Yong Zhang

Many authors investigated the problem about the linear combination of two polygonal numbers being a perfect square, i.e., the Diophantine equation mPₖ(x)+nPₖ(y)=z², where Pₖ(x) denotes the x-th k-polygonal number and m, n are positive integers. In this note, we continue the study of this problem in another direction and consider three Diophantine equations mPₖ(x)-1=z², mPₖ(x)-nPₖ(y)=z², mPₖ(x)-nPₖ(y)=1. By the theory of Pell equations and congruences, we obtain some conditions such that the above three Diophantine equations have infinitely many positive integer solutions.

Back-illuminated spot mapping is essential for observations with modern spectroscopic survey telescopes. In the fiber position detection system, after the fiber view camera (FVC) detects the positions of the light spots from the robotic fiber positioners (RFPs), these positions must be mapped to the RFPs individually to complete subsequent calculations of their operational data. The back-illuminated images captured by the FVC show thousands of small light spots, each only hundreds of microns in size, randomly distributed on a completely black background, making accurate one-to-one mapping to the actual RFP positions highly challenging due to the lack of additional RFP features. This study constructs an artificial probability potential field to calculate the likelihood of each light spot appearing in the patrol area of an RFP, employing a global probability potential energy optimization method to establish the optimal mapping relationship. The proposed method successfully maps all light spots in the closed-loop fiber positioning task and is ultimately applied to the fiber positioning system of the Large Sky Area Multi-Object Fiber Spectroscopic Telescope, achieving accurate mapping of 4000 RFPs on the focal plane. This algorithm effectively addresses challenges such as multitarget interference, uniform background, and RFP failures, providing a reliable approach for mapping the back-illuminated points of tens of thousands of dual-rotation RFPs without requiring special backlight designs in future spectral survey telescopes.

With new artificial intelligence (AI) technologies and application scenarios constantly emerging, AI technology has become widely used in astronomy and has promoted notable progress in related fields. A large number of papers have reviewed the application of AI technology in astronomy. However, relevant articles seldom mention telescope intelligence separately, and it is difficult to understand the current development status of and research hotspots in telescope intelligence from these papers. This paper combines the development history of AI technology and difficulties with critical telescope technologies, comprehensively introduces the development of and research hotspots in telescope intelligence, conducts a statistical analysis of various research directions in telescope intelligence, and defines the merits of these research directions. A variety of research directions are evaluated, and research trends in each type of telescope intelligence are indicated. Finally, according to the advantages of AI technology and trends in telescope development, potential future research hotspots in the field of telescope intelligence are given.

Artificial intelligence technology has been widely used in astronomy, and new artificial intelligence technologies and application scenarios are constantly emerging. There have been a large number of papers reviewing the application of artificial intelligence technology in astronomy. However, relevant articles seldom mention telescope intelligence separately, and it is difficult to understand the current development status and research hotspots of telescope intelligence from these papers. This paper combines the development history of artificial intelligence technology and the difficulties of critical technologies of telescopes, comprehensively introduces the development and research hotspots of telescope intelligence, then conducts statistical analysis on various research directions of telescope intelligence and defines the research directions' merits. All kinds of research directions are evaluated, and the research trend of each telescope's intelligence is pointed out. Finally, according to the advantages of artificial intelligence technology and the development trend of telescopes, future research hotspots of telescope intelligence are given.

In this work, we consider the following two Diophantine problems that arise from the search of pairs of rational triangles whose areas and perimeters are in certain relations. The first problem is about finding pairs of rational triangles whose products between area and perimeters are the same. The second one is to find the pairs of rational triangles sharing the same perimeter and whose product of areas is a perfect square. We show that if certain elliptic curves have rational points which satisfy some conditions, then there are infinitely many pairs of such triangles.

Space-based and ground-based telescopes have extensively documented the impact of satellites on astronomical observations. With the proliferation of satellite mega-constellation programs, their influence on astronomical observations has become undeniable. It is crucial to quantify the impact of satellites on telescopes. To address this need, we have enhanced the circular orbit model for satellites and introduced a methodology based on two-line element (TLE) orbit data. This involves constructing a satellite probability distribution model to evaluate the impact of satellites on telescopes. Using our method, we assessed the satellite impact on global observatories. The results indicate that the regions most severely affected by satellite interference currently are those near the equator, with latitudes around 50 and 80 degrees experiencing the most significant impact from low Earth orbit satellites. Furthermore, we validated the reliability of our method using imaging data obtained from the focal surface acquisition camera of the LAMOST telescope.

Realizing accurate positioning with the Large Sky Area Multi-Object Fiber Spectroscopic Telescope (LAMOST) closed-loop system depends on accurate high-precision calibration of the visual measurement system, which has a great impact on collision avoidance and accurate positioning. We designed fiducial fibers for the calibration of the LAMOST closed-loop system to provide accurate fiducial positions for visual measurement. The benchmark position accuracy of the fiducial fibers is a key factor affecting the accuracy of the visual measurement system; the more accurate the fiducial fiber positions are, the higher the visual measurement correction accuracy. In this study, three measurement methods were used to obtain the fiducial fiber positions, namely, measuring the hole positions using a coordinate measuring machine, imaging the fiducial fibers using a calibrated photographic system, and directly measuring the fiducial fiber spatial positions using a laser tracker. By evaluating the fiber positions obtained via the three methods, we can obtain a stable and reliable fiducial fiber position benchmark. A fiducial fiber positions evaluation method based on an optimal residual criterion is proposed, and the optimal residual solution for a small calibration target (SCT) is used to evaluate the optimal fiducial fiber measurement method. Specifically, the fiducial positions obtained via each of the three methods are used to invert the camera calibration parameters, which are then used to calculate the physical position of an SCT. Finally, the residual value between the calculated and theoretical positions is taken as the standard for evaluating the fiducial fiber measurement benchmark performance. The results show that the fiducial fiber positions measured using the laser tracker can be applied to effectively calibrate the photographic system, enabling the LAMOST vision measurement system to achieve a positioning accuracy of nearly 10 μm with the camera 20 m from the focal surface, whereas the accuracy is within 20 μm for ∼95 % of the measurement points.

The imaging quality of a telescope directly affects the reliability of astronomical research. Through the monitoring and diagnosis of imaging quality, the cause of the deterioration of imaging quality can be found in time, which is essential for ensuring the peaking performance of the telescope and high-quality imaging. Moreover, these operations are complex and crucial for achieving high-quality imaging of future giant telescope systems involving active optics, adaptive optics, and other advanced techniques. We propose a three-component method based on cutting-edge artificial intelligence technology to real-time monitor and efficiently diagnose the telescope image quality. The first component, an image quality monitoring system, monitors and outputs the telescope’s image quality. The second component is a query system with a knowledge graph, which outputs the node chains as the possible cause of poor image quality based on the input. The third component, a final estimator, uses the node parameter, which contains historical fault data and real-time updated data from sensors, to give the probability of each node chain. We construct and test the system in the Large Sky Area Multi-Object Fiber Spectroscopy Telescope.