Patterns of human growth established for one population have rarely been tested in other populations. In a previous study, three growth curves from puberty were modelled for each sex in a longitudinal study of a Caucasian population based on stature, age at peak of growth and biological maturation. Each curve represents the canalisation of growth associated with the type of puberty. The high precision (± 3 cm) of individual adult stature predictions shows that growth kinetics are already set up at puberty and are canalised depending on biological maturity. Our aim is to assess whether this model can be extrapolated to other populations to test whether growth canalisation is a population-dependent phenomenon or if the model reflects a canalisation pattern specific to our species. The modelled curves predicted adult stature with the same high degree of precision in basketball players and the Baka pygmies. Therefore, (1) the relationship between growth kinetics and age at maturity is similar in all populations and (2) growth according to pubertal stages follows the same canalisation patterns in the populations despite the wide differences in their average adult statures. It suggests that morphological diversity in modern humans results from processes taking place in early development. Maturity in Humans takes twice as long as in chimpanzees, and present accelerated growth at puberty that is not observed in our closest relative. Thus, the two main and distinctive traits of the human life cycle are the long period of growth (closely linked to the long human lifespan) and the presence of a growth spurt in adolescence1. These aspects have been observed in standard populations as well as in populations at the extremes of morphological variation, such as Pygmies2. In other words, despite the polymorphism of our species, there are aspects of growth and development that are similar between populations. The distinction between factors that are constant within our species and aspects that vary between populations allows us (1) to better understand what characterises our species compared to other species and (2) to establish the aspects that change according to the adaptation of different populations to the constraints in their environment.Among primates, humans are distinctive in two main respects, their polymorphism and their very particular growth pattern in absolute and relative terms, which defines what is called the ‘human life-cycle’. Despite differences in size and shape among modern human populations, human growth characterises by the presence of a childhood phase, the long lap of time between birth and age at first reproduction and relatively long brain development10. It is important to note that the degree of biological maturity is accompanied by different growth kinetics, so that children with early or late puberty have different kinetics to children with standard puberty12. The correlation observed between adult stature and stature in early adolescence has introduced the concept of growth canalisation15.The characterisation of growth in particular populations is therefore of particular interest from the evolutionary point of view. Many studies have been devoted to characterising the growth of different populations. Longitudinal studies of growth have used two methodologies, one based on the relationship between the increase in anthropometric characteristics (e.g. stature and weight) according to chronological age and the other relative to the biological maturity of individuals. In the latter, pubertal stages are defined from the age at the peak of growth and individuals are grouped into three categories according to whether secondary sexual characters show early, standard or late puberty according to specific sample19, but models have rarely been tested on populations with a different adult morphology. It has not been assessed whether growth canalisation established for one population can be extrapolated to track growth in another population, in other words, whether growth channels are population-dependent or whether the canalisation process reflects a larger phenomenon specific to our species, which can be modelled but is independent of phenotypic variations between populations.Studies on growth canalisation are based on and widely applied to studies of so-called standard Caucasian populations e.g.20, we modelled growth curves based on age and biological maturation in a so-called standard (Caucasian) population, from longitudinal data on stature, chronological age, age of peak of growth and secondary pubertal stages  age at maturity plays a key role in or is closely related to the type of growth kinetics the individual is following and (2) growth canalisation follows the same pathways, both aspects being independent of the average adult stature of populations.The stature and pubertal stages between 160 and 164 months of age in males, and the stature and age at menarche in female between 159 and 178 months are given in Table The values of the correlation coefficient between estimated and actual stature are very high for the two populations studied Table . In addi23. Sperlich et al.24 performed a stature prediction for boys affected by untreated constitutional stunting using bone age methods. The prediction for 16 out of the 49 boys tested (32.6%) deviates from the final stature by more than 5 cm. Ostojic25 established a model for predicting the stature of young Caucasian boys practising basketball, football, volleyball and swimming using the Tanner Whitehouse method based on hand X-rays. This method resulted in a prediction of adult stature within -5.8 cm and + 4.5 cm in 95% of cases. Ali et al.26 developed a model for predicting stature from curvilinear regression without exposing individuals to X-rays. Their method includes age at peak growth and is therefore only applicable beyond 12 years of age for females and 14 years for males. They obtained a prediction within − 3.3 and + 5 cm with a 90% confidence interval. Lee et al.27 made a fairly accurate prediction of adult stature from a multiple regression that includes biological parameters. However, they introduced into the prediction equation the age immediately preceding the age at the peak of growth on the growth curve. For this reason, this method is only applicable at the beginning of the pubertal period. Sherar et al.28 validated and demonstrated how adult stature can be predicted using reference values obtained from sex-specific and cumulative velocity curves. Individuals were classified as having late, standard, or early maturation based on their age at the peak of growth. This method gives a rather inaccurate prediction of the stature as adults at ± 5.3 cm in 95% of cases observed in 224 males and ± 6.81 cm in 95% of cases in 120 females. The non-invasive method of predicting stature based on skeletal age, suggested by Beunen et al.29, makes it possible to differentiate the maturation of individuals. The mean deviation of the predicted stature is 2.3 cm with a standard estimation error of 4.7 cm at the age of 13 years. More recently, Beunen et al.30 proposed a model for predicting the adult stature of females without using skeletal maturation. Adult height is estimated based on stature at age 13 or 14, leg length, sitting height, and age at menarche. In 95% of cases, the prediction is in the range of ± 5.4 cm at 13 years and ± 4.6 cm at 14 years. Our previous reference study20 showed that associating biological maturation with chronological age makes it possible to differentiate three groups of individuals according to pubertal stages: early, standard and late puberty. These three distinct maturation groups were used to model new mean growth kinetics curves, all three of which converge to 177 cm in males and 165 cm in females, suggesting that the final average stature reached does not depend on the different stages of puberty31. These growth curves describing canalisation have allowed individual growth in males and females to be monitored and show high efficiency in predicting adult stature. With previous models, the difference between actual and predicted adult stature is greater than 3 cm in 95% of cases, whereas our model predictions are within ± 3 cm of the actual adult stature in 95% of cases, and are therefore more accurate.Many models have been proposed to predict adult statureIn previous studies, models to predict adult size were obtained from industrialised populations and eventually applied to other populations whose morphology is close to that used to establish it. They have never been applied to populations with a very different adult morphology. In our study, we applied the model established from the study of a European population to two populations whose morphology lies at the extremes of current human variation, a group of basketball players and a population of Baka pygmies. The Baka pygmies is a seminomadic population of hunter-gatherers. Although the sample size in our study is limited, having 40 hunter-gatherer individuals monitored for longitudinal growth from adolescence to the adult stage based on a precise chronology is a unique case. The accuracy in the prediction of adult stature for these two populations was surprisingly similar to that for the original population, at ± 3 cm. This means that (1) the relationship between growth kinetics and age at maturity (around 11–15 years in boys and around 10–14 years in girls) is similar in all three populations and (2) that the growth depending on pubertal stages follows the same channels in all three populations despite the very wide difference in average adult stature between the Baka pygmies 153.7 cm) and the basketball players (197.8 cm). The accuracy of the model in predicting adult stature suggests that adult stature is determined as from puberty, that the stature differences between the populations at puberty Tables , 7 are r cm and tThe applicability of our model to populations with different adult statures means that the stature difference is already present at puberty. It may be suggested that the relationship between biological maturity and stature at puberty is specific to each population and may differ between populations from the adaptation of each one to its environment. This implies that the changes in growth in relation to environmental adaptation had to take place before puberty; any change after this stage would result in a change in the relationship between pubertal growth kinetics and adult stature, and the hypothesis of similar canalisations in different populations would therefore have limitations. Few studies have been done on growth processes related to the morphological adaptation of a given population to its environment. Although the morphology of Inuit, Nilotic (Nuer. Dinka. Turkana) and of other groups with a small average adult stature has often been interpreted as adaptations to the environment, the growth processes that accompany morphology have unfortunately been very little explored.32. In a study on growth in the Aeta (Philippines), Migliano et al.33 proposed that their small adult stature could be due to an early cessation of growth due to high mortality. If these results are confirmed, they would suggest that the small stature of the Aeta is not in itself an adaptation to life in the forest but a by-product of high mortality. The particularity in the Aeta would then be an early arrest of growth to ensure reproduction in a high risk environment. Since it occurs after the pubertal stage and thus probably without affecting age and stature at puberty, the hypothesis of a relationship common to all populations between growth kinetics at puberty and adult stature cannot be upheld and the possibility of extrapolating the model would be seriously limited. However, the lack of calibration of individual growth with chronology and some flaws in the statistical methods and interpretation of results cast serious doubts on the relevance of these conclusions35. In their study on Pume foragers, Kramer and Graves36 suggest that girls reach a higher proportion of their adult stature at puberty and that growth follows at low velocity in adolescence, and even after their first pregnancy, to reach a standard stature. They established that 93% of adult stature is already acquired at menarche. However, Walker et al.37 have estimated that 77 to 92% of adult stature is reached by 10 years of age in several forager and horticulturalist populations, so that 93% of adult stature at 12.5 years of age in the Pume does not seem to be of particular note. Furthermore, in basketball players, the proportion of adult stature at menarche varies from 97.3% in females with late-onset puberty to 99.1% in females with early-onset puberty  on births among Efe pygmies suggested that the Efe already have a small stature at birth. The pygmy phenotype therefore expresses a changing growth pattern well before puberty.The phenotype of Pygmies and other groups outside Africa with a small average stature and living in a forest environment has been interpreted as an adaptation to life in the forest40. This concept suggests that in mammals, irreversible developmental pathways are taken early in life in order to cope with environmental conditions and maximise reproductive fitness as adults. This concept mainly concerns processes occurring in the embryonic, foetal and perinatal periods and corresponds to the processes described for Pygmies. What is certain is that the earlier these changes take place during development, the less energy is required to produce them. Our results lead us to suggest that growth from puberty follows canalisation patterns that are shared by populations with very different adult morphologies and that each population is characterised by a particular relationship between stature and age at biological maturity. If so, the morphological diversity of our species would result from processes that take place in early development, no later than childhood, most probably as adaptive responses to environmental constraints.The idea that changes during the developmental period before puberty can ensure adaptation of a population to a particular environment is broadly in agreement with the concept of predictive adaptive responsesThe growth canalisation established in the study of a Caucasian population made it possible to predict the adult stature of each individual with a very small discrepancy (± 3 cm) (SI). If the result is also satisfactory for other populations with different average adult statures, this means that the kinetics are the same and therefore do not depend on the population analysed: they are independent of the diversity of adult morphology among populations and canalisations can therefore be extrapolated.To test whether the growth kinetics established from a standard reference population can be extrapolated to other populations with different adult morphologies, the mean growth curves obtained for the reference population Fig.  were use41. Pygmies share a common ancestor and split from Bantu-speaking populations at around 60,000 years BP42. The genetic bases of this very specific phenotype are polygenic. They include genes from the growth hormone axis (GH – IGFI) as well as genomic and gene regions that relate to the traits themselves45. The pygmy phenotype itself is usually interpreted as an adaptation to life in equatorial rain forests32.The second sample is made up of 40 individuals  from Moange le Bosquet, a Baka pygmy village in south-eastern Cameroon. African pygmies, who include many different groups, live in equatorial rain forests and grow to an average adult stature of < 155 cm2. Birth records had been kept for many years in Moange le Bosquet by nuns in a medical centre run by the catholic mission. They are available from 1980 to 1983 and from December 1987 to the present. We conducted one field study each year in May–June from 2007 to 2017, and an additional field study in October from 2011 to 2015. We systematically measured the stature and weight of Baka individuals of known age (3–25 years of age) living in the locality. Thanks to the birth records, we were able to describe growth and life history variables based on an absolute chronology46. The individuals included in the present study are a subset of those included in the previous one. We must remember that the Baka pygmies are semi-nomadic hunter-gatherers and although our field works were carried out systematically, performing the same measurements on the same individuals for many years is a very difficult task as either the individuals are not present in the locality at the time of our stay, they have established residence in a camp in the middle of the forest or in another locality. The stature of the 40 individuals included in our study was measured systematically for 8–10 years from adolescence to adulthood (from 12 to 19 years of age). Although this is a limited sample, it represents the only case of hunter-gatherers in which a longitudinal study could be conducted by taking measurements at adolescence and adult based on an absolute chronological calibration thanks to birth records . The secondary sex characteristics of the males and the age at menarche of the females were not recorded, but the age at peak growth in stature was determined from longitudinal data, making it possible to identify whether biological maturation was late, standard or early according to the criteria set out in Tables The Baka pygmies have an average adult stature of 146.7 cm (sd = 4.7) in females and 153.5 cm (sd = 6.2) in malesPrior informed consent was obtained from all participants and from both parents of any participants aged under 18. All methods are non-invasive and were carried out in accordance with relevant guidelines and regulations. The study obtained approval from the French Centre National de la Recherche Scientifique (CNRS), Agence National de la Recherche (ANR) and the Institut de Recherche pour le Développement (IRD) and was carried out under an international agreement between the IRD and the Cameroon Ministry of Scientific Research and Technology.2 + 1.66 t – 32.2 . By applying the growth kinetics curves and the projection of the Z-score obtained with the reference sample, we proposed a prediction of adult (18-year-old) stature in both cohorts. The first step was to determine whether puberty was late, standard or early in each individual. The Z score for stature was then calculated from the age and mean stature value on the corresponding mean curve as a function of biological maturity Fig. , S5. Fin2 values and standard estimation errors (SEE) obtained. The values of the lower and upper limits within a 95% confidence interval on either side of the mean deviation are given. For all tests, α = 0.05. The statistical analysis was carried out using Statistica software . If the prediction of adult stature in basketball players and the Baka pygmies is effective, the growth kinetics and stature at the beginning of puberty, always taking biological maturity into account, will be similar for all populations and will follow the same canalisation pattern regardless of adult morphology.The individual differences between the resulting estimates and the actual adult stature values were compared and the specified rSupplementary information.


Smart environments and technology used for elder care, increases independent living time and cuts long-term care costs. A key requirement for these systems consists in detecting and informing about abnormal behavior in users’routines. In this paper, our objective is to automatically observe the elderly behavior over time and detect anomalies that may occur on the long term. Therefore, we propose a learning method to formalize a normal behavior pattern for each elderly people related to his Activities of Daily Living (ADL). We also adopt a temporal similarity score between activities that allows to detect behavior changes over time. In change behavior period we focus on each activity to detect anomalies. A use case with real datasets are promising. With the growing elderly population, research in elderly living and well-being has been aimed toward medical analysis and supporting independent living of elderly people. Elderly people are often disabled by several interacting problems, such as loss of function and social and environmental factors. All these factors, separately or together, determine the elderly person’s level of independence and influence his/her quality of life.In this context, most researchers aim to improve the living of elderly people with medical issues, such as diabetes and cognitive disabilities, by analyzing the behavior of residents within sensor-based environments. The progress of technology  enables the development of effective solutions to help older people to live independently in their homes.The smart home concept includes homes equipped with simple environmental sensors and more complex systems including audio, video and biometric systems. The raw information captured by the sensors can obviously not be shared as such with the medical staff or used directly to detect changes in behavior automatically. On the contrary, extracted knowledge could be used to enrich the information displayed to the medical staff and improve the precision of early detections. There is evidence that opportunistic home surveillance prevents in some cases hospitalization.In this paper, we focus on the problem of learning from smart home sensor data describing elderly’s activities. Our objective in this work is to propose an approach to identify periods of time when behavior changes occur and detect anomalies in this period . Our contributions in this paper can be summarized as follows.We model a behavior pattern using training dataset, defined as the user’s usual activities in his/her daily routine.We calculate a daily score by comparing activity patterns. This daily score variation provides a global vision of the behavior of the elderly person over a period of time.We detect anomalies related to every activity in the period of behavioral deviation.The rest of this paper is organized as follows. In Sect. With the use of smart homes, the daily activities and behavioral patterns of residents can now be monitored through sensors embedded within various areas in the home. This allows elderly people to be more independent while providing assistance to their family and caregivers. In this section, we describe some research works regarding the analysis of behavior and health monitoring for elderly people in the smart home context.Works in  use anom  and 3]  3] studyGjoreski et al.  have proAnother detection strategy was proposed by Sprint et al. . First, Anomaly detection systems for detecting abnormal behavior has been surveyed and reviewed in –8 implicPrevious works describe existing research regarding the analysis of behavior and health monitoring from a smart home. A set of these works , 4 only The overall objective of this study is to analyze the daily behavior of elderly people in their apartment through ambient sensors. In the following, we introduce our model to characterize the normal behavior pattern for elderly people. This normal behavior pattern can then be used to detect behavior changes over time by comparing the current behavioral data of an elderly with her/his usual behavior pattern.Activities of Daily Living (ADLs) is a term used by healthcare professionals to refer to the basic self-care tasks an individual does on a day-to-day basis. These fundamental activities are crucial for maintaining independence. They are used by health professionals as a way of measuring an individual’s functional status, especially for elderly people.The importance of this issue has led to the development of numerous solutions that can monitor activities . Basic Location: the specific place where an activity occurs, for example, “eating” takes place in the kitchen.Time: the duration and occurring time of an activity. The user may perform a same activity at different times  but some activities only occur at specific times of the day . The start time and duration of each activity instance may be logged by the user, or better detected by an activity recognition system based on in-home sensors.Our notion of activity comprises two key criteria used also in  that areThe daily behavior pattern involves several activity patterns. It defines order constraints on them and introduces eventual temporal delays. The daily behavior pattern describes how the user performs her/his activities at different times and models links between them. The daily behavior pattern is represented by a sequence of usual activities. It can be built from data derived from sensors in a smart home.Let A = {routine behavior or regular behavior, based on training data to model regularities in every individual activity. The normal behavior consists of the list of activities that a resident performs in her/his house, with time of the day and the duration. Thus, it captures the repetitive daily routines and deviations from the normal behavior may indicate changes of lifestyle or loss of capacity.The first step of any behavior anomaly detection system is to characterize the normal behavior, also called We first use training data collected during the previous period which was treated as a baseline behavior period. We then follow an unsupervised learning approach: clustering to find point anomalies. To address this, we cluster instances of each activity based on start time and duration without considering the day of the week. For clustering, we use the DBSCAN algorithm  which isWe eliminate point anomaly and we calculate the average start time and duration for each activity in training data.To build the routine behavior model, we follow the following steps: n hours, of the activity “going to the toilets”.The activity “going to the toilets” that occurs several times a day is treated separately. There are usually regular schedules for this activity. It is not essential to be very precise on the realization time of this activity. We then choose to study its frequency rather than the occurring time. Using the same training data to model regularities in the three studied activities, we calculate the frequency, per Once computed, the normal behavior pattern can be used to detect anomalies by comparing the current behavioral data of an elderly with her/his normal behavior pattern. The basic idea of our behavioral deviation detection system is to estimate the similarity between both patterns using a score. We therefore consider three criteria for the activities: the time, duration and chronological order of the activities in the sequence.Behavior Modeling Using a Daily Activity Score: Intuitively, a particular activity is similar to a pattern if its start time, duration and location are similar to the ones defined by the pattern. The similarity of the time and duration for each activity is estimated by a score.a in a day d is calculated by the formula  may be an initial signal of decline and should generate a notification to the caregivers or the family members.For regular activities, occurring several times a day, we compare the frequency between a routine day and an observed day. By simply plotting the daily score along time, it is possible to identify certain days with unusual activities , or trends of evolution that indicate deviations from the previous activity routine.A subsequent work detailed in Sect. Anomaly Detection in Behavior Change Period: In this section, we investigate the accuracy of anomaly detection: per day and per hour of the day, in the behavior change period. Activities in this period has been mapped and compared with normal life pattern.point anomaly [collective anomaly [contextual anomaly [point anomaly .In the domain of at-home activities, anomalies can be classified as point, collective and contextual anomalies. The  anomaly –14 consi anomaly  consider anomaly , 17 consFor the activity “going to the toilet” which occurs several times per day, we model and compare frequencies between a routine day and the observed day in the behavior change period. A frequency is provided by the normal behavior pattern. This will be illustrated in the following section.In this section, we present in The dataset used for our analysis is provided by Washington State University’s CASAS programWe use one public data set (named HH120)  which wamin_pts which is the minimum number of points in a cluster, and the other is Eps which is the maximum distance between two data points for them to be considered in the same cluster. While learning, data out of Eps would be considered out of clusters so marked anomalous. Trained results are depicted in Fig. For building the normal behavior pattern from the training data set, we follow the different learning steps presented in the Sect. Figure We then eliminate point anomalies and calculate, for every activity in the training data set representing one month of collected data, the average start time and average duration. Figure In the first stage of our experiments, we computed the daily scores introduced in Sect. In the second stage of our experiment, we focus on the deviation period (days with decreasing/increasing scores) to detect point anomalies due to a missing activity or activities with unusually long/short durations. To do this, we plot in Figs. At days 13 and 14, Figs. As mentioned previously, the activity “going to the toilet” that occurs several times a day is treated separately. As for the other activities, anomalies related to duration are eliminated using DBSCAN as illustrated in Fig. All these anomalies may be a signal of sickness  so our system may send an alert message to inform remote caregivers.In this article, we presented our research work to detect behavioral changes in the elderly’s usual behavior. Our solution relies on the construction of a behavior model. Thanks to this scheme and based on the detection of anomalies, we are able to detect changes in the elderly behavior, not only sudden changes such as a fall or a temporary illness, but also changes over time. For example, the elderly sleep less and less every month, which can be worrying and cause many problems. In the future, we would like to integrate other activities into our model, such as “going outside”, which are an important aspect for characterizing the elderly’s health. This information should also be coupled with contextual elements such as weather conditions. Other information on health conditions can also be used to refine our detection of behavioral changes. Finally, when our system detects changes in behavior, it must make a decision on the appropriate solution: for example, if it is a sudden and persistent change, it may be an emergency solution and if it is a degradation of the subject’s daily routine, a visit from the caregiver would be necessary. It is difficult to make the “best” decision so we plan at this point to talk to healthcare professionals to configure our decision support system.


Volcanic eruptions trigger ENSO response through shifts in the ITCZ and extratropical-to-tropical teleconnections. The mechanisms through which volcanic eruptions affect the El Niño–Southern Oscillation (ENSO) state are still controversial. Previous studies have invoked direct radiative forcing, an ocean dynamical thermostat (ODT) mechanism, and shifts of the Intertropical Convergence Zone (ITCZ), among others, to explain the ENSO response to tropical eruptions. Here, these mechanisms are tested using ensemble simulations with an Earth system model in which volcanic aerosols from a Tambora-like eruption are confined either in the Northern or the Southern Hemisphere. We show that the primary drivers of the ENSO response are the shifts of the ITCZ together with extratropical circulation changes, which affect the tropics; the ODT mechanism does not operate in our simulations. Our study highlights the importance of initial conditions in the ENSO response to tropical volcanic eruptions and provides explanations for the predominance of posteruption El Niño events and for the occasional posteruption La Niña in observations and reconstructions. The El Niño–Southern Oscillation (ENSO) phenomenon arises spontaneously from the coupling between the atmosphere and the ocean in the tropical Pacific  gradient along the equatorial Pacific. The reduced zonal SST gradient causes a relaxation of the trade winds, which, in turn, yields a temporary weakening of the ocean upwelling in the eastern Pacific that is then amplified by the Bjerknes feedback, leading to an El Niño. This so-called “ocean dynamical thermostat” (ODT) mechanism emerged in simulations with an idealized model and an imposed uniform radiative cooling  circulation  cannot explain the posteruption ENSO evolution (in particular for TrSH experiments), other previously unidentified mechanisms must be at play. In particular, we noted that the volcanically induced radiative forcing can produce an extratropical circulation response that may, in turn, affect the tropics. Extratropical responses to radiative forcing—including meridional shifts in jets and storm tracks—are complex and often consist of a “tug of war” between opposing effects , then one would expect to see changes in the jet similar to those expected for a La Niña event. However, the jet changes in TrSHENS01 are opposite to those associated with a La Niña event  and are similar to the SLP and surface wind anomalies of the summer preceding an El Niño event  weaken the volcanically induced direct extratropical response (El Niño like) in TrSHENS01, while only the volcanically induced direct extratropical response (El Niño like) is expected in the TrSHENS02 simulation. Hence, in the TrSHENS01 experiment, the northward ITCZ shift wins out on the extratropical forcing leading to La Niña–like anomalies, while the extratropical forcing is larger in the TrSHENS02 than in the TrSHENS01 experiment and cancels the effects of the ITCZ displacement.Furthermore, the initial response of the equatorial Pacific may feed back into the midlatitudes. Hence, teleconnections from the tropics into the extratropics also need to be accounted for, which likely depend on the initial ENSO state and/or the spatial aerosol distribution. Such tropical-to-extratropical teleconnections can then modify the direct extratropical forced response. In the TrNHrNHENS01 . In a siFurther modeling studies are needed to test this mechanism as potential biases in the volcanically induced changes in the atmospheric circulation in the troposphere may have been introduced by a poor representation of the circulation in the stratosphere .In this study, we designed and performed a set of experiments with an Earth system model to illuminate the short-term response of ENSO to volcanic forcing. Our results demonstrate that the ENSO response is due to the combined impact of two processes: (i) a meridionally displaced ITCZ due to hemispherically asymmetric cooling and (ii) extratropical responses affecting the tropics. In response to NH eruptions, the ITCZ is displaced meridionally toward the equator and, hence, leads to El Niño–like response anomalies 4 to 6 months later. On the other hand, SH eruptions shift the ITCZ away from the equator and, hence, trigger a La Niña–like response. Eruptions in either hemisphere change the meridional temperature gradient and the land-ocean temperature contrast in the extratropics of that hemisphere, causing a regional atmospheric response in the extratropical Pacific that favors the development of El Niño conditions. Hence, summertime NH eruptions robustly cause El Niño–like anomalies about 6 months later, while the response to a summertime SH eruption depends on which of the two processes is dominant. In both hemispheres, the amplitude of the ENSO response depends on the initial conditions in the tropical Pacific and the spatial structure of the applied volcanic forcing, which is also affected by the initial conditions.These results demonstrate the importance of the ITCZ shift in triggering an ENSO response not only in high-latitude eruptions  over a period of 3 days to mimic a Tambora-like eruption. Although the injection height was likely much higher than what we used [likely more than 40 km : 1 June 1923. In one experiment, a volcanic eruption was simulated in the subtropics (17°N) of the NH (TrNH), and in the other experiment, an eruption was simulated in the subtropics (17°S) of the SH , but in the absence of an eruption is trending to La Niña conditions 3 months later . The prescribed SO4 mixing ratio for the EqPAC simulations has been created to have a peak anomaly in radiative forcing in July over the equatorial Pacific similar to the asymmetric simulations but with approximately double their strength (fig. S1). The prescribed SO4 gradient at the borders of the forcing region  is not dissimilar from the gradients that develop in experiments where the volcanic plume is allowed to evolve [see We have also performed a highly idealized experiment to test the ODT, in which the SOTo test whether the ODT mechanism is at play in response to a tropical volcanic eruption, we prescribed an aerosol forcing that is symmetric about the equator in the tropical Pacific rather than simulate a volcanic eruption at the equator. This experimental design precludes extratropical forcing and hemispheric asymmetry in forcing that would accompany any tropical eruption by way of the Brewer-Dobson circulation (ENS01 or VENS02) and the unperturbed climate state (NVENS01 or NVENS02): Δvolc* = VENS0* – NVENS0*, where * stands for the ensemble identifier (1 or 2). Each mean climate state  is defined for each given ensemble  as the ensemble average of all its members.The volcanically induced anomalies (Δvolc) are calculated as the difference between the climate state induced by the eruption (V4 peak concentration and e-folding time against the Pinatubo eruption (June 1991), which is currently the best observed large tropical eruption . NorESM is able to reproduce the main features of the Pinatubo eruption, which is sufficient for the scope of this study, that is, to delve into the mechanisms that trigger an ENSO response following a uniform radiative forcing.We have tested the ability of NorESM in simulating the SO4 produced and the strength of the forcing are larger for the TrNH eruptions  the representativeness of the chosen forcing as the 1815 Tambora eruption corresponds to a strong and rare volcanic event and (ii) the limitations of the single model used here and its skill in simulating ENSO and the mean climate state . The 1815 Tambora eruption is a test case to study the climatic response to volcanic forcing . We apply a 5-month running mean to remove intraseasonal variations in SST. An El Niño event is defined when the Niño3.4 index exceeds 1 SD (+0.4°C) for at least 6 consecutive months. Unless otherwise noted, all differences discussed in this study are significant at the 95% confidence level using a Student’s


We analysed the coronavirus disease 2019 epidemic curve from March to the end of April 2020 in Germany. We use statistical models to estimate the number of cases with disease onset on a given day and use back-projection techniques to obtain the number of new infections per day. The respective time series are analysed by a trend regression model with change points. The change points are estimated directly from the data. We carry out the analysis for the whole of Germany and the federal state of Bavaria, where we have more detailed data. Both analyses show a major change between 9 and 13 March for the time series of infections: from a strong increase to a decrease. Another change was found between 25 March and 29 March, where the decline intensified. Furthermore, we perform an analysis stratified by age. A main result is a delayed course of the pandemic for the age group 80 + resulting in a turning point at the end of March. Our results differ from those by other authors as we take into account the reporting delay, which turned out to be time dependent and therefore changes the structure of the epidemic curve compared to the curve of newly reported cases. The first phase of the coronavirus disease 2019 (COVID-19) pandemic in Germany was managed relatively successfully in comparison to other countries in Europe. Therefore, it is worth taking a closer look at the course of the pandemic in Germany, which has already led to controversial discussions in the public. This particularly concerns the important question about the effectiveness of various control measures. Several publications discuss the effects of control measures in different countries, see, e.g., . As 4]  4] pointIn a recent paper on Germany , the autIn our analysis we focus on the curve of infection at two geographical levels: the federal state of Bavaria and all of Germany and on four age-strata. The paper is organised as follows. In section ‘Data and methods’, we present the data and the strategy of estimating the relevant daily counts. Then the segmented regression model, which is the basis for further analyses, is presented. The penultimate section presents the results followed by a discussion.For the analysis of the Bavarian data, we use the COVID-19 reporting data of the Bavarian State Office for Health and Food Safety (LGL) that is collected within the framework of the German Infection Control Act (IfSG). At the case level, these data include the reporting date (the date at which the case was reported to the LGL) as well as the time of disease onset (here: symptom onset). However, the latter is not always known: partly because it could not be determined and partly because the case did not (yet) have any symptoms at the time of entry into the database. A procedure for imputation of missing values regarding the disease onset has been developed by , using aFor the German data, no individual case data were available, so instead we used publicly available aggregated case reporting data published on a daily basis by the RKI . These dWe estimated the imputation model for the German data based on all cases reported up until 1 June. The percentage of imputed values was 37% for the Bavarian data and 28% for the German data. Since this percentage is rather high, we performed a sensitivity analysis using (1) only data with a documented disease onset and (2) utilising the reporting date of cases as disease onset date when the actual onset date is unknown.To interpret the course of the pandemic and possible effects of interventions, case-based data on time of infection is essential. However, as such data are generally not available, one simple approach is to shift the curve of disease onsets to the past by the average incubation period. The average incubation period for COVID-19 is about five days . A more tY is the number of detected COVID-19 cases by time t of infection, K is the number of change points and x+ = max is the positive part of x. The change points kCP, k = 1, …, K are used to partition the epidemic curve tY into K + 1 phases. These are characterised by different growth parameters. In the phase before the first change point CP1 the growth is characterised by the parameter β1, in the 2nd phase between CP1 and CP2 by β2 = β1 + γ1. The next change is then at time CP2. In the 3rd phase between CP2 and CP3 the growth parameter is given by β3 = β1 + γ1 + γ2. This applies accordingly until the last phase after KCP. The quantities exp(jβ), j = 1, …, K + 1 can be interpreted as daily growth factors. Since we use estimated, non-integer values tY from the back-projection procedure as outcome, we assume a  Gaussian distribution for log(tY). Furthermore, we assume an AR(1) error term for the regression model, since serial correlation occurs due to smoothing in the backprojection procedure.To analyse the temporal course of the infection, we use the following regression model and change points see , 18):1wh:1wh18]):K is increased from K = 1 up to a maximum of K = 6. It is examined whether the increase of the number of change points leads to a lower value of the Bayesian information criterion (BIC). Since the considered time series consist of only 61 data points, we exclude models with more than 6 change points, since they are hardly interpretable and the danger of overfitting is high.Since model (1) is a generalised linear model given the change points, the parameters of the model (including the change points) can be estimated by minimising the respective likelihood function. However, due to the estimation of the change points, the numerical optimisation problem is not straightforward. For the estimation of the model we use the R package segmented, see . The staWe apply the segmented regression model to time series of the estimated daily counts of infections for detected COVID-19 cases in Bavaria and all of Germany. Since the back-projection algorithm yields an estimate for the expected values of the number of daily infections and does so by inducing a smoothing effect, as a sensitivity analysis for the location of the breakpoints, we also apply a regression model to the time series of the daily counts of disease onsets. The results of this sensitivity analysis are presented in the supplementary material. Furthermore, as a more detailed analysis, we apply our procedure to data stratified by age groups. A special focus is on the age group 80 + , as this group has the highest risk for a critical course of the disease.x-axis, instead there is also a notable change in the structure of the curves. The curve relating to the date of infection is clearly smoothed due to the back-projection procedure (cf. Section 2.2) and has a clear maximum both for Bavaria and Germany.In The overall Bavarian model includes five change points. The result can be seen in The resulting six phases are:1st phase: There is a substantial increase in new infections with a high daily multiplication factor of 1.25. The first phase ends on 6th March.2nd phase: The increase slows down to a daily multiplication factor of 1.17.This phase lasts to the 10th of March.3rd phase: For a short time the multiplication factor further goes down to 1.064th phase: The change point on 12th March marks a clearly visible change in the course of the pandemic. It is the turning point of the curve (change of the multiplication factor to 0.97).5th phase: A further change point is found around 26th March. There is an accelerated decrease in the infections with a daily factor of 0.94.6th phase: On 25th April the estimated number of infections of reported cases is rather low and there is no further decrease after this change point (multiplication factor close to 1).The age-stratified analysis gives some interesting further insights about the course of the pandemic, see The results for the German data are presented in 1st phase: There is a strong increase (multiplication factor 1.32 per day) in new infections in the beginning of the pandemic until 5th March, where the increase decreases.2nd phase: From 5th March to 10th March there is still a substantial increase of infections with a daily multiplication factor around 1.1.3rd phase: After the change point on 10th March, there is a clearly visible change in the course of the pandemic. There is change from an increasing to a decreasing curve with a daily multiplication factor of about 0.98. This phase lasts until 27th March.4th phase: An acceleration of the decrease can be seen from 27th March onward. The daily multiplication factor is 0.94.5th phase After 21st April, the number of infections is rather low with a slow decrease.The results of the age stratified analysis are similar to the results for Bavaria  and a back-projection to the date of infections, and therefore, despite its limitations, is better suited to describe the course of the pandemic than the more common analysis of daily or cumulative reported case numbers. In the analysis of the Bavarian and the German data our main result is the change point, where the exponential growth was stopped: this clearly happened already between 9 and 13 March. The timing of this change point coincides with the implementation of the first control measures: the partial ban of mass events with more than 1000 people. Furthermore, in a press conference on March 11th chancellor Merkel and the president of the RKI appealed to self-enforced social distancing . FurtherIn Bavaria and in Germany, the change point at 26/27th March of the infection date is apparent. This change point is associated with different measures taken in March (closing of schools and stores on 16th March and the shut- down including contact ban on 22nd March in Germany including Bavaria). Other measures were similar in timing in Bavaria and all over Germany. In Bavaria, some measures were implemented a little earlier. Since there were many measures administered simultaneously and since – as described above – other factors beyond the measures itself contributed, we do not think is not possible to quantify the effect of individual measures to the development of the epidemic curve.The results for the 80 + age group indicate an infection curve which is delayed by about one week compared to the other age groups. This is possibly due to the fact that the disease was first introduced into Germany by younger holiday makers and business travellers. Hence, it likely took some additional generations of transmission before the infections mitigated into the 80 + group. Furthermore, many infections in the age group 80 + are due to outbreaks in nursing homes and homes for the elderly, where very different mechanisms of contact occur compared to the rest of the population. Many of the restrictions were targeted at the younger age groups , hence, the effect of these interventions is only indirect for the 80 + group and thus the impact is delayed. This underlines the need for more direct measures for this group. For the age group of 0–14 the breakpoints are similar to the other age groups. The much lower infection rate could be partly due to lower case detection ratio, since infected children show less symptoms, see e.g. .et al. ..2]. Our The temporal connection between the change points in our analysis and various control measures should be interpreted as an association, rather than a direct causal relationship. In the end many other explanations exists and from a simple time-series analysis it is not possible to say to what extent the population already had changed their behaviour voluntarily, as for example observed in mobility data , 31, andTime-series data including only a limited amount of explanatory factors has to be interpreted with care and with respect to the many uncertainties which remain regarding COVID-19 .R(t). Furthermore, the reproduction rate does not contain information about how many people are currently affected, or whether the infected persons belong to risk groups. The course of the time-varying reproduction number calculated by us for Bavaria fits well with the change point analysis [R(t) > 1 corresponds to a rate of increase >1, noting that the time delays in the interpretation of R(t) must be kept in mind.Despite the limitations of the approach, we argue that it is advantageous and important to directly interpret the epidemic curve and the absolute number of cases, rather than indirect measures like the analysis . A valueet al. [It should be noted, that the presented analysis is retrospective. Control measures have to be decided based on a completely different level of information than what the retrospectively established epidemic curve suggests. The simple observation of the course of the reported case numbers by reporting date is also problematic because this course is strongly influenced by the reporting behaviour and the methods and capacities of the test laboratories. Typically, substantially fewer cases are reported at weekends than during the week. Therefore, the estimation proposed by Günther et al.  is an imSince the impact of the measures also depends on how they are implemented by the population (compliance), the results cannot be directly transferred to the future. Nevertheless, it remains a remarkable result that the clear turning point of the early COVID-19 infection data in Germany is associated with non-drastic measures (no shutdown) and strong appeals by politicians.


The goal of this article is to discuss the Simple Equations Method (SEsM) for obtaining exact solutions of nonlinear partial differential equations and to show that several well-known methods for obtaining exact solutions of such equations are connected to SEsM. In more detail, we show that the Hirota method is connected to a particular case of SEsM for a specific form of the function from Step 2 of SEsM and for simple equations of the kinds of differential equations for exponential functions. We illustrate this particular case of SEsM by obtaining the three- soliton solution of the Korteweg-de Vries equation, two-soliton solution of the nonlinear Schrödinger equation, and the soliton solution of the Ishimori equation for the spin dynamics of ferromagnetic materials. Then we show that a particular case of SEsM can be used in order to reproduce the methodology of the inverse scattering transform method for the case of the Burgers equation and Korteweg-de Vries equation. This particular case is connected to use of a specific case of Step 2 of SEsM. This step is connected to: (i) representation of the solution of the solved nonlinear partial differential equation as expansion as power series containing powers of a “small” parameter  The methodology was extended [Nature and human society are rich sources of complex systems extended  and applextended ,61,62). extended  used varextended ,64,65 anextended ,68,69,70m and Recently we presented an algorithm for obtaining exact and approximate solutions of nonlinear partial differential equations called Simple Equations Method (SEsM) ,72,73,74dkgdξkl=∑In the course of time, we extended the algorithm of the Modified Method of Simplest Equation. Here we are going to discuss the last version which is connected to the possibility of use of more than one simple equation. This modification is called SEsM—Simple Equations Method. The reason for the use of this name is that the used simple equations are more simple than the solved nonlinear partial differential equation, but these simple equations in fact can be quite complicated. Thus we have to substitute the word “simplest” by the word “simple”. A variant of SEsM based on two simple equations was applied in  and the We note that SEsM is not a universal algorithm for obtaining exact analytical solutions of nonlinear differential equations. With very large probability there are many exact solutions of nonlinear differential equation which can not be obtained by SEsM. The presence of such solutions is of large interest for us as it shows possible ways for further development of SEsM. We note also that some steps of SEsM are not rigidly fixed. Examples are the choice of transformations in Step 1 of SEsM and the selection of the simple equation in Step 5. This flexibility increases the number of equations which can be treated by SEsM but the set of these equations is still a subclass of the class of nonlinear differential equations that possess exact solutions.The organization of the text below is as follows. We describe SEsM in N nonlinear differential equations by the use of solutions of M simple equations. The most applications of the algorithm up to now are for obtaining solutions of 1 nonlinear differential equations by the use of solutions of (a) one simple equation or (b) more than one simple equation  is an algorithm for obtaining exact and approximate solutions of nonlinear differential equations. In general  the algoe, e.g., ,93,94,95SEsM has seven steps which are shown in We consider a system of nonlinear partial differential equations(1.) We apply transformationse, e.g., ,76,77,78In numerous particular cases, one may skip this step (then we have just ation of  to 2) l lui f and g are connected to solutions of some differential equations  which are more simple than Equation . The forms of the functions F can have the formF can be different from (The functions Equation . We notethe formF=α+∑i1=1the formF=α+∑i1=1the formF=α+∑i1=1(3.) In general the functions used in In many cases  We assume that the functions A, B, …. Thus, different relationships are possible. Often one uses a finite-series relationship, for example,Note that SEsM does not prescribe the forms of the functions (5.) The functions (6.) The application of the steps 1–5 to Equation  transfor(7.) We may obtain a nontrivial solution of Equation  if all cu of the solved equation and at step 1 of SEsM we transform u by means of a transformation T. The solved equation is transformed into an equation for the function F and at the following steps of SEsM F is represented by functions T) and F is represented as a power series of the solution f of a single simple equation. The substitution of these power series in the solved equation reduces this equation to a system of nonlinear algebraic equations and each nontrivial solution of this system leads to an exact particular solution of the solved nonlinear differential equation.Below we age going to discuss the relation among SEsM and several much-used methods for obtaining exact solutions of nonlinear partial differential equations.Hirota method is a very popular method for obtaining soliton solutions of integrable nonlinear partial differential equations. One of the first applications of the method was in the famous article of Hirota  and detaf is called sometimes the The function the formf=α+ϵf1+ϵIn order to deal with Step 4 of his method, Hirota introduced the famous bilinear operators, follows (11)Dxm  f by meaNote that this is a linear equation. The equations for The equations for ution of  we startThe substitution of  in 19)  19) leadfWe note that no balance procedure is required here . The substitution of f1 in  leads tof leads to the one-soliton solution for u by use of   19) leadf becomesThe substitution of  in 20)  20) leadof f2 in  leads toAgain the ation of  leads toIn order to obtain the three-soliton solution of the Korteweg-de Vries equation, we start with the solution for The substitution of  in 19)  19) leadution of  in 20)  (33)ωi+λf isThe substitution of the obtained solutions for nd f2 in  leads toation of  we obtaiN-soliton solution of the Korteweg-de Vries equation can be obtained by means of a particular case of the SEsM methodology. We use N simple equations for exponential functions and begin by solution N-soliton solution of the Korteweg-de Vries equation.The procedure can be easily continued and the F and G. The connection of SEsM with the Hirota’s method makes this task of obtaining the two-soliton solution quite easy. The nonlinear Schrödinger equation isq is a parameter. We consider the case Now we show how SEsM leads to the two-soliton solution of the nonlinear Schrödinger equation. Here the transformation at Step 1 of SEsM leads to equations for two functions ation isi∂ψ∂t+∂2ψNote that the transformation  is diffeThe substitution of  to 37)  37) leadThis can be written asF and G by functions which will be then constructed by solutions of simple equations . These representations are made by the following expansionsIntroducing the constant basis of (41)[iDtpansionsF=1+ϵf1+ϵ+ϵ2f2+…,G=g0k is a reution of  and 42)k is a reFor the terms of order of For the terms of order of Next we write relationships if the functions uations.  and 49)f1,…,g1,…ction of  and (49)ction of  in 48)  f1,…,g1,This choice of In order to obtain the two-soliton solution of the nonlinear Schrödinger equation, we start with solutions of  and 49)49) whichf1=exp(η1ution of  and (49)The substitution of , 58) in in58) inThe relationships for More complicated soliton solutions of the nonlinear Schrödinger equation can be obtained too. One has just to start by solution constructed by 3,4,… solutions of simple differential equations for the exponential function.exterior (wedge) product of corresponding vectors. Step 1 of SEsM is connected with the transformation of the nonlinearity of the studied equation [The last example is connected to the area of solid state physics: dynamics of spin chains. The Ishimori equation  is (2+1)equation ,102. BefThe Ishimori equation becomesf and g are complex functions. This allows us to write the Ishimori equation in the bilinear form by means of the Hirota operatorsAt Step 1 of SEsM, we perform the transformationf and one expansion for g: At Step 2 of SEsM, we use two expansions: one expansion for for N-soliton solution we start by solution constructed by solutions of simple equations for exponential functions as followsM is an arbitrary complex constant, After substitution of  in 67)  67) we aIn a similar way we can obtain two-soliton solution, three-soliton solution, etc. of the Ishimori equation.Below we shall discuss the connection between SEsM and IST method for the case of Korteweg-de Vries equation. First of all we briefly remember application of the IST for the case of the KdV equation:Researchers tried to solve this equation by means of some transformation. Almost 50 years ago Gardner, Greene, Kruskal, and Miura  tried thu depends on x and t. However, t can be treated as a parameter and then   73) lead∞ leads to the conclusion that t; (ii) For t. Then C does not depend on x. Then, ∣x∣→∞. Thnd from  of  has the k,x) of ∼exsolution . Thus, oows from  and 80)u coAt The solution of the scattering problem at behaviorψ∼xp(ikx). when μ=kFinally, at step 4, one has to solve the inverse scattering problem. One has to find We note that the parameters We shall discuss below the connection between SEsM and IST for the case of Korteweg-de Vries equation. The main points are as follows. We skip Step 1 of SEsM and consider the particular case where no transformation of the solved equation is performed. Then we represent the searched solution asu and the powers of This is Step 2 of SEsM. One initially treats ction of  in the s of SEsM ). Then bRosales  used theThe Burgers equation isx and we shall no longer require We skip Step 1 of SEsM: the transformation of nonlinearity and at Step 2 of SEsM we write the function t) as in . For nowThe equations for ution of  in 92)  un Now step by step we can solve the Equation  and we cAt Steps 3–5 of SEsM, we are going to represent ransformu1=∫Cdλ(kentation  is a linThe choice of solution for u1 as  influencWe search for solution of  as(99)uThe substitution of  in 97)  97) and Next we assume the general form of the solution of the equations for un to be:un=∫ution of . Then weu asBy means of the notationWe use the formula for summation of the power seriesly it to  with q=ζx.We remember that addition  is exactaddition  is a solLet us now consider a more complicated equation: the Korteweg-de Vries equation.We consider the Korteweg-de Vries equationWe skip Step 1 of SEsM (the transformation of the nonlinearity in ). At Stetionship  where ϵ ution of  in 108)u byAt Step 3 of SEsM, we shall use solutions of the obtained equations for The solutions for The substitution of  in 112)112) leadThe substitution of  in 113)113) leadIn addition we can obtain for the parameters of the solution The continuation of the calculations leads us toThe solution of the Korteweg-de Vries equation becomesThe straight summation of the infinite series in the sum in  is imposWe write  as(120)We consider first the case of discrete ral from  can be rp is the column vector of all P is the square matrix of all pThenu=∂∂x∑n=1P is real symmetric and positive definite. Then u from .P is in general a polynomial function of its arguments, w and setting its coefficient to zero, one obtains an ordinary differential equation for k degree homogeneous functions in various derivatives of k is the order of k degree homogeneous equation in partial derivatives of The homogeneous balance method was discussed by Wang and et al. ,107,108.P,[f and the function ons from ). The usF which has particular form to be a linear combination of the functions from ,wheredvdξ=k, just one function u is used and the particular form  is used u is used. At Step 3 of SEsM we consider the particular case where we search for a traveling wave solution of the solved equation. At Step 4 of SEsM we assume a particular case of the relationship between the functions u and v, namely  and to show that this method is connected to a particular case of SEsM. In the MMEHBM we do not use the transformation of the nonlinearity of the solved equation. We use arbitrary number of functions Assumption 5.The MMEHBM is connected to a particular case of SEsM where there is no transformation of the nonlinearity of the equation (Step 1 of SEsM is skipped), arbitrary number of functions i-th solved equations. At Step 3 of SEsM, we consider the particular case where we search for a traveling wave solution of the solved equation and the traveling waves associated with the different functions In general, MMEHBM will be used for obtaining exact traveling wave solutions of systems of nonlinear partial differential equations. We consider SEsM and impose restrictions on it in order to reduce SEsM to the particular case connected to MMEHBM. First, we consider the particular case of SEsM where no transformation of the nonlinearity of the solved equation is made at Step 1 of SEsM. At Step 2 of SEsM, we consider again a particular case where we use one function the kind . We notethe kind . Steps 6The summary of the auxiliary equation method is as follows . One conThen traveling wave variable Equation  becomesThe solution of  is searcEquations  and 142142 are iequation .Now we show that the auxiliary equation method is connected to a particular case of SEsM.Assumption 6.The auxiliary equation method is connected to a particular case of SEsM where there is no transformation of the nonlinearity of the equation (Step 1 of SEsM is skipped); Function F at Step 2 of SEsM has particular form—); just oF—. At Step 2 of SEsM we use one of the possible forms of the function F—. We skipthe form . In suchF instead of  is connected to a particular case of SEsM where there is no transformation of the nonlinearity of the equation (Step 1 of SEsM is skipped); Function F at Step 2 of SEsM has particular form and this is the corresponding form used by GAEM; just one simple equation is used that this simple equation has the particular form which is the corresponding form used by GAEM.F—the corresponding form used when GAEM is applied. This possible form is just one of the many forms that can be used in SEsM. We skip Steps 3 and 4 of SEsM as the function We start from SEsM, impose restrictions on it and reduce SEsM to a particular case connected to GAEM. At Step 1 of SEsM we do not transform the nonlinearity of the solved equation (we skip this step). At Step 2 of SEsM we use a possible form of the function An illustration of a particular case of GAEM can be seen in .We show first that the Jacobi elliptic function expansion method (JEFEM) is connected to a particular case of SEsM. Then we describe general Jacobi elliptic function expansion method (GJEFEM) and show that it is connected to a particular case of SEsM. Finally, we list several methods used in the literature which are particular cases of GJEFEM.k and c are parameters. m is the modulus of the function The classic form of JEFEM is as given by . One conThis is a generalization of the tanh-method because for ution of  and (147ution of  can leadution of .Assumption 8.The Jacobi elliptic function expansion method (JEFEM) is connected to a particular case of SEsM where there is no transformation of the nonlinearity of the equation (Step 1 of SEsM is skipped); Function F at Step 2 of SEsM has particular form—); just oF—. Additional restriction is that we search for a traveling wave solution of the solved equation . At StepF—. This foF— in the JN nonlinear partial differential equations and search for traveling wave solutions based on different coordinates Next we formulate general Jacobi elliptic function expansion method (GJEFEM). By this method we solve in general a system of We show below that the GJEFEM is connected to a particular case of SEsM.Assumption 9.The general Jacobi elliptic function expansion method is connected to a particular case of SEsM where, there is no transformation of the nonlinearity of the equation (Step 1 of SEsM is skipped); Functions lar form—); and thlar form—.We start from SEsM, impose restrictions on it and reduce SEsM to JEFEM. At Step 1. of SEsM we do not transform the nonlinearity of the solved equation (we skip this step). An additional restriction is that we search for a traveling wave solution of the solved Equation . At Stepui—. We skipui— in the Jui—. This isBelow, we list several particular cases of GJEFEM.U is a power series of the function JEFEM is a particular case of GJEFEM for the case of just one solved nonlinear partial differential equation and when the simple equation is the equation for the Jacobi elliptic function Parks et al.  and Fu e kind of .Fan and Zhang  present Another particular case of GJEFEM is applied by Yan  who treathe kind ). 150)  ui, i=1,U isAnother particular case of GJEFEM is used in . The simLiu and Fan  apply paYe at al.  extend  and use Other variants for g et al. , Chen ang et al. , Lü [124g et al. , El-Sabbg et al. ,127.Another particular case of GJEFEM is the F-expansion method which have the same ideology as JEFEM but only the form of the simple equations for the Jacobi elliptic functions are not specified. In the different variants of the F- expansion method one uses different particular cases for the functions m GJEFEM ,130,131.According to  the modiu(z)PAk are coNow we show that the modified simple equation method is connected to a particular case of SEsM.Assumption 10.The modified simple equation method is connected to a particular case of SEsM where there is no transformation of the nonlinearity of the equation (Step 1 of SEsM is skipped); Function F at Step 2 of SEsM has particular form—) and jusz—. An additional restriction is that we search for solution of the solved equation which depends on a single coordinate z—. At Stepz— of the fion from  in the JThe trial function method is described in ,134 and N; Function F at Step 2 of SEsM has particular form—F which is constructed by means of trial function. In most cases, F is presented by a finite power series of the trial function. The trial function is a solution to one simple equation. Thus, by means of the restrictions above, we reduce SEsM to a particular case connected to the trial function method.We start from SEsM, impose restrictions on it and reduce SEsM to a particular case connected to the trial function method. At Step 1 of SEsM we do not transform the nonlinearity of the solved equation (we skip this step). An additional restriction is that we search for a solution of the solved equation which depends on a single coordinate which can be a traveling wave coordinate or another kind of coordinate. At Step 2 of SEsM we use a particular form of the function The first integral method for obtaining exact solutions of nonlinear partial differential equations is as follows . One wanIn order to do this one converts  to an ord writes  as systeThe solution is obtained by the assumption that the derivative of the relationship her with  allows cAssumption 12.The first integral method is connected to a particular case of SEsM for the case when equations of the kindare considered, there is no transformation of the nonlinearity of the equation (Step 1 of SEsM is skipped); one simplest equation is used and this simplest equation is determined by the condition .X and plays the role of the implicit simple equation which together with  for obtaining exact solutions of nonlinear differential equations and discuss its connections with other methods for obtaining exact solutions of nonlinear differential equations. Special importance is given to the connection to two of the most famous method for obtaining exact solutions of integrable nonlinear partial differential equations - method of Hirota and the inverse scattering transform method. We show that one particular case of SEsM is connected to the Hirota method. This means that the SEsM is a useful tool for obtaining exact solutions of nonlinear integrable differential equations and can lead to multisoliton solutions of such equations. Then we show that SEsM can be connected to the famous inverse scattering transform method. This is done in the case of the Korteweg-de Vries equation and much more research on this connection will be presented in the future. In addition to solutions of integrable nonlinear differential equations SEsM can lead also to exact particular solutions of nonintegrable nonlinear differential equations. Actually, the simplest version of SEsM  was We believe that the SEsM algorithm is useful one and it presents a fruitful way for obtaining solutions of nonlinear differential equations. The development of SEsM is far from finished. More research on the possible classes of transformations can be done as well as additional simple equations can be used. We are sure that over the course of the years this algorithm will be further refined. Our intention to go in this direction and we are going to extend and to apply SEsM for obtaining solutions of many nonlinear differential equations of practical significance for natural and social sciences.


The surface of a mixture of silica and carbon black powder became brown after laser beam irradiation for a few tens of seconds, and clear peaks of crystalline silicon were observed by Raman shift measurements, confirming the successful carbothermal reduction of silica. The influence of process parameters, including the laser beam intensity, radiation time, nitrogen gas flow in a reaction chamber, and the molar ratios of silica/carbon black of the mixture, on the carbothermal reduction process is explained in detail.We report the extraction of silicon via a carbothermal reduction process using a CO Silicon is one of the most essential materials for electronic devices. Integrated circuit chips fabricated on a silicon wafer are currently used in almost all devices, making our daily lives more convenient than ever before. Moreover, the demand for silicon wafers is increasing due to the photovoltaic industry in which crystalline silicon solar cells play a major role, and the production cost of silicon wafers is critical in the commercialization of silicon solar cells. Indeed, advances in silicon technologies have revolutionized modern society in terms of the development of ubiquitous electronics and energy technology and associated industries.1. In industry, carbothermal reduction using an electric arc furnace is adopted to produce pure silicon from silica because this system is considered to be the most economical route in large-scale production4. Carbothermal reduction using an arc furnace has been thoroughly investigated to date8; however, the traditional carbothermal reduction process requires large-scale facilities to achieve high thermal efficiency during the high-temperature process and requires a stabilization process for normal operation after process initiation3. Although carbothermal reduction using an arc furnace has contributed to the large-scale production of metallurgical silicon, it does not seem to be a general solution for obtaining silicon through the reduction of silica, especially when the reduction process should occur in a specific region or on a small scale.Silicon is an abundant element in Earth’s crust; however, it exists as silicon oxides, mostly as silica. It requires considerable time and energy to produce pure silicon from silica by reduction because of the high bonding energy between silicon and oxygen atoms11. This electrochemical reduction process enabled the process temperature to be less than 1000 °C and enabled the reduction reaction to take place at a contact point with an electrode. Other approaches, such as carbothermal reduction of silica using concentrated solar energy12 and reduction of silica from rice husks, have been reported14. Although these attempts incorporate additional metals or chemicals other than silica and carbon, they involve meaningful routes for producing silicon without using traditional silica reduction processes.A few novel approaches for the reduction of silica have been reported to date. Successful electrochemical reduction of silica to silicon has been reported2 laser beam. A CO2 laser is considered an excellent energy source in terms of heat flux; however, no report has yet studied the carbothermal reduction of silica using a CO2 laser beam. We observed that carbothermal reduction took place within a few seconds of the laser beam illuminating the silica/carbon mixture. The intrinsic peaks of crystalline silicon were detected from the silica/carbon mixture after laser beam irradiation by Raman spectroscopy. Our experimental results confirmed that the intensity of the laser beam and the N2 gas flow into the reaction chamber largely influenced the carbothermal reduction process.In this paper, for the first time, we report evidence of reduced silicon from silica by ultrafast carbothermal reduction using a CO2 laser beam with a 10.6 µm wavelength  was used as an energy source. A raw laser beam with a 6 ± 0.5 mm diameter defined by 1/e2 intensity from a Gaussian profile was used without any focusing lens. A simple reactor was designed to isolate chemical reactions upon a laser beam from ambient air. The reactor had a window made of ZnSe for the laser beam entrance and two gas flow lines for the N2 gas inlet and outlet. A rotary vacuum pump was connected to the outlet to keep fumes and gases draining to the outlet for all the experiments. A mixture of silica  and carbon black  powder was loaded in a 13 mm inner diameter graphite crucible. The loading weight of the mixture was 200 mg, and the molar ratio of silica to carbon was 1:4. The mixture was compressed into the crucible with a 5 kg force using a steel block and a customized tool to make the powder surface even and compact. The crucible with the mixture was heated on a hot plate at 100 °C for an hour to remove residual moisture in the sample. A scanning electron microscopy (SEM) image of the pristine mixture is shown in Fig. 2 gas flow was regulated using a mass flow controller (MFC) during the process to control the atmosphere of the reactor and to protect the surface of the ZnSe window from contamination due to vaporized substances. The N2 gas flow rate was 5000 standard cubic centimeters per minute (sccm), and the pressure inside the reactor was approximately 100 Torr.A schematic of the experiment is depicted in Fig. The laser beam radiation caused instantaneous heating of the surface of the mixture, which induced radical phase changes and chemical reactions within a local area before reaching the equilibrium status of the global system. Figure −1 are seen from the spectrum of point 1, which is contiguous to the crater15. It has been widely reported that SiC plays an essential role in the carbothermal reduction of silica as an intermediate compound17. The SiC peaks disappeared and a crystalline silicon peak was detected at point 2, where the brown color was observed, implying that silicon has been successfully reduced in this area. The silicon peak was located at 511 cm−1, which was shifted from the peak of stress-free, bulk crystalline silicon at 521 cm−1. We propose that the shift arises from small silicon crystallites, grain boundaries and thermal stress during the process20. It was also observed that the broad amorphous silicon peak at approximately 480 cm−1 contributes to the shoulder of the peak. The intensity of the crystalline silicon peak was reduced as the measured point shifted outwards, as shown in the spectra of points 3, 4 and 5, which is consistent with the changes in the color of the sample. The tendencies based on the Raman spectra, including clear peaks of SiC and Si according to the different areas of the sample, may imply that there is a favorable reaction zone or conditions for carbothermal reduction.A small piece of brown powder was investigated using a micro-Raman spectrometer. The Raman spectra were collected with a 532 nm wavelength, 2 mW power laser beam. The Raman spectra in Fig. 6 W/m2 when the total power was 50 W, provided that the laser beam followed a Gaussian profile, and this intensity was considered the critical intensity for carbothermal reduction in our experimental setup. A laser beam intensity of ~ 2.15 × 106 W/m2 at 25 W total power condition was sufficient to melt the silica beads; however, it was not sufficient to initiate the reduction reaction. Therefore, it is reasonable to assume that the threshold intensity for the carbothermal reduction reaction using a CO2 laser beam is close to 4.29 × 106 W/m2. Given that the laser beam intensity followed a Gaussian profile, the temperature decreased as the measurement point shifted further from the center of the irradiated area. The temperature at the center of the irradiated spot was measured using an infrared pyrometer during laser beam radiation. The maximum temperature reached 1780 and 2100 °C for 25 and 50 W total laser power conditions, respectively. The temperature reached its maximum as soon as laser beam illumination began, and this is considered the most beneficial characteristic when the laser beam is used as a heat source at a local area. Note that the laser beam radiation time was 10 s. While the in situ diagnosis of the reaction mechanism within the chamber will not be discussed due to a lack of proper technique in the present paper, the carbothermal reduction reaction seemed to take place on the order of seconds. We believe that this is the fastest carbothermal reduction reaction of silica to silicon ever reported.The maximum intensity of the laser beam was estimated to be ~ 4.29 × 102 gas flow, silicon, which was shown as the brown area, was not observed at any laser power, radiation time or molar ratio of silica and carbon. If the temperature of the local area under the laser beam radiation was the necessary and sufficient factor, a signal of the silicon from the sample would have been observed under vacuum condition. This provides insight into the reduction mechanism at the confined area in our system, suggesting that mass transport of the gas phase is highly involved in the reaction21. It is well understood that silicon is not produced by the direct reduction of silica by carbon. Several reactions take place during the carbothermal reduction of silica to silicon as follows3A few process parameters were controlled to investigate their effects on the reduction reaction. When the experiment was performed under vacuum conditions without N2. Under vacuum conditions without a N2 gas flow in our experiments, it was assumed that not only CO but also gaseous SiO were drained to the outlet without undergoing chemical reaction (2 gas was introduced to the reactor. We expect that the N2 gas flow might have influenced gaseous SiO by swirling around the irradiated area. Then, Si could be produced around the favorable temperature area by a chemical reaction (SiC is inevitably involved in Si production as an intermediate. The production of Si is determinative of the chemical reaction . If thesreaction . In contreaction , as show23. We prepared mixture samples with molar ratios of silica to carbon black varying from 1:1 to 1:10 and conducted carbothermal reduction using a laser beam. Interestingly, silicon was successfully produced from all the samples. Radical and localized heating upon laser beam irradiation might account for the results; otherwise, the vapor pressures of the generated gases according to a thermally equilibrated state might be influenced by the molar ratio of silica to carbon in the typical carbothermal reduction system.During traditional carbothermal reduction, the ratio of silica and carbon is important for the production of silicon in terms of the process temperature of the crucible and the production yield−1. The peaks at 295 and 941 cm−1 can be assigned to the second-order acoustic phonon mode and the second-order transverse optical phonon in crystalline silicon25. In addition, a higher power of laser beam radiation yielded the expected results. Clear peaks of crystalline silicon in the Raman spectra were observed for the 100 W irradiated samples; however, the diameter of the crater was increased, and the depth of the crater reached the bottom of the crucible after less than 40 s of irradiation. It is concluded that a power higher than 75 W is not desirable to obtain intact silicon in our experiments because the brown silicon band was damaged due to vigorous chemical reactions under high-power irradiation. The detailed experimental results regarding the process parameters, including the laser power, radiation time and molar ratio of the mixture, can be found in the Supplementary Information.The brown-colored area was widened as the irradiation time increased. After 80 s of irradiation with a 50 W power condition, most of the surface area of the mixture was brown, except for the crater area. The depth of the crater reached the bottom of the crucible. Figure 2 laser beam has been utilized as a heat source in the carbothermal reduction of silica to obtain silicon for the first time. Crystalline silicon was confirmed from the mixture of silica and carbon black under laser beam irradiation within a few seconds, as supported by the Raman spectra. We found that the laser beam could supply heat energy in the form of intensified heat flux to the mixture of silica and carbon black at an intensity above ~ 4.29 × 106 W/m2 for carbothermal reduction. The intensity of the laser beam and N2 gas flow during the process were critical to obtaining silicon, and an increase in the intensity of the peaks of crystalline silicon was observed in the Raman spectra upon extending the radiation time. The beneficial aspects of the CO2 laser beam as a heat source in terms of feasibility were explored for the carbothermal reduction process for the first time. Although quantitative analysis of the conversion yield of silicon from silica is still in question at this time, we believe that our findings support one new method for extracting silicon from silica and carbon materials and may become the starting point for further process optimization and other silicon applications in the future.In this report, a COSupplementary information 1.Supplementary information 2.Supplementary information 3.Supplementary information 4.Supplementary information 5.


The use of high‐fidelity simulation practice as an educational tool is becoming increasingly prevalent in nursing education. Despite the learning effects of simulation practice, students have been shown to experience high levels of stress and anxiety during simulation. In recent years, peer learning has been defined as an acquisition of knowledge and skills through active support and support among equal or equal peers and has been shown to be an effective educational intervention for clinical health science students.The purpose of this study was to incorporate peer learning into simulation learning and to clarify the differences between stress and anxiety during personal and peer simulations.Third‐grade undergraduate students in a four‐year course at two nursing universities participated in this study. In this study, the simulated patient was a 53‐year‐old man who had undergone gastrectomy for the treatment of gastric cancer. The scenario was that the patient had completely recovered consciousness in the operating room, and his tracheal tube had been removed one hour before the students examined him. Stress while simulation training was evaluated with heart rate variability. Anxiety was evaluated by the STAI after the simulations were complete.n = 50) and peer simulation practice  was conducted. The personal group included 7 male students, and the peer group included 12 male students; the difference in male proportion was not significant. At the first patient assessment phase, stress of heart rate variability components at the peer group significantly increased relative to that of the personal. In addition, the personal had a significantly higher state anxiety score after simulation than the peer.Personal simulation practice (personal group; This study shows that in the face‐to‐face scene involving vital sign measurements, the presence of peers did not objectively alleviate stress. In previous studies, simulation‐based nursing education interventions have been shown to have a strong educational effect, especially in the psychomotor region  has been found to be increased during simulation practice . In 2016, 2017 and 2018, the peer learning simulation practice was conducted . The individual simulation was performed only for this study, and the peer learning simulation was included in one of the practices. All students were recruited before the two simulations started. In the peer learning simulation, only those willing to participate in this study were included through peer matching.Third‐grade undergraduate students in a four‐year course at two nursing universities participated in this study. We recruited participants by disclosing this study protocol to all students. Only students who indicated their intention to participate were included as participants. All participating students had completed their basic lecture‐based units in basic medicine and basic nursing such as nursing theory and nursing skills before the beginning of this study. None of the students had previously used a high‐fidelity manikin. Each had practiced daily care, such as bathing and shampooing, of one patient over 2 weeks during clinical practice. The students were not accustomed to observing actual perioperative patients. Therefore, the educators prepared simple scenarios that were easy for the students to address. Participants with hypertension, heart disease, diabetes, kidney disease, or reliance on daily medications were excluded from the study. All participants avoided alcohol and caffeine the day before the study and were asked to attempt to sleep well the night before. In addition, participants provided verbal confirmation of the quality of their sleep the day before their study participation. In 2014 and 2015, the individual simulation practice was performed . The manikin can emit (and attenuate) a breathing sound and can be set to maintain or change a specific HR, blood pressure and respiratory rate. In this study, the simulated patient was a 53‐year‐old man who had undergone gastrectomy for the treatment of gastric cancer. The scenario was that the patient had completely recovered consciousness in the operating room and his tracheal tube had been removed one hour before the students examined him. When the student examined the patient, he had a central venous catheter, an oxygen mask with a flow rate of 3 L/min, a nasogastric tube, a urethral catheter, a dressing to protect his abdominal wounds and an indwelling abdominal drain. The educator set the physiological parameters of the ECG monitor and manikin as follows: HR 62–70 beats per minute (bpm), blood pressure 120 to 128/66 to 70 mm Hg and respiratory rate 16–20 respirations per minute. In both types of simulation practices, students were given instructions for the simulation scenario before the simulation started. Table 2.4Before the simulation practices began, the students received an explanation of the characteristics of the high‐fidelity manikin and were free to touch it. The researcher connected each student to the Holter ECG inspection system . Participants were given a questionnaire to evaluate their anxiety (State‐Trait Anxiety Inventory‐Form JYZ (STAI‐JYZ)) after they agreed to participate in the study. They answered the questions without the educator present. Later, the students were given the break to stabilize their autonomic nervous systems. During the break, they sat on a chair in a quiet room with the curtains drawn and were advised to take deep breaths. After the break, the students entered the patient area with the educators and began the first patient assessment phase. After the first patient assessment, the students mainly confirmed the patient's breathing frequency, breathing sounds, cyanosis state and peripheral circulation. In addition, they confirmed that the oxygen tube was not bent and that the dose of oxygen was correct. In the reporting phase, the students who participated in the individual simulation reported the condition of the patient to educators and the students who participated in the peer simulation confirmed their observations among the peer members and shared the information using a large whiteboard. The debriefings of both simulations were conducted in a different room and were based on the following five guidelines by Decker et al. : the deb2.5Heart rate variability (HRV), one of the measurements that can be used for objective stress assessment, is calculated from the change in the R‐R wave interval in successive normal ECG signals measured by the Holter electrocardiographic recording system. HRV is widely accepted as an indicator of autonomic nervous activity . The HR, HF and LF/HF components in the two phases (the break and patient assessment phases) were compared between the individual and peer simulation practices. The statistical significance level was set at p < .05.Data from the two groups were compared using the chi‐square test and the Mann–Whitney 2.8The educator obtained informed consent from all participants and explained the ethical considerations of participating in the study as follows: participation was voluntary, and the students could withdraw from the study at any time without affecting their grades. This study was conducted with the approval of the research ethics committee of the university that conducted the study.3p = .453). The median age of participants in the two groups was 21 years (range 20–22 years), and no significant differences were observed between the two groups in terms of background      students participated in the individual simulation and 59/318 (18.5%) students participated in the peer simulation. The individual group included 7 male and 43 female students and the peer group included 12 male and 47 female students; the difference in the proportion of males between the two groups was not significant (.5% studep = .352, 0.077; Figure p = .010; Figure The scores of state and trait anxiety before practice as obtained by the STAI did not differ between the individual and peer simulations (4In this study, during the first patient assessment phase, no significant difference in HR was found between the two types of simulation, but the HF index was significantly decreased and the LF/HF index was significantly increased in the peer simulation group relative to the individual simulation group. Thus, the peer simulation students were exposed to greater stress at the patient assessment phase than were the individual simulation students. In previous studies, nursing students were found to experience greater stress and pressure during simulation practice than during passive learning (Boostel et al., Preparing for stressful situations can be achieved not only by the practice of skills and knowledge but also by positive cognitive preparation and encouragement from team members (vonRosenberg, In this study, there was no difference between the two groups in state or trait anxiety, as measured using the STAI, before undergoing simulation. However, in postsimulation state anxiety, the individual simulation students scored significantly higher than the peer simulation students. This result indicates that the anxiety level in the individual simulation group was higher than that in the peer simulation group postsimulation. Previous studies have shown that anxiety adversely affects learning (Jiang et al., This study had the following strengths and limitations. This study showed that nursing students increased stress and reduced anxiety from peer simulation practices. Therefore, even in hospital nursing education practice, nursing students may be able to enhance their learning effectiveness through active support and support among peers in the same nursing student positions. In addition, in the field of nursing practice, peer nursing may help nurses provide care without anxiety. The present study does not reveal any differences in performance between the individual and peer simulations. Future research should focus on the differences in learning effects between different methods.5This study compared student stress during individual and group simulation practices. During the patient assessment phase, the stress level of the peer‐learning simulation group was significantly increased relative to that of the individual simulation group. This study showed that in face‐to‐face interactions involving vital sign measurements, the presence of other students does not objectively reduce stress. In addition, the anxiety level of the peer‐learning simulation group was significantly decreased relative to that of the individual simulation group. The findings indicate that the type of simulation method affects the stress situation experienced by students and student anxiety.The authors declare no conflict of interest associated with this manuscript.