近似计算独立多面体截断松弛的体积

截断分数匹配多胞体体积的确定性近似

Deterministic approximation for the volume of the truncated fractional matching polytope

该研究基于簇展开技术，提出了图截断分数匹配多胞体体积的确定性FPTAS，并将结果推广至超图情形，统一了已有相关结论。

确定性地近似Kostka多面体的体积

Deterministically approximating the volume of a Kostka polytope

本文提出了一种多项式时间确定性算法，用于近似Kostka多面体的体积，其乘法因子为exp(O(n log n))，并基于连续Schur多项式的一个连续类别的分区函数解释。

Approximating the Volume of Tropical Polytopes is Difficult

We investigate the complexity of counting the number of integer points in tropical polytopes, and the complexity of calculating their volume. We study the tropical analogue of the outer parallel body and establish bounds for its volume. We deduce that there is no approximation algorithm of factor $α=2^{\text{poly}(m,n)}$ for the volume of a tropical polytope given by $n$ vertices in a space of dimension $m$, unless P$=$NP. Neither is there such an approximation algorithm for counting the number of integer points in tropical polytopes described by vertices. If follows that approximating these v…

Geometric realization of $γ$-vectors of 2-truncated cubes

This paper continues investigation of the class of flag simple polytopes called 2-truncated cubes. It is an extended version of the short note Volodin (2012). A 2-truncated cube is a polytope obtained from a cube by sequence of truncations of codimension 2 faces. Constructed uniquely defined function which maps any 2-truncated cube to a flag simplicial complex with $f$-vector equal to $γ$-vector of the polytope. As a corollary we obtain that $γ$-vectors of 2-truncated cubes satisfy Frankl-Furedi-Kalai inequalities.

Approximating sums by integrals only: multiple sums and sums over lattice polytopes

The Euler--Maclaurin (EM) summation formula is used in many theoretical studies and numerical calculations. It approximates the sum $\sum_{k=0}^{n-1} f(k)$ of values of a function $f$ by a linear combination of a corresponding integral of $f$ and values of its higher-order derivatives $f^{(j)}$. An alternative (Alt) summation formula was recently presented by the author, which approximates the sum by a linear combination of integrals only, without using high-order derivatives of $f$. It was shown that the Alt formula will in most cases outperform, or greatly outperform, the EM formula in terms…

多元样条与多面体

Multivariate Splines and Polytopes

本文利用多元样条研究多面体的体积，给出了多元截断幂的显式公式，证明了多项式在多面体上的积分可通过多元截断幂处理，并基于箱样条与立方体切片体积的联系给出了Good猜想的简单证明。