Research suggests that Black youth are less likely to use e-cigarettes than their white counterparts, yet little is known as to why. We examined perceptions of e-cigarettes among Black young adults (ages 18–25) to explore the meanings these youth ascribe to e-cigarettes and the role that identity plays in how these devices are viewed. Analysis of in-depth interviews with 36 Black smokers and non-smokers in the San Francisco Bay Area suggests that Black youth perceive e-cigarettes as serving distinct, yet overlapping roles: a utilitarian function, in that they are recognized as legitimate smoking cessation tools, and a social function, insofar as they serve to mark social identity, specifically a social identity from which our participants disassociated. Participants described e-cigarette users in highly racialized and classed terms and generally expressed disinterest in using e-cigarettes, due in part perhaps to the fact that use of these devices would signal alignment with a middle class, hipster identity. This analysis is discussed within a highly charged political and public health debate about the benefits and harms associated with e-cigarette use. Use of electronic nicotine delivery systems (ENDS), e.g., e-cigarettes, is increasing among youth in the United States . While tThough e-cigarettes appear to be increasing in popularity among youth and young adults generally, e-cigarette initiation and use nevertheless varies across youth groups. Similar to patterns of conventional cigarette use , Black aSimilar trends are present among young adults. McMillen and colleagues 2015) conducted nationally representative surveys each year between 2010 and 2013. Among young adults aged 18 to 24, current e-cigarette use increased in prevalence from 0% to 14% during this time 5 conduct. AnalysiFrom a public health perspective, lower e-cigarette uptake among Black youth may be regarded positively or negatively . Though To understand how Black youth perceive e-cigarettes, we draw on social science theories regarding social and cultural identity and the use of commodities to communicate those identities . Nearly Cultural attitudes towards products are shaped by the degree to which products are associated with utilitarian functions , as well as social identity functions  . In thisnon-helmet-wearing, superseded the unequivocally positive utilitarian function [Individuals may also choose not to use a product, or actively avoid certain objects or behaviors in order to disassociate with social groups . In thisfunction . Thus, dBetween April 2014 and July 2015, we conducted multiple method in-depth interviews with 46 Black (ages 18–25) smokers and non-smokers in the San Francisco Bay Area for a study designed to understand perceptions of tobacco control policies and the intersections of tobacco-related stigma with other social identity stigmas . Of the 46 participants in the study, 36 participants discussed these devices in their interviews, and 15 had used e-cigarettes at some point in the past.Participants were recruited on the street, through Facebook and Craigslist advertising, and by referrals. The interview guide was informed by (1) existing qualitative research on smoking among youth as well as by (2) exploratory focus groups we conducted with Black youth prior to data collection in an effort to identify potentially salient issues . E-cigarFollowing the interviews, the interviewer completed field notes to record preliminary analytical ideas emergent from interviews and to contextualize interviews with descriptive information not captured in audio recordings. Interviews were digitally recorded and professionally transcribed. Two research staff coded transcripts to distill data into manageable analytical segments, using ATLAS.ti , a qualitative data management system. During the coding process, researchers wrote memos to capture their analytical thinking about the data. Memos serve to contextualize coded narrative data with researchers’ emergent interpretations that are based on the entire context of the interview. The code list was extensive to capture main study objectives, but for the purposes of the present analysis, only one relevant code was included  which indexed any narrative in interviews that either referenced e-cigarettes directly or was relevant to understanding perceptions of e-cigarettes. After the coding process, the lead author conducted a second more directed yet exploratory pattern-level analysis of all narrative segments coded as ENDS and all related memos. Patterns of emergent themes were identified primarily by examining the frequency and omission of topics across interviews . FrequenAnalysis of the patterns of participants’ narratives illustrated that Black youth in this study perceive e-cigarettes as serving distinct, yet overlapping roles: a utilitarian function, in that they are recognized as legitimate smoking cessation tools, and a social function, insofar as they serve to mark social identity, specifically a social identity from which our participants disassociated. The majority of participants in this study were of low socio-economic status, with a household income of less than $35,000 USD, technically classified as living in poverty in the San Francisco Bay Area. Twenty-four young women and 22 young men comprised the sample. Twenty-five participants were current tobacco or nicotine users defined by past 30-day use. Among the 15 ever e-cigarette users, 12 had used e-cigarettes in the previous year, and among these, all reported smoking cigarettes in the previous 30 days. Among the 36 respondents whose interviews included discussion of e-cigarettes, 21 were current smokers  and 15 were non-smokers .“A higher-income person, (someone) who goes to the doctor and feels guilty about smoking cigarettes and tobacco, and is into technology and (chuckles) maybe rides a bike.”Overall, participants in our study described the stereotypical e-cigarette user as someone racially, socio-economically, and/or socially different from themselves. Participants emphasized that e-cigarettes “weren’t for them,” ascribing social meanings to e-cigarette users from which participants in our study sought to disassociate. For example, “Z”, a 24-year-old non-smoking man, who has not used e-cigarettes, identified a “typical” e-cigarette user as:“I feel like a lot of hipsters use e-cigarettes. A hipster is someone who tries to fit in with a certain image because it looks cool. E-cigs are very trendy so they’re like ‘Ooh I look so cool with my e-cig.’”The perception of a typical e-cigarette user as someone who is affluent was common among our participants, as was the concept of e-cigarettes being popular among “hipsters” and “techies” as is alluded to in the quote above and reinforced in other participants’ narratives. For example, “Jazzy”, an 18-year-old non-smoking woman, who does not use e-cigarettes, succinctly positioned e-cigarettes as a marker of social identity when describing them as products for people who are trying to look cool or trendy:“…non-Black people… So similarly with the cigarette smoking, actually having a cigarette, that is mostly White people. Then secondary, non-Black people, non-White people. But with the e-cigarettes, I suppose it would just be non-Blacks. I’ve seen Asians smoking as well.”Although hipster culture is notoriously difficult to define ,30, it i“Jane Doe”, a 24-year-old woman who smokes, also stated that e-cigarettes are not something that she sees in the Black community:“I sure don’t see Black people smoking them… I see Black people smoking real cigarettes. I never see Black people with e-cigarettes or vapors…” She considered trying e-cigarettes but did not.“Who do I see smoking e-cigarettes? Upper class individuals. I haven’t seen a person of color smoking those e-cigarettes ‘cause they’re kind of expensive.’”Meanwhile “Dee”, a 24-year-old woman who smokes, but does not use e-cigarettes, highlighted socioeconomic factors that may contribute to the association of e-cigarettes with the “commodification and consumption (that) is intrinsically connected to being a hipster” .“Who doShe went on to state that: “Usually I see affluent people smoking those… I don’t think they sell those in marginalized communities…”“Let’s just be straightforward. Particularly, I’ve seen them in the hands of European Americans, white folks… I worked this tech show last week in San Francisco and I saw people with it.”“Keeno”, a non-smoking, non-e-cigarette-using man aged 24, reinforces the notion that e-cigarettes signal membership in an external group (“tech show” participants) typically associated with a hipster style of consumption. Noting increasing e-cigarette use in Oakland and San Francisco, Keeno observes:“I see a lot more business people with vape pens nowadays taking a… break outside instead of cigarettes. So that’s a good thing, in my opinion… Maybe young black kids will prefer those over… cigarettes at some point in time.”From a public health perspective, the disassociation of e-cigarettes among Black youth who smoke may be problematic due to the potential health benefits of switching from conventional smoking to e-cigarettes. One participant, “Lee”, a non-smoking man aged 21, commented that he hoped e-cigarettes would become more than just a subcultural phenomenon, raising the possibility of harm reduction through e-cigarette use for Black youth who smoke. Lee had tried e-cigarettes in the past, but did not like them, and only occasionally “takes a few puffs” from friends’ devices.Lee’s comment points to how e-cigarettes can be positively perceived for their utilitarian functions but still rejected when they are socially positioned as cultural commodities that signal membership in a group from which a particular youth group may be disassociated. In addition to the social identity signaling of e-cigarette use, analysis of the narrative data suggests that e-cigarettes were perceived to serve important utilitarian functions for some people—i.e., for smoking reduction or cessation or for minimizing cravings when smoking is prohibited. According to “Lee”, above, e-cigarettes are targeted “towards an older audience, you know, people that are trying to quit smoking or that are looking for an alternative for whatever reason”. “Black Tar Rebellion”, for example, a 21-year-old non-smoking woman, summarizes the typical e-cigarette user as: “Somebody who wants to quit or somebody who doesn’t want to smell like cigarettes.” Black Tar Rebellion, who has never smoked, briefly used a nicotine-free vape pen as a way to look rebellious, but gave it up after experiencing negative health effects. “Some people who actually had a habit of smoking cigarettes, who are trying to be better, they use them. I respect them way more.”Participants generally regarded people who used e-cigarettes for their utilitarian function as opposed to their social identity function more positively, and their perceptions of the typical utilitarian user did not tend to carry connotations of race and class. “Jazzy”, for example, who mocked the typical e-cigarette user as a hipster who just wants to look “cool”, acknowledged a utilitarian role of e-cigarettes and stated that she has more respect for people who use e-cigarettes to quit smoking, as opposed to adopting a new trend:“People who smoke e-cigarettes are trying to get off cigarettes and they just can’t go cold-turkey, so that’s like their in-between… But it’s very ineffective on that in-between to get you off of it.”Though participants in this study tended to disassociate from vapers and vaping, 15 participants nevertheless had experimented with e-cigarettes. In general, their initial motivation for trying the devices stemmed from health concerns about smoking, and they had heard that e-cigarettes might help to facilitate cessation. Notably, however, there was skepticism about the effectiveness of e-cigarettes as cessation tools, and not just among the participants who had tried the devices. For example, “Karoline”, a non-smoking woman aged 19, believed that e-cigarettes were primarily for individuals who cannot quit smoking, though she questioned the extent to which they work.“He had an e-cig, ‘cause my mom bought it for him I think‘ cause she wanted to help him. She was happy that he was trying to be better, so in order to help him on his journey, she got him some e-cigs. He would charge it. Everything was cool. He stopped using cigarettes. Now he’s just using his e-cig. Then after that, the next step would have been, ‘Okay, now you’re not even using an e-cig and you’re good. You’re fine. You don’t even need it.’… He ended up relapsing.”Similarly, “Jazzy” described an acquaintance who used e-cigarettes to quit smoking, but who was ultimately unsuccessful.“It’s just weird that they say it’s not smoke, but then I learned that they about have the same, like, bad stuff as cigarettes. I thought it was healthier. That’s why I was doing it. But there’s no difference.”In addition to skepticism about the effectiveness of e-cigarettes for smoking cessation, participants in this study who smoked were generally not interested in switching to e-cigarettes due to questions about their safety. For example, “Marley”, a 22-year-old smoker who had tried e-cigarettes to quit smoking, believes that e-cigarettes are just as harmful as cigarettes.Similarly, “Jade,” a 22-year-old smoker, questioned the point of switching from cigarettes to e-cigarettes: “This makes me wanna know a lot more about e-cigarettes, like what’s in them, why it’s okay to switch to e-cigarettes and it’s okay to maintain that habit, but not regular tobacco cigarettes. So I wanna know what the differences are, but then also, why is the solution to stopping one habit starting another…”“I smoke like two cigarettes a day. But I’ll try the vape pen most of the time. Like today, I forgot my vape pen, so I’m like, damn it, I’m gonna have to go get a cigarette. I try to do as much of the vape pen as I can.”Notably, however, among the few participants who reported regular use of e-cigarettes was “Nefarious,” a 21-year-old woman who was using e-cigarettes to cut down on smoking and to minimize what she perceives to be negative consequences of smoking.“The smell (of cigarettes). That’s why I got the vape pen… I don’t smell like anything. It’s all sweet and fruity. I still smoke like two cigarettes, but I’ve been working since I was homeless to not smell like cigarettes. I was smelling like straight up cigarettes and God knows what else when I was homeless. I did not like that. My friends didn’t like it.”She goes on to say that she originally purchased a vaping device because of:Here Nefarious’ use of e-cigarettes in combination with cigarettes can be described as “dual use,” about which researchers in public health have expressed great concern due to the potential of dual use to exacerbate nicotine addiction and reinforce smoking ,32,33. HFew participants had adopted positions similar to Nefarious, however. Instead, participants in this study generally expressed safety concerns and/or skepticism about the effectiveness of e-cigarettes for cessation and few participants were using the devices. Research by Camenga and colleagues, who conducted focus groups with middle school, high school, and college students, also found that adolescents and young adults were skeptical about the efficacy of e-cigarettes in successful smoking cessation. Nevertheless, their participants generally perceived e-cigarettes to be less harmful than cigarettes . ConversFindings should be considered in light of the following limitations. Importantly, this exploratory analysis is based on the narratives of 36 Black women and men who were selected purposively for a study on perceptions of tobacco control strategies and experiences with multiple stigmas. Narrative surrounding e-cigarettes emerged occasionally throughout interviews, and after conducting 10 interviews for the study, we integrated a few open-ended questions about perceptions of e-cigarettes. As such, this study was not designed around understanding perceptions of e-cigarettes among Black youth, and, therefore, findings should be considered in light of this important limitation. Additionally, the study is limited in scope by the relatively small, non-random sample. Although this study is not necessarily a representative sample of Black youth in the San Francisco Bay Area, we did attempt to recruit a diverse range of participants  by using a multi-tiered recruitment strategy to reach participants from a variety of social contexts and networks. Notably, qualitative research does not aim for traditional conceptualizations of generalizability or representativeness. Instead, it aims to maximize the variation of perspectives among participants to develop theory that can be tested in future large-scale studies. A final limitation to note here is that the theoretical ideas discussed in this paper are based on our research team’s interpretation of participants’ narratives, and multiple interpretations of these narratives are possible. However, we integrate standard procedures into our study designs in an effort to reduce threats to the valid interpretation of data and enhance reliability to justify analytical generalizability, i.e., the likelihood that the theoretical contribution from our study is applicable to another situation. As Douglas and Isherwood wrote, “Goods that minister to physical needs… are no less carriers of meaning than ballet or poetry”  (p. 49).In particular, study participants offered clear individuating descriptions of a typical e-cigarette user, such as “hipster” and “rides a bike”. There is evidence of a hipster association with e-cigarettes in the literature. Tobacco companies have actively courted hipsters , and thenon-use of e-cigarettes also communicates meaning about social identities. It may be that our participants express disinterest in e-cigarettes for several reasons, but arguably an underlying context of this disassociation is related to social identity. We theorize that the rejection of e-cigarettes among our study participants who smoke is, in part, a function of dissociative signaling from a white, middle class, hipster social identity. Using e-cigarettes, even for their utilitarian function, would nevertheless signal membership in a racial, social, and socioeconomic status group that does not represent the identities of participants.Inasmuch as the use of e-cigarettes signals group membership, Research in consumer psychology has explored why consumers will refuse to use or cease to use a product when it is associated with an unfavorable group ,39. For In our study, participants clearly viewed e-cigarettes as a tool that could aid in smoking cessation. However, study participants who smoked, some of whom indicated an interest in quitting or reducing their smoking, nevertheless expressed disinterest in e-cigarettes due to their association with other youth cultural groups. As “Lee” points out above, e-cigarettes may be inaccessible to some Black youth, which may harm them in the long run. This situation is problematic from a range of harm reductionist viewpoints, which may incorporate claims that continue to be a source of controversy among public health advocates , as Whether or not e-cigarettes are a viable long-term strategy for tobacco-related harm reduction has yet to be determined . Some puThough we are more cautious in our interpretation, we do acknowledge that studies among adults suggest that e-cigarettes can be an effective smoking cessation tool, and the majority of adult users are current or former smokers ,49. Stud


S-transform and chaotic analysis is proposed. There are BAFS , local spectral characteristics MSCC , and chaotic features (12). Finally, radar chart and F-score are proposed to optimize the features by the hierarchical visual fusion. The feature set could be optimized from 526 to 96 dimensions based on NKI-CCRT corpus and 104 dimensions based on SVD corpus. The experimental results denote that new features by support vector machine (SVM) have the best performance, with a recognition rate of 84.4% on NKI-CCRT corpus and 78.7% on SVD corpus. The proposed method is thus approved to be effective and reliable for pathological speech intelligibility evaluation.Pathological speech usually refers to speech distortion resulting from illness or other biological insults. The assessment of pathological speech plays an important role in assisting the experts, while automatic evaluation of speech intelligibility is difficult because it is usually nonstationary and mutational. In this paper, we carry out an independent innovation of feature extraction and reduction, and we describe a multigranularity combined feature scheme which is optimized by the hierarchical visual method. A novel method of generating feature set based on  S-transform in 1996, which can be regarded as the combination of wavelet transform and short time Fourier transform  can be expressed as x[n] = x(nΔT), n = 0,1, 2,…, N − 1. The x[n]  S-transform can learn from discrete Fourier transform calculation. The x[n] Fourier transform isk = 0,1 …, N − 1.Let S-transform x[n] is computed by FFT:h, m = 0,1, 2,…, N − 1.The discrete x[n] of continuous signal x(t) is converted into the N∗N complex time-frequency matrix by S-transform from x[n] in FrameLen.Framing: framing (2)S-Transform: transform matrix S is got by S-transform (ransform .(3)Energy spectrum: energy spectra are obtained based on step (2).(4)Bandpass filter: the 26 filter banks are constructed.Log energy is calculated for each time in each filter bank:S is spectrum by S-transform in hΔT, x′ is the m filter output in hΔT, and Hm(k) is the frequency response of triangle filters.where (5)L MSCC coefficients is got:Discrete cosine transform (DCT): discrete time mapping cepstrum domain in the MSCC extraction process is as follows; The chaotic-based features are presented in the previous sections, and anomalies in pathological voices stem from malfunctions of some parts of the voice production system. Speech signal has fractal characteristics; chaotic phenomena can occur during speech production when the vocal organ is within a lesion. Traditional acoustic parameters are very effective to analyze cycle speech signal, which have certain limitations on analyzing noncycle and chaotic signals. Chaotic features provide useful information on distinguishing normal and pathological voices. Therefore, three nonlinear chaotic features (CF) can be extracted, which are the largest Lyapunov exponent to measure the speech signal chaotic degree, approximate entropy to measure speech signal complexity, and Lempel-Ziv complexity which is another complexity index , 17, wheIn this article, the largest Lyapunov exponent extraction process as an example is introduced. In order to guarantee the largest Lyapunov exponent reliability, the classic small data set algorithm is used; we get 4 statistics , and the 526-dimensional feature set was composed of 4 statistics and the other 522 features.x(t) = {x1, x2, x3,…, xN} is a one-dimensional time series, where N is the total number of time series; phase space is reconstructed as follows:m is embedding dimension, τ is delay, M is the total number of phase points, and M = N − (m − 1)∗τ.Pathological speech signal (1)p: the spectrum is obtained by the Fourier transform. The corresponding frequency is got in the maximum amplitude. This averaging period is the reciprocal of the frequency.Calculated time series averaging period (2)X(i), the nearest neighbor X(j) is found in the case of restrictions brief separation:In the phase space p is the average period of time series.where ‖·‖ represents two-norm value and (3)dj(i) is the distance between X(j) and i discrete time:For each reference point, (4)λ1 is got by the reference point Xj and the nearest neighbor The Lyapunov exponents represent the initial closed orbit exponential divergence of phase space; it is assumed that the exponential divergence Both sides of the equation were taken as the logarithm:i ~ lndj(i) meet the linear relation of the slope λ1Δt; thus,As can be seen above, q is the number of nonzero dj(i) and Δt is sample sampling period.where (5)λ1 is the slope of this line:Linear regression is done using the least square, and the largest Lyapunov exponent The specific calculation steps of the small data set method are as follows:The 526-dimensional feature set is constructed by the above three features' extraction, which are BAFS (430), MSCC (84), and CF (12).F-score method to sort the features.A set of high-dimensional data is obtained after pathological speech signal feature extraction. Visual techniques and multi-information fusion idea are a high-dimensional data reduction approach; at the same time, they depict the internal structural relationship of features, which is beneficial to data classification. Radar chart has good interaction, which is able to reflect the trend of changes in a feature set and every dimensional situation. In order to express the structural characteristics among attributes, radar chart information visualization graphical feature is extracted. According to radar chart uniqueness theorem, radar chart must be unique if the input feature is restricted to a specified alignment. Therefore, the extraction of graph feature is closely related to the feature order; we introduce F-score is a measure to distinguish the two types of samples [xk ∈ Rm,  k = 1,2,…, n, l  (l ≥ 2), is the number of the sample category and nj is the sample number in the j class, j = 1,2,…, l. The F-score of i is defined in the training samplesi feature of the whole training set, i feature of the j class, and xk,ij is the i feature of the first k sample data in j class. samples , given tRadar graphic information is called graphical feature , 18. GraM-dimensional radar chart is constructed by sample data r1, r2,…, ri,…, rM, M polygon  is composed of arbitrary continuous adjacent m-dimensional variables, and the center of M polygon by geometric algebra isw = 2π/M is the angle of the adjacent features, absim is the amplitude of M polygonal center , and  anglei is angle direction of M polygonal center .An x1, x2,…, xi,…, xn), xi is the first i feature, and n is feature dimension; the features fusion and reduction algorithm are as follows: input is original feature . Output is feature Re_fea after reduction.(1)F-score value: Fi of xi  F-score according to formula ( formula .(2)x1, x2,…, xi,…, xn) by the F-score value; then  is got, and sort the F-score value , where F1′ ≥ F2′ ≥ ⋯≥Fi′ ≥ ⋯≥Fn′.Feature sort: sort all features ((3)F_Mean is F-score average; the first F_first is less than F_Mean, so F1′ ≥ F2′ ≥ ⋯≥FF_first−1′ ≥ F_Mean, F_Mean < FF_first′; the first lay is ; compute F_Mean2 of the average F-score from F_first to n; then F_second is less than F_mean2; the second level is ; the third level is .Slicing: (4)m = 2, to obtain the center of gravity when m = 2, m = 3, and m = 4, original feature set S′ is constructed by three feature set fusions.Visual features fusion: specifically, in , if m = (5)S′ is repeated to do steps (1), (2), and (3). Re_fea is got.In this work, we used the hierarchical visual technique for feature optimization. There are two hierarchical fusions in the process. In each level, firstly, the main aim is to sort the high-dimensional features. Secondly, the effecting features are got, which are grouped together as input to the next level, and the process is repeated to get fusion feature. Process is shown in F-score and radar chart. Finally, the speech intelligibility is evaluated by SVM classifier.In In the classification problems, SVM follows a certain procedure to find the separating hyperplane with the largest margin of two classes. Radial basis function (RBF), a kernel, is used in this article. The sensitivity, specificity, accuracy, and UA are an index. As a classification tool to evaluate the NKI-CCRT corpus by different feature sets, SVM algorithm constructs a set of reference vectors in role of boundaries that minimize the number of misclassifications. Therefore, it represents a low-cost, accurate, and automatic tool for pathological voice classification in contrast with other tools, such as Gaussian mixture model .Z, quantified as 16 b. The corpus distribution is in The NKI-CCRT corpus  is recorrecordings of vowels /a/, /i/, and /u/ produced at normal, high, low, and low-high-low pitch;recordings of sentence “Guten Morgen, wie geht es Ihnen?” .SVD  is the fThat makes a total of 13 files per session. In addition, the electroglottogram (EGG) signal is also stored for each case in a separate file. The length of the files with sustained vowels is between 1 and 3 seconds. All recordings are sampled at 50 kHz and their resolution is 16 bits. 71 different pathologies are contained, including both functional and organic. The corpus distribution is in Further analysis is required to study the effect of various features of each subsystem.S-transform is a time-frequency analysis method by Stock Well which combines the advantage of wavelet transform with short time Fourier transform [S-transform. MSCC is compared with the traditional MFCC in the NKI-CCRT and SVD corpus. Recognition results are shown in Tables ransform , which sransform . TherefoF-score values to evaluate MSCC and MFCC; MSCC shows better performance by F-score in x-axis represents feature dimension. y-axis represents F-score values. MFCC is generally less than 0.2; the average is at 0.09. The maximum F-score of MSCC is nearly 0.8, and the average is about 0.39. The results of F-score indicate that the MSCC feature is stronger in the pathology classification.For example, in NKI-CCRT corpus, each value index is improved, where UA is increased from 51.58% to 64.76% and accuracy is increased from 50.54% to 63.67%. Thus, MSCC contains more pathological information than MFCC. Meanwhile, in order to show the contrast that MSCC contains more information than MFCC directly, we use Firstly, MSCC is compared with basis acoustic features (430) by support vector machine (SVM). As it can be seen in Tables In our continued investigation, we design an automatic pathological speech intelligibility evaluation system by information visualization optimization method. Furthermore, this hierarchical method is experimented with in NKI-CCRT corpus. The classification accuracy of 84.44% can be achieved. Recognition results are shown in In our study, The signal characteristics of pathological speech have been studied widely in the literature. A previous study showed that changes from articulatory manner are associated with pathological speech, while variability in articulatory place occurs to both normal and pathological speech. Therefore, the results of this research show that MSCC acoustic features fed to other pathology common features can be used together with invasive methods as complementary tools for pathological speech intelligibility evaluation. Furthermore, results of classification demonstrated that optimized feature set has great capability for classification of pathological voices to normal ones compared with the other feature that is examined in this research. Therefore, efficient combination of this work is composed of acoustic long-time features, MSCC, chaotic features, and SVM, which yield sensitivity of 84.4%. This structure significantly improves the results of pathological speech recognition in comparison with the proposed algorithm found in the references .F-score. Further analysis is required to study the effect of fusion difference classifiers. In addition, we would also like to study the effectiveness of other features and reduction methods like particle swarm optimization. In a word, the proposed method has greatly improved the pathological speech intelligibility evaluation performance and can provide important theoretical bases of the clinical application of speech pathology, which can be applied to other areas.Feature extraction and pattern classification are a key of pathological speech recognition. This study proposes a new feature set and feature fusion method. The basis acoustic feature, precise time-frequency feature, and chaotic feature showed discriminating power for binary classification based fusion method . Featur


However, the sources of DDT in eastern China are mainly from historic application of technical DDTs and dicofol. The inverse dependence of o,p’-DDT and p,p’-DDE on temperature evidences the transformation of parent DDT to its metabolites. Initial usage, abiotic parameters and microbial communities are found to be the main factors influencing the migration and transformation of DDT isomers and their metabolites in soils. In addition, a prediction equation of DDT concentrations in soils based on stepwise multiple regression analysis is developed. Results from this study offer insights into the migration and transformation pathways of DDTs in Chinese arable soils, which will allow data-based risk assessment on their use.Due to the adverse impact of DDTs on ecosystems and humans, a full fate assessment deems a comprehensive study on their occurrence in soils over a large region. Through a sampling campaign across China, we measured the concentrations, enantiomeric fractions (EFs), compound-specific carbon isotope composition of DDT and its metabolites, and the microbial community in related arable soils. The geographically total DDT concentrations are higher in eastern than western China. The EFs and δ Dichlorodiphenyltrichloroethanes (DDTs) were widely used in the world as effective insecticides for vector-borne diseases control and for crop protectiono,p’-DDT consists of one pair of enantiomers. The enantiomeric fraction (EF) can be changed during biodegradation and is a good indicator of past versus fresh chiral pollutant inputs8in situ biodegradation processes of organic pollutants10121114o,p’-DDT is also rather limited, especially at a species level of bacteria. It is necessary to well understand their environmental behaviors under real complicated environment using the techniques mentioned above.Historical applications of technical DDTs and dicofol, which usually contains high levels of DDT compounds as impurities in the production process, are the two main sources of DDTs in the environment. DDTs can migrate from soils or transform to their metabolites via various mechanisms, including degradation, volatilization, leaching, run-off, adsorption and plant removal413C) of DDT and its main metabolites in arable soils. High-throughput techniques and network analysis were employed to characterize the microbial communities and explore their co-occurrence patterns with DDT and its metabolites in soils. The obtained data were used to estimate the migration and transformation of DDTs on a large scale and to explore the underlying influencing physicochemical and microbiological factors in arable soils after 30 years of DDT ban in China. Furthermore, results from this study would provide basic scientific data for the contamination management and risk avoidance of DDTs in Chinese soils.Therefore, in this study, we conducted a nationwide farmland sampling campaign across China and measured the concentrations, EFs and carbon isotope compositions  ranged from 0.025 to 211 ng/g, with a mean value of 8.06 ng/g. Among the six DDT components, p,p’-DDE, which is the main metabolite of DDT, had the highest mean concentration (3.29 ng/g). Regional variation of DDT concentrations and compositions in arable soils across China was mapped in p < 0.0001, p < 0.05) was also identified between the residual levels of DDTs and socioeconomic indicators (SIs), including gross regional product (GPR) and population (P)  17. Amongrn China . p,p’-DDConsidering complex factors impacting on the residue levels of DDTs, stepwise multiple regression analysis was employed to explore the dominant factors and develop a prediction model. Based on the results of correlation analysis above, variables including organic matter, temperature, longitude and latitude were included. However, after significance and multicollinearity testing, temperature was excluded in the analysis. The significant prediction equation is as follows:where, C is the residue concentrations of ∑DDTs in agricultural soils (ng/g); OM is the content of organic matter (%); LONG is the longitude of sampling site (°E); LAT is the latitude of sampling site (°N).p,p’-DDT, 15% o,p’-DDT, 5% p,p’-DDE, <0.5% p,p’-DDD, <0.5% o,p’-DDD, <0.5% o,p’-DDE and <0.5% unidentified compoundsp,p’-DDE+ p,p’-DDD)/p,p’-DDT > 1 and of o,p’-DDT/p,p’-DDT < 0.2 /p,p’-DDT was much higher in soils with higher organic matter contents (>2.3%) and higher temperatures (>14 °C)  . In addi(>14 °C) .o,p’-DDT levels were selected for enantiomeric analysis. In general, the EF values of o,p’-DDT in soils varied greatly, ranging from 0.066 to 1 as the initial stable carbon isotope composition of DDTs applied for agriculture. The δ13C of p,p’-DDE ranged from −24.81 ± 0.12‰ to −30.26 ± 0.06‰ and deviated either positively or negatively from those of the initial product  but with a wider range  -o,p’-DDT and o,p’-DDT showed positive correlations with Devosia, Adhaeribacter and Cellulomonadaceae, but negative correlations with Gp3, Clostridium_sensu_stricto, Nitrospiraceae and Chloroflexi. The abundance of both o,p’-DDT and its (−)-enantiomer has negative correlations with Caedibacter_caryophilus and positive correlations with Skermanella_aerolata, Brevundimonas_alba, Cellvibrio, Brevundimonas, Devosia, Rhodobacteraceae, and Caulobacterales.Nineteen bacterial phyla were identified via Illumina sequencing from 28 samples across China. bacteria . Network < 0.01) . Bacterintiomers . Strong p,p’-DDT, p,p’-DDE is more persistent than other DDT componentsResidues of DDTs are found to be ubiquitous in agricultural soils across China even after 30-year ban of their application. Being the main metabolite of o,p’-DDT/p,p’-DDT in technical DDTs is 0.19, which is much lower than that in dicofol (7.0)p,p’-DDT component with time compared to our previous study9o,p’-DDT varied greatly in arable soils across China. Similar to previous studies919o,p’-DDT in western China also deviated from that of the racemic compound. This might be due to long-range transport from other places and subsequent wet or dry deposition. The enantiomeric signatures of chiral chemicals may retain racemic or nearly racemic in the airo,p’-DDT (mean = 0.016) compared to those at other sites further supports a significant DDT source from atmospheric deposition in western China. In addition, the different physiochemical and biological features of the soil environments might also contribute to the different EF values among the four geographic regions across China. Other studies also obtained EF values of o,p’-DDT either > or <0.5, indicating a complex degradation preference91919o,p’-DDT, but not on the enantiomer preference. It implies that the preferential degradation of o,p’-DDT enantiomers may be modulated by the integrated effects of multiple environmental factors in soil environment. However, the absolute degree of the enantioselective degradation of o,p’-DDT was found to be slightly elevated by temperature. It might be due to the higher enantioselective degradation rates related to higher microbial activities in sites with higher temperature.Historical usage is a crucial factor influencing the DDT concentrations in the environment. In this study, the spatial distribution pattern of ∑DDTs in arable soils roughly overlaps with the usage inventory map of technical DDTs in 1951–1983 and dicofol in 1984–2003 in China dicofol .0p,p’-DDin situ degradation of POPs in various environmentset al.13C difference of at least 2‰ for reliably interpreting the sources of different sites considering environmental complexity. Different δ13C of p,p’-DDE and o,p’-DDT were observed in different soils in this study  and its own dissipation (13C-enriched process). The fact that the Shanghai soil with an abnormally high concentration of p,p’-DDT (15.9 ng/g) bearing the lowest δ13C of p,p’-DDE is consistent with this explanation . In addition, DDE residues in eastern China are mainly from the degradation of DDT. Their residues in colder western China, where less amount of DDTs were used than eastern China, are mostly from the long-range transport via air from eastern China13C is found for p,p’-DDE in western China, which could partly account for the negative correlation between its δ13C and local temperature. This interpretation is also suitable for the positive correlation between the δ13C of o,p’-DDT and temperature. All these hypotheses should be further tested with controlled experiments or additional sampling. Milosevic et al.13C of BTEX at contaminated landfill with lower concentrations, which was due to their microbial degradation and subsequent enrichment of 13C. Likewise, this trend was also observed for p,p’-DDE in this study. However, the possible reason for our findings might be different due to the different initial usage amounts of DDTs across China. More technical DDTs and dicofol have been applied in eastern China than western China. Therefore, the relative newer and higher concentrations of DDT residues might be the main reason for the more negative δ13C of p,p’-DDE in eastern China.Compound-specific isotope compositions can offer insights into the pathways of is study . o,p’-DDlanation . The poso,p’-DDT. Both the EF and δ13C values of o,p’-DDT in the measured samples differentiated from those of racemic standard products, further suggesting active metabolism of DDTs in soils. Interestingly, the stable carbon isotope compositions of o,p’-DDT showed different characteristics in soils with EF > 0.5 or <0.5. The δ13C of o,p’-DDT in soils with EF > 0.5 displayed more negative values than those in soils with EF < 0.5. Bashir et al. observed higher stable carbon isotope fractionation for (+)- than (−)-α-HCH during biodegradationo,p’-DDT also have the similar carbon isotope fractionation, the observed different isotope fractionation patterns between EF > 0.5 -enantiomer) and <0.5 -enantiomer) could be explained. This prediction can be further tested using enantiomer-specifics table isotope analysis in future laboratory study.The combination of the enantiomer and stable carbon isotope fractionation, in a two-dimensional approach, may aid to further understand the degradation pathways of chiral et al.91619p,p’-DDT to p,p’-DDE and p,’p-DDD by soil OM and temperature was observed in this study. It might be associated with the higher microbial activities in sites with higher soil OM and temperature. The R2 value of the prediction equation based on stepwise multiple regression analysis is 0.222. It means that 22% of the total variations in DDT concentrations are explained by OM, longitude and latitude. The content of OM is the principal factor among these variables. Although the prediction model is significantly developed, the R2 value is still a little small. The initial usage amounts of technical DDTs might be the most predominant factor influencing the distribution and residue levels of DDTs in agricultural soils across China.The initial use of chemicals can lead to a primary distribution pattern, which is controlled by the usage amount of chemicalsSerratia marcescens DT-1P, Pseudomonas fluorescens and Alcaligenes sp. KK, Sphingobacterium sp. D6 and Chryseobacterium sp. PYR2 and Bacillus sp. BHD-4, have been documented to be capable of converting DDTs to their metabolites3233343536Chloroflexi, Coriobacteriaceae (belong to Actinobacteria) and Syntrophobacter, Thermodesulfovibrio, Caedibacter_caryophilus (belong to Gammaproteobacteria), whose abundances correlate negatively with the fraction of p,p’-DDT and positively with that of p,p’-DDE, may be capable of biodegrading DDTs. The significant positive correlation between the degree of DDT degradation and the abundance of Actinobacteria and Gammaproteobacteria was also identified by Sun et al.3940Syntrophobacter and Thermodesulfovibrio, p,p’-DDT but positively with that of its aerobic metabolite p,p’-DDE. It suggests that these bacteria may have the potency to dechlorinate DDT to DDE in aerobic environments. Clostridia are obligate anaerobes and found to be affected by the pollution of metal and petroleum hydrocarbons42o,p’-DDD-polluted soil across China as suggested by Guo et al.o,p’-DDT with Alphaproteobacteria  and Sphingobacteria (Adhaeribacter found in this study) were also observed in Guo et al.26Peredibacter_starrii and Rhodobacteraceae correlate positively with both (+)-o,p’-DDT and (−)-o,p’-DDT, indicating that different degradation rates by the same bacteria may be one possible explanation for the enantioselective dissipation of o,p’-DDT in soils. This result is consistent with those reported by Lewis et al.Gp3, Clostridium_sensu_stricto, Nitrospiraceae, Chloroflexi and Caedibacter_caryophilus, are only capable of degrading (+)- or (−)-enantiomer of o,p’-DDT. Thus, different bacteria can degrade different enantiomers which could also result in the EF values deviating from 0.5. Generally, the taxa we found to correlate well with the stereoisomers and metabolites of DDTs in this study are mostly at genus and species level, not of higher levels. Our results demonstrate that DDT residues can influence the specific microbes and they could in turn impact the redistributions of DDTs in Chinese arable soils.Soil microbial communities, which are influenced by environmental conditions, play critical roles in the degradation of contaminants. Various bacteria, such as A total of 123 surface soils (0–20 cm) were collected from agricultural fields covered 31 provinces across China in April and May 2013. Detailed information on sampling locations and strategy was described in our previous study and briefly presented in the o,p’-DDE, p,p’-DDE, o,p’-DDD, p,p’-DDD, o,p’-DDT and p,p’-DDT) (their molecular structures are listed in 2SO4), neutral silica gel, alumina, and florisil. Then, the target analytes eluted by hexane/DCM (7:3) were further concentrated for analysis.A mix of 6 DDT compounds . The determination of enantiomeric fractions (EFs) of chiral o,p’-DDT were performed by a gas chromatograph-mass spectrometry (Agilent 7890A GC-5975C MS). A larger quantity of soils was used to obtain sufficient amounts of targets for isotope analysis. The stable carbon isotope composition (δ13C) of DDT and its metabolites was measured using an Agilent 7890A GC coupled to a GV Isoprime IRMS  (GC-C-IRMS) via a modified GC5 combustion interface at College of Environmental and Resource Sciences, Zhejiang University. Due to the limitation of concentration and the existence of interfering substances, only the δ13C of p,p’-DDE and o,p’-DDT were determined.The concentrations of DDT and its metabolites were measured on an Agilent 7890A gas chromatograph  equipped with BGB-172 chiral capillary column  based on the instructions of the manufacturer. 16S rRNA gene amplicons were pair-end sequenced at Shenzhen Huada Genomics Institute by using Illumina Miseq (300 bp-PE). The detailed procedures were described in Blank controls were ministered for every 15 soil samples to check potential cross-contaminations and interferences. The recoveries of DDTs were in the range of 82.8% to 105%. The mean recoveries of surrogates, including TCmX and PCB209 were 93.9 ± 12% and 89.2 ± 15%, respectively. Three times of signal-to-noise ratio were calculated as the limits of detection (LOD), which ranged from 0.003 to 0.005 ng/g for target compounds. Potential thermal degradation of DDTs was checked daily and a ratio less than 6% was satisfied. Duplicate experiments were performed in the whole procedure.o,p’-DDT are defined as the portion of the (+)-enantiomer to the sum of the (−)- and (+)-enantiomers. Racemic standard of o,p’-DDT with every 10 samples were injected and the mean EF value was 0.498 ± 0.005. Data was deemed acceptable only if the EF differences between the two monitored ions were within 0.05. The o,p’-DDT in samples was considered to be racemic when the EF value was within the 95% confidence interval (CI) of the standard.The EF values of How to cite this article: Niu, L. et al. Enantiomer signature and carbon isotope evidence for the migration and transformation of DDTs in arable soils across China. Sci. Rep.6, 38475; doi: 10.1038/srep38475 (2016).Publisher's note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.


Meanwhile, an analogous result is computed for the p-adic Lebesgue product spaces with power weights. In addition, we characterize a sufficient and necessary condition which ensures that the weighted p-adic Hardy type operator is bounded on the p-adic Lebesgue product spaces. Furthermore, the p-adic weighted Hardy-Cesàro operator is also obtained.In this paper, we introduce the  Howeveeader to –5. In rep, let p-adic numbers. It is defined as the completion of the field of rational numbers p-adic norm x is represented as γ is an integer and the integers m, n are indivisible by p, then (i)(ii)(iii)For a prime number p-adic analysis, we know that any non-zero element x of p-adic norm because p-adic integrals in It is well known that p-adic norm on The space Denote by dx on dx such thatSince f is a nonnegative integrable function on The most fundamental averaging operator is Hardy operator defined byto Hardy , states f is a nonnegative measurable function on For the multidimensional case ussed in –14. \docn-dimensional spherical averaging operator, which was introduced by Christ and Grafakos in [The other definition is the fakos in  as folloet al. [n-dimensional p-adic Hardy operator:f is a nonnegative measurable function on p-adic Hardy operator on Lebesgue spaces with power weights.In 2012, Fu et al.  defined et al. [f is a nonnegative measurable function on In 2013, Lu et al.  gave thep-adic product spaces as follows and discuss the boundedness and best bound on the product of p-adic Lebesgue spaces.Next, we will introduce the definition of Hardy type operator on the higher-dimensional f be a nonnegative measurable function on p-adic Hardy type operator is defined byLet ψ. For Example, when In 1984, Carton-Lebrun and Fosset  defined p-adic Hardy operator ψ is a nonnegative function defined on ψ for which p-adic Hardy operator In 2006, Rim and Lee  defined et al. [ψ defined on In 2013, Fu et al.  introducp-adic product spaces.Next, we will extend the operator f be a nonnegative measurable function on p-adic weighted Hardy type operator is defined byφ is a nonnegative measurable function on Let It follows from the Fubini theorem that we can easily formulate the dual operator of f be a nonnegative measurable function on p-adic weighted Hardy type operator is defined byφ is a nonnegative measurable function on Let In 2014, Chuong and Hung[f on Let p-adic analysisp-adic analog of the weighted Hardy-Cesàro operator on weighted Lebesgue spaces and weighted BMO spaces. In each case, the corresponding operator norms are obtained. In 2016, Chuong et al. [p-adic weighted Hardy-Cesàro operators and their commutators on weighted functional spaces of Morrey type, and the corresponding operator norms are also computed.In 2014, Hung  considerg et al.  considerp-adic weighted Hardy-Cesàro operator.Motivated by these famous results, first we will give a higher-dimensional version of the f be a nonnegative measurable function on p-adic weighted Hardy-Cesàro type operator is defined byLet p-adic Lebesgue product spaces with power weights. In Section p-adic weighted Hardy-Cesàro type operator also has the corresponding conclusion. In Section The paper is organized as follows. Section LetIfthen thep-adic Hardy type operatoris bounded onmoreover, the norm ofcan be obtained as follows:We provide the following weighted extension of this result.LetIfwhereandthen thep-adic Hardy type operatoris bounded onmoreover, the norm ofcan be obtained as follows:p-adic Hardy type operator will be easy to get on the p-adic Lebesgue product spaces, so we only provide the proof of Theorem When Without loss of generality, we consider only the situation when p-adic Lebesgue product spaces with power weights. We haveIt is clear that f is a radial function, then we have f is a radial function.This implies the claim that if Substituting the variable f is a radial function, we have Using the generalized Minkowski’s inequality and noting that Now let us prove that our estimate is sharp. For Letφis a nonnegative measurable function onIfthen thep-adic weighted Hardy type operatoris bounded onif and only ifSince the case We first prove the sufficiency of Theorem This completes the proof of the sufficiency of Theorem Next we prove the necessary of Theorem Now, for any Obviously, k approach ∞, then ε approaches 0 and Since the inequalitySince the operator Letandφbe a nonnegative measurable function onIfthen thep-adic weighted Hardy type operatoris bounded onif and only ifUsing methods similar to Theorem We note that in , the autp-adic Lebesgue product spaces.Inspired by the paper, we will consider sufficient and necessary conditions of the weighted Hardy-Cesàro type operator SupposeLetbe a measurable function such thatfor some realβand almost everywhereIfthen thep-adic weighted Hardy-Cesàro type operatoris bounded onif and only ifp-adic change of variables. It is worth pointing out that we need to make some appropriate modifications for the necessity. Applying this condition t, so it is easy to get the desired results by adopting the same method as Theorem Noting that Theorem p-adic Hardy type operator and considered the problem of finding a bound for the norm of it from p-adic Lebesgue product spaces with power weights to itself. Moreover, we characterized a sufficient and necessary condition which ensures that the weighted p-adic Hardy type operator is bounded on the p-adic Lebesgue product spaces. In addition, we also extended the results to the p-adic weighted Hardy-Cesàro operator.In the present study, we introduced a class of


Inequal. Appl. 2013:514, The resolvent operator approach is applied to address a system of generalized ordered variational inclusions with ⊕ operator in real ordered Banach space. With the help of the resolvent operator technique, Li  This theory provides us with a framework to understand and solve many problems arising in the field of economics, optimization, transportation, elasticity and applied sciences. A lot of work considered with the ordered variational inequalities and ordered equations was done by Li al.; see , 6, 8, 9The fundamental goal in the theory of variational inequality is to develop a streamline algorithm for solving a variational inequality and its various forms. These methods include the projection method and its novel forms, approximation techniques, Newton’s methods and the methods derived from the auxiliary principle techniques.H-accretivity [H-monotonicity [e.g., [It is widely known that the projection technique cannot be applied to solve variational inclusion problems and thus one has to use resolvent operator techniques to solve them. The beauty of the iterative methods involving the resolvent operator is that the resolvent step involves the maximal monotone operator only, while other parts facilitate the problem’s decomposition. Most of the problems related to variational inclusions are solved by maximal monotone operators and their generalizations such as retivity , H-monottonicity  and manyy [e.g., –17 and tEssentially, using the resolvent technique, one can show that the variational inclusions are commensurate to the fixed point problems. This equivalent formation has played a great job in designing some exotic techniques for solving variational inclusions and related optimization problems.We initiate a study of a system of generalized ordered variational inclusions in real ordered Banach space. We design an iterative algorithm based on the resolvent operator for solving system of generalized ordered variational inclusions. We prove an existence as well as a convergence result for our problem. For more details of related work, we refer to , 18 and X be a real ordered Banach space endowed with norm θ and partial order ≤ defined by the normal cone C with a normal constant In the paper, assume that The following well-known definitions and results are essential to achieve the goal of this paper.C (≠ϕ) be a closed, convex subset of X. C is said to be a cone if (i)for (ii)p and if Let C is called a normal cone iff there exists a constant C.X. The real Banach space X endowed with the ordered relation ≤ defined by C is called an ordered real Banach space.For arbitrary elements p and q are called comparable to each other and this is denoted by For arbitrary elements β-ordered comparison map, if it is a comparison mapping andA map Ifpandqare comparable to each other, thenandexist, andLetCbe a normal cone with normal constantinX, then for eachwe have the relations: (i)(ii)(iii)(iv)ifthen, 4–6Let ≤ be a partial order relation defined by the coneCwith a normal constantinXin Definition Then hereinafter relations survive: allowλto be real, thenifp, qandwcan be comparative to each other, thenpresumeexists, and ifandthenifp, q, r, wcan be compared with each other, thenifandthenifthenifthenifandandthenandfor allandA is called a γ-order non-extended mapping if there exists a constant A is called a strongly comparison map if it is a comparison mapping and Allow M is called a weak-comparison map, if for M is called an α-weak-non-ordinary difference map associated with A, if it is weak comparison and for each M is called a λ-order different weak-comparison map associated with A if ∃ a M, a weak-comparison map, is called an ordered α-weak-non-ordinary difference map and a λ-order different weak-comparison map associated with A, and Allow γ-order non-extended map and an α-non-ordinary difference mapping with respect to A, respectively. The resolvent operator A and M is defined byLet I is the identity map on X.A map Ifandare anα-weak-non-ordinary difference map associated withAand aγ-order non-extended map, respectively, withthenis anα-weak-non-ordinary difference map associated withAand the resolvent operatorofis a single-valued fori.e., ofholds.Letandbe aweak-ANODD set-valued map and a strongly comparison map associated withrespectively. Then the resolvent operatoris a comparison map.Letbe an orderedweak-ANODD map andbe aγ-ordered non-extended map associated withforrespectively. Then the following relation holds:X to be a real ordered Banach space and C a normal cone having the normal constant Allow For some X), M and N are single-valued mappings and If  problem  reduces mentclass2pt{minimM is a single-valued mapping, then problem . Here, we discuss some special cases of SGIOVI . If \docNow, we mention the fixed point formulation of SGIOVI .The set of elementsbecome a solution of SGIOVI , under some apt conditions.LetCbe a normal cone having a normal constantin a real ordered Banach spaceX. Letbe single-valued mappings such thatAis acompression mapping, is acompression andis acompression andare comparison mappings, respectively. Letbe single-valued mappings such thatis anrestricted-accretive mapping w.r.t. andis anrestricted-accretive mapping w.r.t. respectively. Supposeare the set-valued mappings such thatMis aweak-ANODD set-valued mapping andis aweak-ANODD set-valued mapping, respectively.In addition, ifand for allthe following conditions are satisfied:andThen the SGIOVI 17), we hnce from , 18) an an18\docq2∥). By , we knowf SGIOVI . □This part of the article is associated with the construction of an iterative scheme as well as the strong convergence of the sequences achieved by the said scheme to the exact solution of SGIOVI .C to be a normal cone with the normal constant X. Let Allow For the initial guess Allowandto be sequences of nonnegative real numbers such that they satisfy(i)and(ii)Thenapproaches zero asnmoves to ∞.AllowX, C, M, N, andto be as in Theorem such that all the assertions of Theorem are valid. Then the sequenceformulated by Algorithm (and (converges strongly to the unique solutionof SGIOVI (lgorithm  and 22)AllowX, f SGIOVI .By Theorem By , 23) an an23) anBy using the same argument as in Theorem ows from  and 24)26\docume−q∥). By , we knowSystem of generalized ordered variational inclusions can be viewed as natural and innovative generalizations of the system of generalized ordered variational inequalities. Two of the most difficult and important problems related to inclusions are the establishment of generalized inclusions and the development of an iterative algorithm. In this article, a system of generalized ordered variational inclusions is introduced and studied which is more general than many existing systems of ordered variational inclusions in the literature. An iterative algorithm is established with the ⊕ operator to approximate the solution of our system, and a convergence criterion is also discussed.We remark that our results are new and useful for further research and one can extend these results in higher dimensional spaces. Much more work is needed in all these areas to develop a sound basis for applications of the system of general ordered variational inclusions in engineering and physical sciences.


Nature Communications 10.1038/s41467-017-01510-7, published online 27 November 2017Correction to: y-axis label in Fig. 2b was incorrectly labelled ‘Snap-in force (μN)’. In the correct version, the axis is labelled ‘Snap-in force (nN)’.The original version of this Article contained an error in Fig. 2b. The In the original version of this Article, the “Butterfly wings” section of the Methods incorrectly stated the name of the butterfly as ‘Golden bird’. In the correct version, the name of the butterfly is ‘Golden birdwing’.These have now been corrected in both the PDF and HTML versions of the article.


N can be “unfolded” to obtain a MUL-tree U(N) and, conversely, a MUL-tree T can in certain circumstances be “folded” to obtain aphylogenetic network F(T) that exhibits T. In this paper, we study properties of the operations U and F in more detail. In particular, we introduce the class of stable networks, phylogenetic networks N for which F(U(N)) is isomorphic to N, characterise such networks, and show that they are related to the well-known class of tree-sibling networks. We also explore how the concept of displaying a tree in a network N can be related to displaying the tree in the MUL-tree U(N). To do this, we develop aphylogenetic analogue of graph fibrations. This allows us to view U(N) as the analogue of the universal cover of a digraph, and to establish a close connection between displaying trees in U(N) and reconciling phylogenetic trees with networks.Phylogenetic networks are rooted, labelled directed acyclic graphswhich are commonly used to represent reticulate evolution. There is a close relationship between phylogenetic networks and multi-labelled trees (MUL-trees). Indeed, any phylogenetic network  N to obtain a MUL-tree U(N) and that, conversely, a MUL-tree T can under certain conditions be “folded” to obtain a phylogenetic network F(T) that exhibits T. We illustrate these operations in Fig. T is a MUL-tree, and the network N is obtained by first inserting vertices into T and then folding up the resulting tree by identifying vertices in T to obtain F(T); the unfolding U(N) of N  is precisely T.Phylogenetic networks are rooted, directed acyclic graphs whose leaves are labelled by some set of species ) is isomorphic to T for any MUL-tree T (Huber and Moulton U and F operations are applied in the opposite order to some network as there are networks N for which F(U(N)) is not isomorphic to N ) is isomorphic to N. We call these stable networks. In Sect. More specifically, although it is always the case that In Sect. We then focus on the problem of displaying trees in networks. In Sect. weakly displayed by a phylogenetic network N if it is displayed by the MUL-tree U(N). Using the concepts developed in Sect. Finally, in Sect. X denote a finite set of size at least two. In addition, all graphs that we consider are connected.Throughout this paper, we let G is a directed acyclic graph with vertex set V(G), arc set A(G) and a single root G, we denote the headv of a by tailu by a and G are parallel if v is a vertex of G, then the in-degree of v, denoted by indeg(v), is the number of incoming arcs of v, and the out-degree of v, denoted by outdeg(v), is the number of outgoing arcs of v. We say that a vertex below a vertex w to v in G. We call a vertex v of G a reticulation vertex of G, if v a tree vertex of G if v a leaf of G, or G by L(G) and the set of interior verticesv of G, that is, v is neither the root nor a leaf of G, by Suppose M is a multi-set onX if the set resulting from M by ignoring the multiplicities of the elements in M is X. Following Huber and Moulton for all (ii)for every leaf In addition, we call MUL-tree onX if the tree T does not have any vertices with in-degree one and out-degree one. If the map T rather than X is of no relevance to the discussion then we will call T a (pseudo) MUL-tree rather than a (pseudo) MUL-tree on X. For example, the tree T in Fig. X that is not a MUL-tree on X. We say that two (pseudo) MUL-trees X are isomorphic if there is a digraph isomorphism We say that a multi-set  Moulton , we defimentclass2pt{minimT is a pseudo MUL-tree. For v a non-root vertex of T, we denote by T(v) the connected subgraph of T that contains v obtained by deleting the incoming arc of v. Clearly T(v) is a pseudo MUL-tree. We call a pseudo MUL-tree pseudo subMUL-tree of T if there exists a non-root vertex v of T such that T(v) and T a MUL-tree we say that a subMUL-tree T is inextendible if there exist distinct vertices v and T such that T(v) and T is inextendible if T contains more than one copy of that subMUL-tree. We say that a subMUL-tree T is maximal inextendible if T that contains Suppose T and its folding b are all inextendible subtrees of T and so are the two leaves labelled c. Each one of two subtrees of T of length two (ignoring the directions of the arcs of T) that have leaf label set To illustrate these definitions consider for example the MUL-tree mentclass2pt{minimX-networkN is a rooted directed acyclic graph, in which parallel arcs are allowed,An (i)N that has in-degree zero and out-degree at least two,there exists a unique root (ii)N except the root is either a reticulation vertex or a tree vertex,every vertex of (iii)there exists no vertex of in-degree one and out-degree one, and(iv)L(N) of leaves of N is X.the set A phylogenetic network ( onX) is an X-network that does not have parallel arcs. A phylogenetic tree onX is a phylogenetic network on X that has no reticulation vertices. We say that a phylogenetic network N is binary if the degree of every reticulation vertex and every non-leaf tree-vertex is three and X-networks N and isomorphic if there exists a bijection N with head u and tail v equals the number of arcs in X.such thatIn this section, we recall the unfolding and folding operations mentioned in the introduction that were first proposed in Huber and Moulton  , andthere is an arc from vertex x in X are labelled by x.the vertices in The MUL-tree obtained by suppressing all in-degree one and out-degree one vertices in U(N). In Huber and Moulton (U(N) is indeed a MUL-tree.We first describe the unfolding operation  Moulton  it is shv of an directed graph G as in Sect. v by ch(v) and say that an X-network Nexhibits a MUL-tree T if the MUL-trees U(N) and T are isomorphic. In particular, any X-network N exhibits the MUL-tree U(N). Note that there exist MUL-trees T for which there is no phylogenetic network that exhibits T .We denote for all vertices F for constructing an X-network F(T) from a MUL-tree T introduced in  are isomorphic (as pseudo MUL-trees). The pseudo MUL-tree obtained once the last element in Now, to obtain F(T), we define an equivalence relation v, w in F(T) be the X-network obtained by taking the quotient of G(T) denote the DAG with vertex set m. The X-network obtained from G(T) by suppressing all vertices of indegree one and outdegree one and defining the leaf labels in the natural way is F(T).Finally, to obtain T in Fig. u in F(T), representing u. Similarly, the two vertices labelled w in F(T) in terms of w. Clearly F(T) from u to w.For example, consider the MUL-tree entclass1pt{minimaT is isomorphic with U(F(T)) (as MUL-trees) (Huber and Moulton T and U(F(T)). Also, note that if T is binary, then F(T) is semi-resolved, that is, every tree vertex in F(T) has out-degree 2. Moreover, in Huber and Moulton (F(T) is semi-resolved, then F(T) has the minimum number of reticulation vertices amongst all phylogenetic networks that exhibit T.Note that any MUL-tree  Moulton , ProposiX need not be a phylogenetic network. For example, consider the MUL-tree U(N) in Fig. F(U(N)) is not a phylogenetic network as it contains parallel arcs. We now characterize those MUL-trees T for which F(T) is a phylogenetic network.In general, the folding of an arbitrary MUL-tree on T is a binary MUL-tree on X. Then F(T) is a phylogenetic network if and only if there is no pair of distinct vertices v, w in T which share a parent in T and are such that T(v) and T(w) are isomorphic.Suppose F(T) is a phylogenetic network then there is no pair of distinct vertices in T with the stated property by establishing the contrapositive. Suppose T is a binary MUL-tree on X that contains two distinct vertices v and w which share a parent in T and are such that T(v) and T(w) are isomorphic. Without loss of generality, we may assume that v and w are such that there exist no two vertices T on the directed paths from the root of T to v and w, respectively, such that T(v) is inextendible. If T(v) is maximal inextendible then F(T) contains a parallel arc and so F(T) is not a phylogenetic network, as required. So, assume that T(v) is not maximal inextendible. Then there must exist some vertex T distinct from v and w such that T(v) is a subMUL-tree of We prove the claim that if v such that T. Then T(v) and T(w) as a subMUL-tree. Thus, T(v) is rendered maximal inextendible at some stage in the construction of F(T). Applying the operation F to T(v) introduces a parallel arc into F(T) and thus F(T) is not a phylogenetic network, as required.Let T is a binary MUL-tree on X such that there exist no two vertices v, w in T which share a parent in T such that T(v) and T(w) are inextendible. Assume for contradiction that F(T) is not a phylogenetic network. Then F(T) must contain parallel arcs a and v is a tree vertex and w is a reticulation vertex of F(T). Let z denote the unique child of w in F(T). Note that since the folding operation implies that F(T) cannot contain an arc both of whose end vertices are reticulation vertices, z must in fact be a tree vertex in F(T).Conversely, suppose that F(T) to z which contain arcs a and U(F(T)) on X by v such that T. Thus, F(T) must be a phylogenetic network. Now, let F can be considered as the reverse of the operation U. However, there exist phylogenetic networks N such that F(U(N)) is not isomorphic to N ) and N are isomorphic.As mentioned above, the folding operation N for which F(U(N)) is isomorphic to N. We call such networks stable.In this section, we shall give a characterization of phylogenetic networks N is a phylogenetic network on X. We call two distinct tree vertices v and w in Nidentifiable if there exist directed paths N to v and w such that the subMUL-trees U(N) are isomorphic. In addition, we say that N is irreducible if it does not contain an identifiable pair of tree vertices. To illustrate, the network N depicted in Fig. v and w are identifiable.We start by recalling the definition of an irreducible network (Huber and Moulton N is a phylogenetic network then let Ret(N) denote the set of reticulation vertices of N. We call Ncompressed if the child of each vertex in Ret(N) is a tree vertex. Note that in Cardona et al. N is stable.(ii)N is compressed and irreducible.(iii)N is compressed and there does not exist a pair of distinct tree vertices v, w in N such that Suppose that v, w in N such that N is semi-resolved we have N is compressed the children a and b, respectively, are tree-vertices of N. Let N to v and w, respectively. Similarly, let N from v and w, respectively. Then the subMUL-trees U(N) are isomorphic and so are the subMUL-trees v. Similarly, let w. Then the MUL-tree obtained from v, w is an identifiable pair in N. Hence N is not irreducible which provides the required contradiction.(ii) N is not irreducible. Then N contains an identifiable pair of vertices v, w. Without loss of generality, we may assume that v and w are such that there are no vertices v and w, respectively, that also form an identifiable pair.(iii) ch(v) and ch(w) are contained in Ret(N). Suppose that u of N let N to u. If s is a leaf of N then, since v and w are an identifiable pair, the MUL-trees X. Hence, s is a leaf of N.To obtain the required contradiction, we first claim that s is a non-leaf tree-vertex of N then, since z of N gives rise to a subset of vertices in the MUL-tree U(N), it follows that there exists a non-leaf tree vertex w such that v and w, we cannot have that s and s is assumed to be a tree vertex of N. Since every non-root vertex of N is either a tree-vertex or a reticulation vertex of N, it follows that If N is compressed, the child s must be a tree-vertex of N. Since r in N below w such that N which is impossible. Hence, r form an identifiable pair in N with v and r below w which is impossible in view of the choice of v and w. Thus, To complete the proof, assume for contradiction that there exists some (i)  Moulton , TheoremN is compressed and irreducible. Let N and F(U(N)), respectively. Since N is irreducible so is N exhibits U(N) so does N is compressed, it follows that N is stable. (ii)  Moulton , CorollaN is a phylogenetic network. A vertex w of N distinct from some vertex v of N is a sibling of v if v and w share the same parent, and a sibling that is a tree vertex is called a tree-sibling vertex. In addition, N is called a tree-child network if every non-leaf vertex of N has a child that is a tree vertex of N  such that X-morphism. We call the inverse image fibre over v. Informally, the fibre over v is the subset of V(T) that is mapped to v under f. For example, for the tree N depicted in Fig. u, v and w in N are given by the vertices of Recall that the head of an arc mentclass2pt{minimmentclass2pt{minimU(N) of a network N. This result is an analogue of Boldi and Vigna  of a MUL-tree T can also give rise to a folding map. In analogy with Boldi and Vigna (V(T) of a pseudo MUL-tree T satisfies the local out-isomorphism property (LOIP) if the following holds for all LOIP: If T with tail v to the set of arcs in T with tail w such that a in T with tail v.We now use the LOIP-property to characterize when equivalence relations on MUL-trees give rise to folding maps (cf. Boldi and Vigna v in a rooted directed tree by pa(v).As we shall now show, the folding nd Vigna , p. 25, X, and V(T). Then the equivalence classes N on X) if and only if (i)LOIP,(ii)for all (iii)for all (iv)for all (v)v.for all Suppose that N is a phylogenetic network on X and f. For each a in T with tail v to the set of arcs in T with tail w by putting N is a phylogenetic network, it is straight-forward to check that (ii) must hold as no vertex in N can have in-degree and out-degree 1, (iii) must hold as every vertex of N that is not the root of N is either a reticulation vertex or a tree vertex (but not both), and that (iv) must hold as all elements in f to a vertex labeled by x which has in-degree 1. Finally, (v) follows from the fact that N does not contain parallel arcs.Assume first that V(T) that satisfies properties (i)–(v). To simplify notation, put u in V(T). Let T by u], [v]) an arc in u] in x in X with X.Conversely, assume that X-morphism that maps each vertex u in V(T) to its equivalence class [u], and each arc  in A(T) to the arc . It is straight-forward to check that f is a folding map as properties (i) and (iv) imply that f yields a well-defined surjective X-morphism from T to Now, define T, consider the equivalence relation F(T) is a phylogenetic network, we obtain a folding map Given a MUL-tree We now state a result that provides additional insight into unfoldings of networks, and that will also be useful in the last section. It can be regarded as a phylogenetic analogue of path lifting in topology ). By the comment following Theorem F(U(N)). As N is isomorphic to F(U(N)), there also exists a folding map from N. By Corollary Suppose entclass1pt{minimaN is isomorphic to Conversely, suppose entclass1pt{minimaF(U(N)) is isomorphic to N is isomorphic to F(U(N)), as required. Now, X-networks in general. We will not pursue this possibility further here, but it could be of interest to understand categorical properties of such maps , u to v). We illustrate this concept in Fig. Following Iersel et al. , we say In Kanj et al.  it is shTreeDisplayingInstance: A binary stable phylogenetic network on X and a binary phylogenetic tree on X.Question: Is T displayed by N?N is a binary phylogenetic network on X and that R is a binary phylogenetic tree on X. Let N, let R, and let N and that R. Then the operation HangLeaves(v) adds the vertices N as well as the arcs R as well as the arcs To establish this fact, we show that this problem is NP-complete when restricted to compressed, binary tree-sibling networks and apply Corollary TreeDisplaying is NP-complete, even when restricted to the class of binary, compressed tree-sibling networks.T be a binary phylogenetic tree on X and let N be a binary phylogenetic network on X. We will  modify N to a binary, compressed tree-sibling network X and, simultaneously, modify T to a binary phylogenetic tree T is displayed by N if and only if By Corollary j et al.  that it HangLeaves to transform N into a compressed tree-sibling network on some yet to be specified leaf set T to a binary phylogenetic tree on N a phylogenetic network e in N whose head is a reticulation vertex of N we subdivide e by a new vertex HangLeaves to X, every reticulation vertex of N has two siblings in The construction of v of N one of its two siblings. Let s denote that sibling. Let s and v in HangLeaves to v in X and all of the leaves added to N this way. Then it is easy to check that the resulting DAG Next, we follow the proof of Iersel et al. , TheoremT is displayed by N if and only if T is displayed by N if and only if HangLeaves, to a vertex v of N.We now establish our claim that T is displayed by N. To see that N that is a subdivision of T. Combined with the fact that the subgraph of Assume first that N that is a subdivision of T. Thus, T is displayed by N which completes the proof of the claim. A repeated application of the last claim implies that T is displayed by N if and only if Conversely, assume that T and a network N on X, we say that T is weakly displayed by N if it is displayed by U(N)  that is a subdivision of T). For example, both of the trees in Fig. N, but the tree in (ii) is not displayed by N. As we shall see, this concept is closely related to the problem of reconciling gene trees with species networks. In Sect. T is displayed by a phylogenetic network N even if it is stable. In this section, we show that, in contrast, one can decide in polynomial time whether or not a tree is weakly displayed by any given phylogenetic network.Given a phylogenetic tree N on X, let V(N) together with the root on N. Following Zhang et al.  in A(T) is associated with a directed path N with initial vertex r(u) and terminal vertex r(v).Before presenting our algorithm, we first derive a characterization for when a tree is weakly displayed by a phylogenetic network in terms of so-called tree reconciliations. Given a phylogenetic network g et al. , a reconr between T and Nlocally separated if for each pair of vertices T that have the same parent v, both We now give the aforementioned characterization for when a tree is weakly displayed by a phylogenetic network. We call a reconciliation N is a phylogenetic network on X. Then a phylogenetic tree T on X is weakly displayed by N if and only if there exists a locally separated reconciliation between T and N.Suppose that r between T and N, then T is weakly displayed by N. We illustrate the main idea of the proof in Fig. r induces an X-morphism T into N, and so, using Theorem X-morphism T is displayed by U(N).We first prove that if there is a locally separated reconciliation r is a locally separated reconciliation between T and N. Since each arc in T is associated with a directed path in N which contains at least one arc, it follows that r induces an X-morphism T to N. By Theorem X-morphism from r is locally separated, it follows that the map More specifically, suppose that X-morphism, each vertex in T. Since U(N) does not contain any in-degree one and out-degree one vertices and U(N) it follows that T is displayed by U(N), and so T is weakly displayed by N, as required.Now, consider the set T is weakly displayed by N. Then there exists a subdivision T such that U(N). Since U(N), there exists a subdivision T) such that X-morphism from N given by Theorem X-morphism N induces a map r from V(T) to V(N) defined by putting r(v) exists because V(U(N)) as X-morphism. Moreover, since X-morphisms it follows that r(v) is a tree vertex of N.Conversely, suppose that entclass1pt{minimau, v) in T, denote the subdivision of  in u to v in r(u) to r(v) in N which contains at least one arc. Then it follows that r is a reconciliation between T and N. Moreover, to see that r is locally separated, consider an arbitrary pair of distinct vertices V(T) that have the same parent v. Denote the initial arcs of the two (necessarily distinct) directed paths a of U(N) put r is locally separated, as required. Now, for every arc  the phylogenetic network obtained from by N by first restricting N to v and all the vertices of N below v and then suppressing any resulting vertices with in-degree one and out-degree one. In addition, we define a function v is not a leaf in V(T), then we set u in N, and (ii) there exists a locally separated reconciliation between T(v) and v is a leaf with label x, then we set u is a leaf in N labeled with x or x is a leaf in N below u. We remark that u is below N.We now present a dynamic programming algorithm to solve this problem. Let T and N if and only if By definition, there exists a locally separated reconciliation between T be a binary phylogenetic tree on X, and N a phylogenetic network on X. Suppose that v is an interior vertex in T with two children u is an interior vertex in N with u, or there exist two distinct children u in N such that Let u then, by the previous remark, u is an interior vertex in N with two children N such that N such that for N obtained by combining the arc We begin by establishing the ‘if’ direction. Note that if v and f is a reconciliation between T(v) and N(u). Combined with the fact that f is also locally separated. Hence, Now consider the map v an interior vertex in T and u as otherwise the proposition clearly follows. Under this assumption, it follows that there exists a locally separated reconciliation f between T(v) and N(u) with Conversely, suppose that v and f is a locally separated reconciliation, it follows that u and that v but distinct initial arcs, it follows that there exist two distinct children u such that Now, let Proposition k be the maximum number of children that any vertex in N may have. Note first that a topological ordering V(T)  such that T implies O(m) time. Similarly, we can compute a topological ordering V(N) in O(kn) time. Now, noting that i, j)th entry is O(mnk) time to fill this matrix and, therefore, to compute T on X has Let entclass1pt{minimaT is a binary phylogenetic tree on X, and that N is a phylogenetic network on X. Then, using a dynamic programming algorithm, it can be decided inT is weakly displayed by N.Suppose that


Following this, Dresher type inequalities for the radial Blaschke-Minkowski homomorphisms with respect to p-radial Blaschke and p-harmonic Blaschke additions are established.In the paper, we first improve the radial Blaschke and harmonic Blaschke additions and introduce the  The surface of B is n-ball is denoted by n-dimensional volume of a body K. Associated with a compact subset K of K will be called a star body. Let δ̃ denote the radial Hausdorff metric, i.e., if The setting for this paper is an n in the The radial Minkowski linear combination, mentclass2pt{minimith dual mixed volume of K and L, If entclass1pt{minimaIK whose radial function satisfies, for v is K. The volume of the intersection body of K is given by Ifandthenwith equality if and only ifKandLare dilates.K and L are dilates. In the following, we establish the Brunn-Minkowski inequality for the p-radial Blaschke addition.Taking t}i=0 in , we haveIfandthenwith equality if and only ifKandLare dilates.p-dual mixed quermassintegral p-radial Blaschke addition and together with inequality   \document}i=0 in . □K and L are star bodies in K and L are dilates.Let us recall the concept of harmonic Blaschke addition defined by Lutwak . Supposehed (see ). If \doIn the section, we give an improved concept of the harmonic Blaschke addition.p-harmonic Blaschke addition of star bodies K and L denoted by For p-harmonic Blaschke addition follows immediately from  a ap-harmoIfandthenwith equality if and only ifKandLare dilates.Ψ is continuous.For all n dimensions.For all A map K and L are star bodies in K and L are dilates.Radial Blaschke-Minkowski homomorphisms are important examples of star body valued valuations. Their natural duals, Blaschke-Minkowski homomorphisms, are an important notion in the theory of convex body valued valuations . In K and L are star bodies in K and L are dilates. The inequality is reversed when If e, e.g., )4.2\docp<0 (see ).K and L are star bodies in K and L are dilates. Taking In 2013, an ished in : If K anent}p=1, .(see )Letbe a radial Blaschke-Minkowski homomorphism. There is a continuous operatorsymmetric in its arguments such that, forandi of K. K.Clearly, Theorem etail in , 24. If (see )A mapis a radial Blaschke-Minkowski homomorphism if and only if there is a measuresuch thatwheredenotes the set of nonnegative zonal measures oni are integers. We now extend the integers i to real numbers, define the Blaschke-Minkowski homomorphism of order p of K.For the mixed radial Blaschke-Minkowski homomorphism induced by Ψ, Schuster  proved tp of K, denoted by Let LetIfandthenwith equality if and only ifandare dilates.K and ΨL are dilates.Taking t}j=n in  and noti(u)=nωn,  becomes t}p=0 in , 5.1) b b\documeLetIfandsatisfythenwith equality if and only ifandare dilates.Taking t}k=n in  and noti(u)=nωn,  becomes t}k=n in , we haveAn extension of Beckenbach’s inequality  Ifandϕis a distribution function, thenwith equality if and only if the functionsfandgare proportional.We are now in a position to prove Theorem LetIfandsatisfythenwith equality if and only ifandare dilates.μ is the generating measure of Ψ from Lemma From , we obtafore, by , we haveTaking ment} in  and \docWe are now in a position to prove Theorem LetIfandsatisfythenwith equality if and only ifandare dilates.From , we obtace, from , we obtai(L). By , we haveTaking ment} in  and \docp-radial addition and the p-harmonic Blaschke addition.In the present study, we first revised and improved the concepts of radial Blaschke addition and harmonic Blaschke addition in an