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25 November 2021, Volume 2021 Issue 6
Review Article
Mathematics
 The structure of 3-Lie-Rinehart algebras Ruipu BAI, Xiaojuan LI 2021, 2021 (6):  15-23.  doi: 10.3969/j.issn.1000-5641.2021.06.002 Abstract ( 5 )   HTML ( 0 )   PDF (674KB) ( 1 )   In this paper, we introduce a class of 3-ary algebras, called the 3-Lie-Rinehart algebra, and we discuss the basic structure thereof. The 3-Lie-Rinehart algebras are constructed using 3-ary differentiable functions, modules of known 3-Lie algebras, and inner derivatives of 3-Lie algebras.
 Finite sums in higher order powers of shifted-harmonic numbers Qinglun YAN, Zhaofen WANG, Juan MI 2021, 2021 (6):  24-32.  doi: 10.3969/j.issn.1000-5641.2021.06.003 Abstract ( 2 )   HTML ( 0 )   PDF (430KB) ( 1 )   In this article, using methods such as the partial fraction method, we study a set of combined identities for an Euler-type summation. We calculate, furthermore, the finite summation form of the product of the high order shifted-harmonic number and the reciprocal of the binomial coefficient. By using special values for the parameters, interesting identities can be obtained.
 Braided vector algebra $V(R',R)$ Hongmei HU 2021, 2021 (6):  33-37.  doi: 10.3969/j.issn.1000-5641.2021.06.004 Abstract ( 7 )   HTML ( 0 )   PDF (472KB) ( 3 )   Braided vector algebras are an important class of Hopf algebras in braided tensor categories. In this paper, it is shown that braided vector algebras are isomorphic to quantum vector spaces as associative algebras; hence, the algebraic structure of braided vector algebras and three equalities of the pair $(R',R)$ are recovered from representations of quantized enveloping algebras $U_q(\mathfrak g)$ .
 Picard-type theorems for entire functions of several complex variables with total derivatives Shengyao ZHOU, Liu YANG 2021, 2021 (6):  38-46.  doi: 10.3969/j.issn.1000-5641.2021.06.005 Abstract ( 5 )   HTML ( 0 )   PDF (607KB) ( 4 )   In this paper, we use the logarithmic derivative lemma for several complex variables to extend the Milloux inequality to differential polynomials of entire functions. As an application, we subsequently apply the concept to two Picard-type theorems: (1) Let $f$ be an entire function in $\mathbb{C}^{n}$ and $a, b\;(\neq 0)$ be two distinct complex numbers. If $f\neq a, {\cal{P}}\neq b,$ then $f$ is constant. (2) If $f^{s}D^{t_{1}}(f^{s_{1}})\cdots D^{t_{q}}(f^{s_{q}})\neq b$ and $s+$ $\sum_{j = 1}^{q}s_{j}\geqslant 2+\sum_{j = 1}^{q}t_{j},$ then $f$ is constant, where $D^{k}f$ is the $k$ -th total derivative of $f$ and ${\cal{P}}$ is a differential polynomial of $f$ with respect to the total derivative.
 Ambrosetti-Prodi results for second-order discrete periodic boundary value problems Rui WANG, Yanqiong LU, Xiaomei YANG 2021, 2021 (6):  47-57.  doi: 10.3969/j.issn.1000-5641.2021.06.006 Abstract ( 6 )   HTML ( 0 )   PDF (801KB) ( 2 )   This paper explores the relationship between the number of solutions and the parameter $s$ of second-order discrete periodic boundary value problems of the form　　　　　　　　　　 $\left\{ \begin{array}{ll} \Delta^{2} u(t-1)+f\Delta u(t)+g(t,u(t)) = s, \;t\in[1,T]_{\mathbb{Z}}, \\ u(0) = u(T-1),\;\Delta u(0) = \Delta u(T-1), \end{array} \right.$ where $g: [1,T]_{\mathbb{Z}}\times \mathbb{R}\to\mathbb{R}$ is a continuous function, $f\geqslant0$ is a constant, $T\geqslant2$ is an integer, and $s$ is a real number. By using the upper and lower solution method and the theory of topological degree, we obtain the Ambrosetti-Prodi type alternatives which demonstrate the existence of either zero, one, or two solutions depending on the choice of the parameter $s$ with fixed constant $s_{0}\in \mathbb{R}$ .
 Singularity indices of hyperelliptic fibrations Zhiming GUO 2021, 2021 (6):  58-64.  doi: 10.3969/j.issn.1000-5641.2021.06.007 Abstract ( 3 )   HTML ( 0 )   PDF (702KB) ( 3 )   Xiao introduced a series of singularity indices to survey hyperelliptic fibrations. However, it remains unknown whether the second singularity index, $s_2$ , is non-negative. In this paper, I demonstrate a series of examples of degeneration of curves where $s_2$ tends to $-\infty$ as the genus $g$ grows. Moreover, I obtain a lower bound for $s_2$ for a given genus $g$ , thereby confirming that the index $s_2$ of fibrations for genus $g=2,3,4$ is non-negative.
Computer Science
 A fast key points matching method for high resolution images of a planar mural Xinye ZHANG, Weiqing TONG, Haisheng LI 2021, 2021 (6):  65-80.  doi: 10.3969/j.issn.1000-5641.2021.06.008 Abstract ( 6 )   HTML ( 0 )   PDF (2053KB) ( 2 )   Existing methods of key points matching were invented for grayscale images and are not suitable for high resolution images. Mural images typically have very high resolution, and there may be areas with the same gray textures and different colors. For this special kind of image, this paper proposes a high-speed algorithm of key points matching for high-resolution mural images (NeoKPM for short). NeoKPM has two main innovations: (1) first, the homography matrix of rough registration for the original image is obtained by downsampling the image, which substantially reduces the time required for key points matching; (2) second, a feature descriptor based on gray and color invariants is proposed, which can distinguish different colors of texture with the same gray level, thereby improving the correctness of key points matching. In this paper, the performance of the NeoKPM algorithm is tested on a real mural image library. The experimental results show that on mural images with a resolution of 80 million pixels, the number of correct matching points per pair of images is nearly 100 000 points higher than that of the SIFT (Scale Invariant Feature Transform) algorithm, the processing speed of key points matching is more than 20 times faster than that of the SIFT algorithm, and the average error of dual images based on a single pixel of the image is less than 0.04 pixels.
 Research on large-field microscopic images based on the best stitching path Yang XU, Hongying LIU, Quanjie ZHUANG 2021, 2021 (6):  81-87.  doi: 10.3969/j.issn.1000-5641.2021.06.009 Abstract ( 8 )   HTML ( 1 )   PDF (861KB) ( 1 )   Image stitching technology is one of the key technologies in the application of large-field microscopic digital images. The existing traditional image stitching method is to stitch in a fixed order after image registration, and once there is an error, it will be accumulated along a fixed path, thereby causing problems such as misalignment of subsequent images. In this study, through experimental analysis, a method for optimizing the stitching path of the large-field image was proposed, which greatly optimized the problems caused by error accumulation and registration failure, and effectively improved the stitching quality of the large-field microscopic digital image. This method can be used not only for the stitching of large-field microscopic images, but also for other types of stitching.