Communications in Contemporary Mathematics

We study the ordinary differential equation εẍ + ẋ + εg(x) = εf(ωt), with f and g analytic and f quasi-periodic in t with frequency vector ω ∈ ℝ^d. We show that if there exists c₀ ∈ ℝ such that g(c₀) equals the average of f and the first non-zero derivative of g at c₀ is of odd order 𝔫, then, for ε small enough and under very mild Diophantine conditions on ω, there exists a quasi-periodic solution close to c₀, with the same frequency vector as f. In particular if f is a trigonometric polynomial the Diophantine condition on ω can be completely removed. This extends results previously available in the literature for 𝔫 = 1. We also point out that, if 𝔫 = 1 and the first derivative of g at c₀ is positive, then the quasi-periodic solution is locally unique and attractive.

We study the horizontal mean curvature flow in the Heisenberg group by using the level-set method. We prove the uniqueness, existence and stability of axisymmetric viscosity solutions of the level-set equation. An explicit solution is given for the motion starting from a subelliptic sphere. We also give several properties of the level-set method and the mean curvature flow in the Heisenberg group.

We use variational methods to study the existence of non-trivial and radially symmetric solutions to the Hénon–Lane–Emden system with weights, when the exponents involved lie on the "critical hyperbola". We also discuss qualitative properties of solutions and non-existence results.

We consider the Cauchy problem for the nonlinear self-focusing Schrödinger equation in ℝ^N with an inhomogeneous smooth damping coefficient and we prove, for suitable initial data, and in the spirit of the seminal work of R. Glassey, a blowup result for the corresponding local solutions. We also give some lower bound estimates for the blowing-up solutions.

In this paper we introduce and study n-point Virasoro algebras, , which are natural generalizations of the classical Virasoro algebra and have as quotients multipoint genus zero Krichever–Novikov type algebras. We determine necessary and sufficient conditions for the latter two such Lie algebras to be isomorphic. Moreover we determine their automorphisms, their derivation algebras, their universal central extensions, and some other properties. The list of automorphism groups that occur is C_n, D_n, A₄, S₄ and A₅. We also construct a large class of modules which we call modules of densities, and determine necessary and sufficient conditions for them to be irreducible.

In this note, we obtain existence results for complete Ricci-flat Kähler metrics on crepant resolutions of singularities of Calabi–Yau varieties. Furthermore, for certain asymptotically flat Calabi–Yau varieties, we show that the Ricci-flat metric on the resolved manifold has the same asymptotic behavior as the initial variety.

In this paper, we translate the problem of extending an associative algebra by another associative algebra into the language of codifferentials. The authors have been constructing moduli spaces of algebras and studying their structure by constructing their versal deformations. The codifferential language is very useful for this purpose. The goal of this paper is to express the classical results about extensions in a form which leads to a simpler construction of moduli spaces of low-dimensional algebras.