We propose an implementation of the data structure presented by Kaplan, Mozes, Nussbaum, and Sharir in [KMNS12] for submatrix maximum queries in Monge matrices. The implementation shows that the average running time of the algorithm is similar to the proved worse case running time: O(log n). We also propose a new efficient and low-memory consuming procedure to generate random Monge matrices.

We discuss the performance of three numerical methods for the fully nonlinear Monge-Ampère equation. The first two are pseudo time continuation methods while the third is a pure pseudo time marching algorithm. The pseudo time continuation methods are shown to converge for smooth data on a uniformly convex domain. We give numerical evidence that they perform well for the nondegenerate Monge-Ampè...

The complexification of the compact group G is the group G whose Lie algebra is the complexification of the Lie algebra g of G and which satisfies π1(G ) = π1(G). The complexification G/K of G/K can be then identified (G-equivariantly) with the tangent bundle of G/K. We also remark that the Kähler form obtained in the Theorem is exact. This result has been proved in [9] for symmetric spaces of ...

This paper introduces a numerical method for the solution of the nonlinear elliptic Monge-Ampère equation. The boundary conditions correspond to the optimal transportation of measures supported on two domains, where one of these sets is convex. The new challenge is implementing the boundary conditions, which are implicit and non-local. These boundary conditions are reformulated as a nonlinear H...

This paper investigates a restricted version of the Quadratic Assignment Problem (QAP), where one of the coefficient matrices is an Anti-Monge matrix with non-decreasing rows and columns and the other coefficient matrix is a symmetric Toeplitz matrix. This restricted version is called the Anti-Monge–Toeplitz QAP. There are three well-known combinatorial problems that can be modeled via the Anti...

We show that the traveling salesman problem with a symmetric relaxed Monge matrix as distance matrix is pyramidally solvable and can thus be solved by dynamic programming. Furthermore, we present a polynomial time algorithm that decides whether there exists a renumbering of the cities such that the resulting distance matrix becomes a relaxed Monge matrix.

We obtainC2,β estimates up to the boundary for solutions to degenerate Monge–Ampère equations of the type det D2u = f in , f ∼ distα(·, ∂ ) near ∂ , α > 0. As a consequence we obtain global C∞ estimates up to the boundary for the eigenfunctions of the Monge–Ampère operator (det D2u)1/n on smooth, bounded, uniformly convex domains in Rn .

We obtain boundary Hölder gradient estimates and regularity for solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are affine invariant analogues of the boundary Hölder gradient estimates of Krylov.