Paintshop, odd cycles and necklace splitting

The following problem has been presented in [T. Epping, W. Hochstättler, P. Oertel, Complexity results on a paint shop problem, Discrete Applied Mathematics 136 (2004) 217-226] by Epping, Hochstättler and Oertel: cars have to be painted in two colors in a sequence where each car occurs twice; assign the two colors to the two occurrences of each car so as to minimize the number of color changes. More generally, the “paint shop scheduling problem” is defined with an arbitrary multiset of colors given for each car, where this multiset has the same size as the number of occurrences of the car; the mentioned article states two conjectures about the general problem and proves its NP-hardness. In a subsequent paper in [P. Bonsma, Th. Epping, W. Hochstättler, Complexity results for restricted instances of a paint shop problem for words, Discrete Applied Mathematics 154 (2006) 1335-1343], Bonsma, Epping and Hochstättler proved its APX-hardness and noticed the applicability of some classical results in special cases.We first identify the problem concerning two colors as a minimum odd circuit cover problem in particular graphs, exactly situating the problem. A resulting two-way reduction to a special minimum uncut problem leads to polynomial algorithms for subproblems, to observing APX-hardness through MAX CUT in 3-regular graphs, and to a solution with at most 3/4th of all possible remaining color changes (when all obliged color changes have been made).For the general problem concerning an arbitrary number of colors, we realize that the two aforementioned conjectures are corollaries of the celebrated “necklace splitting” theorem of Alon, Goldberg and West.     

A simple bijection between a subclass of 2-binary trees and ternary trees

A bijection between binary trees, with nodes labelled black or white, and a black node never having a white right child, and ternary trees is given. It is simpler than another bijection that has recently appeared in this journal.     

Vague集的表现定理

在Fuzzy集理论和现有Vague集理论的基础上,引入二元集合套的概念,讨论它的代数性质。在此基础上,结合Vague集的分解定理,建立Vague集的表现定理,同时得到一系列相关结果,进一步揭示Vague集与经典集合之间的联系。     

几类基于L*上二元聚合算子的蕴涵

L*是区间值模糊与Atanassov意义下的直觉模糊集的基本格。本文首先基于单位区间上的三角模与三角余模,引入L*上两组对偶的二元聚合算子,然后,类似于剩余蕴涵与强蕴涵的构成方法,利用引入的对偶聚合算子生成几类L*上的蕴涵,并对其性质进行讨论。     

High-Pressure Fabrication of Binary Organic Solar Cells with High Molecular Weight D18 Yields Record 19.65 % Efficiency

In this work, inspired by the principles of a pressure cooker, we utilized a high-pressure method to address the processing challenges associated with high molecular weight polymers. Through this approach, we successfully dissolved high molecular weight D18 in chloroform at 100 °C within a pressure-tight vial. The increased steam pressure raised the boiling point and dissolving capacity of chloroform, enabling the creation of a hybrid film with superior properties, including more ordered molecular arrangement, increased crystallinity, extended exciton diffusion length, and improved phase morphology. Organic solar cells (OSCs) based on D18 : L8-BO prepared using this high-pressure method achieved an outstanding power conversion efficiency of 19.65 %, setting a new record for binary devices to date. Furthermore, this high-pressure method was successfully applied to fabricate OSCs based on other common systems, leading to significant enhancements in device performance. In summary, this research introduces a universal method for processing high molecular weight D18 materials, ultimately resulting in the highest performance reported for binary organic solar cells.     

A note on the distribution of weights of fixed-rank matrices over the binary field

Let M be a random $m\times n$ rank-r matrix over the binary field ${\mathbb{F}}_2$, and let $\mathrm{wt}(\mathbf{M})$ be its Hamming weight, that is, the number of nonzero entries of M.We prove that, as $m,n\to +\infty$ with r fixed and $m/n$ tending to a constant, we have that$\frac{\mathrm{wt}(\mathbf{M})-\frac{1-2^{-r}}{2}mn}{\sqrt{\frac{2^{-r}(1-2^{-r})}{4}(m+n)mn}}$ converges in distribution to a standard normal random variable.