184 research outputs found
Multiple closed geodesics on Finsler -dimensional sphere
In 1973, Katok constructed a non-degenerate (also called bumpy) Finsler
metric on 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 . In this paper, we proved this
conjecture for bumpy Finsler 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 -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
-structure on the cochain complex is also determined via a
generalised version of higher derived brackets. The equivalence between
-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 -operated algebra is a an (associative) algebra equipped
with a set of linear operators which might satisfy certain operator
identities such as the Leibniz rule. A free -operated algebra can
be generated on an algebra similar to a free algebra generated on a set. If
has a Gr\"{o}bner-Shirshov basis and if the linear operators
satisfy a set of operator identities, it is natural to ask when the
union is a Gr\"{o}bner-Shirshov basis of . A previous work
answers this question affirmatively under a mild condition, and thereby obtains
a canonical linear basis of .
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 and . For every or , we construct a single-element
generated nonsymmetric operad with Gelfand-Kirillov dimension . 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
In this paper, we prove that for every non-degenerate compact
star-shaped hypersurface in which carries no prime
closed characteristic of Maslov-type index or no prime closed
characteristic of Maslov-type index , there exist at least three prime
closed characteristics on .Comment: 30 pages. arXiv admin note: text overlap with arXiv:2205.07082,
arXiv:1510.08648, arXiv:2205.1478
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