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Relaxing the assumption of constant transition rates in a multi-state model in hospital epidemiology
Micki Hill1,
Paul C. Lambert1,2 &
Michael J. Crowther1,2
Multi-state models are being increasingly used to capture complex disease pathways. The convenient formula of the exponential multi-state model can facilitate a quick and accessible understanding of the data. However, assuming time constant transition rates is not always plausible. On the other hand, obtaining predictions from a fitted model with time-dependent transitions can be challenging. One proposed solution is to utilise a general simulation algorithm to calculate predictions from a fitted multi-state model.
Predictions obtained from an exponential multi-state model were compared to those obtained from two different parametric models and to non-parametric Aalen-Johansen estimates. The first comparative approach fitted a multi-state model with transition-specific distributions, chosen separately based on the Akaike Information Criterion. The second approach was a Royston-Parmar multi-state model with 4 degrees of freedom, which was chosen as a reference model flexible enough to capture complex hazard shapes. All quantities were obtained analytically for the exponential and Aalen-Johansen approaches. The transition rates for the two comparative approaches were also obtained analytically, while all other quantities were obtained from the fitted models via a general simulation algorithm. Metrics investigated were: transition probabilities, attributable mortality (AM), population attributable fraction (PAF) and expected length of stay. This work was performed on previously analysed hospital acquired infection (HAI) data. By definition, a HAI takes three days to develop and therefore selected metrics were also predicted from time 3 (delayed entry).
Despite clear deviations from the constant transition rates assumption, the empirical estimates of the transition probabilities were approximated reasonably well by the exponential model. However, functions of the transition probabilities, e.g. AM and PAF, were not well approximated and the comparative models offered considerable improvements for these metrics. They also provided consistent predictions with the empirical estimates in the case of delayed entry time, unlike the exponential model.
We conclude that methods and software are readily available for obtaining predictions from multi-state models that do not assume constant transition rates. The multistate package in Stata facilitates a range of predictions with confidence intervals, which can provide a more comprehensive understanding of the data. User-friendly code is provided.
Multi-state models are being increasingly used to investigate complex disease pathways, for example, when interest lies in subsequent and/or intermediate events as well as a primary event. This unified approach facilitates a better understanding of the whole disease profile and provides clinically relevant predictions, for example, transition probabilities and expected duration in each state. One example is in breast cancer, where the time to intermediate events, such as local recurrence and distant metastases, is of interest as well as overall survival [1]. Another example is repeated hospitalisations in patients with heart failure, where interest lies in the time spent in hospital (during each episode and in total) [2]. Further applications include other cancers (colorectal [3, 4], ovarian [5] and acute myeloid leukemia [6]), progression to diabetes [7], health-care associated urinary tract infections [8] and pleural effusion following allogeneic hematopoietic stem cell transplantation [9].
Semi- and non-parametric methods have been frequently used to analyse multi-state models, however, interest is growing in parametric approaches. Although a variety of complex parametric models can relatively easily be fitted to each transition; the difficulty lies in obtaining the corresponding predictions from the full multi-state model. Assuming an exponential Markov model allows for direct calculation of the transition probabilities, as the Kolmogorov forward equations can be solved analytically, however, assuming constant transition rates can be restrictive. Piecewise exponential models relax this assumption [10], however, discontinuous transition rates may not be biologically plausible. Another suggestion has been to model the transition rates of a Markov model with quadratic B-splines and obtain predictions by numerically solving the Kolmogorov forward equations [11]. This paper focuses on a general simulation algorithm to obtain predictions from a range of fitted parametric models [12], including Royston-Parmar models [13]. In terms of implementation, available software includes: mstate in R [14] for semi- and non-parametric methods; msm in R [10] for exponential and piecewise exponential models; flexsurv in R [15] for fitting models and obtaining predictions by numerically solving the Kolmogorov forward equations; and flexsurv in R [15] or multistate in Stata [12] for the general simulation algorithm, the latter following model fitting by merlin [16, 17].
Von Cube et al. [18] recommended the exponential model as an accessible approach to obtain a quick, general understanding of the data. The authors demonstrated this method on hospital acquired infection (HAI) data: an extended illness-death model where a patient can have a HAI (intermediate event) and then/or be discharged or die (competing risks, the death/discharge with HAI are distinct from those without, resulting in 6 states). Von Cube et al. [18] acknowledged the potential implausibility of time constant transition rates (saying that this assumption is rarely met in practice) and recommended more sophisticated methods if the assumption was violated.
In this paper, we compare the method described in von Cube et al. [18] to the approach described in Crowther and Lambert [12]. We demonstrate that there is accessible software - namely the multistate package in Stata [12] - to obtain transition probabilities from multi-state models with transition-specific (and time-dependent) distributions. We then extend the previous analysis [18] by presenting, and highlighting the importance of, uncertainties and by estimating length of stay.
This paper is organised as follows: the "Methods" section introduces the multi-state process, illustrative example, metrics of interest and analysis approaches. The "Results" section displays the quantities of interest graphically, including comparisons between the approaches and confidence intervals for the predictions. The "Discussion" section provides recommendations for future analyses. User-friendly code for the illustrative example is provided in Additional file 1.
Multi-state models
Following Fiocco et al. [19], consider a stochastic process Y(t),t≥0 with a finite state space \( \mathcal {Z} = \{1,\ldots,Z\} \) and process history up to time \( s, \mathcal {H}_{s} = \{ Y(u); 0 \leq u \leq s \} \). The transition probabilities can then be defined as:
$$ P(Y(t)=b|Y(s)=a,\mathcal{H}_{s-}) $$
This is the probability that a patient in state a at time s moves to state b by time t, conditional on the process history up until the time just before \( s, \mathcal {H}_{s-} \), where \( a, b \in \mathcal {Z} \). This can be simplified to a Markov model, which makes the assumption that the probability in Eq. 1 is only conditional on the state at time s and no other process history:
$$ {}P\left(Y(t) = b | Y(s) = a, \mathcal{H}_{s-}\right) = P(Y(t) = b | Y(s) = a) $$
Henceforth, let Pab(s,t) represent the transition probability given in Eq. 2. This paper focuses on Markov models.
The transition rate, or transition hazard, from state a to state b at time t is:
$$h_{ab}(t) = \underset{\delta t\to 0}{\lim} \frac{P(Y(t+\delta t) = b|Y(t) = a)}{\delta t} $$
This represents the instantaneous failure rate of moving from state a to b and is analogous to the hazard function in the standard survival setting. The collection of transition rates governs the rate at which patients move between states and therefore the multi-state model. For a tutorial in multi-state models see Putter et al. [20].
Another useful measure is the restricted length of stay in a state. This is analogous to restricted mean survival in the standard survival setting [21]. The residual, restricted expected length of stay in state b given a patient is in (non-absorbing) state a at time s is:
$$e_{ab} (s,t) = \int_{s}^{t} P(Y(u) = b | Y(s) = a) du $$
See Grand and Putter [22] for more details on expected length of stay.
The extended illness-death model for HAIs
This paper considers a multi-state model in the context of hospital acquired infections (HAIs), as previously described by von Cube et al. [18]. When a patient is admitted to hospital, they are at risk of acquiring a HAI, which could lead to an increased hospital stay or increased risk of (hospital) death. An extended illness-death model with six states and five transitions, as illustrated in Fig. 1, has been used to investigate the risks and consequences of HAIs. The time scale is days since hospital admission. All patients begin in state 1 at time 0, where the patient has been admitted to hospital but does not have an infection. The patient will then either become infected (state 2), be discharged without an infection (state 3) or die without an infection (state 4). If the patient acquires an infection, they will then either be discharged (state 5) or die (state 6) with an infection. The ith transition rate from state ai to bi has been denoted as hi(t) where:
$$\begin{array}{*{20}l} \left\{a_{1},a_{2},a_{3},a_{4},a_{5}\right\} = \{1,1,1,2,2\} \\ \{b_{1},b_{2},b_{3},b_{4},b_{5}\} = \{2,3,4,5,6\} \end{array} $$
Extended illness-death model for discharge and death with and without a hospital acquired infection (HAI)
Metrics of interest
The first metric of interest was transition probabilities from state 1 at time 0, P1b(0,t),b={1,2,3,4,5,6}. By definition, HAIs take at least three days to develop [18] and so there were no HAI events prior to time 3 (3 days after hospital admission). Therefore, transition probabilities from state 2 at time 3, P2b(3,t),b={2,5,6}, were also estimated.
Following the formulas from Schumacher et al. [23] and von Cube et al. [18], let P1D(s,t) denote the probability of dying (states 4 or 6) by time t, given a patient was in hospital without a HAI (state 1) at time s. Let P16|+(s,t) denote the probability of dying with a HAI (state 6) by time t, given a patient was in hospital without a HAI (state 1) at time s and had become infected (states 2, 5 or 6) by time t. Similarly, let P14|−(s,t) denote the probability of dying without a HAI (state 4) by time t, given a patient was in hospital without a HAI at time s and in a non-infected state (states 1, 3 or 4) at time t. The quantities can be calculated as follows:
$$\begin{array}{*{20}l} P_{1D}(s,t) & = P(Y(t)\in \{4,6\} | Y(s) = 1) \\ & = P_{14}(s,t) + P_{16}(s,t) \end{array} $$
$$\begin{array}{*{20}l} {}P_{16|+}(s,t) & = P(Y(t)=6|Y(s)=1, Y(t) \in \{2,5,6\}) \\ & = \frac{P_{16}(s,t)}{P_{12}(s,t) + P_{15}(s,t) + P_{16}(s,t)} \end{array} $$
$$\begin{array}{*{20}l} {\kern-6.2pt}P_{14|-}(s,t) & = P(Y(t)=4|Y(s)=1, Y(t) \in \{1,3,4\}) \\ & = \frac{P_{14}(s,t)}{P_{11}(s,t) + P_{13}(s,t) + P_{14}(s,t)} \end{array} $$
The second set of metrics of interest were attributable mortality (AM) and population attributable fraction (PAF). AM and PAF can be used to investigate the excessive risk of dying due to HAIs, see references for discussion [18, 23].
$$ AM(s,t) = P_{16|+}(s,t) - P_{14|-}(s,t) $$
$$ PAF(s,t) = \frac{P_{1D}(s,t) - P_{14|-}(s,t)} { P_{1D}(s,t) } $$
AM and PAF were estimated from time 0 (i.e. s=0 in Eqs. 3 and 4).
The third metric of interest was expected length of stay. Given a patient started in state 1 at time 0, the following quantities were estimated: restricted length of stay in state 1 (e11(0,t)), in state 2 (e12(0,t)) and overall in hospital (e11(0,t)+e12(0,t)). Due to the three day delay in developing a HAI, the residual, restricted length of stay in state 2, conditional on having entered state 2 by time 3, was also of interest (e22(3,t)). These were calculated up until the last event time (from any transition).
Analysis approaches
The three (set of) metrics were obtained from three models. The predictions from the different models were compared against each other and against non-parametric Aalen-Johansen estimates. Note that we use the terms "Aalen-Johansen" and "empirical" interchangeably to refer to the non-parametric estimates. The Aalen-Johansen estimator generalises the Kaplan-Meier estimator to Markov multistate processes. The metrics are obtained analytically, see Additional file 2 for brief details or the following references for more comprehensive details [24–26].
The first approach was to fit an exponential model to each of the transitions and obtain the metrics analytically, as was demonstrated by von Cube et al. [18]. This approach was referred to as the "Exp" model. The parametrisation of all three models is presented in Additional file 2.
The second approach was to select the best fitting distribution for each transition based on the Akaike Information Criterion (AIC), henceforth denoted the "AIC" model. Following Crowther and Lambert [12], to each transition the following parametric models were applied: exponential, Weibull, Gompertz, log-logistic, log-normal, generalised gamma and Royston-Parmar models [13] with 2 to 5 degrees of freedom. The confidence intervals for the transition rates were obtained using the delta method. Once the multi-state model was fitted analytically, with the best fitting distribution for each transition, the general simulation algorithm was applied to obtain the metrics, see the next section for more details.
The third approach was to fit a Royston-Parmar model with 4 degrees of freedom to each of the transitions, henceforth denoted the "RP(4)" model. This was chosen as a reference parametric model for comparison purposes, as it should have sufficient flexibility to capture most complex hazard shapes. A recent sensitivity analysis of Royston-Parmar models [27] suggested that 4 degrees of freedom can adequately capture the baseline hazard. The analysis was performed in a relative survival setting; however, the conclusions can be applied to standard survival [27]. Once the multi-state model was fitted analytically, the general simulation algorithm was applied to obtain the metrics.
Simulation algorithm
The simulation algorithm works by projecting a patient through the multi-state model in order to create their full event history. This is done a large number of times and the metrics of interest are calculated empirically from the large complete set of histories. The process will now be described in brief, for further details see Crowther and Lambert [12]. Let a be the starting state, entered at time ta. If desired, specify a maximum follow-up time tmax. For each simulated patient, following the algorithm of Fiocco et al. [19] and Crowther and Lambert [12], repeat the following:
Let \( \mathcal {B} \) be the set of states that can be reached from state a and let Na be the cardinality of set \( \mathcal {B} \). If Na=0 (i.e. a is an absorbing state), stop. Otherwise, for each state \( b \in \mathcal {B} \), let hab(t) represent the transition rate from a→b.
For each state \( b \in \mathcal {B} \), use hab(t) to simulate event times \( t^{*}_{ab} \) conditional on entering state a at time ta. Event times are simulated using the general inversion method described in Crowther and Lambert [28].
The observed event time is then \( t^{*} = min\left \{t^{*}_{a1},\ldots,t^{*}_{aN_{a}},t_{max}\right \} \). If t∗=tmax, stop.
Set a=c where \( t^{*} = t^{*}_{ac}, c \in \mathcal {B} \) and set ta=t∗.
The algorithm above is repeated for a large N number of patients. The transition probabilities are then estimated by calculating the proportion of simulated patients in each state at each time point of interest. The full event history of the simulated patients is known and therefore extended predictions can easily be obtained. For example, expected length of stay can be calculated by averaging the time spent in each state (up to each time point of interest) over all patients.
Let \( \hat {\mathbf {b}} \) be the vector of parameter estimates and \( \hat {\mathbf {V}} \) be the variance-covariance matrix from the fitted multi-state model (note that it is \( \hat {\mathbf {b}} \) that is used to obtain the transition rates hab(t) in the algorithm above). Confidence intervals can be obtained by drawing from a multivariate normal distribution with mean \( \hat {\mathbf {b}} \) and variance \( \hat {\mathbf {V}} M \) times [12, 19]. For each draw m, the simulation algorithm above is repeated using the sampled \({\hat {\mathbf {b}}_{m}}\) instead of \( \hat {\mathbf {b}} \) to calculate the transition rates (and therefore event times). The variance of the M sets of estimates is then calculated and used to produce confidence intervals via normal approximation.
When the general simulation approach was utilised, 1000000 simulated patients were used (N=1000000) for the point estimates. 100000 simulated patients (N=100000) were repeated 500 times (M=500) for the corresponding confidence intervals. These values were chosen as a balance to minimise the Monte Carlo error and computational time. To produce the confidence intervals for the transition probabilities from the "AIC" model, where probabilities were calculated at 165 equally spaced time points, took 48.5 minutes on a standard HP laptop with i5 processor and 8 GB of RAM.
All analyses were performed in Stata version 15.1 and the code can be found in Additional file 1. The parametric transition rates were obtained using merlin, version 1.12.0 dated 20/09/2020. All predictions and confidence intervals obtained via the general simulation algorithm were achieved using predictms, version 4.0.0 dated 28/10/2020, from the multistate package. The Aalen-Johansen estimates were obtained using msaj, version 1.0.1 dated 11/09/2020, also part of the multistate package.
The analysis was performed on the publicly available los.data from the R package etm [24]. This is a sample from an observational cohort study conducted to analyse the burden of HAIs in intensive care, see Beyersmann et al. [29] for details. 756 patients were admitted to hospital (all patients started in state 1 at time 0). 632 patients did not acquire an infection during the study, of which 475 were discharged and 157 died. 124 patients did acquire an infection, of which 90 were discharged and 34 died. There was no censoring in this sample and the last event occurred 82 days after admission.
Transition rates
Table 1 gives the AIC for each distribution fitted to each transition separately. The AIC indicated that the following models gave the best fit for each transition and were therefore chosen for the "AIC" model:
Transition 1: Royston-Parmer model with 4 degrees of freedom.
Table 1 AIC for each parametric model fitted to each transition separately (to determine the "AIC" model)
Transition 2: Generalised gamma model.
Transition 4: Log-normal model.
Figure 2 illustrates the transition rates from the "AIC" model. The point estimates and confidence intervals are shown from the time of the first event until the last event for each transition by a solid line. The corresponding intervals were [3,40],[3,82],[3,69],[5,78] and [6,54] for transitions 1-5, respectively. The point estimates were extrapolated to cover the interval [0,82] with a dashed line. It was evident that the transition rates were not constant over time and transition 2 (admission to discharge without HAI) appeared to deviate most drastically from this assumption. Additional file 3: Figure S1 compares the transition rates from the "AIC" model to the "Exp" model, "RP(4)" model and to non-parametric estimates obtained using the Epanechnikov kernel smoother. The smoothed non-parametric estimates varied depending on the kernel type and bandwidth used, however, in all cases, the transition rates were clearly not constant.
Transition rates from the "AIC" model with 95% confidence intervals (shaded region). Point estimates and confidence intervals were defined from the time of the first event until the last event for each transition (solid lines). The point estimates were extrapolated to cover the interval [0,82] (dashed line)
Transition probabilities
Figure 3 compares the transition probability estimates from the three approaches with empirical estimates (starting in state 1 at time 0). The predictions from the "AIC" and "RP(4)" models had high concordance with the Aalen-Johansen estimates. As von Cube et al. noted [18], despite clear departures from the constant transition rates assumption, the "Exp" model performed well for states 4, 5 and 6 (death without HAI, discharge with HAI and death with HAI). There were some discrepancies with states 1, 2 and 3 (hospital admission without HAI, with HAI and discharge with HAI) up to 30 days after admission. Importantly, the predictions obtained from the "AIC" and "RP(4)" models captured the three day delay in acquiring a HAI (and in fact the delay in experiencing any event, as the minimum event time was 3 days since admission), which the "Exp" model could not capture. Where estimates are shown comparing approaches in the main body of the paper, Additional file 3: Figures S2, S5 and S6 show the corresponding graphs for the "AIC" model with confidence intervals.
Transition probabilities from state 1 at time 0 to each state for the different approaches: "AJ" (black), "Exp" (blue), "RP(4)" (red) and "AIC" (green). Note that there is considerable overlap between the "RP(4)" and "AIC" estimates
Additional file 3: Figure S3 illustrates the transition probabilities conditional on being in state 2 (in hospital with a HAI) by time 3. Predictions from the "AIC" and "RP(4)" models were slightly more consistent with the empirical estimates than those from the "Exp" model (especially in the first 20 days). Additional file 3: Figure S4 is the corresponding graph for the "AIC" model with confidence intervals.
Attributable mortality and population attributable fraction
About 20-25 days after hospital admission, AM was above 0 for the "AIC" and "RP(4)" models. This suggested that after 25 days, the probability of dying was greater for those with an infection than for those without. AM can be interpreted as, for example: an individual that acquired a HAI by time 10 had a 4.7 percentage point decreased probability of dying by time 10 compared to an individual who did not acquire a HAI. Alternatively, an individual that acquired a HAI by time 30 had a 1.4 percentage point increased probability of dying by time 30. These results were similar for PAF, suggesting that after 25 days the occurrence of a HAI increased the risk of death and therefore the overall probability of dying. PAF can be interpreted as, for example: the proportion of individuals dying by time 10 would have increased by 4.2% if there were no HAIs. Alternatively, the proportion of individuals dying by time 30 would have decreased by 0.9% if there were no HAIs (all predictions from the "AIC" model). Schumacher et al. [23] describes the phenomenon of the AM and PAF initially being lower than 0.
The relative differences in predictions between the models were greatest for AM and PAF, see Fig. 4 (although the absolute differences were small). The "Exp" model appeared to overestimate (until 15 days after admission) and then underestimate AM and PAF considerably; whereas both "AIC" and "RP(4)" predictions appeared to approximate the empirical estimates well. There was a slight inconsistency with the "AIC" model and the empirical estimates towards the end of the time window, however, this should not be over-interrupted due to the small number of events occurring past 50 days.
Attributable mortality (AM) and population attributable fraction (PAF) of HAIs for the different approaches: "AJ" (black), "Exp" (blue), "RP(4)" (red) and "AIC" (green)
Figure 5 illustrates the restricted expected length of stay for the three models with empirical estimates (starting in state 1 at time 0). The graph can be interpreted as follows: 82 days since hospital admission, on average a patient spent 8.68 (95% CI 8.04,9.39) days in hospital without a HAI and 1.98 (95% CI 1.53,2.58) days in hospital with a HAI (estimates taken from the "AIC" model). The empirical estimates were slightly better approximated by the "AIC" and "RP(4)" models (especially in the first 20 days). The confidence intervals for total hospital stay for the "AIC" model are shown in Additional file 3: Figure S7.
Length of stay in hospital without (state 1, left) and with (state 2, right) a HAI from state 1 at time 0 for the different approaches: "AJ" (black), "Exp" (blue), "RP(4)" (red) and "AIC" (green)
Figure 6 illustrates the residual, restricted expected length of stay, conditional on being in state 2 (in hospital with a HAI) by time 3. Between the interval [3,82] days, a patient would have spent on average 13.61 (95% CI 11.30,16.26) days in hospital with a HAI, given they had a HAI and were in hospital by time 3 (estimates taken from the "AIC" model). While predictions obtained from the "AIC" and "RP(4)" models were consistent with empirical results, Fig. 6 shows a large discrepancy between the latter and the "Exp" model. In the context of health economics, such differences could be non-trivial when translated into costs.
Residual length of stay in hospital with a HAI (state 2) from state 2 at time 3. In the left panel, for the different approaches:"AJ" (black), "Exp" (blue), "RP(4)" (red) and "AIC" (green). In the right panel, for the "AIC" model (solid line) with 95% confidence intervals (shaded region)
Assuming constant transition rates in a multi-state model can facilitate a basic understanding of the data and this approach has been well demonstrated by von Cube et al. [18]. However, this assumption may not always be plausible and, as a result, predictions may be misspecified. In the example shown, despite the transition rates not being constant (Fig. 2), the transition probabilities from the "Exp" approach were similar to the Aalen-Johansen estimates. However, this was not the case for some functions of the transition probabilities, for example, AM and PAF (Fig. 4). In addition, predictions from the "Exp" model starting in state 2 at time 3 had poorer concordance with the non-parametric estimates (Additional file 3: Figure S3 and Fig. 6).
This paper compared the "Exp" model to two parametric approaches, where predictions from the latter were obtained from the fitted model via a general simulation algorithm. The "AIC" and "RP(4)" predictions were more consistent with the Aalen-Johansen estimates than the "Exp" model for all metrics. The greatest improvements were seen in AM and PAF and when considering delayed entry (predictions starting from state 2 at time 3).
As with any parametric approach, assumptions need to be made regarding the most appropriate distribution for each transition. A balance needs to be sought in terms of parsimony and sufficient parameters to appropriately capture the hazard shapes. This work has highlighted the challenges of model selection as the "AIC" model (i.e. the collection of the best fitting distributions for each transition in terms of AIC) did not always have better concordance with the non-parametric estimates compared to the reference "RP(4)" model. Regardless of approach, we would always recommend sensitivity analyses around the assumptions of the baseline hazard. It is important to note that both approaches still performed better than the "Exp" model. For this data example, a conditional parametric model would have been more appropriate for any transitions that could not have happened before day 3 (by definition or design of the study). We chose not to consider a conditional model to be consistent with, and allow easier comparison with, the motivating paper by von Cube et al. [18].
In addition to being able to model the transitions with a range of parametric distributions, the general simulation algorithm has other advantages. It easily lends itself to extended predictions, such as length of stay, the probability of ever visiting a state and disease specific quantities, such as AM and PAF. Uncertainties can easily be obtained, which can facilitate a more comprehensive understanding of the data. There is also great flexibility available when modelling covariate effects, including time-dependent effects [12]. The approach generalises to more complex multi-state models, i.e. models with a greater number of states, greater number of transitions and backward transitions. It can also be applied to non-Markov models, unlike methods that rely on solving the Kolmogorov forward equations to obtain predictions.
A disadvantage of the general simulation algorithm is computational time. Although point estimates can be obtained relatively quickly, confidence intervals can require a considerable amount of time, especially in the case of more complicated user-defined functions. A balance between computational time and Monte Carlo error is therefore needed when choosing N (number of simulations) and M (number of repetitions for confidence intervals). One possible alternative would be to use a hybrid approach when calculating predictions, where the transition rates obtained through parametric methods are substituted into the non-parametric Aalen-Johansen estimator [15, 30]. This approach would greatly decrease computational time, however, is only applicable to Markov models.
We conclude that methods and software are readily available for obtaining predictions from multi-state models that do not assume constant transition rates. The multistate package in Stata facilitates a range of predictions with confidence intervals obtained from a fitted multi-state model via a general simulation algorithm. We agree that assuming constant transition rates can provide a quick, basic understanding of the data; however, we recommend a more sophisticated parametric approach for a comprehensive understanding that includes uncertainty.
The dataset analysed is publicly available in the Comprehensive R Archive Network package etm [24]. Stata code is available in Additional file 1 of this paper.
AIC:
Akaike information criterion
Attributable mortality
HAI:
PAF:
Population attributable fraction
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Grand MK, Putter H. Regression models for expected length of stay. Stat Med. 2016; 35(7):1178–92.
Schumacher M, Wangler M, Wolkewitz M, Beyersmann J. Attributable mortality due to nosocomial infections. A simple and useful application of multistate models. Methods Inf Med. 2007; 46(5):595–600.
Allignol A, Schumacher M, Beyersmann J. Empirical transition matrix of multi-state models: The etm package. J Stat Softw. 2011; 38(4):1–15.
Aalen OO, Johansen S. An empirical transition matrix for non-homogeneous Markov chains based on censored observations. Scand J Stat. 1978; 5:141–50.
Andersen PK, Borgan O, Gill RD, Keiding N. Statistical models based on counting processes. New York: Springer-Verlag; 1993.
Syriopoulou E, Mozumder SI, Rutherford MJ, Lambert PC. Robustness of individual and marginal model-based estimates: A sensitivity analysis of flexible parametric models. Cancer Epidemiol. 2019; 58:17–24.
Crowther MJ, Lambert PC. Simulating biologically plausible complex survival data. Stat Med. 2013; 32(23):4118–34.
Beyersmann J, Gastmeier P, Grundmann H, Bärwolff S, Geffers C, Behnke M, Rüden H, Schumacher M. Use of multistate models to assess prolongation of intensive care unit stay due to nosocomial infection. Infect Control Hosp Epidemiol. 2006; 27(5):493–9.
Cook RJ, Lawless JF. Statistical issues in modeling chronic disease in cohort studies. Stat Biosci. 2014; 6:127–61.
MJC was funded by a MRC New Investigator Research Grant, Grant/Award Number: MR/P015433/1. PCL was funded by the Swedish Research Council [2017-01591]. The funders had no role in data collection and analysis, decision to publish or preparation of the manuscript.
Biostatistics Research Group, Department of Health Sciences, University of Leicester, George Davies Centre, University Road, Leicester, LE1 7RH, UK
Micki Hill, Paul C. Lambert & Michael J. Crowther
Department of Medical Epidemiology and Biostatistics, Karolinska Institutet, Box 281, Stockholm, 17177, Sweden
Paul C. Lambert & Michael J. Crowther
Micki Hill
Paul C. Lambert
Michael J. Crowther
MH performed the data analysis, wrote the Stata code and drafted the manuscript. PCL and MJC supported the analysis and provided revisions and improvements to the manuscript. All authors read and approved the final manuscript.
Correspondence to Micki Hill.
Additional file 1
Stata code for predictions. A Word document (.docx) with Stata code to analyse the HAI data. It formats the los.data dataset (once imported from the R package etm), fits the parametric models and obtains the predictions from the different approaches (Aalen-Johansen estimates, "Exp" model, "AIC" model and "RP(4)" model). The predictions include all those calculated in this paper: transition probabilities, AM and PAF and expected length of stay.
Further details of the analysis approaches. A PDF giving details about how the Aalen-Johansen estimates were calculated and showing the functional forms of the transition rates for the three parametric approaches.
Additional figures. A PDF including supplementary figures from the HAI data analysis.
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visithttp://creativecommons.org/licenses/by/4.0/. The Creative Commons Public Domain Dedication waiver (http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated in a credit line to the data.
Hill, M., Lambert, P.C. & Crowther, M.J. Relaxing the assumption of constant transition rates in a multi-state model in hospital epidemiology. BMC Med Res Methodol 21, 16 (2021). https://doi.org/10.1186/s12874-020-01192-8
Competing risks
Markov processes
Expected length of stay


text:
Przełęcz Krzyżówka (także Przełęcz Huta, 745 m) – przełęcz na granicy Beskidu Niskiego i Beskidu Sądeckiego, położona powyżej Krynicy-Zdroju.

Na przełęczy znajduje się skrzyżowanie dróg nr 75 oraz nr 981.

Szlaki turystyczne
  – niebieski szlak: Krynica-Zdrój – Przełęcz Krzyżówka (745 m) – Kamianna

Przełęcze Beskidu Niskiego
Polskie przełęcze Beskidu Sądeckiego

redpajama_set_name: RedPajamaWikipedia


Hackers Trying to Steal Drug Formulas for Counterfeiters Can Wreak Havoc - MWA CONSULTING, INC.
IT expert advocating for stronger pharma IT security.
This article highlights the threat that hackers and cyber-criminals pose to drug and device manufacturing systems. MWA assesses the security of your IT systems during audits, particularly those that consider Part 11 compliance. If you feel that your organization is protected, read this article and share it with your IT personnel. You may want to reconsider your conclusions.
Hackers trying to steal drug formulae for counterfeiters can wreak havoc says an IT expert advocating stronger security for networked pharmaceutical manufacturing facilities.


text:
CF Manny López #59
Columbus Coalition BSA
Age: 23 | Bats: R | Throws: R | Morale: Very Unhappy
BNN Index | BSA | SCORES | STANDINGS | STATS | TEAMS | PLAYERS | TRANSACTIONS | HISTORY
CF MANNY LÓPEZ #59 - AGE: 23 - BATS: R - THROWS: R - MORALE: VERY UNHAPPY
70 168 36 12 0 0 12 14 39 .214 .287 .286 4 -0.2
Range: - 11 18
Arm: 1 13 17
1st Base: 13 Center Field: 18
2nd Base: 9 Right Field: 20
3rd Base: 8
08/20/2035 NAS Loss, 1-10 2 0 0 0 0 1 0 0
08/17/2035 @ ORL Loss, 2-3 3 0 3 1 0 0 0 0
08/14/2035 NEV Loss, 6-7 3 0 0 0 0 1 1 0
08/12/2035 @ BOS Loss, 2-4 4 0 1 0 0 0 0 0
08/11/2035 @ BOS Win, 7-6 0 0 0 0 0 0 0 0
08/08/2035 @ NAS Loss, 2-4 4 0 1 1 0 0 2 0
08/04/2035 NEW Win, 5-3 3 0 0 0 0 0 0 0
08/01/2035 @ SCS Win, 6-0 1 1 1 0 0 0 0 0
Soft-spoken and goes about his business.
Born in: Zaraza, GUA
Nationality: Venezuelan
Salary: $400,000
Major Service: 1 Year, 118 Days
Service This Yr: 148 Days
40-Man Service: 4 Years, 145 Days
Drafted: -
BASEBALL STARS ASSOCIATION 2035 BATTING STATS
Overall 49 188 4 0 0 3 0 2 48 12 4 2 66.7 14.5 2.83 -0.2
Versus Left 53 14 7 0 0 2 8 4 9 0 0 1 .264 .328 .396 .724
Versus Right 115 22 5 0 0 10 8 10 30 2 0 3 .191 .269 .235 .504
Last Week 2 0 0 0 0 0 0 1 0 0 0 0 .000 .333 .000 .333
April 64 9 4 0 0 7 5 4 19 2 0 1 .141 .197 .203 .400
May 22 6 2 0 0 1 4 1 2 0 0 1 .273 .333 .364 .697
Spring Training 34 10 2 1 0 3 2 7 8 0 0 0 .294 .415 .412 .826
Last Year 230 51 10 2 4 23 35 20 53 1 0 5 .222 .297 .335 .632
Career 398 87 22 2 4 35 52 34 92 3 0 9 .219 .293 .314 .607
Home 67 14 6 0 0 7 5 9 15 1 0 1 .209 .308 .299 .606
Road 101 22 6 0 0 5 12 5 24 1 0 3 .218 .273 .277 .550
Close/Late 43 10 2 0 0 0 5 4 10 0 0 2 .233 .327 .279 .606
Inning 7-9 52 11 3 0 0 0 5 2 9 0 0 2 .212 .268 .269 .537
Extra Innings 6 2 1 0 0 0 2 3 1 0 0 0 .333 .556 .500 1.056
1 Outs 50 14 5 0 0 8 6 7 6 2 0 1 .280 .367 .380 .747
Bases Empty 99 22 9 0 0 0 12 9 25 0 0 2 .222 .300 .313 .613
Runner on 2nd 14 2 0 0 0 0 1 0 2 0 0 0 .143 .143 .143 .286
Runner on 3rd 5 0 0 0 0 3 0 0 2 2 0 1 .000 .125 .000 .125
Runner on 1st & 3rd 3 2 0 0 0 2 0 0 0 0 0 0 .667 .667 .667 1.333
Runner on 2nd & 3rd 8 3 1 0 0 4 1 1 0 0 0 0 .375 .444 .500 .944
Two Strikes 70 10 4 0 0 6 7 6 39 1 0 2 .143 .228 .200 .428
First Pitch 24 6 3 0 0 1 3 0 0 1 0 1 .250 .269 .375 .644
Count 1-0 17 8 3 0 0 0 4 0 0 0 0 0 .471 .471 .647 1.118
Batting Leadoff 10 0 0 0 0 0 0 0 4 0 0 1 .000 .091 .000 .091
Batting 2nd 17 8 5 0 0 1 6 3 3 0 0 2 .471 .591 .765 1.356
Low Leverage 76 21 9 0 0 3 11 5 16 0 0 2 .276 .337 .395 .732
Medium Leverage 63 9 2 0 0 4 5 7 15 2 0 2 .143 .243 .175 .418
High Leverage 29 6 1 0 0 5 0 2 8 0 0 0 .207 .258 .241 .499
2034 Columbus - BSA 22 82 230 51 10 2 4 23 35 20 5 1 53 8 6 .222 .297 .335 .632 73 71 0.9
2035 Columbus - BSA 23 70 168 36 12 0 0 12 17 14 4 2 39 4 2 .214 .287 .286 .573 57 57 -0.2
Total BSA
2030 Jasper - A 18 63 200 50 10 1 7 25 23 12 8 0 58 7 4 .250 .318 .415 .733 94 96 0.0
2031 Jasper - A 19 149 537 118 31 5 9 79 61 41 15 5 180 17 12 .220 .291 .346 .637 67 65 0.0
2032 Yellowknife - R 20 62 253 77 18 0 14 47 40 14 6 3 48 3 3 .304 .351 .542 .893 139 137 0.0
2033 Nantucket - A 21 45 127 33 3 2 5 16 14 12 2 0 41 2 3 .260 .333 .433 .766 102 97 0.0
2035 Venezuela - TRN 23 4 0 0 0 0 0 0 0 0 0 0 0 0 1 .000 .000 .000 .000 -100 100 0.0
Total PIO (24 & Under)
2031 Jasper - PIO (24 & Under) 19 4 9 3 1 0 0 3 1 2 2 1 4 1 0 .333 .500 .444 .944 0 0 0.0
2032 Yellowknife - BR (22 & Under) 20 3 11 3 1 0 1 4 3 2 0 0 4 1 0 .273 .385 .636 1.021 0 0 0.0
2030 Jasper - A 1B 2 0 2 0 0 2 0 1.000 2.0 9.00 0.0 .000
2030 Jasper - A 2B 1 0 1 1 0 2 0 1.000 2.0 9.00 -0.0 .987
2030 Jasper - A LF 2 0 0 0 0 0 0 .000 3.0 0.00 0.0 .000
2030 Jasper - A CF 7 2 4 0 0 4 0 1.000 29.0 1.24 +0.7 1.311
2030 Jasper - A RF 58 58 88 6 1 96 2 .979 462.0 1.83 +6.3 1.106
2031 Jasper - A 3B 2 0 0 0 0 0 0 .000 4.0 0.00 0.0 .000
2031 Jasper - A LF 2 0 2 0 0 2 0 1.000 5.0 3.60 +0.1 1.042
2031 Jasper - A CF 147 146 322 14 4 340 4 .988 1259.0 2.40 +15.6 1.065
2032 Yellowknife - R 1B 0 0 0 0 0 0 0 .000 0.0 0.00 0.0 .000
2032 Yellowknife - R LF 14 14 21 3 1 24 0 1.000 119.0 1.82 +2.0 1.176
2032 Yellowknife - R CF 50 48 119 1 0 120 0 1.000 413.2 2.61 +8.5 1.095
2032 Yellowknife - R RF 1 0 0 0 0 0 0 .000 1.0 0.00 0.0 .000
2033 Nantucket - A C 12 11 92 10 0 102 0 1.000 102.2 8.94 -1.2 .562 5 13 5 38.5
2033 Nantucket - A 2B 1 0 0 1 1 1 0 1.000 1.0 9.00 -0.0 .999
2033 Nantucket - A 3B 1 0 0 0 0 0 0 .000 1.0 0.00 0.0 .000
2033 Nantucket - A LF 5 4 7 0 0 7 0 1.000 34.0 1.85 +0.3 1.063
2033 Nantucket - A CF 21 11 26 1 0 28 1 .964 117.0 2.08 +1.5 1.080
2033 Nantucket - A RF 5 2 5 0 0 5 0 1.000 24.0 1.88 +0.2 1.063
2034 Columbus - ML C 1 1 7 0 0 7 0 1.000 8.0 7.88 -0.1 .000 0 1 0 0.0
2034 Columbus - ML 1B 2 0 4 2 0 6 0 1.000 3.1 16.20 +0.1 1.086
2034 Columbus - ML 2B 2 0 0 1 0 1 0 1.000 5.0 1.80 +0.0 1.003
2034 Columbus - ML 3B 2 0 1 0 0 1 0 1.000 3.0 3.00 0.0 .000
2034 Columbus - ML LF 5 1 0 0 0 0 0 .000 15.0 0.00 0.0 .000
2034 Columbus - ML CF 65 63 187 3 1 192 2 .990 558.1 3.06 +11.3 1.081
2034 Columbus - ML RF 3 2 1 0 0 1 0 1.000 18.0 0.50 +0.0 1.053
2035 Columbus - ML C 2 2 9 0 0 9 0 1.000 18.0 4.50 -0.3 .000 0 5 0 0.0
2035 Venezuela - TRN 1B 1 0 0 0 0 0 0 .000 2.0 0.00 0.0 .000
2035 Columbus - ML 2B 3 2 3 5 3 8 0 1.000 19.0 3.79 +0.6 1.216
2035 Columbus - ML 3B 8 0 1 0 0 1 0 1.000 10.0 0.90 -0.7 .000
2035 Columbus - ML LF 1 1 3 0 0 3 0 1.000 8.0 3.38 +0.8 1.551
2035 Venezuela - TRN LF 3 0 1 0 0 1 0 1.000 4.0 2.25 +0.0 1.018
2035 Columbus - ML CF 32 26 89 2 0 92 1 .989 245.1 3.34 +0.7 1.010
2035 Columbus - ML RF 19 18 39 1 0 41 1 .976 146.1 2.46 +1.0 1.044
03/16/2028 Signed a minor league contract out of Venezuela with the Baffin Island Beavers organization.
01/01/2029 OSA scouting updated ratings (potential): Contact: - (3); Power: - (11); Eye: - (10).
01/01/2030 OSA scouting updated ratings (potential): Contact: - (7); Power: - (6); Eye: - (13).
06/03/2030 Released by the Baffin Island organization.
06/19/2030 Signed a minor league contract with the Irvine Gang Green organization.
04/17/2031 Injured (Wrist soreness), day-to-day for 1-2 weeks.
05/01/2031 Injured (Bruised forearm), day-to-day for 2 days.
06/09/2031 Injured (Strained back), day-to-day for 4 days.
10/11/2031 Injured (Knee contusion), day-to-day for 3 days.
10/29/2031 Wins the 2031 PIO (24 & Under) PIO Glove Wizard Award at CF.
02/14/2032 Was claimed off waivers by Philadelphia while with Irvine.
07/29/2032 Injured (Strained back), day-to-day for one week.
08/05/2032 Injured (Bruised knee), day-to-day for 3 days.
08/10/2032 Injured (Hamstring soreness), day-to-day for 4 days.
08/15/2032 Was selected to the 2032 BSA Rookie League All-Star Game.
10/23/2032 Finished 2nd in 2032 BR (22 & Under) SL1 Most Valuable Player Award voting.
06/16/2033 Injured (Strained hamstring), out for 5-6 weeks.
01/12/2034 Was claimed off waivers by Atlanta while with Philadelphia.
04/16/2034 Was claimed off waivers by Kansas City while with Atlanta.
05/01/2034 Was claimed off waivers by Atlanta while with Kansas City.
05/15/2034 Was claimed off waivers by Columbus while with Atlanta.
06/23/2034 Injured (Bruised thumb), day-to-day for 1 day.
02/12/2035 Named to the 2035 World Baseball Classic for Venezuela.
08/11/2035 Injured (Sprained knee), day-to-day for 3 weeks.

redpajama_set_name: RedPajamaCommonCrawl


Over the past three decades, the volume of the international coal trade has quadrupled, rising from 291,000 tonnes in 1982 to 1,283,000 tonnes in 2012. With that growth has come an increase in shipping accidents, resulting in loss of life to crews; damage to harbors, reefs, and other marine ecosystems; and financial losses for shippers and insurers.
December 22 2001 150 km off the Azores island of Graciosa, Atlantic Ocean Colombia Christopher Puerto Bolivar Redcar, United Kingdom 160000 ship sank with loss of 27 crew.
June 8 2007 Nobbys Beach, Newcastle (Article) Australia Pasha Bulker Newcastle empty at time of accident grounded on beach, salvaged, repaired and renamed MV Drake.
February 5 2010 Barbers Point Harbor, Hawaii, United States United States Vogetrader unknown Hawaii, United States unknown The ship grounded on a coral reef Barbers Point Harbor, Hawaii. In 2014 it was announced that the shipowner, Denak Ship Management and Vogetrader Shipping Inc., would pay $840,000 to the National Oceanic and Atmospheric Administration for the damage to the reef.
December 24 2010 Trenton Channel in Detroit United States McKee Sons unknown Detroit Edison Pier approx 13,000 tonnes The 579-foot ship, which was being pushed by the tug Invincible, ran aground on Christmas Eve. It was was freed two days later with no damage reported.
August 3 2011 near Transylvania, Louisiana United States unnamed barge unknown barge sank after breaking free from tug.
August 4 2011 off the coast of Mumbai (Article) India MV Rak Lubuk Tutung, Indonesia Dahej, Gujarat, India 60000 Sank, oil spill of up to 290 tones of fuel oil.
September 27 2011 Yangtze River, Wuhan City, Hubei Province China unnamed barge 4080 barge sank, salvaged, 11 crew rescued.
December 9 2012 Westshore Terminals Canada Cape Apricot Westshore Terminals unknown Coal ship destroyed 100 metres of coal conveyer system; under control of pilot.
January 13 2013 Puerto Drummond (Article) Colombia unnamed barge Puerto Drummond approx 2,000 tonnes Barge threatened with sinking; company dumped coal overboard.
February 16 2013 off the coast of Pangasinan province Philippines M/V Harita Bauxite Indonesia China ship sank, Reported that 1 crewman dies and 14 were missing.
March 24 2013 Mississippi River near Cape Girardeau, Missouri United States 3 unnamed barges unknown 313 laden coal barges break loose from tug, 3 damaged and partially sunk.
August 19 2013 off Richards Bay (Article) South Africa MV Smart Richards Bay China 127000 Ship sank. Crew rescued.
January 20 2014 Batanghari River, Jambi, Sumatra Indonesia Unnamed barge and tug Barge and tug sank, 2 crew missing.
July 14 2014 Port Rupert, British Columbia (Article) Canada Amakusa Island Port Rupert 80000 Grounded, ripped 2 metre hole in ourter hull, temporary repairs and continued.
September 20 2014 near Bayfront Festival Park, Duluth Harbor, Minnesota United States Paul R. Tregurtha Duluth, Minnesota 62,000 tonnes Grounded when it failed to make a turn. No damage or injuries.
November 3 2014 near bridge connecting Grosse Ile and Trenton, Detroit River, Michigan United States unknown Toledo Trenton, Michigan 10-15,000 tonnes The ship grounded near the bridge connecting Grosse Ile and Trenton; bridge closed briefly; no damage and no injries.
July 27 2015 Qisha port, Guangxi Zhuang Autonomous Region, China China Names of 5 ships unknown unknown Qisha, China unknown Five Vietnamese-registered coal ships sank, one was grounded. It was reported that 48 crew from the six vessels were rescued.
October 2015 Passur Bangladesh unknown unknown unknown unknown A fully loaded coal barge sank in Passur near the Sundarbans.
February 2016 Faux Cap, southwestern coast Madagascar New Mykonos Richards Bay, South Africa Vizag port, India 160,000 tonnes The New Mykonos rammed into a sandbar and got stuck off Faux Cap. After days of pounding by the sea, the boat broke in two and sank, with some coal spilling into the sea and some structure remaining visible.
January 2017 south of Hiron Point, Bay of Bengal (Article) Bangladesh MV Aij Gati unknown Noapara, Bangaldesh 1,000 tonnes MV Aij Gati, carrying more than 1,000 tonnes of coal, sank 10-15 kilometres south of Hiron Point and about 100km from Mongla Port. The site of the sinking was reported as being only 22km from the Sundarbans, the world's largest mangrove forest.
May 24, 1966: The MV Kaitawa sank off Cape Reinga in New Zealand with the loss of all 29 crew. The ship was carrying 2957 tons of coal from Westport on the South Island to Portland at Whangarei on New Zealand's North Island.
February 12, 1983: The Marine Electric sank off the coast of Chincoteague, Virgina while loaded with approximately 24,800 tons of coal which were destined for a power station at Somerset, Massachussetts. 31 crew died with only three surviving.
December 25,1983: A valve on an unnamed barge froze open allowing one of three barges in a lock on the Monongahela River in Pennsylvania to fill with water and sink. The barge was carrying 900 tons of coal.
September 8, 2000: The Eurobulker IV ran aground off the coast of Portovesme, Sardinia, Italy. The ship, which was registered in Saint Vincent and the Grenadines, was carrying approximately 14,000 tons of coal. The ships crew were evacuated at the time of the grounding. Efforts were made to remove the cargo from the ships holds but the operation was suspended due to bad weather. In a storm the ship split in two and sank on October 2.
December 22, 2001: The Christopher sank near the Azores, Portugal killing all 27 crew members. The ship was carrying 160,000 tonnes of coal from Colombia to the United Kingdom.
January 15, 2008: Taiwan: a Panamanian cargo ship transporting coal from Hong Kong was capsized by a wave near Taiwan. All eight crew members died.
February 5, 2010: The Vogetrader grounded on a reef on its way to Barbers Point Harbor, Hawaii, United States. It was reported that the ship's grounding damaged over 100,000 coral colonies. In 2014 it was announced that the shipowner, Denak Ship Management and Vogetrader Shipping Inc., would pay $840,000 to the National Oceanic and Atmospheric Administration for the damage to the reef.
April 3, 2010: On 3 April 2010, Shen Neng 1 was transiting from Gladstone, Queensland to China carrying a cargo of 65,000 tonnes of coal, she ran aground 70km off Great Keppel Island, Australia. She was severely damaged on her port side, and a 3 km long oil slick was later reported to have been seen. The accident occurred in a restricted area which forms part of the Great Barrier Reef Marine Park, a World Heritage Site designated by UNESCO. Two investigations were held into the grounding, one by the Australian Maritime Safety Authority (AMSA) and the Australian Transport Safety Bureau (ATSB). The preliminary ATSB investigation found fatigue as the major contributor to the grounding. (See Shen Neng 1 bulk carrier crashing into the Great Barrier Reef for more details).
approx December 16, 2010: Bloomberg report that a barge owned by Vale sank "this week". The report states that the barge was carrying 70 tonnes of coal belonging to Vale and 3000 tonnes belonging to "another exporter".
December 24, 2010: The 579-foot ship McKee Sons, which was being pushed by the tug Invincible, ran aground in the Trenton Channel in Detroit on Christmas Eve on its way to the Detroit Edison Pier. It was was freed two days later with no damage reported.
May 8, 2011: A ship loaded with coal ran aground and destroyed a large portion of a marine sanctuary in the southern Philippines, damaging coral reefs in the Sarangani Bay. The Panamanian-registered ship with 21 Filipino crew members was headed to India from Australia carrying 65,000 tons of coal. An investigation showed the ship's crew miscalculated its path and sailed into shallow waters.
August 4, 2011: The Sally Bromfield was towing 35 barges on the Mississippi River near Transylvania in north Louisiana. A coal barge sank in the river channel. It was reported that another barge was partially submerged and eight others damaged. However, it was unclear whether these too were carrying coal.
August 5, 2011: The Panamanian-registered MV Rak, which was carrying 60,000 tonnes of Indonesian coal, sank approximately 20 nautical miles off the Mumbai. The inspector general of the Coast Guard, Mr SPS Basra, stated that the suspected cause was leaks in the hull allowing water into the ship's holds. The ship was also carrying 290 tons of fuel oil and 50 tons of diesel at the time of sinking. (See The sinking of the MV Rak off Mumbai, India for more details).
September 27, 2011: A coal barge travelling on the Yangtze River in Wuhan City, Hubei Province in China sank with 4500 tons of coal on board. The eleven crew were rescued and salvage of the ship undertaken.
August 16, 2012: The MV Marine 7 sank near Mongla Port in Bangladesh after colliding with two other ships. The ship was carrying 400 tonnes of coal.
December 4, 2012: The Volgo-Balt 199 sank off the coast of Sile in Turkey during a storm. The ship was carrying 33 tons of Russian coal to Antalya, Turkey. It was reported that 1 of the crew died while three rescuers also died when their ship sank. Twelve crew from the two ships were reported missing.
December 9, 2012: When the coal carrier the Cape Apricot was docking at Westshore Terminals in Roberts Bank it destroyed a section of the offshore coal conveyor system and roadway. The Vancouver Sun reported that the accident happened at 1 a.m. when the ship "slammed into a trestle, the only link between the berth and the terminal, destroying more than 100 metres of it. The ship went right through the causeway, taking a road, the coal-carrying conveyor belt, and electric and water lines with it." The ship was under the control of a pilot at the time and an unspecified quantity of coal spilled into the water.
January 13, 2013: the crew of a barge owned by the Drummond Company dumped 2,000 tonnes of coal into the ocean in an unsuccessful bid to avoid the barge sinking. Drummond did not report the sinking to Colombian environmental authorities until 17 days after the accident even though its environmental licence required the company to report such an event within a maximum of three days. Director of the Colombian Environmental Licensing Agency (ANLA), Luz Helena Sarmiento, accused Drummond of covering the accident up. Drummond claimed the coal was dumped overboard to save lives but Sarmiento said that the company's licence did not allow it to dump into the ocean to save the barge. In 2011 Drummond exported approximately 25 million tons of coal from its mines near La Loma through Puerto Drummond. (See Drummond dumps coal overboard to prevent barge sinking for more details).
February 16, 2013: A Myanmar ship carrying 24 crewmen and a cargo of coal lost engine power and later sank off the coast of Pangasinan province in northwestern Philippines. The ship was en route to China with a load of Indonesian coal. It was reported that one crew member had died and fourteen others were missing. Nine crew were rescued with two of them injured.
March 24, 2013: The tug Sheila Johnson was towing 25 coal-laden barges the Mississippi River near Cape Girardeau, Missouri and was reported to have hit something in the water resulting in 13 barges breaking loose. Three barges were damaged and partially sank.
August 19, 2013 : A Greek-owned but Panamanian registered ship, MV Smart, was carrying an estimated 140,000 tonnes of South African coal when it was grounded on a sandbank after loading at the Richards Bay Coal Terminal. The MV Smart was carrying a shipment of coal to the Fangcheng port in China. After the ship became stranded three tug boats attempted to pull the vessel off but later in the day it buckled and then broke in two. The crew of 23 were airlifted off the ship. The MV Smart was reported to be carrying up to 2,000 tonnes of oil. The Department of Environmental Affairs said that the priority of an emergency response team was to protect the Mhlathuze Sanctuary south of the wreck. (See Sinking of the MV Smart near Richards Bay, South Africa for more details).
January 2014: A tugboat and a coal barge sank in the Batanghari River in Jambi, central Sumatra, Indonesia. The accident occurred when the tugboat became unstable due to the strong river current, capsized, hit the barge and sand. The crew of two were missing.
July 14, 2014: The side of a Japanese-owned coal ship was ripped open and will take two weeks to repair after striking rocks in the outer harbour of Prince Rupert in British Columbia. At the time of the accident the 228-metre Amakusa Island, which was loaded with 80,000 tonnes of coal, had a pilot on board and was being moved in good weather from a berth at the Ridley Terminal's coal terminal to an anchorage. (See The grounding of the Amakusa Island ship in Prince Rupert harbour, British Columbia, Canada for further details).
August 28, 2014: The Phu Cuong 0148, carrying 2,700 tonnes of coal sank offshore Vietnam's northern Hai Phong city approximately 90 kilometres east of the Hanoi. According to Vietnam's state-run news agency VNA the Phu Cuong 0148 ship sank five nautical miles southeast of Ha Mai island near Bach Long Vy Island in Hai Phong with six crew members onboard. It was reported that the ship sank in rough seas caused by a gale. It was later reported that the ship sank at 3.40pm and that a rescue operation had been launched for the missing crew members. The six crew members were later rescued.
September 20, 2014: The Paul R. Tregurtha, the largest ship operating on the Great Lakes, ran aground in front of the Bayfront Festival Park in Duluth Harbour, Minnesota, United States. The ship reportedly grounded at approximately 3pm and was freed later that evening. Following the accident the ship was moved to the Seaway Port Terminal for a damage inspection by the U.S. Coast Guard. At the time of the accident the ship was loaded with 62,000 tonnes of coal. Eyewitnesses stated that the grounding occurred when the Paul R. Tregurtha sought to avoid a smaller vessel in the shipping lane. (See The grounding of the Paul R. Tregurtha ship in Duluth Harbour, Minnesota, United States for further details).
November 3, 2014: The ship grounded near the bridge connecting Grosse Ile and Trenton on its way to DTE Energy's Trenton Channel Power Plant from Toledo. The bridge was closed for approximately half an hour until the ship was freed. No damage was reported and there were no injuries. (See The grounding of the of an unnamed coal ship near Trenton power station in Detroit River, Michigan, United States for further details).
July 27, 2015: In a monsoonal storm five Vietnamese-flagged coal carriers sank and another ran aground near South China's Qisha port in Guangxi Zhuang Autonomous Region. It was reported that 48 crew from the six vessels were rescued. Qisha is the main port for coal imports from Vietnam.
October, 2015: A fully loaded coal barge sank in Passur near the Sundarbans.
February, 2016: The Panamanian -registered New Mykonos rammed into a sandbar and got stuck off Faux Cap. After days of pounding by the sea, the boat broke in two and sank, with some coal spilling into the sea and some structure remaining visible.
January 13, 2017: MV Aij Gati, carrying more than 1,000 tonnes of coal, sank 10-15 kilometres from Fairboya and 100km from Mongla Port. The site of the sinking was reported as being only 22km from the Sundarbans, the world's largest mangrove forest. The ship's crew of 12 were rescued. "The ship sank after suffering a rupture to its bottom when loading coal at Fairboya," said Golam Faruk, the ships owner.
↑ "Total Coal Exports (Thousand Short Tons)," U.S. Energy Information Administration, accessed September 2014 (figures have been converted from short tons to metric tons or tonnes).
↑ Michael Cox, "Anger lingers over Kaitawa sinking", Waikata Times, October 24, 2010.
↑ "Coal ship which sank had patched hull", Observer Reporter, February 18, 1983.
↑ "Freak accident sinks coal barge", Observer Reporter, December 27, 1983.
↑ "Coal Carrier Sinks Off Sardinian Coast", Marine Link.com, October 4, 2000.
↑ "Accident Investigation: MV Christopher", Ministry of Communications and Works, Department of Merchant Shipping, Lemesos, approx 2003.
↑ Independent investigation into the grounding of the Panamanian registered bulk carrier Pasha Bulker on Nobbys Beach, Newcastle, New South Wales on 8 June 2007, Australian Transport Safety Bureau, May 23, 2008.
↑ "Shipwrecks Since 1833", Infoplease.
↑ "Damage Hawaii reef, pay $840,000: Cargo ship ran aground on reef off Oahu, harming coral", Seattle Times, November 25, 2014.
↑ Jackie Cowhig, Jonathan Saul, Henning Gloystein and Denise Luna, "Weather sinks Colombia coal barge, exports suffer", Reuters, December 17, 2010.
↑ US Coast Guard, "Coast Guard responds to barge aground in Trenton Channel, Detroit", News Release, December 24, 2010.
↑ US Coast Guard, "Barge no longer aground in Trenton Channel, Detroit", News Release, December 26, 2010.
↑ "Part of Mississippi River closed after barge sinks in north Louisiana", Nola.com, August 4, 2011.
↑ "Coal Carrier Accident Stops Shipping for Three Hours on Yangtze River in Wuhan", NTDTV.com, September 27, 2011.
↑ "Barges Break Loose Overnight On Mon River: All bridges have now reopened", CBS, January 19, 2012.
↑ Casey Conley, "Runaway coal barge sinks after tow strikes Pittsburgh bridge in high wind, swift current", Professional Mariner, April 24, 2012.
↑ "Coal-laden Indian vessel sinks near Mongla Port", Financial Express, August 16, 2012.
↑ "1,000-foot coal freighter stuck in St. Marys River", The Detroit News, August 15, 2012.
↑ Casey Conley, "Largest laker runs aground bow-to-stern, blocking St. Marys River", November 8, 2012.
↑ "Four die, while 12 remain missing as cargo ship sinks near Turkey", RT, December 04, 2012.
↑ Gordon Hamilton and Tiffany Crawford, "Ship crashes into dock at Westshore Terminals, spilling coal into water", Vancouver Sun, December 9, 2012.
↑ "1 dead, 14 missing in ship sinking off Pangasinan", Sun Star, February 17, 2013.
↑ Jen Baird, "Coal removed from barges that partially sank on Mississippi River", KFVS12, March 25, 2013.
↑ Lauren Rawlins, "Fears for habitat as ship left stranded", IOL News, August 20 2013.
↑ "Two missing after tugboat barge sink", Jakarta Post, January 22 2014.
↑ "Prince Rupert ship grounding highlights risk of oil spill, critics warn: Bulk carrier ripped hole in hull while moving to anchorage from terminal", Vancouver Sun, July 16, 2014.
↑ "Collier sinks offshore northern Vietnam, leaving 6 crew members missing", ShanghaiDaily.com, August 29,2014.
↑ "Breakfast @ Tuoi Tre News – August 31", Tuoitrenews.vn, August 31, 2014.
↑ "Large ship runs aground in Duluth Harbor", Kare11, September 20, 2014.
↑ "Paul R Tregertha Cleared After Running Aground", Fox 21, September 21, 2014.
↑ "Freighter runs aground at Grosse Ile free bridge", Detroit News, November 3, 2014.
↑ "Tug boats to rescue as ship breaks moorings", The Observer, September 10, 2014.
↑ Ebony Battersby, "Grounded coal ship paralyses harbour", The Observer, September 11, 2014.
↑ Angela Yu, "Five vessels sink, one runs aground in South China", IHS Maritime 360, July 29, 2015.
↑ Hedait Hossain Molla, "Coal-laden vessel sinks near Sundarbans", Dhaka Tribune, January 13, 2017.
↑ Ben Cubby and Edmund Tadros, "Ships ignored warning to leave", Sydney Morning Herald, June 9, 2007.
↑ "Salvage team boards beached ship", Sydney Morning Herald, June 9, 2007.
↑ "ATSB releases final Pasha Bulker report", Australian Transport Safety Bureau, May 23, 2008.
↑ 52.0 52.1 "Great Barrier Reef oil disaster fear from stricken ship", BBC News Online (4 April 2010). Retrieved on 4 April 2010.
↑ Owens, Jared (5 April 2010). "Leaking ship's owners face $1m fine", The Australian. Retrieved on 4 April 2010.
↑ Marine Safety Investigation Report - Preliminary, Australian Transport Safety Bureau, 3 May 2010.
↑ "Ship carrying coal destroys Philippine coral reef", San Diego Union-Tribune, May 12, 2011.
↑ "Capesize coal shipment set for China sinks off South Africa's Richards Bay", Platts, August 20, 2013.
↑ "Two missing after tugboat barge sink", Jakarta Post, January 2.2 2014.
↑ "Six missing from cargo ship sunk off Hai Phong", VietnamNet, August 31, 2014.
↑ Billy Wagness, "Paul. R Tregurtha, largest in Great Lakes fleet, freed after running aground in Duluth Harbor", NNCNOW.com, September 20, 2014.
Bob Burton, "Where coal ships go, trouble follows", Huffington Post, August 20, 2014.
Wikipedia also has an article on Coal transport accidents. This article may use content from the Wikipedia article under the terms of the GFDL.


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Hello Joliet
Joliet Ill...
Joliet Illinois Museums
A new museum honoring musicians and bands with ties to Illinois is planned for Joliet and today announced two upcoming day trips that invite participants to explore Illinois history and culture by motor bus. The museum, which is described as a "vibrant and dynamic cultural centre of all ages, races and musical styles," will be completed by the end of the year, according to a press release.
In partnership with the city, the museum has been identified as a partnership between the Illinois State Historical Society and the Joliet Museum of Art. The well-known reformer Dorothea Dix personally lobbied the Illinois General Assembly for the construction of a new penitentiary after an inspection of Alton Prison in 1847. The Gothic-style mansion was designed by Chicago architect and architect John F. Kennedy, who also designed Chicago's Water Tower as part of his "Castellate Gothic" architecture, according to the press release.
To this end, the organization intends to run a world-class museum that collects works by artists who have achieved national and international recognition, while offering educational programs. The ISM website, including the Hill, is carefully edited by the Illinois State Historical Society and the Joliet Museum of Art to support its future development. For more information about the museum, call (815) 927-1540 or visit www.art.org.
The Joliet Area Historical Museum hopes visitors will pay $20 to walk through the museum and listen to a presentation by the Illinois State Historical Society about the museum's history and history. The cost of the trip Lockport - to Joliet is $10 for adults and $5 for children under 12 and is covered by a $1,000 grant from the Illinois Department of Natural Resources.
With such a long list of artists and bands, it would be possible to update the museum's exhibits from time to time. In addition to the permanent collections, seasonal or special exhibitions could also be shown to highlight local artists, such as local musicians, musicians and musicians from the past or present.
The information on this website has been compiled from a variety of sources and is subject to change without notice by the City of Joliet as a result of updates and corrections. The information on the Site is provided only as provided by our Site and as such, you should always consult the official records of the City where legal reliability of this information is required.
Peerbolte said the museum intends to release the updated information once the state's public health benchmarks are approved. Peerbollo said they intend to announce the update as early as possible, as long as they allow a state benchmark for public health. The museum plans to announce the updated information in the next few weeks, he said: "The museum expects and intends to announce this update in early April, just as quickly as our state - the - public health standards - permit.
Those interested can visit the museum's website to see the 20 nominees in each of the acting categories. Moore said the leadership, led by a former prison guard, is being considered by museum staff as part of their ongoing research into public health issues in the prison system.
The first stop will be Dellwood Park, the first plated and registered African American in the city, so buy a family pass before you head down. On the way to the prison, just down the road, drive past Dellwoods Park (which is actually in Lockport) and stop for a picnic in the park, complete with picnic tables and picnic chairs.
Since the museum opened in autumn 2002, it has served as its most important exhibition gallery. The funny thing is that the historic ironworks district made its first visit to the Joliet Area Historical Museum when I took my toddler. Since then, the city has been waiting for it to take a lead role in running the site, and we have a timetable.
The Illinois State Museum (ISM) is a system of museums and galleries serving the people of Illinois. The Illinois State Museum Board is responsible for the day-to-day operation of our museum and its galleries. This is the position of the NGO and reflects the commitment of the Board at all levels to achieve national, national and international recognition of the museum.
The reality of the financial impact of COVID-19 is doubly significant for the Illinois State Museum and its members. The benefits of membership include access to the museum and all opening hours, as well as a number of other benefits. PeerBolting commented: "The Chicago Public Library, the Chicago Museum of Natural History and the University of Illinois at Urbana-Champaign.
The museum is currently in a membership campaign and will use the proceeds for its realization. Romero said he was hopeful the museum could open this year, but only if all the donations are included. There will be a full-day FOIA Fest conference dedicated to public information and an open day on Friday, June 16, from 2-4 p.m. at the Illinois State Museum.
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Join us at our Annual Dinner 2020 at @ColwickHall. We have former Olympian Greg Rutherford MBE as guest speaker, discussing his journey to the top and the steps he took towards cementing and maintaining his success. Book tickets here. 👉 bit.ly/39WKfmz pic.twitter.com/t9lOTv0wRi
This year's conference will discuss the strategies, efficiencies and support on offer to business for the creation of competitive yet sustainable manufactured goods. To view the full agenda and book, please click here bit.ly/ME-Conf @DerbyUni @CarbonDerby @derbycollege pic.twitter.com/OE9Ni3uXWT
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New Brexit checklist for business launched
Marking a year before the UK's formal exit from the EU, the Chamber has launched a 'Brexit checklist' setting out the main issues firms need to be considering now to best position themselves to succeed in a post-Brexit environment.
The document has been produced by the Chamber's newly established Brexit Advisory Group, which aims to help East Midlands businesses navigate their way through the many questions and changes that Brexit will create.
"Despite there being only a year to go, considerable doubt remains over what the future holds for business, with things to consider ranging from customs arrangements through employing EU nationals to contracts and Intellectual Property rights." said Scott Knowles, Chief Executive at the Chamber.
"In business terms, a year is a very short time in which to go through all the documents, processes and programmes currently in use and revise all those that need it so that trading can continue post-Brexit,"
He added: "The uncertainty is the hardest thing to deal with because you don't know what outcome you're aiming for, so you have to plan for the worst case scenario while hoping for the best.
"What we have done with our briefing paper is set out what we see, and what our members tell us, are some of the more critical elements of the Brexit process that companies need to be looking at today to ensure they can continue to trade globally tomorrow."
The Brexit Advisory Group has been set up in collaboration with Chamber Strategic Partner RSM, one of the UK's leading providers of audit, tax and consulting services.
It is chaired by Dr Nik Kotecha OBE, Chief Executive at Loughborough-based Morningside Pharmaceuticals, which has supplied quality pharmaceuticals and medical device products to global markets for over 25 years.
The group will shape and help deliver the Chamber's wider Brexit influence and support programme, including its ongoing lobbying activities based on business intelligence and needs.
It will lead MPs briefings regionally and at Westminster, help to shape the Chamber's 2018 Manifesto on Brexit and provide up-to-date information to businesses through seminars, workshops and the media about potential outcomes of the UK leaving the EU.
Scott said: "Whether you voted for it or against it, Brexit is happening. What we need to do today is make sure we can, as far as possible, future-proof against any potentially negative impact of the divorce.
"The aim of the Brexit Advisory Group is to provide a conduit between business and Westminster so that the creators of jobs and wealth can influence the direction of Government and, at the same time, be accurately informed about at the earliest opportunity about the decisions and agreements being made that will affect them."
Kevin Harris, Office Managing Partner at RSM, a global company with local offices at Leicester and Nottingham, said: "We are pleased to be working with the Chamber to provide businesses with a voice in Westminster and the facts they need to make informed choices for a successful withdrawal from the EU and post-Brexit success.
"We'll get only one shot at a successful Brexit so it's important that we get right first time."
The Brexit briefing paper published today by the Chamber and available for free electronic download from its website is a checklist of things businesses should be focusing on now to minimise future risks resulting from Brexit.
Scott added: "The briefing paper is not a definitive list but it is a guide to what businesses need to be considering today. There will be updates as we get closer and closer to Brexit day, a year from today, but the message we need to get out there is that businesses can't sit back and wait to see what happens, they have to seize the initiative and plan for all eventualities if they are to succeed post-Brexit.
The Brexit Checklist for Business pdf briefing paper can be found here.
The Brexit Checklist word document is here.
Both are free downloads.
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The Blount County Community Action Agency is governed by a diverse group of people who represent the people served by its programs, local government and the community at large. Community Action Agencies (CAAs) are nonprofit private and public organizations established under the Economic Opportunity Act of 1964 to fight America's War on Poverty. Community Action Agencies assist low-income, needy and/or elderly individuals to help themselves in achieving self-sufficiency.


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Hamsik remains in Naples "Agreements not respected"
Resounding! Marek Hamsik's transfer to Dalian has jumped in the past few minutes. With a tweet, the company communicated its decision. "Napoli football has decided to suspend the sale of Marek Hamsik to the Chinese because the methods of payment of the agreed amount do not coincide with the agreements previously reached". It seems that the Chinese club wanted to pay in installments the 20 million agreed to define the purchase of the player. A solution that Aurelio De Laurentiis did not want to accept. Yesterday, the transfer formula had also been called into question: Naples would have imposed a permanent sale, while the Dalian would have wanted the loan with a redemption obligation.
The news came unexpectedly, last night, Hamsik had been celebrated by his companions, a way for the team to show the captain their love. But the statement issued by Aurelio De Laurentiis, Monday morning, during the presentation of the withdrawal of Dimaro, had created some perplexities: "Paying money, see camel", the president had said about the negotiation with the Chinese club. Probably, he already knew about Dalian's claims and the proposal to defer payment.


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redpajama_set_name: RedPajamaGithub


text:
Elon Musk is reportedly trying to raise billions of dollars to get Twitter out of debt: Musk's statement
Using NVIDIA's New 'Eye Contact' Feature in a Way You Never Thought: What It Would Look Like If Characters in Movies Watched You
What is the difference between USB-C and Thunderbolt connection type?
How to distinguish the types of USB-C standardThe organization...
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Hermann Schwarz
For the philosopher, see Hermann Schwarz (philosopher). For the company founder, see Rohde & Schwarz.
Not to be confused with Laurent Schwartz.
Karl Hermann Amandus Schwarz
(1843-01-25)25 January 1843
Hermsdorf, Silesia, Prussia
30 November 1921(1921-11-30) (aged 78)
Gewerbeinstitut
Cauchy–Schwarz inequality
University of Halle
Swiss Federal Polytechnic
Göttingen University
Karl Weierstrass
Ernst Kummer
Lipót Fejér
Harris Hancock
Gerhard Hessenberg
Paul Koebe
Leon Lichtenstein
Heinrich Maschke
Robert Remak
Theodor Vahlen
Ernst Zermelo
Karl Hermann Amandus Schwarz (German: [ˈhɛʁman ˈʃvaʁts]; 25 January 1843 – 30 November 1921) was a German mathematician, known for his work in complex analysis.
Life[edit]
Schwarz was born in Hermsdorf, Silesia (now Jerzmanowa, Poland). On 30 June 1912 he married Marie Kummer, who was the daughter to the mathematician Ernst Eduard Kummer[1] and Ottilie née Mendelssohn (a daughter of Nathan Mendelssohn's and granddaughter of Moses Mendelssohn). Schwarz and Kummer had six children, including his daughter Emily Schwarz.[1]
Schwarz originally studied chemistry in Berlin but Ernst Eduard Kummer and Karl Theodor Wihelm Weierstrass persuaded him to change to mathematics.[2] He received his Ph.D. from the Universität Berlin in 1864 and was advised by Ernst Kummer and Karl Weierstraß.[3] Between 1867 and 1869 he worked at the University of Halle, then at the Swiss Federal Polytechnic.[4] From 1875 he worked at Göttingen University,[4] dealing with the subjects of complex analysis, differential geometry and the calculus of variations. He died in Berlin.
Schwarz's works include Bestimmung einer speziellen Minimalfläche, which was crowned by the Berlin Academy in 1867 and printed in 1871, and Gesammelte mathematische Abhandlungen (1890).
Among other things, Schwarz improved the proof of the Riemann mapping theorem,[5] developed a special case of the Cauchy–Schwarz inequality, and gave a proof that the ball has less surface area than any other body of equal volume.[6] His work on the latter allowed Émile Picard to show solutions of differential equations exist (the Picard–Lindelöf theorem).[2]
In 1892 he became a member of the Berlin Academy of Science and a professor at the University of Berlin, where his students included Lipót Fejér, Paul Koebe and Ernst Zermelo. In total, he advised 20 Ph.D students.[3]
His name is attached to many ideas in mathematics,[1] including:
Additive Schwarz method
Schwarz alternating method
Schwarzian derivative
Schwarz lantern
Schwarz lemma
Schwarz's list
Schwarz minimal surface
Schwarz theorem (also known as Clairaut's theorem)
Schwarz integral formula
Schwarz–Christoffel mapping
Schwarz–Ahlfors–Pick theorem
Schwarz reflection principle
Schwarz triangle
Schwarz triangle map
Cauchy–Schwarz inequality.
Publications[edit]
Schwarz, H. A. (1871), Bestimmung einer speziellen Minimalfläche, Dümmler
Schwarz, H. A. (1972) [1890], Gesammelte mathematische Abhandlungen. Band I, II, Bronx, N.Y.: AMS Chelsea Publishing, ISBN 978-0-8284-0260-6, MR 0392470
^ a b c Agarwal, Ravi; Sen, Syamal (2014-11-11). Creators of Mathematical and Computational Sciences. Springer. pp. 297–298. ISBN 9783319108704.
^ a b O'Connor, J. J.; Robertson, E. F. "Schwarz biography". www-gap.dcs.st-and.ac.uk. The MacTutor History of Mathematics. Archived from the original on 2016-06-05. Retrieved 2016-05-22.
^ a b "The Mathematics Genealogy Project - Hermann Schwarz". www.genealogy.math.ndsu.nodak.edu. Retrieved 2016-05-22.
^ a b Chang, Sooyoung (2011-01-01). Academic Genealogy of Mathematicians. World Scientific. pp. 77–78. ISBN 9789814282291.
^ Bottazzini, Umberto (2003-04-30). "Algebraic truths vs geometric fantasies: Weierstrass' Response to Riemann". arXiv:math/0305022.
^ Schwarz, Hermann Amandus (1884). "Proof of the theorem that the ball has less surface area than any other body of the same volume". News of the Royal Society of Sciences and the Georg-August-Universität Göttingen. 1884: 1–13.
Wikimedia Commons has media related to Hermann Schwarz.
O'Connor, John J.; Robertson, Edmund F., "Hermann Schwarz", MacTutor History of Mathematics archive, University of St Andrews .
Hermann Schwarz at the Mathematics Genealogy Project
MGP: 7487
SNAC: w6kh13ct
Trove: 969739
WorldCat Identities (via VIAF): 61145857791723020403
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20th-century German mathematicians
ETH Zurich faculty
University of Göttingen faculty
Martin Luther University of Halle-Wittenberg faculty
Humboldt University of Berlin faculty
Corresponding Members of the St Petersburg Academy of Sciences
People from the Province of Silesia
People from Głogów County
Wikipedia articles with HDS identifiers
Wikipedia articles with MGP identifiers
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