On the Problem of Covering a 3-D Terrain

We study the problem of covering a 3-dimensional terrain by a sweeping robot that is equipped with a camera. We model the terrain as a mesh in a way that captures the elevation levels of the terrain; this enables a graph-theoretic formulation of the problem in which the underlying graph is a weighted plane graph. We show that the associated graph problem is NP-hard, and that it admits a polynomial time approximation scheme (PTAS). Finally, we implement two heuristic algorithms based on greedy approaches and report our findings.

Davenport–Zannier polynomials over ℚ

In this paper, we study pairs of polynomials with a given factorization pattern and such that the degree of their difference attains its minimum. We call such pairs of polynomials Davenport–Zannier pairs (DZ-pairs). The paper is devoted to the study of DZ-pairs with rational coefficients. In our earlier paper [F. Pakovich and A. K. Zvonkin, Minimum degree of the difference of two polynomials over [Formula: see text], and weighted plane trees, Selecta Math., (N.S.) 20(4) (2014) 1003–1065], in the framework of the theory of dessins d’enfants, we established a correspondence between DZ-pairs and weighted bicolored plane trees. These are bicolored plane trees whose edges are endowed with positive integral weights. When such a tree is uniquely determined by the set of black and white degrees of its vertices, it is called unitree, and the corresponding DZ-pair is defined over [Formula: see text]. In our cited paper above, we classified all unitrees. In this paper, we compute all the corresponding polynomials. We also present some additional material concerning the Galois theory of DZ-pairs and weighted trees.

Vertex operators, t-boson model and weighted plane partitions in finite boxes

We consider two different subjects: the algebra of Hall–Littlewood functions and t-boson model. Tsilevich and Sułkowski, respectively, give that the creation operator [Formula: see text] in the monodromy matrix of t-boson model can be represented by [Formula: see text], where [Formula: see text] and [Formula: see text] are vertex operators closely related to the Hall–Littlewood functions. In this paper, we obtain that the annihilation operator [Formula: see text] in the monodromy matrix and other relations of t-boson model can also be realized in the algebra of Hall–Littlewood functions. Meanwhile, we get that the generating functions of weighted plane partitions in finite boxes can be obtained from the operators [Formula: see text].

Optimization Design for Size of Footwear Outer Packaging Boxes

According to the problems of the footwear packaging boxes presented in Chinese manufacturing enterprises, this research proposes an optimization strategy to design a series of standard footwear packaging boxes. This strategy consists of two main parts, the optimization design of plane size and the optimization design of height size. The first part is targeted to maximize the comprehensive weighted plane utilization rate and meanwhile meet the total amount limitation of plane size. During this process, the plane size of each inner shoebox, the K-medoids algorithm and bin packing patterns are used. The second part is targeted to maximize the comprehensive weighted space utilization rate within the limited boxes amount and in this part, multiple lengths of inner shoeboxes and linear 0-1 programming is used for the optimization design. Finally, with the data obtained from enterprise production practices, the empirical study shows the optimization strategy's effectiveness and adaptability to solve practical problem.

GRAPHICAL CALCULI FOR THE DUBROVNIK POLYNOMIAL WITH APPLICATIONS

We introduce a polynomial for plane graphs, which is proposed to equal the Dubrovnik polynomial of the corresponding alternating link diagrams via the medial construction. Then using this polynomial we define another polynomial for doubly edge-weighted plane graphs, which has a natural connection with Dubrovnik polynomial of links formed from plane graphs by edge-tangle replacements. In the final, via this connection we give an explanation for a result due to Lipson and compute the Dubrovnik polynomials of classical pretzel links.