_k^s\right)\times dcf(k), $ derived from the variance model 1 in [21], where $ dcf(k)=\lambda \left({t}_k^e-{t}_k^s\right)/\left[ exp\left(-\lambda {t}_k^s\right)- exp\left(-\lambda {t}_k^e\right)\right] $ is the decay correction function and λ is the decay constant of the isotope. Given the independence of the frames, W is diagonal and can be calculated as W kk = 1/σ 2(k). The basis is typically overcomplete (N > F − 1), leading to an under-determined set of equations that basis pursuit denoising solves with the addition of a 1-norm penalty term [20] $$ { \min}_{\varPhi }{\left\Vert {\mathbf{W}}^{1/2}\mathbf{Y}-{\mathbf{W}}^{1/2}\varPsi \varPhi \right\Vert}_2^2+\mu {\left\Vert \varPhi \right\Vert}_1 $$ Here, μ > 0 is a regularisation parameter which balances the approximation error and sparseness of Φ and imposes a unique solution. To avoid the difference in the scales, the basis functions are normalised here so that ‖Ψ i ‖2 = 1 for all i. We previously proposed an efficient way to determine the value for μ [17], and based on this approach, we use a value of μ = 8.69 for [11C]-(+)-PHNO. This general reference tissue kinetic model is used as the pharmacokinetic model in the GIR method. It constrains the registration in a groupwise fashion by using the temporal information. The complete algorithm is summarised in Fig. 1. Step 1 initialises the algorithm using the identity function for T. In step 2, the discrete reference data is first extracted from the motion-corrected PET data Y(T − 1(x), t) as a regional time activity curve from the anatomical reference region of choice, and the reference input function C R (t) is generated using linear interpolation. For the purpose of describing the tracer kinetics, rather than estimating the absolute parameters for binding or uptake, a region with low specific binding is mathematically appropriate for deriving the reference input. Step 3 calculates the basis functions which are convolutions of the reference tissue input and the pre-defined exponentials. Step 4 solves the kinetic model fitting via basis pursuit denoising. In step 5, the original motion-corrupted PET data Y(x, t) is registered to the model-predicted PET data Y Φ to update the motion estimation T and motion-corrected PET data Y(T − 1(x), t). Steps 2–5 repeat until convergence, and the algorithm returns the motion-corrected PET data Y(T − 1(x), t) in step 6. Schematic illustration of the proposed groupwise image registration (GIR) algorithm for motion correction. With the initialisation in step 1, the algorithm repeats steps 2–5 until convergence and returns the final motion corrected dynamic PET data in step 6 In this work, for kinetic modelling in the GIR motion framework, the reference input function was derived from the grey matter of the cerebellum for the [11C]-(+)-PHNO PET data. The reference region was delineated via nonlinear registration using SPM8 (http://www.fil.ion.ucl.ac.uk/spm) of a predefined brain atlas [22] to the subject's MRI (aligned to the PET image) to propagate the segmentation. Regional calculation of [11C]-(+)-PHNO BPND Regional analysis of BPND and occupancy analysis were performed after (1) no motion correction, (2) FBF motion correction and (3) GIR motion correction with reference tissue input. The [11C]-(+)-PHNO kinetics were analysed for six regions-of-interest (ROIs): substantia nigra (SN), globus pallidus (GP), ventral striatum (VST), dorsal caudate (CD), dorsal putamen (PU) and thalamus (TH). These target ROIs were defined manually according to guidelines described previously [22]. GP, VST, CD, PU and TH were drawn on each subject's structural T1-weighted magnetic resonance imaging (MRI). The MR T1-weighted image was registered to the time-weighted integral of the dynamic PET images following motion correction using the rigid registration function in SPM8 with a mutual information cost function. SN was defined on each subject's baseline PET integral image given the insufficient contrast available from MRI data. Regional time-activity curves (TACs) were then derived for each ROI. Subsequently, a two-tissue compartmental (2TC) plasma input model was applied to the regional time activity curves to appropriately quantify regional [11C]-(+)-PHNO volume of distribution (V T ) estimates in the basal ganglia ROIs [19]. This included a fixed blood volume correction of 5 %. Regional $ {\mathrm{BP}}_{\mathrm{ND}}^{\mathrm{ROI}} $ estimates were then derived for each of the target regions using the cerebellum as the reference region, $$ {\mathrm{BP}}_{\mathrm{ND}}^{\mathrm{ROI}}=\frac{V_T^{\mathrm{ROI}}-{V}_T^{\mathrm{CER}}}{V_T^{\mathrm{CER}}} $$ Competitive binding of drug and PHNO The [11C]-(+)-PHNO occupancy study was designed to measure the dopamine D3 and D2 receptor occupancy of GSK618334 and requires the application of a two-site competitive binding model [19]. Given the baseline binding potential, $ {\mathrm{BP}}_{\mathrm{ND}}^{\mathrm{base}} $, the binding potential following drug administration, $ {\mathrm{BP}}_{\mathrm{ND}}^{\mathrm{drug}} $, and the plasma concentration of the drug (GSK618334), $ {C}_p^{\mathrm{drug}} $, then, $$ {\mathrm{BP}}_{\mathrm{ND}}^{\mathrm{drug}}={\mathrm{BP}}_{\mathrm{ND}}^{\mathrm{base}}\left(\frac{f_{\mathrm{PHNO}}^{\mathrm{D}3}}{1+\frac{C_p^{\mathrm{drug}}}{{\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}3}}}+\frac{1-{f}_{\mathrm{PHNO}}^{\mathrm{D}3}}{1+\frac{C_p^{\mathrm{drug}}}{{\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}2}}}\right), $$ where $ {f}_{\mathrm{PHNO}}^{\mathrm{D}3} $ is the regional fraction of baseline [11C]-(+)-PHNO BPND corresponding to D3 binding with values of 0.87 for SN, 0.66 for GP, 0.39 for VST, 0.69 for TH, 0.21 for CD and 0.14 for PU [19], $ {\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}3} $ and $ {\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}2} $ are the plasma concentrations of the drug (GSK618334) that would result in 50 % occupancy of the D3 and D2 receptors, respectively. Recent work has demonstrated that for true quantification, it is necessary to account for mass effects of [11C]-(+)-PHNO itself and a small displaceable specific signal in the cerebellum in addition to competitive binding of the drug at D3 and D2 sites [19]. For the actual fitting of the [11C]-(+)-PHNO BPND data, we used an extension of the competitive binding model in Eq. 6 that includes corrections for PHNO mass dose effect at the D3 sites and cerebellum specific binding [19]. This model is given in Additional file 1. Note that when modelling the competitive binding, all regions were fitted simultaneously and the $ {\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}3} $ and $ {\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}2} $ parameters assumed constant across all regions and subjects [19]. Motion correction of [11C]-(+)-PHNO data The GIR and FBF motion correction algorithms were applied on reconstructed [11C]-(+)-PHNO PET data to address inter-frame misalignment caused by subject motion. PET data that had already been attenuation corrected was used due to the clinical pipeline and the use of PET data without attenuation correction will be considered in the "Discussion" section. Firstly, visual inspection using the movie mode in FSLview (http://fsl.fmrib.ox.ac.uk/fsl/fslview/) was performed on the 24 scans obtained from the eight healthy subjects without motion correction to derive an initial qualitative assessment of motion. In four scans, there was severe motion with up to 10° rotations or 40 mm translations. In eight scans, there was motion at the level of the voxel size (2 mm), and in 12 scans, the motion was difficult to detect. For the motion correction algorithms, a metric summarising the displacement was calculated for each time frame by using the estimated translation and rotation parameters, $$ \mathrm{Displacement}=\sqrt{\frac{1}{M}{{\displaystyle {\sum}_{j=1}^M\left\Vert \mathbf{T}\left({\mathbf{x}}_j\right)-{\mathbf{x}}_j\right\Vert}}^2}, $$ where T is the rigid transformation determined by the translations and rotations, M is the number of all the voxels and x j is the coordinate of voxel j. Figure 2 shows a summary of the displacements introduced by both FBF and GIR motion correction algorithms for the 24 scans. Whilst the individual reference frames for both the FBF and GIR methods may be different, we are interested in comparing the distributions of the displacements which should be insensitive to this. Summary of displacements introduced by FBF and GIR MC methods for [11C]-(+)-PHNO PET occupancy study data. In each box, the central mark denotes the median, the edges of the box are the 25th and 75th percentiles, the whiskers extend to the most extreme data points not considered to be outliers, and outliers are plotted individually using the symbol +. Subjects (S1–S8) exhibited various degrees of motion during the scans. The scans marked with grey background had visually negligible motion following assessment by an observer viewing the data in the movie mode in FSLview. For these scans, the FBF method introduced up to 5 mm displacement, whereas the displacement introduced by the GIR method was at a sub-voxel level. Plasma concentration of GSK618334 is also shown for each scan The computation time of the GIR motion correction algorithm depended on the amplitude of the motion and the image noise. On a desktop workstation (CPU 3.20 GHz, 16 GB RAM) with MATLAB 7.7, the GIR algorithm took between 20 and 90 min of computation time for each 26-frame dynamic image; the FBF algorithm took in general 60 min. No GPU or parallel computing was applied in this work. Figures 3 and 4 illustrate the performance of the GIR and FBF algorithms when registering [11C]-(+)-PHNO PET data from subjects with visually obvious motion. The motion artefact was well corrected by the GIR algorithm as indicated by the sagittal view of the PET data. Furthermore, for the baseline scan, the voxel-based TACs from dorsal caudate and globus pallidus shown on the sagittal slice are displayed before and after GIR motion correction together with the normalised population TACs for these ROIs. The population TACs were generated by averaging the baseline [11C]-(+)-PHNO PET data after motion correction over the eight healthy subjects and were scaled according to dose and subject weight. The consistency with the normalised population data after MC by the GIR algorithm provides supporting evidence for the successful removal of the inter-frame misalignment caused by motion. Selected temporal frames from a sagittal slice from subject 2's baseline [11C]-(+)-PHNO data. Times are mid-frame times. a Before motion correction, b after motion correction by the conventional FBF algorithm and c after motion correction by the proposed GIR algorithm. Units are kBq/ml. d TACs from voxels in dorsal caudate and globus pallidus, depicted in colours corresponding to the voxels shown in a, b and c, which are spatially fixed to demonstrate the displacement. The subject exhibited obvious rotation of ~10°, as shown on the sagittal slices in a, which was corrected by the proposed method, as shown in c. The TACs for these regions were also obtained from all eight healthy subjects' baseline data after motion correction, and the population TACs were generated by averaging TACs normalised for [11C]-(+)-PHNO dose and subject weight. These population TACs were scaled to match subject 2's baseline data and are shown in d. The tracer kinetics showed consistency with the population data after GIR motion correction. The scan had no GSK618334 taken, and the [11C]-(+)-PHNO injected activity was 384.9 MBq (injected mass 4.24 μg) Selected temporal frames from a sagittal slice from subject 2's follow-up [11C]-(+)-PHNO data. Times are mid-frame times. a Before motion correction, b after motion correction by the conventional FBF algorithm and c after motion correction by the proposed GIR algorithm. Units are kBq/ml. The GSK618334 plasma concentration in the scan was 58.2 ng/ml, and the [11C]-(+)-PHNO injected activity was 139.8 MBq (injected mass 2.35 μg) Binding potential of [11C]-(+)-PHNO Regional estimates of [11C]-(+)-PHNO BPND were derived for baseline and post-GSK618334 PET scans for SN, GP, VST, CD, PU and TH before and after motion correction. During the 2TC kinetic parameter estimation, two unrealistically large V T values were obtained for the data with no motion correction, three unrealistically large V T values were obtained after motion correction by FBF, whereas none were obtained after MC by GIR. For the scans where there were unrealistic values of V T , the motion correction error of FBF was not always visually detectable, suggesting that small residual motion can introduce significant errors into V T particularly for regions with slower kinetics such as the globus pallidus and ventral striatum. Corresponding BPND values for baseline and post-dose scans are shown in Fig. 5a with unrealistic values shown above the line breaks. After excluding the unrealistic data points considered as convergence failure, the inter-subject variability was assessed on baseline BPND by the coefficient of variation (CV) Competitive binding analysis of [11C]-(+)-PHNO data. a Fits (shown as curves) to the baseline and post-dose data before and after motion correction by FBF and GIR (shown as circles, crosses and stars) using a competitive binding model with unweighted BPND data. The unweighted sum of squared differences (SSQ) of the competitive model fitting was calculated for each BPND data set and was then scaled to the SSQ of data before motion correction so that SSQ_motion = 1. Other than for the GIR approach, all methods resulted in some unrealistic estimates of BPND that affected the fits. b Competitive model fits of BPND data points derived from PET data before MC with removal of unrealistic values; after MC by FBF with the removal of unrealistic values; after only MC by the proposed GIR $$ \mathrm{C}\mathrm{V}=\frac{\sigma \left({\mathrm{BP}}_{\mathrm{ND}}\right)}{\mu \left({\mathrm{BP}}_{\mathrm{ND}}\right)} $$ where σ and μ are the standard deviation and mean across the eight subjects, respectively. The CV values before motion correction, after motion correction by the FBF algorithm and by the proposed GIR algorithm are shown in Table 1, together with the ROI size in cubic centimetres. These data provide further evidence for improved registration with GIR through the significant reduction in the CV of baseline BPND data across all regions. It was also apparent that the conventional FBF algorithm could lead to subsequent convergence problems in the kinetic fitting, whereas the proposed GIR algorithm avoided such problems, thus eliminating the need to subjectively exclude outliers due to unsuccessful motion correction. The conventional FBF algorithm produced less of a reduction in CV (even after excluding unrealistic data). Table 1 The inter-subject variability (CV) in baseline BPND before and after motion correction (MC) Occupancy and estimation of the EC50 of GSK618334 The BPND of [11C]-(+)-PHNO measured in the follow-up PET data, after dosing with GSK618334, was modelled using the extension of Eq. 6 that is provided in the Additional file 1. The two-site competitive binding model, including correction for PHNO mass on D3 binding and a small specific signal in the cerebellum, was applied to the measured data before and after motion correction. The BPND values obtained before motion correction, and after applying the FBF and GIR algorithms, are shown in Fig. 5a for each of the six target ROIs, together with the competitive binding model fits. Motion correction using the proposed GIR algorithm avoided the convergence problems that led to data points with unrealistic values, which occurred with uncorrected and FBF corrected PET data. In practice, to maintain the integrity at the study level, it is possible (though not ideal) to remove the outliers with appropriate testing. Here, we considered BPND values greater than ten as outliers. In Fig. 5b, the competitive binding model fits are shown using BPND before motion correction with outliers excluded, BPND after FBF with outliers excluded, and BPND estimates obtained directly from the GIR algorithm with no exclusions. Even with all this extra help for the other methods, GIR still produces the best fit to the competitive binding data as judged by its ability to achieve the smallest SSQ. The primary outcome measures of this study, $ {\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}3} $ and $ {\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}2} $ of GSK618334, estimated using Eq. 6, are presented along with 95 % confidence intervals in Table 2. Table 2 $ {\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}3} $ and $ {\mathrm{EC}}_{50}^{\mathrm{drug},\mathrm{D}2} $ for GSK618334 estimated before and after motion correction (MC) The EC50 estimates obtained following GIR are in the range of the values obtained before motion correction with the removal of unrealistic BPND data points (the ones with significant subject motion) and have smaller confidence intervals. Image-based registration methods are frequently used in brain PET studies to minimise the impact of subject movement on the outcome measures derived from tracer kinetic analysis. In this work, we have investigated and evaluated the applicability of two such motion correction algorithms for dynamic PET data obtained as part of a clinical dopamine D3/D2 receptor occupancy study with GSK618334. This involved a more traditional FBF approach along with a novel GIR method that we have recently introduced. The GIR approach incorporates a pharmacokinetic model into the registration process so as to provide additional temporal constraints in the registration process over and above just spatial image similarity maximisation. The input function for the pharmacokinetic model is derived directly from the tomographic PET data using a reference region and therefore the GIR method does not necessarily require any arterial blood sampling. We hypothesised that the application of a spatio-temporal (GIR) method that makes better use of the available data would lead to improved results over the purely spatial (FBF) method. The performance of the FBF and GIR methods was evaluated using data from a dopamine D3/D2 receptor occupancy study in humans with [11C]-(+)-PHNO. In the PET data, there were different levels of subject motion and a range of signal-to-noise ratios (SNR) due to competitive binding of the drug at varying doses. The performance was assessed directly by visual inspection of the PET data and indirectly by assessing the inter-subject variability in baseline BPND, convergence and residuals of the competitive binding modelling and the drug EC50 estimation. In addition to the visually improved removal of subject motion, the GIR method led to more reliable BPND estimates with reduced variation and bias at baseline and when modelling competitive binding of GSK618334 as compared to the FBF method. It also provided estimates of GSK618334 EC50 that were consistent with a previously published study that had employed outlier removal techniques but with reduced confidence intervals [19]. These convergent data all provide evidence that the proposed GIR method yields improved registration for dynamic PET data. On the study level, it increases the statistical power by reducing the motion-introduced variability, and in practice, less PET scans would be required to achieve the same outcome parameter precision once the motion correction is accurately conducted using the GIR method. The proposed GIR method uses the full dynamic data in addition to spatial similarity, and from a theoretical point of view, it should perform better than the FBF method. For the brain D3/D2 images, whilst there is limited binding data outside the striatum, there is still information available from the delivery and washout to regions just containing free and non-specifically bound tracer. Similarly, for other tracers with different distributions, areas of relatively low signal may still contribute usefully to the motion correction process. The pharmacokinetic model employed by the GIR approach is generic allowing for different compartmental topologies at individual voxels and thus should not only handle the kinetics displayed by a broad range of tracers but even different kinetic behaviour in different regions of the image. Furthermore, it is not necessary for the reference region to be devoid of the target biology, as the data can be quantified subsequently, and a region with the fastest kinetics could be used as the reference region. Thus, the method should be generally applicable to dynamic brain PET data (except perhaps when significant metabolite components contribute to the data). The generalised reference tissue model employed is able to describe a range of different behaviours, which will likely include regions outside the brain. Qualifying how well it is able to describe such data is not strictly necessary in order to assess the performance of the approach (for instance, it would not matter if it did not describe these regions particularly well if the algorithm provides improved performance in image registration over existing approaches). Future studies will explore the utility of the approach with other tracers, and further extension to the deformation motion model could also allow application to dynamic imaging outside the brain as well. In this paper, we have employed filtered back projection (FBP) for the reconstruction of the dynamic image sequence. We fully acknowledge that the application of iterative reconstruction algorithms could have improved the performance of both the FBF and GIR approaches, but an assessment of this was beyond the scope of this paper. Future studies will evaluate the impact of the reconstruction algorithm in more detail. Our hypotheses are, firstly, that the application of iterative reconstruction algorithms to the FBF method would bring its performance closer to that of the current GIR (FBP) method and, secondly, that the application of iterative reconstruction to the GIR method would further increase its performance. The reconstructed PET data used in this work represents a very common clinical workflow. In practice, the subject motion introduces mismatched attenuation and scatter correction in the reconstruction, which are theoretically challenging to eliminate with post reconstruction approaches. Addressing these issues, however, requires access to the raw PET emission data and extra fast-processing hardware/software that would impose an undesirable cost for clinical use. The approach proposed in this work is directly based on the reconstructed PET images and in the presence of intra-frame motion artefacts (attenuation, scatter etc.), it demonstrates an improvement in the kinetic analysis of dynamic data compared to alternative image-based methods. Further extension of this approach to fully account for attenuation/scatter mismatch in the PET reconstruction framework will be explored and evaluated in future work. Besides the image-based motion correction methods discussed in this work, which estimate and eliminate the subject motion using only the measured PET data, subject motion can also be tracked and corrected using additional hardware. However, such motion-tracking systems are not always available in typical clinical settings, and additional processing and calibration are required to ensure the mapping of the motion parameters from the motion-tracking space into the PET image space is accurate. The image-based methods presented provide a more accessible and less demanding way to remove the subject motion in the majority of PET studies. In summary, we have demonstrated the applicability of a novel groupwise-based imaged registration for improving the quality of data obtained from PET receptor occupancy studies, using only measured PET data. The generic nature of the incorporated pharmacokinetic model means that this should have wide utility across PET neuroimaging studies. Groupwise image-based registration of dynamic brain PET data provides an improved method to correct for subject motion. Incorporation of a reference input-based general pharmacokinetic model that requires no arterial blood sampling allows for wide applicability of the technique. The approach has value for increasing the integrity of both individual scan data and outcome measures from clinical studies involving a series of scans enabling increased precision or reduction in the required number of scans. 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Neuroimage. 2011;54(1):264–77. doi:10.1016/J.Neuroimage.2010.06.044. We would like to acknowledge GlaxoSmithKline for the provision of the data. Jieqing Jiao has received funding from the Chinese Ministry of Education—University of Oxford Scholarships, the GlaxoSmithKline Clinical Imaging Centre, and the Cancer Research UK/EPSRC Cancer Imaging Centre at Oxford. Department of Engineering Science, Institute of Biomedical Engineering, University of Oxford, Oxford, UK Jieqing Jiao, Julia A. Schnabel & Roger N. Gunn Imanova Limited, Hammersmith Hospital, 2nd Floor, Burlington Danes Building, London, UK Jieqing Jiao, Graham E. Searle & Roger N. Gunn Department of Medicine, Imperial College London, Du Cane Road, London, W12 0NN, UK Roger N. Gunn Jieqing Jiao Graham E. Searle Julia A. Schnabel Correspondence to Roger N. Gunn. Image processing and data analysis were performed by JJ and GES. JAS and RNG conceived the study, participated in its design and coordination and helped to draft the manuscript. All authors read and approved the final manuscript. Schnabel and Gunn jointly directed this work Two-site competitive binding model with corrections for PHNO mass on D3 binding and specific signal in the cerebellum. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/. Jiao, J., Searle, G.E., Schnabel, J.A. et al. Impact of image-based motion correction on dopamine D3/D2 receptor occupancy—comparison of groupwise and frame-by-frame registration approaches. EJNMMI Phys 2, 15 (2015). https://doi.org/10.1186/s40658-015-0117-0 Motion correction Groupwise registration Pharmacokinetic modelling Receptor occupancy studies	CommonCrawl
Retinal vessel model fabricated on a curved surface structure for a simulation of microcannulation Takeshi Hayakawa†1Email authorView ORCID ID profile, Ippei Kato†1, Fumihito Arai†1, Mamoru Mitsuishi2, Naohiko Sugita2, Kanako Harada2, Shinichi Tanaka2, Yasuo Noda3 and Takashi Ueta3 Received: 29 April 2016 Accepted: 31 August 2016 A remarkable number of vitreoretinal surgeries are performed each year despite their difficulty. As a result, a high demand exists for a mock-up simulator of retinal vessels to simulate these surgeries. Thus, we propose an artificial retinal vessel model for simulating microcannulation surgery. Using laser lithography, polydimethylsiloxane molding, and hydraulic transfer techniques, we fabricated microchannels approximately ≃10 μm on a 24-mm-diameter curved surface structure that mimics the human eye. In the fabrication, the channel size and wall thickness were controlled to mimic a touch of retinal vessels, which gives important information for microcannulation. We succeeded in fabrication of the proposed model and liquid circulates within the microchannels of this model without leaking. Furthermore, we demonstrated a simulation of microcannulation and measurement of applied force to the model during the simulation using a force sensor placed at the bottom of the model. The results of such experiments are useful to quantitatively evaluate medical techniques. Surgical simulator Retinal vessel 3D microfabrication Vitreoretinal surgeries have become quite common over the last few decades. For example, approximately 300,000 operations are conducted annually in Japan, and this number is predicted to increase because of the aging population. One such operation is called microcannulation, which is proposed to patients with central retinal vein occlusion [1]. In microcannulation, a surgeon inserts a micropipette into the eyeball and reaches the eye fundus, which is a curved concave structure (Fig. 1a). Next, the surgeon finds the occluded vein on the fundus and, via the micropipette, injects a thrombolytic drug into the vein to dissolve the clots. This operation requires mature surgical skills because the retinal vessel is less than 100 μm in diameter (Fig.1b), This technique is used despite human hand tremors measuring approximately 100 μm [2, 3]. Furthermore, the retinal vessels lie on the curved surface and surgeons must find the target vessel based on their senses of vision and touch. Therefore, obtaining the skill required to perform microcannulation surgery takes a long time, and the consequent long evaluation and training of young doctors is one of the drawbacks of microcannulation. Another drawback of this approach is the evaluation of new medical equipment. Several medical equipment for this difficult surgery, such as surgical robots, have been proposed in recent years [4–6]. However, large barriers currently prevent commercialization of these new instruments, including the lengthy period required for clinical trials. Thus, a demand exists for methods to reduce the time to market for such equipment and shorten the training and evaluation periods required for surgeons. Central retinal vein occlusion: a schematic of microcannulation and b optical-microscope photograph of retinal vessels At present, training and evaluation of this medical technique are done with animal samples such as swine eyes or chick embryos [4–10]. However, structural differences between individual animal samples are significant, so the conditions of evaluation or training differ for each trial. Furthermore, using such animal samples in an actual operating room presents contamination risks. Therefore, a high demand exists for a reproducible method of evaluating vitreoretinal surgery techniques that can be used in an actual operating room. One response to this demand is a surgical simulator. Basically, two types of surgical simulators exist: a computer-based virtual reality (VR) simulator [11–15], and a mock-up simulator made of artificial materials [16–20]. Once electronic data from the target parts are acquired, the VR simulator can construct the target parts with high reproducibility. However, mimicking the physical sensation of touch or the texture of a retinal vessel with the VR simulator is very difficult, and such sensations are critical for the surgeon performing vitreoretinal surgery. Additionally, the effect of new medical equipment on the human body is difficult to evaluate because interactions between the human body and medical equipment are too complex to completely reproduce with current computer technology. Thus, VR simulators are not suitable for evaluating medical techniques. The second approach of a mock-up simulator has the advantage that it can be physically touched, allowing the user to learn the requisite sensations of touch or texture. Mock-up simulators use individual samples that are highly reproducible because of their well-controlled fabrication methods. Mimicking blood flow can also be implemented by circulating a liquid in the vessel models by connecting them to external tubes and pumps. Furthermore, selection of appropriate materials allows the simulator to be sterilized and facilitate the reproduction of the effects of such medical equipment on the human body. Thus, mock-up simulators can be used in actual operating rooms to evaluate the performance of medical equipment. Therefore, the goal of this research is to develop a mock-up surgical simulator to allow evaluation of medical techniques (e.g., surgical skills) or the performance of medical equipment. Several studies have already been published discussing mock-up simulators for various sections of blood vessels [17–20]. For large vessels (in mm) such as the coronary artery or arteries in the brain, such simulators can benefit from 3D printing technology. To simulate catheterization surgery, Ikeda et al. fabricated a millimeter-sized 3D vessel model made of polydimethylsiloxane (PDMS) using the lost-wax method [17, 18]. However, fabricating a model of fine vessels of size below 100 μm (e.g., retinal vessels) is difficult with this method. Photolithography-based fabrication techniques can be applied to fabricate models of smaller vessels. For example, Nakano et al. fabricated a microchannel that mimics a fine blood vessel with sizes down to about 10 μm using photolithography techniques and PDMS molding [19]. Although photolithography offers sufficient resolution for the retinal vessel model, it can only be applied to 2D structures. The target of the present study, however, is to model retinal vessels of size about 100 μm and that lie on the eye fundus, which forms a concave curved surface. Therefore, both the above-mentioned fabrication methods are difficult to apply in this case (i.e., a retinal vessel model with microchannels on a curved surface). We thus propose a retinal vessel model with microchannels smaller than 100 μm that lie on a concave curved surface. This retinal vessel model is fabricated by combining laser-lithography-based fabrication techniques, PDMS molding, and hydraulic transfer techniques. Furthermore, the microchannel wall thickness is controlled with 10 μm accuracy to mimic the physical human sensation of touching retinal vessels. We circulate liquid through the fabricated microchannels and simulate a microcannulation procedure. We also measure the force applied to the model during the simulated surgery to demonstrate the usefulness of the proposed model not only for qualitative evaluation but also for quantitative evaluation of medical techniques. Figure 2 shows the concept of the proposed retinal vessel model, which consists of three layers. The bottom layer is a curved surface structure (curvature diameter of 24 mm) that mimics the human eye. The middle layer is a PDMS sheet with microchannels less than 100 μm in size, which mimics the structure of human retinal vessels. Finally, the top layer consists of a thin PDMS sheet, whose thickness is controlled to within about 10 μm to mimic the sensation of touching human retinal vessels. Conceptual image of retinal vessel model on curved surface: a overview of model, b side view showing line A-A', and c close-up of side view First, microchannels in the middle layer are fabricated on a PDMS sheet by laser lithography techniques and PDMS molding. Next, this middle layer is transferred to the concave bottom layer using a hydraulic transfer technique. The result is a fine vessel structure (vessels less than 100 μm in diameter) superposed on the concave surface. Third, again using hydraulic transfer, the 10-μm-thick top layer is laid on the middle layer. Thus, the wall thickness of the top of the microchannels can be controlled to mimic the sensation of touching a retinal vessel. Furthermore, the mechanical properties of the model, such as Youngs modulus or tear strength, can be tuned by changing the mixing ratio of the base resin and the curative reagent for PDMS. Therefore, this model offers both the physical structure of human retinal vessels and the tactile sensation of touching them. All parts of the model consist of PDMS or glass, which can withstand high temperatures and sterilization. As a result, the model can be used in actual operating rooms. Additionally, the microchannels can be connected to external tubes, as shown in Fig. 2a, allowing liquid to be circulated through the microchannels for mimicking blood flow. This enables us to circulate liquid in the microchannel for mimicking blood flow. Thus, the surgical procedure of injecting thrombolytic drugs into veins with a micropipette can be simulated with the proposed model. Furthermore, the force applied during the simulation can be measured using a force sensor placed underneath the model. The force information can be used to quantitatively evaluate medical techniques. We thus expect the proposed model to strongly contribute to evaluating surgical techniques or medical equipment and in the training of surgeons for retinal vessel surgery. Design of microchannel First, we designed microchannels for mimicking retinal vessels based on a shape of real retinal vessels. We simplified shape of real retinal vessels as shown in Fig. 3 and determined sized of vessels according to Murray's law [21, 22]. Murray's law states that the cube of radius of a parent vessel equals the sum of the cubes of the radii of the daughters, as shown in Fig. 3a and following equation. $$\begin{aligned} R^3_0 = R^3_1 + R^3_2 \end{aligned}$$ We determined the width of first channel ($W_1$) as 150.0 μm (radius: $R_1 = 75.0$ μm), and calculated the width of second channel ($W_2$) as 119.1 μm (radius: $R_2 = 59.5$ μm) according to Murray's law. Similarly, we calculated the width of third and fourth channels as $W_3 = 94.5$ μm (radius: $R_3 = 47.2$ μm), and $W_4 = 75.0$ μm (radius: $R_4 = 37.5$ μm), respectively. Design of microchannel. a Schematic figure of Murray's law, and b actual design and sizes of microchannel Mechanical characteristics of PDMS Second, we confirmed that the mechanical characteristics of PDMS may be suitably controlled by changing the mixing ratio of the main resin and curative reagent. In this study, we focus on Youngs modulus and tear strength because our target is to simulate microcannulation, and these two parameters are thought to be strongly related to the sensation of touching retinal vessels and of feeling a puncture. Park et al. discussed the controllability of the mechanical characteristics of PDMS and used this approach to evaluate surgical skills for a coronary-artery bypass graft [23]. Although the coronary-artery target differs from the target of this study (i.e., retinal vessels), we use the same Youngs modulus and tear strength in our model as a reference (i.e., 0.13 ± 0.02 MPa and 0.6 ± 0.13 N/mm, respectively). The Youngs modulus and tear strength were evaluated based on the Japanese Industrial Standards (JIS) K6251 and K6252, which are equivalent to the International Organization for Standardization (ISO) 37 and 34. The dumb-bell test piece 7 and the angle test piece were used as samples. Each sample was pulled at 200 mm/min and 500 mm/min. We fabricated samples with various mixing ratios (ratio of curative reagent to main resin) of 10, 20, 33, 50 and 66 wt%. The tensile tests were done three times for each mixing ratio. Figure 4a and b show the measured Young's modulus and tear strength, respectively. We tuned the Youngs modulus from 0.14 to 1.14 MPa and the tear strength from 0.68 to 1.70 N/mm. Considering the target values, we chose a mixing ratio of 66 wt% for this study. Furthermore, we performed preliminary sensory testing by ophthalmologist about sensation of touch and puncture. By using flat vessel model, we confirmed that the PDMS with mixing ratio of 66 wt% are the most suitable for retinal vessel model. Measured a Youngs modulus and b tear strength as a function of mixing ratio of the curative agent. The red areas indicate the target values Hydraulic transfer of PDMS sheet To fabricate a retinal vessel structure on a curved surface, we hydraulically transferred the PDMS pattern. Hydraulic transfer is generally used for printing on curved surfaces [24, 25]. By floating a printed film on water and pressing a curved surface onto the film, uniform printing on the curved surface is obtained. Retinal vessels on a curved surface are obtained by overlaying patterned PDMS sheets on a curved surface, as shown in Fig. 5. With this hydraulic transfer process, we can realize the fine microchannel with size of $\simeq 10$ μm on a curved surface, which is difficult to achieve with conventional fabrication techniques. The details of fabrication are as follows: Pattern SU-8 photoresist (Nippon Kayaku Co. Ltd, Tokyo, Japan) on a silicon surface by laser lithography. This pattern is used as a mold for the microchannels and the size of the microchannels can be locally controlled by adjusting the exposure conditions. Spincoat LOR (Nippon Kayaku Co. Ltd., Tokyo, Japan) and PDMS (Silpot 184, Dow Corning Toray Co. Ltd., Tokyo, Japan) onto the cured PDMS. Push SU-8 mold ointo spin-coated PDMS and heat the ensemble to 85 °C for 10 min with a hot plate. Dissolve LOR with ethanol and remove PDMS sheet. Transfer PDMS sheet to base structure by using hydraulic transfer. The base structure is made by a 3D printer (EDEN250, Stratasys Ltd.). The diameter of the base structure is 24 mm, which is the average diameter of a human eye. Transfer the PDMS sheet to the concave PDMS made by 3D-printer mold. Prior to this step, treat both bonding surfaces with O2 plasma for surface-activated bonding of PDMS. Remove base structure. Pattern a connection channel at the bottom of the curved PDMS surface. The mold for the connection channel is fabricated by photolithography. Punch holes into microchannel and connection channel to connect external tubes. For a cover layer, bond the thin PDMS sheet to the curved surface. The polyvinyl alcohol (PVA) and thin PDMS sheet are coated onto the base structure by dip coating. The PVA is used to demold the thin PDMS sheet from the base structure. Dissolve PVA with hot water and remove base structure. Bond bottom glass to model to seal the connection channel and connect the external tubes. Fabrication process for proposed retinal vessel model In this fabrication process, the 10-μm-sized microchannel can be fabricated by using laser lithography (step 1 in Fig. 5). The patterning on a curved structure is done by using hydraulic transfer (steps 5 and 6). Furthermore, the thickness of the cover layer is controlled by changing the dip-coating conditions (step 10). This means that the wall thickness of the microchannels can be controlled to best mimic the sensation of touching retinal vessels. According to Ref. [26], the wall thickness of retinal vessels ranges in humans from 10 to 20 μm. We tentatively confirmed that this range can be covered by changing the drawing speed of the dip-coating process. In this study, we use a 20 μm cover layer. According to our proposed fabrication process, the cross section of the fabricated vessel model is square because the cross-section of the patterned photoresist is square. We have already fabricated the vessel model with circular vessel cross sections by using a reflow process with the patterned photoresist [19]. However, this approach requires additional processing steps, wihch increases the cost of fabrication. Therefore, this approach is not favored for commercial versions of this model. In addition, we made preliminary evaluations of both the square- and circular-cross-section microchannels. According to qualitative evaluations of these two models by medical doctors, no significant difference is apparent between the models with vessels of different cross-sectional shape. We therefore used vessels with square cross-sections in this study. The fabricated retinal vessel model is shown in Fig. 6. The channel was neither broken nor collapsed after the hydraulic transfer, as shown in Fig. 6b and c. We varied the width and height of the vessels from 75.0 to 119.1 μm to mimic the size range of actual retinal vessels. Details of the design values and the measured results for the three typical parts of the samples are shown in Table 1. The aspect ratio of these vessels is approximately 1.0. In addition, we evaluate the thickness of the cover-layer PDMS sheet, which is related to sensation of touching a retinal vessel. We measured the thickness at five points on a sample and the average value of these five thicknesses is 19.3 ± 0.3 μm, which is close to the target value of 20 μm. The standard deviation of the thickness is less than 2 % of the target value. Thus, we conclude that the proposed model can be fabricated with good reproducibility. Measurements of fabricated model (N = 5) Design value Fabricated results W (μm) H (μm) 119.3 ± 2.5 93.7 ± 1.7 Photographs of fabricated model: a overview, b cross-sectional view cut along line A-A', and c enlarged image of cross-sectional view Next, we tested the circulation of liquid in the microchannel by injecting a blue liquid into the microchannel via the external tube connected to the model. The injected liquid circulated through the channel without leakage, as shown in Fig. 7 (Additional file 1). Successive photographs of the circulation of liquid through the microchannel: a start of circulation and b 10 s later. The black arrows point to the air-liquid interface, whereas the blue dotted arrows show the direction of flow Based on these results, we conclude that the proposed retinal vessel model was successfully fabricated on a concave surface. This fabrication method can thus be used to fabricate microchannels on concave surfaces. Simulation of microcannulation Finally, we report a microcannulation procedure simulated by the fabricated retinal vessel model. The standard microcannulation procedure consists of the following four steps [27]: Approach target retinal vessel with micropipette. Puncture target vessel with micropipette. Inject thrombolytic drug via the micropipette and hold micropipette in place for approximately 30 s. Withdraw micropipette from retinal vessel. Additionally, we measured the vertical force applied to the model during this procedure by using a load cell placed underneath the model. Photographs taken before and after puncturing (step 2) are shown in Fig. 8a and b, respectively. The micropipettes successfully puncture the microchannel and inject the liquid into the channel (Additional file 2). The vertical force applied during the procedure is shown in Fig. 8c and varies from approximately −150 to 180 mN. Here, negative values indicate that the force is applied downward towards the model (i.e., a pushing force). Similarly, positive values indicate a pulling force. Thus this model allows quantitative measurements of the force applied during the simulation of a microcannulation procedure. Results of microcannulation simulation: photographs of retinal vessel model a before and b after puncture of retinal vessel by micropipette. These photographs were acquired by using an eye-surgery microscope. c Measured force applied during microcannulation procedure We propose herein a retinal vessel model and describe how to fabricate the model. Further functionalization of the proposed model is also discussed. Simulating entire steps of microcannulation surgery requires a whole-eye model. In this study, we focus on simulating the retinal vessels at the posterior segment of the eye. Our model can be used to simulate the puncture and injection processes in microcannulation surgery. However, other processes in an actual operation, such as insertion of forceps into the eyeball, should also be simulated. To do so would require a whole-eye model that can mimic the structures and mechanical characteristics of whole parts of the human eyeball. The fabrication of such a complex 3D structure with the appropriate materials is planned in future work. Furthermore, we demonstrate the measurement of applied force with the proposed model, as shown in Fig. 8c. Integrating such sensing functions into a mock-up surgical simulator is of great use for quantitatively evaluating medical techniques. Other surgical procedures use thermal or electrical effects for treatments. Simulating the associated sensing functions (e.g., thermal or electrical sensors) is required to evaluate these surgical procedures. Such highly-functionalized surgical simulators will be reported in the near future. We propose herein a retinal vessel model fabricated on a concave surface to simulate microcannulation. The structure of the proposed model is very reproducible and can be sterilized for use in actual operating rooms. The model is fabricated by using laser lithography, PDMS molding, and hydraulic transfer, and can create vessels as small as 10 μm in size. Furthermore, we confirmed that liquid can be circulated through the fabricated microchannel. In addition, we simulate the puncture and injection processes of microcannulation. We also measure the applied force by using a force sensor placed underneath the model. 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Research Article \| Open \| Published: 31 July 2015 HaploPOP: a software that improves population assignment by combining markers into haplotypes Nicolas Duforet-Frebourg1,2,3, Lucie M. Gattepaille4, Michael G.B Blum1,2 & Mattias Jakobsson4,5 In ecology and forensics, some population assignment techniques use molecular markers to assign individuals to known groups. However, assigning individuals to known populations can be difficult if the level of genetic differentiation among populations is small. Most assignment studies handle independent markers, often by pruning markers in Linkage Disequilibrium (LD), ignoring the information contained in the correlation among markers due to LD. To improve the accuracy of population assignment, we present an algorithm, implemented in the HaploPOP software, that combines markers into haplotypes, without requiring independence. The algorithm is based on the Gain of Informativeness for Assignment that provides a measure to decide if a pair of markers should be combined into haplotypes, or not, in order to improve assignment. Because complete exploration of all possible solutions for constructing haplotypes is computationally prohibitive, our approach uses a greedy algorithm based on windows of fixed sizes. We evaluate the performance of HaploPOP to assign individuals to populations using a split-validation approach. We investigate both simulated SNPs data and dense genotype data from individuals from Spain and Portugal. Our results show that constructing haplotypes with HaploPOP can substantially reduce assignment error. The HaploPOP software is freely available as a command-line software at www.ieg.uu.se/Jakobsson/software/HaploPOP/. Molecular markers provide powerful approaches in forensic science and ecology to assign individuals into predefined populations [1, 2]. With the advent of new sequencing technologies, the number of available molecular markers in different species is rapidly increasing. At the same time, dense datasets tend to contain increasingly correlated markers because Single Nucleotide Polymorphisms (SNPs) that are physically close on a chromosome, often are in Linkage Disequilibrium (LD). Such correlations are usually perceived as a nuisance factor in statistical analyses since it violates a common assumption of independence among markers. This statistical nuisance can be overcome by pruning SNPs using for example the software PLINK [3]. However much information may be lost because of the pruning process. Another approach is to explicitly model the correlation between markers to control for LD [4–6], or to include the pruning process in the statistical analysis [7]. In addition, it has been shown that it can be useful to combine correlated markers into haplotypes to augment the information about population structure at a finer scale [8]. Such an approach is valuable for assignment methods when the level of genetic differentiation among groups is low [9]. Gattepaille and Jakobsson [10] introduced the Gain of Informativeness for Assignment (GIA), which is a statistic measuring the gain in information for population assignment by combining two markers into haplotypes. GIA is based upon an ancestry information criterion that measures to what extent a molecular marker is informative about population assignment [11]. GIA is defined as the difference between the ancestry information carried by two markers and the ancestry information carried by the haplotypes resulting from the combination of the two markers. Building haplotypes with GIA increases correct assignment to predefined populations [10]. However, a major combinatorial challenge arises when using GIA because of the prohibitively large number of pairs of markers that can be combined into haplotypes. In this article, we present a new algorithm that efficiently uses GIA to build informative haplotypes for population assignment. The algorithm needs reference individuals whose population of origin is known. Based on these reference individuals, the algorithm uses GIA to construct informative haplotypes. To handle large numbers of markers, we provide a heuristic approach where only markers located within the same genomic region can be combined to form haplotypes. Combining markers into haplotypes is a recursive process so that haplotypes can result from the combination of two or more markers. The raw genotype data are recoded into multi-allelic haplotype data and the new data file containing both genotypic and haplotypic information can be used to assign individuals to populations based on for instance Principal Component Analysis, or model-based assignment approaches [12, 13]. Because the construction of haplotypes uses predefined populations, there is a risk of overfitting. For example if the evaluation of population assignment is performed with the same individuals that were used to construct the haplotypes, the assignment errors may be underestimated. Additionally, the constructed haplotypes can generate artificial population structure although there is no true stratification among the predefined populations. Both problems arise because the construction of haplotypes can exaggerate the differentiation among populations. To get a fair evaluation of population assignment, we implement a split-validation approach where we use different individuals to construct the haplotypes and to evaluate assignment [14]. Haplotypes are built using a subset of the individuals, consisting of a training set. The quality of population assignment can then be assessed using the remaining individuals (the validation set). If the individuals in the validation set cluster with individuals in the training set, there is evidence for some level of population structure, which may not have been detected based only on genotype markers. Our new algorithm for combining markers into haplotypes is implemented in the software HaploPOP. The software is a command-line program written in C. We give examples of how to use haploPOP to perform population assignment with SNP data that were simulated from a population divergence model. We also show that HaploPOP improves assignment of individuals from Spain and Portugal using the POPRES dataset that contains 447,245 SNPs [15]. Gain of informativeness for assignment The Gain of Informativeness for Assignment (GIA) is a one-dimensional statistic that provides a criterion to decide whether markers should be combined into haplotypes in order to improve population assignment [10]. It is based on the Informativeness for Assignment (IA) statistic, which measures how informative a marker is for assigning individuals to different populations [11]. The more different the allele frequencies are in a set of predefined populations, the more informative the marker is to assign individuals of unknown origin to their source population, and the larger is the IA statistic. Denoting by K the number of populations, by N the number of alleles of the marker under consideration, by $p_{j}^{(i)}$ the frequency of allele j in population i, and by $\overline {p}_{j}$ the average frequency of allele j across all populations, the IA statistic is computed as follows [11] $$ \textit{IA} = \sum\limits_{j=1}^{N}\left(-\overline{p}_{j}\log \overline{p}_{j} + \sum\limits_{i=1}^{K}\frac{p_{j}^{(i)}}{K}\log p_{j}^{(i)}\right). $$ Given two multi-allelic markers M 1 and M 2, the question is whether combining M 1 with M 2 into a haplotype marker H improves the assignment of individuals to predefined populations. GIA computes the difference between the informativeness for assignment of H and the sum of the informativeness of M 1 and M 2 $$ \textit{GIA} = \textit{IA}(H)-(\textit{IA}(M_{1})+\textit{IA}(M_{2})). $$ If GIA is positive, it suggests that population assignment is improved by considering haplotype H instead of using the two markers separately. However, if GIA is negative, there is no advantage of combining the two markers into a haplotype. In particular, it can be shown that if the two markers are in linkage equilibrium, GIA is expected to be non-positive [10]. Maximizing the informativeness for assignment We assume that genotype data are available for n individuals at l molecular markers (M 1,...M l ). We also assume that the dataset has been phased, where all individuals have been phased together in one go to avoid introducing any haplotype difference due to phasing (note that there may still be switch errors from the phasing, but these should affect all individuals similarly). The approach implemented in the software HaploPOP builds a set of haplotypes that increases the total informativeness for assignment contained in the genotype data. To find the optimal haplotype set Γ 0, we address the maximization problem $$ \left\{ \begin{array}{l} \displaystyle \Gamma_{0} = arg \max_{\Gamma} \sum\limits_{H \in \Gamma} \textit{IA}(H) \\ \Gamma \in Part(M_{1} \dots M_{l}) \end{array} \right. $$ where P a r t(M 1...M l ) is the set of all possible partitions of the l markers. The number of partitions in a dataset of l markers is given by Bell's number [16]. Because this number is large, we cannot evaluate the objective function for all possible partitions. A commonly used heuristic is to apply a greedy strategy, although it can perform arbitrarily good. In the case of increasing Informativeness for Assignment, the resulting haplotypes always provide genetic data with augmented informativeness. Because the cost of this algorithm increases rapidly with the number of genetic markers, we limit potential combinations of markers within windows of fixed size. In a first step, the algorithm constructs haplotypes from the phased genotype file of individuals with known origin and returns a haplotype coding file that provides the correspondence between haplotypes and initial markers. This is the LEARN option of HaploPOP. The construction of haplotypes is constrained by a predefined window-size. The set (M 1,…M l ) of markers is divided into subsets of contiguous markers corresponding to the genomic windows. Haplotypes are constrained to be combinations of markers of the same window. The window size is chosen by the user and can be defined based on number of markers, on base pairs, or genetic distance. By choosing genetic distances, one can account for non-uniform recombination rates. In every window, the GIA statistic is computed for all pairs of markers, and the pair with the greatest GIA value is merged to form a haplotype. Combinations proceed recursively until there is no pair of markers for which GIA >0 (or a certain positive user-defined threshold). A particular haplotype-loci formed by a combination of markers is thereafter treated as a (potentially multi-allelic) marker of the particular window and can be combined with other markers in a recursive manner. We denote by n the number of reference individuals whose population of origin is known, and by l the total number of initial markers. The greedy algorithm proceeds as follows: divide the 2n×l data matrix in contiguous windows. for every window do Calculate GIA for all pair of markers. while for all markers M and M', $\displaystyle {max}_{M, M^{\prime }} (\textit {GIA}(M, M^{\prime }) > 0)$, do $\displaystyle (M_{0},M^{\prime }_{0}) = {argmax}_{M, M^{\prime }} \textit {GIA}(M,M^{\prime }) > 0$. Combine the markers M 0 and $M^{\prime }_{0}$ to form a haplotype marker H 0. Remove the GIA statistics involving M 0 and $M^{\prime }_{0}$ and compute the new GIA statistics with pairs of markers that include H 0. At every end of the inner loop, the algorithm partitions the markers into a set of haplotypes that increases the score of the objective function (3). It stops when no additional pairwise combination improves the total score of the partition. A warning is raised when the number of haplotype-alleles reaches the number of chromosomes 2n making haplotypes useless because they become private to every individual and do not provide any useful information for assignment. In a second step, HaploPOP combines the SNPs in the initial genotype file into haplotypes according to the combinations of markers constructed at the first step, and generates the haplotype data file. The genotype file can contain individuals of unknown origin that the user is trying to determine, as well as the individuals of known or suggested origin used to construct the haplotypes. This corresponds to the APPLY function of HaploPOP. When two markers are combined, the resulting haplotype-alleles are coded in a range from 0 to the number of haplotype-alleles minus one, in order of appearance in the list of individuals. Window size A key parameter of the method is the window size. This parameter is important for both speed of the algorithm and level of informativeness of the haplotypes. The choice of window size governs the number of operation performed by the algorithm. In the case of a fixed window of S markers, the number of windows is n window=l/S, and the cost C of the algorithm in number of operations is $$ C(n_{window}, S, n, K) = O(n_{window}\times(2nS^{2}K + S^{3})) $$ where K is the number of populations in the data. The algorithm scales very well for genome wide datasets, since for a given window size S, the cost of the algorithm is proportional to the number of markers l in the data. The term proportional to S 3 corresponds to the iterative maximum search in all possible pairs of the window. The term S 3 is an upper bound for a search that is done in the worst case S times in a matrix of size S×S or less. In the event of choosing a large window size, there may be a large number of haplotype-alleles, which could fit closely to the distribution of haplotype-alleles of individuals in the training set. Such a set of haplotypes would likely perform poorly for other sets of individuals from the same reference population, and reduces accuracy of population assignment. We refer to this phenomenon as overfitting. Limiting the size of the window is one way to avoid overfitting. We demonstrate in the Results section that the window size has a strong impact on the performance of the created combinations of markers, and an optimal value generally exists. The optimal window size depends on multiple factors, including the effective population sizes of the investigated groups and the extent of Linkage Disequilibrium in the groups. Split-validation To validate the gain in assignment accuracy provided by the constructed haplotypes, we implement a split-validation technique [14]. For each population, we randomly split the set of individuals into two subsets consisting of the training subset used to learn the haplotypes and the validation subset used to compute assignment accuracy. It is important that the division between validation and training set is done after phasing. Phasing performed on the two datasets separately could introduce haplotypic differences and weaken the informativeness for assignment of the haplotypes built by the algorithm. To assign individuals to populations, we use Principal Component Analysis (PCA) as implemented in the software EIGENSOFT [12]. For each of the constructed haplotype-loci, we enumerate all haplotype-alleles present in the dataset. We use a presence/absence coding for each haplotype-allele. In particular, we add one column per haplotype-allele and note 1 for a chromosome carrying the allele, and 0 otherwise. The number of principal components we consider equals the number of populations used for constructing the haplotypes minus one [12]. We determine the PC axes using individuals from the training and the validation set. For each individual of the validation set, we compute Euclidean distances on the PC space between this individual and the barycentric coordinates of each population computed from the training set of individuals. We assign individuals to the population that has the closest barycenter. Because the origins of individuals in the validation sets were known for all examples (see below), we can measure the number of incorrectly assigned individuals in these examples. Note that the assessment of individuals to populations depends on the assignment procedure itself (here we use PCA) and that different assignment procedures may lead to different assignment errors (see [10] for a comparison of different assignment strategies). However, since we are primarily interested in the comparison between assignment using the raw genotype data and assignment using combined markers found with HaploPOP, we focus on a single assignment approach based on PCA. With this assignment approach each haplotype-allele is treated as a unique allele with the same relationship to all other alleles. We evaluate the performance of the approach and the HaploPOP software on both simulated and empirical data. Application to simulated data We evaluated the assignment approach and the HaploPOP software for simulated data generated by the software ms [17]. We simulated 200 kb sequences from a 3-population divergence model. We set the effective population sizes of all populations to N e =1,000, the mutation rate to μ=0.012, so that θ=48, and we considered a sample of 100 individuals in each population. The population divergence between population 2 and population 3 was set to occur at T 1=0.025 coalescent time units (or 100 generations) in the past and the population divergence between population 2 and population 1, was set to occur at T 2=0.05 coalescent time units (or 200 generations) in the past. We generated four datasets for a hypothetical 200 kb region with effective recombination rates of ρ=30,60,120, and 240, and replicated this procedure 10 times for each value of the recombination rate. To assess the assignment accuracy provided by the haplotypes constructed by HaploPOP, we used a split-validation technique. The training set and the validation set contained each 50 randomly chosen individuals in each population. Figure 1 a shows that assignment accuracy improves by constructing haplotypes. The assignment error decreases as the window size increases (up to a certain level). However, most of the improvement occurs when moving from genotypes to haplotypes spanning up to 50 kb (Fig. 1 a). Compared to the error of assignment obtained with genotype data, constructing haplotypes decreases the error by 20−70 % depending on the recombination rate. Constructing haplotypes for 200 kb windows compared to 50 kb windows only reduces the error by at most an additional 8 %. Mean percentage of Incorrect Assignment (MIA) for simulated data from a divergence model with 3 populations (see main text for details on simulations). Panel a: x axis represents the window sizes. Note that window size = 1 corresponds to using SNP genotype data to assign individuals to populations. Panel b: x axis represents the proportion of individuals in the samples that are used in the training set. The mean incorrect assignment of individuals is evaluated with individuals from the validation set that were not used to construct the haplotypes Furthermore, we find that the mean incorrect assignment is lower with greater recombination rates. This emphasizes the fact that strongly correlated polymorphisms tend to carry less information for assignment than the same number of independant polymorphisms. Since simulated sequences have on average the same number of SNPs, sequences with a greater recombination rate carry more informativeness for assignment. For a fixed window size of 50 kb, we construct the haplotypes for different sizes of training sets ranging from 2 to 60 individuals. When comparing the mean incorrect assignment of individuals from the validation set, we find a decay of MIA with increasing numbers of individuals in the training set for all recombination rates. However, when using a fraction of individuals greater than 10 % of the overall population, the change in MIA is minimal (Fig. 1 b). Hence, even a fairly small fraction of individuals in a sample can be used to accurately train the algorithm. Application on human data We investigate to what extent constructing haplotypes with HaploPOP improves population assignment of respectively 133 and 125 self-reported Spanish and Portuguese individuals from the POPRES dataset, which contains 447,245 SNPs [15]. We first phased the data using fastPhase [10, 18]. No pruning of SNPs were performed because our aim is to capture the information for assignment for all markers, including markers in LD. All markers were therefore retained and used to build haplotypes. Considering the first two PCs based on all SNP-genotype data, we found that the two populations cannot be distinguished (Fig. 2). A thorough PCA exploration of all European individuals of the POPRES collection was further unable to distinguish between Spanish and Portuguese individuals [19]. Using HaploPOP, we constructed the haplotypes that are informative to discriminate between Spanish and Portuguese individuals. We randomly selected 67 Spanish individuals and 63 Portuguese individuals for constructing the training set. We then performed PCA based on the haplotype markers generated by HaploPOP and compute the PC scores for the individuals from the training set and the validation set (Fig. 3 a). On PC1, the Spanish and Portuguese samples from the training set are clearly separated. By contrast, the 66 Spanish individuals and 62 Portuguese individuals from the validation set overlap but a large majority of these individuals are pulled in the direction of their population of origin. Principal Component Analysis on 447,245 SNPs for Spanish and Portuguese samples from POPRES Principal Component Analysis of the Spanish and Portuguese samples from POPRES using the haplotypes found with HaploPOP. The haplotypes were built from 447,245 SNPs using a window size of 150 kb. For constructing haplotypes, the training sets consist of the Portuguese and Spanish individuals (Panel a) or a mix of Portuguese and Spanish individuals in both sets 'A' and 'B' (Panel b) To show that the population labels, Portugal and Spain, correspond to true population differentiation, we generated a control training set. We arbitrarily assign each individual of the training set to a label A or B and by construction, individuals labeled by A (or B) contain both individuals from Spain and Portugal. Using this training set of half the individuals in the dataset, we learn the haplotypes that are informative for discriminating between A and B. Using the validation set, we find that the haplotypes learned with populations A and B cannot distinguish between Spanish and Portuguese ancestry (Fig. 3 b). This analysis shows that the above demonstrated separation between Spanish and Portuguese individuals corresponds to true population differentiation and that the separation is not a consequence of overfitting. These two examples emphasize two important features of HaploPOP. First, the haplotypes constructed based on the training set are very efficient in separating individuals of the training set, regardless of any true stratification between candidate populations. Comparing only the individuals from the training set on a PC plot can either lead to the wrong conclusion that two populations can be distinguished (Fig. 3 b) or at least exaggerate the ability to distinguish between the two populations (Fig. 3 a) because of overfitting. Second, evidence for population structure comes from the ability of the constructed haplotypes to distinguish between individuals that were not used in the training process. If a validation set of individuals can be assigned to the candidate populations, it is a good indication of fine-scale level of stratification between the candidate populations that might be difficult to detect using SNPs only. We show that two populations that were not distinguishable with raw genotype data can be separated based on haplotypes. This highlights how HaploPOP can be used to study samples where prior belief suggests that there is population structure but SNP-genotype data fail to detect it. When computing the error for assignment of the validation set for the POPRES data, we find that there is an optimal window size (Fig. 4) at which the assignment error can be reduced by 45 %. Intuitively, combining SNPs into haplotypes can only improve the power to assign individuals to groups up to some level: for too large window sizes, we run into overfitting problems where trained haplotypes are well suited to separate the particular individuals of the training set but not of the individuals in the validation set. The optimal value of the window size depends on many factors, such as the extent of linkage disequilibrium in the groups, or the degree of genetic differentiation between groups. The strategy we advocate for choosing the window size is to try different window sizes and to find the minimal assignment error as estimated with a split-validation approach. Such a strategy is computationally costly and requires for each of the chosen window size a run of HaploPOP, where the cost will be dependent of the window size S as described in equation (4). Mean percentage of Incorrect Assignment (MIA) when distinguishing the Spanish and Portuguese samples from POPRES. The error is evaluated with a split-validation approach Recently, many model-based methods have been proposed to assign unlabeled individuals to populations [13, 20–22]. These methods can be used together with HaploPOP to reduce the proportion of incorrect assignment, as it is shown with the software Structure [23] in a previous article [10]. In this article we focus on using Principal Component Analysis and, from a statistical point of view, model-based approaches and PCA are related [20]. In the case of assigning individuals to labeled populations, we expect that most of these methods will result in similar assignment accuracy. In this article, we present a new algorithm that uses the GIA statistic to construct haplotypes with a window-based approach. The algorithm is implemented in the command-line software HaploPOP. The software allows users to apply a 2-step procedure. First, HaploPOP constructs haplotypes that are informative about population assignment from a training set of individuals. Second, HaploPOP recodes the genotype data to haplotypes. These new haplotype data can then be used to assign unknown individuals to candidate populations or investigate fine-scale population structure using e.g. PCA. We have shown how constructing haplotypes with HaploPOP can substantially reduce mis-assignment of individuals to candidate populations. For SNP data simulated in a 3-population divergence model, the assignment-error was reduced by 20 % to 70 %. Using the 447,245 SNPs of the POPRES data, the assignment-error was reduced by 45 % when trying to distinguish Portuguese from Spanish individuals. Constructing Haplotypes with HaploPOP is a promising approach to assign individuals into populations in forensic science and ecology. 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Mathieson I, McVean G. Differential confounding of rare and common variants in spatially structured populations. Nat Genet. 2012; 44(3):243–6. This work was supported by a grant from the Swedish Foundation for International Cooperation in Research and Higher Education (STINT) awarded to Mattias Jakobsson and Michael Blum. A grant from the French national research agency provided support to Michael Blum and Nicolas Duforet-Frebourg (DATGEN project, ANR-2010-JCJC-1607-01) and a grant from the Swedish research council provided support to Mattias Jakobsson. The POPRES data were obtained from dbGaP (accession number phs000145.v1.p1). Univ. Grenoble Alpes, TIMC-IMAG, Grenoble, F-38000, France Nicolas Duforet-Frebourg & Michael G.B Blum CNRS, TIMC-IMAG, Grenoble, F-38000, France Department of Integrative Biology, University of California Berkeley, Berkeley, 94720-3140, California, USA Department of Evolutionary Biology, Evolutionary Biology Centre, Uppsala University, Uppsala, Sweden Lucie M. Gattepaille & Mattias Jakobsson Science for Life Laboratory, Uppsala University, Uppsala, Sweden Mattias Jakobsson Search for Nicolas Duforet-Frebourg in: Search for Lucie M. Gattepaille in: Search for Michael G.B Blum in: Search for Mattias Jakobsson in: Correspondence to Nicolas Duforet-Frebourg or Mattias Jakobsson. LG, and MJ designed the project and set the requirements of the software. NDF wrote the source code. NDF and LG ran simulations and analyzed data. All authors wrote, read and approved the final manuscript. Assignment Accuracy Assignment Approach Population Assignment Assignment Error Sequence analysis (methods)	CommonCrawl
Search SpringerLink Limitations of cognitive control on emotional distraction – Congruency in the Color Stroop task does not modulate the Emotional Stroop effect Elisa Ruth Straub1,2, Constantin Schmidts1, Wilfried Kunde1, Jinhui Zhang1, Andrea Kiesel1 & David Dignath1 Cognitive, Affective, & Behavioral Neuroscience (2021)Cite this article Emotional information receives prioritized processing over concurrent cognitive processes. This can lead to distraction if emotional information has to be ignored. In the cognitive domain, mechanisms have been described that allow control of (cognitive) distractions. However, whether similar cognitive control mechanisms also can attenuate emotional distraction is an active area of research. This study asked whether cognitive control (triggered in the Color Stroop task) attenuates emotional distraction in the Emotional Stroop task. Theoretical accounts of cognitive control, and the Emotional Stroop task alike, predict such an interaction for tasks that employ the same relevant (e.g., color-naming) and irrelevant (e.g., word-reading) dimension. In an alternating-runs design with Color and Emotional Stroop tasks changing from trial to trial, we analyzed the impact of proactive and reactive cognitive control on Emotional Stroop effects. Four experiments manipulated predictability of congruency and emotional stimuli. Overall, results showed congruency effects in Color Stroop tasks and Emotional Stroop effects. Moreover, we found a spillover of congruency effects and emotional distraction to the other task, indicating that processes specific to one task impacted to the other task. However, Bayesian analyses and a mini-meta-analysis across experiments weigh against the predicted interaction between cognitive control and emotional distraction. The results point out limitations of cognitive control to block off emotional distraction, questioning views that assume a close interaction between cognitive control and emotional processing. Emotions affect behavior, cognition, and physiology. Negative emotional stimuli, for instance, are considered to have preferential access to awareness and receive prioritized processing over concurrent cognitive processes (Carretié, 2014; Ohman & Mineka, 2001; Yang & Pourtois, 2018). While this is beneficial if emotional information is task-relevant, emotional information that is task-irrelevant interferes with other processes, resulting in impaired task performance (Schmidts et al., 2020; Verbruggen & De Houwer, 2007; Vuilleumier & Huang, 2009). A prominent task to investigate the interference produced by emotional stimuli is the Emotional Stroop task (Watts et al., 1986). This task comprises negative (e.g., "WAR") and neutral (e.g., "HOUSE") words written in different print-colors, and participants have to name the print-color of the words. Emotional Stroop effects were expressed in longer latencies and higher error rates when responding to the print-color of negative compared to neutral words (i.e., emotional distraction) and have been quantified in two ways (Frings et al., 2010; McKenna & Sharma, 2004). First, negative valent words impair responses within the current Emotional Stroop trial. Second, the influence of negative words persists in time and impairs performance across trials subsequent to negative word stimuli, even when emotional words are not perceptually present anymore. The Emotional Stroop paradigm allows researchers to quantify the costs of emotional distraction, which is why it has been widely used as a diagnostic tool in various psychopathologies (e.g., depression, see Mogg & Bradley, 1998; anxiety disorders, see Bar-Haim et al., 2007; social phobia, see Andersson et al., 2006; alcohol, see Lusher et al., 2004; panic disorder, see Harber et al., 2019, for a meta-analysis on fMRI studies on clinical and healthy subjects, see Feng et al., 2018). Current research provides two different theoretical accounts for emotional distraction instigated by Emotional Stroop tasks. According to the attention account (Williams et al., 1996), negative words pull attention towards the irrelevant dimension (i.e., word-processing) which impairs attention for the relevant dimension (i.e., naming the print-color). Alternatively, according to the threat account, negative stimuli have a freezing effect, which slows down all ongoing activities (Algom et al., 2004). For instance, Stolicyn et al. (2017) suggested that emotional words activate the amygdala, which supports representations of task-related stimuli via projections to the medial prefrontal cortex and the orbitofrontal cortex at the expense of other ongoing tasks (Stolicyn et al., 2017). Cognitive Control The Emotional Stroop effect indicates the vulnerability of goal-directed behavior by showing the costs of emotional distraction. However, adaptive human behavior critically relies on the ability to shield current goals from such distraction. Typically, it is assumed that our cognitive system has evolved dedicated control mechanisms that suppress interference from task-irrelevant information (Miller & Cohen, 2001). In the lab, response-interference-tasks (i.e., tasks that activate multiple response options) serve to examine cognitive control processes. A prominent task is the Color Stroop task (Stroop, 1935) in which participants show a delay in reaction times (RTs) and increased error rates when they respond to the print-color (i.e., relevant target dimension) of color words that can be congruent (e.g., the word "BLUE" printed in blue color) or incongruent (e.g., the word "BLUE" printed in red color) to the word's meaning (i.e., irrelevant distractor dimension). Performance differences between incongruent and congruent trials have been termed congruency effects. Notably, conflict in response-interference-tasks (such as the Color Stroop task) have been found to operate on different timescales (Braver, 2012). Cognitive conflict triggers control in two modes: a reactive control mode that influences the most recent events and permits control during conflict (Scherbaum et al., 2011; Weichart et al., 2020) and a proactive control mode, which operates on a longer timescale following conflict across subsequent trials (Hubbard et al., 2017; Pastötter et al., 2013). Interaction of cognitive control and emotion Is control confined to "cognitive" disturbances or also effective to shield against emotional distraction? Traditionally, cognitive control and emotion have been often described as independent, relying on separate mental faculties (Zajonc, 1980). This view received support from studies showing behavioral (Soutschek & Schubert, 2013) and neural (Egner et al., 2008) evidence for a double dissociation between those tasks that tap into cognitive control and others that tap into emotional processing. More specifically, results from an fMRI study by Egner et al. (2008) revealed that a lateral prefrontal "cognitive control" circuitry that resolved nonemotional conflict can be separated from a rostral anterior cingulate "emotional control" circuitry that resolved emotional conflict (Egner et al., 2008). These findings of a functional segregation were in line with neuroimaging research that showed a separation between the rostral anterior cingulate cortex that is primarily involved in affective processing and regions of the dorsal anterior cingulate cortex and the lateral prefrontal cortex that were associated with nonemotional cognitive processes (Bush et al., 2000). In contrast, other theories question this separation and propose a close interaction between emotion and control (Dignath et al., 2020; Inzlicht et al., 2015; Pessoa, 2008; Shackman et al., 2011; Vermeylen et al., 2020). For instance, Pessoa (2008) suggested that brain regions, such as the amygdala, the orbitofrontal cortex, and the anterior cingulate cortex, function as central hubs that integrate emotional and cognitive information (Pessoa 2008; for a meta-analysis see Shackman et al., 2011). Lesion studies support this view demonstrating that dorsal anterior cingulate cortex lesions cause deficits in the recognition of negative emotions and cognitive response-interference-tasks (Tolomeo et al., 2016). Experimental studies suggest that the anterior cingulate cortex responds similarly to cognitive conflict and negative pictures (Braem et al., 2017). Furthermore, using multivariate pattern classification, Vermeylen et al. (2020) showed overlapping activity for cognitive conflict and negative affect in the medial frontal cortex. Together, neurophysiological studies corroborate the idea that cognitive control and negative affect share a common functional architecture (see also Song et al., 2017). On a behavioral level, studies show that exerting control can attenuate emotional distraction suggesting that cognitive control may block-off task-irrelevant emotional stimuli (Cohen et al., 2012, 2015; see also Straub et al., 2020). More specifically, Cohen et al. (2012) induced conflict by an arrow flanker task (i.e., participants are required to identify the direction of an arrow which is surrounded by flanking arrows that are either congruent or incongruent with the direction of the arrow in the center) and emotional distraction by negative (vs. neutral) valent pictures. Presenting the response-interference-task and the pictures alternatingly, they found reduced emotional distraction from negative valent picture stimuli in incongruent response-interference-tasks (i.e., reactive control attenuated emotional distraction, see Cohen et al., 2012). In another experiment, the authors used a task-switching design and presented first a response-interference-task that was followed by a color-discrimination task devoid of conflict which measured the impact of a task-irrelevant negative valent picture stimuli presented between both tasks. Results showed that control from incongruent response-interference tasks reduced emotional distraction from negative valent pictures measured in the discrimination task (i.e., proactive control attenuates emotional distraction, see Cohen et al., 2012, 2015, see Goldsmith, 2018 for failure to replicate). However, other studies that addressed the impact of task-irrelevant emotional stimuli on control in response-interference tasks found rather mixed evidence (Fruchtman-Steinbok et al., 2017; Goldsmith, 2018; Hart et al., 2010). This ambiguity of findings is illustrated by Liu et al. (2017) who concluded that "[ …] negative affect has been found to improve, impair, or have no effect on conflict resolution" (Liu et al., 2017, p.69), questioning how generalizable the interplay between cognitive control and emotion is. For instance, Ahmed and Sebastian (2019) suggested that predictability of task conditions (i.e., incongruent or congruent stimuli in conflict tasks, negative or neutral stimuli in emotional tasks) might be a critical moderator for the interaction between cognitive and emotional tasks. They argue that top-down anticipation in cognitive and emotional domains (e.g., when incongruent and congruent as well as negative and neutral stimuli are presented in blocks) is required for conflict-triggered modulation of emotional distraction (Ahmed & Sebastian, 2019). Against this background, theoretical models are needed that (i) allow a more detailed understanding of a hypothesized interaction of cognitive control and emotion and (ii) thus make testable predictions on how control should modulate emotional distraction. For instance, Cohen et al. (2012, 2015) referred to the conflict monitoring theory to account for their findings that reactive and proactive control reduced emotional distraction (caused by irrelevant negative valent picture stimuli, Cohen et al., 2012, 2015). The conflict monitoring theory (Botvinick et al., 2001) describes control as a feedback loop within a connectionist model. A monitoring unit outputs a measure of competing response activation during a trial, which then scales the activation level of a task demand unit. Increased activation of task demands, representing the current task-set, leads to a change in weightings of related stimulus information and response activation. As a consequence, processing in the next trial is biased towards more relevant information relative to irrelevant information, alleviating further distraction. This model received empirical support from behavioral and neuroimaging studies showing that after incongruent trials, processing of relevant information is enhanced (Egner & Hirsch, 2005) and irrelevant information is suppressed (Stürmer et al., 2002). However, the model as described above, addressed response-interference tasks devoid of emotional stimuli. It remains unclear, therefore, how the original conflict monitoring proposal can account for the empirical findings of Cohen et al. (2012, 2015). Interestingly, an extension by Wyble et al. (2008) allowed to simulate performance both in the Color Stroop task and in the Emotional Stroop task. Here, emotional distraction is modeled by adding an additional "negative emotional node" that exerts an inhibitory influence on the task demand unit and thereby decreases the activation level of the current task representation. As a consequence, task-irrelevant emotional information impairs performance by reducing cognitive control. Model simulations showed that "an incongruent [Color Stroop] trial reduces the impact of a following negative emotional stimulus by suppressing the emotional node […]" (Wyble et al., 2008, p. 19). Based on the architecture of the model, we derived predictions about how performance in the Emotional Stroop task and the Color Stroop task should interact. Please note that although this prediction is based on simulated data, it corresponds closely to the empirical observation of reduced emotional distraction of pictures after incongruent response-interference-tasks in Cohen's studies (2012, 2015). Thus, if correct, the model would allow a mechanistic explanation of how cognitive control and emotions might interact in terms of the conflict monitoring theory. However, the tasks used in previous empirical work differ in many aspects from the simulated data in Wyble et al. (2008). For instance, cognitive control measured in Stroop and flanker tasks differs on a behavioral (De Houwer, 2003), physiological (Tillman & Wiens, 2011), and theoretical level (Kornblum et al., 1990; Schuch et al., 2019). Therefore, the goal of the present research is to provide a direct empirical test of the prediction that cognitive control from Color Stroop tasks modulates emotional distraction instigated by Emotional Stroop tasks. The present research We tested whether conflict in the Color Stroop task interacts with emotional distraction in the Emotional Stroop task, as suggested by the model of Wyble et al. (2008). Stimuli in both tasks vary on the same relevant dimension (i.e., naming the print-color in the Color and the Emotional Stroop task). This is important, because it allows to describe the interaction between cognitive control and emotional distraction in terms of the conflict monitoring model. More specifically, in incongruent Color Stroop tasks, attentional weights of the relevant dimension (i.e., naming the print-color) are increased and conflict from Color Stroop tasks should facilitate print-color-naming in Emotional Stroop tasks. Accordingly, proactive and reactive cognitive control from incongruent trials in the Color Stroop tasks should decrease emotional distraction instigated by the Emotional Stroop task. However, in congruent Color Stroop stimuli, attention is directed towards the irrelevant word-meaning and the relevant print-color (because both predict the correct response) and print-color-naming in Emotional Stroop tasks is not facilitated. These considerations are in line with empirical evidence of studies showing that cognitive control generalizes across cognitive tasks that share the same relevant dimension (Kunde & Wühr, 2006; Notebaert & Verguts, 2008). Furthermore, the two tasks used in our design share not only the same relevant dimension but also the irrelevant stimulus dimension (i.e., ignoring the semantic meaning of the carrier word in the Color Stroop and the Emotional Stroop task). Critically, tasks differ according to the response set associated with the irrelevant dimension. While the Color Stroop task's irrelevant dimension affords a response that either matches or mismatches the correct response afforded by the relevant stimulus dimension, this is not true for the Emotional Stroop task. Naming the color of a negative valent or neutral word (i.e., irrelevant stimulus dimension) is not mapped to any response option and thus it can neither be congruent nor incongruent with color-naming (e.g., the word "ILLNESS" is not mapped to any distinct print-color). Based on this, emotional distraction instigated by Emotional Stroop tasks and congruency effects from Color Stroop tasks can be considered as two functional distinct effects (i.e., a threat induced slow-down in the Emotional Stroop task versus competition between incongruent stimuli in the Color Stroop task, see Algom et al., 2004). To probe the hypothesized interaction between cognitive control and emotional distraction, we intermixed Color and Emotional Stroop tasks. On each trial, participants had to name the print-color of a colored word (same relevant dimension in both tasks). To create Color Stroop and Emotional Stroop conditions, the type of colored words alternated on each trial between color words and negative/neutral valent words (both tasks vary on the same irrelevant dimension). Critically, while negative/neutral valent words were not related to any print-color, all color words presented referred to the same set of colors as used for print-color (different irrelevant response set between both tasks). Color Stroop trials, but not Emotional Stroop trials, created congruent (e.g., "RED" printed in red) and incongruent (e.g., "RED" printed in green) conditions that allowed to express performance as a congruency effect (incongruent – congruent trials). In reverse, Emotional Stroop trials, but not Color Stroop trials, created neutral (e.g., "HOUSE" printed in red) and negative valent (e.g., "WAR" printed in red) conditions that allowed to express performance as an Emotional Stroop effect (negative – neutral trials). This alternating-runs design (i.e., Color Stroop task - Emotional Stroop task - Color Stroop task - Emotional Stroop task…) allowed us to test whether emotional distraction is modulated by both, proactive and reactive control. As explained above, emotional distraction occurs in the Emotional Stroop task and also persists in time and occurs in the task subsequent to the negative word stimuli and thus we can measure the emotional distraction at two different times. Accordingly, this design tested (i) how proactive control from the Color Stroop task modulates emotional distraction in the subsequent Emotional Stroop task ('proactive control on emotional distraction') and (ii) how reactive cognitive control from the Color Stroop task modulates emotional distraction that persists in time and stems from the previous Emotional Stroop task ('reactive control on emotional distraction'). We expected that cognitive control reduces emotional distraction and thus hypothesized that (i) proactive cognitive control from Color Stroop tasks reduces emotional distraction in subsequent Emotional Stroop tasks and (ii) that reactive cognitive control reduces emotional distraction instigated by the previous Emotional Stroop task (Figure 1). Proactive and Reactive Control on emotional distraction. Note. Example trial sequence with Color Stroop stimuli and Emotional Stroop stimuli. In 'proactive control on emotional distraction', control from the Color Stroop task modulates emotional distraction in the subsequent Emotional Stroop task and in 'reactive control on emotional distraction' control from the Color Stroop task modulates emotional distraction from the previous Emotional Stroop task. In terms of predictability (Ahmed & Sebastian, 2019), we presented congruency (congruent vs. incongruent stimuli in Color Stroop tasks) and valence (negative vs. neutral stimuli in Emotional Stroop tasks) conditions in different configurations across the Experiments (i.e., Experiment 1a & Experiment 1b blocked valence and congruency, Experiment 2 manipulated congruency trialwise, Experiment 3 manipulated valence trialwise). Experiment 1a In Experiment 1a, congruency in Color Stroop and valence in Emotional Stroop tasks was manipulated blockwise, which predicts the largest Emotional Stroop effects (McKenna & Sharma, 2004; Phaf & Kan, 2007). We aimed to find proactive and reactive control on emotional distraction. Previous research that investigates how congruency effects generalize across different tasks that share the same relevant dimension observed effect sizes ranging from dz = 0.956 (Notebart & Verguts, 2008) to dz = 2.11 (see Kunde & Wühr, 2006, Experiment 1). Power analyses using G*Power suggested a minimum sample size of N = 14 to detect an effect of dz = 0.956 (with α = 0.05 and 1-β = 0.9) for the within-groups comparison between emotional distraction in congruent and incongruent conditions. The present study tested 40 participants and was completed at the University of Freiburg, Germany. All participants reported normal or corrected-to-normal vision and were compensated with either course credit or money. Exclusion criteria were identical for all experiments and defined a priori based on conventions of our workgroup. We excluded participants with (i) random responses (error rate above 50%) and (ii) with an error rate above three standard deviations (SDs) from the remaining sample after excluding random responses. In Experiment 1a, no participant was excluded due to random responses. Data of one participant were excluded due to error rates above three standard deviations (SDs). Hence, we analyzed data of 39 participants (1 left-handed, 27 female, Mage = 26.6 years). Apparatus, Procedure, and Stimuli The experiment was programmed and presented with e-Prime software 2.0, E-Studio (version: 2.0.10.252; Schneider et al., 2002). Responses were collected with standard German QWERTZ-keyboard. After providing informed consent, participants were instructed to respond to the four different print-colors in which words on the screen were presented via four previously assigned keys (i.e., green, red, yellow, blue). Color-to-key mapping was counterbalanced across participants. Each trial started with a Color Stroop task (i.e., a color word ("RED," "GREEN," "BLUE," "YELLOW") presented in either green (RGB = 0, 128, 64), red (255, 0, 0), blue (0, 255, 255), or yellow (255, 255, 0) print-color). Participants had to classify the print-color (target) and ignore the word-meaning of the carrier word (distractor). Stimuli were either congruent to the word-meaning (e.g., "GREEN" written in green print-color) or incongruent (e.g., "GREEN" written in yellow print-color). For incongruent stimuli, all possible color-word combinations were used (e.g., the word "BLUE" can be printed in red, yellow, or green color). In each trial, a Color Stroop task was followed by an Emotional Stroop task (i.e., a word with either negative (distractor) or neutral valence presented in green, red, yellow, or blue print-color (target). Participants had to respond to the print-color with the previously assigned keys. Stimuli comprised 20 neutral and 20 negative words taken from the Berlin Affective Word List (BAWL, Võ et al., 2009). According to the database, mean arousal was rated on a 6-point scale ranging from 0 to 5 (0 = not arousing to 5 = highly arousing) and valence was rated on a 7-point scale ranging from −3 (very negative) through 0 (neutral) to 3 (very positive). Ratings of the arousal of stimuli used in our experiment corresponded to mean ratings of 4.25 (SD = 0.32) in negative stimuli and 1.76 (SD = 0.09) in neutral stimuli, t(38) = 33.128, p < 0.01, and ratings of the valence of stimuli correspond to mean ratings of −2.66 (SD = 0.17) in negative stimuli and 0.03 (SD = 0.11) in neutral stimuli, t(38) = −59.290, p < 0.01 , respectively. We matched the number of letters in negative and neutral words but not word frequency of both conditions (i.e., in our stimulus set, frequency of use is lower in negative compared to neutral words), which have been shown to increase Emotional Stroop effects (Kahan & Hely, 2008; Larsen et al., 2006). All words were written in capital letters and one letter subtended 1.72° (width) × 1.33° (height) of visual angle, measured from a viewing distance of 60 cm. In Experiment 1a, both congruency (congruent or incongruent) of the Color Stroop task and valence (negative or neutral) of the word stimuli in the Emotional Stroop task were manipulated blockwise resulting in four block conditions (i.e., (i) incongruent color-word stimuli and negative word stimuli, (ii) congruent color word stimuli and negative word stimuli, (iii) incongruent color-word stimuli and neutral word stimuli, and (iv) congruent color word stimuli and neutral word stimuli). Each block was presented 3 times, resulting in 12 blocks in total. Within these 12 blocks, the 4 different block conditions were presented in random order. Each block comprised 20 trials in which each negative or neutral word was presented once, resulting in 240 trials in total. Stimuli within one block were presented in random order. Participants made self-paced rests after each block. A trial started with the presentation of a word stimulus that remained on the screen for 3,000 ms or until a response was registered, followed by a 500-ms Inter-Trial-Interval. A trial sequence was identical for the Color and the Emotional Stroop task. Participants received accuracy feedback after each task and were asked to respond as quickly and accurately as possible. They started with a practice session, including accuracy feedback of 24 trials. After finishing the Experiment participants completed a German version of the Emotion Regulation Questionnaire (ERQ) (Gross & John, 2012; Loch et al., 2011) and a German version of Trait Anxiety Inventory (STAI-T) (Laux, Glanzmann et al., 1981; Spielberger & Sydeman, 1994). Mean RTs and error rates were calculated separately for Color and Emotional Stroop tasks. Data were analyzed with a repeated-measures Analysis of Variance (ANOVA) with the within-subject factors congruency (incongruent vs congruent) and valence (negative vs. neutral). Testing 'proactive control on emotional distraction´, independent variables were previous congruency (pre-congruency) of Color Stroop tasks (i.e., congruent or incongruent) and valence (valence) of Emotional Stroop tasks (i.e., negative or neutral). RTs and error rates in Emotional Stroop tasks served as dependent variables. Testing 'reactive control on emotional distraction', independent variables were previous valence (pre-valence) of Emotional Stroop tasks and congruency (congruency) of Color Stroop tasks. RTs and error rates in Color Stroop tasks served as dependent variables. Participant's trait anxiety (STAI-T score) and the emotion regulation strategies (ERQ-Score) were correlated with modulation of emotional distraction by cognitive control. Bonferroni corrected p-values were calculated via dividing the desired alpha level (i.e., α = 0.05) by the number of comparisons (i.e., ERQ and 'proactive control on emotional distraction', ERQ and 'reactive control on emotional distraction', STAI-T and 'proactive control on emotional distraction', STAI-T and 'reactive control on emotional distraction'; n = 4), resulting in a least significant difference p-value of 0.05/4 = 0.0125. Relevant null-effects (i.e., nonsignificant interaction effects between congruency and valence) were further analyzed by calculating Bayes Factors (BF) using JASP (Jarosz & Wiley, 2014). The Bayesian approach is a model selection procedure that compares the likelihood of the data considered under both, the null- and the alternative-hypothesis via calculation of the BF01. The BF01 gives an index of how strong data are in favor of the null-hypothesis. The default setting of JASP Bayesian statistics paired t-test was used as prior, which consists of a Cauchy distribution (i.e., a t-distribution with a single degree of freedom) with its parameter set to r = 0.707. Trials in which participants committed an error in the Color or the Emotional Stroop task and all trials following an error trial were excluded (5.8% and 4.3% of responses in the Color and the Emotional Stroop task, respectively). Furthermore, RTs that deviate more than three SDs for each participant and each condition (i.e., each cell of the ANOVA design) were removed from RT analyses (1.4% and 1.8% of responses in the Color and the Emotional Stroop task, respectively). Impact of proactive cognitive control on emotional distraction Results of the two-way ANOVA with the within-subject factors pre-congruency and valence and performance in the Emotional Stroop task serving as dependent variable revealed no significant main effects of pre-congruency and valence and no interaction of pre-congruency × valence, (Fs < 1). Impact of reactive cognitive control on emotion distraction Results of the two-way ANOVA with pre-valence and congruency as within-subject factors and performance in the Color Stroop task as dependent variable revealed a significant main effect of pre-valence F(1,38) = 5.919, p = 0.020, $ {\eta}_p^2 $= 0.135, indicating prolonged RTs in Color Stroop tasks that were preceded by negative word stimuli (M = 760 ms, SE = 25 ms) compared with RTs in Color Stroop tasks that were preceded by neutral word stimuli (M = 740 ms, SE = 25 ms), a significant main effect of congruency, F(1,38) = 85.177, p < 0.001, $ {\eta}_p^2 $= 0.692 demonstrating faster responses in congruent Color Stroop tasks (M = 683 ms, SE = 22 ms) compared with incongruent Color Stroop tasks (M = 817 ms, SE = 29 ms) but no interaction of pre-valence × congruency (F < 1). Bayesian Analysis Quantification of the results by BFs assumes that in 'proactive control on emotional distraction', the null-hypothesis indicates that emotional distraction within Emotional Stroop tasks preceded by incongruent Color Stroop tasks is not smaller compared with emotional distraction within Emotional Stroop tasks preceded by congruent Color Stroop tasks. The alternative-hypothesis indicates that emotional distraction within Emotional Stroop tasks preceded by incongruent Color Stroop tasks is smaller compared with emotional distraction within Emotional Stroop tasks preceded by congruent Color Stroop tasks. The corresponding BF provides positive evidence for the null-hypothesis relative to the alternative-hypothesis (BF01 = 3.194) and indicates that the data are three times more likely under the null-hypothesis than under the alternative-hypothesis (Schönbrodt & Wagenmakers, 2018). In the analysis of 'reactive control on emotional distraction', the null-hypothesis indicates that emotional distraction in Color Stroop tasks instigated by preceding negative Emotional Stroop tasks is not smaller in incongruent compared with congruent Color Stroop tasks. The alternative-hypothesis indicates that emotional distraction in Color Stroop tasks instigated by preceding negative Emotional Stroop tasks is smaller in incongruent compared with congruent Color Stroop tasks. The corresponding BF01 provides positive evidence for the null-hypothesis relative to the alternative-hypothesis (BF01 = 7.086) and indicates that the data are seven times more likely under the null-hypothesis than under the alternative-hypothesis. Error Rates Analogous analyses were performed on error rates. Results of the two-way ANOVA with pre-congruency and valence serving as within-subject factors and performance in the Emotional Stroop task serving as dependent variable revealed no significant main effect of valence, F(1,38) = 1.779, p = 0.190, $ {\eta}_p^2 $= 0.045. The main effect of pre-congruency and the interaction effect of pre-congruency and valence were not significant (Fs < 1). Impact of reactive cognitive control on emotional distraction Results of the two-way ANOVA with pre-valence and congruency as within-subject factors and performance in the Color Stroop task serving as dependent variable revealed no significant main effects of pre-valence and congruency (Fs < 1), but a significant interaction effect, F(1,38) = 6.669, p = 0.014, $ {\eta}_p^2 $= 0.149 with more emotional distraction in congruent Color Stroop tasks (M = 8.5%, SE = 3.7%) compared with incongruent Color Stroop tasks (M = −7.7%, SE = 4.2%). The corresponding Bayesian Analysis indicates that in 'proactive control on emotional distraction' data are seven times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 6.769). BFs were calculated for non-significant interaction effects only. Correlations between ERQ- and STAI-ScoresFootnote 1 and modulation of emotional distraction were not significantFootnote 2 ('proactive control on emotional distraction', ERQ: r(38) = −0.030, p = 0.858, STAI-T: r(39) = 0.200, p = 0.222, 'reactive control on emotional distraction', ERQ: r(38) = 0.186, p = 0.264, STAI-T: r(39) = 0. 097, p = 0.561). Experiment 1b We aimed to boost interference effects from emotional stimuli to further analyze the modulation of emotional distraction by cognitive control. Therefore, we changed the duration and inter-trial-intervals of stimuli. The blank time between tasks was set to 6.9 msFootnote 3 and the duration of stimuli presentation was reduced to 2000 ms. This modification was based on studies by McKenna (1986) and McKenna & Sharma (1995), who observed reliable interference effects in Emotional Stroop tasks under time pressure (McKenna, 1986; McKenna & Sharma, 1995). We used the same stimuli and procedure as in Experiment 1a but added four word stimuli from the BAWL database to the negative and the neutral word categories, resulting in 24 trials per block. Each block was presented five times, resulting in 20 blocks and 480 trials in total. We added the constraint that the four block conditions (i.e., (i) incongruent Color Stroop stimuli and negative words, (ii) congruent Color Stroop stimuli and negative words, (iii) incongruent Color Stroop stimuli, and neutral words (iv) congruent Color Stroop stimuli and neutral words) did not repeat throughout the experiment. Furthermore, stimuli within one block were presented randomly with the constraint that the task-relevant dimension (i.e., print-color of the words) did not repeat in two consecutive trials. In the practice session, we presented 100 words of randomly mixed letters in the four colors and accuracy feedback was provided so that participants learned the color-to-key mapping. Study site, results of the power analyses, number of recruited participants, as well as inclusion and exclusion criteria for participants were the same as in Experiment 1a. No participant was excluded due to random responses. Data of one participant was excluded due to error rates of three SDs above the mean error rates. Hence, we analyzed data of 39 participants (5 left-handed, 30 female, Mage = 24.70 years). Results of the two-way ANOVA with the within-subject factors pre-congruency and valence and performance in the Emotional Stroop task serving as dependent variable revealed a significant main effect of pre-congruency, F(1,38) = 6.268, p = 0.017, $ {\eta}_p^2 $= 0.142, demonstrating faster responses after congruent Color Stroop tasks (M = 876 ms, SE = 17 ms) compared with incongruent Color Stroop tasks (M = 860 ms, SE = 15 ms) and valence, F(1,38) = 9.462, p = 0.004, $ {\eta}_p^2 $= 0.199, indicating emotional distraction demonstrated in prolonged RTs in negative Emotional Stroop tasks (M = 877 ms, SE = 16 ms) compared with neutral Emotional Stroop tasks (M = 859 ms, SE = 15 ms) but no interaction of pre-congruency × valence, F(1,38) = 0.001, p = 0.973, $ {\eta}_p^2 $ < 0.001. Results of the two-way ANOVA with the within-subject factors pre-valence and congruency and performance in the Color Stroop task serving as dependent variable revealed a significant main effect of pre-valence, F(1,38) = 5.132, p = 0.029, $ {\eta}_p^2 $= 0.119, indicating emotional distraction demonstrated in prolonged RTs in Color Stroop tasks that were preceded by negative word stimuli (M = 875 ms, SE = 17 ms) compared with RTs in Color Stroop tasks that were preceded by neutral word stimuli (M = 860 ms, SE = 16 ms). Furthermore, there was a significant main effect of congruency, F(1,38) = 150.488, p < 0.001, $ {\eta}_p^2 $= 0.798 demonstrating faster responses in congruent (M = 806 ms, SE = 15 ms) compared with incongruent Color Stroop tasks (M = 929 ms, SE = 20 ms), but no interaction of pre-valence × congruency, F(1,38) = 0.149, p = 0.702, $ {\eta}_p^2 $= 0.004. The corresponding Bayesian Analysis indicates that in 'proactive control on emotional distraction', data are six times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 5.948). In 'reactive control on emotional distraction', data are four times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 4.189). Results of the two-way ANOVA with the within-subject factors pre-congruency and valence and performance in the Emotional Stroop task serving as dependent variable revealed no significant main and interaction effects (all Fs < 1). Results of the two-way ANOVA with the within-subject factors pre-valence and congruency and performance in the Color Stroop task serving as dependent variable revealed no significant main effect of pre-valence, F(38) = 1.566, p = 0.218 , $ {\eta}_p^2 $= 0.040, but a significant main effect of congruency, F(38) = 13.827, p = 0.001, $ {\eta}_p^2 $= 0.267, demonstrating less errors in congruent tasks (M = 6.6%, SE = 0%) compared to incongruent tasks (M = 8.7%, SE = 1%). Interaction effects between pre-valence and congruency were not significant (F < 1). The corresponding Bayesian Analysis indicates that in 'proactive control on emotional distraction', data are eight times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 8.133). In 'reactive control on emotional distraction', data are seven times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 6.860). Correlations between ERQ- and STAI-T scores and modulation of emotional distraction were not significant ('proactive control on emotional distraction', ERQ: r(39) = 0.017, p = 0.919, STAI-T: r(39) = −0.392, p = 0.014Footnote 4, 'reactive control on emotional distraction', ERQ: r(39) = 0.013, p = 0.938, STAI-T: r(39) = −0.070, p = 0.674). Modifications in the design of Experiment 1a revealed emotional distraction within and subsequent to the Emotional Stroop tasks in Experiment 1b. However, contrary to our predictions, we did not find any evidence for 'proactive control on emotional distraction' or 'reactive control on emotional distraction'. We consider two possible reasons for the null-effects that implicate further manipulations in the following experiments. First, we suggest that predictability of congruency (i.e., conflict tasks were presented in blocks with either congruent or incongruent stimuli in Experiments 1a and 1b) may play a role in the activation of top-down anticipatory control mechanism (Ahmed & Sebastian, 2019; Grimshaw et al., 2018). We removed the predictability of the Color Stroop task's congruency in Experiment 2 by manipulating congruency trialwise. Procedure and Stimuli were the same as in Experiment 1b. In each block incongruent (50%) and congruent (50%) Color Stroop stimuli were presented in random order and valence of the Emotional Stroop stimuli was either negative or neutral within one block. There were 20 blocks in total, each comprising 24 trials. The two different block conditions (i.e., (i) incongruent and congruent Color stimuli with negative words, and (ii) incongruent and congruent Color stimuli with neutral words) were presented in random order. The study was completed at the University of Würzburg, Germany. Power analyses, inclusion, and exclusion criteria for participants were the same as in Experiments 1a and 1b. A total of 41 participants completed the study. No participants were excluded due to random answering or error rates of 3 SDs above the mean error rates. Hence, we analyzed data of 41 participants (3 left-handed, 28 females, Mage = 24.37 years). Trials in which participants committed an error in the Color or the Emotional Stroop task and all trials following an error trial were excluded (8.9% and 8.1% of responses in the Color and the Emotional Stroop task, respectively). Furthermore, RTs that deviate more than three SDs for each participant and each condition (i.e., each cell of the ANOVA design) were removed from RT analyses (1.0% and 1.3% of responses in the Color Stroop task and the Emotional Stroop task, respectively). Results of the two-way ANOVA with the within-subject factors pre-congruency and valence and performance in the Emotional Stroop task serving as dependent variable revealed significant main effects of pre-congruency, F(1,40) = 4.103, p = 0.050, $ {\eta}_p^2 $= 0.093, demonstrating faster responses after congruent (M = 864 ms, SE = 16 ms) compared with incongruent Color Stroop tasks (M = 878 ms, SE = 16 ms) and a valence, F(1,40) = 9.860, p = 0.003, $ {\eta}_p^2 $= 0.198, indicating emotional distraction demonstrated in prolonged RTs in negative (M = 878 ms , SE = 16 ms) compared with neutral Emotional Stroop tasks (M = 864 ms, SE = 16 ms) but no interaction of pre-congruency × valence, F(1,40) = 0.027, p = 0.871, $ {\eta}_p^2 $ < 0.001. Results of the two-way ANOVA with the within-subject factors pre-valence and congruency and performance in the Color Stroop task serving as dependent variable revealed significant main effects of pre-valence F(1,40) = 11.246, p = 0.002, $ {\eta}_p^2 $= 0.219, indicating emotional distraction demonstrated in prolonged RTs in Color Stroop tasks that were preceded by negative Emotional Stroop tasks (M = 893 ms, SE = 17 ms) compared with RTs in Color Stroop tasks that were preceded by neutral word stimuli (M = 873 ms, SE = 15 ms) and congruency, F(1,40) = 236.559, p < 0.001, $ {\eta}_p^2 $= 0.855, demonstrating faster responses in congruent (M = 818 ms, SE = 16 ms) compared with incongruent Color Stroop tasks (M = 948 ms, SE = 17 ms), but no interaction of pre-valence × congruency, F(1,40) = 3.214 , p = 0.081, $ {\eta}_p^2 $= 0.074. Emotional distraction was descriptively larger in incongruent Color Stroop tasks (M = 28 ms, SE = 8 ms) compared with congruent Color Stroop tasks (M = 13 ms, SE = 8 ms), which is against the prediction (see also Bayesian Analysis below). The corresponding Bayesian Analysis indicates that in 'proactive control on emotional distraction', data are five times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 5.204). In 'reactive control on emotional distraction', data are 15 times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 15.357). Results of the two-way ANOVA with the within-subject factors pre-valence and congruency and performance in the Color Stroop task serving as dependent variable revealed a significant main effect of pre-valence, F(1,40) = 6.090, p = 0.018, $ {\eta}_p^2 $= 0.132, indicating emotional distraction demonstrated in more errors in Color Stroop tasks that were preceded by negative Emotional Stroop tasks (M = 8.4%, SE = 1%) compared with Color Stroop tasks that were preceded by neutral Emotional Stroop tasks (M = 7.3%, SE = 1%). Furthermore, there was a significant main effect of congruency, F(1,40) = 20.828, p < 0.001, $ {\eta}_p^2 $= 0.342, demonstrating less errors in congruent (M = 6.5%, SE = 1) compared with incongruent Color Stroop tasks (M = 9.2%, SE = 1%) and an interaction of pre-valence × congruency, F(1,40) = 5.376, p = 0.026, $ {\eta}_p^2 $ = 0.118 with more emotional distraction in incongruent Color Stroop tasks (M = 2.1%, SE = 0.7%) compared with congruent Color Stroop tasks (M = 0.1%, SE = 0.5%), which is against the expected results. The corresponding Bayesian Analysis indicates that in 'proactive control on emotional distraction', data are ten times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 10.483). Correlations between the questionnaires and modulation of emotional distraction by congruency were not significant ('proactive control on emotional distraction', ERQ: r(41) = 0.237, p = 0.136. STAI-T: r(41) = 0.016, p = 0.921, 'reactive control on emotional distraction', ERQ: r(41) = −0.161, p = 0.316, STAI-T: r(41) = 0.126 , p = 0.431). In Experiments 1b and 2, we found emotional distraction within and subsequent to the Emotional Stroop tasks but no evidence for 'proactive control on emotional distraction' or 'reactive control on emotional distraction'. We consider predictability of valence (i.e., Emotional Stroop tasks were presented in blocks with either negative or neutral valent stimuli in all previous experiments) as a second possible reason for the null-effects. Hence, we removed the predictability of valence in Emotional Stroop tasks. The procedure and stimuli used in Experiment 3 were the same as in Experiment 2 with the exception that the valence in Emotional Stroop tasks was manipulated trialwise. We found no differences in the size of congruency effects between blocked or trialwise presentation of Color Stroop stimuli (Experiment 1b blockwise, MCongruencyeffect = 125.77, SD = 63.36, Experiment 2MCongruencyeffect = 129.25, SD = 54.87, t(80) = −0.263, p = 0.793) and thus presented Color Stroop stimuli trialwise due to larger congruency effects in Experiment 2 compared with Experiment 1b. Within each block, congruency of the Color Stroop task and valence of the Emotional Stroop task were presented randomly. There were 14 blocks in total; each included 48 trials. We included four catch trials in each block (i.e., two randomly chosen words of both, the negative and the neutral categories printed in gray) and instead of indicating the word's print-color, participants had to categorize the meaning of these words into "negative" or "neutral" via keypress. The study was completed at the University of Freiburg. Results of the Power analyses and inclusion criteria for participants were the same as in Experiments 1a, b, and 2. Participants were excluded if the error rate in the categorization of the catch trial word's meaning exceeded 50%. A total of 42 participants completed the study. No participants were excluded due to random answering, one participant was excluded due to error rates of three SDs above the mean error rates, and one participant was excluded due to an error rate above 50% in the catch trials. Hence, we analyzed data of 40 participants (4 left-handed, 25 females, Mage = 27.73 years). Trials in which participants committed an error in the Color or the Emotional Stroop task and all trials following an error trial were excluded (9.5% and 9.1% of responses in the Color Stroop task and the Emotional Stroop task, respectively). Furthermore, RTs that deviate more than three SDs for each participant and each condition (i.e., each cell of the ANOVA design) were removed from RT analyses (0.9% and 1.1% of responses in the Color Stroop task and the Emotional Stroop task, respectively). Results of the two-way ANOVA with the within-subject factors pre-congruency and valence and performance in the Emotional Stroop task serving as dependent variable revealed a significant main effect of pre-congruency, F(1,39) = 11.378, p = 0.002, $ {\eta}_p^2 $= 0.226, demonstrating faster responses after congruent (M = 873 ms, SE = 18 ms) compared with incongruent Color Stroop tasks (M = 890 ms, SE = 17 ms). Results revealed no significant main effect of valence, F(1,39) = 0.158, p = 0.693, $ {\eta}_p^2 $= 0.004 and no significant interaction of pre-congruency × valence, F(1,39) = 0.787, p = 0.381, $ {\eta}_p^2 $ = 0.020. Results of the two-way ANOVA with the within-subject factors pre-valence and congruency and performance in the Color Stroop task serving as dependent variable revealed a significant main effect of pre-valence, F(1,39) = 13.138, p = 0.001, $ {\eta}_p^2 $= 0.252 indicating emotional distraction demonstrated in prolonged RTs in Color Stroop tasks that were preceded by negative Emotional Stroop tasks (M = 901 ms, SE = 18 ms) compared with RTs in Color Stroop tasks that were preceded by neutral word stimuli (M = 887 ms, SE = 18 ms), a significant main effect of congruency, F(1,39) = 216.185, p < 0.001, $ {\eta}_p^2 $= 0.847 demonstrating faster responses in congruent (M = 831 ms, SE = 18 ms) compared with incongruent Color Stroop tasks (M = 957 ms, SE = 19 ms), but no interaction of pre-valence × congruency, F(1,39) = 0.179 , p = 0.674, $ {\eta}_p^2 $= 0.005. The corresponding Bayesian Analysis indicates that in 'proactive control on emotional distraction', data are three times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 2.530) and in 'reactive control on emotional distraction', data are eight times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 7.883). Results of the two-way ANOVA with the within-subject factors pre-congruency and valence and performance in the Emotional Stroop task serving as the dependent variable revealed no significant main or interaction effects (Fs < 1). Results of the two-way ANOVA with the within-subject factors pre-valence and congruency and performance in the Color Stroop task serving as dependent variable revealed no significant main effect of pre-valence, F(1,39) = 4.071, p = 0.051, $ {\eta}_p^2 $= 0.095. Descriptively, error rates in Color Stroop tasks that were preceded by negative Emotional Stroop tasks were higher (M = 7.9%, SE = 1%) compared with Color Stroop tasks that were preceded by neutral Emotional Stroop tasks (M = 7.4%, SE = 1%). Furthermore, there was a significant main effect of congruency, F(1,39) = 14.709, p < 0.001, $ {\eta}_p^2 $= 0.274, demonstrating less errors in congruent (M = 6.5%, SE = 1%) compared with incongruent Color Stroop tasks (M = 8.8%, SE = 1%) but no significant interaction of pre-valence × congruency (F < 1). The corresponding Bayesian Analysis indicates that in 'proactive control on emotional distraction', data are five times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 5.180) and in 'reactive control on emotional distraction', data are four times more likely under the null-hypothesis than under the alternative-hypothesis (BF01 = 3.527). Correlations between ERQ- und STAI-T-scores and modulation of emotional distraction by congruency were not significant ('proactive control on emotional distraction', ERQ: r(40) = −0.244 , p = 0.129, STAI-T: r(40) = −0.175, p = 0.279, 'reactive control on emotional distraction', ERQ: r(40) = −0.012, p = 0.944, STAI-T: r(40) = 0.220, p = 0.172). Mini-Meta-Analysis We subjected the four experiments to a meta-analytic summary to get an overall estimate of the influences of cognitive control on emotional distraction. We wanted to attain an average true effect in the set of our four studies, which are methodologically similar and thus used a fixed-effect approach in which the mean effect size was weighted by sample size. At first, individual effect sizes were calculated separately for RTs and errors with the following formula (Cumming, 2014): $$ {d}_{av}=\frac{M_{diff}}{\ {S}_{AV}\ }. $$ Mdiff refers to the difference in emotional distraction between congruent and incongruent conditions. A positive value indicates that emotional distraction in incongruent conditions is smaller compared with congruent conditions. A negative value indicates that emotional distraction in congruent conditions is smaller compared to incongruent conditions. $$ {M}_{diff}={M}_{EDcon}-{M}_{EDinc,} $$ SAV refers to the pooled standard deviation of emotional distraction in congruent and incongruent conditions, $$ {S}_{AV}=\sqrt{\frac{SD_{EDcon}^2+{SD}_{EDinc}^2}{2}}. $$ Then, dz was corrected with Hedges's method (Hedges, 1982): $$ {g}_z={d}_z\times \left(1-\frac{3}{4\times \left(N-1\right)-1}\right), $$ in which N represents the sample size. The sampling variance (vi) was calculated according to Cumming (2014). $$ {v}_i=\left(\frac{1}{N}+\frac{dz^2}{2N}\right)\times {\left(1-\frac{3}{4\times \left(N-1\right)-1}\right)}^2. $$ Mini meta-analyses were conducted separately for RTs and errors in R (R Core Team, 2015) using the metafor package (Viechtbauer, 2010). Results showed no effects of congruency on emotional distraction in 'proactive control on emotional distraction' and 'reactive control on emotional distraction'. Specifically, the estimate for 'proactive control on emotional distraction' in RTs (M = 0.09, 95% confidence interval [CI] = [−0.06, 0.24]) was statistically nonsignificant, z = 1.16, p = 0.25, nor was the estimate for 'reactive control on emotional distraction' for RTs (M = −0.10, 95% CI = [−0.26, 0.06]), z = −1.30, p = 0.19. Furthermore, the estimate for 'proactive control on emotional distraction' in error rates (M = −0.08, 95% CI = [−0.23, 0.07]) was statistically nonsignificant, z = −0.99, p = 0.32, nor was the estimate for 'reactive control on emotional distraction' for error rates (M = 0.06, 95% CI = [−0.10, 0.22]), z = 0.78, p = 0.44, see Figure 2 for individual and overall estimates for RTs and for error rates). The observation that all effect-size confidence intervals include the null strengthens our observation that conflict in Color Stroop tasks does not modulate Emotional Stroop effects. Effect size estimates of proactive reactive control on emotional distraction. Note. Observed effect size estimates of the difference of emotional distraction between congruency conditions in 'proactive control on emotional distraction' and 'reactive control on emotional distraction' in RT and error analysis of Experiments 1a,b,2,3 and the overall effect sizes (represented with a diamond) with their 95% confidence intervals The study asked whether cognitive control (triggered in the Color Stroop task) attenuates emotional distraction in the Emotional Stroop task. This investigation was motivated by previous theoretical and empirical studies, suggesting that conflict-triggered activation of top-down monitoring processes suppress the effect of irrelevant emotional stimuli and thus attenuate emotional distraction (Cohen et al., 2012, 2015). We presented Color and Emotional Stroop tasks in an alternating-runs design and tested (i) how proactive control from the Color Stroop task modulates emotional distraction in the subsequent Emotional Stroop task ('proactive control on emotional distraction') and (ii) how reactive cognitive control from the Color Stroop task modulates emotional distraction that stems from the previous Emotional Stroop task and persists in time ('reactive control on emotional distraction'). We predicted that proactive and reactive cognitive control reduces emotional distraction. In three experiments (Experiments 1b, 2, 3) that manipulated predictability of congruency level (Experiment 2) and predictability of emotional content (Experiment 3), we found reliable congruency effects in Color Stroop tasks and emotional distraction within the Emotional Stroop task when emotional content was predictable. Moreover, in all three experiments, we also found a spillover of congruency effects and emotional distraction to the other task. More specifically, we observed prolonged responses in Emotional Stroop tasks following incongruent relative to congruent Color Stroop trials (i.e., conflict slowing effect, see Ullsperger et al., 2005; Verguts et al., 2011), and we observed prolonged responses in the Color Stroop tasks following negative relative to neutral Emotional Stroop trials, indicating that effects from both, incongruent Color Stroop trials, and negative Emotional Stroop trials cross task boundaries across trials to the other task. Consistent with previous literature (McKenna & Sharma, 2004; Phaf & Kan, 2007), Emotional Stroop effects that were instigated by a previous Emotional Stroop task were more robust than Emotional Stroop effects within the Emotional Stroop task, which were mainly limited to conditions with the blockwise presentation of emotional words (Experiment 1b and 2). However, against our hypotheses, we did not find reliable modulations of emotional distraction by cognitive control. This absence of an interaction was supported by Bayesian evidence for the null-model "emotional distraction is not smaller in incongruent compared with congruent conditions" for 13 of 14 tests (BFs ranging from BF01 = 2.530 to BF01 = 15.357) and by a mini-meta-analysis across all reported experiments, suggesting that the confidence interval of the overall effect size for a modulation of emotional distraction by cognitive control includes the null. In the following, we will discuss the present results with regard to theoretical accounts and previous empirical findings. Implications for theories A computational model of adaptive attentional control by Wyble et al. (2008) provides a detailed account of how cognitive control in Color and Emotional Stroop tasks interacts in terms of the conflict monitoring theory. It is assumed that a monitoring unit detects cognitive conflict and increases the activation level of a task demand unit, which represents the current task set. Consequently, processing of the task-relevant dimension is enhanced and distraction from the irrelevant task dimension in a consecutive trial is reduced. Wyble et al. (2008) propose that interference from task-irrelevant emotion stems from a "negative emotional node", which exerts an inhibitory influence on current task representations (Wyble et al., 2008; see also Stolicyn et al., 2017 for a neurobiologically inspired model). According to their simulated data, cognitive control in incongruent Color Stroop trials suppresses the emotional node and reduces the impact of task-irrelevant emotional information on subsequent Emotional Stroop trials. In the present study, we provide an empirical test of this prediction. While we observed that (i) interference effects occur in incongruent Color Stroop tasks and emotional distraction occurs within Emotional Stroop tasks and (ii) effects operate across trials to other tasks (i.e., conflict slowing effects and emotional distraction in tasks subsequent to the Emotional Stroop task), our results do not support the hypothesis that emotional distraction instigated by Emotional Stroop tasks is modulated by cognitive control from Color Stroop tasks. This suggests that changes in attentional weights for the relevant dimension in Color Stroop trials following or during conflict do not affect how strongly the irrelevant dimension in the Emotional Stroop task (i.e., the meaning of the negative word) distracts performance. This limitation of cognitive control is incompatible with model simulations put forward by Wyble et al. (2008) and predictions that we derived from the model. Furthermore, the observed limitation of cognitive control to block-off emotional distraction questions a domain-general view of cognitive and emotional control and suggests that cognitive control from Color Stroop tasks may not reflect the same conflict-triggered adaptation processes required to reduce emotional distraction. This may be the case even if both tasks share the same relevant and irrelevant dimension (but differ in terms of conflict instigated by cognitive tasks [competing response activation] and emotional tasks [general slow-down]). This dovetails with research that contrasts mechanisms involved in cognitive and emotional processes on behavioral and neural levels. These studies show that cognitive and emotional tasks are processed on different, domain-specific levels (Imbir et al., 2020; Kunde et al., 2012; Soutschek & Schubert, 2013) and are dissociable on a neural level (i.e., a lateral prefrontal cognitive control mechanism and a rostral anterior cingulate emotional control mechanism (Egner et al., 2008). Furthermore, our results indicate that Emotional Stroop tasks may not evoke cognitive control through the need for suppression of the emotional distraction (Okon-Singer et al., 2013), because this would potentially lead to an interaction with control from Color Stroop tasks (yet see an alternative interpretation of Vermeylen et al., 2020). The present results also might be of interest to the debate whether the Emotional Stroop effect is a special type of Stroop effect (Dalgleish, 2005) or is better characterized as a distinct phenomenon (Algom et al., 2004). Algom et al. (2004) present empirical data showing that Emotional Stroop effects behave differently than Color Stroop effects and argue in a conceptual analysis that the mechanisms underlying Color and Emotional Stroop tasks differ structurally and qualitatively. The results of the present research testing trial-by-trial combinations of Emotional and Color Stroop tasks show that these tasks do not interact, which strongly weights in favor of the independence of both tasks. It further indicates that Emotional Stroop tasks lack properties of Color Stroop tasks and vice versa and both effects are most likely distinct phenomena. Our results complement Algom et al.'s (2004) empirical observations and conceptual analysis by providing further evidence that Emotional and Color Stroop effects behave differently. Furthermore, by testing predictions that we derived from a computational model of the Emotional Stroop task, this research suggests that theoretical accounts of Emotional Stroop effects most likely require a cognitive architecture that differs from Cognitive Stroop models. Finally, the present findings also are of interest for research on the scope of control in response-interference-tasks. It has been debated whether cognitive control in one task generalizes across trials to other tasks (see Braem et al., 2014 for a review). A critical boundary condition that has been put forward assumes that if two different tasks share the same relevant dimension, control acts across trials to other tasks (Notebaert & Verguts, 2008). The authors proposed that processing of relevant dimensions is enhanced in conflict trials, which improves processing of (the same) relevant dimension in another task. While the tasks in our study meet this criterion (i.e., Color and Emotional Stroop tasks vary the same relevant dimension) our results show no evidence for a generalization of cognitive control across trials. Speculatively, emotional tasks may represent an exception to this proposal; emotional stimuli distract the processing of a task's relevant dimension in a conflict trial and thereby lever out any effects of cognitive control across tasks. Relation to previous research The present study used different tasks and stimulus material to induce conflict and emotional distraction (i.e., Color and Emotional Stroop task) compared to previous research (i.e., Flanker or Simon tasks and emotional pictures, see Cohen et al., 2012, 2015). As outlined above, we chose these tasks to test predictions that we derived from the computational model of Wyble et al. (2008). This new combination of cognitive and emotional tasks showed that cognitive control from Color Stoop tasks does not modulate emotional distraction instigated by Emotional Stroop tasks. This observation contrast with previous research showing that under specific circumstances cognitive control seems to attenuate emotional distraction (e.g., Cohen et al., 2012, 2015). In the following, we speculate how differences in response-interference-tasks and control mechanisms, timing and emotional distraction could account for these differences. First, the 4-choice Color Stroop tasks used in the present research differs from previous 2-choice flanker tasks in their complexity and thus resulted in different overall RTs (e.g., 4-choice Stroop: 700-1,000 ms vs. two-choice flanker: 400-800 ms, see Cohen et al., 2012, 2015; Straub et al., 2020). Many accounts hold that control takes time to develop suggesting that congruency effects differ between faster and slower responses (Ridderinkhof, 2002, see also Nieuwenhuis & de Kleijn, 2013 for the impact of alertness on cognitive control). Possibly, differences in overall RTs could explain the difference between the present and previous research. We tested this assumption in two post-hoc analyses that compared a possible interaction between cognitive control and emotional distraction across the RT distribution of each participant (within-subject comparison across percentiles) and averaged across all participants (between-subject comparison of relatively fast and slow participants). Both analyses replicated Color and Emotional Stroop effects but found no interaction between both, independent of the overall RT level. Thus, different overall RT levels do not viably explain the diverging results of our and previous studies. Second, the type of conflict and consequently, conflict resolution mechanisms, differ between response-interference tasks. While semantic and response conflict (De Houwer, 2003), as well as task conflict (Goldfarb & Henik, 2007), contribute to Stroop interference, flanker tasks create stimulus, and response conflict (van Veen & Carter, 2005). Furthermore, in the Stroop task, which has been used in the present research, control is concerned with feature-based attention (color vs. word), whereas in flanker tasks used in most of the studies by Cohen and colleagues, control has been attributed to changes in spatial attention (Wendt et al., 2012). Regarding mechanisms of conflict resolution, it has been suggested that in the Stroop task control leads to an amplification of the task-relevant dimension (Egner et al., 2007; Egner & Hirsch, 2005), whereas control in the Simon task (e.g., used in combination with affect in Fruchtman-Steinbok et al., 2017) is usually described as an inhibition of automatic response activation by the irrelevant dimension (Stürmer et al., 2002). Although largely speculative, differences between tasks could account for heterogeneous findings whether cognitive control does or does not modulate affective processing. Third, differences in emotional stimulus material and processing of emotional information could account for the discrepant results. In contrast to many previous studies that presented emotional pictures, the present research used emotional words, which have been criticized as ecologically invalid (Schimmack & Derryberry, 2005), and there has been a debate whether affective responses differ between words and pictures (Hinojosa et al., 2009; Kensinger & Schacter, 2006). Furthermore, it has been suggested that cognitive control over emotional distraction is limited to situations in which emotional stimuli are either task-relevant (i.e., when participants evaluated the valence of pictures) or entirely task-irrelevant (i.e., when participants ignored pictures completely). However, responding to a feature of the picture different than valence failed to produce an interaction between cognitive control and emotional distraction (see Cohen et al., 2016). Possibly, the lack of an interaction in the present study could be explained by the "implicit processing" account (see Cohen et al., 2016 for a detailed description of the account) of emotional stimuli (i.e., subjects respond to the print-color of the emotional words and not to the word's emotional content). However, other research suggested that attention to emotional stimuli (but not task-relevance) is a necessary condition for emotional distraction ( Kanske, 2012; Okon-Singer et al., 2007) and cognitive control over emotional stimuli (Kanske & Kotz, 2011). Therefore, future research should test specific moderators of cognitive control - emotion interactions. Our experiments demonstrated that while Color and Emotional Stroop effects specific to one task impact on the other task, conflict-triggered control in the Color Stroop does not modulate emotional distraction at different timescales. This constrains theoretical accounts of control in cognitive and emotional tasks that predict such an interaction for tasks that recruit the same relevant dimension. Rather, the present results point out severe limitations of cognitive control to generalize across tasks when cognitive tasks were intermixed with emotional tasks. ERQ data of one participant were missing. Alpha-corrected p-value = 0.0125 Corresponding to the minimum refresh rate (144 Hz) of the screen. Note that p-value > 0.0125 were not significant according to the Bonferroni correction explained in the Data Analysis section. Ahmed, S. P., & Sebastian, C. L. (2019). Emotional interference during conflict resolution depends on task context. Cognition and Emotion, 1–15. https://doi.org/10.1080/02699931.2019.1701417 Algom, D., Chajut, E., & Lev, S. (2004). A Rational Look at the Emotional Stroop Phenomenon: A Generic Slowdown, Not a Stroop Effect. 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G., Mathews, A., & MacLeod, C. (1996). The emotional Stroop task and psychopathology. Psychological Bulletin, 120(1), 3–24. https://doi.org/10.1037/0033-2909.120.1.3 Wyble, B., Sharma, D., & Bowman, H. (2008). Strategic regulation of cognitive control by emotional salience: A neural network model. Cognition and Emotion, 22(6), 1019–1051. https://doi.org/10.1080/02699930701597627 Yang, Q., & Pourtois, G. (2018). Conflict-driven adaptive control is enhanced by integral negative emotion on a short time scale. Cognition and Emotion, 1–17. https://doi.org/10.1080/02699931.2018.1434132 Zajonc, R. B. (1980). Feeling and thinking: Preferences need no inferences. American Psychologist, 35(2), 151–175. https://doi.org/10.1037/0003-066X.35.2.151 This research was supported by a grant within the Priority Program, SPP 1772 from the German Research Foundation (Deutsche Forschungsgemeinschaft, DFG), grant no DI 2126/1-2 to DD. The authors thank the student assistants for data collection. They are grateful to Dr. Lisa Hüther-Pape for improving the English and to Moritz Schiltenwolf for helpful discussions. Open Access funding enabled and organized by Projekt DEAL. Department of Psychology, University of Freiburg, Schleichstraße 4, 72076, Tübingen, Germany Elisa Ruth Straub, Constantin Schmidts, Wilfried Kunde, Jinhui Zhang, Andrea Kiesel & David Dignath Department of Psychology, Cognition, Action, and Sustainability, Engelbergerstr. 41, 79085, Freiburg, Germany Elisa Ruth Straub Constantin Schmidts Wilfried Kunde Jinhui Zhang Andrea Kiesel David Dignath Corresponding authors Correspondence to Elisa Ruth Straub or David Dignath. Open Practices Statements The data and materials for all experiments can be retrieved from the Open Science Framework (https://osf.io/pcsba/) and experiments were not preregistered. Table 1. Words from Berlin affected word list (Võ et al., 2009) Table 2 Interaction effects and main effects of Experiment 1-3 Straub, E.R., Schmidts, C., Kunde, W. et al. Limitations of cognitive control on emotional distraction – Congruency in the Color Stroop task does not modulate the Emotional Stroop effect. Cogn Affect Behav Neurosci (2021). https://doi.org/10.3758/s13415-021-00935-4 Cognitive conflict Emotional distraction Emotional Stroop effect Over 10 million scientific documents at your fingertips Switch Edition Academic Edition Corporate Edition Not affiliated © 2022 Springer Nature Switzerland AG. Part of Springer Nature.	CommonCrawl
Sina Torfi 3D Convolutional Neural Networks for Audio-Visual Recognition Computer Vision, NumPy, Scikit-Learn, Speech Technology, TensorFlow The essential problem is to find the correspondence between the audio and visual streams, which is the goal of this work. We proposed the utilization of a coupled 3D Convolutional Neural Network (CNN) architecture that can map both modalities into a representation space to evaluate the correspondence of audio-visual streams using the learned multimodal features. Check relevant links: [Paper, GitHub, Project Page] This repository projects the implementation of our paper: 3D Convolutional Neural Networks for Cross Audio-Visual Matching Recognition. The input pipeline must be prepared by the users. This code is aimed to provide the implementation for Coupled 3D Convolutional Neural Networks for audio-visual matching. Lip-reading can be a specific application for this work. The Problem and the Approach How to leverage 3D Convolutional Neural Networks? Input Pipeline for this work Audio-visual recognition (AVR) has been considered as a solution for speech recognition tasks when the audio is corrupted, as well as a visual recognition method used for speaker verification in multi-speaker scenarios. The approach of AVR systems is to leverage the extracted information from one modality to improve the recognition ability of the other modality by complementing the missing information. The proposed architecture will incorporate both spatial and temporal information jointly to effectively find the correlation between temporal information for different modalities. By using a relatively small network architecture and much smaller dataset, our proposed method surpasses the performance of the existing similar methods for audio-visual matching which use CNNs for feature representation. We also demonstrate that an effective pair selection method can significantly increase the performance. In the visual section, the videos are post-processed to have an equal frame rate of 30 f/s. Then, face tracking and mouth area extraction are performed on the videos using the dlib library [dlib]. Finally, all mouth areas are resized to have the same size and concatenated to form the input feature cube. The dataset does not contain any audio files. The audio files are extracted from videos using the FFmpeg framework [ffmpeg]. The processing pipeline is the below figure. The proposed architecture utilizes two non-identical ConvNets which use a pair of speech and video streams. The network input is a pair of features that represent lip movement and speech features extracted from 0.3 seconds of a video clip. The main task is to determine if a stream of audio corresponds with a lip motion clip within the desired stream duration. In the two next sub-sections, we are going to explain the inputs for speech and visual streams. Speech Net On the time axis, the temporal features are non-overlapping 20ms windows which are used for the generation of spectrum features that possess a local characteristic. The input speech feature map, which is represented as an image cube, corresponds to the spectrogram as well as the first and second-order derivatives of the MFEC features. These three channels correspond to the image depth. Collectively from a 0.3-second clip, 15 temporal feature sets (each forms 40 MFEC features) can be derived which form a speech feature cube. Each input feature map for a single audio stream has a dimensionality of $15 \times 40 \times 3$. This representation is depicted in the following figure: The speech features have been extracted using the SpeechPy package. Visual Net The frame rate of each video clip used in this effort is 30 f/s. Consequently, 9 successive image frames form the 0.3-second visual stream. The input of the visual stream of the network is a cube of size 9x60x100, where 9 is the number of frames that represent the temporal information. Each channel is a 60×100 gray-scale image of the mouth region. The architecture is a coupled 3D convolutional neural network in which two different networks with different sets of weights must be trained. For the visual network, the lip motions' spatial information alongside the temporal information are incorporated jointly and will be fused for exploiting the temporal correlation. For the audio network, the extracted energy features are considered as a spatial dimension, and the stacked audio frames form the temporal dimension. In the proposed 3D CNN architecture, the convolutional operations are performed on successive temporal frames for both audio-visual streams. I am an expert in Machine Learning (ML) and Artificial Intelligence (AI) making ML accessible to a broader audience. I am also an entrepreneur who publish tutorials, courses, newsletter, and books. Know more about me Explore through my blog Questions? Contact Me « May Apr » Address : Blacksburg, VA, 24060 Email : [email protected] Copyright © 2022 Sina Torfi \| Credits Powered by Sina Torfi	CommonCrawl
chaoticsstrategies.com What Is A Scale Factor Of 5? What is the scale factor of 4 and 16? What is the scale factor of 5? What is a scale factor in math in 7th grade? What is a scale copy? What is a scale in math? How do u find the scale factor? What is the scale factor of 12? What does a scale of 5 1 mean? What is a scale factor math is fun? What is a scale factor example? What is a scale factor of a triangle? How do you calculate a scale? What does a scale mean? How do you calculate scale size? What's the difference between a scale and a scale factor? Now we want to figure out the scale factor. To do this, we simplify the ratio using the greatest common factor. The greatest common factor of 4 and 16 is 4. The scale factor is \begin{align}\frac{1^{\prime\prime}}{4\ ft}\end{align}.. 3. Scale factor with area and volume. If a figure is enlarged by a factor of five, the length will be five times greater, the width will be five times greater and the height will be five times greater. The area will therefore be five times five times greater. The size of an enlargement/reduction is described by its scale factor. For example, a scale factor of 2 means that the new shape is twice the size of the original. A scale factor of 3 means that the new shape is three times the size of the original. ● Scale Factor: The ratio of any two corresponding lengths in two similar. geometric figures. A scale copy of a figure is a figure that is geometrically similar to the original figure. This means that the scale copy and the original figure have the same shape but possibly different sizes. … In real life, a scale copy is often smaller than the original figure. In math, a scale in graphs can be defined as the system of marks at fixed intervals, which define the relation between the units being used and their representation on the graph. … Each small interval or division measures 100 ml. Fun Facts. A ruler is often called a scale. To find the scale factor, locate two corresponding sides, one on each figure. Write the ratio of one length to the other to find the scale factor from one figure to the other. In this example, the scale factor from the blue figure to the red figure is 1.6 : 3.2, or 1 : 2. Architectural ScalesDrawing ScaleScale FactorViewport Scale3/8″ = 1′-0″321/32xp1/2″ = 1′-0″241/24xp3/4″ = 1′-0″161/16xp1″ = 1′-0″121/12xp7 more rows•Aug 12, 2018 A 50mm line is to be drawn at a scale of 5:1 (ie 5 times more than its original size). The measurement 50mm is multiplied by 5 to give 250mm. A 250mm line is drawn. more … The ratio of the length in a drawing (or model) to the length on the real thing. Example: in the drawing anything with the size of "1" would have a size of "10" in the real world, so a measurement of 150mm on the drawing would be 1500mm on the real horse. A scale factor is a number which multiplies ("scales") a quantity. For example,the "C" in y = Cx is the scale factor for x. If the equation were y = 5x, then the factor would be 5. When two triangles are similar, the reduced ratio of any two corresponding sides is called the scale factor of the similar triangles. … The ratios of corresponding sides are 6/3, 8/4, 10/5. These all reduce to 2/1. It is then said that the scale factor of these two similar triangles is 2 : 1. To scale an object to a smaller size, you simply divide each dimension by the required scale factor. For example, if you would like to apply a scale factor of 1:6 and the length of the item is 60 cm, you simply divide 60 / 6 = 10 cm to get the new dimension. Definition of scale (Entry 5 of 7) 1 : a graduated series of musical tones ascending or descending in order of pitch according to a specified scheme of their intervals. 2 : something graduated especially when used as a measure or rule: such as. To convert a measurement to a larger measurement simply multiply the real measurement by the scale factor. For example, if the scale factor is 1:8 and the measured length is 4, multiply 4 × 8 = 32 to convert. The scale can be written as a scale factor, which is the ratio of the length or size of the drawing or model to the length of the corresponding side or part on the actual object. Scale Factor needs to be the SAME UNITS! … A scale is the ratio between two sets of measurements. Quick Answer: How Much Does A Game Tester Make UK? How much do video game testers make 2020? Quick Answer: What Does The Pipe Mean In Probability? What does a vertical line mean in probability? Question: What Apps Are Compatible With Chromebook? What are Chromebooks good for? In essence, Chromebooks What Paint Do You Use On Canvas? Do you need to prime canvas for acrylic paint? Can You Sell 3d Models Online? How much do 3d prints sell for? The $3 is broken down What Paint Is Safe For Babies? How do I make paint safe for my baby? 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Fluids and Barriers of the CNS Changes in the cerebrospinal fluid circulatory system of the developing rat: quantitative volumetric analysis and effect on blood-CSF permeability interpretation Jean-François Ghersi-Egea1,2, Anaïd Babikian1, Sandrine Blondel1 & Nathalie Strazielle2,3 Fluids and Barriers of the CNS volume 12, Article number: 8 (2015) Cite this article The cerebrospinal fluid (CSF) circulatory system is involved in neuroimmune regulation, cerebral detoxification, and delivery of various endogenous and exogenous substances. In conjunction with the choroid plexuses, which form the main barrier site between blood and CSF, this fluid participates in controlling the environment of the developing brain. The lack of comprehensive data on developmental changes in CSF volume and distribution impairs our understanding of CSF contribution to brain development, and limits the interpretation of blood-CSF permeability data. To address these issues, we describe the evolution of the CSF circulatory system during the perinatal period and have quantified the volume of the different ventricular, cisternal and subarachnoid CSF compartments at three ages in developing rats. Immunohistofluorescence was used to visualize tight junctions in parenchymal and meningeal vessels, and in choroid plexus epithelium of 19-day fetal rats. A quantitative method based on serial sectioning of frozen head and surface measurements at the cutting plane was used to determine the volume of twenty different CSF compartments in rat brain on embryonic day 19 (E19), and postnatal days 2 (P2) and 9 (P9). Blood-CSF permeability constants for sucrose were established at P2 and P9, following CSF sampling from the cisterna magna. Claudin-1 and claudin-5 immunohistofluorescence labeling illustrated the barrier phenotype acquired by all blood–brain and blood-CSF interfaces throughout the entire CNS in E19 rats. This should ensure that brain fluid composition is regulated and independent from plasma composition in developing brain. Analysis of the caudo-rostral profiles of CSF distribution and of the volume of twenty CSF compartments indicated that the CSF-to-cranial cavity volume ratio decreases from 30% at E19 to 10% at P9. CSF compartmentalization within the brain changes during this period, with a major decrease in CSF-to-brain volume ratio in the caudal half of the brain. Integrating CSF volume with the measurement of permeability constants, adds to our understanding of the apparent postnatal decrease in blood-CSF permeability to sucrose. Reference data on CSF compartment volumes throughout development are provided. Such data can be used to refine blood-CSF permeability constants in developing rats, and should help a better understanding of diffusion, bulk flow, and volume transmission in the developing brain. The cerebrospinal fluid represents 50% of the brain extracellular fluid in adult mammals including humans. Recent findings indicate that the CSF circulatory system is not a mere passive drainage system for the brain, but is involved in neuroimmune regulation [1-3], cerebral detoxification [4], and cerebral delivery of various endogenous or exogenous molecules [5-7]. CSF circulation is complex. In adult, the fluid moves fast within the ventricular system, and reaches the cisterna magna and lateral recesses of the fourth ventricle through the foramina of Magendie and Luschka. From there it slows down while flowing through the subarachnoid and cisternal spaces, which occurs in three main directions. The CSF circulates 1) in a caudo-rostral direction, either dorsally around the cerebellum and cortex, or centrally through the internal cisterns, which in rodents are mainly formed by the ambient and quadrigeminal cisterns located between the midbrain and hippocampi/cortices, 2) in a caudo-rostral direction and ventrally through the cerebellopontine, interpeduncular, optic tract, and laminae terminalis cisterns, and around the olfactory bulbs, and 3) caudally along the spinal cord. In places such as the velum interpositum, ventricular and cisternal spaces are separated from each other only by a thin membrane which allows direct exchanges of material between the two fluid compartments [8,9]. The CSF is secreted and protected from peripheral harmful molecules by the choroid plexuses which develop from the neuroepithelium to form the major site of the blood-CSF barrier. Choroid plexuses also have the capacity to secrete into CSF different bioactive substances including guidance molecules as well as growth and differentiation factors. Owing to the early fetal development of the choroidal tissue, the choroid plexus-CSF system is likely to fulfill essential functions for brain development [10-12]. The current lack of comprehensive data on developmental changes in CSF spaces still impairs our understanding of CSF contribution to brain development, and limits the interpretation of blood-CSF permeability data. To address the early role of the CP-CSF system we first examined whether tight junction proteins are present at all brain barriers including CSF-bathed meningeal and cisternal vessels in embryonic day 19 (E19) rat fetuses, a requirement for generating a brain fluid environment which is controlled and independent from plasma composition before birth. Using a quantitative method to measure the volumes of all CSF compartments in rats at E19 and postnatal day 2 (P2) and 9 (P9), we describe the evolution of the CSF circulatory system in the developing brain during the perinatal period. We then applied some of these volume data to refine the interpretation of blood-CSF permeability constants in developing rats. Tissue collection Animal care and procedures were conducted according to the guidelines approved by the French ethical committee (decree 87–848), and by the European Community (directive 86-609-EEC). Sprague–Dawley rats, either pregnant time-dated females, or females with their litter, were obtained from Janvier (Le Genest Saint Isle, France). All animals were kept under identical conditions in standard cages, with free access to food and tap water under a controlled environment (12 h day-light cycles). Timed-pregnant female rats were anesthetized with inhaled isoflurane (5%) and body temperature was maintained with a heated pad. E19 animals were removed one by one from the mother and frozen in 2-methylbutane cooled at −50°C, in a position that allowed serial sectioning of the brain. P2 and P9 animals were decapitated and the severed head immediately frozen in 2-methylbutane cooled at −50°C. Limited expansion of the brain at the time of freezing led occasionally to a partial crack in the cranial bones. These brains were discarded from the analysis. Animal heads were kept at −80°C until use for serial sectioning. Immunohistochemical analysis of claudins in E19 rat brains (n = 3) was performed as previously described [13], using anti-claudin-1 polyclonal rabbit antibody 51–9000 and anti-claudin-5 mouse monoclonal antibody 35–2500 (Invitrogen, Carlsbad, CA, USA). Primary antibodies were diluted and used overnight at 4°C at a final concentration of 0.625 μg/ml for claudin-1, and 2 μg/ml for claudin-5. Alexa 555®-conjugated goat anti-mouse antibody A-21424 and Alexa 488®-conjugated goat anti-rabbit antibody A-11034 (Invitrogen) were used at a final concentration of 2 μg/ml at room temperature for 1 hour. Diamidine-2-phenylindole-dihydrochloride (DAPI, 236276, Roche Diagnostics, Manheim, Germany) was used as a fluorescent nuclear stain (0.1 μg/ml in saline phosphate buffer for 10 minutes at room temperature). Immunofluorescence was viewed and analyzed using an Imager Z1 fluorescence microscope equipped with a MozaiX motorized module, a Z-stack apotome system and a Digital Camera (Zeiss, Jena, Germany). Images were acquired using the AxioVs40 V 4.8 software (Zeiss). Morphometric analysis The whole head (n = 4 for E19 and P9, n = 5 for P2) with intact meninges and CSF spaces was cut into 35-μm-thick serial sections using a CM 3050S cryostat (Leica, Nussloch, Germany). After every fifth section, a photograph of the remaining tissue block in the cryostat was taken with a sharp focus on the cutting plane using a VR-320 camera (Olympus, Tokyo, Japan). On these photographs, the tissue could be easily distinguished from the crystal-clear CSF present in the cisternal, subarachnoid, and ventricular spaces. Each photograph included a scale bar added in the cutting plane to enable quantification of the surface area of the cerebral structures and spaces of interest. In addition, every fifth section was collected on a glass slide and stained with hematoxylin and floxin. Micrographs of the sections were taken using a AxioCam ERc5s camera (Zeiss) connected to a SMZ800 stereotaxic microscope (Nikon, Amsterdam, Netherlands). The surface areas of the cranial cavity and of the CSF spaces of interest were delineated on the photographs and determined using the ZEN 2012 software (Zeiss). Micrographs of the stained sections were used to determine the surface area of the smaller intraparenchymal CSF-filled compartments (such as the third ventricle), when the borders were not sharp enough to be outlined in the tissue block photographs. The total brain tissue volume and the volume of each fluid compartment were calculated by building the area under the surface area-distance curve, between the section where the given structure appears and the section where it disappears. Calculation was done using the composite trapezoidal rule for integration. CSF compartment volumes and tissue volumes were corrected by factors of 0.910 and 0.920, respectively, to account for water expansion at the time of freezing. The correction factor for tissue was adjusted for a water content in neonatal rat brain tissue of 88.5% [14,15]. The extremities of the brain cavity were defined as the appearance of the cerebellar subarachnoid space, and the disappearance of the olfactory bulbs. Protein measurement Choroid plexuses from individual animals (n = 6 for both P2 and P9) were micro-dissected under a stereomicroscope and digested in 1 M sodium hydroxide. Protein concentration was determined by the method of Peterson [16] using bovine serum albumin as reference protein for the standard curve. Blood-CSF permeability measurement to [14C]-sucrose Radio labelled [14C]-sucrose (435 mCi/mmol, Hartmann Analytic, Braunschweig, Germany) was administrated by intraperitoneal injection (12.5nCi/g) under slight gas anaesthesia, to prevent backflow. Blood was sampled from different animals of the same litter (n = 5 for P2 and n = 9 for P9) by cardiac puncture under pentobarbital anesthesia at times ranging from 3 to 30 minutes after injection. Additional animals from the litter were injected with 50nCi/g [14C]-sucrose. In these animals, blood was sampled twenty minutes after injection, rapidly followed by CSF sampling as follows. The skin was incised above the cistern magna, and a glass pipette was introduced in the cistern for CSF collection. The CSF was transferred in a microtube and the sample volume measured. Blood was collected in a heparinized tube and centrifuged at 5000 rpm for 5 min. Plasma and CSF samples were analyzed for radioactivity in a 1600 TR Packard scintillation counter, using Ultima-gold (Perkin-Elmer, Waltham, MA) as scintillation cocktail. A litter-based area under the plasma [14C]-sucrose concentration-time curve (AUC) [17] was built from 0 to 30 minutes for each developmental stage using the composite trapezoidal rule for integration. Two permeability constants K in csf and K w csf were calculated as follows: $$ {{\mathrm{K}}_{\mathbf{in}}}_{\mathrm{csf}}={\mathrm{C}}_{\mathrm{t}}/\mathrm{AU}{\mathrm{C}}_{0\to \mathrm{t}} $$ where Ct is the [14C]-sucrose concentration in CSF at the time of sampling t (ranging from 20 to 23.5 minutes), AUC 0→t the AUC recalculated from the litter-based experimental AUC and from the plasma concentration measured before CSF sampling. K in csf is similar to an influx constant as defined previously for brain tissue. $$ {{\mathrm{K}}_{\mathbf{w}}}_{\mathrm{csf}}={{\mathrm{K}}_{\mathbf{in}}}_{\mathrm{csf}}\times {\mathrm{V}}_{\mathrm{csf}} $$ where Vcsf is the total volume of CSF in the brain. K w csf represents the plasma volume cleared in the whole CSF of the animal. Antibodies against claudin-5, the hallmark constituent of tight junctions at the blood–brain barrier stained the entire microvascular network throughout the brain as well as endothelial junctions of meningeal vessels in E19 rat fetuses (Figure 1). The epithelial layer in all choroid plexuses was immunoreactive for claudin-1, the major protein of tight junctions at the blood-CSF barrier. This highlights that the barrier phenotype is acquired by blood–brain interfaces throughout the entire CNS in rat before birth. This stage can be compared to mid-gestation in humans. Staining of tight junction proteins in choroid plexus and meningeal vessels, but not in ependyma, infers that CSF-filled compartments are an integral part of the CNS at this developmental stage. In line with previous reports [11,13], it also suggests that the brain fluid composition is tightly regulated during development and is independent from plasma composition, a prerequisite for the CSF to fulfill its function in brain maturation. To better appreciate the extent of the CSF circulatory system and its implication in CNS physiology during perinatal development, we undertook a quantitative morphometric analysis of its compartments at E19, P2 and P9. Immunohistofluorescence analysis of claudin-1 and claudin-5 in E19 rat brain. A and B: forebrain. The vascular network is labeled by claudin-5 antibodies (red), while both the third and lateral ventricles choroid plexuses are labeled by claudin-1 antibodies (green). C and D: Hindbrain. The cerebellar and medullar vascular networks are labeled by claudin-5 antibodies, while both the central and lateral parts of the fourth ventricle choroid plexus are labeled by claudin-1 antibodies. Note the intercellular junction labeling of the meningeal vessels (long arrows) such as that located in the ambient cistern (enlarged in B). Claudin-1 immunoreactivity at the periphery of choroidal epithelial cells is clearly seen in the enlarged fourth ventricle area (D). Claudin-5 immunoreactivity is found in penetrating choroidal vessels (arrowhead), but not in the terminal vascular loops of choroidal villi. The ependyma (Ep) is devoid of staining. Other abbreviations: AmbCi: ambient cistern, LV, 3 V, 4 V: Lateral, third and fourth ventricles, respectively. Methodology of the quantitative morphometric analysis Brain ventricles of developing rats have previously been visualized by magnetic resonance imaging in animals suffering from hydrocephalus [18]. They were not observable in control animals at that time. Although the technique has since been adapted to small animal brain analysis, resolution remains a limiting factor to distinguish and to quantify the volumes of the various cisternal, subarachnoid, and ventricular compartments in normal developing rodents. We therefore conducted our morphometric study by a conventional approach based on the analysis of brain sections, with two distinct features. First, we generated these sections from whole frozen heads. Because of this unique aspect in the procedure, CSF is trapped within the subarachnoid, cisternal, and ventricular compartments. The shape and volumes of these spaces remain unchanged, with the exception of limited expansion of water at the time of freezing (Figure 2 and [8]). Second, photographs of the frozen tissue block taken at the cutting plane were preferred to micrographs of stained sections for surface area measurement, because on the latter the larger fluid compartments were easily distorted in the process of cutting and drying. Stained sections were used only for quantifying small fluid spaces when their boundaries could not be easily outlined on photographs. Examples of both cutting planes and stained sections covering a selection of CSF compartments from cerebellar to olfactory bulb subarachnoid spaces are shown in Figure 2. The size of CSF compartments changed between stages. To quantify these changes, we measured the total volume of the cranial cavity (from the cistern magna to the end of the olfactory bulbs), and the volume of twenty ventricular, cisternal, and subarachnoid compartments (listed in Table 1) at the three developmental stages. These two sets of data were used to calculate brain tissue volumes. All volumes were then corrected for water expansion at the time of freezing (see Method section). Brain tissue volume measured using this method was not statistically different from brain volume deduced from the weight of fresh brain sampled from age- and weight- matched animals, assuming an overall density of 1 (data not shown). This correlation provides a good index of measurement accuracy. The complete set of volume data obtained for the twenty compartments at the three developmental stages is displayed in a table see Additional file 1. It should be noted that with the exception of the ventricular system and lateral recesses of the fourth ventricle, all spaces of interest contain, in addition to CSF, arachnoid trabeculae, immune cells, and vessels traveling through the spaces before entering the brain. When appearing on sections, the major venous sinuses were excluded from the cisternal surface area measurements (e.g. Figure 2, A1 and C1). The CSF contained in the Virchow-Robin spaces which are located around penetrating vessels and are connected to the main subarachnoid and cisternal spaces could not be included, as the resolution of the method does not allow their visualization. Histological images of all the CSF spaces analyzed in this paper can be found in previous publications for the adult rat [3,8,19]. Selected examples of cutting-plane photographs and histological sections used for morphometric analysis. Images are from E19 (A1-A3), P2 (B1-B3) P9 (C1-C3) animals. Fluid spaces and cranial cavity are delineated by a blue line. See Table 1 for abbreviations. Scale bar: 1 mm. Table 1 CSF compartments identified and analyzed in the study, and their corresponding abbreviations Developmental changes in CSF compartment volumes The caudo-rostral profiles of the cranial cavity, brain tissue, and overall CSF spaces are shown for all three developmental stages in Figure 3. The volume values generated for these three parameters are listed in the adjacent tables. While the cranial cavity increased 1.9- and 5 times from E19 to P2 and P9, respectively, the total CSF volume increased only moderately over the same period. This translates into a decrease of the CSF-to-cranial cavity volume ratio from 28% at E19 to 20% at P2 and 10% at P9. The profiles show that CSF compartmentalization changes during development. There is a substantial decrease in the CSF space compared to the cranial cavity, which is mostly notable in the caudal half of the brain. We subdivided the CSF spaces in 7 main compartments to appreciate more precisely the geographic distribution of CSF over the developmental period (Table 2). Data expressed as a percentage of cranial cavity (left columns) indicated that the cerebellar subarachnoid spaces, ventricles, internal cisterns, and caudal part of the cortical subarachnoid spaces contribute most of the developmental decrease in CSF-to-cranial cavity volume ratio. Results expressed as a percentage of total CSF (right columns) show how the fluid distributes at each developmental stage. CSF distribution among the seven compartments differed more between E19 and P2 than between P2 and P9. A large part of the CSF was found in the internal cisterns at all stages. The cerebellar subarachnoid spaces and the caudal part of cortical subarachnoid spaces were also major CSF compartment at E19. The remote forebrain subarachnoid spaces and the hindbrain spaces mostly formed by the lateral recesses of the fourth ventricle became prominent in P9 animals. This resulted from the decrease in volume of other CSF compartments (relative to the cranial cavity volume). The lateral recesses of the fourth ventricle formed relatively large CSF spaces. They are generally not observable when the brain is sampled by conventional procedure, but were easily visualized by the method used in this study. These spaces are transitional areas between the ventricular and subarachnoid/cisternal system, in that no trabeculae or vessels pass through, and the border between the CSF and neuropil is not ependyma but the glia limitans. They extend dorsally along the medulla to become the spinal cord subarachnoid space. The latter also was relatively large in developing animals. In the anterior part of the spinal cord it ranged on average from 40% (E19) to 30% (P9) of the spinal canal volume (data not shown). Caudo-rostral profiles of brain tissue volume and CSF space in the developing rat brain. Upper panel: 19-day-old embryo (E19), middle panel: 2-day-old rat (P2), lower panel: 9-day old rat (P9). Graphs show typical profiles combined from at least 3 animals. Positions of the cerebellum (Cb) and olfactory bulb (Ob) are indicated. Cranial cavity, brain tissue and CSF space profiles are represented in blue, red and green, respectively. Note scale difference between stages. Volumes corresponding to the three spaces are reported in the apposed tables as means ± SD, n = 4 (E19, P9), and n = 5 (P2). Table 2 CSF compartmentalization in the different fluid-filled spaces of the developing brain The relative importance of hindbrain CSF spaces in E19 and their decrease throughout development is accounted for by the delayed development of the cerebellum, and the growth of the caudal part of cortex. Of note, cortical subarachnoid CSF is mainly found around this caudal part of the cortex and in the rhinal fissure, as the space between the surface of the cortex and the dura mater is narrow at all stages. This suggests that in the rodent, which does not display cortical circumvolution (but for the rhinal fissure), CSF flow is very limited between the pia-covered cerebral cortex and the arachnoid/dura mater. This is likely to be different in human, in which cortical foldings develop mainly between week 25 and 30 of gestation [20], resulting in changes in the pattern of flow at the cortical surface. Finally the data show that the size of the ventricular system remains modest at all stages. The volume of the ventricles changed only moderately between E19 and P9, with a slight decrease for the lateral ventricles, an increase for the third ventricle and no change for the fourth ventricle (Additional file 1). By contrast, the size of choroid plexuses continues to enlarge during this period. The protein content of the lateral ventricle choroid plexuses increased from 100 ± 9 to 177 ± 13 μg between P2 and P9, and that of the fourth ventricle choroid plexus increased from 69 ± 9 to 131 ± 19 μg (mean ± SD of six animals at both ages). This increase in protein content parallels the change in size of dissected tissues observed under the stereomicroscope. This suggests there is increased flow of CSF through the ventricular system, which will be further enhanced by the concurrent increase in the rate of CSF secretion by the choroidal cells. Overall, the volume data provided in this study for twenty compartments of the rat CSF system at different developmental stages (see Additional file 1) form reference figures which can be used in future studies to better delineate the role of the fluid environment in brain developmental processes. Taking into account the total CSF volume can also help to better interpret blood-CSF permeability measurements as exemplified below. Developmental changes in blood-CSF permeability constants for sucrose used as a marker of blood-CSF barrier efficiency We measured [14C]-sucrose permeability constants by sampling CSF 20 min after injection. Blood concentration continuously increased during this period, minimizing the impact of tracer back flux. To account for changes in [14C]-sucrose plasma concentration over time we calculated the integrated plasma concentration-time product. We then generated influx constants K in csf with the assumption that the concentration measured in CSF drawn through the cisterna magna (5 to 10% of total CSF) is representative of the overall total CSF concentration. K in csf values did not differ significantly between P2 and P9 animals (Table 3). A new CSF permeability constant K w csf which takes into account the total volume of CSF in which the tracer is distributed (tables in Figure 3), was then calculated for each age (Table 3). K w csf was also not statistically different between the two stages, but became significantly lower at P9 as compared to P2 when values were normalized for choroid plexus protein content (Table 3). Table 3 Blood-CSF permeability constants for [ 14 C]-sucrose in the developing rat By using a short time point of 20 minutes, it is expected that a large proportion of the tracer reaches the CSF through the choroidal blood-CSF barrier and possibly across the meningeal vessel endothelium, another blood-CSF barrier site. The route across the blood–brain barrier proper is likely involved only to a limited extent. The presence of tight junctions that link the barrier cells of all choroidal epithelium, meningeal vessels, as well as parenchymal vessels ([11,13] and Figure 1) explains the limited blood-CSF permeability of [14C]-sucrose at both stages. The apparent permeability as assessed by K in csf measurement remains however higher than for adult rat. In the latter, K in csf was measured using a 30-minute time-point for inulin (a large molecule expected to diffuse 4 times less than sucrose across brain barriers), was 0.17 × 10−3 min−1 [21]. The constants Kw csf provide different information when comparing blood-CSF permeability at multiple developmental stages. K w csf represents the virtual volume of plasma from which the tracer is cleared into the CSF per unit of time, which was about 0.2 μl.min−1 at both stages. Total cerebral blood flow across a 10-day-old rat brain, averaged from previous papers, is around 400 μl.min−1.g−1 tissue [22,23]. This is equivalent to 340 μl.min−1.brain−1 in P9 animals (brain volume taken from Figure 3). Therefore, only 0.06% of the tracer flowing through the brain vasculature reaches the CSF at P9. Data on cerebral blood flow in P2 animals and data on choroidal blood flow at both stages would have been useful to further analyze the data, but are not available to our knowledge. K w csf measured in P9 animals becomes significantly lower than K w csf measured in P2 animals when the values are normalized for choroid plexus protein content (Table 3). Assuming this protein content is proportional to the surface area of exchange across the blood-CSF barrier, this change in normalized K w csf may be explained by a decrease in the paracellular permeability of the choroidal epithelium. The developmental expression profile of tight junction proteins, the similar claudin immunolocalization pattern in the choroid plexus at both ages [13], and restriction of a polar tracer by tight junctions observed by electron microscopy [11] suggest that junctions are already efficient at both ages. In addition, factors other than paracellular diffusion could also be involved in the apparent decrease in blood-CSF permeability between P2 and P9. A change in CSF turnover and an increase in CSF-to-tissue diffusion may affect CSF sucrose concentration differently at the two developmental stages. The capacity of the transcellular route for transport for plasma material across selected epithelial cells of the choroid plexus [24] may decrease during the postnatal period in rat. Hence, these factors, all independent from CSF volume, can possibly explain the higher CSF concentration of polar tracers measured at steady-state, several hours after injection, in P2 rats as compared to older animals [25]. By contrast, changes in the CSF distribution volume during the embryonic period appeared to have a substantial influence on apparent blood-to-CSF permeability. Thus, a 60% decrease in the CSF/plasma ratio for sucrose measured in conditions approaching steady-state in rat between E13 and E18 was attributed to the concomitant increase in CSF volume rather than changes in paracellular permeability of the choroidal epithelium [26]. The presence of tight junctions at blood–brain/CSF interfaces in 19-day rat embryos and the [14C]-sucrose permeability constants measured in 2- and 9-day-old rats confirm that the brain fluid environment is controlled and independent from plasma composition during development. Volumes of the different CSF fluid compartments measured during pre- and postnatal development in rat indicate not only a decrease in CSF-to-brain volume ratio, but also a geographical redistribution of the fluid especially around birth in rat. The volume data can be used to refine blood-CSF permeability constants in developing animals. The description of the different ventricular, cisternal and subarachnoid spaces and their respective developmental volume profile compared to total brain volume can be used to better understand cerebral fluid dynamic and diffusional/bulk flow movement of solutes in the developing brain. These fluid spaces are involved in volume transmission, clearance of potentially deleterious molecules, and immune cell migration within the brain These results will enable a better appreciation of the role of fluid compartments in brain development, neuroimmune surveillance of the infant, and in neonatal injuries. AUC: Area under the curve CSF: Cerebrospinal fluid CNS: Central nervous system, anatomical abbreviations: see Table 1 Kivisakk P, Mahad DJ, Callahan MK, Trebst C, Tucky B, Wei T, et al. Human cerebrospinal fluid central memory CD4+ T cells: evidence for trafficking through choroid plexus and meninges via P-selectin. Proc Natl Acad Sci U S A. 2003;100:8389–94. Onaivi ES, Schanz N, Lin ZC. Psychiatric disturbances regulate the innate immune system in CSF of conscious mice. Transl Psychiatry. 2014;4:e367. Schmitt C, Strazielle N, Ghersi-Egea JF. Brain leukocyte infiltration initiated by peripheral inflammation or experimental autoimmune encephalomyelitis occurs through pathways connected to the CSF-filled compartments of the forebrain and midbrain. J Neuroinflammation. 2012;9:187. Iliff JJ, Wang M, Liao Y, Plogg BA, Peng W, Gundersen GA, et al. A paravascular pathway facilitates CSF flow through the brain parenchyma and the clearance of interstitial solutes, including amyloid beta. Sci Transl Med. 2012;4:147ra111. Redzic ZB, Preston JE, Duncan JA, Chodobski A, Szmydynger-Chodobska J. The choroid plexus-cerebrospinal fluid system: from development to aging. Curr Top Dev Biol. 2005;71:1–52. Chodobski A, Silverberg G, Szmydynger-Chodobska J. The role of the choroid plexus in transport and production of polypeptides. In: Zeng W, Chodobski A, editors. The blood-cerebrospinal fluid barrier. Boca Raton, Fl, USA: CRC Press; 2005. p. 237-74. Schmitt C, Strazielle N, Richaud P, Bouron A, Ghersi-Egea JF. 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Johansson PA, Dziegielewska KM, Ek CJ, Habgood MD, Liddelow SA, Potter AM, et al. Blood-CSF barrier function in the rat embryo. Eur J Neurosci. 2006;24:65–76. This work was supported by ANR-10-IBHU-0003 CESAME grant. The authors thank Joseph Fenstermacher for stimulating discussion and Chantal Watrin for her skillful technical assistance. BIP Platform, Faculté de Médecine RTH Laennec, INSERM U 1028, CNRS UMR5292, Lyon Neuroscience Research Center, Rue Guillaume Paradin, Cedex 08, 69372, Lyon, France Jean-François Ghersi-Egea , Anaïd Babikian & Sandrine Blondel Oncoflam Team, INSERM U1028, CNRS UMR5292, Lyon Neuroscience Research Center, Lyon, France & Nathalie Strazielle Brain-i, Lyon, France Nathalie Strazielle Search for Jean-François Ghersi-Egea in: Search for Anaïd Babikian in: Search for Sandrine Blondel in: Search for Nathalie Strazielle in: Correspondence to Jean-François Ghersi-Egea. JFGE and NS conceived and designed the study. AB carried out the morphometric measurements and analyzed the data. SB performed the immunofluorescence study. JFGE drafted the manuscript. JFGE and NS edited and revised the manuscript. All authors organized the data, read, and approved the final manuscript. Volumes of the CSF compartments in 19-day-old rat embryos (E19), 2-day old rats (P2), and nine-day old rats (P9). Mean and standard deviations for the volumes of all the twenty compartments measured as listed in Table 1. Ghersi-Egea, J., Babikian, A., Blondel, S. et al. Changes in the cerebrospinal fluid circulatory system of the developing rat: quantitative volumetric analysis and effect on blood-CSF permeability interpretation. Fluids Barriers CNS 12, 8 (2015). https://doi.org/10.1186/s12987-015-0001-2 Ventricle Influx constant Blood–brain barrier Claudin	CommonCrawl
On the existence and uniqueness of a generalized solution of the Protter problem for $(3+1)$-D Keldysh-type equations Nedyu Popivanov1, Tsvetan Hristov1, Aleksey Nikolov2 & Manfred Schneider3 A $(3+1)$-dimensional boundary value problem for equations of Keldysh type (the second kind) is studied. Such problems for equations of Tricomi type (the first kind) or for the wave equation were formulated by M.H. Protter (1954) as multidimensional analogues of Darboux or Cauchy-Goursat plane problems. Now, it is well known that Protter problems are not correctly set, and they have singular generalized solutions, even for smooth right-hand sides. In this paper an analogue of the Protter problem for equations of Keldysh type is given. An appropriate generalized solution with possible singularity is defined. Results for uniqueness and existence of such a generalized solution are obtained. Some a priori estimates are stated. In the present paper we consider an analogue of the Protter problems for $(3+1)$-D Keldysh-type equations. For $m \in \mathbf{R}$, $0< m<2$, we study some boundary value problems (BVPs) for the weakly hyperbolic equation $$ L_{m}[u]\equiv u_{x_{1}x_{1}}+u_{x_{2}x_{2}}+u_{x_{3}x_{3}} - \bigl(t^{m}u_{t} \bigr)_{t}=f(x,t), $$ expressed in Cartesian coordinates $(x,t)=(x_{1},x_{2},x_{3},t) \in \mathbf{R}^{4}$ in a simply connected region $$\Omega_{m}:= \biggl\{ ( x,t ) : 0< t< t_{0}, \frac{2}{2-m} t^{\frac{2-m}{2}}< \sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}< 1- \frac{2}{2-m} t^{\frac{2-m}{2}} \biggr\} , $$ bounded by the ball $\Sigma_{0}:=\{(x,t): t=0, \sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}<1\}$, centered at the origin $O=(0,0,0,0)$ and by two characteristic surfaces of equation (1.1) $$\begin{aligned}& \Sigma_{1}^{m}:= \biggl\{ (x,t):0< t< t_{0}, \sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}=1- \frac {2}{2-m} t^{\frac{2-m}{2}} \biggr\} , \\& \Sigma_{2}^{m}:= \biggl\{ (x,t): 0< t< t_{0}, \sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}= \frac{2}{2-m} t^{\frac{2-m}{2}} \biggr\} , \end{aligned}$$ where $t_{0}:= ( \frac{2-m}{4} ) ^{\frac{2}{2-m}}$. In this work we are interested in finding sufficient conditions for the existence and uniqueness of a generalized solution of the following problem. Problem PK Find a solution to equation (1.1) in $\Omega_{m}$ that satisfies the boundary conditions $$ u\|_{\Sigma_{1}^{m}}=0; \quad\quad t^{m}u_{t} \rightarrow0, \quad \text{as } t\to+0. $$ The adjoint problem to PK is as follows. Problem $\mathbf {PK}^{\ast }$ Find a solution to the self-adjoint equation (1.1) in $\Omega_{m}$ that satisfies the boundary conditions $$u\|_{\Sigma_{2}^{m}}=0;\quad \quad t^{m}u_{t}\rightarrow0,\quad \text{as } t\to+0. $$ First, we present a brief historical overview here and provide an extensive list of references. Protter arrived at similar problems, but for Tricomi-type equations, while studying BVPs which describe transonic flows in fluid dynamics. It is well known that most important boundary value problems that, in the case of linear mixed-type equations, appear in hodograph plane for two-dimensional transonic potential flows are the classical Tricomi, Frankl', and Guderley-Morawetz problems. The first two for flows in nozzles and jets and the third one as an approximation in flows about airfoils. For such connections, see the paper of Morawetz [1]. About sixty years ago Murray Protter [2] stated a multidimensional variant of the famous Guderley-Morawetz plane problem for hyperbolic-elliptic equations that had been studied by Morawetz [3], Lax and Phillips [4]. This problem now is known as the Protter-Morawetz problem. A result for uniqueness was obtained by Aziz and Schneider [5] in the case of Frankl-Morawetz problem. However, the multidimensional case is rather different, and there is no general understanding of the situation. Even the question of well posedness is not completely resolved. For different statements of multidimensional Darboux-type problems or some related Protter-Morawetz problems for mixed-type equations, see [1, 6–13]. Some Tricomi problems for the Lavrent'ev-Bitsadze equation are studied in [14–16]. On the other hand, different problems for elliptic-hyperbolic equations of Keldysh type have specific applications in plasma physics, optics, and analysis on projective spaces (see Otway [17, 18] and Otway and Marini [19]). Various statements of problems for mixed equations of Tricomi or Keldysh type can be found in Oleı̌nik and Radkevič [20], Nakhushev [21], and several applications of such problems in the study of transonic flows are described in Chen [22], Čanić and Keyfitz [23]. Let us also mention some results in the thermodynamic theory of porous elastic bodies given in [24, 25]. In order to analyze the spatial behavior of solutions, some appropriate estimates and similar procedures are used there. In relation to the mixed-type problems, Protter also formulated and studied some BVPs in the hyperbolic part of the domain for the wave equation [26] and degenerated hyperbolic (or weakly hyperbolic) equations of Tricomi type [2]. In that case the Protter problems are multidimensional analogues of the plane Darboux or Cauchy-Goursat problems (see Kalmenov [27] and Nakhushev [28]). The equations are considered in $(3+1)$-D domain, bounded by two characteristic surfaces and noncharacteristic plane region. The data are prescribed on one characteristic and on a noncharacteristic boundary part. Protter considered [2, 26] Tricomi-type equations or the wave equation ($m \in\mathbf{R}$, $m \geq0$) $$ \tilde{L}_{m}[u]:= t^{m}[u_{x_{1}x_{1}}+u_{x_{2}x_{2}}+u_{x_{3}x_{3}}]- u_{tt}=f(x,t) $$ in the domain $$ \tilde{\Omega}_{m}:= \biggl\{ ( x_{1},x_{2},x_{3},t ) : t>0, \frac{2}{m+2} t^{\frac{m+2}{2}}< \sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}< 1- \frac{2}{m+2} t^{\frac{m+2}{2}} \biggr\} , $$ bounded by $\Sigma_{0}$ and two characteristics surfaces of (1.2) $$\begin{aligned}& \tilde{\Sigma}_{1}^{m}= \biggl\{ t>0, \sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}=1- \frac{2}{m+2} t^{\frac{m+2}{2}} \biggr\} , \\& \tilde{\Sigma}_{2}^{m}= \biggl\{ t>0, \sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}= \frac{2}{m+2} t^{\frac{m+2}{2}} \biggr\} . \end{aligned}$$ He proposed four problems, known now as Protter problems. Protter problems Find a solution of equation (1.2) in the domain $\tilde{\Omega}_{m}$ with one of the following boundary conditions: $$\begin{aligned}& P1:\quad u\|_{\Sigma_{0}\cup\tilde{\Sigma}_{1}^{m}}=0, \quad \quad P1^{\ast}:\quad u\|_{\Sigma_{0}\cup\tilde{\Sigma}_{2}^{m}}=0; \\& P2:\quad u\|_{\tilde{\Sigma}_{1}^{m}}=0,\quad\quad u_{t}\|_{\Sigma_{0}}=0,\quad\quad P2^{\ast}:\quad u\|_{\tilde{\Sigma}_{2}^{m}}=0,\quad\quad u_{t}\|_{\Sigma_{0}}=0. \end{aligned}$$ The boundary conditions in problem P1∗ (respectively P2∗) are the adjoint boundary conditions to problem P1 (respectively P2) for (1.2) in $\tilde{\Omega}_{m} $. It turns out that instead of both boundary conditions given in problems P1 on $\tilde{\Sigma}_{1}^{m}$, $\Sigma_{0}$ and in P2 on $\tilde{\Sigma}_{2}^{m}$, $\Sigma_{0}$ for the Tricomi-type equation (1.2), in the case of Keldysh-type equation (1.1), they are reduced to only one boundary condition on the characteristic $\Sigma_{1}^{m}$ and a condition on the growth of possible singularity of the derivative $u_{t}$ as $t \to+0$. We mention some known results for Protter problems in the Tricomi case that make the investigation of such problems interesting and reasonable. Garabedian [29] obtained a result for the uniqueness of classical solution to problem P1 for the wave equation (i.e., equation (1.2) with $m=0$). It is interesting that contrary to their plane analogues, the 3-D Protter problems are not well posed (see [30, 31] and the monograph of Bitsadze [32]). The reason is that the adjoint homogeneous problems P1∗ and P2∗ have an infinite number of linearly independent nontrivial classical solutions. On the other hand, the unique generalized solutions of 3-D problems P1 and P2 could have strong power-type singularity on the $\tilde{\Sigma}^{m} _{2}$ even for smooth right-hand sides (see [31, 33, 34]). Behavior of the singular solutions to 3-D problems P1 and P2 is studied in [35, 36]. Such results are announced for the 4-D case as well [37]. Didenko [38] studied problems P1 and P1∗ for the Tricomi equation ($m = 1$) in the symmetric case. Aldashev [39] studied some multidimensional analogues of Protter problems for equation (1.2), but he did not mention any possible singular solutions. These known results for Protter problems for Tricomi-type equations and many interesting applications of different boundary value problems for equations of Keldysh type motivate us to study problems PK and $\mathit {PK}^{\ast}$ and to try to find out new effects that appear. In [40] ill-posedness of 3-D Protter problems for Keldysh-type equations in the frame of classical solvability is discussed, and the results for uniqueness of quasi-regular solutions are obtained. Existence and uniqueness of generalized solutions to problem PK in that case are obtained in [41], and some singular generalized solutions are announced in [42]. In [31, 33] we study Protter problems for Tricomi-type equations. For 3-D Keldysh-type equation in [43], we formulate a new Protter problem and announce some results for the existence and uniqueness of a generalized solution in the case $0< m<1$. In [44] we announce analogical results for $(3+1)$-D equations of Keldysh type in a more general case $0< m<4/3$ and claim the existence of infinitely many classical smooth solutions of the $(3+1)$-D homogeneous problem $\mathit {PK}^{\ast}$. Now, in the present paper we work in the case $0< m<4/3$. Using an appropriate Riemann-Hadamard function, we find an exact integral representation of the generalized solution and prove the results announced in [44]. To avoid an infinite number of necessary conditions in the frame of classical solvability, we give a notion of a generalized solution to problem PK which can have some singularity at the point O. In order to deal successfully with the encountered difficulties for $\varepsilon\in(0,1)$, we introduce the region $$\Omega_{m,\varepsilon}:=\Omega_{m} \cap \bigl\{ \vert x\vert > \varepsilon \bigr\} , $$ where $\vert x\vert =\sqrt{x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}$. We give the following definition of a generalized solution of problem PK in the case $0< m<4/3$. We call a function $u(x,t)$ a generalized solution of problem PK in $\Omega_{m}$, $0< m<\frac{4}{3}$, for equation (1.1) if: $u,u_{x_{j}} \in C(\bar{\Omega}_{m} \setminus O)$, $j=1,2,3$, $u_{t} \in C(\bar{\Omega}_{m} \setminus\bar{\Sigma}_{0})$; $u\|_{\Sigma_{1}^{m}}=0$; For each $\varepsilon\in(0,1)$ there exists a constant $C(\varepsilon)>0$ such that in $\Omega_{m,\varepsilon}$ $$ \bigl\vert u_{t}(x,t) \bigr\vert \leq C( \varepsilon)t^{-\frac{3m}{4}}; $$ The identity $$\begin{aligned} \int_{\Omega_{m}} \bigl\{ t^{m} u_{t}v_{t}-u_{x_{1}}v_{x_{1}}- u_{x_{2}}v_{x_{2}}-u_{x_{3}}v_{x_{3}}-fv \bigr\} \,dx_{1}\,dx_{2}\,dx_{3}\,dt =0 \end{aligned}$$ holds for all v from $$V_{m}:= \bigl\{ v(x,t):v \in C^{2}(\bar{\Omega}_{m}), v\|_{\Sigma_{2}^{m}} =0, v \equiv0 \text{ in a neighborhood of }O \bigr\} . $$ Remark 1.1 We mention that all the first derivatives of the generalized solutions of 3-D Protter problems in the Tricomi case can have singularity on the boundary of the domain (see [31, 33]). Actually, this fact corresponds to the analogical situation in a 2-D case of the Darboux problem (see [27]). While in the Keldysh case, according to Definition 1.1, the derivative $u_{t}$ can be unbounded when $t\to+0$, but $u_{x_{1}}$, $u_{x_{2}}$ and $u_{x_{3}}$ are bounded in each $\bar{\Omega}_{m,\varepsilon}$, $\varepsilon >0$. In this paper, first, we prove results for the uniqueness of a generalized solution to problem PK. Theorem 1.1 If $m \in(0,\frac{4}{3})$, then there exists at most one generalized solution of problem PK in $\Omega_{m}$. Further, we use the three-dimensional spherical functions $Y_{n}^{s}(x)$ with $n \in\mathbf{N} \cup\{0\}$; $s=1,2, \ldots,2n+1$. The functions $Y_{n}^{s}(x)$ are defined usually on the unit sphere $S^{2}:=\{(x_{1},x_{2},x_{3}):x_{1}^{2}+x_{2}^{2}+x_{3}^{2}=1\}$, and $Y_{n}^{s}$ form a complete orthonormal system in $L_{2}(S^{2})$ (see [45]). For convenience of discussions that follow, we extend the spherical functions out of $S^{2}$ radially, keeping the same notation for the extended functions $Y_{n}^{s}(x):=Y_{n}^{s}(x/\vert x\vert )$ for $x \in\mathbf {R}^{3}\setminus\{0\}$. Let the right-hand side function $f(x,t)$ of equation (1.1) be fixed as a "harmonic polynomial" of order l with $l \in\mathbf{N} \cup\{0\}$, and it has the following representation: $$ f(x,t)=\sum_{n=0}^{l}\sum _{s=1}^{2n+1}f_{n}^{s} \bigl( \vert x\vert ,t \bigr)Y_{n}^{s}(x) $$ with some coefficients $f_{n}^{s}(\vert x\vert ,t)$. In this special case we give an existence result as well. Let $m \in(0,\frac{4}{3})$. Suppose that the right-hand side function $f(x,t)$ has the form (1.5) and $f \in C^{1}(\bar{\Omega}_{m})$. Then the unique generalized solution $u(x,t)$ of problem PK in $\Omega_{m}$ exists and has the form $$ u(x,t)=\sum_{n=0}^{l}\sum _{s=1}^{2n+1}u_{n}^{s} \bigl( \vert x\vert ,t \bigr)Y_{n}^{s}(x). $$ Actually, when the right-hand side function $f(x,t)$ has the form (1.5) in Theorem 1.2, we find explicit representations for the functions $u_{n}^{s}(\vert x\vert ,t)$ in (1.6). These representations involve appropriate hypergeometric functions. In the case when the right-hand side function $f(x,t)$ has the form (1.5), we give an a priori estimate for the generalized solution of problem PK in $\Omega_{m}$ as well. Let the conditions in Theorem 1.2 be fulfilled. Then the unique generalized solution of problem PK in $\Omega_{m}$ has the form (1.6) and satisfies the a priori estimate $$ \bigl\vert u(x ,t) \bigr\vert \leq c \Bigl( \max _{\bar{\Omega}_{m}} \vert f\vert \Bigr) \vert x\vert ^{-l-1}, $$ with a constant $c>0$ independent of f. Estimate (1.7) shows the maximal order of possible singularity at point O, when the right-hand side function $f(x,t)$ is a "harmonic polynomial" of fixed order l. We will point out that a similar a priori estimate for generalized solutions to 3-D Protter problem P1 in the Tricomi case is obtained in [36], while an estimate from below in this case is given in [31]. The present paper contains the introduction and five more sections. In Section 2, the Protter problem PK is considered in a model case when the right-hand side function $f(x,t)$ of equation (1.1) is fixed as a "harmonic polynomial" (1.5) of order l. In that case we formulate the 2-D boundary value problems $\mathit {PK}_{1}$ and $\mathit {PK}_{2}$, corresponding to the $(3+1)$-D problem PK. We give a notion for a generalized solution of Cauchy-Goursat problem $\mathit {PK}_{2}$, and in Section 3, using the Riemann-Hadamard function associated to this problem, we find an integral representation for a generalized solution. Further, we obtain existence and uniqueness results for a generalized solution of problem $\mathit {PK}_{2}$. Actually, this is the essential result in this paper and has the most difficult proof. Using the results of the previous section, in Section 4 we prove the main results in this paper, i.e., Theorem 1.1, Theorem 1.2 and Theorem 1.3. In Appendix A we give the main properties of the Riemann-Hadamard function associated to the Cauchy-Goursat problem $\mathit {PK}_{2}$. In Appendix B some auxiliary results, needed for the study of the generalized solution to problem $\mathit {PK}_{2}$, are proven. Two-dimensional Cauchy-Goursat problems corresponding to problem PK Using spherical coordinates $(r,\theta, \varphi, t)\in \mathbf{R}^{4}$, $0 \leq\theta< \pi$, $0 \leq\varphi< 2 \pi$, $r>0 $ with $$x_{1}=r \sin\theta\cos\varphi, \quad\quad x_{2}=r \sin\theta\sin \varphi, \quad\quad x_{3}=r \cos\theta $$ problem PK can suitably be treated. Written in the new coordinates, equation (1.1) becomes $$ L_{m}u=\frac{1}{r^{2}} \bigl(r^{2} u_{r } \bigr)_{r }+ \frac{1}{r^{2} \sin\theta} (\sin\theta u_{\theta})_{\theta} +\frac{1}{r ^{2} \sin^{2} \theta}u_{\varphi\varphi}- \bigl(t^{m}u_{t} \bigr)_{t}=f. $$ We consider equation (2.1) in the region $$\Omega_{m}= \biggl\{ (r,\theta, \varphi, t):0< t< t_{0}, 0 \leq\theta< \pi, 0 \leq\varphi< 2 \pi, \frac {2}{2-m}t^{\frac{2-m}{2}} < r < 1-\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} , $$ bounded by the following surfaces: $$\begin{aligned}& \Sigma_{0}= \bigl\{ (r,\theta, \varphi, t):t=0, 0 \leq\theta< \pi, 0 \leq\varphi< 2 \pi, r< 1 \bigr\} , \\& \Sigma_{1}^{m}= \biggl\{ (r,\theta, \varphi, t):t>0, 0 \leq \theta< \pi, 0 \leq\varphi< 2 \pi, r=1-\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} , \\& \Sigma_{2}^{m}= \biggl\{ (r,\theta, \varphi, t):t>0, 0 \leq\theta< \pi, 0 \leq\varphi< 2 \pi, r=\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} . \end{aligned}$$ Problem PK becomes the following one: find a solution to equation (2.1) with the boundary conditions $$u\|_{\Sigma_{1}^{m} } =0; \quad\quad t^{m} u_{t} \to0,\quad \text{as }t \to+0. $$ The two-dimensional spherical functions, expressed in terms of θ and φ in the traditional definition (see [45]), are $Y_{n}^{s}(\theta,\varphi):=Y_{n}^{s}(x)$, $x \in S^{2}$, $n \in\mathbf{N} \cup\{0\}$, $s=1,2,\ldots,2n+1$, and satisfy the differential equation $$ \frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \biggl(\sin\theta \frac{\partial}{\partial\theta}Y_{n}^{s} \biggr)+\frac{1}{\sin^{2} \theta} \frac{\partial^{2}}{\partial\varphi^{2}}Y_{n}^{s}+n(n+1)Y_{n}^{s}=0. $$ In the special case when the right-hand side function $f(x,t)$ of equation (2.1) has the form $$ f(r ,\theta, \varphi,t)=f_{n}^{s}(r,t)Y_{n}^{s}( \theta, \varphi), $$ we may look for a solution of the form $$ u(r ,\theta, \varphi,t)=u_{n}^{s}(r,t)Y_{n}^{s}( \theta, \varphi) $$ with unknown coefficient $u_{n}^{s}(r,t)$. For the coefficients $u_{n}^{s}(r ,t)$ which correspond to the right-hand sides $f_{n}^{s}(r ,t)$, we obtain the 2-D equation $$\begin{aligned} u_{rr}+\frac{2}{r}u_{r}- \bigl(t^{m}u_{t } \bigr)_{t}-\frac{n(n+1)}{r ^{2}}u=f \end{aligned}$$ $$G_{m }= \biggl\{ (r ,t):0< t< t_{0}, \frac{2}{2-m}t^{\frac{2-m}{2}}< r < 1-\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} , $$ which is bounded by the segment $S_{0} =\{( r ,t): 0 < r <1, t=0\}$ and the characteristics $$\begin{aligned}& S_{1}^{m}:= \biggl\{ (r ,t):0< t< t_{0}, r =1-\frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} , \\& S^{m}_{2}:= \biggl\{ (r ,t):0< t< t_{0}, r = \frac{2}{2-m}t^{\frac{2-m}{2}} \biggr\} . \end{aligned}$$ In this case, for $u(r ,t)$, the 2-D problem corresponding to PK is the problem $$ \mathit {PK}_{1} \quad \textstyle\begin{cases} u_{rr}+\frac{2}{r}u_{r}-(t^{m}u_{t })_{t}-\frac{n(n+1)}{r ^{2}}u=f(r,t) \quad\text{in } G_{m }, \\ u\|_{S_{1}^{m}}=0; \quad\quad t^{m} u_{t} \to0,\quad \text{as }t \to+0. \end{cases} $$ When the right-hand side function $f(x,t)$ has the form (1.5), it is enough to take test functions $v \in V_{m}$ in the identity (1.4) to have the form $v=w(\vert x\vert ,t)Y_{n}^{s}(x)$, $n \in\mathbf{N} \cup\{0\}$, $s=1,2,\ldots,2n+1$ and $$\begin{aligned} w \in V_{m}^{(1)}:= \bigl\{ w(r,t): w \in C^{2}( \bar{G}_{m}), w\|_{S_{2}^{m}}=0, w \equiv0 \text{ in a neighborhood of }(0,0) \bigr\} . \end{aligned}$$ The generalized solution of problem $\mathit {PK}_{1}$ is defined as follows. We call a function $u(r,t)$ a generalized solution of problem $\mathit {PK}_{1}$ in $G_{m}$ ($0< m<\frac{4}{3}$) if: $u, u_{r} \in C(\bar{G}_{m}\setminus(0,0))$, $u_{t}\in C(\bar {G}_{m}\setminus\bar{S}_{0})$; $u\|_{S_{1}^{m}}=0$; For each $\varepsilon\in(0,1)$ there exists a constant $C(\varepsilon)>0$ such that the estimates $$\bigl\vert u_{t}(r,t) \bigr\vert \leq C(\varepsilon)t^{-\frac{3m}{4}} $$ hold in $G_{m,\varepsilon}:=G_{m} \cap\{r> \varepsilon\}$; $$ \int_{G_{m}} \biggl\{ u_{r}v_{r}-t^{m} u_{t}v_{t}+\frac{n(n+1)}{r^{2}}uv+fv \biggr\} r^{2} \,dr \,dt =0 $$ holds for all $v \in V_{m}^{(1)}$. Substituting the new characteristic coordinates $$ \xi=1-r -\frac{2}{2-m}t^{\frac{2-m}{2}}, \quad \quad\eta=1-r + \frac{2}{2-m}t^{\frac{2-m}{2}} $$ and the new functions $$ \begin{aligned} &U(\xi,\eta)=r(\xi,\eta) u \bigl(r (\xi,\eta ),t(\xi, \eta ) \bigr), \\ &V(\xi,\eta)=r(\xi,\eta) v \bigl(r (\xi,\eta),t(\xi,\eta) \bigr), \\ &F(\xi,\eta)=\frac{1}{8}(2-\xi-\eta)f \bigl(r (\xi,\eta),t(\xi,\eta ) \bigr), \end{aligned} $$ from problem $\mathit {PK}_{1}$, we get the 2-D Cauchy-Goursat problem $$D:= \bigl\{ (\xi,\eta):0< \xi< \eta< 1 \bigr\} \subset\mathbf{R}^{2}, $$ and the parameter $\beta=\frac{ m}{ 2(2-m)}\in(0,1)$ since $0< m<\frac{4}{3}$. We call a function $U(\xi,\eta )$ a generalized solution of problem $\mathit {PK}_{2}$ in D ($0<\beta<1$) if: $U, U_{\xi}+U_{\eta} \in C(\bar{D} \setminus(1,1))$, $U_{\xi}-U_{\eta} \in C(\bar{D}\setminus \{\eta=\xi\})$; $$ U(0,\eta)=0; $$ For each $\varepsilon\in(0,1)$ there exists a constant $C(\varepsilon)>0$ such that $$ \bigl\vert (U_{\xi}-U_{\eta}) (\xi,\eta) \bigr\vert \leq C(\varepsilon) (\eta-\xi)^{-\beta}\quad \text{in }\bar{D}_{\varepsilon} \setminus\{\eta=\xi\}, $$ where $D_{\varepsilon}:=D \cap\{\xi< 1-\varepsilon\}$; $$ \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{\xi}V_{\eta}+U_{\eta}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}UV+2FV \biggr\} \,d\xi \,d\eta=0 $$ holds for all $$V \in V^{(2)}:= \bigl\{ V(\xi,\eta): V \in C^{2}(\bar{D}), V (\xi,1)= 0, V \equiv0 \text{ in a neighborhood of }(1,1) \bigr\} . $$ Existence and uniqueness of a generalized solution to the Cauchy-Goursat plane problem $\mathit {PK}_{2}$ In this section we prove the existence and uniqueness of a generalized solution to problem $\mathit {PK}_{2}$. In order to do this, we use the Riemann-Hadamard function associated to problem $\mathit {PK}_{2}$ to find an integral representation for a generalized solution of this problem in D. According to Gellerstedt [46] and the results of Nakhushev mentioned in the book of Smirnov [47], this function has the form $$ \Phi(\xi,\eta; \xi_{0} ,\eta_{0} ) = \textstyle\begin{cases} \Phi^{+}(\xi,\eta; \xi_{0},\eta_{0} ), & \eta> \xi_{0}, \\ \Phi^{-}(\xi,\eta; \xi_{0},\eta_{0} ), & \eta< \xi_{0}, \end{cases} $$ for $(\xi_{0},\eta_{0}) \in D$ and $(\xi,\eta) \in \bar{T} \cup\bar{\Pi} \setminus\{\eta=\xi_{0}\}$, where $$\begin{aligned} T:= \bigl\{ (\xi,\eta): 0 < \xi< \eta< \xi_{0} \bigr\} , \quad\quad \Pi:= \bigl\{ ( \xi,\eta): 0 < \xi< \xi_{0}, \xi_{0} < \eta< \eta_{0} \bigr\} . \end{aligned}$$ The Riemann-Hadamard function $\Phi(\xi,\eta;\xi_{0},\eta_{0})$ should have the following main properties (see [46, 47]): The function Φ as a function of $(\xi_{0},\eta_{0})$ satisfies $$ \begin{aligned}[b] E_{\xi_{0},\eta_{0}}[\Phi]&:= \frac{\partial^{2} \Phi }{\partial\xi_{0}\, \partial \eta_{0}}+\frac{\beta}{\eta_{0}-\xi_{0}} \biggl( \frac{\partial \Phi}{\partial\xi_{0}}-\frac{\partial\Phi}{\partial \eta_{0}} \biggr)-\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}\Phi \\ &=0 \quad\text{in } D, \eta\neq\xi_{0} \end{aligned} $$ and with respect to the first pair of variables $(\xi,\eta)$ $$ \begin{aligned}[b] E^{\ast}_{\xi,\eta}[\Phi]&:= \frac{\partial^{2} \Phi }{\partial\xi\,\partial \eta}- \frac{\partial}{\partial\xi} \biggl(\frac{\beta \Phi}{\eta-\xi} \biggr)+ \frac{\partial}{\partial\eta} \biggl( \frac{\beta \Phi}{\eta-\xi} \biggr)-\frac{n(n+1)}{(2-\xi-\eta)^{2}}\Phi \\ &=0 \quad\text{in } D, \eta\neq \xi_{0}; \end{aligned} $$ $\Phi^{+}(\xi_{0},\eta_{0};\xi_{0},\eta_{0})=1$; $\Phi^{+}(\xi,\eta_{0};\xi_{0},\eta_{0})= ( \frac{\eta_{0}-\xi}{\eta_{0}-\xi_{0}} )^{\beta}$; $\Phi^{+}(\xi_{0},\eta;\xi_{0},\eta_{0})= ( \frac{\eta-\xi_{0}}{\eta_{0}-\xi_{0}} )^{\beta}$; The jump of the function Φ on the line $\{\eta=\xi_{0}\}$ is $$\begin{aligned}{} [ [\Phi] ] :=& \lim_{\delta\to +0} \bigl\{ \Phi^{-}(\xi,\xi_{0}-\delta;\xi_{0}, \eta_{0})- \Phi^{+}(\xi,\xi_{0}+\delta; \xi_{0},\eta_{0}) \bigr\} \\ =& \cos(\pi\beta) \lim_{\delta\to +0} \bigl\{ \Phi^{+}( \xi,\xi_{0}+\delta;\xi_{0},\xi_{0}+\delta) \Phi^{+}(\xi_{0},\xi_{0}+\delta; \xi_{0},\eta_{0}) \bigr\} \\ =& \cos(\pi\beta) \biggl(\frac{\xi_{0}-\xi}{\eta_{0}-\xi_{0}} \biggr)^{\beta}; \end{aligned}$$ $\Phi^{-}$ vanishes on the line $\{\eta=\xi\}$ of power 2β. Actually, the function $\Phi^{+}$ is the Riemann function for equation (2.5). In the case $\boldsymbol{0<\beta<1/2} $ and $\boldsymbol{F(\xi,\eta)=(\eta-\xi)^{-4 \beta}f(\xi,\eta)}$, where $f \in C(\bar{D})$, a generalized solution of problem $\mathit {PK}_{2}$ has an explicit integral representation (see [46] and [47]). We find an integral representation in the case $\boldsymbol{0<\beta<1}$ and $F\in C(\bar{D})$ using the properties of the Riemann-Hadamard function $\Phi(\xi,\eta;\xi_{0},\eta_{0})$. The existence of a function $\Phi(\xi,\eta;\xi_{0},\eta_{0})$ with properties $\mbox{(i)} \div\mbox{(vi)}$ is shown in Appendix A (see also [44]). Let $0<\beta<1$ and $F\in C(\bar{D})$. Then each generalized solution of problem $\mathit {PK}_{2}$ in D has the following integral representation: $$ U (\xi_{0},\eta_{0}) = \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) \Phi(\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi. $$ Let $U(\xi,\eta)$ be a generalized solution of problem $\mathit {PK}_{2}$ in D. For any arbitrary function $\psi(\xi,\eta)$ from $C_{0}^{\infty}(D)$, we have $\psi\in V^{(2)}$, and from (2.9) we obtain the identity $$ \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{\xi\eta}+ \frac{\beta}{\eta-\xi}(U_{\xi}-U_{\eta}) -\frac{n(n+1)}{(2-\xi-\eta)^{2}}U-F \biggr\} \psi \,d\xi \,d\eta=0, $$ where $U_{\xi\eta}$ is the weak derivative of U. Therefore $$U_{\xi\eta}=F +\frac{n(n+1)}{(2-\xi-\eta)^{2}}U-\frac{\beta}{\eta-\xi}(U_{\xi}-U_{\eta }) \in C(D) $$ since $F, U, U_{\xi}-U_{\eta} \in C(D)$. From this it follows that $U_{\xi\eta}$ is a classical derivative of U and $U(\xi,\eta)$ satisfies the differential equation (2.5) in D in a classical sense. Now, using the properties of the Riemann-Hadamard function $\Phi(\xi,\eta;\xi_{0},\eta_{0})$, we obtain the integral representation (3.4) for a generalized solution of problem $\mathit {PK}_{2}$ by integrating the identity $$\Phi(\xi,\eta;\xi_{0},\eta_{0}) E_{\xi,\eta} \bigl[U( \xi,\eta) \bigr] -U(\xi,\eta) E^{\ast}_{\xi,\eta} \bigl[\Phi(\xi, \eta; \xi_{0},\eta_{0}) \bigr] = F(\xi,\eta) \Phi (\xi,\eta; \xi_{0},\eta_{0}) $$ over a triangle $$T_{\delta}:= \bigl\{ (\xi,\eta):0 < \xi< \xi_{0}-2 \delta, \xi+ \delta< \eta< \xi_{0} - \delta \bigr\} $$ and then over the rectangle $$\Pi_{\delta}:= \bigl\{ (\xi,\eta): 0 < \xi< \xi_{0}-2 \delta, \xi_{0} + \delta< \eta< \eta_{0} \bigr\} $$ with $\delta>0$ small enough, and finally letting $\delta\to0$. □ Theorem 3.1 claims the uniqueness of a generalized solution to problem $\mathit {PK}_{2}$. Next, we prove that if $F \in C^{1}(\bar{D})$ and $U(\xi,\eta)$ is a function defined by (3.4) in D, then $U(\xi,\eta)$ is a generalized solution to problem $\mathit {PK}_{2}$ in D. In order to do this, we introduce the notation $$M_{F}:=\max \Bigl\{ \max_{\bar{D}_{0}}\vert F\vert , \max_{\bar {D}_{0}}\vert F_{\xi}+F_{\eta} \vert \Bigr\} , $$ and we mention that, according to Lemma A.1 (see Appendix A below), the Riemann-Hadamard function $\Phi(\xi,\eta;\xi_{0},\eta_{0})$ can be decomposed in the following way: $$\Phi(\xi,\eta;\xi_{0},\eta_{0})= H(\xi,\eta; \xi_{0},\eta_{0})+G(\xi,\eta;\xi_{0}, \eta_{0}), $$ where $H(\xi,\eta;\xi_{0},\eta_{0})$ is the Riemann-Hadamard function (A.12) associated to problem $\mathit {PK}_{2}$ in the case $n=0$ and $G(\xi,\eta;\xi_{0},\eta_{0})$ is an additional term. Therefore we can rewrite representation (3.4) in the form $$ U(\xi_{0},\eta_{0})=U^{H}( \xi_{0},\eta_{0})+U^{G}(\xi_{0}, \eta_{0}), $$ $$ U^{H}(\xi_{0},\eta_{0}):= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) H (\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi $$ $$ U^{G}(\xi_{0},\eta_{0}):= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) G (\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi. $$ Firstly, we will study the function $U^{H}(\xi_{0},\eta_{0})$. To do this, we use the estimates for some integrals involving function $H(\xi,\eta;\xi_{0},\eta_{0})$ obtained in Appendix B. Let $0 < \beta<1$ and $F \in C^{1}(\bar{D})$. Then, for the function $U^{H}(\xi_{0},\eta_{0})$, we have $U^{H}, U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}}\in C(\bar{D}\setminus(1,1))$, $U^{H}_{\eta_{0}}\in C(\bar{D}\setminus\{\eta_{0}=\xi_{0}\})$ and the following estimates hold: $$\begin{aligned}& \bigl\vert U^{H}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{1} M_{F} \xi_{0}\quad \textit{in } \bar{D}\setminus(1,1), \\& \bigl\vert U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}} \bigr\vert (\xi_{0},\eta_{0}) \leq K_{1} M_{F} \eta_{0} \quad \textit{in } \bar{D}\setminus(1,1), \\& \bigl\vert U^{H}_{\eta_{0}}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{1} M_{F} \xi_{0}(\eta_{0}-\xi_{0})^{-\beta} \quad \textit{in } \bar{D}\setminus\{\eta_{0}=\xi_{0}\}, \end{aligned}$$ where $K_{1}>0$ is a constant independent of F. Step 1. From (3.6) and (B.1) from Lemma B.1 (see Appendix B) we obtain $$ \bigl\vert U^{H}(\xi_{0}, \eta_{0}) \bigr\vert \leq M_{F} \int_{0}^{\xi_{0}} \int_{\xi }^{\eta_{0}} H (\xi,\eta;\xi_{0}, \eta_{0}) \,d\eta \,d\xi =M_{F}I^{1}( \xi_{0},\eta_{0}) \leq k_{1} M_{F} \xi_{0}. $$ Step 2. Differentiating (3.6) with respect to $\eta_{0}$ and using (B.4) from Lemma B.2, we obtain $$ \begin{aligned} \bigl\vert U^{H}_{\eta_{0}}( \xi_{0},\eta_{0}) \bigr\vert &= \biggl\vert \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) H_{\eta_{0}} (\xi, \eta;\xi_{0},\eta_{0}) \,d\eta \,d\xi+ \int_{0}^{\xi_{0}}F(\xi,\eta_{0}) \frac{(\eta_{0}-\xi)^{\beta}}{(\eta_{0}-\xi_{0})^{\beta}} \,d\xi \biggr\vert \\ & \leq M_{F} \biggl\{ \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}} \bigl\vert H_{\eta_{0}} (\xi, \eta;\xi_{0},\eta_{0}) \bigr\vert \,d\eta \,d\xi+ \int_{0}^{\xi_{0}} \biggl(\frac{\eta_{0}-\xi}{\eta_{0}-\xi_{0}} \biggr)^{\beta} \,d\xi \biggr\} \\ &= M_{F} \biggl\{ I^{2}(\xi_{0}, \eta_{0})+ \int_{0}^{\xi_{0}} \biggl(\frac{\eta_{0}-\xi}{\eta_{0}-\xi_{0}} \biggr)^{\beta} \,d\xi \biggr\} \\ &\leq M_{F}(k_{2}+1) \xi_{0}( \eta_{0}-\xi_{0})^{-\beta}. \end{aligned} $$ Step 3. According to Remark A.1, the derivatives $H_{\xi_{0}}^{+}$, $H_{\xi_{0}}^{-}$ have singularities of order $\vert \eta-\xi_{0}\vert ^{-1}$ on the line $\{\eta=\xi_{0}\}$. Gellerstedt [46] and Moiseev [48] consider the case $n= 0$ and suggest differentiating (3.6) after appropriate substitutions of variables. In that way one can find integral representations for the first derivatives of the solution which do not involve the first derivatives of function H. In order to do this, following Moiseev [48], we introduce new variables $$ \tilde{\xi}:=\frac{\xi_{0}-\xi}{\eta_{0}-\xi_{0}}, \quad\quad \tilde{\eta}:= \frac{\eta_{0}-\eta}{\eta_{0}-\xi_{0}}. $$ We define $$\tilde{H}^{+}(\tilde{\xi},\tilde{\eta}):=H^{+}(\xi,\eta; \xi_{0},\eta_{0}), \quad\quad \tilde{H}^{-}(\tilde{\xi}, \tilde{\eta}):=H^{-}(\xi,\eta;\xi_{0},\eta_{0}), $$ from (A.12) we obtain $$\begin{aligned}& \tilde{H}^{+} (\tilde{\xi},\tilde{\eta})= (1-\tilde{ \eta}+\tilde{\xi})^{\beta}F \biggl(\beta,1-\beta,1;\frac{\tilde{\xi} \tilde{\eta}}{1-\tilde{\eta}+\tilde{\xi}} \biggr),\quad\tilde{\eta}< 1, \\& \tilde{H}^{-}(\tilde{\xi},\tilde{\eta}) = \frac{k (1-\tilde{\eta}+\tilde{\xi}) ^{2\beta}}{\tilde{\xi}^{\beta}\tilde{\eta}^{\beta}} F \biggl(\beta,\beta,2\beta;\frac{1-\tilde{\eta}+\tilde{\xi}}{\tilde{\xi} \tilde{\eta}} \biggr),\quad\tilde{\eta}>1. \end{aligned}$$ Then we have $$\begin{aligned} &U^{H}(\xi_{0}, \eta_{0}) \\ &\quad =(\eta_{0}-\xi_{0})^{2} \int_{0}^{\frac{\xi_{0}}{\eta_{0}-\xi_{0}}} \biggl\{ \int_{0}^{1+\tilde{\xi}}F \bigl(\xi_{0}-( \eta_{0}-\xi_{0})\tilde{\xi},\eta _{0}-( \eta_{0}-\xi_{0})\tilde{\eta} \bigr) \tilde{H }(\tilde{\xi}, \tilde{\eta}) \,d\tilde{\eta} \biggr\} \,d\tilde{\xi}, \end{aligned} $$ $$ \begin{aligned} & \bigl(U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}} \bigr) (\xi_{0},\eta_{0}) \\ &\quad = (\eta_{0}-\xi_{0})^{2} \int_{0}^{\frac{\xi_{0}}{\eta_{0}-\xi_{0}}} \int_{0}^{1+\tilde{\xi}}(F_{\xi}+F_{\eta}) \bigl(\xi_{0}-(\eta_{0}-\xi_{0})\tilde{\xi}, \eta_{0}-(\eta_{0}-\xi_{0})\tilde{\eta} \bigr) \tilde{H} (\tilde{\xi},\tilde{\eta}) \,d\tilde{\eta} \,d\tilde{\xi} \\ &\quad\quad{} +(\eta_{0}-\xi_{0}) \int_{0}^{\frac{\eta_{0}}{\eta_{0}-\xi_{0}}}F \bigl(0,\eta_{0}-( \eta_{0}-\xi_{0})\tilde{\eta} \bigr) \tilde{H} \biggl( \frac{\xi_{0}}{\eta_{0}-\xi_{0}},\tilde{\eta} \biggr) \,d\tilde{\eta}. \end{aligned} $$ Now the inverse transform of (3.8) gives $$ \begin{aligned} \bigl(U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}} \bigr) (\xi_{0},\eta_{0})&= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}(F_{\xi}+F_{\eta}) (\xi,\eta) H (\xi,\eta;\xi_{0},\eta_{0}) \,d\eta \,d\xi \\ &\quad{} + \int_{0}^{\eta_{0}}F(0,\eta)H(0,\eta;\xi_{0}, \eta_{0}) \,d\eta. \end{aligned} $$ Now (B.1) from Lemma B.1 and (B.6) from Lemma B.3 give $$\bigl\vert \bigl(U^{H}_{\xi_{0}}+U^{H}_{\eta_{0}} \bigr) (\xi_{0},\eta_{0}) \bigr\vert \leq M_{F} \bigl\{ I^{1}(\xi_{0},\eta_{0})+I^{3}( \xi_{0},\eta_{0}) \bigr\} \leq M_{F}(k_{1}+k_{3}) \eta_{0}. $$ Let the conditions in Theorem 3.2 be fulfilled. Then for the function $U^{G}(\xi_{0},\eta_{0})$ we have $U^{G}, U^{G}_{\xi_{0}}, U^{G}_{\eta_{0}} \in C(\bar{D}\setminus(1,1))$, and the following estimates hold in $\bar{D}\setminus(1,1)$: $$\begin{aligned}& \bigl\vert U^{G}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{2}M_{F} \xi_{0}(2-\xi_{0}-\eta_{0})^{-n}, \end{aligned}$$ $$\begin{aligned}& \bigl\vert U^{G}_{\xi_{0}}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{2}M_{F} \xi_{0}(2-\xi_{0}-\eta_{0})^{-n-1}, \end{aligned}$$ $$\begin{aligned}& \bigl\vert U^{G}_{\eta_{0}}(\xi_{0}, \eta_{0}) \bigr\vert \leq K_{2}M_{F} \xi_{0}(2-\xi_{0}-\eta_{0})^{-n-1}, \end{aligned}$$ Using estimates (A.26) and (A.27), from (3.7) we obtain estimate (3.9): $$ \begin{aligned} \bigl\vert U^{G}(\xi_{0}, \eta_{0}) \bigr\vert &= \biggl\vert \int_{0}^{\xi_{0}} \int_{\xi}^{\xi_{0}}F(\xi,\eta) G^{-}(\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi \\ &\quad{} + \int_{0}^{\xi_{0}} \int_{\xi_{0}}^{\eta_{0}}F(\xi,\eta) G^{+}(\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi \biggr\vert \\ &\leq C_{G}M_{F}\xi_{0} \biggl\{ \frac{1}{2}\xi_{0}(2-\xi_{0}-\eta_{0})^{-n}+( \eta_{0}-\xi_{0})^{1-\beta} \biggr\} \\ &\leq K_{2}M_{F}\xi_{0}(2-\xi_{0}- \eta_{0})^{-n}. \end{aligned} $$ Now we calculate $$U^{G}_{\xi_{0}}(\xi_{0},\eta_{0})= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta ) G_{\xi_{0}}(\xi, \eta,\xi_{0},\eta_{0}) \,d\eta \,d\xi. $$ Here we do not have integrals on the boundaries because $Y=0$ on the line $\{\xi=\xi_{0}\}$, and the function $G(\xi,\eta,\xi_{0},\eta_{0})$ has no jump on the line $\{\eta=\xi_{0}\}$ (see Appendix A). Applying estimates (A.30) and (A.31) to this integral, we have $$ \begin{aligned} \bigl\vert U^{G}_{\xi_{0}}( \xi_{0},\eta_{0}) \bigr\vert &\leq\frac{M_{F}C_{G}}{(2-\xi_{0}-\eta_{0})^{n+1}} \int_{0}^{\xi_{0}} \int_{\xi}^{\xi_{0}} (\xi_{0}- \eta)^{-\beta} \,d\eta \,d\xi \\ &\quad{} +\frac{M_{F}C_{G}}{2-\xi_{0}-\eta_{0}} \int_{0}^{\xi_{0}} \int_{\xi_{0}}^{\eta_{0}}(\eta-\xi_{0})^{-\beta} \,d\eta \,d\xi \\ &\leq\frac{M_{F} C_{G}}{(2-\xi_{0}-\eta_{0})^{n+1}} \bigl(I_{1}^{1}+2^{n}I_{2}^{1} \bigr). \end{aligned} $$ Now (B.2) and (B.3) from Lemma B.1 give estimate (3.10). Further, we calculate $$U^{G}_{\eta_{0}}(\xi_{0},\eta_{0})= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta ) G_{\eta_{0}}(\xi, \eta;\xi_{0},\eta_{0}) \,d\eta \,d\xi, $$ where we used that $Y=0$ on the line $\eta=\eta_{0}$. Analogously, applying estimates (A.28) and (A.29), which are even better than (A.30) and (A.31), to the last integral for the derivative $G_{\eta_{0}}$, we obtain estimate (3.11). □ As a direct consequence of Theorem 3.2 and Theorem 3.3, in view of $U=U^{H}+U^{G}$, we have the following theorem. Let $0 < \beta<1$ and $F \in C^{1}(\bar{D})$. Then, for the function $U(\xi,\eta)$ defined by (3.4), we have $U, U_{\xi}+U_{\eta} \in C(\bar{D} \setminus(1,1))$, $U_{\eta} \in C(\bar{D}\setminus \{\eta=\xi\})$ and for some constant $K_{3}>0$ the estimates below hold $$ \begin{gathered} \bigl\vert U(\xi,\eta) \bigr\vert \leq K_{3}M_{F}\xi(2-\xi-\eta)^{-n} \quad\textit{in }\bar{D} \setminus(1,1), \\ \bigl\vert (U_{\xi}+U_{\eta}) (\xi,\eta) \bigr\vert \leq K_{3}M_{F}(2-\xi-\eta)^{-n-1} \quad\textit{in } \bar{D} \setminus (1,1), \\ \bigl\vert U_{\eta}(\xi,\eta) \bigr\vert \leq K_{3}M_{F} \xi(\eta-\xi)^{-\beta}(2-\xi- \eta)^{-n-1} \quad\textit{in } \bar{D}\setminus\{ \eta=\xi\}. \end{gathered} $$ Now, we are able to prove the following existence result. Let $0 < \beta<1$ and $F \in C^{1}(\bar{D})$. Then there exists one and only one generalized solution to problem $\mathit {PK}_{2}$ in D, which has integral representation (3.4), and it satisfies estimates (3.12). Let $U(\xi,\eta)$ be the function known from Theorem 3.4. Therefore $U, U_{\xi}+U_{\eta} \in C(\bar{D} \setminus(1,1))$, $U_{\eta} \in C(\bar{D}\setminus \{\eta=\xi\})$, and it satisfies estimates (3.12) in Definition 2.2. But in view of (3.12) it is obvious that condition (2.7) and estimate (2.8) hold. To prove that $U(\xi,\eta)$ satisfies identity (2.9) in Definition 2.2, we need several steps as follows. Step 1. We prove that $U(\xi,\eta)$ satisfies the differential equation (2.5) in a classical sense and $\frac{\partial}{\partial\eta} (U_{\xi} ) \in C(D)$. (1.i) Following Smirnov [47], we find another representation formula for the function $U^{H}(\xi,\eta)$. Let us introduce the function $$ R_{0}(\xi,\eta;\xi_{0},\eta_{0}):= \textstyle\begin{cases} R_{0}^{+}(\xi,\eta;\xi_{0},\eta_{0}), &\eta>\xi_{0},\\ R_{0}^{-}(\xi,\eta;\xi_{0},\eta_{0}), &\eta< \xi_{0}, \end{cases} $$ $$ \begin{gathered} R_{0}^{+}(\xi,\eta;\xi_{0}, \eta_{0}) := \biggl(\frac{\eta_{0}-\eta}{\eta_{0}-\xi_{0}} \biggr)^{\beta} \biggl( \frac{\eta_{0}-\eta}{\eta_{0}-\xi} \biggr)^{1-\beta}F_{1} \biggl(1-\beta,\beta,1- \beta,2;\frac{\eta_{0}-\eta}{\eta_{0}-\xi_{0}},\frac{\eta_{0}-\eta}{\eta_{0}-\xi} \biggr), \\ R_{0}^{-}(\xi,\eta;\xi_{0},\eta_{0}) :=\gamma \biggl(\frac{\eta-\xi}{\xi_{0}-\xi} \biggr)^{\beta} \biggl(\frac{\eta-\xi }{\eta_{0}-\xi} \biggr)^{\beta}F_{1} \biggl(\beta,\beta,\beta,1+2 \beta; \frac{\eta-\xi}{\xi_{0}-\xi},\frac{\eta-\xi}{\eta_{0}-\xi} \biggr). \end{gathered} $$ Here $\gamma=- \frac{\Gamma(\beta)}{\Gamma(1-\beta) \Gamma(1+2 \beta )}$ and $F_{1}(a,b_{1},b_{2},c;x,y)$ is the hypergeometric function (A.8) of two variables (see Appendix A). In [47] the case $0<\beta<1/2$ is considered, but here we find that in a more general case $0<\beta<1$ the function $R_{0}(\xi,\eta;\xi_{0},\eta_{0})$ solves $$ \begin{gathered} \frac{\partial R_{0}}{\partial\eta}=-(\eta- \xi)^{-1}H( \xi,\eta;\xi_{0},\eta_{0})\quad\text{for }(\xi, \eta) \in\Pi\cup T, \\ R_{0}\|_{\eta=\eta_{0}}=0,\quad\quad R_{0}\|_{\eta=\xi}=0, \end{gathered} $$ where $(\xi_{0},\eta_{0}) \in D$ and $H(\xi,\eta;\xi_{0},\eta_{0})$ is function (A.12). Using (3.13), integration by parts and $$\bigl[R_{0}^{+} - R_{0}^{-} \bigr]\big\|_{\eta=\xi_{0}}= \frac{1}{\beta} $$ leads to the integral representation $$ \begin{aligned}[b] U^{H}(\xi_{0}, \eta_{0}) &:= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}\frac{\partial}{\partial\eta} \bigl[(\eta-\xi)F( \xi,\eta) \bigr] R_{0} (\xi,\eta;\xi_{0}, \eta_{0}) \,d\eta \,d\xi \\ &\quad{} +\frac{1}{\beta} \int_{0}^{\xi_{0}}(\xi_{0}-\xi)F(\xi, \xi_{0}) \,d\xi. \end{aligned} $$ (1.ii) Differentiating (3.14) we obtain that $U^{H}$ satisfies the differential equation $$ \bigl(U^{H}_{\xi_{0}} \bigr)_{\eta_{0}}+ \frac{\beta}{\eta_{0}-\xi_{0}} \bigl( U^{H}_{ \xi_{0}}- U^{H}_{\eta_{0}} \bigr)=F(\xi_{0},\eta_{0}), $$ where all derivatives are in a classical sense and they are continuous in D. (1.iii) Since $H(\xi,\eta;\xi_{0},\eta_{0})$ satisfies the differential equation (3.2) with $n=0$ and $\Phi =H+G$ satisfies (3.2) with $n \geq0$ for the difference $G=\Phi- H$, we obtain $$G_{\xi_{0}\eta_{0}}+\frac{\beta}{\eta_{0}-\xi_{0}} ( G_{ \xi_{0}}- G_{\eta_{0}} )- \frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}G=\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}H. $$ Now, using integral representation (3.7) for $U^{G}(\xi_{0},\eta _{0})$, we calculate $$ \begin{aligned}[b] & \bigl(U^{G}_{\xi_{0}} \bigr)_{\eta_{0}}+\frac{\beta}{\eta_{0}-\xi_{0}} \bigl( U^{G}_{ \xi_{0}}- U^{G}_{\eta_{0}} \bigr)-\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}U^{G} \\ &\quad = \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) \biggl[G_{\xi_{0}\eta_{0}}+ \frac{\beta}{\eta_{0}-\xi_{0}} ( G_{ \xi_{0}}- G_{\eta_{0}} ) \\ &\quad\quad{} -\frac {n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}G \biggr](\xi,\eta;\xi_{0},\eta_{0}) \,d\eta \,d\xi \\ &\quad =\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}} \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}}F(\xi,\eta) H(\xi,\eta; \xi_{0},\eta_{0}) \,d\eta \,d\xi \\ &\quad =\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}U^{H}, \end{aligned} $$ (1.iv) Since $U=U^{H}+U^{G}$, summing up equations (3.15) and (3.16), we obtain the differential equation $$(U_{\xi_{0}} )_{\eta_{0}}+\frac{\beta}{\eta_{0}-\xi_{0}} ( U_{ \xi_{0}}- U_{\eta_{0}} )-\frac{n(n+1)}{(2-\xi_{0}-\eta_{0})^{2}}U=F(\xi_{0},\eta_{0}) $$ in a classical sense. But, since $F, U, U_{\xi_{0}}-U_{\eta_{0}} \in C(D)$, it follows that $(U_{\xi_{0}} )_{\eta _{0}}\in C(D)$. Step 2. We will prove that identity (2.9) holds for all $V(\xi,\eta) \in V^{(2)}$. (2.i) Let $V(\xi,\eta) \in V^{(2)}$ and in addition $V(\xi,\eta)\equiv0$ in a neighborhood of $\{\eta=\xi\}$ and in a neighborhood of $\{\eta=1\}$. From Step 1 we know that $U(\xi,\eta)$ satisfies the differential equation (2.5), where all derivatives are in a classical sense, continuous in D. Let us consider $$ I_{V}:= \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{\xi}V_{\eta}+U_{\eta}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}UV+2FV \biggr\} \,d\xi \,d\eta. $$ Now we integrate by parts in $I_{V}$ in the following way: in the term $U_{\xi}V_{\eta}$, we move the derivative from $V_{\eta}$ to $U_{\xi}$ and obtain the term $(U_{\xi} )_{\eta}V$: $$ \int_{D}(\eta-\xi)^{2 \beta}U_{\xi}V_{\eta} \,d\xi \,d\eta= - \int_{D}(\eta-\xi)^{2 \beta} \biggl[ (U_{\xi} )_{\eta}+\frac{2\beta}{\eta-\xi}U_{\xi} \biggr]V \,d\xi \,d\eta; $$ in the term $U_{\eta}V_{\xi}$, we move the derivative from $U_{\eta}$ to $V_{\xi}$ and obtain the term $U (V_{\xi} )_{\eta}$: $$\int_{D}(\eta-\xi)^{2 \beta}U_{\eta}V_{\xi} \,d\xi \,d\eta= - \int_{D}(\eta-\xi)^{2 \beta} \biggl[ (V_{\xi} )_{\eta}+\frac{2\beta}{\eta-\xi}V_{\xi} \biggr]U \,d\xi \,d\eta. $$ There are not integrals on the boundary of D because $U(0,\eta)=0$, $V(\xi,\eta)\equiv0$ in a neighborhood of $\{\eta=\xi\}$ and in a neighborhood of $\{\eta=1\}$. since $V \in C^{2}(\bar{D})$, we have $(V_{\xi} )_{\eta}= (V_{\eta } )_{\xi}$; in the term $(V_{\eta} )_{\xi}U$, we move the derivatives from $(V_{\eta} )_{\xi}$ to U and obtain the term $(U_{\xi} )_{\eta}V$: $$ \begin{aligned}[b] \int_{D}(\eta-\xi)^{2 \beta}U_{\eta}V_{\xi} \,d\xi \,d\eta&= - \int_{D}(\eta-\xi)^{2 \beta} \biggl[ (V_{\eta} )_{\xi}+\frac{2\beta}{\eta-\xi}V_{\xi} \biggr]U \,d\xi \,d\eta \\ &= \int_{D}(\eta-\xi)^{2 \beta} \biggl[U_{\xi}V_{\eta}- \frac{2\beta}{\eta-\xi}(V_{\xi}+V_{\eta})U \biggr] \,d\xi \,d\eta \\ &=- \int_{D}(\eta-\xi)^{2 \beta} \biggl[ (U_{\xi} )_{\eta}-\frac{2\beta}{\eta-\xi}U_{\eta} \biggr]V \,d\xi \,d\eta. \end{aligned} $$ Again there are not integrals on the boundary of D, and putting (3.18) and (3.19) into (3.17), we get $$ I_{V}= -2 \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ (U_{\xi} )_{\eta}+\frac{\beta}{\eta-\xi}(U_{\xi}-U_{\eta}) - \frac{n(n+1)}{(2-\xi-\eta)^{2}}U-F \biggr\} V \,d\xi \,d\eta=0. $$ (2.ii) Let $V(\xi,\eta) \in V^{(2)}$ and $\Psi(s)$ be a function having the properties $\Psi(s) \in C^{\infty}(\mathbf{R}^{1})$, $\Psi(s)=1$ for $s\geq2$, $\Psi(s)=0$ for $s\leq1$. If $k, l \in\mathbf{N}$, then according to (2.i) (see (3.17) and (3.20)) for the functions $$V_{k,l}(\xi,\eta):=V(\xi,\eta)\Psi \bigl(k [1-\eta] \bigr)\Psi \bigl(l [ \eta-\xi] \bigr) $$ identity (2.9) holds. Therefore we have $$ \begin{aligned}[b] 0&= \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{\xi}V_{\eta}+U_{\eta}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}UV+2FV \biggr\} \\ &\quad{}\times \Psi \bigl(k [1-\eta] \bigr) \Psi \bigl(l [ \eta-\xi] \bigr) \,d\xi \,d\eta \\ &\quad{} + \int_{D}l(\eta-\xi)^{2 \beta} \{U_{\xi}-U_{\eta} \}\Psi \bigl(k [1-\eta] \bigr)\Psi' \bigl(l [\eta-\xi] \bigr) V \,d \xi \,d\eta \\ &\quad{} - \int_{D}k(\eta-\xi)^{2 \beta}U_{\xi} \Psi' \bigl(k [1-\eta] \bigr) \Psi \bigl(l [\eta-\xi] \bigr)V \,d\xi \,d\eta \\ &=: I_{1,kl}+I_{2,kl}+I_{3,kl}. \end{aligned} $$ Obviously, $I_{1,kl} \to I_{V}$, as $k, l \to\infty$. We know that $V \equiv0$ in a neighborhood of $(0,0)$ and $\operatorname {supp}\Psi'(l[\eta-\xi])$ is contained in $\{1\leq l[\eta-\xi]\leq2 \}$. Using estimate (3.12) we find that on $\operatorname {supp}\Psi'(l[\eta-\xi])$ the functions $$W_{k,l}(\xi,\eta):=l(\eta-\xi)^{2 \beta} \{U_{\xi}-U_{\eta} \}\Psi \bigl(k [1-\eta] \bigr)\Psi' \bigl(l [\eta-\xi] \bigr) V $$ satisfy the estimates $$ \bigl\vert W_{k,l}(\xi,\eta) \bigr\vert \leq \mathit{const.} ( \eta-\xi)^{\beta-1}. $$ Since, obviously, the sequence ${W_{k,l}}$ converges pointwise almost everywhere to zero and it is dominated by a Lebesgue integrable function in D for $0<\beta<1$ (see (3.22)). Thus, according to the Lebesgue dominated convergence theorem, $I_{2,kl} \to0$ as $k, l \to \infty$. Since $V(\xi,1)=0$, we have $$k \bigl\vert V(\xi,\eta) \bigr\vert \bigl\vert \Psi' \bigl(k [1-\eta] \bigr) \bigr\vert = k(1-\eta) \bigl\vert V_{\eta}(\xi, \tilde{ \eta}) \bigr\vert \bigl\vert \Psi' \bigl(k [1-\eta] \bigr) \bigr\vert \leq c_{v}, $$ where $c_{v}$ is a constant and $\eta<\tilde{\eta}<1$. Therefore $I_{3,kl} \to0$ as $k, l \to\infty$. Thus, letting $k, l \to\infty$ in (3.21), we obtain that identity (2.9) holds for $V \in V^{(2)}$. Consequently, the function $U(\xi,\eta)$ is a generalized solution to problem $\mathit {PK}_{2}$. □ Proof of the main results In this section we give the proofs of Theorem 1.1, Theorem 1.2 and Theorem 1.3 formulated in Section 1. Proof of Theorem 1.1 Let and $u_{1}$ and $u_{2}$ be two generalized solutions of problem PK in $\Omega_{m}$. Then the function $u:=u_{1}-u_{2}$ solves the homogeneous problem PK. We will show that the Fourier expansion $$u(r ,\theta, \varphi ,t)=\sum_{n=0}^{\infty} \sum_{s=1}^{2n+1}u_{n}^{s}(r,t)Y_{n}^{s}( \theta, \varphi) $$ has zero Fourier-coefficients $$\begin{aligned} u_{n}^{s}(r ,t):= \int_{0}^{\pi} \int_{0}^{2\pi} u(r,\theta,\varphi,t)Y_{n}^{s}( \theta,\varphi)\sin\theta \,d\varphi \,d\theta \end{aligned}$$ in $G_{m}$, i.e., $u \equiv0$ in $\Omega_{m }$. For u we know that the identity $$\begin{aligned} \int_{\Omega_{m}} \bigl\{ t^{m} u_{t}v_{t}-u_{x_{1}}v_{x_{1}}-u_{x_{2}}v_{x_{2}}-u_{x_{3}}v_{x_{3}} \bigr\} \,dx_{1}\,dx_{2}\,dx_{3}\,dt =0 \end{aligned}$$ holds for all test functions $v(x,t)=w(r,t)Y_{n}^{s}(x)$ described in Remark 2.1. Therefore from (4.1) we derive $$ \int_{G_{m}} \biggl\{ u_{n,r}^{s}w_{r}-t^{m} u_{n,t}^{s}w_{t}+\frac{n(n+1)}{r^{2}}u_{n}^{s}w \biggr\} r^{2} \,dr \,dt =0 $$ for all $w(r,t)\in V_{m}^{(1)}$ (see Definition 2.1), $n \in\mathbf{N}\cup\{0\}$, $s=1,2,\ldots, 2n+1$. Since $u(x,t)$ satisfies conditions (1), (2) and (3) in Definition 1.1, the functions $u_{n}^{s}(r,t)$ satisfy conditions (1), (2) and (3) in Definition 2.1, and therefore they are generalized solutions of problem $\mathit {PK}_{1}$. Using (2.3) we see that the functions $V(\xi,\eta):=r(\xi,\eta)w(r(\xi,\eta),t(\xi,\eta)) \in V^{(2)}$. Now from (4.2) we obtain that for the functions $U_{n}^{s}(\xi,\eta):=r(\xi,\eta)u_{n}^{s}(r(\xi,\eta),t(\xi,\eta))$ the identity $$ \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{n,\xi}^{s}V_{\eta}+U_{n,\eta }^{s}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}U_{n}^{s}V \biggr\} \,d\xi \,d\eta=0 $$ holds for all $V(r,t)\in V^{(2)}$ (see Definition 2.2), $n \in\mathbf{N}\cup\{0\}$, $s=1,2,\ldots, 2n+1$. The functions $U_{n}^{s}(\xi,\eta)$ satisfy conditions (1), (2) and (3) in Definition 2.2 and, consequently, $U_{n}^{s}(\xi,\eta)$ are generalized solutions of the 2-D homogeneous problem $\mathit {PK}_{2}$. Theorem 3.1 gives $U_{n}^{s}(\xi,\eta) \equiv0$ in D. Therefore $u_{n}^{s}(r,t)\equiv0$ in $G_{m}$ and thus $u=u_{1}-u_{2} \equiv0$ in $\Omega_{m}$. □ From Theorem 1.1 it follows that there exists at most one generalized solution of problem PK in $\Omega_{m}$. Since $f(x,t)$ has the form (1.5), we look for a generalized solution of the form (1.6), i.e., $$u(x,t)=\sum_{n=0}^{l}\sum _{s=1}^{2n+1}u_{n}^{s} \bigl( \vert x\vert ,t \bigr)Y_{n}^{s}(x). $$ To find such a solution means to find functions $u_{n}^{s}(r,t)$ that satisfy the identities $$\begin{aligned} \int_{G_{m,\varepsilon}} \biggl[u_{n,r }^{s}v_{r }-t^{m} u_{n,t}^{s}v_{t} +\frac{n(n+1)}{r^{2}}u_{n}^{s}v+f_{n}^{s}v \biggr]r^{2} \,dr \,dt =0 \end{aligned}$$ for all $v \in V_{m}^{(1)}$ and satisfy the corresponding conditions (1), (2) and (3) in Definition 2.1. In view of (2.3) to find such functions means to find functions $$U_{n}^{s}(\xi,\eta)=r(\xi,\eta)u_{n}^{s} \bigl(r(\xi,\eta),t(\xi,\eta) \bigr), $$ such that for $F_{n}^{s}(\xi,\eta):=\frac{1}{4}r(\xi,\eta)f_{n}^{s}(r(\xi,\eta),t(\xi,\eta))$ the identity $$ \int_{D}(\eta-\xi)^{2 \beta} \biggl\{ U_{n,\xi}^{s}V_{\eta}+U_{n,\eta }^{s}V_{\xi} +\frac{2n(n+1)}{(2-\xi-\eta)^{2}}U_{n}^{s}V+2F_{n}^{s}V \biggr\} \,d\xi \,d\eta=0 $$ holds for all $V(\xi,\eta)=r(\xi,\eta)v(r(\xi,\eta),t(\xi,\eta)) \in V^{(2)}$ and satisfies the corresponding conditions (1), (2) and (3) in Definition 2.2. Theorem 3.5 gives the existence of such functions $U^{s}_{n}(\xi,\eta)$ which are generalized solutions of problem $\mathit {PK}_{2}$ in D. In that way we found functions $u_{n}^{r}(r,t)=r^{-1}U_{n}^{s}(\xi(r,t),\eta(r,t))$ which are generalized solutions of problem $\mathit {PK}_{1}$ in $G_{m}$. Therefore the function $u(x ,t)$, given by (1.6), is a generalized solution of problem PK in $\Omega_{m}$. □ Theorem 1.1 and Theorem 1.2 claim the existence and uniqueness of generalized solutions $u(x ,t)$ of problem PK in $\Omega_{m}$, which has the form (1.6). Using (2.3) for functions $U_{n}^{s}(\xi,\eta)=r(\xi,\eta)u_{n}^{s}(r(\xi,\eta),t(\xi,\eta))$ and $F_{n}^{s}(\xi,\eta)=\frac{1}{4}r(\xi,\eta)f_{n}^{s}(r (\xi,\eta),t(\xi ,\eta))$, we obtain the 2-D problem $\mathit {PK}_{2}$. According to Theorem 3.5, estimates (3.12) hold, and in view of (3.6) and (3.7) we see that the estimate for $\vert U_{n}^{s}(\xi,\eta)\vert $ holds with ${ ( \max_{\bar{G}_{m}} \vert f_{n}^{s}\vert )}$ instead of $M_{F}$: $$\bigl\vert U_{n}^{s}(\xi,\eta) \bigr\vert \leq K \Bigl( \max_{\bar{G}_{m}} \bigl\vert f_{n}^{s} \bigr\vert \Bigr) (2-\xi-\eta)^{-n} $$ with a constant $K>0$ independent of $f_{n}^{s}$. 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AIP, New York (2014) Hristov, T, Popivanov, N, Schneider, M: Protter problem for 3-D Keldysh type equations involving lower order terms. In: Applications of Mathematics in Engineering and Economics: AMEE'15, Sozopol, Bulgaria, June 2015. AIP Conf. Proc., vol. 1690, pp. 04002001-04002012. AIP, New York (2015) Hristov, T, Nikolov, A, Popivanov, N, Schneider, M: Generalized solutions of Protter problem for $(3+1)$-D Keldysh type equations. In: Pasheva, V, Venkov, G, Popivanov, N (eds.) Applications of Mathematics in Engineering and Economics: AMEE'16, Sozopol, Bulgaria, June 2016. AIP Conf. Proc., vol. 1789, pp. 04000701-04000713. AIP, New York (2016) Jones, M: Spherical Harmonics and Tensors for Classical Field Theory. Research Studies Press, Letchworth (1986) Gellerstedt, S: Sur une équation linéaire aux dérivées partielles de type mixte. Ark. Mat. Astron. Fys. A 25(29), 1-23 (1937) Smirnov, M: Degenerating Hyperbolic Equations. Vysheishaia shkola Publ. House, Minsk (1977) [in Russian] Moiseev, E: Approximation of the classical solution of a Darboux problem by smooth solutions. Differ. Equ. 20, 59-74 (1984) Bateman, H, Erdelyi, A: Higher Transcendental Functions, vol. 1. McGraw-Hill, New York (1953) Smirnov, M: Mixed Type Equations. Vishaya shkola Publ. House, Moskow (1985) [in Russian] Volkodavov, V, Zaharov, V: Tables of Riemann and Riemann-Hadamard Functions for Some Differential Equations in n-Dimensional Euclidean Spaces. Samara State Teacher's Training University, Samara (1994) [in Russian] Meredov, M: The unique solvability of Darboux's problem for a degenerating system. Differ. Equ. 10, 63-70 (1975) The authors thank the anonymous referees for making several helpful suggestions. The research of Tsvetan Hristov, Aleksey Nikolov and Nedyu Popivanov was partially supported by the Sofia University Grant 152/2016. Faculty of Mathematics and Informatics, University of Sofia, Sofia, 1164, Bulgaria Nedyu Popivanov & Tsvetan Hristov Faculty of Applied Mathematics and Informatics, Technical University of Sofia, Sofia, 1000, Bulgaria Aleksey Nikolov Faculty of Mathematics, Karlsruhe Institute of Technology, Karlsruhe, 76131, Germany Manfred Schneider Nedyu Popivanov Tsvetan Hristov Correspondence to Nedyu Popivanov. All authors contributed equally to the writing of this paper. The authors read and approved the final manuscript. Appendix 1: The Riemann-Hadamard function $\Phi(\xi,\eta,\xi_{0},\eta_{0})$ Firstly, to aid the reader, we briefly recall some known properties of the hypergeometric function of Gauss $F(a,b,c;\zeta)$ that we will use. If $c \ne0,-1,-2,\ldots$ , then $$ F(a,b,c;\zeta) := \sum_{i=0}^{\infty}\frac{(a)_{i} (b)_{i} }{i! (c)_{i} } \zeta^{i}, $$ (A.1) with $(a)_{i} =\Gamma(a+i)/\Gamma(a)$, where Γ is the Euler gamma function of Euler. For $i\in\mathbf{N}$, one has $(a)_{i}=a(a+1)\cdots (a+i-1)$, $(a)_{0}=1$. The series (A.1) converges absolutely for $\zeta\in \mathbf{C}$ with $\vert \zeta \vert < 1$ and also for $\vert \zeta \vert = 1$ if $\operatorname{Re}(c-a-b) > 0$. If $-1 < \operatorname{Re}(c-a-b) <0$, then the series converges conditionally for $\vert \zeta \vert =1$ with $\zeta\neq1$. We mention the following properties of the hypergeometric function (see [31, 49, 50]): $$ F(a,b,c;\zeta)=\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)} \int_{0}^{1} t^{a-1} (1-t)^{c-a-1} (1-\zeta t)^{-b} \,dt $$ for $\zeta\in\mathbf{C}$, $0<\operatorname{Re}(a) < \operatorname{Re}(c)$, $\vert \arg(1-\zeta )\vert <\pi$. In the case $c-a-b>0$: $$ \bigl\vert F(a,b,c;\zeta) \bigr\vert \leq \mathit{const.}, \quad\quad F(a,b,c;1) = \frac{\Gamma(c)\Gamma(c-a-b)}{\Gamma(c-a)\Gamma(c-b) }; $$ resp. $c-a-b<0$: $$ F(a,b,c;\zeta) = (1-\zeta)^{c-a-b} F(c-a,c-b,c;\zeta) $$ $$ \bigl\vert F(a,b,c;\zeta) \bigr\vert \leq \mathit{const.}(1-\zeta )^{c-a-b}; $$ resp. $c-a-b=0$: For each $\alpha>0$, there exists a constant $c(\alpha)>0$ such that $$\begin{aligned}& \bigl\vert F(a,b,c;\zeta \bigr\vert \leq c(\alpha) (1-\zeta )^{-\alpha }, \end{aligned}$$ $$\begin{aligned}& \frac{d}{d\zeta} F(a,b,c;\zeta) = \frac{ab}{c} F(a+1,b+1,c+1;\zeta). \end{aligned}$$ The hypergeometric function of two variables is defined by $$ F_{1}(a,b_{1},b_{2},c;x,y):=\sum _{i=0}^{\infty}\sum _{j=0}^{\infty} \frac{(a)_{i+j}(b_{1})_{j}(b_{2})_{i}}{(c)_{i+j}i!j!}x^{j}y^{i}. $$ The series converges absolutely for $x,y \in\mathbf{C}$ with $\vert x\vert <1$, $\vert y\vert <1$ (for more properties of $F_{1}$, see [49], pp.224-228). Now, in the case $n \in\mathbf{N}\cup\{0\}$, we construct the following Riemann-Hadamard function of the form (3.1) associated to problem $\mathit {PK}_{2}$: For $(\xi_{0},\eta_{0}) \in D$ $$ \begin{gathered} {\Phi^{+} = \biggl(\frac{\eta-\xi}{\eta_{0}-\xi_{0} } \biggr)^{\beta}F_{3}(\beta,n+1,1-\beta,-n,1;X,Y)}, \quad \eta> \xi_{0}, \\ {\Phi^{-} = k \biggl(\frac{\eta-\xi}{\eta_{0}-\xi_{0} } \biggr)^{\beta}X^{-\beta} H_{2} \biggl(\beta,\beta,-n,n+1,2\beta; \frac{1}{X},-Y \biggr)}, \quad \eta< \xi_{0}, \end{gathered} $$ $$\begin{aligned}& k= \frac{\Gamma(\beta)}{\Gamma(1-\beta)\Gamma(2\beta)}, \\& X = X(\xi,\eta,\xi_{0},\eta_{0}):= \frac{(\xi_{0}-\xi) (\eta_{0}-\eta) }{(\eta- \xi) (\eta_{0} - \xi_{0}) }, \\& Y=Y(\xi,\eta,\xi_{0},\eta_{0}):=-\frac{(\xi_{0}-\xi)(\eta_{0}-\eta)}{(2-\xi-\eta )(2-\xi_{0}-\eta_{0})}. \end{aligned}$$ Here $F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y)$ is the Appell series $$ F_{3}(a_{1},a_{2},b_{1},b_{2},c;x,y):= \sum_{i=0}^{\infty}\sum _{j=0}^{\infty} \frac{(a_{1})_{j}(a_{2})_{i}(b_{1})_{j}(b_{2})_{i}}{(c)_{i+j}i!j!}x^{j}y^{i} $$ (A.10) which converges absolutely for $x,y \in\mathbf{C}$ with $\vert x\vert <1$, $\vert y\vert <1$ (see [49], pp.224-228) and $H_{2}(a_{1},a_{2},b_{1},b_{2},c;x,y)$ is the Horn series $$ H_{2}(a_{1},a_{2},b_{1},b_{2},c;x,y):= \sum_{i=0}^{\infty}\sum _{j=0}^{\infty} \frac{(a_{1})_{j-i}(a_{2})_{j}(b_{1})_{i}(b_{2})_{i}}{(c)_{j}i!j!}x^{j}y^{i} $$ which converges absolutely for $x,y \in\mathbf{C}$ with $\vert x\vert <1$, $\vert y\vert (1+\vert x\vert )<1$ (see [49], pp.224-228). We mention that for $(\xi_{0},\eta_{0})\in D$ we have $\vert X\vert <1$ in Π̄ and $1/\vert X\vert <1$ in T̄, while $\vert Y\vert <1$ in Π̄ but $\vert Y\vert $ could be greater than 1 in T. However, the function Φ is well defined because $n \in\mathbf{N}\cup\{0\}$, since $b_{1}=-n$, and we have a finite sum with respect to i in the function $H_{2}$ (see (A.11)), which appears in (A.9). We will fix all these properties a little bit later. Let, for $(\xi_{0},\eta_{0}) \in D$ and $(\xi,\eta) \in\bar{T} \cup\bar{\Pi} \setminus\{\eta=\xi_{0}\}$, us introduce the functions $$ H(\xi,\eta; \xi_{0} ,\eta_{0} ) = \textstyle\begin{cases} H^{+}(\xi,\eta; \xi_{0},\eta_{0} ), & \eta> \xi_{0}, \\ H^{-}(\xi,\eta; \xi_{0},\eta_{0} ), & \eta< \xi_{0}, \end{cases} $$ $$ \begin{gathered} H^{+}(\xi,\eta;\xi_{0}, \eta_{0}) = \biggl(\frac{\eta-\xi}{\eta_{0}-\xi_{0} } \biggr)^{\beta}F( \beta,1-\beta,1;X), \\ H^{-}(\xi,\eta;\xi_{0},\eta_{0}) = k \biggl( \frac{\eta-\xi}{\eta_{0}-\xi_{0} } \biggr)^{\beta}X^{-\beta} F \biggl(\beta,\beta,2 \beta;\frac{1}{X} \biggr) \end{gathered} $$ $$ G(\xi,\eta; \xi_{0} ,\eta_{0} ) = \textstyle\begin{cases} G^{+}(\xi,\eta; \xi_{0} ,\eta_{0} ) , &\eta> \xi_{0}, \\ G^{-} (\xi,\eta; \xi_{0} ,\eta_{0} ), & \eta< \xi_{0}, \end{cases} $$ $$\begin{aligned}& {G^{+}(\xi,\eta;\xi_{0}, \eta_{0}):= \biggl(\frac{\eta-\xi }{\eta_{0}-\xi_{0} } \biggr)^{\beta}\sum _{i=1}^{n} c_{i} Y^{i} F(\beta, 1-\beta,i+1; X)}, \end{aligned}$$ $$\begin{aligned}& {G^{-}(\xi,\eta;\xi_{0},\eta_{0}):=k \biggl( \frac{\eta-\xi }{\eta_{0}-\xi_{0} } \biggr)^{\beta}X^{-\beta}\sum _{i=1}^{n} d_{i} Y^{i} F \biggl(\beta-i, \beta,2\beta; \frac{1}{X} \biggr)}, \\& c_{i}:=\frac{(n+1)_{i}(-n)_{i}}{i! i!}, \quad\quad d_{i}:=\frac{(n+1)_{i}(-n)_{i}}{(1-\beta)_{i} i!}. \end{aligned}$$ Now we prove the following important lemma. Lemma A.1 The function $\Phi(\xi,\eta;\xi_{0},\eta_{0})$ has the following decomposition: $$ \Phi(\xi,\eta;\xi_{0},\eta_{0})=H(\xi,\eta; \xi_{0},\eta_{0})+G(\xi,\eta;\xi_{0}, \eta_{0}). $$ (i) In view of (A.10) we have $$F_{3}(\beta,n+1,1-\beta,-n,1;X,Y)=\sum_{i=0}^{n} \sum_{j=0}^{\infty}\frac{(\beta)_{j}(1-\beta)_{j}(n+1)_{i}(-n)_{i}}{(1)_{i+j} i! j!}X^{j}Y^{i}. $$ Since $(1)_{i+j}=(i+j)!=i! (i+1)_{j}$ for $i,j \in\mathbf{N} \cup \{0\}$, we obtain from (A.1) and (A.9) $$\begin{aligned} \Phi^{+}(\xi,\eta;\xi_{0}, \eta_{0}) &= \biggl(\frac{\eta-\xi }{\eta_{0}-\xi_{0} } \biggr)^{\beta}\Biggl\{ \sum_{j=0}^{\infty}\frac{(\beta)_{j}(1-\beta)_{j}}{(1)_{j} j!}X^{j} + \sum_{i=1}^{n} c_{i}Y^{i} \sum_{j=0}^{\infty}\frac{(\beta)_{j}(1-\beta)_{j}}{(i+1)_{j} j!}X^{j} \Biggr\} \\ &= \biggl(\frac{\eta-\xi}{\eta_{0}-\xi_{0} } \biggr)^{\beta}\Biggl\{ F(\beta,1-\beta,1;X) + \sum_{i=1}^{n} c_{i}Y^{i} F(\beta,1-\beta,i+1;X) \Biggr\} \\ &=H^{+}(\xi,\eta;\xi_{0},\eta_{0})+G^{+}( \xi,\eta;\xi_{0},\eta_{0}). \end{aligned} $$ (ii) In view of (A.11) we have $$H_{2} \biggl(\beta,\beta,-n,n+1,2 \beta;\frac{1}{X},Y \biggr):= \sum_{i=0}^{n}\sum _{j=0}^{\infty} \frac{(\beta)_{j-i}(\beta)_{j}(-n)_{i}(1-n)_{i}}{(2 \beta)_{j}i!j!}X^{-j}(-Y)^{i}. $$ We mention that for $0<\beta<1$ and $i,j \in\mathbf{N} \cup\{0\}$ $$(\beta)_{j-i}=\frac{\Gamma(\beta+j-i)}{\Gamma(\beta)}=\frac{\Gamma(\beta -i)}{\Gamma(\beta)}( \beta-i)_{j}, $$ and using the relation $\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)}$ we calculate $$\begin{aligned} (\beta)_{j-i}(1-\beta)_{i}&=( \beta-i)_{j}\frac{\Gamma(\beta-i)\Gamma(1-\beta +i)}{\Gamma(\beta)\Gamma(1-\beta)} \\ &=(\beta-i)_{j}\frac{\sin(\beta\pi)}{\sin((\beta-i)\pi)} \\ &=(-1)^{i}(\beta-i)_{j}. \end{aligned} $$ Now, from (A.1) and (A.9) we obtain $$\begin{aligned} \Phi^{-}(\xi,\eta;\xi_{0}, \eta_{0}) &= k \biggl(\frac{\eta-\xi }{\eta_{0}-\xi_{0} } \biggr)^{\beta}X^{-\beta} \Biggl\{ \sum_{j=0}^{\infty} \frac{(\beta)_{j}(\beta)_{j}}{(2 \beta)_{j} j!}\frac{1}{X^{j}} + \sum_{i=1}^{n} d_{i}Y^{i} \sum_{j=0}^{\infty} \frac{(\beta-i)_{j}(\beta)_{j}}{(2 \beta)_{j} j!}\frac {1}{X^{j}} \Biggr\} \\ &=k \biggl(\frac{\eta-\xi}{\eta_{0}-\xi_{0} } \biggr)^{\beta}X^{-\beta} \Biggl\{ F \biggl(\beta,\beta,2\beta;\frac{1}{X} \biggr) + \sum _{i=1}^{n} d_{i}Y^{i} F \biggl( \beta-i,\beta,2 \beta;\frac{1}{X} \biggr) \Biggr\} \\ &=H^{-}(\xi,\eta;\xi_{0},\eta_{0})+G^{-}( \xi,\eta;\xi_{0},\eta_{0}). \end{aligned} $$ We mention here that function (A.9) is closely connected to the Riemann-Hadamard function announced in [51], p.25, example 7, which is associated to a Cauchy-Goursat problem for an equation connected with (2.5) with some appropriate substitutions. Actually, the function $H(\xi,\eta; \xi_{0} ,\eta_{0} )$ is the Riemann-Hadamard function associated to problem $\mathit {PK}_{2}$ in the case $n=0$ (see Gellerstedt [46] and Smirnov [47]). It is well known that the function $H(\xi,\eta; \xi_{0} ,\eta_{0} )$ has the properties $\mbox{(i)} \div\mbox{(vi)}$ listed in Section 3. It is not difficult to check that in the case $n \geq0$ function $\Phi(\xi,\eta; \xi_{0} ,\eta_{0} )$ has the same properties. Using the systems of differential equations that $F_{3}$ and $H_{2}$ satisfy (see [49], pp.233-234), with a straightforward calculation we check that the function $\Phi(\xi,\eta; \xi_{0} ,\eta_{0} )$ satisfies equations (3.2) and (3.3). Further, since $X(\xi_{0},\eta,\xi_{0},\eta_{0})=X(\xi,\eta_{0},\xi_{0},\eta_{0})=0$, $Y(\xi_{0},\eta,\xi_{0},\eta_{0})=Y(\xi,\eta_{0},\xi_{0},\eta_{0})=0$, we see that the function Φ has the properties (ii), (iii) and (iv). We also have $X(\xi,\xi,\xi_{0},\eta_{0})=0$, and therefore the function $G(\xi,\eta; \xi_{0} ,\eta_{0}) $ vanishes on the line $\{\eta=\xi\}$ of power 2β. Therefore the function Φ has the properties (vi). Let us calculate the jump of the function Φ on the line $\{\eta=\xi_{0}\}$. We will show that the function G has no jump on the line $\{\eta=\xi_{0}\}$. Using (A.3) and the relation $\Gamma(i)=(i-1)!$ for $i \in\mathbf{N}$, we calculate $$c_{i} F(\beta, 1-\beta,i+1; 1)=k d_{i} F(\beta-i, \beta,2 \beta; 1)=\frac{(n+1)_{i}(-n)_{i}}{i \Gamma(1-\beta+i)\Gamma(\beta+i)}. $$ In view of (A.14) and (A.15) we have $$\begin{aligned} G^{+}(\xi,\xi_{0}; \xi_{0},\eta_{0})&=G^{-}(\xi,\xi_{0}; \xi_{0},\eta_{0}) \\ &= \biggl(\frac{\xi_{0}-\xi}{\eta_{0}-\xi_{0} } \biggr)^{\beta}\sum _{i=1}^{n} \frac{(n+1)_{i}(-n)_{i}}{i \Gamma(1-\beta+i) \Gamma(\beta+i)} Y^{i}(\xi, \xi_{0},\xi_{0},\eta_{0}). \end{aligned} $$ Therefore the jump $[[G]]=0$, and in view of (A.16) we have $[[\Phi]]=[[H]]$. Consequently, the function Φ has the property (v) since $[[H]]=\cos(\pi\beta) (\frac{\xi_{0}-\xi}{\eta_{0}-\xi_{0}} )^{\beta}$ (see Gellerstedt [46]). 1.1 The function $H(\xi,\eta,\xi_{0},\eta_{0})$ Using the properties of a hypergeometric function mentioned above and the relations $$\begin{aligned}& 1-X = \frac{(\eta_{0}-\xi) (\eta-\xi_{0}) }{(\eta- \xi) (\eta_{0} - \xi_{0}) }, \quad\quad 1- \frac{1}{X} = \frac{(\eta_{0}-\xi) (\xi_{0}-\eta) }{(\xi_{0}- \xi) (\eta_{0} - \eta) }, \\& X_{\xi_{0}}=\frac{(\eta_{0}-\xi) (\eta_{0}-\eta) }{(\eta -\xi) (\eta_{0} - \xi_{0})^{2} }, \quad\quad X_{\eta_{0}}= \frac{(\eta-\xi_{0}) (\xi_{0}-\xi) }{(\eta-\xi) (\eta_{0} - \xi_{0})^{2} }, \end{aligned}$$ $$\begin{aligned}& \biggl(\frac{1}{X} \biggr)_{\xi_{0}}= \frac{(\xi-\eta_{0}) (\eta-\xi) }{(\eta_{0}-\eta) (\xi_{0} - \xi)^{2} }, \quad\quad \biggl(\frac{1}{X} \biggr)_{\eta_{0}}= \frac{(\xi_{0}-\eta) (\eta-\xi) }{(\xi_{0} - \xi)(\eta_{0}-\eta)^{2} } , \end{aligned}$$ we prove the following lemma. Let $0 < \beta<1$ and $0 < \xi_{0} < \eta _{0} <1$. Then there exists a constant $C_{H}>0$ such that $$\begin{aligned}& \bigl\vert H^{+}(\xi,\eta;\xi_{0}, \eta_{0}) \bigr\vert \leq C_{H}(\eta- \xi_{0})^{-\beta},\quad(\xi,\eta) \in\Pi, \end{aligned}$$ $$\begin{aligned}& \bigl\vert H^{-}(\xi,\eta;\xi_{0},\eta_{0}) \bigr\vert \leq C_{H}(\xi_{0}-\eta)^{-\beta}, \quad( \xi,\eta) \in T, \end{aligned}$$ $$\begin{aligned}& \bigl\vert H^{+}_{\eta_{0}}(\xi,\eta; \xi_{0},\eta_{0}) \bigr\vert \leq C_{H} \frac{(\eta-\xi_{0})^{-\beta}}{\eta_{0}-\xi_{0}}, \quad(\xi,\eta) \in\Pi, \end{aligned}$$ $$\begin{aligned}& \bigl\vert H^{-}_{\eta_{0}}(\xi,\eta;\xi_{0}, \eta_{0}) \bigr\vert \leq C_{H}\frac{(\xi_{0}-\eta)^{-\beta}}{\eta_{0}-\eta }, \quad (\xi,\eta) \in T. \end{aligned}$$ (i) Using (A.6) we find that for each $\alpha>0$ there exists a constant $c(\alpha)>0$ such that $$ \begin{gathered} \bigl\vert H^{+}(\xi, \eta; \xi_{0},\eta_{0}) \bigr\vert \leq c(\alpha) \biggl( \frac{\eta-\xi }{\eta_{0}-\xi_{0} } \biggr)^{\beta}(1-X)^{-\alpha} \\ \hphantom{\bigl\vert H^{+}(\xi, \eta; \xi_{0},\eta_{0}) \bigr\vert }=c(\alpha)\frac{(\eta-\xi)^{\alpha +\beta}(\eta_{0}-\xi_{0})^{\alpha-\beta}}{(\eta-\xi_{0})^{\alpha}(\eta_{0}-\xi )^{\alpha}} , \\ \bigl\vert H^{-}(\xi,\eta;\xi_{0},\eta_{0}) \bigr\vert \leq c(\alpha) \frac{ (\eta-\xi) ^{2\beta}}{(\xi_{0}-\xi)^{\beta}(\eta_{0}-\eta)^{\beta} } \biggl(1-\frac{1}{X} \biggr)^{-\alpha} \\ \hphantom{\bigl\vert H^{-}(\xi,\eta;\xi_{0},\eta_{0}) \bigr\vert } =c(\alpha)\frac{(\eta-\xi)^{2\beta}(\xi_{0}-\xi)^{\alpha-\beta}(\eta _{0}-\eta)^{\alpha-\beta}}{(\eta_{0}-\xi)^{\alpha}(\xi_{0}-\eta)^{\alpha}}. \end{gathered} $$ From here, choosing $\alpha=\beta$, we obtain estimates (A.19), (A.20). (ii) In view of (A.17), (A.18) for the derivatives with respect to $\eta_{0}$, using (A.4) and (A.7), we obtain $$\begin{aligned} \bigl\vert H^{+}_{\eta_{0}} \bigr\vert &= \biggl\vert - \frac{\beta}{\eta_{0} -\xi_{0}} H^{+} +\beta(1-\beta) \biggl(\frac{\eta-\xi }{\eta_{0}-\xi_{0} } \biggr)^{\beta}X_{\eta_{0}}F(1+\beta,2-\beta,2;X) \biggr\vert \\ &=\frac{\beta(\eta- \xi)^{\beta}}{(\eta_{0} - \xi_{0})^{1+\beta}} \biggl\vert -F(\beta,1-\beta,1;X)+(1-\beta) \frac{\xi_{0}-\xi}{\eta_{0}-\xi} F(1-\beta,\beta,2;X) \biggr\vert \\ &\leq c(\alpha) \frac{(\eta-\xi)^{\beta}}{(\eta_{0}-\xi_{0})^{1+\beta} }(1-X)^{-\alpha}\leq c(\alpha) \frac{(\eta_{0}-\xi_{0})^{\alpha-\beta-1}}{(\eta-\xi_{0})^{\alpha}}, \end{aligned} $$ $$\begin{aligned} \bigl\vert H^{-}_{\eta_{0}} \bigr\vert &= \biggl\vert \frac{\beta H^{-}}{\eta-\eta_{0}} + \frac{k \beta(\eta-\xi) ^{2\beta }}{2(\xi_{0}-\xi)^{\beta}(\eta_{0}-\eta)^{\beta}} \biggl(\frac{1}{X} \biggr)_{\eta_{0}}F \biggl(1+\beta,1+\beta,1+2\beta;\frac{1}{X} \biggr) \biggr\vert \\ &= \frac{k\beta(\eta-\xi) ^{2\beta}}{(\xi_{0}-\xi)^{\beta}(\eta_{0}-\eta)^{1+\beta}} \biggl\vert -F \biggl(\beta,\beta,2\beta; \frac{1}{X} \biggr)+ \frac{1}{2}\frac{\eta-\xi}{\eta_{0}-\xi}F \biggl(\beta, \beta,1+2\beta;\frac{1}{X} \biggr) \biggr\vert \\ &\leq c(\alpha) \frac{ (\eta-\xi) ^{2\beta}}{(\xi_{0}-\xi)^{\beta}(\eta_{0}-\eta)^{1+\beta} } \biggl(1-\frac{1}{X} \biggr)^{-\alpha}\leq c(\alpha)\frac{(\eta_{0}-\eta)^{\alpha-\beta -1}}{(\xi_{0}-\eta)^{\alpha}}. \end{aligned} $$ Now we choose $\alpha=\beta$ to obtain the desired estimates (A.21), (A.22). □ Remark A.1 In the same manner, for the derivatives with respect to $\xi_{0}$, we obtain $$ H^{+}_{\xi_{0}}=\beta\frac{(\eta- \xi)^{\beta}}{(\eta _{0} - \xi_{0})^{1+\beta}} \biggl[F(\beta,1-\beta,1;X)+(1-\beta)\frac{\eta_{0}-\eta}{\eta-\xi_{0}} F(1-\beta,\beta,2;X) \biggr] $$ $$ H^{-}_{\xi_{0}}=- \frac{k\beta(\eta-\xi) ^{2\beta}}{(\xi_{0}-\xi)^{1+\beta} (\eta_{0}-\eta)^{\beta}} \biggl[F \biggl(\beta,\beta,2\beta;\frac{1}{X} \biggr)+ \frac{1}{2} \frac{\eta-\xi}{\xi_{0}-\eta}F \biggl(\beta,\beta,1+2\beta;\frac{1}{X} \biggr) \biggr]. $$ In the case $0 <\beta< 1/2 $, Smirnov [47] and Meredov [52] claim $$ \bigl\vert H^{+} _{\xi_{0}}(\xi,\eta; \xi_{0},\eta _{0}) \bigr\vert \leq \frac{(\eta-\xi)(\eta_{0}-\xi_{0})^{-2\beta}}{(\eta_{0}-\xi) ^{1-\beta} (\eta-\xi_{0}) ^{1-\beta}}, $$ i.e., $H^{+} _{\xi_{0}}(\xi,\eta;\xi_{0},\eta_{0})$ has integrable singularity on $\{\eta=\xi_{0}\}$. As we see from (A.24) and (A.25), the derivative with respect to $\xi_{0}$ of function H has not integrable singularity on $\{\eta=\xi_{0}\}$. 1.2 The function $G(\xi,\eta,\xi_{0},\eta_{0})$ In this section we prove some properties of the function $G(\xi,\eta;\xi_{0},\eta_{0})$ defined by (A.13). Let $0 < \beta<1$ and $0 < \xi_{0} < \eta _{0} <1$. Then there exists a constant $C_{G}>0$ such that $$\begin{aligned}& \bigl\vert G^{+}(\xi,\eta;\xi_{0}, \eta_{0}) \bigr\vert \leq C_{G} (\eta_{0}- \xi_{0})^{-\beta},\quad(\xi,\eta) \in\Pi, \end{aligned}$$ $$\begin{aligned}& \bigl\vert G^{-}(\xi,\eta;\xi_{0},\eta_{0}) \bigr\vert \leq C_{G} (2-\xi_{0}-\eta_{0})^{-n}, \quad (\xi,\eta) \in T , \end{aligned}$$ $$\begin{aligned}& \bigl\vert G^{+}_{\eta_{0}}(\xi,\eta, \xi_{0},\eta_{0}) \bigr\vert \leq C_{G} \frac{(\eta_{0}-\xi_{0})^{-\beta}}{2-\xi_{0}-\eta_{0}},\quad(\xi,\eta)\in\Pi, \end{aligned}$$ $$\begin{aligned}& \bigl\vert G^{-}_{\eta_{0}}(\xi,\eta,\xi_{0}, \eta_{0}) \bigr\vert \leq C_{G}\frac{(\eta_{0}-\eta)^{-\beta}}{(2-\xi _{0}-\eta_{0})^{n+1}}, \quad( \xi,\eta)\in T, \end{aligned}$$ $$\begin{aligned}& \bigl\vert G^{+}_{\xi_{0}}(\xi,\eta; \xi_{0},\eta_{0}) \bigr\vert \leq { C_{G} \frac{(\eta-\xi_{0})^{-\beta}}{(2-\xi_{0}-\eta_{0}) }}, \quad(\xi,\eta)\in \Pi, \end{aligned}$$ $$\begin{aligned}& \bigl\vert G^{-}_{\xi_{0}}(\xi,\eta;\xi_{0}, \eta_{0}) \bigr\vert \leq {C_{G}\frac{(\xi_{0}-\eta)^{-\beta}}{(2-\xi _{0}-\eta_{0})^{n+1} } }, \quad(\xi,\eta)\in T. \end{aligned}$$ First, we mention that $$Y_{\xi_{0}}=-\frac{(2-\xi-\eta_{0})(\eta_{0}-\eta)}{(2-\xi -\eta)(2-\xi_{0}-\eta_{0})^{2}}, \quad\quad Y_{\eta_{0}}=- \frac{(2-\xi_{0}-\eta)(\xi_{0}-\xi)}{(2-\xi-\eta)(2-\xi_{0}-\eta_{0})^{2}}. $$ (i) Let $(\xi,\eta) \in\Pi$. Then we have $$ \begin{gathered} \vert X_{\xi_{0}}\vert \leq \frac{\eta_{0}-\xi}{(\eta-\xi)(\eta_{0}-\xi_{0})}, \\ \vert Y\vert < 1, \quad \quad\frac{\vert Y\vert }{\eta_{0}-\xi _{0}}\leq\frac{\vert Y\vert (\eta_{0}-\xi)}{(\eta-\xi)(\eta_{0}-\xi _{0})}\leq \frac{1}{2-\xi_{0}-\eta_{0}}, \\ \vert Y_{\xi_{0}}\vert \leq\frac{1}{2-\xi_{0}-\eta_{0}}, \vert Y_{\eta_{0}}\vert \leq\frac{2}{2-\xi_{0}-\eta_{0}}. \end{gathered} $$ According to (A.3), $\vert F(\beta, 1-\beta,i+1; X)\vert \leq \mathit{const.}$, $i=1,2,\ldots,n$, in expression (A.14) for $G^{+}$. Therefore estimate (A.26) holds. With use of (A.7) we calculate the derivative with respect to $\xi_{0}$ $$\begin{aligned} G^{+}_{\xi_{0}}&= \biggl( \frac{\eta-\xi}{\eta_{0}-\xi_{0}} \biggr)^{\beta}\Biggl\{ \sum _{i=1}^{n} c_{i} \biggl[ \frac{\beta Y^{i}}{\eta_{0}-\xi_{0}} +i Y^{i-1}Y_{\xi_{0}} \biggr] F(\beta, 1- \beta,i+1; X) \\ &\quad{} + \beta(1-\beta)\sum_{i=1}^{n} \frac{c_{i}}{i+1} Y^{i} X_{\xi_{0}} F(\beta+1, 2-\beta,i+2; X) \Biggr\} \end{aligned} $$ and the derivative with respect to $\eta_{0}$ $$\begin{aligned} G^{+}_{\eta_{0}}&= \biggl( \frac{\eta-\xi}{\eta_{0}-\xi_{0} } \biggr)^{\beta}\Biggl\{ \sum _{i=1}^{n}c_{i} \biggl[ - \frac{\beta Y^{i}}{\eta_{0}-\xi_{0}}+i Y^{i-1}Y_{\eta_{0}} \biggr] F(\beta, 1- \beta,i+1; X) \\ &\quad{} + \beta(1-\beta)\sum_{i=1}^{n} \frac{c_{i}}{i+1} Y^{i} X_{\eta_{0}} F(\beta+1, 2-\beta,i+2; X) \Biggr\} . \end{aligned} $$ According to (A.3) and (A.6), for the hypergeometric functions in the expressions for $G^{+}_{\xi_{0}}$ and $G^{+}_{\eta_{0}}$, we have $$ \begin{gathered} \bigl\vert F(\beta+1, 2-\beta,3; X) \bigr\vert \leq c( \alpha)\frac{(\eta_{0}-\xi_{0})^{\alpha} (\eta-\xi)^{\alpha} }{(\eta-\xi_{0})^{\alpha} (\eta_{0} - \xi)^{\alpha} }\leq c(\alpha) \biggl(\frac{\eta_{0}-\xi_{0}}{\eta-\xi_{0}} \biggr)^{\alpha} , \quad\alpha>0, \\ \bigl\vert F(1+\beta, 2-\beta,i+2; X) \bigr\vert \leq \mathit{const.},\quad i=2,3, \ldots,n. \end{gathered} $$ Therefore in view of (A.17) we have $$\begin{aligned}& \bigl\vert G^{+}_{\xi_{0}} \bigr\vert \leq \frac{C_{1}(\alpha)}{(\eta_{0}-\xi _{0})^{\beta}} \biggl\{ \frac{\vert Y\vert }{\eta_{0}-\xi_{0}}+\frac{1}{2-\xi_{0}-\eta_{0}} + \frac{\vert Y\vert (\eta_{0}-\xi)}{(\eta-\xi)(\eta_{0}-\xi_{0})} \biggl(\frac{\eta_{0}-\xi_{0}}{\eta-\xi_{0}} \biggr)^{\alpha} \biggr\} , \\& \bigl\vert G^{+}_{\eta_{0}} \bigr\vert \leq \frac{C_{2}(\alpha)}{(\eta_{0}-\xi _{0})^{\beta}} \biggl\{ \frac{\vert Y\vert }{\eta_{0}-\xi_{0}}+\frac{2}{2-\xi_{0}-\eta_{0}} + \frac{\vert Y\vert (\xi_{0}-\xi)}{(\eta-\xi)(\eta_{0}-\xi_{0})} \biggl(\frac{\eta-\xi_{0}}{ \eta_{0} - \xi_{0}} \biggr)^{1-\alpha} \biggr\} . \end{aligned}$$ Now, taking $\alpha=\beta\in(0,1)$ and using (A.32), we obtain estimates (A.30) and (A.28). (ii) Let $(\xi,\eta) \in T$. Then we have $$ \begin{gathered} \biggl\vert \biggl( \frac{1}{X} \biggr)_{\xi_{0}} \biggr\vert \leq\frac{\eta_{0}-\xi}{(\xi_{0}-\xi)(\eta _{0}-\eta)}, \\ \vert Y\vert < \frac{1}{2-\xi_{0}-\eta_{0}}, \quad \quad\frac{\vert Y\vert }{\eta_{0}-\eta}\leq \frac{1}{2-\xi_{0}-\eta_{0}}, \\ \frac{\vert Y\vert }{\xi_{0}-\xi}\leq\frac{\vert Y\vert (\eta_{0}-\xi)}{(\xi_{0}-\xi)(\eta_{0}-\eta)}\leq\frac{1}{2-\xi_{0}-\eta_{0}}, \\ \vert Y_{\xi_{0}}\vert \leq\frac{1}{(2-\xi_{0}-\eta_{0})^{2}}, \vert Y_{\eta_{0}}\vert \leq\frac{1}{(2-\xi_{0}-\eta_{0})^{2}}. \end{gathered} $$ According to (A.3) $\vert F(\beta-i, \beta,2\beta ;1/X)\vert \leq \mathit{const.}$, $i=1,2,\ldots,n$, in expression (A.15) for $G^{-}$. Since $$\biggl(\frac{\eta-\xi}{\eta_{0}-\xi_{0}} \biggr)^{\beta}X^{-\beta}= \frac{(\eta-\xi)^{2\beta}}{(\xi_{0}-\xi)^{\beta}(\eta_{0}-\eta)^{\beta}}, $$ we see that estimate (A.27) holds. $$\begin{aligned} G^{-}_{\xi_{0}}&= \frac{k (\eta-\xi)^{2\beta}}{(\xi_{0}-\xi)^{\beta}(\eta_{0}-\eta)^{\beta}} \Biggl\{ \sum_{i=1}^{n}d_{i} \biggl[-\frac{\beta Y^{i}}{\xi_{0}-\xi}+i Y^{i-1} Y_{\xi_{0}} \biggr] F \biggl(\beta-i,\beta,2\beta; \frac{1}{X} \biggr) \\ &\quad{} + \frac{1}{2}\sum_{i=1}^{n} ( \beta-i)d_{i} Y^{i} \biggl(\frac{1}{X} \biggr)_{\xi_{0}} F \biggl(\beta-i+1,\beta+1,2\beta+1; \frac{1}{X} \biggr) \Biggr\} \end{aligned} $$ $$\begin{aligned} G^{-}_{\eta_{0}}&= \frac{k (\eta-\xi)^{2\beta}}{(\xi_{0}-\xi)^{\beta}(\eta_{0}-\eta)^{\beta}} \Biggl\{ \sum_{i=1}^{n}d_{i} \biggl[-\frac{\beta Y^{i}}{\eta_{0}-\eta}+i Y^{i-1}Y_{\eta_{0}} \biggr] F \biggl( \beta-i,\beta,2\beta; \frac{1}{X} \biggr) \\ &\quad{} + \frac{1}{2}\sum_{i=1}^{n} ( \beta-i)d_{i} Y^{i} \biggl( \frac{1}{X} \biggr)_{\eta_{0}} F \biggl(\beta-i+1,\beta+1,2\beta+1; \frac{1}{X} \biggr) \Biggr\} . \end{aligned} $$ According to (A.3) and (A.6), for the hypergeometric functions in the expressions for $G^{-}_{\xi_{0}}$ and $G^{-}_{\eta_{0}}$, we have $$ \begin{aligned} & \biggl\vert F \biggl(\beta, \beta+1,2\beta+1; \frac{1}{X} \biggr) \biggr\vert \leq c(\alpha)\frac{(\eta_{0}-\eta )^{\alpha} (\xi_{0}-\xi)^{\alpha} }{(\xi_{0}-\eta)^{\alpha} (\eta_{0} - \xi)^{\alpha} }\leq c( \alpha) \biggl(\frac{\eta_{0}-\eta}{\xi_{0}-\eta} \biggr)^{\alpha}, \quad \alpha>0, \\ & \bigl\vert F(\beta-i+1,1+ \beta,1+2\beta;1/X) \bigr\vert \leq \mathit{const.},\quad i=2,3, \ldots,n. \end{aligned} $$ Now using (A.18) we calculate $$\begin{aligned} \bigl\vert G^{-}_{\xi_{0}} \bigr\vert &\leq\frac{C_{3}(\alpha)}{(\eta_{0}-\eta )^{\beta}} \biggl\{ \frac{\vert Y\vert }{\xi_{0}-\xi}+\frac{1}{(2-\xi_{0}-\eta_{0})^{2}} + \frac{\vert Y\vert (\eta_{0}-\xi)}{(\xi_{0}-\xi)(\eta_{0}-\eta)} \biggl(\frac{\eta_{0}-\eta}{\xi_{0}-\eta} \biggr)^{\alpha} \biggr\} \sum_{i=1}^{n}\vert Y\vert ^{i-1} \\ & \leq3 C_{3}(\alpha)\frac{(\eta_{0}-\eta)^{\alpha- \beta}}{(\xi_{0}-\eta )^{\alpha}} \sum _{i=1}^{n} (2-\xi_{0}- \eta_{0})^{-i-1} \end{aligned} $$ and for $0<\alpha<1$ $$\begin{aligned} \bigl\vert G^{-}_{\eta_{0}} \bigr\vert &\leq\frac{C_{4}(\alpha)}{(\eta_{0}-\eta )^{\beta}} \biggl\{ \frac{\vert Y\vert }{\eta_{0}-\eta}+\frac{1}{(2-\xi_{0}-\eta _{0})^{2}} \\ & \quad{} +\frac{\vert Y\vert (\eta-\xi)}{(\xi_{0}-\xi)(\eta_{0}-\eta)} \biggl(\frac{\xi_{0}-\eta}{\eta_{0}-\eta} \biggr)^{1-\alpha} \biggr\} \sum_{i=1}^{n}\vert Y\vert ^{i-1} \\ &\leq \frac{3C_{4}(\alpha)}{(\eta_{0}-\eta)^{\beta}}\sum_{i=1}^{n} (2-\xi _{0}-\eta_{0})^{-i-1}. \end{aligned} $$ Therefore, taking $\alpha=\beta\in(0,1)$, we obtain estimates (A.29) and (A.31). □ Appendix 2: Auxiliary results Lemma B.1 Suppose $0< \beta< 1$ and $0 < \xi_{0} < \eta _{0} <1$. Then $$ I^{1}(\xi_{0},\eta_{0}):= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}} H (\xi,\eta;\xi_{0}, \eta_{0}) \,d\eta \,d\xi\leq k_{1} \xi_{0}. $$ (B.1) From (A.19) and (A.20) we obtain $$\begin{aligned} I^{1} &\leq C_{H} \biggl\{ \int_{0}^{\xi_{0}} \int_{\xi}^{\xi_{0}}(\xi_{0}- \eta)^{-\beta} \,d\eta \,d\xi+ \int_{0}^{\xi_{0}} \int_{\xi_{0}}^{\eta_{0}}(\eta-\xi_{0})^{-\beta} \,d\eta \,d\xi \biggr\} \\ &=:C_{H} \bigl\{ I_{1}^{1}+I_{2}^{1} \bigr\} . \end{aligned} $$ Now we obtain $$ I_{1}^{1}=\frac{\xi_{0}^{2-\beta}}{(1-\beta)(2-\beta)} $$ $$ I_{2}^{1}= \frac{1}{1-\beta} \xi_{0}(\eta_{0}-\xi_{0})^{1-\beta}. $$ Therefore estimate (B.1) holds. □ Suppose $0< \beta< 1$ and $0 < \xi_{0} < \eta_{0} <1$. Then $$ I^{2}(\xi_{0},\eta_{0}):= \int_{0}^{\xi_{0}} \int_{\xi}^{\eta_{0}} \bigl\vert H_{\eta_{0}} (\xi, \eta;\xi_{0},\eta_{0}) \bigr\vert \,d\eta \,d\xi\leq k_{2} \xi_{0}(\eta_{0}-\xi_{0})^{-\beta}. $$ $$\begin{aligned} I^{2}&\leq C_{H} \biggl\{ \int_{0}^{\xi_{0}} \int_{\xi}^{\xi_{0}}\frac{(\xi_{0}-\eta)^{-\beta}}{\eta_{0}-\eta} \,d\eta \,d\xi + \int_{0}^{\xi_{0}} \int_{\xi_{0}}^{\eta_{0}}\frac{(\eta-\xi_{0})^{-\beta}}{\eta_{0}-\xi_{0}} \,d\eta \,d\xi \biggr\} \\ &=:C_{H} \biggl\{ I_{1}^{2}+\frac{I^{1}_{2}}{\eta_{0}-\xi_{0}} \biggr\} . \end{aligned} $$ In $I_{1}^{2}$ we substitute $\eta=\xi+(\xi_{0}-\xi)\sigma$ and, according to (A.2), we get $$\begin{aligned} I_{1}^{2} =& \int_{0}^{\xi_{0}} \biggl[ \int_{0}^{1} (1-\sigma)^{-\beta} \biggl(1- \frac{\xi_{0} -\xi}{\eta_{0}-\xi}\sigma \biggr)^{-1} \,d\sigma \biggr] \frac{(\xi_{0}-\xi)^{1-\beta}}{\eta_{0}-\xi} \,d\xi \\ =&\frac{\Gamma (1)\Gamma(1-\beta)}{\Gamma(2-\beta)} \int_{0}^{\xi_{0}}F(1, 1,2 -\beta;\zeta) \frac{(\xi_{0}-\xi)^{1-\beta}}{\eta_{0}-\xi} \,d\xi, \end{aligned}$$ where $\zeta=\frac{\xi_{0} -\xi}{\eta_{0}-\xi}$. Since $c-a-b=-\beta <0$, according to (A.5), the hypergeometric function $\vert F\vert \leq \mathit{const.} (1-\zeta)^{-\beta}$. Therefore $$ I_{1}^{2} \leq c_{1} ( \eta_{0}-\xi_{0})^{-\beta} \int_{0}^{\xi_{0}} \biggl(\frac{\xi_{0} -\xi}{\eta_{0}-\xi} \biggr)^{1-\beta} \,d\xi\leq c_{1} \xi_{0}( \eta_{0}-\xi_{0})^{-\beta}. $$ Now (B.3) and (B.5) give estimate (B.4). □ $$ I^{3}(\xi_{0},\eta_{0}):= \int_{0}^{\eta_{0}} H (0,\eta;\xi_{0}, \eta_{0}) \,d\eta\leq k_{3} \eta_{0}. $$ Using (A.23) with $\alpha=\beta$, we obtain $$ \begin{aligned} I^{3} &\leq c(\beta)\eta_{0}^{-\beta} \biggl\{ \int_{0}^{\xi_{0}}\frac{\eta^{2\beta}}{(\xi_{0}-\eta)^{\beta}} \,d\eta+ \int_{\xi_{0}}^{\eta_{0}}\frac{\eta^{2\beta}}{(\eta-\xi_{0})^{\beta}} \,d\eta \biggr\} \\ & \leq c(\beta)\eta_{0}^{-\beta} \biggl\{ \xi_{0}^{2\beta} \int_{0}^{\xi_{0}}(\xi_{0}- \eta)^{-\beta} \,d\eta+ \eta_{0}^{2\beta} \int_{\xi_{0}}^{\eta_{0}}(\eta-\xi_{0})^{-\beta} \,d\eta \biggr\} \\ &=\frac{ c(\beta)}{1-\beta}\eta_{0}^{-\beta} \bigl\{ \xi_{0}^{1+\beta}+\eta_{0}^{2\beta}( \eta_{0}-\xi_{0})^{1-\beta} \bigr\} \\ &\leq k_{3} \eta_{0}. \end{aligned} $$ Popivanov, N., Hristov, T., Nikolov, A. et al. On the existence and uniqueness of a generalized solution of the Protter problem for $(3+1)$-D Keldysh-type equations. Bound Value Probl 2017, 26 (2017). https://doi.org/10.1186/s13661-017-0757-1 35D30 weakly hyperbolic equations generalized solutions behavior of solution Recent Advances in PDE and Their Applications	CommonCrawl
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Applied Water Science September 2018 , 8:133 \| Cite as Hydrogeochemical characteristics and groundwater quality assessment for drinking and irrigation purposes in the Mazar-i-Sharif city, North Afghanistan Ali Mahaqi Mohammad Mehdi Moheghi Marzieh Mehiqi Mohmmad Anvar Moheghy First Online: 06 August 2018 The Mazar-i-Sharif city is part of the Balkh province, north of Afghanistan, and its groundwater resources are developed for water supply and irrigation purposes. The main lithological units consist chiefly of evaporite, conglomerate, sandstone, siltstone, and loess. In order to evaluate the quality of groundwater in the study area, 28 samples were collected and analyzed for various ions. Chemical indices like sodium adsorption ratio, the percentage of sodium, residual sodium carbonate and permeability index were calculated. Based on the analytical results, groundwater in the area is generally very hard, brackish, high to very high saline, and alkaline in nature. The abundance of the major ions is as follows: Cl− > HCO3− > SO42− > NO3− and Na+ > Ca2+ > Mg2+ > K+. The dominant hydrochemical facies of groundwater is the Na–Cl type, and alkalis (Na+, K+) and strong acids (Cl−, SO42−) are slightly dominating over alkali earths (Ca2+, Mg2+) and weak acids (HCO3−, CO32−). About 67% of the samples were showing a high concentration of nitrate, exceeding permissible limit of WHO (50 mg/l). The sources of NO3− in the study area are anthropogenic activities (domestic wastewater infiltration from the cesspits) and intense agricultural practices in nearby areas (Balkh district) of the city that utilize nitrogen and sulfate fertilizers. The chemical quality of groundwater is related to the dissolution of minerals, ion exchange, anthropogenic activities, and the residence time of the groundwater in contact with rock materials. The results of calculation saturation index by computer program PHREEQC show that nearly all of the water samples were supersaturated with respect to carbonate minerals (calcite and dolomite) and under-saturated with respect to sulfate minerals (gypsum and anhydrite). Assessment of water samples from various methods indicated that groundwater in the study area is chemically unsuitable for drinking and agricultural uses. Groundwater quality Mazar-i-Sharif city Hydrogeochemical processes Saturation index Deficiency of water is a problem in many countries. This problem has recently become more serious due to the disparity of rainfall caused by global warming. Utilization of well water for drinking will, therefore, become more important in the future (Kato et al. 2016). The access to safe drinking water is essential to human health (Karanth 1987); it is one of the basic human rights and the constituent of an effective health protection policy (Shygonskyi and Shygonska 2016). Afghanistan is rich in water even though it is at least mainly semiarid steppe lands or desert, mainly because at the top of the watershed in this region, the Kohi Baba, Hindu Kush, and Afghan Pamir are covered by snow in winter and provide vital meltwaters. Over 80% of the country's water resources have their origin in mountains > 2000 m, which function as a natural storage of snow and ice that supports perennial flow in all major rivers in summer (Shroder 2014). Underground water is a critical resource for the continued development of the Afghanistan economy and for improvements to the health of the Afghan people, both within the growing urban population and the agricultural areas. The economic lifeblood of Afghanistan is agrarian based, with more than 80% of the Afghan people working in agriculture with arable land comprising only 12% of the total land area (Shroder and Ahmadzai 2016). The Mazar-i-Sharif city (MSC) is located in the northern part of Afghanistan (within Afghan-Tajik Basin) about 425 km north of the country's capital, Kabul. It is surrounded by Uzbekistan in the north, Sari Pul and Jowzjan Provinces in the southwest and Kunduz Province in the east. The study area covers an area of 100 km2. Topographically, the state ranges in elevation from 320 to 420 m asl, with a gentle slope toward the Amu Darya River. MSC's population has increased greatly during the last 10 years and accommodates 650,000 inhabitants (USAID 2009). Some geological and hydrological studies were carried out in MSC city area by Miskin (1968), Russian Scientists (1969–1997) and Radojicic and Arsalan (1979), Afghan Geological Survey (AGS), as well as other international and local NGOs mainly for the development of water supplies and similar purposes. Nevertheless, most of the information that emanated from those studies had been lost over the years due to the prolonged wars. The developing residential in the study area and the increasing population growth has caused serious problems, particularly groundwater pollution and over-abstraction. In the all supply wells within MSC, particularly during the summer season, the water level drops. As a result, residents of MSC facing serious drinking water problems. In this study, for the first time, major ions of groundwater resources of MSC were investigated and interpreted thoroughly. Also, the potential hazard of nitrate (NO3−) and fluoride (F−) content in MSC's wells was analyzed. This paper could help international experts in environmental geochemistry to have a preliminary imagination of hydrochemistry status in North Afghanistan. Also, because of major water quality projects in Afghanistan were done by international NGOs and consulting firms, this will provide a rewarding database for their future projects and researchers in this subject. Geology and hydrogeology setting According to Ruleman et al. 2007, Afghanistan is situated in the southern margin of the Eurasian plate postulated during the Permian–Triassic times. It comprises three thick sedimentary rock regions, namely: Northern Afghanistan Basin (NAB), Southwestern Afghanistan Basin, and Southeastern Katawaz region. The area of MSC is mostly situated within the NAB. The Afghan–Tajik Basin is located in the desert and semidesert areas of southwest Uzbekistan, southeastern Tajikistan, and Northern Afghanistan (Ulmishek 2004). Northern Afghanistan has a pre-Jurassic unconformable basement overlain by Jurassic to Paleocene oil- and gas-bearing terrigenous and carbonate rocks, which in turn are unconformable and overlain by Neocene orogenic continental clastic rocks (Brookfield and Hashmat 2001). The MSC area contains Mesozoic to Cenozoic rocks of approximately seven or eight kilometers thick (Dastyar et al. 1990). The lithological characteristics of the Quaternary sediments and rocks in MSC are shown in Table 1. Geological units and their characteristics in the study area Geological unit Conglomerate and sandstone; alluvium, detrital sediments, gravel and sands Q4sm Mud, silt, clay, more abundant than sand; limestone, gypsum, and salt Holocene—late Pleistocene Conglomerate and sandstone; detrital sediments, gravel and sands Late Pleistocene Q2loe Loess; loess content more than sand and clay Middle Pleistocene The geological units of direct interest to the study area are the Quaternary-aged alluvial, proluvial deposits and salt marsh sediments, non-marine sediments and have formed when runoff gathered in swamp-like areas so because of high temperature and following precipitation formed them, covering Afghan North Plain (Fig. 1). According to Ashworth 2005, the Quaternary deposits in the study area can be divided into two main units, like alluvium deposits (Hilly area) and comprise terrace sands and gravels, with occasional cobbles and proluvial deposits cover most of the area. Open image in new window Geological map of Mazar-I-Sharif city (USGS 2005) The prevailing climate in MSC is semiarid and arid (Fig. 2). There is little rainfall throughout the year. In MSC, the average annual temperature is 21 °C and annual precipitation is 170 mm. With an average of 36.5 °C, July is the hottest month. The lowest average temperature in the year occurs in December, when it is around 4.3 °C (Ashworth 2005). Temperature and precipitation in MSC In MSC, water for irrigation and drinking is provided from two main sources, namely groundwater from shallow and deep aquifers and surface water from the Nahri-Shahi stream. The shallow aquifer is considered to be divided into two zones brackish water and freshwater. In the north of Shrine Ali Tomb toward the north salty desert, the aquifer was found to be mostly salty and unsuitable for human consumption, but toward the Balkh River (southwest of MSC), the aquifer has freshwater and is suitable for human consumption. The Balkh River flows into the area from the southwest. It is the main surface water source of MSC. This River and its tributaries flow through a series of gorges. The main headwater of the River is the BaBah Mountain (Bande Amir), which consists of five lakes. It has an altitude of 3750 m, and the upper side of this River is very narrow. From this area, the trajectory of the Balkh River changes from south to west (Favre and Kamal 2004). The Balkh River is the source of many water streams within the alluvial fan area, like the Nahri-Shahi stream which carries the water supply from the Balkh River to the city. The Balkh River water used for irrigation and water supply purposes. Sampling was carried out by water supply department from 28 hand-pumped wells during July 2014. Water samples were collected in 150 ml dark glass bottles. pH, total dissolved solids (TDS) and electrical conductivity (EC) were measured in the field by multiparameter VWR symphony SP90M5. Calcium (Ca2+) and magnesium (Mg2+) [measured by titration with ethylenediaminetetraacetic acid (EDTA)], sodium (Na+) and potassium (K+) (determined by flame photometry), chloride (Cl−) (measured by Argentometric method) and sulfate (SO42−) (determined by turbidimetric methods), bicarbonate (HCO3−) (determined by titration with HCl), nitrate (NO3−) (determined by Spectrophotometer methods), and F− (determined by ion-sensitive electrodes) were analyzed in the Irrigation Department of MSC, where cations and anions are expressed in mequiv/l. All analyses reported in this study have ion balance < 5%. Sodium absorption ratio (SAR), sodium percent (%Na), saturation index (SI), residual sodium carbonate (RSC) and permeability index (PI) were calculated using equations in Table 2. Saturation indices of the water samples were calculated for different minerals using the PHREEQC code (Parkhurst and Appelo 1999). Equations used in calculating SAR, %Na, SI, RSC, and PI ${\text{SAR}} = \frac{{{\text{Na}}^{ + } }}{{\sqrt {\frac{{{\text{Ca}}^{2 + } + {\text{Mg}}^{2 + } }}{2}} }}$ Richard (1954) $\% {\text{Na}} = \frac{{({\text{Na}} ^{ + } + {\text{K}}^{ + } )}}{{( {\text{Ca}}^{2 + } + {\text{Mg}}^{2 + } + {\text{Na}}^{ + } + {\text{K}}^{ + } )}}100$ Wilcox (1955) ${\text{SI}} = { \log }\frac{{\text{IAP}}}{{\text{KT}}}$ Garrels and Mackenzie (1967) ${\text{RSC}} = \left( {{\text{CO}}_{ 3}^{ 2- } + {\text{HCO}}_{ 3}^{ - } } \right) - ( {\text{Ca}}^{ 2+ } + {\text{Mg}}^{ 2+ } )$ Eaton (1950) ${\text{PI}} = 1 0 0 \left[ {\left( {\left[ {{\text{Na}}^{ + } } \right] + \left[ {{\text{HCO}}_{ 3}^{ - } } \right] 1 / 2} \right)} \right]/\left[ {{\text{Na}}^{ + } } \right] + \left[ {{\text{Ca}}^{ 2+ } } \right] + \left[ {{\text{Mg}}^{ 2+ } } \right]$ Ragunath (1987) Groundwater chemistry The statistical characteristics of all the groundwater samples are presented in Table 3. The overall groundwater pH and EC values of the study area are ranged from 6.6 to 8.4 and 1003 to 6235 μS/cm, respectively. The large variation in EC is mainly attributed to geochemical processes prevailing in this region. TDS in the study area vary from 667 to 4021 mg/l. The groundwater in the study area falls under fresh (TDS < 1000 mg/l) to brackish (TDS > 1000 mg/l) types of water (Freeze and Cherry 1997). The total hardness (as CaCO3) ranges from 327 to 1719 mg/l. Cation concentrations and ratios can trace water–rock interaction processes, such as mineral weathering and cation exchange (Han et al. 2009). In the study area, the Na+ and K+ concentrations in groundwater range from 71 to 1120 and 1.13 to 32 mg/l, respectively. High concentrations of Na+ in the groundwater are attributed to cation exchange among minerals. The concentrations of calcium range from 57 to 293 mg/l, which is derived from calcium-rich minerals like calcite, dolomite, and gypsum. The major source of Mg2+ in the groundwater is due to ion exchange of minerals in rocks and soils by water. The concentrations of Mg2+ found in the groundwater samples vary in the range 34–199 mg/l. The CO32− and HCO3− concentration in groundwater are derived from carbonate weathering as well as the dissolution of carbonic acid in the aquifers (Kumar et al. 2009; Eq. 1). $$\begin{aligned} & {\text{CaCO}}_{3} + {\text{CO}}_{2} + {\text{H}}_{2} {\text{O}} \to 2{\text{HCO}}_{3}^{ - } \\ & {\text{CO}}_{2} + {\text{H}}_{2} {\text{O}} \to {\text{H}}^{ + } + {\text{HCO}}_{3}^{ - } \\ \end{aligned}$$ Summary statistics of the analytical data and groundwater samples of the study area exceeding the permissible limits prescribed by WHO for drinking purposes Guideline values WHO (2011) EC (µS/cm) TDS (mg l−1) Ca2+ (mg l−1) Mg2+ (mg l−1) Na+ (mg l−1) K+ (mg l−1) HCO3− (mg l−1) Cl− (mg l−1) SO 4 −2 (mg l−1) NO3− (mg l−1) CO32− F− %Na − 32.4 − 6.46 SI calcite SI dolomite SI gypsum SI anhydrate HCO3− in the study area ranges from 207 to 1493 mg/l. The concentration of Cl− ranges from 119 to 1607 mg/l and increases from the recharge to discharge area. SO42− varies from 119 to 1254 mg/l. High values of SO42− in groundwater indicated that SO42− derived from chemical fertilizers constituted an additional sulfate source. Furthermore, dissolution of sulfate-bearing minerals, especially gypsum and anhydrite that present in the formations is another cause of high content of SO42− in groundwater samples. In the study area, NO3− concentrations in samples ranged from 42 to 96 mg/l. The major source of NO3− in the groundwater is agricultural activities, septic tanks, and human and animal wastes. Because there is no systematic sewage collection and treatment or refuse collection, the groundwater is affected by considerable contamination and the associated hygiene problems. In terms of land use pattern, in MSC and specifically adjacent area, there are vast farmlands and intense agricultural activity. Also, animal husbandry is one of the main activities among people, and then these practices have a significant contribution on NO3− content of water samples. The mean F− of samples is 0.063 mg/l with values ranging from 0.009 to 0.063 mg/l. These values are found to be within the prescribed limit of WHO guideline, 1.5 mg/l, so there is no potential hazard of F− content in groundwater samples in MSC. Figure 3 shows that Na+ and Cl− are dominant cations and anion, respectively. A further illustration of this is shown in Fig. 3, where the median values of Cl− exceeded 50% of total anions in the mille-equivalent unit. The abundance of the major ions in groundwater is in the following order: Na+ > Ca2+ > Mg2+ > K+ and Cl− > HCO3− > SO42− > NO3−. Pie diagram of median values of major ions Minimum, maximum, and average values of physical and chemical parameters of groundwater samples are presented in Table 3. Hydrochemical evaluation The geochemical variations in the ionic concentrations in the groundwater can easily be understood when they are plotted along an X–Y coordinate (Guler and Thyne 2004). Results from the chemical analyses were used to identify the geochemical processes and mechanisms in the groundwater aquifer system. The chemical data of the groundwater samples are plotted for Ca2+ + Mg2+ versus HCO3− − CO32− diagram (Fig. 4a). The majority of data fall above the equiline (1:1), which suggests that an excess of alkalinity in the water has been balanced by alkalis (Na+ + K+), while the sample points lie below the equiline in a plot of Ca2+ + Mg2+ versus total cation (TC) (Fig. 4b). The graph of Ca2+ + Mg2+ versus TC shows most of the samples far below the theoretical line (1:1; Fig. 4b), indicating an increasing contribution of alkalis to the major ions caused by silicate weathering (Subba Rao 2008). In a plot of Na+ + K+ versus TC (Fig. 4c), the chemical data of the samples fall below the equiline and above the Na+ + K+/0.50 TC line. This leads to infer that the supply of cations via silicate weathering and/or soil salts is more significant (Stallard and Edmond 1983), whereas the increase in alkalis with a simultaneous increase in Cl− + SO42− (Fig. 4d) reflects a common source for these ions from the dissolution of soil salts (Sarin et al. 1989; Datta and Tyagi 1996). The observed excess of Na+ over K+ is because of the greater resistance of K+ to chemical weathering and its adsorption on clay minerals (Subba Rao 2008). Most of the samples have a Na+/Cl− ratio around or above 1, indicating that an ion exchange process is prevalent in the study area (Fig. 4e) (Kumar et al. 2006). The evidence for ion exchange in the development of salinization can lead to the release of Na+ from clay products, replacing Ca2+ that is present in groundwater. Figure 4f shows the ion exchange reactions, where Na+ is plotted against Ca2+, in which Ca2+ levels are observed between 2.3 and 28 meq/l, while Na+ levels are found between 2.8 and 54 meq/l. Hence, the ion exchange process appears as responsible for contributing higher concentration of Na+ in the groundwater. If the ion exchange is the only controlling process of groundwater composition, the relation between (Ca2+ + Mg2+) − (SO42− + HCO3−) and Na+–Cl− should show the negative linear trend with a slope of unity, considering the participation of cations in the ion exchange reaction (Fisher and Mullican 1997). Graphs of different parameters (solid and dashed lines denotes 1:1 and 1:0.5, respectively) In Fig. 4g, the samples show a trend of (Ca2+ + Mg2+) − (SO42− + HCO3−) versus Na+− Cl− with a negative slope of less than unity, but they spread above and below the linear trend. This suggests that the controlling of groundwater quality depends not only on the involvement of ion exchange process but also on the involvement of other processes. Otherwise, the spreading of sample points above and below the linear trend should not be expected. The graph of Ca2+ + Mg2+ versus SO42− + HCO3− will feature a nearly 1:1 line if dissolutions of calcite, dolomite, and gypsum are the dominant reactions in the system (Srivastava and Ramanathan 2008). Ion exchange tends to shift the points right because of the excess of SO42− + HCO3− ions, which may be due to anthropogenic input in the groundwater system (Cerling et al.1989; Fisher and Mullican 1997). The graph of Ca2+ + Mg2+ versus SO42− + HCO3− (Fig. 4h) shows that nearly all of the samples fall above the 1:1 ratio line and show a deficiency of Ca2+ + Mg2+ relative to SO42− + HCO3−. Therefore, Na+ must balance the excess of the negative charge of SO42− and HCO3− ions. Higher concentration of Na+ in the groundwater is an index of ion exchange process. Hydrochemical facies The term hydrochemical facies is used to describe the bodies of groundwater in an aquifer that differ in their chemical composition. The facies is a function of the lithology, solution kinetics, and flow patterns of the aquifer (Raju et al. 2009). Large tables of analytical data are usually difficult to interpret regarding the variations in water quality. Graphs are useful for this purpose, and several specialized types are in use. Piper diagram is one of them. It provides a convenient method to classify and compare water types based on the ionic composition of different water samples (Chkirbenea et al. 2009). The values obtained from the groundwater sample analyzing and their plot on Piper's diagrams (Piper 1944) reveal that the major cation is Na+ and the anion is Cl−. In the study area, the major groundwater types are Na–Cl and mixed Ca, Mg–Cl types that the alkalis (Na+, K+) are significantly dominating over the alkaline earth metals (Ca2+, Mg2+; Fig. 5). The Na–Cl water type in the study area is due to the low velocity of groundwater, ion exchange, long time contacts of water, and formations as well as the type of the rocks. Chemical facies of groundwater in Piper diagram Saturation index Saturation indices are used to evaluate the degree of equilibrium between water and minerals. Changes in saturation state are useful to distinguish different stages of hydrochemical evolution and help identify which geochemical reactions are important in controlling water chemistry (Coetsiers and Walraevens 2006; Drever 1997; Langmuir 1997). The saturation indices were determined using the hydrogeochemical equilibrium model, PHREEQC for Windows (Parkhurst and Appelo 1999). The saturation index of a mineral is obtained from Table 2c equation (Garrels and Mackenzie 1967), where IAP is the ion activity product of the dissociated chemical species in solution and Kt is the equilibrium solubility product for the chemical involved at the sample temperature. An index (SI), less than zero, indicating that the groundwater is under-saturated with respect to that particular mineral. Such a value could reflect the character of water from a formation with an insufficient amount of the mineral for the solution or short residence time. An index (SI), greater than zero, specifies that the groundwater being supersaturated with respect to the particular mineral phase and, therefore, incapable of dissolving more of the mineral. Such an index value reflects groundwater discharging from an aquifer containing ample amount of the mineral with sufficient resident time to reach equilibrium. Nonetheless, supersaturating can also be produced by other factors that include incongruent dissolution, common ion effect, evaporation, the rapid increase in temperature, and CO2 exsolution (Appelo and Postma 1996; Langmuir 1997). In Table 1, the SI for calcite, dolomite, anhydrite, and gypsum is shown. Figure 6 shows the plots of SI for all the investigated water. Nearly, all water samples were supersaturated with respect to calcite and dolomite and all samples under-saturated with respect to gypsum and anhydrite, suggesting that these carbonate mineral phases may have influenced the chemical composition of the study area. In Na–Cl water type, the mean values of SIcal, SIdol, SIgyp, SIanhy are 0.35, 0.85, − 1.17, and − 1.34, respectively. Plots of saturation indices with respect to calcite, dolomite, gypsum, and anhydrate minerals Drinking and irrigation water quality The analytical results have been evaluated to ascertain the suitability of groundwater in the study area for drinking and agricultural uses. The drinking water quality is evaluated by comparing with the specifications of TH and TDS set by the WHO (2011). According to WHO (2011) specification, TDS up to 500 mg/l is the highest desirable and up to 1500 mg/l is maximum permissible (Table 4). Based on this classification, 27% of samples belong to maximum permissible category, and the remaining samples are exceeding the maximum allowable limits. The classification of groundwater based on total hardness (TH) (Sawyer and McCartly 1967) shows that all of the groundwater samples fall in the very hard water category (Table 4). Maximum allowable limit of TH for drinking is 500 mg/l, and the most desirable limit is 100 mg/l as per the WHO international standard. Based on this classification, it indicates that 73% samples exceed the maximum allowable limits; such water (water with TH greater than 80 mg/l) cannot be used for domestic purposes because it coagulates soap lather. Classification of groundwater based on total hardness (TH), electrical conductivity (EC), chloride concentration, sodium adsorption ratio (SAR), sodium percent (%Na), and residual sodium carbonate (RSC) Water class Percent of samples Na % (Wilcox 1955) SAR (Richards 1954) EC (Wilcox 1955) RSC (Richards 1954) < 1.25 TH (Sawyer and McCarthy 1967) < 75 Moderately hard Very hard Cl− Classification (Stuyfzand 1989) Extremely fresh Very fresh Fresh brackish brackish 28.21–282.06 Brackish salt > 564.13 Hypersaline The development and maintenance of successful irrigation projects involve not only the supplying of irrigation water to the land but also the control of salt and alkali in the soil (Haritash et al. 2008). Salinity and indices such as SAR, %Na, RSC, and PI are important parameters for determining the suitability of groundwater for agricultural uses (Srinivasa Gowd 2005; Raju 2007). EC is a good measure of salinity hazard to crops as it reflects the TDS in groundwater. The US Salinity Laboratory (1954) classified groundwater on the basis of EC (Table 4). Based on this classification, 22% of samples belong to the permissible category, 53% doubtful category, and 25% unsuitable category. Stuyfzand (1989) classified water on the basis of Cl− ion concentration into eight divisions as shown in Table 3. Based on this classification, 10% of groundwater samples were fresh, 27% fresh brackish, 42 and 20% were brackish salt and salt on the basis of Cl− concentration. SAR is an important parameter for determining the suitability of groundwater for irrigation because it is a measure of alkali/sodium hazard to crops (Subramani et al. 2005). SAR is defined by using Table 1a equation, where all ionic concentrations are expressed in meq/l. The SAR values range from 1.2 to 19.2. According to the Richards (1954) classification based on SAR values (Table 4), 83% of samples belong to the excellent category, 11% of them belong to good category and the remaining samples belong to the doubtful category. SAR can indicate the degree to which irrigation water tends to enter cation exchange reactions in soil. Sodium replacing adsorbed calcium and magnesium is a hazard as it causes damage to the soil structure and becomes compact and impervious (Raju 2007). In all natural waters, percent of sodium content is a common parameter to assess its suitability for agricultural purposes (Wilcox 1948). The %Na is obtained by using Table 1b equation, where all ionic concentrations are expressed in meq/l, according to the Wilcox (1955) classification based on %Na values (Table 4), 5% of samples belong to the good category, 62% of them belong to permissible category, and the remaining samples belong to the doubtful category. RSC has been calculated to determine the hazardous effect of carbonate and bicarbonate on the quality of water for agricultural purposes and has been determined by using Table 1d equation, where all ionic concentrations are expressed in meq/l (Eaton 1950). The classification of irrigation water according to the RSC values is water containing more than 2.5 meq/l of RSC are not suitable for irrigation, while those having 1.25–2.5 meq/l are doubtful and those with less than 1.25 meq/l are good for irrigation (Table 4). Based on this classification, 5% samples belong to the good category, 16% samples belong to the doubtful category, and 79% belong to unsuitable category. The PI values also indicate that the groundwater is unsuitable for irrigation. It is defined by using Table 1e equation, where all the ions are expressed in meq/l (Ragunath 1987). WHO (2011) uses a criterion for assessing the suitability of water for irrigation based on PI. According to PI values, the groundwater in the study area can be designated as class II (25–75%) indicating that the groundwater is unsuitable for irrigation excepting the two samples, which was classified as class I (> 75%). In summary, west and north of MSC, albeit high concentration of dissolved ions, are suitable locations for drinking water wells, although east and north of MSC are relative appropriate place for irrigational wells. According to these results, authorities could manage groundwater resources properly for various demands in the study area. Interpretation of hydrochemical analysis reveals that the groundwater in the study area is very hard, fresh to brackish and alkaline in nature. The sequence of the abundance of the major ions is in the following order: Na+ > Ca2+ > Mg2+ > K+ and Cl− > HCO3− > SO42− > NO3−. Alkalis slightly exceed alkali earths, and strong acids exceed weak acids. The compositional relations and mineral saturation states indicate that the dominant control of associated hydrogeochemical processes is the dissolution of evaporite minerals such as halite and gypsum, ion exchange, and the residence time of the groundwater in contact with rock materials. Hydrochemical studies indicate that the concentration of nitrate is higher than permissible limit (50 mg/l) in most of the groundwater collected from hand-pumped wells. The chief sources of nitrate pollution in the study area are agricultural activities, septic tanks, and human and animal wastes. Septic systems, animal waste, and fertilizer are all potential sources of nitrate contamination. High values of SO42− in samples indicated that SO42− derived from chemical fertilizers and dissolution of sulfate-bearing minerals, especially gypsum and anhydrite, that present in the lithological units. The F− values of samples are within the prescribed limit of WHO guideline, 1.5 mg/l, so there is no potential hazard of F− content in groundwater samples in MSC. In the study area, the dominant hydrochemical facies of groundwater is Na–Cl. Ionic concentrations, TDS, EC, and water quality suggest that groundwater residence time is primarily controlled by the occurrence of different hydrochemical facies. Assessment of water samples according to exceeding the permissible limits prescribed by WHO for drinking purposes indicated that groundwater in the study area is chemically unsuitable for drinking uses. 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In: Proceedings of the 18th TWSA Water Working, Testing and Research Institute. KIWA, NieuwegeinGoogle Scholar Subramani T, Elango L, Damodarasamy SR (2005) Groundwater quality and its suitability for drinking and agricultural use in Chithar River Basin, Tamil Nadu, India. Environ Geol. https://doi.org/10.1007/s00254-005-1243-0 Google Scholar Ulmishek GF (2004) Petroleum geology and resources of the Amu Darya Basin, Turkmenistan, Uzbekistan, Afghanistan and Iran. USGS BulletinGoogle Scholar USAID (2009) Commercialization of Afghanistan water and sanitation activity (CAWSA), assessment report MSC water supply (CAWSA). Retrieved from: www. ijens. org/Vol_12_I_06/1214406-8585-IJCEEIJENS.pdfGoogle Scholar USGS (2005) Geological map of quadrangles (3666), Balkh (219), Mazar-i- Sharif (220), Qarqin (213) and Hazara Toghal (214) quadrangles map, Afghanistan. http://pubs.usgs.gov/of/2005/1093/A/index.html WHO (2011) Guidelines for drinking-water quality. World Health Organization, GenevaGoogle Scholar Wilcox LV (1948) The quality of water for irrigation use. U.S. Department of Agriculture, WashingtonGoogle Scholar Wilcox LV (1955) Classification and use of irrigation water. US Geol Dept Agri Arc 969:19Google Scholar © The Author(s) 2018 Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. 1.Department of GeologyFerdowsi University of MashhadMashhadIran 2.Agriculture FacultyHerat UniversityHeratAfghanistan 3.Faculty of AgricultureUniversity of TehranTehranIran 4.Department of GeologyBamiyan UniversityBamiyanAfghanistan Mahaqi, A., Moheghi, M.M., Mehiqi, M. et al. Appl Water Sci (2018) 8: 133. https://doi.org/10.1007/s13201-018-0768-9 Received 04 December 2016 Accepted 17 July 2018 First Online 06 August 2018 Publisher Name Springer Berlin Heidelberg Published in cooperation with King Abdulaziz City for Science and Technology Not logged in Not affiliated 54.172.234.236	CommonCrawl
Tag: Andreatta Until now, we've looked at actions of groups (such as the $T/I$ or $PLR$-group) or (transformation) monoids (such as Noll's monoid) on special sets of musical elements, in particular the twelve pitch classes $\mathbb{Z}_{12}$, or the set of all $24$ major and minor chords. Elephant-lovers recognise such settings as objects in the presheaf topos on the one-object category $\mathbf{M}$ corresponding to the group or monoid. That is, we look at contravariant functors $\mathbf{M} \rightarrow \mathbf{Sets}$. Last time we've encountered the 'Cube Dance Grap' which depicts a particular relation among the major, minor, and augmented chords. Recall that the twelve major chords (numbered for $1$ to $12$) are the ordered triples of tones in $\mathbb{Z}_{12}$ of the form $(n,n+4,n+7)$ (such as the triangle on the left). The twelve minor chords (numbered from $13$ to $24$) are the ordered triples $(n,n+3,n+7)$ (such as the middle triangle). The four augmented chords (numbered from $25$ to $28$) are the triples of the form $(n,n+4,n+8)$ (such as the rightmost triangle). The Cube Dance Graph relates two of these chords when they share two tones (pitch classes) whereas the remaining tones differ by a halftone. Picture modified from this post. We can separate this symmetric binary relation into three sub-relations: the extension of the $P$ and $L$-operations on major and minor chords to the augmented ones (these are transformations), and the remaining relation $U$ which connects the major and minor chords to the augmented chords (and which is not a transformation). Binary relations on the same set can be composed, so we get a monoid $\mathbf{M}$ generated by the three relations $P,L$ and $U$. The action of $\mathbf{M}$ on the $28$ chords no longer gives us an ordinary presheaf (because $U$ is not a transformation), but a relational presheaf as in the paper On the use of relational presheaves in transformational music theory by Alexandre Popoff. That is, the action defines a contravariant functor $\mathbf{M} \rightarrow \mathbf{Rel}$ where $\mathbf{Rel}$ is the category (actually a $2$-category) of sets, but with binary relations as morphisms (that is, $Hom(X,Y)$ is all subsets of $X \times Y$), and the natural notion of composition of such relations. The $2$-morphism between relations is that of inclusion. To compute with monoids generated by binary relations in GAP one needs to download, compile and load the package semigroups, and to represent the binary relations as partitioned binary relations as in the paper by Martin and Mazorchuk. This is a bit more complicated than working with ordinary transformations: P:=PBR([[-13],[-14],[-15],[-16],[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-9],[-10],[-11],[-12],[-25],[-26],[-27],[-28]],[[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[25],[26],[27],[28]]); L:=PBR([[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-13],[-14],[-15],[-16],[-9],[-10],[-11],[-12],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-25],[-26],[-27],[-28]],[[17],[18],[19],[20],[21],[22],[23],[24],[13],[14],[15],[16],[9],[10],[11],[12],[1],[2],[3],[4],[5],[6],[7],[8],[25],[26],[27],[28]]); U:=PBR([[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-17,-21,-13,-4,-8,-12],[-5,-1,-9,-18,-14,-22],[-2,-6,-10,-15,-23,-19],[-24,-16,-20,-11,-3,-7]],[[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[25],[25],[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[17,21,13,4,8,12],[5,1,9,18,14,22],[2,6,10,15,23,19],[24,16,20,11,3,7]]); But then, GAP quickly tells us that $\mathbf{M}$ is a monoid consisting of $40$ elements. gap> M:=Semigroup([P,L,U]); gap> Size(M); The Semigroups-package can also compute Green's relations and tells us that there are seven such $R$-classes, four consisting of $6$ elements, two of four, and one of eight elements. These are also visible in the Cayley graph, exactly as last time. Or, if you prefer the cleaner picture of the Cayley graph from the paper Relational poly-Klumpenhouwer networks for transformational and voice-leading analysis by Popoff, Andreatta and Ehresmann. This then allows us to compute the Heyting algebra of the subobject classifier, and all the Grothendieck topologies, at least for the ordinary presheaf topos of $\mathbf{M}$-sets, not for the relational presheaves we need here. We can consider the same binary relation on the larger set of triads when we add the suspended triads. These are the ordered triples in $\mathbb{Z}_{12}$ of the form $(n,n+5,n+7)$, as in the rightmost triangle below. There are twelve suspended chords (numbered from $29$ to $40$), so we now have a binary relation $T$ on a set of $40$ triads. The relation $T$ is too coarse, and the art is to subdivide $T$ is disjoint sub-relations which are musically significant, between major and minor triads, between major/minor and augmented triads, and so on. For each such partition we can then consider the monoids generated by these sub-relations. In his paper, Popoff suggest relevant sub-relations $P,L,T_U,T_V$ and $T_U \cup T_V$ of $T$ which in our numbering of the $40$ chords can be represented by these PBR's (assuming I made no mistakes…ADDED march 24th: I did make a mistake in the definition of L, see comment by Alexandre Popoff, below the corect L): P:=PBR([[-13],[-14],[-15],[-16],[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-9],[-10],[-11],[-12],[-25],[-26],[-27],[-28],[-36],[-37],[-38],[-39],[-40],[-29],[-30],[-31],[-32],[-33],[-34],[-35]],[[13],[14],[15],[16],[17],[18],[19],[20],[21],[22],[23],[24],[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[25],[26],[27],[28],[34],[35],[36],[37],[38],[39],[40],[29],[30],[31],[32],[33]]); L:=PBR([[-17],[-18],[-19],[-20],[-21],[-22],[-23],[-24],[-13],[-14],[-15],[-16],[-9],[ -10],[-11],[-12],[-1],[-2],[-3],[-4],[-5],[-6],[-7],[-8],[-25],[-26],[-27],[-28],[-29], [-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40]],[[17], [18], [19], [ 20],[21],[22],[23],[24],[13],[14],[15],[16],[9],[10],[11],[12],[1],[2],[3],[4],[5], [6], [7],[8],[25],[26],[27],[28],[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40] ]); TU:=PBR([[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-25],[-26],[-27],[-28],[-4,-8,-12,-13,-17,-21],[-1,-5,-9,-14,-18,-22],[-2,-6,-10,-15,-19,-23],[-3,-7,-11,-16,-20,-24],[],[],[],[],[],[],[],[],[],[],[],[]],[[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[25],[25],[26],[27],[28],[25],[26],[27],[28],[25],[26],[27],[28],[4,8,12,13,17,21],[1,5,9,14,18,22],[2,6,10,15,19,23],[3,7,11,16,20,24],[],[],[],[],[],[],[],[],[],[],[],[]]); TV:=PBR([[-29],[-30],[-31],[-32],[-33],[-34],[-35],[-36],[-37],[-38],[-39],[-40],[-36],[-37],[-38],[-39],[-40],[-29],[-30],[-31],[-32],[-33],[-34],[-35],[],[],[],[],[-1,-18],[-2,-19],[-3,-20],[-4,-21],[-5,-22],[-6,-23],[-7,-24],[-8,-13],[-9,-14],[-10,-15],[-11,-16],[-12,-17]],[[29],[30],[31],[32],[33],[34],[35],[36],[37],[38],[39],[40],[36],[37],[38],[39],[40],[29],[30],[31],[32],[33],[34],[35],[],[],[],[],[1,18],[2,19],[3,20],[4,21],[5,22],[6,23],[7,24],[8,13],[9,14],[10,15],[11,16],[12,17]]); TUV:=PBR([[-26,-29],[-27,-30],[-28,-31],[-25,-32],[-26,-33],[-27,-34],[-28,-35],[-25,-36],[-26,-37],[-27,-38],[-28,-39],[-25,-40],[-25,-36],[-26,-37],[-27,-38],[-28,-39],[-25,-40],[-26,-29],[-27,-30],[-28,-31],[-25,-32],[-26,-33],[-27,-34],[-28,-35],[-4,-8,-12,-13,-17,-21],[-1,-5,-9,-14,-18,-22],[-2,-6,-10,-15,-19,-23],[-3,-7,-11,-16,-20,-24],[-1,-18],[-2,-19],[-3,-20],[-4,-21],[-5,-22],[-6,-23],[-7,-24],[-8,-13],[-9,-14],[-10,-15],[-11,-16],[-12,-17]],[[26,29],[27,30],[28,31],[25,32],[26,33],[27,34],[28,35],[25,36],[26,37],[27,38],[28,39],[25,40],[25,36],[26,37],[27,38],[28,39],[25,40],[26,29],[27,30],[28,31],[25,32],[26,33],[27,34],[28,35],[4,8,12,13,17,21],[1,5,9,14,18,22],[2,6,10,15,19,23],[3,7,11,16,20,24],[1,18],[2,19],[3,20],[4,21],[5,22],[6,23],[7,24],[8,13],[9,14],[10,15],[11,16],[12,17]]); The resulting monoids are huge: gap> G:=Semigroup([P,L,TU,TV]); gap> H:=Semigroup([P,L,TUV]); gap> Size(H); In Popoff's paper these monoids have sizes respectively $473,293$ and $994,624$. Strangely, the offset is in both cases $144=12^2$. (Added march 24: with the correct L I get the same sizes as in Popoff's paper). Perhaps we should try to transform such relational presheaves to ordinary presheaves. One approach is to use the Grothendieck construction and associate to a set with such a relational monoid action a directed graph, coloured by the elements of the monoid. That is, an object in the presheaf topos of the category \xymatrix{C & E \ar[l]^c \ar@/^2ex/[r]^s \ar@/_2ex/[r]_t & V} \] and then we should consider the slice topos over the one-vertex bouquet graph with one loop for each element in the monoid. If you want to have more details on the musical side of things, for example if you want to know what the opening twelve chords of "Take a Bow" by Muse have to do with the Cube Dance graph, here are some more papers: A categorical generalization of Klumpenhouwer networks, A. Popoff, M. Andreatta and A. Ehresmann. From K-nets to PK-nets: a categorical approach, A. Popoff, M. Andreatta and A. Ehresmann. From a Categorical Point of View: K-Nets as Limit Denotators, G. Mazzola and M. Andreatta.	CommonCrawl
Help with math Visual illusions Cut the knot! What is what? Inventor's paradox Math as language Outline mathematics Analogue gadgets Proofs in mathematics Things impossible Index/Glossary Fast Arithmetic Tips Stories for young Make an identity Hadamard's Determinant Inequalities and Applications II Hadamard's Second Theorem A square matrix $A=(a_{ij})$ is said to be positive if, for all $i=1,2,\ldots,n,$ $a_{ii}\ge 0$ and $\det A\ge 0.$ If $A=(a_{ij})\in M_n(\mathbb{R})$ is positive, then $\displaystyle \det A\le\prod_{j=1}^na_{ii}.$ (E.g., Finbarr Holland, Another Proof of Hadamard's Determinantal Inequality, Irish Math. Soc. Bulletin 59 (2007), 61-64) Consider the $n\times n$ matrix $\displaystyle M_n=\left(\begin{array}{ccccc}a&1&1&\cdots&1\\1&a&1&\cdots&1\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&1&1&\cdots&a\end{array}\right).$ We are going to prove that $\det M_n=(n + a - 1)(a - 1)^{n-1},$ which immediately implies the required inequality. The proof is by induction by induction. Let's denote $\det M_n=D_n$ and $\displaystyle N_n=\left(\begin{array}{ccccc}1&1&1&\cdots&1\\1&a&1&\cdots&1\\\vdots&\vdots&\vdots&\ddots&\vdots\\1&1&1&\cdots&a\end{array}\right).$ By subtracting the first row from the rest $\det N_n=(a-1)^{n-1}.$ Further, $D_2=a^2-1=(a+2-1)(a-1)^{2-1}.$ Assume that indeed $D_n=(a+n-1)(a-1)^{n-1}.$ Then, by expanding along the first column and swapping the columns to reduce several determinants to one, $\begin{align}D_{n+1}&=aD_n-n\det N_n=a(a+n-1)(a-1)^{n-1}-n(a-1)^{n-1}\\ &=(a-1)^{n-1}(a^2+an-a-n)\\ &=(a-1)^{n-1}(a(a+n)-(a+n))\\ &=(a-1)^{n}(a+n)\\ &=(a+(n+1)-1)(a-1)^{(n+1)-1}, \end{align}$ as required. Let $b=a-1.$ Then the required inequality reduces to $(b+1)^n\ge b^{n-1}(n+b)=b^n+nb^{n-1},$ which is obviously true since the right-hand side is just a part of the binomial expansion of $(b+1)^n.$ The above is based on an article Application of Hadamard's Theorems to inequalities by Dan Sitaru and Leo Giugiuc that appeared in the Crux Mathematicorum (v 44, n 1, pp 25-27). I am very much indebted to Dan Sitaru for bringing this article to my attention. Linear Algebra Tools for Proving Inequalities $\;\left(\displaystyle\left(\frac{a}{b-c}\right)^2+\left(\frac{b}{c-a}\right)^2+\left(\frac{c}{a-b}\right)^2\ge 2\right)$ Linear Algebra Tools for Proving Inequalities: Cauchy-Binet Formula $\;\left(\displaystyle\left(\sum_{i=1}^{n}\frac{x_i^2}{a_i}\right)\cdot\left(\sum_{1\le i\lt j\le n}a_ia_j(x_iy_j-x_jy_i)^2\right)\ge \sum_{i=1}^{n}a_iy_i^2\right)$ An Inequality from Gazeta Matematica, March 2016 (If $a^2+b^2+c^2=3\,$ then $(a+c)(1+b)\le 4)$ An Inequality from Gazeta Matematica, March 2016 II (If $x^2+y^2+z^2+t^2=1\,$ then $\;(x+z)(y+t)\le 4)$ An Inequality from Gazeta Matematica, March 2016 III $\;(a^2+b^2+1\ge a+ab+b)$ An Inequality from Gazeta Matematica, March 2016 IV (If $a^2+b^2+c^2=1\,$ then $a+ac+b\le 2)$ Problem 3980 from Crux Mathematicorum $\;\left(\displaystyle\sum_{cycl}\frac{a+b}{a-b}\prod_{cycl}\frac{a+b}{a-b}\lt\frac{1}{3}\right)$ NonSquare Matrix as a Tool for Proving an Inequality $\;\left(2(a + b + c)((a + 2b + 3c) \ge (\sqrt{b(a+b)} + 2\sqrt{c(b+c)} + \sqrt{a(c+a)})^2\right)$ An Inequality in Parallelogram of Unit Area $\;\left(a^2+b^2+c^2+d^2+ac+bd\ge\sqrt{3}\right)$ An Inequality from a Vietnamese Problem Book $\;\left(\displaystyle \frac{a^3+2}{b+2c}+\frac{b^3+2}{c+2a}+\frac{c^3+2}{a+2b}\ge 3\right)$ Hadamard's Determinant Inequalities and Applications I $\left((2-a-b-c+abc)^2\le (a^2+2)(b^2+2)(c^2+2)\right)$ \|Contact\| \|Front page\| \|Contents\| \|Algebra\| Copyright © 1996-2018 Alexander Bogomolny	CommonCrawl
Find exact value of $\sum_{n=1}^{\infty} \frac{1}{2^{n}} \tan \left( \frac{\pi}{2^{n+1}} \right)$ How can I compute the series $$\sum_{n=1}^{\infty} \frac{1}{2^{n}} \tan \left( \frac{\pi}{2^{n+1}} \right)$$ I just guess using half-angle formula to compute this series, but I can't do any approaches. How should I do to solve this infinite sum? real-analysis calculus sequences-and-series asked May 9 '19 at 1:24 bFur4listbFur4list $\begingroup$ Wolfram Alpha doesn't evaluate it exactly, and gets $0.6366197723...$ $\endgroup$ $\begingroup$ if it's not $2/ \pi$, then $2/ \pi$ is remarkably close. I wouldn't know how to prove it, though $\endgroup$ – Rob Bland $\begingroup$ Is there a reason to suspect this has a closed form? $\endgroup$ – Clayton $\begingroup$ I would doubt any closed form. There doesn't seem to be any nice way to make it telescope, which would be the only way I would expect it may converge. Numerically computing this isn't so bad though, since the $n$th partial sum has $\mathcal O(4^{-n})$ error. $\endgroup$ – Simply Beautiful Art Let $$\begin{align}S(x)&=\sum_{n=1}^\infty\frac1{2^n}\tan\left(\frac{x}{2^n}\right)\\&=\frac{d}{dx}\sum_{n=1}^\infty\ln\left(\sec\left(\frac{x}{2^n}\right)\right)\\&=\frac{d}{dx}\ln\left(\prod_{n=1}^\infty\sec\left(\frac{x}{2^n}\right)\right)\\&=\frac{d}{dx}\ln\left(\lim_{m\to\infty}\left\{2^m\csc(x)\sin(2^{-m}x)\right\}\right)\\&=\frac{d}{dx}\ln(x\csc(x))\\&=\frac{(1-x\cot(x))\csc(x)}{x\csc(x)}\\&=\frac1x-\cot(x)\end{align}$$ Then we want to compute $$S(\pi/2)=\frac1{π/2}-0=\bbox[5px,border:2px solid black]{\frac2{\pi}}$$ This agrees with the numerical result on Wolfram Alpha. Some other details: I used the step $$\prod_{n=1}^m\sec(2^{-n}x)=2^m\csc(x)\sin(2^{-m}x)$$ in the computation. As Yuta pointed out in the comments, this can be proved by multliplying through by $\csc(2^{-m} x)$ and using $$\begin{align}\csc(2^{-m} x)\prod_{n=1}^m\sec(2^{-n}x)&=\frac1{\sin(2^{-m} x)}\prod_{n=1}^m\frac1{\cos(2^{-n}x)}\\&=2\frac1{\sin(2^{-m+1}x)}\prod_{n=1}^{m-1}\frac1{\cos(2^{-n}x)}\\&=\cdots\\&=2^m\frac1{\sin(x)}\end{align}$$ The limit was evaluated by doing a Taylor expansion. John DoeJohn Doe $\begingroup$ For $\prod_{n=1}^m\sec(2^{-n}x)=2^m\csc(x)\sin(2^{-m}x)$, multiplying both sides by $\csc(2^{-m}x)$ and apply the formula $\sin 2\theta=2\sin\theta\cos\theta$ repeatedly should work. $\endgroup$ – Yuta $\begingroup$ @Yuta Ah yes, that's neat :) $\endgroup$ $$\cot A-\tan A=2\cot2A\implies\dfrac12\tan A=\dfrac12\cot A-\cot2A$$ Using Telescoping series $$S(m)=\sum_{n=1}^m\dfrac1{2^n}\tan\dfrac x{2^{n+1}}=\dfrac1{2^m}\cot\dfrac x{2^{m+1}}-\dfrac1{2^{1-1}}\cot\dfrac x{2^1}$$ $$\lim_{m\to\infty}S(m)=\dfrac2x-\cot\dfrac x2$$ setting $\dfrac x{2^m}=y,m\to\infty,y\to0$ and $\lim_{y\to0}\dfrac{\sin y}y=1$ Here $x=\pi$ lab bhattacharjeelab bhattacharjee $$\sum_{n=1}^\infty \frac1{2^n}\tan\left(\fracπ{2^{n+1}}\right) = 2\frac d {dπ}\sum_{n=1}^\infty \ln\left(\sec\left(\fracπ{2^{n+1}}\right)\right) = 2\frac d {dπ}\ln\left(\frac1{\prod_{n=1}^\infty\cos\left(\fracπ{2^{n+1}}\right)}\right)$$ But $$\prod_{n=1}^\infty\cos\left(\fracπ{2^{n+1}}\right) = \cos\left(\fracπ4\right)\cos\left(\fracπ8\right)\cos\left(\fracπ{16}\right)\cos\left(\fracπ{32}\right)... = \frac{\sqrt2}2\frac{\sqrt{2+\sqrt2}}2\frac{\sqrt{2+\sqrt{2+\sqrt2}}}2... = \frac2π$$ (This is Viète's formula.) Then our original sum is $$2\frac d {dπ}\ln\left(\frac1{\frac2π}\right) = 2\frac d {dπ}\ln\left(\fracπ2\right) = 2\frac{\frac12}{\fracπ2} = \frac2π$$ BelowAverageIntelligenceBelowAverageIntelligence $\begingroup$ (+1) Nice alternative way of calculating the product, haven't seen that before. $\endgroup$ $\begingroup$ $\displaystyle \frac{d}{d\pi}???$ $\endgroup$ – Unit $\begingroup$ @Unit: think of $\pi$ as a variable and differentiate with respect to it (this part is essentially the same as John Doe's answer...the evaluation of the product is where the two answers really differ). $\endgroup$ $\begingroup$ @Clayton I get it, it's just terrible. $\endgroup$ Not the answer you're looking for? Browse other questions tagged real-analysis calculus sequences-and-series or ask your own question. How to show that $ \int_{-\frac{\pi}{6}}^{\frac{\pi}{6}} \ln\left(\tan(x)+\tan\left(\frac{\pi}{6}\right)\right)\tan(x)\space dx=\frac{\zeta(2)}{6} $ Find the value of $\sum_{m=1}^\infty \tan ^ {-1}\frac{2m}{m^4+m^2+2}$ Find the exact value of $\tan^{-1}\left(\frac{12}{5}\right)$ On convergence of the series. $\sum_{n=1}^{\infty}\left[\frac{1}{n}-\tan^{-1}\left(\frac{1}{n}\right)\right]^{a}$ Computing the sum $\sum_\limits{n=2}^\infty \left(\frac{1}{(n-1)!}-\frac{1}{n!}\right)\frac{1}{n+1}$ Is there a way to sum $\sum_{n=1}^{\infty}\left[ \frac{y}{n} - \tan^{-1}\left( \frac{y}{n} \right)\right]$ Convergence of $\sum_{n=1}^{+\infty}\tan \left( \frac{\pi}{n}\right )$ Convergence of $\sum_{n=1}^{+\infty}n\tan \left( \frac{\pi}{2^{n+1}}\right )$	CommonCrawl
What's the difference between a realization, a representation and an implementation in metrology? In a recent answer, a metrologist casually used the terms 'realization' and 'implementation' of an SI unit as if they were different, which looks very strange to the untrained eye. Some further digging (example) also throws up uses of 'representation' of an SI unit as a technical term with its own distinct meaning. What is the precise meaning of these terms in metrology, and what are the precise differences between them? What are good examples of currently implemented realizations and representations vs implementations? terminology units si-units metrology Emilio PisantyEmilio Pisanty The reference document for metrological terms is the International Vocabulary of Metrology (VIM). Definitions there are carefully crafted, but frequently they might seem a bit obscure to non metrologists and further remarks might be needed. For what concerns realization and reproduction (representation is also found in the literature for reproduction), their meaning is found under the term measurement standard: Realization of the definition of a given quantity, with stated quantity value and associated measurement uncertainty, used as a reference. In particular, the related notes 1 and 3 say: NOTE 1 A "realization of the definition of a given quantity" can be provided by a measuring system, a material measure, or a reference material. NOTE 3 The term "realization" is used here in the most general meaning. It denotes three procedures of "realization". The first one consists in the physical realization of the measurement unit from its definition and is realization sensu stricto. The second, termed "reproduction", consists not in realizing the measurement unit from its definition but in setting up a highly reproducible measurement standard based on a physical phenomenon, as it happens, e.g. in case of use of frequency-stabilized lasers to establish a measurement standard for the metre, of the Josephson effect for the volt or of the quantum Hall effect for the ohm. The third procedure consists in adopting a material measure as a measurement standard. It occurs in the case of the measurement standard of 1 kg. Therefore, the terms realization and reproduction denote an object or an experiment with specific properties. To illustrate the difference between a strict realization and a reproduction, let's take the example of a specific quantity, the unit ohm (note that a unit is a quantity, albeit a specially chosen one). First, we have to define what this quantity is: this can be done in words, possibly with the help of mathematical relationships involving other quantities, and by adding specifications on influence quantities. The ohm in the SI is defined as follows [CIPM, 1946: Resolution 2]: The ohm is the electric resistance between two points of a conductor when a constant potential difference of 1 volt, applied to these points, produces in the conductor a current of 1 ampere, the conductor not being the seat of any electromotive force. So far, so good, or, at least, it seems. Actually we are a bit stuck because we can realize the ampere and the volt respectively through current and voltage balances, but the reproducibility of the ohm realized in this way would be low, roughly at the $10^{-6}$ level. And the procedure would be rather complex. We are saved in 1956 by Thompson and Lampard who discovered a new theorem in electrostatics [1], which is the electrostatic dual of the van der Pauw theorem [2,3]. This theorem essentially says that you can build a standard of capacitance (that is, realize the farad or one of its submultiples), whose capacitance can be accurately calculated (something you cannot do with a parallel plate capacitor, for instance). If we have a standard of capacitance, through the relationships $Y = \mathrm{j}\omega C$ and $Z = 1/Y$, we have the standards of admittance and impedance, that is, we have the siemens and the ohm, however in the AC regime. Thus, the strict SI realization of the ohm, as standard of resistance, is roughly the following: You build a calculable capacitor (and ten years of your life are gone). Typically a 1 m long calculable capacitor has a capacitance of around 1 pF, which at kHz frequency correspond to a quite high impedance (for a short bibliography on the calculable capacitor, see this page). By means of impedance bridges, you scale the capacitance to higher values (e.g., 1 nF). By means of a quadrature impedance bridge, you compare the impedance value of a standard resistor with calculable AC-DC behaviour to that of the scaled capacitance. You calculate the DC value of the resistance. You scale down the resistance to 1 ohm by means of a resistance bridge. Once you have all the experiments working (after many years), the realization of the ohm through the above chain of experiments can take more than one month, but the most important issue is that the reproducibility of the ohm realized in this way, though better than that obtainable through the realizations of the volt and the ampere, is just at the $10^{-7}$-$10^{-8}$ level. Then it arrives the quantum Hall effect (QHE). A QHE element under conditions of low temperature and high magnetic field, realizes a four terminal resistance (or transresistance) with resistance value $R_\mathrm{H} = R_\mathrm{K}/i$, where $R_\mathrm{K}$ is a constant, the von Klitzing constant, and $i$ is an integer, called plateau index (typically we use the plateau corresponding to $i=2$). By the end of the 1980s it was clear that QHE elements could provide resistance standards with much better reproducibility than the other methods described above: at the time, of the order of $10^{-8}$-$10^{-9}$; nowadays, of the order of $10^{-10}$-$10^{-11}$ (two-three order of magnitudes better than that obtainable with a calculable capacitor). It turns out, also, that the von Klitzing constant is linked to two fundamental constants, the Planck constant and the elementary charge, $R_\mathrm{K} = h/e^2$. The situation in the late 1980s is thus the following: A QHE experiment is much easier to implement than that of a calculable capacitor (and much less expensive). The resistance realized by a QHE experiment has a much better reproducibility than that realizable by a calculable capacitor experiment. The accuracy of the von Klitzing constant, however, is only at the level of the SI ohm realization, that is, of about $10^{-7}$, and the relationship $R_\mathrm{H} = R_\mathrm{K}/i = h/(e^2 i)$ has not yet enough sound theoretical foundation to be exploited. The first two points suggest the adoption of a conventional unit of resistance, by defining a conventional value of the von Klitzing constant [CIPM, 1988: Recommendation 2]. This conventional value of the von Klitzing constant is denoted by $R_{\mathrm{K}-90}$ (because it was adopted in 1990) and has value $$R_\mathrm{K-90} = 25\,812.807\,\Omega\quad \text{(exact)}.$$ The conventional unit of resistance is the $\mathit{\Omega}_{90}$, defined as 1 $$\mathit{\Omega}_{90} = \frac{R_\mathrm{K}}{\{R_\mathrm{K-90}\}} = \frac{R_\mathrm{K}}{25\,812.807}.$$ At present, virtually all national resistance scales are traceable to this conventional unit. It is now worth pointing out that the quantity $\mathit{\Omega}_{90}$ has no links to the SI ohm: it's close to (the relative discrepancy is of the order of $10^{-8}$), but quite not the same thing. Thus, the $\mathit{\Omega}_{90}$ is called a reproduction (or a representation) of the ohm, because it realizes somehow the ohm, but not according to its definition. At present, this is not the only reproduced unit: the volt is currently reproduced by means of the Josephson effect through a conventional value of the Josephson constant, and the thermodynamic temperature scale is reproduced through two conventional temperature scales, the International Temperature Scale of 1990 (ITS-90) and the Provisional Low Temperature Scale of 2000 (PLTS-2000). Instead, with the forthcoming revision of the International System of Units, the so called "new SI", the quantum Hall effect and the Josephson effect will really provide SI realizations of the ohm and the volt (see this draft of the mise en pratique of the electrical units). Finally, for what concerns the term implementation, as far as I know, it has no specific technical meaning within the community of metrologists, and it is used in the common English meaning (whereas realization has a somehow different connotation). Thus, for instance, we can speak of two different implementations of a quantum Hall resistance experiment (because some details might be different). 1 A note on notation: the quantity $\mathit{\Omega}_{90}$ is typeset in italics because it's not an SI unit; braces denote the numerical value of a quantity, according to the notation $Q = \{Q\}[Q]$ [4,5, and this question]. [1] A. M. Thompson and D. G. Lampard (1956), "A New Theorem in Electrostatics and its Application to Calculable Standards of Capacitance", Nature, 177, 888. [2] L.J. van der Pauw (1958), "A method of measuring specific resistivity and Hall effect of discs of arbitrary shape", Philips Research Reports, 13, 1–9. [3] L.J. van der Pauw (1958), "A method of measuring the resistivity and Hall coefficient on lamellae of arbitrary shape", Philips Technical Review, 20, 220–224. [4] E. R. Cohen et al. (2008), Quantities, Units and Symbols in Physical Chemistry, IUPAC Green Book, 3rd Edition, 2nd Printing, IUPAC & RSC Publishing, Cambridge [Online] [5] E R Cohen and P. Giacomo (1987), Symbols, Units, Nomenclature and Fundamental Constants in Physics, IUPAP SUNAMCO Red Book, 1987 revision, IUPAP & SUNAMCO, Netherlands [Online] Massimo OrtolanoMassimo Ortolano $\begingroup$ You know, I was just about to start a bounty on this one, but I realized that it's likely that you're the only one here who can answer this, so I was wondering whether to start it right away or to tell flippiefanus to bug you, to see whether you were back ;-). So, here goes, but I still want to know about that QHE stuff. $\endgroup$ – Emilio Pisanty Aug 25 '16 at 13:06 $\begingroup$ I also want to quiz you on the quantum metrological triangle, but it'll take a bit of thinking to phrase a good question. $\endgroup$ – Emilio Pisanty Aug 25 '16 at 13:08 $\begingroup$ @EmilioPisanty I went back just yesterday ;-) QHE will arrive (historically it's not the first reproduction of the ohm, because of course we somehow had the ohm before the calculable capacitor, but I wanted to start from the strict realization). However, I'm a slow writer, I also want to complete the other answer: keep your other questions for later ;-) $\endgroup$ – Massimo Ortolano Aug 25 '16 at 13:11 $\begingroup$ @EmilioPisanty Added the QHE part: let me know if there is something unclear or some part that you would like to see expanded (in reference to the original question). $\endgroup$ – Massimo Ortolano Aug 26 '16 at 12:52 Not the answer you're looking for? Browse other questions tagged terminology units si-units metrology or ask your own question. Square bracket notation for dimensions and units: usage and conventions What is a base unit in the new SI, and why is the ampere one of them? What's the difference between inclusive and exclusive decays? What's the difference between constitutive laws and governing equations? Difference between nautical and terrestrial miles What's the difference between hopping and tunneling? What are the proposed realizations in the New SI for the kilogram, ampere, kelvin and mole? What's the difference between frequency, spectral and cepstral domains? Base unit definition and realization under relativistic conditions	CommonCrawl
Structural basis of transcriptional regulation by a nascent RNA element, HK022 putRNA Promoter-proximal elongation regulates transcription in archaea Fabian Blombach, Thomas Fouqueau, … Finn Werner Structural basis for transcription antitermination at bacterial intrinsic terminator Linlin You, Jing Shi, … Yu Zhang Structural basis of SNAPc-dependent snRNA transcription initiation by RNA polymerase II Srinivasan Rengachari, Sandra Schilbach, … Patrick Cramer Pausing controls branching between productive and non-productive pathways during initial transcription in bacteria David Dulin, David L. V. Bauer, … Achillefs N. Kapanidis Structure of paused transcription complex Pol II–DSIF–NELF Seychelle M. Vos, Lucas Farnung, … Patrick Cramer Transcription activation by a sliding clamp Jing Shi, Aijia Wen, … Yu Feng Transcription factors modulate RNA polymerase conformational equilibrium Chengjin Zhu, Xieyang Guo, … Albert Weixlbaumer Structural basis for intrinsic transcription termination Linlin You, Expery O. Omollo, … Yu Zhang A ligand-gated strand displacement mechanism for ZTP riboswitch transcription control Eric J. Strobel, Luyi Cheng, … Julius B. Lucks Seungha Hwang ORCID: orcid.org/0000-0002-0028-01671, Paul Dominic B. Olinares ORCID: orcid.org/0000-0002-3429-66182, Jimin Lee1, Jinwoo Kim ORCID: orcid.org/0000-0002-7205-91701, Brian T. Chait ORCID: orcid.org/0000-0003-3524-557X2, Rodney A. King3 & Jin Young Kang ORCID: orcid.org/0000-0002-8493-78901 Bacteriophages Cryoelectron microscopy Transcriptional regulatory elements Transcription, in which RNA polymerases (RNAPs) produce RNA from DNA, is the first step of gene expression. As such, it is highly regulated either by trans-elements like protein factors and/or by cis-elements like specific sequences on the DNA. Lambdoid phage HK022 contains a cis-element, put, which suppresses pausing and termination during transcription of the early phage genes. The putRNA transcript solely performs the anti-pausing/termination activities by interacting directly with the E.coli RNAP elongation complex (EC) by an unknown structural mechanism. In this study, we reconstituted putRNA-associated ECs and determined the structures using cryo-electron microscopy. The determined structures of putRNA-associated EC, putRNA-absent EC, and σ70-bound EC suggest that the putRNA interaction with the EC counteracts swiveling, a conformational change previously identified to promote pausing and σ70 might modulate putRNA folding via σ70-dependent pausing during elongation. Most viruses utilize the host transcription apparatus to express their genes, and viral genomes contain assorted cis- and trans-elements that manipulate the host transcription machineries1,2. Most early genes of lambdoid phages are preceded by transcription terminators; therefore, the host transcription apparatus must be converted to a terminator-resistant form to promote full gene expression of viral genes1,3,4. For λ, anti-termination is promoted by the virus-encoded N protein, which binds to the cis-acting nut sites and suppresses transcription termination5,6,7. By contrast, bacteriophage HK022, discovered in Hongkong in the early 1970s, is related to λ phage but only requires cis-acting RNAs, named put (polymerase-utilization), to promote read through of transcription terminators without any dedicated trans-acting protein factors8,9. Put-mediated anti-termination is efficient and robust, and has been shown to suppress both intrinsic and ρ-dependent transcription terminations9,10,11. The HK022 genome contains two putRNAs—putL and putR, located downstream of the early promoters PL and PR, respectively11. Both putRNAs are ~70-nt long, share sequence similarity, and are composed of two stem-loop structures12. Whereas putL is located just downstream of the PL promoter, putR is located ~270-nt downstream of the PR transcription start site11. The anti-termination activity of the putRNAs persists even on terminators located at least 10 kb away and is not dependent on the tethering between the putRNA and the elongation complex (EC) via the nascent transcript, suggesting that putRNA itself can remain stably associated with the EC as elongation proceeds13. The activity of putRNA is blocked by the host RNA polymerase (RNAP) mutations that are located exclusively in β' zinc-binding domain (β'ZBD)10,14,15. This genetic evidence together with biochemical evidence suggests that putRNA interacts with the RNAP via the β'ZBD16. In addition to anti-termination activity, putRNA exhibits anti-pausing activity. PutL inhibits backtracking at a pause located 21-nt downstream of the second stem (stem II) of putL, and this activity is abrogated by the insertion or deletion of several bases between the putL and the pausing site17. The dependency on the distance between putRNA and its location of action suggests that the anti-pausing and anti-termination activities of putRNA differ mechanistically. Interestingly, putRNA reduces both backtrack and hairpin-dependent pauses like RfaH18. RfaH, a paralog of NusG, recognizes an ops (operon polarity suppressor) sequence on the non-template DNA strand loaded onto the RNAP EC, changes its C-terminal helices into a β-sheet KOW domain fold to become active, and inhibits transcriptional pausing by resisting RNAP swiveling19,20. A structural analysis of the putRNA-associated EC is required to understand the molecular mechanism of anti-pausing and anti-termination activities of the putRNA. In this study, we synthesized the putL RNA using the Eco RNAP σ70-holoenzyme initiated from the native HK022 PL promoter, and captured the modified ECs using a transcription roadblock for cryo-EM analysis. We observed putRNA-associated EC (putEC), putRNA-absent EC (put-less EC), and σ70-bound EC that contains intact putRNA (σ70-bound putEC). Comparison between putEC and put-less EC structures revealed that the putRNA binding to the β'ZBD hinders pausing by reducing the swiveling motion of the EC. Additionally, the σ70-bound putEC structure suggested that σ70 binding to EC might facilitate RNA folding as well as play a role in transcription modulation. Preparation and examination of putEC Because an active form of HK022 putRNA can be produced only by the enzymatic synthesis using host RNAP, we prepared the putRNA-associated EC by initiating RNA synthesis with Eco RNAP holoenzyme and stalling the synthesis using a roadblock protein LacI (lac repressor), as previously described with some optimization for cryo-EM study (Fig. 1)16. Briefly, we first synthesized a DNA scaffold including HK022 PL promoter, the putL sequence, and the lacO sequence (LacI binding sequence) (Fig. 1a). Hereafter, we will use 'putRNA' to denote putL RNA for convenience. Holoenzyme containing Eco core RNAP and σ70 was added to the DNA scaffold to form an open complex, and LacI was added to bind to the lacO sequence on the DNA as a roadblock (Fig. 1b). Upon rNTP addition, RNAP synthesized the putRNA and stalled on the DNA at the roadblock. Excess rNTP was then removed by gel filtration column to prevent further RNA synthesis, and isopropyl β-D-thiogalactopyranoside (IPTG) was added to release the LacI from the DNA scaffold. The complexes were concentrated for cryo-EM grid preparation as well as native mass spectrometry (nMS) analysis. Fig. 1: Scaffold design, reconstitution strategy, and evaluation for the putEC. a The nucleic acid scaffold in the reconstituted putEC. The DNA was synthesized by PCR, and the RNA is synthesized in vitro from Eco RNAP holoenzyme by adding rNTP as described in b. The transcription regulation elements including −35 element, −10 element, putRNA, −10-like sequence, the pausing site, and lacO sequence for roadblocking are shown. Transcription bubble is marked with blue dotted box. Transcript numbering is written for reference. b A schematic diagram of the putEC reconstitution. First, Eco holoenzyme containing Eco RNAP and σ70 binds to the promoter region of the DNA forming RPo. LacI is added to the RPo to make a roadblock at the lacO site and rNTP is added to initiate the transcription. After transcribing putRNA, the RNAP stalls due to the roadblocking LacI. Then, free rNTP is removed by using size-exclusion column before LacI is detached by IPTG to prevent further RNA synthesis. The prepared, active putEC is used for cryo-EM and nMS analyses. c Radiolabeled transcription assay reveals that the wild-type (WT) put sequence (from pRAK3111) did not display paused transcript at the pausing site (lane 2) while the put− sequence showed the paused transcript band (lane 3). The designed DNA scaffold, 7-nt, did not show pausing in the absence of LacI, but stalled at the pausing site when LacI was added (Lane 4, 5). For the lanes 2–5, the transcription reaction was done for 2 min at 37 °C. To show the roadblocked EC is still capable of transcription, we added 2 mM IPTG, incubated for 2 min, and resumed the transcription by adding additional rNTP. The resumed reaction mixture was quenched after 2 min and loaded onto the gel (Lane 6, labeled 'Resume'). Details are in the Method section. RO stands for run-off. d The RNA extracted from the reconstituted putEC is analyzed by nMS. The result revealed two main peaks of C94 and U95, labeled as 94-mer and 95-mer, respectively. The ratio between 94-mer and 95-mer was approximately 2:3. Partial run-off products were observed at higher mass region. Source data are available as a Source Data file. To stall the EC at a site where the RNAP pauses in the absence of putRNA, we generated multiple DNA scaffolds having various distances between the pausing site and the lacO sequence, and performed radiolabeled transcription assay with these scaffolds. For screening, we used a put− DNA scaffold that allows transcriptional pausing at the pausing site as a control (Fig. 1c, Supplementary Note 1, Supplementary Fig. 1, Supplementary Table 1)11. From the screening, we chose a scaffold that has a 7-nt spacer between the pausing site and the lacO sequence (7-nt scaffold), and then attempted to analyze the assembled putEC by nMS using the same workflow as in our previous nMS studies of bacterial ECs21,22,23. However, we were unable to observe the fully assembled putEC due most likely to sample instability or sample heterogeneity and adduction on the long, exposed nucleic acid scaffold during nMS analysis. Nevertheless, nMS analysis of the RNAs extracted from the reconstituted putEC revealed two main populations—one RNA was synthesized until the known pausing site (C94), and the other RNA extended by 1-nt (U95) (Fig. 1d)11,17. The quantity of the U95 RNA was roughly 1.5 times more than the C94 RNA; therefore, we modeled the U95 RNA placing the U95 nucleotide at the i + 1 site. Since the C94 and U95 nucleotides are located at the i and i + 1 sites, the modeled structures would exhibit the same RNA-DNA register both for the C94 and the U95 RNA-containing ECs at the post- and pre-translocated states, respectively (See below). Cryo-EM structures of the putEC in three conformations Cryo-EM analysis of the putEC prepared by promoter-dependent transcription initiation, transcription elongation, and LacI roadblocking revealed three EC populations at sub-4 Å resolution: (1) the putEC (3.2 Å-resolution, 36.9% of the EC particles) that contains well-folded putRNA, (2) the put-less EC (3.6 Å, 22.4%) that does not display any well-defined putRNA density and (3) the σ70-bound putEC (3.6 Å, 40.7%) that contains both σ70 and putRNA (Table 1, Supplementary Figs. 2 and 3). We also observed a population consisting of the RNAP holoenzyme loaded onto the template DNA. This population probably resulted from abortive initiation and generated a 3.0 Å-resolution map. This complex is not discussed here because the structures of the holoenzyme open complex have been described in previous reports24,25,26. Table 1 Cryo-EM data collections, processing, and validation In the cryo-EM structure of the putEC, the putRNA was located at the opening of the RNA exit channel of the EC adjacent to the β'ZBD (Fig. 2a). This location is consistent with the put-inactivating RNAP mutations and potentially would restrict the RNA hairpin formation in the adjoining RNA exit channel via electrostatic repulsion (Supplementary Fig. 4)10,27. The quality of the cryo-EM map allowed us to build the highly-structured putRNA de novo (estimated local resolution of the cryo-EM map around the putRNA was ~3.5 Å; Fig. 2b, c, Supplementary Note 2, Supplementary Figs. 5–7). The modeled putRNA from U2* to U74* contains twenty-one Watson-Crick (WC) base pairs and five non-canonical base pairs, A9-G35 (Saenger class VIII), G12-U32 (Saenger class XXVIII), G43-A64 (Saenger class XI), G42-U65 and C44-A63 (Supplementary Fig. 8). To distinguish the nucleic acid residues of the putRNA from the amino acid residues in the RNAP, we have added an asterisk () to the residue number of the putRNA throughout this manuscript. Surprisingly, the cryo-EM structure of the putRNA was different from previously published data11,12 as follows (Fig. 2b–d): First, the 5'-end of the putRNA is not C10 but A3. In the structure, the putRNA region from A3 to G7* makes an RNA duplex with the opposite strand from U19* to C15. Interestingly, this corresponds to the result of the putL V1 RNase reaction, which suggested the presence of RNA duplex in the upstream of C1012. Furthermore, this RNA duplex interacts with another RNA strand from U21* to C25* forming an unexpected minor groove RNA triplex structure28. The deletion of the third RNA strand, Δ20−23, decreased the anti-termination activity of the putRNA by ~50%12, indicating that the triple helix region has a significant effect on the function of the putRNA. Second, G35, which was expected to be located at the bifurcation point of the two stem regions in an unpaired state, base pairs with A9. This G35-A9 base pair provides a platform for β'ZBD binding and stabilizes the overall structure of the putRNA. This base pair explains why the G35U mutation retained 70% of the anti-termination activity while the G35A mutation completely abolished the activity12. Meanwhile, A9C mutation abolished the in vivo anti-termination activity, implying complicated effects of mutations on the G35-A9* base pair12. Third, the putRNA contains a bulged loop region (from C26* to G29) in the middle of stem I, in contrast to the prediction that stem I has a loop region at the end of the stem I ranging from C18 to C26. This region also provides an interface for binding to the RNAP. At last, the middle region of stem II exhibits distinct base pairings compared to the predicted structure. The middle region of stem II in the cryo-EM structure contains three non-canonical base pairs with three unpaired bases instead of having one non-canonical base pair with five unpaired bases in the predicted structure. This region has relatively high local resolution indicating its structural stability, and makes interfaces with RNAP and the stem I of the putRNA. Fig. 2: The putEC structure. a The cryo-EM map of the putEC (3.2 Å-resolution) is rendered as semi-transparent surface, colored as labeled, and superimposed with the putEC model. The RNAP domain that binds to putRNA, β'ZBD, is highlighted in magenta. The putRNA boxed in red is re-drawn in cartoon format and colored from blue to red according to the residue numbers. b The putRNA is drawn in an orientation revealing its overall structure more clearly. The putRNA figure contains 73 nucleotides from U2 to U74* (the first and the last base-pairing residues are A3* and C72, respectively), and has 'V'-shape with two long stem-loop structures. The first stem region forms an RNA triplex. c A schematic diagram of the putRNA structure and its potential interactions with RNAP β' and β subunits. The putRNA bases forming canonical WC base pairs and the unpaired bases are colored in light green. The bases forming non-canonical base pairs are colored in light purple. The RNA triplex region is marked in gray color, and the bases in the triplex are colored in orange. The pink and blue boxes represent β' and β residues, respectively, and the lines between the RNAP residues and the putRNA nucleotides indicate potential hydrogen bond or salt bridge interactions. The ribose, shown as pentagon, and the phosphates, shown as circles, are colored in gray and the residue numbers of the putRNA are written in the ribose. Transcription bubble is marked with blue shade. d (Left) The predicted putRNA structure11. The pausing site and the put− mutation that abolishes the anti-termination activity are marked. Non-canonical base pairs are represented with red dotted line. (Right) The putRNA structure modeled based on the cryo-EM map. Triple helix region is marked by a gray box. Non-canonical base pairs are represented with red dotted lines. Transcription bubble region and the pausing site are marked. Interactions between the putRNA and the EC In the putEC structure, the 'V'-shaped putRNA binds to the prominent β'ZBD by its pothole formed in the center of the 'V' (Fig. 3a). The β'ZBD fits snuggly to the putRNA surface, generating a 1130.3 Å2 interface area formed by ~33% of the total putRNA residues29. At the backside of the putRNA-β'ZBD interface, the N-terminal loop of the β flap-tip helix makes significant contact with the putRNA with an interface area of 281.6 Å2. Most of the potential interactions between the putRNA and the RNAP comprise polar interactions such as salt bridges, hydrogen bonds, cation-π interactions, and long-range ionic interactions. Although the resolution of the map is not sufficient to specify these short-range interactions, we suggested possible interactions for reference (Figs. 2c, 3b, Supplementary Table 2). At the bifurcation point of the two stem structures, β'R77 locates like a wedge to separate G35 and G36* and forms a cation-π interaction with G35. This cation-π interaction is often found between the terminal, exposed base of a nucleic acid bound to a protein and the protein loop that confines the nucleic acid. In addition, β'L78 and β'K79 are located between G35 and G36, stabilizing the separation of stem I and stem II of the putRNA. Fig. 3: Interaction between the putRNA and the RNAP. a The binding interface between the putRNA and β'ZBD is split and each region is rotated by 90° in the opposite direction to display the binding surface. The cryo-EM map is rendered as semi-transparent surface, colored as labeled, and superimposed with the structure. The RNA and amino acid residues in contact with the other partner are drawn in stick format. The amino acid residues forming polar interactions via side chains are colored and labeled in magenta while the ones interacting via peptide backbone are in light pink (detailed in b). The β'ZBD binding region mapped onto the putRNA is marked by dotted line. b (Left two boxes) Close-up views of the putRNA-β'ZBD interface with potential interactions - salt bridges (red dotted line) and hydrogen bonds (green dotted line). The labels of the amino acids forming hydrogen bonds via backbone peptides are underlined on their label. The putRNA residues are colored according to their residue numbers. (Right box) Close-up view of the putRNA-β flap-tip helix. The cryo-EM density is superimposed on the model in all three boxes. c Measurement of the anti-pausing activities of the putRNA mutants. From the radiolabeled transcription assay, the relative anti-pausing activity of each mutant is calculated and plotted (details in Methods). The mutants were classified into three groups according to the activities and labeled as 'Inactivating' (<20%), 'Inert' (>90%), and 'Others' (between 20% and 90%). The relative activity of 0.9, which separates the inert and other mutations, is marked by a dotted line. The assays were done in triplicates (n = 3 independent experiments), and data are presented as mean values ± SEM. All data points are plotted as dots in different colors for clarity. d Conservation of putRNA residues and the location of the putRNA mutations in c. The sequence alignment is done in LocARNA server30. The conservation score in each residue is measured from the alignment and colored on the structure as indicated in the color bar in the figure. Source data are available as a Source Data file. A mutant named put−, or mutant G, has the sequence A43GAUC47 and does not exhibit anti-termination activity11. Our transcription assay revealed that put− also has poor anti-pausing activity (Fig. 1c). In the structure, this region does not directly interact with the RNAP, however, its counter-strand, from the 59th to 64th residues, forms a central area of the binding interface. Therefore, the base substitutions in the put− mutant likely change the structure of the binding interface and disrupt putRNA binding to the RNAP. It is also possible that these mutations interfere with the proper folding of the putRNA as well. To test the validity of the structure of the putEC modeled in the cryo-EM density, we introduced assorted mutations in the template DNA and performed in vitro radiolabeled transcription assays to examine the effects of the mutations on the anti-pausing activity (Fig. 3c, Supplementary Fig. 9). For the quantification of the anti-pausing activities of the mutants, the anti-pausing activities of wild-type put and put− were set to 1 and 0, respectively, and the anti-pausing activity of each mutant was located on a linear scale accordingly (Details are in the Methods section). To display the location of the mutated residues as well their conservation, the conservation of the putRNA residues was calculated from the sequence alignment with ten known put sequences and marked by color (Fig. 3d, Supplementary Fig. 10a)30,31. Among the twenty-three mutations we generated, eleven mutants showed ≤ 20% anti-pausing activity (named 'inactivating' mutations) and three mutants showed ≥ 90% anti-pausing activity (named 'inert' mutations). The inactivating mutations, Δ3−7, U28A, U28C, G35A, G35U, G35C, G45A, A64G, G35C/A9G, G35A/A9G, and G35A/A9U, suggest that (1) the 5'-region (from A3* to G7) is essential for the anti-pausing activity. A3GACG7 and its base-pairing region, U19CUGC15 have relatively high conservation scores of (6,6,4,9,9) and (7,7,5,10,10), respectively. This region is the first RNA duplex formed during the putRNA synthesis, and therefore, may provide a platform for further RNA folding. (2) U28, which protrudes toward the β'ZBD and binds to a small pocket is essential for the function. Interestingly, while U28A and U28C abolish the anti-pausing activity, U28G retained ~60% of the activity. From the structure, we substituted the U28 with the other bases and found that G can form three hydrogen bonds with the surrounding β' residues while A and C form two and one potential hydrogen bonds, corroborating the result of the mutational study (Supplementary Fig. 10b). Interestingly, the original put residue, U28* forms fewer hydrogen bonds than guanine and adenine, but exhibits better activity than these, implying that U28* might have additional role(s) besides binding to the RNAP, or the mutants might have different structures from the modeled ones (Discussed below). (3) All of the G35* mutations we generated abrogated the anti-pausing activity of putRNA. We expected that the double mutants, G35C/A9G, and G35A/A9G might have some activity because they preserve the predicted base-pairing of G35-A9 in the structure. However, mutating G35* to any base abrogated the anti-pausing activity and this was not recovered by the mutation of the base-pairing partner, implying that G35, and possibly its base-pairing partner A9, may have sequence-specific roles in the anti-pausing activity. We noticed that G35U exhibited ~70% anti-termination activity in vivo12. This discrepancy could come from the different conditions encountered in vivo vs. in vitro. For example, the G35U might form some intact or partially active putRNA in vivo, possibly aided by an unknown cellular factor(s) whereas in vitro synthesized putRNA containing G35U* could be inactive. (4) We also found that A64* is critical for the anti-pausing activity. This result is also consistent with the structural data because it contacts the stem I region of putRNA and the RNAP. All the inert mutations are of U20, which lacks any significant interaction with other residues, supporting our structure. The remaining nine mutants exhibited moderate activities suggesting a significant, but not critical role of the residues (A8, U21, C25, U32, G43). In summary, our mutagenesis study supports our cryo-EM structure of the putEC. The comparison of the putEC, the put-less EC, and other ECs To determine if putRNA binding to the EC changes the conformation of the EC to suppress transcriptional pausing, we aligned the putEC with multiple EC structures including non-paused EC (PDB 6ALF), RNA hairpin-paused EC (PDB 6ASX), backtracked PEC (paused EC) (PDB 6RIP), and the put-less EC determined here (Fig. 4a, b, Supplementary Table 3)21,32,33. We assume that the put-less EC contains a roadblocked but unfolded RNA because (1) the majority (>~70%) of the sample was roadblocked properly (Fig. 1d), and both the putEC and the put-less EC together comprise ~60% of the EC population in the cryo-EM data, (2) the third EC class, σ70-bound putEC shows extra RNA duplex density connected to the putRNA suggesting that this class was not properly roadblocked, and (3) the put-less EC map contains some weak RNA density around the RNA exit channel and the β'ZBD, implying that the RNA is present, but it is not well-structured. We suggest this put-less EC could serve as a good negative-control model as shown in a previous study32. Fig. 4: Comparison of putEC, put-less EC, and other ECs. a The cryo-EM map of the put-less EC (3.6 Å) is rendered as semi-transparent surface, colored as labeled, and superimposed with the final put-less EC model. Different with the putEC map, no structured RNA density was observed except that of an RNA-DNA hybrid. b The swiveling of the putEC and the put-less EC are compared with that of other representative ECs containing non-paused EC (PDB 6ALF), backtracked PEC (swiveled, PDB 6RIP), and hisPEC (PDB 6ASX)21,32,33. All five ECs were aligned according to the core module. (Left) Front view of the aligned ECs. Non-paused EC is drawn in semi-transparent surface and loop formats in gray color. Other ECs are drawn in loop format and colored as labeled. βSI2 domains show distinct conformations between non-paused EC/putEC and put-less EC/hisPEC, with ~15 Å displacement. βSI2 of the backtracked PEC is located in the middle of the two groups. (Right) Top view. Only the swivel modules (clamp, shelf, jaw, SI3 domains and C-terminal region of β' subunit) of the aligned ECs are drawn for clarity. The swiveling angles relative to the non-paused EC are shown. c Comparison of the RNA-DNA hybrids near RNAP active site with the putEC, the put-less EC, and the post-translocated, non-paused EC (PDB 6ALF)21. The RNA-DNA hybrid, bridge helix (BH), and Mg2+ ions are colored as labeled. The cryo-EM densities of the putEC and put-less EC hybrids are shown in mesh, and their Mg2+ ions are shown as semi-transparent spheres. The base-pair locations are marked with background boxes, and the BH beside the i + 1 site is drawn for reference. d The putEC contains 11-bp RNA-DNA, one base pair longer than other reported bacterial EC structures. The lid and bridge helix (BH) of β' are colored in light pink, and each nucleotide is numbered from downstream to upstream template DNA. We first examined the swiveled states of the ECs (Fig. 4b). Swiveling indicates the rigid-body rotation of a set of domains—the clamp, dock, shelf, jaw, SI3, and the C-terminal region of the β' subunit—about an axis parallel to the bridge helix toward the RNA exit channel, and known to interfere with the proper folding of the trigger-loop which is required for efficient nucleotide addition to the nascent RNA. Swiveling was first introduced from the structural study of hisPEC, and later revealed in the backtracked PEC, implying that the swiveling motion potentially plays an important role in both RNA hairpin pause and backtrack pause32,33,34. The alignment of the EC structures according to the core module revealed that the putEC structure is most similar to the non-paused, active EC conformation, having the lowest RMSD values between Cɑ-carbons of domains as well as the smallest swiveling angle of 1.2° (Fig. 4b, Supplementary Table 3). The put-less EC is more swiveled than the putEC, having a swivel angle of 1.8°, although the swiveling angle of the put-less EC was less than that of the hisPEC or backtracked PEC (3.1° and 2.6°, respectively; Supplementary Table 3). Interestingly, the conformational difference between the putEC and put-less EC is more noticeable in the βSI2 (or βi9) region with 15 Å-distance between the Cɑ atoms of βE1006, which is located at the end of the βSI2 domain. While the swiveling motions of the aligned ECs are relatively continuous with the rotation angles from 1.2° to 3.1°, the arrangement of the βSI2 is more discrete – the βSI2 in the putEC overlaps with that of the non-paused EC while the βSI2 of the put-less EC is in the same location with the hisPEC. Interestingly, the βSI2 of the backtracked PEC is located between the two conformations. These conformational features suggest that the proper folding and binding of the putRNA to the EC moved the EC toward the non-swiveled, active state, aiding pause escape or omission. The strength of the RNA-DNA hybrid influences pausing and termination35,36. Therefore, we compared the RNA-DNA hybrid of the putEC and the put-less EC (Fig. 4c, left). In the putEC, the active site region of the RNA-DNA hybrid exhibited a post-translocated state similar to the non-paused EC at the high threshold value of the map. As the threshold value decreases, the putEC map revealed a density blob for a nucleotide base that base pairs with the template DNA base at the i + 1 site. This density became connected to the nascent RNA at the lower density threshold. As stated above, we suspect that this results from the mixed population of the nascent RNAs roadblocked at either +94 or +95 position, having either post- or pre-translocated states, respectively. However, we did not observe any classes having a folded trigger-loop with the SI3 domain shifted closer to the βlobe domain as in the Eco RNAP structure of the pre-translocated state24. In addition, the putEC contained 11 template DNA bases in the RNA-DNA hybrid, in contrast to other reported EC structures (Fig. 4d). To contain one additional nucleotide in the main channel, the lid, which is known to aid the unwinding of the RNA-DNA hybrid, is pushed by about 2.6 Å (by the Cɑ atom of β'256D) compared to the known non-paused EC (Supplementary Fig. 11a)19,21,33. However, it is not certain if this 11-nt hybrid is just an alternative conformation of an EC, or a specific conformation in the putEC. The put-less EC showed distinct RNA-DNA base-pairing at the i + 1 site (Fig. 4c, right). In the put-less EC, the template DNA base at the i + 1 site is more tilted toward the RNA base at the i site; therefore, it is not optimally placed for substrate binding. In fact, the RNA base at the i site is more closely associated with the DNA base at the i + 1 site than that of the i site. Consequently, the base-pairing hydrogen bonds are broken between the template DNA base and the nascent RNA base at the i site. The remaining region of the RNA-DNA hybrid of the put-less EC overlaps well with that of the non-paused, active EC as in the putEC. The conformational difference of the nucleotides at the active site between the putEC and the put-less EC indicates that putRNA binding to the β'ZBD influences the active site conformation, even though the catalytic magnesium ion is ~62.5 Å away from the zinc ion in the β'ZBD. This was also shown in the hisPEC structure, where the pause hairpin placed in the RNA exit channel has an influence on the active site as well as the bridge helix32,34,37. The length of the template DNA in the RNA-DNA hybrid of the put-less EC was also 11-nt, implying that this longer RNA-DNA hybrid is not caused by the putRNA. In addition to these changes, we also observed that the RNAP domains of the putEC have similar locations to those of the non-paused EC while the domains in the put-less EC have a similar arrangement with those of backtracked PEC (Supplementary Table 4). Although we could not find any density for the backtracked RNA in the put-less EC, the pausing site was expected to have a backtrack pause. In summary, from the structures of the putEC, put-less EC, and other ECs, we found that the putRNA binding to the EC leads to the anti-pausing activity by promoting the active, non-swiveled conformation of the EC. σ70-bound putEC structure The third EC population, σ70-bound putEC, contains a σ70 bound to the clamp helices in addition to the well-folded putRNA as in the putEC (Fig. 5a, Supplementary Fig. 2). In contrast to the holoenzyme structure, the σ70-bound putEC map reveals only σ1.2, σNCR and a part of σ2, indicating that the σ2 binding to the EC is relatively stable while the other σ domains are very mobile as predicted in a prior study38. It has been reported that σ70 can remain associated with RNAP after promoter escape and the association is enhanced when the non-template DNA contains a −10 element-like sequence in the promoter-proximal region that induces σ-dependent pausing39,40,41. In particular, σ-dependent pausing provides a time and space window for the anti-termination λQ protein to bind to the EC and read through the intrinsic terminator42,43. Recently, cryo-EM structures of σ70-bound ECs were reported in the context of 21Q-, λPR'-, and Qλ-associated ECs44,45,46,47,48. While these complexes are at the paused state in that the σ2 domain interacts with a −10-like sequence, our σ70-bound putEC is not in a σ-dependent paused state and contains > 100 base-long RNA having a σ70 in a different conformation from those in other σ70-bound ECs (Fig. 1d). Fig. 5: The σ70-bound putEC structure. a The cryo-EM map of the σ70-bound putEC (3.6 Å) is rendered as semi-transparent surface, colored as labeled, and superimposed with the final σ70-bound putEC model. The unexpected RNA hairpin found in the RNA exit channel is zoomed, rotated by ~30°, and re-drawn with the density in a red box. The downstream duplex DNA was not observed in the main channel. Instead, an unknown blob, colored in light brown, was present in the location. b Comparison of the σ70-RNAP binding between the RPo and the σ70-bound putEC. Nucleic acids were not drawn for clarity. β' clamp helices (β'CH), the main structural element interacting with σ70 in the RPo, are highlighted in magenta color. σ70 is colored by domains as labeled. β'-clamp-toe, which interacts with σNCR in the σ70-bound putEC, is colored green. In the σ70-bound putEC structure, we noticed that the RNAP contains an open clamp (79.3 Å opening), which is ~20 Å larger than the non-paused EC23. This suggests that the σ70-bound putEC is in an inactive state. We suspect that this class might represent the partial run-off EC population that appeared in the nMS analysis (Fig. 1d) because (1) the main channel of the RNAP did not contain downstream duplex DNA while the RNA-DNA hybrid was present and (2) an RNA duplex density, which is connected to both putRNA and the RNA-DNA hybrid, was observed in the RNA exit channel, indicating that the RNA was transcribed beyond the roadblock site (Fig. 5a, Supplementary Fig. 11b). We used the RNAfold Server to search potential RNA secondary structures in the template DNA and found that it contains a potential RNA hairpin sequence downstream of the roadblock site (Supplementary Fig. 11b)49. We, therefore, modeled the RNAP and nucleic acid scaffold into the map and found that the potential RNA hairpin matches well with the extra density observed in the RNA exit channel (Fig. 5a, Supplementary Fig. 11b). The location of this extra RNA duplex overlaps with the pause hairpin in the hisPEC32,34. The putRNA density in the σ70-bound putEC was at a lower resolution than that in the putEC; however, the putRNA map region was identical to that in the putEC. The RMSD of the whole atoms in the putRNA region in the σ70-bound putEC and the putEC was only 0.839 Å. To compare the σ2-RNAP interaction in the initiation and the elongation stages, we aligned the RNAP clamp-σ2 domain regions from the σ70-bound putEC and the recently published RPo (RNAP-promoter open complex) structure25. For the σ70-bound putEC, we only modeled the visible part for the σ70 (σ70 residues 112–151 and 214–447). Then, we compared the two structures only via the modeled σ70 regions and other σ domains were excluded in the comparison discussed below. Not surprisingly, the binding interface between the σ70 and the RNAP, in particular, the β'clamp domain, was different between the RPo and the σ70-bound putEC (Fig. 5b). The binding interface between the β' subunit and the σ70 was 812 Å2 in the RPo and the interface mostly occurs on the β'clamp helices. By contrast, in the σ70-bound putEC, the interface area was 1287 Å2. This unexpected increase in the binding area results from the newly-formed interface between β'-clamp-toe domain (ranging 144–179)50 and the σ70NCR, the non-conserved σ70 region between σ1.2 and σ2.1 (ranging 274–307 and 359–374 in the structure, Fig. 5b) that does not participate in the RNAP-σ70 interface in the RPo. Since both β'-clamp-toe and σ70NCR are conserved in the γ-proteobacteria, the interaction between these two domains might be specific for the bacteria class. In addition, the shifted position of the σ2 domain in the σ70-bound putEC is more suitable for the σ70 to associate with the progressing EC because this conformation provides space for the upstream DNA to rewind and exit from the main channel of the RNAP. If the σ70 is bound to the RNAP as in the holoenzyme, σ70 would clash with the exiting upstream duplex DNA. However, at the moment, further investigation would be required to see whether these new interactions between the σ70 and the RNAP in the σ70-bound putEC are due to the transcription stage transition from initiation to elongation, or to the clamp opening which inactivates the transcription activity of the RNAP. Additionally, we found a low-resolution blob in the main channel for the downstream DNA (Fig. 5a). The DNA scaffold used in the study spans to +122 position while the RNA modeled in this map ends at +105. nMS analysis revealed three RNA populations of 110-mer, 114-mer and 116-mer (Fig. 1d). Therefore, there should be some downstream duplex DNA around the RNAP. However, the low-resolution of the blob prevents us from locating any specific molecule in the density. We suspect that the blob could be either from the downstream duplex DNA, which is very mobile due to the open clamp conformation, or from the σ701.1 because the σ701.1 is known to bind at the position in the holoenzyme before the enzyme binds to promoter DNA. We would need further investigation to confirm this speculation. In this study, we extended prior studies on the putRNA by determining its three-dimensional structure when complexed with RNAP. Our result corroborates previous analyses suggesting a two-stem structure with multiple indents and bulges. However, cryo-EM structures also revealed new and unexpected features such as an unexpected boundary of the put transcript, a short triple RNA helix in the putL stem I, and alternative base pairs. The importance of many of these features is strengthened by the observed effect of specific put mutations11,12. The structure provides clear physical evidence that the putRNA binds to the β'ZBD, a result that is strongly supported by prior genetic and biochemical experiments on putRNA. The structure also revealed a mechanistic explanation for the anti-pausing activity promoted by putRNA-RNAP interaction. When putRNA is bound to the β'ZBD, the RNAP is held in a non-swiveled, active conformation, which is associated with anti-pausing activity as previously shown in the RfaH-associated EC19. In contrast, a put-less EC exhibited a swiveled conformation suggesting that the EC is in a paused state when transcription elongation is physically blocked at a pausing site. Together, the structures revealed that putRNA promotes RNA synthesis by resisting swiveling. We were surprised to observe a putEC population that retained σ70 even though the EC had progressed about 100 nucleotides from the start of transcription. The occurrence of the σ70-bound putEC and its structure suggests a few intriguing points. First, the σ70-bound EC successfully folded putRNA, even more efficiently than a complex lacking σ70. We found that the ratio between the putEC and the put-less EC is roughly 2:3 from the number of particles in each class, presumably reflecting the success rate of putRNA folding in vitro. Curiously, there was no put-less σ70-bound EC, suggesting that the presence of σ70 aided the proper folding and stabilization of putRNA. We found that the putL sequence contains a weak −10-like sequence (NANNAT) located at positions +23 to +28 relative to the start of the putL transcript, which lies on the third strand of RNA triple helix and a bulge region of stem I (Figs. 1a, 2b, c). A −10-like sequence is known to induce σ-dependent pausing by engaging its non-template DNA region with the σ2 domain51. We suggest that this sequence may cause σ-dependent pausing which facilitates putL folding by providing more time. Notably, among the ten putRNA sequences we aligned, all the putLs contain the identical −10-like sequences while all putRs do not (Supplementary Figs. 10a, 11c). Therefore, we speculate that σ-dependent pausing may be necessary for putL folding but not for putR which is located further downstream of its promoter. In addition, U28* is completely conserved in putL and the critical 6th residue of the −10-like sequence. Although U28G exhibited intermediate activity in our mutagenesis study, U28A and U28C nearly abolished activity, supporting the existence and importance of σ-dependent pausing at this position. Furthermore, we examined the ratio between σ70-bound EC and σ70-unbound EC from both the putEC sample and the put−-EC to see if the presence of put affects the σ70 retention (Supplementary Fig. 12). The put−-EC was prepared in exactly the same way as the putEC preparation except the put− template was used as the DNA scaffold and did not show any well-folded putRNA density in the cryo-EM maps. From the cryo-EM data analysis, the percentages of σ70-bound EC in the putEC (having intact put) and the put−-EC sample were ~40.7% and ~44.2%, respectively, suggesting that the presence of put does not affect σ70 retention. Second, the σ70-bound EC was resistant to the LacI roadblock. The σ70-bound putEC revealed an extra density for a duplex RNA in the RNA exit channel, suggesting that the retained σ70 modified the EC to overcome the roadblock during elongation (Figs. 1d, 5a, Supplementary Fig. 11b). Structural studies on prokaryotic anti-termination complexes including λN, Q21, Xoo P7, Qλ, and HK022 put suggest general strategies for anti-termination7,44,45,47,48. (1) The anti-termination factors inhibit RNA hairpin formation by either narrowing the channel or hindering the RNA hairpin folding (Supplementary Figs. 4, 13). The RNA exit channel is thought to aid RNA hairpin formation by its positively-charged residues located inside the channel32. In Q21, Qλ and Xoo P7 anti-termination complex, the anti-termination factors, Q21, Qλ, and P7 proteins bind at the mouth of the RNA exit channel and confine the channel (Supplementary Fig. 13). The narrowed RNA exit channel only allows single-stranded RNA to move through it and restricts nascent RNA folding for hairpin-dependent pausing and intrinsic termination. In λN anti-termination complex, λN binding to the EC remodels the bound NusA and NusE to destabilize the RNA hairpin folding. In addition, the rearranged Nus factors bind to β flap-tip, which stabilizes RNA hairpin pause, possibly preventing the flap-tip from assisting RNA hairpin pausing and termination52. Like λN, HK022 put also does not narrow the RNA exit channel directly. Instead, the phosphate backbone of the putRNA is located near the RNA exit channel, prohibiting the RNA hairpin formation with its negative-charged surface. Modeling an RNA duplex in the RNA exit channel of putEC shows that the phosphate backbones of the modeled RNA duplex and the putRNA are just ~5 Å apart from each other (Supplementary Fig. 4). In addition, the β flap-tip binds to the putRNA, possibly sequestering it from assisting RNA hairpin pause as in the λN-anti-termination complex. (2) In general, anti-termination proteins stabilize the elongation-proficient conformation of EC. λN transverses the RNAP hybrid cavity stabilizing the active form of the EC and binds to the upstream duplex DNA, enhancing the anti-backtracking and anti-swiveling activity of NusG. In the Q21-EC structure, Q21 binding is not compatible with swiveled conformation. Therefore, Q21 counteracts swiveling, leading to anti-pausing47. Our data suggest that putRNA also reduces swiveling. This stabilization of the active form of an EC may consolidate the RNA exit channel so that it can no longer accommodate the folding of secondary structures that promote pausing and termination53,54,55. Komissarova et al.17, found that ΔU68 does not suppress termination, but retains anti-pausing activity in vitro. U68* is located at the lower region of stem II like a wedge, forming no base-pairing. According to our modeling, the presence of U68* kinks the stem II ~19° (Supplementary Fig. 14). This perturbation might weaken the stability of the putRNA folding by widening the space between the two stem-loop structures. In addition, the structural change would affect the interface between the putRNA and the RNAP because the interface is composed of putRNA residues from both stem I and II. Therefore, putRNA without U68* might be well-folded and reduces pausing immediately after synthesis but may unfold or dissociate from the RNAP before encountering terminators located further downstream. Alternatively, the mutant RNA may not be able to adopt an anti-terminating structure which could be different from the anti-pausing structure in vitro. In λ phage paradigm, the anti-termination factor λN plays the role of gatekeeper for the infection process. In other words, λN accumulation is required to transcribe early genes of the genome. HK022, instead, has Nun protein, which competes with λN and blocks λ transcription. In addition, the HK022 genome harbors the put element in the place for the λ nut (N-utilization) sites, which are required for the action of λN. By substituting the N protein with Nun, HK022 acquired immunity against its competitor, λ. HK022, instead, lacks a λN-like anti-termination factor, but relies solely on the putRNA to promote full expression of its early genes. These differences benefit HK022 survival, without increasing transcription regulation complexity. In this study, we investigated the anti-pausing mechanism of putRNA. Since transcriptional pausing is a prerequisite of transcriptional termination, our results provide important insights into the mechanism of putRNA action. It remains possible that putRNA may adopt different structures and/or interactions with RNAP to promote anti-termination as prior studies indicate that anti-pausing and anti-termination activities differ. To deepen our understanding of these events, structural studies on the putRNA-associated EC at a terminator sequence would be required. Eco RNAP was prepared as described previously25,56. Briefly, pET-based plasmid that contains rpoA (α), rpoB (β), rpoC (β') with C-terminal deca-histidine tag and rpoZ (ω) was co-expressed with a pACYCDuet-1 plasmid contained rpoZ in BL21(DE3) (Novagen). The cells were grown at 37 °C in the LB broth media in the presence of 100 μg/mL Ampicillin and 34 μg/mL Chloramphenicol, and transferred to 30 °C when the OD600 reached 0.3. Protein expression was induced at an OD600 of 0.6–0.8 with 1 mM IPTG for 4 hours. Cells were harvested, resuspended in lysis buffer (50 mM Tris pH 8.0, 5% glycerol, 1 mM EDTA (pH 8.0), 1 mM ZnCl2, 10 mM DTT, home-made protease inhibitor cocktail), and lysed by French Press (Avestin) at 4 °C. The lysate was precipitated by adding polyethyleneimine (PEI, 10% (w/v), Sigma Aldrich) to a final concentration of 0.6% (w/v) dropwise. The pellets were washed three times with wash buffer containing TGED (10 mM Tris pH 8.0, 5% glycerol, 0.1% EDTA pH 8.0, 10 mM DTT) + 0.5 M NaCl, and the RNAP was eluted from the pellet with elution buffer (TGED + 1 M NaCl). The eluted RNAP was precipitated by adding ammonium sulfate (35 g per 100 ml solution) and eluted again with chelating buffer (20 mM Tris pH 8.0, 1 M NaCl, 5% glycerol) to be loaded onto Hitrap IMAC HP columns (Cytiva) for purification by Ni2+-affinity chromatography. The pulled protein by adding imidazole gradient was dialyzed in TGED + 100 mM NaCl buffer and loaded onto a Biorex-70 column (Bio-rad) for ion exchange chromatography. Eluted RNAP by NaCl gradient was concentrated by Amicon Ultra centrifugal filter (Merck Millipore), and loaded onto HiLoad 16/600 Superdex 200 pg column (Cytiva) equilibrated SEC buffer (TGED + 0.5 M NaCl) for size-exclusion chromatography. The purified protein was supplemented by 15% glycerol, flash-frozen in liquid nitrogen, and stored at −80 °C until use. Full-length Eco σ70 was expressed from pET21-based expression vector encoding an N-terminal hexa-histidine tag followed by a PreScission protease (GE healthcare) cleavage site. The full-length Eco σ70 plasmid was transformed BL21(DE3) cells and grown at 37 °C. Protein expression was induced at an OD600 of 0.7 with 1 mM IPTG and incubated for 4 hours at 30 °C. Cells were harvested, resuspended in σ70 lysis buffer (20 mM Tris pH 8.0, 500 mM NaCl, 5% Glycerol, 5 mM Imidazole, home-made protease inhibitor cocktail) and lysed by French Press. The supernatant was loaded to Hitrap IMAC HP column (Cytiva) equilibrated with 20 mM Tris pH 8.0, 500 mM NaCl, 5% glycerol. The eluted protein by adding imidazole gradient was concentrated using Amicon Ultra centrifugal filter (Merck Millipore) and injected to HiLoad 16/600 Superdex 200 pg (Cytiva) equilibrated in TGED + 500 mM NaCl. The final elution was flash-frozen using liquid nitrogen after adding 15% glycerol. Lac repressor (LacI) was purified as described previously57. LacI-containing pBAD plasmid with Kanamycin resistance (pBAD_Kan-LacI) was obtained from Addgene (plasmid #79826). BL21(DE3) cells that were transformed with the plasmid were grown overnight at 37 °C in 2X YT media containing 50 μg/mL Kanamycin. The seed culture was added to 2× YT media containing 50 μg/mL Kanamycin at 1:100 ratio, grown at 32 °C for 2 hours, and moved to 16 °C. Protein expression was induced with 0.2% l-arabinose for 16 hours incubation right after changing the temperature to 16 °C. Cells were harvested and lysed by French Press in lysis buffer (50 mM sodium phosphate buffer pH 8.0, 500 mM NaCl, 20 mM Imidazole, 2.5% glycerol, 1 mM DTT, 10 mM MgCl2, 0.1% Tween-20, 1 mg/mL lysozyme, home-made protease inhibitor cocktail). The lysate was added by 1000 U of DNaseI, and centrifuged to remove cell debris. The supernatant was loaded onto Hitrap IMAC HP (Cytiva) that pre-equilibrated with 50 mM sodium phosphate buffer (pH 8.0), 500 mM NaCl, 20 mM imidazole, 2.5% glycerol, and 0.2 mM DTT. Protein was eluted with 20 mM sodium phosphate buffer (pH 7.4), 300 mM NaCl, imidazole gradient from 30 to 300 mM and concentrated using 30 K MWCO Amicon Ultra Centrifugal Filter (Merck Millipore). The concentrated protein was injected onto HiLoad 16/600 Superdex 200 pg (Cytiva) gel filtration column equilibrated with 20 mM Tris-HCl (pH 8.0), 150 mM KCl, 5 mM MgCl2, and 1 mM DTT. The final eluted protein was added by 15% glycerol, flash-frozen, and stored at −80 °C until use. Radiolabeled in vitro transcription assay In vitro transcription assay is performed as described previously58. Holoenzyme was reconstituted by mixing Eco RNAP and Eco σ70 with 1:2 molar ratio, and incubating for 15 min at 37 °C. Holoenzyme and DNA were mixed with 4:1 molar ratio in glutamate-based T buffer (20 mM Tris-glutamate pH 8.0, 10 mM Mg-glutamate, 150 mM K-glutamate, 5 mM DTT), and incubated at 37 °C for 10 min to make RPo. RPo and LacI were mixed with 1:10 molar ratio and incubated at 37 °C for 10 min. The final concentration of holoenzyme and template DNA in the reaction mixture was 50 nM and 12 nM, respectively. Transcription was started by adding rNTP mix to final concentrations of 200 µM ATP, 200 µM UTP, 200 µM GTP, 25 µM CTP (Cytiva) and 0.05 µM α-32P-CTP (PerkinElmer) at 37 °C, and quenched after 2 min by adding 2× loading buffer (10 M Urea, 50 mM EDTA pH 8.0, 0.05% bromophenol blue, 0.05% xylene cyanol). To show the roadblocked EC is capable of further transcription, the roadblocked EC was added by 2 mM IPTG, incubated for 2 min at 37 °C for LacI dissociation, and added additional rNTP to final concentrations of 162 µM ATP, 162 µM UTP, 162 µM GTP, 75 µM ATP and 0.15 µM α-32P CTP. The samples were loaded on 10% Urea-PAGE gel and ran in 1X TBE. The gel was exposed to an imaging plate (Fujifilm) for 2 hr, and the imaging plate was scanned to get an image (TyphoonTM FLA 7000). For the mutational study, 50 nM holoenzyme and 12 nM template DNA were used for the transcription assay without roadblocking. In addition, the transcription reaction was quenched at 0-, 0.5-, and 2-min time point, and the data at 0.5 min were used to estimate the relative anti-pausing activity plotted in Fig. 3c although using 2-min data also showed similar result (data not shown). For the transcription reaction, 200 µM ATP, 200 µM UTP, 200 µM GTP, 25 µM CTP, and 0.05 µM α-32P CTP were used. For the estimation of the relative anti-pausing activity, we measured the intensities of the paused and the run-off transcripts of the put constructs, and calculated the fraction of the paused transcripts by dividing the intensity of the paused transcript by the sum of the intensities of the paused and run-off transcripts (Supplementary Fig. 9). The fractions of the paused transcripts were calculated for the wild-type put, inactive put−, and mutant put constructs, and their relative anti-pausing activities were calculated by the equation below and plotted: $${{{{{\rm{Relative}}}}}}\,{{{{{\rm{anti}}}}}}\mbox{-}{{{{{\rm{pausing}}}}}}\,{{{{{\rm{activity}}}}}}\,{{{{{\rm{of}}}}}}\,{{{{{\rm{mutant}}}}}}\,{{{{{\rm{x}}}}}}=1-\frac{({{{{{{\rm{P}}}}}}}_{{{{{{\rm{X}}}}}}}-{{{{{{\rm{P}}}}}}}_{{{{{{\rm{WT}}}}}}})}{({{{{{{\rm{P}}}}}}}_{put-}-{{{{{{\rm{P}}}}}}}_{{{{{{\rm{WT}}}}}}})}\\ ({{{{{{\rm{P}}}}}}}_{{{{{{\rm{x}}}}}}}={{{{{\rm{the}}}}}}\,{{{{{\rm{fraction}}}}}}\,{{{{{\rm{of}}}}}}\,{{{{{\rm{the}}}}}}\,{{{{{\rm{paused}}}}}}\,{{{{{\rm{band}}}}}}\,{{{{{\rm{of}}}}}}\,{{{{{\rm{mutant}}}}}}\,{{{{{\rm{x}}}}}})$$ For the paused fraction quantification, the intensities for the run-off and paused transcripts were calibrated according to the number of cytosines the transcripts contain. The assay was done in triplicate (n = 3 independent experiments). Native mass spectrometry analysis The RNA portion of the de novo reconstituted putEC was prepared by phenol/chloroform extraction, resuspended in RNase-free water and flash-frozen in liquid nitrogen. Prior to analysis, the sample was thawed and then buffer-exchanged into nMS solution (500 mM ammonium acetate, 0.01% Tween-20, pH 7.5) using Zeba desalting microspin columns (Thermo Fisher). The buffer-exchanged sample was diluted to 5 µM with nMS solution and was loaded into a gold-coated quartz capillary tip that was prepared in-house. The sample was then electrosprayed into an Exactive Plus EMR instrument (Thermo Fisher Scientific) using a modified static nanospray source59. The MS parameters used were similar from previous work22: spray voltage, 1.2 kV; capillary temperature, 150 °C; S-lens RF level, 200; resolving power, 8750 at m/z of 200; AGC target, 1 × 106; number of microscans, 5; maximum injection time, 200 ms; in-source dissociation, 10 V; injection flatapole, 10 V; interflatapole, 7 V; bent flatapole, 6 V; high energy collision dissociation, 85 V; ultrahigh vacuum pressure, 6.6 × 10−10 mbar; total number of scans, 100. Mass calibration in positive EMR mode was performed using cesium iodide. Raw nMS spectra were visualized using Thermo Xcalibur Qual Browser (version 4.2.47). Data processing and spectra deconvolution were performed using UniDec version 4.2.060,61. The UniDec parameters used were m/z range: 2000–7000; mass range: 25,000–45,000 Da; sample mass every 0.5 Da; smooth charge state distribution, on; peak shape function, Gaussian; and Beta softmax function setting, 20. The expected masses for the de novo synthesized RNA include 94-mer (30,630 Da), 95-mer (30,936 Da), 110-mer (35,776 Da), 114-mer (37,038 Da), and 116-mer (37,671 Da). The mass deviations of the measured masses from the expected masses were within 1 Da or less. PutEC preparation and cryo-EM grid freezing Holoenzyme was formed by mixing Eco RNAP and Eco σ70 with 1:2 molar ratio and incubating for 15 min at 37 °C, and purified in Superdex 200 Increase 10/300 Increase GL column (Cytiva). Template DNA was amplified in thermocycler. pRAK31 plasmid62 was used as template DNA for the PCR reaction. The forward and reverse primer sequences (from Macrogen) for the reaction are as follows; Forward primer-5'-GCATGAATTCCTATTGGTACTTTACATTAA-3', Reverse primer-5'-CGAATTGTGAGCGCTCACAATTCTAAAAGCAAAAAAGCCTTC-3'. Holoenzyme and template DNA were mixed and incubated for 10 min at 37 °C to form RPo. After RPo reconstitution, LacI, which is also purified by size-exclusion chromatography before use, was added and incubated for 10 min for roadblocking. To the mixture, 1 mM rNTP (Cytiva) was added and incubated for 2 min at 37 °C for transcription. The sample was loaded onto zeba spin desalting column (Thermo Fisher) to remove free rNTP, and 2 mM IPTG was added to the complex. After 2 min incubation at 37 °C, the mixture was concentrated using 30 K MWCO Amicon Ultra Centrifugal Filter (Merck Millipore) up to 5–10 μM. The final buffer condition for all cryo-EM samples was 20 mM Tris-glutamate (pH 8.0), 10 mM Mg-glutamate, 150 mm K-glutamate, 5 mM DTT. 0.5% CHAPSO was added to the sample right before grid freezing. For cryo-EM grid freezing, Quantifoil R 1.2/1.3 Cu 400 grids were glow discharged at negative polarity, 0.26 mbar, 15 mA, 25 sec. Using a Vitrobot Mark IV (Thermo Fisher), grids were blotted and plunge-frozen into liquid ethane with 100% chamber humidity at 22 °C. Cryo-EM data acquisition and processing Micrographs were taken using a 300 keV Krios G4 (Thermo Fisher Scientific) with a K3 BioQuantum direct electron detector (Gatan) with 20 eV energy filter slit width. Images were recorded with EPU with a pixel size of 1.06 Å/pix over a defocus range of −0.8 µm to −2.6 µm. Total dose given to the data set is 42.16 e−/Å2 and total frame number was 55. The movies were drift-corrected, summed, and dose-weighted using MotionCor2 in RELION3.163. The contrast transfer function (CTF) was estimated using Gctf64, and the summed images were sorted based on CTF max resolution (<10 Å) and CTF figure of merit (>0.01). The sorted images were transferred to cryoSPARC v3.2.0 for further process65. First, 411.9k particles were picked using blob picker from 2000 movies, extracted with 320 pixels box size, and 2D classified to make picking templates. Then, 1447.1k particles were picked using template picker from 8174 images. The particles were 2D classified twice, and the selected 863.1k particles from 43 classes were used as templates for Topaz picker. From Topaz train, 1202.5k particles were picked and extracted from 8162 images. The particles were 2D classified into 100 classes and 90 classes were selected. The selected particles were divided into five classes in heterogeneous refinement. Among the five templates, three are from the previous data set collected from Glacios, two are from EMDB EMD-8585, a non-paused EC map. Among five classes, three classes were subjected to homogeneous refinement. Each homogeneous-refined class was further heterogeneous-refined into two classes, resulting in total of four significant EC classes—RPo, putEC, put-less EC, and σ70-bound putEC. All particles of the four classes were imported to RELION3.1 for further refinements. The particles belonged to holoenzyme structure were 3D auto-refined, particle-polished three times, and 3D classified into three classes. Among the three classes, the major class was 3D auto-refined and post-processed yielding 3.0 Å-resolution map. The putEC particles were 3D auto-refined, particle-polished three times, and subjected to focused classification onto putRNA region into three classes. Among the three classes, two classes are combined, 3D auto-refined and post-processed yielding 3.2 Å-resolution map. The put-less particles were 3D auto-refined, particle-polished three times, and post-processed yielding 3.6 Å-resolution map. The σ70-bound putEC particles were 3D auto-refined, particle-polished three times, and 3D classified into three classes. Among the three classes, one best class was further refined and post-processed yielding 3.6 Å-resolution map. Model building, refinement, and validation The local resolution estimation and filtration were done by blocres and blocfilt commands in bsoft package (version 2.0.5), respectively66. For the EC structures, EC coordinates including RNAP, DNA, and RNA are used from PDB 6C6T because this is modeled from the high-resolution EC map (3.5 Å). For σ70-bound putEC, the recently published high-resolution RPo model (PDB 6OUL) was used. In the model building, the models were first fitted onto the final cryo-EM map by using UCSF Chimera (version 1.11.12)67. Then, the RNAP domains were rigid-body refined in PHENIX (version 1.18.2)68, and the nucleic acid were mutated to have the correct sequence in Coot69. The structures were then real-space refined in PHENIX, manually modified in Coot, and iterated this process until satisfied. The putRNA was manually built into the map de novo. A .eff file that includes restraints maintaining the nucleic acid base pairing and stacking interactions was provided for each real-space refinement run. For the final refinement run, the nonbonded_weight parameter value was set to 500 (default value: 100) to improve the MolProbity and clash scores. 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Acta. Crystallogr. D Biol. Crystallogr. 60, 2126–2132 (2004). We thank Dr. Jin-Seok Choi at the KAIST Analysis Center for Research Advancement for help with cryo-EM grid screening, Dr. Bum Han Ryu at the Research Solution Center of the Institute of Basic Science (IBS) for help with cryo-EM data collection, Dr. Hanseong Kim at Institute of Membrane Proteins in Pohang for help with preliminary cryo-EM data collection, and Dr. Seth A. Darst and Dr. Elizabeth A. Campbell at the Rockefeller University, and Dr. Max E. Gottesman at the Columbia University for the manuscript reading and helpful advice. Computational works for this research were performed on the data analysis hub, Olaf in the IBS Research Solution Center. This work was supported by the National Research Foundation of Korea, NRF-2019R1F1A1064026, NRF-2019M3E5D6066058, NRF-2021R1C1C100656011 to JYK, and National Institutes of Health P41 GM109824 and P41 GM103314 to BTC. This manuscript is dedicated to the memory of Dr. Robert Weisberg, who laid the intellectual and experimental groundwork for these studies, and whose untimely passing deprived him of the satisfaction of seeing these come to fruition. Department of Chemistry, Korea Advanced Institute of Science and Technology (KAIST), Daejeon, Republic of Korea Seungha Hwang, Jimin Lee, Jinwoo Kim & Jin Young Kang Laboratory of Mass Spectrometry and Gaseous Ion Chemistry, The Rockefeller University, 1230 York Avenue, New York, NY, USA Paul Dominic B. Olinares & Brian T. Chait Biology Department, Western Kentucky University, Bowling Green, KY, USA Rodney A. King Seungha Hwang Paul Dominic B. Olinares Jimin Lee Jinwoo Kim Brian T. Chait Jin Young Kang S.H. purified Eco RNAP, reconstituted putEC, performed radiolabeled transcription assay, collected and analyzed cryo-EM data, and wrote the manuscript. P.D.B.O. performed nMS experiment, analyzed the result, and wrote the manuscript. J.L. helped S.H. in radiolabeled transcription assay, cryo-EM grid preparation, and cryo-EM data collection. J.K. helped S.H. in cryo-EM data analysis. B.T.C. provided nMS facility and funding and reviewed the manuscript. R.A.K provided materials, analyzed the data, and wrote the manuscript. J.Y.K. conceived the project, analyzed the cryo-EM data with S.H., wrote the manuscript with R.A.K., S.H., and P.D.B.O., and obtained funding for the project. Correspondence to Jin Young Kang. Nature Communications thanks the anonymous reviewers for their contribution to the peer review of this work. Peer reviewer reports are available. Hwang, S., Olinares, P.D.B., Lee, J. et al. Structural basis of transcriptional regulation by a nascent RNA element, HK022 putRNA. Nat Commun 13, 4668 (2022). https://doi.org/10.1038/s41467-022-32315-y Received: 17 February 2022	CommonCrawl
Abiogenic petroleum origin Abiogenic petroleum origin is a body of hypotheses which propose that petroleum and natural gas deposits are mostly formed by inorganic means, rather than by the decomposition of organisms. Thomas Gold's deep gas hypothesis states that the origin of some natural gas deposits were formed out of hydrocarbons deep in the earth's mantle. Theories explaining the origin of petroleum as abiotic, however, are generally not well accepted by the scientific community. [1][why?] Earlier studies of mantle-derived rocks from many places have shown that hydrocarbons from the mantle region can be found widely around the globe. However, the content of such hydrocarbons is in low concentration.[2] While there may be large deposits of abiotic hydrocarbons, globally significant amounts of abiotic hydrocarbons are deemed unlikely.[3] Overview hypothesesEdit Some abiogenic hypotheses have proposed that oil and gas did not originate from fossil deposits, but have instead originated from deep carbon deposits, present since the formation of the Earth.[4] Additionally, it has been suggested that hydrocarbons may have arrived on Earth from solid bodies such as comets and asteroids from the late formation of the Solar System, carrying hydrocarbons with them.[5][6] The abiogenic hypothesis regained some support in 2009 when researchers at the Royal Institute of Technology (KTH) in Stockholm reported they believed they had proven that fossils from animals and plants are not necessary for crude oil and natural gas to be generated.[7][8] In his 2014 publication Chemistry of the Climate System,[9] German chemist Detlev Moller documents sufficient reliable evidence to show that both processes can be shown to co-exist, that they're not mutually exclusive. An abiogenic hypothesis was first proposed by Georgius Agricola in the 16th century and various additional abiogenic hypotheses were proposed in the 19th century, most notably by Prussian geographer Alexander von Humboldt,[when?] the Russian chemist Dmitri Mendeleev (1877[10]) and the French chemist Marcellin Berthelot.[when?] Abiogenic hypotheses were revived in the last half of the 20th century by Soviet scientists who had little influence outside the Soviet Union because most of their research was published in Russian. The hypothesis was re-defined and made popular in the West by Thomas Gold who developed his theories from 1979 to 1998, and published his research in English. Abraham Gottlob Werner and the proponents of neptunism in the 18th century regarded basaltic sills as solidified oils or bitumen. While these notions proved unfounded, the basic idea of an association between petroleum and magmatism persisted. Alexander von Humboldt proposed an inorganic abiogenic hypothesis for petroleum formation after he observed petroleum springs in the Bay of Cumaux (Cumaná) on the northeast coast of Venezuela.[11] He is quoted as saying in 1804, "the petroleum is the product of a distillation from great depth and issues from the primitive rocks beneath which the forces of all volcanic action lie".[citation needed] Other early prominent proponents of what would become the generalized abiogenic hypothesis included Dmitri Mendeleev[12] and Berthelot. In 1951, the Soviet geologist Nikolai Alexandrovitch Kudryavtsev proposed the modern abiotic hypothesis of petroleum.[13][14] On the basis of his analysis of the Athabasca Oil Sands in Alberta, Canada, he concluded that no "source rocks" could form the enormous volume of hydrocarbons, and therefore offered abiotic deep petroleum as the most plausible explanation. (Humic coals have since been proposed for the source rocks.[15]) Others who continued Kudryavtsev's work included Petr N. Kropotkin, Vladimir B. Porfir'ev, Emmanuil B. Chekaliuk, Vladilen A. Krayushkin, Georgi E. Boyko, Georgi I. Voitov, Grygori N. Dolenko, Iona V. Greenberg, Nikolai S. Beskrovny, and Victor F. Linetsky. Astronomer Thomas Gold was a prominent proponent of the abiogenic hypothesis in the West until his death in 2004. More recently, Jack Kenney of Gas Resources Corporation has come to prominence,[16][17][18] supported by studies by researchers at the Royal Institute of Technology in Stockholm.[7] Foundations of abiogenic hypothesesEdit Within the mantle, carbon may exist as hydrocarbons—chiefly methane—and as elemental carbon, carbon dioxide, and carbonates.[18] The abiotic hypothesis is that the full suite of hydrocarbons found in petroleum can either be generated in the mantle by abiogenic processes,[18] or by biological processing of those abiogenic hydrocarbons, and that the source-hydrocarbons of abiogenic origin can migrate out of the mantle into the crust until they escape to the surface or are trapped by impermeable strata, forming petroleum reservoirs. Abiogenic hypotheses generally reject the supposition that certain molecules found within petroleum, known as biomarkers, are indicative of the biological origin of petroleum. They contend that these molecules mostly come from microbes feeding on petroleum in its upward migration through the crust, that some of them are found in meteorites, which have presumably never contacted living material, and that some can be generated abiogenically by plausible reactions in petroleum.[17] Some of the evidence used to support abiogenic theories includes: Gold The presence of methane on other planets, meteors, moons and comets[19][20] Gold, Kenney Proposed mechanisms of abiotically chemically synthesizing hydrocarbons within the mantle[16][17][18] Kudryavtsev, Gold Hydrocarbon-rich areas tend to be hydrocarbon-rich at many different levels[4] Kudryavtsev, Gold Petroleum and methane deposits are found in large patterns related to deep-seated large-scale structural features of the crust rather than to the patchwork of sedimentary deposits[4] Gold Interpretations of the chemical and isotopic composition of natural petroleum[4] Kudryavtsev, Gold The presence of oil and methane within non-sedimentary rocks upon the Earth[21] Gold The existence of methane hydrate deposits[4] Gold Perceived ambiguity in some assumptions and key evidence used in the conventional understanding of petroleum origin.[4][16] Gold Bituminous coal creation is based upon deep hydrocarbon seeps[4] Gold Surface carbon budget and oxygen levels stable over geologic time scales[4] Kudryavtsev, Gold The biogenic explanation does not explain some hydrocarbon deposit characteristics[4] Szatmari The distribution of metals in crude oils fits better with upper serpentinized mantle, primitive mantle and chondrite patterns than oceanic and continental crust, and show no correlation with sea water[22][23] Gold The association of hydrocarbons with helium, a noble gas[clarification needed][4] Recent investigation of abiogenic hypothesesEdit This section's representation of one or more viewpoints about a controversial issue may be unbalanced or inaccurate. Please improve the article or discuss the issue on the talk page. (July 2009) As of 2009[update], little research is directed towards establishing abiogenic petroleum or methane, although the Carnegie Institution for Science has reported that ethane and heavier hydrocarbons can be synthesized under conditions of the upper mantle.[24] Research mostly related to astrobiology and the deep microbial biosphere and serpentinite reactions, however, continue to provide insight into the contribution of abiogenic hydrocarbons into petroleum accumulations. rock porosity and migration pathways for abiogenic petroleum[25] mantle peridotite serpentinization reactions and other natural Fischer-Tropsch analogs[1] Primordial hydrocarbons in meteorites, comets, asteroids and the solid bodies of the Solar System[citation needed] Primordial or ancient sources of hydrocarbons or carbon in Earth[5][6] Primordial hydrocarbons formed from hydrolysis of metal carbides of the iron peak of cosmic elemental abundance (chromium, iron, nickel, vanadium, manganese, cobalt)[26] isotopic studies of groundwater reservoirs, sedimentary cements, formation gases and the composition of the noble gases and nitrogen in many oil fields the geochemistry of petroleum and the presence of trace metals related to Earth's mantle (nickel, vanadium, cadmium, arsenic, lead, zinc, mercury and others) Similarly, research into the deep microbial hypothesis of hydrocarbon generation is advancing as part of the attempt to investigate the concept of panspermia and astrobiology, specifically using deep microbial life as an analog for life on Mars. Research applicable to deep microbial petroleum theories includes Research into how to sample deep reservoirs and rocks without contamination Sampling deep rocks and measuring chemistry and biological activity[27] Possible energy sources and metabolic pathways which may be used in a deep biosphere[28][29] Investigations into the reworking of primordial hydrocarbons by bacteria and their effects on carbon isotope fractionation Proposed mechanisms of abiogenic petroleumEdit This section may be too technical for most readers to understand. Please help improve it to make it understandable to non-experts, without removing the technical details. (July 2008) (Learn how and when to remove this template message) Primordial depositsEdit Thomas Gold's work was focused on hydrocarbon deposits of primordial origin. Meteorites are believed to represent the major composition of material from which the Earth was formed. Some meteorites, such as carbonaceous chondrites, contain carbonaceous material. If a large amount of this material is still within the Earth, it could have been leaking upward for billions of years. The thermodynamic conditions within the mantle would allow many hydrocarbon molecules to be at equilibrium under high pressure and high temperature. Although molecules in these conditions may disassociate, resulting fragments would be reformed due to the pressure. An average equilibrium of various molecules would exist depending upon conditions and the carbon-hydrogen ratio of the material.[30] Creation within the mantleEdit Russian researchers concluded that hydrocarbon mixes would be created within the mantle. Experiments under high temperatures and pressures produced many hydrocarbons—including n-alkanes through C10H22—from iron oxide, calcium carbonate, and water.[18] Because such materials are in the mantle and in subducted crust, there is no requirement that all hydrocarbons be produced from primordial deposits. Hydrogen generationEdit Hydrogen gas and water have been found more than 6,000 metres (20,000 ft) deep in the upper crust in the Siljan Ring boreholes and the Kola Superdeep Borehole. Data from the western United States suggests that aquifers from near the surface may extend to depths of 10,000 metres (33,000 ft) to 20,000 metres (66,000 ft). Hydrogen gas can be created by water reacting with silicates, quartz, and feldspar at temperatures in the range of 25 °C (77 °F) to 270 °C (518 °F). These minerals are common in crustal rocks such as granite. Hydrogen may react with dissolved carbon compounds in water to form methane and higher carbon compounds.[31] One reaction not involving silicates which can create hydrogen is: Ferrous oxide + water → magnetite + hydrogen 3FeO + H2O → Fe3O4 + H2[5][dubious – discuss] The above reaction operates best at low pressures. At pressures greater than 5 gigapascals (49,000 atm) almost no hydrogen is created.[5] Thomas Gold reported that hydrocarbons were found in the Siljan Ring borehole and in general increased with depth, although the venture was not a commercial success.[32] However, several geologists analysed the results and said that no hydrocarbon was found.[33][34][35][36][37] Serpentinite mechanismEdit In 1967, the Ukrainian scientist Emmanuil B. Chekaliuk proposed that petroleum could be formed at high temperatures and pressures from inorganic carbon in the form of carbon dioxide, hydrogen and/or methane. This mechanism is supported by several lines of evidence which are accepted by modern scientific literature. This involves synthesis of oil within the crust via catalysis by chemically reductive rocks. A proposed mechanism for the formation of inorganic hydrocarbons[38] is via natural analogs of the Fischer-Tropsch process known as the serpentinite mechanism or the serpentinite process.[22][39] C H 4 + 1 2 O 2 → 2 H 2 + C O {\displaystyle \mathrm {CH_{4}+{\begin{matrix}{\frac {1}{2}}\end{matrix}}O_{2}\rightarrow 2H_{2}+CO} } ( 2 n + 1 ) H 2 + n C O → C n H 2 n + 2 + n H 2 O {\displaystyle \mathrm {(2n+1)H_{2}+nCO\rightarrow C_{n}H_{2n+2}+nH_{2}O} } Serpentinites are ideal rocks to host this process as they are formed from peridotites and dunites, rocks which contain greater than 80% olivine and usually a percentage of Fe-Ti spinel minerals. Most olivines also contain high nickel concentrations (up to several percent) and may also contain chromite or chromium as a contaminant in olivine, providing the needed transition metals. However, serpentinite synthesis and spinel cracking reactions require hydrothermal alteration of pristine peridotite-dunite, which is a finite process intrinsically related to metamorphism, and further, requires significant addition of water. Serpentinite is unstable at mantle temperatures and is readily dehydrated to granulite, amphibolite, talc–schist and even eclogite. This suggests that methanogenesis in the presence of serpentinites is restricted in space and time to mid-ocean ridges and upper levels of subduction zones. However, water has been found as deep as 12,000 metres (39,000 ft),[40] so water-based reactions are dependent upon the local conditions. Oil being created by this process in intracratonic regions is limited by the materials and temperature. Serpentinite synthesisEdit A chemical basis for the abiotic petroleum process is the serpentinization of peridotite, beginning with methanogenesis via hydrolysis of olivine into serpentine in the presence of carbon dioxide.[39] Olivine, composed of Forsterite and Fayalite metamorphoses into serpentine, magnetite and silica by the following reactions, with silica from fayalite decomposition (reaction 1a) feeding into the forsterite reaction (1b). Reaction 1a: Fayalite + water → magnetite + aqueous silica + hydrogen 3 F e 2 S i O 4 + 2 H 2 O → 2 F e 3 O 4 + 3 S i O 2 + 2 H 2 {\displaystyle \mathrm {3Fe_{2}SiO_{4}+2H_{2}O\rightarrow 2Fe_{3}O_{4}+3SiO_{2}+2H_{2}} } Reaction 1b: Forsterite + aqueous silica → serpentinite 3 M g 2 S i O 4 + S i O 2 + 4 H 2 O → 2 M g 3 S i 2 O 5 ( O H ) 4 {\displaystyle \mathrm {3Mg_{2}SiO_{4}+SiO_{2}+4H_{2}O\rightarrow 2Mg_{3}Si_{2}O_{5}(OH)_{4}} } When this reaction occurs in the presence of dissolved carbon dioxide (carbonic acid) at temperatures above 500 °C (932 °F) Reaction 2a takes place. Olivine + water + carbonic acid → serpentine + magnetite + methane ( F e , M g ) 2 S i O 4 + n H 2 O + C O 2 → M g 3 S i 2 O 5 ( O H ) 4 + F e 3 O 4 + C H 4 {\displaystyle \mathrm {(Fe,Mg)_{2}SiO_{4}+nH_{2}O+CO_{2}\rightarrow Mg_{3}Si_{2}O_{5}(OH)_{4}+Fe_{3}O_{4}+CH_{4}} } or, in balanced form: 18 M g 2 S i O 4 + 6 F e 2 S i O 4 + 26 H 2 O + C O 2 {\displaystyle \mathrm {18Mg_{2}SiO_{4}+6Fe_{2}SiO_{4}+26H_{2}O+CO_{2}} } → 12 M g 3 S i 2 O 5 ( O H ) 4 + 4 F e 3 O 4 + C H 4 {\displaystyle \mathrm {12Mg_{3}Si_{2}O_{5}(OH)_{4}+4Fe_{3}O_{4}+CH_{4}} } However, reaction 2(b) is just as likely, and supported by the presence of abundant talc-carbonate schists and magnesite stringer veins in many serpentinised peridotites; Olivine + water + carbonic acid → serpentine + magnetite + magnesite + silica ( F e , M g ) 2 S i O 4 + n H 2 O + C O 2 → M g 3 S i 2 O 5 ( O H ) 4 + F e 3 O 4 + M g C O 3 + S i O 2 {\displaystyle \mathrm {(Fe,Mg)_{2}SiO_{4}+nH_{2}O+CO_{2}\rightarrow Mg_{3}Si_{2}O_{5}(OH)_{4}+Fe_{3}O_{4}+MgCO_{3}+SiO_{2}} } The upgrading of methane to higher n-alkane hydrocarbons is via dehydrogenation of methane in the presence of catalyst transition metals (e.g. Fe, Ni). This can be termed spinel hydrolysis. Spinel polymerization mechanismEdit Magnetite, chromite and ilmenite are Fe-spinel group minerals found in many rocks but rarely as a major component in non-ultramafic rocks. In these rocks, high concentrations of magmatic magnetite, chromite and ilmenite provide a reduced matrix which may allow abiotic cracking of methane to higher hydrocarbons during hydrothermal events. Chemically reduced rocks are required to drive this reaction and high temperatures are required to allow methane to be polymerized to ethane. Note that reaction 1a, above, also creates magnetite. Reaction 3: Methane + magnetite → ethane + hematite n C H 4 + n F e 3 O 4 + n H 2 O → C 2 H 6 + F e 2 O 3 + H C O 3 + H + {\displaystyle \mathrm {nCH_{4}+nFe_{3}O_{4}+nH_{2}O\rightarrow C_{2}H_{6}+Fe_{2}O_{3}+HCO_{3}+H^{+}} } Reaction 3 results in n-alkane hydrocarbons, including linear saturated hydrocarbons, alcohols, aldehydes, ketones, aromatics, and cyclic compounds.[39] Carbonate decompositionEdit Calcium carbonate may decompose at around 500 °C (932 °F) through the following reaction:[5] Hydrogen + calcium carbonate → methane + calcium oxide + water 4 H 2 + C a C O 3 → C H 4 + C a O + 2 H 2 O {\displaystyle \mathrm {4H_{2}+CaCO_{3}\rightarrow CH_{4}+CaO+2H_{2}O} } Note that CaO (lime) is not a mineral species found within natural rocks. Whilst this reaction is possible, it is not plausible. Evidence of abiogenic mechanismsEdit Theoretical calculations by J.F. Kenney using scaled particle theory (a statistical mechanical model) for a simplified perturbed hard-chain predict that methane compressed to 30,000 bars (3.0 GPa) or 40,000 bars (4.0 GPa) kbar at 1,000 °C (1,830 °F) (conditions in the mantle) is relatively unstable in relation to higher hydrocarbons. However, these calculations do not include methane pyrolysis yielding amorphous carbon and hydrogen, which is recognized as the prevalent reaction at high temperatures.[17][18] Experiments in diamond anvil high pressure cells have resulted in partial conversion of methane and inorganic carbonates into light hydrocarbons.,[41][42] Biotic (microbial) hydrocarbonsEdit The "deep biotic petroleum hypothesis", similar to the abiogenic petroleum origin hypothesis, holds that not all petroleum deposits within the Earth's rocks can be explained purely according to the orthodox view of petroleum geology. Thomas Gold used the term the deep hot biosphere to describe the microbes which live underground.[4][43][44] This hypothesis is different from biogenic oil in that the role of deep-dwelling microbes is a biological source for oil which is not of a sedimentary origin and is not sourced from surface carbon. Deep microbial life is only a contaminant of primordial hydrocarbons. Parts of microbes yield molecules as biomarkers. Deep biotic oil is considered to be formed as a byproduct of the life cycle of deep microbes. Shallow biotic oil is considered to be formed as a byproduct of the life cycles of shallow microbes. Microbial biomarkersEdit Thomas Gold, in a 1999 book, cited the discovery of thermophile bacteria in the Earth's crust as new support for the postulate that these bacteria could explain the existence of certain biomarkers in extracted petroleum.[4] A rebuttal of biogenic origins based on biomarkers has been offered by Kenney, et al. (2001).[17] Isotopic evidenceEdit Methane is ubiquitous in crustal fluid and gas.[29] Research continues to attempt to characterise crustal sources of methane as biogenic or abiogenic using carbon isotope fractionation of observed gases (Lollar & Sherwood 2006). There are few clear examples of abiogenic methane-ethane-butane, as the same processes favor enrichment of light isotopes in all chemical reactions, whether organic or inorganic. δ13C of methane overlaps that of inorganic carbonate and graphite in the crust, which are heavily depleted in 12C, and attain this by isotopic fractionation during metamorphic reactions. One argument for abiogenic oil cites the high carbon depletion of methane as stemming from the observed carbon isotope depletion with depth in the crust. However, diamonds, which are definitively of mantle origin, are not as depleted as methane, which implies that methane carbon isotope fractionation is not controlled by mantle values.[33] Commercially extractable concentrations of helium (greater than 0.3%) are present in natural gas from the Panhandle-Hugoton fields in the USA, as well as from some Algerian and Russian gas fields.[45][46] Helium trapped within most petroleum occurrences, such as the occurrence in Texas, is of a distinctly crustal character with an Ra ratio of less than 0.0001 that of the atmosphere.[47][48] The Chimaera gas seep, near Antalya (SW Turkey), new and thorough molecular and isotopic analyses including methane (~87% v/v; D13C1 from -7.9 to -12.3 ‰; D13D1 from -119 to -124 ‰), light alkanes (C2+C3+C4+C5 = 0.5%; C6+: 0.07%; D13C2 from -24.2 to -26.5 ‰; D13C3 from -25.5 to -27 ‰), hydrogen (7.5 to 11%), carbon dioxide (0.01-0.07%; D13CCO2: -15 ‰), helium (~80 ppmv; R/Ra: 0.41) and nitrogen (2-4.9%; D15N from -2 to -2.8 ‰) converge to indicate that the seep releases a mixture of organic thermogenic gas, related to mature Type III kerogen occurring in Paleozoic and Mesozoic organic rich sedimentary rocks, and abiogenic gas produced by low temperature serpentinization in the Tekirova ophiolitic unit.[49] Biomarker chemicalsEdit Certain chemicals found in naturally occurring petroleum contain chemical and structural similarities to compounds found within many living organisms. These include terpenoids, terpenes, pristane, phytane, cholestane, chlorins and porphyrins, which are large, chelating molecules in the same family as heme and chlorophyll. Materials which suggest certain biological processes include tetracyclic diterpane and oleanane.[citation needed] The presence of these chemicals in crude oil is a result of the inclusion of biological material in the oil; these chemicals are released by kerogen during the production of hydrocarbon oils, as these are chemicals highly resistant to degradation and plausible chemical paths have been studied. Abiotic defenders state that biomarkers get into oil during its way up as it gets in touch with ancient fossils. However a more plausible explanation is that biomarkers are traces of biological molecules from bacteria (archaea) that feed on primordial hydrocarbons and die in that environment. For example, hopanoids are just parts of the bacterial cell wall present in oil as contaminant.[4] Trace metalsEdit Nickel (Ni), vanadium (V), lead (Pb), arsenic (As), cadmium (Cd), mercury (Hg) and others metals frequently occur in oils. Some heavy crude oils, such as Venezuelan heavy crude have up to 45% vanadium pentoxide content in their ash, high enough that it is a commercial source for vanadium. Abiotic supporters argue that these metals are common in Earth's mantle, but relatively high contents of nickel, vanadium, lead and arsenic can be usually found in almost all marine sediments. Analysis of 22 trace elements in oils correlate significantly better with chondrite, serpentinized fertile mantle peridotite, and the primitive mantle than with oceanic or continental crust, and shows no correlation with seawater.[22] Reduced carbonEdit Sir Robert Robinson studied the chemical makeup of natural petroleum oils in great detail, and concluded that they were mostly far too hydrogen-rich to be a likely product of the decay of plant debris, assuming a dual origin for Earth hydrocarbons.[30] However, several processes which generate hydrogen could supply kerogen hydrogenation which is compatible with the conventional explanation.[50] Olefins, the unsaturated hydrocarbons, would have been expected to predominate by far in any material that was derived in that way. He also wrote: "Petroleum ... [seems to be] a primordial hydrocarbon mixture into which bio-products have been added." This has however been demonstrated later to be a misunderstanding by Robinson, related to the fact that only short duration experiments were available to him. Olefins are thermally very unstable (that is why natural petroleum normally does not contain such compounds) and in laboratory experiments that last more than a few hours, the olefins are no longer present.[citation needed] The presence of low-oxygen and hydroxyl-poor hydrocarbons in natural living media is supported by the presence of natural waxes (n=30+), oils (n=20+) and lipids in both plant matter and animal matter, for instance fats in phytoplankton, zooplankton and so on. These oils and waxes, however, occur in quantities too small to significantly affect the overall hydrogen/carbon ratio of biological materials. However, after the discovery of highly aliphatic biopolymers in algae, and that oil generating kerogen essentially represent concentrates of such materials, no theoretical problem exists anymore.[citation needed] Also, the millions of source rock samples that have been analyzed for petroleum yield by the petroleum industry have confirmed the large quantities of petroleum found in sedimentary basins. Empirical evidenceEdit Occurrences of abiotic petroleum in commercial amounts in the oil wells in offshore Vietnam are sometimes cited, as well as in the Eugene Island block 330 oil field, and the Dnieper-Donets Basin. However, the origins of all these wells can also be explained with the biotic theory.[1] Modern geologists think that commercially profitable deposits of abiotic petroleum could be found, but no current deposit has convincing evidence that it originated from abiotic sources.[1] The Soviet school saw evidence of their[clarification needed] hypothesis in the fact that some oil reservoirs exist in non-sedimentary rocks such as granite, metamorphic or porous volcanic rocks. However, opponents noted that non-sedimentary rocks served as reservoirs for biologically originated oil expelled from nearby sedimentary source rock through common migration or re-migration mechanisms.[1] The following observations have been commonly used to argue for the abiogenic hypothesis, however each observation of actual petroleum can also be fully explained by biotic origin:[1] Lost City hydrothermal vent fieldEdit The Lost City hydrothermal field was determined to have abiogenic hydrocarbon production. Proskurowski et al. wrote, "Radiocarbon evidence rules out seawater bicarbonate as the carbon source for FTT reactions, suggesting that a mantle-derived inorganic carbon source is leached from the host rocks. Our findings illustrate that the abiotic synthesis of hydrocarbons in nature may occur in the presence of ultramafic rocks, water, and moderate amounts of heat."[51] Siljan Ring craterEdit The Siljan Ring meteorite crater, Sweden, was proposed by Thomas Gold as the most likely place to test the hypothesis because it was one of the few places in the world where the granite basement was cracked sufficiently (by meteorite impact) to allow oil to seep up from the mantle; furthermore it is infilled with a relatively thin veneer of sediment, which was sufficient to trap any abiogenic oil, but was modelled as not having been subjected to the heat and pressure conditions (known as the "oil window") normally required to create biogenic oil. However, some geochemists concluded by geochemical analysis that the oil in the seeps came from the organic-rich Ordovician Tretaspis shale, where it was heated by the meteorite impact.[52] In 1986–1990 The Gravberg-1 borehole was drilled through the deepest rock in the Siljan Ring in which proponents had hoped to find hydrocarbon reservoirs. It stopped at the depth of 6,800 metres (22,300 ft) due to drilling problems, after private investors spent $40 million.[34] Some eighty barrels of magnetite paste and hydrocarbon-bearing sludge were recovered from the well; Gold maintained that the hydrocarbons were chemically different from, and not derived from, those added to the borehole, but analyses showed that the hydrocarbons were derived from the diesel fuel-based drilling fluid used in the drilling.[34][35][36][37] This well also sampled over 13,000 feet (4,000 m) of methane-bearing inclusions.[53] In 1991–1992, a second borehole, Stenberg-1, was drilled a few miles away to a depth of 6,500 metres (21,300 ft), finding similar results. Bacterial matsEdit Direct observation of bacterial mats and fracture-fill carbonate and humin of bacterial origin in deep boreholes in Australia are also taken as evidence for the abiogenic origin of petroleum.[54] Example proposed abiogenic methane depositsEdit Panhandle-Hugoton field (Anadarko Basin) in the south-central United States is the most important gas field with commercial helium content. Some abiogenic proponents interpret this as evidence that both the helium and the natural gas came from the mantle.[47][48][55][56] The Bạch Hổ oil field in Vietnam has been proposed as an example of abiogenic oil because it is 4,000 m of fractured basement granite, at a depth of 5,000 m.[57] However, others argue that it contains biogenic oil which leaked into the basement horst from conventional source rocks within the Cuu Long basin.[21][58] A major component of mantle-derived carbon is indicated in commercial gas reservoirs in the Pannonian and Vienna basins of Hungary and Austria.[59] Natural gas pools interpreted as being mantle-derived are the Shengli Field[60] and Songliao Basin, northeastern China.[61][62] The Chimaera gas seep, near Çıralı, Antalya (southwest Turkey), has been continuously active for millennia and it is known to be the source of the first Olympic fire in the Hellenistic period. On the basis of chemical composition and isotopic analysis, the Chimaera gas is said to be about half biogenic and half abiogenic gas, the largest emission of biogenic methane discovered; deep and pressurized gas accumulations necessary to sustain the gas flow for millennia, posited to be from an inorganic source, may be present.[49] Local geology of Chimaera flames, at exact position of flames, reveals contact between serpentinized ophiolite and carbonate rocks.[citation needed] Fischer-Tropsch process can be suitable reaction to form hydrocarbon gases. Geological argumentsEdit Incidental arguments for abiogenic oilEdit Given the known occurrence of methane and the probable catalysis of methane into higher atomic weight hydrocarbon molecules, various abiogenic theories consider the following to be key observations in support of abiogenic hypotheses: the serpentinite synthesis, graphite synthesis and spinel catalysation models prove the process is viable[22][39] the likelihood that abiogenic oil seeping up from the mantle is trapped beneath sediments which effectively seal mantle-tapping faults[38] outdated[citation needed] mass-balance calculations[when?] for supergiant oilfields which argued that the calculated source rock could not have supplied the reservoir with the known accumulation of oil, implying deep recharge.[13][14] the presence of hydrocarbons encapsulated in diamonds [63] The proponents of abiogenic oil also use several arguments which draw on a variety of natural phenomena in order to support the hypothesis: the modeling of some researchers shows the Earth was accreted at relatively low temperature, thereby perhaps preserving primordial carbon deposits within the mantle, to drive abiogenic hydrocarbon production[citation needed] the presence of methane within the gases and fluids of mid-ocean ridge spreading centre hydrothermal fields. [38][42] the presence of diamond within kimberlites and lamproites which sample the mantle depths proposed as being the source region of mantle methane (by Gold et al.).[30] Incidental arguments against abiogenic oilEdit Oil deposits are not directly associated with tectonic structures. Arguments against chemical reactions, such as the serpentinite mechanism, being a source of hydrocarbon deposits within the crust include: this[clarification needed] is contradicted by numerous studies which have documented the existence of hydrologic systems operating over a range of scales and at all depths in the continental crust.[64] the lack of any hydrocarbon within the crystalline shield[clarification needed] areas of the major cratons, especially around key deep seated structures which are predicted to host oil by the abiogenic hypothesis.[33] See Siljan Lake. lack of conclusive proof[clarification needed] that carbon isotope fractionation observed in crustal methane sources is entirely of abiogenic origin (Lollar et al. 2006)[29] drilling of the Siljan Ring failed to find commercial quantities of oil,[33] thus providing a counter example to Kudryavtsev's Rule[clarification needed][34] and failing to locate the predicted abiogenic oil. helium in the Siljan Gravberg-1 well was depleted in 3He and not consistent with a mantle origin[65] The Gravberg-1 well only produced 84 barrels (13.4 m3) of oil, which later was shown to derive from organic additives, lubricants and mud used in the drilling process.[34][35][36] Kudryavtsev's Rule has been explained for oil and gas (not coal)—gas deposits which are below oil deposits can be created from that oil or its source rocks. Because natural gas is less dense than oil, as kerogen and hydrocarbons are generating gas the gas fills the top of the available space. Oil is forced down, and can reach the spill point where oil leaks around the edge(s) of the formation and flows upward. If the original formation becomes completely filled with gas then all the oil will have leaked above the original location.[66] ubiquitous diamondoids in natural hydrocarbons such as oil, gas and condensates are composed of carbon from biological sources, unlike the carbon found in normal diamonds.[67] Extraterrestrial argumentEdit The presence of methane on Saturn's moon Titan and in the atmospheres of Jupiter, Saturn, Uranus and Neptune is cited as evidence of the formation of hydrocarbons without biological intermediate forms,[1] for example by Thomas Gold.[4] (Terrestrial natural gas is composed primarily of methane). Some comets contain massive amounts of organic compounds, the equivalent of cubic kilometers of such mixed with other material;[68] for instance, corresponding hydrocarbons were detected during a probe flyby through the tail of Comet Halley in 1986.[69] Drill samples from the surface of Mars taken in 2015 by the Curiosity Rover's Mars Science Laboratory have found organic molecules of benzene and propane in 3 billion year old rock samples in Gale Crater.[70] Eugene Island block 330 oil field Fischer-Tropsch process Nikolai Alexandrovitch Kudryavtsev ^ a b c d e f g Höök, M.; Bardi, U.; Feng, L.; pang, X. (2010). "Development of oil formation theories and their importance for peak oil". Marine and Petroleum Geology. 27 (10): 1995–2004. doi:10.1016/j.marpetgeo.2010.06.005. hdl:2158/777257. Retrieved 5 October 2017. ^ Sugisuki, R.; Mimura, K. (1994). "Mantle hydrocarbons: abiotic or biotic?". Geochimica et Cosmochimica Acta. 58 (11): 2527–2542. 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Tracking a Foam Front in a 3D, Heterogeneous Porous Medium C. S. Boeije ORCID: orcid.org/0000-0002-1208-74361,2, C. Portois2, M. Schmutz2 & O. Atteia2 Transport in Porous Media volume 131, pages23–42(2020)Cite this article Foam is to be used as a blocking agent for confining a pollutant source zone and avoid spreading in an aquifer. To this end, it is necessary to determine where injected foam flows and stays inside a porous medium. This study examines the use of electrical resistivity tomography for this purpose. Foam is injected in a large-scale 3D heterogeneous porous medium (0.84 × 0.84 × 0.84 m). During the injection, electrical resistivity tomography measurements are performed. We show that combining a large number of measurements with inversion techniques allows for the monitoring of a foam front in 3D during the injection process. Foam injection is used in petroleum engineering applications as an enhancing oil recovery method. More conventional gas injection processes suffer from poor volumetric sweep (i.e., the portion of the reservoir that is contacted by the gas is low). This is caused by differences in physical properties between the injected gas and the displaced fluid (i.e., viscosity, density). The injected gas has a very low viscosity which leads to fingering through the resident fluid or channel through high-permeability zones. Also its density is low causing it to flow to the top of the reservoir bypassing a large portion of the fluids. Foam can be used to enhance the oil recovery as it traps the injected gas in small bubbles which are separated by thin liquid films known as lamellae. This trapping means that the gas is no longer free to flow and thus cannot easily bypass the resident fluids. Therefore, foam injection processes can provide a much more stable displacement front than conventional gas injection. Most studies on foam in porous media available in the literature focus on petroleum applications including experimental work (e.g., Chabert et al. 2014; Singh and Mohanty 2016; Batôt et al. 2016) and modeling studies (e.g., Ma et al. 2014; Masoudi et al. 2015). In more recent years, foam injection has been applied for environmental remediation purposes. It can be used (i) as a selective reductive permeability agent (e.g., Hirasaki et al. 1997; Bertin et al. 2017), (ii) to increase the mobilization of contaminant (Mulligan and Eftekhari 2003; Huang and Chang 2000; Wang and Chen 2014; Longpré-Girard et al. 2016; Maire and Fatin-Rouge 2017), (iii) to increase the contact between remedial agent and contaminant (Rothmel et al. 1998; Shi et al. 2018) or (iv) as a confining agent (Portois et al. 2018). Only few attempts were done in the field (Hirasaki et al. 1997; Maire et al. 2018; Portois et al. 2018), and all authors highlighted the difficulties to predict foam behavior in 3D dimensions. Two main metrics were used to monitor the foam propagation, namely breakthrough of foam in observation wells (Hirasaki et al. 1997; Maire et al. 2018) and the use of modeling to approximate the progressing foam front (Portois et al. 2018). Foam remains an unstable fluid, and its behavior is quite difficult to predict in the field, even more in the case of environmental remediation where optimal conditions such as a high injection pressure cannot be achieved. As a result due to the lower apparent viscosity (compared to oil production), foam can also suffer from gravity override and lead to a relatively poor sweep efficiency of the porous media. Injection pressure is the commonly tracked parameter during an injection. Considering radial flow in a 2D or a 3D medium, pressure decreases drastically with increasing radial distance from the injection well, leading to foam collapse and subsequent gravity override of the free gas. Thus, injection pressure alone does not provide sufficient information of the foam behavior far away from the injection well. Apart from pressure monitoring, X-ray computed tomography is one of the most metrics used to monitor foam behavior in 1D (Du et al. 2007; Nguyen et al. 2007; Simjoo et al. 2013), but this method has received limited attention for 2D or 3D injection cases. The work of Bertin et al. (1999) did feature 3D cross-flow effects, but the physical scale of the porous medium they used was limited. Tsai et al. (2009) studied the propagation of foam in a laboratory sandbox using iron powder as a tracer. They were able to draw gas saturation contours (identified as foam) right after the foam injection, but this method is highly invasive and requires the collection of several soil samples. Also it does not allow real-time tracking of the foam propagation. Portois et al. (2018) highlighted the use of hydraulic tests (pumping test combined with several observation wells) to monitor a foam front in a real aquifer. As stated by the authors, the extent to which foam propagates horizontally and vertically can only be roughly estimated. This study focuses on the monitoring of a foam front during an injection process using electrical resistivity tomography, which can provide 3D noninvasive imaging of the foam inside the porous medium during the injection process. We performed a foam injection process in a large-scale 3D heterogeneous porous medium, and the resistivity measurements were carried out at various stages during the injection process to track the position of the foam front. Background Theory Electrical resistivity tomography (ERT) is a method for characterizing the spatial distribution of electrical charge carriers, such as ions in mineralized water, clay and metal particles. ERT for the application of monitoring fluid flows in contaminated sites has been the subject of several studies (e.g., Wilkinson et al. 2010; Van Dam et al. 2014) due to its noninvasive nature. Electrodes can be placed toward the edge of the porous medium therefore not interfering with the fluid flows inside it. The method is suitable for monitoring dynamic processes as the measurements can be carried out during very short intervals. This is known as time-lapse ERT and has been used extensively for a number of applications ranging from the monitoring of surface water–groundwater interaction (e.g., Slater et al. 2010) to monitoring the salinization of freshwater (Wagner et al. 2013), water injection processes (Kuras et al. 2009), interactions between surface water and groundwater (Slater et al. 2010) and soil–plant interactions (Cassiani et al. 2015, 2016). Analysis of time-lapse ERT data requires data inversion techniques where the time is included explicitly in the procedure (Kim et al. 2009; Karaoulis et al. 2014) and where there is specific attention paid to the sensitivity changes of the parameters over time (Karaoulis et al. 2014) in order not to misinterpret the results. Another application which has been studied extensively is the use of ERT for monitoring the presence of dense non-aqueous phase liquids (DNAPLs) in the subsurface (e.g., Naudet et al. 2011; Chambers et al. 2004; Power et al. 2015). Lucius et al. (1992) found that DNAPLs tend to have a highly resistive nature compared to ground water, thus using ERT allows for easy distinction between DNAPLs and the ground water. Revil et al. (2011) determine that DNAPLs could have various electrical signatures ranging from highly resistive for fresh and unalterated DNAPL, to very conductive for alterated DNAPL. Also the use of time-lapse ERT has proven useful in monitoring remediation processes of DNAPL spills (e.g., Power et al. 2014). Tracking the movement of foams in porous media using ERT has not been studied extensively thus far, but uniformity (Wang and Cilliers 1999) and liquid content (Karapantsios and Kostoglou 2011) of bulk foams have been studied previously with ERT. Furthermore, in recent years the storage of CO2 in geological formations has received extensive interest. Several studies (e.g., Carrigan et al. 2013; Yang et al. 2015; Pezard et al. 2015; Sauer et al. 2014) use ERT to monitor the movement of CO2 inside these formations, indicating the feasibility of tracking gas fronts using this method. Experimental Setup A wooden structure consisting of 28-mm-thick boards is used to house an unconsolidated sand pack (0.84 × 0.84 × 0.84 m), which functions as the porous medium in these experiments. An impervious plastic liner is added to the inside of the wooden container to prevent leaks. The porous medium is created by adding sand and water into the structure, thereby creating a medium that is initially fully saturated with water. Most of the sand used is fairly coarse sand (MI 0.4–0.9) with a mean grain diameter of 677 μm. However, to determine how the foam is affected by heterogeneity we added a cylindrical lens of fine sand in the center of the porous medium consisting of finer sand (MI 31) with a mean grain diameter of 300 μm. This fine sand lens is 54 cm in diameter and 27 cm in height. These grain sizes result in permeabilities of 70 and 20 D for the coarse and fine sand regions, respectively. The porous medium has a total pore volume of 208 L and a porosity of 0.353. A vertical injection well (64 mm diameter) is placed in the center of the porous medium, and four production wells (30 mm diameter) are placed vertically in each of the corners of the structure. A clay layer with a thickness of approximately 80 mm is placed on top of the sand pack which prevents leakage of gas from the top of the sand pack. An additional wooden plate is placed on top of the clay layer and is used to compact the entire medium. A global overview of the sand pack inside the structure is given in Fig. 1. Schematic overview showing vertical (left) and horizontal (right) cross sections of the porous medium inside the wooden structure including dimensions given in millimeters. The darker area in the middle of the medium is the heterogeneity that consists of finer sand (k = 20 D), which is surrounded by coarser sand (k = 70 D) A schematic of the injection system used in these experiments is shown in Fig. 2. A Cole-Parmer Masterflex L/S pump and a SMG gas metering system are installed for the injection of surfactant solution and gas, respectively. These are co-injection experiments, so both fluids are injected simultaneously. A tube runs down the injection well in the center of the porous medium and is connected to a model porous medium which functions as a bubble generator. This ensures that foam is pre-generated within the well itself during the co-injection process before it enters the porous medium, which ensures a strong foam right at the well. Injection system showing liquid and gas injection systems connected to injection well. A bubble generator (model porous medium) at the bottom of the well ensures strong foam generation inside the well during co-injection of gas and liquid A further close-up schematic of the injection well is shown in Fig. 3. This figure shows that the well is screened over the entire height within the sand pack, but not within the clay layer near the top of the well. This means that fluids can be injected over the entire height of the sand pack. Two inflatable packers are installed above and below the bubble generator that can block parts of the well. This allows for injection into certain segments of the injection well rather than the entire height. Cross-sectional schematic view of the injection well showing screened well within the sand pack. By placing two inflatable packers inside the injection well, the injection region can be controlled During the injection process, liquid that is initially in place is produced from the four production wells that are placed vertically in all four corners of the setup. The Cole-Palmer pump that is used for the injection of liquid is also used for the liquid production by installing a second pump head. This head is connected to four tubes which are placed inside the production wells and continuously sucks out fluid from the wells. The bottom of these tubes is aligned with the top of the sand pack, so the water level in the production wells can never exceed the top of the sand pack which prevents flooding of water from the top of the wells. This system is shown schematically in Fig. 4. Aspiration system drawing liquid from the vertical production wells in all four corners of the porous medium Resistivity Measurement System and Electrode Layout The ERT measurements carried out to track the foam front inside the porous medium use a series of electrodes placed inside and along the sides of the sand pack, which are used for the current injection and potential measurement. In total, 128 electrodes are placed inside the porous medium. Sixty-four of these are positioned along vertical lines in the corners of the sand pack, approximately 10 cm away from the production wells. The other 64 electrodes are placed inside the porous medium on horizontal and vertical lines. These electrodes are spread out to all sides of the pilot to ensure that combinations of the electrodes can cover the entire sand pack. The electrodes used in this experiment are brass rods approximately 1 cm in length and 3 mm in diameter. The electrodes on the inside of the pilot may influence the flow somewhat, but due to their small physical size, their influence is limited. Good contact between electrodes and the medium is ensured by the fully saturated medium. Figure 5 shows the position of the electrodes inside the pilot. The blue cube represents the porous medium, and the small spheres represent the electrodes. The figure shows that the electrodes are positioned on several lines along the sides and inside the pilot. Each line has 16 electrodes on it. Since the sand pack has a thickness of 84 cm, the chosen distance between the electrodes is 5 cm to cover the entire sand pack. Positions of electrodes within the sand pack indicated by colored dots. Different colors represent different lines of electrodes. White dots are electrodes placed along vertical lines near the corners of the porous medium, 10 cm away from the production wells, black dots are electrodes placed on horizontal, diagonal lines at heights of 15, 40 and 65 cm and red dots form a vertical line of electrodes closer to the well to provide better resolution near the center of the sand pack In order to allow for 3D imaging of the fluids inside the porous medium, a large number of resistivity measurements are required combined with inversion techniques. A single measurement uses four electrodes (quadripole) two of which are used for the current injection and the other two for the simultaneous measurement of potential difference that is caused by the injected current through the medium. The measurement of DC resistivity ρ is based on Ohm's law, with the resistivity being equal to the ratio of the measured potential U (V) to the product of the injected current I (A) and the geometrical factor K (m). The latter is based on the positions of the electrodes used in the measurements and the boundary conditions of the overall geometry of the porous medium. In order to reach a high sensitivity all over the medium, we have to combine a high number of electrodes well distributed all over the medium, as well as a high number of quadripoles also well distributed. Once all the quadripole combinations (called a sequence) are acquired, the resistivity distribution can be estimated through inversion of the combined measurements. In total, the first measurement sequence contains more than 5000 measurement quadripoles. This includes reciprocal measurements (i.e., quadripoles where the current injection and potential measurement electrodes are inversed) which are used to check the quality of the measured data. Inversion techniques are required to turn the measured results into a 3D representation of the resistivity distribution. In this study, the software R3t (Binley 2009) is used to invert the resistivity data. This software allows for inversion on bounded, unstructured, 3D geometries such as the porous medium used here. The software uses a 3D finite element model of the porous medium. The model was created using Gmsh (Geuzaine and Remacle 2009) and uses a fully unstructured tetrahedral mesh. Electrode positions are integrated into the model and appear as nodes between certain elements. Here, we use null flux conditions on the outer bounds of the geometry of the porous medium to serve as boundary conditions which are used in the inversion process. The simulation uses an initial estimate of the porous medium as a starting point for the inversion. Here, we assume that the initial state of the porous medium is homogeneous with a resistivity of 100 Ωm, which is a representative value for a porous medium that is fully saturated with water. The inversion process will introduce a certain degree of error (defined as root-mean-square (RMS) error between calculated apparent resistivity and measured apparent resistivity) and equivalence. The error is caused both by the finite number of measurements and the finite element model used in the inversion process which is not a perfect realization of the actual porous medium. By equivalence, we mean the uncertainty in resistivity value at each inverted point that is inherent to potential inversion methods due to the nonlinearity of the equation and to ill-posed nature of the problem (less measurements compared to parameters to estimate). Also, the simulation performs iterative calculations until a certain error threshold is reached. R3t allows for setting both offset and relative errors. In these simulations, only a relative error is used of 0.05. An additional series of measurements are performed using electrode quadripoles that are placed in the center of the pilot (i.e., using the electrodes shown as black and red dots in Fig. 5). These measurements allow for directly measuring the resistance at their location. These measurements only provide local information of the medium's resistance and are combined with the measurements from the larger sequence to increase the accuracy of the inversion process. In total, more than 2000 quadripoles were used for every inversion. The measurements are performed using an SYSCAL PRO (Iris Instrument) resistivity measurement multielectrodes and multichannels system. This system allows for the simultaneous connection of up to 72 electrodes, meaning that a single sequence can use 72 electrodes allowing the use of all possible quadripole combinations. Since two different sequences are used in these measurements, two different subsets of 72 electrodes out of the total of 128 electrodes that are present in the porous medium are used. Figure 6 shows which electrodes are used during the sequences. Note that some electrodes are used in both sequences. For one identical couple of injection electrodes, this system allows the simultaneous acquisition of 10 quadripoles. Electrode layout within the porous medium. Color of the dots indicates in which sequence the electrodes are used. White is used in the first sequence aimed at providing 3D distribution of resistivity, black is used in the second sequence aimed at reducing uncertainty in the results on the plane of measurement, and red is used in both sequences Salt Solution Injection Before the foam injection, injection of salt (NaCl, 0.5 g/L) solution was performed to analyze the porous medium. Adding salt to the water increases the number of ions inside the fluid, thus the amount of electrical charge carriers and thus the conductivity of the fluid, thereby reducing the resistivity of the porous medium. The salt solution used here has a conductivity of 1330 µS/cm versus 420 µS/cm for the fresh water that it replaces, and thus the salt solution is more than three times more conductive than the water that is originally in place. This is expected to be sufficient contrast for the salt solution to be detectable. Running the ERT measurement sequences at various stages during the salt injection process (i.e., the injection is stopped, while the measurements are being performed) allows for determining where the injected fluids flow as the injected salt solution will show up as regions of lower resistivity compared to the water that is initially in place. The effect of the heterogeneity of the porous medium on the flow can be studied in this way. It is also a good method for testing the measurement approach and inversion method as the salt solution can be flushed out of the porous medium afterward to reverse the porous medium back to its initial state, which is not possible after the injection of foam. A total of volume 150 L (= 0.72 PV) of salt solution was injected into the porous medium at a rate of 600 mL/min. ERT measurements were performed at 5 stages during the injection process, namely after 0, 25, 60, 100 and 150 L (or 0, 0.12, 0.29, 0.48 and 0.72 PV). During the entire salt solution injection process, the injection well is screened over its entire height, so no packers are installed to block off certain sections of the well. After the salt injection, fresh water was injected to remove the salt solution from the porous medium. Foam Injection Approach The main experiment is a foam injection process. Initially, 2 PV of surfactant solution (ammonium alcohol ether sulfate, Stepan Petrostep ES-65A) is injected into the porous medium to satisfy the adsorption criterion. The surfactant concentration used throughout this experiment is 0.1 wt% (wt./wt.) active content. This is followed by performing the ERT measurement sequence, which is used as the baseline result (so a porous medium without any gas present in it). Thereafter, the foam injection process is started by co-injecting gas (air from a bottle supplied by Air Liquide) and surfactant solution at a total flow rate of 440 mL/min with foam quality (= gas fractional flow) of 0.9. A preliminary foam injection experiment in a column showed that the decent foaming performance (i.e., mobility reduction) can be expected using this surfactant concentration and foam quality. Both fluids flow combined through the filter at the end of the injection tube which serves as a bubble generator that ensures strong foam is formed inside the injection well. The three-dimensional nature of these experiments means that some concessions need to be made in order to show presentable figures in this manuscript. For clarity reasons, the majority of results are a 2D plane representation of the complete inverted data which are in 3D. The chosen plane is a vertical diagonal plane through the center of the porous medium (so also through the injection well and two out of the four production wells). This is also the plane in which the electrodes on the horizontal diagonal lines (cf. black dots in Fig. 5) are located. This plane was chosen as it allows for observation of the radial and vertical position of the foam front as it progresses from the injection well. Also, since it has the most electrodes directly on the plane, it is likely to have the lowest level of uncertainty within the 3D medium. Figure 7 shows the chosen plane on which most of the results are shown. Vertical diagonal clipping plane through the center of the porous medium (so also through the injection well). The majority of the resistivity distribution results in the remainder of this text are shown on this plane. The white dots indicate the positions of the electrodes that are inside the pilot (so excluding the ones near the corners). Note that the electrodes on the horizontal lines (which are also shown as black dots in Fig. 5) are on this vertical plane, thereby minimizing the uncertainty within the results shown in the remainder of this text Injection of Salt Solution Figure 8 shows the initial distribution of resistivity on the vertical diagonal plane which was obtained by inverting the measured ERT data, so before the injection process has started. Large variations of resistivity are observed at this stage. These may have several reasons. First of all, the region of low resistivity near the top of the figure (blue region) is caused by the presence of the clay layer on top of the sand pack. Note that the simulation model only intended to capture the sand pack and not the clay layer on top, but the clay layer apparently affects the inverted resistivity distribution in the top region of the model. Another possibility is that the clay layer is actually physically thicker than intended and extends further down into the sand pack. Base case resistivity before the salt injection. The low-resistivity region near the top represents the clay layer on top of the porous medium. The remainder of the porous medium shows a significant variation in resistivity The high-resistivity regions in the top left and right portions of the figures are likely to be caused by a lack of electrodes in those regions. The far-left and far-right portions of this and subsequent figures represent regions that near the corners of the medium, and there are no electrodes present within them (cf. Fig. 5 which shows that electrodes are at least 10 cm away from the corners). Errors in the obtained resistivity distribution are to be expected in these regions, because there are no possible electrode quadripoles that can capture the resistivity here. The remainder of the figure also shows large variation in resistivity. These may have been caused by dissolution of minerals from the sand during the construction of the sand pack which may have affected the resistivity distribution. Figure 9a–d shows the resistivity distribution on the vertical plane for various stages during the injection of salt solution. These figures show the advancing salt solution front as it progresses through the medium. As expected, the regions where the salt solution has swept show up as regions with a significantly lower resistivity than the remaining regions. Hand-drawn yellow curves are added to the figures which provide an estimate of the advancing salt front. The clay layer on top makes it hard to judge the position of the salt in the top region of these figures, but the salt can be accurately tracked in the remainder of the medium. It is clear that the coarser sand near the top and bottom of the medium cause preferential flow due to its higher permeability and the salt solution progresses a lot slower through the finer sand in the center. After injection of 0.72 PV of solution, breakthrough of salt solution in the production wells was observed by measuring increased conductivity of the produced water, so the injection process was stopped. ERT measurements on a vertical cross section along the diagonal of the pilot showing resistivity distributions at the indicated points in time in during the salt injection process. Note that the color axis is the same as that shown in Fig. 8 for all of these figures. The yellow lines are hand-drawn and represent an estimate of the advancing salt front in the center portion of the medium. at = 0.12 PV, bt = 0.29 PV, ct = 0.48 PV and dt = 0.72 PV The salt solution injection process has shown that the utilized ERT measurements are capable of tracking fluids with intrinsically different resistivity values than the fluids that are being displaced. The proposed sequences are sufficient for tracking the front of injected fluid. In the case of the porous medium analyzed here, the heterogeneity formed by the low-permeability lens in the center of the sand pack influences the flow such that there is limited flow going through the center section and most flow is diverted to the coarse sand regions. The Foam Injection Process After the salt solution injection process finished, the salt is flushed out of the system by first injecting fresh water and afterward 2 PV of surfactant solution. This provides the initial state for the foam injection process, and its resistivity distribution is shown in Fig. 10. This is once again the resistivity distribution on the vertical diagonal plane that was also used for showing the results of the salt injection along with an identical color axis to allow direct comparison with previous results. Note that the initial state shown here appears far more homogeneous than that found before the salt injection had started (cf. Fig. 8). Some slight variations in resistivity are still observed here, but not nearly as severe as that found before. Those were already implied to be at least partially the result of dissolution of minerals from the sand and clay. The distribution shown in Fig. 10 was taken directly after fully saturating the porous medium with surfactant solution, so the distribution was expected to be homogeneous. Still the contrast between the two figures is very stark, and the complete reason behind the large variations found in Fig. 8 is unknown. Base case resistivity before foam injection. The low-resistivity region near the top represents the clay layer on top of the porous medium. Overall a fairly uniform distribution of resistivity is measured within the porous medium although some regions of higher resistivity can be identified Figure 11a–h shows the resistivity distribution on the vertical diagonal plane at various stages during the foam injection process. For the initial stages of the foam injection process, two packers were used to block off certain sections of the injection well allowing for targeted injection into specific sections of the porous medium. This was done to determine whether foam can be used to block off certain regions within the porous medium. The idea was to inject into the top and bottom regions first which contain coarse sand and afterward inject over the entire height to see whether the blocking property of the foam in the coarse sand layer is sufficient for diverting flow to the lower-permeability lens in the center of the porous medium. Afterward, foam was injected over the entire height of the injection well for a while, but some additional injection in the top and bottom layers proved necessary to improve the blocking effect. In the end, foam injection over the entire height of the well was carried out for a prolonged period of time to try and sweep the entire porous medium with foam. This injection procedure is outlined in Table 1 which also includes the corresponding figures for each of the injection stages. ERT measurements on a vertical cross section along the diagonal of the pilot showing resistivity distributions at the indicated points in time in during the injection process. Note that the color axis is the same as that for the salt injection shown in Figs. 8 and 9 and the base resistivity shown in Fig. 10 for all of these figures. at = 0.04 PV, bt = 0.08 PV, ct = 0.12 PV, dt = 0.20 PV, et = 0.28 PV, ft = 0.40 PV, gt = 0.58 PV and ht = 0.76 PV Table 1 Outline of the foam injection procedure Figure 11a, b shows the result after the initial injection in the top and bottom layers. When comparing these results to the initial state shown in Fig. 10, there are clear regions of increased resistivity which can be observed near the top and the bottom of the medium. This is as expected due to the increased resistivity of the foam swept regions. The next step was injecting over the entire height of the injection well, and its result is shown in Fig. 11c. This figure shows only a very slight increase in resistivity in the fine sand lens in the center of the pilot, but there is a significant increase that is observed near the top of the medium, which implies that gas is flowing there due to gravity override. This means that the foam's ability to reduce mobility is insufficient and the foam that was already in place in the coarse sand layers was also not enough to prevent the gas flowing to the top. To improve on this, additional foam is injected in the top and bottom layers. The result of this is shown in Fig. 11d in which an extension of the high-resistivity regions can be observed near the top and bottom of the medium. After this, the injection over the entire height of the well was performed for an extended period of time. The injection was halted at various points in time to perform the ERT measurements. The results of these are shown in Fig. 11e–h. Subsequent figures show an ever growing expansion of the foam swept region, but the gravity override could not be prevented, so in each figure the preferential gas flow to the top of the medium can be clearly observed. The foam strength was found to be insufficient to provide a stable radial displacement front. During the experiment, the injection pressure was monitored, which serves as an additional measure of the foaming performance. However, since only the injection pressure is monitored and no pressure data are available further away from the injection well, this can only describe the foaming performance in the near-well region. Figure 12 shows the injection pressure during 0.4 and 0.58 PV of injection, which corresponds to the period of time between Fig. 11f, g. This injection timespan was chosen as it represents a pressure profile which is fairly typical for this experiment. During other injection stages within the foam injection process, the resulting pressure profile may vary slightly from this due to the foam front having swept a smaller or larger region, but not to such an extent that it alters the conclusion drawn from the calculations below. Injection pressure profile during one of the foam injection stages As shown in Fig. 12, the pressure increases gradually during the injection until it reaches a plateau. We will use this plateau value (roughly 5.35 × 104 Pa) to base our calculations on. It should be noted that this elevated injection pressure did not compromise the structure of the apparatus. No deformations in the structure were observed during the experiment. As mentioned above, these experiments were carried out using a fixed total flow rate of 440 mL/min. During the injection stage for which the pressure profile is given in Fig. 12, the injection occurs over the full height of the injection well. This means that injection occurs in both the coarse and fine sand layers, and thus we need to make use of the arithmetic average of the permeability to find an average permeability value. This average value is based on the thickness of the fine and coarse sand layers (see Eq. 1) and amounts to 52 D. $$ k_{\text{avg}} = \frac{{k_{\text{fine}} \cdot H_{\text{fine}} + k_{\text{coarse}} \cdot H_{\text{coarse}} }}{H} = 52\,{\text{D}} $$ where kavg is the average permeability of the medium near the well, kfine is the fine sand permeability of 20 D, Hfine is the height of the fine sand layer of 30 cm, kcoarse is the coarse sand permeability of 70 D, Hcoarse is the height of the coarse sand of 54 cm, and H is the overall height of the sand pack of 84 cm. Using this value of permeability, the expected pressure drop is calculated if only water is flowing inside the medium using Darcy's law in radial form (Eq. 2). This value can then be compared to the measured value of the injection pressure to determine the foam apparent viscosity in the near-well region. $$ \Delta p_{\text{w}} = \frac{{Q\mu_{\text{w}} \ln \left( {{{R_{\text{e}} } \mathord{\left/ {\vphantom {{R_{\text{e}} } {R_{\text{w}} }}} \right. \kern-0pt} {R_{\text{w}} }}} \right)}}{{2\pi Hk_{\text{avg}} }} = 30.4\,{\text{Pa}} $$ where Δpw is the expected pressure drop for a single-phase water flood, Q is the overall injection flow rate of 440 mL/min, μw is the dynamic viscosity of water of 1 mPa s, Re is the distance between the injection and the production well, and Rw is the injection well radius. This implies that the injection pressure (i.e., the pressure needed to overcome the viscous forces due to the generated foam) is 5.35 × 104/30.4 = 1.76 × 103 times higher than that expected for a single-phase water flood. This means that, at least near the well and since the viscosity of water equals 1 mPa s, the foam apparent viscosity is 1.76 Pa s. So the foam's ability to reduce mobility near the well should be able to provide a stable displacement front. Another measure which determines whether the foam front is stable is a comparison between the viscous and gravitational forces. The viscous pressure is equivalent to the injection pressure as it is this pressure that drives the flow of foam. However, in radial flow configurations such as this, the pressure quickly decreases with increasing distance from the injection well. This means that even though foam may initially be stable, further away from the well, gravitational forces may become dominant leading to gravity override and foam collapse. The maximum possible pressure due to gravity is equal to the hydrostatic pressure, which for the injection of gas in a water-saturated medium is given in Eq. 3. $$ \hbox{max} \left( {\Delta p_{\text{h}} } \right) = \left( {\rho_{\text{w}} - \rho_{\text{g}} } \right)gH = 8.23 \times 10^{3} \;{\text{Pa}} $$ Thus, near the well, the viscous pressure gradient can be assumed to be higher than the gravitational one, due to the injection pressure being substantially (more than six times) higher than the maximum possible hydrostatic pressure, so the main driving force of the flow is the viscous pressure. However, if purely radial flow of foam is assumed and a constant foam apparent viscosity is assumed, a pressure profile throughout the pilot as a function of its distance from the injection well can also be estimated. The result of this is given in Fig. 13, which is based on Eq. 2, but using the foam apparent viscosity rather than the water viscosity. This figure shows that pressure quickly decreases as a function of radius and at a distance of 0.31 m from the well center, the pressure has decreased so much that the gravitational forces will become dominant and gravity override could be expected for distances larger than this. Note that this is an ideal scenario where the foam strength is constant and no foam collapse occurs, so in reality, probably gravity override can be expected even sooner. This result shows that the gravity override that was found in the experiment is not unexpected. Estimated pressure profile as a function of radial distance from the well assuming purely radial flow and the presence of foam with a foam apparent viscosity of 1.76 Pa s The resistivity data can also be used to estimate the liquid saturation within the foam swept region by applying Archie's law (Archie 1942) (Eq. 4). $$ \rho = \rho_{\text{l}} \cdot \phi^{ - m} \cdot S_{\text{l}}^{ - n} $$ where ϕ is the porosity, which was measured during the preparation of the sand pack and is equal to 0.35, ρ is the resistivity of the formation, ρl is the resistivity of the liquid, which in this case is the surfactant solution with a conductivity measured at 1330 μS/cm which corresponds to 7.5 Ωm, Sl is the fluid saturation of the medium (fully saturated Sl = 1), m is the cementation factor estimated at 0.6, and n is the saturation exponent estimated at 2. Thus, for a fully saturated medium the resistivity is about 15 Ωm. As observed in the resistivity figures (Fig. 11), the foam swept region has resistivity values upward of 200 Ωm. When using this value, we find a liquid saturation of Sl = 0.27. Note that the color axis in Fig. 11 is clipped, so higher resistivity values are found within the foam swept region, and thus the liquid saturation value may vary. In addition to the results on the vertical diagonal plane, an attempt has been made to visualize the results in three dimensions by means of iso-resistivity contour plots. Three figures showing the last three stages of the foam injection process are shown in Fig. 14a–c. These are 3D representations of the advancing foam front, and they correspond to Fig. 11f–h, respectively, which show results at the same time, but only the in-plane results. The red contours shown in Fig. 14a–c are iso-resistivity contours with a value of 200 Ωm. This value was empirically chosen as it is significantly higher than the resistivity values obtained for a medium fully saturated with water. Therefore, it is a qualitative indication of the region where the foam has swept. These figures allow determining whether there are preferential flow paths in radial direction. Preferential flow appears to be the case in Fig. 14a which shows the foam front after 0.40 PV of injection. The front has already progressed to the top left corner of the medium at this stage, but has not quite reached the other corners. This might be the result of preferential flow, but could also be that the resistivity on the right side of the figure is just slightly below the contour value of 200 Ωm. The latter appears to be the case when comparing the results from the Fig. 14b, c which shows improved axial symmetry, thus not indicating preferential flow toward a particular corner of the medium. Also, these latter two figures are directly comparable to the 2D distributions shown in Fig. 13g, h which implies that for axisymmetric flow the 2D representation provides sufficient information to characterize the advancing foam front inside the medium. Another explanation for the observed asymmetry in Fig. 14a is the constant rate from the production wells. If the permeability of the porous medium was not fully axisymmetric, this would lead to preferential flow toward one of the production well and the constant rate production would mean that more water was pulled out of one of the production wells. Changing to a constant pressure rather than constant rate production could prevent such preferential flow. 3D iso-resistivity contours for the indicated injection time. The red contours are iso-contours with a resistivity of 200 Ωm. at = 0.40 PV, bt = 0.58 PV and ct = 0.76 PV One further observation from Fig. 14c is that the foam swept region adopts an almost spherical shape. Kovscek et al. (1997) found that this is the expected shape for a region that is swept with strong foam. It appears that in this case a region of strong foam does form near the well, but further away from the well the foam strength is reduced which leads to foam collapse and the subsequent flow of free gas to the top of the medium as a result of gravity override. Foam flow can be accurately monitored inside large-scale porous media using ERT due to the large contrast of resistivity between liquid- and foam-saturated porous media. A large number of measurement quadripoles are required to generate the inverted image showing the resistivity distribution in 3D. This puts some constraints on the experimental process. The injection has to be stopped during the measurement sequence to minimize changing conditions during the entire sequence. Still, foam may flow even when the injection is halted due to the higher pressure near the point of injection and possible gravity override from the gas. This need for intermittent injection may negatively impact the foaming performance inside the porous medium. In the current experiment, the time needed to perform the measurements was similar to the duration of injection and thus the injection was halted about half the time. Both the foam flow and the ERT results are affected by this. Internal electrodes appeared as anomalies on the inverted ERT results. These are local anomalies though and the global flow of foam (or salt solution) can still be tracked. However, the result is qualitative, meaning that results are limited to tracking the foam front due to the large contrast in resistivity between water- and foam-saturated regions. Basic calculations, such as the ones presented here based on injection pressure and resistivity values, can provide initial estimates into more quantitative data, like foam apparent viscosity and liquid saturation, that can help explain the foam performance. For further, more accurate calculations of these data, additional measurements of properties that influence the results are required. These properties include, but are not limited to, the following: salt concentration of the liquid, transfer of water to the gas phase and adsorption of surfactant. The foam injection performed in this study was not entirely successful. The foam front proved to be unstable and suffered from severe gravity override. This implies that the mobility reduction was not sufficient enough and the volumetric sweep of the medium was therefore limited. Overall, it can be stated that the foam injection process needs to be improved upon in order to prevent the gravity override. However, when generating a stronger foam in the injection well which can be better suited to reduce overall mobility of the fluids, the injection pressure will have to increase as well or the flow rate needs to be lowered. Operating pressure is a major constraint for aquifer application of foam, and thus lower flow rates with a stronger, more effective foam, may be the preferred choice. Another method of injection may be to use a fixed pressure rather than a fixed injection rate. In this way, we can ensure that allowable operating pressure is not exceeded. This method has been used in previous studies by Hirasaki et al. (1997) and Maire and Fatin-Rouge (2017). Furthermore, Shan and Rossen (2004) found that, for oil reservoir conditions, maintaining maximum allowable injection pressure minimizes foam segregation and gravity. Other injection schemes such as surfactant-alternating-gas (SAG) injection could also be considered for this purpose as it would increase injectivity compared to co-injection methods. Archie, G.E.: The electrical resistivity log as an aid in determining some reservoir characteristics. Trans. 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Geophys. 70(4), 268–276 (2010) Yang, X., Lassen, R.N., Jensen, K.H., Looms, M.C.: Monitoring CO2 migration in a shallow sand aquifer using 3D crosshole electrical resistivity tomography. Int. J. Greenhouse Gas Control 42, 534–544 (2015) École nationale d'ingénieurs de Saint-Étienne (ENISE), Saint-Étienne, France C. S. Boeije Bordeaux INP – ENSEGID, Pessac, France , C. Portois , M. Schmutz & O. Atteia Search for C. S. Boeije in: Search for C. Portois in: Search for M. Schmutz in: Search for O. Atteia in: Correspondence to C. S. Boeije. Boeije, C.S., Portois, C., Schmutz, M. et al. Tracking a Foam Front in a 3D, Heterogeneous Porous Medium. Transp Porous Med 131, 23–42 (2020) doi:10.1007/s11242-018-1185-0 Porous media	CommonCrawl
Directional limits on persistent gravitational waves using data from Advanced LIGO's first two observing runs We perform an unmodeled search for persistent, directional gravitational wave (GW) sources using data from the first and second observing runs of Advanced LIGO. We do not find evidence for any GW signals. We limit the flux emitted at 25~Hz from point sources with a power law spectrum to be $F_{\alpha,\Theta} (0.05-25)\times 10^{-8} ~\unit[]{erg\,cm^{-2}\,s^{-1}\,Hz^{-1}}$ and the (normalized) energy density spectrum in GWs at 25 Hz from extended sources to be $\Omega_{\alpha}(\Theta) (0.19-2.89)\times 10^{-8} ~\unit[]{sr^{-1}}$ where $\alpha$ is the spectral index of the energy density spectrum. These represent improvements of $2.5-3\times$ over previous limits. We also consider point sources emitting GWs at a single frequency, targeting the directions of Sco X-1, SN 1987A, and the Galactic Center. The best upper limits on the strain amplitude of a potential source in these three directions range from $h_0 (3.6-4.7)\times 10^{-25}$, 1.5$\times$ better than previous limits set with the same analysis method. We also report on a marginally significant outlier at 36.06~Hz. This outlier is not consistent with a persistent gravitational-wave source as its significance diminishes when combining all of the available data. Main pdf product. (sph_O2_main.pdf, 1.9 MB) Referenced by: LIGO-T1900079-v7: Data for Directional limits on persistent gravitational waves using data from Advanced LIGO's first two observing runs	CommonCrawl
MSC Classifications MSC 2010: Field theory and polynomials 12Jxx Refine listing Only show content I have access to (14) Only show open access (8) Last 3 months (2) Last 12 months (6) Last 3 years (11) Proceedings of the Edinburgh Mathematical Society (5) Mathematika (4) Bulletin of Symbolic Logic (2) Compositio Mathematica (2) Journal of the Institute of Mathematics of Jussieu (2) The Journal of Symbolic Logic (2) Bulletin of the Australian Mathematical Society (1) Ergodic Theory and Dynamical Systems (1) Forum of Mathematics, Pi (1) Forum of Mathematics, Sigma (1) Glasgow Mathematical Journal (1) Journal of K-Theory (1) Journal of the Australian Mathematical Society (1) LMS Journal of Computation and Mathematics (1) LMS (7) Association for Symbolic Logic (4) Australian Mathematical Society Inc (2) Forum of Mathematics (2) 25 results in 12Jxx NOTES ON ATKIN–LEHNER THEORY FOR DRINFELD MODULAR FORMS Arithmetic algebraic geometry Discontinuous groups and automorphic forms Topological fields TARUN DALAL, NARASIMHA KUMAR Journal: Bulletin of the Australian Mathematical Society , First View Published online by Cambridge University Press: 15 November 2022, pp. 1-19 We settle a part of the conjecture by Bandini and Valentino ['On the structure and slopes of Drinfeld cusp forms', Exp. Math.31(2) (2022), 637–651] for $S_{k,l}(\Gamma _0(T))$ when $\mathrm {dim}\ S_{k,l}(\mathrm {GL}_2(A))\leq 2$ . We frame and check the conjecture for primes $\mathfrak {p}$ and higher levels $\mathfrak {p}\mathfrak {m}$ , and show that a part of the conjecture for level $\mathfrak {p} \mathfrak {m}$ does not hold if $\mathfrak {m}\ne A$ and $(k,l)=(2,1)$ . Cohomology of algebraic varieties over non-archimedean fields Homology and cohomology theories Arithmetic problems. Diophantine geometry Model theory Pablo Cubides Kovacsics, Mário J. Edmundo, Jinhe Ye Journal: Forum of Mathematics, Sigma / Volume 10 / 2022 Published online by Cambridge University Press: 21 October 2022, e94 We develop a sheaf cohomology theory of algebraic varieties over an algebraically closed nontrivially valued nonarchimedean field K based on Hrushovski-Loeser's stable completion. In parallel, we develop a sheaf cohomology of definable subsets in o-minimal expansions of the tropical semi-group $\Gamma _{\infty }$ , where $\Gamma $ denotes the value group of K. For quasi-projective varieties, both cohomologies are strongly related by a deformation retraction of the stable completion homeomorphic to a definable subset of $\Gamma _{\infty }$ . In both contexts, we show that the corresponding cohomology theory satisfies the Eilenberg-Steenrod axioms, finiteness and invariance, and we provide natural bounds of cohomological dimension in each case. As an application, we show that there are finitely many isomorphism types of cohomology groups in definable families. Moreover, due to the strong relation between the stable completion of an algebraic variety and its analytification in the sense of V. Berkovich, we recover and extend results on the singular cohomology of the analytification of algebraic varieties concerning finiteness and invariance. On the analyticity of WLUD∞ functions of one variable and WLUD∞ functions of several variables in a complete non-Archimedean valued field Other (nonclassical) types of functional analysis Non-Archimedean analysis Approximations and expansions Real functions: Miscellaneous topics Khodr Shamseddine Journal: Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 3 / August 2022 Published online by Cambridge University Press: 07 July 2022, pp. 691-704 Let $\mathcal {N}$ be a non-Archimedean-ordered field extension of the real numbers that is real closed and Cauchy complete in the topology induced by the order, and whose Hahn group is Archimedean. In this paper, we first review the properties of weakly locally uniformly differentiable (WLUD) functions, $k$ times weakly locally uniformly differentiable (WLUD$^{k}$ ) functions and WLUD$^{\infty }$ functions at a point or on an open subset of $\mathcal {N}$ . Then, we show under what conditions a WLUD$^{\infty }$ function at a point $x_0\in \mathcal {N}$ is analytic in an interval around $x_0$ , that is, it has a convergent Taylor series at any point in that interval. We generalize the concepts of WLUD$^{k}$ and WLUD$^{\infty }$ to functions from $\mathcal {N}^{n}$ to $\mathcal {N}$ , for any $n\in \mathbb {N}$ . Then, we formulate conditions under which a WLUD$^{\infty }$ function at a point $\boldsymbol {x_0} \in \mathcal {N}^{n}$ is analytic at that point. Revisiting closed asymptotic couples Differential and difference algebra Ordered structures Asymptotic theory Matthias Aschenbrenner, Lou van den Dries, Joris van der Hoeven Journal: Proceedings of the Edinburgh Mathematical Society / Volume 65 / Issue 2 / May 2022 Published online by Cambridge University Press: 20 June 2022, pp. 530-555 Every discrete definable subset of a closed asymptotic couple with ordered scalar field ${\boldsymbol {k}}$ is shown to be contained in a finite-dimensional ${\boldsymbol {k}}$ -linear subspace of that couple. It follows that the differential-valued field $\mathbb {T}$ of transseries induces more structure on its value group than what is definable in its asymptotic couple equipped with its scalar multiplication by real numbers, where this asymptotic couple is construed as a two-sorted structure with $\mathbb {R}$ as the underlying set for the second sort. Hensel minimality I Birational geometry Diophantine equations Raf Cluckers, Immanuel Halupczok, Silvain Rideau-Kikuchi Journal: Forum of Mathematics, Pi / Volume 10 / 2022 Published online by Cambridge University Press: 16 May 2022, e11 We present a framework for tame geometry on Henselian valued fields, which we call Hensel minimality. In the spirit of o-minimality, which is key to real geometry and several diophantine applications, we develop geometric results and applications for Hensel minimal structures that were previously known only under stronger, less axiomatic assumptions. We show the existence of t-stratifications in Hensel minimal structures and Taylor approximation results that are key to non-Archimedean versions of Pila–Wilkie point counting, Yomdin's parameterization results and motivic integration. In this first paper, we work in equi-characteristic zero; in the sequel paper, we develop the mixed characteristic case and a diophantine application. DEFINABILITY OF HENSELIAN VALUATIONS BY CONDITIONS ON THE VALUE GROUP Topological rings and modules Connections with logic LOTHAR SEBASTIAN KRAPP, SALMA KUHLMANN, MORITZ LINK Journal: The Journal of Symbolic Logic , First View Published online by Cambridge University Press: 28 April 2022, pp. 1-19 Given a Henselian valuation, we study its definability (with and without parameters) by examining conditions on the value group. We show that any Henselian valuation whose value group is not closed in its divisible hull is definable in the language of rings, using one parameter. Thereby we strengthen known definability results. Moreover, we show that in this case, one parameter is optimal in the sense that one cannot obtain definability without parameters. To this end, we present a construction method for a t-Henselian non-Henselian ordered field elementarily equivalent to a Henselian field with a specified value group. Algebraic and Model Theoretic Properties of O-minimal Exponential Fields Lothar Sebastian Krapp Journal: Bulletin of Symbolic Logic / Volume 27 / Issue 4 / December 2021 An exponential $\exp $ on an ordered field $(K,+,-,\cdot ,0,1,<)$ is an order-preserving isomorphism from the ordered additive group $(K,+,0,<)$ to the ordered multiplicative group of positive elements $(K^{>0},\cdot ,1,<)$ . The structure $(K,+,-,\cdot ,0,1,<,\exp )$ is then called an ordered exponential field (cf. [6]). A linearly ordered structure $(M,<,\ldots )$ is called o-minimal if every parametrically definable subset of M is a finite union of points and open intervals of M. The main subject of this thesis is the algebraic and model theoretic examination of o-minimal exponential fields $(K,+,-,\cdot ,0,1,<,\exp )$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp (0) = 1$ . This study is mainly motivated by the Transfer Conjecture, which states as follows: Any o-minimal exponential field $(K,+,-,\cdot ,0,1,<,\exp )$ whose exponential satisfies the differential equation $\exp ' = \exp $ with initial condition $\exp (0)=1$ is elementarily equivalent to $\mathbb {R}_{\exp }$ . Here, $\mathbb {R}_{\exp }$ denotes the real exponential field $(\mathbb {R},+,-,\cdot ,0,1,<,\exp )$ , where $\exp $ denotes the standard exponential $x \mapsto \mathrm {e}^x$ on $\mathbb {R}$ . Moreover, elementary equivalence means that any first-order sentence in the language $\mathcal {L}_{\exp } = \{+,-,\cdot ,0,1, <,\exp \}$ holds for $(K,+,-,\cdot ,0,1,<,\exp )$ if and only if it holds for $\mathbb {R}_{\exp }$ . The Transfer Conjecture, and thus the study of o-minimal exponential fields, is of particular interest in the light of the decidability of $\mathbb {R}_{\exp }$ . To the date, it is not known if $\mathbb {R}_{\exp }$ is decidable, i.e., whether there exists a procedure determining for a given first-order $\mathcal {L}_{\exp }$ -sentence whether it is true or false in $\mathbb {R}_{\exp }$ . However, under the assumption of Schanuel's Conjecture—a famous open conjecture from Transcendental Number Theory—a decision procedure for $\mathbb {R}_{\exp }$ exists (cf. [7]). Also a positive answer to the Transfer Conjecture would result in the decidability of $\mathbb {R}_{\exp }$ (cf. [1]). Thus, we study o-minimal exponential fields with regard to the Transfer Conjecture, Schanuel's Conjecture, and the decidability question of $\mathbb {R}_{\exp }$ . Overall, we shed light on the valuation theoretic invariants of o-minimal exponential fields—the residue field and the value group—with additional induced structure. Moreover, we explore elementary substructures and extensions of o-minimal exponential fields to the maximal ends—the smallest elementary substructures being prime models and the maximal elementary extensions being contained in the surreal numbers. Further, we draw connections to models of Peano Arithmetic, integer parts, density in real closure, definable Henselian valuations, and strongly NIP ordered fields. Parts of this thesis were published in [2–5]. Abstract prepared by Lothar Sebastian Krapp E-mail: [email protected] URL: https://d-nb.info/1202012558/34 Chaotic behavior of the p-adic Potts–Bethe mapping II Difference and functional equations Difference equations Topological dynamics OTABEK KHAKIMOV, FARRUKH MUKHAMEDOV Journal: Ergodic Theory and Dynamical Systems / Volume 42 / Issue 11 / November 2022 Published online by Cambridge University Press: 30 September 2021, pp. 3433-3457 Print publication: November 2022 The renormalization group method has been developed to investigate p-adic q-state Potts models on the Cayley tree of order k. This method is closely related to the examination of dynamical behavior of the p-adic Potts–Bethe mapping which depends on the parameters q, k. In Mukhamedov and Khakimov [Chaotic behavior of the p-adic Potts–Behte mapping. Discrete Contin. Dyn. Syst. 38 (2018), 231–245], we have considered the case when q is not divisible by p and, under some conditions, it was established that the mapping is conjugate to the full shift on $\kappa _p$ symbols (here $\kappa _p$ is the greatest common factor of k and $p-1$ ). The present paper is a continuation of the forementioned paper, but here we investigate the case when q is divisible by p and k is arbitrary. We are able to fully describe the dynamical behavior of the p-adic Potts–Bethe mapping by means of a Markov partition. Moreover, the existence of a Julia set is established, over which the mapping exhibits a chaotic behavior. We point out that a similar result is not known in the case of real numbers (with rigorous proofs). SURREAL ORDERED EXPONENTIAL FIELDS Ordered sets PHILIP EHRLICH, ELLIOT KAPLAN Journal: The Journal of Symbolic Logic / Volume 86 / Issue 3 / September 2021 Published online by Cambridge University Press: 13 August 2021, pp. 1066-1115 Print publication: September 2021 In 2001, the algebraico-tree-theoretic simplicity hierarchical structure of J. H. Conway's ordered field ${\mathbf {No}}$ of surreal numbers was brought to the fore by the first author and employed to provide necessary and sufficient conditions for an ordered field (ordered $K$ -vector space) to be isomorphic to an initial subfield ( $K$ -subspace) of ${\mathbf {No}}$ , i.e. a subfield ( $K$ -subspace) of ${\mathbf {No}}$ that is an initial subtree of ${\mathbf {No}}$ . In this sequel, analogous results are established for ordered exponential fields, making use of a slight generalization of Schmeling's conception of a transseries field. It is further shown that a wide range of ordered exponential fields are isomorphic to initial exponential subfields of $({\mathbf {No}}, \exp )$ . These include all models of $T({\mathbb R}_W, e^x)$ , where ${\mathbb R}_W$ is the reals expanded by a convergent Weierstrass system W. Of these, those we call trigonometric-exponential fields are given particular attention. It is shown that the exponential functions on the initial trigonometric-exponential subfields of ${\mathbf {No}}$ , which includes ${\mathbf {No}}$ itself, extend to canonical exponential functions on their surcomplex counterparts. The image of the canonical map of the ordered exponential field ${\mathbb T}^{LE}$ of logarithmic-exponential transseries into ${\mathbf {No}}$ is shown to be initial, as are the ordered exponential fields ${\mathbb R}((\omega ))^{EL}$ and ${\mathbb R}\langle \langle \omega \rangle \rangle $ . Transfer Principles in Henselian Valued Fields Pierre Touchard Journal: Bulletin of Symbolic Logic / Volume 27 / Issue 2 / June 2021 Published online by Cambridge University Press: 16 September 2021, pp. 222-223 Print publication: June 2021 In this thesis, we study transfer principles in the context of certain Henselian valued fields, namely Henselian valued fields of equicharacteristic $0$ , algebraically closed valued fields, algebraically maximal Kaplansky valued fields, and unramified mixed characteristic Henselian valued fields with perfect residue field. First, we compute the burden of such a valued field in terms of the burden of its value group and its residue field. The burden is a cardinal related to the model theoretic complexity and a notion of dimension associated to $\text {NTP}_2$ theories. We show, for instance, that the Hahn field $\mathbb {F}_p^{\text {alg}}((\mathbb {Z}[1/p]))$ is inp-minimal (of burden 1), and that the ring of Witt vectors $W(\mathbb {F}_p^{\text {alg}})$ over $\mathbb {F}_p^{\text {alg}}$ is not strong (of burden $\omega $ ). This result extends previous work by Chernikov and Simon and realizes an important step toward the classification of Henselian valued fields of finite burden. Second, we show a transfer principle for the property that all types realized in a given elementary extension are definable. It can be written as follows: a valued field as above is stably embedded in an elementary extension if and only if its value group is stably embedded in the corresponding extension of value groups, its residue field is stably embedded in the corresponding extension of residue fields, and the extension of valued fields satisfies a certain algebraic condition. We show, for instance, that all types over the power series field $\mathbb {R}((t))$ are definable. Similarly, all types over the quotient field of $W(\mathbb {F}_p^{\text {alg}})$ are definable. This extends previous work of Cubides and Delon and of Cubides and Ye. These distinct results use a common approach, which has been developed recently. It consists of establishing first a reduction to an intermediate structure called the leading term structure, or $\operatorname {\mathrm {RV}}$ -sort, and then of reducing to the value group and residue field. This leads us to develop similar reduction principles in the context of pure short exact sequences of abelian groups. Abstract prepared by Pierre Touchard. E-mail: [email protected] URL: https://miami.uni-muenster.de/Record/a612cf73-0a2f-42c4-b1e4-7d28934138a9 A NOTE ON p-ADIC SIMPLICIAL VOLUMES Low-dimensional topology STEFFEN KIONKE, CLARA LÖH Journal: Glasgow Mathematical Journal / Volume 63 / Issue 3 / September 2021 We define and study generalizations of simplicial volume over arbitrary seminormed rings with a focus on p-adic simplicial volumes. We investigate the dependence on the prime and establish homology bounds in terms of p-adic simplicial volumes. As the main examples, we compute the weightless and p-adic simplicial volumes of surfaces. This is based on an alternative way to calculate classical simplicial volume of surfaces without hyperbolic straightening and shows that surfaces satisfy mod p and p-adic approximation of simplicial volume. DEFINABLE SETS OF BERKOVICH CURVES Pablo Cubides Kovacsics, Jérôme Poineau Journal: Journal of the Institute of Mathematics of Jussieu / Volume 20 / Issue 4 / July 2021 Published online by Cambridge University Press: 11 October 2019, pp. 1275-1339 In this article, we functorially associate definable sets to $k$-analytic curves, and definable maps to analytic morphisms between them, for a large class of $k$-analytic curves. Given a $k$-analytic curve $X$, our association allows us to have definable versions of several usual notions of Berkovich analytic geometry such as the branch emanating from a point and the residue curve at a point of type 2. We also characterize the definable subsets of the definable counterpart of $X$ and show that they satisfy a bijective relation with the radial subsets of $X$. As an application, we recover (and slightly extend) results of Temkin concerning the radiality of the set of points with a given prescribed multiplicity with respect to a morphism of $k$-analytic curves. In the case of the analytification of an algebraic curve, our construction can also be seen as an explicit version of Hrushovski and Loeser's theorem on iso-definability of curves. However, our approach can also be applied to strictly $k$-affinoid curves and arbitrary morphisms between them, which are currently not in the scope of their setting. Existence of valuations with smallest normalized volume Local theory Harold Blum Journal: Compositio Mathematica / Volume 154 / Issue 4 / April 2018 Published online by Cambridge University Press: 08 March 2018, pp. 820-849 Print publication: April 2018 Li introduced the normalized volume of a valuation due to its relation to K-semistability. He conjectured that over a Kawamata log terminal (klt) singularity there exists a valuation with smallest normalized volume. We prove this conjecture and give an explicit example to show that such a valuation need not be divisorial. Generalized Euler characteristic in power-bounded T-convex valued fields Yimu Yin Journal: Compositio Mathematica / Volume 153 / Issue 12 / December 2017 We lay the groundwork in this first installment of a series of papers aimed at developing a theory of Hrushovski–Kazhdan style motivic integration for certain types of nonarchimedean $o$-minimal fields, namely power-bounded $T$-convex valued fields, and closely related structures. The main result of the present paper is a canonical homomorphism between the Grothendieck semirings of certain categories of definable sets that are associated with the $\text{VF}$-sort and the $\text{RV}$-sort of the language ${\mathcal{L}}_{T\text{RV}}$. Many aspects of this homomorphism can be described explicitly. Since these categories do not carry volume forms, the formal groupification of the said homomorphism is understood as a universal additive invariant or a generalized Euler characteristic. It admits not just one, but two specializations to $\unicode[STIX]{x2124}$. The overall structure of the construction is modeled on that of the original Hrushovski–Kazhdan construction. A Generalization of the Eisenstein–Dumas–Schönemann Irreducibility Criterion Algebraic number theory: global fields General field theory Bablesh Jhorar, Sudesh K. Khanduja Journal: Proceedings of the Edinburgh Mathematical Society / Volume 60 / Issue 4 / November 2017 In 2013, Weintraub gave a generalization of the classical Eisenstein irreducibility criterion in an attempt to correct a false claim made by Eisenstein. Using a different approach, we prove Weintraub's result with a weaker hypothesis in a more general setup that leads to an extension of the generalized Schönemann irreducibility criterion for polynomials with coefficients in arbitrary valued fields. Defining Coarsenings of Valuations Arithmetic rings and other special rings Franziska Jahnke, Jochen Koenigsmann We study the question of which Henselian fields admit definable Henselian valuations (with or without parameters). We show that every field that admits a Henselian valuation with non-divisible value group admits a parameter-definable (non-trivial) Henselian valuation. In equicharacteristic 0, we give a complete characterization of Henselian fields admitting a parameter-definable (non-trivial) Henselian valuation. We also obtain partial characterization results of fields admitting -definable (non-trivial) Henselian valuations. We then draw some Galois-theoretic conclusions from our results. A DEFINABLE $p$-ADIC ANALOGUE OF KIRSZBRAUN'S THEOREM ON EXTENSIONS OF LIPSCHITZ MAPS Algebraic number theory: local and $p$-adic fields Raf Cluckers, Florent Martin Journal: Journal of the Institute of Mathematics of Jussieu / Volume 17 / Issue 1 / February 2018 Published online by Cambridge University Press: 20 October 2015, pp. 39-57 Print publication: February 2018 A direct application of Zorn's lemma gives that every Lipschitz map $f:X\subset \mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$ has an extension to a Lipschitz map $\widetilde{f}:\mathbb{Q}_{p}^{n}\rightarrow \mathbb{Q}_{p}^{\ell }$. This is analogous to, but easier than, Kirszbraun's theorem about the existence of Lipschitz extensions of Lipschitz maps $S\subset \mathbb{R}^{n}\rightarrow \mathbb{R}^{\ell }$. Recently, Fischer and Aschenbrenner obtained a definable version of Kirszbraun's theorem. In this paper, we prove in the $p$-adic context that $\widetilde{f}$ can be taken definable when $f$ is definable, where definable means semi-algebraic or subanalytic (or some intermediary notion). We proceed by proving the existence of definable Lipschitz retractions of $\mathbb{Q}_{p}^{n}$ to the topological closure of $X$ when $X$ is definable. Complexity of OM factorizations of polynomials over local fields Computational number theory Jens-Dietrich Bauch, Enric Nart, Hayden D. Stainsby Journal: LMS Journal of Computation and Mathematics / Volume 16 / October 2013 Let $k$ be a locally compact complete field with respect to a discrete valuation $v$. Let $ \mathcal{O} $ be the valuation ring, $\mathfrak{m}$ the maximal ideal and $F(x)\in \mathcal{O} [x] $ a monic separable polynomial of degree $n$. Let $\delta = v(\mathrm{Disc} (F))$. The Montes algorithm computes an OM factorization of $F$. The single-factor lifting algorithm derives from this data a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$, for a prescribed precision $\nu $. In this paper we find a new estimate for the complexity of the Montes algorithm, leading to an estimation of $O({n}^{2+ \epsilon } + {n}^{1+ \epsilon } {\delta }^{2+ \epsilon } + {n}^{2} {\nu }^{1+ \epsilon } )$ word operations for the complexity of the computation of a factorization of $F(\mathrm{mod~} {\mathfrak{m}}^{\nu } )$, assuming that the residue field of $k$ is small. Orderings and signatures of higher level on multirings and hyperfields Forms and linear algebraic groups Paweł Gładki, Murray Marshall Journal: Journal of K-Theory / Volume 10 / Issue 3 / December 2012 Multirings are objects like rings but with multi-valued addition. In the present paper we extend results of E. Becker and others concerning orderings of higher level on fields and rings to orderings of higher level on hyperfields and multirings and, in the process of doing this, we establish higher level analogs of the results previously obtained by the second author. In particular, we introduce a class of multirings called ℓ-real reduced multirings, define a natural reflection A ⇝ Qℓ-red(A) from the category of multirings satisfying to the full subcategory of ℓ-real reduced multirings, and provide an elementary first-order description of these objects. The relationship between ℓ-real reduced hyperfields and the spaces of signatures defined by Mulcahy and Powers is also examined. The Ultrametric Corona Problem and spherically complete fields Alain Escassut Journal: Proceedings of the Edinburgh Mathematical Society / Volume 53 / Issue 2 / June 2010 Published online by Cambridge University Press: 30 April 2010, pp. 353-371 Let K be a complete ultrametric algebraically closed field and let A be the Banach K-algebra of bounded analytic functions in the 'open' unit disc D of K provided with the Gauss norm. Let Mult(A,‖ · ‖) be the set of continuous multiplicative semi-norms of A provided with the topology of simple convergence, let Multm(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal and let Multa(A, ‖ · ‖) be the subset of the φ ∈ Mult(A, ‖ · ‖) whose kernel is a maximal ideal of the form (x − a)A with a ∈ D. We complete the characterization of continuous multiplicative norms of A by proving that the Gauss norm defined on polynomials has a unique continuation to A as a norm: the Gauss norm again. But we find prime closed ideals that are neither maximal nor null. The Corona Problem on A lies in two questions: is Multa(A, ‖ · ‖) dense in Multm(A, ‖ · ‖)? Is it dense in Multm(A, ‖ · ‖)? In a previous paper, Mainetti and Escassut showed that if each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖), then the answer to the first question is affirmative. In particular, the authors showed that when K is strongly valued each maximal ideal of A is the kernel of a unique φ ∈ Mult(m(A, ‖ · ‖). Here we prove that this uniqueness also holds when K is spherically complete, and therefore so does the density of Multa(A, ‖ · ‖) in Multm(A, ‖ · ‖).	CommonCrawl
Nutrition Research and Practice The Korean Nutrition Society (한국영양학회) Agriculture, Fishery and Food > Food and Nutrition Science Nutrition Research and Practice (NRP) is an official journal, jointly published by the Korean Nutrition Society and the Korean Society of Community Nutrition since 2007. The journal had been published quarterly at the initial stage and has been published bimonthly since 2010. NRP aims to stimulate research and practice across diverse areas of human nutrition. The Journal publishes peer-reviewed original manuscripts on nutrition biochemistry and metabolism, community nutrition, nutrition and disease management, nutritional epidemiology, nutrition education, institutional food service in the following categories: Original Research Articles, Notes, Communications, and Reviews. Reviews will be received by the invitation of the editors only. Statements made and opinions expressed in the manuscripts published in this Journal represent the views of authors and do not necessarily reflect the opinion of the Societies. This journal is indexed/tracked/covered by PubMed, PubMed Central, Science Citation Index Expanded (SCIE), SCOPUS, Chemical Abstracts Service (CAS), CAB International (CABI), KoreaMed, Synapse, KoMCI, CrossRef and Google Scholar. http://www.nrpesubmit.org/ KSCI KCI SCOPUS SCIE Short-term impact of sugar consumption on hunger and ad libitum food intake in young women Penaforte, Fernanda R.O.;Japur, Camila C.;Pigatto, Leticia P.;Chiarello, Paula G.;Diez-Garcia, Rosa W. 77 https://doi.org/10.4162/nrp.2013.7.2.77 PDF KSCI The hypothesis of this study was that greater sugar consumption at breakfast promotes a stronger sensation of hunger and a later increase in energy consumption. The objective was to assess the relation between sugar consumption in a meal and the subsequent sensations of hunger and ad libitum food consumption. Sixteen women consumed a breakfast accompanied by 2 drinks sweetened ad libitum with sugar. After 3 h, a lunch was offered to evaluate ad libitum food consumption. During the period from breakfast to lunch, hunger sensations were evaluated at 30 min intervals. Women were divided according to the median amount of sugar used to sweeten the breakfast drinks (20 g). The group who consumed sugar above the median showed a greater hunger sensation in the preprandial period, and a greater ad libitum intake at lunch ($390{\pm}130g{\times}256{\pm}67g$, P = 0.002), compared to the group who had a lower sugar consumption. The amount of sugar consumed at breakfast was correlated positively with the sensation of preprandial hunger and food intake at lunch. We concluded that foods with a high glycemic index can modulate the appetite within a short period of time. Chemical composition of nuts and seeds sold in Korea Chung, Keun Hee;Shin, Kyung Ok;Hwang, Hyo Jeong;Choi, Kyung-Soon 82 Eleven types of nuts and seeds were analyzed to determine their energy (326-733 mg), moisture (1.6-18.3 mg), carbohydrate (8.8-70.9 mg), protein (4.9-30.5 mg), lipid (2.5-69.8 mg), and ash (1.2-5.5 mg) contents per 100 g of sample. Energy content was highest in pine nuts (733 mg/100 g), carbohydrate level was highest in dried figs (70.9 mg/100 g) and protein was highest in peanuts (30.5 mg/100 g). The amino acid compositions of nuts and seeds were characterized by the dominance of hydrophobic (range = 1,348.6-10,284.6 mg), hydrophilic (range = 341.1-3,244.3 mg), acidic (range = 956.1-8,426.5 mg), and basic (range = 408.6-4,738.5 mg) amino acids. Monounsaturated fatty acids (MUFA) were highest in macadamia nuts (81.3%), whereas polyunsaturated fatty acids (PUFA) were highest in the walnuts (76.7%). Macadamia nuts did not contain any vitamin E, whereas sunflower seeds contained the highest level (60.3 mg/kg). Iron (Fe) content was highest in pumpkin seeds ($95.85{\pm}33.01$ ppm), zinc (Zn) content was highest in pistachios ($67.24{\pm}30.25$ ppm), copper (Cu) content was greatest in walnuts ($25.45{\pm}21.51$ ppm), and lead (Pb) content was greatest in wheat nuts ($25.49{\pm}4.64$ ppm), significantly (P < 0.05). In conclusion, current commercial nuts and seeds have no safety concerns, although further analysis of Pb contents is necessary to ensure safety. Inorganic sulfur reduces cell proliferation by inhibiting of $ErbB_2$ and $ErbB_3$ protein and mRNA expression in MDA-MB-231 human breast cancer cells Ha, Ae Wha;Hong, Kyung Hee;Kim, Hee Sun;Kim, Woo Kyoung 89 Dietary inorganic sulfur is the minor component in our diet, but some studies suggested that inorganic sulfur is maybe effective to treat cancer related illness. Therefore, this study aims to examine the effects of inorganic sulfur on cell proliferation and gene expression in MDA-MB-231 human breast cancer cells. MDA-MB-231 cells were cultured the absence or presence of various concentrations (12.5, 25, or 50 ${\mu}mol/L$) of inorganic sulfur. Inorganic sulfur significantly decreased proliferation after 72 h of incubation (P < 0.05). The protein expression of $ErbB_2$ and its active form, $pErbB_2$, were significantly reduced at inorganic sulfur concentrations of 50 ${\mu}mol/L$ and greater than 25 ${\mu}mol/L$, respectively (P < 0.05). The mRNA expression of $ErbB_2$ was significantly reduced at an inorganic sulfur concentration of 50 ${\mu}mol/L$ (P < 0.05). The protein expression of $ErbB_3$ and its active form, $pErbB_3$, and the mRNA expression of $pErbB_3$ were significantly reduced at inorganic sulfur concentrations greater than 25 ${\mu}mol/L$ (P < 0.05). The protein and mRNA expression of Akt were significantly reduced at an inorganic sulfur concentration of 50 ${\mu}mol/L$ (P < 0.05), but pAkt was not affected by inorganic sulfur treatment. The protein and mRNA expression of Bax were significantly increased with the addition of inorganic sulfur concentration of 50 ${\mu}mol/L$ (P < 0.05). In conclusion, cell proliferation was suppressed by inorganic sulfur treatment through the ErbB-Akt pathway in MDA-MB-231 cells. Inhibitory effects of Capsicum annuum L. water extracts on lipoprotein lipase activity in 3T3-L1 cells Baek, Jongmi;Lee, Jaesung;Kim, Kyoungkon;Kim, Taewoo;Kim, Daejung;Kim, Cheonan;Tsutomu, Kanazawa;Ochir, Sarangowa;Lee, Kooyeon;Park, Cheol Ho;Lee, Yong-Jik;Choe, Myeon 96 Obesity, an intractable metabolic disease, currently has no medical treatment without side effects, so studies have been actively carried out to find natural compounds that have anti-obesity activity with minimum side effects. In this study, the anti-obesity effects of water extracts of seven Capsicum annuum L. varieties being Putgochu (Pca), Oyee gochu (Oca), Kwari putgochu (Kca), Green pepper (Gca), Yellow paprika (Yca), Red paprika (Rca) and Cheongyang gochu (Cca), were examined through the evaluation of lipoprotein lipase (LPL) mRNA expression level in 3T3-L1 cells (mouse pre-adipocytes). After capsaicin elimination by chloroform defatting, freeze-dried powder of Cca was treated to 3T3-L1 cells and anti-obesity effects were examined by determining the LPL mRNA level using the RT-PCR method. Of the primary fractions, only proven fractions underwent secondary and tertiary refractionating to determine anti-obesity effects. From seven different Capsicum annuum L., there was a significant decrease of the LPL mRNA expression level of 50.9% in Cca treatment compared to the control group. A significant decrease of the LPL mRNA expression level was shown in primary fractions (Fr) 5 (36.2% decrease) and 6 (30.5% decrease) of the Cca water extracts. Due to the impurities checked by UPLC chromatography, Fr 5 and 6 were refractionated to determine the LPL mRNA expression level. Treatment of Fr 6-6 (35.8% decrease) and Fr 5-6 (35.3% decrease) showed a significant decrease in the LPL mRNA expression level. When analyzed using UPLC, major compounds of Fr 6-6 and Fr 5-6 were very similar. Subsequently, we refractionated Fr 6-6 and Fr 5-6 to isolate the major peak for structure elucidation. Treatment of Fr 5-6-1 (26.6% decrease) and Fr 6-6-1 (29.7% decrease) showed a significant decrease in the LPL mRNA expression level. Consequently, the fractions may have a possibility to ameliorate obesity through the decrease of the LPL mRNA expression level. The antidiabetic effects of an herbal formula composed of Alnus hirsuta, Rosa davurica, Acanthopanax senticosus and Panax schinseng in the streptozotocin-induced diabetic rats Hu, Weicheng;Yeo, Jin-Hee;Jiang, Yunyao;Heo, Seong-Il;Wang, Myeong-Hyeon 103 https://doi.org/10.4162/nrp.2013.7.2.103 PDF KSCI A folk prescription consisting of Alnus hirsuta, Rosa davurica, Acanthopanax senticosus and Panax schinseng has been used in the treatment of diabetes mellitus. The aim of the present investigation was to evaluate the antidiabetic effects of the herb formula extract (HFE) composed of Alnus hirsuta, Rosa davurica, Acanthopanax senticosus and Panax schinseng in the streptozotocin (STZ)-induced diabetic rats. The HFE was mixed in the food supply of the healthy and STZ-induced diabetic male Sprague-Dawley rats, and its effects on the body weight, water and food intake, hyperglycemia, hypolipidemic and islet structure were studied. The treatment of the rats with STZ for 6 weeks resulted in marasmus, polydipsia, polyphagia, hyperglycemia and hypoinsulinemia. In addition, the diabetic rats showed an apparent decrease in the insulin immunoreactivity and the number of ${\beta}$-cells in the pancreas. The addition of the HFE to the rats' food supply significantly lowered the serum glucose and the serum triglycerides level and preserved the normal histological appearance of the pancreatic islets. These results indicate that the HEF have a strong antidiabetic potential along with the significant hypoglycemic and hypolipidemic effects, which may be applicable in the pharmaceutical industry. Effects of vitamin C and E supplementation on oxidative stress and liver toxicity in rats fed a low-fat ethanol diet Lee, Soo-Jung;Kim, Seon-Young;Min, Hyesun 109 We compared the preventive capacity of high intakes of vitamin C (VC) and vitamin E (VE) on oxidative stress and liver toxicity in rats fed a low-fat ethanol diet. Thirty-two Wistar rats received the low fat (10% of total calories) Lieber-DeCarli liquid diet as follows: either ethanol alone (Alc group, 36% of total calories) or ethanol in combination with VC (Alc + VC group, 40 mg VC/100 g body weight) or VE (Alc + VE group, 0.8 mg VE/100 g body weight). Control rats were pair-fed a liquid diet with the Alc group. Ethanol administration induced a modest increase in alanine aminotransferase (ALT), aspartate aminotransferase (AST), conjugated dienes (CD), and triglycerides but decreased total radical-trapping antioxidant potential (TRAP) in plasma. VE supplementation to alcohol-fed rats restored the plasma levels of AST, CD, and TRAP to control levels. However, VC supplementation did not significantly influence plasma ALT, AST, or CD. In addition, a significant increase in plasma aminothiols such as homocysteine and cysteine was observed in the Alc group, but cysteinylglycine and glutathione (GSH) did not change by ethanol feeding. Supplementing alcohol-fed rats with VC increased plasma GSH and hepatic S-adenosylmethionine, but plasma levels of aminothiols, except GSH, were not influenced by either VC or VE supplementation in ethanol-fed rats. These results indicate that a low-fat ethanol diet induces oxidative stress and consequent liver toxicity similar to a high-fat ethanol diet and that VE supplementation has a protective effect on ethanol-induced oxidative stress and liver toxicity. Characterization of food allergies in patients with atopic dermatitis Kwon, Jaryoung;Kim, Jungyun;Cho, Sunheui;Noh, Geunwoong;Lee, Sang Sun 115 We examined the characteristics of food allergy prevalence and suggested the basis of dietary guidelines for patients with food allergies and atopic dermatitis. A total of 2,417 patients were enrolled in this study. Each subject underwent a skin prick test as well as serum immunoglobulin E (IgE) measurement. A double-blind, placebo-controlled food challenge was conducted using milk, eggs, wheat, and soybeans, and an oral food challenge was performed using beef, pork, and chicken. Food allergy prevalence was found among 50.7% in patients with atopic dermatitis. Among patients with food allergies (n = 1,225), the prevalence of non-IgE-mediated food allergies, IgE-mediated food allergies, and mixed allergies was discovered in 94.9%, 2.2%, and 2.9% of the patients, respectively. Food allergy prevalence, according to food item, was as follows: eggs = 21.6%, milk = 20.9%, wheat = 11.8%, soybeans = 11.7%, chicken = 11.7%, pork = 8.9% and beef = 9.2%. The total number of reactions to different food items in each patient was also variable at 45.1%, 30.6%, 15.3%, 5.8%, 2.2%, and 1.0% for 1 to 6 reactions, respectively. The most commonly seen combination in patients with two food allergies was eggs and milk. The clinical severity of the reactions observed in the challenge test, in the order of most to least severe, were wheat, beef, soybeans, milk, pork, eggs, and chicken. The minimum and maximum onset times of food allergy reactions were 0.2-24 hrs for wheat, 0.5-48 hrs for beef, 1.0-24 hrs for soybeans, 0.7-24 hrs for milk, 3.0-24 hrs for pork, 0.01-72 hrs for eggs, and 3.0-72 hrs for chicken. In our study, we examined the characteristics of seven popular foods. It will be necessary, however, to study a broader range of foods for the establishment of a dietary guideline. Our results suggest that it may be helpful to identify food allergies in order to improve symptoms in patients with atopic dermatitis. Prevalence of child malnutrition in agro-pastoral households in Afar Regional State of Ethiopia Fentaw, Rabia;Bogale, Ayalneh;Abebaw, Degnet 122 Based on data generated from 180 randomly selected households with children age under five years old in Aysaita district of Afar region of Ethiopia, this study explored prevalence of malnutrition and scrutinized household characteristics, maternal characteristics, specifics of the child and economic variables associated with child malnutrition. The height-for-age Z-scores (HAZ), weight-for-height Z-scores (WHZ) and weight-for-age Z-scores (WAZ) were used to measure the extent of stunting, wasting and underweight, respectively. The results revealed that prevalence of long term nutritional imbalance and malnutrition status indicator (i.e. stunting) was 67.8%. The short term measure (wasting) was found to be 12.8% and underweight was found to be 46.1%. Moreover, children in households which are headed by women, and characterized by more dependency ratio, less access to assets, health services and institutions are more likely to be undernourished. Assessing the children's views on foods and consumption of selected food groups: outcome from focus group approach Ishak, Sharifah Intan Zainun Sharif;Shohaimi, Shamarina;Kandiah, Mirnalini 132 The food choices in childhood have high a probability of being carried through into their adulthood life, which then contributes to the risk of many non-communicable diseases. Therefore, there is a need to gather some information about children's views on foods which may influence their food choices for planning a related dietary intervention or programme. This paper aimed to explore the views of children on foods and the types of foods which are usually consumed by children under four food groups (snacks, fast foods, cereals and cereal products; and milk and dairy products) by using focus group discussions. A total of 33 school children aged 7-9 years old from Selangor and Kuala Lumpur participated in the focus groups. Focus groups were audio-taped, transcribed and analyzed according to the listed themes. The outcomes show that the children usually consumed snacks such as white bread with spread or as a sandwich, local cakes, fruits such as papaya, mango and watermelon, biscuits or cookies, tea, chocolate drink and instant noodles. Their choices of fast foods included pizza, burgers, French fries and fried chicken. For cereal products, they usually consumed rice, bread and ready-to-eat cereals. Finally, their choices of dairy products included milk, cheese and yogurt. The reasons for the food liking were taste, nutritional value and the characteristics of food. The outcome of this study may provide additional information on the food choices among Malaysian children, especially in urban areas with regard to the food groups which have shown to have a relationship with the risk of childhood obesity. Gender specific effect of major dietary patterns on the metabolic syndrome risk in Korean pre-pubertal children Park, Soo Jin;Lee, Seung Min;Kim, Seon Mee;Lee, Myoungsook 139 There is a lack of data on metabolic risk factors during pre-puberty, which is important for identifying the subgroups of youth, at whom early interventions should be targeted. In this study, we evaluated the prevalence of metabolic risk factors and its subsequent relations with dietary patterns in Korean pre-pubertal children through a cross-sectional sample (n = 1,008; boys = 513) of pre-pubertal children (aged 8-9 years) from a sub-study of the Korea Metabolic Syndrome Research Initiatives (KMSRI) in Seoul, Korea. Measures of anthropometry and blood pressure as well as fasting blood samples were used in the analysis. A three-day food records were collected. The metabolic syndrome was defined according to the age-adjusted National Cholesterol Education Program Adult Treatment Panel III guidelines. An added metabolic risk score was calculated for each subject by summing the quintile values of the five individual risk factors. Among the 5 risk components of metabolic syndrome, high waist circumference (WC) was the major factor (P < 0.001). A significant increasing trend of the added metabolic syndrome risk score was observed with the increase of WC (P (trend) < 0.001) among both genders. The cutoff point for high WC for pre-pubertal children was 61.3 cm for boys and 59.9 cm for girls. The prevalence of high triglyceride (TG) values was significantly higher in girls than it was in boys (P < 0.01). Girls in the highest quintile of balanced dietary pattern scores had lower TG values (P (trend) = 0.032) than did those in the lowest quintile. Moreover, girls in the highest quintile of western dietary pattern scores showed increasing trend for the added metabolic risk score (P (trend) = 0.026) compared with those in the lowest quintile. Adverse associations exist between western dietary patterns and the accumulation of metabolic risks among girls, not in boys, even during pre-puberty. Erratum: Lutein decreases oxidative stress and inflammation in liver and eyes of guinea pigs fed a hypercholesterolemic diet Kim, Jung Eun;Clark, Richard M.;Park, Youngki;Lee, Jiyoung;Fernandez, Maria Luz 146	CommonCrawl
Methodology Article Robust sparse canonical correlation analysis Ines Wilms1 & Christophe Croux1 BMC Systems Biology volume 10, Article number: 72 (2016) Cite this article Canonical correlation analysis (CCA) is a multivariate statistical method which describes the associations between two sets of variables. The objective is to find linear combinations of the variables in each data set having maximal correlation. In genomics, CCA has become increasingly important to estimate the associations between gene expression data and DNA copy number change data. The identification of such associations might help to increase our understanding of the development of diseases such as cancer. However, these data sets are typically high-dimensional, containing a lot of variables relative to the number of objects. Moreover, the data sets might contain atypical observations since it is likely that objects react differently to treatments. We discuss a method for Robust Sparse CCA, thereby providing a solution to both issues. Sparse estimation produces canonical vectors with some of their elements estimated as exactly zero. As such, their interpretability is improved. Robust methods can cope with atypical observations in the data. We illustrate the good performance of the Robust Sparse CCA method by several simulation studies and three biometric examples. Robust Sparse CCA considerably outperforms its main alternatives in (1) correctly detecting the main associations between the data sets, in (2) accurately estimating these associations, and in (3) detecting outliers. Robust Sparse CCA delivers interpretable canonical vectors, while at the same time coping with outlying observations. The proposed method is able to describe the associations between high-dimensional data sets, which are nowadays commonplace in genomics. Furthermore, the Robust Sparse CCA method allows to characterize outliers. Canonical correlation analysis (CCA), introduced by [1], identifies and quantifies the associations between two sets of variables. CCA searches for linear combinations, called canonical variates, of each of the two sets of variables having maximal correlation. The coefficients of these linear combinations are called the canonical vectors. The correlations between the canonical variates are called the canonical correlations. CCA is used to study associations in, for instance, genomic data [2], environmental data [3], or biomedical data [4]. For more information on canonical correlations analysis, see e.g. [5], Chapter 10. Sparse canonical vectors are canonical vectors with some of their elements estimated as exactly zero. The canonical variates then only depend on a subset of the variables, those corresponding to the non-zero elements of the estimated canonical vectors. Hence, the canonical variates are easier to interpret, in particular for high-dimensional data sets. Sparse estimation shows good performance in analyzing, for instance, genomic data (e.g. [6–8]), or biological data (e.g. [9, 10]). Examples of CCA for high-dimensional data sets can be found in, for example, genetics [11–13] and machine learning [14]. Different approaches for sparse CCA have been proposed in the literature. Parkhomenko et al. [15] use a sparse singular value decomposition to derive sparse singular vectors. Witten et al. [16] develop a penalized matrix decomposition, and show how to apply it for sparse CCA. Waaijenborg et al. [17], Lykou and Whittaker [18], An et al. [19] and Wilms and Croux [20] convert the CCA problem into a penalized regression framework to produce sparse canonical vectors. Chen et al. [21] and Gao and Zhou [22] discuss theoretical properties for sparse CCA. All these methods are not robust to outliers. A common problem in multivariate data sets, however, is the frequent occurrence of outliers. In genomics, for instance, some patients can react very differently to treatments because of their individual-specific genetic structure. Therefore, the possible presence of outlying observations should be taken into account. Several robust CCA methods have been introduced in the literature. Dehon and Croux [23] considers robust CCA using the Minimum Covariance Determinant estimator [24]. Asymptotic properties for CCA based on robust estimators of the covariance matrix are discussed in [25]. Branco et al. [26] use a robust alternating regression approach to obtain the canonical variates. CCA can also be considered as a prediction problem, where the canonical variates obtained from the first data set serve as optimal predictors for the canonical variates of the second data set, and vice versa. As such, [27] use a robust M-scale to evaluate the prediction quality, whereas [28] use a robust estimator of the multivariate linear model. None of these methods, however, are sparse. This paper proposes a CCA method that is sparse and robust at the same time. As such, we deal with two important topics in applied statistics: sparse model estimation and the presence of outliers in the data. We use an alternating robust, sparse regression framework to sequentially obtain the canonical variates. Robust Sparse CCA has clear advantages: (i) it provides well interpretable canonical vectors since some of the elements of the canonical vectors are estimated as exactly zero, (ii) it is still computable for high-dimensional data sets, where the sample size exceeds the number of variables in each data set, (iii) it can cope with outliers in the data, which are even more likely to occur in high dimensions, and (iv) it provides an interesting way to characterize these outliers. Simulation studies were performed to investigate the performance of Robust Sparse CCA. These simulations show that Robust Sparse CCA achieves a substantially better performance compared to its leading alternatives CCA, Robust CCA and Sparse CCA. We illustrate the application of the Robust Sparse CCA method to an environmental data set and two genomic data sets. Robust Sparse CCA provides easy interpretable results. Moreover, we use Robust Sparse CCA to detect outlying observations in such high-dimensional data sets. First, we consider the robust and sparse estimator for the CCA problem. Next, we discuss the algorithm. Finally, we discuss the simulation designs and performance measures used to compare the performance of Robust Sparse CCA to standard CCA, Robust CCA and Sparse CCA. The estimator We consider the CCA problem in a regression framework ([29, 30]). Given a sample of n observations $\mathbf {x}_{i} \in \mathbb {R}^{p}$ and $\mathbf {y}_{i} \in \mathbb {R}^{q}$ (i=1,…,n). The two data matrices are denoted as X=[x 1,…,x n ]T and Y=[y 1,…,y n ]T. We assume the data matrices are robustly centered using the median. The estimated canonical vectors are collected in the columns of the matrices $\widehat {\mathbf {A}} \in \mathbb {R}^{p \times r}$ and $ \widehat {\mathbf {B}} \in \mathbb {R}^{q \times r}$. Here r is the number of canonical vectors. The columns of the matrices $\mathbf {X}\widehat {\mathbf {A}}$ and $\mathbf {Y}\widehat {\mathbf {B}}$ contain the estimates of the realizations of the canonical variates, and we denote their j th column by $\hat {\mathbf {u}}_{j}$ and $\hat {\mathbf {v}}_{j}$, for 1≤j≤r. The objective function defining the canonical vector estimates is $$ (\widehat{\mathbf{A}}, \widehat{\mathbf{B}}) = \underset{(\mathbf{A}, \mathbf{B}) }{\operatorname{argmin}} \ \sum_{i=1}^{n} \|\| \mathbf{A}^{T}x_{i} - \mathbf{B}^{T} \mathbf{y}_{i} \|\|^{2}. $$ The objective function in (1) is minimized under the restriction that each canonical variate $\hat {\mathbf {u}}_{j}$ is uncorrelated with the lower order canonical variates $\hat {\mathbf {u}}_{k}$, with 1≤k<j≤r. Similarly for the canonical vectors within the second set of variables. For identification purpose, a normalization condition requiring the canonical vectors to have unit norm is added. Typically, the canonical vectors are obtained by an eigenvalue analysis of a certain matrix involving the inverses of sample covariance matrices. But if n<max(p,q), these inverses do not exist. We estimate the canonical vectors with an alternating regression procedure. If the matrix A in (1) is kept fixed, the matrix B can be obtained from a Least Squares regression of the canonical variates on y (and vice versa for estimating A keeping B fixed). The standard Least Squares estimator, however, is not sparse, nor robust to outliers. Therefore, we replace it by the sparse Least Trimmed Squares (sparse LTS) estimator [31]. The sparse LTS estimator can be applied to high-dimensional data and is robust to outliers. We use a sequential algorithm to derive the canonical vectors. First canonical vector pair. Denote the first canonical vector pair by (A 1,B 1). Assume that the value of A 1 is known. Denote the vector of squared residuals by $\mathbf {r}^{2}({{\mathbf {B}}_{1}})= \left ({r_{1}^{2}},\ldots,{r_{n}^{2}}\right)^{T}$, with ${r_{i}^{2}} = \left (\mathbf {A}_{1}^{T}\mathbf {x}_{i} - \mathbf {B}_{1}^{T} \mathbf {y}_{i}\right)^{2}, i=1,\ldots,n$. The estimate of B 1 is obtained as $$ \widehat{\mathbf{B}}_{1}\|{\mathbf{A}}_{1} = \underset{\mathbf{B}_{1}}{\operatorname{argmin}} \sum_{i=1}^{h} \left(\mathbf{r}^{2}({{\mathbf{B}}_{1}}) \right)_{i:n} + h\lambda_{B_{1}} \sum_{j=1}^{q} \|{b_{1}}_{j}\|, $$ where $\lambda _{B_{1}}>0$ is a sparsity parameter, b 1 j is the j th element, j=1,…,q, of the first canonical vector B 1, and (r 2(B 1))1:n ≤…≤(r 2(B 1)) n:n are the order statistics of the squared residuals. The canonical vector $\widehat {\mathbf {B}}_{1}$ is normed to length 1. The solution to (2) equals the sparse LTS estimator with X A 1 as response and Y as predictor. Regularization by adding a penalty term to the objective function is necessary since the design matrix Y can be high-dimensional. Sparse model estimates are obtained by adding an L 1 penalty to the LTS objective function, similar as for the lasso regression estimator [32]. The sparse LTS estimator is computed with trimming proportion 25 %, so size of the subsample h=⌊0.75n⌋. To increase efficiency, we use a reweighting step. Further discussion and more detail on the sparse LTS estimator is provided in Additional file 1. As such, we get a robust sparse estimate $\widehat {\mathbf {B}}_{1}$. Analogously, for a fixed value B 1, denote the vector of squared residuals by $\mathbf {r}^{2}({{\mathbf {A}}_{1}})= ({r_{1}^{2}},\ldots,{r_{n}^{2}})^{T}$, with ${r_{i}^{2}} = \left (\mathbf {B}_{1}^{T}\mathbf {y}_{i} - \mathbf {A}_{1}^{T} \mathbf {x}_{i}\right)^{2}, i=1,\ldots,n$. The sparse LTS regression estimate of A 1 with Y B 1 as response and X as predictor is given by $$ \widehat{\mathbf{A}}_{1}\|{\mathbf{B}}_{1} = \underset{\mathbf{A}_{1}}{\operatorname{argmin}} \sum_{i=1}^{h} \left(\mathbf{r}^{2}({{\mathbf{A}}_{1}}) \right)_{i:n} + h\lambda_{A_{1}} \sum_{j=1}^{p} \|{a_{1}}_{j}\|, $$ where $\lambda _{A_{1}}>0$ is a sparsity parameter, a 1 j is the j th element, j=1,…,p of the first canonical vector A 1, and (r 2(A 1))1:n ≤…≤(r 2(A 1)) n:n are the order statistics of the squared residuals. The canonical vector $\widehat {\mathbf {A}}_{1}$ is normed to length 1. This leads to an alternating regression scheme, updating in each step the estimates of the canonical vectors until convergence. After convergence of the algorithm, the values of A 1 and B 1 in subsequent iterations remain stable, and the same observations will be detected as outliers in regressions (2) and (3). Higher order canonical vector pairs. We use deflated data matrices to estimate the higher order canonical vector pairs (see e.g. [26]). For the second canonical vector pair, the deflated matrices are $\mathbf {X}^{}_{2}$, the residuals of a column-by-column LTS regression of X on all lower order canonical variates, $\hat {\mathbf {u}}_{1}$ in this case; and $\mathbf {Y}^{}_{2}$, the residuals of a column-by-column LTS regression of Y on $\hat {\mathbf {v}}_{1}$. Since these regressions only involve a small number of regressors, the standard LTS estimator with λ=0 can be used. The second canonical variate pair is then obtained by alternating between the following regressions until convergence: $$ \widehat{\mathbf{B}}_{2}^{}\|{\mathbf{A}}^{}_{2} = \underset{\mathbf{B}^{}_{2}}{\operatorname{argmin}} \sum_{i=1}^{h} \left(\mathbf{r}^{2}({{\mathbf{B}}_{2}^{}}) \right)_{i:n} + h\lambda_{B_{2}^{}} \sum_{j=1}^{q} \|{b_{2}^{}}_{j}\|, $$ where $\mathbf {r}^{2}({{\mathbf {B}}_{2}^{\star }})= \left ({r_{1}^{2}},\ldots,{r_{n}^{2}}\right)^{T}$, with ${r_{i}^{2}} = \left (\mathbf {A}_{2}^{T}\mathbf {x}_{2,i}^{\star } - \mathbf {B}_{2}^{\star T}\mathbf {y}_{2,i}^{\star }\right)^{2}, i=1,\ldots,n$. $$ \widehat{\mathbf{A}}_{2}^{}\|{\mathbf{B}}^{}_{2} = \underset{\mathbf{A}_{2}^{}}{\operatorname{argmin}} \sum_{i=1}^{h} \left(\mathbf{r}^{2}({{\mathbf{A}}_{2}^{}}) \right)_{i:n} + h\lambda_{A_{2}^{}} \sum_{j=1}^{p} \|{a_{2}^{}}_{j}\|, $$ where $\mathbf {r}^{2}({{\mathbf {A}}_{2}^{\star }})= ({r_{1}^{2}},\ldots,{r_{n}^{2}})^{T}$, with ${r_{i}^{2}} = (\mathbf {B}_{2}^{T}\mathbf {y}_{2,i}^{\star } - \mathbf {A}_{2}^{\star T} \mathbf {x}_{2,i}^{\star })^{2}, i=1,\ldots,n$. The canonical vectors $\widehat {\mathbf {B}}_{2}^{}$ and $\widehat {\mathbf {A}}_{2}^{}$ are both normed to length 1. We obtain $\hat {\mathbf {u}}^{}_{2}=\mathbf {X}^{}_{2}\widehat {\mathbf {A}}_{2}^{}$ and $\hat {\mathbf {v}}^{}_{2}=\mathbf {Y}^{}_{2}\widehat {\mathbf {B}}_{2}^{}$. Finally, the second canonical vector needs to be expressed as linear combinations of the columns of the original data matrices, and not the deflated ones. Since we want to allow for zero coefficients in these linear combinations, a sparse approach is needed. To obtain a sparse $\widehat {\mathbf {A}}_{2}$, we regress $\hat {\mathbf {u}}^{}_{2}$ on X using the sparse LTS estimator, yielding the fitted values $\hat {\mathbf {u}}_{2}=\mathbf {X}\widehat {\mathbf {A}}_{2}$. To obtain a sparse $\widehat {\mathbf {B}}_{2}$, we regress $\hat {\mathbf {v}}^{}_{2}$ on Y using the sparse LTS estimator, yielding the fitted values $\hat {\mathbf {v}}_{2}=\mathbf {Y}\widehat {\mathbf {B}}_{2}$. The higher order canonical variate pairs are obtained in a similar way. We perform alternating sparse LTS regressions as in (4) and (5), followed by a final sparse LTS step to retrieve the estimated canonical vectors $({\widehat {\mathbf {A}}}_{l},{\widehat {\mathbf {B}}}_{l})$. It is not really necessary to use a sparse approach in regressions (4) and (5), other penalty functions can be used. A schematic representation of the complete algorithm is provided in Additional file 2. Finally, note that as in other sparse CCA proposals (e.g. [15–17, 20]) the canonical variates are in general not uncorrelated. The robust sparse canonical vectors we obtain yield an interpretable basis of the space spanned by the canonical vectors. This basis can be made orthogonal (but not sparse) after suitable rotation if one desires so. Initial value. A starting value for A 1 is required to start up the algorithm. We compute the first robust principal component of Y, denoted z 1. The first robust principal component is calculated from the first eigenvector of the robustly estimated covariance matrix. For this aim, we use the spatial sign covariance estimator [33]. We regress z 1 on X using the sparse LTS. The estimated regression coefficient matrix of this regression is used as initial value for A 1. To obtain an initial estimate for the higher order canonical vectors A l , for l=2,…,r, we use the first robust principal component of the deflated data matrix and proceed analogously. We performed several numerical experiments to investigate the sensitivity of the outcome of the algorithm to the choice of initial value. In low-dimensional settings, the choice of initial value is not important. In high-dimensional settings, a good initial value is more important. Note that the initial value should exist and be easily computable in all settings, which holds for our proposal. Number of canonical variates to extract. To decide on the number of canonical variates r to extract, we use the maximum eigenvalue ratio criterion of [19]. We apply the Robust Sparse CCA algorithm and calculate the robust correlations $\hat {\rho }_{1}, \ldots,\hat {\rho }_{\text {rmax}} $, with rmax=min(p,q,10). For high-dimensional data sets, we consider a maximum of 10 canonical correlations, since in practice, more than 10 canonical vector pairs are never used. Each $\hat \rho _{j}$ is obtained by computing the correlation between $\hat {\mathbf {v}}_{j}$ and $\hat {\mathbf {u}}_{j}$ from the bivariate Minimum Covariance Determinant (MCD) estimator with 25 % trimming. Let $\hat {k}_{j} = \hat {\rho }_{j}/\hat {\rho }_{j+1}$ for j=1,…,rmax−1. We extract r pairs of canonical variates, where $r=\text {argmax}_{j} \hat {k}_{j}$. Convergence criterion. In each step of the alternating regression algorithm we update the estimates of the canonical vectors $\widehat {\mathbf {B}}_{l}^{} $ and $\widehat {\mathbf {A}}_{l}^{} $, for l=1,…,r. We iterate until the relative change in the value of the convergence criterion in two successive iterations1 is smaller than the convergence tolerance value ε=10−2. As convergence criterion, we consider $$ \text{Convergence criterion} = \frac{1}{h} \sum_{i=1}^{h} \left(\mathbf{r}^{2}\left(\widehat{{\mathbf{A}}}_{l}^{}, \widehat{{\mathbf{B}}}_{l}^{} \right) \right)_{i:n}, $$ for l=1,…,r, where $\mathbf {r}^{2}\left (\widehat {{\mathbf {A}}}_{l}^{}, \widehat {{\mathbf {B}}}_{l}^{}\right)= \left ({r_{1}^{2}},\ldots,{r_{n}^{2}}\right)^{T}$, with ${r_{i}^{2}} = \left (\widehat {{\mathbf {A}}}_{l}^{T}\mathbf {x}_{l,i}^{\star } - \widehat {{\mathbf {B}}}_{l}^{T} \mathbf {y}_{l,i}^{\star }\right)^{2}, i=1,\ldots,n$. $\mathbf {X}^{}_{l}$ and $\mathbf {Y}^{}_{l}$ are the original data sets for l=1, and the deflated data matrices for l=2,…,r. In the simulations we conducted, convergence was almost always reached.2 For data sets with n=100,p=q=10, on average 6 iterations per canonical vector pair are needed to converge. For n=50,p=q=100, on average 10 iterations are needed to converge. Choice of the sparsity parameter. The sparsity parameters controlling the penalization on the regression coefficient matrices are selected with the Bayesian Information Criterion (e.g. [34]). We use a range of values for the sparsity parameters and select the one with the lowest value of $$ \text{BIC}_{\lambda_{\widehat{\mathbf{A}}^{}_{l}}} = n \cdot\text{log}\left(\frac{1}{h} \sum_{i=1}^{h} \left(\mathbf{r}^{2}({{\widehat{\mathbf{A}}}_{l}^{}}) \right)_{i:n} \right) + df_{\lambda_{\widehat{\mathbf{A}}^{}_{l}}} \cdot \log(n), $$ $$ \text{BIC}_{\lambda_{\widehat{\mathbf{B}}^{}_{l}}} = n \cdot\text{log}\left(\frac{1}{h} \sum_{i=1}^{h} \left(\mathbf{r}^{2}({\widehat{\mathbf{B}}_{l}^{}}) \right)_{i:n} \right) + df_{\lambda_{\widehat{\mathbf{B}}^{}_{l}}} \cdot \log(n), $$ for l=1,…,r, with $df_{\lambda _{\widehat {\mathbf {A}}^{}_{l}}}$ and $df_{\lambda _{\widehat {\mathbf {B}}^{}_{l}}}$ the respective number of non-zero estimated regression coefficients. Computation time. All computations are carried out in R version 3.2.1. The code of the algorithm is made available on a webpage of the first author (http://feb.kuleuven.be/ines.wilms/software). For data sets with n=100,p=q=10, on average 10 seconds are needed to extract one canonical vector pair on an Intel Xeon E5-2699 v3 @ 2.30GHz machine. For n=50,p=q=100, we need 540 seconds on average, for n=100,p=q=10000, computation time increases to 11 hours on average. But even in high dimensions, the number of iterations remains lows (8 on average). The high computing time needed for p=q=10000 is mainly due to the sparse LTS estimator, taken from the R-package robustHD [35]. By including a variable screening step [36] preceding the computation of the sparse LTS estimator, one could reduce the total computation time considerably. Simulation designs To investigate the performance of Robust Sparse CCA, we conduct a simulation study. We consider several simulation designs. In the "Uncorrelated Sparse Low-dimensional" and "Correlated Sparse Low-dimensional" design, there is one canonical variate pair and the canonical vectors have a sparse structure. The variables within each data set are uncorrelated in the first design, and correlated in the second design. In the "NonSparse Low-dimensional" design, there are two canonical variate pairs and the canonical vectors are non-sparse. The remaining three designs are high-dimensional with a lot of variables compared to the sample size. Only Sparse CCA and Robust Sparse CCA can be computed. In the "Sparse High-dimensional 1" design with n=100,p=100,q=4, there is one canonical variate pair and the canonical vectors are sparse. In the "Sparse High-dimensional 2" design with n=100,p=q=100, there is one canonical variate pair and each canonical vector contains ten non-zero elements. In the "Sparse Ultra High-dimensional" design there are much more variables (p=q=10000) than observations (n=100). There is one canonical variate pair and each canonical vector contains ten non-zero elements. The number of simulations for each design except the last one is M=1000. For the "Sparse Ultra High-dimensional design" M=100 to reduce computational burden. For each design, the following settings are considered No contamination. We generate data matrices X and Y according to a multivariate normal distribution N p+q (0,Σ), with covariance matrix $$ {\boldsymbol{\Sigma}} = \left[\begin{array}{ll} {\boldsymbol{\Sigma}}_{xx} & {\boldsymbol{\Sigma}}_{xy} \\{\boldsymbol{\Sigma}}_{xy}^{T} & {\boldsymbol{\Sigma}}_{yy} \end{array}\right] $$ described in Table 1. Table 1 Simulation designs t-distribution. We generate data matrices X and Y according to a multivariate t-distribution with three degrees of freedom t 3(0,Σ). Contamination. 90 % of the data are generated from N p+q (0,Σ), and 10 % of the data are generated from N p+q (2,Σ cont), with $$ {\boldsymbol\Sigma_{\text{cont}}} = \left[\begin{array}{ccc} \boldsymbol \Sigma_{xx} && \mathbf{0} \\ \mathbf{0} && \boldsymbol \Sigma_{yy} \end{array}\right]. $$ Similar conclusions can be drawn from other contamination settings (e.g. where only one of the two data sets is contaminated) and are available from the authors upon request. In our simulation study, the estimators are evaluated on their estimation accuracy and sparsity recognition performance. For evaluating estimation accuracy, we compute for each simulation run m, with m=1,…,M, the angle $\theta ^{m}(\hat {\mathbf {A}}^{m},\mathbf {A})$ between the subspace spanned by the estimated canonical vectors (contained in the columns of $\hat {\mathbf {A}}^{m}$) and the subspace spanned by the true canonical vectors (contained in the columns of A). We proceed analogously for the matrix B. The average angles, measuring the estimation accuracy, are given by $$\begin{aligned} \bar{\theta}(\hat{\mathbf{A}},\mathbf{A}) &= \frac{1}{M} \sum_{m=1}^{M} \theta^{m}(\hat{\mathbf{A}}^{m},\mathbf{A}) \text{\ \ \ and \ \ \ \ }\\ \bar{\theta}(\hat{\mathbf{B}},\mathbf{B}) &= \frac{1}{M} \sum_{m=1}^{M} \theta^{m}(\hat{\mathbf{B}}^{m},\mathbf{B}). \end{aligned} $$ For evaluating sparsity, we use the true positive rate and the true negative rate $$ \begin{aligned} \text{TPR}(\hat{\mathbf{A}}^{m},\mathbf{A}) &= \frac{\# \{ (i,j):{{\widehat{\mathbf{A}}_{ij}}}^{m} \neq 0 \text{and} {\mathbf{A}}_{ij}\neq 0 \}}{ \# \{ (i,j): {\mathbf{A}}_{ij} \neq 0 \} }\\ \text{TNR}(\hat{\mathbf{A}}^{m},\mathbf{A}) &= \frac{\# \{ (i,j):{ {\widehat{\mathbf{A}}_{ij}}}^{m}= 0 \text{and}{\mathbf{A}}_{ij} = 0 \}} {\# \{ (i,j): {\mathbf{A}}_{ij} = 0\}}. \end{aligned} $$ We proceed analogously for the matrix B. A true positive is a coefficient that is non-zero in the true model, and is estimated as non-zero. A true negative is a coefficient that is zero in the true model, and is estimated as zero. Both should be as high as possible for a sparse estimator. Note that the false positive rate is the complement of the true negative rate (i.e. FPR=1-TNR). A sparse estimator should control the FPR, which can be seen as a false discovery rate, at a sufficiently low level. In our empirical applications, to decide on the number of canonical variate pairs to extract, we use the maximum eigenvalue ratio criterion, as discussed in the "Methods" Section. To compare the performance of the CCA approaches, we perform a leave-one-out cross-validation exercise and compute the cross-validation score $$ CV = \frac{1}{r} \frac{1}{h} \sum_{i=1}^{h} \|\| \widehat{\mathbf{A}}^{T}_{-i}\mathbf{x}_{i} - \widehat{\mathbf{B}}^{T}_{-i}\mathbf{y}_{i} \|\|^{2}, $$ where $\widehat {\mathbf {A}}^{T}_{-i}$ and $\widehat {\mathbf {B}}^{T}_{-i}$ contain the estimated canonical vectors when the i th observation is left out of the estimation sample and h=⌊n(1−α)⌋, with α=0 (0 % Trimming) or α=0.1 (10 % Trimming). We use trimming to eliminate the effect of outliers in the cross-validation score. Simulation study We compare the performance of the Robust Sparse CCA method with (i) standard CCA, (ii) Robust CCA, and (iii) Sparse CCA. The alternating regression algorithm is used for all four estimators, for ease of comparability. Robust CCA uses LTS instead of sparse LTS, and corresponds to the alternating regression approach of [26]. Sparse CCA uses the lasso instead of sparse LTS, Pearson correlations for computing the canonical correlations, and ordinary PCA for getting the initial values. The sparsity parameters for sparse CCA are selected with BIC. Standard CCA is like sparse CCA, but using the LS instead of the lasso. Summary results for the estimator $\widehat {\mathbf {A}}$ are in Table 2. The results for $\widehat {\mathbf {B}}$ are similar and, therefore, omitted. Standard errors around the average angles, TPRs and TNRs are in almost all cases smaller than 6 % of the reported numbers in Table 2. Table 2 Simulation results. Average of the angles between the space spanned by the true and estimated canonical vectors; average true positive rate and true negative rate are reported for each method First we discuss the results from the "Uncorrelated Sparse Low-dimensional" design. In the scenario without contamination, the sparse estimators Sparse CCA and Robust Sparse CCA achieve a much better average estimation accuracy than the non-sparse estimators CCA and Robust CCA. As expected, a sparse method results in increased estimation accuracy when the true canonical vectors have a sparse structure. Looking at sparsity recognition performance, Sparse CCA and Robust Sparse CCA perform equally good in retrieving the sparsity in the data generating process. In the contaminated simulation setting, the robust estimators maintain their accuracy. Robust Sparse CCA performs best and clearly outperforms Robust CCA: for instance, Robust Sparse CCA achieves an average estimation accuracy of 0.05 against 0.15 for the contamination setting, see Table 2. The non-robust estimators CCA and Sparse CCA are clearly influenced by the outliers, as reflected by the much higher values of the average angle $\bar {\theta }(\hat {\mathbf {A}},\mathbf {A})$ in Table 2. Sparse CCA now performs even worse than Robust CCA. The considered contamination induces overfitting in Sparse CCA, reflected in the low values of the true negative rate, or alternatively, the high values of the false positive rate. In an unreported simulation study, we investigated the effect of the signal strength on the results. We vary the value of the true canonical correlation in the first design from 0.1 to 0.9, thereby increasing the signal strength. If outliers are present, Robust Sparse CCA always performs best. The margin by which it outperforms Sparse CCA is larger if the signal is stronger. If no outliers are present, Sparse CCA performs best for weak signal levels below 0.6. Similar conclusions can be drawn from the "Correlated Sparse Low-dimensional" and "NonSparse Low-dimensional" design. Note that the true negative rate in Table 2 is omitted for the "NonSparse Low-dimensional" design since the true canonical vectors are non-sparse. In the situation without contamination, the price the sparse methods pay in the "NonSparse Low-dimensional" design is a decreased estimation accuracy, as measured by the average angle. For Robust Sparse CCA compared to Robust CCA this decrease is marginal. In the contaminated settings, the robust methods perform best and show similar performance. For the high-dimensional designs, only Sparse CCA and Robust Sparse CCA are computable. For the "Sparse High-dimensional 1" design, Robust Sparse CCA is competitive to Sparse CCA if no outliers are present. When adding outliers, the performance of Sparse CCA gets distorted. For the heavier tailed t-distribution, the average estimation accuracy of Robust Sparse CCA compared to Sparse CCA is much better: 0.56 against 0.70. For the contamination setting, the average estimation accuracy of Robust Sparse CCA is even more than twice as good as the average estimation accuracy of Sparse CCA. Similar conclusions hold for the second high-dimensional design. In the "Sparse Ultra High-dimensional" design, Sparse CCA performs best if no outliers are present. For the heavier tailed t-distribution, Robust Sparse CCA and Sparse CCA perform comparable in terms of estimation accuracy. But in the presence of outliers, Robust Sparse CCA improves estimation accuracy of Sparse CCA by about 22 %. Moreover, Robust Sparse CCA achieves a good balance between the TPR and the TNR, while Sparse CCA suffers from a low TPR if outliers are present. In sum, Robust Sparse CCA shows the best overall performance in this simulation study. It performs best in sparse contaminated settings. In sparse non-contaminated settings, Robust Sparse CCA is competitive to Sparse CCA. In contaminated non-sparse settings, Robust Sparse CCA is competitive to Robust CCA. Comparison of Robust Sparse CCA to other CCA alternatives We compare the performance of Robust Sparse CCA to the sparse CCA methods of [15, 16] and [17]. The sparsity parameters of all methods are selected as proposed by the respective authors. Note that these methods are not robust. sparse CCA applied on pre-processed data. As a pre-processing step to remove outliers, we transformed the data towards normality by replacing them by their normal scores (see e.g. [37], page 150). sparse CCA using the robust initial value for the algorithm as Robust Sparse CCA. Summary results for the estimator $\widehat {\mathbf {A}}$ are in Table 3. For reasons of brevity, we only report the results from the "Sparse High-dimensional 2" design. Similar conclusions are obtained from the other designs and are available from the authors upon request. Table 3 As in Table 3, comparing Robust Sparse CCA to other alternatives in the "Sparse High-dimensional 2 design" If no outliers are present, (i) Robust Sparse CCA is competitive to the sparse CCA methods of [15, 16] and [17]. (ii) Robust Sparse CCA performs comparable to Sparse CCA on pre-processed data. (iii) Sparse CCA with the same initial value as Robust Sparse CCA performs comparable to Sparse CCA. If outliers are present, (i) Robust Sparse CCA outperforms the sparse CCA methods of [15, 16] and [17]. (ii) Robust Sparse CCA outperforms Sparse CCA on pre-processed data. Sparse CCA on pre-processed data performs better than Sparse CCA. (iii) Robust Sparse CCA outperforms Sparse CCA with the same initial value. Here, differences in performance between Robust Sparse CCA and Sparse CCA stem from the use of the sparse LTS instead of the lasso regressions. Hence, the use of the sparse LTS estimator in the alternating regression scheme is essential. We consider three biometric applications. The first data set is low-dimensional and often used in Robust Statistics. The other two data sets are high-dimensional and have been used before in papers on sparse CCA. We show that the performance of Robust Sparse CCA on these data sets is much better than the performance of Sparse CCA. Evaporation data set We analyze an environmental data set from [38]. Two sets of environmental variables have been measured on n=46 consecutive days from June 6 until July 21.3 The first set contains p=3 soil temperature variables (maximum, minimum and average soil temperature). The second set contains q=7 environmental variables (maximum, minimum and average air temperature; maximum, minimum and average daily relative humidity; and total wind). The aim is to find and quantify the relations between the soil temperature variables and the remaining variables. As a first inspection of the data, we use the Distance-Distance plot [39] in Fig. 1. The Distance-Distance plot displays the robust distances versus the Mahalanobis distances. The vertical and horizontal lines are drawn at values equal to the square root of the 97.5 % quantile of a chi-squared distribution with 10 degrees of freedom. Points beyond those lines would be considered as outliers. The Distance-Distance plot reveals some outliers: objects 31 and 32, for example, are extreme outliers. This suggests the need for a robust CCA method. Table 4 reports the cross-validation scores from Eq. (6) for the four CCA methods. For all methods two canonical variate pairs are extracted. The method that achieves the lowest cross-validation score has the best out-of-sample performance. Robust Sparse CCA achieves the best cross-validation score. Evaporation data set: Distance-Distance plot Table 4 Evaporation data set: Cross-validation score for standard CCA, Robust CCA, Sparse CCA and Robust Sparse CCA Table 5 shows the estimated canonical vectors for the Robust CCA and Robust Sparse CCA method. By adding the penalty term, the number of non-zero coefficients in the two canonical vectors is reduced from a total of 20 for Robust CCA to 10 for Robust Sparse CCA. The price to pay for the sparseness is a slight decrease in the estimated canonical correlations (computed using the bivariate MCD estimator, see "Methods" Section): they drop from 0.93 to 0.87 for the first one, and from 0.56 to 0.48 for the second canonical correlation. We find this decrease acceptable, given the gained sparsity in the canonical vectors. The sparse structure of the canonical vectors facilitates interpretation. The first canonical variate in the soil temperature data set, for instance, is uniquely determined by the variable AVST. Table 5 Evaporation data set: Estimated canonical vectors using Robust CCA and Robust Sparse CCA Nutrimouse data set This genetic data set is publicly available in the R package CCA [11]. Two sets of variables, i.e. gene expressions and fatty acids, are available for n=40 mice. The first set contains expressions of p=120 genes measured in liver cells. The second set of variables contains concentrations of q=21 hepatic fatty acids (FA). In this experiment, there are two groups of mice (wild-type and PPAR α deficient mice) that receive a specific diet (five possible diets). More details on how the data were obtained can be found in [40]. The aim is to identify a small set of genes that are correlated with the fatty acids. In this data set, the number of experimental units is smaller than the number of variables. Therefore, standard CCA nor Robust CCA can be performed. Robust Sparse CCA and Sparse CCA can be applied in this high-dimensional setting and produce interpretable, sparse canonical vectors. For both methods, one canonical variate pair is extracted. The cross-validation scores from Eq. (6) are reported in Table 6. Robust Sparse CCA outperforms Sparse CCA. The cross-validation scores are reduced by about 90 % when using the robust method. Table 6 Nutrimouse data set: Cross-validation score for Sparse CCA and Robust Sparse CCA Given its better out-of-sample performance, we discuss the estimated canonical vectors obtained using Robust Sparse CCA. The top panel of Fig. 2 displays the coefficients of the selected genes, i.e. those genes with non-zero estimated coefficients, in the first canonical vector: 24 out of 120 variables are selected. The solution is very sparse, facilitating interpretation. Martin et al. [40] find a consistent reduction of Cyp3a11 in PPAR α livers on the one hand, and an overexpression of CAR1 on the other hand. Both genes are selected and have among the highest (absolute) coefficients. The coefficients of the selected fatty acids are displayed in the bottom panel of Fig. 2: 13 out of 21 fatty acid variables are selected. The fatty acids C22:6n-3, C22:5n-3, C22:5n-6, C22:4n-3 and C20:5n-3 are related to the effect of the five diets used in this experiment. From Fig. 2, we see that four out of these five fatty acids are selected. Nutrimouse data set: Coefficients of selected genes (top) and coefficients of selected fatty acids (bottom) in the first canonical vector pair Breast cancer data set The genetic data set is described in [41] and available in the R package PMA [42]. Two sets of data, i.e. gene expression data (19 672 variables) and comparative genomic hybridization (CGH) data (2149 variables) are available for n=89 patients, and this for 23 chromosomes. We analyze the data for each of the chromosomes separately, each time using the CGH and gene expression variables for that particular chromosome. Depending on the chromosome, either 1, 2, 3, or 4 canonical vector pairs are extracted. The aim is to identify a subset of CGH variables that are correlated with a subset of gene expression variables. Results of the cross-validation scores of Eq. (6) are reported in Fig. 3. For each of the 23 chromosomes, we plot the value of the cross-validation score (0 % trimming) for Robust Sparse CCA (horizontal axis) and Sparse CCA (vertical axis). Results when using 10 % trimming are similar and, therefore, omitted. The cross-validation scores of Robust Sparse CCA are much better than those of Sparse CCA: all points are lying above the 45°-line. For chromosomes 1, 3, 4, and 11, for instance, the cross-validation scores of Robust Sparse CCA are more than 10 times lower than those of Sparse CCA. Since Robust Sparse CCA performs much better, outliers might be present for these chromosomes. Hence, it is safer to use Robust Sparse CCA instead of Sparse CCA. Breast cancer data set: 23 cross-validation scores (one for each chromosome) for Robust Sparse CCA (horizontal axis) and Sparse CCA (vertical axis). The dashed line is the 45°-line The Robust Sparse CCA method yields an interesting way to characterize the outliers. To this end, we create the Residual Distance plot of the residuals $\mathbf {X}{\widehat {\mathbf {A}}} - \mathbf {Y}{\widehat {\mathbf {B}}}$, and this for each of the 23 chromosomes. The Residual Distance plot displays the robust distance of the residuals (vertical axis) versus the observation number (horizontal axis). Points above the horizontal black line are marked as outliers. Results for chromosome 3 and 8 are displayed in Fig. 4, results for the other chromosomes are available upon request. For some chromosomes, like chromosome 3, the difference in cross-validation scores of Robust Sparse CCA and Sparse CCA in Fig. 3 is outspoken, suggesting that outliers might be present. We use the Residual Distance plot (Fig. 4, left panel) to detect which patients are outlying. In the Residual Distance plot of chromosome 3 a lot of patients are marked as outliers. For chromosome 8, on the other hand, the cross-validation scores of Sparse CCA and Robust Sparse CCA are nearly identical, which might suggest that there are no outliers. Looking at the Residual Distance Plot of chromosome 8 (Fig. 4, right panel), no outliers are indeed detected. Breast cancer data set: Residual Distance plot for chromosome 3 (left) and chromosome 8 (right) Robust Sparse CCA has three important advantages over Robust CCA. (i) Robust Sparse CCA improves model interpretation since only a limited number of variables, those corresponding to the non-zero elements of the canonical vectors, enter the estimated canonical variates (cfr. evaporation application), (ii) if the number of variables approaches the sample size, the estimation precision of Robust CCA suffers, and (iii) if the number of variables exceeds the sample size, Robust CCA can not even be performed. Robust Sparse CCA can still be computed (cfr. nutrimouse and breast cancer application). The key ingredient of the Robust Sparse CCA algorithm is the sparse LTS proposed by [31]. The choice of the subsample size h, see Eq. (2) involves a trade-off between robustness and estimation accuracy. We use h=⌊0.75·n⌋, as recommended by [31]. This guarantees a sufficiently high estimation accuracy and a good robustness/accuracy trade-off. If the researcher thinks that the proportion of outliers in one of the two data sets is larger than 25 %, one could consider higher values of h. Our Robust Sparse CCA algorithm starts by robustly centering each variable using the coordinatewise median. The spatial median (e.g. [37], page 251) could serve as an alternative to the coordinatewise median. Several questions are left for future research. One could use a joint selection criterion for the number of canonical variate pairs and the sparsity parameter. This would, however, increase computation time substantially. To obtain sparse canonical vectors, we use a Lasso penalty. Other penalty functions such as the Adaptive Lasso [43] could be considered. The Adaptive Lasso is consistent for variable selection, whereas the Lasso is not. Furthermore, we use a regularized version of the LTS estimator. One could also use a regularized version of the S-estimator or the MM-estimator to increase efficiency. Up to our knowledge, however, the sparse LTS is the only robust sparse regression estimator for which efficient code [35] is available. Sparse Canonical Correlation Analysis delivers interpretable canonical vectors, with some of its elements estimated as exactly zero. Robust Sparse CCA retains this advantage, while at the same time coping with outlying observations. Typically, the canonical vectors are based on the sample versions of the covariance matrices. One could think of estimating those covariance matrices with an estimator that is robust and sparse at the same time, and then, to compute the eigenvectors. This approach would result in canonical vectors being robust, however, not sparse. To circumvent this pitfall, we reformulate the CCA problem in a regression framework. Nowadays, high-dimensional data sets where the researcher suspects contamination to be present are commonplace in genetics. 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Both authors read and approved the final manuscript. Leuven Statistics Research Centre (LStat), KU Leuven, Naamsestraat 69, Leuven, 3000, Belgium Ines Wilms & Christophe Croux Ines Wilms Christophe Croux Correspondence to Ines Wilms. Additional file 1 Sparse LTS estimator. (PDF 133 kb) Robust Sparse CCA algorithm. (PDF 150 kb) Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated. Wilms, I., Croux, C. Robust sparse canonical correlation analysis. BMC Syst Biol 10, 72 (2016). https://doi.org/10.1186/s12918-016-0317-9 Canonical correlation analysis Penalized estimation Robust estimation	CommonCrawl
DL-3-n-butylphthalide improved physical and learning and memory performance of rodents exposed to acute and chronic hypobaric hypoxia Gang Xu1,2,3, Yi-Kun Shi1,2,3, Bin-Da Sun1,2,3, Lu Liu1,2,3, Guo-Ji E.1,2,3, Shu He1,2,3, Jian-Yang Zhang1,2,3, Bao Liu1,2,3, Qiu Hu1,2,3, Jian Chen1,2,3, Yu-Qi Gao1,2,3 & Er-Long Zhang ORCID: orcid.org/0000-0003-4940-35161,2,3 Studies have revealed the protective effect of DL-3-n-butylphthalide (NBP) against diseases associated with ischemic hypoxia. However, the role of NBP in animals with hypobaric hypoxia has not been elucidated. This study investigated the effects of NBP on rodents with acute and chronic hypobaric hypoxia. Sprague-Dwaley rats and Kunming mice administered with NBP (0, 60, 120, and 240 mg/kg for rats and 0, 90, 180, and 360 mg/kg for mice) were placed in a hypobaric hypoxia chamber at 10,000 m and the survival percentages at 30 min were determined. Then, the time and distance to exhaustion of drug-treated rodents were evaluated during treadmill running and motor-driven wheel-track treadmill experiments, conducted at 5800 m for 3 days or 20 days, to evaluate changes in physical functions. The frequency of active escapes and duration of active escapes were also determined for rats in a shuttle-box experiment, conducted at 5800 m for 6 days or 27 days, to evaluate changes in learning and memory function. ATP levels were measured in the gastrocnemius muscle and malonaldehyde (MDA), superoxide dismutase (SOD), hydrogen peroxide (H2O2), glutathione peroxidase (GSH-Px), and lactate were detected in sera of rats, and routine blood tests were also performed. Survival analysis at 10,000 m indicated NBP could improve hypoxia tolerance ability. The time and distance to exhaustion for mice (NBP, 90 mg/kg) and time to exhaustion for rats (NBP, 120 and 240 mg/kg) significantly increased under conditions of acute hypoxia compared with control group. NBP treatment also significantly increased the time to exhaustion for rats when exposed to chronic hypoxia. Moreover, 240 mg/kg NBP significantly increased the frequency of active escapes under conditions of acute hypoxia. Furthermore, the levels of MDA and H2O2 decreased but those of SOD and GSH-Px in the sera of rats increased under conditions of acute and chronic hypoxia. Additionally, ATP levels in the gastrocnemius muscle significantly increased, while lactate levels in sera significantly decreased. NBP improved physical and learning and memory functions in rodents exposed to acute or chronic hypobaric hypoxia by increasing their anti-oxidative capacity and energy supply. Exposure to a high altitude environment may lead to substantial decreases in work and cognitive performances [1, 2]. At an altitude of 4500 m, the maximum working capacity was found to be reduced to 50% of that observed at low altitude [2]. People exposed to hypobaric hypoxic conditions display significant alterations in cognitive processes, including attention span, short-term memory, decision making, simple and complex reaction times, and mood [1]. The changes that occur in physical and cognitive functions in response to hypobaric hypoxia may greatly affect work and normal life. Therefore, the development of drugs and methods to alleviate physical and cognitive impairment is imperative for those located at high altitudes for extended periods. High altitude acclimatization is a physiological process that comprises a number of responses by different systems in the body, which take place upon exposure to plateau hypoxia. Many studies have revealed that the changes involved in acclimatization occur in various systems and with varying time courses. Many people can maintain the homeostasis needed for normal bodily function through acclimatization at a low oxygen partial pressure [3, 4]. However, acute and chronic high altitude illness will occur in people with poor acclimatization and can significantly damage diverse functionalities [5, 6]. Therefore, studying the molecular mechanisms of high altitude diseases may identify pathways that will enhance human capabilities at high altitude. Recent studies found that disorders of the inflammation response [7] and metabolism [8] as well as oxidative stress [9] were closely associated with high altitude illnesses. Agents that modulate these processes have the potential to improve human activities at high altitude. DL-3-n-butylphthalide (NBP), a racemic mixture of an optical isomer extracted from the seeds of Apium graveolens Linn. (Fig. S1), is widely used to treat patients with ischemic stroke [10]. NBP is thought to inhibit inflammation and oxidative as well as endoplasmic reticulum stress and promote angiogenesis in animals and humans with cerebral ischemia [11]. NBP was recently shown to exert neuroprotective effects by alleviating vascular cognitive impairment [12] and promoting neuroplasticity and motor recovery after cerebral ischemia [13] and chronic intermittent hypoxia-hypercapnia [14]. Whether NBP exerts beneficial effects under other hypoxic conditions such as hypobaric hypoxia is, however, unclear. In this work, we investigated the effects of NBP on the physical and cognitive abilities of rodents under hypobaric hypoxia conditions (equivalent to 5800 m). The effects of NBP on rodents' behavior were evaluated through exhaustive exercise and shuttle-box experiments. We evaluated the potential mechanism of NBP by collecting muscle and blood samples from treated rodents and analyzing the levels of ATP, malonaldehyde (MDA), superoxide dismutase (SOD), hydrogen peroxide (H2O2), lactate, and glutathione peroxidase (GSH-Px) as well as by performing routine blood tests. Experimental rodents and NBP administration Male pathogen-free Sprague-Dawley (SD) rats (6 to 8 weeks old, weighing 180–220 g) and male pathogen-free Kunming mice (6 weeks old, weighing 18–20 g) were used in this study. Rats and mice were obtained from the Laboratory Animal Center of Army Medical University, and the animal study protocol was approved by the Animal Care and Use Committee Guidelines of the Army Medical University. NBP (purity, 99.6%) was obtained from Shijiazhuang Pharma Group NBP Pharmaceutical Co., Ltd. (Shijiazhuang, Hebei, China). A total of 200 rats and 224 mice were used for seven separate experiments in our study. Rats or mice for each experiment were randomly divided into four groups respectively: control group (administered an equivalent volume of corn oil), NBP low dose-treated group (60 mg/kg for rats and 90 mg/kg for mice), NBP intermediate dose-treated group (120 mg/kg for rats and 180 mg/kg for mice), and NBP high dose-treated group (240 mg/kg for rats and 360 mg/kg for mice). The detailed plan for each experiment was described as followed. NBP was intragastrically administered once or twice every day. Rodents had free access to food and water. Acute hypoxia survival experiment at 10,000 m in a hypobaric hypoxia chamber The purpose of this experiment was to evaluate the effect of NBP on the hypoxia tolerance of rats and mice exposed to acute hypobaric hypoxia. Forty rats were randomly divided to four experimental groups with 10 rats in each group. Sixty four mice were randomly divided to four experimental groups with 16 mice in each group. NBP was given to SD rats and Kunming mice by intragastric administration at 1 ml/100 g body weight for 7 days at 300 m altitude (Chongqing altitude). At 1.14 h after the last administration, the rodents were placed in the hypobaric hypoxia chamber. The high altitude of the chamber ascended to 10,000 m at a rate of approximately 1000 m/min. The survival of rodents was observed and recorded [15] and each experiment was stopped after 30 min at 10,000 m (Fig. 1a). Diagrams depicting hypoxia tolerance experiments and experiments conducted under conditions of acute hypoxia. (a) Acute hypoxia survival experiment in a 10,000 m hypobaric hypoxia chamber; (b) standard hypoxia tolerance experiment for mice; (c) motor-driven wheel-track treadmill fatigue experiments for acute hypoxic mice; (d) treadmill running experiment for acute hypoxic rats; (e) shuttle-box experiment for acute hypoxic rats Standard hypoxia tolerance experiment of mice This experiment was used to evaluate the effect of NBP on the hypoxia tolerance of mice under conditions of normobaric hypoxia. One hundred and twenty Kunming mice were separately used for experiments with NBP intragastric administration for 3 days, 5 days or 7 days, and forty mice were used for each time point. Forty mice of each time were randomly divided into four experimental groups with 10 mice in each group. At 1.14 h after the last administration, each mouse was put into a 125-ml bottle and 5 g of soda lime were added, which was used to absorb the carbon dioxide and water vapor produced by breathing. Then, the lid was tightly sealed until the mouse's breathing movements stopped. The survival time (ST, min), i.e., the time from when the mouse was sealed in the bottle to its death, was recorded (Fig. 1b) [15], and standard hypoxia tolerance time (STT, min/ml) was calculated according to the following formula [16]: $$ \mathrm{STT}=\mathrm{ST}/\left(V-\mathrm{BW}/0.94\right) $$ where V is the bottle volume (ml) and BW is the body weight of the mouse (g). Motor-driven wheel-track treadmill experiments for acute hypoxic mice This experiment was used to evaluate the effect of NBP on the physical ability of mice under conditions of acute hypobaric hypoxia. Forty mice were randomly divided to four experimental groups with 10 mice in each group. Kunming mice were intragastrically treated with NBP twice per day for 6 days. Mice were raised at 300 m altitude for 3 days. On day 4, mice were placed at 5800 m altitude and exercised on a motor-driven wheel-track treadmill (YLS-10B, Shandong Academy of Medical Sciences, Jinan, China) for 3 days under the following conditions: 10 min/d, electronic shock current: 0.8 mA, electronic shock time: 3 s, 30 s rest after electronic shock, and five or more times rest in 10 min as exhausted standard [17]. On day 7, mice were performed exhaustion test on the motor-driven wheel-track treadmill 1.14 h after NBP administration at 5000 m altitude and the distance to exhaustion was recorded (Fig. 1c). Treadmill running experiment for acute hypoxic rats This experiment was used to evaluate the effect of NBP on the physical ability of rats to run on a treadmill under conditions of acute hypobaric hypoxia. Forty rats were randomly divided to four experimental groups with 10 rats in each group. NBP was intragastrically administered to SD rats for 3 days at an altitude of 300 m. Rats were then subjected to treadmill exercise every day as follows: 5 min for adaptation, followed by 15 m/min for 10 min and 20 m/min for 20 min. On day 4, rats were placed in a 5800 m hypobaric hypoxia chamber, administered NBP and subjected to the above exercise regimen for 3 days. On day 7, an exhaustion test was conducted by extending the treadmill running experiment after NBP administration at 5000 m altitude. The experimental plan included 5 min for adaptation, followed by 15 m/min for 3 min, 20 m/min for 3 min, 25 m/min for 24 min, and then 30 m/min up to exhausted status [18]. The total running time was used to evaluate physical ability under acute hypoxia exposure. Rats were anaesthetized and their arterial blood was collected for analysis of MDA, SOD, H2O2, lactate, and GSH-Px levels and routine blood tests were also performed. In addition, the gastrocnemius muscle tissues were excised and used for ATP detection. MDA, SOD, H2O2, lactate, and GSH-Px levels were detected using Nanjing Jiancheng assay agents (A003–1, A001–3, A064–1-1, A020–1-2, and A005, respectively). The ATP level was analyzed using a Beyotime ATP assay kit (S0026). Routine blood routine tests were performed at the Xinqiao Hospital, Chongqing, China (Fig. 1d). Treadmill running experiment for chronic hypoxic rats This experiment was used to evaluate the effect of NBP on the physical ability of rats under conditions of chronic hypobaric hypoxia. Forty rats were randomly divided to four experimental groups with 10 rats in each group. Rats were placed in a hypobaric hypoxia chamber at 5800 m and intragastrically administered NBP for 13 days from day 8. Rats were exercised on the treadmill as above for 3 days from day 18. On day 21, an exhaustion test was conducted using the treadmill running experiment after NBP administration at 5000 m altitude. At exhausted status, rats were anaesthetized and their arterial blood samples were obtained to analyze MDA, SOD, H2O2, lactate, and GSH-Px levels and perform routine blood tests. Further, the gastrocnemius muscles were excised for ATP detection (Fig. 2a). Diagrams depicting experiments conducted under conditions of chronic hypoxia. (a) Treadmill running experiment for chronic hypoxic rats; (b) shuttle-box experiment for chronic hypoxic rats Shuttle-box experiment for acute hypoxic rats To investigate the effects of NBP on learning and memory ability under conditions of acute hypoxia, the behavior of rats was evaluated in a shuttle-box (RD1106-SB-R, Shanghai Mobile Datum Information Technology Company, Shanghai, China). Forty rats were randomly divided to four experimental groups with 10 rats in each group. Rats were placed in a hypobaric hypoxia chamber at 5800 m and administered NBP via the intragastric route for 6 days. On day 4, rats were subjected to an exercise program. Each rat underwent 50 trials daily after a 5-min adaptation period. Rats were exposed to a 10-s sound and light signal followed by a 10-s foot shock (2.2 mA) and a 10-s interval in each trial. If the rat moved to the other chamber with the onset of sound and light, the behavior was counted as an active escape and the sound and light signal was turned off with no exposure to foot shock. The time from the start of an active escape to the expected foot shock was regarded as the duration of the active escape. If the rat did not change chambers during a trial, the behavior was counted as an error and the rat received a 10-s foot shock. In each series of 50 trials, the frequency and duration of active escapes were used as the measure of learning and memory [19]. On day 7, the frequency and duration of active escapes of rats were recorded (50 trials) in the shuttle-box experiment 1.14 h after NBP administration in the 5000 m hypobaric hypoxia chamber. Rats were then anaesthetized and their arterial blood was collected to measure MDA, SOD, H2O2, and GSH-Px levels (Fig. 1e). Shuttle-box experiment for chronic hypoxic rats This experiment was used to evaluate the effect of NBP on learning and memory ability of rats under chronic hypobaric hypoxia. Forty rats were randomly divided to four experimental groups with 10 rats in each group. Rats were placed in a hypobaric hypoxia chamber at 5800 m and administered NBP via the intragastric route for 20 days from day 8. From day 25, rats were subjected to the exercise program described in section 2.7 for 3 days. On day 28, the frequency and duration of active escapes of rats were recorded (50 trials) using the shuttle-box experiment 1.14 h after NBP administration in the 5000 m hypobaric hypoxia chamber. At the end of the experiment, rats were anaesthetized and arterial blood was collected for the analysis of MDA, SOD, H2O2, and GSH-Px levels (Fig. 2b). Statistical analysis was carried out using the SPSS 19.0 software. For the survival analysis, a Kaplan-Meier curve was generated using the log-rank test. Other experiments were analyzed using One-way ANOVA test. Data represented mean values of at least three independent experiments ± SD (standard deviation). A value of P < 0.05 was considered statistically significant. NBP improved the hypoxia tolerance of rats and mice We firstly studied the effect of NBP on survival time at 10,000 m with 30 min as the stop time. The survival curves of the experiments showed that administration of 120 and 240 mg/kg NBP significantly improved the survival times of rats. Compared with the control groups (rats 100%, mice 81.3%), the death percentages at 30 min for the 120 (90.0%) and 240 mg/kg group (80.0%) for rats and the 180 mg/kg group (62.5%) for mice had declined (Fig. 3). Moreover, the standard tolerance time of mice under conditions of closed hypoxia was also markedly extended by 360 mg/kg NBP administration for 5 days (control group, 12.18 ± 2.00 min/(100 ml·g); 360 mg/kg group, 13.94 ± 1.54 min/(100 ml·g); P = 0.048; Table S1). These results indicated that NBP could improve the hypoxia tolerance of rats and mice. Survival analysis of rats and mice with administration of NBP at 10,000 m exposure. The survival curve of rats (a) and mice (b) at 10,000 m exposure after NBP administration for 7 days with 30 min as the stop time. Death percentages of rats (c) and mice (d) after 30 min exposure at 10,000 m NBP improved physical abilities under conditions of acute and chronic hypoxia Under acute hypoxic conditions, 90 mg/kg NBP significantly improved the time to exhaustion (control group, 6.79 ± 8.72 min; 90 mg/kg group, 16.79 ± 14.00 min; P = 0.012) and distance to exhaustion (control group, 101.62 ± 130.78 m; 90 mg/kg group, 251.49 ± 210.07 m; P = 0.011) of mice (Fig. 4a-b) and NBP treatment at 120 and 240 mg/kg concentrations significantly increased the time to exhaustion of rats (control group, 38.28 ± 17.85 min; 120 mg/kg group, 61.80 ± 19.30 min; P = 0.024; 240 mg/kg group, 73.26 ± 26.89 min; P = 0.001; Fig. 4c). Moreover, NBP at 60, 120, and 240 mg/kg concentrations significantly increased the time to exhaustion of rats under conditions of chronic hypoxia (control group, 38.59 ± 16.83 min; 60 mg/kg group, 66.83 ± 28.10 min; P = 0.014; 120 mg/kg group, 75.00 ± 32.34 min; P = 0.002; 240 mg/kg group, 79.10 ± 26.33 min; P = 0.001; Fig. 4d). These results suggest that NBP may improve the physical ability of rodents under conditions of acute and chronic hypoxia. Effects of NBP on the time and distance to motor exhaustion of mice and rats under conditions of acute and chronic hypoxia. Time (a) and distance (b) to exhaustion of mice in the forced exercise wheel-track treadmill experiment at 5800 m for 3 days. Time to exhaustion of rats in the treadmill running experiment at 5800 m for 3 days (c) and 20 days (d). P < 0.05 compared with control group; #P < 0.05 compared with 60 mg/kg group To clarify the mechanism underlying the NBP-mediated acceleration in physical activity under acute and chronic hypoxic conditions, we evaluated the levels of MDA, H2O2, SOD, GSH-Px, and lactate in the blood and ATP in the gastrocnemius muscle of rats. The levels of MDA (control group, 38.83 ± 19.59 nmol/ml; 60 mg/kg group, 16.89 ± 14.48 nmol/ml; P < 0.001; 120 mg/kg group, 9.98 ± 6.95 nmol/ml; P < 0.001; 240 mg/kg group, 11.11 ± 9.17 nmol/ml; P < 0.001) and H2O2 (control group, 67.84 ± 36.43 mmol/L; 120 mg/kg group, 42.44 ± 12.26 mmol/L; P = 0.031; 240 mg/kg group, 42.36 ± 28.25 mmol/L; P = 0.027) decreased but that of SOD (control group, 148.47 ± 24.80 U/ml; 120 mg/kg group, 173.78 ± 29.33 U/ml; P = 0.020) increased with no obvious change of GSH-Px levels under conditions of acute hypoxia following treatment with various doses of NBP (Fig. 5a). Under conditions of chronic hypoxia, levels of MDA (control group, 8.84 ± 2.06 nmol/ml; 120 mg/kg group, 5.88 ± 1.75 nmol/ml; P = 0.002; 240 mg/kg group, 4.86 ± 1.27 nmol/ml; P < 0.001) and H2O2 (control group, 351.13 ± 111.57 mmol/L; 120 mg/kg group, 245.28 ± 76.66 mmol/L; P = 0.037) decreased, but GSH-Px expression was upregulated (control group, 457.01 ± 326.03 U/mg protein; 120 mg/kg group, 976.35 ± 462.23 U/mg protein; P = 0.014; 240 mg/kg group, 1210.58 ± 294.57 U/mg protein; P = 0.002) with no obvious change of SOD levels (Fig. 5b). Thus, NBP may exert opposite effects by increasing the anti-oxidant capacity of the rats and decreasing the oxidant capacity. The lactate levels significantly decreased under conditions of acute hypoxia (control group, 32.79 ± 8.63 mmol/L; 240 mg/kg group, 16.83 ± 13.14 mmol/L; P = 0.002) and chronic hypoxia (control group, 23.50 ± 13.03 mmol/L; 60 mg/kg group, 13.38 ± 5.15 mmol/L; P = 0.010; 120 mg/kg group, 8.90 ± 5.01 mmol/L; P < 0.001; 240 mg/kg group, 12.48 ± 8.65 mmol/L; P = 0.008), whereas the levels of ATP (control group, 686.55 ± 129.51 μmol/g protein; 60 mg/kg group, 1228.51 ± 364.34 μmol/g protein; P < 0.001) significantly increased under conditions of chronic hypoxia but not acute hypoxia (Fig. 5a-b). These observations suggest NBP promoted ATP production via oxidative phosphorylation instead of glycolysis. NBP at a dose of 240 mg/kg significantly decreased the red blood cell (P = 0.007), hemoglobin (P = 0.010), and platelet counts (P = 0.002) as well as the hematocrit level (P = 0.025) under conditions of acute hypoxia (Table S2). However, NBP significantly increased the white blood cell count at a dose of 60 mg/kg (P = 0.044) under conditions of chronic hypoxia (Table S3). Effects of NBP on the energy metabolism and oxidative stress of exhausted rats under conditions of acute and chronic hypoxia. MDA, H2O2, SOD, GSH-Px, and lactate levels in the serum and ATP levels in the gastrocnemius muscle of acute (a) and chronic (b) hypoxic rats were detected. P < 0.05 compared with control group; #P < 0.05 compared with 60 mg/kg group; ∆P < 0.05 compared with 120 mg/kg group NBP improved learning and memory abilities under conditions of acute and chronic hypoxia The cognitive functions of the rats were evaluated using a shuttle-box experiment. Under conditions of acute hypoxia, NBP at a dose of 240 mg/kg significantly increased the frequency of active escapes (control group, 1.44 ± 1.50; 240 mg/kg group, 3.13 ± 2.17; P = 0.046) with no effect on the duration of active escapes (Fig. 6a). And NBP showed no effects on the frequency and duration of active escapes under conditions of chronic hypoxia (Fig. 6b). The beneficial effects of NBP on learning and memory function were not as consistent as those on physical activity. Furthermore, MDA, H2O2, SOD, and GSH-Px levels were analyzed in rodent blood samples. The expression of GSH-Px (control group, 5999.26 ± 659.23 U/mg protein; 240 mg/kg group, 6704.55 ± 451.45 U/mg protein; P = 0.045) was upregulated with no effects on MDA, H2O2 and SOD levels (Fig. 7a) under conditions of acute hypoxia. Under conditions of chronic hypoxia, the levels of MDA H2O2, SOD and GSH-Px levels showed no obvious change (Fig. 7b). Thus, NBP increased the anti-oxidant capacity of the rats during cognitive tests. Effects of NBP on rats under conditions of acute and chronic hypoxia in the shuttle-box experiment. Frequency of active escapes and duration of active escapes for rats at 5800 m for 6 days (a) and 27 days (b) in the shuttle-box experiments. P < 0.05 compared with control group Effects of NBP on the oxidative stress of rats under conditions of acute and chronic hypoxia after the shuttle-box experiment. Levels of MDA, H2O2, SOD, and GSH-Px in the serum of acute (a) and chronic (b) hypoxia rats were detected. P < 0.05 compared with control group; ∆P < 0.05 compared with 120 mg/kg group High altitude is characterized by a decline in air pressure and partial oxygen pressure. Physical and cognitive functions may be severely affected following exposure to high altitude. Nowadays, the number of people who remain at high altitude for prolonged periods is increasing although it is acknowledged that their work performance and daily life are greatly affected by high altitude. Therefore, there is an urgent need to develop methods that alleviate the physical and cognitive impairments that occur at high altitude in a hypobaric hypoxic environment. Several studies have demonstrated that NBP ameliorates ischemic injury and promotes neuroplasticity and recovery from injury. The therapeutic mechanisms underlying these effects are closely associated with various processes, including anti-oxidant and anti-inflammation activities, angiogenesis, anti-thrombosis, neurogenesis, and metabolic reprogramming. According to our results above, the effects of NBP on hypobaric hypoxia were mainly related to its anti-oxidant properties and metabolic reprogramming. The failure of human physiological responses to hypobaric hypoxia may result in increased inflammation, oxidative stress, and metabolic adjustment. Our study found that exposure to high altitude resulted in upregulation in the expression of inflammatory cytokines and downregulation in the expression of anti-inflammatory cytokines [7]. Moreover, hypobaric hypoxia induced oxidative damage and decreased antioxidative functions [20]. Metabolic modulation, may cause glycolytic capacity to be promoted and oxidative metabolism to be suppressed in response to hypoxia [21, 22]. A recent study demonstrated the relationship between oxidative stress and accelerated cognitive decline in chronic mountain sickness [23], suggesting that the molecular changes induced by hypobaric hypoxia may lead to behavioral abnormality. Given the effect of NBP on rodents with hypobaric hypoxia, we conclude that NBP reverses the alterations in hypoxia-induced oxidative stress and metabolism, thereby possibly reversing physical and learning and memory changes. Whether NBP could improve the effect of hypoxia on inflammation or not warrants further study. The effects and mechanisms of acute and chronic hypoxia on humans and animals are remarkably different [24,25,26,27], and treatments for reducing damage or diseases due to acute and chronic hypoxia also show great discrepancies. However, in this work, there were no obvious differences in the effects of NBP on acute and chronic hypoxia. There could be a common mechanism underlying acute and chronic hypoxia, so that some therapies and agents, such as NBP, could improve certain abilities under conditions of both acute and chronic hypoxia. Moreover, the effects of hypoxia on the brain display region-specific characteristics [28, 29]. We studied the effects of NBP on learning and memory functions [30, 31], which commonly decline under these conditions, to evaluate the effects of this agent on cognitive ability. The effects of NBP on other cognitive functions under conditions of hypobaric hypoxia remain to be investigated. While NBP played an important role in improving the physical and learning and memory functions of rodents under hypobaric hypoxia, it would be interesting to assess the effects of NBP on diverse cognitive functions at high-altitude (and after exposure). However, the dosage administered in this study was relatively higher than those employed in other studies. Therefore, it is imperative to refine the NBP treatment and develop new therapeutics for reducing hypobaric hypoxia-induced physical and cognitive decline. In summary, we demonstrated the NBP-mediated improvement in physical and learning and memory abilities of rodents under conditions of acute and chronic hypobaric hypoxia, as evidence from the elevation in anti-oxidative functions and promotion of oxidative phosphorylation rather than glycolysis. BW: GSH-Px: Glutathione peroxidase H2O2 : MDA: Malonaldehyde NBP: DL-3-n-butylphthalide SOD: Superoxide dismutase Survival time STT: Standard hypoxia tolerance time Yan X. 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Region-specific effects on brain metabolites of hypoxia and hyperoxia overlaid on cerebral ischemia in young and old rats: a quantitative proton magnetic resonance spectroscopy study. J Biomed Sci. 2010;17(1):14. https://doi.org/10.1186/1423-0127-17-14. Zhu M, Xu M, Zhang K, Li J, Ma H, Xia G, Li X, Zhang B, Shi H. Effect of acute exposure to hypobaric hypoxia on learning and memory in adult Sprague-Dawley rats. Behav Brain Res. 2019;367:82–90. https://doi.org/10.1016/j.bbr.2019.03.047. Qaid E, Zakaria R, Sulaiman SF, Yusof NM, Shafin N, Othman Z, et al. Insight into potential mechanisms of hypobaric hypoxia-induced learning and memory deficit - lessons from rat studies. Hum Exp Toxicol. 2017;36(12):1315–25. https://doi.org/10.1177/0960327116689714. This work was supported by Grants from the National Science and Technology Major Project (2014ZX09J14102-05B and 2018ZX09J18109). Institute of Medicine and Equipment for High Altitude Region, College of High Altitude Military Medicine, Army Medical University, Number 30, Gaotanyan Street, District of Shapingba, Chongqing, 400038, China Gang Xu, Yi-Kun Shi, Bin-Da Sun, Lu Liu, Guo-Ji E., Shu He, Jian-Yang Zhang, Bao Liu, Qiu Hu, Jian Chen, Yu-Qi Gao & Er-Long Zhang Key Laboratory of Extreme Environmental Medicine, Ministry of Education of China, Chongqing, China Key Laboratory of High Altitude Medicine, People's Liberation Army, Chongqing, China Gang Xu Yi-Kun Shi Bin-Da Sun Lu Liu Guo-Ji E. Shu He Jian-Yang Zhang Bao Liu Qiu Hu Yu-Qi Gao Er-Long Zhang GX performed experiments, interpreted the data, and wrote the manuscript. YKS, BDS, GJE, and QH raised animals and performed animal experiments. LL, SH, and JYZ participated in experiments of blood samples. BL and JC participated in data analysis and manuscript writing. YQG and ELZ conceived the study design, experimental plan, and manuscript writing. All authors read and approved the final manuscript. Correspondence to Yu-Qi Gao or Er-Long Zhang. All co-authors have read and approved of its submission to this journal. Additional file 1: Figure S1. Structure of DL-3-n-butylphthalide. Standard tolerance times (min/(100 ml·g)) of mice under conditions of closed hypoxia and administration of NBP and closed hypoxia at 3, 5, 7 days (mean ± SD). P < 0.05 compared with control group. Effects of NBP on routine blood tests of exhausted rats under conditions of acute hypoxia (mean ± SD). WBC. White blood cell; HCT. Hematocrit; RBC. Red blood cell; MCH. Mean corpuscular hemoglobin; HGB. Hemoglobin; MCHC. Mean corpuscular hemoglobin concentration; MCV. Mean corpuscular volume; PLT. Platelet count. P < 0.05 compared with control group; #P < 0.05 compared with 60 mg/kg group; ∆P < 0.05 compared with 120 mg/kg group. Effects of NBP on routine blood tests of exhausted rats under conditions of chronic hypoxia (mean ± SD). WBC. White blood cell; HCT. Hematocrit; RBC. Red blood cell; MCH. Mean corpuscular hemoglobin; HGB. Hemoglobin; MCHC. Mean corpuscular hemoglobin concentration; MCV. Mean corpuscular volume; PLT. Platelet count. *P < 0.05 compared with control group; #P < 0.05 compared with 60 mg/kg group. Xu, G., Shi, YK., Sun, BD. et al. DL-3-n-butylphthalide improved physical and learning and memory performance of rodents exposed to acute and chronic hypobaric hypoxia. Military Med Res 8, 23 (2021). https://doi.org/10.1186/s40779-021-00314-7 Hypobaric hypoxia Learning and memory function Oxidative stress	CommonCrawl
How to translate matrix back into Dirac notation? In Circuit composition and entangled states section of Wikipedia's article on Quantum logic gates the final result of a combined Hadamard gate and identity gate on $\|\Phi^{+}\rangle$ state is: $ M \frac{\|00\rangle + \|11\rangle}{\sqrt{2}} = \frac{1}{2} \begin{bmatrix} 1 & 0 & 1 & 0 \\ 0 & 1 & 0 & 1 \\ 1 & 0 & -1 & 0 \\ 0 & 1 & 0 & -1\end{bmatrix} \begin{bmatrix}1 \\ 0 \\ 0 \\ 1\end{bmatrix} = \frac{1}{2} \begin{bmatrix} 1 \\ 1 \\ 1 \\ -1 \end{bmatrix} = \frac{\|00\rangle + \|01\rangle + \|10\rangle - \|11\rangle}{2}$ How exactly does the $\begin{bmatrix} 1 \\ 1 \\ 1 \\ -1 \end{bmatrix}$ translate into $\|00\rangle + \|01\rangle + \|10\rangle - \|11\rangle$ states? I had no problems translating $\|00\rangle$ into $\begin{bmatrix}1 \\ 0 \\ 0 \\ 0\end{bmatrix}$ and $\|11\rangle$ into $\begin{bmatrix}0 \\ 0 \\ 0 \\ 1\end{bmatrix}$ but I'm not exactly sure how do you reverse the process in this example. quantum-state mathematics Michał ZającMichał Zając First read about the standard representation of qubit systems of and the basics of bra-ket notation. I had no problems translating $\|00\rangle$ into $\begin{bmatrix}1 & 0 & 0 & 0\end{bmatrix}^T$ and $\|11\rangle$ into $\begin{bmatrix}0 & 0 & 0 & 1\end{bmatrix}^T$ but I'm not exactly sure how do you reverse the process in this example. Cool. Then you should also be able to understand that $\|00\rangle + \|11\rangle \equiv \begin{bmatrix}1 & 0 & 0 & 0\end{bmatrix}^T + \begin{bmatrix}0 & 0 & 0 & 1\end{bmatrix}^T = \begin{bmatrix}1 & 0 & 0 & 1\end{bmatrix}^T$. Yes? Read on. How exactly does the $\begin{bmatrix} 1 & 1 & 1 & -1 \end{bmatrix}^T$ translate into $\|00\rangle + \|01\rangle + \|10\rangle - \|11\rangle$ states? The standard basis states of a $2$-qubit sytem $\|00\rangle,\|01\rangle,\|10\rangle,\|11\rangle$ i.e. standard basis elements of $\Bbb C^2\times \Bbb C^2$ can be mapped to the four $4\times 1$ column vectors $$\begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}^T, \begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix}^T,\begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix}^T \& \ \begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}^T.$$ This is essentially an isomorphism from $\Bbb C^2\times \Bbb C^2$ to $\Bbb R^4$. You say that you already know the following mappings: $$\begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}^T \to \|00\rangle$$ $$\begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix}^T \to \|01\rangle$$ $$\begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix}^T \to \|10\rangle$$ $$\begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}^T \to \|11\rangle$$ Now you simply need to expand the column vector $\begin{bmatrix} 1 & 1 & 1 & -1 \end{bmatrix}^T$ in terms of its basis elements, as follows: $$\begin{bmatrix} 1 & 1 & 1 & -1 \end{bmatrix}^T$$ $$= \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}^T + \begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix}^T + \begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix}^T + \begin{bmatrix} 0 & 0 & 0 & -1 \end{bmatrix}^T$$ $$= \begin{bmatrix} 1 & 0 & 0 & 0 \end{bmatrix}^T + \begin{bmatrix} 0 & 1 & 0 & 0 \end{bmatrix}^T + \begin{bmatrix} 0 & 0 & 1 & 0 \end{bmatrix}^T - \begin{bmatrix} 0 & 0 & 0 & 1 \end{bmatrix}^T$$ $$\implies \begin{bmatrix} 1 & 1 & 1 & -1 \end{bmatrix}^T \equiv \|00\rangle + \|01\rangle + \|10\rangle - \|11\rangle$$ In case you can't understand the last couple of steps, review Matrix addition. And you're done! Note: $T$ stands for transpose. edited Jan 23 at 0:07 Sanchayan DuttaSanchayan Dutta $\begingroup$ Oh, okay, it was way simpler than I expected. Thanks! $\endgroup$ – Michał Zając Jan 23 at 0:11 This is using vector addition. $\|00\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix}$ $\|00\rangle + \|01\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 0 \\ 0 \end{bmatrix}$ by the rules of vector addition. Adding in the other two terms: $\|00\rangle + \|01\rangle + \|10\rangle - \|11\rangle = \begin{bmatrix} 1 \\ 0 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 1 \\ 0 \\ 0 \end{bmatrix} + \begin{bmatrix} 0 \\ 0 \\ 1 \\ 0 \end{bmatrix} - \begin{bmatrix} 0 \\ 0 \\ 0 \\ 1 \end{bmatrix} = \begin{bmatrix} 1 \\ 1 \\ 1 \\ -1 \end{bmatrix}$ The reason you might want to write a state vector this way is to see how measurement of one qbit affects the other, for example (see here). ahelwerahelwer Not the answer you're looking for? Browse other questions tagged quantum-state mathematics or ask your own question. How does bra-ket notation work? How does measurement of one qubit affect the others? What exactly is a phase vector? How is a single qubit fundamentally different from a classical coin spinning in the air? How exactly is the stated composite state of the two registers being produced using the $R_{zz}$ controlled rotations? How do you represent the output of a quantum gate in terms of its basis vectors? Outcome of Hadamard transformation on a complex state Grover operator as a rotation matrix Grover algorithm for more than one element What is the meaning of the state $\|1\rangle-\|1\rangle$? Writing the transformation matrix for the following in terms of Kronecker products of elementary 2-qubit gates Making a maximally mixed 2-qubit state in the IBM Q	CommonCrawl
Carbon Balance and Management December 2018 , 13:10 \| Cite as Estimating urban above ground biomass with multi-scale LiDAR Phil Wilkes Mathias Disney Matheus Boni Vicari Kim Calders Andrew Burt Urban Carbon Fluxes Urban trees have long been valued for providing ecosystem services (mitigation of the "heat island" effect, suppression of air pollution, etc.); more recently the potential of urban forests to store significant above ground biomass (AGB) has also be recognised. However, urban areas pose particular challenges when assessing AGB due to plasticity of tree form, high species diversity as well as heterogeneous and complex land cover. Remote sensing, in particular light detection and ranging (LiDAR), provide a unique opportunity to assess urban AGB by directly measuring tree structure. In this study, terrestrial LiDAR measurements were used to derive new allometry for the London Borough of Camden, that incorporates the wide range of tree structures typical of an urban setting. Using a wall-to-wall airborne LiDAR dataset, individual trees were then identified across the Borough with a new individual tree detection (ITD) method. The new allometry was subsequently applied to the identified trees, generating a Borough-wide estimate of AGB. Camden has an estimated median AGB density of 51.6 Mg ha–1 where maximum AGB density is found in pockets of woodland; terrestrial LiDAR-derived AGB estimates suggest these areas are comparable to temperate and tropical forest. Multiple linear regression of terrestrial LiDAR-derived maximum height and projected crown area explained 93% of variance in tree volume, highlighting the utility of these metrics to characterise diverse tree structure. Locally derived allometry provided accurate estimates of tree volume whereas a Borough-wide allometry tended to overestimate AGB in woodland areas. The new ITD method successfully identified individual trees; however, AGB was underestimated by ≤ 25% when compared to terrestrial LiDAR, owing to the inability of ITD to resolve crown overlap. A Monte Carlo uncertainty analysis identified assigning wood density values as the largest source of uncertainty when estimating AGB. Over the coming century global populations are predicted to become increasingly urbanised, leading to an unprecedented expansion of urban land cover. Urban areas will become more important as carbon sinks and effective tools to assess carbon densities in these areas are therefore required. Using multi-scale LiDAR presents an opportunity to achieve this, providing a spatially explicit map of urban forest structure and AGB. Above ground biomass Urban forest Airborne LiDAR Terrestrial LiDAR Allometry above ground biomass airborne laser scanning projected crown area balanced iterative reducing and clustering using hierarchies canopy surface model diameter at breast height DBSCAN density based spatial clustering and noise maximum crown height individual tree detection light detection and ranging QSM quantitative structure model RMSE root means square error terrestrial laser scanning UK EA United Kingdom Environment Agency tree volume Urban districts are often namesakes of the forests they have since replaced; in London for example, Norwood, Oakwood, Colliers Wood and Hainault were all once forests. Although the forest has long been cleared (some remnant individual trees may remain), urban landscapes still incorporate significant trees and areas of woodland as tree-lined streets, public and private gardens and parkland; collectively known as the urban forest. The ecosystem services provided by urban forests have long been recognised [1], for example, mitigating the urban "heat island" effect [2], providing habitat for city dwelling flora and fauna [3] and abating air pollution [4] (although see [5]) as well as aesthetic and well-being benefits [6]. These services have been valued at nearly $1 million km2 per annum [7] and individual urban trees can have a replacement value of up to £450,000 (~ $600,000) [8]. Another important ecosystem service provided by urban vegetation is the sequestration of carbon from the atmosphere. This is absorbed into plant tissue through photosynthesis and stored (sometimes for centuries) in woody tissues as biomass. Urban vegetation plays a disproportionate role in sequestrating anthropogenic carbon emissions as it is proximate to major sources i.e. vehicle emissions, as well as providing shade for buildings which reduce energy consumption [9, 10]. This biogenic sequestration of carbon by urban trees has been valued at £4.8 M ($6.3 M) per annum or £17.80 per tree in Greater London [10] and $2 bn per annum in the USA [11]. Large trees are of particular importance as they have the capacity to sequester more carbon than their smaller counterparts [9, 12]. Currently, however, the contribution of urban forests in the global carbon cycle is given little consideration, owing to their relatively small spatial area in terms of global forest cover [13]. Yet, as urban area is predicted to increase as a fraction of total land cover [14, 15], tools to accurately assess and monitor carbon stored in urban vegetation are required. Particularly as urban vegetation can be highly dynamic e.g. higher mortality [16] and faster growth rates [17] than natural forests, and methods designed for natural ecosystems may not be transferable to urban areas [18]. Above ground biomass (AGB) is defined as "the aboveground standing dry mass of live or dead matter from tree or shrub (woody) life forms, expressed as a mass per unit area" [19], typically Mg ha–1. Urban trees can account for up to 97% of urban AGB [20]. AGB can only be directly measured with destructive harvesting, an expensive and time-consuming approach that precludes remeasurement and is rarely practical beyond a handful of trees. For these reasons, AGB is often inferred through the use of allometric equations that associate more easily-measured parameters, such as diameter-at-breast-height dbh (usually measured at 1.3 m above the ground), tree height e.g. maximum crown height H or projected crown area Ar, with either stem volume V or AGB. To scale up estimates of AGB beyond the tree level, inventory techniques are applied in both traditional forestry and urban studies [11, 20] where a representative sample of trees are measured. However, data acquisition for field inventory can be expensive, time-consuming and is often incomplete e.g. restricted to public lands; large area estimates then rely on scaling factors and land cover maps. Further, inventory data does not provide a spatially explicit map of the tree canopy and its attributes, which is useful for mapping other ecosystem services e.g. habitat extents, pollution dispersal etc. Remote sensing presents an opportunity to capture synoptic, temporally frequent (every few days to weeks), fine spatial resolution data. This has already been widely applied to estimate AGB, across a range of scales, using both active and passive sensors from space based and aerial platforms [21, 22, 23]. In particular, light detection and ranging (LiDAR) techniques provide an unprecedented opportunity to capture high resolution, 3D information on tree and forest structure, such as canopy height, crown size and stem density [24, 25]. LiDAR instruments can be mounted on a range of platforms (hand held, tripods, vehicles, aeroplanes, satellites, etc.) that provide different scales information and detail. Two commonly referred to technologies are terrestrial and airborne laser scanning (aka TLS and ALS respectively); the former provides high fidelity information over a small spatial extents (10's to 100's of metres) whereas the latter offers synoptic data over large regional areas. Both TLS [26, 27, 28] and ALS [23, 29, 30, 31] have been used to estimate individual tree and stand level AGB. Remote sensing methods for estimating AGB can be categorised into (i) area-based and (ii) individual tree detection (ITD) methods, where the latter are considered the state-of-the-art [30, 32]. Area-based methods use summary statistics of canopy structure to develop statistical associations with field inventory data, whereas ITD methods measure crown scale metrics to be used directly with allometry. LiDAR based ITD approaches can be grouped into two further categories dependent on data dimensionality; (i) image analysis of the rasterised canopy surface model (CSM), and (ii) cluster analysis of higher dimension datasets, typically $\mathbb {R}^3$ where the point cloud xyz coordinates are used. Image analysis often detect local maxima within the CSM, followed by expansion or watershed analysis to delineate crowns [16, 33]. Urban areas pose a particular challenge with regard to remote sensing of vegetation, where occlusion by tall buildings, high species diversity and heterogeneous and highly dynamic land cover add complexity to analysis. Tigges and Lakes [34] provide a review of the state-of-the-art of remote sensing to estimate urban AGB. In urban areas, ITD has been achieved by combining ALS with hyperspectral imagery to identify trees [35], tree species [36, 37] and estimate leaf area index [38]. Regarding AGB, ITD has been applied to RapidEye [16] and Quickbird imagery [39] where crowns were subsequently attributed with LiDAR derived H to estimate AGB. Using a solely LiDAR based approach, Singh et al. [40] derived area-based AGB estimates from LiDAR predictor variables. Suggested advantages of a LiDAR derived ITD method to estimate AGB in urban area (as opposed to one from imagery) are (i) LiDAR data are more information rich [41] e.g. 3-dimensional and higher resolution (e.g. > 1 sample m–2), (ii) data is often acquired with greater overlap, including multiple viewing geometries, mitigating occlusion by tall buildings, and (iii) the 3D information inherent in LiDAR data can be used to segment trees based on their morphology as well as directly measure crown shape. A common factor amongst the research discussed above is the use of high pulse density LiDAR data (e.g. > 10 pulses m–2), often acquired with complementary high resolution hyperspectral imagery, acquired over small spatial domains. Recently, government agencies and local authorities world-wide have opened their archives of spatial data, including ALS, under open data licence agreements. Harnessing this freely available resource could allow for large scale maps of urban vegetation attributes, such as AGB, to be computed without the cost of acquisition. Additionally, data is often acquired at regular temporal intervals that would allow for a Life Cycle Assessment of urban AGB [34]. However, a comprise of using these data is that it they are often captured for a different purpose e.g. flood-risk mapping, at a lower resolution and without coincident imagery. Therefore, newly developed techniques have to be adaptable and robust to differences in data quality. As mentioned, allometric equations have long been used to estimate AGB, including in urban forests [9, 18]. However, the reliability of allometry (and it's associated uncertainties) has been questioned owing to small, unrepresentative sample of destructively harvested trees or application outside the domain of observations (particularly diameter and mass) [42]. McHale et al. [18] compared allometry derived from trees grown in natural forest to that derived specifically for urban areas, noting large variability in AGB particularly at the tree scale. Vaz Monteiro et al. [43] computed allometry to estimate H and Ar from dbh for different UK cities; allometry for smaller trees were transferable between cities, whereas larger trees were prone to greater uncertainty. Further, understanding the range of allometric properties of urban trees, which tend to be grown under a wider range of pressures and constraints (water, space etc.) and display greater morphological plasticity (open-grown vs. closed canopy, management etc.), may help better understand the range of allometric variations in natural forests. Recently, TLS methods have developed to accurately estimate the volume of individual trees; an approach known as quantitative structure modelling (QSM) [44, 45]. These methods have been shown to estimate tree AGB to within 10% of destructively harvested trees compared to up > 35% underestimation when applying species specific allometry [26, 27]. Further, as TLS is non-selective of trees captured, the allometry captures a range of structural conditions, including that of large trees. Lefsky and McHale [44] applied this approach to urban trees, reporting good agreement between QSM and field measured stem diameter. Here we demonstrate a multi-scale LiDAR based approach to determine urban tree AGB for the London Borough of Camden, UK (Fig. 1). A new ALS ITD method is presented to identify and attribute individual trees with structure metrics. TLS is used to derive new allometry at four locations across the Borough, transferable tree structure metrics are identified and used to model tree volume. The new allometry is subsequently applied to the ALS segmented tree crowns to generate a Borough-wide map of AGB. To the best of our knowledge, LiDAR based ITD, to derive structural information for use in allometry, has not been previously applied in an urban context. A map of the London Borough of Camden and location in UK (right). Field locations are identified in italics. Contains OS data ©Crown copyright and database right (2018) TLS scanning location and description Leaf status Dominant species 51°31′18.0″N P. acerifolia 0°07′33.5″W Malet Street St Pancras Old Churchyard and Church and Road 0°07′50.5"W Adjacent street Highgate Cemetery F. excelsior The London Borough of Camden is located in inner north west London and comprises an area of 21.8 km2 (Fig. 1). The area was once forested but was extensively developed during the nineteenth and twentieth centuries to a mix of residential and industrial land use. Camden was chosen as it is typical of inner London Boroughs, containing a range of urban land cover types ("unmanaged" urban forest, large managed parks, tree-lined streets, private gardens, industrial areas and transport infrastructure e.g. train lines) encompassing a broad range of tree and forest management strategies, age structures, species composition and municipal functions. Camden also has good coverage of recent UK Environment Agency (UK EA) ALS. The Borough contains the suburbs of Camden Town and Hampstead, large areas of park land, including Hampstead Heath, and a number of smaller public squares and private gardens. The Borough is home to ~ 28,000 street trees with an additional 10–15 K trees in parks and nature reserves [46]; however, this does not include trees located in City of London managed parks as well as other private land. For example, there are an estimated 30 K additional trees on Hampstead Heath in the north of the Borough (pers. comm. David Humphries, Trees Management Officer, City of London). Street tree species are dominated by Platanus x acerifolia (London Plane) 15% and Tilia europaea (Common Lime) 7%; all other species ($N=242$) comprise ≤ 4% each. To derive new allometry for the Borough, four locations were scanned with TLS (Fig. 1 and Table 1). The locations were chosen for their representativeness of park and street trees in Camden, Highgate Cemetery was chosen after preliminary analysis suggested the area contained very high AGB. TLS acquisition and processing TLS was captured with a RIEGL VZ-400 laser scanner (RIEGL Laser Measurement Systems GmbH) which has a beam divergence of 0.35 mrad, a pulse repetition rate of 300 KHz, a maximum range of 600 m and can record multiple returns. For all locations, the scanning resolution was set to an angular step of 0.04° as this has previously proved sufficient for tree extraction and QSM modelling [47]. As the RIEGL VZ-400 captures data in a panoramic field of view (100° in zenith when the scanner is upright), it is necessary to tilt the scanner by 90° to capture the full hemisphere. To capture data from multiple viewing positions and reduce the effects of occlusion, a number of scan positions were captured at each location (Table 2). To co-register scan positions it is necessary to have tie-points between scans that are easily identified in post-processing, here this was achieved using cylindrical retro-reflective targets mounted on poles [47]. Survey pattern was different for each location based upon tree density, leaf status, access and time constraints; mean distance between scan locations are presented in Table 2. Details of TLS scanning Scan positions Mean distance between positions (m) Captured area (m2) Area refers to the convex hull computed for the extracted trees Point clouds from each scan were co-registered using RIEGL RiSCAN Pro software. Individual trees were then identified and extracted using the treeseg software library [48]. V was estimated using the QSM approach of Raumonen et al. [45], where the patch size variable $d_{min}$, which controls the size of cover sets used to generate cylinders (and ultimately the topological detail captured), was iterated over [48]. As the initialisation of each of QSM reconstruction is stochastic, 10 reconstructions for each tree point cloud and for each $d_{min}$ value were generated [26], this resulted in up to 160 reconstructions per tree. The set of reconstructions with the largest value of $d_{min}$ that produced satisfactory results [48] were chosen, from these the reconstruction with a volume closest to the mean was retained. To reduce uncertainty in tree volume and subsequent allometry, point clouds and QSMs had to meet certain quality criteria to be considered for use in allometry development. These criteria were; (i) the mean nearest neighbour distance (computed as the mean Euclidean distance between a point and its four closest neighbours [47]) computed for each 1 m slice through a tree point cloud had to be ≤ 5 cm (excluding the uppermost slice), (ii) the 95% confidence level for the 10 QSM reconstructions for each tree point cloud had to be ≤ 10% of volume, and (iii) the point cloud had to be unaffected by wind i.e. no shadowing of branches visible in the point cloud. The set of trees that fulfilled this criteria, referred to as QSM trees, were used to construct allometric equations (see below). TLS extracted trees could not be reliably mapped to a tree species, instead a mean wood density value for the dominant species on a per location basis (Table 1) were taken from the Global Wood Density Database [49]. ALS acquisition and processing The UK EA capture ALS data over England primarily for flood risk mapping, this is distributed through an Open Government Licence by the UK Environment Agency as 1 km2 .las tiles [50]. Data for the area covering Camden were acquired on 6th February, 2015, at a pulse density of 2 pulses m–2 (calculated as the density of first returns in an open area) where for each outgoing pulse a maximum of 4 returns were recorded. Environment agency LiDAR data are captured to a vertical accuracy of ± 5 cm and a horizontal accuracy of ± 40 cm [51]. Data for the area intersecting the Camden Borough boundary were extracted from the global dataset. 5% of the Borough extent fell outside of the LiDAR footprint, previous UK EA acquisitions have been preprocessed to remove the majority of vegetation returns (Alastair Duncan, UK EA, pers comm) and were therefore unsuitable for filling gaps. Data were ground-normalised using the LAStools lasheight tool [52] so that z values were relative to the ground plane. A filter to remove points where $z \le 1$ m was then applied to remove ground and other low returns. Segmenting trees from Airborne LiDAR Clustering techniques group individual data points into features sets that share some commonality. With regard to LiDAR data, features are often identified as groups of points connected in 3D space, such as street furniture [53] or tree crowns as discussed here. Some techniques require the number of features a priori e.g. k-means clustering, local maxima identified in the CSM are used to prime the algorithms as well as seed points from which clustering is initiated [29, 54]. Examples of cluster approaches that rely solely on the 3D point data included the Mean Shift algorithm [55] which uses a variable kernel to determine the search window size for which points are clustered and PTrees [56] which uses a multi-scale segmentation selecting the most likely segments as crown clusters. However, both of these approaches have only been applied to small forest plots and may not scale to large city-wide datasets owing to their complexity. Here we demonstrate a LiDAR point cloud based clustering approach that identifies individual tree crowns without additional imagery and that is scalable to large urban areas (Fig. 2). Individual tree detection work flow (i–vi) for segmenting ALS data into tree crowns, the bottom panel shows a TLS derived crown map as a comparison. Letters in panels 4 and 5 refer to common issues with the ITD crown segmentation where; A a small crown subsumed into a larger one, B remaining building points increasing crown area, C over segmentation of crowns, D commission errors, E under segmentation of crowns, and F omission errors (particularly of suppressed trees). Presented data is of Malet Street (Table 1) A point cloud D contains points p where $D = \{p^N\}$ and $N = \|D\|$. Each $p \in D$ is a set of coordinates and other metadata associated with the .las format, for simplicity we need only consider $\{\mathbf {a}, rn\}$ where $\mathbf {a}$ = (x, y, z) coordinate vector and rn refers to the "Number of Returns" metafield [57]. The aim is to compute a set of clusters $C = \{c^N\}$ where cluster c corresponds to an individual tree crown. Each cluster $c =\{P, H, Ar, r\}$, where P is the point cloud that corresponds to the tree crown, H is the maximum $p_z \in P$, Ar is the projected crown area calculated as a 2D convex hull $\forall p \in P$ [58] and $r=\root \of {\dfrac{Ar}{\pi }}$, r was derived to simplify regression of crown dimensions with H (see below). As urban areas are a patchwork of buildings, roads, trees, other green spaces etc., not all non-ground LiDAR returns are backscattered from tree crowns; therefore, $D = C + \epsilon$ where $\epsilon$ needs to be filtered before clustering can commence. This was achieved by firstly filtering D so that $\forall p \in D: p_{rn} > 1$ [59, 60]. This step removes the majority of buildings and other hard surfaces, which tend to backscatter a single return i.e. $p_{rn} = 1$ (Fig. 2ii). The majority of remaining points were resultant from vegetation backscatter, as well as from building edges, roof mounted air conditioning units and aerials, cranes etc [60]. This step also vastly reduces data volume, decreasing processing time in subsequent steps. D was segmented into C using a two-step cluster approach. Here we use Density-Based Spatial Clustering of Applications with Noise (DBSCAN) [61] as a low pass filter to identify discrete tree crowns and canopies (Fig. 2iii) followed by Balanced Iterative Reducing and Clustering using Hierarchies (BIRCH) [62] to extract individual trees from canopy segments (Fig. 2iv). DBSCAN and BIRCH were both implemented using Python Scikit-Learn [63]. DBSCAN is suited to ITD from LiDAR point data as (i) \|C\| is not required as an a priori input, (ii) features can be of an arbitrary shape and size, (iii) outliers $\epsilon$ are removed, examples here include linear features e.g. building edges, where points do not fulfil the criteria (i.e. density) to form a cluster, and (iv) efficient scaling to large datasets. Ayrey et al. [64] used DBSCAN to identify and remove understorey shrubs from an ALS dataset captured over a conifer forest. DBSCAN requires two parameters, a neighbourhood radius eps and a minimum number of points min_sample so that c is considered a cluster when $\|c_P\| > min\_sample$ and $p \in c_P$ if $\Vert p - q\Vert < eps$. Values for eps and $min\_sample$ are a function of crown morphology and the ALS point density, $min\_sample$ increases monotonically with eps. If eps is too small, crowns tend to be split into sub-crown components (both horizontally and vertically) as well as an increase in false positive. If eps is too large then features of interest are ignored. Here, eps and $min\_sample$ were set to 3.5 m and 20 points respectively, this allows for smaller features to be identified ($\root \of {\pi 3.5}\approx 38$ m2) where point density ~ 2 points m–2. DBSCAN will concatenate adjacent, or density-connected, points into larger clusters that have a radius $>eps$ [61]. This is desirable as it allows c to have an arbitrary shape and size capturing the idiosyncrasies of a tree crown. However, this behaviour also leads to the merging of c into canopies, where points from adjacent crowns are in close enough proximity (Fig. 2). This is further exacerbated by low LiDAR point density that require lower values of $min\_sample$. BIRCH is therefore applied to further segment the output of DBSCAN into its constituent crowns if: $$\begin{aligned} \beta + \alpha (c_{H}) < c_{r} \end{aligned}$$ where $\alpha$ and $\beta$ were determined empirically from a regression of TLS derived maximum canopy height with the 95${\mathrm {th}}$ percentile prediction interval of crown radius (Fig. 3). Prediction interval was chosen as the dependent variable to avoid segmenting larger crowns. Local and Borough-wide thresholds for initiating BIRCH as well as the Borough-wide $B_t$ regression. Crowns that fall within the shaded area were further segmented with BIRCH BIRCH is a hierarchical clustering algorithm that has two parameters; maximum radius of a cluster $B_t$ (if $c_r > B_t$ the cluster is split) and the total number of clusters $B_N$. $B_t$ was calculated in a similar way to the left hand side of Eq. 1 where instead crown radius was the dependent variable in the regression. $$\begin{aligned} B_t = \beta + \alpha (c_{H}) \end{aligned}$$ Once BIRCH was initiated, it ran as a loop iteratively dividing c into smaller clusters for which $B_t$ was recalculated. Division of clusters ceased when $c_r \ge \beta + \alpha (c_H)$ for all new clusters. For each iteration of BIRCH was run twice; for the first run $B_N$ was not set allowing BIRCH to return a non-optimal set of clusters constrained only by $B_t$. For the second run $B_N$ is set to the number of crowns identified in the first iteration, this producing an optimal segmentation [63]. ALS ITD models were developed using the set of QSM trees from each location ('local') and using all QSM trees ('Borough-wide'). For each model, the functions that were used to split large c and determine $B_t$ were computed as illustrated in Fig. 3. Upscaling TLS volume estimates to ALS Individual tree volume can not be directly measured with low pulse density ALS in a similar way to the TLS methods described above. Instead, ALS derived tree structure metrics are often used to infer volume and AGB. However, regression models computed using a suite of ALS variables can be idiosyncratic and only suitable for the domain in which they were derived [30]. In an urban context, there are a number of different forest types and scenarios which may preclude empirical modelling with multiple parameters. Further, as the aim is to extract and measure individual trees from both TLS and ALS instruments, metrics need to have an analogue for both measurement techniques. Considering these factors, maximum crown height H and projected crown area Ar were used as independent variables in the development of allometric equations [31, 33]. C was computed using the Borough-wide ALS model and exported as polygon vector layer of 2D crown envelopes attributed with Ar and H. Some cleaning was required ($<3\%$ of polygons) to remove duplicate trees (usually vertically offset) as well as false positives e.g building edges, cranes etc., these were easily identified as having maximum crown heights greater than expected. Polygons with an area < 10 m2 were also removed as the tended to coincide with building edges. TLS derived allometric equations were then applied to estimate V for each polygon. To convert V to AGB, an estimate of mean wood density was derived by mapping the trees in the Camden Council street tree database to a wood density value in the Global Wood Density Database [49]. Trees were first mapped at the species level ($N=9526$) and then, if no match was found, at the genus level ($N=10,973$); 287 trees could not be matched at either level and were disregarded. A mean wood density of 537 kg m–3 (s.d. 0.08 kg m–3) was used to convert V to AGB. Allometry uncertainty analysis A Monte Carlo (MC) approach was used to identify and quantify uncertainties in allometry-derived AGB estimates [65, 66]. MC methods allow for complex and non-linear uncertainty to propagate to estimates of AGB. Estimates of uncertainty are computed by running the model N times where for each iteration the model input parameters are drawn from a probability density function (PDF) that characterises the uncertainty. Individual inputs can be also be isolated by freezing the other inputs, allowing for an estimate of their contribution to overall uncertainty. Three potential sources of error were identified in the derivation and application of the allometry: (1) QSM estimates of V, (2) ALS-derived H and Ar, and (3) wood density values. Variability in TLS-derived tree structure parameters (H and Ar) were tested by random subsampling of TLS points clouds ($N=100,$ $\sigma =0.75$); RMSE for H was < 0.05 and < 1.8 m for Ar; therefore, TLS-derived structure was not considered in the MC analysis. QSM uncertainty was estimated on a per tree basis using the 10 reconstructions, the mean and standard deviation of V were used to parametrise a Gaussian PDF. A sample of $c \subset C$ ($N=250$) was used to estimate uncertainty in ALS derived crown structure. $c_P$ were randomly subsampled ($N=100$, $\sigma =0.75$) where H and Ar were calculated for each iteration. The standard deviation of H and Ar were then used to generate PDFs of measurement uncertainty for each extracted crown in C. Finally, a non-parametric PDF of wood density was constructed using wood density values mapped to each tree in the Camden street tree database. For different scenarios, different sources of uncertainty were considered. When computing TLS AGB, wood density values were set to that of the dominant species, therefore, only QSM uncertainty was considered. When calculating ALS derived AGB at each of the TLS locations wood density was again assumed known and uncertainty in QSM and ALS measurements were computed. When computing AGB estimates for the entire Borough all sources of uncertainty were considered. For all scenarios, 100 MC simulations were run. TLS derived tree structure and AGB A total of 385 trees were identified and extracted from the TLS data across the four locations. Of these, 99 trees (referred to as QSM trees) met the criteria for estimating tree volume (Table 3). A large number of trees were discarded from the QSM tree set for reasons including; (i) scanning domain did not cover the complete region of interest, therefore, trees on the periphery suffered from low point density, (ii) scan pattern were too sparse, particularly for St Pancras where leaf-on conditions resulted in high occlusion and low point density towards the top of the canopy and (iii) wind effects. Even light winds can produce "ghosting" in the point cloud which leads to an underestimation in stem volume, particularly towards the top of the canopy where poorly resolved branches are not identified in the QSM (see Fig. 11). Wind was not deemed to significantly impact Ar. Of the QSM trees, the largest by height and volume were both Platanus x acerifolia located in Russell Square (RS-54 and RS-31 in Fig. 4 respectively). TLS measurements provided precise estimates of tree volume, particularly when captured in leaf-off conditions where 95% confidence level in QSM volume $\le 4\%$ (Table 3). Tree form is highly dependent on location and context e.g. trees that are found in street canyons have a strongly asymmetric crown shape (e.g. MS-25 and MS-7 in Fig. 4). Trees also vary in shape when grown in open parkland compared to those found in closed canopy forest, $\overline{Ar}$ is an order of magnitude smaller for closed-canopy forest trees (compare Highgate Cemetery and Russell Square trees in Fig. 4). Summary statistics of the extracted trees are presented in Table 3. Tree structure metrics and AGB estimates generated from TLS Tree height (m) Projected crown area (m2) Tree volume (m3)a AGB (Mg ha–1)c $\rho$ (ha$^{-1}$) $\sum$ Meanb 12.3 ± 4% 201.6 ± 2.2 6.4 ± 3% 244.7 ± 10.5 As the volume of all TLS extracted trees could not be calculated confidently, AGB values are calculated using locally and Borough-wide derived allometry. It should be noted that AGB values are for inter-comparison purposes only and are not representative of likely AGB densities in a 1 ha area, for example, Malet Street is located in a highly developed area which is mostly devoid of trees a Tree volumes are for QSM trees only b 95% confidence level c P. acerifolia wood density value used except for Highgate Cemetery where F. excelsior was more common Profile (left) and plan (right) views of tree point clouds extracted from the TLS data. Tree codes refer to individual trees from Russell Square (RS), St. Pancras (SP), Malet Street (MS) and Highgate Cemetery (HS) Allometry was derived using the set of QSM trees from each location ('local') and all QSM trees ('Borough-wide'). Considering all QSM trees, V and dbh, Ar and $ab^H$ (where $ab^H$ is an exponential function, see Fig. 5) all showed $r^2>0.7$. A multiple linear regression was computed with Ar and $ab^H$ as independent variables ($p<0.001$) which explained 93.9% of variance in V (RMSE = 3.2 m3), the intercept was forced through the origin to avoid negative V for smaller trees. The allometric equation was subsequently applied to the polygon layer to estimate Borough-wide AGB. For the local allometry, $ab^H$ was not a statistically significant term ($p>0.01$). Regression between AGB and dbh (left), H (centre) and Ar (right). The top panel is combined frequency for all locations, the centre panel is regression of independent variable with V and the bottom panel are regression residuals A comparison of TLS and allometry derived V (Fig. 6) shows that local allometry produced more accurate results than the Borough-wide equation (compare Malet Street trees in Fig. 6). The Borough-wide allometry tends to under and overestimate V of large trees ans small trees respectively. Large differences in allometry-derived AGB estimates are evident for Highgate Cemetery (Table 3) where the addition of H in the Borough-wide allometry significantly increases estimated AGB. This is due to the differing crown structure between open-grown and closed-canopy trees, where the former is dominant in the Borough-wide allometry i.e. open grown trees of a similar H have a much greater AGB. A comparison of trees with similar heights (e.g. MS-25 and HC-98 in Fig. 4) reveals that AGB for closed canopy trees can be a factor of ~ 5 less. A comparison of QSM derived and allometry estimated V for the QSM trees. a Allometry was derived for each location ('local') and b using all QSM trees ('Borough-wide'). Horizontal error bars represent the 95th percentile confidence level of tree volume from the 10× QSM model reconstructions and the vertical error bars represent prediction error from the regression. Inset panels magnify V between 0 and 10 m3 As all of the large trees (H > 10 m, $N = 26$) along Malet Street were successfully extracted from the TLS, a direct comparison of QSM computed and allometry estimated volume and AGB can be drawn. QSM derived AGB was 92.5 Mg, compared to local and Borough-wide derived allometry values of 93.8 Mg ± 1.1 Mg and 135.8 Mg ± 2.3 Mg respectively, suggesting allometry for this site overestimates AGB by 1.4 and 46.8% respectively. The overestimate of Malet Street V by the Borough-wide allometry can be seen in Fig. 6b. Applying allometry for P. acerifolia street trees from the US [67] estimates a growing stock volume of 80.5 m3 for Malet Street, compared to 165.6, 172.6 and 231.0 m3 for QSM, local and Borough-wide allometry; highlighting the requirement for caution when applying allometry derived for different circumstances. A comparison of TLS and ALS derived tree structure and AGB Summary statistics of ALS-derived crown metrics for each location are presented in Table 4 and a comparison of crown envelopes produced using TLS and local and Borough-wide ALS models is presented in Fig. 7. Both local and Borough-wide ALS models underestimate AGB by ≤ 25% compared TLS calculated values, where local parametrisation is slightly more accurate. The exception is Highgate Cemetery where AGB is underestimated by up to 55%. Both local and Borough-wide ALS models underestimate $\sum {Ar}$ as they are unable to resolve crown overlap (Fig. 7). When a model underestimates N trees, $\overline{Ar}$ is often overestimated to compensate and vice versa (Table 4). ALS derived crown structure and AGB estimates where N is number of crowns, $\overline{Z}$ is mean height, $\overline{Ar}$ is mean projected crown area, $\sum {Ar}$ is sum of projected crown area ${\overline{Z}}$ (m) ${\sum _{Ar}}$ (m$^{2}$) ${\overline{Ar}}$ (m$^{2}$) AGB (Mg ha–1) 88 (1.44) 15137.9 (0.95) 172.0 (0.70) 189.9 ± 12.1 (0.94) 164.7 ± 9.0 (0.85) 3905.1 (0.80) 96.6 ± 6.7 (0.81) Values in parentheses indicate the fraction compared to TLS estimated values where > 1 suggests an ALS overestimate and vice versa Different parameters sets (derived locally or Borough-wide) were used when segmenting crowns (see Eqs. 1, 2 and Fig. 3) ALS derived tree crown polygons for local (red) and Borough-wide (black) ALS models, compared with TLS derived crowns (grey) At Highgate Cemetery, forest structure is not characterised well with either the local or Borough-wide ALS models. For example, N trees is underestimated by 14 and 64% respectively compared to the TLS estimate and Ar coefficient of variation is ~ 32% for both ALS models, compared to 100% for TLS-derived Ar. Differences between ALS and TLS identified crowns are caused by an uneven age structure of a mix of older trees with large crowns and younger trees filling canopy gaps (Fig. 7). All trees have similar H however, therefore, BIRCH will compute a similar crown radius during segmentation (Eq. 2). Other suggested reasons for poor characterisation include low ALS pulse density not characterising individual crown morphology and a relatively small capture area that compounds scaling errors. Borough wide estimate of AGB Camden has an estimated median AGB density of 51.7 Mg ha–1 (s.d. 68.5 Mg ha–1) and a maximum density of 376.5 Mg ha–1 situated in the Hampstead Heath area (Fig. 8). Maximum values are likely to be an overestimate owing to the poor representation in the allometry as discussed previously. A total of 84,282 individual tree crowns were identified across the Borough, median and maximum tree densities were 36 and 215 trees ha–1 respectively. High AGB areas are concentrated to the north of the Borough (Fig. 8) and are coincident with areas of maximum tree density. ALS-derived tree density values for the forested areas is likely to be an underestimate as TLS estimates for tree count in Highgate Cemetery are 385 trees ha–1 (Tables 3 and 4). Borough-wide maps of ALS derived AGB density (a), tree density (b) and absolute (c) and relative uncertainty (d) Trees in non-forest areas where $10< H < 15$ m account for ≥ 25% of trees and ~ 20% of total AGB (Fig. 9). Trees in forested areas account for 38% of total AGB where forested areas account for $<8\%$ of total land cover. Large trees i.e. trees where H $\ge$ 30 m, account for < 2% of total AGB, these large trees are more common in non-forest areas in the south of the Borough. The tallest and largest volume trees identified in the ALS were 36.0 m and 35.0 m3 respectively, both were located in Gray's Inn Fields. Histograms of tree count (left), sum of crown area (centre) and proportion of AGB (right) as a function of tree height class. Trees have been classified into forest and non-forest using the OSGB forest extent map (see Fig. 1) Uncertainty in AGB can be > 100 Mg ha–1 (95% confidence level); however, greatest uncertainty as a proportion of AGB occurs in areas of low AGB (Fig. 8). MC simulations indicate AGB is estimated to ± 30%, the largest source of uncertainty is wood density which accounts for ~ 65% of overall uncertainty. ALS measurement uncertainty and QSM volume uncertainty account for 30 and 5% respectively. Urban areas as a carbon sink To inter-compare carbon (C) densities with other cities and ecotones, AGB values are converted to C by multiplying by a factor of 0.471 [68]. Median carbon density for Camden is 24.3 Mg C ha–1, this is significantly higher than previously published estimates for inner (16.1 Mg C ha–1) and Greater London (14.8 Mg C ha–1) [10]. The distribution of AGB is likely skewed to the right by an overestimate of "forest" C density calculated with the Borough-wide allometry (Table 3), although Camden does have a greater proportion of park land compared to inner London [69]. For non-forest areas, median C density is 18.9 Mg C ha–1 which is again higher than reported inner London values. The ALS predicted number of trees is much less than the mean value previously reported for London (51 tree ha–1) [10] and the mean value for UK towns (58.4 tree ha–1) [1]; reasons for this include smaller trees being either subsumed into or occluded by larger trees using ALS ITD, whereas the i-Tree Eco and other protocols record all trees where dbh >7 cm [1, 10]. Compared to other UK towns, Leicester has a much higher C density (31.6 Mg ha–1) [20] whereas Edinburgh (16 Mg C ha–1) [70] and Torbay (15.4 Mg C ha–1 [69] are considerably lower. A comparison with other European cities suggests that Camden has a much higher biomass density, for example, Barcelona [71] and Berlin [34] have mean C densities of Berlin 7.3 and 11.2 Mg ha–1 respectively. Lower densities for Berlin could be due to smaller mean tree size where mean tree mass is 372 kg compared to 882 kg in Camden. A comparison with cities globally; major cities in the US have a mean C density of of 7.7 Mg C ha–1 [72] and major Chinese cities have a mean of 21.3 Mg C ha–1 [73]. Considering "woodland" areas, using the locally calibrated TLS data, estimated C density for Highgate Cemetery is 132.4 Mg C ha–1. This compares to Leicester which has a C density of 280.6 Mg C ha–1 for mixed ownership woodland and 287.6 Mg C ha–1 for public ownership [20] which are considerably higher. UK forest and woodlands have a mean density of 53.6 Mg C ha–1 [74]; therefore, forested areas of Camden could be considered AGB "hotspots". In the US, the forests surrounding Seattle have a density of 104 Mg C ha–1 for mixed forest and 166 Mg C ha–1 for conifer forest [75]. US forests have a mean density of 53.5 Mg C ha–1 [76]. A comparison with C sinks from different ecotones is presented in Fig. 10. This shows that, although the contribution of urban areas to global AGB maybe relatively small owing to the limited spatial extent, some urban forests have AGB density comparable to tropical and temperate forests. Therefore the importance of conserving these areas as AGB sinks can not be understated, particularly locally. A comparison of median C density for different ecotones [92] with TLS and ALS derived values for Camden. AGB was converted to C using a conversion factor of 0.471 [68] It should be noted that values presented above were computed using very different data processing and analysis methods which may hinder inter comparison [41]. For example, techniques vary from using ALS (this study), interpretation of satellite imagery [16] or aerial photos [77], field inventory where plots are located per land class [20] or along transects [75]. As a comparison, mean C density for Leicester is estimated as 31.6 Mg ha–1 using a stratified sample of inventory plots in conjunction with published allometry [20]. Applying the method presented here to 2014 UK EA ALS data captured for the same area (and using the Borough-wide allometry) computes a much lower C density of 9.1 Mg ha–1. Using TLS to estimate AGB and derive allometry This study highlights the importance of applying allometric equations in the correct context and with prior knowledge of their derivation. For example, a difference of >200 Mg ha–1 was computed at Highgate Cemetery by applying location specific and Borough-wide (yet still local) allometric equations. A large difference in total V was also noted when applying an equation from the literature [67], compared with local and Borough-wide allometry for Malet Street. Computing locally applicable allometric equations is not always feasible, however, as demonstrated by Calders et al. [26] and Gonzalez de Tanago Menaca et al. [27], as well as here, TLS measurement can be used to derive unbiased allometry quickly and non-destructively. Widely applied allometric equations (e.g. Chave et al. [78]) often include a dbh term, due in part to theoretical scaling laws of tree mass [79] as well as ease of measurement. From an airborne or satellite remote sensing perspective, dbh can only be inferred and is therefore modelled as a function of other variables such as H and Ar [31]. As demonstrated here, a linear combination of $ab^H$ and Ar explained 93.9% variance in V and was therefore suitable for deriving new allometry that excludes a dbh term. Others have also omitted a dhb term, using H and Ar to estimate V and AGB from airborne LiDAR [33, 66]. In fact, both $ab^H$ and Ar explained more variance than dbh for the QSM trees; however, this may be unique to urban trees where tree management e.g. pollarding, may cause deviation from a theoretical ideal. The strong linear association between V and Ar can be explained by the relativity high proportion of V distributed in the tree crown (Fig. 11), particularly for small diameter branches (ø ≤ 20 cm) which can constitute 20–40 % of AGB. Goodman et al. [80] noted a similar trend for trees in tropical forests. Vertical profiles of QSM derived tree volume classified into small (0.05–0.1 m diameter) and large (> 0.1 m) branches. Solid lines ($N_{QSM}$) are produced using QSM trees only, dashed lines ($N_{ALL}$) are for all QSM models (regardless of quality). Number in parentheses are the percentage of total AGB. Branches with a diameter of < 0.05 m were removed from analysis Using the Borough-wide allometry, RMSE for predicted tree level AGB was 1.8 Mg where model residuals show a degree of heteroskedasticity. This is likely due to plasticity in crown shape caused by location (open park land, closed canopy forest, street canyon) as well as factors of competition for space and resources (artificial watering), pollution exposure, management strategies etc. Vaz Monteiro et al. [43] conclude that applying allometry to large trees grown in different locations across the UK results in significant uncertainties. Here, however, error (as a proportion of tree volume) is more evident in smaller trees (AGB < 10 Mg). This is due to taller QSM trees having similar characteristics (open-grown) whereas there a larger number of small trees with a high degree of variability in tree structure. To convert V to AGB requires an estimate of wood density, this represented the largest uncertainty when estimating AGB. Here a mean value was applied to all trees derived from the Camden street tree database. However, in Highgate Cemetery (and most likely other wooded areas) the most common species were Fraxinus excelsior, fortunately this has a similar wood density to the mean of 560 kg m–3 [49]. Fusion of LiDAR and spectral data may allow for more accurate identification of tree species and from which to derive wood density values [34, 37]. Airborne LiDAR to estimate tree volume Considering ITD methods, applicability of either cluster analysis or CSM based methods is likely to be forest type (e.g. tree density) and sensor/data dependent [30, 81, 82, 83, 84]. Currently is dense tropical forests, a CHM approach proved more reliable [30]. However, cluster analysis are increasing in popularity owing to new techniques, increased computing power and instrument capability [48]. A cluster approach was developed here that utilises the unique characteristics of trees when scanned with LiDAR, such as multiple interceptions of LiDAR pulses and predictable tree morphology. An advantage of DBSCAN is that it is responsive to tree morphology without a priori information of canopy structure. BIRCH, on the other hand, segments larger canopy clusters into crowns of similar sizes where H is similar regardless of underlying morphology, this caused errors in the representation of crown structure e.g. Highgate Cemetery (Fig. 7). If higher pulse density ALS was available, the BIRCH step could possibly be replaced by a CSM watershed based approach to identify crown extents from canopy clusters. Regardless, it is suggested that future urban studies first discard points where $p_{rn} = 1$ to facilitate the identification of vegetation. When compared to TLS estimated canopy and crown structure, ALS tended to underestimate crown height and projected crown area (Table 4). Underestimation of H is a common error associated with ALS as pulses often miss the apex of the tree [24], an issue exacerbated by low pulse density. Underestimation of crown area is caused by ALS not being able to delineate overlapping crowns satisfactorily (Fig. 7). Increased crown overlap is common in urban areas owing to tree management practices e.g. closer tree spacing than naturally occurring, reduced resource competition, pollarding etc. Tigges et al. [16] reported an underestimate of tree numbers (~20%) when applying ITD to Rapideye captured over Berlin. Our approach was more accurate for street and park trees (Table 4) as smaller (i.e. Ar < 100 m2) and sub-dominant trees were identified [aided by a winter (leaf-off) ALS capture]. In "forest" areas ALS ITD performed less well, underestimating the number of trees and overestimating their mass. Overestimated mass was caused by under-representation of closed-canopy forest in the Borough-wide allometry. Applying a land-cover classification and computing land-cover specific allometry may reduce errors in AGB estimates; however, errors may be exacerbated by poor classification or land cover definitions. The ALS ITD method satisfactorily identified and attributed individual trees, despite the relatively low pulse density of the data. Maps of individual tree structure are not only useful for estimating AGB, but could also be applied to pollution dispersion [85] and habit extent modelling, for example. The utility of open-access, large area LiDAR datasets is yet to be fully realised for vegetation mapping, particularly LiDAR in urban areas. In England for example, 70% of the land area is covered by airborne LiDAR data (although see earlier comments regarding processing level) with multi-temporal coverage available for certain areas. Recent advances in LiDAR technology, such as the ability to record full waveform backscatter, has also allowed for more accurate mapping of urban vegetation i.e. identifying understorey and suppressed trees [86, 87]. However, full-waveform LiDAR capture at a city wide scale is still experimental, expensive to capture and store and complex to analyse [87]. Alternatively, data fusion of passive (e.g. multi- and hyperspectral sensors) and active sensors (including mobile scanners [88]), as well as inclusion of open source or freely available data (e.g. Google Street View [89, 90]) could be used. Multiple data streams could create a temporally rich analysis that allows for an urban AGB Life Cycle Assessment [34] as well as for application in protocols (i.e. i-Tree Eco protocol [91]) which combine meteorological data with tree structure metrics to determine a suite of ecosystem services. Increasingly, urban trees are being valued for all the ecosystem services they can provide, including as an AGB sink. Although urban areas are currently a small proportion of total land cover, urbanisation is predicted to increase long into the century; therefore, an effective tool set to measure urban AGB, as well as other tree structure metrics, is required. Advances in remote sensing technology are allowing for new methods to more accurately map forest AGB. In particular, LiDAR technologies, both terrestrial and airborne, allow for highly detailed information on tree structure to be derived over large areas, surpassing the capabilities of traditional inventory or image analysis techniques. Urban areas pose particular challenges for remote sensing of tree structure, this is due to a heterogeneous and complex land cover as well as a wide range of potential tree structures. Here we presented methods and results for a new ALS Individual Tree Detection (ITD) method that is robust to a heterogeneous tree layer, allowing attribution of structure metrics from which AGB could be estimated. TLS provides highly accurate representations of tree structure and estimates of volume which were then used to develop local allometry. However, derivation of representative allometry for larger areas, including wood density values, continue to be a major source of uncertainty in estimating AGB, both in natural and urban forest. It should be noted that the ALS and TLS methods can be applied independently of each other, for example, literature allometry could be applied to the ITD method if TLS methods were unavailable. Owing to their proximity and inherent variabilities and idiosyncrasies in tree structure, urban forests provide an excellent testing ground for new methods and technologies to assess tree AGB. PW and MD conceived the project; PW, MBV, KC and MD conducted the field work; PW, AB and MBV processed the data, PW developed the manuscript and all authors contributed to editing. All authors read and approved the final manuscript. The authors would like to thank David Houghton from Camden Council for sharing data and granting access to scan in public parks, Ian Dungavell and Frank Cano for allowing us to scan at Highgate Cemetery and their assistance, David Humpheries for sharing data and Paul Wood (@TheStreetTree) and other Twitter users who crowd sourced the names of once wooded London suburbs. Airborne LiDAR can be accessed from the UK government data portal [50] and terrestrial LiDAR data are available from the authors upon request. A .kml file of tree crown polygons can be accessed with the DOI https://doi.org/10.6084/m9.figshare.5630305.v1. All authors consent to the publication of this manuscript. PW is funded by the NERC National Centre for Earth Observation (NCEO). MD acknowledges the support of the NCEO and funding awards NE/N00373X/1, NE/P011780/1 and NE/K002554/1 as well as support by the EU Horizon2020 project (BACI project funded by the EU's Horizon 2020 Research and Innovation Programme under grant agreement 640176). MBV is funded through Science Without Borders from the National Council of Technological and Scientific Development—Brazil (Process number 233849/2014-9). KC was funded through the Metrology for Earth Observation and Climate project (MetEOC-2), grant number ENV55 within the European Metrology Research Programme (EMRP). The EMRP is jointly funded by the EMRP participating countries within EURAMET and the European Union. AB was funded by NERC award NE/N00373X/1. Britt C, Johnston M. 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Cross Validated is a question and answer site for people interested in statistics, machine learning, data analysis, data mining, and data visualization. Join them; it only takes a minute: Hidden Markov models and expectation maximization algorithm Can somebody clarify how hidden Markov models are related to expectation maximization? I have gone through many links but couldn't come up with a clear view. markov-process expectation-maximization hidden-markov-model thchandthchand The EM algorithm (expectation maximization) is a general algorithm for optimization of the likelihood function in cases where the model is specified probabilistically in terms of an observed and an unobserved (latent) component. HMMs (hidden Markov models) are models of this form because they have an unobserved component, the hidden states, and the actual observations are often called the emissions in the HMM terminology. Hence, HMMs form a class of models for which the EM algorithm can be useful. In generel, if the model consists of two components $(X,Y)$, which we assume take values in a finite space for simplicity, and if the probabilistic model specification consists of the joint point probabilities $p_{\theta}(x,y)$, parametrized by $\theta$, then the likelihood when observing only $X = x$ is $$L_x(\theta) = \sum_{y} p_{\theta}(x,y).$$ Though the sum looks innocent, it is not. For HMMs the sum will be over all possible transitions between the hidden states, which quickly becomes a formidable number when the length of the observed sequence grows. Fortunately there are algorithms for HMMs (forward-backward) for fast computation of the likelihood, and the likelihood could then, in principle, be plugged into any general purpose optimization algorithm for maksimum-likelihood estimation of $\theta$. The alternative is the EM-algorithm. This is an algorithm that iteratively alternates between the E-step, which is a computation of a conditional expectation given the observed $x$ under the current estimate of $\theta$ the M-step, which is a maximization The EM-algorithm makes most sense if the two steps above can be implemented in a computationally efficient way, e.g. when we have closed form expressions for the conditional expectation and the maximization. Historically, the general EM-algorithm is credited to Dempster, Laird and Rubin, who proved in their 1977 paper, among other things, that the algorithm leads to a sequence of parameters with monotonely increasing likelihood values. They also coined the term "EM-algorithm". Interestingly, the EM-algorithm for HMMs was described already in 1970 by Baum et al., and is also often referred to as the Baum-Welch algorithm in the HMM literature (I don't know precisely what Welch did ...). NRHNRH $\begingroup$ Welch invented what is now called Baum-Welch algorithm (he call it "the easy part"); Baum proves mathematically that algorithm works ("the hard part"). See courses.cs.tamu.edu/rgutier/cpsc689_s07/welch2003baumWelch.pdf for exact details. $\endgroup$ – Mikhail Korobov Mar 25 '13 at 23:01 $\begingroup$ @MikhailKorobov, thanks for this informative reference. $\endgroup$ – NRH Mar 26 '13 at 19:38 Expectation Maximization is an iterative method used to perform statistical inference on a variety of different generative statistical models, for example a mixture of Gaussians, and other Bayesian network type models. The only connection is that HMMs are also Bayesian networks. But one would probably not use EM on HMMs because there is an exact algorithm for inference within HMMs called the Viterbi algorithm. So although one could use EM to perform inference on a HMM, you wouldn't because there's no reason to. WilliamWilliam $\begingroup$ This is not entirely accurate because you mix up two different kinds of "inference". EM is an algorithm for estimation of unknown parameters, Viterbi is the algorithm for computing the most probable sequence of hidden states. You would, indeed, use EM for HMMs for parameter estimation. I have given more details on the EM-algorithm with historical references explaining the relation between HMMs and EM in my answer. $\endgroup$ – NRH Sep 14 '11 at 19:39 In HMM, we try to estimate mainly three parameters: The initial state probabilities. This is a vector with $K$ elements, where $K$ is the number of states. The transition matrix. This is a square matrix of size $K\times K$. The conditional probabilities of observing an item, conditioned of some state. This is also a matrix of size $K\times N$, where $N$ is the number of observations. Now, the EM part comes when you try to estimate the quantities/parameters stated above. Starting with some random guess, the likelihood of the observations are evaluated and the parameters are adjusted iteratively until we get maximum likelihood. So, through HMM, we model some process and for that we need to introduce some parameters. To estimate the parameters, EM is rendered. This is a very brief answer. Implementing EM requires a bunch of other sub-problems to solve via a series of techniques. For in depth understanding, Rabiner classic tutorial paper is highly recommended. Riaz KhanRiaz Khan Thanks for contributing an answer to Cross Validated! Not the answer you're looking for? Browse other questions tagged markov-process expectation-maximization hidden-markov-model or ask your own question. Resources for learning Markov chain and hidden Markov models Hidden markov models: output observations defined by a (non-hidden) markov model? Hidden Markov models and anomaly detection Can first-order Markov chain be considered a special case of a hidden Markov model? Use of Hidden Markov Models for Clustering Markov Chain vs Hidden Markov Model Why Expectation Maximization is important for mixture models? What is the difference between a Hidden Markov Model and a Mixture Markov Model? What is an Expectation Maximisation Algorithm for Markov chains? Hidden Markov Models states	CommonCrawl
About IAI Quantitative PCR-Based Competitive Index for High-Throughput Screening of Salmonella Virulence Factors Hyunjin Yoon, Phillipe Gros, Fred Heffron A. J. Bäumler, Editor Hyunjin Yoon Department of Molecular Microbiology and Immunology, Oregon Health & Science University, Portland, Oregon 97239 Phillipe Gros Department of Biochemistry, McGill University, Montreal, Quebec H3G 0B1, Canada Fred Heffron For correspondence: [email protected] A. J. Bäumler DOI: 10.1128/IAI.00873-10 Salmonella enterica serovar Typhimurium is an intracellular pathogen and a main cause of food-borne illness. In this study, a quantitative PCR (qPCR)-based competitive index (CI) method was developed to simultaneously compare the growth of multiple Salmonella strains. This method was applied to a mixture of 17 Salmonella mutants lacking regulator genes, and their survival ratios were compared based on expression of natural resistance-associated macrophage protein 1 (Nramp1). Nramp1, as a major host innate immune component, controls the intracellular replication of pathogens. Deletion strains containing unique DNA barcodes in place of regulator genes were mixed with the parental control, and the bacteria were inoculated into congenic mice differing only at Nramp1. Most of the deletion strains were outcompeted by wild-type bacteria in either mouse strain, and the lack of Nramp1 didn't increase the tested strain/parent control replication ratios. When the same collection of mutants was tested in congenic mouse-derived primary macrophages, a major Nramp1-expressing cell type, six strains (ΔhimD, ΔphoP/phoQ, ΔrpoE, ΔrpoS, ΔompR/envZ, and Δhfq strains) grew better in Nramp1−/− than in Nramp1+/+ macrophages, suggesting that these six regulators may play roles in overcoming Nramp1-mediated bactericidal activity in primary macrophages. The discrepancy in survival of macrophages and that of mice suggests either that there are differences in macrophage populations or that other cell types expressing Nramp1 control Salmonella proliferation in the host. The method described allows competitive infection analysis to be carried out on complex mixtures of bacteria and provides high reproducibility from independent biological replicates. Salmonella enterica serovar Typhimurium, hereafter S. Typhimurium, causes gastroenteritis and a self-limiting disease in humans but a typhoid fever-like systemic disease in mice. S. Typhimurium has been studied as a model for typhoid fever because the host range of S. enterica serovar Typhi is limited to humans. Salmonella has been equipped with a plethora of virulence factors to resist hostile host intracellular milieus. Genes that encode virulence factors are widely distributed around the entire chromosome of pathogenic Salmonella, and their expression is tightly controlled by at least 20 different regulators that sense environmental cues during infection. In a previous study, we identified 17 of 83 regulator genes tested to be required for systemic infection in mice (61). Salmonella strains deleted in those 17 regulator genes were significantly attenuated in virulence in BALB/c mice (61). The virulence phenotype can be influenced by several parameters, including the route of administration, the inoculation dose, the organs examined, and the genotype of the host animal (25, 34, 53, 54). The BALB/c strain used in the previous virulence study lacks a functional Nramp1 protein, a major host innate immune component, and cannot control Salmonella replication, thereby succumbing to low infectious doses (27, 43). We reasoned that the Salmonella regulator mutants that were attenuated in BALB/c mice might exhibit different phenotypes in the presence of Nramp1. Nramp1 (also known as Slc11a1), a highly hydrophobic protein with 12 transmembrane domains, is expressed in cells of myeloid origin and is localized mainly to the phagosomal membrane of macrophages, neutrophils, and dendritic cells (11, 22, 55). This protein is required for resistance against taxonomically unrelated pathogens, including Mycobacterium, Salmonella, and Leishmania (28, 58, 60, 64). The mechanism by which Nramp1 restrains pathogens from proliferating within host tissue cells is likely to be linked to its role as an iron and manganese antiporter, because these are essential nutrients promoting the growth of microorganisms (17, 28, 64). Besides depletion of divalent metals from the phagosomal space, Nramp1 has been reported to exert a variety of other functions. Nramp1 increases major histocompatibility complex (MHC) class II expression and antigen presentation (33, 51) and induces rapid proinflammatory responses such as upregulation of gamma interferon (IFN-γ), interleukin 1β (IL-1β), tumor necrosis factor alpha (TNF-α), and keratinocyte chemoattractant (KC) (31, 32, 46, 55, 56). As well as inducing higher production of cytokines and chemokines, Nramp1 facilitates the formation of reactive oxygen and nitrogen species as an antimicrobial defense mechanism (2, 5, 21). Expression of Nramp1 in macrophages also increases expression of Salmonella pathogenicity island 2 (SPI-2)-associated virulence genes, providing increased bacterial defenses to counteract host immunity (63). To better understand the interaction between Salmonella and host innate immune responses mediated by Nramp1, we compared replication of a variety of Salmonella regulator mutants in mice with or without Nramp1, as well as primary macrophages derived from these same mice. A traditional method to compare growth between wild-type and mutant strains has been the competitive index (CI) assay (3, 18, 52). The conventional CI test is performed by infecting animals or cells with a mixture of mutants and wild-type bacteria that can be distinguished based on specific phenotypic differences. The number of each strain is enumerated in the input inoculum and in the output organ to compare persistence between test strains and the wild-type strain. The competitive index has become a standard for measuring virulence because it is more sensitive than the 50% lethal dose (LD50) assay and less prone to animal-to-animal differences. However, it has several disadvantages, including the excessive animal usage and the limited selection markers between the strains tested. The phenotypic traits, such as antibiotic resistance and metabolic characteristic, must be able to distinguish parent from mutant bacteria without influencing bacterial virulence (3, 18, 52). In this study, we developed a novel competitive index method using DNA barcode-tagged mutant strains, thus enabling us to determine the survival rates of numerous mutants in a single experiment by quantitative PCR (qPCR). Using the qPCR-based competitive index method (CIqPCR), we identified six Salmonella regulator mutants whose growth was more attenuated than that of the wild type in response to Nramp1 in primary macrophages. Bacterial strains and plasmids.All Salmonella strains used in this study are Salmonella enterica serovar Typhimurium 14028s and its isogenic derivatives. Mutant strains with deletions in regulator genes were constructed using modified pKD13 (pKD13-mod) plasmids (pKD13; GenBank accession no. AY048744), which were designed to replace genes of interest with 135-nucleotide (nt) barcode sequences following homologous recombination. Linearized PCR products amplified from a pKD13-mod plasmid contain a kan cassette in the middle and a 40-nt sequence at each terminus. The 40-nt termini are homologous to a gene of interest and facilitate homologous recombination at the correct chromosomal location (15). Replacement of a target gene with a kan cassette was confirmed by PCR. Prior to elimination of the kan cassette, the mutant allele was transferred to a "clean" genetic background by using P22 transduction, and the location of the kan cassette in the transductant was verified by PCR. Expression of FLP recombinase in trans removes the kan cassette via site-specific recombination (15), resulting in in-frame, nonpolar deletions of the target genes. The sequences inserted in place of a target gene are shown in Fig. 1. Regulator genes deleted in this study are listed in Table S1 in the supplemental material. Scar sequences in deletion strains. The 135-nt scar sequences, replacing coding sequences of a target gene, are shown. The scar site or FLP recognition target (FRT) site left after FLP-mediated excision of the kan cassette is highlighted in dark gray. Barcode sequences inserted via SacI and AvrII are composed of 24 random nucleotides and are outlined in the schematic diagram. N, any nucleotide with either a purine or pyrimidine base; V, any nucleotide except thymine-based nucleotides. Sequences recognized by primers scarF and scarR in nested PCR are underlined and indicated by priming sites 4 and 1, respectively. T7 promoter sequences are shown between square brackets. Isolation and culture of bone marrow-derived macrophages from mouse.Bone marrow was extracted from the femurs of 5-week-old female 129SvJ mice as previously described (14) and incubated for 7 days in Dulbecco's modified Eagle's medium (DMEM; Invitrogen), which was complemented with 10% fetal bovine serum (Invitrogen) and 20% L929 cell supernatant containing macrophage colony-stimulating factor. Bone marrow-derived macrophages (BMDM) were counted and 2 × 106 cells were seeded in each well of a 6-well plate 1 day before infection. BMDM were incubated in DMEM containing 10% fetal bovine serum at 37°C with 5% CO2 overnight. CIqPCR in macrophages.Salmonella strains were grown individually in Luria-Bertani (LB) broth overnight and equivalent amounts of each strain were mixed as an inoculum. A mix of bacteria strains was opsonized with 10% mouse serum (Innovative Research) for 20 min prior to infection (8). Bacterial cells were added to BMDM monolayers at an input multiplicity of infection (MOI) of 10, and infection was initiated by centrifugation at 1,000 × g for 5 min. In order to estimate the threshold cycle (CT) value of each strain in the input, the same dose of bacteria mix, which was inoculated into a single macrophage well, was washed with distilled water and used as a template in nested PCR (2 × 107 CFU/PCR). The resulting PCR products were used as template DNAs in quantitative PCR after a serial dilution. Following 30 min of incubation at 37°C with 5% CO2, the medium was replaced with DMEM containing 100 μg/ml gentamicin, and cells were incubated for 1 h to remove extracellular bacteria. After treatment with 100 μg/ml gentamicin, BMDM were washed with PBS twice and overlaid with DMEM containing 20 μg/ml gentamicin for the remainder of the experiment. In order to enumerate intracellular bacteria at appropriate time points after infection, BMDM were lysed with 1% Triton X-100 for 10 min after several PBS washes, and the lysate was spun to collect bacterial cells at 4,500 × g for 5 min. A pellet of bacterial mix from each well was washed three times with PBS and resuspended in 30 μl of distilled water. A bacterial mix from a single well of a 6-well plate was regarded as an individual output sample, and the cells in suspension from a single well (30 μl) were used as a template in nested PCR. Serially diluted nested PCR products were subjected to quantitative PCR to measure the CT value of each strain in the output samples. In a nested PCR of input and output samples, two primers recognizing priming sites 4 and 1 (see Fig. 1), scarF (5′-ATTCCGGGGATCCGTCGACCT-3′) and scarR (5′-GTGTAGGCTGGAGCTGCTCC-3′), respectively, were used to amplify 24-nt barcode sequences from a variety of deletion strains. Nested PCR was performed by one cycle of 95°C for 10 min, 30 cycles of 95°C for 30 s, 50°C for 30 s, and 72°C for 30 s, and one cycle of 72°C for 5 min. Serially diluted nested PCR products were used as template DNAs in quantitative PCR using primer scarF and primers specific to the barcode sequences. Duplex DNA products resulting from quantitative PCR were detected with SYBR green reagent (Applied Biosystems) by using a StepOnePlus real-time PCR instrument (Applied Biosystems). Real-time PCR was carried out by 50 cycles of 95°C for 15 s and 60°C for 1 min, following 95°C for 10 min. In order to normalize CT values, the amplification efficiency of each barcode primer during PCR was calculated from the standard curve of each strain by using serially diluted template DNAs. Mutant strain/wild-type strain survival ratios were quantified using PCR amplification efficiency (E) and CT values of strains in input and output qPCR analyses by using the CIqPCR formula below (42, 62). At threshold, Qwt × $$mathtex$$$E_{\mathrm{wt}}^{CT_{\mathrm{wt}}}$$$mathtex$$= Qmutant × $$mathtex$$$E_{\mathrm{mutant}}^{CT_{\mathrm{mutant}}}$$$mathtex$$(where Q is template DNA quantity and wt is wild type). $$mathtex$$\[\ \mathrm{CI}_{\mathrm{qPCR}}{=}\frac{\frac{(\mathrm{Q}_{\mathrm{mutant}})_{\mathrm{output}}}{(\mathrm{Q}_{\mathrm{wt}})_{\mathrm{output}}}}{\frac{(\mathrm{Q}_{\mathrm{mutant}})_{\mathrm{input}}}{(\mathrm{Q}_{\mathrm{wt}})_{\mathrm{input}}}}{=}\frac{\frac{(\mathrm{E}_{\mathrm{wt}})^{(CT_{\mathrm{wt}})_{\mathrm{output}}}}{(\mathrm{E}_{\mathrm{mutant}})^{(CT_{\mathrm{mutant}})_{\mathrm{output}}}}}{\frac{(\mathrm{E}_{\mathrm{wt}})^{(CT_{\mathrm{wt}})_{\mathrm{input}}}}{(\mathrm{E}_{\mathrm{mutant}})^{(CT_{\mathrm{mutant}})_{\mathrm{input}}}}}\]$$mathtex$$(1) $$mathtex$$\[\ \mathrm{CI}_{\mathrm{qPCR}}{=}\frac{(\mathrm{E}_{\mathrm{mutant}})^{(CT_{\mathrm{mutant}})_{\mathrm{input}}{-}(CT_{\mathrm{mutant}})_{\mathrm{output}}}}{(\mathrm{E}_{\mathrm{wt}})^{(CT_{\mathrm{wt}})_{\mathrm{input}}{-}(CT_{\mathrm{wt}})_{\mathrm{output}}}}\]$$mathtex$$(2) $$mathtex$$\[\ \mathrm{CI}_{\mathrm{qPCR}}{=}\frac{(\mathrm{E}_{\mathrm{mutant}})^{{\Delta}CT_{\mathrm{mutant}}}}{(\mathrm{E}_{\mathrm{wt}})^{{\Delta}CT_{\mathrm{wt}}}}\]$$mathtex$$(3) CIqPCR in mouse.Salmonella strains were grown individually in LB medium overnight as described above and mixed equivalently in phosphate-buffered saline (PBS) based on optical density at 600 nm (OD600) values. A bacterial mixture was washed with PBS and diluted to 105 CFU/ml. Female 4- to 5-week-old Nramp1+/+ 129SvJ (Jackson Laboratory) mice and congenic Nramp1−/− 129SvJ (57, 58) mice were intraperitoneally (i.p.) infected with 100 μl of the mixed Salmonella strains at a final dose of 104 CFU/mouse. The number of injected bacteria was confirmed by plating diluted inoculum on LB agar plates. A portion of the inoculation mixture, corresponding to 2 × 108 CFU, was resuspended in distilled water and used in nested PCR to evaluate CT values of the strains used as the input. At desired time points after infection, mice were euthanized to compare bacterial persistence between strains. The liver and spleen were homogenized and plated on LB agar to isolate intracellular Salmonella. Colonies were scraped and collected in a tube with PBS. In order to assess CT values in the output, a mixture of bacterial cells corresponding to 2 × 108 CFU was resuspended in distilled water and used as the template for nested PCR (2 × 108 CFU/PCR). Barcode DNAs amplified from nested PCRs of input and output samples served as template DNAs in the qPCR step, as above. As an alternative, total DNAs were isolated from the spleen or liver homogenates using GeneElute bacterial genomic DNA kit (NA2110, Sigma) and used as templates in nested PCR instead of isolating bacterial colonies on agar plates. The CIqPCR values were comparable between two methods, although interestingly a larger deviation was observed between specimens when a small piece of liver was applied, as reviewed by Mastroeni et al. (38). Phenotype-based conventional CI.Bacterial strains were cultivated individually overnight in LB prior to infection. Each strain was washed and diluted in PBS at 2 × 105 CFU/ml. The test strain was mixed with a reference strain (MA6054) at a ratio of 1:1, and 100 μl (104 CFU/mouse) of the mixed cells was used to infect Nramp1+/+ 129SvJ mice intraperitoneally. The inoculum, used in infection, was diluted in PBS and spread on plates to enumerate the injected dose of bacteria. The reference strain, MA6054, produces an arabinose-inducible β-galactosidase and can be distinguished from test strains on LB agar containing X-Gal (5-bromo-4-chloro-3-indolyl-β-d-galactopyranoside; 40 μg/ml) and arabinose (1 mM) (26). Infected mice were sacrificed at days 2, 5, and 7 postinfection to isolate the spleens. The spleens were homogenized by mechanical disruption, and the suspensions were plated on LB agar plates with X-Gal and arabinose. The conventional competitive index was then calculated as [number of test strains/number of reference strains]output/[number of test strains/number of reference strains]input (3, 18, 52). Ethics statement.Mouse experiments were approved under the protocol of Oregon Health & Science University Institutional Animal Care and Use Committee (OHSU IACUC; no. A085/2008) and performed in accordance with the guidelines for the Care and Use of Laboratory Animals of the National Institutes of Health to minimize animal suffering. Strategy to tag deletion mutants with 24-nt barcodes.Homologous sequence-mediated recombination methods have been widely used to delete genes of interest in eukaryotes and prokaryotes (12, 15, 16). Using the bacteriophage λ Red (γ, β, exo) recombinase system established by Datsenko and Wanner, a target Salmonella gene is replaced with a linear PCR fragment consisting of an antibiotic resistance gene flanked with 40-nt sequences identical to the 5′ and 3′ ends of the gene to be deleted (15, 16). Once antibiotic-resistant clones are obtained, the antibiotic marker gene is eliminated by FLP recombinase, leaving a scar sequence containing a single FLP recombinase target (FRT). We modified plasmid pKD13, which is one of the PCR template plasmids used in the original λ Red recombination method, for use with our method (15). Two restriction enzyme recognition sites (SacI and AvrII) were added between priming site 1 and the FRT site. Then, double-stranded DNAs containing a T7 promoter and a random 24-nt sequence synthesized without T in the first position of each codon were cloned between the SacI and AvrII sites. More than 150 synthetic DNAs were cloned in the pKD13 derivative plasmid and sequenced to ensure that the insert did not have stop codons or frameshift mutations. Finally, 102 modified pKD13 (pKD13-mod) plasmids harboring unique barcodes were constructed and tested in qPCRs using primers specific to the inserted barcodes. A pool containing equivalent DNA from each pKD13-mod was analyzed in qPCR to examine the efficiency of the barcodes during PCR. Of the 102 pKD13-mod derivatives containing 24-nt barcodes, 88 were found to perform well in test qPCRs, producing comparable CT values at a threshold level. Thereafter, these 88 pKD13-mod plasmids were employed as templates in the construction of deletion strains. The allelic replacement strategy was to replace all codons between the translational initiation codon and the last seven codons with a 135-nt sequence containing a DNA barcode and T7 promoter. The final sequence following FLP-mediated recombination encodes a 45-amino-acid sequence without stop codons that is in frame and therefore not likely to be polar upon expression of downstream genes in the same operon. The T7 promoter upstream of the barcode sequences can be used to identify strains using in vitro transcription from the T7 promoter. Small labeled RNAs produced via AvrII digestion followed by in vitro transcription can be applied to microarray hybridization for quantification of strains. An additional advantage of this approach was the ability to leave the last 7 amino acids of the deleted gene to avoid the frequent problem of overlapped gene coding sequences (B. L. Wanner, personal communication). The scar sequence remaining after FLP-mediated excision of the kan cassette is shown in Fig. 1. Using the pKD13-mod library with a variety of barcodes, genes of interest were deleted by λ Red recombination and replaced with a 135-nt sequence, including a unique barcode. For the parental control in competitive infection studies, we inserted a barcode in the pseudogene STM0314, which did not affect Salmonella virulence (see Fig. S1 in the supplemental material). Calculation of the competitive index utilizing quantitative PCR in mixed populations.Quantitative reverse transcription-PCR (qRT-PCR) has been a strong tool for measuring transcription levels and expression changes of genes of interest (40, 51, 61). We applied qRT-PCR to a mixed bacterial population to enumerate each bacterial strain and compare growth and survival between strains during a single infection. Each mutant strain was distinguished via a 24-nt barcode with a specific primer in the mixed population. The quantity of PCR products is theoretically proportional to the quantity of initial template DNAs under the exponential phase of increase, when PCR reagents are not limited (24). The amounts of PCR products are deducible from the initial template quantities and amplification efficiencies of barcode primers and are equivalent between strains at a threshold level. If the PCR efficiency is ideal, the amount of PCR product will double for each cycle during the exponential phase of PCR amplification (37). However, due to differences in primer specificity among the barcode sequences, the amount of PCR products will not be increased twice every cycle (29, 44). Therefore, it was necessary to take account of the PCR efficiency of each barcode primer in enumeration by using RT-PCR (44, 62). The relative ratio of a mutant strain to the wild-type strain was determined from the efficiency-calibrated mathematical model, which has been broadly used for relative quantification in RT-PCR (42, 62). To calibrate CT values between strains, the amplification efficiency (E) of each barcode primer was determined using the slope of the standard curve of each strain, as demonstrated in Fig. S2 in the supplemental material (9, 42). As shown in the equation in Materials and Methods, a competitive index formula using qPCR (CIqPCR) was computed based on PCR amplification efficiency and CT values of strains in input and output samples. In order to rule out the possibility of cross-reactivity between different barcode primers, a bacterial mixture composed of equivalent amounts of 8 barcode-tagged strains was subjected to qPCR-based relative quantification using 8 cognate barcode primers, specific to the 8 mixed strains and 8 unrelated primers, chosen at random but not specific to the 8 test strains (see Fig. S3 in the supplemental material). The unrelated barcodes showed a ΔCT value (reference barcode CT − test barcode CT) of less than −15 cycles (−25.6 ≤ ΔCT ≤ −15.2), whereas barcodes used in the mixed strains exhibited CT values similar to that of the reference barcode (see Fig. S3 in the supplemental material), indicating that these 8 unrelated barcodes gave a signal that was essentially undetectable compared to the signal from relevant barcodes. Description of the CIqPCR method.A flowchart of the CIqPCR method in mice is shown in Fig. 2. A reference strain (the ΔSTM0314 strain) and deletion strains, each with a unique barcode, were cultivated individually prior to infection. Strains were mixed equally in the inoculum and 104 CFU of the mixed cells were intraperitoneally administered to each mouse. The inoculum was subjected to qPCR-based quantification to calculate initial CT values (CTinput) for each strain. At day 5 postinfection, the liver and spleen were harvested, homogenized, and plated on agar plates to grow the intracellular bacterial population. Bacterial strains were collected and combined, and the mix was used to measure CToutput values of the deletion strains. CTinput and CToutput values were entered in the CIqPCR equation to assess the competitive indices of the strains tested. As an alternative to plating the spleen and liver homogenates, total DNA was prepared from the infected organs and used as a template in quantitative PCR. Similar results were observed with both methods. Schematic of the qPCR-based CI method in a mouse model. Bacterial strains were cultivated separately overnight (1) and mixed equivalently based on OD600 measurement (2). Mixed cells corresponding to 2 × 108 CFU were used directly in nested PCR to amplify the barcode sequences from the input inoculum (3). Product DNAs from nested PCR were used as templates in the following input qPCR (4). In order to compare survival rates between strains, mice were i.p. injected with 104 CFU consisting of an equal mix of each of the mutant strains and the parent strain (5). The inoculum was verified by plating and counting the number of CFU. The mice were euthanized to isolate Salmonella-infected organs at the desired time point after infection (6). Organs were homogenized and spread on LB agar to grow bacteria (7). Bacterial colonies were scraped and collected the next day. The mixture was diluted in PBS buffer, and 2 × 108 CFU were used as the template for nested PCR (8). Bacterial strains from the output populations were analyzed by qPCR using barcode-specific primers (9). CIqPCR was calculated by comparing CT values between a wild-type strain and a mutant strain in input and output qPCRs, taking into account that the amplification efficiency of individual barcodes ranged from 1.53 to 1.94. The quantitative PCR was performed using a two-step PCR procedure described in detail in Materials and Methods. In an initial nested PCR step, two outside primers corresponding to either end of the 135-bp scar sequences were used to amplify the barcode sequences. The products, each of which had the same ends but a unique barcode in the middle, were serially diluted and used as templates in a second round of amplification: qPCR in which one of the outside primers and the barcode-specific primer were used. By using nested PCR rather than qRT-PCR directly on the mixture, the specificity was greatly increased, as has been shown in other studies and observed in our laboratory (1, 23, 49). The reliability of the CIqPCR assay was validated by comparison with the traditional CI method (Fig. 3). Four strains, including a reference strain (the ΔSTM0314 strain) and three barcode-labeled strains (ΔSTM2209, ΔSTM3096, and ΔSTM4333 strains), were mixed equivalently, and the bacteria were inoculated intraperitoneally into mice (104 CFU/mouse). Intracellular bacteria residing in the spleens were enumerated using CIqPCR at days 2, 5, and 7 postinfection. In parallel with the CIqPCR assay, each deletion strain was mixed with a reference strain (MA6054 [26]), which has an arabinose-inducible β-galactosidase gene, at a 1:1 ratio, and 104 CFU of the mixed bacteria was inoculated into each mouse. Mice were sacrificed at days 2, 5, and 7 postinfection, and the spleen homogenates were plated on LB agar with X-Gal and arabinose. The conventional CI was calculated using the equation described in Materials and Methods. The CIqPCR assay exhibited results similar to those of the conventional CI assay. ΔSTM3096 and ΔSTM4333 strains showed a tendency to decrease more in the CIqPCR assay than in the traditional CI. This difference may be due to the lower bacterial number for each strain at inoculation in the CIqPCR assay (2.5 × 103 CFU in CIqPCR versus 5 × 103 CFU in CI), which may be cleared faster by the host immune system. However, the differences were not statistically significant (P values were all >0.15). Comparison of CIqPCR with the traditional CI. In the CIqPCR assay, a reference strain (the ΔSTM0314 strain) and three deletion strains (ΔSTM2209, ΔSTM3096, and ΔSTM4333 strains) with barcodes were mixed equivalently, and the mix was used to infect a group of three Nramp1+/+ 129SvJ mice at 104 CFU/mouse. For the traditional CI infection, a mixture containing a reference strain (MA6054) and each single strain at a 1:1 ratio was used to infect a group of three Nramp1+/+ 129SvJ mice at 104 CFU/mouse. Mice were sacrificed at days 2, 5, and 7 postinfection, and the intracellular bacteria residing in the spleens were enumerated using the traditional CI formula or CIqPCR formula as described in Materials and Methods. There was no statistically significant difference between the two methods based on Student's t test. Survival of Salmonella mutants lacking virulence regulators in mice (Nramp1+/+ and Nramp1−/− 129SvJ).In order to confirm the validity of the CIqPCR method as a high-throughput screening tool, the persistence of 17 mutants lacking regulator genes was compared within Nramp1+/+ and Nramp1−/− mice using quantitative PCR. Our previous work identified 17 regulators required for growth in BALB/c mice by either intragastric (i.g.) or intraperitoneal (i.p.) infection (61). While these regulators were clearly required for systemic infection in BALB/c mice, which is Nramp1−/−, we thought it possible that some of these mutant strains might have a differential effect on survival of 129SvJ Nramp1+/+ and 129SvJ Nramp1−/− mice. The 17 regulators required for survival in Nramp1−/− BALB/c mice are SpvR, FruR, YbdM, HimD, PhoP/PhoQ, SsrA/SsrB, SlyA, Hnr, STM2575, RpoE, SmpB, CsrA, STM2912, RpoS, Crp, OmpR/EnvZ, and Hfq (see Table S1 in the supplemental material). Two genes (STM2575 and STM2912) annotated as regulators but not characterized further were included. Genes encoding these 17 regulators were replaced with unique barcode DNAs listed in Table S2 in the supplemental material. A mixture containing equal amounts (around 556 CFU/strain) of the 17 mutants and the reference strain (the ΔSTM0314 strain) was inoculated i.p. into mouse strain 129SvJ with or without Nramp1 at 104 CFU/mouse. Bacterial cells in the input inoculum and the output splenic homogenates were subjected to a qPCR-based CI test as described above, and the survival rate of each strain was enumerated at day 5 postinfection (Fig. 4). Considering that a CIqPCR value of 1 indicates a comparable growth between the parent strain and a mutant, most of the deletion strains showed growth attenuation in both congenic Nramp1+/+ and Nramp1−/− mice, which is consistent with our single-infection results (see Table S1 in the supplemental material) (61) and the results reported elsewhere by other investigators. Some mutant strains, including ΔspvR, ΔhimD, ΔSTM2912, ΔrpoS, and ΔompR/envZ strains, were outcompeted by wild-type bacteria more strongly in Nramp1−/− mice than in Nramp1+/+mice. The mutant strains with lower CIqPCR values in Nramp1−/− mice are likely to be attenuated in intracellular replication independent of Nramp1, whereas the wild-type reference bacteria proliferate better in Nramp1−/− mice. CIqPCR analysis of 18 strains in Nramp1+/+ and Nramp1−/− mice. Groups of five mice (Nramp1+/+ and Nramp1−/− 129SvJ) were infected with equal mixtures of 17 strains, containing mutations in the genes indicated, as well as the wild-type control. The total number of infecting bacteria was about 104. The spleens were extracted at day 5 postinfection, and persistence levels of Salmonella strains in the spleen were compared via qPCR using the formula shown. CIqPCR values were averaged from the five mice in each group and shown in white (Nramp1+/+) and gray (Nramp1−/−). Strains with CIqPCR values greater than 1 indicate that they outcompeted the wild-type strain; strains with values less than 1 indicate that they were outcompeted by the wild-type strain. A Student t test was applied for statistical analysis of the results, and strains showing significant changes between Nramp1+/+ and Nramp1−/− mice (P < 0.05) are labeled with an asterisk. Virulence-attributed phenotypes can vary depending on the origin of host cells (7). After i.p. administration into host animals, Salmonella migrates quickly to the filtering organs, the spleen and liver (45, 47). To assess whether the survival and growth of the regulator mutant strains differed in the spleen and liver, the same mixture of 17 deletion strains and a reference strain was injected into Nramp1+/+ or Nramp1−/− mice, and the persistence in the spleen was compared to the persistence in the liver in each mouse strain (see Fig. S4 in the supplemental material). With the exception of 6 strains (ΔphoP/phoQ, ΔssrA/ssrB, ΔslyA, ΔcsrA, ΔSTM2912, and Δhfq strains), the regulator mutants exhibited similar growth profiles between the two organs, suggesting that the spleen and liver provide a similar milieu to the survival of the tested strains. Notably, the ΔssrAB strain had a lower CIqPCR in Nramp1−/− liver than in Nramp1+/+ liver but comparable values in the spleen. Overall, strains (including ΔhimD, ΔslyA, Δhnr, ΔrpoE, ΔcsrA, and Δhfq strains) appeared to be more defective for growth in the liver than in the spleen. The lower CIqPCR in the liver might be attributable to a more restrictive environment in the liver for the growth of some mutants, as observed in the infection with Listeria monocytogenes (10, 13). The more significant attenuation of the ΔslyA strain in the liver has been noted before and is consistent with our observation (35). Survival of Salmonella mutants lacking virulence regulators in primary bone marrow-derived macrophage from Nramp1+/+ and Nramp1−/− 129SvJ mice.Comparing the persistence of Salmonella mutants lacking regulators in Nramp1+/+ and Nramp1−/− mice to that of the parent control, we did not find any regulator mutants that survived better in Nramp1−/− than in Nramp1+/+ mice (see Fig. 4). Most of the deletion strains were defective for growth in either mouse strain. After phagocytosis of pathogenic bacteria, Nramp1 is targeted to the membrane of the pathogen-containing phagosome in macrophages, neutrophils, and myeloid-derived dendritic cells (11, 22, 48, 55). In order to further investigate the effects of the 17 regulators on Salmonella survival in the presence of Nramp1, we also performed the assay in a cell culture model of infection. BMDM were prepared from both Nramp1+/+ and Nramp1−/− 129SvJ mice and infected with equal mixtures of the 17 deletion strains and the parent strain at an input MOI of 10 to 1. At 30 min, 6 h, and 18 h after infection, primary macrophages were lysed and the intracellular bacteria were subjected to two-step qPCR to enumerate each deletion strain (Fig. 5; see also Fig. S5 in the supplemental material). CIqPCR values at 18 h postinfection are shown in Fig. 5. Some Salmonella strains, including the ΔhimD, ΔphoP/phoQ, ΔrpoE, ΔrpoS, ΔompR/envZ, and Δhfq strains, survived better (i.e., a higher CIqPCR value) in Nramp1−/− macrophages than in Nramp1+/+ macrophages, indicating that these strains were more sensitive to the effects of Nramp1. Additionally, Salmonella strains not expressing SlyA, RpoE, CsrA, and Hfq were attenuated more than 5-fold compared to the parent strain in both Nramp1+/+ and Nramp1−/− mouse-derived cells, implying significant roles of SlyA, RpoE, CsrA, and Hfq during bacterial survival in macrophages. However, two mutant strains, the ΔspvR and ΔfruR strains, replicated significantly better than a wild-type strain in both Nramp1+/+ and Nramp1−/− macrophages. Different phenotypes between macrophages and animal models may be attributed to the function of other cell types in controlling replication of pathogenic bacteria in the whole animal, as described further in Discussion. CIqPCR analysis of 18 strains in Nramp1+/+ and Nramp1−/− BMDM. Equal mixtures of the deletion strains indicated above were used to infect Nramp1+/+ and Nramp1−/− bone marrow-derived macrophages that were approximately 50% confluent at an input MOI of 10. Macrophages were lysed at18 h postinfection, and the survival ratios of Salmonella mutants to the reference strain were compared as described in Materials and Methods. CIqPCR values from three independent BMDM infections were averaged, and CIqPCR in Nramp1+/+ (white) and Nramp1−/− (gray) are shown. Strains with CIqPCR values greater than 1 indicate that they outcompeted the wild-type bacteria in survival; strains with values less than 1 indicate that they were outcompeted by the wild type. Mutant strains showing significant differences in CIqPCR values for Nramp1+/+ and Nramp1−/− cells are denoted by an asterisk (P < 0.05 in the Student t test). In this work, we demonstrate the efficacy of a novel qPCR-based CI method to distinguish the effect of Salmonella regulators in the presence or absence of Nramp1. By tagging mutant strains with unique DNA sequences, we were able to evaluate growth of multiple strains by quantitative PCR. Using far fewer mice than what would be required for LD50 experiments, the qPCR-based competitive index method was robust and resulted in reduced variation between animals (average standard deviation of 0.41 among mice and 0.36 among macrophage cultures). As in other high-throughput screening methods, the CIqPCR method required several parameters to be optimized for successful reproducibility in an animal model inoculated with the same pool of mutant strains. Using too high of an inoculum may overwhelm the host immune response, quickly killing the mice and leading to the growth of avirulent mutants that could otherwise be attenuated. However, using too small of an inoculum may result in complete clearance of individual mutants, leading to a spuriously low competitive index. Higher-dose inoculation has been suggested to obtain consistent phenotypic characteristics of mutant strains unless it causes a physical burden irrelevant to phenotypic results (39). We used a mixture of 104 CFU as an inoculum in Nramp1+/+ or Nramp1−/− mice. The larger the number of strains to be tested together, the smaller the inoculum of any single mutant. Consistent results were obtained regardless of the number of tested strains when the number of mixed strains was between 2 and 36 in the inoculum of 104 CFU (data not shown). However, titration of optimal inoculum should be carried out depending on the genotype of the host animal, the route of administration, and the period of infection. Some mutant strains are attenuated for growth in intragastric infection but not in intraperitoneal infection or vice versa (see Table S1 in the supplemental material). Bacterial persistence profiles can also be influenced by the infection period as well. We observed that some Salmonella mutant strains (ΔSTM2281, ΔhilD, and ΔbarA strains) were attenuated at day 7 postinfection but were comparable with a wild-type strain at day 2 postinfection (data not shown). A short infection period may not provide a sufficient time for some mutants to exhibit their phenotype if the deleted genes play roles in long-term systemic infection in the host, such as resistance to the adaptive immune system. The composition of strains in the mixed population might also affect their survival phenotypes. The possibility of trans complementation has been an issue in previous coinfection experiments, in which one bacterial mutant could complement another during infection of the same mouse. Trans complementation is more likely to occur in high-dose inoculations where multiple bacteria may infect a common host cell. Recently, outer membrane vesicles have been highlighted due to their role in transferring virulence factors between adjacent bacteria and even between host cells (6, 30, 59) (Yoon et al., submitted for publication). Regulator mutants defective in the expression of virulence factors, which are secreted by outer membrane vesicles, might be complemented, at least in principle, by vesicles produced from coinfecting parental bacteria. However, in comparing individual infections to infections with a mixture, we have not yet observed the possibility of trans complementation. The importance of dose, combination and ratio of strains, infection time, site of recovery, and infection route has been previously emphasized in mixed infections using two or more strains (4, 53). Nramp1 is expressed exclusively in dendritic cells, macrophages, and neutrophils—all cells of myeloid origin (22, 48, 51). Comparing bacterial survival between Nramp1+/+ and Nramp1−/− macrophages, we identified 6 regulators (HimD, PhoP/PhoQ, RpoE, RpoS, OmpR/EnvZ, and Hfq) whose absence more strongly attenuated Salmonella survival in Nramp1+/+ than in Nramp1−/− macrophages. These 6 regulators may regulate virulence factors necessary for resistance to Nramp1-mediated bactericidal activities within macrophages. Salmonella has been reported to increase expression of Salmonella pathogenicity island 2 (SPI-2)-associated virulence genes in response to Nramp1, presumably as a bacterial defense (63). In a previous study, we observed that these 6 regulators coordinately activated SPI-2 genes under acidic minimal media that partially mimic the macrophage intracellular milieu (61). Accordingly, these 6 regulators may have a role in the induction of SPI-2 expression responding to Nramp1. In the acute mouse infection model, Salmonella cells are rapidly disseminated to the spleen and liver and are present in several cell types (19). However, the survival phenotype within macrophages has been regarded as a barometer to infer virulence in the animal; therefore, we anticipated that there would be little difference between Nramp1+/+/Nramp1−/− mice and primary macrophages derived from the same strains of mice. Interestingly, Salmonella strains with deletions in spvR, himD, phoP/phoQ, ssrA/ssrB, smpB, crp, and ompR/envZ were significantly attenuated in mice at day 5 postinfection (Fig. 4) and at a shorter infection time (less than 4 days [data not shown]) but not in bone marrow-derived macrophages (Fig. 5). This discrepancy might be attributable to several possibilities. The microbicidal activity of macrophages may vary between the sources, as described in reference 36. Buchmeier and Heffron reported that the survival of Salmonella mutant strains is influenced by the origin of macrophages (7). Peritoneal macrophages are more microbicidal than splenic and bone marrow-derived macrophages toward Salmonella (7). Salmonella strains inoculated into the peritoneal cavities of the mice appear to be cleared more efficiently than Salmonella in primary macrophage infections. Differential survival within other tested organs has been observed in infection with a variety of pathogenic bacteria, and thus the target organ has been another parameter in considering the survival ability of a mutant in the host (7, 10, 13, 20). Another possible explanation for the discrepancy in survival between macrophages and mice is the contribution of cell types other than macrophages in controlling bacterial proliferation. Geddes et al. determined the cell types targeted by Salmonella using flow cytometry and found that ∼55% and ∼18% of the infected cells were neutrophils and monocytes, respectively, whereas macrophages containing Salmonella were hardly detected (19). Furthermore, Salmonella was able to replicate intracellularly within neutrophils despite the short half-life and bactericidal properties of these cells. Surprisingly ∼23% of infected splenic cells were B and T cells (19). Dendritic cells play a role in shuttling Salmonella across the intestinal epithelial barrier and are the only cell type capable of stimulating naïve T cells (41, 50). Recently, it was reported that Nramp1 expression is increased in intestinal, splenic, and bone marrow-derived dendritic cells upon Salmonella infection, although the increased expression did not trigger increased clearance of bacteria (55). Based on these diverse locations of Salmonella, the roles of Nramp1 in other phagocytic cells needs to be defined to understand the interaction between Nramp1 and Salmonella. Survival in cells other than macrophages may be important for Salmonella to either manipulate host cellular functions or subvert host immune responses and cause systemic disease. In conclusion, our results demonstrate that the CIqPCR method is a novel technique that can be employed to enumerate multiple strains in mixed pools of bacteria in a short time within the same animal, decreasing animal-to-animal variation. This new approach to investigating the role of bacterial regulators is applicable to other pathogenic bacteria and will help elucidate how virulence is coordinated in relation to specific host factors. 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Chandran, … Kishore Kumar Deepak Harutoyo Hirano ORCID: orcid.org/0000-0002-0634-18051, Renjo Takama2, Ryo Matsumoto2, Hiroshi Tanaka2, Hiroki Hirano2, Zu Soh ORCID: orcid.org/0000-0001-7182-93833, Teiji Ukawa4, Tsuneo Takayanagi4, Haruka Morimoto4, Ryuji Nakamura ORCID: orcid.org/0000-0001-8074-47875, Noboru Saeki ORCID: orcid.org/0000-0002-1967-08785, Haruki Hashimoto6, Shogo Matsui6, Shinji Kishimoto ORCID: orcid.org/0000-0002-5109-78057, Nozomu Oda6, Masato Kajikawa7, Tatsuya Maruhashi6, Masashi Kawamoto5, Masao Yoshizumi8, Yukihito Higashi7,9 na1 & Toshio Tsuji3 na1 Scientific Reports volume 8, Article number: 9263 (2018) Cite this article Diagnostic markers This paper proposes a novel non-invasive method for assessing the vascular endothelial function of lower-limb arteries based on the dilation rate of air-cuff plethysmograms measured using the oscillometric approach. The principle of evaluating vascular endothelial function involves flow-mediated dilation. In the study conducted, blood flow in the dorsal pedis artery was first monitored while lower-limb cuff pressure was applied using the proposed system. The results showed blood flow was interrupted when the level of pressure was at least 50 mmHg higher than the subject's lower-limb systolic arterial pressure and that blood flow velocity increased after cuff release. Next, values of the proposed index, %ezFMDL, for assessing the vascular endothelial function of lower-limb arteries were determined from 327 adult subjects: 87 healthy subjects, 150 subjects at high risk of arteriosclerosis and 90 patients with cardiovascular disease (CAD). The mean values and standard deviations calculated using %ezFMDL were 30.5 ± 12.0% for the healthy subjects, 23.6 ± 12.7% for subjects at high risk of arteriosclerosis and 14.5 ± 15.4% for patients with CAD. The %ezFMDL values for the subjects at high risk of arteriosclerosis and the patients with CAD were significantly lower than those for the healthy subjects (p < 0.01). The proposed method may have potential for clinical application. Arteries are composed of three layers: the tunica aventitia, the tunica media and the tunica intima. The vascular endothelium is the innermost layer of the tunica intima, and consists of monostratal vascular endothelial cells that produce various vasodilators such as nitric oxide (NO) to adjust for the coarctation of vascular vessels and maintain flexibility1. When vascular endothelial function deteriorates as a result of risk factors such as diabetes and hypertension, the flexibility of vascular vessels becomes impaired and arteriosclerosis develops2,3. Arteries in the lower-limbs are sites of predilection for peripheral artery disease (PAD), which is a type of advanced arteriosclerosis4 caused by vascular endothelial dysfunction and can result in subcutaneous tissue necrosis if treatment is delayed. Arteriosclerosis of the lower-limb arteries is reportedly related to the progression of systemic cardiovascular conditions such as coronary arterial disease and carotid stenosis5. Accordingly, quantitative assessment of vascular endothelial function in these arteries may support the early recognition and treatment of arteriosclerosis. Numerous methods for assessing vascular endothelial function in lower-limb arteries have been proposed. As an example of an invasive approach, blood flow variations were measured using plethysmography after the patient was given a NO agonist drug or a NO antagonist drug6,7. However, invasive methods carry risk, cause patient discomfort and are problematic to apply because a NO agonist drug or a NO antagonist drug must be delivered arterially via a catheter. Meanwhile, flow-mediated dilation (FMD) is often used as a non-invasive method for assessing upper-limb vascular endothelial function8,9. FMD allows vascular endothelial function evaluation based on vascular diameter variations measured with an ultrasound device. In FMD testing, shear stress caused by increased blood flow after arterial avascularization and release applies a stimulus to the vascular endothelium, which in turn produces NO. However, the test requires an ultrasonologist with a high level of technical ability for accurate measurement of the patient's vascular diameter accurately over an extended period, and its results depend on the patient's blood pressure10. In addition, it is difficult to accurately measure the vascular diameter of lower-limb arteries with an ultrasound device because they are located deep under the skin and ultrasonic signals do not propagate well through biological tissues. It is therefore technically challenging to apply FMD testing to lower-limb arteries. With this in mind, the our research group proposed a novel method for determining the vascular endothelial function of upper-limb arteries easily without the use of an ultrasound device11,12,13. This technique, known as the enclosed zone FMD (ezFMD), involves the application of oscillometry (as used with commercial automated sphygmomanometers at patients' homes and in hospitals) rather than vascular diameter measurement using an ultrasound device. In the ezFMD approach, plethysmograms are extracted using an air cuff attached to an upper limb, and the vascular endothelial function of the upper-limb arteries is assessed by calculating the difference in the maximum amplitudes of the extracted plethysmograms before and after cuff occlusion. The ezFMD is a noninvasive and simple method for assessing endothelial function compared to the FMD. Our previous studies showed that the ability of ezFMD to assess endothelial function at the upper arm was equal to or greater than that of FMD at the forearm and has good potential for screening with minimal technical requirements to assess endothelial function because there was wide variability in both ezFMD and FMD12,13. However, the ezFMD method has not been applied to lower-limb arteries, which are sites of predilection for arteriosclerosis. Therefore, the authors propose a novel noninvasive and easy method for assessing the vascular endothelial function of lower limb arteries based on the ezFMD approach, which does not use an ultrasound device. A total of 327 adults (236 males, 91 females; mean age ± S.D.: 55.9 ± 21.2 yrs) were used for the lower-limb ezFMD measuring experiment. The subject breakdown is as follows: 87 healthy subjects (78 males and 9 females; mean age ± S.D.: 30.8 ± 14.9 yrs), 150 subjects at high risk of arteriosclerosis (90 males and 60 females; mean age ± S.D.: 62.0 ± 16.2 yrs) and 90 patients with cardiovascular disease (CAD) (68 males and 22 females; mean age ± S.D.: 70.2 ± 10.3 yrs). Healthy subjects had no history of cardiovascular diseases, liver diseases, renal diseases, autoimmune diseases, or malignant diseases and had no coronary risk factors, including hypertension, dyslipidemia, diabetes mellitus, and current smoking. The subjects at high risk of arteriosclerosis were defined as people who were affected by one or more of the following: hypertension, diabetes, dyslipidemia, or current cigarette smoking. Hypertension was defined as systolic blood pressure of >140 mmHg or diastolic blood pressure of > 90 mmHg, in a sitting position, on at least three different occasions14. Diabetes mellitus was defined according to the American Diabetes Association15. Dyslipidemia was defined according to the third report of the National Cholesterol Education Program16. Current smokers were defined as smokers who had smoked ≥ 1 pack-year, 1 pack-year being defined as 20 cigarettes per day for 1 year. CAD was defined as coronary artery disease, cerebrovascular disease, and PAD. Coronary artery disease included angina pectoris, myocardial infarction, and unstable angina. Cerebrovascular disease included ischemic stroke, hemorrhagic stroke, and transient ischemic attack. Patients with intermittent claudication or revascularization (surgery or catheter-based interventions), or limb amputation were diagnosed as PAD. Framingham risk score was calculated with points for the following risk factors: age, total cholesterol level, high-density lipoprotein cholesterol level, systolic blood pressure, and smoking status17. Estimated glomerular filtration rate was calculated using the following equation: 194 × serum creatinine−1.0949 × age−0.287 (× 0.739 for women)18. In addition, the 8 males of the above healthy subjects (mean age ± S.D.: 23.1 ± 0.8 yrs) were randomly chosen for measuring the lower-limb blood flow and the other 5 males of the above healthy subjects (mean age ± S.D.: 22.2 ± 0.8 yrs) were randomly selected to measure the lower-limb ezFMD of continuous days. The FMD testing was performed for 89 randomly chosen subjects: 8 healthy subjects (6 males and 2 females; mean age ± S.D.: 38.2 ± 16.1 yrs), 49 subjects at high risk of arteriosclerosis (33 males and 16 females; mean age ± S.D.: 60.2 ± 16.1 yrs), and 32 patients with CAD (29 males and 3 females; mean age ± S.D.: 66.1 ± 11.1 yrs). Experiments were conducted in accordance with the Declaration of Helsinki. Informed consent was obtained from all study participants before the experiments were performed and the study was approved by the ethics committee of Hiroshima University (https://upload.umin.ac.jp. Unique identifier: UMIN000004902). The proposed system requires fulfillment of the following four criteria for application of ezFMD to lower-limb arteries: The system can be used to perform lower-limb arterial blood flow occlusion and prompt post-occlusion reactive hyperemia. The proposed method's evaluation index %ezFMD L is highly reproducible. The %ezFMD L index shows statistically significant differences between healthy subjects and subjects at high risk of arteriosclerosis and between healthy subjects and patients with cardiovascular disease. The %ezFMD L index shows a significant correlation to %ezFMD B (the index for evaluating vascular endothelial function at the brachial artery) for the same healthy subjects because these subjects are considered to have normal vascular endothelial function in all arteries. Lower-limb blood flow measuring experiment Blood flow was observed during lower-limb ezFMD measurement to confirm the fulfillment of requirement (1). The target arteries for evaluation of vascular endothelial function in terms of lower-limb ezFMD are the anterior tibial artery, the posterior tibial artery and the fibular artery. However, as it is difficult to observe blood flow in these arteries with ultrasound due to their location deep below the skin, flow in the dorsal pedis artery (located in the periphery of the anterior tibial artery) was observed instead. Blood flow volume F b (t) is often measured for observation of reactive hyperemia. The equation used here is: $${F}_{b}(t)=\frac{\pi }{4}{V}_{b}(t)\{{D}_{b}(t{)\}}^{2},$$ where V b (t) and D b (t) are blood flow velocity and vascular diameter, respectively. Vascular diameter needs to be measured to determine blood flow volume F b (t) as Equation (1). However, it is technically difficult to track immediate variations in vascular diameter with a Doppler ultrasound blood flowmeter based on the echo tracking method while measuring cuff oscillation and during the period immediately after the release of cuff occlusion. It was therefore verified that the proposed system can be used to perform blood flow occlusion and prompt post-occlusion reactive hyperemia in lower-limb arteries. The chosen 8 healthy subjects were instructed to abstain from eating for at least eight hours beforehand to eliminate the effects of numerous factors that influence vasodilatation8. All subjects were instructed to assume a resting prone position for five minutes before blood flow velocity was measured. The velocity V b (t) in the dorsal pedis artery was measured with a Doppler ultrasound blood flowmeter (QFM-21, Hadeco) while lower-limb ezFMD was measured with an ezFMD measuring device (FMD tester, Nihon Kohden). Blood flow velocity V b (t), cuff pressure and oscillation were measured at 1,000 Hz using an analog-to-digital converter (CSI-360116, Interface), and the data obtained were saved to a PC. The root mean square (RMS) of blood flow velocity is calculated to evaluate blood flow variations, and RMS values calculated using the following equation from data collected before, during and after cuff occlusion were compared: $${V}_{brms}=\sqrt{\frac{1}{{T}_{2}-{T}_{1}}{\int }_{{T}_{1}}^{{T}_{2}}{\{{V}_{b}(t)\}}^{2}dt},$$ where T1, and T2 are the RMS calculation start and end times, respectively. The time ranges for calculation were 30 seconds before occlusion, 3 minutes during occlusion and 30 seconds immediately after occlusion. Bonferroni correction was applied for comparison of the means of all subjects' RMS values calculated from data collected before, during and after cuff occlusion. Lower-limb ezFMD measuring experiment Lower-limb ezFMD was measured using the protocol outlined in Methods Chapter to verify requirements (2)–(4). Blood pressure in these arteries was measured with a physiological monitor (OPV1510, Nihon Kohden), and an ezFMD measuring device was used to measure oscillation and apply occlusion. The cuff pressure and oscillation were measured at 1000 Hz using an analog-to-digital converter and saved to a PC. Paired t-testing was first performed to compare the mean of oscillation amplitudes in data collected before and after cuff occlusion. The testing was conducted for each mean value for the healthy subjects and the subjects at high risk of arteriosclerosis and those with CAD. Tukey-Kramer testing was also applied to compare the means calculated from all subjects' %ezFMD L values among the healthy subjects and the subjects at high risk of arteriosclerosis and those with CAD to verify requirement (3). Next, the %ezFMD L s of the selected 5 healthy males were measured on five consecutive days to verify requirement (2). The subjects were instructed to abstain from eating for at least eight hours before each measurement. The coefficient of variation value CV was calculated using %ezFMD L data collected over the five days, and the mean CV value for the five subjects was also calculated. CV value was determined using the following equation: $$CV=\frac{SD}{\bar{x}},$$ where SD and $\bar{x}$ are the standard variation and mean, respectively, of the %ezFMD L values measured over the five days. To evaluate the proposed system's capacity for repeatability, the mean CV value for the lower-limb ezFMD measured from the five subjects was compared with the corresponding value for the brachial artery as measured from five other healthy males (mean age ± S.D.: 23.2 ± 0.7 yrs) using an ezFMD measuring device designed for this purpose19. Welch's t-test was used to compare the mean CV value for the lower-limb ezFMD with the brachial values. There is no index with which %ezFMD L can be compared to verify the effectiveness of the proposed system because no research group has proposed a non-invasive method for quantitative evaluation of lower-limb vascular endothelial function. However, as healthy subjects are considered to have normal vascular endothelial function in all arteries, a correlation between lower-limb and upper-limb artery function can be assumed (requirement (4)). In this experiment, both the ezFMD of the brachial artery and the lower-limb arterial ezFMD were measured in all subjects, and the measured %ezFMD L was statistically compared with %ezFMD B 12 (representing the rate of change in cuff oscillation amplitude in the brachial artery as determined using an ezFMD measuring device designed for this purpose). Here, upper-limb blood pressure was measured with a physiological monitor. Cuff pressure and oscillation were measured at 1,000 Hz using an analog-to-digital converter and saved to a PC. The correlation coefficient and regression lines were calculated from each subject's %ezFMD L and %ezFMD B to verify requirement (4). Testing for no correlation was conducted to evaluate the calculated correlation coefficient. In addition, receiver operating characteristic (ROC) analysis20 was performed for the calculated %ezFMD L and %ezFMD B values to evaluate lower-limb vascular endothelial function screening capacity. %FMD, the de facto standard index for estimating vascular endothelial function, and the ankle brachial pressure index21 were also measured with an ultrasound device for FMD testing (UNEXEF-18G, Unex) and a blood pressure pulse wave inspection device (form PWV/ABI BP-203, Omron Colin), respectively. Lower-limb vascular endothelial function assessment method Intravascular pressure and external pressure originating from the cuff of the automated sphygmomanometer are applied to the arterial wall while blood pressure is measured. The maximum variation of the cuff's plethysmogram signals, which shows the difference between systolic and diastolic pressures, is determined under a condition in which intravascular pressure is approximately equal to the external pressure originating from the cuff because arterial wall compliance is maximized in this state. Cuff plethysmogram signals are related to vascular vessel volume. Assuming that the temperature in the cuff is constant, the relationship between the internal cuff volume V and the internal cuff pressure P can be expressed as $$P\times V=const\mathrm{.,}$$ where the vascular volume increases by ΔV, the cuff volume decreases by ΔV and the cuff pressure increases by ΔP in relation to cuff oppression caused by increased cuff volume. The oscillation amplitude ΔP can be expressed as $$(P+{\rm{\Delta }}P)\times (V-{\rm{\Delta }}V)=P\times V\mathrm{.}$$ Assuming that ΔP × ΔV is minute compared with the other terms, Equation (5) can be described as $${\rm{\Delta }}P=\frac{{\rm{\Delta }}V\times P}{V}\mathrm{.}$$ Therefore, if the cuff volume V and the pressure in the cuff P are constant, the oscillation amplitude ΔP is proportional to the vascular volume variation ΔV. Here, vasodilation is evaluated from the oscillation amplitude ΔP. Blood flow increases sharply after the vascular vessel is released from cuff occlusion with a level of pressure higher than that of systolic blood pressure for a given length of time. This physiological phenomenon is called reactive hyperemia22. Shear stress is generated at the boundary of the vascular endothelium in relation to the increased blood flow in the area of cuff occlusion. Vascular endothelial cells are stimulated due to the release of various vasodilators such as NO, and this NO causes increased compliance of the vascular smooth muscle in the tunica media. The relationship linking vascular volume variation ΔV, pulse pressure PP and arterial compliance C can be expressed as following23: $${\rm{\Delta }}V=C\times PP\mathrm{.}$$ Assuming that blood pressure is equal before and after cuff occlusion, the following Equation (8) can be obtained from Equations (6) and (7): $$\frac{{\rm{\Delta }}{P}_{2}}{{\rm{\Delta }}{P}_{1}}=\frac{{\rm{\Delta }}{V}_{2}}{{\rm{\Delta }}{V}_{1}}=\frac{{C}_{2}}{{C}_{1}},$$ where ΔV1 and ΔV2 are the vascular volume variation before and after cuff occlusion, C1 and C2 represent arterial compliance before and after cuff occlusion, and ΔP1 and ΔP2 show the oscillation amplitude before and after cuff occlusion. Vascular compliance variations can thus be measured by comparing the oscillation amplitude before and after cuff occlusion. Figure 1 shows the proposed lower-limb ezFMD measurement system, which consists of three parts: (i) a measurement part for determination of cuff oscillation using the oscillometric approach; (ii) an analysis part for determination of vascular feature quantity from the measured oscillation; (iii) an assessment part for evaluation of vascular endothelial function in lower-limb arteries based on the feature quantity determined in (ii). Overview of the proposed lower-limb ezFMD measuring system. In the measurement part, the standard cuff used with an oscillometric automatic sphygmomanometer is attached to the ankle of the subject at rest in a prone position. Oscillation measurement and arterial occlusion are performed with the standard cuff and the mean pressure method24 result in improved repeatability in the upper limbs during cuff depressurization. Occlusion pressure is adjusted to a level at least 50 mmHg higher than the subject's systolic arterial pressure so that arteries are completely occluded8. Cuff oscillation is first measured once, and occlusion is then applied for five minutes. Cuff oscillation is measured again five times at 30-second intervals after the release of the occlusion. In the analysis part, the maximum amplitude per beat is determined from each oscillation measurement. With the proposed method, the effects of lower-limb arterial blood pressure are eliminated by applying cuff pressure corresponding to the subject's blood pressure, and the vascular compliance characteristics of the arterial wall are extracted without the influence of this pressure. The method therefore enables evaluation of the maximum vascular compliance based on calculation to determine beat amplitudes in each oscillation measurement. In the assessment part, the rate of change in lower-limb arterial amplitude %ezFMD L is calculated using the maximum amplitudes observed before and after cuff occlusion in the analysis part and the following equation: $$ \% ezFM{D}_{L}=(\frac{{A}_{{\rm{post}}}}{{A}_{{\rm{pre}}}}-1)\times \mathrm{100,}$$ where Apre and Apost are the maximum amplitude before the cuff occlusion and the average of the measured maximum amplitude during the period from the third to the fifth measured oscillations after the cuff occlusion in consideration of vasodilatation response time25, respectively. After occlusion, vascular volume variation is considered to increase and oscillation waves are considered to expand because the NO released causes relaxation of the smooth muscle in the tunica media. As a result, if vascular endothelial function is normal, the maximum amplitude will increase after occlusion and the calculated rate of change in amplitude will be positive. The author thus defined the index %ezFMD L for the evaluation of vascular dilatation variations before and after occlusion, and the proposed index was used for quantitative evaluation of lower-limb vascular endothelial function. Baseline clinical characteristics The baseline clinical characteristics of the subjects are summarised in Table 1. The age range was 21–90 years. Of the 327 subjects, 236 (72.2%) were males and 91 (22.3%) were females. 173 (58.1%) had hypertension, 158 (53.0%) had dyslipidemia, 73 (24.5%) had diabetes mellitus and 39 (13.2%) were smokers. The mean value of ezFMD was 23.0 ± 14.5%. No significant difference was found between males and females (23.1 ± 14.9 v.s. 22.9 ± 13.0, p = 0.90). Table 1 Clinical characteristics of the Subjects. Figure 2(a) shows the blood flow velocity V b (t) measured while the lower-limb ezFMD of a healthy male subject (Sub. A) was measured. Figure 2(b) shows the calculated RMSs of blood flow velocity ${V}_{{b}_{RMS}}$ before, during and after cuff occlusion as measured from a healthy subject (Sub. A). The measured blood flow velocity V b (t) varied during the period before and after cuff occlusion due to the effect of the cuff pressure applied for plethysmogram extraction. Figure 2(a) shows that measured blood pressure V b (t) was approximately 0 cm/sec during cuff occlusion and increased sharply immediately afterward. Figure 2(b) shows that the RMS of the measured blood pressure ${V}_{{b}_{RMS}}$ was close to 0 cm/sec during occlusion and increased afterward. Figure 2(c) shows the means and standard deviations of the RMS of measured blood pressure ${V}_{{b}_{RMS}}$ for all subjects. It can be seen that latter value was close to 0 cm/sec during cuff occlusion and significantly increased afterward compared with the corresponding value before occlusion for all subjects (p < 0.01). Blood flow velocity during lower-limb ezFMD measurement: (a) time variation of blood flow velocity (Sub. A), (b) RMS of blood flow velocity (Sub. A), (c) RMS of blood flow velocity (all subjects). Figure 3(a,b) show the measured cuff oscillation waveform and calculated amplitude value from beat to beat before and after cuff occlusion for a healthy male subject (Sub. A), where the amplitude was calculated from the beats within the shaded area. It can be seen that the amplitude was higher after occlusion than before. Figure 3(c–e) show the mean values and standard deviations calculated from the oscillation amplitude before and after cuff occlusion based on measurements from the healthy subjects, subjects at high risk of arteriosclerosis and patients with CAD, respectively. Figure 3(c–e) indicated that the amplitudes after the cuff occlusion were significantly higher than those before the cuff occlusion among the healthy subjects, the subjects at high risk of arteriosclerosis and the patients with CAD (healthy subjects: p = 1.2 × 10−39, subjects at high risk of arteriosclerosis: p = 2.5 × 10−46, patients with CAD: p = 5.4 × 10−13). Measured results of cuff oscillation: (a) air-cuff oscillation and maximum amplitude measured from a healthy subject (Sub. A) before cuff occlusion, (b) those after cuff occlusion, (c) comparison of maximum amplitude between before and after cuff occlusion in healthy subjects, (d) that in subjects at high risk of arteriosclerosis, (e) that in patients with CAD. Figure 4 shows that the mean values and standard deviations calculated from %ezFMD L for the healthy subjects, the subjects at high risk of arteriosclerosis and those with CAD were 30.5 ± 12.0%, 23.6 ± 12.7% and 14.5 ± 15.4%, respectively. Significant differences were confirmed between the %ezFMD L of the healthy subjects and that of the subjects at high risk of arteriosclerosis, between that of the healthy subjects and that of the patients with CAD and between that of the subjects at high risk of arteriosclerosis and that of the patients with CAD (healthy subjects v.s. subjects at high risk of arteriosclerosis: p = 4.1 × 10−5, healthy subjects v.s. patients with CAD: p = 1.0 × 10−12, subjects at high risk of arteriosclerosis v.s. patients with CAD: p = 5.7 × 10−6). The mean values and standard deviations of measured %FMD for the healthy subjects, subjects at high risk of arteriosclerosis, and those with CAD were 7.83 ± 4.42%, 4.81 ± 4.03% and 3.43 ± 2.27%, respectively. A significant difference was confirmed between the healthy subjects and the patients with CAD (p = 2.6 × 10−2), whereas no significant differences were confirmed between the healthy subjects and those at a high risk of arteriosclerosis or between those at a high risk of arteriosclerosis and those with CAD (healthy subjects vs. those at high risk of arteriosclerosis: p = 1.0 × 10−1, subjects at high risk of arteriosclerosis vs. patients with CAD: p = 5.4 × 10−2). In contrast, mean values of %ezFMD L for the same subjects who underwent FMD testing were 35.8 ± 10.6%, 24.1 ± 12.7%, and 17.6 ± 15.4%, respectively. Significant differences were confirmed among all subject groups (healthy subjects vs. subjects at high risk of arteriosclerosis: p = 1.8 × 10−2; healthy subjects vs. patients with CAD: p = 1.2 × 10−3; and subjects at high risk of arteriosclerosis vs. patients with CAD: p = 4.9 × 10−2). Results of comparison for calculated %ezFMD L among healthy subjects, subjects at high risk of arteriosclerosis and patients with CAD. Figure 5(a) shows variations in %ezFMD L over five consecutive days for all subjects. Figure 5(b) shows that the mean value and standard deviations of CV for the healthy subjects calculated from %ezFMD L and those of CV measured with the ezFMD measuring device for the brachial artery were 0.23 ± 0.15 and 0.22 ± 0.04, respectively. The CV calculated from %ezFMD L was thus similar to that from the brachial artery. No statistically significant difference was seen between the CV from %ezFMD L and that from the brachial artery. Five-day variation of measured %ezFMD L in each healthy subject: (a) time variations in %ezFMD L , (b) comparison of calculated coefficients of variation. Figure 6(a,b) show the correlation between %ezFMD B (representing the vascular endothelial function of the brachial artery) and %ezFMD L and between %FMD and %ezFMD L , respectively. The ratio of %ezFMD L to %ezFMD B for the healthy subjects, that for the subject at high risk of arteriosclerosis and that for the patient with CAD were 0.48, 0.31, and 0.46, respectively (p < 0.001). Statistical comparison of indices: (a) comparison between the ezFMD of the upper and lower limbs and (b) comparison between FMD and ezFMD testing of the lower limbs. Figure 7(a,b) show the results of the %ezFMD L and %ezFMD B that ROC curve analysis performed to discriminate the subjects at high risk of arteriosclerosis from the healthy subjects and AUCs calculated from the full area under these ROC curves. Figure 7(b) indicates that the AUCs of %ezFMD L and %ezFMD B were 0.66 and 0.65, respectively. Figure 7(c,d) show the results of the %ezFMD L and %ezFMD B that ROC curve analysis performed to discriminate the patients with CAD from the healthy subjects and AUCs calculated from the full area under these ROC curves. Figure 7(d) indicates that the AUCs of %ezFMD L and %ezFMD B were 0.78 and 0.80, respectively. Results of ROC analysis: (a) ROC curves for healthy subjects and subjects at high risk of arteriosclerosis, (b) calculated AUC of %ezFMD L and %ezFMD B for healthy subjects and subjects at high risk of arteriosclerosis, (c) ROC curves for healthy subjects and patients with CAD, (d) calculated AUC of %ezFMD L and %ezFMD B for healthy subjects and patients with CAD. The blood flow was interrupted by the applied cuff pressure at least 50 mmHg higher than the subject's lower-limb systolic arterial pressure as shown the measured blood flow velocity V b (t) of a healthy male subject (Sub. A) in Fig. 2(a). In addition, shear stress caused by the sharp increase in blood flow velocity V b (t) immediately after occlusion release may have caused vascular endothelial cell stimulation. The proposed system is capable of interrupting lower-limb arterial blood flow, as the mean RMS of measured blood pressure ${V}_{{b}_{RMS}}$ during cuff occlusion for all healthy subjects was sufficiently close to 0 cm/sec as shown the mean values and standard deviations of the RMS of measured blood pressure ${V}_{{b}_{RMS}}$ for all healthy subjects in Fig. 2(c). Reactive hyperemia in the lower-limb vascular endothelium is also shown due to the significant increase observed in measured blood flow V b (t) after cuff occlusion using the proposed system. These outcomes show that lower-limb blood flow is interrupted and the lower-limb vascular endothelium is stimulated using the proposed system. The lower-limb arterial dilation after occlusion for a healthy male subject as shown in Fig. 3(a,b). The amplitude values measured from the healthy subjects, the subjects at high risk of arteriosclerosis and the patients with CAD significantly increased just after cuff occlusion release as shown in Fig. 3(c–e). This is because the lower-limb vascular compliance of the healthy subjects increased well after cuff occlusion release in association with sufficient release of NO, and additionally because that of the subjects at high risk of arteriosclerosis and the patients with CAD also increased in association with NO release even though their lower-limb vascular endothelial function may have been weaker than that of the healthy subjects. The %ezFMD L of the patients with CAD was significantly lower than that of the healthy subjects and that of the subjects at high risk of arteriosclerosis was significantly lower than that of the healthy subjects as shown in Fig. 4, which satisfies requirement (3) listed in Study Protocol section. This result indicates that the amount of NO produced after cuff occlusion was reduced because of lower-limb vascular endothelial dysfunction. The CV of %ezFMD L over a period of five consecutive days exhibited large variations among the healthy subjects as shown in Fig. 5. However, it can be seen that the repeatability of the proposed system is almost equal to that of the previous index ezFMD B . The CV of the proposed system was also better than that of %FMD, which is the current de facto standard for non-invasive evaluation of vascular endothelial function. The proposed system therefore satisfied requirement (2) listed in Study Protocol section and may be considered useful in the assessment of lower-limb vascular endothelial function. The index %ezFMD L exhibited a moderate correlation with %ezFMD B in the healthy subjects and the patients with CAD as shown in Fig. 6. These outcomes indicate that %ezFMD L satisfied requirement (4) listed in Study Protocol section and that ezFMD can be used for quantitative assessment of lower-limb vascular endothelial function. The effectiveness of using %ezFMD L to discriminate patients with CAD from healthy subjects was equal to that of %ezFMD B as shown in Fig. 7. The discrimination method between patients with CAD and healthy subjects using %ezFMD L for the diagnosis is to use the cutoff value calculated from the ROC analysis because it is an indicator to distinguish between positive and negative test outcomes. A %ezFMD L value of 22.6% was optimal for distinguishing between patients with CAD and healthy individuals in this study. Peripheral artery disease is known as an occlusive arterial disease that most commonly affects the legs. It is thus important to measure the vascular endothelial function at the lower-limb arteries. the proposed system is highly promising for assessment to discriminate lower-limb vascular endothelial dysfunction from normal vascular endothelial function in lower-limb arteries. In conclusion, this paper proposes a novel system for evaluating vascular endothelial function in lower-limb arteries based on ezFMD. The results reported that the system is capable of prompting reactive hyperemia and stimulating the vascular endothelium in lower-limb arteries. A lower-limb ezFMD measuring experiment was conducted on healthy subjects whose vascular endothelial function was normal. The results showed that the measured amplitude of cuff oscillation was significantly higher after cuff occlusion than before for all healthy subjects, and that the repeatability of the proposed index for assessing vascular endothelial function was equal to or better than that of the previous index. 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Reactive hyperemia test in a random sample of the general population. Journal of Vascular Surgery 17, 479–486 (1993). Ukawa, T. et al. Improvement of novel noninvasive measurement of endothelial function: ezfmd. In 2011 IEEE/SICE International Symposium on System Integration, 446–451 (2011). Ukawa, T. et al. Repeatability of novel non-invasive measurement of endothelial function: ezfmd. Therapeutic. Research 33, 403–411 (2012). Kihara, D. et al. Estimation of arterial viscoelastic properties during the flow-mediated dilation test. Transactions of the Society of Instrument and Control Engineers 49, 1029–1036 (2013). Node, K. et al. Plasma nitric oxide end products are increased in the ischemic canine heart. Biochemical and Biophysical Research Communications 211, 370–374 (1995). This work was supported by JSPS KAKENHI Grant Number 16K21076. Yukihito Higashi and Toshio Tsuji contributed equally to this work. Academic Institute, College of Engineering, Shizuoka University, Hamamatsu, 739-8527, Japan Harutoyo Hirano Graduate School of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan Renjo Takama, Ryo Matsumoto, Hiroshi Tanaka & Hiroki Hirano Department of System Cybernetics, Faculty of Engineering, Hiroshima University, Higashi-Hiroshima, 739-8527, Japan Zu Soh & Toshio Tsuji Nihon Kohden corporation, Tokorozawa, 359-8580, Japan Teiji Ukawa, Tsuneo Takayanagi & Haruka Morimoto Department of Anesthesiology and Critical Care, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, 734-8553, Japan Ryuji Nakamura, Noboru Saeki & Masashi Kawamoto Department of Cardiovascular Medicine, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, 734-8553, Japan Haruki Hashimoto, Shogo Matsui, Nozomu Oda & Tatsuya Maruhashi Department of Cardiovascular Regeneration and Medicine, Research Institute for Radiation Biology and Medicine, Hiroshima University, Hiroshima, 734-8553, Japan Shinji Kishimoto, Masato Kajikawa & Yukihito Higashi Department of Cardiovascular Physiology and Medicine, Graduate School of Biomedical and Health Sciences, Hiroshima University, Hiroshima, 734-8553, Japan Masao Yoshizumi Division of Regeneration and Medicine, Medical Center for Translational and Clinical Research, Hiroshima University Hospital, Hiroshima, 734-8551, Japan Yukihito Higashi Renjo Takama Ryo Matsumoto Hiroshi Tanaka Hiroki Hirano Zu Soh Teiji Ukawa Tsuneo Takayanagi Haruka Morimoto Ryuji Nakamura Noboru Saeki Haruki Hashimoto Shogo Matsui Shinji Kishimoto Nozomu Oda Masato Kajikawa Tatsuya Maruhashi Masashi Kawamoto Toshio Tsuji H.H, R.T., Y.H. and T.T. drafted the manuscript and managed this study. H.H, R.T., H.H., T.U., T.T., H.M. and T.T. developed the system. R.T., R.M., H.T., H.H., S.M., S.K., N.O., M.K. and T.M. measured the data and analyzed the results. Z.S., R.N., N.S., M.K. and M.Y. revised the article critically for important intellectual content. All authors reviewed the manuscript. Correspondence to Harutoyo Hirano, Yukihito Higashi or Toshio Tsuji. Teiji Ukawa, Tsuneo Takayanagi, Haruka Morimoto and Ryo Matsumoto are employees of Nihon Kohden Corporation. Harutoyo Hirano, Renjo Takama, Hiroshi Tanaka, Hiroki Hirano, Zu Soh, Ryuji Nakamura, Noboru Saeki, Haruki Hashimoto, Shogo Matsui, Shinji Kishimoto, Nozomu Oda, Masato Kajikawa, Tatsuya Maruhashi, Masashi Kawamoto, Masao Yoshizumi, Yukihito Higashi and Toshio Tsuji declare no potential conflict of interest. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Hirano, H., Takama, R., Matsumoto, R. et al. Assessment of Lower-limb Vascular Endothelial Function Based on Enclosed Zone Flow-mediated Dilation. Sci Rep 8, 9263 (2018). https://doi.org/10.1038/s41598-018-27392-3 Accepted: 01 June 2018	CommonCrawl
Eigenvector of gravity gradient tensor for estimating fault dips considering fault type Shigekazu Kusumoto1 The dips of boundaries in faults and caldera walls play an important role in understanding their formation mechanisms. The fault dip is a particularly important parameter in numerical simulations for hazard map creation as the fault dip affects estimations of the area of disaster occurrence. In this study, I introduce a technique for estimating the fault dip using the eigenvector of the observed or calculated gravity gradient tensor on a profile and investigating its properties through numerical simulations. From numerical simulations, it was found that the maximum eigenvector of the tensor points to the high-density causative body, and the dip of the maximum eigenvector closely follows the dip of the normal fault. It was also found that the minimum eigenvector of the tensor points to the low-density causative body and that the dip of the minimum eigenvector closely follows the dip of the reverse fault. It was shown that the eigenvector of the gravity gradient tensor for estimating fault dips is determined by fault type. As an application of this technique, I estimated the dip of the Kurehayama Fault located in Toyama, Japan, and obtained a result that corresponded to conventional fault dip estimations by geology and geomorphology. Because the gravity gradient tensor is required for this analysis, I present a technique that estimates the gravity gradient tensor from the gravity anomaly on a profile. In recent years, gravity gradiometry surveys have been widely conducted to obtain detailed subsurface structure data (e.g., Jekeli 1988; Dransfield 2010; Chowdhury and Cevallos 2013; Braga et al. 2014). Data collected by these surveys is the gravity gradient tensor defined by second derivatives of the gravity potential, and its response to subsurface structures is more sensitive than the gravity anomaly. At present, gravity gradiometry surveys have mainly been performed using a helicopter. Consequently, their observation interval is about 3 m on the flight profile, and the observation density is very high. The gravity gradiometry surveys allowed for high observation density, high resolution, and high sensitivity to the subsurface structures; therefore, these surveys contribute greatly to the earth science and resource engineering fields in terms of being useful and powerful tools for the estimation of subsurface structures. Various analysis techniques using gravity gradient tensors have been suggested and discussed (e.g., Zhang et al. 2000; Beiki 2010; Martinez et al. 2013; Cevallos 2014; Li 2015). These are considered to be so-called inversion techniques. A semi-automatic interpretation method that can extract subsurface structure characteristics without geological and geophysical data input has also been developed and applied to field data (e.g., Cooper 2012; Ma 2013; Ferreira et al. 2013). A typical semi-automatic interpretation method is an edge emphasis technique that uses extraction techniques to find locations (namely, edge) where the potential field changes abruptly due to density variations. The horizontal gravity gradient method and vertical gravity gradient method (e.g., Evjen 1936; Elkins 1951; Tsuboi and Kato 1952; Blakely and Simpson 1986) are classic edge emphasis techniques. In recent years, higher and keener extraction techniques have been suggested (e.g., Miller and Singh 1994; Cooper and Cowan 2006; Sertcelik and Kafadar 2012; Zhang et al. 2014). In addition, attention has been paid to techniques that evaluate the shape of the potential field (e.g., Koenderink and van Doorn 1992; Robert 2001; Zhou et al. 2013; Cevallos 2014). Among these methodologies, a technique for estimating the dip of geological boundary using the gradient tensor of the potential fields has been developed (e.g., Beiki 2013). Beiki and Pedersen (2010) showed that the maximum eigenvector of the gravity gradient tensor points to the causative body (Fig. 1a). Since this property is common in the potential fields, Beiki (2013) applied it to a magnetic anomaly in the Åsele area (Sweden) and obtained useful information on the dip of the dike swarms. Kusumoto (2015), considering that the basement consists of an aggregate of high-density prisms (Fig. 1b), applied Beiki's technique (Beiki and Pedersen 2010; Beiki 2013) to the estimation of fault dips. This method provided results wherein the fault dip estimated by the gravity gradient tensor harmonized with the dip observed from seismic surveys (Kusumoto 2015, 2016a). In addition, the dip of an earthquake source fault of the Kumamoto Earthquake that occurred in April 2016 estimated from the gravity gradient tensor also corresponded with the dip of the fault model (normal fault of 60°), thus explaining the crustal movement observed by GNSS (Global Navigation Satellite System) (Kusumoto 2016b). The range for which this method is applicable is wide from low dip to high dip (e.g., Beiki 2013; Kusumoto 2015, 2016a, 2016b), although it has some numerical instability to the vertical fault (e.g., Kusumoto 2015). Schematic illustration of the maximum eigenvectors for two-dimensional (2D) structures such as dykes and faults. a Basic model. In this figure, v 1 is the maximum eigenvector of the gravity gradient tensor and points to the causative body. The angle α between the surface and the maximum eigenvector is the dip of the causative body. b Fault model. A basement consists of an aggregate of high-density prisms, and the angle, α, indicates the fault dip Although analyses using the gravity gradient tensor have yielded excellent results in subsurface structure estimations and edge detections, gravity gradiometry surveys have been conducted in only a few areas, limiting the tensor data available. If we were to carry out these analyses in areas where gravity gradiometry surveys have not been conducted yet, we would have to use the tensor estimated from existing gravity anomaly data. The procedure for estimating the gravity gradient tensor from gravity anomaly data has already been suggested by Mickus and Hinojosa (2001). This technique estimates the gravity gradient tensor from spatial distribution of gravity anomalies by the Fourier transform. Since the database of gravity anomalies has been prepared, studies using the gravity gradient tensor estimated by Mickus and Hinojosa's method will progress in the future. On the other hand, it is difficult to apply this method directly to gravity anomalies obtained by gravity surveys conducted on a profile employed frequently in active fault research. In dense gravity surveys researching fault structures in detail, profiles were set perpendicular to the fault and short-spaced gravity observations were taken along the profiles (e.g., Iwano et al. 2001; Inoue et al. 2004). It is important to find the fault shape, especially its dip, in these studies because the fault dip affects the area of disaster occurrence (e.g., Abrahamson and Somerville 1996; Takemura et al. 1998) and is an important parameter in numerical simulations for hazard map creation (e.g., Irikura and Miyake 2011). Consequently, in two-dimensional gravity surveys for faults, a fault dip estimated from the eigenvectors of the gravity gradient tensor calculated from the gravity anomaly would be of additional value. In addition, since this analysis technique does not require vast calculation times, I expect it will be an effective new technique for analyzing high-resolution data obtained densely, i.e., through dense gravity surveys for fault research and also airborne gravity gradiometry surveys. In this study, I first introduce the technique for the estimation of the gravity gradient tensor from a gravity anomaly on the profile. After that, I discuss the relationship between fault dips and eigenvectors of the gravity gradient tensor and apply its result to gravity anomaly data obtained on the profile crossing the Kurehayama Fault in Toyama, Japan. Methods/Experimental Gravity gradient tensor on the profile Gravity gradient tensor Γ on the profile is defined as follows (e.g., Beiki and Pedersen 2011) $$ \varGamma =\left[\begin{array}{cc}\hfill {g}_{xx}\hfill & \hfill {g}_{xz}\hfill \\ {}\hfill {g}_{zx}\hfill & \hfill {g}_{zz}\hfill \end{array}\right] $$ Here, g xx , g xz , g zx , and g zz are each component of the tensor and are defined as the first derivative of gravity vector components g x and g z for each direction. In addition, gravity vectors g x and g z are given by the first derivative of gravity potential, W, namely, g x = ∂W/∂x and g z = ∂W/∂z. As the gravity potential satisfies the Laplace equation, ∂ 2 W/∂x 2 + ∂ 2 W/∂z 2 = g xx + g zz = 0, we find the relationship g zz = −g xx . Also, the relationship is known to be g xz = g zx because the gravity gradient tensor is a symmetric tensor (e.g., Torge 1989). Relationship between subsurface structure and gravity anomaly In the two-dimensional analyses, a structure in one direction is assumed to be infinite. Although this assumption is not realistic, it is a good approximation in fault structure analyses and gives us some practical analysis techniques. In calculations of the gravity gradient tensor from the gravity anomaly, we need gravity anomaly values at different heights. Consequently, I will show the relationship between two-dimensional subsurface structures and gravity anomalies in this subsection before estimating the gravity gradient tensor from the gravity anomaly. As the simplest subsurface model, I set a two-dimensional double layer model consisting of a sedimentary layer and a basement (Fig. 2). Horizontal positions are given by x, and vertical positions are given by z. Depth is zero (z = 0) on the surface, and z increases with depth. As shown in Fig. 2, an average boundary depth between the sedimentary layer and basement is defined as z = D (>0). When the boundary surface at point Q(x') deviates by h(x') from the average boundary depth (Fig. 2), gravity anomaly g z (x) at the point P(x) on the surface caused by this deviation is given by the following equation (e.g., Blakely 1996). Model of subsurface structure. A double-layer model consisting of a sedimentary layer and a basement is assumed here. D is the average depth of the stratum boundary, and h(x') is the deviation of the boundary from the average. Here, the deviation is assumed to be very small, i.e., h(x') << D $$ {g}_z(x)=2\gamma \varDelta \rho {\displaystyle {\int}_{-\infty}^{\infty }{\displaystyle {\int}_D^{D+ h\left( x\hbox{'}\right)}\frac{z\hbox{'}}{{\left( x-{x}^{\prime}\right)}^2+ z{\hbox{'}}^2}} d{x}^{\prime } d{z}^{\prime }} $$ where γ is the gravitational constant and Δρ is the density contrast between the sedimentary layer and basement. The integration on z in Eq. (2) is solved as: $$ {\displaystyle {\int}_D^{D+ h\left( x\hbox{'}\right)}\frac{z\hbox{'}}{{\left( x-{x}^{\prime}\right)}^2+ z{\hbox{'}}^2}} d{z}^{\prime }=\frac{1}{2} \log \left[\frac{{\left( x-{x}^{\prime}\right)}^2+{\left( D+ h\left({x}^{\prime}\right)\right)}^2}{{\left( x-{x}^{\prime}\right)}^2+{D}^2}\right] $$ here, if h(x') is much smaller than D, namely, h(x') << D, (D + h)2 is {D[1 + (h/D)]}2 ≈ D 2(1 + 2 h/D) = D 2 + 2Dh, Eq. (3) would be rewritten as follows: $$ {\displaystyle {\int}_D^{D+ h\left( x\hbox{'}\right)}\frac{z\hbox{'}}{{\left( x-{x}^{\prime}\right)}^2+ z{\hbox{'}}^2}} d{z}^{\prime}\approx \frac{1}{2} \log \left[\frac{{\left( x-{x}^{\prime}\right)}^2+{D}^2+2 Dh\left({x}^{\prime}\right)}{{\left( x-{x}^{\prime}\right)}^2+{D}^2}\right]=\frac{1}{2} \log \left[1+\frac{2 Dh\left({x}^{\prime}\right)}{{\left( x-{x}^{\prime}\right)}^2+{D}^2}\right] $$ In general, if −1 < ξ ≤ 1 in log(1 + ξ), we have the following approximation (e.g., Gradshteyn and Ryzhik 2007) $$ \log \left(1+\xi \right)=\xi -\frac{1}{2}{\xi}^2+\frac{1}{3}{\xi}^3-\frac{1}{4}{\xi}^4+\cdots ={\displaystyle \sum_{p=1}^{\infty }{\left(-1\right)}^{p+1}\frac{\xi^p}{p}} $$ The second term, 2Dh/[(x − x')2 + D 2], in Eq. (4) is small because D > > h. We can use Eq. (5) to derive a linear approximate equation of Eq. (4). By neglecting higher terms of ξ, Eq. (3) or (4) is rewritten as follows: $$ {\displaystyle {\int}_D^{D+ h\left( x\hbox{'}\right)}\frac{z\hbox{'}}{{\left( x-{x}^{\prime}\right)}^2+ z{\hbox{'}}^2}} d{z}^{\prime}\approx \frac{ D h\left({x}^{\prime}\right)}{{\left( x-{x}^{\prime}\right)}^2+{D}^2} $$ Consequently, we obtained the following equation. $$ {g}_z(x)\approx 2\gamma D\varDelta \rho {\displaystyle {\int}_{-\infty}^{\infty}\frac{h\left({x}^{\prime}\right)}{{\left( x-{x}^{\prime}\right)}^2+{D}^2} d{x}^{\prime }} $$ Here, I introduce a new function, ϕ, defined by: $$ \varphi (x)=\frac{1}{x^2+{D}^2} $$ and Eq. (7) is rewritten as follows: $$ {g}_z(x)=2\gamma D\varDelta \rho {\displaystyle {\int}_{-\infty}^{\infty}\varphi \left( x-{x}^{\prime}\right) h\left({x}^{\prime}\right) d{x}^{\prime }} $$ This form is convoluted, and we obtain Eq. (10) by applying the Fourier transformation to Eq. (9) $$ {G}_z=2\gamma D\varDelta \rho \varPhi H $$ where, G z , Φ, and H are Fourier transforms of g z (x), ϕ(x), and h(x), respectively. As is well known, the Fourier transform of Eq. (8) is (e.g., Blakely 1996; Gradshteyn and Ryzhik 2007) $$ \varPhi =\frac{\pi}{D}{e}^{- D\left\| k\right\|} $$ Here, \|k\| = ik z = \|k x \| (e.g., Blakely 1996) and k x is the wave number in the x direction. Here, I employed the Fourier transform, F, of a function f(x) defined as follows (e.g., Blakely 1996): $$ F={\displaystyle {\int}_{-\infty}^{\infty } f(x){e}^{- ikx} dx} $$ By Eq. (11), Eq. (10) is rewritten as: $$ {G}_z=2\pi \gamma \varDelta \rho H{e}^{- D\left\| k\right\|} $$ This is the relationship between gravity anomaly on the profile and two-dimensional subsurface structure. Relationship between gravity anomaly and gravity gradient tensor As shown in the previous section, the gravity gradient tensor is given by the second derivative of the gravity potential. The relationship between gravity anomaly g z and gravity potential W is $$ W=-{\displaystyle \int {g}_z dz} $$ From Eq. (13), the equation giving the gravity anomaly at point P'(x) of an arbitrary height z from the surface (Fig. 2) is obtained in the Fourier domain as follows: $$ {G}_z=2\pi \gamma \varDelta \rho H{e}^{-\left( D+ z\right)\left\| k\right\|} $$ By integrating this equation to z and substituting z = 0, we obtain the gravity potential at the surface. If the Fourier transform of the gravitational potential is represented by U, from these calculations, the U would be given by G z as follows: $$ U=\frac{1}{\left\| k\right\|}{G}_z $$ As the x direction component of gravity anomaly is given by the first derivative in the x direction of the gravity potential W, the g x in the Fourier domain, G x , would be given by a differential formula in the Fourier domain (e.g., Blakely 1996) as follows: $$ {G}_x= i{k}_x U $$ From Eq. (16), we obtained $$ {G}_x=\frac{i{ k}_x}{\left\| k\right\|}{G}_z $$ g xx in the Fourier domain is given by $$ {G}_{x x}=\frac{-{k}_x^2}{\left\| k\right\|}{G}_z $$ We can obtain g xx by applying the inverse Fourier transform to G xx , and g zz would be obtained from the relationship of g zz = −g xx . The other component g zx (=g xz ) would be given by: $$ {G}_{z x}={G}_{x z}= i{k}_x{G}_z $$ where G zx and G xz are the Fourier transform of g zx and g xz . Here, although I showed a technique to calculate the gravity gradient tensor in the Fourier domain, there is another technique to calculate the tensor by a simple finite-difference method (e.g., Blakely 1996) of gravity vectors g x and g z in the space domain. Relationship between subsurface structures and eigenvectors As indicated by Beiki and Pedersen (2010), the maximum eigenvector of the gravity gradient tensor points to the causative body of the gravity anomaly (Fig. 1a). They also pointed out that the minimum eigenvector of the tensor indicates the strike direction of structures such as dikes in three-dimensional analyses. Since there are two perpendicular eigenvectors of the gravity gradient tensor in the two-dimensional analyses, it is expected that the minimum eigenvector of the tensor will point to the low-density causative body or medium if the maximum eigenvector of the tensor points out high-density causative bodies such as a dike in a low-density layer such as a sedimentary layer. To clear this inference, I calculated the gravity gradient tensor on the profile caused by the model shown in Fig. 3 and investigated the dips of the maximum and minimum eigenvectors of the tensor. The model shown in Fig. 3 has a width and height of 0.25 and 2.0 km, respectively. Model of subsurface structures. Here, rectangular causative body of width and height of 0.25 and 2.0 km, respectively, is assumed. Densities of a medium and a causative body are ρ 2 and ρ 1, respectively Each component of the gravity gradient tensor caused by two-dimensional structures such as the dike shown in Fig. 3 is given by Telford et al. (1990). The relationship between eigenvectors and structural boundaries will be discussed widely in this study; I therefore employed calculation formulas given by Talwani et al. (1959). Talwani et al. (1959) show well-known calculation formulas giving g x and g z for two-dimensional arbitrary structures closed by a polygon. In this study, I obtained g zx (=g xz ) and g xx components by the numerical differentiation of g z and g x , and the g zz component was given by g zz = −g xx . A simple finite-difference method (e.g., Blakely 1996) was employed for these numerical differentiations. In addition, the dip of each eigenvector (α) was calculated by $$ \alpha = \arctan \left(\frac{v_z}{v_x}\right) $$ where v x and v z are x and z components of each eigenvector. Density structures and eigenvectors Figure 4a shows distributions of the maximum (red) and minimum (blue) eigenvectors of the gravity gradient tensor caused by the model structure (Fig. 3) whose density contrast (Δρ = ρ 1 − ρ 2) is 200 kg/m3. Figure 4b shows distributions of the maximum (red) and minimum (blue) eigenvectors of the tensor caused by the model structure (Fig. 3) whose density contrast (Δρ) is −200 kg/m3. In each figure, the lengths of all the eigenvectors are the same. Eigenvectors of the gravity gradient tensor caused by the subsurface model shown in Fig. 3. The maximum eigenvector and minimum eigenvector are shown by red and blue, respectively. a Eigenvectors on the high-density causative body (light green). When the high-density causative body was given in the low-density medium, the maximum eigenvector of the gravity gradient tensor points to the causative body and the minimum eigenvector points to the low-density medium. b Eigenvectors on the low-density causative body (light yellow). When the low-density causative body was given in the high-density medium, the minimum eigenvector of the tensor points to the causative body and the maximum eigenvector points to the high-density medium From Fig. 4a, it is found that the maximum eigenvector of the gravity gradient tensor points to a high-density causative body if the body is embedded in the low-density medium. In this case, the minimum eigenvector of the tensor points to the low-density medium around the high-density body. On the other hand, the minimum eigenvector of the gravity gradient tensor points to a low-density causative body if the body is embedded in the high-density medium. In this case, the maximum eigenvector of the tensor points to the high-density medium around the low-density body. From these results, in the two-dimensional analyses, it was shown that the maximum eigenvector points to a high-density causative body and the minimum eigenvectors points to a low-density causative body. In Fig. 4, there are vectors pointing to the area z < 0. This indicates that α is negative. Structures exist underground, and the negative α is not realistic. Consequently, I will add π to α if α is negative. Fault types and eigenvectors In calderas and/or sedimentary basins, high-density and low-density materials are in contact with each other via normal faults and/or reverse faults. In gravity anomalies and gravity gradient tensors, differences in fault type are defined as differences in density structure. As it was shown that the behavior of each eigenvector is dependent on the density structure in the previous subsection, I investigated the relationship between eigenvectors and fault type by the simplified sedimentary basin models. Figure 5a is a simplified sedimentary basin model in which the sedimentary layer is in contact with the basement by normal faults, and Fig. 5b is a simplified sedimentary basin model in which the sedimentary layer is in contact with the basement by reverse faults. Density contrast between sedimentary layer and basement is assumed to be −200 kg/m3. Simplified sedimentary basin model. Light yellow and white areas indicate sedimentary layer and basement, respectively. a Sedimentary basin model where sedimentary layer is in contact with the basement by normal faults of 45° dip. b Sedimentary basin model where the sedimentary layer is in contact with the basement by reverse faults of 45° dip In Fig. 6, I showed distributions of the maximum (red) and minimum (blue) eigenvectors of the gravity gradient tensor caused by these models. In each figure, the lengths of all the eigenvectors are the same, because we are interested in the fault dip and only angle information is necessary for this study. Eigenvectors of the gravity gradient tensor caused by the simplified sedimentary basin models shown in Fig. 5. The maximum eigenvector and minimum eigenvector are indicated by red and blue, respectively. a Eigenvectors on the sedimentary basin formed by normal faults. When the sedimentary layer is in contact with the basement by normal fault, the dip of the maximum eigenvector follows the dip of the normal fault. b Eigenvectors on the sedimentary basin are formed by reverse faults. When the sedimentary layer is in contact with the basement by reverse fault, the dip of the minimum eigenvector follows the dip of the reverse fault From Fig. 6a, it is found that the dip of the maximum eigenvector of the gravity gradient tensor closely follows the dip of the normal fault. When the basement distributes near the surface, the maximum eigenvector points in the vertical direction to the high-density basement. The effect of the high-density basement is weak in the sedimentary layer area, while the effect of the low-density sedimentary layer is strong; therefore, the minimum eigenvector points in the vertical direction to the low-density sediment and the maximum eigenvector points in the horizontal direction. When the boundary is a reverse fault, from Fig. 6b, it is found that the dip of the minimum eigenvector of the gravity gradient tensor indicates the dip of the fault well. The maximum eigenvector on the basement points in the vertical direction to the high-density basement, and the minimum eigenvector points in the horizontal direction. Since the low-density sediment distributes near the surface in the sedimentary layer area, the minimum eigenvector points vertically. From these results, it was concluded that if the structural boundary is a normal fault, its dip can be estimated from the dip of the maximum eigenvector of the gravity gradient tensor, and if the boundary is a reverse fault, its dip can be estimated from the dip of the minimum eigenvector of the tensor. In addition, in the area away from the boundary, it was found that the maximum eigenvector on the basement and the minimum eigenvector on the sediment point in the vertical direction, and the maximum eigenvector on the sediment and the minimum eigenvector on the basement point in the horizontal direction, regardless of whether the boundary is a normal fault or reverse fault. Subsurface structures and eigenvectors By simple numerical simulations, it was found that the maximum eigenvector of the gravity gradient tensor points to a high-density causative body and that the minimum eigenvector points to a low-density causative body. In addition, it was found that the dip of the maximum eigenvector of the tensor closely follows the dip of the normal fault and that the dip of the minimum eigenvector closely follows the dip of the reverse fault. As mentioned above, Beiki and Pedersen (2010) have already pointed out that the maximum eigenvector of the gravity gradient tensor points to the high-density causative body. The result in Fig. 4a confirms that their results are true for the two-dimensional analyses as well. When the basement distributes near the surface, the maximum eigenvector points in the vertical direction. This property also shows that Beiki and Pedersen (2010) are correct, and the idea of the basement as an aggregate of high-density prisms (Fig. 1b), suggested by Kusumoto (2015, 2016b), would not be incorrect. As to why the dip of normal fault was given by the dip of the maximum eigenvector of the gravity gradient tensor, I considered that the lower part of the boundary structure (fault) exists inside the low-density area more than its upper part. Therefore, because the gravity gradient tensor is most sensitive to the subsurface structures near the surface, the structure shown in Fig. 5a was considered a high-density body that intruded into the low-density layer, and the dip of the normal fault was given by the dip of the maximum eigenvector. I believe Kusumoto (2015, 2016a, 2016b) was able to obtain results that coincided with seismic surveys since he estimated the fault dip in normal fault regions by the maximum eigenvector of the tensor. On the other hand, when the maximum eigenvector points to high-density causative bodies embedded in low-density medium or low-density causative bodies embedded in a high-density medium, the minimum eigenvector points to the low-density mediums or to the causative bodies. Beiki and Pedersen (2010) have not explicitly referred to analyses of low-density causative bodies using eigenvectors. Since it is necessary to analyze anomalies caused by low-density bodies in the field, it seems that the result, in which the minimum eigenvector points to the low-density bodies, would play an important role in subsurface structure estimation, although this is the result of two-dimensional analysis. In addition, it was found that the dip of the minimum eigenvector of the gravity gradient tensor gave the dip of the reverse fault. As to the reason why the dip of reverse fault was given by the minimum eigenvector of the gravity gradient tensor, I considered that the lower part of the boundary structure (fault) exists inside the high-density area more than its upper part. Namely, because this structure was considered a low-density body that intruded into the high-density layer, the dip of the reverse fault was given by the dip of the minimum eigenvector of the gravity gradient tensor. As is understood from the results and discussions obtained in this study, selecting a suitable eigenvector for estimating the fault dip is important. If the study area is not too wide and prior geological information is available, the eigenvector that should be employed for estimating the fault dip correctly would be selected based on the information. If the study area was a fault area where normal faults were mainly distributed, the maximum eigenvector of the gravity gradient tensor would be employed for estimating the fault dip. If the study area was a fault area where reverse faults were mainly distributed, the minimum eigenvector would be employed. In the three-dimensional study for high-density causative bodies, it is pointed out that the minimum eigenvector is parallel to the strike direction of the structure (Beiki and Pedersen 2010; Beiki 2013). However, in the two-dimensional analyses, the strike direction of the structure is perpendicular to x- and z-axes and does not appear in the analyses. As it is difficult to directly compare the properties of the minimum eigenvector obtained in different dimensions, in the future, it would be necessary to discuss detailed properties of the minimum eigenvector. Application to field data As an application of the techniques, I estimated the dip of the Kurehayama Fault located in Toyama, Japan. The Kurehayama Fault is a reverse fault located at the center of the Toyama basin, and it strikes in the NNE-SSW direction (Fig. 7). The length of the fault is about 22 km, and the fault dip is about 45° (e.g., The Headquarters for Earthquake Research Promotion 2008; Toyama City 2013). The Toyama City has carried out seismic surveys and dense gravity surveys crossing this fault (Toyama City 2013). Toyama City (2013) set three profiles crossing the Kurehayama Fault, and the dense gravity surveys of 50 m spaced measurements have been conducted on these profiles, although spacing of several hundred meters has been usually employed for these surveys. Here, I used gravity anomaly data on the profile located at the shoreline. Figure 8 shows the Bouguer anomaly in which the Bouguer density of 2260 kg/m3 was assumed (Toyama City 2013). The indication "Kurehayama Fault" shown in this figure indicates a rough fault location. Location map of the study area. Kurehayama Fault is a reverse fault located in the center of the Toyama Basin, Toyama Prefecture, Japan. Its location has been estimated by topographic, geological, and geophysical data. The red line and brown lines denote the estimated location of the Kurehayama Fault, Toyama City (Toyama City 2013), and The Headquarters for Earthquake Research Promotion (The Headquarters for Earthquake Research Promotion 2008), respectively. Blue line a - b indicates the dense gravity survey profile, which has gravity observation points at about 50 m intervals Bouguer anomalies on the profiles (after Toyama City 2013). The Bouguer density of 2260 kg/m3 is assumed. The "Kurehayama Fault" shown in this figure indicates a rough fault location by Toyama City (Toyama City 2013). The unit of the gravity anomaly is given in milligal, and mGal = 10−5 m/s2 I applied the techniques to the Bouguer anomaly and obtained the gravity gradient tensor shown in Fig. 9. Figure 10 shows distributions of the maximum eigenvector (red) and the minimum eigenvector (blue) of the gravity gradient tensor. Since the Kurehayama Fault is a reverse fault, I focus on the dip of the minimum eigenvector. From Fig. 10, it is found that the dip (α) of the Kurehayama Fault was about 138°. Since the angle α is measured clockwise from the surface (x-axis), it seems that the obtained dip indicates the dip of the reverse fault of 42°. This fault dip is consistent with conventional data. Gravity gradient tensor (g xx , g xz (=g zx ), g zz ) on the profile. These are estimated from the Bouguer anomalies on the profile shown in Fig. 8. The component of g xx and g xz is calculated by a finite-difference method of gravity vectors g x and g z in the space domain. The "Kurehayama Fault" shown in this figure indicates a rough fault location by Toyama City (Toyama City 2013). The unit of the gravity gradient tensor is given in E (Eötvös), and 1 E = 0.1 mGal/km Eigenvectors of the gravity gradient tensor on the profile shown in Fig. 7. The maximum eigenvector and minimum eigenvector are indicated by red and blue, respectively. The dips of eigenvectors are given clockwise from x-axis to z-axis. Since it is known that the Kurehayama Fault is a reverse fault, we focus on the minimum eigenvector of the tensor. The "Kurehayama Fault" shown in this figure indicates a tentative fault location in Toyama City (Toyama City 2013). The average dip of the minimum eigenvector in the Kurehayama Fault zone shown by a rectangle with dashed lines is about 138°, and this angle indicates that the Kurehayama Fault would be a reverse fault of 42°. In addition, the maximum eigenvectors on the right side of the "Kurehayama Fault" shown in this figure point to the vertical direction, and the minimum eigenvectors in the left side of the "Kurehayama Fault" point to the vertical direction The estimated fault dip would be the dip near the surface because the method employs the gravity gradient tensor, which is sensitive to subsurface structures near the surface. Since it is important to know quantitatively which depth the estimated fault dip is, in the future, it would be necessary to develop a technique estimating the depth of the estimated dip or the dip in the arbitrary depth. In this study, I showed techniques for estimating the gravity gradient tensor from gravity anomalies on the profile and for estimating the fault dip by eigenvector of the observed or calculated gravity gradient tensor on the profile. I also investigated its properties by numerical simulations. From numerical simulations, it was found that the maximum eigenvector of the tensor points to a high-density causative body and that the dip of the maximum eigenvector closely follows the dip of the normal fault. In addition, if the basement distributes near the surface, the maximum eigenvector points to the vertical direction. They have been pointed out already in previous studies, and the results shown in here confirmed that their results are true for the two-dimensional analyses as well. On the other hand, it was found that the minimum eigenvector of the tensor points to a low-density causative body and that the dip of the minimum eigenvector closely follows the dip of the reverse fault. Since eigenvector analyses of the anomalies caused by the low-density causative body have not been discussed explicitly in previous studies, these results would play an important role in estimations of subsurface structures in the future. From these results, it was found that the eigenvector of the gravity gradient tensor for estimating fault dips is determined by fault type, and we would estimate the fault dip correctly if we were to employ suitable eigenvectors based on prior information. As an application of suggestions, I estimated the dip of the Kurehayama Fault located in Toyama, Japan, and obtained the fault dip of about 42° as the dip of the minimum eigenvector of the gravity gradient tensor because the fault is the reverse fault. This dip harmonized with conventional geological information. Since the analysis technique shown in this study does not require complex calculations and vast calculation times, it will be an effective technique for analyzing high-resolution data obtained densely by not only dense gravity surveys for fault research but also airborne gravity or gravity gradiometry surveys. Abrahamson NA, Somerville P (1996) Effects of the hanging wall and footwall on ground motions recorded during the Northridge earthquake. Bull Seism Soc Am 86:S93–S99 Beiki M (2010) Analytic signals of gravity gradient tensor and their application to estimate source location. Geophysics 75:I59–I74 Beiki M (2013) TSVD analysis of Euler deconvolution to improve estimating magnetic source parameters: an example from the Asele area, Sweden. J Appl Geophys 90:82–91 Beiki M, Pedersen LB (2010) Eigenvector analysis of gravity gradient tensor to locate geologic bodies. 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Zisin 2(57):45–54 (in Japanese with English abstract) Irikura K, Miyake H (2011) Recipe for predicting strong ground motion from crustal earthquake scenarios. Pure Appl Geophys 168:85–104. doi:10.1007/s00024-010-0150-9 Iwano S, Fukuda Y, Ishiyama T (2001) An estimation of fault related structures by means of one-dimensional gravity surveys - case studies at the Katagihara Fault and the Fumotomura Fault. J Geog 110:44–57 (in Japanese with English abstract) Jekeli C (1988) The gravity gradiometer survey system (GGSS). EOS Trans AGU 69:105 Koenderink JJ, van Doorn AJ (1992) Surface shape and curvature scales. Im Vis Comp 10:557–564 Kusumoto S (2015) Estimation of dip angle of fault or structural boundary by eigenvectors of gravity gradient tensors. Butsuri-Tansa 68:277–287 (in Japanese with English abstract) Kusumoto S (2016a) Structural analysis of caldera and buried caldera by semi-automatic interpretation techniques using gravity gradient tensor: a case study in central Kyushu Japan. In: Nemeth K (ed) Updates in Volcanology - From volcano modelling to volcano geology. InTech, Rijeka Kusumoto S (2016b) Dip distribution of Oita-Kumamoto Tectonic Line located in central Kyushu, Japan, estimated by eigenvectors of gravity gradient tensor. Earth Plan Space 68:153. doi:10.1186/s40623-016-0529-7 Li X (2015) Curvature of a geometric surface and curvature of gravity and magnetic anomalies. Geophysics 80:G15–G26 Ma G (2013) Edge detection of potential field data using improved local phase filter. Expl Geophys 44:36–41 Martinez C, Li Y, Krahenbuhl R, Braga MA (2013) 3D inversion of airborne gravity gradiometry data in mineral exploration: a case study in the Quadrilatero Ferrifero, Brazil. Geophysics 78:B1–B11 Mickus KL, Hinojosa JH (2001) The complete gravity gradient tensor derived from the vertical component of gravity: a Fourier transform technique. 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Cambridge University Press, Cambridge The Headquarters for Earthquake Research Promotion (2008) Evaluations of Tonami Fault zone and Kurehayama Fault zone. In: The Headquarters for Earthquake Research Promotion web site. http://www.jishin.go.jp/main/chousa/katsudansou_pdf/56_tonami_kureha_2.pdf (in Japanese). Accessed 21 Nov 2016 Torge W (1989) Gravimetry. Walter de Gruyter, Berlin Toyama City (2013) Research report on Kurehayama Fault (2). Toyama-shi, Toyama (in Japanese) Tsuboi C, Kato M (1952) The first and second vertical derivatives of gravity. J Phys Earth 1:95–96 Zhang C, Mushayandebvu MF, Reid AB, Fairhead JD, Odegrad ME (2000) Euler deconvolution of gravity tensor gradient data. Geophysics 65:512–520 Zhang X, Yu P, Tang R, Xiang Y, Zhao C-J (2014) Edge enhancement of potential field data using an enhanced tilt angle. Expl Geophys 46:276–283. doi:10.1071/EG13104 Zhou W, Du X, Li J (2013) The limitation of curvature gravity gradient tensor for edge detection and a method for overcoming it. J Appl Geophys 98:237–242 The author is most grateful to the two anonymous reviewers for their constructive reviews and comments on the manuscript. In addition, the author is most grateful to Yuichi Hayakawa for his editorial advices and cooperation. The manuscript was improved by these reviewers' comments and suggestions. This work was supported partially by JSPS (Japan Society for the Promotion of Science) KAKENHI Grant Numbers 15K14274, 16H05651, 17K01325. The author is grateful to JSPS. This work was supported partially by JSPS KAKENHI Grant Numbers 15K14274, 16H05651, and 17K01325. SK planned this study and conducted all the calculations and discussion. He also drafted this manuscript. SK is an associate professor at the University of Toyama. The author declares no competing interests. Graduate School of Science and Engineering for Research (Science), University of Toyama, 3910 Gofuku, Toyama, 930-8555, Japan Shigekazu Kusumoto Search for Shigekazu Kusumoto in: Correspondence to Shigekazu Kusumoto. Kusumoto, S. Eigenvector of gravity gradient tensor for estimating fault dips considering fault type. Prog. in Earth and Planet. Sci. 4, 15 (2017) doi:10.1186/s40645-017-0130-0 Fault dip Gravity gradient tensor Normal fault Reverse fault Kurehayama Fault 3. Human geosciences High-definition topographic and geophysical data in geosciences	CommonCrawl
Philosophy of Mind > Philosophy of Consciousness > Consciousness and Materialism > Mind-Body Problem, General Mind-Body Problem, General Edited by David Chalmers (New York University) The Knowledge Argument (334) Zombies and the Conceivability Argument (367) Kripke's Modal Argument Against Materialism (98) Arguments from Disembodiment (62) Other Anti-Materialist Arguments (128) Qualia and Materialism (239) Consciousness and Materialism, Misc (184) Phenomenal Concepts (262) Dualism about Consciousness (131) Metaphysics of Consciousness, Misc (55) Metaphysics of Mind (0) Postdoctoral Research Fellow, Digital Minds Project The Mind/Brain Identity Theory: A Critical Appraisal.Leslie Allan - manuscriptdetails The materialist version of the mind/brain identity theory has met with considerable challenges from philosophers of mind. The author first dispenses with a popular objection to the theory based on the law of indiscernibility of identicals. By means of discussing the vexatious problem of phenomenal qualities, he explores how the debate may be advanced by seeing each dualist and monist ontology through the lens of an evolutionary epistemology. The author suggests that by regarding each ontology as the core of a (...) scientific research programme, each of these logically irrefutable hypotheses can be tested rationally. (shrink) Mind-Body Problem, General in Philosophy of Mind Mind-Brain Identity Theory in Philosophy of Mind Physicalism about the Mind, Misc in Philosophy of Mind Qualia and Materialism in Philosophy of Mind Not Communication.Marc Burock - manuscriptdetails Informational ontologies more and more envelop the natural sciences. The growth of communication technologies and social networking characterize our age. Instead of seeing our world solely as matter in motion, as did Democritus, we now imagine living in a world composed of flowing information. Bits of information have since replaced atoms of matter, and the space-time movement of bits is now called communication. This work is partly a criticism of the materialism and idealism that gave birth to today's worldview, and (...) especially a criticism of the concepts of communication and information as they arise in science and language. Not that I have any interest in proving these concepts false; I agree that each is useful in its domain. Rather, these criticisms may open one up to a less confined communication that is not bounded by science or language but perfuses each. This communication nourishes the paradoxical connection between separated things. (shrink) Intentionality, Misc in Philosophy of Mind Linguistic Communication in Philosophy of Language Causal-Logical Ontology.Johan Gamper - manuscriptdetails In this paper we begin categorizing a plurality of possible worlds on the basis of permitting or not permitting ontologically different things to be causally connected. We build the work on the dual principle that all universes are causally closed either because no universe causes anything outside itself or because no universe has anything in it that is caused by another universe. The Unsolvability of the Mind-Body Problem Liberates the Will.Scheffel Jan - manuscriptdetails The mind-body problem is analyzed in a physicalist perspective. By combining the concepts of emergence and algorithmic information theory in a thought experiment employing a basic nonlinear process, it is argued that epistemically strongly emergent properties may develop in a physical system. A comparison with the significantly more complex neural network of the brain shows that also consciousness is epistemically emergent in a strong sense. Thus reductionist understanding of consciousness appears not possible; the mind-body problem does not have a reductionist (...) solution. The ontologically emergent character of consciousness is then identified from a combinatorial analysis relating to system limits set by quantum mechanics, implying that consciousness is fundamentally irreducible to low-level phenomena. In the perspective of a modified definition of free will, the character of the physical interactions of the brain's neural system is subsequently studied. As an ontologically open system, it is asserted that its future states are undeterminable in principle. We argue that this leads to freedom of the will. (shrink) Consciousness and Physics, Misc in Philosophy of Cognitive Science Explaining Consciousness, Misc in Philosophy of Mind Free Will and Physics in Philosophy of Action Nonreductive Materialism in Philosophy of Mind Philosophy of Mind, General Works in Philosophy of Mind The Explanatory Gap in Philosophy of Mind Relativistic Implications for Physical Copies of Conscious States.Andrew Knight - manuscriptdetails The possibility of algorithmic consciousness depends on the assumption that conscious states can be copied or repeated by sufficiently duplicating their underlying physical states, leading to a variety of paradoxes, including the problems of duplication, teleportation, simulation, self-location, the Boltzmann brain, and Wigner's Friend. In an effort to further elucidate the physical nature of consciousness, I challenge these assumptions by analyzing the implications of special relativity on evolutions of identical copies of a mental state, particularly the divergence of these evolutions (...) due to quantum fluctuations. By assuming the supervenience of a conscious state on some sufficient underlying physical state, I show that the existence of two or more instances, whether spacelike or timelike, of the same conscious state leads to a logical contradiction, ultimately refuting the assumption that a conscious state can be physically reset to an earlier state or duplicated by any physical means. Several explanatory hypotheses and implications are addressed, particularly the relationships between consciousness, locality, physical irreversibility, and quantum no-cloning. (shrink) Personal Identity, Misc in Metaphysics Philosophy of Consciousness, Misc in Philosophy of Mind Philosophy of Physics, Misc in Philosophy of Physical Science Science of Consciousness, Foundations in Philosophy of Cognitive Science The Existential Passage Hypothesis. [REVIEW]David Robert - manuscriptdetails [Excerpt from "Section 1: Summary of the conclusions"] In Chapter 9, Stewart defends the thesis that if non-reductive physicalism is true, then, contrary to a widespread belief, death does not bring about eternal oblivion, a permanent cessation of the stream of consciousness at the moment of death. Stewart argues that the stream of consciousness continues after death—devoid of the body's former memories and personality traits—and it does so as the stream of consciousness of new, freshly conscious bodies (other humans, animals, (...) etc., that are conceived and develop consciousness). And so, any permanent cessation of the stream of consciousness at the moment of death is impossible as long as new, freshly conscious bodies come to exist. Consciousness is defined here as awareness, and is not limited to self-awareness (i.e., the recognition of one's awareness). This general thesis does not specify when in the future those new, freshly conscious bodies must have come into being. This thesis has been independently defended by several authors. (shrink) Death and Dying, Misc in Applied Ethics What Matters in Survival in Metaphysics A Question Concerning Descartes: The Method of Doubt and The Mind/Body Problem.Christopher P. Satoor - manuscriptdetails Metaphysics of Mind in Philosophy of Mind Philosophy of Consciousness in Philosophy of Mind René Descartes in 17th/18th Century Philosophy Mind-Body problemets olösbarhet frigör viljan.Jan Scheffel - manuscriptdetails Mind-body problemet analyseras i ett reduktionistiskt perspektiv. Genom att kombinera emergensbegreppet med algoritmisk informationsteori visas i ett tankeexperiment att ett starkt epistemiskt emergent system kan konstrueras utifrån en relativt enkel, ickelinjär process. En jämförelse med hjärnans avsevärt mer komplexa neurala nätverk visar att även medvetandet kan karakteriseras som starkt epistemiskt emergent. Därmed är reduktionistisk förståelse av medvetandet inte möjlig; mind-body problemet har alltså inte en reduktionistisk lösning. Medvetandets ontologiskt emergenta karaktär kan därefter konstateras utifrån en kombinatorisk analys; det är därmed (...) principiellt oreducerbart till lägre-nivå-fenomen. I perspektivet av en modifierad definition av fri vilja diskuteras den fysiska växelverkan som äger rum i hjärnans neurala system. Trots att enskilda neurala lägre-nivå-processer är deterministiska, kan globala processer visas vara icke- deterministiska i ontologisk mening. Vi argumenterar för att detta leder till viljans frihet. (shrink) Emergence in Cognitive Science in Philosophy of Cognitive Science Theories of Free Will, Misc in Philosophy of Action Methodological Note: Bio-Psycho-Social Being, What Does It Mean?Marcos Wagner Da Cunha - manuscriptdetails The different approaches of the mind-body problem a fortiori have implications on the foundations of Psychology, Psychopathology and Psychiatry, leading to many clashing theories about the determinants of "normal" human behavior, as well of the mental illnesses. These schools of research on the human mind may on a first approach be divided in two main branches: 1) the neurogenetic ones; 2) the psychogenetic ones. This paper sprang up from a lifelong pondering on its subject by its author, while working as (...) a Clinical Psychiatrist and conducting a Ph D in Philosophy. (shrink) Consciousness and Neuroscience, Foundational Issues in Philosophy of Cognitive Science Consciousness and Psychology, Foundational Issues in Philosophy of Cognitive Science Mental Illness in Philosophy of Cognitive Science Philosophy of Psychiatry and Psychopathology, Misc in Philosophy of Cognitive Science A Geneticist's Roadmap to Sanity.Gilbert B. Côté - details World news can be discouraging these days. In order to counteract the effects of fake news and corruption, scientists have a duty to present the truth and propose ethical solutions acceptable to the world at large. -/- By starting from scratch, we can lay down the scientific principles underlying our very existence, and reach reasonable conclusions on all major topics including quantum physics, infinity, timelessness, free will, mathematical Platonism, happiness, ethics and religion, all the way to creation and a special (...) type of multiverse. -/- This article amounts to a summary of my personal Theory of Everything. -/- DOI: 10.13140/RG.2.2.36046.31049. (shrink) Abstract Objects, Misc in Metaphysics Entanglement in Philosophy of Physical Science Ethics and Science in Value Theory, Miscellaneous Happiness in Normative Ethics Mathematical Platonism in Philosophy of Mathematics Multiple Universes in Philosophy of Physical Science Origin of the Universe in Philosophy of Physical Science Philosophy of Information, Misc in Philosophy of Computing and Information Why is there Something? in Philosophy of Physical Science Our Fundamental Problem: A Revolutionary Approach to Philosophy.Nicholas Maxwell - June 2020 - Montreal, Canada: Mcgill-Queen's University Press.details How our human world can exist and best flourish even though it is embedded in the physical universe. Natural Sciences, Misc in Natural Sciences Physicalism, Misc in Metaphysics Wisdom in Epistemology Lightweight and Heavyweight Anti-Physicalism.Damian Aleksiev - forthcoming - Synthese.details I define two metaphysical positions that anti-physicalists can take in response to Jonathan Schaffer's ground functionalism. Ground functionalism is a version of physicalism where explanatory gaps are everywhere. If ground functionalism is true, arguments against physicalism based on the explanatory gap between the physical and experiential facts fail. In response, first, I argue that some anti-physicalists are already safe from Schaffer's challenge. These anti-physicalists reject an underlying assumption of ground functionalism: the assumption that macrophysical entities are something over and above (...) the fundamental entities. I call their position "lightweight anti-physicalism." Second, I go on to argue that even if anti-physicalists accept Schaffer's underlying assumption, they can still argue that the consciousness explanatory gap is especially mysterious and thus requires a special explanation. I call the resulting position "heavyweight anti-physicalism." In both cases, the consciousness explanatory gap is a good way to argue against physicalism. (shrink) Conceptual Analysis and A Priori Entailment in Philosophy of Mind Emergence in Metaphysics Grounding in Metaphysics Metaphysical Levels in Metaphysics Other Anti-Materialist Arguments in Philosophy of Mind Physicalism in Metaphysics Zombies and the Conceivability Argument in Philosophy of Mind Mind-Body Trends & Innovations Body Trends & Innovations From Around the Globe From Around the Globe.Lawrence Biscontini - forthcoming - Mind.details Consciousness and Causality: Dharmakīrti Against Physicalism.Christian Coseru - forthcoming - In Birgit Kellner, McAllister Patrick, Lasic Horst & McClintock Sara (eds.), Reverberations of Dharmakīrti's Philosophy: Proceedings of the Fifth International Dharmakīrti Conference, Heidelberg August 26 to 30, 2014. Vienna: Austrian Academy of Sciences. pp. 21-40.details This paper examines Dharmakīrti's arguments against Cārvāka physicalism in the Pramāṇasiddhi chapter of his magnum opus, the Pramāṇavārttika, with a focus on classical Indian philosophical attempts to address the mind-body problem. The key issue concerns the relation between cognition and the body, and the role this relation plays in causal-explanatory accounts of consciousness and cognition. Drawing on contemporary debates in philosophy of mind about embodiment and the significance of borderline states of consciousness, the paper proposes a philosophical reconstruction that builds (...) on two important features of the Buddhist account: an expanded conception of causality and a robust account of phenomenal content. (shrink) Arguments from Disembodiment in Philosophy of Mind Buddhist Logic in Asian Philosophy Schlick, Carnap and Feigl on the Mind-Body Problem.Sean Crawford - forthcoming - In Thomas Uebel and Christoph Limbeck (ed.), The Routledge Handbook of Logical Empiricism. London, UK: pp. 238-247.details Behaviorism in Philosophy of Mind Carnap: Physicalism in 20th Century Philosophy Logical Empiricism in 20th Century Philosophy Panpsychism in Philosophy of Mind Russellian Monism in Philosophy of Mind Chancy Covariance and The Mind-Body Problem.Benjamin Eva - forthcoming - Oxford Studies in Philosophy of Mind.details Most agree that mental properties depend in some way on physical properties. While phys- icalists describe this dependence in terms of deterministic synchronic relations like identity or supervenience, some dualists prefer to think of it in terms of indeterministic dynamic relations, like causation. I'm going to develop a third conception of the dependence of the mental on the physical that falls somewhere between the deterministic synchronic dependence relations of the physicalist and the indeterministic diachronic dependence relations advocated by some dualists. (...) I'll then use this new conception of metaphysical dependence to formulate a novel approach to the mind body problem that (i) posits a necessary, metaphysically robust synchronic dependence of the mental on the physical, (ii) satisfies several of the key motivations of both non-reductive physicalism and naturalistic dualism, (iii) is consistent with both the causal efficacy of the mental and the causal closure of the physical, and (iv) is capable of reconciling determinism about the physical world with indeterminism about the mental world. (shrink) $87.02 used $90.97 new $98.92 from Amazon (collection) Amazon page Embodied Higher Cognition: Insights From Merleau-Ponty's Interpretation of Motor Intentionality.Jan Halák - forthcoming - Phenomenology and the Cognitive Sciences:1-29.details This paper clarifies Merleau-Ponty's original account of "higher-order" cognition as fundamentally embodied and enacted. Merleau-Ponty's philosophy inspired theories that deemphasize overlaps between conceptual knowledge and motor intentionality or, on the contrary, focus exclusively on abstract thought. In contrast, this paper explores the link between Merleau-Ponty's account of motor intentionality and his interpretations of our capacity to understand and interact productively with cultural symbolic systems. I develop my interpretation based on Merleau-Ponty's analysis of two neuropathological modifications of motor intentionality, the case (...) of the brain-injured war veteran Schneider, and a neurological disorder known as Gerstmann's syndrome. Building on my analysis of Schneider's sensorimotor compensatory performances in relation to his limitations in the domains of algebra, geometry, and language usage, I demonstrate a strong continuity between the sense of embodiment and enaction at all these levels. Based on Merleau-Ponty's interpretations, I argue that "higher-order" cognition is impaired in Schneider insofar as his injury limits his sensorimotor capacity to dynamically produce comparatively more complex differentiations of any given phenomenal structure. I then show how Merleau-Ponty develops and specifies his interpretation of Schneider's intellectual difficulties in relation to the ambiguous role of the body, and in particular the hand, in Gerstmann's syndrome. I explain how Merleau-Ponty defends the idea that sensorimotor and quasi-representational cognition are mutually irreducible, while maintaining that symbol-based cognition is a fundamentally enactive and embodied process. (shrink) Maurice Merleau-Ponty in Continental Philosophy Psychopathology in Philosophy of Cognitive Science The Body in Metaphysics Alien Subjectivity and the Importance of Consciousness.Geoffrey Lee - forthcoming - In Adam Pautz & Daniel Stoljar (eds.), Themes from Block. MIT Press.details The Value of Consciousness in Philosophy of Mind The Super Justification Argument for Phenomenal Transparency.Kevin Morris - forthcoming - Inquiry: An Interdisciplinary Journal of Philosophy.details In Consciousness and Fundamental Reality, Philip Goff argues that the case against physicalist views of consciousness turns on "Phenomenal Transparency", roughly the thesis that phenomenal concepts reveal the essential nature of phenomenal properties. This paper considers the argument that Goff offers for Phenomenal Transparency. The key premise is that our introspective judgments about current conscious experience are "Super Justified", in that these judgments enjoy an epistemic status comparable to that of simple mathematical judgments, and a better epistemic status than run (...) of the mill perceptual judgments. After presenting the key ideas in the "Super Justification Argument", I distinguish two Super Justification theses, which vary according to the kind of introspective judgments that they take to be Super Justified. I argue that Goff's case requires "Strong Super Justification", according to which a wide range of introspective judgments about conscious experience are Super Justified. Unfortunately, it turns out that Strong Super Justification is implausible and not well-supported by examples. In contrast, a weaker Super Justification thesis does not require anything like Phenomenal Transparency and, indeed, can be explained by physicalistic accounts of phenomenal concepts. (shrink) Phenomenal Concepts in Philosophy of Mind Phenomenology and Consciousness in Philosophy of Cognitive Science Qualia in Philosophy of Mind Non-Eliminative Reductionism: The Basis of a Science of Conscious Experience?Dennis Nicholson - forthcoming - Philosophical Psychology.details A physicalist view of qualia labelled non-eliminative reductionism is outlined. If it is true, qualia and physicalism can co-exist without difficulty. First, qualia present no particular problem for reductionist physicalism - they are entirely physical, can be studied and explained using the standard scientific approach, and present no problem any harder than any other scientists face. Second, reductionist physicalism presents no particular problem for qualia – they can be encompassed within an entirely physicalist position without any necessity, either to reduce (...) them to non-existence, or to treat them as new fundamental properties. It is suggested that the position also has sufficient explanatory power to successfully deal with the 'why like anything – why does experience exist at all' question and to counter both Chalmers' Conceivability Argument and Jackson's Knowledge Argument. (shrink) The Knowledge Argument in Philosophy of Mind `Hard' and `Easy' Problems in Philosophy of Mind Body, Mind, and Other Scottish Concordances.Charles Stewart-Robertson - forthcoming - Rivista di Storia Della Filosofia.details Mind–Body Interaction and Modern Physics.Charis Anastopoulos - 2021 - Foundations of Physics 51 (3):1-27.details The idea that mind and body are distinct entities that interact is often claimed to be incompatible with physics. The aim of this paper is to disprove this claim. To this end, we construct a broad mathematical framework that describes theories with mind–body interaction as an extension of current physical theories. We employ histories theory, i.e., a formulation of physical theories in which a physical system is described in terms of a set of propositions about possible evolutions of the system (...) and a probability assignment to such propositions. The notion of dynamics is incorporated into the probability rule. As this formulation emphasises logical and probabilistic concepts, it is ontologically neutral. It can be used to describe mental 'degrees of freedom' in addition to physical ones. This results into a mathematical framework for psycho-physical interaction. Interestingly, a class of ΨΦ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Psi \Phi$$\end{document}I theories turns out to be compatible with energy conservation. (shrink) Interactionism in Philosophy of Mind Quantum Mechanics, Misc in Philosophy of Physical Science Consciousness and the Laws of Physics.Sean M. Carroll - 2021 - Journal of Consciousness Studies 28 (9-10):16-31.details We have a much better understanding of physics than we do of consciousness. I consider ways in which intrinsically mental aspects of fundamental ontology might induce modifications of the known laws of physics, or whether they could be relevant to accounting for consciousness if no such modifications exist. I suggest that our current knowledge of physics should make us skeptical of hypothetical modifications of the known rules, and that without such modifications it's hard to imagine how intrinsically mental aspects could (...) play a useful explanatory role. Draft version of a paper submitted to Journal of Consciousness Studies, special issue responding to Philip Goff's Galileo's Error: Foundations for a New Science of Consciousness. (shrink) The Body and Embodiment: A Philosophical Guide.Frank Chouraqui - 2021 - Rowman & Littlefield Publishers.details Perfect for use at advanced undergraduate and graduate level, this is the first text to offer students a unified narrative regarding the place of the body in Western thinking. The body is simultaneously active and passive, powerful and vulnerable and as such, it fundamentally informs ontological, political, ethical and epistemological issues. Feminism: The Body in Philosophy of Gender, Race, and Sexuality Husserl: Embodiment and Action in Continental Philosophy Racial Identity, Misc in Philosophy of Gender, Race, and Sexuality The Modal Argument Improved.Brian Cutter - 2021 - Analysis 80 (4):629-639.details The modal argument against materialism, in its most standard form, relies on a compatibility thesis to the effect that the physical truths are compatible with the absence of consciousness. I propose an alternative modal argument that relies on an incompatibility thesis: The existence of consciousness is incompatible with the proposition that the physical truths provide a complete description of reality. I show that everyone who accepts the premises of the standard modal argument must accept the premises of the revised modal (...) argument, but not vice versa. (shrink) Epistemological Solipsism as a Route to External World Skepticism.Grace Helton - 2021 - Philosophical Perspectives 35 (1):229-250.details I show that some of the most initially attractive routes of refuting epistemological solipsism face serious obstacles. I also argue that for creatures like ourselves, solipsism is a genuine form of external world skepticism. I suggest that together these claims suggest the following morals: No proposed solution to external world skepticism can succeed which does not also solve the problem of epistemological solipsism. And, more tentatively: In assessing proposed solutions to external world skepticism, epistemologists should explicitly consider whether those solutions (...) extend to knowledge of other minds. Finally, and also tentatively: epistemological solipsism warrants more philosophical attention than it currently enjoys. (shrink) Abduction and Other Minds in Philosophy of Mind Artificial Intelligence Safety in Philosophy of Cognitive Science Machine Consciousness in Philosophy of Cognitive Science Other Minds, Misc in Philosophy of Mind Psychological Behaviorism in Philosophy of Cognitive Science Simulation Hypothesis in Philosophy of Computing and Information Social Epistemology in Epistemology Varieties of Skepticism, Misc in Epistemology The Roots of Occasionalism? Causation, Metaphysical Dependence, and Soul-Body Relations in Augustine.Tamer Nawar - 2021 - Vivarium 59:1-27.details It has long been thought that Augustine holds that corporeal objects cannot act upon incorporeal souls. However, precisely how and why Augustine imposes limitations upon the causal powers of corporeal objects remains obscure. In this paper, the author clarifies Augustine's views about the causal and dependence relations between body and soul. He argues that, contrary to what is often thought, Augustine allows that corporeal objects do act upon souls and merely rules out that corporeal objects exercise a particular kind of (...) causal power (that of efficient or sustaining causes). He clarifies how Augustine conceives of the kind of causal influence exercised by souls and bodies. (shrink) Augustine in Medieval and Renaissance Philosophy Causal Occasionalism in Metaphysics Mental Causation, Misc in Philosophy of Mind Neoplatonists in Ancient Greek and Roman Philosophy Stoics in Ancient Greek and Roman Philosophy The Soul in Philosophy of Religion Theories of Causation, Misc in Metaphysics Modal Arguments Against Materialism.Michael Pelczar - 2021 - Noûs 55 (2):426-444.details Kripke's Modal Argument Against Materialism in Philosophy of Mind The Hard Problem Isn'T Getting Any Easier: Thoughts on Chalmers' "Meta-Problem".Ben White - 2021 - Philosophia 49:495-506.details Chalmers' meta-problem of consciousness is the problem of explaining "problem reports"; i.e. reports to the effect that phenomenal consciousness has the various features that give rise to the hard problem. Chalmers suggests that solving the meta-problem will likely "shed significant light on the hard problem." Against this, I argue that work on the meta-problem will likely fail to make the hard problem any easier. For each of the main stances on the hard problem can provide an account of problem reports, (...) and we have no way of deciding which of these accounts gives the correct explanation of an individual's problem reports without presupposing a stance on the hard problem. We thus cannot determine which of the available solutions to the meta-problem is correct without having already solved the hard problem. (shrink) Magical Thinking.Andrew M. Bailey - 2020 - Faith and Philosophy 37 (2):181-201.details According to theists, God is an immaterial thinking being. The main question of this article is whether theism supports the view that we are immaterial thinking beings too. I shall argue in the negative. Along the way, I will also explore some implications in the philosophy of mind following from the observation that, on theism, God's mentality is in a certain respect magical. Material Through and Through.Andrew M. Bailey - 2020 - Philosophical Studies 177 (8):2431-2450.details Materialists about human persons think that we are material through and through—wholly material beings. Those who endorse materialism more widely think that everything is material through and through. But what is it to be wholly material? In this article, I answer that question. I identify and defend a definition or analysis of 'wholly material'. Formulating Physicalism in Philosophy of Mind Persons, Misc in Metaphysics Disillusioned.Katalin Balog - 2020 - Journal of Consciousness Studies 27 (5-6):38-53.details In "The Meta-Problem of Consciousness", David Chalmers draws a new framework in which to consider the mind-body problem. In addition to trying to solve the hard problem of consciousness – the problem of why and how brain processes give rise to conscious experience –, he thinks that philosophy, psychology, neuro-science and the other cognitive sciences should also pursue a solution to what he calls the "meta-problem" of consciousness – i.e., the problem of why we think there is a problem with (...) consciousness. My claim is that, while Chalmers's project is generously ecumenical as well as beautiful in its meticulous detail, it is mistaken in its core assumption that the meta-problem can be formulated as an "easy problem" for science to solve. Furthermore, the project tilts the field toward illusionism against Type-B materialism, as far as physicalist solutions to the hard problem and the meta-problem are concerned. I will argue that Type-B materialism emerges unscathed from this dialectic. (shrink) Philosophy of Consciousness, Miscellaneous in Philosophy of Mind Life, the Universe and Consciousness: An Introduction to the Theory of Universal Life.A. T. Bollands - 2020 - Oxford, UK: Bollands Publishing.details We live in a world full of mysteries. How do our brains create consciousness? Which animals are conscious, and which are not? How can we have free-will in a deterministic universe? What are the fundamental Laws of Nature? What caused the Big Bang? How can we make sense of Quantum Mechanics? Why is the universe so finely-tuned for Life? And how did Life begin? Despite investigating such mysteries for decades or more, scientists and philosophers are no closer to finding clear (...) and convincing answers that everyone can agree upon. The big question is: why do such mysteries exist, and can they ever be solved? -/- In Life, the Universe and Consciousness, A. T. Bollands introduces us to twelve so-called intractable problems, demonstrating how each one arises from a conflict between beliefs. And yet, beliefs that are in conflict cannot all be true, which means that one or more of them must be false. To solve each problem, therefore, all we need to do is work out which of our beliefs are false and correct them. This may sound easy, but these beliefs include the most fundamental assumptions of modern science. If that wasn't enough, it turns out that these twelve intractable problems are related, such that we cannot solve any one of them satisfactorily until we have solved all twelve within the same coherent worldview. -/- Examining each of these problems in turn, we finally arrive at our twelfth intractable problem: What is Life? A. T. Bollands proposes a new definition for Life, and with it a new way of understanding the physical world – a scientific worldview he calls Universal Life. Armed with this new worldview, he proceeds to solve all twelve intractable problems one by one, thereby developing the Theory of Universal Life and the solution to the "Big Problem of 20th Century Science". By successfully solving each problem, he shows that Universal Life offers us a simpler, more natural, more coherent and hence more credible explanation for ourselves and the world we live in than our current scientific worldview ever can. (shrink) Complex Systems in Philosophy of Physical Science History of Physics in Philosophy of Physical Science Interpretation of Quantum Mechanics in Philosophy of Physical Science Quantum Mechanics, Miscellaneous in Philosophy of Physical Science Space and Time in Philosophy of Physical Science $15.51 new $19.50 from Amazon Amazon page The Backside of Habit: Notes on Embodied Agency and the Functional Opacity of the Medium.Maria Brincker - 2020 - In Fausto Caruana & Italo Testa (eds.), Habits: Pragmatist Approaches from Cognitive Neuroscience to Social Science by Caruana F. & Testa I. (Eds.). Cambridge University Press. Cambridge University Press. pp. 165-183.details In this chapter what I call the "backside" of habit is explored. I am interested in the philosophical implications of the physical and physiological processes that mediate, and which allow for what comes to appear as almost magic; namely the various sensorimotor associations and integrations that allows us to replay our past experiences, and to in a certain sense perceive potential futures, and to act and bring about anticipated outcomes – without quite knowing how. Thus, the term "backside" is meant (...) to refer both the actual mediation and the epistemic opacity of these backstage intermediaries that allow for the front stage magic. The question is if the epistemic complexities around sensorimotor mediation gives us valuable insights into the nature of human agency and further how it might begin to show us new ways to think of the mind as truly embodied yet not reducible to any finite body-as-object. (shrink) Action Theory, Miscellaneous in Philosophy of Action American Pragmatism, Misc in Philosophy of the Americas Habits in Philosophy of Action The Structure of Action in Philosophy of Action Varieties of Action, Misc in Philosophy of Action Causation in Psychology.John Campbell - 2020 - Harvard University Press.details "A blab droid is a robot with a body shaped like a pizza box, a pair of treads, and a smiley face. Guided by an onboard video camera, it roams hotel lobbies and conference centers, asking questions in the voice of a seven-year-old. "Can you help me?" "What is the worst thing you've ever done?" "Who in the world do you love most?" People pour their hearts out in response. This droid prompts the question of what we can hope from (...) social robots. Might they provide humanlike friendship? Philosopher John Campbell doesn't think so. He argues that, while a social robot can remember the details of a person's history better than some spouses can, it cannot empathize with the human mind, because it lacks the faculty for thinking in terms of singular causation. Causation in Psychology makes the case that singular causation is essential and unique to the human species. From the point of view of practical action, knowledge of what generally causes what is often all one needs. But humans are capable of more. We have a capacity to imagine singular causation. Unlike robots and nonhuman animals, we don't have to rely on axioms about pain to know how ongoing suffering is affecting someone's ability to make decisions, for example, and this knowledge is not a derivative of general rules. The capacity to imagine singular causation, Campbell contends, is a core element of human freedom and of the ability to empathize with human thoughts and feelings"--. (shrink) Causal Reasoning, Misc in Epistemology Manipulability Theories of Causation in Metaphysics Philosophy of Psychology, Misc in Philosophy of Cognitive Science Reasons and Causes in Philosophy of Action Singular Causation in Metaphysics Collapse of the Ontological Gradient.Ted Dace - 2020 - Философия И Космология 24:70-82.details Because an unmeasured quantum system consists of information — neither tangible existence nor its complete absence — no property can be assigned a definite value, only a range of likely values should it be measured. The instantaneous transition from information to matter upon measurement establishes a gradient between being and not-being. A quantum system enters a determinate state in a particular moment until this moment is past, at which point the system resumes its default state as an evolving superposition of (...) potential values of properties, neither strictly being nor not-being. Like a "self-organized" chemical system that derives energy from breaking down environmental gradients, a quantum system derives information from breaking down the ontological gradient. An organism is a body in the context of energy and a mind in the context of information. (shrink) Physics of Time in Philosophy of Physical Science Wave-Particle Duality in Philosophy of Physical Science Causal Closure of the Physical, Mental Causation, and Physics.Dejan R. Dimitrijević - 2020 - European Journal for Philosophy of Science 10 (1):1-22.details The argument from causal closure of the physical is usually considered the most powerful argument in favor of the ontological doctrine of physicalism. Many authors, most notably Papineau, assume that CCP implies that physicalism is supported by physics. I demonstrate, however, that physical science has no bias in the ontological debate between proponents of physicalism and dualism. I show that the arguments offered for CCP are effective only against the accounts of mental causation based on the action of the mental (...) forces of a Newtonian nature, i.e. those which manifest themselves by causing accelerations. However, it is conceivable and possible that mental causation is manifested through the redistribution of energy, momentum and other conserved quantities in the system, brought about by altering the state probability distribution within the living system and leading to anomalous correlations of neural processes. After arguing that a probabilistic, interactionist model of mental causation is conceivable, which renders the argument from causal closure of the physical ineffective, I point to some basic features that such a model must have in order to be intelligible. At the same time, I indicate the way that conclusive testing of CCP can be done within the theoretical framework of physics. (shrink) The Exclusion Problem in Philosophy of Mind Thermodynamics and Statistical Mechanics in Philosophy of Physical Science The Oxford Handbook of the Philosophy of Consciousness.Uriah Kriegel (ed.) - 2020 - Oxford: Oxford University Press.details The Oxford Handbook of the Philosophy of Consciousness provides the most comprehensive overview of current philosophical research on consciousness. Featuring contributions from some of the most prominent experts in the field, it explores the wide range of types of consciousness there may be, the many psychological phenomena with which consciousness interacts, and the various views concerning the ultimate relationship between consciousness and physical reality. It is an essential and authoritative resource for anyone working in philosophy of mind or interested in (...) states of consciousness. (shrink) Philosophy of Consciousness, General Works in Philosophy of Mind Theories of Consciousness, Miscellaneous in Philosophy of Mind $125.79 used $132.41 new $143.55 from Amazon Amazon page The Mind-Body Problem(s) in Descartes' "Meditations" and Husserl's "Crisis" (Part1).Andrii Leonov - 2020 - Filosofska Dumka 4:91-100.details The main topic of this paper is the mind-body problem. The author analyzes it in the context of Hus- serlian phenomenology. The key texts for the analysis and interpretation are Descartes' magnum opus "Meditations on the First Philosophy" and Husserl' last work "The Crisis of European Sciences and Transcendental Phenomenology". The author claims that already in Descartes' text instead of one mind-body problem, one can find two: the ontological mind-body problem (mind-brain relation) and conceptual one ("mind" and "body" as concepts). (...) In Descartes' "Meditations", the ontological level is explicit, while the conceptual level is implicit. In Husserl's "Crisis", on the other hand, the situation is different: the conceptual level of the problem (as the opposition between transcendental phenom- enology and natural sciences) is explicit, while the ontological level is implicit. Nevertheless, it seems that Husserl has answers to both the "traditional" as well as the "conceptual" mind-body problems. (shrink) Husserl and Other Philosophers, Misc in Continental Philosophy Husserl: Crisis in Continental Philosophy Husserl: Philosophy of Mind, Misc in Continental Philosophy Reducing Uncertainty: Understanding the Information-Theoretic Origins of Consciousness.Garrett Mindt - 2020 - Dissertation, Central European Universitydetails Ever since the hard problem of consciousness (Chalmers, 1996, 1995) first entered the scene in the debate over consciousness many have taken it to show the limitations of a scientific or naturalist explanation of consciousness. The hard problem is the problem of explaining why there is any experience associated with certain physical processes, that is, why there is anything it is like associated with such physical processes? The character of one's experience doesn't seem to be entailed by physical processes and (...) so an explanation which can overcome such a worry must (1) explain how physical processes give rise to experience (explain the entailment), (2) give an explanation which doesn't rely on such physical processes, or (3) show why the hard problem is misguided in some sense. Recently, a rather ambitious and novel theory of consciousness has entered the scene – Integrated Information Theory (IIT) of Consciousness (Oizumi et al., 2014; Tononi, 2008; Tononi et al., 2016) – and proposes that consciousness is the result of a specific type of information processing, what those developing the theory call integrated information. The central aim of this dissertation is to philosophically investigate IIT and see whether it has the ability to overcome the hard problem and related worries. I then aim to use this philosophical investigation to answer a set of related questions which guide this dissertation, which are the following: Is it possible to give an information-theoretic explanation of consciousness? What would the nature of such an explanation be and would it result in a novel metaphysics of consciousness? In this dissertation, I begin in chapter one by first setting up the hard problem and related arguments against the backdrop of IIT (Mindt, 2017). I show that given a certain understanding of structural and dynamical properties IIT fails to overcome the hard problem of consciousness. I go on in chapter two to argue that a deflationary account of causation is the best view for IIT to overcome the causal exclusion problem (Baxendale and Mindt, 2018). In chapter three, I explain IIT's account of how the qualitative character of our experience arises (qualia) and what view of intentionality (the directedness of our mental states) IIT advocates. I then move on in chapter four to show why the hard problem mischaracterizes structural and dynamical properties and misses important nuances that may shed light on giving a naturalized explanation of consciousness. In the last and fifth chapter, I outline a sketch of a novel metaphysics of consciousness that takes the conjunction of Neutral Monism and Information-Theoretic Structural Realism to give what I call Information-Theoretic Neutral-Structuralism. (shrink) Consciousness and Neuroscience, Misc in Philosophy of Cognitive Science Nevědomí jako dvojznačné vědomí. Merleau-Ponty o psychoanalýze.Jan Puc - 2020 - Ostium 16 (1).details Merleau-Ponty's attitude to psychoanalysis was ambiguous. On the one hand, he realized that the phenomena psychoanalysis deals with require to go beyond the area of act intentionality, and that, from a different angle, psychoanalysis addresses the same problem as Gestalt psychology, which played the central role in Merleau-Ponty's philosophical project. On the other hand, he explicitly rejected the terms used by Freud for conveying his discoveries. Merleau-Ponty replaced unconscious mental contents, which act on conscious behavior, by ambiguous consciousness. In the (...) Structure of Behavior and Phenomenology of Perception, he used the terms "habit," "bad faith," "bodily expression," "affective intentionality," and "body schema" to specify his notion of experiential "ambiguity." This paper aims to present the key concepts of Merleau-Ponty's early interpretation of the psychoanalytic concept of the unconscious. A partial task is to show how Merleau-Ponty's concept of bad faith differs from that of Sartre, and to distinguish existential ambiguity from the psychoanalytic concept of overdetermination. (shrink) Bodily Experience in Philosophy of Mind Embodied Memory in Philosophy of Mind Jean-Paul Sartre in Continental Philosophy Unconscious and Conscious Processes in Philosophy of Cognitive Science Causal Efficiency of Intentional Acts.Maria A. Sekatskaya - 2020 - Epistemology and Philosophy of Science 57 (1):79-95.details Causal Accounts of Mental Content, Misc in Philosophy of Mind Free Will and Neuroscience in Philosophy of Action Free Will and Psychology in Philosophy of Action The Will in Philosophy of Action Recensione di 'The Outer Limits of Reason' (I limiti esterni della ragione) di Noson Yanofsky 403p (2013)(revisto 2019).Michael Richard Starks - 2020 - In Benvenuti all'inferno sulla Terra: Bambini, Cambiamenti climatici, Bitcoin, Cartelli, Cina, Democrazia, Diversità, Disgenetica, Uguaglianza, Pirati Informatici, Diritti umani, Islam, Liberalismo, Prosperità, Web, Caos, Fame, Malattia, Violenza, Intellige. Las Vegas, NV, USA: Reality Press. pp. 182-196.details Io do una recensione dettagliata di 'The Outer Limits of Reason' di Noson Yanofsky da una prospettiva unificata di Wittgenstein e psicologia evolutiva. Inditesto che la difficoltà con questioni come il paradosso nel linguaggio e nella matematica, l'incompletezza, l'indecidibilità, la computabilità, il cervello e l'universo come computer ecc., derivano tutto dall'incapacità di guardare attentamente al nostro uso del linguaggio nel contesto appropriato e quindi alla mancata separazione delle questioni di fatto scientifico dalle questioni di come funziona il linguaggio. Discuto le (...) opinioni di Wittgenstein sull'incompletezza, la paracoerenza e l'indecidibilità e il lavoro di Wolpert sui limiti del calcolo. Per riassumere: L'Universo Secondo Brooklyn---Buona Scienza, Non Così Buona Filosofia. Coloro che desiderano un quadro aggiornato completo per il comportamento umano dalla moderna vista a due systems possono consultare il mio libro 'La struttura logica dellafilosofia, psicologia, Mind e il linguaggio in Ludwig Wittgenstein e John Searle' 2nd ed (2019). Coloro che sono interessati a più dei miei scritti possono vedere 'TalkingMonkeys--Filosofia, Psicologia, Scienza, Religione e Politica su un Pianeta Condannato--Articoli e Recensioni 2006-2019 3rd ed (2019) e Suicidal Utopian Delusions in the 21st Century 5th ed (2020) . (shrink) Liar Paradox in Logic and Philosophy of Logic Paradoxes, Miscellaneous in Logic and Philosophy of Logic Chalmers V Chalmers.Daniel Stoljar - 2020 - Noûs 54 (2):469-487.details This paper brings out an inconsistency between David Chalmers's dualism, which is the main element of his philosophy of mind, and his structuralism, which is the main element of his epistemology. The point is ad hominem , but the inconsistency if it can be established is of considerable independent interest. For the best response to the inconsistency, I argue, is to adopt what Chalmers calls 'type‐C Materialism', a version of materialism that has been much discussed in recent times because of (...) its promise to move us beyond the stand‐off between standard versions of materialism and dualism. In turn, if that version of materialism is true, both dualism and structuralism should be rejected. (shrink) The Epistemic Approach to the Problem of Consciousness.Daniel Stoljar - 2020 - In Uriah Kriegel (ed.), The Oxford Handbook of the Philosophy of Consciousness. Oxford, UK:details Neutral Monism in Philosophy of Mind $125.79 used $132.41 new $143.55 from Amazon (collection) Amazon page Panpsychism and Non-Standard Materialism: Some Comparative Remarks.Daniel Stoljar - 2020 - In William Seager (ed.), The Routledge Handbook of Panpsychism. New York, NY, USA:details Much of contemporary philosophy of mind is marked by a dissatisfaction with the two main positions in the field, standard materialism and standard dualism, and hence with the search for alternatives. My concern in this paper is with two such alternatives. The first, which I will call non-standard materialism, is a position I have defended in a number of places, and which may take various forms. The second, panpsychism, has been defended and explored by a number of recent writers. My (...) main goals are: (a) to explain the differences between these positions; and (b) to suggest that non-standard materialism is more plausible than panpsychism. (shrink) Panpsychism, Misc in Philosophy of Mind The Combination Problem for Panpsychism in Philosophy of Mind The (Real) Theory of Everything.Ilexa Yardley - 2020 - Intelligent Design Center.details Yin and Yang (ancient) is Zero and One (modern). $200.00 new Amazon page Consciousness and World. A Neurophilosophical and Neuroethical Account.Federico Zilio - 2020 - Pisa: Edizioni ETS.details "What is consciousness?" "What is the relationship between consciousness and the world?" Contemporary consciousness studies are dominated by a neurocentric paradigm that tends to reduce our mind to a mere product of the brain, thus impeding the complete understanding of the multifaceted nature of consciousness. It is therefore necessary to change the direction of research, focusing no more on the isolated brain or on the disembodied mind, rather on an interdisciplinary and nonreductive approach to experience that intertwines philosophy, phenomenology, and (...) neuroscience. This neurophilosophical and neuroethical perspective will reconsider consciousness not merely in terms of the brain- or mind-relationship, but as intrinsically world-related. (shrink) Consciousness and Neuroscience in Philosophy of Cognitive Science Neuroethics in Applied Ethics Neurophilosophy in Philosophy of Cognitive Science Phenomenology in Continental Philosophy Against the Middle Ground: Why Russellian Monism is Unstable.Brian Cutter - 2019 - Analytic Philosophy 60 (2):109-129.details Chalmers' Principle of Organizational Invariance Makes Consciousness Fundamental but Meaningless Spectator of its Own Drama.Danko Georgiev - 2019 - Activitas Nervosa Superior 61 (4):159-164.details The principles of classical physics, including deterministic dynamics and observability of physical states, are incompatible with the existence of unobservable conscious minds that possess free will. Attempts to directly accommodate consciousness in a classical world lead to philosophical paradoxes such as causally ineffective consciousness and possibility of alternate worlds in which functional brain isomorphs behave identically but lack conscious experiences. Here, we show that because Chalmers' principle of organizational invariance is based on a deficient nineteenth century classical physics, it is (...) inherently flawed and implies evolutionary inexplicable epiphenomenal consciousness. Consequently, if consciousness is a fundamental ingredient of physical reality, no psychophysical laws such as Chalmers' principle of organizational invariance are needed to establish correspondence between conscious experiences and brain function. Quantum mechanics is the most successful and only modern physical theory capable of naturally accommodating consciousness without violation of physical laws. (shrink) Functionalism and Qualia in Philosophy of Mind	CommonCrawl
Why the value of Gaussian curve drop to 1/19 at 2 standard deviation? Taken from Guide to DSP where it says: ... at two ... standard deviations from the mean, the value of the Gaussian curve has dropped to about 1/19 ... It seems to be a straight forward calculation but my math just didn't work out to 1/19. The Gaussian probability distribution function is given by: $$ P(x) = {1 \over \sqrt{2\pi} \sigma } e^{-{{(x - \mu)^2} \over {2 \sigma^2}}} $$ At the mean (where maximum probability occurs), $$ \begin{align} P(\mu) &= {1 \over \sqrt{2\pi} \sigma } e^{-{{(\mu - \mu)^2} \over {2 \sigma^2}}} \\ &= {1 \over \sqrt{2\pi} \sigma } e^{0} = {1 \over \sqrt{2\pi} \sigma} \end{align} $$ And at 2 standard deviation, or $ x = \mu +2\sigma $ $$ \begin{align} P(\mu + 2\sigma) &= {1 \over \sqrt{2\pi} \sigma } e^{-{{(\mu + 2\sigma - \mu)^2} \over {2 \sigma^2}}} \\ &= {1 \over \sqrt{2\pi} \sigma } e^{-{{4\sigma^2} \over {2 \sigma^2}}} = {1 \over \sqrt{2\pi} \sigma } e^{-2} \end{align} $$ To see how much probability has dropped from max probability at $ \mu $ to probability at $ 2\sigma $, I can simply take the ratio: $$ { P(\mu + 2 \sigma) \over P(\mu) } = { {{1 \over {\sqrt{2\pi}\sigma}} e^{-2}} \over {{1 \over {\sqrt{2\pi}\sigma}}}} = e^{-2} = 0.135 \approx {2 \over 15} \neq {1 \over 19} $$ gaussian probability-distribution-function KMCKMC $\begingroup$ Yeah, that seems like a mistake. Since he publishes errata as well, maybe drop the author an email. $\endgroup$ – mmmm $\begingroup$ The blog must have referred to the standard normal distribution, i.e. the Gaussian curve with $\mu=0$ and $\sigma=1$. In that case, the values of the PDF at 2, 4, and 6 are 1/19, 1/7563, and 1/166,666,666 respectively. I agree that the wording of the blog is confusing. $\endgroup$ – AlexTP You've computed the most general case and have shown it's always 2/15, thus 1/19 is incorrect, at least if interpreting "1/x-th of value" as $f(\text{value})/f_\text{max}$. This simulation confirms it. Edit: I took a guess, 1/19 is the value of the Gaussian at two standard deviations for $(\mu, \sigma) = (0, 1)$, which does qualify per exact wording of "the value drops to 1/19". But I agree your measure is more meaningful as it's independent of $(\mu, \sigma)$ and interprets as "the value drops to 2/15 of its peak value". OverLordGoldDragonOverLordGoldDragon As noted by AlexTP and OverLordGoldDragon, the confusion is that the actual value of the Gaussian probability density curve is 1/19, while the value compared to the 0.4 peak value is 2/15. When talking about probability distributions, the middle or peak value does not have a particular significance. The absolute value of PDF is more useful in practice than the ratio to peak value. For example, a distribution that has a wide flat peak with sharp edges would have a lower peak probability, for example 0.2. If it has density of 1/19 at 2 sigma, it has the same probability of having outliers that far as the Gaussian distribution would. But if you used the comparison to peak value, you would think it was less sharp drop than the Gaussian, just because the middle is more flat. As an example, let's compare the Laplace distribution to Gaussian (normal) distribution: (Image source) The ratio of value at 2 standard deviations compared to peak value is 0.1 for both distributions. The actual probability of events outside 2 sigma is lower for Gaussian distribution, while Laplace distribution does have steeper descent at the middle. The discussion in the original article was that in Gaussian distribution the "tails drop toward zero very rapidly", where tails means the long portion further away from middle. jpajpa $\begingroup$ 1/19 by itself provides no information on extent of decay, whereas 2/15 does. Only if another absolute value is provided can one infer about rate of decay. $\endgroup$ – OverLordGoldDragon $\begingroup$ @OverLordGoldDragon Also, what about a uniform distribution whose support is $[-M\sigma,+M\sigma]$ where $\sigma^2$ is the variance of the normal distribution of interest. The ratio "mean to (every point of the support)" is $1$, and I can choose $M$ to make the PDF value as small as I want. The ratio is not useful without assuming a class of distribution. $\endgroup$ $\begingroup$ @OverLordGoldDragon (1) No, the OP's linked context wants to justify "In practice, the sharp drop of the Gaussian pdf dictates that these extremes almost never occur." by the rate of decay, which is confusing and simply misleading. In the context of the OP's question and the referenced blog, the ratio is only useful to compare two normal distributions. (2) I don't say that the ratio is always useless because if you want to use it, it is useful. But, again, not in the context of this specific question of the OP. $\endgroup$ $\begingroup$ @AlexTP Absolute value at $s$ standard deviations changes, but not ratio to peak; since this holds at all points and for every $(\mu, \sigma)$, it illustrates invariance. $\endgroup$ $\begingroup$ @OverLordGoldDragon and (2) Isn't the invariance just a consequence of "the affine transformation of a normally distributed random variable is a normally distributed random variable"? For (2), to avoid writing down rigorous proof, just think that distribution is fully defined by its PDF. I mean your statement about the invariance may be a nice property, but not many find it useful, sorry. You may want to create a new question or answer to avoid extending this comment section. $\endgroup$ Not the answer you're looking for? Browse other questions tagged gaussian probability-distribution-function or ask your own question. Choice of Gaussian kernel parameters when lowpass filtering before image resampling? Gaussian random generator How to determine variance/std deviation of Gaussian noise from measured data From Uniform to 2D gaussian Gaussian window and standard deviation Why should an image be blurred using a Gaussian Kernel before downsampling? Time derivative of signal - effect on noise distribution Scaling a wavelet in continuous wavelet transform	CommonCrawl
Clinical evaluation of corneal changes after phacoemulsification in diabetic and non-diabetic cataract patients, a systematic review and meta-analysis Changes in corneal endothelial cell density after initial Ex-PRESS drainage device implantation and its relating factors over 3 years Yurika Aoyama, Rei Sakata, … Makoto Aihara Evaluation of corneal hysteresis after pars plana vitrectomy combined phacoemulsification and intraocular lens implantation Manami Ohta, Makiko Wakuta, … Kazuhiro Kimura Corneal endothelial cell loss after trabeculectomy and phacoemulsification in one or two steps: a prospective study María Isabel Soro-Martínez, Juan Antonio Miralles de Imperial-Ollero, … María Paz Villegas-Pérez Risk Factors for Corneal Endothelial Cell Loss in Patients with Pseudoexfoliation Syndrome Takanori Aoki, Koji Kitazawa, … Chie Sotozono Intraocular lens power calculations in eyes with pseudoexfoliation syndrome Aleksandra Wlaź, Agnieszka Kustra, … Tomasz Żarnowski Femtosecond laser-assisted cataract surgery after corneal refractive surgery Hyunmin Ahn, Ikhyun Jun, … Tae-im Kim Intraocular Lens power calculation after laser refractive surgery: A Meta-Analysis Hui Chen, Xinyi Chen, … Ke Yao Changes in the corneal thickness and limbus after 1 year of scleral contact lens use Beatriz de Luis Eguileor, Arantxa Acera, … Jaime Etxebarria Ecenarro Posterior Chamber Phakic Intraocular Lens Implantation in Eyes with an Anterior Chamber Depth of Less Than 3 mm: A Multicenter Study Kazutaka Kamiya, Kimiya Shimizu, … Kazuo Ichikawa Yizhen Tang1,2, Xinyi Chen1,2, Xiaobo Zhang1,2, Qiaomei Tang1,2, Siyu Liu1,2 & Ke Yao1,2 Scientific Reports volume 7, Article number: 14128 (2017) Cite this article Corneal diseases Eye manifestations Corneal endothelium morphological abnormalities result in fluid imbalance, stromal swelling, and loss of transparency, thus impairing visual function. Recently, growing number of studies have focused on diabetic corneal abnormalities after cataract surgery and its comparison with non-diabetic patients, the results remain conflicting. Thus, to evaluate the effect of phacoemulsification on the corneal properties in diabetic and non-diabetic patients, prospective studies were comprehensively searched through PubMed, EMBASE, and Cochrane databases updated to Jan 2017. A meta-analysis of the 13 identified studies was performed using weighted mean difference (WMD) and 95% confidence interval (CI). For the dynamic changes between preoperative and postoperative values, significant differences were identified between the two groups in endothelial cell density (ECD) and hexagon cells (HC%) at 1 day, 1 week, 1 month, and 3 months postoperatively, in central corneal thickness (CCT) at 1 month postoperatively, and in coefficient variation (CV) at 1 week and 1 month postoperatively. However, no significant differences were observed in CCT at 1 day, 1 week and 3 months postoperatively or in CV at 1 day and 3 months postoperatively. Diabetic corneas are more vulnerable to stress and trauma, resulting in greater morphological abnormalities and longer recovery time. As of 2015, an estimated 415 million people had diabetes worldwide1, which is almost 1.5 times greater than in 2010 (285 million). The global prevalence of diabetes is growing much faster than earlier forecasts predicted (366 million by the year 2030)2,3. Obviously, diabetes mellitus (DM) is becoming more prevalent and threatening than was previously thought. It has been acknowledged that DM leads to various complications such as nephropathy, neuropathy, cardiovascular issues, and several ocular complications like diabetic retinopathy, diabetic cataract, diabetic keratopathy, and diabetic optic nerve diseases4. Although the cornea may appear disease free in the diabetic, an awareness of the marked biochemical and ultrastructural abnormalities in the diabetic enables us to prevent more overt complications. The diabetic cornea suffers from endothelium cellular dysfunction and dysfunctional repair mechanisms including corneal edema, delayed wound healing, and so on5,6. Over the past decades, the pathology of the diabetic corneal endothelium dysfunction has become understood in more detail7. The state of hyperglycemia results in an increase in aldose reductase activity, the expression of metalloproteinase (MMP), and the formation of advanced glycation end products (AGEs). Evidence has shown that the inhibition of aldose reductase reduces dysmorphological changes in the corneal endothelium8,9. Enhanced MMPs can damage the basement membrane and limit cell migration, resulting in poor healing10. Moreover, the accumulation of AGEs can lead to an abnormality of cell adhesion11. The cornea is likely to be more vulnerable to stress and trauma in diabetic patients than in non-diabetics. Phacoemulsification has become the predominant treatment procedure for cataract, the leading cause of blindness and visual impairment worldwide12,13. Although most cataract patients achieve decent recovery, unfortunately, in a complex disease environment, cataract surgery with phacoemulsification and lens implantation leads to larger endothelial cell loss in diabetic corneas14,15,16,17,18. Furthermore, the corneal endothelium can be adversely affected by surgery due to factors like lens nuclear sclerosis, effective phacoemulsification time (EPT), phacoemulsification energy, and IOL implantation15,19,20,21. These factors coupled with the effect of DM indicate a great risk of long-term endothelium cell dysfunction with decompensation and the development of bullous keratopathy22. Accelerated losses of corneal endothelial cells have been reported to continue even 10 years after surgery23. To evaluate the corneal state, corneal thickness and endothelial cell morphology are the top two clinical concerns. Central corneal thickness (CCT), as an indicator of the physiological condition of the corneal endothelium, is generally used in diagnoses like keratoconus, Fuchs' dystrophy, and glaucoma. Recognizing CCT is important because it can mask an accurate reading of intraocular pressure (IOP)24, causing doctors to unnecessarily treat for a condition that may not exist. Endothelial morphological changes in corneas, including endothelial density (ECD), coefficient of variation (CV), and percentage of hexagonal cells (HC%), can alter the cornea's ability to function. Abnormal corneal endothelial cell morphology coupled with increased CCT25 is another marker of endothelial cell dysfunction, which results in fluid imbalance, stromal swelling, and loss of transparency, thus impairing visual function. Recently, a growing number of studies have focused on the importance of diabetic corneal abnormalities, which were commonly found in patients after cataract surgery due to factors including diabetic state and surgical procedures26. However, there is still conflict concerning the differences in corneal properties between diabetic and non-diabetic patients after phacoemulsification. According to our knowledge, there has been no comprehensive review or meta-analysis regarding corneal changes after phacoemulsification for diabetic and non-diabetic groups so far. Under the circumstances, this article is set to evaluate the effect of phacoemulsification on ECD, HC%, CV, and CCT in diabetic patients and non-diabetic patients. Literature selection The workflow chart in Fig. 1 shows the literature selection process. After duplication removal, a total of 361 studies were retrieved from the databases. 330 studies were excluded by scanning titles and abstracts. Furthermore, 18 studies were excluded after full-text reading: four on extracapsular cataract extraction (ECCE), two on manual small incision cataract surgery (MSICS) which is different from phacoemulsification from either the incision size or the surgical method, five retrospective studies, five with unavailable data (e.g. no postoperative data, no cohort, or an unmatched comparison group), one that was a poster, and one that was a review. Finally, 13 studies17,19,27,28,29,30,31,32,33,34,35,36,37 meeting all of the predefined criteria were identified. The characteristics of the included studies are shown in Table 1. The details of N OS scale are shown in Table S6. Workflow chart of the literature selection process. Table 1 Characteristics of included studies. NA = not available, DM = diabetes mellitus, NOS = The Newcastle–Ottawa scale. Meta-analysis of the outcomes A total of 13 prospective studies including 1923 eyes (941 in the non-diabetic group and 982 in the diabetic group) were identified. The surgical parameters (e.g. phaco-time and phaco-energy) and lens nucleus hardness were reported with no significant differences between the DM and non-DM groups, as shown in Tables S1–S4. Endothelial cell density There were thirteen studies reporting the outcome of ECD. The analysis was made at 1 day, 1 week, 1 month, and 3 months postoperatively. It was found that diabetic patients have a significantly lower ECD at preoperative and all postoperative time points than the non-diabetic group (baseline: WMD = −98.60, 95% CI: −181.39 to −15.82, P = 0.02; 1 day postoperative: WMD = −129.29, 95% CI: −149.47 to −109.10, P < 0.001; 1 week postoperative: WMD = −192.17, 95% CI: −267.04 to −117.29, P < 0.001; 1 month postoperative: WMD = −205.53, 95% CI: −258.30 to −152.76, P < 0.001; 3 months postoperative: WMD = −229.83, 95% CI: −283.54 to −176.12, P < 0.001; Fig. 2). Furthermore, the percentage of the loss of ECD (ECL%, difference between preoperative and postoperative), which was calculated from equations (1–5), was also evaluated to see the effect caused by phacoemulsification. There are significant differences in ECL% at all postoperative times for the DM group compared to the non-DM group (1 day postoperative: WMD = 3.40, 95% CI: 1.82 to 4.97, P < 0.001; 1 week postoperative: WMD = 3.45, 95% CI: 2.64 to 4.25, P < 0.001; 1 month postoperative: WMD = 3.80, 95% CI: 1.84 to 5.75, P < 0.001; 3 months postoperative: WMD = 4.85, 95% CI: 1.60 to 8.10, P = 0.003; Fig. 3). Forest plot comparison of the corneal endothelial cell density (ECD) after phacoemulsification in diabetic and non-diabetic patients. Forest plot comparison of the corneal endothelial cell loss in percentage (ECL%) after phacoemulsification in diabetic and non-diabetic patients. There were eight studies reporting the outcome of coefficient of variation. The analysis was made at 1 day, 1 week, 1 month, and 3 months postoperatively. It was observed that diabetic patients had a significantly higher CV at preoperative and all postoperative time points than the non-DM group (baseline: WMD = 2.62, 95% CI: 1.41 to 3.83, P < 0.001; 1 week postoperative: WMD = 5.09, 95% CI: 2.68 to 7.51, P < 0.001; 1 month postoperative: WMD = 6.71, 95% CI: 3.60 to 9.81, P < 0.001; 3 months postoperative: WMD = 6.65, 95% CI: 3.14 to 10.15, P < 0.001), except at 1 day postoperatively (WMD = 4.75, 95% CI: −1.51 to 11.00, P = 0.14, Fig. 4). The increase of CV (dCV, difference between preoperative and postoperative) appears significantly larger in the DM group compared to the non-DM group at 1 week postoperatively (WMD = 2.02, 95% CI: 0.66 to 339, P = 0.004) and at 1 month postoperatively (WMD = 3.68, 95% CI: 0.47 to 6.88, P = 0.02), while no significant differences were found at 1 day postoperatively (WMD = 3.27, 95% CI: −3.02 to 9.56, P = 0.31) and 3 months postoperatively (WMD = 3.24, 95% CI: −1.06 to 7.53, P = 0.14, Fig. 5). Forest plot comparison of the coefficient of variation (CV) of corneal endothelial cells after phacoemulsification in diabetic and non-diabetic patients. Forest plot comparison of the change of coefficient of variation (dCV) of corneal endothelial cells after phacoemulsification in diabetic and non-diabetic patients. Hexagonal cell percentage There were 11 studies reporting the outcome of hexagonal cells. The analysis was made at 1 day, 1 week, 1 month, and 3 months postoperatively. It was found that diabetic patients have a significantly smaller HC% at preoperative and all postoperative time points (all: P < 0.001, Fig. 6) and a significantly larger HC% loss (difference of preoperative and postoperative) at all postoperative times (all: P < 0.001, Fig. 7) compared to the non-DM group. Forest plot comparison of the percentage of hexagonal corneal endothelial cells (HC%) after phacoemulsification in diabetic and non-diabetic patients. Forest plot comparison of the loss of percentage of hexagonal corneal endothelial cells (dHC%) after phacoemulsification in diabetic and non-diabetic patients. Central corneal thickness There were four studies reporting the outcome of CCT. The analysis was made at 1 day, 1 week, 1 month, and 3 months postoperatively. No significant difference was observed in CCT between diabetic patients and non-diabetic patients. (baseline: WMD = 2.86, 95% CI: −3.65 to 9.37, P = 0.39, 1 day postoperative: WMD = 13.37, 95% CI: −5.24 to 31.99, P = 0.16). After phacoemulsification, the CCT of the DM group was significantly higher than that of the non-DM group at all postoperative time points (1 week postoperatively: WMD = 17.96, 95% CI: 5.24 to 30.68, P = 0.006; 1 month postoperatively: WMD = 22.59, 95% CI: 10.23 to 34.94, P < 0.001; 3 months postoperatively: WMD = 12.92, 95% CI: 9.22 to 16.63, P < 0.001; Fig. 8). Moreover, the percent increase of CCT (dCCT%, difference between preoperative and postoperative percentages) showed no significant difference between the DM group and the non-DM group at 1 day postoperatively (WMD = 2.26, 95% CI −1.82 to 6.43, P = 0.28), 1 week postoperatively (WMD = 2.81, 95% CI: −0.36 to 5.98, P = 0.08) and 3 months postoperatively (WMD = 1.56, 95% CI: −0.57 to 3.70, P = 0.15), but a significantly larger dCCT% was found in diabetic patients at 1 month postoperatively compared to the non-diabetic ones (WMD = 3.86, 95% CI: 1.28 to 6.45, P = 0.003, Fig. 9). Forest plot comparison of the central corneal thickness (CCT) after phacoemulsification in diabetic and non-diabetic patients. Forest plot comparison of the increased central corneal thickness percentage (dCCT%) after phacoemulsification in diabetic and non-diabetic patients. Sensitivity analysis and publication bias Since some of the results show heterogeneity (${I}^{2} > 50$), a random-effect meta-regression model was chosen to analyze this condition. No publication bias was found through Begg's and Egger's test as shown in Table S5. Outcomes included in only 3 studies are too small to do a sensitivity analysis so as to exclude trials at a high risk of bias. Thus, one-study-removed analyses were conducted for all remaining outcomes. The sensitivity analysis revealed that there are two pooled outcomes lacking stability: the outcome of ECD at the baseline and the outcome of dCV at 1 month postoperatively. Yan and Chen (2014)29, whose patients had a long duration of DM, is the source of the statistical heterogeneity in the meta-analysis of the preoperative ECD result as shown in Table 2. When this outlier study is removed, the outcome is stable, indicating that the outcome is still rational and reliable. The heterogeneity of the second sensitive outcome may be due to several design differences among the studies that affect the response degree and recovery duration, such as duration of diabetes and blood glucose control. No significant publication bias was demonstrated in the funnel plot. Table 2 One-study-removed analysis for the outcomes of ECD preoperatively and dCV at 1 month postoperatively. The results of the present meta-analysis provided robust evidence that the effect of phacoemulsification on corneal changes in diabetics is greater than for non-diabetics. Significant differences have been observed between the diabetic and non-diabetic groups in terms of ECD, HC%, CV, and CCT preoperatively and 1 day, 1 week, 1 month, and 3 months postoperatively, except CV at 1 day postoperatively and CCT preoperatively. For the changes between the preoperative and postoperative state, significant differences were identified in ECL% and HC% loss at 1 day, 1 week, 1 month, and 3 months postoperatively, in dCCT% at 1 month postoperatively, and in dCV at 1 week and 1 month postoperatively between the two groups. Nevertheless, no significant differences were observed in dCV and dCCT% at 1 week and 3 months postoperatively. Corneal endothelial cell morphology It was once considered controversial that diabetes could affect corneal endothelium morphology preoperatively. Inoue et al.38 investigated 1394 patients before cataract surgery and their multiple regression analysis revealed that age instead of DM was the only variable relevant to ECD, CV, and HC%. However, Lee et al.39 reported that corneal endothelium morphology was significantly different between DM and non-DM patients, and CV is significantly correlated with diabetes duration. Taking multiple studies into account, the results of this review offer the judgment that DM patients have lower ECD and HC%, but higher CV, than non-DM patients (CV: P < 0.001; ECD: P = 0.02; HC%: P < 0.001; Figs 2, 4 and 6) before phacoemulsification. The fragility of the corneal endothelium in the eyes of diabetic patients might be explained by several mechanisms. With an enhanced polyol pathway and the accumulated sugar alcohol in cells converted by excessive glucose, the osmotic pressure goes up, causing the fragility of diabetic corneal endothelial cells. Morphological abnormalities in the corneal endothelium have been reported8,40 to improve after administration of an aldose reductase inhibitor in the polyol pathway, which supports its involvement in the corneal endothelial abnormalities of patients with DM. Furthermore, the enhanced accumulation of AGEs in diabetic corneas provides strong evidence22 that nuclear oxidative DNA damage caused by the accumulation of AGEs is responsible for the apoptotic damage of corneal endothelial cells in diabetic patients, which also results in decreased ECD. Diabetes also reduces the activity of Na+/K+-ATPase of the corneal endothelium41, which plays a key role in the maintenance of its structure. This causes morphological and functional changes, including increased CV and decreased HC% in diabetic corneas. Since a regular hexagonal pattern of the corneal endothelium provides the most stable covering plane, deviation from this pattern leads to a less stable monolayer. In the diabetic endothelium, there is a greater surface tension on the monolayer caused by the loss of the regular hexagonal pattern and the increasingly irregular shapes of the corneal endothelium, which makes the corneas of diabetic patients more fragile26,42. According to quantitative morphometric analysis, the corneas of patients with DM may be at risk in intraocular surgical procedures. Clinical observations have indicated that the corneal endothelium is capable of compensation to prevent complicated diseases of the cornea, such as bullose keratopathy22, unless the cell density reaches a very low threshold of 400–500 cells/mm2, at which point the cornea cannot maintain its normal physiological function43. Typically, it is widely accepted that 1000 cells/mm2 is the minimum preoperative value to prevent corneal decompensation after surgery. The diabetic cornea, which is more fragile and vulnerable to trauma, possesses a weaker compensatory capacity. The study by Furuse et al.44 reportedfwhich point the cornea cannot that there is no significant difference in the ECL% and CV values beween the two groups postoperatively. However, a recent study by Dhasmana et al.45 showed a severe increase in ECL% in the DM group compared to the control after cataract surgery. The results of our meta-analysis showed that DM patients have a significantly greater ECL% than non-DM patients from the first day to 3 months postoperatively (P < 0.01, Fig. 3), confirming that diabetic patients are more susceptible to corneal endothelial damage after phacoemulsification. Coefficient of variation and hexagonal cell percentage It's common practice that ECD is used to evaluate the state of corneas after phacoemulsification, but it cannot reflect the dynamic of the healing process for trauma. The change in morphology has a closer relationship with the dynamic of the corneal recovery process. The loss of endothelial cells, as an immediate response to surgery, leads to some defeats. Unlike the corneal epithelium, the cells of the endothelium do not regenerate. Instead, the remaining cells enlarge and stretch to cover the posterior corneal surface in order to fill the space. Ideally, the earliest phenomena should be an increase in cell size coupled with an enlargement in CV and a decrease in HC%. After a period of rearrangement, the defects would diminish, and CV and HC% would return to preoperative values as well46. Gradually, the cells return to stability to maintain the physiological function of corneas. Although many studies28,29,30,32,34,35,36,37,45,47 have mentioned that a significant difference in CV and HC% were found between DM and non-DM groups or between postoperative and preoperative periods, quite a few studies discuss the comparison of the dynamic change of CV and HC% between the two groups. Our meta-results showed that dCV follows a pattern of maximum increase between 1 day and 1 week postoperatively and then slowly reduces for at least 3 months. On the first day after phacoemulsification, both groups have an enlarged inhomogeneity in cell size, which gives two large dCV values without a significant difference (P = 0.31). The ascending process lasts for 1 day17,34 to 1 month32 before cell shape starts to compensate to be uniform. Significant differences between the two groups started at 1 week postoperatively (P = 0.004), peaked at 1 month postoperatively (P = 0.02, Fig. 7), and then vanished at 3 months postoperatively with the recovery of the diabetic patient (P = 0.14, Fig. 5). However, a sensitivity analysis showed an unstable outcome at 1 month postoperatively. It can be inferred that this is because the measurements were done at some critical point when the significant difference started to appear or disappear. What's more, different durations of DM29 and blood glucose control34,48 might affect the result. This kind of recovery was not uniform for all corneal morphological properties. For HC% loss, there was always a significant difference between the DM and non-DM groups postoperatively (P < 0.001, Fig. 7). Once the endothelial cells lose their hexagonal structure, the stretching and rearrangement process will not easily give a second chance for the cells to stabilize into hexagons again. The time of the HC% recovery in diabetic corneas should be longer than 3 months. For the preoperative state, Kotecha et al.49 found no significant difference in CCT between the DM and non-DM groups. However, Lee et al.18,39 reported that CCT, which is strongly correlated with DM duration, is slightly greater in diabetic patients than in non-diabetic patients. The analysis revealed a slightly greater CCT in patients with DM, but no significant difference preoperatively (P = 0.39, Fig. 8). The effect of DM on CCT is still ambiguous. There are several possible explanations, such as the inhibition of endothelial pumping, growing stromal swelling pressure, and increasing endothelial permeability caused by diabetic metabolism50,51,52,53. Briefly, the normal corneal endothelium plays a key role in keeping the cornea moist and transparent as well as maintaining integrity to prevent stromal swelling. Tight apical junctions on the endothelial cells function as physical barriers. The movement of water outward from the corneal stroma into the anterior chamber is increased due to ion pumps in the endothelial cells. Thus, corneal edema can be caused by a breakdown of either the anatomical barrier or the pump function of the corneal endothelial cells, representing an increase in CCT. This effect depends on the pathology insults of DM18 and the severity of the physical trauma. After phacoemulsification, the increase in CCT was maximum at 1 day and 1 week postoperatively and then gradually decreased for at least 3 months17,26,32,33,45,47. Altintas et al.26 demonstrated that corneal thicknesses were greater in both diabetic and non-diabetic patients 1 week postoperatively than in later follow up, while there were no differences in corneal thickness according to phaco-time or diabetic status. Nevertheless, most studies17,32,33,45,47 mentioned a delayed recovery of postoperative corneal edema in diabetics compared to normal controls. This analysis observed significant differences between the two groups at all postoperative times from 1 day to 3 months. For the postoperative changes, the dCCT% followed the same trend as dCV. Severe postoperative responses and long recovery times in the diabetic group are shown by the analysis in Fig. 3. There was a significant difference found at 1 month postoperatively. It can be inferred that this is because, at an early time after phacoemulsification, both DM and non-DM patients have severe responses and sharp increases due to the breakdown of corneal endothelial function caused by the surgical procedure, thus making the difference between the two groups too small to be distinguished (P = 0.28, P = 0.08, Fig. 9). Furthermore, studies have proven that hyperglycemia enhance the expression of MMPs54, the production and activity of which is likely to damage the basement membrane, including type IV collagen, and limit epithelial cell migration, resulting in poor epithelial healing55. Gradually, the corneas of non-diabetic patients start to heal more quickly, showing a smaller CCT compared to the DM group, especially at 1 month postoperatively (P = 0.003, Fig. 9). Finally, differences due to the effect of surgery vanish 3 months postoperatively (P = 0.15, Fig. 9), which means DM patients can take almost 3 months to recover. Besides all of the changes in the cornea, visual rehabilitation is still the top concern for patients undergoing phacoemulsification. Best corrected visual acuity (BCVA) is one of the best parameters for evaluating the quality and efficiency of a surgical technique. Although CCT values were significantly different between the two groups after phacoemulsification, there was no difference in visual acuity in the long-term comparison as reported27,33,45. However, the BCVA of the non-diabetic group was better at 1 week postoperatively33, indicating that the diabetic achieve worse vision recovery, which is consistent with the CCT results. Eventually, patients in both groups had better postoperative visual acuity at the end of the follow-up period, which indicates that phacoemulsification should be considered as a safe procedure for cataract extraction in the diabetic. As a consequence, greater efforts and concentration should be made by the surgeon to minimize surgical trauma, especially for the diabetic. To achieve that, phaco-power near the cornea should be avoided. A viscoelastic agent could be generously used to cushion the endothelium as well. What's more, close postoperative observation and intervention is suggested in patients with transient corneal edema and decompensation, as it can be a predicted factor for the development of pseudophakic cystoid macular edema56. Since the duration of DM and blood glucose have proven to be associated with the severity of corneal damage caused by phacoemulsification29,48, diabetic patients are recommended to choose the proper timing when good glycemic and HbA1c control is achieved for cataract surgery35,48, thereby preventing further complications and minimizing visual loss. Femtosecond laser-assisted cataract surgery is currently reported to be safer and more effective in reducing endothelial cell loss and postoperative central corneal thickening and achieving better visual and refractive outcomes compared to conventional phacoemulsification surgery. Therefore, it might be a potentially better choice for the diabetic. Further studies are needed to validate this hypothesis. To the best of our knowledge, this is the first meta-analysis and review of corneal changes after phacoemulsification in diabetic and non-diabetic patients. Not only the postoperative state but also the changes between preoperative and postoperative states, which have an advantage over the former, are evaluated. In addition, the dynamic healing process and the changes of these parameters are carefully demonstrated in this study with at least three studies in each analysis to give a rational analysis result. We offer a systematic evaluation as well as possible mechanisms and treatment to the clinic. Inevitably, this meta-analysis has several limitations. First, the limitations came from the clinical trial itself. It was not possible to have a randomized control trial (RCT) as diabetic patients are aware of their condition and diabetic group naturally exists. Cohort studies, which are not as reliable as RCTs, were therefore included in this meta-analysis. Secondly, the duration of diabetes for the diabetic patients, the surgical conditions, the surgeons, and the data collection techniques all work together to make some of the outcomes not uniform. Thirdly, most of the patients in the included studies are from Asia, which might be a potential source of deviations as corneal biomechanics may vary among races57,58. Search strategy PubMed, EMBASE, and the Cochrane Controlled Trials Register were searched for relevant literature dated up to Jan 2017. The following terms were used to search for prospective studies in the selected databases: ((diabete OR diabetes) AND cataract surgery) AND corneal; (('cataract extraction'/exp OR 'cataract extraction') AND 'diabetes mellitus'/exp OR 'diabetes mellitus') AND corneal; "diabetes" AND "cataract surgery" AND "corneal". The bibliographies of the relevant review and original research articles were also scanned for potential trials that may have been missed in the primary searches. Study selection The inclusion criteria for our selection process were set as follows: 1) Prospective controlled study, 2) the study included DM patients and normal patients who underwent phacoemulsification and IOL implantation, 3) the study reported at least one basic dataset of corneal properties, such as ECD, CV, HC%, and CCT, 4) the patients in the trials were absent of additional underlining diseases or eye disorders other than diabetes and cataract. Patients with other complications that could affect corneal state (e.g. severe liver or kidney dysfunction, glaucoma, iritis, or eye injury) were excluded. Screening process Two reviewers, working independently from each other, first conducted preliminary reviews of the titles and abstracts; then, the full articles were analyzed to select the studies that met our predefined criteria. Disagreements between the two reviewers were resolved through careful discussion, involving a third or fourth reviewer when necessary, until a consensus was reached. The Newcastle–Ottawa scale (NOS) was used for quality assessment. The NOS contains eight items (nine scores in total) which fit into three categories: selection (four scores), comparability (two scores), and exposure of a case-control study or outcome of a cohort study (three scores). A score ≥6 indicates good quality. Data extraction process The patient data was extracted from the selected studies via a standard form: first author, country (province), year of publication, age of patient, sex of patient, follow up duration, quality control, and preoperative diabetes condition. The second reviewer double-checked all data. The measurement of corneal properties included corneal endothelial cell density (ECD), corneal endothelial hexagon percentage (HC%), corneal endothelial coefficient of variation (CV), and central corneal thickness (CCT). Most of the studies only reported the absolute values of the outcomes at a preoperative baseline and postoperative time points. The mean and standard deviation outcomes of the corneal changes were calculated as follows: $${\rm{\Delta }}mean=mea{n}_{post}-mea{n}_{pre}$$ $$S{D}_{{\rm{\Delta }}mean}=\sqrt{S{D}_{pre}^{2}+S{D}_{post}^{2}-2\rho S{D}_{pre}S{D}_{post}}$$ where ρ is the covariance coefficient, pre and post are short for preoperative and postoperative state. Generally, ρ was treated as ~0.5. The mean change percentages were calculated according to: $${\rm{\Delta }}mean \% =\frac{{\rm{\Delta }}mean}{mea{n}_{pre}},\,S{D}_{{\rm{\Delta }}mean \% }=\frac{S{D}_{{\rm{\Delta }}mean}}{mea{n}_{pre}}$$ For situations when the selected study included multiple groups, a group-combining method59 was used in this meta-analysis to create a single pair-wise comparison. Considering a two-group combining process in which group 1 has a sample size of N 1, a mean outcome of M 1, and a standard derivation of SD1, and group 2 has similar (N2, M2, and SD 2), the combined group 1 + 2 would be calculated as: $${N}_{1+2}={N}_{1}+{N}_{2};{M}_{1+2}=\frac{{N}_{1}{M}_{1}+{N}_{2}{M}_{2}}{{N}_{1}+{N}_{2}}$$ $$S{D}_{1+2}=\sqrt{\frac{({N}_{1}-1)S{D}_{1}^{2}+({N}_{2}-1)S{D}_{2}^{2}+\frac{{N}_{1}{N}_{2}}{{N}_{1}+{N}_{2}}{({M}_{1}-{M}_{2})}^{2}}{{N}_{1}+{N}_{2}-1}}$$ If there were more than two groups to combine, the strategy was to repeat this method sequentially (i.e. combine group 1 and group 2 to create group 1 + 2, and then combine group 1 + 2 and group 3 to create group 1 + 2 + 3, and so on). The statistical analysis was performed using Review Manager 5.3. The weighted mean difference (WMD) and 95% confidence interval (CI) were calculated from selected outcomes. P < 0.05 was considered statistically significant. 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Comparison of corneal biomechanical properties between healthy blacks and whites using the Ocular Response Analyzer. American journal of ophthalmology 150, 163–168, e161 (2010). Higgins, j. P. & Green, S. Cochrane Handbook for Systematic Reviews of Interventions (2011). This study was supported by Program of National Natural Science Foundation of China (No. 81371001), Program of National Natural Science Foundation (No. 81570822), Zhejiang Key LaboratoryFund of China (No. 2011E10006), Project of National Clinical Key Discipline of Chinese Ministry of Health, Zhejiang Province Key Research and Development Program (2015C03042). Eye Center, Second Affiliated Hospital, School of Medicine, Zhejiang University, Hangzhou, Zhejiang, P.R. China Yizhen Tang, Xinyi Chen, Xiaobo Zhang, Qiaomei Tang, Siyu Liu & Ke Yao Key Laboratory of Ophthalmology of Zhejiang Province, Hangzhou, P.R. China Yizhen Tang Xinyi Chen Xiaobo Zhang Qiaomei Tang Siyu Liu Ke Yao K.Y. and Y.Z.T. designed this study. Y.Z.T., X.Y.C., X.B.Z., Q.M.T. and S.Y.L. collected and double checked the data. Y.Z.T. and X.Y.C. carried out the statistical analysis. Y.Z.T. wrote the paper. K.Y. and X.Y.C. provided critical revision to the article. All author participated in revision and approved the final version for submission. Correspondence to Ke Yao. Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Supplementary Tables S1-S8 Tang, Y., Chen, X., Zhang, X. et al. Clinical evaluation of corneal changes after phacoemulsification in diabetic and non-diabetic cataract patients, a systematic review and meta-analysis. Sci Rep 7, 14128 (2017). https://doi.org/10.1038/s41598-017-14656-7 Received: 07 April 2017 The effect of diabetes on corneal endothelium: a meta-analysis Kaikai Zhang Liangliang Zhao Meisheng Zhao BMC Ophthalmology (2021) Accelerated corneal endothelial cell loss in two patients with granulomatosis with polyangiitis following phacoemulsification Fang-Chi Hsiao Hung-Ta Chen Hung-Chi Chen Endothelial cell density changes in diabetic and nondiabetic eyes undergoing phacoemulsification employing phaco-chop technique Erika Fernández-Muñoz Rocío Zamora-Ortiz Roberto Gonzalez-Salinas International Ophthalmology (2019)	CommonCrawl
$\displaystyle \bigwedge_{i=1}^n [i \ne (i+1)]$, $\displaystyle \neg \left(\bigvee_{i=1}^n P_i \right) = \bigwedge_{i=1}^n \neg P_i$, $\mathrm{N} \mathbf{x}$, $\nexists \mathbf{x}$, $\mathrm{N}x P(x) \equiv \\ \forall x \, \neg P(x)$, $\exists_3 x \in \mathbb{Z}\, (5 < x < 9)$, $\exists_{\le 10} x \, (x^2 \le 100) \equiv$, $\mathbf{\alpha}[\mathbf{x}/\mathbf{t_0}]$. \veebar, ≢ ∧ Set symbols of set theory and probability with name and definition: set, subset, union, intersection, element, cardinality, empty set, natural/real/complex number set Also, the → symbol is often used to denote "changed to", as in the sentence "The interest rate changed. Originally founded as a Montreal-based math tutoring agency, Math Vault has since then morphed into a global resource hub for people interested in learning more about higher mathematics. In philosophy and mathematics, logic plays a key role in formalizing valid deductive inferences and other forms of reasoning. For all formulas $\alpha$ and $\beta$, $\alpha \land \beta \equiv \beta \land \alpha$. As logicians are familiar with these symbols, … ⊥ (read "falsum") is required as a fixed propositional symbol. Required fields are marked, Get notified of our latest developments and free resources. For all terms $\mathbf{t}_1$ and $\mathbf{t}_2$, '$f(\mathbf{t}_1, \mathbf{t}_2)$' is a term. , and the existential quantifier as Basic math symbols; Geometry symbols; Algebra symbols; Probability & statistics symbols; Set theory symbols; Logic symbols; Calculus & analysis symbols; Number symbols; Greek symbols; Roman numerals; Basic math symbols In logic, a set of symbols is commonly used to express logical representation. What does this symbol mean? If $\mathcal{L}$ is a language with equality and constant $a$, then '$a = a$' is a formula in $\mathcal{L}$. Choose from 500 different sets of honors geometry a logic flashcards on Quizlet. \iff \! The following table documents the most notable of these symbols — along with their respective meaning and example. Conditional: a conditional is something which states that one statement implies another. Is the mathematical symbols keyboard working well on your computer? Similar to other fields in mathematics, variables are used as placeholder symbols for varying entities in logic. Given a few mathematical statements or facts, we would like to be able to draw some conclusions. Play this game to review Geometry. {\displaystyle \wedge } Example ... Geometry symbols. More specifically, geometry and logic uses a precise kind of declarative sentence that is either definitely true or false; such declarative sentences are called statements. $\forall x \, (x \ge 1) \! 1. p is the hypothesis. An operand of a conjunction is a conjunct. ... tion symbols (also called predicate symbols). The logic of Aristotle and the geometry of Euclid are universally recognized as towering scientific achievements of ancient Greece. Some Symbols from Mathematical Logic ∴ (three dots) means "therefore" and first appeared in print in the 1659 book Teusche Algebra ("Teach Yourself Algebra") by Johann Rahn (1622-1676). A mathematical symbol is a figure or a combination of figures that is used to represent a mathematical object, an action on mathematical objects, a relation between mathematical objects, or for structuring the other symbols that occur in a formula.As formulas are entierely constitued with symbols of various types, many symbols are needed for expressing all mathematics. $(\forall x \, \alpha)^{\sigma} = \top$ if and only if for all $u$ in the universe of discourse $U$, $\alpha^{\sigma (x/u)} = \top$. 0-ary relation symbols are called propositional symbols. In logic, constants are often used to denote definite objects in a logical system. Mathematical symbols and signs of basic math, algebra, geometry, statistics, logic, set theory, calculus and analysis In logic, mathematics and linguistics, And (∧) is the truth-functional operator of logical conjunction; the and of a set of operands is true if and only if all of its operands are true. $P \downarrow Q \equiv \\ (\neg P \land \neg Q)$, $(P \to Q) \land \\ (P \not\leftarrow Q)$. Accept. Apart from classical logic we will also deal with more constructive logics: minimal and intuitionistic logic. The following table documents the most notable of these — along with their respective example and meaning. However, it is much more common Before one can start to understand logic, and thereby begin to prove geometric theorems, one must first know a few vocabulary words and symbols. Meaning / definition. Leave me suggestions and feedbacks. ∧ is true if and only if is true and is true. o What conclusion can you draw, using all of the following statements? Overline is also a rarely used format for denoting, This page was last edited on 27 November 2020, at 22:06. {\displaystyle \parallel } Operators are symbols used to denote mathematical operations, which serve to take one or multiple inputs to a similar output. Some of the worksheets for this concept are Math symbol matching work, Ms work 132 153 geometry 06, 2 information in geometric diagrams, Basic geometric terms, Basic geometry terms, The meaning of sacred geometry, Logic and conditional statements, Teaching geometry according to the common core standards. Symbol. For all variables $\mathbf{x}_1$ and $\mathbf{x}_2$, '$\mathbf{x}_1 = \mathbf{x}_2$' is a formula. In logic, a set of symbols is commonly used to express logical representation. March 20% → April 21%". List of logic symbols From Wikipedia, the free encyclopedia (Redirected from Table of logic symbols) See also: Logical connective In logic, a set of symbols is commonly used to express logical representation. Set theory symbols. (8 ÷ 4) ÷ 2 = 2 ÷ 2 = 1, but 8 ÷ (4 ÷ 2) = 8 ÷ 2 = 4. Algebra symbols. This dynamic geometry is only understood by the symbol itself. We … This geometry video tutorial explains how to write the converse, inverse, and contrapositive of a conditional statement - if p, then q. Perform the operations inside the parentheses first. \not\equiv, ≡ If $\Box P$, then $\neg \Diamond \neg P$. Chapter 3 Symbolic Logic and Proofs. Your email address will not be published. For lists of symbols categorized by type and subject, refer to the relevant pages below for more. ∨ Effects within symbols act to dynamically alter the geometry before the graphical aspects of the symbol are applied. Hyperbolic functions The abbreviations arcsinh, arccosh, etc., are commonly used for inverse hyperbolic trigonometric functions (area hyperbolic functions), even though they are misnomers, since the prefix arc is the abbreviation for arcus, while the prefix ar stands for area. Relational Symbols. In logic, a disjunction is a compound sentence formed using the word or to join two simple sentences. \equiv, :⇔ For example, an acorn will grow into an oak tree and nothing else. \parallel, ⊻ Part 1 of a brief rundown of the basic principles of the subject of logic. (whenever you see $$ ν $$ read 'or') When two simple sentences, p and q, are joined in a disjunction statement, the disjunction is expressed symbolically as p $$ ν$$ q. Write p. 2. For more, see about us. You can see our cookies policy, here - If you continue browsing this site, you are accepting its use. Unicode Technical Report #25 provides comprehensive information about the character repertoire, their properties, and guidelines for implementation. We'll also keep you informed of our latest developments and freebies! {\displaystyle \vee } Brief introduction to the symbols of logic. [1] The last column provides the LaTeX symbol. [7][8] The same applies for Germany.[9][10]. Additionally, the third column contains an informal definition, the fourth column gives a short example, the fifth and sixth give the Unicode location and name for use in HTML documents. Your email address will not be published. Previous Next . \sim, ∥ Symbol Name. These symbols are sorted by their Unicode value: The following operators are rarely supported by natively installed fonts. Einführung in die mathematische Logik: klassische Prädikatenlogik. For readability purpose, these symbols are categorized by their function into tables. Early childhood education is the key to the betterment of society. Conditional statements can be written using logic symbols, which represent different concepts.. An arrow (→) connects the hypothesis to the conclusion in if-then statements.A double headed arrow (↔) connects the hypothesis to the conclusion in biconditionals.A negation is symbolized by (~).We can write conditional statements in different ways using these logic symbols. If $\Diamond P$, then $\Diamond \Diamond P$. Math Symbols List. You may also want to … For the master list of symbols, see mathematical symbols. List of notation used in Principia Mathematica, Mathematical operators and symbols in Unicode, Wikipedia:WikiProject Logic/Standards for notation, https://en.wikipedia.org/w/index.php?title=List_of_logic_symbols&oldid=991029932, Short description is different from Wikidata, Articles lacking reliable references from May 2020, All articles with specifically marked weasel-worded phrases, Articles with specifically marked weasel-worded phrases from July 2020, Articles containing potentially dated statements from 2014, All articles containing potentially dated statements, Creative Commons Attribution-ShareAlike License, The statement ⊥ is unconditionally false. The symbol for this is $$ ν $$ . Symbolic logic differs from traditional logic in its extensive use of symbols similar to those used in mathematics, in its lack of concern with the psychology and epistemology of knowledge, ... the letter p is often used in geometry to represent a point. In logic, relational symbols play a key role in turning one or multiple mathematical entities into formulas and propositions, and can occur both within a logical system or outside of it (as metalogical symbols). P can then be used to describe line segments, intersections, and other geometric concepts. Privacy Policy Terms of Use Anti-Spam Disclosure DMCA Notice. {\displaystyle :\Leftrightarrow } \beta$, $\neg (P \to Q) \equiv \\ P \land \neg Q$. In logic, these operators include logical connectives from propositional/modal logic, quantifiers from predicate logic, as well as other operators related to syntactic substitution and semantic valuation. In logic, a set of symbols is commonly used to express logical representation. Mathematics Enhanced Scope and Sequence – Geometry ... Logic and Conditional Statements Reporting Category Reasoning, Lines, ... Use symbols p, q, , and ~ and math vocabulary to answer this question. {\displaystyle \not \equiv } As of 2014[update] in Poland, the universal quantifier is sometimes written Logic Gate Symbols (Digital Electronic). Logic symbols. Basic Math Symbols – Geometry, Algebra, Greek, Logic, Number; Basic Math Symbols – Geometry, Algebra, Greek, Logic, Number. Logic is the study of consequence. If $\Phi \models \phi$, then $\Phi \cup \Psi \models \phi$. The ⇒ symbol is often used in text to mean "result" or "conclusion", as in "We examined whether to sell the product ⇒ We will not sell it". The following table lists many common symbols, together with their name, pronunciation, and the related field of mathematics. SAT Math Test Prep Online Crash Course Algebra & Geometry Study Guide Review, Functions,Youtube - Duration: 2:28:48. Below is the complete list of Windows ALT codes for Math Symbols: Logical Operators, their corresponding HTML entity numeric character references, and when available, their corresponding HTML entity named character references, and Unicode code points.This list is comprised of logical & set operators, modal logic operators and logical ands & ors. ∼ Geometry Symbols - Displaying top 8 worksheets found for this concept.. 3 (the such that sign) means "under the condition that". Learn honors geometry a logic with free interactive flashcards. If $\phi^{\sigma} = \top$, then $\sigma \models \phi$. $\mathbf{x}, \mathbf{y}, \mathbf{w}, \mathbf{z}$. Definitive resource hub on everything higher math, Bonus guides and lessons on mathematics and other related topics, Where we came from, and where we're going, Join us in contributing to the glory of mathematics. {\displaystyle \veebar } For example, "It is purple" is a declarative sentence, but we don't know what "it" is, so we cannot argue its truth or falsehood. A negation of a statement has the opposite meaning of a truth value. The following table features the most notable of these — along with their respective example and meaning. Calculus & analysis symbols. Formal theories for mathematics Letters symbols Logic & Theory Geometry Equivalence & Proportion Operators Other symbols Uncheck all - Check all. The following … A comprehensive collection of the most notable symbols in formal/mathematical logic, categorized by function into tables along with each symbol's meaning and example. Symbol geometry logic. The largest collection of schematic electric and electronic symbols on the Internet. In logic, relational symbols play a key role in turning one or multiple mathematical entities into formulas and propositions, and can occur both within a logical system or outside of it (as metalogical symbols). Just drop in your email and we'll send over the 26-page free eBook your way! If $P \not\to Q$, then $P \not\leftrightarrow Q$. {\displaystyle \equiv } :\Leftrightarrow. This website uses cookies. Mathematics Instructional Plan – Geometry Virginia Department of Education ©2018 7 Logic and Conditional Statements, Part 1 Name Date Use the following conditional statement to complete 1-11: "If elephants fly, then fish don't swim." Each answer should be a complete sentence, not symbols. In symbols: is . In Boolean logic, $\mathbb{B} = \{ 0 ,1\}$. Insert details about how the information is going to be processed. (The symbol ⊥ may also refer to. '$\neg \left(1 = s(1) \right)$' is a formula in the language of first-order arithmetic. In most mathematical notation, a conditional is often written in the form p ⇒ q, which is read as "If p, then q" wh… A conditional contains two parts: the condition and the conclusion, where the former implies the latter. Mathematical operators and symbols are in multiple Unicode blocks.Some of these blocks are dedicated to, or primarily contain, mathematical characters … Springer-Verlag, 2013. {\displaystyle \sim } So, for students of logic, the following table lists many common symbols together with … Also, you can find specific mathematical symbols with their sign, and meaning. Probability and statistics symbols. logic, Boolean algebra direct sum The direct sum is a special way of combining several one modules into one general module (the symbol ⊕ is used, is only for logic). Other comprehensive lists of symbols — as categorized by subject and type — can be also found in the relevant pages below (or in the navigational panel). WHITE CONCAVE-SIDED DIAMOND WITH LEFTWARDS TICK, WHITE CONCAVE-SIDED DIAMOND WITH RIGHTWARDS TICK, Although this character is available in LaTeX, the. The following is a comprehensive list of the most notable symbols in logic, featuring symbols from propositional logic, predicate logic, Boolean logic and modal logic. Here you will get a list of basic math symbols. The Unicode Standard encodes almost all standard characters used in mathematics. . List of all mathematical symbols and signs - meaning and examples. The following table documents the most notable of these symbols — along with their respective meaning and example. As logicians are familiar with these symbols, they are not explained each time they are used. Working with logic A true-false statement is any sentence that is either true or false but not both. \implies 1 \ge 1$, $\alpha \equiv \beta$, $\alpha \Leftrightarrow \beta$, $\alpha \! Awesome! The logical connective that represents this operator is typically written as ∧ or ⋅ . Get the master summary of mathematical symbols in eBook form — along with each symbol's usage and LaTeX code. __CONFIG_colors_palette__{"active_palette":0,"config":{"colors":{"b7b07":{"name":"Main Accent","parent":-1},"a1fa2":{"name":"Main Lighter","parent":"b7b07","lock":{"saturation":1,"lightness":1}}},"gradients":[]},"palettes":[{"name":"Default","value":{"colors":{"b7b07":{"val":"var(--tcb-skin-color-0)"},"a1fa2":{"val":"rgb(232, 230, 227)","hsl_parent_dependency":{"h":39,"l":0.9,"s":0.1}}},"gradients":[]},"original":{"colors":{"b7b07":{"val":"rgb(57, 164, 210)","hsl":{"h":198,"s":0.62,"l":0.52}},"a1fa2":{"val":"rgb(228, 232, 233)","hsl_parent_dependency":{"h":192,"s":0.1,"l":0.9}}},"gradients":[]}}]}__CONFIG_colors_palette__, __CONFIG_group_edit__{"k5p7rda8":{"name":"All Text(s)","singular":"-- Text %s"}}__CONFIG_group_edit__, {"email":"Email address invalid","url":"Website address invalid","required":"Required field missing"}, Definitive Guide to Learning Higher Mathematics, Comprehensive List of Mathematical Symbols. 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By their function into tables want to … logic Gate symbols ( also called predicate symbols ) symbols categorized their... logic geometry symbols Bissell Proheat 2x Revolution Pet Pro, Cracker Barrel Kettle Corn, St Augustine Confessions, Taco Bueno Prices, Mckinsey Valuation Workbook Pdf, Can We Drink Coke After Eating Jackfruit, Best Wood Bats, logic geometry symbols 2020	CommonCrawl
A posteriori eigenvalue error estimation for a Schrödinger operator with inverse square potential Error analysis for numerical formulation of particle filter July 2015, 20(5): 1355-1375. doi: 10.3934/dcdsb.2015.20.1355 Euler-Maclaurin expansions and approximations of hypersingular integrals Chaolang Hu 1, , Xiaoming He 2, and Tao Lü 1, College of Mathematics, Sichuan University, Chengdu,Sichuan, 610064, China, China Department of Mathematics and Statistics, Missouri University of Science and Technology, Rolla, MO 65401, United States Received March 2013 Revised January 2015 Published May 2015 This article presents the Euler-Maclaurin expansions of the hypersingular integrals $\int_{a}^{b}\frac{g(x)}{\|x-t\|^{m+1}}dx$ and $\int_{a}^{b}% \frac{g(x)}{(x-t)^{m+1}}dx$ with arbitrary singular point $t$ and arbitrary non-negative integer $m$. These general expansions are applicable to a large range of hypersingular integrals, including both popular hypersingular integrals discussed in the literature and other important ones which have not been addressed yet. The corresponding mid-rectangular formulas and extrapolations, which can be calculated in fairly straightforward ways, are investigated. Numerical examples are provided to illustrate the features of the numerical methods and verify the theoretical conclusions. Keywords: mid-rectangular quadrature formula, Euler-Maclaurin expansion, arbitrary singular point, extrapolation., Hypersingular integral. Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3. Citation: Chaolang Hu, Xiaoming He, Tao Lü. Euler-Maclaurin expansions and approximations of hypersingular integrals. Discrete & Continuous Dynamical Systems - B, 2015, 20 (5) : 1355-1375. doi: 10.3934/dcdsb.2015.20.1355 I. V. 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Communications on Pure & Applied Analysis, 2013, 12 (6) : 2465-2495. doi: 10.3934/cpaa.2013.12.2465 Chaolang Hu Xiaoming He Tao Lü	CommonCrawl
Strict feasibility of variational inclusion problems in reflexive Banach spaces Tianyi Li , System Dynamics Group, Sloan School of Management, MIT, Cambridge, MA 02139, USA * Corresponding author: Tianyi Li Received November 2018 Revised January 2019 Published September 2020 Early access May 2019 Figure(4) Personality heterogeneity is an important topic in team management. In many working groups, there exists certain type of people that are talented but under-disciplined, who could occasionally make extraordinary contributions for the team, but often have less satisfactory overall performance. It is interesting to investigate whether the existence of such people in the team does help improve the overall team performance, and if it does so, what are the conditions for their existence to be positive, and through which channel their benefits for the team are manifested. This study proposes an analytical model with a simple structure that sets up an environment to study these questions. It is shown that: (1) feedback learning could be the mechanism through which outliers' benefits to the team are established, and thus could be a prerequisite for outliers' positive existence; (2) different types of teamwork settings have different outlier-positivity conditions: a uniform round-wise punishment for teamwork failures could be the key idea to encourage outliers' existence; for two specific types of teamwork, teamwork that highlights assistance in interactions are more outliers-friendly than teamwork that consists internal competitions. These results well match empirical observations and may have further implications for managerial practice. Keywords: Team dynamics, personality heterogeneity, mathematical modeling, feedback learning, monte Carlo simulations. Mathematics Subject Classification: 90B50, 90B70. Citation: Tianyi Li. Does the existence of "talented outliers" help improve team performance? Modeling heterogeneous personalities in teamwork. 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Liu, Application of K-means clustering algorithm for classification of NBA guards, International Journal of Science and Engineering Applications, 5 (2016), 1-6. Google Scholar ">Figure 1. Summary of the model and major assumptions. Teamwork is conducted in a multi-round game setting. In the team, normal players (empty nodes) and "talented" outliers (the solid node) behave differently (orange box). Outliers have worse average performance but a greater performance potential than normal players. Moreover, unlike normal players, outliers do not adjust his performance according to feedbacks from past interactions Figure 3. Testing two utility types and two distributions of individual performance. Left: $ \Delta Q_g $ for individual MC runs; right: average $ \Delta Q_g $ for all 200 MC runs. Utility function: equation (3): U1; equation (14): U2. Performance distribution: uniform: D1; Gaussian: D2. $ \{\gamma, m, K, L\} $ is chosen such that $ H<0 $ (blue; D1U1) and $ H'>0 $ (red; D1U2). Proposition 3 is demonstrated since $ \Delta Q_g(\mathit{\boldsymbol{D\mathit{1}U\mathit{2}}})>0>\Delta Q_g(\mathit{\boldsymbol{D\mathit{1}U\mathit{1}}}) $. The Gaussian distribution of individual performance is more stringent for outlier's positive existence than the uniform distribution Figure 2. The significance of the feedback learning mechanism as a potential prerequisite for the positivity condition of outliers' existence. Left: no feedback. MC simulation results are consistent with equation (7). Right: the feedback mechanism activated. Inset: average all-round team performance as a function of the number of outliers in the team $ N_o $. Results show that $ N_o = 1 $ produces the best outcome of teamwork under this parameter set, which satisfies $ H'>0 $ Figure 4. Feedback learning from multiple past rounds. Results show $ \Delta Q_g $ (as in Figure 2) as a function of $ w $, which ranges from 1 to 5. The results from a few individual runs are plotted together, for all four combinations of $ U $ and $ D $. No conclusion could be drawn from this test Chjan C. Lim, Joseph Nebus, Syed M. Assad. Monte-Carlo and polyhedron-based simulations I: extremal states of the logarithmic N-body problem on a sphere. Discrete & Continuous Dynamical Systems - B, 2003, 3 (3) : 313-342. doi: 10.3934/dcdsb.2003.3.313 Olli-Pekka Tossavainen, Daniel B. Work. Markov Chain Monte Carlo based inverse modeling of traffic flows using GPS data. Networks & Heterogeneous Media, 2013, 8 (3) : 803-824. doi: 10.3934/nhm.2013.8.803 Zhiyan Ding, Qin Li. Constrained Ensemble Langevin Monte Carlo. Foundations of Data Science, 2021 doi: 10.3934/fods.2021034 Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291 Guillaume Bal, Ian Langmore, Youssef Marzouk. Bayesian inverse problems with Monte Carlo forward models. Inverse Problems & Imaging, 2013, 7 (1) : 81-105. doi: 10.3934/ipi.2013.7.81 Ajay Jasra, Kody J. H. Law, Yaxian Xu. Markov chain simulation for multilevel Monte Carlo. Foundations of Data Science, 2021, 3 (1) : 27-47. doi: 10.3934/fods.2021004 Theodore Papamarkou, Alexey Lindo, Eric B. Ford. Geometric adaptive Monte Carlo in random environment. Foundations of Data Science, 2021, 3 (2) : 201-224. doi: 10.3934/fods.2021014 Stephen Pankavich, Christian Parkinson. Mathematical analysis of an in-host model of viral dynamics with spatial heterogeneity. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1237-1257. doi: 10.3934/dcdsb.2016.21.1237 Michael B. Giles, Kristian Debrabant, Andreas Rössler. Analysis of multilevel Monte Carlo path simulation using the Milstein discretisation. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3881-3903. doi: 10.3934/dcdsb.2018335 Jiakou Wang, Margaret J. Slattery, Meghan Henty Hoskins, Shile Liang, Cheng Dong, Qiang Du. Monte carlo simulation of heterotypic cell aggregation in nonlinear shear flow. Mathematical Biosciences & Engineering, 2006, 3 (4) : 683-696. doi: 10.3934/mbe.2006.3.683 Adélia Sequeira, Rafael F. Santos, Tomáš Bodnár. Blood coagulation dynamics: mathematical modeling and stability results. Mathematical Biosciences & Engineering, 2011, 8 (2) : 425-443. doi: 10.3934/mbe.2011.8.425 Joseph Nebus. The Dirichlet quotient of point vortex interactions on the surface of the sphere examined by Monte Carlo experiments. Discrete & Continuous Dynamical Systems - B, 2005, 5 (1) : 125-136. doi: 10.3934/dcdsb.2005.5.125 Mazyar Zahedi-Seresht, Gholam-Reza Jahanshahloo, Josef Jablonsky, Sedighe Asghariniya. A new Monte Carlo based procedure for complete ranking efficient units in DEA models. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 403-416. doi: 10.3934/naco.2017025 Abhinav Tandon. Crop - Weed interactive dynamics in the presence of herbicides: Mathematical modeling and analysis. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021244 M.A.J Chaplain, G. Lolas. Mathematical modelling of cancer invasion of tissue: dynamic heterogeneity. Networks & Heterogeneous Media, 2006, 1 (3) : 399-439. doi: 10.3934/nhm.2006.1.399 Benjamin Steinberg, Yuqing Wang, Huaxiong Huang, Robert M. Miura. Spatial Buffering Mechanism: Mathematical Model and Computer Simulations. Mathematical Biosciences & Engineering, 2005, 2 (4) : 675-702. doi: 10.3934/mbe.2005.2.675 Zhisheng Shuai, P. van den Driessche. Impact of heterogeneity on the dynamics of an SEIR epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (2) : 393-411. doi: 10.3934/mbe.2012.9.393 Avner Friedman, Wenrui Hao. Mathematical modeling of liver fibrosis. Mathematical Biosciences & Engineering, 2017, 14 (1) : 143-164. doi: 10.3934/mbe.2017010 Shuo Wang, Heinz Schättler. Optimal control of a mathematical model for cancer chemotherapy under tumor heterogeneity. Mathematical Biosciences & Engineering, 2016, 13 (6) : 1223-1240. doi: 10.3934/mbe.2016040 Gang Bao. Mathematical modeling of nonlinear diffracvtive optics. Conference Publications, 1998, 1998 (Special) : 89-99. doi: 10.3934/proc.1998.1998.89 Tianyi Li	CommonCrawl
The Boss Guide To Binary Options Trading Financial exotic pick with an all-or-zero payoff binary selection is a financial exotic option in which the payoff is either some fixed budgetary amount or nothing at all.[1] The two main types of binary options are the cash-or-zip binary selection and the asset-or-zero binary option. The former pays some fixed corporeality of cash if the choice expires in-the-money while the latter pays the value of the underlying security. They are likewise called all-or-nil options, (more than common in forex/interest rate markets), and (FROs) (on the American Stock Commutation).[3] While binary options may be used in theoretical asset pricing, they are prone to fraud in their applications and hence banned by regulators in many jurisdictions every bit a form of gambling.[4] Many binary pick outlets have been exposed every bit fraudulent.[v] The U.S. FBI is investigating binary option scams throughout the world, and the Israeli police have tied the industry to criminal syndicates.[6] The European Securities and Markets Authorisation (ESMA) have banned retail binary options trading.[9] Australian Securities and Investments Committee (ASIC) considers binary options as a "high-risk" and "unpredictable" investment choice, and finally besides banned binary options sale to retail investors in 2021.[11] The FBI estimates that the scammers steal United states$10 billion annually worldwide.[12] The use of the names of famous and respectable people such as Richard Branson to encourage people to buy fake "investments" is frequent and increasing.[13] paper explain the fraud in detail, using the feel of former insiders such equally a job-seeker recruited by a fake binary options broker, who was told to "get out [his] conscience at the door".[xiv] Following an investigation past The Times of Israel, State of israel's chiffonier approved a ban on auction of binary options in June 2017,[16] and a police banning the products was canonical by the Knesset in October 2017.[17] [xviii] On January xxx, 2018, Facebook banned advertisements for binary options trading also as for cryptocurrencies and initial coin offerings (ICOs).[19] Google and Twitter announced like bans in the post-obit weeks.[21] Binary options "are based on a unproblematic 'aye' or 'no' proposition: Will an underlying asset be to a higher place a certain price at a certain time?"[22] Traders place wagers every bit to whether that will or will not happen. If a customer believes the cost of an underlying nugget will be to a higher place a certain price at a ready time, the trader buys the binary choice, but if he or she believes it will be below that price, they sell the selection. In the U.S. exchanges, the price of a binary is e'er under $100.[22] Investopedia described the binary options trading process in the U.S. thus: [A] binary may be trading at $42.50 (bid) and $44.50 (offering) at one p.m. If y'all buy the binary option right then you will pay $44.l, if you decide to sell right and so you'll sell at $42.50. Let'southward presume you make up one's mind to buy at $44.fifty. If at 1:30 p.grand. the price of golden is above $1,250, your pick expires and it becomes worth $100. You brand a profit of $100 – $44.50 = $55.50 (less fees). This is chosen being "in the coin". But if the price of aureate is below $ane,250 at ane:30 p.m., the choice expires at $0. Therefore you lose the $44.fifty invested. This is chosen existence "out of the money". The bid and offer fluctuate until the option expires. Yous tin can close your position at any fourth dimension before death to lock in a profit or a reduce a loss (compared to letting it expire out of the coin).[22] In the U.South., every binary option settles at $100 or $0, $100 if the bet is correct, 0 if it is not.[22] In the online binary options manufacture, where the contracts are sold past a banker to a customer in an OTC style, a different choice pricing model is used. Brokers sell binary options at a fixed price (e.g., $100) and offering some fixed per centum return in case of in-the-money settlement. Some brokers, as well offer a sort of out-of-coin reward to a losing customer. For example, with a win reward of 80%, out-of-coin reward of 5%, and the choice cost of $100, ii scenarios are possible. In-the-coin settlement pays back the choice cost of $100 and the reward of $80. In instance of loss, the pick price is not returned but the out-of-money reward of $5 is granted to the customer.[23] On non-regulated platforms, client money is not necessarily kept in a trust account, equally required by regime financial regulation, and transactions are not monitored by third parties in guild to ensure off-white play.[24] Binary options are often considered a form of gambling rather than investment because of their negative cumulative payout (the brokers have an edge over the investor) and because they are advertised as requiring petty or no knowledge of the markets. Gordon Pape, writing in Forbes.com in 2010, called binary options websites "gambling sites, pure and simple", and said "this sort of thing can quickly become addictive… no one, no matter how knowledgeable, can consistently predict what a stock or article will do within a short time frame".[25] Pape observed that binary options are poor from a gambling standpoint as well considering of the excessive "house edge". I online binary options site paid $71 for each successful $100 trade. "If you lose, you get back $15. Let's say you lot make 1,000 "trades" and win 545 of them. Your turn a profit is $38,695. But your 455 losses volition toll you $38,675. In other words, yous must win 54.5% of the time but to break even".[25] The U.South. Commodity Futures Trading Committee warns that "some binary options Cyberspace-based trading platforms may overstate the average return on investment by advertising a higher average return on investment than a customer should expect given the payout structure."[26] Black–Scholes valuation In the Blackness–Scholes model, the price of the option can be found by the formulas below.[27] In fact, the Blackness–Scholes formula for the cost of a vanilla call option (or put selection) can exist interpreted by decomposing a call option into an asset-or-null telephone call pick minus a cash-or-nothing call option, and similarly for a put – the binary options are easier to analyze, and represent to the ii terms in the Blackness–Scholes formula. is the time to maturity, is the dividend rate, denotes the cumulative distribution function of the normal distribution, {\displaystyle \Phi (ten)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{x}e^{-z^{two}/ii}dz.} {\displaystyle d_{1}={\frac {\ln {\frac {S}{One thousand}}+(r-q+\sigma ^{two}/two)T}{\sigma {\sqrt {T}}}}.} {\displaystyle d_{2}=d_{one}-\sigma {\sqrt {T}}.} Cash-or-cipher telephone call This pays out one unit of cash if the spot is above the strike at maturity. Its value now is given by {\displaystyle C=east^{-rT}\Phi (d_{2}).\,} This pays out one unit of measurement of cash if the spot is below the strike at maturity. Its value now is given by {\displaystyle P=eastward^{-rT}\Phi (-d_{ii}).\,} Asset-or-zilch call This pays out 1 unit of asset if the spot is above the strike at maturity. Its value now is given past Nugget-or-nothing put This pays out one unit of measurement of asset if the spot is below the strike at maturity. Its value at present is given by: {\displaystyle P=Se^{-qT}\Phi (-d_{ane}).\,} American binary put with G = 100, r = 0.04, σ = 0.two, T = 1 An American option gives the holder the correct to exercise at any betoken up to and including the decease fourth dimension {\displaystyle Thousand} {\displaystyle K\geq S} {\displaystyle K\leq S} ), the respective American binary put (resp. call) is worth exactly one unit. Let {\displaystyle a={\frac {1}{\sigma }}\ln(Yard/S){\text{, }}\xi ={\frac {r-q}{\sigma }}-{\frac {\sigma }{ii}}{\text{, and }}b={\sqrt {\xi ^{2}+2r}}.\,} {\displaystyle M<S} {\displaystyle Thou>Southward} {\displaystyle {\frac {1}{2}}eastward^{a\left(\xi -b\right)}\left\{1+\operatorname {sgn} (a)\operatorname {erf} \left({\frac {bT-a}{\sqrt {2T}}}\right)+e^{2ab}\left[1-\operatorname {sgn} (a)\operatorname {erf} \left({\frac {bT+a}{\sqrt {2T}}}\right)\right]\right\}\,} denotes the sign function. The above follows immediately from expressions for the Laplace transform of the distribution of the conditional outset passage time of Brownian motion to a particular level.[28] Foreign substitution If we denote by the FOR/DOM exchange charge per unit (i.eastward., i unit of foreign currency is worth South units of domestic currency) we can detect that paying out 1 unit of the domestic currency if the spot at maturity is above or below the strike is exactly like a cash-or nothing call and put respectively. Similarly, paying out ane unit of the foreign currency if the spot at maturity is higher up or below the strike is exactly like an asset-or nothing telephone call and put respectively. Hence if we at present take , the domestic interest rate, and the residual every bit above, we go the following results. In case of a digital telephone call (this is a call FOR/put DOM) paying out 1 unit of the domestic currency we get equally present value, {\displaystyle C=e^{-r_{\mathrm {DOM} }T}\Phi (d_{two})\,} In example of a digital put (this is a put FOR/telephone call DOM) paying out ane unit of the domestic currency we get every bit present value, While in case of a digital phone call (this is a call FOR/put DOM) paying out one unit of the foreign currency we become as present value, and in instance of a digital put (this is a put FOR/call DOM) paying out i unit of the foreign currency nosotros get as present value, {\displaystyle P=Se^{-r_{\mathrm {FOR} }T}\Phi (-d_{1})\,} In the standard Black–Scholes model, 1 can interpret the premium of the binary pick in the take a chance-neutral world as the expected value = probability of beingness in-the-money * unit, discounted to the present value. The Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset. Marketplace makers adjust for such skewness by, instead of using a unmarried standard divergence for the underlying asset across all strikes, incorporating a variable ane {\displaystyle \sigma (Yard)} where volatility depends on strike price, thus incorporating the volatility skew into business relationship. The skew matters because it affects the binary considerably more than the regular options. A binary phone call option is, at long expirations, similar to a tight call spread using 2 vanilla options. One can model the value of a binary greenbacks-or-nothing option, G, as an infinitesimally tight spread, where {\displaystyle C_{5}} {\displaystyle C=\lim _{\epsilon \to 0}{\frac {C_{v}(G-\epsilon )-C_{five}(Grand)}{\epsilon }}} Thus, the value of a binary call is the negative of the derivative of the price of a vanilla telephone call with respect to strike cost: {\displaystyle C=-{\frac {dC_{v}}{dK}}} is a function of {\displaystyle 1000} {\displaystyle C=-{\frac {dC_{v}(K,\sigma (K))}{dK}}=-{\frac {\partial C_{v}}{\fractional K}}-{\frac {\partial C_{five}}{\partial \sigma }}{\frac {\fractional \sigma }{\partial K}}} The first term is equal to the premium of the binary option ignoring skew: Chiliad {\displaystyle -{\frac {\partial C_{v}}{\partial K}}=-{\frac {\fractional (South\Phi (d_{1})-Ke^{-rT}\Phi (d_{2}))}{\partial Thou}}=eastward^{-rT}\Phi (d_{2})=C_{\mathrm {noskew} }} {\displaystyle {\frac {\fractional \sigma }{\fractional G}}} is sometimes called the "skew slope" or just "skew". Skew is typically negative, so the value of a binary phone call is higher when taking skew into business relationship. due north Human relationship to vanilla options' Greeks Since a binary call is a mathematical derivative of a vanilla call with respect to strike, the price of a binary call has the aforementioned shape as the delta of a vanilla call, and the delta of a binary call has the same shape as the gamma of a vanilla phone call. Many binary option "brokers" have been exposed as fraudulent operations.[29] In those cases, there is no real brokerage; the customer is betting against the broker, who is acting as a bucket shop. Manipulation of price data to cause customers to lose is common. Withdrawals are regularly stalled or refused by such operations; if a client has good reason to expect a payment, the operator volition but stop taking their phone calls.[14] Though binary options sometimes trade on regulated exchange, they are generally unregulated, trading on the Net, and decumbent to fraud.[3] About of the binary options brokers are registered in Saint Vincent and the Grenadines and offer their services globally.[ The land's Financial Services Authority has issued a warning to the general public nigh unlicensed Forex and binary options trading provided by entities registered in Saint Vincent and the Grenadines.[thirty] On 23 March 2018, The European Securities and Markets Authority, a European Union fiscal regulatory establishment and European Supervisory Authority located in Paris, agreed to new temporary rules prohibiting the marketing, distribution or sale of binary options to retail clients.[9] The Australian Securities and Investments Commission (ASIC) warned Australian investors on 13 February 2015 against Opteck, an unlicensed binary option provider.[31] The ASIC later began a focused effort to command unlicensed derivative providers, including "review" websites, broker affiliates, and managed service providers related to binary option products.[32] ASIC finally released a ban on auction of binary options to retail clients in 2021.[11] In August 2016, Belgium'southward Fiscal Services and Markets Say-so banned binary options schemes, based on concerns almost widespread fraud.[33] No firms are registered in Canada to offer or sell binary options, so no binary options trading is currently allowed. Provincial regulators take proposed a complete ban on all binary options trading include a ban on online advertising for binary options trading sites.[34] A complete ban on binary options trading for options having an expiration less than thirty days was announced on September 28, 2017.[35] On May 3, 2012, the Republic of cyprus Securities and Exchange Committee (CySEC) announced a policy change regarding the classification of binary options as fiscal instruments. The issue is that binary options platforms operating in Republic of cyprus, where many of the platforms are now based, would accept to be CySEC regulated within half-dozen months of the date of the announcement. CySEC was the first Eu MiFID-member regulator to treat binary options equally financial instruments.[36] In 2013, CySEC prevailed over the disreputable binary options brokers and communicated intensively with traders in order to prevent the risks of using unregulated financial services. On September xix, 2013, CySEC sent out a printing release warning investors against binary options broker TraderXP, who was non and had never been licensed by CySEC.[37] On Oct 18, 2013, CySEC released an investor warning about binary options broker NRGbinary and its parent company NRG Capital (CY) Ltd., stating that NRGbinary was non and had never been licensed by CySEC.[38] CySEC also temporarily suspended the license of the Cedar Finance on December xix, 2013, because the potential violations referenced appeared to seriously endanger the interests of the visitor's customers and the proper performance of capital markets, as described in the official issued printing release. CySEC besides issued a warning against binary option broker PlanetOption at the cease of the year and some other alert against binary pick broker LBinary on Jan 10, 2014, pointing out that it was not regulated by the Commission and the Commission had non received whatsoever notification past any of its counterparts in other European countries to the result of this business firm beingness a regulated provider. The Republic of cyprus regulator imposed a punishment of €15,000 against ZoomTrader. OptionBravo and ChargeXP were likewise financially penalized. CySEC also indicated that information technology had voted to decline the ShortOption license application.[39] In 2016, the regulator fined Banc De Binary Ltd once again for violation of its legislation. The broker has come to a settlement of €350,000.[41] In August 2016, France's Sapin Two bill on transparency was appear by the Autorité des Marchés Financiers (AMF), seeking to outlaw all financial derivatives advertising. The AMF stated that it would ban the ad of certain highly speculative and risky fiscal contracts to private individuals by electronic means.[42] The document applies specifically to binary options, and to contracts for deviation (CFDs), and financial contracts on currencies. The French regulator is determined to cooperate with the legal authorities to have illegal websites blocked.[44] The police also prohibits all forms of sponsorship and partnership that results in direct or indirect advertising of the financial products information technology covers. This ban was seen past industry watchers equally having an impact on sponsored sports such as European football clubs.[45] The Cyprus-based company 24Option[46] was banned from trading in France past AMF earlier in 2016.[47] They had sponsored a well-known Irish gaelic mixed martial artist, Conor McGregor, who in turn promoted the company through social media.[48] Frg German Federal Financial Supervisory Authority (BaFin) has been regularly publishing investor warnings. On November 29, 2018, BaFin announced that it is planning to "prohibit the marketing, distribution and sale of binary options to retail clients at a national level".[49] According to the Commodity Futures Trading Regulatory Agency (CoFTRA) in Indonesia, also known as BAPPEBTI, binary options are considered a course of online gambling and is illegal in the state. The motility to delegalize binary options stems from concerns that the public may exist swayed by misleading advertisements, promotions, and offers to participate in fraudulent practices that operate under the guise of binary options trading.[fifty] As of 2 February 2022, at least 92 binary options websites, including Binomo, IQ Option, and Olymp Merchandise, have been classified as unlicensed operators and blocked past the Indonesian government.[51] In March 2016 binary options trading inside State of israel was banned by the Israel Securities Authorization, on the grounds that such trading is essentially gambling and non a grade of investment management. The ban was extended to overseas clients likewise in Oct 2017.[17] It was canonical by the Knesset in October, despite potent opposition from the binary options industry.[18] ran several manufactures on binary options fraud. "The wolves of Tel Aviv: Israel's vast, amoral binary options scam exposed" revealed that the industry is a scam.[14] A second article describes in item how a binary options salesman fleeced clients. "According to one ex-employee of a firm that employs over one,000 people in a high-rise office edifice in Tel Aviv, losses are guaranteed because the 'dealing room' at the binary options firm controls the trading platform — similar the kleptomaniacal ownership of a rigged casino manipulating the roulette bicycle".[15] In July 2016 the Israeli binary option firms Vault Options and Global Trader 365 were ordered by the U.Due south. District Court for the Northern Commune of Illinois to pay more than $four.v million for unlawful off-exchange binary options trading, fraud, and registration violations. The companies were also banned permanently from operating in the United States or selling to U.S. residents.[53] In November 2016 the Israel Securities Dominance carried out a raid on the Ramat Gan offices of binary selection broker iTrader. The CEO and six other employees were charged with fraud, providing unlicensed investment advice, and obstruction of justice.[54] On May 15, 2017, Eliran Saada, the owner of Express Target Marketing, which has operated the binary options companies InsideOption and SecuredOptions, was arrested on suspicion of fraud, simulated accounting, forgery, extortion, and bribery. The case involves a Singaporean adult female who claims to have lost over $500,000 to the firm.[55] In August 2017 Israeli police superintendent Rafi Biton said that the binary trading industry had "turned into a monster". He told the Israeli Knesset that criminal investigations had begun.[eight] In September 2017, the FBI arrested Lee Elbaz, CEO of binary options trading company Yukom Communications, upon her inflow in the United states of america. They arrested her for wire fraud and conspiracy to commit wire fraud.[57] In Feb 2019, the FBI arrested Austin Smith, Founder of Wealth Recovery International, after his arrival in the Us. Smith was arrested for wire fraud due to his involvement as an employee of Binarybook.com.[59] Republic of malta In March 2013 the Republic of malta Financial Services Authority (MFSA) announced that binary options regulation would be transferred away from Malta'due south Lottery and Gaming Authorization.[sixty] On 18 June 2013 MFSA confirmed that in their view binary options barbarous under the telescopic of the Markets in Financial Instruments Directive (MiFID), which made Republic of malta the second Eu jurisdiction to regulate binary options as a financial instrument. This required providers to obtain a category 3 Investment Services license and conform to MiFID's minimum capital requirements; firms could previously operate from the jurisdiction with a valid Lottery and Gaming Authority license.[61] In Apr 2017, New Zealand's Financial Markets Authority (FMA) announced that all brokers that offer curt-term investment instruments that settle within three days are required to obtain a license from the bureau.[62] This is intended to cover binary options also as contracts for difference (CFDs). In the UK, binary options were regulated past the Gambling Commission rather than the Financial Acquit Authority (FCA).[63] This regulation, however, applied only to firms that accept gambling equipment in the UK.[64] The FCA in 2016 did propose bringing binary options under its jurisdiction and restricting them. They stated that binary options "did not appear to see a genuine investment need".[65] In March 2017, Action Fraud issued a warning on binary options.[66] The Isle of man, a self-governing Crown dependency for which the UK is responsible, has issued licenses to companies offer binary options as "games of skill" licensed and regulated nether fixed odds betting by the Island of Man Gambling Supervision Commission (GSC).[67] This positions binary options as a form of gambling, and the ambassador of the trading equally something akin to a casino, as opposed to an commutation or brokerage house. On October 19, 2017, London police raided 20 binary options firms in London.[65] On Jan 3, 2018, the FCA took over regulation of binary options from the Gambling Commission.[63] In December 2018, FCA has proposed new rules which would permanently ban the auction, marketing and distribution of binary options to retail consumers.[68] Fraud inside the market is rife, with many binary options providers using the names of famous and respectable people without their knowledge. According to a national fraud and cybercrime reporting centre Action Fraud, 664 binary options frauds were reported in 2015/sixteen, increasing to i,474 in 2016/17. The Metropolis of London police force in May 2017 said that reported losses for the previous financial year were £13 million, increased from £2 million the yr before.[13] In the start half of 2017, 697 people reported losses totaling over £xviii million.[65] In the United states of america, the Securities and Commutation Commission (SEC) approved exchange-traded binary options in 2008.[69] Trading commenced on the American Stock Exchange (AMEX) and the Chicago Board Options Exchange (CBOE) in May and June 2008.[seventy] AMEX (now NYSE American) offers binary options on some exchange-traded funds and a few highly liquid equities such as Citigroup and Google. On the commutation binary options were called "fixed return options" (FROs). To reduce the threat of market place manipulation of single stocks, FROs utilize a "settlement alphabetize" defined equally a book-weighted average of trades on the expiration twenty-four hours. AMEX and Donato A. Montanaro submitted a patent application for exchange-listed binary options using a volume-weighted settlement index in 2005.[71] CBOE offers binary options on the South&P 500 (SPX) and the CBOE Volatility Alphabetize (VIX).[72] NADEX, a U.S.-based Commodity Futures Trading Commission (CFTC) regulated exchange, launched binary options for a range of Forex, commodities, and stock indices' markets in June 2009,.[75] NADEX have since offered binary options trading between buyers and sellers. They practise not participate in the trades.[77] On June vi, 2013, the U.S. CFTC and the SEC jointly issued an Investor Alert to warn well-nigh fraudulent promotional schemes involving binary options and binary options trading platforms. The two agencies said that they had received numerous complaints of fraud about binary options trading sites, "including refusal to credit client accounts or reimburse funds to customers; identity theft; and manipulation of software to generate losing trades". Other binary options operations were violating requirements to register with regulators.[26] In June 2013, U.S. regulators charged Israeli-Cypriot visitor Banc De Binary with illegally selling binary options to U.S. investors.[26] In Feb 2016, the company reached an $eleven meg settlement with U.S. authorities. Regulators institute the company used a "virtual part" in New York'south Trump Tower in pursuit of its scheme, evading a ban on off-exchange binary pick contracts. The company neither admitted nor denied the allegations.[79] In November 2016, SEC published yet another Investor Alert on binary options websites.[fourscore] reported that the FBI was conducting an active international investigation of binary option fraud, emphasizing its international nature, proverb that the bureau was "not limited to the United states". Victims from around the globe were asked to contact an FBI field office or the FBI's Internet Crime Complaint Center. The investigation is non limited to the binary options brokers, but is comprehensive and could include companies that provide services that permit the industry to operate. Credit card issuers will be informed of the fraudulent nature of much of the industry, which could perchance allow victims to receive a chargeback, or refund, of fraudulently obtained money.[6] On March 13, 2017, the FBI reiterated its warning, declaring that the "perpetrators behind many of the binary options websites, primarily criminals located overseas, are only interested in one thing—taking your coin". They too provide a checklist on how to avert being victimized.[81] [vii] At that place is also a popular binary options recovery services scam, where fraudsters promise to "hunt" down the binary options scammers and think the coin from them through legal methods.[82] Choice (finance) Breeden, D. T., & Litzenberger, R. H. (1978). "Prices of country-contingent claims implicit in option prices". The volatility surface: a practitioner'south guide Binary Pick Definition Investopedia. Retrieved 2013-06-thirty. Tsipori, Tali (15 Dec 2016). "Binary options worth $1.25b to Israel'southward GDP in 2016". Weinglass, Simona (February 15, 2017). "FBI says it's investigating binary options fraud worldwide, invites victims to come forward". Feb 15, Weinglass, Simona (March 15, 2017). "FBI places public warning confronting 'Binary Options Fraud' at top of its main news". Appelberg, Shelly (2017-08-03). "In First, State of israel Police Admit Criminal offense Syndicates Are Behind Binary Options Industry". www.esma.europa.european union "Binary Options Trading In Australia: How Safety Is Information technology?". International Business organisation Times AU. 2018-05-14. Retrieved "21-064MR ASIC bans the auction of binary options to retail clients". www.asic.gov.au Weinglass, Simona (March 4, 2017). "As Israel-based financial fraud soars, police swoop on 20 suspects equally office of global, FBI-led sting". The Times of State of israel Press Association (18 May 2017). "Richard Branson says scammers are using his proper name to dupe investors". Weinglass, Simona & Horovitz, David (Apr seven, 2016). "Ex-binary options salesman: Hither'south how nosotros fleece the clients". Tova Cohen (June 18, 2017), "Israel cabinet approves ban on sale of binary options", Weinglass, Simona (Oct 23, 2017). "State of israel bans binary options industry, finally closing vast, 10-year fraud". The Times of Israel. Retrieved Tomer, Uri (2017-10-24). "State of israel Bans Binary Options Manufacture That Defrauded Millions". 2017-x-24 Frier, Sarah; Verhage, Jules (January xxx, 2018). "Facebook Bans Ads Associated With Cryptocurrencies". February vii, Cornish, Chloe (January 30, 2018). "Facebook and regulators motion to halt cryptocurrency scams". February 7, Weinglass, Simona (March 28, 2018). "European Marriage bans binary options, strictly regulates CFDs". The Times of State of israel. Retrieved Mitchell, Cory (11 June 2014). "A Guide To Trading Binary Options In The U.Due south." "Banker's Edge Reckoner". (PDF). Financial Market place Potency (Austria). Retrieved Pape, Gordon (27 July 2010). "Don't Gamble On Binary Options". www.cftc.gov. U.S. Commodity Futures Trading Committee. Retrieved Options, Futures and Other Derivatives. Prentice Hall. ISBN0-13-149908-4. Airtight-form expressions for perpetual and finite-maturity American binary options . parsiad.ca (2015-03-01). Retrieved on 2016-07-xviii. "Investor Alert Binary Options and Fraud" (PDF). U.Southward. Securities and Exchange Commission. Retrieved "UNLICENSED FOREX/BINARY OPTIONS – Financial Services Authority". "15-024MR ASIC warns of Opteck and other unlicensed binary option providers". Retrieved Weinglass, Simona (August thirteen, 2016). "In European first, Kingdom of belgium bans binary options". Shecter, Barbara (April 26, 2017). "Canadian watchdogs motion to ban binary options as fraudulent schemes fleece investors, steal identities". "Securities regulatory grouping announces ban on brusk-term binary options". four June "The projects of the CySEC regulator in terms of binary options in 2014". Archived from the original on 15 October 2017. Retrieved "Ban on the ad of forex products, binary options and some CFDs: AMF launches consultation on changes to its General Regulation", (printing release), August 1, 2016, archived from the original on June 17, 2019, retrieved Andrew Saks-McLeod (9 January 2017). "IG Grouping officially responds to French FX and CFD advert ban". Maria Nikolova (10 January 2017). "AMF toughens its stance on advertising post-obit public consultation". Finance Feeds. Archived from the original on 7 May 2017. Retrieved Maria Nikolova, "French football clubs rush to end binary options partnerships", Under Article 77 of the Sapin Two law, football game teams have until the end of June this year to finish all sponsorships "24OPTION – Cyprus Business Proper name – CyprusRegistry.com". Fifty'Autorité des marchés financiers interdit à Rodeler Limited (" 24option ") de fournir des services sur le territoire français Barry O'Halloran (November 19, 2016), "Conor McGregor sponsor barred from operating in France", "BaFin plans national prohibition of binary options for retail clients". Federal Financial Supervisory Authorization. November 29, 2018. Retrieved Dirgantara, Hikma (2022-02-03). "Bappebti Blokir 92 Entitas Binary Option di 2021, Termasuk Binomo, IQ Pick". Weinglass, Simona (June 18, 2017). "Israeli ministers approve bill to outlaw unabridged binary options industry". "Federal Court Orders Israeli Web-based Firms, Vault Options, Ltd. and Global Trader 365, to Pay More than $4.5 One thousand thousand for Unlawful Off-Substitution Binary Options Trading, Fraud, and Registration Violations". Bolt and Futures Trading Commission. July 28, 2016. Weinglass, Simona; Horovitz, David (Nov 12, 2016). "Israeli authorities raid binary options firm, abort CEO, salespeople". Weinglass, Simona (May xvi, 2017). "Tel Aviv binary options boss nabbed for fraud, extortion; visitor raided, computers confiscated". May 16, Cohen, Tova (May sixteen, 2017). "Israeli police abort possessor of binary options business firm accused of fraud". Staff, Tori (September 24, 2017). "FBI arrests Israeli binary options CEO equally she disembarks El Al flight at JFK". "Former CEO of Israeli Visitor Sentenced to 22 Years in Prison for Orchestrating Major International Binary Options Fraud Scheme". Department of Justice. December 19, 2019. Retrieved Baronial nineteen, "Gaming or Trading? That is the Question – MFSA To Regulate Binary Options as a Financial Product", "FMA confirms short-term derivatives to exist licensed \| FMA". 2017-04-eighteen Spud, Hannah (January 12, 2018). "FCA warns on illegal binary options providers". Fantato, Damian (November xv, 2017). "FCA warns investors confronting binary options". Murphy, Hannah (October 19, 2017). "Police force launch raids on 20 City of London businesses". "What is Binary Options Fraud and what are the police enforcement doing about it?". Activeness Fraud. March 31, 2017. Retrieved gov.im. Isle of mann Government. Retrieved Fantato, Damian (December 7, 2018). "FCA to permanently restrict binary options sales". Frankel, Doris (June 9, 2008). "CBOE to list binary options on S&P 500, VIX". Reuters. Retrieved leaprate.com. January five, 2017. Retrieved "System and methods for trading binary options on an exchange" Archived 2009-02-xviii at the Wayback Automobile, Earth Intellectual Property Organization filing. Retrieved on 2013-01-12. (PDF). Chicago Board Options Commutation. September 10, 2008. Archived from the original on September x, 2008. Retrieved "^BSZ (CBOE Binary Options On The S&P 500 Index)". Chicago Lath Options Commutation. Dec 8, 2018. Retrieved "^BVZ (CBOE Binary Options Volatility)". Chicago Board Options Commutation. December 8, 2018. Retrieved "Nadex, Due north American derivatives exchange rolls out new trading platform, products" on Apr i, 2012. Retrieved "Regulated & Legal for US Residents". "SEC Warns Investors About Binary Options and Charges Cyprus-Based Company with Selling Them Illegally in U.S." U.S. Securities and Commutation Commission. Retrieved Patrick, Margot (February 25, 2016). "Banc de Binary Reaches $11 Million Settlement with U.S. Government". "Investor Alert: Binary Options Websites May Exist Used for Fraudulent Schemes". U.Due south. Securities and Commutation Commission. November 10, 2016. Retrieved FBI.gov. Federal Agency of Investigation. March xiii, 2017. Retrieved Kay, Martin (December v, 2018). "Binary Options Recovery Services – The Real Story". "U.S. charges two over fraud featuring bogus SEC employees". Reuters. Jan 24, 2018. Retrieved Fisher, Jenna (Baronial xvi, 2018). "Man Gets 5+ Years Prison For Boston, Dominican SEC Electronic mail Scam". Horovitz, David; Weinglass, Simona (10 July 2019). "Alee of fraud trial, US lays bare 'multitude of lies' of Israeli binary options". The Times of Israel. Retrieved x July Levy-Weinrib, Ela (February 2, 2017). "Regulator seeks blanket binary options ban". Globes English. Retrieved Baca juga: End Of Day Forex Trading Strategy	CommonCrawl

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