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Black Americans and HIV/AIDS: The Basics
Black Americans have been disproportionately affected by HIV/AIDS since the epidemic's beginning, and that disparity has deepened over time.1,2
Although they represent only 12% of the U.S. population, Blacks account for a much larger share of HIV diagnoses (43%), people estimated to be living with HIV disease (42%), and deaths among people with HIV (43%) than any other racial/ethnic group in the U.S.3,4
Among Black Americans, Black women, youth, and gay and bisexual men have been especially hard hit.5,6
A number of challenges contribute to the epidemic among Blacks, including poverty, lack of access to health care, higher rates of some sexually transmitted infections and smaller sexual networks, lack of awareness of HIV status, and stigma.7
Despite this impact, recent data indicate some encouraging trends, including declining new HIV diagnoses among Blacks overall, especially among women, and a leveling off of new diagnoses among Black gay and bisexual men.8,9,10 However, given the epidemic's continued and disproportionate impact among Blacks, a continued focus is critical to addressing HIV in the United States.
Figure 1: New HIV Diagnoses & U.S. Population, by Race/Ethnicity, 2017
Today, there are more than 1.1 million people living with HIV/AIDS in the U.S., including 468,800 who are Black.11,12,13
Although Black Americans represent only 12% of the U.S. population,14 they accounted for 43% of new HIV diagnoses in 2017 (see Figure 1) and an estimated 42% of people living with HIV.15
The rate of new HIV diagnoses per 100,000 among Black adults/adolescents (41.1) was 8 times that of whites (5.1) and more than twice that of Latinos (16.1) in 2017 (see Figure 2).16 The rate for Black men (77.6) was the highest of any group, more than twice that of Latino men (36.4), the second highest group. Black women (24.9) had the highest rate among women.17
Figure 2: Rates of New HIV Diagnoses per 100,000, by Race/Ethnicity, 2017
The latest data indicate declines in both the number and rate of annual new diagnoses among Blacks in recent years.18,19 Black women, as a subgroup, have experienced a sizable drop in new diagnoses over the years, decreasing by more than 40% between 2008 and 2015.20
Blacks accounted for more than 4 in 10 (43%) deaths among people with an HIV diagnosis (deaths may be due to any cause) in 2016.21 The number of deaths among Blacks with an HIV diagnosis decreased 11% between 2012 and 2016; deaths among Latinos and whites decreased as well (by 2% and 2%, respectively).22 Despite these declines, HIV was the 6th leading cause of death for Black men ages 25-34 and 5th for Black women ages 35-44 in 2016, ranking higher than for their respective counterparts in any other racial/ethnic group.23
HIV death rates (deaths for which HIV was indicated as the leading cause of death) are highest among Blacks. In 2016, Blacks had the highest age-adjusted HIV death rate per 100,000 – 7.2, compared to 0.8 per 100,000 whites.24
Transmission25,26
Transmission patterns vary by race/ethnicity. While male-to-male sexual contact accounts for the largest share of HIV cases among both Blacks and whites, proportionately, fewer Blacks are infected this way and heterosexual sex plays a bigger role among Blacks compared with whites. Among Blacks, 59% of new HIV diagnoses in 2017 were attributable to male-to-male sexual contact and 34% were attributable to heterosexual sex; among whites, 70% of new HIV diagnoses in 2017 were attributable to male-to-male sexual contact and 14% were attributable to heterosexual sex. The remainder of HIV diagnoses in each group were attributable to other causes, including injection drug use.
HIV transmission patterns among Black men vary from those of white men as well. Although both groups are most likely to have been infected through sex with other men, white men are more likely to have been infected this way. Heterosexual transmission accounts for a greater share of new diagnoses among Black men than white men.
Black women are most likely to have been infected through heterosexual transmission, the most common transmission route for women overall. Black women are less likely to have been infected through injection drug use than white women.
Women and Young People
Among all women, Black women account for the largest share of new HIV diagnoses (4,397, or 59% in 2017), and the rate of new diagnoses among Black women (24.9) is nearly 15 times the rate among white women and nearly 5 times the rate among Latinas.27 Black women also accounted for the largest share of women living with an HIV diagnosis at the end of 2016.28
Although new HIV diagnoses continue to occur disproportionately among Black women, data show a 38% decrease in new diagnoses for Black women between 2008 and 2016.29
In 2017, Black women represented about one quarter (26%) of new HIV diagnoses among all Blacks – a higher share than Latinas and white women (who represented 12% and 15% of new diagnoses among their respective groups).30
In 2017, Black teens and young adults, ages 13-24, represented more than half (52%) of new HIV diagnoses in that age group, the majority of whom were young gay and bisexual men.31
According to a national survey of young adults ages 18-30, about three times as many Blacks (46%) as whites (15%) say HIV today is a "very serious" concern for people they know. Almost twice as many Black young adults (30%) say they know someone who is living with, or has died of, HIV/AIDS, compared to whites (16%).32
Gay and Bisexual Men
Among gay and bisexual men, Blacks have been disproportionately affected by HIV and Blacks account for 38% of HIV diagnoses attributable to male-to-male sexual contact.33
In 2017, male-to-male sexual contact accounted for more than half (59%) of new HIV diagnoses among Blacks overall and a majority (80%) of new diagnoses among Black men.34
Young Black gay and bisexual men are particularly affected, with those ages 13-24 representing over half (52%) of new HIV diagnoses among all gay and bisexual men in that age group.35
In addition, newly diagnosed Black gay and bisexual men are younger than their white counterparts, with those ages 13-24 accounting for 35% of new HIV diagnoses among Black gay and bisexual men in 2017, compared to 17% among whites.36
Annual new infections among Black gay and bisexual men have remained stable in recent years, as have new infections among gay and bisexual men overall.37
A study in 20 major U.S. cities found that about 36% of Black gay and bisexual men were infected with HIV, compared to 22% of gay and bisexual men overall, and awareness of infection among Black gay and bisexual men was lower compared with gay and bisexual men in the study overall.38
Although HIV diagnoses among Blacks have been reported throughout the country, the impact of the epidemic is not uniformly distributed.
Regionally, the South accounts for the majority of Blacks newly diagnosed with HIV (63% in 2017) and Blacks living with an HIV diagnosis at the end of 2016 (58%).39
As with the nation as a whole, HIV diagnoses among Blacks are clustered in a handful of states, with the 10 states with the highest number of Blacks living with an HIV diagnosis accounting for the majority (67%) of cases at the end of 2016 (see Figure 3). Florida and New York top the list. In addition, the District of Columbia had the highest rate of Blacks living with an HIV diagnosis at the end of 2016 (3,939.5 per 100,000).40
Figure 3: Number of Black Adults/Adolescents Estimated to be Living with an HIV Diagnosis, Top 10 States, year-end 2016
HIV Testing and Access to Prevention & Care
Three quarters (76%) of Blacks over age 18 report ever having been tested for HIV and they are more likely than Latinos or whites to report having been tested (58% and 50%, respectively).41
Among those who are HIV positive, 20% of Blacks were tested for HIV late in their illness – that is, were diagnosed with AIDS within 3 months of testing positive for HIV; by comparison, 22% of both whites and Latinos were tested late.42
Looking across the spectrum of access to care, from HIV diagnosis to viral suppression, reveals missed opportunities for reaching Blacks. While many Blacks (84%) are diagnosed, 46% remain in regular care, and fewer are virally suppressed (43%).43 Blacks also may be less likely to sustain viral suppression and may experience longer periods of time with higher viral loads, compared to other groups.44
CDC. Special Data Request; 2006.
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CDC. HIV Surveillance Report, Diagnoses of HIV Infection in the United States and Dependent Areas, 2017, Vol. 29; November 2018. HIV diagnosis data are estimates from 50 states, the District of Columbia, and 6 U.S. dependent areas. Estimates for 2017 are preliminary and are not included in trend calculations.
CDC. HIV Surveillance Supplemental Report, Monitoring Selected National HIV Prevention and Care Objectives by Using HIV Surveillance Data, Vol. 23, No. 4; June 2018.
CDC. Fact Sheet: HIV Among African Americans; July 2018.
CDC. NCHHSTP Atlas Plus. Accessed January 2019.
CDC. HIV Surveillance Supplemental Report, Estimated HIV Incidence and Prevalence in the United States 2010-2015, Vol. 23, No. 1; March 2018.
Includes those diagnosed as well as estimates for those undiagnosed.
KFF. State Health Facts; accessed January 2019.
CDC. HIV Surveillance Report, Diagnoses of HIV Infection in the United States and Dependent Areas, 2017, Vol. 29; November 2018. Estimates for 2017 are preliminary and are not included in trend calculations.
CDC. NCHHSTP Atlas Plus. Accessed February 2018.
CDC. NCHS. Deaths: Leading Causes for 2016; July 2018.
CDC. NCHS. Health, United States, 2017; September 2018.
CDC. Slide Set: HIV Surveillance by Race/Ethnicity (through 2017).
CDC. HIV Surveillance Report, Diagnoses of HIV Infection in the United States and Dependent Areas, 2017, Vol. 29; November 2018. HIV diagnosis data are estimates from 50 states, the District of Columbia, and 6 U.S. dependent areas.
KFF. National Survey of Young Adults on HIV/AIDS; November 2017.
CDC. HIV Incidence: Estimated Annual Infections in the U.S., 2008-2014; February 2017.
CDC. HIV Surveillance Special Report, HIV Infection Risk, Prevention, and Testing Behaviors among Men Who Have Sex With Men. National HIV Behavioral Surveillance, 20 U.S. Cities, 2014, No. 15; January 2016.
KFF. Kaiser Health Tracking Poll; July 2014.
CDC. National Black HIV/AIDS Awareness Day Social Media Resources; accessed February 2019.
CDC. Racial and Ethnic Disparities in Sustained Viral Suppression and Transmission Risk Potential Among Persons Receiving HIV Care — United States, 2014; MMWR, Vol. 67, No. 4; February 2018.
HIV/AIDS in U.S.
Key Facts on Health and Health Care by Race and Ethnicity
Donor Government Funding for HIV in Low- and Middle-Income Countries in 2018
The U.S. Government Engagement in Global Health: A Primer
National Survey of Young Adults on HIV/AIDS


The Warrior From West Belfast song lyrics and guitar chords, a tribute to Brendan Hughes. Written by Maire McNally Brian Maguire and preformed by The Irish Brigade.
The Warrior From West Belfast Guitar Chords And Lyrics.


Epicondylitis is a topic covered in the Diagnosaurus.
Zeiger, Roni F.. "Epicondylitis." Diagnosaurus, 4th ed., McGraw-Hill Education, 2014. Emergency Central, emergency.unboundmedicine.com/emergency/view/Diagnosaurus/114024/all/Epicondylitis.
Zeiger RF. Epicondylitis. Diagnosaurus. 4th ed. McGraw-Hill Education; 2014. https://emergency.unboundmedicine.com/emergency/view/Diagnosaurus/114024/all/Epicondylitis. Accessed April 18, 2019.
Zeiger RF. Epicondylitis [Internet]. In: Diagnosaurus. McGraw-Hill Education; 2014. [cited 2019 April 18]. Available from: https://emergency.unboundmedicine.com/emergency/view/Diagnosaurus/114024/all/Epicondylitis.


text:
Sell Flowers (F. & S. II.66)
Flowers (F. & S. II.66) Andy Warhol
You are seeing this page because we currently don't have this work for sale, but we do have an active buyer seeking this work. Please click the link below if you have Flowers (F. & S. II.66) by Andy Warhol to sell.
This signed screen print from 1970 is a limited edition of 300 from Andy Warhol's Flower series. Flowers (F. & S. II.66) shows four hibiscus flowers in pink and orange hues, layered on top of a contrasted electric green background of undergrowth.
ANDY WARHOL, FLOWERS (F. & S. II.66), FLOWERS SERIES, SIGNED SCREEN PRINT, 1970
The screen print Flowers (F. & S. II.66), from Andy Warhol's Flowers series (1970), shows the artist's famous flower motif, rotated, rendered in this print with soft pink and oranges hues against a starkly contrasted grass background. With the Flowers series, Warhol exhibits his unrivalled skill in the screen print process by using the same photographic motif for each print and rendering it in a multitude of variations of colour and composition.
Taken from a photograph by Patricia Caulfield found in a 1964 issue of Modern Photography, Warhol deliberately appropriates and repeats the image excessively to mirror the mechanical forms of reproduction found in mass-media that he was so fascinated by. This idea of assembly-line production was reinforced by Warhol's 'Factory' that opened in New York in 1964, where he produced many of his screen prints, noting: 'Mechanical means are today and using them I can get more art to more people. Art should be for everyone.'
Flowers (F. & S.II.66) reworks the traditional art historical genre of flower painting, by appropriating an image from a magazine and reproducing it in a 'machine-like' manner, to challenge ideas of fine art, authorship and creativity. Warhol directly participates in appropriation and image dissemination. Consciously banal and synthetic. He rejects hierarchical compositions in favour of flattened perspective and abolishes complex colour harmonies for monochrome planes of flat colour and artificially bright ink.
Find out more about Flowers by Andy Warhol.
ABOUT ANDY WARHOL
Andy Warhol (born Andrew Warhola) is a name synonymous with the celebrity culture and mass consumerism which coloured the boulevards of New York City in the Post World War II era. Born into a working class immigrant family in the urban landscape around the bustling metropolis, Warhol's early life was characterised by a climb up the capitalist rungs of society.
The artist himself noted, "buying is more American than thinking, and I'm as American as they come", this quote came to demarcate Warhol's artistic practice as he embraced the commodification of the American Dream. The Pop artist's beginnings in the business sector gave him the practical skill set to experiment with a more commercial approach to art throughout his career, particularly with regards to screen printing. Read more about Andy Warhol.
Details Of Renaissance Paintings (Piero Della Francesca Madonna Del Duca Da Montefeltro)
Blackglama (Judy Garland) (F. & S. II.351)
Ladies And Gentlemen (F. & S. II.131)
Flowers (black and white)
Hammer And Sickle (F. & S. II.63)
Camouflage (F. & S. II.411)
Space Fruit
A Gold Book
La Recherche du Shoe Perdu

redpajama_set_name: RedPajamaCommonCrawl


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Improved AGN light curve analysis with the z-transformed discrete correlation function
Tal Alexander
Department of Particle Physics & Astrophysics
POB 26, Rehovot 76100, Israel
The cross-correlation function (CCF) is commonly employed in the study of AGN, where it is used to probe the structure of the broad line region by line reverberation, to study the continuum emission mechanism by correlating multi-waveband light curves and to seek correlations between the variability and other AGN properties. The z-transformed discrete correlation function (ZDCF) is a new method for estimating the CCF of sparse, unevenly sampled light curves. Unlike the commonly used interpolation method, it does not assume that the light curves are smooth and it does provide errors on its estimates. The ZDCF corrects several biases of the discrete correlation function method of Edelson & Krolik (1988) by using equal population binning and Fisher's z-transform. These lead to a more robust and powerful method of estimating the CCF of sparse light curves of as few as 12 points. Two examples of light curve analysis with the ZDCF are presented. 1) The ZDCF estimate of the auto-correlation function is used to uncover a correlation between AGN magnitude and variability time scale in a small simulated sample of very sparse and irregularly sampled light curves. 2) A maximum likelihood function for the ZDCF peak location is used to estimate the time-lag between two light curves. fortran 77 and fortran 95 code implementations of the ZDCF and the maximum likelihood peak location algorithms are freely available (see http://www.weizmann.ac.il/weizsites/tal/research/software/).
The problem of analysing sparse, unevenly sampled light curves is frequently encountered in the study of AGN, notably in line reverberation mapping and in multi-wavelength variability studies. In many cases, despite great observational efforts, the light curves can not be reliably analysed by Fourier inversion or other inversion methods (Maoz, 1994) . There is, to date, no satisfactory alternative but to carry the analysis in the time domain. A widely used tool for extracting information from such light curves is the cross-correlation function (CCF) Blandford & McKee (1982)
CCF(τ)=E[(a(t)−Ea)(b(t+τ)−Eb)]√VaVb, (1)
where a and b are the two light curves, τ the time-lag, E the expectation value and V the variance over the light curve.
The CCF of AGN continuum and emission line light curves is used in line reverberation mapping to estimate the size and geometry of the broad line region Peterson (1994, and references therein) . The CCF between different AGN continuum wave bands is used for studying the continuum emission mechanism by looking for causal connection between the various wavebands (e.g. Korista et al. 1995). Other uses of the CCF include the determination of the Hubble constant from the time lag between variations of the multiple images of a macro-lensed QSO (e.g. Vanderriest et al. 1989) and the linear reconstruction of discretely sampled light curves, whose input is the time-lag dependence of the auto-correlation function (ACF) Rybicki & Press (1992) .
In practice, finite segments of the light curves are sampled discretely with measurement errors, at unevenly spaced intervals. The expectation values and variances of the light curves are unknown and must also be estimated from the observations. Three conditions are implicitly assumed in the process of estimating the correlation function: 1) statistical stationarity, 2) the ergodic assumption (i.e. that the ensemble average of all the light curves, which are statistically similar to the observed one, is equivalent to the time average over a single infinite light curve) and 3) random sampling of the light curves. If these are satisfied, then the data at hand can indeed yield an estimate for the population correlation between signals that are statistically similar to the observed light curves. The observed data are the result of a two level sampling process. First, Nature 'draws' a light curve from the continuum of possible light curves. Second, the observations sample the continuum of points in a finite segment of this given light curve (See Fig. 1). Conditions (1) and (2) relate to first sampling level. They must usually be assumed, since they cannot be inferred from the observed data. The ZDCF (like the two other CCF estimators that are described below) deals only with the second level of sampling, where the parent distribution is the continuous distribution of the points in the given finite segments.
There are in general two approaches for dealing with the uneven sampling: interpolation and the discrete correlation function (DCF). The interpolation method of Gaskell & Peterson (1987) calculates the CCF by averaging the cross-correlation obtained by pairing the observed a(ti) with the interpolated value b(ti−τ), and that obtained by pairing the observed b(tj) with the interpolated a(tj+τ). The DCF method Edelson & Krolik (1988, EK) initially uses the observed points to calculate
DCFij=(ai−¯a′)(bj−¯b′)s′as′b (2)
where ai, bj are the observed fluxes at times ti and tj, respectively, and ¯a′, ¯b′, s′2a and s′2b are the sample means and variances over the entire light curves. Subsequently, {DCFij} are binned by their associated time-lag, τij=ti−tj, into equal width bins, τ±δτ. The bin average is used to estimate CCF(τ) and the bin's standard deviation to estimate the error.
Figure 1: The two observed points, with time lag τi, sample the correlation at this lag between the bold intervals of the two continuous segments of the light curves.
The two methods were compared by several authors Rodriguez-Pascual, Santo-Lleó & Clavel (1989); White & Peterson (1994) in the context of AGN variability studies, and the general conclusion seems to be that the interpolation method is equal or superior to the DCF in most cases. Nevertheless, interpolation has several obvious drawbacks. Adding interpolated points between those actually observed amounts to inventing data or assuming that the light curve varies smoothly. The interpolation method does not give error estimates on the reconstructed CCF, which are necessary for model fitting. The DCF concept is a more cautious approach to reconstructing the CCF. However, the original DCF method has several problems, which are discussed in detail below. These problems can have adverse effects on the DCF in the realistic case of poorly and unevenly sampled light curves.
This work does not attempt to deal with the many potential pitfalls in the statistical interpretation of the CCF and its significance, which may arise from incorrect assumptions of stationarity or ergodicity or from an incorrect measurement error model (e.g. Box & Newbold 1971). The use of the CCF in AGN light curve analysis is dictated by the present-day quality of the data and the nature of current AGN research. In this given situation, it is important to try and develop more reliable methods for estimating the CCF. The validity of the underlying assumptions in the case of AGN will be ultimately justified if the accumulated results from many observation campaigns converge into a coherent physical picture.
The approach taken here is empirical, in that analytical arguments and approximate assumptions are used only as motivation for the proposed improvements, whose final test is by simulations with AGN-like light curves. Care is taken to verify that in cases where the assumptions do not hold, the CCF estimate is conservative rather than deceptively significant. Several changes in the original DCF method are suggested, the major ones being the use of the z-transform and equal population binning. These result in the z-transformed discrete correlation function (ZDCF), which is an improved, more robust method for reconstructing the CCF under realistically unfavourable conditions.
The ZDCF method is described and contrasted with the DCF in section 2. Their performance is compared by simulations in section 3. Section 4 presents two sample applications of the ZDCF. The usefulness of the ZDCF in extracting information from a small number of observations is demonstrated in section 4.1, where it is used to find a correlation between the variability time scale and magnitude in a small sample of simulated AGN light curves. Section 4.2 presents a maximum likelihood method for estimating the time-lag between two light curves. The results are discussed and summarised in section 5.
2 The z-transformed Discrete Correlation Function
2.1 Estimating the correlation with the z-transform
Let n be the number of {ai,bi} pairs in a given time-lag bin. CCF(τ) is estimated by the correlation coefficient
r=∑ni(ai−¯a)(bi−¯b)/(n−1)sasb. (3)
where ¯a, ¯b are the estimators of the bin averages, and sa, sb the estimators of the standard deviations
s2a=1n−1n∑i(ai−¯a)2, (4)
with s2b is similarly defined. Note, in contrast, that the DCF is not normalized correctly since the moments of the full light curves, which appear in DCFij (equation 2), are used to normalize the individual bins. The correct normalization, as well as ensuring that |r|≤1, reduces the errors when the light curves are non-stationary by limiting the summation to those points that actually contribute to the bin White & Peterson (1994) . The sampling distribution of r is known to be highly skewed and far from normal and therefore estimating its sampling error by the sample variance sr can be very inaccurate. When a and b are drawn from the bivariate normal distribution it is possible to transform r into an approximately normally distributed random variable, Fisher's z (e.g. Kendall & Stuart 1969, p. 390, Kendall & Stuart 1973, p. 486 and references therein). Defining
z=12log(1+r1−r),ζ=12log(1+ρ1−ρ),r=tanhz, (5)
the mean and variance of z are approximately equal to
¯z = ζ+ρ2(n−1)× (6)
[1+5+ρ24(n−1)+11+2ρ2+3ρ48(n−1)2+⋯],
s2z=1n−1[1+4−ρ22(n−1)+22−6ρ2−3ρ46(n−1)2+⋯], (7)
where ρ is the unknown population correlation coefficient of the bin. In order to estimate ¯z and sz, the ansatz ρ=r is assumed. Transforming back to r, the interval corresponding to the normal ±1σ error interval can be estimated by
δr±=|tanh(¯z(r)±sz(r))−r|. (8)
The validity of this ansatz was verified by simulations with normal bivariate distributions of different correlation coefficients and n = 11. The results show that the probability of ρ lying outside the empirical interval r±δr± is 1.03 to 1.08 times higher than the normal value of 0.3174, implying a slight under-estimation of the errors. Unlike the r±sr error interval of the DCF, the z-transformed error interval satisfies |r±δr±|≤1. As |r| tends to 1 it becomes both asymmetric around r and smaller than r±sr, thus improving the error estimates in the physically interesting extrema of the CCF. The normality of z is known to hold down to n=11 for an exact normal bivariate distribution or for ρ=0 and otherwise tends asymptotically with n towards normality. As will be discussed below, the dependence of the z-transform on the shape of the parental distribution plays an important role in determining the properties and limitations of the ZDCF. The effects of deviations from binormality, and especially from mesokurtosis, can be significant Gayen (1951) . While the resulting bias in ¯z is moderate and tends to zero as n increases, the bias in sz may be non-negligible and its behaviour as n increases varies depending on the underlying bivariate distribution. Nevertheless, the n=11 lower limit still holds as long as the deviations from binormality are moderate. There exist also further refinements of the z transform, z∗ and z∗∗ Hotelling (1953) . Their effect on the ZDCF was checked by simulations, similar to those described in section 3 below and the improvement in the results was found to be negligible and not worth the added calculational complexity.
2.2 The binning method
The ZDCF binning method differs from that of EK in both the binning criterion and the treatment of interdependent pairs in the bin. The ZDCF bins by equal population and as a result the bins are not equal in time-lag width. Each bin contains at least nmin=11 points, (the minimum for a meaningful statistical interpretation) and does not contain interdependent pairs, which are discarded. Consequently, light curves of less then 12 observations cannot be analysed by this method.
2.2.1 Statistical properties
The main source of bias in estimating the bin's correlation coefficient is the existence of inner correlations within the sets {ai} and {bi}. The mean time between the observations in a bin associated with time-lag τ is
¯¯¯¯¯¯¯Δtτ=T−τnmin (9)
where T is the duration of the two observed light curves (for simplicity, it will be assumed from this point on that the two light curves were observed over the same period). If the light curve is significantly auto-correlated over a coherence time scale τ0, then when ¯¯¯¯¯¯¯Δtτ<τ0, the sampling is no longer random since consecutive points in the bin are not independent111The coherence time scale which is relevant here is that of finite segments of length T and not that of infinite light curves, which may be larger. This distinction is relevant, for example, for light curves with power-law power spectra, where τ0 increases with T as the lower frequency, higher amplitude variations become more dominant.. ¯¯¯¯¯¯¯Δtτ should not be confused with the mean time between observations, ¯¯¯¯¯¯¯Δt=T/(nobs−1). ¯¯¯¯¯¯¯Δtτ is not a function of the number of observations, nobs, and therefore equal population binning automatically stabilises the results to changes in nobs.
It is instructive to discuss the effect of auto-correlation on the z transform in terms of its marginal parental distributions. The irregularly spaced observation times, {ti}, can be viewed as sampling the distribution of a random variable t. The observed points are therefore distributed as functions of the random variable t, namely a(t) and b(t). Since these distributions are not necessarily normal, the z transform may be biased, even when the signals are not auto-correlated. The bias caused by auto-correlation has a distinctive signature which can be demonstrated by studying smooth light curves in the limit of short T. In this case the coherence time scales of both light curves are of the order of T itself, the observed segments are almost linear and the distributions of {ai} and {bi} are approximately linear functions of the distribution of t. Suppose further that t is uniformly distributed, so that kurt(t)=−1.2. Since the kurtosis is invariant under linear transformations of the random variable, the kurtosis of a and b will also be -1.2. Negative kurtosis in a or b can contribute to significant over-estimation of sz Gayen (1951, eqs. 81–84) , since
s2z,bias = s2z,true− (10)
(n−3−ρ2)(n−1)2ρ2(kurt(a)+kurt(b))4(1−ρ2)2+⋯
This qualitative explanation of the tendency of the ZDCF to over-estimate the errors of auto-correlated signals is supported by the simulations in section 3. The DCF is also known to suffer from this bias.
Astronomical observations are often clustered on daily or seasonal timescales. One simple model for this distribution is that of N periods, each divided into two sub-periods. At the first sub-period, of length T1, the object can be observed, and the observing times are uniformly distributed. At the second sub-period, of length T2, the object cannot be observed. It can be shown that the kurtosis of this distribution is also negative, kurt=−1.2F(N,T2/T1), where the function F is bracketed between 1 and 5/3 and approaches 1 very rapidly with increasing N. Such sampling patterns will therefore also lead to over-estimation of the sampling errors.
The robustness of ¯z and sz to deviations from the assumed binormality of the parental distributions determines their usefulness in analysing auto-correlated light curves. The nature and magnitude of these deviations depend on the ratio ¯¯¯¯¯¯¯Δtτ/τ0 and on the specific statistical properties of the light curve. The simulations below show that in the case of AGN-like data sets, these deviations are still small enough for the z-transform to be of practical use.
Another source of bias is the occurrence of interdependent pairs, such as {ai,bj} and {ai,bk}, in the same bin. If, for example, a and b are fully correlated at time-lag τ, a(t)∝b(t+τ), then r(τ) will be biased towards zero because the repeated occurrences of the point ai are equivalent to substituting the true a by a constant light curve. Although the frequency of these multiple occurrences can be lowered by decreasing the bin width, this is inconsistent with the requirement that the bin contain at least 11 points. The ZDCF addresses this problem directly by discarding the interdependent pairs. This is to be contrasted with the suggestion of EK that the effect of these pairs can be taken into account by heuristically increasing the DCF error estimates beyond the actual scatter in the bin.
The binned correlation coefficient associates with the mean bin lag, ¯τ, an estimate of the bin average of the correlation coefficient. This is not the same as estimating ρ(¯τ). The approximation involved in using the binned average and its interpretation are discussed in appendix A.
2.2.2 The binning algorithm
The binning is implemented by ordering all the possible pairs {ai,bj} by their associated time-lag τ=ti−tj. The ordered list is then divided, bin by bin, into bins of nmin pairs. In the process of adding pairs to a bin, a new pair whose a or b points have previously appeared in that bin, is discarded. The artificial separation of pairs of very close time-lags into adjacent bins is prevented by defining a small parameter ϵ so that a new pair with time-lag τi+1 will be added to the bin as long as τi+1−τi<ϵ, even if nmin is exceeded. Most of the information in the ZDCF is at the time lags where the overlap between the two light curves is large (cf. Fig. 6). It is therefore desirable that the binning algorithm minimize the bin widths at these lags. This can be achieved by starting the allocation of pairs to bins at the lag where the pairs are densest. For the auto-correlation function, this means that the binning proceeds from τ=0 up to τmax. For the CCF, the binning proceeds from the median τ up to τmax and then from the median τ down to τmin.
The allocation of pairs to bins depends on the order in which the binning is carried out and on the choice of the interdependent pairs which are discarded. While this does not affect the average statistical properties of this method, different choices may lead to different ZDCF results for a given poorly sampled light curve. This ambiguity can be resolved when the ZDCF points are used for fitting a model CCF. In this case the fit score can be averaged over the possible binning choices.
2.3 Measurement errors
The measured light curves a, b are the sum of the true light curves x, y and random noises na, nb
a(ti)=x(ti)+na(ti),b(tj)=y(tj)+nb(tj). (11)
As the measurement errors propagate into the CCF in a complicated, non-linear manner, it is suggested that their effect be estimated by Monte Carlo simulations. This is performed by assuming a distribution for the measurement errors (usually the normal distribution), adding a randomly drawn error value to each point and recalculating the ZDCF. The Monte Carlo average of z is then used to obtain r and δr±. The resulting ZDCF is conservative in that it estimates the mean CCF that is consistent with the observations and √2 times the quoted measurement errors. The over-estimated errors are an unavoidable consequence of having no model for the light curve but the observed points themselves. The errors on the ZDCF points are the sampling errors and should not be interpreted as estimating the deviation from the CCF of the true signals due to the measurement errors. For example, the ZDCF of very noisy light curves tends to zero irrespective of the correlation between the true signals. It should be emphasized that the data points are not weighted by their errors. It is therefore recommended that points with atypically high measurement errors (relative to the mean error) be judiciously discarded from the light curve before the ZDCF is applied.
This approach to estimating the effects of the measurement errors is safer than that suggested by EK, which diverges in the limit of a small number of observations. This is discussed in more detail in appendix B
Figure 2: Two examples of simulated light curves and their ZDCF. The time differences between each successive pair of the 20 observations (black dots) are drawn from the uniform distribution. No measurement errors added. Top 2 panels: flicker-noise light curve (P(ν)∝1/ν) and its ACF. Bottom 2 panels: soft spectrum light curve (P(ν)∝1/ν2) and its ACF. The simulated light curve is shown in solid line and the discrete observations in dots. The true ACF is shown in solid line and the auto-correlation function calculated by the ZDCF method is shown as points with error bars.
Figure 3: The fraction of outlying points, fout as function of the number of simulated observations, nobs. The results are the average over 100 simulations. Top: flicker-noise light curves. Bottom: soft, P(ν)∝1/ν2, light curves. The even sampled bins were thinned out so as to have only 11 points per bin. No measurement errors added.
The properties of AGN power spectra are currently well known only in the X-ray range, where the power spectrum can be approximated by a power law, P(ν)∝ν−α, with 1≲α≲2 Green, McHardy & Lehto (1993) . In order to compare the performance of the DCF and ZDCF on AGN-like light curves, the two methods were applied to two types of simulated light curves (Fig 2): Noisy light curves with a P(ν)∝ν−1 power spectrum (flicker noise) and softer light curves with a P(ν)∝ν−2 power spectrum. Both were generated by choosing random phases for frequencies in the range νmin=1/4, νmax=100 at increments of Δν=1/4 (in arbitrary inverse time units). A light curve was then calculated from t=0 to 1.
The simulations consisted of drawing eight sets of 100 light curves, one for each of the two power spectra and nobs=15,20,25 and 30. For each of these light curves, the times of the nobs simulated observations were randomly drawn so that the time difference between two successive observations was uniformly distributed. The light curve was then sampled at these random times and the discrete auto-correlation function was calculated by both the DCF and ZDCF methods. In order to have a common basis for comparing the two methods, the number of bins used by the DCF was adjusted to equal the mean one used by the ZDCF. The ZDCF was calculated with nmin=11 which resulted in ⟨nbin⟩=2,6,14 and 22 for nobs=15,20,25 and 30, respectively. In order to isolate the possible effects of the binning procedure on the results, eight additional sets of simulated light curves were similarly generated and sampled at evenly spaced time intervals. The ZDCF was then calculated after the resulting zero width bins were thinned down to 11 randomly chosen points per bin to avoid possible biases due to the z transform's n dependence. The fit of the ZDCF to the true ACF was evaluated by the fraction of outlying bins, fout, whose error interval does not include ρbin, which was calculated from the full light curve for each bin. Similarly, for the DCF the comparison was to ¯ρ, averaged over the interval τi±δτ. For each of the two light curve types the average kurtosis of a and b and a χ2 fit of the marginal distributions to the normal distribution were also calculated to asses the deviations from the idealized assumptions of the z transform.
Figure 3 shows the fraction of outliers, fout, as a function of nobs for the two light curve types and for the ZDCF, DCF and evenly sampled ZDCF cases. In all the simulations checked here, both methods over-estimate the errors, but the ZDCF results are always closer than the DCF's to the normal value fout=0.3173. There is a marked trend of an increase in the over-estimation for softer power spectra. This is correlated with the deviations from normality. The flicker noise average kurtosis is kurt=−0.27 and the χ2 fit probability for the normality of the marginal distributions is P(χ2)∼0.3 whereas for the soft power spectrum the values are kurt=−0.80 and P(χ2)∼0.03, respectively. The evenly sampled ZDCF dependence on the underlying power spectrum is similar to that of the binned ZDCF, confirming that this effect is not due to the binning procedure.
As anticipated by equation 9, the behaviour of the ZDCF, unlike that of the DCF, is not affected by the number of observations. The difference between the two methods is most marked in the case of the flicker noise power spectrum, where the DCF becomes highly biased as the number of observations decreases.
Qualitatively similar results were obtained for simulations performed with Gauss-Markov random signals Brown & Hwang (1992) and signals described by Gaussian shaped peaks of random height and width, randomly superimposed on a high order polynomial.
4.1 Correlations involving variability time scales
The physical origin of AGN variability is poorly understood. One line of approach to this problem is to look for correlations between the variability and other AGN properties, such as the luminosity (e.g. Hook et al. 1994). The following example is a simplified, simulated version of such an analysis, which was performed on observed data by Netzer et al. (1996) . The ZDCF bias depends on the statistical properties of the light curves, which are generally not known in advance. The following example demonstrates that the reduced bias of the ZDCF, even when it cannot be quantified, makes it more efficient in extracting information from the data than the DCF and thus useful even when applied to sparsely sampled light curves.
The observational situation was simulated as follows. A sample of 30 light curves with power law power spectra, P(ν)∝ν−α were generated as described in section 3. The 30 light curves consisted of three light curves for each of the 10 values α=1.1, 1.2, …, 2.0. All the light curves were then sampled at the same 15 times (this simulates the situation where the AGN sample is recorded simultaneously on one photographic plate). The sampling times (Fig. 4) have a clustered pattern typical of astronomical observations and were taken from the observed light curve shown in Fig. 6 (The original light curve was diluted down to 15 points by throwing very close observing times). Normally distributed 3% measurement errors were added to the simulated observations. A toy model for the magnitude–power spectrum relation was used to associate each light curve with a magnitude
M(α)=26+α+ϵ, (12)
where ϵ is a gaussian variate with zero mean and a standard deviation of 0.1. ϵ represents an additional scatter in the AGN luminosity that is independent of the power spectrum. This may be due to measurement errors in M or to a hidden dependence of M on some physical parameters other than α.
The ACF of each of the light curves was estimated by both the DCF and ZDCF. The 15 observations yielded only two ZDCF bins, and accordingly, the DCF was also calculated with only two bins. The coherence timescale, τ0, is a measure of the typical variability timescale. Following Netzer et al. (1996) , τ0 was formally defined as the shortest lag where ACF(τ0)=0 and was estimated by minimal squares fitting of the straight line r=1−τ/τ0 through the DCF and ZDCF points. The fit took into account both the δr and δτ errors. Finally, Spearman's rank order correlation coefficient between τ0 and M was calculated for the 30 AGN. Note that the rank order correlation is insensitive to the exact functional form of M(α), and is only affected by the ratio between the scatter in M due to the distribution of α in the sample and that due to ϵ.
Figure 4: Two of the simulated light curves used for studying the correlation between variability timescale and magnitude. Both light curves are sampled simultaneously 15 times with a clustered sampling pattern and have a power law index of α=2 (open circles) and α=1 (filled circles).
Figure 5: The sample distribution of Spearman's rank order correlation coefficient between τ0 and M (top) and its probability (bottom) calculated with the DCF (thin line) and ZDCF (bold line). Also listed are the fractions of results with positive correlation and probabilities greater than 0.90, 0.95 and 0.99.
This entire procedure was repeated with 1000 random samples of 30 light curves each. Fig. 5 shows the distribution of the calculated correlation coefficient and of the proability that the correlation is not random, for both the ZDCF and DCF. For power law power spectra, τ0 increases with α and therefore correct estimates of τ0 should yield significant positive correlation coefficients. Fig. 5 shows that the ZDCF is much more efficient than the DCF and has a significantly better chance of uncovering the underlying correlation even in a small sample of sparsely sampled light curves. For example, in 40% of the ZDCF simulations, a positive correlation was detected at the 0.99 significance, as compared to only 6% for the DCF. In 76% of the ZDCF simulations, a positive correlation was detected at the 0.90 significance, as compared to only 23% for the DCF.
4.2 Time lag estimation by maximum likelihood
Figure 6 shows the R and B light curves of the optically violent variable quasar 3C 454 (Netzer et al., 1996) and their ZDCF. The time lag between the two bands, and the question whether it is consistent with zero, are important for testing models of AGN continuum emission. In the following example, the ZDCF is used to estimate the uncertainty in the peak position by using, for the first time in this context, the fiducial interpretation of the likelihood function.
Figure 6: Top: The R band (triangles) and B band (circles) light curves of the optically violent variable quasar 3C 454. The measurement errors are smaller than the point size. Bottom: The central peak of the ZDCF, calculated with nmin=11 and 100 Monte Carlo draws for estimating the effects of the measurement errors.
The likelihood that point i is the maximum, Li, is approximately the product of the probabilities for point i being larger than point j
Li = ∫∞−∞dζi1√2πsz,iexp[−12(zi−¯zisz,i)2]× (13)
∏j≠i∫ζi−∞dζj1√2πsz,jexp[−12(zj−¯zjsz,j)2],
where ¯z and sz are functions of ζ through ρ and z is a function of r, the empirical correlation. The approximation results from neglecting the possible correlations between the ZDCF points (e.g. Box & Jenkins 1970) and from the fact that the ZDCF points are only approximately normally distributed. It is more convenient to express Li as an integral over ρ
Li = ∫+1−11√2πsz,iexp[−12(zi−¯zisz,i)2]× (14)
∏j≠iΦ(zj,ρi)dρi(1−ρ2i),
Φ(z,ρ)=∫ρ−11√2πszexp[−12(z−¯zsz)2]dρ′(1−ρ′2). (15)
Li can be easily calculated numerically. Li reaches its maximum at the highest ZDCF point, so that the maximum likelihood (ML) estimate coincides with the ZDCF peak. The sampling distribution of ML estimators is generally known only in the large sample limit, where it is Gaussian. Here, having a large sample means having many pairs of R and B observations of the same continuous light curve, observed simultaneously by different observers. The actual data set consists of only one such pair, and it is therefore impossible to assign a a confidence interval for the peak.
There is an alternative approach for obtaining interval estimates from the likelihood function, which is related, but not identical, to Bayesian statistics. The normalized likelihood function, defined as the fiducial distribution, is interpreted as expressing the "degree of belief" in the possible value of the estimated parameter, and the 68% interval around the likelihood function's maximum is defined as the 68% fiducial interval (e.g. Frodesen, Skjeggestad & Tøfte 1979). The fiducial function for the peak is calculated by interpolating between the points of the likelihood function. The position of the fiducial distribution's maximum is the maximum likelihood estimate of the time-lag222When the fiducial function is interpolated linearly, the most likely peak position is that of the highest ZDCF point. and the fiducial interval is that which includes 0.3414 (normal 1σ) of the area left and right of the maximum, respectively. In practice, the fiducial interval cannot be narrower than the bin width δτ±(rmax). The fiducial interval can be interpreted as the interval where 68% of the likelihood-weighted ensemble of all possible CCFs reach their peaks. It should be emphasized that a fiducial interval is conceptually different from a confidence interval, and one should not misinterpret it as meaning that if the light curves are resampled and their ZDCF peak calculated, then on the long run, the true peak will lie inside the fiducial intervals 68% of the time. A detailed discussion of the fiducial interval and its relation to Bayesian statistics can be found in Kendall & Stuart (1973) .
The fiducial estimate of the peak has several advantages over other approaches that were used in AGN variability studies. Unlike methods that use simulations to estimate the uncertainty on the time-lag (e.g. Maoz & Netzer, 1989) , the fiducial estimator does not assume anything about the mechanism that generates the light curves. Another commonly used method for estimating the peak location is to fit the CCF peak to a peaked function, such as a parabola or a Gaussian. Some parameterization of the function's width is then used as a measure of the uncertainty in the peak position. The arbitrary choice of the function introduces an unwanted degree of freedom to the final result. It is also often the case that the peak, unlike the fitted functions, is asymmetric and this leads to skewed estimates of the peak location. In contrast, The fiducial estimator does not assume anything about the shape of the peak and thus avoids these problems.
These advantages come at the price of having to make the approximations mentioned above in calculating Li, as well as the approximation involved in interpolating the likelihood function333Note that these approximations are also shared by the function-fitting method, and that methods that use the centroid to parameterize the time-lag also ignore the biases that may be introduced by the correlations. Another unavoidable consequence of this approach is the assumption of a Bayesian prior, namely the uniform distribution of the CCF points in z-space.
The CCF peak of 3C 454 lies between time-lags τ=−200 to 200 days. The maximum likelihood estimate that is calculated for the points in this interval is 0.1+10.0−23.7 days (R lags after B). This is consistent with the hypothesis that the R and B bands of 3C 454 vary simultaneously.
5 Discussion and summary
Monitoring light curves of astronomical objects over long periods of time requires great observational efforts. Inevitable limitations and difficulties stand in the way of obtaining regular, well sampled light curves and severely restrict the reliability of spectral analysis. The result is that in many cases the light curves are analysed directly in the time domain by the CCF. Such a situation calls for an effort, on par with the observational one, to develop improved methods for estimating the CCF, even at the cost of added computational complexity. In many cases the only way to extract information from meager data is by modeling. It is therefore important to have error estimates that can be used for fitting theoretical CCF models to the data and estimate the location and significance of the CCF extrema.
The ZDCF method attempts to correct the biases that affect the original DCF. The simulations show that the ZDCF performs better or at least as good as the DCF under a variety of conditions. The performance of the ZDCF depends on the ratio between ¯¯¯¯¯¯¯Δtτ, the typical time between observations in a bin, and τ0, the coherence timescale. As long as ¯¯¯¯¯¯¯Δtτ>τ0, the z-transform biases are small and the error estimates are realistic. Since ¯¯¯¯¯¯¯Δtτ does not depend on nobs, it is possible to increase the time lag resolution of the ZDCF without increasing its bias by increasing sampling rate (i.e increasing nobs at a constant total time T). When ¯¯¯¯¯¯¯Δtτ<τ0, the error estimates of the z-transform may be over-estimated. However, in this case interpolation is justified. This is seen by noting that ¯¯¯¯¯¯¯Δtτ is a decreasing function of τ, which is approximately bounded from below by
It then follows that if ¯¯¯¯¯¯¯Δtτ<τ0, then ¯¯¯¯¯¯¯Δt<τ0 and therefore that the light curves are well sampled. Even in this case the ZDCF offers a more conservative alternative than interpolation. ¯z, unlike sz, is relatively unbiased and therefore rbin estimates the CCF much better than implied by the formal error estimates.
The simulations were limited to the uniform random sampling pattern. Sampling patterns which are far from uniform may introduce complicated biases to the ZDCF, which can be traced to the fact that the Fourier transform of the underlying signal is convolved with that of the sampling pattern itself. This is a fundamental problem that lies beyond the scope of the ZDCF. A partial solution is to divide the light curve into segments which are roughly uniformly sampled and perform the ZDCF on the union of these segments, at the price of estimating the correlation function only for small values of τ.
To summarize, the calculation of the ZDCF involves the following steps:
Atypically noisy measurements should preferably be discarded from the light curve .
All possible pairs of observations, {ai,bj}, are sorted according to their time-lag ti−tj and binned into equal population bins of at least 11 pairs. Multiple occurrences of the same point in a bin are discarded so that each point appears only once per bin (section 2.2).
Each bin is assigned its mean time-lag and the intervals above and below the mean that contain 1σ (normal) of the points each (section 2.2).
The correlation coefficients of the bins are calculated (equation 3) and z-transformed (equation 5). The error is calculated in z-space (equation 7) and transformed back to r-space (equation 8).
The effect of measurement errors is estimated by Monte Carlo runs, where at each step a random error is added to each point according to its quoted error. The ZDCF is averaged in z-space and the average is transformed back to r-space (section 2.3).
The resulting CCF error bars are roughly equivalent to normal 1σ errors. The simulations performed here show a consistent trend towards over-estimation of the errors and suggest that the error interval may be as large as 1.4σ for strongly auto-correlated light curves. A fortran 77 code of the ZDCF method is available from the author on request.
Appendix A Properties of the binned average
The binned correlation coefficient is an average of the correlation coefficient over the bin's time lag interval. The exact nature of this average can be expressed in terms of the mixture of time-lags {τi}, associated with the observed pairs {ai,bi} in the bin. Let the weight ωi be the fraction of the pairs with time-lag τi (usually ωi=1/nmin). Define E−a(τi), V−a(τi) to be the mean and variance over the continuous light curve a from tmin to tmax−τi, E+b(τi), V+b(τi) to be the mean and variance over the continuous light curve b from tmin+τi to tmax and C±a,b(τi) to be the covariance of the two light curves with the intervals similarly defined (See Fig. 1). It is straightforward to show that the binned variance and covariance over the continuous light curves are
Va = n∑i=1ωiV−a(τi)+∑i<jωiωj(E−a(τi)−E−a(τj))2,
Vb = n∑i=1ωiV+b(τi)+∑i<jωiωj(E+b(τi)−E+b(τj))2, (17)
Ca,b = n∑i=1ωiC±a,b(τi)+ (18)
∑i<jωiωj(E−a(τi)−E−a(τj))(E+b(τi)−E+b(τj)).
It is clear that the sample binned correlation coefficient estimates the quantity
ρbin=Ca,b√VaVb, (19)
which is not identical to the simple weighted arithmetic mean
¯ρ=n∑i=1ωiC±a,b(τi)√V−a(τi)V+b(τi), (20)
although both values approach each other as the bin width is decreased. r estimates an averaged value of ρ, weighted by the density distribution of the time-lag points in the bin. In order to reflect this, the time-lag associated by the ZDCF with the bin is estimated by the mean lag ¯τ. The uncertainty in τ is estimated by the intervals δτ± that contain 0.3414 (normal 1σ) of the points in the bin above and below ¯τ, respectively. The resulting error bars are asymmetric and, unlike usual error bars, do not give the ±1σ uncertainty on the 'true' value ¯τ but rather describe the interval over which ∼2/3 of the averaging was performed. Simulations of sparsely sampled light curves, where τmax is small and E(τi), V(τi) and C(τi) are therefore calculated on largely overlapping stretches of the light curves, confirm that ρbin≃¯ρ to a very good approximation.
Appendix B The effect of measurement errors on the DCF
EK suggest that the sample CCF of the true signals, rx,y, can be recovered from ra,b by a simple analytic correction of replacing equation 3 with
r′x,y=∑ni(ai−¯a)(bi−¯b)/(n−1)√s2a−s2(na)√s2b−s2(nb), (21)
where s(na) and s(nb) are the typical measurement errors. |r′x,y|>|ra,b| by definition, and in the limit of low S/N, |r′x,y| may become arbitrarily large (even larger than 1), despite the fact that the observed light curves are almost pure noise. The origin of this problem is that this correction applies only in the limit of an infinitely sampled, infinite light curve. This is demonstrated by writing ρx,x explicitly in the case of auto-correlation. Assuming for simplicity that the bin contains only a single time-lag τ, the correlation coefficient over a continuous finite segment of the light curve is
ρx,x(τ)=C±a,a−C±x,n−C±n,x−C±n,n√(V−a−V−n−2C−x,n)(V+a−V+n−2C+x,n). (22)
The errors are assumed to be uncorrelated with the signal and with themselves, and therefore in the limit of an infinite light curve,
C±x,n(τ)=C±n,x(τ)=C+x,n(τ)=C−x,n(τ)=0, (23)
and for τ≠0,
C±n,n(τ)=0 (24)
also holds. r′x,x approaches ρx,x only in this limit. Moreover, even in cases where this is an adequate approximation, it is still unclear whether this estimator has the required statistical properties (i.e. consistency, efficiency and lack of bias).
I am grateful to Rick Edelson and Julian Krolik for comprehensive discussions of issues relating to the DCF and ZDCF. Stimulating conversations with Shai Kaspi, Shai Zucker and Eyal Neistein are much appreciated. This work was partially supported by the US-Israel Binational Science Foundation grant no. 89-00179.
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Kate Bollinger is a folk musician from Virginia signed to the label Ghostly International. She recently the seal of approval from Atwood Magazine after receiving the editor's pick on one of her tunes, and has also served as an inspiration to Kanye West, who sampled one of her songs on his album Donda.
"J'aime les filles" (translation: "I love girls") was written by Jacques Dutronc, and was released as a single in 1967. Dutronc is known for his contributions to French rock, particularly French psychedelic-rock, and Yé-yé, a Beatles-y genre. The tune details all the girls the character loves, including Castel, Regine, and girls in magazines. The original is jazzy with lots of piano, vocal scoops, and snare.
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J'aime les filles by Kate Bollinger
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News Tagged with: Jacques Dutronc, Kate Bollinger
2 Responses to "Kate Bollinger Covers '60s French Tune "J'aime les filles""
Noel in Atlanta says:
Well, that was just delightful!
If French rock fans have always appreciated Jacques Dutronc, and even today, his music has never really been part of "French rock", and even less of "psychedelic-rock"!
As for the composition of J'aime les Filles, Dutronc wrote the music, but for the text it is his faithful and inimitable lyricist, Jacques Lanzmann. By the time, the song remained three weeks number 1 in the charts.
Scary Pockets Transform "Purple Haze" Into a Funk Jam Admiral Releases Sedate Folk "Pleasant Valley Sunday" off Monkees Covers Album


Mathew Ryan gave more away after the match than Albion did throughout their FA Cup semi-final examination against the team described by Chris Hughton as "arguably the best in the world".
The Australian keeper, maintaining his habit of spending time with supporters and handing out mementos, eventually walked off at Wembley wearing just his shorts.
Albion could have been similarly exposed after conceding what turned out to be the only goal by Gabriel Jesus after just four minutes.
They were not because, as Pep Guardiola observed, they are "incredible defensively", a remark which can also be applied to the Spaniard's quadruple pursuers.
City are feted for their attacking talents, less so for their work without the ball. Like all great sides, they have both aspects of the game in their armoury.
They have only conceded 21 goals in the Premier League this season, one more than title rivals Liverpool, 13 fewer than the next-best defences of Spurs and Chelsea.
There are occasions when it has been valid to question a toothless-looking Albion attack. This was not one of them.
Hughton said: "It's the two sides of the game. What we have to remember about the top teams like Man City is that they have such incredibly wonderful offensive players but also how good they are defensively. What good athletes they are, the big distances they cover.
"When we worked to the halfway line there are two or three players around the ball. They work incredibly hard.
"When the best teams play the best they are good enough to work out situations. When they are being pressed they have the quality to work it out.
"It's difficult for us. In the areas where we were being pressed, with the quality we've got, it's hard to retain possession. That would be no different to other teams not in the top six."
This assessment of the reality of facing City was overlooked by Jermaine Jenas, who informed the nation on BBC's live coverage that Albion had "missed an opportunity" because City were so "poor and sloppy in certain parts of their game".
It is worth noting there was not a defender amongst the BBC studio panel. If there had been, the inquest might have been more balanced. It was not a pulse-racer but, for the purist, the contest was a lesson in the undervalued art of defending by both teams and the expertise of Guardiola.
Albion, Guardiola admitted, had City worried for a period at the start of the second half. His response, bringing on Fernandinho for goal-provider Kevin De Bruyne, settled the hot favourites down again.
Hughton said: "He is an outstanding manager. When you watch him on the touchline, it doesn't matter whether he is playing against a top team or a team of our level. he is animated, very passionate. He wants to win and you can tell that by everything he does and he makes good enough decisions.
"We had a bit of momentum, certainly in the second period, albeit mostly from set plays and wide free-kicks which we have been very good at. It looked like we might get something, but it's very hard. They'll make the changes that they need to."
It was not for the want of trying on Hughton's behalf. He brought on Florin Andone, Jose Izquierdo and Jurgen Locadia in the final quarter of the game, the latter's introduction prompting a switch to 4-4-2.
Both teams were mean in conceding chances. Most people would have predicted a landslide when City pounced so early. Albion did extraordinarily well from there to be the only side apart from Saturday's visitors Bournemouth to stop them scoring more than once in their last 12 Premier League, Champions League and FA Cup encounters.
Ryan was so well protected that he was only seriously called upon twice, denying Raheem Sterling with a diving save from his second half shot and then from a close range volley on the break in stoppage time as Albion chased improbable parity.
Ryan's opposite number Ederson was required even less, a shot from Izquierdo in the Colombian's quest for sharpness after severe injury setbacks giving the Brazilian a routine catch.
Aymeric Laporte, bought from Athletic Bilbao for more than Albion's entire summer spending, epitomised the high class in Guardiola's ranks.
It was the Frenchman's cross field diagonal that launched City's stunning decider. A cushioned volley inside by Bernardo Silva, a wonderful cross bent in by De Bruyne for Jesus, arriving behind Shane Duffy, to convert with a sprawling header as the ball bounced with pinpoint accuracy into his path.
Jesus is some deputy for the injured Sergio Aguero.
Hughton said: "With any goal you concede you can always pick out somebody who could have dealt with it better, but ultimately they have such quality that it is very difficult to stop those moments."
Albion had their moments as well, two of them in the 'if only' category. Laporte's outstretched boot under his own crossbar prevented Glenn Murray hooking in Duffy's header from the outstanding Anthony Knockaert's corner.
The only defender that let himself down was Kyle Walker. The England full-back's touching of heads with Alireza Jahanbakhsh after the Iranian caught him could easily have resulted in a red card from Anthony Taylor, rather than the yellow confirmed by lenient VAR Paul Tierney.
The difference between a very similar incident when Richarlison was sent-off at Bournemouth for Everton earlier in the season was that Adam Smith reacted as if hurt. Jahanbakhsh, a gentleman like his manager, did not.
Walker was replaced at the interval, Guardiola said due to injury rather than the risk of dismissal.
It is hard enough as it is when you are such huge underdogs against City. Albion needed the marginal moments like that to go in their favour, but if they reproduce anything like this level of performance in the next five matches within a fortnight they will be confronting City again next season with plenty to spare.


The ICC will grant all matches in multi-team one-day tournaments ODI status when they include teams with and without international status, beginning with the Asia Cup in the UAE this month, amid a wider review of the application of status in one-day cricket. The decision will also extend to World Cup Qualifiers in the future.
Hong Kong, who don't hold ODI status, booked their place in the Asia Cup by beating UAE in the qualifying final and it raised the potential of some matches in the event being ODIs and others not. Hong Kong is the only team among the six participating teams without an ODI status.
Dave Richardson, the ICC chief executive, also confirmed a review of ODI status is being undertaken with an outcome expected in the next few months. Earlier this year, all T20 matches between member countries across the men's and women's game were awarded T20I status.
"This decision was informed by our review of the ICC Cricket World Cup Qualifier earlier this year where there was a mix of games with and without international status which proved to be confusing for fans," Richardson said.
"Of course all T20 matches between ICC Members now have T20I status and we are in the midst of reviewing the whole issue of ODI status, which review should be completed in the next few months."


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Home / COP21
The 2015 United Nations Climate Change Conference
To achieve a legally binding and universal agreement on climate,
from all the nations of the world.
This is an exciting event globally, and to the students, faculty and alumni of GMC, it is a momentous occasion to be tracked closely, discussed at length and wrapped into our ongoing studies! Here we'll keep updated with the latest information on talks as they unfold. Our correspondent on the ground in Paris is Adjunct Professor and '10 Alumna Mindy Blank, attending the talks on behalf of HELIO International, where she works as a Project Coordinator.
An Agreement has been Reached!
Francois Mori/AP
Check out the Coverage here.
Dec 11: A Live Update from our Correspondent
Adjunct professor Mindy Blank takes a moment outside the conference to update us on negotiations as they progress.
On the Ground Gallery
Adjunct professor Mindy Blank is sending us photos as Week 2 of Cop21 unfolds. She's our GMC Alumna ('10) and Professor behind the scenes!
Dec 9: New Draft, Steady Progress
The new Draft Paris Outcome text was released on Wednesday, giving Parties an overview of the progress made so far, a starting point for proposing compromise, and outlining open questions that still need more work to be done (without intending to prejudge any conclusions on the contentious issues needing political decision). It is still fully open for negotiation, and UNFCC Executive Secretary, Christiana Figueres, stressed today that this should not be seen as the final text of the agreement. Compared to the previous version from Saturday, the number of pages has been reduced from 43 (21 draft agreement + 22 draft decision) to 29 (14 draft agreement + 15 draft decision). The number of pairs of square brackets in the text has been reduced from 940 to 367! As predicted by many, the topics still in need of much work are Mitigation and Finance, which are currently heavily bracketed, messy pages.
Dec 7-8: Sustainable Innovation Forum
At the 6th annual Sustainable Innovation Forum, focus was on trillion dollar investment opportunities that would allow countries to transition towards sustainable energy sources and away from fossil fuels while simultaneously boosting their economies. More than 750 delegates and over 80 speakers from around the world attended the event to discuss how their respective countries can work together to create clean investment opportunities for the future. Among the many innovations being discussed, there are a handful that are particularly noteworthy:
~ CarbonWealth announced a profitable method of removing carbon from the atmosphere to store it in the earth and convert the remainder into fuel.
~ Impossible Foods announced the development of a burger made entirely from plants that looks and tastes like meat (aptly named, the "Impossible Burger") to combat the detrimental effects of large-scale meat production.
~ Autodesk discussed how their cutting-edge software can be used to design comprehensive 3D models of cities that take into account the many factors needed to mitigate the effects of climate change, including greenhouse gas emissions, energy use, and carbon prices.
~ BMW discussed ways their vehicles can help reduce emissions, both through electric powered engines and their car-sharing service, DriveNow.
While technology will play a large role in the discussions, the conference will also showcase countries that are becoming leaders in green innovation, such as Morocco, Iceland, and the U.S., as well as particular organizations that are leading the movement towards a green economy, including The European Investment Bank and Inter-American Development Bank (IDB), the Carbon Trade Exchange (CTX), and The Carbon Tracker Initiative. The delegates will also discuss how the growth of such an economy can be sustained, and how potential problems with the transition can be avoided. The Sustainable Innovation Forum will present a key opportunity for business leaders, entrepreneurs, and politicians to demonstrate how technology and global networking can effectively catalyze economic growth without exacerbating the effects of climate change. Eben Schumacher, Environmental Ethics
Dec 7: In Search of an Energy Transformation
World-wide, governments, businesses, and cities are making strong commitments to improve energy efficiency and invest in renewable technologies. This day at COP21, three underlying goals were announced:
~ Utilize all available technologies and push forward new research and development
~ Increase ambition to change among all actors in all regions of the world
~ Mobilize the funds needed to enable the transition
Dec 5: Coffee and Agro-ecology Innovation
On December 5, 2015 at Palais de Congres de Paris, Humanist Institution for Co-operation with Developing Countries (HIVOS) and International Coffee Organization (ICO) participated "in an event that looks at the role of agro-ecology for helping smallholder coffee farmers becoming more resilient to the effects of climate change." (goldstandard.org) The ICO is the main intergovernmental organization for coffee, tackling the challenges facing the world coffee sector through international cooperation. (ico.org) They discussed "the role of agro-ecology in exploring innovative, viable adaptation measures for resilient smallholder coffee landscapes." Coffee is an important commodity that small growers in developing countries depend on for income. "Over 90% of coffee production takes place in developing countries with 25 million small producers relying on coffee for a living worldwide." (The Latte Revolution) Unroasted coffee is one of the most traded agriculture commodities in the world and developing countries need help in learning modern faming techniques to produce the most products while also being sustainable and taking care of the soil. Jamie Perlstein, Environmental Ethics See More: http://www.ico.org http://unfccc.int/meetings/paris_nov_2015/meeting/8926/php/view/seors.php http://us2.campaign-archive1.com/?u=710e1af2e5aa5a1629e0baa2c&id=01c8246ddf&e=9e84a4f7f3
Dec 4: Making it easier to #Divest
Have you heard of Portfolio Decarbonization? Your local banker has – and it's making #divestment from fossil fuel giants not only the right thing to do, but a feasible option as well. On December 2nd, gofossilfree.com announced that more than 500 institutions representing over $3.4 trillion in assets have made some form of divestment commitment.
Read More from the LPAA »
New on the #LPAA website: "Divest-Invest" movement, accelerating clean energy: https://t.co/GPTMtMbaK9 #LPAA pic.twitter.com/0TEtUoUpHI
— LPAA Live (@LPAA_Live) November 25, 2015
Learn More about GoFossilFree.org »
Big news -->Divestment Commitments Pass the $3.4 trillion Mark at #COP21 https://t.co/Co7UsoV2Mb #DivestForParis pic.twitter.com/1BIIHh6pK7
— 350 dot org (@350) December 2, 2015
Dec 3: How we get around, why it's hurting us and what we'll do about it
Recognizing that transportation has the highest growth of CO2 emissions of any industrial sector, transport representatives present 10 initiatives at COP21 in Paris today that will reduce emissions and build more sustainable transport systems.
~ 65 countries committed to improve vehicles efficiency, a high impact opportunity!
~ Companies, Cities, National Governments, and Associations converge to commit to electro-mobility and call for others to join
~ Sustainable Urban Mobility Plans in 100 fast-growing cities and 20 countries
~ 50 partners embark on transforming freight, the fastest growing contributor to transport emissions.
~ New efforts in the initiatives announced at the New York Climate Summit on aviation, public transport and rail
The Art of Progress
From The Creator's Project:
For a project called 1 Heart 1 Tree, artist, architect and projection mapping pioneer Naziha Mestaoui is planning a benevolent takeover of the city's major landmarks to send the UN a strong message about sustainability and deforestation in modern policy. "The aim is to reconnect us with ourselves, others and our environment and give the power to anyone to act to change the world," she tells The Creators Project.
Dec 2: New International Initiatives Increase Resilience Protecting Millions of People
Fast action is needed to protect those most vulnerable from climate impacts; to that end, major international partnerships under the Lima to Paris Action Agenda are mobilizing large-scale financing, some one billion dollars, to cover the full spread of peoples´ needs as they face increasing climate impacts. Solving for one issue while ignoring others has not worked, and resilience needs to be approached in a comprehensive fashion in order to be effective!
So, let's get comprehensive – the initiatives set forth by the LPAA include:
~ Early warning systems for over 50 least developed countries and small island states
~ Providing access to insurance to 400 million vulnerable people in 5 years
~ Increase resilience for local communities in the Sahara and Sahel
~ US$ 150 million partnership looks to mobilize more funding for Africa and Asia
~ EU mobilizes €125 million for countries affected by 'El Niño'
~ UN initiative strengthens ability to anticipate, absorb and "reshape" climate impacts
We could save some money spent coping with weather disasters if we invest in resilience measures now. #COP21 #LPAA pic.twitter.com/OeWSpGscLx — Global Goals (@GlobalGoalsUN) November 25, 2015
Dec 1: Global Agricultural Revolution?
At the LPAA Focus on Agriculture, Governments and food and agriculture organizations met to discuss the future of agriculture, and agreed upon six major initiatives that would support farmers through impending Climate Change impacts. From the Press Release:
The "4/1000 Initiative: Soils for Food Security and Climate": A small increase of 4/1000 per year of the soil carbon stock is a major leverage in order to improve soil fertility, resilience of farmers and contribute to the long-term objective of keeping the global average temperature increase below 2 degrees. Read More »
Live Beef Carbon: Promoting innovative livestock farming systems and associated practices will ensure the technical, economic, environmental and social sustainability of beef farms, and thus to reduce the beef carbon footprint by 15% over 10 years in France, Ireland, Italy and Spain. Read More »
Adaptation for Smallholder Agriculture Programme (ASAP): Partners commit to investing climate finance in poor smallholder farmers in developing countries and, by 2034, funding through this initiative will avoid or sequester 80 million tons of GHG emissions (CO2e) and strengthen the resilience of 8 million smallholders.
15 West-African Countries Transitioning to Agro-ecology: This impact-full initiative delivers both adaptation and emission mitigation benefit, and will allow the adoption of agro-ecological practices by 25 million households by 2025. Read More »
The Blue Growth Initiative (BGI): These actions aim at a 10% reduction in carbon emissions from fishery value chains in 10 target countries within 5 years (and 25% within 10 years), and the reduction of overfishing by 20% in the target countries within 5 years (50% within 10 years). Read More »
The SAVE FOOD Initiative: Altogether, this initiative should allow a major reduction in agriculture emissions, as global food waste and loss account for 3.3Gt of CO2 equivalent per year. Read More »
The Consumer Goods Forum: The Consumer Goods Forum, an organization of the global retail industry's leaders, pledged to achieve zero net deforestation by 2020, and are also committed to reduce food waste within their value chain and make consumers aware of this issue. Read More »
Global agricultural revolution under way to boost resilience, reduce emissions https://t.co/E2aXw7AHoo #COP21 #LPAA pic.twitter.com/4nIDKYd5N5 — UN Climate Action (@UNFCCC) December 1, 2015
Hanson @WorldResources to join Save Food Initiative – tackling food loss & waste across value chain #LPAA #COP21 pic.twitter.com/XtIkJbwC2e — LPAA Live (@LPAA_Live) December 1, 2015
Nov 30: $248 USD million pledged to GEF climate fund for most vulnerable countries
During the opening day of the #COP21 talks, Canada, Denmark, Finland, France, Germany, Ireland, Italy, Sweden, Switzerland, United Kingdom, and the United States of America announced their intended contributions to the Least Developed Countries Fund (LDCF). In his statement, President Barack Obama spoke about the most vulnerable countries bearing the most impact from global climate change. "For some, particularly island nations […], climate change is a threat to their very existence. That's why today, in concert with other nations, America confirms our strong and ongoing commitment to the Least Developed Countries Fund. And tomorrow, we'll pledge new contributions to risk insurance initiatives that help vulnerable populations rebuild stronger after climate related disasters". Continue reading the press release here.
Nov 30: The COP21 Conference Officially Opens
"That future is one that we have the power to change. Right here. Right now. But only if we rise to this moment. As one of America's governors has said, "We are the first generation to feel the impact of climate change, and the last generation that can do something about it." With bold statements by President Obama, the talks have officially begun in Paris. Read the full text of statements made during the Leaders Event at the Paris Climate Change Conference here, and get inspired!
Addressing climate change takes all of us, especially the private sector going all-in on clean energy worldwide. https://t.co/qQ6wCZUR1Z — President Obama (@POTUS) November 30, 2015


A new online virtual equipment finder for the processing industry has been launched, allowing manufacturing professionals to quickly and easily identify the most suitable separation solution for their factory and application.
The Russell Finex Interactive Application Finder guides users through various areas of a factory and different production stages, and suggests the screening and/or filtration solution that best matches their process.
The layout of the online tool is clear and easy to use, helping steer users quickly to the most appropriate range of separation equipment. Users are initially presented with a factory divided into three sections: raw material handling, primary and secondary manufacturing processes and end-of-line packaging. Once a section is selected, the finder presents various processes and suggests specific equipment solutions, which link directly to more detailed information if the user wants to find out more.
This new interactive processing equipment finder is the first in its industry and is aimed at educating manufacturing professionals, providing the best application, and potentially easing the machine selection and buying process.
"At Russell Finex, we strive to offer the best customer experiences and support. Our new online Interactive Application Finder was formed to help educate our customers and connect Russell Finex equipment with their manufacturing production lines," states Richard Kay, Business Development Manager. "Although this tool is specific, it is general enough to apply to all business, no matter what industry you are in," concludes Kay.
Russell Finex has been supplying separation solutions for over 80 years and has the widest range of products on the market. As a result, this easy to use tool helps navigate users to the right product for the right application quickly and easily. Try the virtual tool today, or contact Russell Finex for more information.


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spectrumauction2016
Study moots rationalisation of taxes, levies to boost stability of debt-laden telecom sector
Simplification and rationalisation of multiple taxes and levies such as spectrum usage charge (SUC) will provide financial stability to India's debt-laden telecom industry which is splurging big money on network rollouts and spectrum purchaseET Bureau | September 28, 2016, 15:44 IST
KOLKATA: Simplification and rationalisation of multiple taxes and levies such as spectrum usage charge (SUC) will provide financial stability to India's debt-laden telecom industry which is splurging big money on network rollouts and spectrum purchase, an Assocham-KPMG joint study said Wednesday.
The rationalisation call, the study said, is reinforced by the fact that telcos reeling under a Rs 3.8 lakh-crore debt burden have been lately hit by a tariff war unleashed with the entry of a major player.
The Assocham-KPMG study does not name Reliance Jio Infocomm but appears to be alluding to the company, which is the sole major player to have recently entered the telecom turf.
Simplification of taxes and levies, it said, would also be in line with the sector regulator's twin suggestions of a reduction in licence fees and Universal Service Obligation Fund (USOF) charges from 8% and 5% to 6% and 3% respectively.
In an official statement, Mr P Balaji, Chairman of Assocham's National Council on Telecommunications, and also Vodafone India's director-regulatory, external affairs & CSR, said the chamber is confident the "government would support" this endeavour.
The joint study builds a case for rationalisation of taxes/levies since telecoms is one of the top contributors to GDP, likely to bring in Rs 14 lakh-crore by 2020.
The telecom sector, it said, is stung by a slew of taxes and other charges, including corporate tax and VAT, adding that SUC rates too in India were higher than other countries.
A quick resolution of these issues would also facilitate ease of doing business and help the telecom industry realise the government's vision on Digital India.
Here are some interesting facts about Indian telecom companies
Tags : Industry, SUC, spectrumauction2016, Reliance Jio, Assocham


My Bookings table has merely 6000 records yet the first query takes approx ~10 seconds which is outrageous. Luckily, we can optimize this using late lookups like in query 2.
In the second query, we select id from Bookings and then join the original table back. This will make each individual row lookup less efficient but the total number of lookups will be reduced by a lot.
Also, if you could pass more condition into the select, it will greatly improve the performance as well. So instead of making your paging url like this example.com/products?page=10, you can use example.com/products?page=10&last_seen=1023. From that, you can pass WHERE id > 1023 into the pagination query making the whole thing a lot faster.
Those are what I used for optimizing pagination. If you know any other, please let me know.


In a statement released last week, Abu Muhammad al Maqdisi, an influential jihadi ideologue aligned with al Qaeda, criticized Turkey's cooperation with Russia against the Islamic State. Maqdisi warned jihadists and Islamists in Syria to rethink their decision to work with Turkey's Operation Euphrates Shield, which has captured significant territory from the so-called caliphate in northern Syria.
The US killed Rifai Ahmed Taha Musa in an airstrike this week. Musa first began working with Osama bin Laden and Ayman al Zawahiri in the 1980s and 1990s. US intelligence officials identified him as a key figure in al Qaeda's so-called "Khorasan Group." According to one of his former comrades, he was in Syria to promote unity between Al Nusrah Front and Ahrar al Sham.


Našli jsme další záznamy k osobě William Clark 'Bill' Broadbent.
Poloha hrobu osoby William Clark "Bill"
William Clark 'Bill' Broadbent je pohřben(a) na hřbitově Leavitt Cemetery v místě zobrazeném na níže uvedené mapě. Tyto informace o GPS jsou k dispozici POUZE na stránkách BillionGraves. Naše technologie vám pomůže najít umístění hrobu a také další členy rodiny, pohřbené poblíž.
William Clark 'Bill' Broadbent was 15 years old when Disneyland Hotel opens to the public in Anaheim, California. The Disneyland Hotel is a resort hotel located at the Disneyland Resort in Anaheim, California, owned by the Walt Disney Company and operated through its Parks, Experiences and Consumer Products division. Opened on October 5, 1955, as a motor inn owned and operated by Jack Wrather under an agreement with Walt Disney, the hotel was the first to officially bear the Disney name. Under Wrather's ownership, the hotel underwent several expansions and renovations over the years before being acquired by Disney in 1988. The hotel was downsized to its present capacity in 1999 as part of the Disneyland Resort expansion.
William Clark 'Bill' Broadbent was 24 years old when The Beatles make their first appearance on The Ed Sullivan Show, performing before a "record-busting" audience of 73 million viewers across the USA. The Beatles were an English rock band formed in Liverpool in 1960. With members John Lennon, Paul McCartney, George Harrison and Ringo Starr, they became widely regarded as the foremost and most influential music band in history. Rooted in skiffle, beat and 1950s rock and roll, the Beatles later experimented with several musical styles, ranging from pop ballads and Indian music to psychedelia and hard rock, often incorporating classical elements and unconventional recording techniques in innovative ways. In 1963, their enormous popularity first emerged as "Beatlemania"; as the group's music grew in sophistication, led by primary songwriters Lennon and McCartney, the band were integral to pop music's evolution into an art form and to the development of the counterculture of the 1960s.
William Clark 'Bill' Broadbent was 32 years old when Munich massacre: Nine Israeli athletes die (along with a German policeman) at the hands of the Palestinian "Black September" terrorist group after being taken hostage at the Munich Olympic Games. Two other Israeli athletes were slain in the initial attack the previous day. The Munich massacre was an attack during the 1972 Summer Olympics in Munich, West Germany, in which the Palestinian terrorist group Black September took eleven Israeli Olympic team members hostage and killed them along with a West German police officer.
William Clark 'Bill' Broadbent was 49 years old when The tanker Exxon Valdez spilled 10.8 million US gallons (260,000 bbl; 41,000 m3) of oil into Prince William Sound, Alaska, causing one of the most devastating man-made maritime environmental disasters. A tanker is a ship designed to transport or store liquids or gases in bulk. Major types of tankship include the oil tanker, the chemical tanker, and gas carrier. Tankers also carry commodities such as vegetable oils, molasses and wine. In the United States Navy and Military Sealift Command, a tanker used to refuel other ships is called an oiler but many other navies use the terms tanker and replenishment tanker.
William Clark 'Bill' Broadbent was 59 years old when Columbine High School massacre: Eric Harris and Dylan Klebold killed 13 people and injured 24 others before committing suicide at Columbine High School in Columbine, Colorado. The Columbine High School massacre was a school shooting that occurred on April 20, 1999, at Columbine High School in Columbine, an unincorporated area of Jefferson County, Colorado, United States, in the Denver metropolitan area. In addition to the shootings, the complex and highly planned attack involved a fire bomb to divert firefighters, propane tanks converted to bombs placed in the cafeteria, 99 explosive devices, and car bombs. The perpetrators, senior students Eric Harris and Dylan Klebold, murdered 12 students and one teacher. They injured 21 additional people, and three more were injured while attempting to escape the school. The pair subsequently committed suicide.
William Clark 'Bill' Broadbent was 64 years old when Facebook, a mainstream online social networking site, is founded by Mark Zuckerberg. Facebook is an American online social media and social networking service company based in Menlo Park, California. Its website was launched on February 4, 2004, by Mark Zuckerberg, along with fellow Harvard College students and roommates Eduardo Saverin, Andrew McCollum, Dustin Moskovitz and Chris Hughes.


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