Theory suggests that individual behavioral responses impact the spread of flu-like illnesses, but this has been difficult to empirically characterize. Social distancing is an important component of behavioral response, though analyses have been limited by a lack of behavioral data. Our objective is to use media data to characterize social distancing behavior in order to empirically inform explanatory and predictive epidemiological models.We use data on variation in home television viewing as a proxy for variation in time spent in the home and, by extension, contact. This behavioral proxy is imperfect but appealing since information on a rich and representative sample is collected using consistent techniques across time and most major cities. We study the April-May 2009 outbreak of A/H1N1 in Central Mexico and examine the dynamic behavioral response in aggregate and contrast the observed patterns of various demographic subgroups. We develop and calibrate a dynamic behavioral model of disease transmission informed by the proxy data on daily variation in contact rates and compare it to a standard (non-adaptive) model and a fixed effects model that crudely captures behavior.We find that after a demonstrable initial behavioral response  at the onset of the outbreak, there was attenuation in the response before the conclusion of the public health intervention. We find substantial differences in the behavioral response across age subgroups and socioeconomic levels. We also find that the dynamic behavioral and fixed effects transmission models better account for variation in new confirmed cases, generate more stable estimates of the baseline rate of transmission over time and predict the number of new cases over a short horizon with substantially less error.Results suggest that A/H1N1 had an innate transmission potential greater than previously thought but this was masked by behavioral responses. Observed differences in behavioral response across demographic groups indicate a potential benefit from targeting social distancing outreach efforts. The series of flu-like outbreaks over the past decade illustrates the ongoing need for refinement of strategies to control and mitigate the impact of infectious diseases, including SARS in 2003 , the 2000.Empirical analysis of the effect of social distancing behavior on epidemiological dynamics is of clear interest, but it has proven difficult to obtain representative data on actual behavioral responses to epidemics. Empirical investigation of the influence of behavior on flu-like transmission dynamics has been largely limited to binary proxies for behavior, specifically pre-scheduled ,15 and eWe use novel data on variation in home television viewing behavior as a proxy for changes in the level of daily social interaction. We find a strong viewing behavior response in Central Mexico associated with the A/H1N1 influenza virus in April and May of 2009. The data reveal that proxy behavioral responses were greatest among children and wealthier socio-economic groups. Furthermore, we couple the behavioral response with an epidemiological model, and show that the A/H1N1 influenza virus was likely more transmissible than previously believed because the transmission potential was masked by behavioral responses.To leverage the television viewing data for exploring the role of behavior during an epidemic, we extend the binary proxy for time varying infectivity in , where bWe contribute to the literature by examining variation in the behavioral response across time and demographic subgroups and by calibrating and analyzing dynamic behavioral disease transmission models. First, we quantify the dynamic nature of the behavioral response to the 2009 A/H1N1 influenza pandemic and public intervention in Central Mexico. We show that the aggregate response is not constant and describe how it varies systematically over time. Next, we unpack the aggregate dynamic into demographic subgroups and show how certain age groups and/or socio-economic groups respond more strongly than others. Turning to the modeling of disease transmission dynamics, we assess whether accounting for daily changes in contacts better accounts for the variation in new cases. We then explore the potential for bias in the standard model from ignoring underlying changes in behavior. Previous simulation analysis has shown that intervention focused on children is particularly effective in reducing the attack rate of influenza . We examt), exposed (infected but not yet infectious), (Et), infectious (It), and recovered (Rt). The transition dynamics between health states are described by a system of difference equations:βt is the transmission rate, κ is the rate at which incubating individuals progress from the exposed to the infectious health status (or the inverse of the latent period) and γ is the recovery rate (or the inverse of the recovery period).We model the 2009 A/H1N1 epidemic in Central Mexico using an SEIR epidemiological model -24. We dt is becomes a constant scalar. This confounds the combined effect of contacts and the probability of transmission from a contact  for the SD model and the vector  for the FE and DB models. Model parameters were estimated by maximum likelihood. We assumed that the observed number of confirmed new infections each day, λt(ρ) given by the number of new observed infections predicted by the disease model, ϕtκEt. The log-likelihood function is:The main coefficients of interest for estimation are the parameters of the transmission rate functions for each of the three models. Let ρ represent the vector of marginal transmission rate parameters, given by the scalar [ρDevelopment of the log-likelihood function is explained in further detail in Appendix C.Because maximum likelihood estimates can be sensitive to the choice of initial values provided to the numerical optimization algorithm, we used a multiple starting point solver in Matlab (version R2013a) designed to identify the global optimum. For each model, the solver was run for each of M different randomly drawn starting vectors for the unknown parameters in ρ. We set M equal to 50 for the standard model (one parameter) and 100 for the alternative models (two parameters). From this set of local maxima, the solution with the greatest likelihood was selected as the estimate for the global maximum. We estimated 95% confidence intervals for the parameters using the likelihood ratio method . To testΔt (percentage deviation from mean ATV) in Mexico City during April and May 2009 in aggregate  indicates that an above average amount of time was spent in home TV viewing and, by inference, in the home. The mean level of Δt over the period τ is positive and, as shown by a one-sample t-test, significantly different from zero at the 1% level for the aggregate population and each subgroup considered here  before gradually tapering off to near zero by the end of the intervention period. This pattern suggests that the population’s capacity for social distancing might be limited in duration; before the public health intervention concluded, there was a substantial decline in the behavioral response relative to the peak in the first week.  After the NPI concluded there was a period of reduced viewing activity in the home (Δt < 0). Specifically, Δt reached its most negative point on May 10th at −10.5%. Outside of the post-intervention dip, Δt dropped below −10% on only one other day. As further evidence that the dip was likely not a coincident random event, we find that this dip persisted at 5% below the non-NPI period mean for four consecutive days—there are no other instances in the data when Δt falls below 95% of the mean for more than a single day. While the causal mechanism behind these dynamics is not known with certainty, one possibility is that this multi-day period of suppressed in-home activity compensated for forgone social and commercial activities from earlier in the intervention period. The observation of reallocation of risky activities in time is common in the public health literature. Following the introduction of antiretroviral treatment for HIV/AIDS [The dynamic path of HIV/AIDS ,36 find HIV/AIDS  find thaΔt presented in Figure t-test.The age class breakdown for The data from IBOPE are disaggregated into three socioeconomic levels (SELs) based on a set of household characteristics, including the size and amenities of the home, appliance ownership, automobile ownership, and level of education Figure C. DuringFinally, we consider variation in the response by time of day, specifically daytime (6 am-6 pm) versus nighttime (6 pm-6 am) Figure D. The meThe time path for each of the subgroups discussed above follows a path that is qualitatively similar to that of the aggregate population, showing a strong initial positive response that largely or entirely decays before the end of the intervention. For each subgroup comparison considered here, there was a significant difference in the average level of the behavioral response.The maximum likelihood parameter estimates for each model are based on T = 41 days of observations, stretching from April 1 through the end of the intervention period on May 11 Table . Figuresentclass1pt{minimaΔt presented in Figure Δt begins to surge on April 24th (day 24) the number of new confirmed cases plateaus. However, when Δt declines in early May while infections are still common, the number of new confirmed cases The degree to which accounting for changes in contacts better accounts for the variation in new cases is one of our core research questions. Results show that the standard model is indeed incomplete—we reject the SD model in favor of both the DB model (p < 0.01) and FE model (p < 0.01). However, we do not find that the DB model outperforms the FE model. In fact we reject the DB model in favor of the FE model (p < 0.01). To see why it might be the case that a simple fixed effect is preferred in this case to the dynamic, data-driven behavioral model, consider the time series for entclass1pt{minima0) in Table th. In Figure 0. We find that estimates are variable but roughly consistent across models through April 24th. This is not surprising given that before the public health intervention began on April 24th our proxy suggests that behavior had yet to shift discernibly. After this point, estimates of ρ0 for the DB and FE models remain roughly stable near 0.064 while the baseline transmission coefficient for the SD model declines monotonically. Thus over the intervention period when behavioral response is strong, the SD estimate of ρ0 falls each day to account for the new factor. In contrast, models that allow for a behavioral shift result in estimates for baseline transmission that are essentially level over time.Given that both the FE and DB models outperform the SD model, we explored the potential for biased estimates of the transmission parameter in the SD model as a potential shortcoming of ignoring behavioral change. Estimates of the baseline transmission rate  is available across this four day lag. From the raw forecast error in Figure In Figure Δt) for children versus adults presented above motivate exploration of age-class heterogeneity. When we modeled children and adults as separate populations , but constrained transmission parameters to be the same for both populations, estimates were not significantly changed. We further tested an extended model in which transmission parameters  were free to vary between the two groups. This model was not statistically significantly different for either the SD (p = 0.31), DB model (p = 0.41), or FE model (p = 0.12) at the 10% level. For this FE model, relative to the homogeneous (baseline) case, the coefficients ρ0 and ρ1 were roughly 50% larger in magnitude for children and 90% smaller in magnitude for adults. This evidence is not conclusive, but hints that infections between children and from children to adults might have been a leading driver of disease dynamics—and also most sensitive to intervention. However, this effect is too small and imprecisely estimated to assert with statistical significance.We estimated the transmission model results above assuming a single homogenous population. However, differences in the behavioral response .th through our baseline window start date of April 24th. For each of these alternative specifications we found that the associated parameter ρ1 was statistically significantly different from zero. However, we also found that the log-likelihood was greatest for the FE window beginning on April 24th (our baseline specification) illustrating that none of the alternative start dates was statistically preferred.For the FE model, we also considered alternative dates for the start of the intervention window, from April 10The final parameter examined in our sensitivity analysis was the mean probability of confirmation. Our baseline level for 0 term by zero . Under the second alternative, to enhance the behavioral response, we multiply the ρ0 term by two . Fitted curves from the unaltered baseline models  and We explore two alternative scenarios in which the behavioral response to the epidemic is either non-existent or enhanced. We present the path of new confirmed cases under these alternatives, along with fitted curves from the baseline models in Figure g Figure B for botWe used novel data on variation in home television viewing behavior as a proxy for changes in the level of daily social interaction in Central Mexico during the 2009 A/H1N1 influenza pandemic. Results from both behavioral models (FE and DB) suggested that social distancing was a key factor in constraining the initial wave of A/H1N1 in Central Mexico. In the absence of a behavioral response, the estimated counterfactual path of new cases escalated rapidly in initial weeks rather than stabilizing and eventually falling as was observed. The assumption of fixed behavior in the standard (SD) model led to shortcomings in estimation and prediction. Estimates of the baseline rate of transmission systematically shifted over time. If the baseline rate of transmission is interpreted as a measure of biological infectivity in the standard model, this is likely to lead to an underestimate of this parameter, as in our setting, given confounding effects of behavioral responses. This suggests that A/H1N1 had an innate transmission potential much greater than previously thought but this was masked by behavioral responses. This has implications for management advice including the allocation of resources between pharmaceutical and nonpharmaceutical interventions. Furthermore, the error in near term predictions of new cases through time was also substantially greater under the standard model compared to the behavioral models. This error was also systematic—the standard model consistently led to over-prediction in the number of new cases.t was due to a greater share of home time allocated to TV viewing, rather than an increase in time spent at home. Or it could be the case that viewing per unit of time spent at home may be declining in time spent at home. Another explanation might be the inability at this time to empirically capture changes in behavior outside the home to reduce contacts or transmission . Bell [Comparing the behavioral models, we found that that the dynamic behavioral model was not preferred to the simpler fixed effect model. One explanation may be the imperfect nature of variation in viewership as a proxy for changes in public contact rate. For example, it is possible that during the public health intervention the observed increase in ATVr). Bell  notes thWe found that the home viewership response was stronger in the high (versus low) socioeconomic level (SEL) subgroup. This finding is suggestive but should be interpreted with care. On the one hand, individuals in the high SEL subgroup are arguably less constrained in adjusting contacts than those in the low SEL subgroup. For example, Kumar et al.  suggesteIn addition to varied responses across groups, we also found differences over time, namely attenuation in the behavioral response before the conclusion of the public health intervention. Furthermore, we found evidence of a rebound effect in which, after a prolonged period of elevated in-home activity there appeared to be period of suppressed activity. This is consistent with the historical analysis of Caley et al.  who founWhile the dynamic behavioral model based on the home viewership proxy did not out-perform the simple fixed effect model, the results represent progress in identifying and unpacking the drivers behind this fixed effect. Going forward, further detailed data on private and public behavior during outbreaks would serve to identify behavioral effects on transmission with greater precision. For example, we did not model the effect of antiviral treatment. Capturing additional behavioral adjustments made outside of the home to reduce effective contacts is likely be important for explicit modeling of the behavior underlying disease transmission. To this end, there is value in allocating resources during an outbreak to consistently gather data on public and private protective actions, such as antiviral use or the use of face masks. Although transitioning from empirical analysis based on fixed effect measures of behavior to fully dynamic responses at finer time scales will require additional investment in data collection, potential benefits include the promise of informing more finely tuned and less costly public health interventions.


Analysis used real-time data capture software. Observational categories distinguished engagement in social and non-social pursuits, hand stereotypies, self-injury and the receipt of attention from a parent, teacher or carer. Associations between participant behaviour and intake variables and receipt of attention were explored. Concurrent and lagged conditional probabilities between behavioural categories and receipt of attention were calculated. Receipt of adult attention was high. Engagement in activity using the hands was associated with a less severe condition and greater developmental age. Engagement in activity, whether using the hands or not, and social engagement were positively associated with receipt of support. The extent of hand stereotypies varied greatly across participants but was independent of environmental events. Six participants self-injured. There was some evidence that self-injury was related to adult attention. Participants appeared to experience a carer and attention rich environment and their levels of engagement seemed high as a result. As in the more general literature, engagement in activity was related to personal development and to social support. Self-injury contrasted with hand stereotypies in having possible environmental function.The aim was to observe the behaviour of a sample of females with RTT and explore how it was organized in relation to environmental events. Ten participants, all with a less severe form of classic ( MECP2) gene, located on the X chromosome at Xq28  affects almost exclusively females, with an incidence of up to one in every 10,000 live female births. Its cause is most often a mutation in the methyl-CpG binding protein-2 . In a prior stage, a national sample of 91 females with RTT had been recruited by contacting families known to the on-going British Isle Rett Syndrome Survey database . The person undertaking the recording tried to be as discrete as possible during the sessions so as not to intrude on the activities of the participants. Parents and carers/teachers were instructed to interact with the person as normal. On some occasions, observation had to be stopped because the participant had a seizure, was not well or the presence of an extra person in the environment was too disruptive.All videos were coded using OBSWIN software . In comparing the timing of key depressions (onsets and offsets) between observers, those occurring within a tolerance of 5 s were considered agreements. Cohen’s kappa was calculated for each variable under observation. However, as kappa becomes problematically stringent if a behaviour occurs very rarely or nearly all of the time, percentage occurrence agreement and percentage non occurrence agreement were also calculated.Table Percentage occurrence of each category was calculated by dividing its cumulative duration across bouts (onset times minus offset times) by the total session time. Variability in participants’ levels of engagement in activity or social interaction together with the levels of adult attention each received was analysed in relation to participants’ simplified severity scores and Vineland age equivalent scores using Spearman non-parametric correlation.2)2(1/a + 1/b + 1/c + 1/d)] where a, b, c, d are cells of a typical 2 × 2 table. The alpha level for the lag analyses was reduced due to the number of tests performed: a z score above 3.09 (p < .001) was considered as a significant level of association.Variability of participant behaviour in relation to receipt of adult attention was analysed using lag analysis. The conditional probability of behaviour and attention occurring together was calculated by using a zero lag. In addition, the conditional probabilities of participant behaviour occurring up to 100 s before and after the onset of adult attention were calculated by grouping one second intervals into 10s bins using a partial interval rationale and performing 10 lags in both directions. Conditional and unconditional probabilities were then used to calculate Yule’s Q, a simple arithmetic transformation of the odds ratio  and positively related to the Vineland motor skills age equivalent score at school/day centre .The most common activities in which participants were engaged were: watching television, listening to music, listening to a story book and early learning activities involving simple manipulative toys, switches and water. The percentages of time that participants were engaged in constructive activities at home and at school/day centre are set out in Table s = .67, p < .05) and at school/day centre to the level of support (assistance/instruction) received from teacher or carer . Engagement in activity either using or not using the hands, (median 46.7 % of the time at home and 46.6 % of the time at school or day centre) was inversely related to the simplified severity score at school/day centre , positively related to the Vineland motor skills age equivalent score at school/day centre  and positively related to receipt of support in both settings .Engagement that did not involve the use of the hands occupied a median of 27.3 % of the time at home  and 16.0 % of the time at school/day centre . Such engagement was not related to either the severity score or Vineland equivalent age scores but was related at home to the level of adult attention . Only three participants  were engaged in activity for the minority of the time. Social engagement levels were similar between child/adolescent and adult participants. There were no significant differences in the level of non-social engagement between child/adolescent and adult participants either but there was a tendency for levels of engagement using the hands to be higher among adult participants.Social engagement occurred for a median of 10.7 % of the time at home  and a median of 14.3 % of the time at school/day centre  and was not related to either the severity score or Vineland equivalent age scores. At home it was positively related to the level of support received . She spent about two-thirds of the time engaging in hand stereotypies, nearly three-fifths of the time in arm splints and was also one of the participants who self-injured. Her self-injury appeared to be attention-seeking. It occurred significantly less often when she was receiving adult attention and the lag analysis also suggested an attention-seeking motivation.Compared to data from residential settings for individuals with a similar severity of intellectual disability (e.g., Emerson et al. Engagement in activity has been much studied in residential settings and there is fairly consistent evidence that there is a primary association with individual ability as measured by an adaptive behaviour scale and a secondary association with care practices, as measured by the extent of practical support individuals receive or the implementation of ‘active support’ (Felce and Perry Stereotyped hand movements are considered to be characteristic features of RTT and among the essential diagnostic criteria. Despite this, the duration of hand stereotypies observed varied across the sample, from a very limited extent to almost all of the time. Variation appeared to be independent of social context. The correlation between the extents of hand stereotypies and adult attention was insignificant and there were no consistent associations between hand stereotypies and adult attention revealed in the concurrent or sequential lag analyses. This apparent independence from environmental influence is consistent with the findings of Wales et al.  who demoSelf-injury is not an essential diagnostic criterion for RTT but has been shown to occur in a reasonably high proportion. Self-injury was found to occur more frequently among individuals with a severity score in the mild range among the national sample from which this sample was drawn (Cianfaglione et al. Despite the inconsistency of results across participants, the study has highlighted the importance of considering the role of the environment in shaping the behaviour of females with RTT. The participants in this study appeared to experience a carer and attention rich environment and their levels of engagement were likely to be higher than in some other studies as a result. As in the more general literature, engagement in activity was related to personal development and to social support. Hand stereotypies are characteristic, although varying considerably across participants, and appear unrelated to environmental events. Self-injury in RTT appears to occur for reasons similar to that in other individuals with intellectual disability. It was found to be associated with overactivity and impulsivity (Cianfaglione et al.


Heavy hydrogenic atoms like the barium ion are test beds for fundamental physics such as atomic parity violation and they also hold the key to understanding nucleo-synthesis in stars. To draw definitive conclusion about possible physics beyond the standard model by measuring atomic parity violation in the barium ion it is necessary to measure the dipole transition probabilities of low-lying excited states with a precision better than 1%. Furthermore, enhancing our understanding of the barium puzzle in barium stars requires branching fraction data for proper modelling of nucleo-synthesis. Our measurements are the first to provide a direct test of quantum many-body calculations on the barium ion with a precision below one percent and more importantly with no known systematic uncertainties. The unique measurement protocol proposed here can be easily extended to any decay with more than two channels and hence paves the way for measuring the branching fractions of other hydrogenic atoms with no significant systematic uncertainties.A new protocol for measuring the branching fraction of hydrogenic atoms with only statistically limited uncertainty is proposed and demonstrated for the decay of the P In particular, the lifetimes of excited states and their branching fractions upon decay to lower energy levels form the basis for our understanding of the electronic structure of atoms and molecules. In the case of hydrogen, with a single valence electron the electronic structure is exactly solvable. However, for atoms beyond hydrogen in the periodic table, the electronic structure has no exact solution due to the many body nature of the problem. Nevertheless, well developed theoretical many-body techniques make it possible to calculate electronic structure with precision below one percent for heavy atoms with an iso-electronic structure to hydrogen, et al.etc. As ion trap technology has progressed, it has allowed single or multiple ions to be confined in a nearly perturbation free environment over long times thereby eliminating fly-by time related systematics errors. Additionally, short pulsed lasers have allowed the development of new techniques for branching fraction measurements+et al. introduced a technique based on photon counting which is more robust in comparison to previous methods, and importantly, it is only statistically dominated+ branching fractions below one percent1/2 state, where there are only two decay channels. However, direct implementation of the same technique to measure the branching fraction from an excited state with total angular momentum j = 3/2 will be dominated by the systematics due to angular dependence of the observed photons on input laser polarization and magnetic field direction15Methods of measuring the branching fraction of an excited atomic state have evolved over time towards better precision and lower systematic uncertainties. The earliest method is based on a ratio measurement in two branches, obtained directly from fluorescence spectroscopy3/2 state decays into three branches D5/2, D3/2 and S1/2 which will henceforth be referred to as 3, 2 and 1 respectively. The upper two P-states with J = 1/2 and J = 3/2 are referred to as 4 and 5 respectively. In this article, the lasers that excite four possible dipole transitions shown in f of the P3/2 state as it decays into the lower atomic states 1, 2 and 3 are proportional to these states’ decay probabilities which are in turn proportional to the square of their respective transitional dipole matrix elements. These branching fractions are constrained asLet us consider the relevant energy levels of hydrogenic atoms as shown in b), the probability ratio r32 for the atom to be found in the other two levels, 3 and 2 after decaying from 5 is expressed asIf one of the three decay channels is blocked, say for example channel 1–5 by continuous blue laser excitation  channel 3–5, it is possible to measure Thus one-colour-photon measurement of r12 and r32 with low systematic uncertainties. Both ratios depend on the probability of finding the atom in state 2, and the population in this state can be measured without any significant systematic error as it is the lowest excited stateε is determined by counting the total number of green photons c cycles where the subscript green refers to the pumping laser. Thus,The main goal of this protocol is to measure the ratios r32 starts with one or a few ions in their ground electronic state 1. A complete transfer to the D-states (2 and 3) is done by applying the blue laser driving the 1 → 5 transition (pumping step). The red laser is then applied to completely transfer any population from state 2 to 1 via state 4 (re-pumping) and the number of emitted green photons (4 → 1) is counted. This sequence repeated for c cycles provides Nblue green photon counts (where the subscript again refers to the pumping laser). Therefore,As an example, the population transfer sequence for the measurement of r12 can be expressed aswhere the efficiency is cancelled out and the ratio is independent of any external parameters used in the experiment. Similarly, starting from state 5 and applying the orange pump laser for complete population transfer to states 1 and 2, the ratio c cycles while blocking the 5 → 3 decay channel by shining an orange laser light.where, 1/2 level only, the total count is insensitive to input laser polarization, or in other words the Hanle effect does not introduce any systematic error, provided the detection setup is circular polarization insensitive which is usually the case. Experimentally, the influence on photon counts due to the input polarization can be quantified. As the scheme relies on photon counting rather than state detection, population transfer is always complete for any given transition. The experiment is not affected by the number of ions provided the ion number remains the same for each measurement cycle, however it is essential to keep the ions in a crystalline state so that the probed ions do not drift out of the field of view of the photon collection optics. In the following section, systematic uncertainties related to our experiment are further elaborated. Most importantly, this scheme is scalable to any decay with more than two channels, since branching fractions can be measured with reference to the lowest excited state where the efficiency is well calibrated. Therefore it can be easily applied to other heavy atoms like Yb+, Sc++, Ra+ and Hg+.Possible systematic effects that may influence the measured branching fractions in any photon counting protocol come from the dead time of the detector, usually a photomultiplier tube. The dead time introduces a systematically lower count than the actual count if the photomultiplier tube is operated in a non-linear response regime where more than one photon arrives at the detector within its dead time. However, in the scheme presented here, this is not possible due to the technique of projective measurement which means that only one green photon can arrive at the detector per cycle, once it is projected by the red laser unless there are chance coincidences due to background photons of same colour. The number of emitted photons from an oscillating dipole depends on the quantization axis (magnetic field direction in our experiment), polarization of the exciting laser field and the total angular momentum of the excited state. Since the proposed protocol measures the photon counts from the P3/2 state is based on counting 493 nm (green) photons that reach the detector within an interval of time in which the red laser is on. An interference filter with narrow bandwidth around 493 nm placed in front of the PMT results in background free measurement of the photon counts. The characteristic feature of this count as a function of time is an exponential decay as shown in green, Nor and Nblue respectively as in r12 and r32, background counts (mostly dark counts) taken within the same time interval as the measurement have been subtracted. The background counts are taken in each cycle just after each measurement so as to ensure similar ambient condition. The result of the three branching fractions measurement are summarized in The measurement technique when applied to the barium ion PIn any precision measurement the most important experimental challenge is to identify and measure all possible systematic uncertainties associated with the experiment. In the following, all possible systematic effects associated with our experiment are discussed in detail:In a single photon counting regime, once a PMT registers a single photon it remains dead for a certain time before it can register a second photon. Therefore in our experiment, if the count rate is high such that successive photons arrive at the detector within the dead time, one must account for the loss in photon counts. It is straight forward to calculate such a loss and the resulting shift of the branching fraction, as well as the uncertainty, which depends on the uncertainty of the measured dead timeIn the measurement scheme employed here, the total number of registered photons are calculated from the observed decay curve after subtracting the background. Since a decay curve ideally extends to infinity and the measurements are taken within a finite time, one needs to account for possible loss of counts. This has been estimated by fitting the decay curves with the proper exponential and conservatively assuming the uncertainty to be the same as the shifts. As shown in J = 3/2 state, this effect would be the largest contributor to the systematic uncertainties1/2 state decay. Since our detection system is identical to that used inIn case the ions are excited by circularly polarized light and the detection system is more sensitive to one type of circular polarization of the emitted photons than the other, the observed count rate will then be a function of the magnetic field provided the field has a component perpendicular to the direction of propagation of light. This is known as the Hanle effect. In the case of a measurement scheme as proposed inμs and the meta-stable states have lifetimes of 30 and 80 seconds, the meta-stable states are populated at most for 20–30 μs during one cycle. The skewing of results due to radiative decay is therefore estimated to be well below statistical uncertainties. Similarly, the collisional decay rate at the experimental working vacuum of 10−10 mbar, is estimated to be negligible. Considering a collisional quenching rate of 84 ± 11 × 103 s−1 Pa−1 due to H2 which is the dominant background gas in our experimentμs. For elements like Ca+ with relatively short lived meta-stable states, it is important to measure this decay as it may add to the systematics. However, the total cycle period can easily be shortened further by faster population transfer with higher laser power. We further verified experimentally that both the D-states do not decay within our experimental shelving times by probing the shelved states after different waiting times.Once a barium ion is excited by the blue light, it decays to two meta-stable D-states and the ground state. The population of the meta-stable states must be measured before the ion decays to the ground state otherwise the measured branching fraction of the meta-stable states will be systematically low. The decay of meta-stable states can be radiative or the states can quench due to collisional energy transfer. These two issues have been a major challenge in the previously performed experiments−7 Hz which means that D-state mixing is negligible in our case. Note that D-state mixing is not negligible for the Ca+ ion, where it may lead to a significant systematic error provided the operational vacuum is worse than 10−7 mbar. In a similar way as mentioned in the previous point, we also verified that the states are not mixing within our experimental cycle time.Collision with background gas molecules not only causes decay of the meta-stable states but also leads to mixing of the fine-structure levelsDuring the off state of any laser, we achieve better than 70 dB suppression as compared to the on state owing to the double pass configuration. Therefore, experimentally we have not observed any population transfer during off state of the lasers.After considering all possible error sources, the final value of the branching fractions and the total uncertainties are tabulated in 11The dipole allowed transition probability between two states depends on the upper state life-time and the branching fraction to the lower state given by:A, f and τ denotes the transition probability expressed as s−1, branching fraction and upper state (k) life time in ns. Moreover, the transition probability is related to the reduced dipole matrix element between the involved states, denoted as R, bywhere gk and λ stand for the multiplicity of the upper state k and the vacuum wavelength expressed in Å. The result of a many-body theory calculation from first principles yields the eigen states of the atom or ion. Therefore, theorist usually provide the value of R or A while experiments provide the measured value of f. Using the measured f and the best known value of τA depends on the uncertainties of both f and τ. From R, for which the best known value of the wavelength, recommended by the Nation Institute of Standard and Technology (NIST), has been used. A comparative study of the transition probabilities A obtained from previous experiments and existing theory calculations, shown in 20212223246A for the transition 5 → 1 is 1.6% which is about 4 times lower than the measured uncertainty of f in that experiment. The transition probability A3←5 however, measured to be higher as compared to ref. where If we compare the present results with those of theoretical calculations it is now possible to clearly distinguish between different theoretical results with a precision of 1%. Theoretical calculations commonly employ relativistic coupled cluster theory to all orders in perturbation, but with different orders in excitations and in some cases with added interactions like Breit corrections. The calculated values of the transition probabilities also reply on the vacuum wavelength indicated in 3/2 state have for the first time allowed the branching fractions to be measured with precision below one percent. The resulting calculated values of the transition probabilities are now only limited by poor knowledge of the P3/2 lifetime. Therefore, for the barium ion these values clearly sets the test bed for high precision many-body atomic structure calculations to be performed with precision below one percent.An improved protocol to measure the branching fractions of a hydrogenic atom with more than two decay channels has been proposed. It is shown that this protocol has no significant systematic uncertainty and hence the final precision of the measurement is only limited by statistics. Using the new protocol, measurements performed on the barium ion P+ are not cancelled out by sum-over-states calculations. Therefore our measurements will help pinning down the accuracies of the theoretical results on the violation of atomic parity in the barium ionA comparison of calculations performed by different theory groups of the dipole matrix elements, with our experimental results, shows discrepancies. These may originate from inaccurate starting wavefunctions which systematically provide energy eigenvalues of hydrogenic atom that differ from experimental values by about 2–3%. Another possibility may be due to underestimated terms in the Hamiltonian. In either case, the availability of measured values with precision better than 1% will help better understand many-body quantum physics. Moreover, these measurements along with the measurements performed in ref. 2 spectra separately while the red (650 nm) laser is phase locked to a reference cavity as is the master (986 nm) laser of the green laser. The setup is identical to the one in ref. 1/2-P3/2 transition at 455 nm (blue) and the other one to drive the D5/2-P3/2 transition at 614 nm (orange). The blue laser is made from a commercially available 450 nm diode which is frequency locked to a Te2 transition which has been observed for the first time and reported in ref. 5/2 using blue and red, orange re-pumping, red measurement, cooling, blue pumping and finally the red laser measurement. The total cycle time is about 320 μs while there are three cooling steps within an individual cycle period to ensure non-melting of the ion chain crystal. In order to obtain the required statistics, each experiment consists of 3,000,000 cycles, and we performed 18 such experiments with similar numbers of cycle in order to analyse systematic uncertainties.The experiment schematically shown in How to cite this article: Dutta, T. et al. An exacting transition probability measurement - a direct test of atomic many-body theories. Sci. Rep. 6, 29772; doi: 10.1038/srep29772 (2016).


There are errors in the third sentence of the Results section of the Abstract. The correct sentence is: The average physical activity  of the FFs was 3953±2688, whereas those of the POs was 2838±2872  and of the SCs 2212±2293 , respectively.


Although the estimation of average arterial blood pressure using oscillometric methods is possible, there are no established methods for obtaining confidence intervals (CIs) for systolic blood pressure (SBP) and diastolic blood pressure (DBP). In this paper, we propose a two-step pseudomaximum amplitude (TSPMA) as a novel approach to obtain improved CIs of SBP and DBP using a double bootstrap approach. The weighted median (WM) filter is employed to reduce impulsive and Gaussian noises in the step of preprocessing. Application of the proposed method provides tighter CIs and smaller standard deviation of CIs than the pseudomaximum amplitude-envelope and maximum amplitude algorithms with Student's  The maximum amplitude algorithm (MAA) based on oscillometric measurement is the most widely used technique to estimate the average arterial blood pressure –5. The Mt-distribution (ST), instead of asymptotic normal distribution, was utilized to obtain the CI of the SBP and DBP T. Thus, the WM filter's output is given by the use of the median operator such thatWi ∈ R denotes the weighted value for i = 1,2,…, K. Note that the weight signs are uncoupled from the magnitude values of BP envelope and are merged with the observation BP envelope sample as follows . This indicates that the number of bootstrap resamples achieves stability , a skewness parameter δ ∈ , and a location parameter β ∈ R. The scale parameter γ is a key factor to generate impulsive noise, which is similar to the variance of the Gaussian distribution. The stability parameter α = 0.5 in our paper measures the thickness of the tails of the distribution. When the skewness parameter is set to δ = 0 in our paper, the stable distribution is symmetric about the location parameter β = 0 [Occasionally, the oscillometric wave signal is contaminated by additive noise such as impulsive and Gaussian noises generated from subject's moving artifact, electronic device, and environmental conditions in the processes of BP measurements. However, it is not well defined with the Gaussian model . Thus, ter β = 0 . Based oγ from 2.0 to 0.5. In this section, we omitted the explanation of white Gaussian noise's generation because it is a basic method. The conventional MAAST and MAAGUM do not work well in white Gaussian noise contaminated environments for all SNRs. However, the proposed TSPMAEWM works well except for SNRs of 5 and 10 dB in white Gaussian noise contaminated scenarios as given in In The goal of this paper is to derive the improved CIs for SBP and DBP estimates when only a small number of blood pressure measurements are available. The degree of error variability between the readings obtained with the proposed method and those obtained with the auscultatory nurse method as the reference  was inveγ = 2.0 to γ = 0.5. However, the proposed TSPMAE also does not work well in low SNRs of 5 and 10 dB white Gaussian noise contaminated scenarios as given in The bottom histograms of In conclusion, we demonstrated that the CI obtained using the proposed method is narrower and has a narrower standard deviation than CIs obtained using other methods. Note that this paper does not focus on accuracy directly while the accuracy in the estimates can be obtained through the standard error from the golden reference. If the standard deviation of the estimate is low and if there is no bias, then the estimates may be deemed to be accurate. The decrease of the standard deviation in the CI results is attributed to the increase in the effective number of samples due to resampling using bootstrap principles. The results indicate that the proposed methodology reduces the standard deviation and consequently improved the accuracy. Our results imply that the proposed methodology is the best available to deal with small samples of blood pressure measurements. Our proposed method outperformed conventional methods for obtaining the CI under regular recording, impulsive, and white Gaussian noisy conditions. Indeed, the proposed technique can be used extensively as a potential application for self- and home-based monitoring scenario. We expect further studies to extend this methodology for older people with stiff arteries and wide pulse pressures and we will present these results in a near future.


It has been observed that psychical suffering (e.g. the feeling of losing a significant person) tends to reduce the physical pain tolerance threshold, as well as to increase the subjective sense of painfulness. The purpose of this study was to assess pain sensation among a group of caregivers of patients with Alzheimer’s disease, and to determine the psychological factors  that contribute to both pain perception and coping with pain via the use of analgesics.The study comprised 127 caregivers of patients with Alzheimer’s disease. Questionnaires were used to elicit pain intensity, strength of emotional relationship between caregiver and patient, sense of painfulness of the loss experienced, depression level, and somatic ailments.A large majority (87.4 %) of participants reported pain complaints, while 93 % took analgesics without a doctor’s recommendation at least once a week; 8 % took painkillers daily. The strongest predictors of both perceived pain and tendency to use analgesics were sense of loss and painfulness of loss in relation to the patient’s deteriorating condition.The pain experienced by caregivers may be connected to social pain resulting from the experience of losing someone they are close to. Caregivers may resort to excessive use of analgesics as a pain-coping strategy.The online version of this article (doi:10.1186/s12888-015-0571-1) contains supplementary material, which is available to authorized users. Alzheimer’s disease (AD) is the most frequently occurring type of dementia. It is forecasted that in 2050 in Europe alone there will be 14.5 million people suffering from the disease . PatientThe burden of giving both physical and mental care to the AD patient results in frequent occurrence of symptoms of depression , 5, chroThe results described are in alignment with the concept of social pain , 9. In tGiven the current research context that associates a more frequent occurrence of somatic symptoms of depression in caregivers who have had positive experiences with the patient and who assess their bond with the patient as very positive and close, we hypothesized that pain sensations of these caregivers would be relatively intensified and the use of analgesics more frequent.Our focus upon pain perception by caregivers looking after AD patients requires introduction of some specific terminology we use to describe that experience. Pain is a multi-dimensional phenomenon, and – in accordance with the definition provided by the International Association for the Study of Pain – an unpleasant sensory and emotional experience associated with actual or potential tissue damage, or described in terms of such damage . This deThe psychic dimension of pain is associated with attachment of an emotionally negative character to pain sensations. Pain is perceived as unpleasant and is connected with fear/anxiety or depression. This response is a substantial constituent of suffering, understood as a construct joining the physical, psychic, and existential dimensions of pain . The affThe aims of the study were to assess pain sensation among a group of caregivers of AD patients, and to determine the psychological factors  that contribute to both pain perception and coping with pain by use of analgesics.The design was cross-sectional. Caregivers obtained data in the form of medical documentation on the severity of dementia symptoms from doctors. The remaining data were collected from caregivers via questionnaire. Participation by caregivers was voluntary, and participants were informed about the opportunity to receive treatment from a psychologist or psychotherapist if they felt that this would be helpful to them. Written informed consent for participation in the study was obtained from all participants. Ethics approval for the study was provided by the Ethics Committee of the University of Silesia.n = 358), 35.5 % agreed to complete the questionnaire. Those who declined to participate cited lack of time (94.4 %) or lack of interest (5.6 %). All those who declined and cited lack of time as their reason were also the sole caregiver of an individual with AD. The sociodemographic and care-related characteristics of participants are provided in Table The sample comprised 127 primary caregivers of patients with AD. The participants were recruited from caregivers who accompanied the persons entrusted to their care for consultation in psycho-geriatric, geriatric, or psychiatric outpatient clinics, as well as in day care centers for patients with AD in several provinces of Poland. The criteria for inclusion in the study were: at least 1 year of providing care, at least 6 h of care every day, and direct contact with the patient, who had at least a moderate level of dementia. Exclusion criteria were: caregiver experiencing serious emotional disturbances and caregiver being a minor. The study was conducted between July and December 2013. Of all the caregivers who met the inclusion criteria  was used , 13. ParThis tool was developed by the authors to collect data on somatic diseases from which the caregiver was suffering, and drugs taken .The STATISTICA 10 package was used for analysis. Descriptive data are presented as means and standard deviations for continuous data and as frequencies and percentages for categorical data.tau-b correlation coefficients were computed to examine the association of the three key participant variables  with the outcome variables of the intensity of pain perceived and frequency of taking analgesics without a doctor’s recommendation.Kendall’s Logistic regression was used to model the probability of taking analgesics as a function of caregivers’ psychophysical condition and the caregiver–patient emotional bond. The dependent variable was established as daily or almost daily (4 to 7 days a week) use of analgesics by the caregiver, without a doctor’s recommendation. Predictor variables included a strong caregiver–patient relationship and substantial ‘painfulness’ of loss , depression , siIt has been found that eighty-two caregivers were offspring of the patient; 39 were partners or spouses; and six were friends. A vast majority of caregivers were sole carers and also lived with the patient. Female caregivers slightly outnumbered males (57.5 %). Most offspring and all friends who acted as caregivers had at least a secondary education.Data on participants’ psychophysical condition and the strength of their relationship to the patient are summarized in Table Nearly half of caregivers (45.7 %) had had at least one chronic disease diagnosed: 32.3 % had been treated for hypertension, 14.2 % for circulatory insufficiency, 15.7 % for respiratory tract diseases, 20.4 % for diabetes, and 9.4 % took medication for depression.SD = 4.28; min = 0; max = 10).Just 28 participants (22 %) had diagnoses of pathological states that included pain sensations, whereas 87.4 % reported having pain complaints during the week prior to the study; most often these were backache or headache. The mean intensity of the pain was 4.13 for the group as a whole . Another participant took analgesics just once or twice a month. The vast majority of participants (93 %) took analgesics without consulting a doctor at least once a week, while 8 % took such medication every day Fig. . Most frIt was demonstrated that the higher the intensity of physical pain perceived, the higher the assessment of painfulness of loss, and the more acute the sense of loss Table . MoreoveThe frequency of taking analgesics without a doctor’s recommendation also correlated positively with painfulness of loss, a sense of loss, and more intense symptoms of depression Table . It is wThe logistic regression analysis indicated that the sense of loss and painfulness of loss were the factors that most profoundly increased the probability of using analgesics without consulting a doctor Table . The paiOur study shows that many caregivers of AD patients report pain complaints and that they frequently take analgesics without consulting a doctor. The sense of loss and painfulness of loss turned out to be the strongest predictors not only of pain intensity but also of using analgesics. This result confirms our hypothesis that pain sensations are relatively intensified and that the use of analgesics is more frequent among this subset of caregivers of AD patients. Conversely, caregivers who did not perceive the loss as emotionally painful also did not experience as much physical pain. Our results therefore also support the concept of social pain , 9.Earlier studies  have indA loss can be perceived as particularly painful when the caregiver lacks support and understanding from others; yet support for caregivers is often substantially limited. This is often a matter of time restrictions, among other factors: caregivers do not have free time to meet other people who could be sources of support . Our participants were often the only person actually providing care to the patient (77.1 %). In the Polish setting, obtaining additional assistance may also be hindered by limited availability of institutional as well as financial support. Another reason for the absence of support may be the limited possibility of honest discussion concerning the difficulties and internal conflicts experienced, for fear of being assessed negatively by others , 20. In The closeness of the caregiver-patient relationship has emerged as an import predictor of the psychophysical condition of the caregiver. Fauth et al.  demonstrMonin and Schulz , 25 statAlso, higher levels of depression are related to lower assessments of health . In our Zhu et al.  demonstrIn light of the facts mentioned here, it is justified that caregivers should be given not only adequate pharmacological assistance but also support in the form of coping measures for working through the grief they experience while the patient is still alive. This need is especially critical when one considers that the process of losing a person to AD often takes several years. This type of assistance is likely to reduce the psychical suffering of caregivers, but also to minimize their tendency for excessive reliance on analgesics.Our study has several limitations. The first is that the group of participants was not representative of caregivers at large. Participants were recruited from Polish caregivers who were town or city dwellers. We should also mention that only 35.5 % of the caregivers invited to participate in the study took part in it. In the future, it would be worthwhile to devote attention to the group of caregivers who refused to participate in the study and who cited lack of time as the reason. One can expect that provision of care by those people is connected with specific bio-social costs. The variable availability of medical procedures and social care in Poland does not allow direct extrapolation of the results to the general population of caregivers of AD patients. It would be worthwhile to repeat the study in different cultural and economic conditions. In future studies, it would also be useful to consider measures of frustration levels associated with the accessibility of medical and social services.Our study was cross-sectional. The results would be worth verifying in a longitudinal study, to confirm whether the dynamics of the caregiver-patient relationship influence the level of pain complaints reported, and whether these complaints are affected by changes in strategy for coping with stress, including changes related to the use of analgesics (which Zhu et al.  observedOur study was based on single-item thermometer scales, which render the subjective level of sensations experienced by caregivers. The interpretation of results from such measures carries the risk of errors associated with excessive generalization and confounding of concepts. Such factors as, for example, the sense of loss are aggregate variables that may comprise several other constructs. In a future study it would be worthwhile to explore more comprehensively notions of loss and of being close to the patient.Future studies should also investigate such variables as adequacy of social support, or levels of distress and efficiency of strategies for coping with stress, as potential buffers to the costs of providing care to an AD patient.Pain is a symptom commonly experienced by caregivers of AD patients. This phenomenon may be connected to social pain resulting from the experience of losing someone to whom they feel close. Caregivers may resort to excessive use of analgesics as a pain-coping strategy.


N” in the formula on page 3. This formula describes the gait phase modulation (GPM) measure. The corrected formula is written below.In the Original Research Article there is a missing normalization by “N is necessary to scale the GPM magnitude in an interval from 0 to 1. In our calculations this normalization was already applied what is represented in the GPM values we reported in Table 1 and Figure 4 in the original publication.In our original article, the description of the GPM formula as well as the reported results are correct. As properly stated in the commentary from Trenado  to our oIn contrast to Trenado  we suggeThe authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.


The bell prover is widely used for gas flowrate measurements by timing a known stroke of the bell as it rises, presumably with a constant speed, from a bath of sealing liquid. A differential equation for the bell motion (from the previous paper of the same title) is used together with the previous equations of motion for the gas and sealing liquid modified here to provide the basis for a computer simulation of the prover performance. Examples of the results of the computations show substantial fluctuations in all of the motions and modified measurement procedures for improved accuracy are discussed. Proposed modifications to the prover itself are shown by the computations to dampen the fluctuations and to improve the measurement accuracy. Only a limited number of changes of the prover design and of the initial conditions were researched for their effect. Vb), and thermal insulation are installed in the inlet pipe. These are suggested to help attenuate the fluctuations of pressure that are generated in the bell and also to assure equality of temperature of the bell and of the gas input. A second essential addition consists of two annular orifice plates mounted in the two sealing liquid spaces between the bell and its tank. It’s expected that the constricted liquid flow area there will help to attenuate the liquid motion which is shown in the subsequent computations herein to be a major source of flow measurement inaccuracy.The bell prover of iV each time a different meter is installed for calibration. A third benefit arises in that the computations that are to follow here will be more likely to represent actual events for this restricted system as compared to the usual large complicated piping system.Three other additions of lesser importance are shown in More than one nozzle (size) may be needed to be able to cover the full range of flow of any one prover. Other basic requirements are that, at the flow rate of choice, the gas velocity in the nozzle throat be equal to the velocity of sound there, and that the gas at the entry point to bell be at the temperature of the bell. Another concept may prove useful here. Simultaneous measurements of temperature and pressure can be made at the sonic nozzles to calibrate them along with the meter. These calibrated nozzles then should prove useful as secondary flow standards for calibration of meters without the necessity of always exercising the bell prover. A second addition to the ordinary prover should include a thermocouple placed at the exit of the inlet pipe so that it can provide a confirmation that the gas and bell are at equal temperatures all during the measurement stroke of the bell. An average of measurements taken at different times and even places in the bell may not be sufficiently accurate if bell and gas temperatures are not equal.The third difference between the modified and ordinary prover is in the location and design of the diverter valve. It is placed on top of the bell instead of on that portion of the inlet pipe that is external to the prover’s tank. It is proposed that it be an electro-mechanical device designed to be able to remotely select and replicate the stroke time and flow rate history of the gas through the valve. A lateral discharge of the gas from four ports spaced 90° around the new valve will be necessary to avoid a variable and disturbing force on the bell during its upward motion. A rotatable collar or other device with equally spaced and appropriately sized ports together with, possibly, a spring and/or an electric and/or a pneumatic motor to rotate the collar, will be needed to complete the essential parts of the valve. It’s not necessary that the new valve be located as shown in The differential equation of motion for the bell from reference , togetheSpecial integrals of the Navier-Stokes (NS) equation will be used  to derive the liquid drag forces for use in the bell’s equation of motion. Forces connected with entrained liquid and effects not included in the special integrals will be estimated for the same application. The NS equation also provides a justification to treat the motion of the sealing liquid as if it is a “solid” and thus a one-dimensional, second-order differential equation is used for the liquid motion to calculate its effect on the gas volume under the bell.Each set of differential equations will be subjected to numerical integration using the Runge-Kutta fourth-order method outlined in .Av then can be calculated as Av=Avi (1–γ). Here γ is used as t/tv, the time t from start of the stroke divided by the total stroke time tv, and Avi is the initial open area. For a constant acceleration stroke, Av=Avi (1−γ2). The latter assumption is used in the subsequent calculations of the flow as its accomplishment seemed to present a simpler valve design problem, and possibly because it would provide a “smoother” acceleration of the bell upward from its float position. A valve might be designed for a still smoother operation to close slowly at both the beginning and end of its stroke, with a port open area represented, for example, by a function such as Av=Avi (1/2−(1/4) cos3πγ+(3/4) cos πγ). The rate of area change with a valve so designed would be ≃0 at the beginning and end. Effects of valve design on prover performance probably will be researched better by calculations than by experiment.The flow measurement process is initiated by a gradual closing of the diverter valve to start the upward motion of the bell from its balanced position. A calculation of the net mass flow into the bell during the stroke of the diverter valve is based on the difference between the initial steady state mass flow out of the valve and the smaller flow out during its stroke. If the ports are rectangular and if the stroke is assumed to be one of constant velocity the orifice port area D, asP is (P−Pa) and ΔPi is (Pi−Pa) and the first term on the right side of FRI. This rate is to be compared to that established by the calculations from the derived motion of the bell and sealing liquid and for the change of state of the gas in the bell. The mass of gas in the bell and inlet pipe is represented by m and ρ is its density.The net mass flow rate into the bell can be written, using the well known equation for the orifice with coefficient of discharge CP to the bell and oil motions so that it can be inserted into Pi as the initial balance pressure, asmt and Vt are the initial mass and gas volume under the bell and inlet pipe, and Ax is the volume change under the bell due to its motion x from x=0. ΔV is the change of gas volume from oil entrainment on the bell as it moves upward plus the volume change caused by motion of the liquid seal. The exponent k may be taken as the ratio of gas specific heats when an apparatus is used that would give adiabatic conditions. For a very small apparatus in which the process might be one of almost constant temperature, k should approach unity. Note here that A is the effective area of a bell prover and is given in ρ in terms of its pressure in the bell and inlet pipe and then the now CD will be specified as equal to zero at the instant that γ= 1 to simulate a closed diverter valve. V into P/Pi which will be used in The gas law is used to express the gas density Vb and is expressed asm under the bell isCD should be made equal to zero at the instant γ is specified to equal unity. Note that (Pbi‒Pi) is the initial pressure drop that is to be specified across the porous plug and that the flow rate through it is proportional to (Pb−P). This requires a plug designed for viscous flow which probably would be more effective than one designed for turbulent flow.A computer simulation of the modified prover’s performance requires that P as expressed by the equationP, m, mi are for conditions under the bell only. For the specified constant gas temperature in Vb, Pb can be written asAn It is intended here to derive the forces on the bell that are associated with the motion of the bell in the sealing liquid. For this purpose some aspects of Couette flow in both steady and unsteady state motion between parallel plates will be discussed. It is also necessary to look at the development of the laminar boundary layer on a flat plate in the event that this could be dominant as to the force it applies to the bell. These considerations might be useful in examining further the forces from oscillatory liquid motion, but these will be ignored as they tend to cancel when both sides of the bell are considered. However, the liquid oscillation would affect the volume of gas under the bell and an attempt will be made to account for this effect. Other force aspects of the liquid motion, such as from entrained oil on the sides of the bell and motion of the oil at its bottom edge are estimated to include them in the bell’s motion equation.r > >2b and (L−x)> >2b. These length conditions would limit the influence of the complex flows that exist at the free surface and at the bottom edge of the bell. Thus the applicable equation is written asy is the distance from the bell’s surface, z is along its circumference and u and υ are the liquid velocities in the x- and y-directions, respectively, as illustrated in u2/∂x2, ∂2u/∂z2, ∂u/∂x, υ and ∂u/∂t in u = U at y =0 (or that the liquid on the bell’s surface is carried up with the bell whose velocity is U) and second that u =0 at y=2b are now used in the integration of P/∂x was assumed to be constant across y in the derivation of The well-known momentum equation of NS may be used to derive the velocity profile for a fully-developed, steady flow between parallel plates. This condition is practically satisfied if, in P/∂x such that the liquid up-flow adjacent to the bell is balanced by a down-flow adjacent to the tank wall. Thus, an integration of the flowrate across y derives the volume rate total which, when set to zero, gives ∂P/∂x ash is derived from µU(L−x)/2b2ρ0g. The up-flow must spill over at the free surface to build a liquid head at the tank wall comparable to Δh to provide for the flow down along the tank wall and for its recirculation at the bottom of the bell. When the value of ∂P/∂x is inserted into u =0 point (y=2b/3), is calculated to be equal toℓ is an average circumference of the bell. Third, this volume rate can be compared to the entrainment rate df/dt of liquid that adheres to the bell as its surface leaves the liquid. Reference  and α [=y/2(v0t)1/2]. Partial differentiation (with respect to y) gives the velocity gradient at the moving wall (y=0) ast→0, and as t→∞, the gradient decays to that for the steady moving wall.With that modified NS equation and using appropriate boundary conditions the derived velocity profile is given as a series of complementary error functions (erfc) of the quantities αt =0 to t=Δt, and to use the “averaged” gradient to calculate force F2 on the bell. The time-averaged value calculates to bet to be used here. This result is taken as most applicable during the acceleration phase of the motion of the bell. Thus It is proposed here to calculate a time-averaged value of the velocity gradient from h3 as used in h3 and h3 by considering the oil as a solid in motion.(b) The “sudden” concept is also regarded as applicable to estimate the motion of the oil seal (as caused by pressure changes in the bell) and the resultant change of b. The integration limits for η are calculated from α1 and α as given previously. A numerical integration of t assumed equal to integration time intervals Δt to be used here does show an almost constant velocity across practically all of the distance 2b. Thus, a value of h3 derived from the “solid” equation of motion should be reduced by a factor derived from an average value of b. It should be noted here that the oil motion effect on the gas volume is expected to have most significance while the diverter valve is closing. If the subsequent motion of the bell is at constant velocity the motion of the oil seal should have insignificant influence. Note the big “if” in the previous sentence.A velocity profile in this “sudden” case is adequately represented  asA. H. Shapiro  presentsF3 calculated below:Sealing liquid that is entrained on the walls of the bell previously was stated to modify the indicated volume of gas collected. Here the liquid acts to modify the mass of the bell by way of the force of gravity on the liquid. This will be the third non-conservative force (b) Liquid at the bottom edge of the bell is sucked upward by it which creates a drag force there which can be stated asThe equation of motion 13) of [3 of [1] xi=0, to calculate h4i at the initial balance condition using an assumed (or specified) value of (Pi−Pa). Second, eqs (15) and (16) of  A3=A4. For the two “solids” case the equation is quite complicated and contains a product of the oil acceleration and elevation and also its velocity squared. This seems an unwarranted complication at the present time and the single “solid” will be used.Two different approaches might be selected to set up the oil motion equation, one regarding the oils on the two sides of the bell as two separate “solids”, or by treating the oil as a single “solid” using the assumption that h4−h4i)≃−(h3−h3i), to giveρ0(A3+A4)(L−x), and that the term containing h4 is the liquid’s back pressure. The factor 2  arises from the pressures acting on the assumed area=(1/2)(A3−A4). A “Force Loss” is included in the equation to include the effect of the turn that the oil must make at the bottom of the bell of the ordinary prover. This loss is assumed to be equal to one-half of the velocity pressure, or (1/2)(1/2) A3, which then gives the last term on the right side of A5 on each side of the bell to modify this part of the prover, a pressure drop will be introduced into the flow there. This drop will arise from the frictional loss of mechanical energy mostly on the downstream side of each constriction. For an area constriction down to about 10 percent, a possibly conservative estimate of the pressure drop would correspond to one velocity head pressure based on the liquid’s average velocity in the constricted area. On this bases the last term of The equation of motion for the single “solid” is derived using the familiar concept that acceleration is given by force per unit mass and that  assumes the use of a constant acceleration diverter valve. P/Pi and V/V1 as follows:Z1 = [(A3−A4)/A3+A4] (Z3) (4πA1)1/2 ν00.644;Z2 = a correction factor for lack of oil “solidity”;Z3 = a constant whose value depends on the system of units that will be used.Note above that h4 has replaced (h3i−h3) because A3 is assumed equal, in A4 and that the motion equations when integrated will give h4=(h4−h4i).The gas flow P/Pi and Pb/Pi, respectively, and A5 is constructed to be equal to 0.1 (A3+A4). With this substitute, last term of The gas flow CD. A3 was assumed equal to A4 with corresponding assumptions f3=f4 as in h3=h4 as in Vb.A recapitulation of the main assumptions used in the derivations of the above equations seems advisable. It does not seem advisable at the present time to develop a change to ρiA (x2−x1) with corrections applied to it for the fluctuations of P and ΔV. Subscripts 1 and 2 are used on x to signify values at the beginning and end of the chosen timing interval, respectively. It seemed preferable to separate the P and ΔV corrections, and to do this the pressure ratio in P−Pi)kP). With some use of algebra, Vb = 0 in the far right-hand term and Vi should be replaced by Vt which is given by (Vi+Vb).The numerical integrations of the equations of motion are used to derive the performance of the provers which will be illustrated by various curves showing the motions of the bell and sealing oil and the derived gas pressure and gas volume correction all expressed as a function of the time. See  to 5. ThV1 of 5 ft3 (0.1415851 m3) and these were used in three computer runs for the first look at the utility of the proposed analytical method. Results of these runs indicate the effectiveness of the integration method as well as demonstrate the size and sources of errors likely to be encountered with the usual laboratory procedures. The prover specifications are listed in Almost complete physical data were available to the authors on a bell prover of nominal capacity The two runs at the high rate of flow  were done with different time intervals in the integration process, namely 0.002 and 0.01 s. Rates of flow derived from use of these two integration processes differed by about 0.01 percent. Comparison of other physical variables also showed differences near 0.01 percent. On the bases of this good agreement it was decided to use the 0.01-s time interval for the low (50 s collection) flow rate calculation. It seemed reasonable to expect much larger differences than 0.01 percent between prover performance at the two rates of flow.Pi from Pi with a relatively long period, initially about 1.8 s and decreasing to about 1.1s. The pressure fluctuation decreases to about ±30 percent of ΔPi from the mean pressure, but the curve near 7 s shows a hint of a possible increase at that point. Higher frequency fluctuations of the pressure, with periods of about 0.3 s, are also evident. The amplitude of the oil motion as converted from values of h4 to pressure is about 120 percent of the pressure fluctuation. It is evident that the pressure fluctuations are strongly modulated by the oil fluctuations. It is also strongly evident from V under the bell fluctuates practically in unison with h4. This fact suggests that failure to measure h4 possibly can be the major source of error.Pi from Pi, and again a relatively long period initially of about 1.8 s which decreases to about 1.4 s. The pressure fluctuation and the pressure level both decrease, the former to ±1.3 percent of ΔPi about the mean pressure which is about 9 percent of ΔPi below Pi. These fluctuation amplitudes are about six percent of those at the high rate of flow, as compared to the flowrate ratio of 10 percent. The amplitude of the sealing liquid motion (not shown) begins at about 115 percent of the pressure, but winds up near the end of the measurement stroke at about twice that of the pressure fluctuation. As at the higher flow rate, the gas volume correction (not ordinarily measured) fluctuates practically in unison with h4 and with the low frequency pressure fluctuations. The high-frequency fluctuations have periods of about 0.19 s which increases to about 0.31 s.At the low rate of flow x as equal to V is the volume of air and x is the motion of the bell about a stationary position. The vibration period for this well known case is The high frequency motion calculated for the bell can be shown to be real by considering the bell to be vibrating as if suspended from a “spring” which can “replace” the air enclosed in the bell. For the case of the very low values of the pressure fluctuations prevailing here, this concept is sufficiently accurate to derive the “spring constant” on An inspection of P and either Δh4 (or ΔV) are accurate at any instant and accurately correlated with the bell positions x1 and x2, If the measurements of ρiA(x2−x1) are necessary if the initial and final timing points  can be identified by the instrumentation when the variables of (P−Pi) and Δh4 both equal to zero at both bell positions (a zero-zero criterion). In the more general case  this criterion does not prevail. However, pressure transducers probably are available with capability to follow the low frequency fluctuations of the pressure (as caused by the motion of the oil) with good accuracy especially at the fluctuation midpoints. If the timing points x1 and x2 are made to coincide in time with those mid-point pressures, the corrections derived with V Fand the ΔP motions are shown here to coincide. These methods should be used but apparently are not because of the practical difficulties. A third procedural compromise, Method 3, is usually used to derive the corrections.AFP) at two preselected, convenient positions x1 and x2. These measured AFP values are used in V (based on AFP) to derive “corrections” to ρiA(x2−x1). The “worst” timing point choices that can be derived from Measurements are made with a water manometer to give (what is believed to be) mid-point values of the high-frequency fluctuating pressures . This should be and is confirmed if some bell positions, particularly near x2, can be found where AFP values or transducer measured values are equal to Pi. These positions need not necessarily be those that are finally used for the rate measurements.It might be possible to implement the zero-zero criterion as described in method 2 to use what is essentially a “correctionless” procedure. Two requirements are necessary for this purpose. First, the prover must be modified to maintain the average of the maximum/minimum pressures, P and Δh4 is the optical encoder used by G. Kulin, P. Huang, and G. Mattingly [V and values of (P−Pi) both were zero. Therefore, presumably, the encoder’s indication of the maximum bell velocity also occurs at the zero-zero condition. This leads to the suggestion that the timing points should be made to coincide with the maximum velocity indications of the optical encoder to try to eliminate the otherwise needed, but in accurate, AFP corrections to derive the true mass of gas collected. As an alternative to the above, more attention should be given to fully implement Method 2.An instrument which might be used for phase indications of Δattingly  to betteAFP procedure of Method 3 which can be modified to measure the true collected mass of gas. Two modifications appear of possible utility, both needing a constant value of Pm=Pi during the stroke of the bell. The first would use computed motions of Δh4 (as described herein) to identify and use the bell positions x1 and x2 where Δh4 values are expected to be zero. AFP values would then also equal Pi and corrections to ρiA(x2−x1) would also equal zero. Results from use of this method could then be compared to results obtained below with the second modified AFP procedure.It appears to be possible to implement the zero-zero criterion as described in Method 2 using the AFP procedure would utilize a multiplicity of strokes of the bell each with positions x1 and x2 systematically moved until both give AFP values equal to Pi. Pm must also be maintained equal to Pi with this method (see Suggestion 2). This method is based on the assumption that AFP values of pressure fluctuate about Pi with a smaller amplitude than those of the high frequency pressure fluctuations, and with a period of the low frequency fluctuations of Δh4. It’s expected that on average five different strokes of the bell would be needed with x1 and x2 each moved each time a distance corresponding to the distance traveled by the bell during 0.05 period of the Δh4 fluctuation. It’s assumed in the experimental situation that the pressure measuring instrument actually does indicate the AFP values. If all of these criteria prevail the modified, multiple AFP method is also correction less. An advantage of these two procedure modifications is that neither requires apparatus changes.The second modified Pi, and tv was reduced to 0.2 s (from 2 s as used in HIFLO and LOFLO). These three changes were made simultaneously and “almost” arbitrarily, and hence the name HOLHOG was chosen. Note that ΔV increased by only about 50 percent over that in HIFLO in spite of the 14-fold increase of ΔP at the end of 0.2 s. These larger upsurges, however are subsequently efficiently dampened. Although the frequencies of the motions of ΔV and ΔP are the same as in the ordinary prover, the practically constant phase difference between the two motions is not evident here.AFP Method 3 is gratifyingly reduced to 0.053 percent from the 0.586 percent needed for the ordinary prover. This possible “worst” result for HOLHOG is improved to a needed remaining correction of only 0.006 percent if a “best” timing choice happens to be made. Phase differences between the ΔV and ΔP motions that change during the stroke of the bell limit suggestions for improvement in methodology to Suggestion 3 above. However, the need for improvement is much, much less than for HIFLO at its worst.The calculated “worst” corrections for HOLHOG listed in the table above demonstrate that the correction remaining and needed after application of the x1 to make a significant improvement. A third preliminary calculation was done to evaluate the effect of a single change in an initial condition, namely a reduction of tv from 2 to 0.2 s in program HIFLO. This change produced a two-fold increase in the initial upsurge of ΔV over the already large increase evident in HOLHOG. This large increase illustrates the importance of the initial condition, and second that the orifice effectively dampened the motion of the oil in HOLHOG and especially the initial upsurge motion.It’s appropriate at this point to discuss the “almost” arbitrary group of apparatus changes (specified for HOLHOG) and the problem of setting the initial conditions. As a preliminary to HOLHOG, computations of the effect of the orifice installation alone, and next of the effect of the porous plug alone, indicated that dampening was not effective enough before the bell reached Vi to Vb to V1, also could have significant effects on the performance of both provers studied here. This discussion should now revert back to the need to use a motorized and stroke-designed diverter valve as previously suggested herein. This discussion also leads to the conclusion that prover performance cannot be adequately specified until the initial conditions are too.Other changes of initial conditions, such as in the flow rate characteristics of the diverter valve, and changes in the relationship of It appears that the equations of motion for the moving elements of the prover system and the procedure used in their numerical integration can produce calculated results which offer a reasonable chance that they can be used to design improvements in the apparatus and measurement procedures. “Dynamic errors” in the measurement of flow rates can arise primarily from the fluctuating motions of the sealing liquid, second from fluctuating velocity of the bell, and third and usually least in magnitude from entrainment of liquid on the bell. Proposed additions to the prover can be used to (1) assure needed equality between gas and bell temperatures and (2) to reduce the amplitude of the fluctuating motions as one way to reduce the magnitude of measurement errors. Three other suggestions for improved measurement procedures include an ideal situation in which pressures and oil motions are accurately measured at any two bell positions of choice. A second more practical suggestion would require that the timing points for the bell motion be properly correlated with indications of the mid-point pressures from a pressure transducer or correlated with indications of the maximums in the fluctuating bell velocity. A third suggestion for improved accuracy entails the utilization of two different approaches both being modifications of that currently in use which relies on manometer reading of the average indications of the high-frequency fluctuating pressures at the two bell positions of choice. Some changes in the initial conditions produced performance changes which emphasizes that blanket performance and accuracy statements cannot be specified unless all initial conditions are also specified.