Let $r\ge k\ge 2$ and $K_{r,n}^{(k)}$ denote the complete $n$-balanced $r$-partite $k$-uniform hypergraph, whose vertex set consists of $r$ parts, each has $n$ vertices, and whose edge set contains all the $k$-element subsets with no two vertices from one part. A decomposition of $K_{r,n}^{(k)}$ is a partition of $E(K_{r,n}^{(k)})$. A perfect matching (resp., Hamilton tight cycle) decomposition of $K_{r,n}^{(k)}$ is a decomposition of $K_{r,n}^{(k)}$ into perfect matchings (resp., Hamilton tight cycles). In this paper, we prove that if $k\mid n$ (resp., $2\nmid k$ and $k\mid n$), then $K_{k+1,n}^{(k)}$ (resp., $K_{k+2,n}^{(k)}$) has a perfect matching decomposition. We also prove that for any integer $k\geq 2$, $K_{k+1,n}^{(k)}$ has a Hamilton tight cycle decomposition. In all cases, we use constructive methods involving number theory. In fact, we confirm two conjectures proposed by Zhang, Lu, and Liu [ Appl. Math. Comput., 386 (2020), 125492].

The decay rate of swells is an important factor in swell propagation. Munk[1] and Snodgrass[2] used buoy observation data along the U.S. Coast to study the swells generated by Pacific storms. Recently, people have begun to use satellite data to study the law of wave attenuation in the oceans. The decay rate of storm generated swells in the Pacific Ocean was estimated by analyzing the synthetic aperture radar data in the west coast of America[3]. By using altimeter data, Young and Babanin et al. [4] analyzed such decay rate in the Southern Ocean, and Wang et al. [5] studied the attenuation law of Pacific swells. The magnitude of the swell decay rate of these studies is close but the values are quite different.

Bat algorithm (BA) is newly proposed bio-inspired metaheuristic algorithm with the inspiration of the echolocation of bats in nature. Several experimental results have proven to the effectiveness and performance of BA. However, BA may fail to find the global optimal solution occasionally. In this paper, a kind of classical search technology, called variable neighborhood search (VNS), is incorporated into BA as a local search tool. An improved version of BA namely variable neighborhood bat algorithm (VNBA), is thus proposed. In VNBA, the classic BA as a global search tool searches the whole space globally, and this can significantly shrink the search space. Subsequently, VNS as a local search tool is implemented to find the final best solution within the small promising area. After that, the VNBA is benchmarked by sixteen standard benchmark functions. The experimental results imply that VNBA takes the absolute advantage over the basic BA.

This paper investigates a minimum spanning tree (MST) problem with fuzzy costs and quadratic cost structure, which we call the fuzzy quadratic minimum spanning tree problem (FQMST). After formulating the FQMST problem as a chance-constrained programming model based on a credibility measure, the deterministic equivalent is proposed when the fuzzy direct costs and fuzzy interactive costs are characterized by trapezoidal fuzzy numbers. Then, a genetic algorithm is designed for solving FQMST problems. Finally, a numerical example is provided for illustrating the effectiveness of the genetic algorithm.