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In this monograph we consider the general setting of conformal graph directed Markov systems modeled by countable state symbolic subshifts of finite type. We deal with two classes of such systems: attracting and parabolic. The latter being treated by means of the former. We prove fairly complete asymptotic counting results for multipliers and diameters associated with preimages or periodic orbits ordered by a natural geometric weighting. We also prove the corresponding Central Limit Theorems describing the further features of the distribution of their weights. These results have direct applications to a wide variety of examples, including the case of Apollonian Circle Packings, Apollonian Triangle, expanding and parabolic rational functions, Farey maps, continued fractions, Mannenville-Pomeau maps, Schottky groups, Fuchsian groups, and many more. This gives a unified approach which both recovers known results and proves new results. Our new approach is founded on spectral properties of complexified Ruelle–Perron–Frobenius operators and Tauberian theorems as used in classical problems of prime number theory.

A ground segmentation method based on line fitting of adjacent points was proposed for accurate and real-time segmentation of non-ground information from the LiDAR point cloud. Firstly, the point cloud is divided into several ordered regions depending upon the distribution characteristics of the LiDAR’s concentric circles. Then, the Euclidean distance between adjacent points and the spatial geometric features of ground point clouds is used for adaptive line fitting of ground point clouds. Finally, the ground points are divided by the distance between the adjacent points and the outer points of the line. The experiment was conducted using a real car and the KITTI open-source dataset. The approach presented in this research substantially enhances the accuracy of ground segmentation while ensuring real-time performance.

A theory of three-way decision concerns the art, science, and practice of thinking, problem solving, and information processing in threes. It explores the effective uses of triads of three things, for example, three elements, three parts, three perspectives, and so on. In this paper, I examine geometric structures, graphical representations, and semantical interpretations of triads in terms of basic geometric notions of dots, lines, triangles, circles, as well as more complex structures derived from these basic notions. I use examples from different disciplines and fields to illustrate the uses of these structures and their physical interpretations for triadic thinking, triadic computing, and triadic processing. Following the principles of triadic thinking, this paper blends together three common ways to think, namely, numerical thinking, textual thinking, and visual thinking.

This paper presents a novel metaheuristic optimization algorithm inspired by the geometrical features of circles, called the circle search algorithm (CSA). The circle is the most well-known geometric object, with various features including diameter, center, perimeter, and tangent lines. The ratio between the radius and the tangent line segment is the orthogonal function of the angle opposite to the orthogonal radius. This angle plays an important role in the exploration and exploitation behavior of the CSA. To evaluate the robustness of the CSA in comparison to other algorithms, many independent experiments employing 23 famous functions and 3 real engineering problems were carried out. The statistical results revealed that the CSA succeeded in achieving the minimum fitness values for 21 out of the tested 23 functions, and the p-value was less than 0.05. The results evidence that the CSA converged to the minimum results faster than the comparative algorithms. Furthermore, high-dimensional functions were used to assess the CSA’s robustness, with statistical results revealing that the CSA is robust to high-dimensional problems. As a result, the proposed CSA is a promising algorithm that can be used to easily handle a wide range of optimization problems.

This research aims to develop teaching materials with didactical phenomenology in math education with the project of making Delta, Diamond and Fighter kites. This research carried out experiments conducted on various types of shapes to make Delta, Diamond, and the most ideal fighter kites based on theoretical of science. In addition, this study tested various materials to make kites. The results of the experiment show that the most ideal shape of the kite in order are, square, circle, rhombus, trapezoid, square, isosceles triangle and kite. While plastic is the most ideal material for making kites that depend on the mass of the material. with the phenomenon of the kites project can be used as teaching material in geometry class.

In this paper, for extended object tracking in the case of mutation of irregular stars convex shape, using Random Hypersurface Model (RHM ) to model the object, then the target shape is expressed as parameter by radial function. Considering that using circle as a prior shape information requires a long filter consumption time, the RHM-CC-IOU-UKF algorithm proposed in this paper uses the center contour method to correct the prior shape information of the object, and then uses the Intersection over Union (IOU) method to improve the object tracking accuracy in the mutation case. The object estimation shape is updated when combined with a simple filtering algorithm. Eventually, the effectiveness of this algorithm is demonstrated by simulation experiments in two scenarios.

This study aims to explore the student's understanding of mathematical concepts semi-relationalist behaviours category. In the related literature, there are three categories of understanding mathematical concepts behaviour, one of the categories is semi-relationalist category. In this study explored the orientation of semi-relationalist category of understanding mathematical concepts behaviour. Data was obtained from one of the second-grade students of the junior high school as a participant in this study. From the results of the study, it was found that the student as categorized to behave as semirelationalist in understanding mathematical concept behaviour. Seven indicators have been used as references in determining the understanding mathematical concept behaviour. Further, five written test questions were given to participants. The written test results are analyzed and supplemented with interviews. From the seven indicators, it was found that the student was categorized as having a semi-relationalist in the understanding mathematical concept behaviour. Based on the findings, the student was included in the participant who had the opportunity to improve their understanding of mathematical concepts behaviour.

This research will provide an implementation of the study about the area in terms of numerical point of view. Regular n-shaped constructions that load a unit circle and load in unit circles can be made for at least 3 points on the unit circle. The irregular n-shaped construction contained in the unit circle can also be made without conditions with a minimum of 3 points on the unit circle. The problem arises in the irregular n-shaped construction that contains the unit circle, apparently it cannot be made by taking 3 points freely. The purpose of this research is to conduct an analysis related to what conditions are needed so that the irregular n terms can be made so that it contains a unit circle. The method used is a literature review, carried out by comparing and analyzing research that has been done relating to irregular and irregular aspects, the analysis process is carried out on irregular n-construction construction with the help of MATLAB for visualization. The results of the research show that one of the requirements for irregular n-terms can be made and contain unit circles found, and the theory test is carried out empirically with the help of MATLAB for visualization. Another result is a theory about the existence of the irregular n-aspect mathematically.

The brain codes continuous spatial, temporal and sensory changes in daily experience. Recent studies suggest that the brain also tracks experience as segmented subdivisions (events), but the neural basis for encoding events remains unclear. Here, we designed a maze for mice, composed of four materially indistinguishable lap events, and identify hippocampal CA1 neurons whose activity are modulated not only by spatial location but also lap number. These ‘event-specific rate remapping’ (ESR) cells remain lap-specific even when the maze length is unpredictably altered within trials, which suggests that ESR cells treat lap events as fundamental units. The activity pattern of ESR cells is reused to represent lap events when the maze geometry is altered from square to circle, which suggests that it helps transfer knowledge between experiences. ESR activity is separately manipulable from spatial activity, and may therefore constitute an independent hippocampal code: an ‘event code’ dedicated to organizing experience by events as discrete and transferable units.Hippocampal neurons track events as abstract units of episodic experience. These event representations can be transferred between different experiences and are separately manipulable from hippocampal representations of continuous changes in space.