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

Formal deformations, cohomology theory and L∞[1]L_\infty[1]-structures for differential Lie algebras of arbitrary weight

Generalising a previous work of Jiang and Sheng, a cohomology theory for differential Lie algebras of arbitrary weight is introduced. The underlying $L_\infty[1]$-structure on the cochain complex is also determined via a generalised version of higher derived brackets. The equivalence between $L_\infty[1]$-structures for absolute and relative differential Lie algebras are established. Formal deformations and abelian extensions are interpreted by using lower degree cohomology groups. Also we introduce the homotopy differential Lie algebras. In a forthcoming paper, we will show that the operad of homotopy (relative) differential Lie algebras is the minimal model of the operad of (relative) differential Lie algebras

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

Three closed characteristics on non-degenerate star-shaped hypersurfaces in R6\mathbf{R}^{6}

In this paper, we prove that for every non-degenerate $C^3$ compact star-shaped hypersurface $\Sigma$ in $\mathbf{R}^{6}$ which carries no prime closed characteristic of Maslov-type index $0$ or no prime closed characteristic of Maslov-type index $-1$, there exist at least three prime closed characteristics on $\Sigma$.Comment: 30 pages. arXiv admin note: text overlap with arXiv:2205.07082, arXiv:1510.08648, arXiv:2205.1478