A triangular exploration

In this Article we study the following problem: Let ΔABC be an acute-angled triangle. Let the points D, E, F on the sides BC, CA and AB, respectively, be such that AD is the median from A, BE is the internal angle bisector of ∠ABC, and CF is the altitude from C. This is shown in Figure 1.

Angle Bisector Algorithm and Modified Dynamic Programming Algorithm for Dubins Traveling Salesman Problem

A Dubin's Travelling Salesman Problem (DTSP) of finding a minimum length tour through a given set of points is considered. DTSP has a Dubins vehicle, which is capable of moving only forward with constant speed. In this paper, first, a worst case upper bound is obtained on DTSP tour length by assuming DTSP tour sequence same as Euclidean Travelling Salesman Problem (ETSP) tour sequence. It is noted that, in the worst case, \emph{any algorithm that uses of ETSP tour sequence} is a constant factor approximation algorithm for DTSP. Next, two new algorithms are introduced, viz., Angle Bisector Algorithm (ABA) and Modified Dynamic Programming Algorithm (MDPA). In ABA, ETSP tour sequence is used as DTSP tour sequence and orientation angle at each point $i_k$ are calculated by using angle bisector of the relative angle formed between the rays $i_{k}i_{k-1}$ and $i_ki_{k+1}$. In MDPA, tour sequence and orientation angles are computed in an integrated manner. It is shown that the ABA and MDPA are constant factor approximation algorithms and ABA provides an improved upper bound as compared to Alternating Algorithm (AA) \cite{savla2008traveling}. Through numerical simulations, we show that ABA provides an improved tour length compared to AA, Single Vehicle Algorithm (SVA) \cite{rathinam2007resource} and Optimized Heading Algorithm (OHA) \cite{babel2020new,manyam2018tightly} when the Euclidean distance between any two points in the given set of points is at least $4\rho$ where $\rho$ is the minimum turning radius. The time complexity of ABA is comparable with AA and SVA and is better than OHA. Also we show that MDPA provides an improved tour length compared to AA and SVA and is comparable with OHA when there is no constraint on Euclidean distance between the points. In particular, ABA gives a tour length which is at most $4\%$ more than the ETSP tour length when the Euclidean distance between any two points in the given set of points is at least $4\rho$.

Angle Bisector Algorithm and Modified Dynamic Programming Algorithm for Dubins Traveling Salesman Problem

A Dubin's Travelling Salesman Problem (DTSP) of finding a minimum length tour through a given set of points is considered. DTSP has a Dubins vehicle, which is capable of moving only forward with constant speed. In this paper, first, a worst case upper bound is obtained on DTSP tour length by assuming DTSP tour sequence same as Euclidean Travelling Salesman Problem (ETSP) tour sequence. It is noted that, in the worst case, \emph{any algorithm that uses of ETSP tour sequence} is a constant factor approximation algorithm for DTSP. Next, two new algorithms are introduced, viz., Angle Bisector Algorithm (ABA) and Modified Dynamic Programming Algorithm (MDPA). In ABA, ETSP tour sequence is used as DTSP tour sequence and orientation angle at each point $i_k$ are calculated by using angle bisector of the relative angle formed between the rays $i_{k}i_{k-1}$ and $i_ki_{k+1}$. In MDPA, tour sequence and orientation angles are computed in an integrated manner. It is shown that the ABA and MDPA are constant factor approximation algorithms and ABA provides an improved upper bound as compared to Alternating Algorithm (AA) \cite{savla2008traveling}. Through numerical simulations, we show that ABA provides an improved tour length compared to AA, Single Vehicle Algorithm (SVA) \cite{rathinam2007resource} and Optimized Heading Algorithm (OHA) \cite{babel2020new,manyam2018tightly} when the Euclidean distance between any two points in the given set of points is at least $4\rho$ where $\rho$ is the minimum turning radius. The time complexity of ABA is comparable with AA and SVA and is better than OHA. Also we show that MDPA provides an improved tour length compared to AA and SVA and is comparable with OHA when there is no constraint on Euclidean distance between the points. In particular, ABA gives a tour length which is at most $4\%$ more than the ETSP tour length when the Euclidean distance between any two points in the given set of points is at least $4\rho$.

Study of the Preferred Orientation of Hydroxyapatite in Ivory from Zimbabwe and Mammoth Ivory from Siberia

In this paper, the pole figures plotted from X-ray diffraction data are employed to analyze the orientation of hydroxyapatite in ivory and mammoth ivory for the first time. The results present evidence of the lamellar structure and the hydroxyapatite appeared as tabular. A preferred orientation of hydroxyapatite was revealed in terms of the calculated orientation factor and the characteristics found in the pole figures. The c-axes of hydroxyapatite are mainly oriented along the growth direction. Both a-axes are on the left of the angle bisector of Retzius. Approximately 25–30° separates the a-axes and the angle bisector of Retzius in ivory, whereas the figure is approximately 10–15° in mammoth ivory. Our work is significant in providing more accurate knowledge of the shapes and organizational state of bio-mineral crystals and providing insight into crystal formation and development in bio-mineralization.

Fuzzy comprehensive evaluation of maritime boundary delimitation schemes based on AHP-entropy weight method

There are 11 provinces along the coastal regions in the Chinese Mainland and the scientific division of marine management boundaries among different provinces within the extents of the territorial sea is of important significance in promoting the sustainable development of the marine economy in China. A fuzzy evaluation index system that includes 14 indices of three subsets was established based on a case study of maritime boundary delimitation between Jiangsu and Shandong Provinces to evaluate the advantages and disadvantages of maritime boundary delimitation and determine the optimal scheme and offset shortages in the current evaluation index system and evaluation methods. Combined weights of indices were calculated through the analytic hierarchy process (AHP) and entropy weight method. The comprehensive evaluation indices were used to evaluate three maritime boundary delimitation schemes, namely, historical boundary delimitation scheme, angle bisector delimitation scheme, and equidistance delimitation scheme. Results show that the equidistance delimitation scheme is relatively superior to the two other schemes. The evaluation index is 0.504761 and the evaluation grade is “good”. The angle bisector delimitation scheme is the second optimal. The evaluation index is 0.361641 and the grade is “moderate”. The historical boundary delimitation scheme is the poorest. The evaluation index is 0.135345 and the grade is “poor”. In the late optimization of maritime boundary delimitation schemes, more consideration should be given to people’s livelihood and protection of national maritime rights and interests. The fuzzy comprehensive evaluation method based on AHP-entropy weight can not only provide a quantitative top-down sequence of schemes in terms of quality and solve estimation problems in maritime boundary delimitation scheme, but also assist decision-makers to select the best scheme. Therefore, it possesses great application values.

Numerical simulation and experimental study of off-axis tensile performance of triaxial woven fabric reinforced rubber composites

In this paper, the off-axis tensile behaviors of triaxial woven fabric reinforced rubber composites (TWFR) was studied experimentally and numerically. The tensile failure morphologies and stress-strain curves of specimens cut at seven different off-axis angles (0°, 15°, 30°, 45°, 60°, 75°, and 90°) were obtained by finite element analysis and experiment. The finite element analysis results were found to be in good agreement with the experimental results. Stress-strain curves indicated the TWFR exhibited nonlinear tensile behavior under off-axial tension loading, and could be divided into the coupling working stage of yarns and rubber matrix in TWFR, and the stressed stage of yarns. Similar tensile properties were observed for specimens cut along the direction of angle bisector of any two sets of adjacent yarns. The tensile failure morphologies indicated that the failure mechanism of TWFR was tensile-shear mixed failure.

Health Condition Assessment of Marine Systems Based on an Improved Radar Chart

Since health assessment plays a significant role in marine systems (MSs), it has caught the attention of researchers. In this study, a powerful evaluation method called an improved radar chart was developed as a means of reliability estimation. General evaluation methods applied in the comprehensive evaluation of MS are slightly insufficient in terms of considering index coordination. However, the application of a radar chart can solve this problem. To improving the shortcomings of a traditional radar chart, the fuzzy centralization statistical theory and the entropy weight were combined in this study to obtain the comprehensive weight. The weight could be converted into an angle, and it could reflect the influence degree of the indexes on the evaluation objects. Additionally, an angle bisector was introduced as an index axis, and the eigenvector was extracted to get the unique evaluation result. The result showed that the proposed method could achieve the continuous online monitoring of the system state, and the reliable and accurate assessment results were able to provide a reference for condition-based maintenance and decision-making.

Comprehensive evaluation model of wind power accommodation ability based on macroscopic and microscopic indicators

This article explores and establishes comprehensive evaluation index system of wind power accommodation ability considering microscopic index and macroscopic index, and the index system includes conventional evaluation indexes such as forecast deviation, simultaneity factor and anti-peak rate, also newly introduced evaluation indexes such as installed capacity, power adequacy and accommodation space. Bayesian weight modified method is used for solving index weights of 8 wind power accommodation indexes. The paper puts forward a comprehensive evaluation model of wind power accommodation ability based on improved radar chart method, and this model changes traditional radar chart fan-shaped sector to quadrilateral evaluation region, and increasing angle bisector can solve the problem that evaluation results are not unique. It constructs new area and perimeter vectors of radar chart, which make the evaluation results give consideration to level of aggregation and balance degree of evaluation objectives, and case study results show that this model has a certain practical value.