关于Minkowski约化基中极小向量的坐标

论非对易闵可夫斯基时空的非平坦性与探针的奇异行为

On the Non-Flatness Nature of Noncommutative Minkowski Spacetime and the Singular Behavior of Probes

本文在扭曲微分同胚非对易引力框架下证明，正则型非对易闵可夫斯基时空在球极坐标和抛物坐标下具有非平凡曲率修正和奇异性，暗示高能探针可转变为黑洞，而笛卡尔坐标和柱坐标情形则无此现象。

Minkowski bases, Korkin-Zolotarev bases and successive minima

Let $λ_k$ denote the $k$-th successive minimum of a lattice $L$. We study properties of the lengths of certain bases of $L$. If $v_1, \dots v_n$ is a basis which is reduced in the sense of Minkowski we show that $\lvert v_k \rvert^2 \leq \frac{k}{4} λ_{k}^2$ for $k = 6, 7$, confirming a conjecture of Schürmann, and obtaining the first improvement of a classical bound by Van der Waerden. We construct a sequences of lattices where $\lvert v_n \rvert$ is significantly longer than the longest vector in a Korkin-Zolotarev reduced basis, answering a question of Schürmann. In an appendix joint with L…

Solid angles associated to Minkowski reduced bases

Given a lattice $Λ\subset \mathbb{R}^n$, we consider its Minkowski reduced basis and the solid angle $Ω$ spanned by the basis vectors. Such a basis satisfies strong near-orthogonality conditions, which allow us to bound from above and below the measure of $Ω$. Sharp upper and lower bounds are derived for all rank $3$ and rank $4$ lattices so that $Ω$ always measures in between. Extreme cases happen when $Λ$ is similar to the rectangular ($\mathcal{R}$) or alternating ($\mathcal{A}$) lattice. This result settles a question raised earlier by Fukshansky and Robins in connection to sphere packings…

关于Minkowski子格的分类

On classifying Minkowskian sublattices

令$Λ$是$n$维欧几里得空间$E$中的格，令$Λ'$是$Λ$的闵可夫斯基子格，即，具有由$Λ$的闵可夫斯基连续最小值的代表构成的基的子格.我们将子$Λ/Λ'$的可能$\Z/d\Z$ -码的分类推广到维数~ $9$，其中$d\Z$是$Λ/Λ'$的零化子.

Eventual quasi-linearity of the Minkowski length

The Minkowski length of a lattice polytope $P$ is a natural generalization of the lattice diameter of $P$. It can be defined as the largest number of lattice segments whose Minkowski sum is contained in $P$. The famous Ehrhart theorem states that the number of lattice points in the positive integer dilates $tP$ of a lattice polytope $P$ behaves polynomially in $t\in\mathbb{N}$. In this paper we prove that for any lattice polytope $P$, the Minkowski length of $tP$ for $t\in\mathbb{N}$ is eventually a quasi-polynomial with linear constituents. We also give a formula for the Minkowski length of c…