We generalize the football shaped extra dimensions scenario to an arbitrary number of branes. The problem is related to the solution of the Liouville equation with singularities and explicit solutions are presented for the case of three branes. The tensions of the branes do not need to be tuned with each other but only satisfy mild global constraints. ∗email: [email protected]

In this paper, we prove the asymptotic stability of the family of solitons of the Benjamin-Ono equation in the energy space. The proof is based on a Liouville property for solutions close to the solitons for this equation, in the spirit of [16], [18]. As a corollary of the proofs, we obtain the asymptotic stability of exact multi-solitons.

The most general vortex solution of the Liouville equation (which arises in nonrelativistic Chern-Simons theory) is associated with rational functions, f(z) = P (z)/Q(z) where P (z) and Q(z) are both polynomials, degP < degQ ≡ N . This allows us to prove that the solution depends on 4N parameters without the use of an index theorem, as well as the flux quantization : Φ = −4Nπ(sign κ).

The objective of the current study is to develop a simple formula to estimate the fundamental vibration period of tall buildings for using in equivalent lateral force analysis specified in building codes. The method based on Sturm-Liouville differential equation is presented here for estimating the fundamental period of natural vibration. The resulting equation, based on the continuum represent...