Spatiotemporal model for the progression of transgressive dunes

Transgressive dune fields, which are active sand areas surrounded by vegetation, exist on many coasts. In some regions like in Fraser Island in Australia, small dunes shrink while large ones grow, although both experience the same climatic conditions. We propose a general mathematical model for the spatiotemporal dynamics of vegetation cover on sand dunes and focus on the dynamics of transgressive dunes. Among other possibilities, the model predicts growth parallel to the wind with shrinkage perpendicular to the wind, where, depending on geometry and size, a transgressive dune can initially grow although eventually shrink. The larger is the initial area the slower its stabilization process. The model’s predictions are supported by field observations from Fraser Island in Australia.     

Revenue in contests with many participants

We show that in a contest with a single prize, the expected effort made by the k$k$th highest valuation participant bounds the sum of the expected efforts made by all of the participants with valuations less than the k$k$th highest valuations. We also show that in the limit case of a contest with m$m$ prizes, the expected effort made by the k$k$th highest valuation participant when the bidders are risk-neutral is greater than the expected effort in the risk-averse case.     

Rigidity Theorems for Complete Sasakian Manifolds with Constant Pseudo-Hermitian Scalar Curvature

The orthogonal decomposition of the Webster curvature provides us a way to characterize some canonical metrics on a pseudo-Hermitian manifold. We derive some subelliptic differential inequalities from the Weitzenböck formulas for the traceless pseudo-Hermitian Ricci tensor of Sasakian manifolds with constant pseudo-Hermitian scalar curvature and the Chern–Moser tensor of the Sasakian pseudo-Einstein manifolds, respectively. By means of either subelliptic estimates or maximum principle, some rigidity theorems are established to characterize Sasakian pseudo-Einstein manifolds among Sasakian manifolds with constant pseudo-Hermitian scalar curvature and Sasakian space forms among Sasakian pseudo-Einstein manifolds, respectively.     

Why do active and stabilized dunes coexist under the same climatic conditions?

Sand dunes can be active (mobile) or stable, mainly as a function of vegetation cover and wind power. However, there exists as yet unexplained evidence for the coexistence of bare mobile dunes and vegetated stabilized dunes under the same climatic conditions. We propose a model for dune vegetation cover driven by wind power that exhibits bistabilty and hysteresis with respect to the wind power. For intermediate wind power, mobile and stabilized dunes can coexist, whereas for low (or high) wind power they can be fixed (or mobile). Climatic change or human intervention can turn active dunes into stable ones and vice versa; our model predicts that prolonged droughts with stronger winds can result in dune reactivation.     

Monotonicity formulae and Liouville theorems of harmonic maps with potential

We use the stress–energy tensor to obtain the monotonicity formula and vanishing theorems for harmonic maps with potential H$H$ under some conditions on H$H$. We also obtain the unique constant solution of the constant Dirichlet boundary value problem on some starlike domain for harmonic maps with potential.     

Ranking asymmetric auctions

We compare the expected revenue in first- and second-price auctions with asymmetric bidders. We consider “close to uniform” distributions with identical supports and show that in the case of identical supports the expected revenue in second-price auctions may exceed that in first-price auctions. We also show that asymmetry over lower valuations has a stronger negative impact on the expected revenue in first-price auctions than in second-price auctions. However, asymmetry over high valuations always increases the revenue in first-price auctions.     

Urban Segregation as a Nonlinear Phenomenon

A new mathematical model that describes segregation dynamics of two distinct populations in a city is presented. The model associates segregation with an instability of a spatially uniform mixed population state. Segregated states correspond to alternating domains of overrepresentation and underrepresentation of a given population. A second instability designates a transition to a stronger form of segregation involving enclaves of pure population. The model is used to study neighborhood change processes such as displacements of transition zones and tipping point phenomena. The main significance of the model lies in the conceptual framework it introduces by relating sociospatial phenomena to dynamical system and pattern formation theories.     

Liouville Theorems for F-Harmonic Maps and Their Applications

We prove several Liouville theorems for F-harmonic maps from some complete Riemannian manifolds by assuming some conditions on the Hessian of the distance function, the degrees of F(t) and the asymptotic behavior of the maps at infinity. In particular, the results can be applied to F-harmonic maps from some pinched manifolds, and can deduce a Bernstein type result for an entire minimal graph.     

On the structure of submanifolds in the hyperbolic space

Monatshefte für Mathematik - Let $$M^n$$ be a complete submanifold in the hyperbolic space $$\mathbb {H}^{n+m}$$ . We show the vanishing of the Betti numbers $$\beta _p(M)$$ , $$1 \le p \le...