Background Vaccination is a cornerstone of the prevention of communicable infectious diseases; however, vaccines have traditionally met with public fear and hesitancy, and COVID-19 vaccines are no exception. Social media use has been demonstrated to play a role in the low acceptance of vaccines. Objective The aim of this study is to identify the topics and sentiments in the public COVID-19 vaccine–related discussion on social media and discern the salient changes in topics and sentiments over time to better understand the public perceptions, concerns, and emotions that may influence the achievement of herd immunity goals. Methods Tweets were downloaded from a large-scale COVID-19 Twitter chatter data set from March 11, 2020, the day the World Health Organization declared COVID-19 a pandemic, to January 31, 2021. We used R software to clean the tweets and retain tweets that contained the keywords vaccination, vaccinations, vaccine, vaccines, immunization, vaccinate, and vaccinated. The final data set included in the analysis consisted of 1,499,421 unique tweets from 583,499 different users. We used R to perform latent Dirichlet allocation for topic modeling as well as sentiment and emotion analysis using the National Research Council of Canada Emotion Lexicon. Results Topic modeling of tweets related to COVID-19 vaccines yielded 16 topics, which were grouped into 5 overarching themes. Opinions about vaccination (227,840/1,499,421 tweets, 15.2%) was the most tweeted topic and remained a highly discussed topic during the majority of the period of our examination. Vaccine progress around the world became the most discussed topic around August 11, 2020, when Russia approved the world’s first COVID-19 vaccine. With the advancement of vaccine administration, the topic of instruction on getting vaccines gradually became more salient and became the most discussed topic after the first week of January 2021. Weekly mean sentiment scores showed that despite fluctuations, the sentiment was increasingly positive in general. Emotion analysis further showed that trust was the most predominant emotion, followed by anticipation, fear, sadness, etc. The trust emotion reached its peak on November 9, 2020, when Pfizer announced that its vaccine is 90% effective. Conclusions Public COVID-19 vaccine–related discussion on Twitter was largely driven by major events about COVID-19 vaccines and mirrored the active news topics in mainstream media. The discussion also demonstrated a global perspective. The increasingly positive sentiment around COVID-19 vaccines and the dominant emotion of trust shown in the social media discussion may imply higher acceptance of COVID-19 vaccines compared with previous vaccines.

Let L m,p (R n ) be the Sobolev space of functions with m th derivatives lying in L p (R n ). Assume that n < p < ∞. For E ⊂ R n , let L m,p (E) denote the space of restrictions to E of functions in L m,p (R n ). We show that there exists a bounded linear map T : L m,p (E) → L m,p (R n ) such that, for any f ∈ L m,p (E), we have Tf = f on E. We also give

Let E ⊂ R n be a compact set, and f : E → R. How can we tell if there exists a convex extensionAssuming such an extension exists, how small can one take the Lipschitz constant Lip(∇F ) := sup x,y∈R n ,x =yWe provide an answer to these questions for the class of strongly convex functions by proving that there exist constants k # ∈ N and C > 0 depending only on the dimension n, such that if for every subset S ⊂ E, #S ≤ k # , there exists an η-strongly convex functionand Lip(∇F ) ≤ CM 2 /η. Further, we prove a Finiteness Principle for the space of convex functions in C 1,1 (R) and that the sharp finiteness constant for this space is k # = 5.

It is well-known that in dimensions at least three semilinear wave equations with null conditions admit global solutions for small initial data. It is also known that in dimension two such result still holds for a certain class of quasi-linear wave equations with null conditions. The proofs are based on the decay mechanism of linear waves. However, in one dimension, waves do not decay. Nevertheless, we will prove that small data still lead to global solutions if the null condition is satisfied.

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