A Two-Dimensional Minkowski ?(x) Function

Differentiability of the Minkowski question mark function

We prove new results on the derivative of the Minkowski question mark function. Some of our theorems are non-improvable.

The Stern Sequence and Moments of Minkowski's Question Mark Function

1 : We use properties of the Stern Sequence for numerical computations of moments $\int^1_0 t^n d?(t)$ associated to Minkowski's Question Mark function.

Hermite versus Minkowski

We compare for an $n$-dimensional Euclidean lattice $\Lb$ the smallest possible values of the product of the norms of $n$~vectors which either constitute a basis for $\Lb$ (Hermite-type inequalities) or are merely assumed to be independent (Minkowski-type inequalities). We improve on 1953 results of van der Waerden in dimensions $6$ to $8$ and prove partial result in dimension~$9$.

关于峡谷函数的优化：Cohn参数化中的Minkowski-Cohn模曲面

On optimization on ravine functions. Minkowski-Cohn moduli surface in Cohn parameterization

本文以Minkowski-Cohn模曲面为例，从优化角度简要概述峡谷函数，并给出点上最小化问题的表示与求解要素。

Two-Dimensional Analogs of the Minkowski ?(x) Function

Two generalizations of the Minkowski ?(x) function are given. As ?(x) maps quadratic irrationals to rational numbers, it is shown that both generalizations send natural classes of pairs of cubic irrational numbers in the same cubic number field to pairs of rational numbers. It is also shown that these functions satisfy an analog to the fact that ?(x), while continuous and increasing, has derivative zero almost everywhere. Both extend earlier work of Beaver-Garrity on the Farey-Bary map.

Multidimensional continued fractions and a Minkowski function

The Minkowski Question Mark function can be characterized as the unique homeomorphism of the real unit interval that conjugates the Farey map with the tent map. We construct an n-dimensional analogue of the Minkowski function as the only homeomorphism of an n-simplex that conjugates the piecewise-fractional map associated to the Monkemeyer continued fraction algorithm with an appropriate tent map.