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We consider a conjecture that identifies two types of base point free divisors on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$. The first arises from Gromov-Witten theory of a Grassmannian. The second comes from first Chern classes of vector bundles associated with simple Lie algebras in type A. Here we reduce this conjecture on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$to the same statement forn= 4. A reinterpretation leads to a proof of the conjecture on$\overline {\text {M}}_{0,n}$${\overline{M}}_{0,n}$for a large class, and we give sufficient conditions for the non-vanishing of these divisors.

 

Abstract Hypothesis Surfactant-driven Marangoni spreading generates a fluid flow characterized by an outwardly moving “Marangoni ridge”. Spreading on thin and/or high viscosity subphases, as most of the prior literature emphasizes, does not allow the formation of capillary waves. On deep, low viscosity subphases, Marangoni stresses may launch capillary waves coupled with the Marangoni ridge, and new dependencies emerge for key spreading characteristics on surfactant thermodynamic and kinetic properties. Experiments and modeling Computational and physical experiments were performed using a broad range of surfactants to report the post-deposition motion of the surfactant front and the deformation of the subphase surface. Modeling coupled the Navier-Stokes and advective diffusion equations with an adsorption model. Separate experiments employed tracer particles or an optical density method to track surfactant front motion or surface deformation, respectively. Findings Marangoni stresses on thick subphases induce capillary waves, the slowest of which is co-mingled with the Marangoni ridge. Changing Marangoni stresses by varying the surfactant system alters the surfactant front velocity and the amplitude – but not the velocity – of the slowest capillary wave. As spreading progresses, the surfactant front and its associated surface deformation separate from the slowest moving capillary wave. 

For many students, trigonometry is a difficult subject because it requires strong spatial visualization abilities. A team at Jackson State University makes the teaching and learning process easer with a new learning tool for mobile phones developed using augmented reality (AR). The results indicated that AR incorporated learning tool has great potential for learning trigonometry.