Multiple closed geodesics on Finsler 33-dimensional sphere

In 1973, Katok constructed a non-degenerate (also called bumpy) Finsler metric on $S^3$ with exactly four prime closed geodesics. And then Anosov conjectured that four should be the optimal lower bound of the number of prime closed geodesics on every Finsler $S^3$. In this paper, we proved this conjecture for bumpy Finsler $S^{3}$ if the Morse index of any prime closed geodesic is nonzero.Comment: 15 pages. arXiv admin note: text overlap with arXiv:1504.07007, arXiv:1510.02872, arXiv:1508.0557

Gr\"obner-Shirshov bases and linear bases for free multi-operated algebras over algebras with applications to differential Rota-Baxter algebras and integro-differential algebras

Quite much recent studies has been attracted to the operated algebra since it unifies various notions such as the differential algebra and the Rota-Baxter algebra. An $\Omega$-operated algebra is a an (associative) algebra equipped with a set $\Omega$ of linear operators which might satisfy certain operator identities such as the Leibniz rule. A free $\Omega$-operated algebra $B$ can be generated on an algebra $A$ similar to a free algebra generated on a set. If $A$ has a Gr\"{o}bner-Shirshov basis $G$ and if the linear operators $\Omega$ satisfy a set $\Phi$ of operator identities, it is natural to ask when the union $G\cup \Phi$ is a Gr\"{o}bner-Shirshov basis of $B$. A previous work answers this question affirmatively under a mild condition, and thereby obtains a canonical linear basis of $B$. In this paper, we answer this question in the general case of multiple linear operators. As applications we get operated Gr\"{o}bner-Shirshov bases for free differential Rota-Baxter algebras and free integro-differential algebras over algebras as well as their linear bases. One of the key technical difficulties is to introduce new monomial orders for the case of two operators, which might be of independent interest.Comment: 27 page

Growth of nonsymmetric operads

The paper concerns the Gelfand-Kirillov dimension and the generating series of nonsymmetric operads. An analogue of Bergman's gap theorem is proved, namely, no finitely generated locally finite nonsymmetric operad has Gelfand-Kirillov dimension strictly between $1$ and $2$. For every $r\in \{0\}\cup \{1\}\cup [2,\infty)$ or $r=\infty$, we construct a single-element generated nonsymmetric operad with Gelfand-Kirillov dimension $r$. We also provide counterexamples to two expectations of Khoroshkin and Piontkovski about the generating series of operads.Comment: 32 pages, 9 figure

The Value of Backers’ Word-of-Mouth in Screening Crowdfunding Projects: An Empirical Investigation

Reward-based crowdfunding is an emerging financing channel for entrepreneurs to raise money for their innovative projects. How to screen the crowdfunding projects is critical for crowdfunding platform, project founder, and potential backers. This study aims to investigate whether backers’ word-of-mouth (WOM) is a valuable input to generate collective intelligence for project screening. Specially, we answer three questions. First, is backers’ WOM an effective signal for implementation performance of crowdfunding projects? Second, how do the WOM help screen projects during the fund-raising process? Third, which kind of comments (positive or negative) is more effective in screening crowdfunding projects? Research hypotheses were developed based on theories of collective intelligence and WOM communication. Using a cross section dataset and a panel dataset, we get the following findings. First, backers’ negative WOM can effectively predict project implementation performance, however positive WOM does not have that prediction power. The prediction power of positive and negative WOM differs significantly. One possible reason is that negative WOM does contain more information of project quality. Second, project with more accumulative negative WOM tend to attract fewer subsequent backers. However, accumulative positive WOM is not helpful for attracting more potential backers. We conclude that negative WOM is useful for project screening project, because it is a signal of project quality, and meanwhile it could prevent backers make subsequent investments

Movie101: A New Movie Understanding Benchmark

To help the visually impaired enjoy movies, automatic movie narrating systems are expected to narrate accurate, coherent, and role-aware plots when there are no speaking lines of actors. Existing works benchmark this challenge as a normal video captioning task via some simplifications, such as removing role names and evaluating narrations with ngram-based metrics, which makes it difficult for automatic systems to meet the needs of real application scenarios. To narrow this gap, we construct a large-scale Chinese movie benchmark, named Movie101. Closer to real scenarios, the Movie Clip Narrating (MCN) task in our benchmark asks models to generate role-aware narration paragraphs for complete movie clips where no actors are speaking. External knowledge, such as role information and movie genres, is also provided for better movie understanding. Besides, we propose a new metric called Movie Narration Score (MNScore) for movie narrating evaluation, which achieves the best correlation with human evaluation. Our benchmark also supports the Temporal Narration Grounding (TNG) task to investigate clip localization given text descriptions. For both two tasks, our proposed methods well leverage external knowledge and outperform carefully designed baselines. The dataset and codes are released at https://github.com/yuezih/Movie101.Comment: Accepted to ACL 202