We prove that if φ : R × R → R, d ≥ 2, is a homogeneous function, smooth away from the origin and having nonzero Monge–Ampere determinant away from the origin, then R#{(n,m) ∈ Z × Z : |n|, |m| ≤ CR;R ≤ φ(n,m) ≤ R+ δ} . max{R 2 d+1 , Rδ}. This is a variable coefficient version of a result proved by Lettington, 2010, extending a previous result by Andrews, 1963, showing that if B ⊂ R, d ≥ 2, is a...

    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...

Recently, following a new approach, the symmetry of the solutions to overdetermined problems has been established for a class of Hessian type operators. In this paper we prove that the radial solution of the overdetermined Dirichlet problem for the Monge-Ampère equation is stable under suitable perturbations of the data.

S. T. Yau has done extremely deep and powerful work in differential geometry and partial differential equations. His resolution of the Calabi conjecture on the existence of KählerEinstein metrics, by solving a complex Monge-Ampère equation on Kähler manifolds, is of fundamental importance in both mathematics and physics. We would like to recall in this article the contributions of S. Y. Cheng a...

1. Background 5 2. Plurisubharmonic functions 8 3. The complex Monge–Ampère operator 10 3.1. Bedford’s and Taylor’s definition of the complex Monge–Ampère operator 11 3.2. Cegrell’s definition of the complex Monge–Ampère operator 12 4. The Dirichlet problem for the complex Monge–Ampère operator 14 4.1. Boundary blow-up problems for the complex Monge–Ampère operator 17 4.2. Comparison principles...

A complete description of the eigenspace structure for a given n × n Monge matrix in a max-plus algebra is presented. Based on the description, an O(n2) algorithm for computing the eigenspace dimension is formulated, which is faster than the previously known algorithms. © 2007 Elsevier B.V. All rights reserved. MSC: primary 04A72; secondary 05C50;15A33

An M x n matrix C is called Mange matrix if c,, + cTT < clr + cr, for all 1 < i -c r < rn, 1 < j < s d n. In this paper we present a survey on Monge matrices and related Monge properties and their role in combinatorial optimization. Specifically, we deal with the following three main topics: (i) fundamental combinatorial properties of Monge structures, (ii) applications of Monge properties to o...

Several thousands of million tons of wastes are generated worldwide every year. Of these, organic solid wastes have the potential to partially satisfy the production of chemicals, whilst the environmental impact can be minimized and the sustainability of the processes increased compared to those based on fossil resources. However, these wastes are complex due to their high heterogeneity, and ne...

Solutions of nonlinear multi-component Euler-Monge partial differential equations are constructed in n spatial dimensions by dimension-doubling, a method that completely linearizes the problem. Nonlocal structures are an essential feature of the method. The Euler-Monge equations may be interpreted as a boundary theory arising from a linearized bulk system such that all boundary solutions follow...