In this paper we prove that a process we call tropical relaxation gives a convergent method for numerically approximating the Monge--Amp\`ere Dirichlet problem on bounded convex planar domains. We approximate the prescribed probability measure by point clouds and construct the least concave tropical roof with zero boun...
N. Kalinin, Ernesto Lupercio, Higinio Serrano et al.· 2 citations
We introduce a matrix divisor function $\tau_n(T,M)$, counting factorisations $AB=M$ for $n\times n$ integer matrices $A,B$ of height at most $T$. For a fixed non-singular $M$, or for the zero matrix $M=O_n$, we prove an asymptotic formula for $\tau_n(T,M)$, as $T\to \infty$, using lattice point counting. We also prove...
Tim Browning, N. Kalinin, Alina Ostafe et al.· 0 citations
We prove a quantitative tropical approximation to the planar Aleksandrov Monge--Amp\`ere equation. Let $\Omega\subset\mathbb R^2$ be a bounded open convex domain, fix $K\Subset\Omega$, and let $F_N=G_{P_N}(0_\Omega)$ be the minimal nonnegative concave tropical series with integral slopes, zero boundary values, and corn...
N. Kalinin, Ernesto Lupercio, Higinio Serrano et al.· 2 citations
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