区间Pythagorean犹豫模糊连续熵及灰色关联度的应用

区间Pythagorean犹豫模糊集,可以更加全面完整地描述决策者给出的决策结果,因此它是一个表示不确定现象的强有力的工具。针对模糊信息下的决策问题,提出了一种基于区间Pythagorean犹豫模糊连续熵的多属性决策方法。提出连续区间Pythagorean犹豫模糊有序加权平均（CIPHFOWA）算子,并提出了区间Pythagorean犹豫模糊连续熵,同时给出了属性权重完全未知和部分已知时的权重确定方法;提出了一种基于区间Pythagorean犹豫模糊连续熵-灰色关联度的多属性决策方法,并通过新型农村医疗制度完善情况评价案例说明该方法的可行性和有效性。     

利用PowerPoint的VBA编程功能制作勾股定理的探索与验证交互性课件

本文介绍了利用PowerPoint 2003的VBA编程功能制作“勾股定理的探索与验证”交互性课件方珐和程序代码。     

Stakeholder requirement evaluation of smart industrial service ecosystem under Pythagorean fuzzy environment for complex industrial contexts: A case study of renewable energy park

This study focuses on ways to systematically evaluate stakeholder requirements when developing a smart industrial service ecosystem (SISE) in a complex industrial context. The SISE development requires considering the service requirement from both the complex industrial context and service ecosystem manners. This study proposes a systematic framework for stakeholder requirement evaluation in SISE. The first part of the framework is the industrial context-viable system model with ecological thinking (IC-VESM) to elicit the service requirements for the SISE, which facilitates a systematic analysis of the service value proposition and service requirement elicitation in the operational lifecycle of an entire industrial context. This second part of the framework proposes a method for evaluating service requirements that is both feasible and systematic. This is achieved by combining the Fuzzy Kano and AHP methods in a Pythagorean fuzzy (PF) environment. The PF Kano computes the categories and determines the weights of service requirements from a consumer perspective, while the PF AHP hierarchically analyzes the service requirements and provides pairwise comparison paths for design experts. Finally, an illustrative case study in a renewable energy context was used to demonstrate the feasibility and effectiveness of the methodology. The proposed theoretical model provides more reliable and systematic outcomes than traditional methods when eliciting service requirements and evaluating complex smart industrial service solutions. The study has practical implications by providing useful insights for companies to recognize key smart service requirements in complex industrial contexts and to improve sustainable development.     

Computing upper and lower bounds of rotation angles from digital images

Rotations in the discrete plane are important for many applications such as image matching or construction of mosaic images. We suppose that a digital image A is transformed to another digital image B by a rotation. In the discrete plane, there are many angles giving the rotation from A to B, which we call admissible rotation angles from A to B. For such a set of admissible rotation angles, there exist two angles that achieve the lower and the upper bounds. To find those lower and upper bounds, we use hinge angles as used in Nouvel and Rémila [Incremental and transitive discrete rotations, in: R. Reulke, U. Eckardt, B. Flash, U. Knauer, K. Polthier (Eds.), Combinatorial Image Analysis, Lecture Notes in Computer Science, vol. 4040, Springer, Berlin, 2006, pp. 199-213]. A sequence of hinge angles is a set of particular angles determined by a digital image in the sense that any angle between two consecutive hinge angles gives the identical rotation of the digital image. We propose a method for obtaining the lower and the upper bounds of admissible rotation angles using hinge angles from a given Euclidean angle or from a pair of corresponding digital images.     

一种视频驱动的三维虚拟人动画实现方法

研究实现三维人体动画具有广泛的应用前景和实用意义,提出了一种二维视频驱动的三维人体动画实现方法。基于动态帧的关键帧提取算法从二维视频中构建了二维关键帧集合;基于二维关键帧构建二维人体骨骼模型;利用小孔成像原理和勾股定理计算得到关节特征点的深度坐标,从而得到了反映人体动画的三维数据。实验结果表明,该方法生成的三维人体动画形象逼真、成本低、提高了运动生成的实时性,能够应用于虚拟现实、计算机游戏、三维视频游戏制作等领域。     

Approximating curves and their offsets using biarcs and Pythagorean hodograph quintics

This paper compares two techniques for the approximation of the offsets to a given planar curve. The two methods are based on approximate conversion of the planar curve into circular splines and Pythagorean hodograph (PH) splines, respectively. The circular splines are obtained using a novel variant of biarc interpolation, while the PH splines are constructed via Hermite interpolation of C¹ boundary data.We analyze the approximation order of both conversion procedures. As a new result, the C¹ Hermite interpolation with PH quintics is shown to have approximation order 4 with respect to the original curve, and 3 with respect to its offsets. In addition, we study the resulting data volume, both for the original curve and for its offsets. It is shown that PH splines outperform the circular splines for increasing accuracy, due to the higher approximation order.     

Pythagorean Triangles and Musical Proportions

Martin Euser researches the factor and its interesting relations between musical proportions and Pythagorean triangles. The simple scheme is also interesting as a generative set of pairs of numbers. This set looks like a prototype for the generative set of pairs of numbers discussed in a previous article by the author.     

On PH quintic spirals joining two circles with one circle inside the other

The paper derives a spiral condition, for a single Pythagorean hodograph quintic transition curve of G² contact, between two circles with one circle inside the other. A spiral is free of local curvature extrema, making spiral designs an interesting mathematical problem with importance for both physical and aesthetic applications. In the construction of highways or railway routes in particular, it is often desirable to have a transition curve from circle to circle. Here, we treat an open problem on planar quintic spiral segments, called transition curve elements, examine techniques for curve design using the new results, and derive lower and upper bounds for the distance between the two circles. The proposed method is applicable for non-tangent and non-concentric circles.     

An analysis of dam-break flow on slope

The one-dimensional steep slope shallow water equations are used to model the dam-break flow down a uniform slope with arbitrary inclination, and analytical solutions are derived by the hodograph transformation and the Riemann＇s method in terms of evaluated integrals. An implicit analytical solution is obtained to evaluate the spatio-temporal distributions of dam-break flood hydrographs along the slope. For convenience, the solution for representative wave profiles and velocity distributions is shown in charts. Comparing with the Dressler＇s solution and WES experimental data, the analytical solution is seen reasonable.     

基于Hamacher区间模糊算法的技术状态评估权重确定

针对复杂装备技术状态评估权重确定过程中存在的专家犹豫和权重确定动态性不足的问题,结合区间值犹豫模糊熵、毕达哥拉斯模糊集和Hamacher算子,提出Hamacher区间模糊算法。综合各专家对各元素重要性的打分,构建区间值犹豫模糊集,改进区间值犹豫模糊熵,并对模糊集中各犹豫模糊元运算得到区间值犹豫模糊熵矩阵;令区间值犹豫模糊熵为隶属度,打分区间差值为犹豫度,得到毕达哥拉斯模糊集,运用Hamacher区间模糊算法计算得到各元素重要度得分函数,将得分函数归一化得到最终元素权重。算例结果表明,该验证方法对技术状态评估具有适用性。