Analytical Chemistry Chemical Cytometry Quantitates Superoxide

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Figure 2. Biplots of the Iris benchmark data set. The scores (representing the samples) are visualized as symbols, whereas the loadings are visualized by the vectors. The loadings are obtained by (a) projection of the first two columns in the loading matrix and (b) projection of the scores of 10 pseudosamples (for each variable). A biplot is a visualization in which the samples and the variables of a data set are represented together. This technique has been introduced by Gabriel in 1971 36 and is based on singular value decomposition (SVD) or principal component analysis (PCA) 37 of the column mean-centered data matrix X, containing n samples and m variables. By SVD X is decomposed into scores and loadings according to T X (n×m) ) U (n×r) Λ (r×r) V (m×r) T ) S (n×r) L (m×r) The positions of the samples (rows of X) in a two-dimensional biplot are subsequently given by the elements in the columns of UΛ, often called the scores S. The variables (columns of X) are usually represented as vectors pointing from the origin to the coordinates that are given by the elements of the columns of V, called the loadings L. To construct a biplot for the first two singular vectors (or principal components, PCs), the elements in the first two columns of S and in the first two columns of L are used. This is illustrated for the Iris data set (see the Iris benchmark Data Set section) in Figure 2a. The coordinates of a new sample in this biplot can be obtained by premultiplying the sample as a row vector with V: x (1×m) V (m×2) ) s (1×2) The key idea leading to the nonlinear biplot is the interpretation of the loadings in the classical biplot (the representation of the variables) as projections of special so-called pseudosamples that carry all their weight in one variable. This means that these pseudosamples have a value of 0 for all variables, except for one variable. Projecting this pseudosample in the two-dimensional PCA plot yields coordinates that are equal to the loadings of the variable whose weight it carries. For example, [1, 0, 0, ..., 0] is a (1 × m) pseudosample with a value 1 for variable 1 and a value 0 for all other variables. Then [1, 0, 0, ..., 0] (1×m) V (m×2) ) s (1×2) ) v (1×2) (36) Gabriel, K. R. Biometrika. 1971, 58, 453–467. (37) Massart, D. L. Handbook of Chemometrics and Qualimetrics: Part A; Elsevier Science Publishers: Amsterdam, 1997. 7002 Analytical Chemistry, Vol. 82, No. 16, August 15, 2010 (2) (3) (4) where v is the first row of V and contains exactly the loading of variable 1, defining its position in the biplot. If the value of 1 is replaced by p different values z, a total of p pseudosamples are obtained. The projection of these pseudosamples in the PCA plot will result in a trajectory along the direction of the variable vector, as illustrated in Figure 2b. Gower et al. 34 extend this idea to the visualization of variable information in principle coordinate analysis. In principal coordinate analysis (or classical metrics scaling) an SVD is performed not on the original rectangular data matrix X, but on the symmetric matrix of squared Euclidean distances. This approach allows the visualization of the relative distances of the objects. It is often remarked that the information of the original variables (such as in a biplot) is lost. Gower et al., however, showed that this can be overcome by using the concept of trajectories of pseudosamples, as explained earlier. To make this approach feasible, one must be able to calculate the squared Euclidean distances of the pseudosamples with all other samples (e.g., data X). By projecting the rows of the resulting distance matrix in the principal component space, the trajectories that represent the variables are obtained. Gower et al. showed that this concept can be extended from Euclidean distances to many nonlinear distance metrics, as long as the same distance metric can be used to calculate the distances of the pseudosamples to the original data samples. In the classical linear biplot, only one pseudosample per variable is sufficient to represent the variables. For a nonlinear distance metric, the trajectory will be curved and multiple pseudosamples per variable are required. Nonlinear Biplots in SVM Classification. We propose to apply the above approach to the kernel-based SVM method, since the kernel is a (nonlinear) distance metric between objects. This procedure comprises the following steps (shortened): 0. Optimize the kernel function and parameter settings for the SVM classification used (resulting in the kernel function K(xi.,xj.)). 1. From the data X calculate the kernel matrix K, by using the optimized kernel function. This matrix is subsequently centered, resulting in matrix K c of size n × n. 2. Apply SVD on K c and construct different score plots (for the various combinations of principal components) for the n samples in K c . Inspect these score plots to find the direction(s)
(principal components, PCs) in which maximum class separation is obtained. Note that this is not necessarily along the first PCs. 3. Construct a matrix Pj for the j th variable which contains p pseudosamples. The range of these pseudosample values zj should vary between the minimum and maximum value of the original variable. 4. Apply the kernel function K(pi.,xj.) to the pseudosample data Pj to obtain the kernel matrix of the pseudosamples C (i.e., calculate the kernel distances of the pseudo samples to the original samples). 5. Project the rows of the centered pseudosample kernel matrix C c in the score plot that was found in step 2. 6. Repeat steps 3-5 for each variable j. The resulting plot contains a trajectory of pseudosamples for each original variable. These trajectories yield information about the relative contribution of the variables to the SVM classification. The curvature of a trajectory indicates a nonlinear kernel transformation. SVM Classification and Validation Procedure. Each data set was analyzed by SVM using the RBF kernel. The RBF kernel was chosen because of its simplicity for optimization. Because the SVM application that we have used is a binary classifier, each different class in a data set that contains more than two classes was analyzed by a one-against-all approach. 38 The kernel parameters parameter σ and the SVM parameter C (C is a cost parameter 23 ) were optimized by leave-10%-out cross-validation. All calculations are performed in the software program Matlab (The MathWorks Inc.) version 6.5 release 13. The nonlinear biplot approach was implemented in Matlab by using the commercially available PLS toolbox from eigenvector Research Inc. Data Sets. To illustrate the applicability of the proposed method we have used several benchmark data sets (synthetic and real) and a metabolomics data set based on Magnetic Resonance Spectroscopic Imaging (MRSI) data. Synthetic Benchmark Data Set. A synthetic data set was constructed based on the data presented in Figure 1, which is generally used to illustrate the power of SVM classification. The data set contains two variables to represent an inner and outer circle, both consisting of 50 objects. The two circles can not be separated in a linear way and therefore a kernel-based classification method is required to separate the data. To construct the Synthetic data set we added five variables containing random noise (normally distributed) to the data. The variance of these five variables was set to be about 10% of the variance of the two variables which represent the circles. Because noise contains no information, the added variables do not contribute to the classification performance. This data set was constructed to confirm that the first two variables (representing the circles) are only (equally) important in the SVM classification. Iris Data Set. A widely used benchmark data set to exemplify discriminant and cluster analysis is the data published by Fisher in 1936. 39 This Iris data set contains fifty specimens of each of the three species Iris Setosa, Iris Versicolor, and Iris Virginica, resulting in a total of 150 samples. Four properties of the species are measured to determine differences between the three classes. (38) Suykens, J. A. K.; Van Gestel, T.; De Brabanter, J.; De Moor, B.; Vandewalle, J. Least Squares Support Vector Machines; World Scientific: Singapore, 1999. (39) Fisher, R. A. Annu. Eugen. 1936, 7, 179–188. These properties (representing four variables) are Sepal Length, Sepal Width, Petal Length, and Petal Width, all measured in millimeters. The Setosa class can easily be separated from the other two classes using only one variable (either Petal Length or Petal Width). The two other classes are partly overlapping, as shown in Figure 2. We will use the Iris data set to demonstrate the applicability of our proposed method to determine the most discriminative variable for the three species. Metabolomics Data Set. To illustrate the proposed method for variable selection on a more complex data set, we have used a metabolomics data set which consists of magnetic resonance spectroscopic (MRS) spectra obtained from MRSI data. 40 This data set was constructed during a European project called Interpret, which was funded by the European Commission to develop new methodologies to automatically classify tumors in the human brain (see http://azizu.uab.es/INTERPRET). Data from a total of 24 patients and 4 volunteers were acquired by MRS at different positions in the brain, according to an acquisition protocol defined by the Interpret Consortium. The study was approved by the ethical committee and followed the rules of the World Health Organization. After reaching consensus about the histopathology, three tumor types were identified according to the World Health Organization classification system. These three classes contained glial tumors with different grades: Grade II (10 cases), Grade III (4 cases), and Grade IV (7 cases). A fourth class consists of spectra acquired from patients with Meningioma (3 cases). Additionally, a class consisting of Healthy tissues was created from patient (4 cases) and volunteer (4 cases) data. For each predefined class a selection of spectra from the different patients was made. Only spectra acquired at regions which clearly consisted of tissue belonging to the particular class were selected. The data for the Healthy class was selected from the volunteers or from the contralateral brain region of the patients. 41 The resulting data set contains 569 spectra, consisting of five different classes. Each spectrum contains 229 data points, covering the chemical shifts between 4.0 and 0.5 ppm. Details about the acquisition parameters and preprocessing of the data are described in Simonetti et al 42 and are beyond the scope of this paper. RESULTS Synthetic Benchmark Data Set. Classification of the Synthetic data set by using SVMs resulted in a leave-10%-out crossvalidated accuracy of 100%. This accuracy (see also SI Table 1) illustrates that SVMs are able to separate the two circles. As noise contains no information, the particular noise-variables should not have contributed to the class separation. To verify the contribution of each variable in the final classification first we have to find and visualize the optimal (linear) separation between the classes in the resulting kernel matrix. Because the Synthetic data set contains one hundred objects, the feature space in which the original data is mapped (i.e., the space of the kernel matrix) consists as a consequence of one hundred dimensions. Therefore PCA is used to reduce the dimensionality of the feature space. Analysis of the score plots of combinations of only the first five (40) Barker, P. B.; Lin, D. D. M. Prog. Nucl. Magn. Reson. Spectrosc. 2006, 49, 99–128. (41) Simonetti, A. W.; et al. Anal. Chem. 2003, 75, 5352–5361. (42) Simonetti, A. W.; et al. NMR Biomed. 2005, 18, 34–43. Analytical Chemistry, Vol. 82, No. 16, August 15, 2010 7003
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Anal. Chem. 2010, 82, 6745-6750 Let
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purchased from Invitrogen-Molecular
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Figure 3. Electropherograms of TPP-
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Anal. Chem. 2010, 82, 6751-6755 Res
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(pH 8.0) with cysteine and cystamin
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Figure 4. Ion mobility mass spectra
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GOx glucose + O298 gluconic acid +
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coater at 2000 rpm for 20 s, and th
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Figure 5. Comparison of cyclic volt
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in channels with either no grooves
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indicators of atmospheric processin
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Figure 1. GC/MS total ion chromatog
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Table 2. Concentrations and Stable
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on a substrate are preferred. 20-24
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Figure 4. SERS analysis of NAADP co
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Anal. Chem. 2010, 82, 6775-6781 Hig
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tion, 2 µL of proprionaldehyde wer
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Figure 4. Analysis of 2a by HPLC-MS
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eaction of 1a with PBH can be condu
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Numerous references had demonstrate
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Figure 1. TEM images of the prepare
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Figure 4. Schematic representation
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esult in a big SPR signal change wi
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also reduces chemical noise, which
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Table 1. Extraction Yields, Liquid
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scanning of AMPP amides of the anal
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Anal. Chem. 2010, 82, 6797-6806 δ
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Figure 1. Schematic view of the pre
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from the specimen and enclosed in a
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Figure 4. (A) IRMS mass-44 chromato
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Table 3. Ambient Measurement Result
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Anal. Chem. 2010, 82, 6807-6813 Dir
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Polymerase (1 U per sample). Reacti
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of which was constant for all ampli
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analytically useful signals at less
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carbon black and RP-C18 for the ext
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solution in an equal volume, and 1
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Table 2. Concentrations and Ratios
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Anal. Chem. 2010, 82, 6821-6829 Mac
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mg/mL protein, followed by separati
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Figure 2. Productivity of SEQUST an
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Figure 4. High-resolution MS/MS spe
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Figure 6. Characterization of PSMs
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containing T-T mismatches. 23 Based
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Figure 1. Extinction spectra of sol
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Figure 3. (A) The value of Ex650 nm
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Figure 5. Extinction spectra and co
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wished to explore the dehydration o
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constant medium for separation; we
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Figure 2. Standards of (Pi)n, n ) 1
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Figure 5. Quantitative calibration
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Anal. Chem. 2010, 82, 6847-6853 Met
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Figure 1. 226 Ra spectrum by liquid
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Table 1. Counting Properties and De
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Table 3. Analysis of 226 Ra in Sedi
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mercial microarray scanner and fabr
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Figure 2. Optical transmission meas
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Figure 5. Volcano plots detailing t
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data as well. The 41 genes in Table
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corresponding compound if its chemi
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Figure 1. Schematic illustration of
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Table 1. Absolute Quantification Re
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ment. Therefore, the long-time drea
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Figure 1. Chemically actuated micro
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Figure 2. Influence of a surfactant
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Figure 5. Device to eject and mix s
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Anal. Chem. 2010, 82, 6877-6886 Imm
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NaCl, phosphate buffer saline (PBS)
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Figure 2. Product ion mass spectra
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-70 °C resolved the problem, givin
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Table 1. Intraday Precision and Acc
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Anal. Chem. 2010, 82, 6887-6894 Fer
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Figure 1. Infrared spectra of (A) u
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Figure 3. Cyclic voltammograms obta
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Figure 6. Calculated charge from ch
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Anal. Chem. 2010, 82, 6895-6903 Ele
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Table 1. Chemical Structure, pKa Va
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pH with a tilted baseline (Figure 1
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Table 2. Linearity and Detection Li
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ascorbic acid (AA), uric acid (UA),
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the multielement capabilities, the
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RESULTS AND DISCUSSION Sulfur Detec
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Table 2. Molecular Properties and C
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Anal. Chem. 2010, 82, 6911-6918 Dir
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dilution and hybridization buffer.
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solution under appropriate incubati
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Figure 4. Standardization curve for
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Anal. Chem. 2010, 82, 6919-6925 Ele
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the ×10 objective, to have a large
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Figure 2. With a suitable removal o
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the NB signal in a much better foot
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Scheme 1. Reactions of Selenium Rea
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Figure 2. (a) ESI-MS spectrum showi
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Figure 4. (a) ESI-MS spectrum showi
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Anal. Chem. 2010, 82, 6933-6939 Dif
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trode 28 by a finite element using
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Figure 4. Comparison between simula
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Figure 6. Comparison between simula
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educed in the vicinity of double bo
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Figure 2. Normalized product ion ab
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Figure 4. EID (a) and IRMPD (b) of
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Anal. Chem. 2010, 82, 6947-6957 Ide
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Figure 1. Schematic flowchart showi
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Page 233 and 234: nucleotide tails. 3-5 Thus, the amo
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Page 263 and 264: CONCLUSIONS In this paper we have i
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Page 287 and 288: Figure 2. SEM images of Au substrat
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Page 293 and 294: Figure 1. (A) FL spectra of BSPOTPE
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Page 299 and 300: Scheme 1. Proposed Mechanism for Fl
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Page 307 and 308: Figure 2. Comparative analysis of d
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