text:
Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

# Understanding hesitancy with revealed preferences across COVID-19 vaccine types

## Abstract

Many countries have secured larger quantities of COVID-19 vaccines than their population is willing to take. The abundance and the large variety of vaccines created not only an unprecedented intensity of vaccine related public discourse, but also a historical moment to understand vaccine hesitancy better. Yet, the heterogeneity of hesitancy by vaccine types has been neglected in the existing literature so far. We address this problem by analysing the acceptance and the assessment of five vaccine types. We use information collected with a nationally representative survey at the end of the third wave of the COVID-19 pandemic in Hungary. During the vaccination campaign, individuals could reject the assigned vaccine to wait for a more preferred alternative that enables us to quantify revealed preferences across vaccine types. We find that hesitancy is heterogenous by vaccine types and is driven by individuals’ trusted source of information. Believers of conspiracy theories are more likely to evaluate the mRNA vaccines (Pfizer and Moderna) unacceptable. Those who follow the advice of politicians are more likely to evaluate vector-based (AstraZeneca and Sputnik) or whole-virus vaccines (Sinopharm) acceptable. We argue that the greater selection of available vaccine types and the free choice of the individual are desirable conditions to increase the vaccination rate in societies.

## Introduction

The fast development of vaccines against the SARS-COV-2 virus will go to history books as an outstanding scientific achievement1,2. Unprecedented efforts and investments resulted in vaccines of various types3: established techniques resulted in vector-based and whole-virus vaccines while cutting-edge technologies pioneered for the mRNA vaccines. However, large populations remained hesitant against vaccination4,5,6,7,8,9 threatening the efficiency of vaccination plans10,11,12,13,14, especially in less developed regions15.

The existing literature on vaccine hesitancy considers the act of taking a vaccine as a decision under uncertainty, where the individual has to compare the potential risks and benefits of getting vaccinated16. As a result, the personal beliefs17,18 and the trusted source of information, besides demographic and socio-economic characteristics19,20,21, play a critical role in vaccine hesitancy. Previous literature shows that individuals who trust the public authorities and the scientists tend to be less hesitant4,22,23, while those who believe in conspiracy theories tend to be more hesitant against the vaccines24,25,26,27.

Despite the intensive public discourse about the vaccines’ efficiency28 and their potential risks29, the driving forces of hesitancy across vaccine types are unclear in the literature. Individuals might not be hesitant against all types of vaccines; instead, they might be open for vaccines that use certain technologies or come from certain countries30. To the best of our knowledge, however, the question of hesitancy by vaccine types has only been researched on theoretical grounds31. In this paper, we show that individual characteristics and the trusted source of information are key factors in explaining the hesitancy about particular vaccine types.

We compare the acceptance and the assessment of five vaccine types in Hungary, the country that applied the greatest diversity of vaccines during the third wave of the COVID-19 pandemic (Vaccine diversity is simply calculated by the number of available vaccine types from data at https://ourworldindata.org/covid-vaccinations). Besides the centralized European Union (EU) purchase, which included the Pfizer-Biontech, Moderna and AstraZeneca vaccines, the Hungarian government imported Sputnik from Russia and Sinopharm from China. Although Janssen was also available in Hungary from May 2021, only a small share of population received it. We excluded this vaccine from the analysis (Official names of the vaccines are listed here. Pfizer-Biontech: Comirnaty/BNT162b2, COVID-19 mRNA vaccine. Moderna: Spikevax/mRNA-1273. AstraZeneca: Vaxzevria, Covishield/AZD1222, ChAdOx1 nCoV-19, ChAdOx1-S, AZD2816. Sputnik: Gam-COVID-Vac. Sinopharm: Sinopharm: BIBP COVID-19 vaccine/Zhong’aikewei, Hayat-Vax. Janssen: Ad26.COV2.S JNJ-78436735 Ad26COVS1 VAC31518). The availability of Sputnik and Sinopharm in the early vaccination period put Hungary among countries with the highest vaccination rate globally. The vaccination rate was still outstanding among Central and Eastern European countries at the beginning of the fourth wave (According to https://ourworldindata.org/covid-vaccinations, vaccination per capita was much higher in May 2021, when our data was collected, in Hungary than in other Central and Eastern European countries where trust in public authorities is at a similar level). The government strongly communicated the safety and efficiency of Sputnik and Sinopharm27,32; while mRNA vaccines were highly rated in the society33. As hesitancy was high in the beginning of the vaccination campaign22 ($$53\%$$ in January 2021 and $$20\%$$ in May 202133), the government introduced a more flexible vaccine assignment policy to increase people’s willingness to vaccinate. As a result, individuals could reject the assigned vaccine and wait for a more preferred alternative.

A vaccine seeking individual (patient) had to go through the following procedure in Hungary over the third wave of the COVID-19 pandemic in the Spring of 2021. First, the patient had to register online at a government portal. The general physician (GP) received the list of patients in its region and contacted them in an order of priority. The priority was determined based on chronic illness, age and occupation. Upon contacting a patient, the GP offered a type of vaccine to the patient. The offer depended on patient characteristics (as explained above) and the availability of vaccines. The patient could either accept or reject the offered vaccine. In case of a rejection, the GP had to reach out to the patient later, when a different type of vaccine became available. This procedure characterized early vaccination until the vast majority of priority groups were vaccinated. Starting from April, when vaccines became increasingly available, patients could register and select available vaccines through a web-application (first AstraZeneca, then Sinopharm and Sputnik, and later other vaccines as well). Besides giving floor for various individual preferences, the Hungarian mechanism might have provided acceptable alternatives for those who got scared from the offered vaccine. For example, the Sinopharm vaccine was offered initially in Hungary to the older generation, despite it was recommended to a younger population. Similarly, AstraZeneca was offered for the middle-aged generation, but many became afraid when the connection between the vaccine and thrombosis cases were discussed.

This historically unique situation enables us to contribute to the ongoing discussion in the following ways. First, we can compare hesitancy across vaccines by technologies (mRNA, vector-based, and whole virus). Second, we are able to identify the individual characteristics that correlate with hesitancy of specific vaccines, such as the role of trusted source of information. Third, we can analyze the role of free vaccine choice for vaccine hesitancy due to the special allocation mechanism.

## Results

The empirical analysis is based on a nationally representative survey (N = 1000) that we conducted in Hungary at the end of the third wave of the pandemic when vaccines were already available for everyone (Number of cases can vary among analyses. For a precise consideration of sample sizes, please see “Methods”). Data collection is described in “Methods”.

### Vaccine assignment and revealed preferences

In our survey, we focused on demographic, socio-economic, and health related questions that could help to understand COVID-19 vaccine hesitancy better. Specifically, the participants had to evaluate each vaccine, identify their trusted sources regarding the vaccination and state whether their assessment has changed over the third wave period. Then, they were asked a list of questions about their COVID-19 infection and their vaccination history including the date of potential positive tests, date and type of the received and the rejected vaccines. This latter information provides new insights into the nature of vaccine hesitancy through the analysis of revealed vaccine preferences. From now on, $$evaluate$$ refers to whether an individual considers the vaccine types acceptable, $$reject$$ refers to the act of not willing to take the assigned vaccine and $$accept$$ refers to the act of taking the assigned vaccine.

Table 1 describes the vaccination related variables by socio-demographic characteristics of the respondents. The evaluation and the rejection of vaccines—which information makes our dataset unique—are presented by sex, education level, chronic illness, and age categories. Here we can distinguish those who rate all vaccine types unacceptable ($$2.3\%$$) from those who evaluate at least one vaccine unacceptable but evaluate another acceptable ($$18.2\%$$). Furthermore, we can compare those who accepted the first vaccine offered ($$62.9\%$$) with those who rejected the first offer ($$10.8\%$$). Apparently, the women, the younger cohorts, the lower educated and those with no chronic disease are more likely to reject a vaccine offer in general. We also find that $$5\%$$ of those respondents who received Pfizer rejected at least one other vaccine. These shares are lower for other vaccine types. Furthermore, we find that $$20.3\%$$ of all respondents have accepted Pfizer without rejecting any vaccines while $$5.8\%$$ waited for Pfizer after rejecting another vaccine.

Figure 1 illustrates the network of revealed preferences across vaccine types from our data. Focusing on those who rejected a vaccine (in total, 18% of our vaccinated respondents rejected an assigned vaccine and later took another type), we observe a high ratio of individuals, who received Pfizer but rejected Sinopharm ($$23\%$$), Sputnik ($$15\%$$) or AstraZeneca ($$9.7\%$$). We also see that a few individuals rejected an mRNA vaccine to accept a less popular vaccine later. For example, $$2.2\%$$ of the hesitant population rejected Pfizer to receive Sinopharm.

Figure 2A demonstrates the dynamics of revealed preferences by vaccine types. The Pfizer was the first vaccine that patients could receive in Hungary starting at the first week of 2021. The Sputnik, the Sinopharm and the AstraZeneca only became available in larger quantities after week 5 of 2021. Most of the Moderna vaccines were accepted after week 9 of 2021. Figure 2B shows the time of the first vaccine and the age of the patients. We grouped the individuals based on chronic illness status and whether they rejected a vaccine in the past. We find that those chronically ill, elder patients who rejected vaccines received a more preferred vaccine in March. The younger healthy individuals who rejected a vaccine received a more preferred vaccine only after mid April. Supporting Information 1 contains further description of the dynamics of vaccination and rejection by age, chronic illness and vaccine types.

### Trusted information sources characterize groups of vaccine preferences

To observe hesitancy by vaccine types, we asked respondents to evaluate vaccine types on a scale from 1 “unacceptable” to 5 “best”. For the reasons of simplicity, here we code all evaluations different from 1 as “acceptable” and analyse rating details in section “Individual selection improves vaccine assessment”. Supporting Information 2 illustrates that Pfizer and Moderna were more likely to get accepted than Sputnik, which is significantly more likely to get accepted than AstraZeneca and Sinopharm. Chi square tests imply that the distributions of vaccine acceptance are not independent from each other. Instead, acceptance is correlated across vaccine types, and especially between Pfizer and Moderna and between Sputnik and Sinopharm.

We use multivariate regression analysis to investigate the heterogeneity of vaccine evaluation by types and its association with the source of trusted information. This latter determinant was used in the previous studies to explain vaccine hesitancy25, while other research focused on the communication channels that individuals collected vaccine-related information from18,34. Accordingly, respondents were asked in our survey to confirm if they follow the vaccine-related advice of doctors, scientists, anti-vaccine propagators, politicians, family, friends, journalists and celebrities. These major actors communicated differently about the vaccines (Doctors and scientists argued for vaccination, see for example the recommendations of Hungarian Academy of Sciences: https://mta.hu/english/recommendation-of-the-hungarian-academy-of-science-on-the-management-of-the-covid-19-epidemic-in-the-short-and-long-term-110630. Anti-vaccine propagators emphasized the non-effectiveness and the side effects of the COVID-19 vaccines. A summary on anti-vax movement in Hungary can be found at https://www.iribeaconproject.org/our-work-analysis-and-insights/2021-09-16/hungarys-anti-vax-movement-alive-and-kicking. Some medical doctors warned against vaccines that are not approved by the EU, while others urged to vaccinate with any vaccine: https://www.reuters.com/article/us-health-coronavirus-hungary-vaccines-idUSKBN2CE2C3. The government communication campaigned for the non-EU-approved vaccines to improve their rate of acceptance by the population: https://www.euronews.com/2021/02/28/hungary-s-pm-viktor-orban-vaccinated-against-covid-with-chinese-sinopharm-vaccine. Yet, a heated debate around vaccine types has fragmented the society: https://hungarytoday.hu/hungarian-vaccination-politics-vaccine-preference-coronavirus-survey-willingness/). The respondents also had to name their most frequently used information channels on pandemic measures such as web news, social media, press, radio, TV, family and friends. These explanatory variables are complemented by demographic and socio-economic characteristics of respondents25. We apply a two-step procedure in the regression analysis. First, we use a machine learning technique to select the most important variables. Second, we run the regressions using the selected variables.

In the first step, we apply a variable selection method. This is important because the source of trusted information and the channel of communication can influence hesitancy simultaneously. The LASSO model, which is a short notation for the least absolute shrinkage and selection operator, enables us to select the most important variables that have high-enough predictive power. We describe the process in the “Methods” section in greater details. The estimation results are reported in Supporting Information 3, Table S2. We find that communication channels are not important determinants of hesitancy as all but one (web news) are dropped from the model. However, we see that both the demographic characteristics and the sources of trusted information on COVID-19 vaccination are important predictors of the outcome variables. We rely on this result when we select the explanatory variables in the linear probability models.

Next, we estimate linear probability models (LPM) to analyze how different factors shape the evaluation of each vaccine individually. Independent variables include “Advice” dummy variables that take value 1 if the individual follows the advice of the denoted source. “Age” is a numeric variable; it is measured in years and is expected to correlate negatively with hesitancy. “Female” takes value 1 if the respondent is woman and zero otherwise. “University” equals 1 if the respondent has a university degree or equivalent and is expected to correlate negatively with hesitancy. We include health-related information as well. “Smoking” and “Chronic Illness” are binary variables as well as “COVID Previously” and “Serious COVID previously”. We present the summary statistics in Supporting Information 4, which also contains a correlation matrix and VIF statistics. The latter tests indicate no issues of multicollinearity. The estimation specification can be found in “Methods”.

Figure 3 shows the point estimates and the $$95\%$$ confidence intervals calculated from the heteroscedasticity-robust standard errors (for detailed results see Table S3 in Supporting Information 5). The results suggest that the driving factors of rejecting a decision differ among vaccine types. Those who trust the advice of anti-vaccine propagators are 18.2 and 20.2 percent more likely to evaluate mRNA vaccines (Pfizer and Moderna) unacceptable. Trusting the doctors’ advice, however, decreases the hesitancy probability by 5.7, 6.1, and 5.4 percent for Pfizer, Moderna, and Sinopharm, respectively. Trusting scientists reduce the likelihood of evaluating a vaccine unacceptable by 3.5, 2.6 and 5.5 per cent for Pfizer, Moderna, and AstraZeneca, respectively. Those who take advice from politicians are 6.4, 7.6 and 8.4 percent more likely to evaluate AstraZeneca, Sputnik and Sinopharm acceptable, respectively. Those who trust their family’s advice are more likely to evaluate Pfizer, Moderna and Sinopharm acceptable by 4.3, 3.8 and 5.8 percent, respectively. The coefficients for the advice of celebrities are either not or only slightly significant. We find that the point estimates are large and positive for all types of vaccines.

Socio-demographic characteristics of respondents correlate with vaccine hesitancy in the following way. Age has negative correlation implying that more vulnerable, older individuals are less likely to evaluate vaccines unacceptable. Although the coefficient of age is small in Fig. 3, the regression on standardized variables reported in Table S3 (in Supporting Information 4) reveals that age is an important factor for evaluating Pfizer, Moderna and AstraZeneca acceptable. Women are more likely to evaluate AstraZeneca unacceptable, probably due to reported cases of females’ blood clot (https://www.reuters.com/business/healthcare-pharmaceuticals/uk-regulator-says-some-evidence-astrazeneca-clots-occur-more-women-than-men-2021-05-06/). Note that a meta-analysis9 found an overall significant gender difference with males being on average 41% more likely to report that they intended to receive a vaccine than females. Our results can indicate that this extra hesitancy for females might come from the different risk-tolerance regarding serious side-effects. University degree, however, decreases hesitancy towards AstraZeneca.

We have carried out the following robustness checks in Supporting Information 5. First, we show that our regression results hold if we use a logistic regression specification instead of ordinary least squares (both of these specifications have been used previously to examine vaccine acceptance35). Second, our results remain similar after including subjective wealth as a control variable. A potential limitation of our study could be from “spreading of the alternatives”36, which means that after a difficult decision making, individuals tend to value the accepted alternative better and the rejected alternative worse. However, our results do not change if we apply the models on a subsample of those who stated that they had stable vaccine assessment over the time.

### Individual selection improves vaccine assessment

To better understand how free choice across vaccines is related to their acceptance, we analyze individuals’ vaccine evaluation values on a scale from $$1$$ to $$5$$ sorted by the received and the rejected vaccines. Figure 4 summarizes the vaccine evaluations in the form of a heat map. A heat map cell corresponds to the average rating of a type of vaccine. The row denotes the average evaluation for that vaccine among the vaccine group of patients denoted by the column. Supporting Information 6 contains summary statistics of vaccine assessments depicted in Fig. 4.

In Fig. 4A, we consider individuals who accepted their first vaccine offer. Notice that Pfizer has the highest average ratings across all columns. This implies that free vaccine choice does not necessarily mean that everyone gets the most preferred vaccine. Instead, individuals make an uncertain decision considering the acceptable alternatives and the unknown waiting times between the sequential vaccine offers. In the Supporting Information 7, Table S13 presents the distribution of the most preferred vaccine when the accepted vaccine is not the most highly-rated vaccine. We infer that patients were willing to accept a vaccine other than the preferred Pfizer or Moderna to avoid the long anticipated waiting times. In Table S14, we show that patients had to wait almost $$2.76$$ weeks on average to get their most preferred vaccine if they rejected a less preferred vaccine. In Table S15, we also present evidence that the anticipated waiting times closely follow the actual waiting times on average. That is, if a patient accepts the first vaccine offer then the following conditions must hold; the patient prefers the accepted vaccine to the other vaccines or she has a belief that a more preferred vaccine, which is likely to be Pfizer and Moderna, will not be offered anytime soon. We provide an extended analysis in the Supporting Information 7.

Figure 4B shows the ratings of those patients who rejected at least one vaccine. We see that the diagonal values in Fig. 4B are slightly higher than the ones in Fig. 4A. This implies that the individuals who rejected a vaccine before value their received vaccine slightly higher than those who accepted their first vaccine. Figure 4B also shows a high level of dislike towards AstraZeneca, Sinopharm and somewhat Sputnik when it comes to rating vaccines other than the accepted vaccine.

Comparing the values on Fig. 4A,B, we can see fundamental differences between the rejected none and rejected any groups. Based on the values on the diagonal, those individuals who rejected at least one vaccine rated the received vaccine higher than those who accepted the first vaccine offer. Looking at the off-diagonal, we see that the rejected any group tends to rate the non-accepted vaccines to be worse. For instance, the mean ratings for AstraZeneca at the off-diagonal are strictly smaller among those individuals who rejected a vaccine at least once than among those who accepted their first offer. We can also see the individuals who received Sinopharm after rejection tend to rate other vaccines worse than those who accepted Sinopharm in their first offer.

Figure 4C highlights the benefits of free choice. We group individuals based on the first vaccine offer rejection denoted by the columns. We implicitly assume here that the vaccine registration is not affected by the government’s decision to allow free choice. Let us point out that the number of observations is small for the Pfizer and the Moderna groups with only $$6$$ and $$2$$ observations, respectively. In other words, patients rarely reject Pfizer or Moderna when these vaccines are offered for the first time. Focusing on the AstraZeneca, Sputnik and Sinopharm columns, we see that the values for the assigned vaccines on the diagonal are strictly smaller than any of alternatives. We can also see a strong preference towards Pfizer and Moderna from the AstraZeneca, the Sputnik and the Sinopharm columns. Figure 4C suggests that free choice significantly increased the efficiency of vaccine allocation. This follows from enabling the re-selection of acceptable vaccines.

We report how assessment of vaccines have changed over the third wave in Supporting Information 7. The ratio of patients who changed their assessment over time for a specific vaccine stays below 5 percent of the total sample. We can only see a large negative change in the case of the AstraZeneca reaching 10 percent of the sample. This is most likely a result of the news about the possible side effects. Finally, Supporting Information 8 contains a preference analysis of those individuals who were not vaccinated but evaluated minimum 2 but maximum 3 of the investigated vaccines acceptable. This enables us to look at co-acceptance among vaccines. In Fig. S7 we find that the most common co-acceptance link is between Pfizer and Moderna. The Sputnik-Sinopharm co-acceptance link is the most frequent one that does not include Pfizer. This suggests that a wide and diverse vaccine portfolio is preferred in a society where vaccine preferences diverge.

## Discussion

Most European countries did not allow individuals to choose between the available vaccines in the first half of 2021. An exception is the case of Spanish essential workers who were allowed to choose AstraZeneca or Pfizer for the second dose37. However, the potential benefits of free vaccine choice have been addressed in the scientific discussion38,39. The main reason for these benefits is that free choice can fulfill heterogeneous individual preferences that in theory, may differ by the technology and the country of origin30. Experiments carried out in Germany have indeed found evidence that free choice across Pfizer and AstraZeneca, the two most different and also the most preferred vaccines available in the country in April 2021, has mitigated vaccine hesitancy40. Another study has shown that the vast majority of Italian college students preferred the mRNA technology to vector vaccines41. A recent survey conducted in eight European countries shows the differences in vaccine hesitancy and the divergent preferences over the available vaccines42.

We add to this line of research by analyzing a representative survey in Hungary. This survey not only asks the preferences of the patients, but also collects information about their actual decisions over accepting or rejecting specific vaccine offers. Hence, the survey participants provide their revealed preferences over the available vaccines.

Our results imply heterogeneous vaccine preferences. Most individuals prefer mRNA vaccines such as the Pfizer and the Moderna. However, a non-negligible portion ($$5.5\%$$) of respondents who rejected an mRNA vaccine preferred a vector-based or a whole virus vaccine. We find evidence for divergent attitudes towards vaccine types depending on the trusted source of advice. Those who trust politicians are more likely to accept less popular vaccines such as the vector-based (AstraZeneca and Sputnik) and the whole-virus vaccines (Sinopharm). Those who are open to conspiracy theories are more likely to reject mRNA vaccines (Pfizer and Moderna) that are the most preferred vaccines in the country. We show that the rejection of non-desired and the re-selection of preferred vaccines generally improve the assessment of the received vaccine.

These findings suggest that the free choice policy in vaccine allocation should be combined with the right communication strategy by the government to achieve higher vaccination rate in societies. In early stages of vaccination campaigns, such as the one during the third wave of the COVID-19 pandemic, vaccines are hard to evaluate. Competing expert rankings of vaccine technologies, country of origin, effectiveness, side effects, approval status by national and international authorities can yield very diverse recommendations43. Consequently, individual hesitancy can vary by vaccine types. The diverse preferences which are present in societies15,42 can change over time30. At times when the scientific evidence about side-effects and effectiveness of vaccines are not available44, the governments and the scientific community must earn the trust of the society. Helping individuals to find their most preferred vaccines and to allow the free choice across vaccine types are therefore desirable for future vaccination campaigns.

Since our results are based on a survey, we might underestimate the rate of the hesitant population due to the social desirability bias45. Those who are more concerned about the COVID-19 situation and more willing to answer the survey are also more likely to accept a vaccine. Furthermore, due to social pressure some may expressed no hesitancy in the survey, whilst in reality they have never taken any vaccine46.

## Methods

### Data collection

The data used in this paper was collected with a CATI telephone survey on a nationally representative sample of the Hungarian adult (18 years or older) population between May 25–31, 2021, which corresponds to the end of the third wave of the pandemic. The questionnaire was designed by the authors with diverse expertise, including epidemiological mathematicians, network scientists, economists, economic geographers and sociologists—all working with COVID-19 related research. The sample size is 1000, which is equal to the conventional sample size for nationally representative samples in the country. The data is representative for the adult Hungarian population by gender, age, education and domicile. Data was collected both over mobile and landline phones. The public opinion research company, who was responsible for the data collection, controlled data quality by comparing the voice responses with the recorded answers on randomly selected calls. This campaign was a part of a larger data collection effort, which started in March 2020 and it is still ongoing. For more details about the data collection effort, see47. The sample was selected by a multi-step, proportionally stratified, probabilistic sampling procedure. Minor deviations of the data from population ratios were corrected by iterative proportional weighting after the data collection. After data collection, only the anonymised and hashed data was shared with people involved in the project after signing non-disclosure agreements. The data collection was fully complying with the actual European and Hungarian privacy data regulations and was approved by the Hungarian National Authority for Data Protection and Freedom of Information48, and also by the Research Ethic Committee of the Medical Research Council of Hungary (resolution number IV/3073-1/2021/EKU). Informed consent was obtained from all survey participants.

### Statistical analysis

In the LASSO model, the dependent variable takes values between 1 and 5, where 1 is “unacceptable” and 5 is “the best”. In models 1–5 presented in Supporting Information 3, we estimate evaluation values by vaccine types. Then, in models 6 to 9, the ratings’ variance, mean, minimum, and maximum serve as dependent variables. In model 10, the dependent variable is binary taking the value of 1 if ever rejected a vaccine offer and 0 otherwise. The dependent variable is 1 in the model, if any vaccine is unacceptable. The lasso coefficients ($$\widehat{\beta }_{\lambda }^{L}$$) minimize the following expression: $$\sum _{i=1}^{n}(y_{i}-\beta _{0}-\sum _{j=1}^{p}\beta _{j}x_{ij})^{2}+\lambda \sum _{j=1}^{p}\left| \beta _j \right|$$ .

Linear probability models are used to predict rejection of vaccines. These regressions are specified by the formula $$P(Y=1|X)=\alpha + \beta X + \varepsilon$$, where Y equals 1 if the vaccine is rated unacceptable, $$\alpha$$ is the intercept, $$\beta$$ is point estimates, X is a vector of independent variables, and $$\varepsilon$$ is heteroscedasticity-robust standard error.

To generate the heat maps, we used the same vaccine ratings on a scale from $$1$$ to $$5$$, as we mentioned above. We clustered the survey responses based on the acceptance decision of the first vaccine offer. In case of rejection, we grouped individuals based on the accepted vaccine and the first rejected vaccine. Then, we used standard summary statistics tools to find the number of observation, mean and the standard deviation for each of the groups. We report the mean ratings in Fig. 4 and the rest of the summary statistics in Supporting Information 6.

The link weights of the directed network presented in Fig. 1 is generated following the aggregation procedure $$W_{ab}=\sum _{a}L_{ab}$$, where $$L_{ab}$$ equals 1 if the individual has rejected vaccine a but later received vaccine b. The undirected co-occurrence network presented in Fig. S7 follows the rule of $$W_{mn}=\sum _{m}L_{mn,m \ne n}$$ where $$L_{mn}$$ equals 1 if the respondent accepts both vaccines m and n.

All methods were performed in accordance with the guidelines and regulations of the Hungarian National Authority for Data Protection and Freedom of Information48 and were approved by the Research Ethic Committee of the Medical Research Council of Hungary (resolution number IV/3073-1/2021/EKU).

## Data availability

For privacy protection we share only the relevant part of the data, which does not allow the reidentification of participants. We exclude sensitive information such as domicile and labels for categorical responses. The data is available at https://doi.org/10.5281/zenodo.5575586.

## Code availability

All codes to analyze data have been written in R (version 4.0.5) and are available at https://doi.org/10.5281/zenodo.5575586.

## References

1. Cohen, J. Vaccine designers take first shots at COVID-19 (2020).

2. Graham, B. S. Rapid COVID-19 vaccine development. Science 368, 945–946 (2020).

3. Corey, L., Mascola, J. R., Fauci, A. S. & Collins, F. S. A strategic approach to COVID-19 vaccine R&D. Science 368, 948–950 (2020).

4. Lazarus, J. V. et al. A global survey of potential acceptance of a COVID-19 vaccine. Nat. Med. 27, 225–228 (2021).

5. Machingaidze, S. & Wiysonge, C. S. Understanding COVID-19 vaccine hesitancy. Nat. Med., 1–2 (2021).

6. Sallam, M. COVID-19 vaccine hesitancy worldwide: A concise systematic review of vaccine acceptance rates. Vaccines 9, 160 (2021).

7. Solís Arce, J. S. et al. COVID-19 vaccine acceptance and hesitancy in low-and middle-income countries. Nat. Med. 27, 1385–1394 (2021).

8. Mahmud, S., Mohsin, M., Khan, I. A., Mian, A. U. & Zaman, M. A. Knowledge, beliefs, attitudes and perceived risk about COVID-19 vaccine and determinants of COVID-19 vaccine acceptance in Bangladesh. PLoS One 16, e0257096 (2021).

9. Zintel, S. et al. Gender differences in the intention to get vaccinated against COVID-19: A systematic review and meta-analysis. J. Public Health, 1–25 (2022).

10. Wouters, O. J. et al. Challenges in ensuring global access to COVID-19 vaccines: Production, affordability, allocation, and deployment. Lancet 397, 1023–1034 (2021).

11. Giordano, G. et al. Modeling vaccination rollouts, SARS-CoV-2 variants and the requirement for non-pharmaceutical interventions in Italy. Nat. Med. 27, 993–998 (2021).

12. Bubar, K. M. et al. Model-informed COVID-19 vaccine prioritization strategies by age and serostatus. Science 371, 916–921 (2021).

13. Matrajt, L., Eaton, J., Leung, T. & Brown, E. R. Vaccine optimization for COVID-19: Who to vaccinate first?. Sci. Adv. 7, eabf1374 (2021).

14. Priesemann, V. et al. Towards a European strategy to address the COVID-19 pandemic. Lancet 398, 838–839 (2021).

15. Mahmud, S., Mohsin, M., Hossain, S., Islam, M. M. & Muyeed, A. The acceptance of COVID-19 vaccine: A global rapid systematic review and meta-analysis. Available at SSRN 3855987 (2021).

16. Rosselli, R., Martini, M. & Bragazzi, N. L. The old and the new: Vaccine hesitancy in the era of the web 2.0. challenges and opportunities. J. Prev. Med. Hyg. 57, E47 (2016).

17. Troiano, G. & Nardi, A. Vaccine hesitancy in the era of COVID-19. Public Health (2021).

18. Freeman, D. et al. Effects of different types of written vaccination information on covid-19 vaccine hesitancy in the UK (oceans-III): A single-blind, parallel-group, randomised controlled trial. Lancet Public Health 6, e416–e427 (2021).

19. Loomba, S., de Figueiredo, A., Piatek, S. J., de Graaf, K. & Larson, H. J. Measuring the impact of COVID-19 vaccine misinformation on vaccination intent in the UK and USA. Nat. Hum. Behav. 5, 337–348 (2021).

20. Paul, E., Steptoe, A. & Fancourt, D. Attitudes towards vaccines and intention to vaccinate against covid-19: Implications for public health communications. Lancet Reg. Health 1, 100012 (2021).

21. Sonawane, K., Troisi, C. L. & Deshmukh, A. A. COVID-19 vaccination in the UK: Addressing vaccine hesitancy. Lancet Reg. Health-Europe 1 (2021).

22. Lindholt, M. F., Jørgensen, F., Bor, A. & Petersen, M. B. Public acceptance of covid-19 vaccines: Cross-national evidence on levels and individual-level predictors using observational data. BMJ Open 11, e048172 (2021).

23. Schernhammer, E. et al. Correlates of COVID-19 vaccine hesitancy in Austria: Trust and the government. J. Public Health (Oxford, England).

24. Salali, G. D. & Uysal, M. S. COVID-19 vaccine hesitancy is associated with beliefs on the origin of the novel coronavirus in the UK and Turkey. Psychol. Med., 1–3 (2020).

25. Murphy, J. et al. Psychological characteristics associated with COVID-19 vaccine hesitancy and resistance in Ireland and the United Kingdom. Nat. Commun. 12, 1–15 (2021).

26. Soveri, A., Karlsson, L. C., Antfolk, J., Lindfelt, M. & Lewandowsky, S. Unwillingness to engage in behaviors that protect against covid-19: The role of conspiracy beliefs, trust, and endorsement of complementary and alternative medicine. BMC Public Health 21, 1–12 (2021).

27. Bíró-Nagy, A. & Szászi, Á. J. The roots of COVID-19 vaccine hesitancy: Evidence from Hungary. J. Behav. Med., 1–16 (2022).

28. Lipsitch, M. & Dean, N. E. Understanding COVID-19 vaccine efficacy. Science 370, 763–765 (2020).

29. Ledford, H. How could a COVID vaccine cause blood clots? Scientists race to investigate. Nature 592, 334–335 (2021).

30. Schwarzinger, M., Watson, V., Arwidson, P., Alla, F. & Luchini, S. Covid-19 vaccine hesitancy in a representative working-age population in France: A survey experiment based on vaccine characteristics. Lancet Public Health 6, e210–e221 (2021).

31. Dror, A. A. et al. Vaccine hesitancy due to vaccine country of origin, vaccine technology, and certification. Eur. J. Epidemiol., 1–6 (2021).

32. Goodwin, R. et al. Psychological factors underpinning vaccine willingness in Israel, Japan and Hungary. medRxiv (2021).

33. Ligeti, A. S., Karsai, M., Koltai, J. & Gergely, R. A covid-19 elleni védőoltással kapcsolatos attitűdök a magyar társadalomban. In Horn, D. & Bartal, A. M. (eds.) Fehér könyv a covid-19-járvány társadalmi-gazdasági hatásairól, chap. 1.3, 33–42 (Eötvös Loránd Kutatási Hálózat Közgazdaság- és Regionális Tudományi Kutatóközpont, Közgazdaságtudományi Intézet, 2022).

34. Wilson, S. L. & Wiysonge, C. Social media and vaccine hesitancy. BMJ Global Health 5, e004206 (2020).

35. Kaplan, A. K., Sahin, M. K., Parildar, H. & Adadan Guvenc, I. The willingness to accept the COVID-19 vaccine and affecting factors among healthcare professionals: A cross-sectional study in Turkey. Int. J. Clin. Pract., e14226. https://doi.org/10.1111/ijcp.14226 (2021).

36. Luo, J. & Yu, R. The spreading of alternatives: Is it the perceived choice or actual choice that changes our preference?. J. Behav. Decis. Making 30, 484–491 (2017).

37. Dal-Ré, R., Farré, M. & Lucena, M. I. Heterologous COVID-19 vaccination in Spain: A case study of individual autonomy in the real world. Value Health 25, 770–772 (2022).

38. Dal-Ré, R., Stephens, R. & Sreeharan, N. Let me choose my COVID-19 vaccine. Eur. J. Intern. Med. 87, 104–105 (2021).

39. Hughes, M. T. et al. Opinion: The importance of offering vaccine choice in the fight against covid-19. Proc. Natl. Acad. Sci. 118 (2021).

40. Sprengholz, P., Eitze, S., Korn, L., Siegers, R. & Betsch, C. The power of choice: Experimental evidence that freedom to choose a vaccine against COVID-19 improves willingness to be vaccinated. Eur. J. Intern. Med. (2021).

41. Salerno, L., Craxì, L., Amodio, E. & Lo Coco, G. Factors affecting hesitancy to mRNA and viral vector COVID-19 vaccines among college students in Italy. Vaccines 9, 927 (2021).

42. Steinert, J. et al. COVID-19 vaccine hesitancy in eight European countries: Prevalence, determinants and heterogeneity. Sci. Adv. (2022).

43. Abdelwahab, S. F., Issa, U. H. & Ashour, H. M. A novel vaccine selection decision-making model (VSDMM) for COVID-19. Vaccines 9, 718 (2021).

44. Mohsin, M. et al. Side effects of COVID-19 vaccines and perceptions about COVID-19 and its vaccines in Bangladesh. medRxiv (2022).

45. Nederhof, A. J. Methods of coping with social desirability bias: A review. Eur. J. Soc. Psychol. 15, 263–280 (1985).

46. Koltai, J., Vásárhelyi, O., Röst, G. & Karsai, M. Monitoring behavioural responses during pandemic via reconstructed contact matrices from online and representative surveys. arXiv preprint arXiv:2102.09021 (2021).

47. Koltai, J., Vásárhelyi, O., Röst, G. & Karsai, M. Reconstructing social mixing patterns via weighted contact matrices from online and representative surveys. Sci. Rep. 12, 1–12 (2022).

## Acknowledgements

The authors acknowledge the help of Szabolcs Tóth-Zs in finalizing the figures. Péter Biró gratefully acknowledges financial support from the Hungarian Academy of Sciences, Momentum Grant (No. LP2021-2). Julia Koltai acknowledges funding from the Premium Postdoctoral Grant of the Hungarian Academy of Sciences. The work of Balazs Lengyel was financially supported by Hungarian National Scientific Fund (OTKA K 138970). Ágnes Szabó-Morvai was supported by the Lendület programme of the Hungarian Academy of Science (LP2018-2/2018) and by the National Research, Development and Innovation Office (OTKA FK131422). Júlia Koltai, Gergely Röst and Márton Karsai acknowledges funding from the National Laboratory for Health Security (RRF-2.3.1-21-2022-00006).

## Funding

Open access funding provided by ELKH Centre for Economic and Regional Studies.

## Author information

Authors

### Contributions

J.K., A.Sz., G.R., M.K., P.B. and B.L. conceived the survey data collection, K.K., A.Sz., P.B., B.L. designed the investigation, K.K., A.Sz., and B.L. analyzed the data. All authors wrote and reviewed the manuscript.

### Corresponding author

Correspondence to Balázs Lengyel.

## Ethics declarations

### Competing interests

The authors declare no competing interests.

### Publisher's note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

## Rights and permissions

Reprints and Permissions

Kutasi, K., Koltai, J., Szabó-Morvai, Á. et al. Understanding hesitancy with revealed preferences across COVID-19 vaccine types. Sci Rep 12, 13293 (2022). https://doi.org/10.1038/s41598-022-15633-5

• Accepted:

• Published:

• DOI: https://doi.org/10.1038/s41598-022-15633-5


text:
# ModelingToolkit.jl: High-Performance Symbolic-Numeric Equation-Based Modeling

ModelingToolkit.jl is a modeling language for high-performance symbolic-numeric computation in scientific computing and scientific machine learning. It then mixes ideas from symbolic computational algebra systems with causal and acausal equation-based modeling frameworks to give an extendable and parallel modeling system. It allows for users to give a high-level description of a model for symbolic preprocessing to analyze and enhance the model. Automatic transformations, such as index reduction, can be applied to the model before solving in order to make it easily handle equations would could not be solved when modeled without symbolic intervention.

## Installation

To install ModelingToolkit.jl, use the Julia package manager:

using Pkg
Pkg.add("ModelingToolkit")

## Citation

@misc{ma2021modelingtoolkit,
title={ModelingToolkit: A Composable Graph Transformation System For Equation-Based Modeling},
author={Yingbo Ma and Shashi Gowda and Ranjan Anantharaman and Chris Laughman and Viral Shah and Chris Rackauckas},
year={2021},
eprint={2103.05244},
archivePrefix={arXiv},
primaryClass={cs.MS}
}

## Feature Summary

ModelingToolkit.jl is a symbolic-numeric modeling package. Thus it combines some of the features from symbolic computing packages like SymPy or Mathematica with the ideas of equation-based modeling systems like the causal Simulink and the acausal Modelica. It bridges the gap between many different kinds of equations, allowing one to quickly and easily transform systems of DAEs into optimization problems, or vice-versa, and then simplify and parallelize the resulting expressions before generating code.

### Feature List

• Causal and acausal modeling (Simulink/Modelica)
• Automated model transformation, simplification, and composition
• Automatic conversion of numerical models into symbolic models
• Composition of models through the components, a lazy connection system, and tools for expanding/flattening
• Pervasive parallelism in symbolic computations and generated functions
• Transformations like alias elimination and tearing of nonlinear systems for efficiently numerically handling large-scale systems of equations
• The ability to use the entire Symbolics.jl Computer Algebra System (CAS) as part of the modeling process.
• Import models from common formats like SBML, CellML, BioNetGen, and more.
• Extendability: the whole system is written in pure Julia, so adding new functions, simplification rules, and model transformations has no barrier.

For information on how to use the Symbolics.jl CAS system that ModelingToolkit.jl is built on, consult the Symbolics.jl documentation

### Equation Types

• Ordinary differential equations
• Stochastic differential equations
• Partial differential equations
• Nonlinear systems
• Optimization problems
• Continuous-Time Markov Chains
• Chemical Reactions (via Catalyst.jl)
• Nonlinear Optimal Control

## Standard Library

For quick development, ModelingToolkit.jl includes ModelingToolkitStandardLibrary.jl, a standard library of prebuilt components for the ModelingToolkit ecosystem.

## Model Import Formats

• CellMLToolkit.jl: Import CellML models into ModelingToolkit
• Repository of more than a thousand pre-made models
• Focus on biomedical models in areas such as: Calcium Dynamics, Cardiovascular Circulation, Cell Cycle, Cell Migration, Circadian Rhythms, Electrophysiology, Endocrine, Excitation-Contraction Coupling, Gene Regulation, Hepatology, Immunology, Ion Transport, Mechanical Constitutive Laws, Metabolism, Myofilament Mechanics, Neurobiology, pH Regulation, PKPD, Protein Modules, Signal Transduction, and Synthetic Biology.
• SBMLToolkit.jl: Import SBML models into ModelingToolkit
• Uses the robust libsbml library for parsing and transforming the SBML
• ReactionNetworkImporters.jl: Import various models into ModelingToolkit
• Supports the BioNetGen .net file
• Supports importing networks specified by stoichiometric matrices

## Extension Libraries

Because ModelingToolkit.jl is the core foundation of a equation-based modeling ecosystem, there is a large set of libraries adding features to this system. Below is an incomplete list of extension libraries one may want to be aware of:

• Catalyst.jl: Symbolic representations of chemical reactions
• Symbolically build and represent large systems of chemical reactions
• Generate code for ODEs, SDEs, continuous-time Markov Chains, and more
• Simulate the models using the SciML ecosystem with O(1) Gillespie methods
• DataDrivenDiffEq.jl: Automatic identification of equations from data
• Automated construction of ODEs and DAEs from data
• Representations of Koopman operators and Dynamic Mode Decomposition (DMD)
• MomentClosure.jl: Automatic transformation of ReactionSystems into deterministic systems
• Generates ODESystems for the moment closures
• Allows for geometrically-distributed random reaction rates
• ReactionMechanismSimulator.jl: Simulating and analyzing large chemical reaction mechanisms
• Ideal gas and dilute liquid phases.
• Constant T and P and constant V adiabatic ideal gas reactors.
• Constant T and V dilute liquid reactors.
• Diffusion limited rates. Sensitivity analysis for all reactors.
• Flux diagrams with molecular images (if molecular information is provided).
• NumCME.jl: High-performance simulation of chemical master equations (CME)
• Transient solution of the CME
• Dynamic state spaces
• Accepts reaction systems defined using Catalyst.jl DSL.
• FiniteStateProjection.jl: High-performance simulation of chemical master equations (CME) via finite state projections
• Accepts reaction systems defined using Catalyst.jl DSL.

## Compatible Numerical Solvers

All of the symbolic systems have a direct conversion to a numerical system which can then be handled through the SciML interfaces. For example, after building a model and performing symbolic manipulations, an ODESystem can be converted into an ODEProblem to then be solved by a numerical ODE solver. Below is a list of the solver libraries which are the numerical targets of the ModelingToolkit system:

• DifferentialEquations.jl
• Multi-package interface of high performance numerical solvers for ODESystem, SDESystem, and JumpSystem
• NonlinearSolve.jl
• High performance numerical solving of NonlinearSystem
• Optimization.jl
• Multi-package interface for numerical solving OptimizationSystem
• NeuralPDE.jl
• Physics-Informed Neural Network (PINN) training on PDESystem
• MethodOfLines.jl
• Automated finite difference method (FDM) discretization of PDESystem

format: plain


text:
• # question_answer   Find an angle $\theta ,\,\,0<\theta <\frac{\pi }{2},$ which increases twice as fast as it sine.

Let $y=\sin \theta$ On differentiating w.r.t.$'t'$, we get $\frac{dy}{dt}=\frac{d\,(sin\theta )}{dt}=\cos \theta \frac{d\theta }{dt}$ According to the question, $\frac{d\theta }{dt}=2\frac{dy}{dt}$ $\therefore$      $\frac{dy}{dt}=\cos \theta \times 2\frac{dy}{dt}$ $\Rightarrow$   $\cos \theta =\frac{1}{2}$         $\Rightarrow$$\theta =\frac{\pi }{3}$ Hence, the required angle is $\frac{\pi }{3}$.


text:
# MINOTAUR

Welcome to the MINOTAUR wiki page.

## Introduction

MINOTAUR is a toolkit for solving mixed-integer nonlinear optimization problems of the form

 minimize $f(x,y)$ subject to $g_i(x,y) \leq 0 \,$ for $i = 1, ..., m \,$ $x \in X \; y \in Y$ integer

where $f(x,y), g_i(x,y)$ are smooth functions of the continuous variables $x$ and the integer variables <maths>y[/itex].


text:
# Degrees of freedom (physics and chemistry)

Last updated

In physics and chemistry, a degree of freedom is an independent physical parameter in the formal description of the state of a physical system. The set of all states of a system is known as the system's phase space, and the degrees of freedom of the system are the dimensions of the phase space.

## Contents

The location of a particle in three-dimensional space requires three position coordinates. Similarly, the direction and speed at which a particle moves can be described in terms of three velocity components, each in reference to the three dimensions of space. If the time evolution of the system is deterministic (where the state at one instant uniquely determines its past and future position and velocity as a function of time) such a system has six degrees of freedom.[ citation needed ] If the motion of the particle is constrained to a lower number of dimensions – for example, the particle must move along a wire or on a fixed surface – then the system has fewer than six degrees of freedom. On the other hand, a system with an extended object that can rotate or vibrate can have more than six degrees of freedom.

In classical mechanics, the state of a point particle at any given time is often described with position and velocity coordinates in the Lagrangian formalism, or with position and momentum coordinates in the Hamiltonian formalism.

In statistical mechanics, a degree of freedom is a single scalar number describing the microstate of a system. [1] The specification of all microstates of a system is a point in the system's phase space.

In the 3D ideal chain model in chemistry, two angles are necessary to describe the orientation of each monomer.

It is often useful to specify quadratic degrees of freedom. These are degrees of freedom that contribute in a quadratic function to the energy of the system.

Depending on what one is counting, there are several different ways that degrees of freedom can be defined, each with a different value. [2]

## Thermodynamic degrees of freedom for gases

External images
https://chem.libretexts.org/@api/deki/files/9669/h2ovibrations.gif?revision=1
https://chem.libretexts.org/@api/deki/files/9668/co2vibrations.gif?revision=1

By the equipartition theorem, internal energy per mole of gas equals cv T, where T is temperature in kelvins and the specific heat at constant volume is cv = (f)(R/2). R = 8.314 J/(K mol) is the universal gas constant, and "f" is the number of thermodynamic (quadratic) degrees of freedom, counting the number of ways in which energy can occur.

Any atom or molecule has three degrees of freedom associated with translational motion (kinetic energy) of the center of mass with respect to the x, y, and z axes. These are the only degrees of freedom for a monoatomic species, such as noble gas atoms.

For a structure consisting of two or more atoms, the whole structure also has rotational kinetic energy, where the whole structure turns about an axis. A linear molecule, where all atoms lie along a single axis, such as any diatomic molecule and some other molecules like carbon dioxide (CO2), has two rotational degrees of freedom, because it can rotate about either of two axes perpendicular to the molecular axis. A nonlinear molecule, where the atoms do not lie along a single axis, like water (H2O), has three rotational degrees of freedom, because it can rotate around any of three perpendicular axes. In special cases, such as adsorbed large molecules, the rotational degrees of freedom can be limited to only one. [3]

A structure consisting of two or more atoms also has vibrational energy, where the individual atoms move with respect to one another. A diatomic molecule has one molecular vibration mode: the two atoms oscillate back and forth with the chemical bond between them acting as a spring. A molecule with N atoms has more complicated modes of molecular vibration, with 3N − 5 vibrational modes for a linear molecule and 3N − 6 modes for a nonlinear molecule. [4] As specific examples, the linear CO2 molecule has 4 modes of oscillation, [5] and the nonlinear water molecule has 3 modes of oscillation [6] Each vibrational mode has two energy terms: the kinetic energy of the moving atoms and the potential energy of the spring-like chemical bond(s). Therefore, the number of vibrational energy terms 2(3N − 5) for a linear molecule and 2(3N − 6) modes for a nonlinear molecule.

Both the rotational and vibrational modes are quantized, requiring a minimum temperature to be activated. [7] The "rotational temperature" to activate the rotational degrees of freedom is less than 100 K for many gases. For N2 and O2, it is less than 3 K. [1] The "vibrational temperature" necessary for substantial vibration is between 103 K and 104 K, 3521 K for N2 and 2156 K for O2. [1] Typical atmospheric temperatures are not high enough to activate vibration in N2 and O2, which comprise most of the atmosphere. (See the next figure.) However, the much less abundant greenhouse gases keep the troposphere warm by absorbing infrared from the Earth's surface, which excites their vibrational modes. [8] Much of this energy is reradiated back to the surface in the infrared through the "greenhouse effect."

Because room temperature (≈298 K) is over the typical rotational temperature but lower than the typical vibrational temperature, only the translational and rotational degrees of freedom contribute, in equal amounts, to the heat capacity ratio. This is why γ5/3 for monatomic gases and γ7/5 for diatomic gases at room temperature. [1]

Because air is dominated by diatomic gases nitrogen and oxygen, its molar internal energy is close to cv T = (5/2)RT, determined by the 5 degrees of freedom exhibited by diatomic gases. [11] See the graph at right. For 140 K < T < 380 K, cv differs from (5/2) Rd by less than 1%. Only at temperatures well above temperatures in the troposphere and stratosphere do some molecules have enough energy to activate the vibrational modes of N2 and O2. The specific heat at constant volume, cv, increases slowly toward (7/2) R as temperature increases above T = 400 K, where cv is 1.3% above (5/2) Rd = 717.5 J/(K kg).

Component degrees of freedom for N atoms (3N total coordinate motions)
Monatomic Linear molecules Non-linear molecules
Translation (x, y, and z)333
Rotation (x, y, and z)023
Vibration (structural variations)0(3N − 5)(3N − 6)

### Counting the minimum number of co-ordinates to specify a position

One can also count degrees of freedom using the minimum number of coordinates required to specify a position. This is done as follows:

1. For a single particle we need 2 coordinates in a 2-D plane to specify its position and 3 coordinates in 3-D space. Thus its degree of freedom in a 3-D space is 3.
2. For a body consisting of 2 particles (ex. a diatomic molecule) in a 3-D space with constant distance between them (let's say d) we can show (below) its degrees of freedom to be 5.

Let's say one particle in this body has coordinate (x1, y1, z1) and the other has coordinate (x2, y2, z2) with z2 unknown. Application of the formula for distance between two coordinates

${\displaystyle d={\sqrt {(x_{2}-x_{1})^{2}+(y_{2}-y_{1})^{2}+(z_{2}-z_{1})^{2}}}}$

results in one equation with one unknown, in which we can solve for z2. One of x1, x2, y1, y2, z1, or z2 can be unknown.

Contrary to the classical equipartition theorem, at room temperature, the vibrational motion of molecules typically makes negligible contributions to the heat capacity. This is because these degrees of freedom are frozen because the spacing between the energy eigenvalues exceeds the energy corresponding to ambient temperatures (kBT). [1]

## Independent degrees of freedom

The set of degrees of freedom X1, ... , XN of a system is independent if the energy associated with the set can be written in the following form:

${\displaystyle E=\sum _{i=1}^{N}E_{i}(X_{i}),}$

where Ei is a function of the sole variable Xi.

example: if X1 and X2 are two degrees of freedom, and E is the associated energy:

• If ${\displaystyle E=X_{1}^{4}+X_{2}^{4}}$, then the two degrees of freedom are independent.
• If ${\displaystyle E=X_{1}^{4}+X_{1}X_{2}+X_{2}^{4}}$, then the two degrees of freedom are not independent. The term involving the product of X1 and X2 is a coupling term that describes an interaction between the two degrees of freedom.

For i from 1 to N, the value of the ith degree of freedom Xi is distributed according to the Boltzmann distribution. Its probability density function is the following:

${\displaystyle p_{i}(X_{i})={\frac {e^{-{\frac {E_{i}}{k_{B}T}}}}{\int dX_{i}\,e^{-{\frac {E_{i}}{k_{B}T}}}}}}$,

In this section, and throughout the article the brackets ${\displaystyle \langle \rangle }$ denote the mean of the quantity they enclose.

The internal energy of the system is the sum of the average energies associated with each of the degrees of freedom:

${\displaystyle \langle E\rangle =\sum _{i=1}^{N}\langle E_{i}\rangle .}$

A degree of freedom Xi is quadratic if the energy terms associated with this degree of freedom can be written as

${\displaystyle E=\alpha _{i}\,\,X_{i}^{2}+\beta _{i}\,\,X_{i}Y}$,

where Y is a linear combination of other quadratic degrees of freedom.

example: if X1 and X2 are two degrees of freedom, and E is the associated energy:

• If ${\displaystyle E=X_{1}^{4}+X_{1}^{3}X_{2}+X_{2}^{4}}$, then the two degrees of freedom are not independent and non-quadratic.
• If ${\displaystyle E=X_{1}^{4}+X_{2}^{4}}$, then the two degrees of freedom are independent and non-quadratic.
• If ${\displaystyle E=X_{1}^{2}+X_{1}X_{2}+2X_{2}^{2}}$, then the two degrees of freedom are not independent but are quadratic.
• If ${\displaystyle E=X_{1}^{2}+2X_{2}^{2}}$, then the two degrees of freedom are independent and quadratic.

For example, in Newtonian mechanics, the dynamics of a system of quadratic degrees of freedom are controlled by a set of homogeneous linear differential equations with constant coefficients.

### Quadratic and independent degree of freedom

X1, ... , XN are quadratic and independent degrees of freedom if the energy associated with a microstate of the system they represent can be written as:

${\displaystyle E=\sum _{i=1}^{N}\alpha _{i}X_{i}^{2}}$

### Equipartition theorem

In the classical limit of statistical mechanics, at thermodynamic equilibrium, the internal energy of a system of N quadratic and independent degrees of freedom is:

${\displaystyle U=\langle E\rangle =N\,{\frac {k_{B}T}{2}}}$

Here, the mean energy associated with a degree of freedom is:

${\displaystyle \langle E_{i}\rangle =\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,p_{i}(X_{i})={\frac {\int dX_{i}\,\,\alpha _{i}X_{i}^{2}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}{\int dX_{i}\,\,e^{-{\frac {\alpha _{i}X_{i}^{2}}{k_{B}T}}}}}}$
${\displaystyle \langle E_{i}\rangle ={\frac {k_{B}T}{2}}{\frac {\int dx\,\,x^{2}\,\,e^{-{\frac {x^{2}}{2}}}}{\int dx\,\,e^{-{\frac {x^{2}}{2}}}}}={\frac {k_{B}T}{2}}}$

Since the degrees of freedom are independent, the internal energy of the system is equal to the sum of the mean energy associated with each degree of freedom, which demonstrates the result.

## Generalizations

The description of a system's state as a point in its phase space, although mathematically convenient, is thought to be fundamentally inaccurate. In quantum mechanics, the motion degrees of freedom are superseded with the concept of wave function, and operators which correspond to other degrees of freedom have discrete spectra. For example, intrinsic angular momentum operator (which corresponds to the rotational freedom) for an electron or photon has only two eigenvalues. This discreteness becomes apparent when action has an order of magnitude of the Planck constant, and individual degrees of freedom can be distinguished.

## Related Research Articles

Diatomic molecules are molecules composed of only two atoms, of the same or different chemical elements. If a diatomic molecule consists of two atoms of the same element, such as hydrogen or oxygen, then it is said to be homonuclear. Otherwise, if a diatomic molecule consists of two different atoms, such as carbon monoxide or nitric oxide, the molecule is said to be heteronuclear. The bond in a homonuclear diatomic molecule is non-polar.

In physics, the Maxwell–Boltzmann distribution is a particular probability distribution named after James Clerk Maxwell and Ludwig Boltzmann.

The quantum harmonic oscillator is the quantum-mechanical analog of the classical harmonic oscillator. Because an arbitrary smooth potential can usually be approximated as a harmonic potential at the vicinity of a stable equilibrium point, it is one of the most important model systems in quantum mechanics. Furthermore, it is one of the few quantum-mechanical systems for which an exact, analytical solution is known.

The ideal gas law, also called the general gas equation, is the equation of state of a hypothetical ideal gas. It is a good approximation of the behavior of many gases under many conditions, although it has several limitations. It was first stated by Benoît Paul Émile Clapeyron in 1834 as a combination of the empirical Boyle's law, Charles's law, Avogadro's law, and Gay-Lussac's law. The ideal gas law is often written in an empirical form:

The kinetic theory of gases is a simple, historically significant classical model of the thermodynamic behavior of gases, with which many principal concepts of thermodynamics were established. The model describes a gas as a large number of identical submicroscopic particles, all of which are in constant, rapid, random motion. Their size is assumed to be much smaller than the average distance between the particles. The particles undergo random elastic collisions between themselves and with the enclosing walls of the container. The basic version of the model describes the ideal gas, and considers no other interactions between the particles.

In physics, a phonon is a collective excitation in a periodic, elastic arrangement of atoms or molecules in condensed matter, specifically in solids and some liquids. A type of quasiparticle, a phonon is an excited state in the quantum mechanical quantization of the modes of vibrations for elastic structures of interacting particles. Phonons can be thought of as quantized sound waves, similar to photons as quantized light waves.

In physics, a partition function describes the statistical properties of a system in thermodynamic equilibrium. Partition functions are functions of the thermodynamic state variables, such as the temperature and volume. Most of the aggregate thermodynamic variables of the system, such as the total energy, free energy, entropy, and pressure, can be expressed in terms of the partition function or its derivatives. The partition function is dimensionless.

A normal mode of an oscillating system is a pattern of motion in which all parts of the system move sinusoidally with the same frequency and with a fixed phase relation. The free motion described by the normal modes takes place at fixed frequencies. These fixed frequencies of the normal modes of a system are known as its natural frequencies or resonant frequencies. A physical object, such as a building, bridge, or molecule, has a set of normal modes and their natural frequencies that depend on its structure, materials and boundary conditions. In music, normal modes of vibrating instruments are called "harmonics" or "overtones".

In classical statistical mechanics, the equipartition theorem relates the temperature of a system to its average energies. The equipartition theorem is also known as the law of equipartition, equipartition of energy, or simply equipartition. The original idea of equipartition was that, in thermal equilibrium, energy is shared equally among all of its various forms; for example, the average kinetic energy per degree of freedom in translational motion of a molecule should equal that in rotational motion.

The Stark effect is the shifting and splitting of spectral lines of atoms and molecules due to the presence of an external electric field. It is the electric-field analogue of the Zeeman effect, where a spectral line is split into several components due to the presence of the magnetic field. Although initially coined for the static case, it is also used in the wider context to describe the effect of time-dependent electric fields. In particular, the Stark effect is responsible for the pressure broadening of spectral lines by charged particles in plasmas. For most spectral lines, the Stark effect is either linear or quadratic with a high accuracy.

A rotational transition is an abrupt change in angular momentum in quantum physics. Like all other properties of a quantum particle, angular momentum is quantized, meaning it can only equal certain discrete values, which correspond to different rotational energy states. When a particle loses angular momentum, it is said to have transitioned to a lower rotational energy state. Likewise, when a particle gains angular momentum, a positive rotational transition is said to have occurred.

In statistical mechanics, the grand canonical ensemble is the statistical ensemble that is used to represent the possible states of a mechanical system of particles that are in thermodynamic equilibrium with a reservoir. The system is said to be open in the sense that the system can exchange energy and particles with a reservoir, so that various possible states of the system can differ in both their total energy and total number of particles. The system's volume, shape, and other external coordinates are kept the same in all possible states of the system.

In statistical mechanics, a canonical ensemble is the statistical ensemble that represents the possible states of a mechanical system in thermal equilibrium with a heat bath at a fixed temperature. The system can exchange energy with the heat bath, so that the states of the system will differ in total energy.

The molar heat capacity of a chemical substance is the amount of energy that must be added, in the form of heat, to one mole of the substance in order to cause an increase of one unit in its temperature. Alternatively, it is the heat capacity of a sample of the substance divided by the amount of substance of the sample; or also the specific heat capacity of the substance times its molar mass. The SI unit of specific heat is joule per kelvin per mole, J⋅K−1⋅mol−1.

In thermal physics and thermodynamics, the heat capacity ratio, also known as the adiabatic index, the ratio of specific heats, or Laplace's coefficient, is the ratio of the heat capacity at constant pressure to heat capacity at constant volume. It is sometimes also known as the isentropic expansion factor and is denoted by γ (gamma) for an ideal gas or κ (kappa), the isentropic exponent for a real gas. The symbol γ is used by aerospace and chemical engineers.

A molecular vibration is a periodic motion of the atoms of a molecule relative to each other, such that the center of mass of the molecule remains unchanged. The typical vibrational frequencies range from less than 1013 Hz to approximately 1014 Hz, corresponding to wavenumbers of approximately 300 to 3000 cm−1 and wavelengths of approximately 30 to 3 µm.

In statistical mechanics, thermal fluctuations are random deviations of a system from its average state, that occur in a system at equilibrium. All thermal fluctuations become larger and more frequent as the temperature increases, and likewise they decrease as temperature approaches absolute zero.

The rotational partition function relates the rotational degrees of freedom to the rotational part of the energy.

Heat transfer physics describes the kinetics of energy storage, transport, and energy transformation by principal energy carriers: phonons, electrons, fluid particles, and photons. Heat is energy stored in temperature-dependent motion of particles including electrons, atomic nuclei, individual atoms, and molecules. Heat is transferred to and from matter by the principal energy carriers. The state of energy stored within matter, or transported by the carriers, is described by a combination of classical and quantum statistical mechanics. The energy is different made (converted) among various carriers. The heat transfer processes are governed by the rates at which various related physical phenomena occur, such as the rate of particle collisions in classical mechanics. These various states and kinetics determine the heat transfer, i.e., the net rate of energy storage or transport. Governing these process from the atomic level to macroscale are the laws of thermodynamics, including conservation of energy.

Molecular symmetry in physics and chemistry describes the symmetry present in molecules and the classification of molecules according to their symmetry. Molecular symmetry is a fundamental concept in the application of Quantum Mechanics in physics and chemistry, for example it can be used to predict or explain many of a molecule's properties, such as its dipole moment and its allowed spectroscopic transitions, without doing the exact rigorous calculations. To do this it is necessary to classify the states of the molecule using the irreducible representations from the character table of the symmetry group of the molecule. Among all the molecular symmetries, diatomic molecules show some distinct features and they are relatively easier to analyze.

## References

1. Reif, F. (2009). Fundamentals of Statistical and Thermal Physics. Long Grove, IL: Waveland Press, Inc. p. 51. ISBN   978-1-57766-612-7.
2. Waldmann, Thomas; Klein, Jens; Hoster, Harry E.; Behm, R. Jürgen (2013). "Stabilization of Large Adsorbates by Rotational Entropy: A Time-Resolved Variable-Temperature STM Study". ChemPhysChem. 14 (1): 162–9. doi:10.1002/cphc.201200531. PMID   23047526.
3. For drawings, see http://www.colby.edu/chemistry/PChem/notes/NormalModesText.pdf
4. For drawings, see https://sites.cns.utexas.edu/jones_ch431/normal-modes-vibration
5. Section 12-7 (pp. 376-379) of Sears and Salinger, 1975: Thermodynamics, Kinetic Theory, and Statistical Thermodynamics. Third edition. Addison-Wesley Publishing Co.
6. "Molecules Vibrate". UCAR Center for Science Education. Archived from the original on 2014-10-10. Retrieved 2021-01-19.
7. Gatley, D. P., S. Herrmann, H.-J. Kretzshmar, 2008: A twenty-first century molar mass for dry air. HVAC&R Research, vol. 14, pp. 655-662.


text:
# Another proof of Hurewicz theorem

In: Tatra Mountains Mathematical Publications, vol. 49, no. 2
Miroslav Repický

## Details:

Year, pages: 2011, 1 - 7
Keywords:
Hurewicz scheme, $D$-proper mapping, analytic set
A Hurewicz theorem says that every coanalytic non-$Gδ$ set $C$ in a Polish space contains a countable set $Q$ without isolated points such that $\overline Q\cap C=Q$. We present another elementary proof of this theorem and generalize it for $κ$-Suslin sets. As a consequence, under Martin's Axiom, we obtain a characterization of $\boldsymbolΣ12$ sets that are the unions of less than the continuum closed sets.


text:
# A Lattice QCD Calculation of Vector Meson Decay Constants

Principal Investigator: Eric B. Gregory, Institut für Theoretische Physik, FB-C, Universität Wuppertal (Germany)
HPC Platform: JUQUEEN of JSC

Researchers from Bergische Universität Wuppertal (BUW) are using lattice quantumchromodynamics (QCD) to calculate vector meson decay constants fV , where V represents a vector meson such as ρ, ω, Φ, etc.

A meson is a composite particle composed of quarks q and antiquarks q- bound together by the strong force which is carried by gluons. In a simplified picture, we might describe a meson as a bound quark-antiquark pair exchanging gluons. Quarks have intrinsic spin of ½. In a vector meson the quark and antiquark are arranged such that the meson has a total spin of J = 1, and the quantum field describing the vector meson is related to its mirror image in the same way that a vector is.

The vector meson decay constant measures the overlap of the wave function of the quark and antiquark. If the meson is a flavor-singlet (e.g., ρ0 , ω, Φ, J/ψ) then the decay constant is related to the rate of leptonic decay such as ρ → e+ e . In the case of non-singlet vector mesons (such as ρ±, K) decay constants are related to the coupling to the weak vector boson W±. This is depicted diagramatically in Figure 1.

Fig. 1.: A vector meson coupled to a photon. The decay constant fV is related to the strength of the coupling at the vertex

In practice, the description of a meson as just a quark and antiquark pair is incomplete. Quantum field theory tells us that to fully account for the properties of the meson, we must consider that in the vicinity of the meson there exists a fluctuating cloud of quark-antiquark pairs. These pairs are created and then annihilate each other on extremely short time scales. Within each pair the antiquark has exactly the opposite spin and charge and color as the quark, so the fluctuations do not change the net quantum numbers of the meson. Nevertheless, these fluctuations have great impact on the overall properties of the composite particle. Figure 2 is an example of how this cloud of fluctuations contributes to the decay constants fV .

Fig. 2.: The same interaction as Figure 1, but showing some of the quantum fluctuations contributing to fV.

These “sea quarks” interact not only with the original “valence” quark and antiquark, but also with each other. A calculation of fV must include these fluctuations and we do this non-perturbatively with the tools of lattice QCD. The calculation has been performed through large-scale simulation initially involving more than 10 million hours of CPU time on the JUQUEEN HPC system at JSC. We simulate the sea of quark and gluon fluctuations in a four-dimensional discretized box of space-time. On stored configurations representing snapshots of these fluctuations, we invert a large, sparse matrix representing the interactions of the quantum fields.

The process is repeated thousands of times to generate an expectation value of a correlation function representing the meson propagating across the lattice, as depicted in Figure 3. Offline, we fit this data to extract both the mass of the vector mesons and their decay constants as fit parameters.

Fig. 3.: Diagram representing the propagation of a ρ meson from x to y.

We can vary the input parameters such as quark mass and the discretization scale. Figure 4 shows results as a function of quark mass.

Fig. 4.: Rho meson decay constants fρ as a function of the pion mass squared, which serves as a proxy for the quark mass. Larger β values equate to finer lattice scales.

© for all images: Bergische Universität Wuppertal, Theoretische Physik, Fachbereich C

Scientific Contact:

Dr. Eric B. Gregory
Bergische Universität Wuppertal
Fachbereich C - Mathematik und Naturwissenschaften
Theoretische Physik
Gaussstrasse 20
D-42097 Wuppertal/Germany
e-mail: gregory@uni-wuppertal.de

January 2015


text:
# Can an ungrounded conductive cavity provide electrostatic shielding?

This comes from Electromagnetic Fields and Waves by Lorrain et al, page 77 on a Hollow, ungrounded conductor enclosing a charged body:

The surface charge density at a given point on the outside surface of the conductor is independent of the distribution of Q in the cavity. It is the same as if the conductor were solid and carried a net charge Q.

1) I don't believe this. As I move +Q around the inside of the cavity, this will affect the distribution of $-Q$ on the inside surface, which must also affect $+Q$ on the outside surface to keep $E=0$ inside the cavity conductor.

Inversely, the field inside the cavity is independent of the field outside the conductor. The conductor then acts as an *electrostatic shield.

2) I don't believe this either for similar reasons. The external fields will affect the distribution of $+Q$ on the outer surface, which must affect the distribution of $-Q$ on the inner surface and therefore the electrostatic field inside the cavity, to maintain $E = 0$ inside the conductor.

So am I correct to disbelieve that an ungrounded conductive cavity can provide electrostatic shielding for the reasons 1. and 2. above?

-

These are a little bit tricky and can be confusing, but no, standard E&M is correct on shielding.

1) Electric charges inside a conductor are shielded from the outside. One does this by giving the shield an equal and opposite charge. This charge will rearrange so as to eliminate the field inside the shield (which is conductive). Thus there will be no electric field in the conductor. And since the total assemblage, (shield plus charges) has zero charge, there is no field outside the shield.

2) First consider a solid conductor in the presence of an external charge. You agree that the conductor ends up with a surface charge which cancels the fields inside the shield (otherwise current would flow which is not a static situation). Now imagine hollowing out the conductor. Since the metal you're removing is entirely from regions where there is no electric field, you haven't changed anything. The interior hole will now have no electric field.

-
You didn't answer my points in the question about the distribution of charge, but thanks anyway:) (1) In my example, there is a charge inside the cavity, so the total charge is Q. (2) There is an opposite charge on the inside surface of the interior hole to that on the surface. –  Larry Harson Feb 1 '12 at 15:00
The question "does the conductor produce a field outside of itself?" is answered simply by "is there a net charge inside of the conductor?" That is, if the charge on the conductor cancels the charge inside it, there will be no net external charge. So for your example, there will be an external field, but its form will be different from that of the interior charge. –  Carl Brannen Feb 2 '12 at 1:52

As I move +Q around the inside of the cavity, this will affect the distribution of −Q on the inside surface, which must also affect +Q on the outside surface to keep E=0 inside the cavity conductor.

The second "must" is wrong. The change in distribution of the -Q on the inside surface alone will cancel out the electric field generated by the cavity charge +Q out of the inside surface.

The external fields will affect the distribution of +Q on the outer surface, which must affect the distribution of −Q on the inner surface and therefore the electrostatic field inside the cavity, to maintain E=0 inside the conductor.

This "must" is wrong too. The total electric field of cavity charges and the inside surface charges is zero in the area outside of the inside surface, and the total electric field of outside electric field and the outside surface charges is zero inside the outside surface. The result is a net zero field in the intersection of these two area, namely, the area between the inside surface and the outside surface, a.k.a the conductor. These two parts change independently.

-


text: Grant number: 13/26444-2 Support type: Scholarships in Brazil - Post-Doctorate Effective date (Start): May 01, 2014 Effective date (End): March 31, 2016 Field of knowledge: Physical Sciences and Mathematics - Physics - Elementary Particle Physics and Fields Principal Investigator: Carola Dobrigkeit Chinellato Grantee: Eva Maria Martins dos Santos Home Institution: Instituto de Física Gleb Wataghin (IFGW). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil Associated research grant: 10/07359-6 - Study of cosmic rays of highest energies with the Pierre Auger Observatory , AP.TEM Abstract The Pierre Auger Observatory is the World's largest detector of cosmic rays with energies between $10^{17.5}\:\mathrm{eV}$ and $10^{20}\:\mathrm{eV}$. Composed of a surface detector, formed by an array of 1660 \u{C}erenkov water detectors arranged on an area of $3000\:mathrm{km^{2}}$, and 5 fluorescence detectors, the Pierre Auger Observatory has among its main objectives, to determine the origin and nature of these cosmic rays with unprecedented statistics. In the energy region of interest, cosmic rays are detected indirectly through the many millions of particles that are produced in a cascade process - called Extensive Air Shower resulting from their interaction in the atmosphere. Among the most recent results published by the Pierre Auger Observatory is the determination of the chemical composition of cosmic rays with energies $E > 10^{18}\:\mathrm{eV}$. From these results, it is observed that as the energy increases, the average chemical composition of cosmic rays evolves from a population mainly dominated by the lightest elements of the periodic table, to a heavier one. There are several variables that are sensitive to the chemical composition of cosmic rays. Among them is the measurement of the muonic component of the shower and, in particular, the measurement of the profile of the muons produced as a function of the atmospheric depth. In this project, the measurements made by the surface detector of the Pierre Auger Observatory of the muonic component of the shower will be used to determine, in an independent way, the nature of cosmic rays. The study will also address the potentialities and the performance of the muon measurements of the detectors which will be installed after 2015 in the Pierre Auger Observatory. (AU)


text:
# Intervals

415人阅读 评论(0)

## Problem 60: Intervals

Time Limit:1 Ms| Memory Limit:10 MB
Difficulty:0

## Description

There is given the series of n closed intervals [ai; b i], where i=1,2,...,n. The sum of those intervals may berepresented as a sum of closed pairwise non−intersecting intervals. The task is to find such representation with the minimal number of intervals. The intervals of this representation should be written in the output file in acceding order. We say that the intervals [a; b] and [c; d] are in ascending order if, and only if a <= b < c <= d.
Task
Write a program which:
.reads from the std input the description of the series of intervals,
.computes pairwise non−intersecting intervals satisfying the conditions given above,
.writes the computed intervals in ascending order into std output

## Input

In the first line of input there is one integer n, 3 <= n <= 50000. This is the number of intervals. In the (i+1)−st line, 1 <= i <= n, there is a description of the interval [ai; bi] in the form of two integers ai and bi separated by a single space, which are respectively the beginning and the end of the interval,1 <= ai <= bi <= 1000000.

## Output

The output should contain descriptions of all computed pairwise non−intersecting intervals. In each line should be written a description of one interval. It should be composed of two integers, separated by a single space, the beginning and the end of the interval respectively. The intervals should be written into the output in ascending order.

5
5 6
1 4
10 10
6 9
8 10

## Sample Output

1 4
5 10

#include <stdio.h>
#include <string.h>
#include <stdlib.h>
#include <iostream>
#include <algorithm>

using namespace std;

typedef struct Elem
{
int start;
int end;
}elem;

elem temp[50001];

bool cmp(elem a, elem b)
{
if(a.start != b.start)
{
return a.start < b.start;
}
else
{
return a.end < b.end;
}
}

int main()
{
int n, i;
scanf("%d", &n);
for(i = 0; i < n; i++)
{
scanf("%d%d", &temp[i].start, &temp[i].end);
}
sort(temp, temp+n, cmp);
elem t = temp[0];
for(i = 1; i < n; i++)
{
if(temp[i].start > t.end)
{
printf("%d %d\n", t.start, t.end);
t = temp[i];
}
else
{
if(temp[i].end > t.end)
{
t.end = temp[i].end;
}
}
}
printf("%d %d\n", t.start, t.end);
return 0;
}

0
0

* 以上用户言论只代表其个人观点，不代表CSDN网站的观点或立场
个人资料
• 访问：69471次
• 积分：2235
• 等级：
• 排名：第16867名
• 原创：162篇
• 转载：3篇
• 译文：0篇
• 评论：2条
文章分类
最新评论


text:
# A simple non-enzymatic strategy for adenosine triphosphate electrochemical aptasensor using silver nanoparticle-decorated graphene oxide

A simple non-enzymatic strategy for adenosine triphosphate electrochemical aptasensor using... In this work, a sensitive electrochemical aptasensor for the detection of adenosine triphosphate (ATP) has been introduced. A simple and non-enzymatic signal amplification strategy is utilized using silver nanoparticle-decorated graphene oxide (AgNPs–GO) as a redox probe. The modified electrode surface was characterized by scanning electron microscopy, FTIR and UV–Vis spectroscopy, and electrochemical impedance spectroscopy. GO provides an excellent substrate for the presence of the large number of AgNPs, so the monitored oxidation signal of AgNPs has been amplified. ATP-specific DNA aptamer is split into two fragments (F1 & F2) in order to design a sandwich-type assay. For the construction of the sensor, the surface of a graphite screen-printed electrode is modified with electrodeposited gold nanoparticles followed by self-assembling a monolayer of 3-mercaptopropionic acid on the electrode surface. The first amino-labeled fragment, F1, is immobilized on the modified electrode via carbodiimide chemistry. The synthesized AgNPs–GO interacts with F1 via $$\pi{-}\pi$$ π - π stacking. In the presence of ATP, the second fragment of the aptamer, F2, forms an associated complex with the immobilized F1 and causes AgNPs–GO to leave the surface. Consequently, a remarkable decrease in the oxidation signal of the AgNPs is observed. The percentage of this decrease has been monitored as an analytical signal, which is proportional to ATP concentration, and delivers a linear response over the range of 10.0 (±0.6) to 850 (±5) nM with a detection limit of 5.0 (±0.2) nM. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of the Iranian Chemical Society Springer Journals

# A simple non-enzymatic strategy for adenosine triphosphate electrochemical aptasensor using silver nanoparticle-decorated graphene oxide

, Volume 14 (9) – May 15, 2017
10 pages

Publisher
Springer Berlin Heidelberg
Subject
Chemistry; Analytical Chemistry; Inorganic Chemistry; Physical Chemistry; Biochemistry, general; Organic Chemistry
ISSN
1735-207X
eISSN
1735-2428
D.O.I.
10.1007/s13738-017-1138-5
Publisher site
See Article on Publisher Site

### Abstract

In this work, a sensitive electrochemical aptasensor for the detection of adenosine triphosphate (ATP) has been introduced. A simple and non-enzymatic signal amplification strategy is utilized using silver nanoparticle-decorated graphene oxide (AgNPs–GO) as a redox probe. The modified electrode surface was characterized by scanning electron microscopy, FTIR and UV–Vis spectroscopy, and electrochemical impedance spectroscopy. GO provides an excellent substrate for the presence of the large number of AgNPs, so the monitored oxidation signal of AgNPs has been amplified. ATP-specific DNA aptamer is split into two fragments (F1 & F2) in order to design a sandwich-type assay. For the construction of the sensor, the surface of a graphite screen-printed electrode is modified with electrodeposited gold nanoparticles followed by self-assembling a monolayer of 3-mercaptopropionic acid on the electrode surface. The first amino-labeled fragment, F1, is immobilized on the modified electrode via carbodiimide chemistry. The synthesized AgNPs–GO interacts with F1 via $$\pi{-}\pi$$ π - π stacking. In the presence of ATP, the second fragment of the aptamer, F2, forms an associated complex with the immobilized F1 and causes AgNPs–GO to leave the surface. Consequently, a remarkable decrease in the oxidation signal of the AgNPs is observed. The percentage of this decrease has been monitored as an analytical signal, which is proportional to ATP concentration, and delivers a linear response over the range of 10.0 (±0.6) to 850 (±5) nM with a detection limit of 5.0 (±0.2) nM.

### Journal

Journal of the Iranian Chemical SocietySpringer Journals

Published: May 15, 2017

## You’re reading a free preview. Subscribe to read the entire article.

### DeepDyve is your personal research library

It’s your single place to instantly
that matters to you.

over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month ### Explore the DeepDyve Library ### Search Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly ### Organize Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place. ### Access Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals. ### Your journals are on DeepDyve Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more. All the latest content is available, no embargo periods. DeepDyve ### Freelancer DeepDyve ### Pro Price FREE$49/month
\$360/year

Save searches from
PubMed

Create lists to

Export lists, citations


text:
# Participation of scientists from ISP NPP in the IAEA workshop

On 12-14 September 2017 there was held a workshop with IAEA experts’ participation within the framework of measures on decommissioning of the 1-3 Units of the Chornobyl NPP. The main task of the IAEA mission consisted in discussion issues of scientific supports concerning the decommissioning of nuclear power generating plants. IAEA experts shared European experience on decommissioning of nuclear power units and spent fuel and radioactive waste management at the workshop that was held in Slavutych and ChNPP.

The workshop activity with foreign and Ukrainian participants, including scientists from the ISP NPP, was oriented towards helping ChNPP to develop a «Scientific Support Program for the Chornobyl NPP Decommissioning». This document should help in solution the pressing technical and technological problems facing the Chornobyl NPP when decommissioning.

During the workshop two reports were presented by employees of the ISP NPP concerning the issues of the Chernobyl NPP decommissioning.

The head of Decommissioning Department Andriy Sizov presented the report «Work experience at the site of the SSE “ChNPP”», in which he represented to participants the engagement of the Institute in the project of final closure and conservation of the 1-3 Units of ChNPP.

The head of Sector for Radiological Defence Optimization, Paskevich Sergiy, familiarized the participants with the environmental consequences after taking out of service the cooling pond of Chernobyl NPP. Report «Environmental aspects at taking out of service the cooling pond of ChNPP» represented results of research the current state of cooling pond and proposals introduced to be implemented under the «Scientific Support Program for the Chornobyl NPP Decommissioning».

In the framework of the IAEA workshop there was conducted a survey of the industrial site of ChNPP and facilities for decommissioning.

Цей запис також доступний: Ukrainian

## Related posts

Insert math as
$${}$$