Let $(X,\omega )$ be a compact Kähler manifold. We prove that all Monge–Ampère capacities are comparable. Using this we give direct proof of the integration by parts formula for non-pluripolar products recently proved M. Xia.

We prove a generalization of the Monge-Cayley-Salmon theorem on osculation and ruled submanifolds using elementary geometric measure theory.

The Travelling Salesman Problem (TSP) is a classical NPhard optimisation problem. There exist, however, special cases of the TSP that can be solved in polynomial time. Many of the well-known TSP special cases have been characterized by imposing special four-point conditions on the underlying distance matrix. Probably the most famous of these special cases is the TSP on a Monge matrix, which is ...

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane–Emden type, including the following two model problems: −∆pu = u + μ, Fk[−u] = u + μ, u ≥ 0, on R, or on a bounded domain Ω ⊂ R. Here ∆p is the pLaplacian defined by ∆pu = div (∇u|∇u|p−2), and Fk[u] is the k-Hessian defined as the sum of k× k principal minors o...

    M. and M.   Abstract: Existence and uniqueness of solution for singular 2-D systems depends on regularity condition. Simple regularity implies regularity and under this assumption, the generalized wave model (GWM) is introduced to cast singular 2-D system of equations as a family of non-singular 1-D models with variable structure.These index dependent models, along with a set of boundary co...

The paper is a short survey around the author’s recent works on topics related to complex Monge-Ampère equations and strictly pseudoconvex pseudo-Hermitian manifolds. 1. Invariant differential operators In complex analysis of one variable, the fact that the invariant property for Laplace operator under holomorphic change of coordinates plays an important role. Namely, Let φ : D1 → D2 be a holom...

This note describes some recent results on the regularity of optimal transport maps. As we shall see, in general optimal maps are not globally smooth, but they are so outside a “singular set” of measure zero. 1. The optimal transportation problem The optimal transportation problem, whose origin dates back to Monge [19], aims to find a way to transport a distribution of mass from one place to an...

We address the Monge problem in the abstract Wiener space and we give an existence result provided both marginal measures are absolutely continuous with respect to the infinite dimensional Gaussian measure γ.