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In the open literature, there are numerous studies on the normal and extreme (flood and drought) behavior of wet and dry periods based on the understanding of the standard precipitation index (SPI), which provides a series of categorizations by considering the standard normal (Gaussian) probability distribution function (PDF). The numerical meaning of each categorization assessment is quite lacking in terms of future predictions of wet and dry period duration based on historical records. This paper presents a new approach for calculating possible formations of future wet and dry period durations based on historical records through an effective fractal geometric forecasting approach. The essence of the proposed methodology is based on the number of dry periods (steps) of non-overlapping monthly duration along consecutive broken line paths in the SPI classification for wet and dry period durations. It has been observed that the plot of periods on double logarithmic paper falls along a straight line against the number of such periods, implying a power function, which is the essence of fractal geometry. Extending the empirically derived straight line provides the number of periods that may occur in the future over a range of SPI levels. This methodology is referred to as SPI fractal (SPIF), and the classic SPI classification is converted into SPIF wet and dry polygons, which provide additional information about the drought period number within a valid polygonal area, compared to the classic SPI results. The wet and dry period features of any hydro-meteorology time series are constrained in SPIF polygons. The application of the methodology was carried out on monthly rainfall records on the European side of the Istanbul Florya meteorological station in Turkey. [ABSTRACT FROM AUTHOR]

Frieze patterns have been introduced by Coxeter [Acta Arith. 18 (1971), pp. 297–310] in the 1970's and have recently attracted renewed interest due to their close connection with Fomin-Zelevinsky's cluster algebras. Frieze patterns can be interpreted as assignments of values to the diagonals of a triangulated polygon satisfying certain conditions for crossing diagonals (Ptolemy relations). Weak friezes, as introduced by Çanakçı and Jørgensen [Adv. in Appl. Math. 131 (2021), Paper No. 102253], are generalizing this concept by allowing to glue dissected polygons so that the Ptolemy relations only have to be satisfied for crossings involving one of the gluing diagonals. To any frieze pattern one can associate a symmetric matrix using a triangular fundamental domain of the frieze pattern in the upper and lower half of the matrix and putting zeroes on the diagonal. Broline, Crowe and Isaacs [Geometriae Dedicata 3 (1974), pp. 171–176] have found a formula for the determinants of these matrices and their work has later been generalized in various directions by other authors. These frieze pattern determinants are the main focus of our paper. As our main result we show that this determinant behaves well with respect to gluing weak friezes: the determinant is the product of the determinants for the pieces glued, up to a scalar factor coming from the gluing diagonal. Then we give several applications of this result, showing that formulas from the literature, obtained by Broline-Crowe-Isaacs, Baur-Marsh [J. Combin. Theory Ser. A 119 (2012), pp. 1110–1122], Bessenrodt-Holm-Jørgensen [J. Combin. Theory Ser. A 123 (2014), pp. 30–42] and Maldonado [ Frieze matrices and infinite frieze patterns with coefficients , Preprint, arXiv: 2207.04120 , 2022] can all be obtained as consequences of our result. [ABSTRACT FROM AUTHOR]

As core algorithms of geographic computing, overlay analysis algorithms typically have computation-intensive and data-intensive characteristics. It is highly important to optimize overlay analysis algorithms by parallelizing the vector polygons after reasonable data division. To address the problem of unbalanced data partitioning in the task decomposition process for parallel polygon overlay analysis and calculation, this paper presents a data partitioning method based on shape complexity index optimization, which achieves data equalization among multicore parallel computing tasks. Taking the intersection operator and difference operator of the Vatti algorithm as examples, six polygon shape indexes are selected to construct the shape complexity model, and the vector data are divided in accordance with the calculated shape complexity results. Finally, multicore parallelism is achieved based on OpenMP. The experimental results show that when a data set with a large amount of data is used, the effect of the multicore parallel execution of the Vatti algorithm's intersection operator and difference operator based on shape complexity division is clearly improved. With 16 threads, compared with the serial algorithm, speedups of 29 times and 32 times can be obtained. Compared with the traditional multicore parallel algorithm based on polygon number division, the speed can be improved by 33% and 29%, and the load balancing index is reduced. For a data set with a small amount of data, the acceleration effect of this method is similar to that of traditional methods involving multicore parallelism. [ABSTRACT FROM AUTHOR]

In this paper, we develop a new Best Uniform Rational Approximation-Semi-Discrete (BURA-SD) method taking into account the singularities of the solution of fractional diffusion problems in polygonal domains. The complementary capabilities of the exponential convergence rate of BURA-SD and the h p FEM are explored with the aim of maximizing the overall performance. A challenge here is the emerging singularly perturbed diffusion–reaction equations. The main contributions of this paper include asymptotically accurate error estimates, ending with sufficient conditions to balance errors of different origins, thereby guaranteeing the high computational efficiency of the method. [ABSTRACT FROM AUTHOR]

The advancement of information technology has led to the widespread application of remote measurement systems, where information in the form of images or videos, serving as measurement results, is transmitted over networks. However, this transmission is highly susceptible to attacks, tampering, and disputes, posing significant risks to the trustworthy transmission of measurement results from instruments and devices. In recent years, many encryption algorithms proposed for images have focused on encrypting the entire image, resulting in resource waste. Additionally, most encryption algorithms are designed only for single-object-type images. Addressing these issues, this paper proposes a multi-object region encryption algorithm based on an adaptive mechanism. Firstly, an adaptive mechanism is employed to determine the strategy for adjusting the sampling rate of encryption objects, achieved through an encryption resource allocation algorithm. Secondly, an improved polygon segmentation algorithm is utilized to separate single-object regions from multi-object images, dynamically adjusting the sequence of encryption objects based on the adaptive mechanism. Finally, encryption is achieved using a chaos fusion XOR encryption algorithm. Experimental validation using instrument images demonstrates that the proposed algorithm offers high efficiency and security advantages compared to other mainstream image encryption algorithms. [ABSTRACT FROM AUTHOR]

In this paper, we prove a new extremal property of the Reuleaux triangle: it maximizes the Cheeger constant among all bodies of (same) constant width. The proof relies on a fine analysis of the optimality conditions satisfied by an optimal Reuleaux polygon together with an explicit upper bound for the inradius of the optimal domain. As a possible perspective, we conjecture that this maximal property of the Reuleaux triangle holds for the first eigenvalue of the p -Laplacian for any p ∈ (1 , + ∞) (this paper covers the case p = 1 whereas the case p = + ∞ was already known). [ABSTRACT FROM AUTHOR]

GeoAI combines artificial intelligence (AI) with geospatial data, science, and technologies. In this paper, a successful case in utilizing GeoAI to improve the data quality in the national topographic database (TDB) of Finland was introduced. The project employed a GeoAI model to identify buildings from the input data of true orthophotos, digital elevation model (DTM), and digital surface model (DSM). The GeoAI-derived buildings served as reference data, enabling a comparison with building polygons from the topographic database (TDB) to reveal TDB building location deviations, missing structures, and demolished buildings. Throughout the project, algorithms were developed to match the TDB building vectors to the GeoAI-derived building polygons. The challenges include i) the differences between the GeoAI-derived building outlines and the TDB build footprints; ii) the reliability of the GeoAI model across data over different environments: urban, suburban, rural, and forest areas. Throughout the project, the GeoAI model was continuously improved by training massive new datasets: corrected vectors from the model prediction. Testing was conducted on datasets from twelve areas of Finland, covering 2204 km2 and including hundreds of thousands of buildings. The test areas covered urban, suburban, rural, and forest areas. The evaluation was conducted by the mapping team in the organization. The results showed that our methods greatly enhanced the quality of the TDB building footprints. The challenges and lessons of the project were addressed in the paper. [ABSTRACT FROM AUTHOR]

We prove that solutions to several shape optimization problems in the plane, with a convexity constraint on the admissible domains, are polygons. The main terms of the shape functionals we consider are either the Dirichlet energy E f ⁢ (Ω) of the Laplacian in the domain Ω or the first eigenvalue λ 1 ⁢ (Ω) of the Dirichlet-Laplacian. Usually, one considers minimization of such functionals (often with measure constraint), as for example for the famous Saint-Venant and Faber-Krahn inequalities. By adding the convexity constraint (and possibly other natural constraints), we instead consider the rather unusual and difficult question of maximizing these functionals. This paper follows a series of papers by the authors, where the leading idea is that a certain concavity property of the shape functional that is minimized leads optimal shapes to locally saturate their convexity constraint, which geometrically means that they are polygonal. In these previous papers, the leading term in the shape functional was usually the opposite of the perimeter, for which the aforementioned concavity property was rather easy to obtain through computations of its second order shape derivative. By carrying classical shape calculus, a similar concavity property can be observed for the opposite of E f ⁢ (Ω) or λ 1 ⁢ (Ω) when shapes are smooth and convex. The main novelty in the present paper is the proof of a weak convexity property of E f ⁢ (Ω) and λ 1 ⁢ (Ω) among planar convex shapes, namely rather nonsmooth shapes. This involves new computations and estimates of the second order shape derivatives of E f ⁢ (Ω) and λ 1 ⁢ (Ω) interesting for themselves. [ABSTRACT FROM AUTHOR]

Approximating Bézier curve by polygon is a traditional technique in computer-aided geometric design, and there are several methods for constructing such polygons. In this paper, we propose to approximate Bézier curve by the least square polygon (LSP). LSP was derived by least square fitting which minimizes the integral of approximating errors. We provide a closed formula for the vertices of LSP. Because of its optimization method, LSP has lower approximation errors than existing approximating polygons. This observation is verified by many numerical examples we tested and several typical examples are included in this paper. [ABSTRACT FROM AUTHOR]

Let X1, X2, . . ., Xn be a sequence of coherent random variables, i.e., satisfying the equalities Xj = P(A|Gj), j = 1, 2, . . ., n, almost surely for some event A. This paper contains the proof of the estimate P (max 1=i

Research on topological interlocking (TI) assemblies indicates that the geometry of blocks plays a significant role in the performance of a configuration. The current TI generation methods can return assemblies of uniform antiprisms, tetrahedra, cubes, and octahedra. However, other shapes (both convex and concave) are well qualified for use in TI assemblies. This paper presents a framework to generate blocks for TI assembly. Starting from a seed polygon, evolution steps translate and reshape the polygon, contracting it eventually to a point, a line segment, or another polygon. Our framework generalizes and unifies previous-generation methods based on tilting angles and height parameters. We show how the proposed method systematically generates novel TI solids and previously reported others. [ABSTRACT FROM AUTHOR]

To find the area of an irregular shape, we break the shape into common shapes. Then we find the area of each shape and add them, but this approach does not always work. In this paper we investigate Gauss’s area formula (for irregular shapes), also known as the shoelace formula or the shoelace algorithm. This theorem is outside the scope of school program. Nevertheless, we provide several applications that emphasize the importance and usefulness of this theorem. Most of the applications provided in this paper are created by the authors. [ABSTRACT FROM AUTHOR]

This paper introduces Dyck free groups that are geometric realizations of polygon cycles, the ratio of either two or three collinear points in the Desargues affine plane. A biproduct of this work is a concise presentation of Desargues affine planar cycles as a free group. In this paper, we include (1) the study of properties for the ratio of two and three points in a line in the Desargues affine plane. Also, we discuss the cases related to the "line-skew field" characteristic, when it is two and when it is different from two. (2) We have constructed the maps for ratio points-set for two and three points and have proved that these maps are bijections of the lines. (3) We observe that the set of ratio points (for two and for three points) with addition and multiplication of points forms a skew field and these skew fields are sub-skew fields of the "line-skew field" in a Desargues affine plane. (4) We also observe that every Dyck polygon cycle in the Desargues affine plane is the realization of a Dyck path cycle. (5) We prove that every Dyck polygon containing collinear ratio vertices has a free group presentation. [ABSTRACT FROM AUTHOR]

In this paper, the authors present selected information on accelerated reliability tests of special off-road vehicles and their parts. The vehicles were tested in standard operation and on a polygon using accelerated durability tests of the bearings of the vehicles. The paper presents the theoretical basis of accelerated testing of the reliability properties of the parts and the possibilities of calculating the acceleration coefficient. The authors assumed the linearity of the acceleration function, which is proved in practice in most cases. The authors also present the possibility of using a special polygon for accelerated tests. [ABSTRACT FROM AUTHOR]

In this paper, we define the momentum amplituhedron in the four-dimensional split-signature space of dual momenta. It encodes scattering amplitudes at tree level and loop integrands for N = 4 super Yang-Mills in the planar sector. In this description, every point in the tree-level geometry is specified by a null polygon. Using the null structure of this kinematic space, we find a geometry whose canonical differential form produces loop-amplitude integrands. Remarkably, at one loop it is a curvy version of a simple polytope, whose vertices are specified by maximal cuts of the amplitude. This construction allows us to find novel formulae for the one-loop integrands for amplitudes with any multiplicity and helicity. The formulae obtained in this way agree with the ones derived via prescriptive unitarity. It makes prescriptive unitarity naturally emerge from this geometric description. [ABSTRACT FROM AUTHOR]

To generate an orthophoto mosaic from a collection of aerial images, the original images are first orthorectified individually using a Digital Surface Model (DSM). Then, they are stitched together along some determined seamlines to form the orthophoto mosaic. Determining appropriate seamlines is a critical process, as it affects the visual and geometric quality of the results. The stitching process can usually be done in frame-to-frame or multi-frame modes. Although the latter is more efficient, both still involve a lot of pre-processing, such as creating individual orthophotos, image registration, and overlap extraction. This paper presents a novel coarse-to-fine approach that directly determines the seamline network without such pre-processing. Our method has been specifically applied for UAV photogrammetry projects where, due to the large number of images and the corresponding overlaps, the orthophoto mosaic generation can be very challenging and time-consuming. We established the seamlines simultaneously for all the images through a two-step process. First, a DSM was generated, and a low-resolution grid was overlayed. Then, for each grid point, an optimal image was selected. Then, the grid cells are grouped into polygons based on their corresponding optimal image. Boundaries of these polygons established our seamline network. Thereafter, to generate the orthophoto mosaic, we overlayed a higher/full resolution grid on the top of the DSM, the optimal image of each point of which was quickly identified via our low-resolution polygons. In this approach, not only seamlines were automatically generated, but also were the need for the creation, registration, and overlap extraction of individual orthophotos. Our method was systematically compared with a conventional frame-to-frame (CF) technique from different aspects, including the number of double-mapped areas, discontinuities across the seamlines network, and the amount of processing time. The outcomes revealed a 46% decrease in orthophoto generation time and a notable reduction in the number of double-mapped areas, sawtooth effects, and object discontinuities within the constructed orthophoto mosaic. [ABSTRACT FROM AUTHOR]

Currently, the Ministry of Land, Infrastructure, Transport, and Tourism (Japan) is in the process of developing an open 3D city model known as PLATEAU. Abundant measurement data related to buildings, including maps produced by private companies and mobile mapping system point clouds, have been collected to enhance the value of the 3D city model. To achieve this, it is necessary to identify the buildings for which measurement data is available. In this study, we propose and evaluate an efficient matching method for various building measurement data, primarily using geometric properties. In Numazu city, PLATEAU IDs were assigned to 88,525 Zenrin buildings as part of a private map. The results indicate that 90.6% of the polygons were matched. For aerial images, 93.6% of the extracted buildings matched the PLATEAU buildings, although only 70.9% of the PLATEAU data was extracted from the images. Using the level of detail 1 and 2 models, 46 textured building files were created from the mobile mapping system point cloud. In addition, the cover ratio for the laser profiling point cloud was mostly greater than 40%, which was higher than that of the mobile mapping system. [ABSTRACT FROM AUTHOR]

A three-dimensional representation of the random distribution of fractures in rock masses, known as the discrete fracture network (DFN), is widely used to analyze the stability of jointed rock slopes. In this paper, a new framework for constructing three-dimensional DFN models in rock masses has been proposed to overcome the limitations of conventional circular or polygon-based models. The framework utilizes the universal elliptical disc (UED) model and integrates it with the discrete element method in 3DEC for the stability evaluation of jointed rock slopes. This paper starts by introducing the basic principles of the UED model. The procedures for constructing a three-dimensional DFN using the UED model is then outlined. In the present study, a case study of a rock slope in Zhejiang Province, China is used to demonstrate the implementation of the proposed framework. A comprehensive comparative study is conducted to investigate the impacts of several UED model parameters, including the ratio of major axis to minor axis and rotation angle, and discontinuity density, on the stability of rock slopes and compared to the conventional Baecher disc model. The results show that the framework can effectively integrate the UED model into 3DEC and provide a realistic representation of the three-dimensional DFN, leading to improved accuracy and efficiency in the stability evaluation of jointed rock slopes. The framework also shed light on the interactive effects of the UED model parameters and discontinuity density on the rock slope stability, providing a strong reference for using the UED model in constructing DFN models for rock slopes. Highlights: Three-dimensional discrete fracture networks (DFNs) of rock masses are successfully constructed using the universal elliptical disc (UED) model. Python subroutines are developed in 3DEC to construct 3D DFNs using the UED model for improved discrete element analyses of jointed rock slopes. The effects of several characteristic parameters of the UED model on the stability of jointed rock slopes are investigated. Based on a key block model, the mechanisms underlying the relationship between jointed rock slope stability and fracture discontinuity are rigorously explained. The advantages of the UED model over the conventional Baecher disc model are comprehensively discussed based on a rock slope case study. [ABSTRACT FROM AUTHOR]

By a fractional quadratic transformation, an indirect-PH curve can have rational offsets. In this paper, I study properties of planar sextic indirect-PH curves, in terms of their B'ezier control polygon legs. With our results, sextic B'ezier curves can be effciently tested whether they are indirect- PH curves. The main strategy to achieve our results is using complex representation of planar parametric curve. Sextic indirect-PH curves can be classified into three classes according to different factorizations of their hodographs. Necessary and suffcient conditions for all classes of sextic indirect- PH curves can be described by non-linear complex systems. By analyzing these non-linear systems, algebraic conditions for a sextic B'ezier curve to be an indirect-PH curve are first discussed, then geometric characteristics in terms of legs of its control polygon are revealed. [ABSTRACT FROM AUTHOR]

Floyd showed that if a sequence of compact hyperbolic Coxeter polygons converges, then so does the sequence of the growth rates of the Coxeter groups associated with the polygons. For the case of the hyperbolic 3-space, Kolpakov discovered the same phenomena for specific convergent sequences of hyperbolic Coxeter polyhedra. In this paper, we show that the growth rate is a continuous function on the space of Coxeter systems. This is an extension of the results due to Floyd and Kolpakov since the convergent sequences of Coxeter polyhedra give rise to that of Coxeter systems in the space of marked groups. [ABSTRACT FROM AUTHOR]

In this paper we provide a review of the concept of center of an n-gon, generalizing the original idea given by C. Kimberling for triangles. We also generalize the concept of central line for n-gons for n ≥ 3 and establish its basic properties. [ABSTRACT FROM AUTHOR]

This paper explores the position of the vertex centroid for a generalisation of Van Aubel’s theorem: specifically we look at what happens to the vertex centroid when directly similar quadrilaterals are placed on the sides of an arbitrary quadrilateral. After giving a simple proof that the position of the vertex centroid remains unchanged, the result is further generalised to directly similar triangles (or other directly similar shapes) on the sides of polygons using vectors. Not only are the results mathematically interesting, but can also provide an appropriate classroom opportunity for dynamic geometry exploration, and to build 2D models with clay and drinking straws (or thin wire) to illustrate and check the theoretical solutions. [ABSTRACT FROM AUTHOR]

The moduli space of planar polygons with generic side lengths is a smooth, closed manifold. It is known that these manifolds contain the moduli space of distinct points on the real projective line as an open dense subset. Kapranov showed that the real points of the Deligne–Mumford–Knudson compactification can be obtained from the projective Coxeter complex of type 퐴 (equivalently, the projective braid arrangement) by iteratively blowing up along the minimal building set. In this paper, we show that these planar polygon spaces can also be obtained from the projective Coxeter complex of type 퐴 by performing an iterative cellular surgery along a subcollection of the minimal building set. Interestingly, this subcollection is determined by the combinatorial data associated with the length vector called the genetic code. [ABSTRACT FROM AUTHOR]

In the separability problem of bichromatic point sets. given two sets of points colored as blue and red, we want to put a special geometric shape in a manner that includes the blue points while avoiding the red ones. This problem has various applications in data mining and other fields. Separability by various shapes, including L-shaped polygons. has been studied in the literature. In this paper. the separability of bichromatic point sets by C-shaped polygons, which are more general than L-shaped polygons, is studied and an 0(n log n)-time algorithm is presented. where n is the total number of points. [ABSTRACT FROM AUTHOR]

In this paper I will establish a special case of a conjecture that intertwines the deep diagonal pentagram maps and Poncelet polygons. The special case is that of the |$3$| -diagonal map acting on affine equivalence classes of centrally symmetric octagons. The proof involves establishing that the map is Arnold-Liouville integrable in this case, and then exploring the Lagrangian surface foliation in detail. [ABSTRACT FROM AUTHOR]

Spatial similarity provides an important basis for geographic information processing and is widely applied in multi-source data fusion and update, data retrieval and query, and cartographic generalization. To address the shape description and similarity measurement of polygon entities, this study presents a new hierarchical shape description approach and examines its application in similarity measurement of polygon entities. Using the rotation and segmentation methods, we first constructed a hierarchical shape description model for target polygon entities, followed by measurement of global and hierarchical shape description of polygon entities, respectively, using the farthest-point-distance and geometric feature description methods. Finally, we constructed a comprehensive similarity measurement model through a weighted integration of position, size, direction, and shape. The hierarchical shape description approach proposed in this paper can be applied to the shape similarity measurement of polygon elements, similarity measurement after spatial object simplification, and multi-scale polygon entity matching. The experimental results showed that the hierarchical shape description approach and similarity measurement model are able to effectively measure spatial similarity between different polygon entities, and have obtained good application results. [ABSTRACT FROM AUTHOR]

Based on Gurtin–Murdoch surface/interface model and complex potential theory, by constructing a new conformal mapping function, the fracture behavior on n nano-cracks emanating from a n-polygon nano-hole under far-field anti-plane mechanical load, inplane electrical load and magnetic load is studied. The analytical solutions of stress intensity factor, electric displacement intensity factor and magnetic induction intensity factor at the crack tip are given under the boundary conditions of magnetoelectrically semi-permeable. In addition, numerical examples show that when considering the surface effect, the stress field intensity factor, electric displacement intensity factor and magnetic induction intensity factor have obvious size-dependence. The results of this paper show that when the defect size increases to a certain extent, the influence of the surface effect begins to decrease, and finally tends to the results of classical elastic theory. [ABSTRACT FROM AUTHOR]

Wen [Some problems in fractal geometry, in Report in Chinese Conference on Fractal Geometry and Dynamical Systems, 2023] posed the following questions: when a quadrangle is a self-similar set with the open set condition (OSC). It is well known that any convex quadrangle is a self-similar set. For the concave quadrangles, this case is more complicated. Loosely speaking, some concave polygons may be not self-similar. In this paper, we prove two results. First, we give a necessary and sufficient condition such that a concave quadrangle is a self-similar set. Moreover, we also give a necessary condition for a concave quadrangle to be a self-similar set with the OSC. For the convex quadrangle, we investigate the trapezoid. We show under some condition that the trapezoid is a self-similar set with the OSC. In particular, for any trapezoid, if the ratio of the lengths of two bases is rational, then the trapezoid is a self-similar set with the OSC. [ABSTRACT FROM AUTHOR]

Thermodynamics as a fundamental branch of physics examines the relationships between heat, work, and energy. The maximum energy extraction can be characterized by using the passive states that have no extracted energy under any cyclic unitary process. In this paper, the focus is on the concept of marginal passive state energy and derive polygon inequalities for multi‐qubit entangled pure states. It is shown that the marginal passive state energies collectively form a convex polytope for each class of quantum states that are equivalent under SLOCC. Multipartite passive state energy criteria is introduced to classify multipartite entanglement under SLOCC. The present result provides a thermodynamic method to witness multipartite entanglement. [ABSTRACT FROM AUTHOR]

According to Ptolemy's theorem, the product of the lengths of the diagonals of a quadrilateral inscribed in a circle on the Euclidean plane equals the sum of the products of the lengths of opposite sides. This theorem has various generalizations. In one of the generalizations on the plane, a quadrilateral is replaced with an inscribed hexagon. In this event the lengths of the sides and long diagonals of an inscribed hexagon is called Ptolemy's theorem for a hexagon or Fuhrmann's theorem. Casey's theorem is another generalization of Ptolemy's theorem. Four circles tangent to this circle appear instead of four points lying on some fixed circle whilst the lengths of the sides and diagonals are replaced by the lengths of the segments tangent to the circles. If the curvature of the Lobachevsky plane is , then in the analogs of the theorems of Ptolemy, Fuhrmann and Casey for the polygons inscribed in a circle or circles tangent to one circle, the lengths of the corresponding segments, divided by 2, will be under the signs of hyperbolic sines. In this paper, we prove some theorems that generalize Casey's theorem and Fuhrmann's theorem on the Lobachevsky plane. The theorems involve six circles tangent to some line of constant curvature. We prove the assertions that generalize these theorems for the lengths of tangent segments. If, in addition to the lengths of the segments of the geodesic tangents, we consider the lengths of the arcs of the tangent horocycles, then there is a correspondence between the Euclidean and hyperbolic relations, which can be most clearly demonstrated if we take a set of horocycles tangent to one line of constant curvature on the Lobachevsky plane. In this case, if the length of the segment of the geodesic tangent to the horocycles is , then the length of the "horocyclic" tangent to them is equal to . Hence, if the geodesic tangents are connected by a "hyperbolic" relation, then the "horocyclic" tangents will be connected by the corresponding "Euclidean" relation. [ABSTRACT FROM AUTHOR]

Technological Pedagogical Content Knowledge (TPACK) is defined as the teacher knowledge needed for effective technology integration. This study aimed to investigate preservice elementary mathematics teachers' TPACK development. TPACK survey was administered to 33 preservice teachers (PSTs), and six of them were selected via maximum variation sampling to represent as different cases as possible based on their mathematical knowledge and technological self-assessment. These six PSTs implemented four technology-based lessons (two microteaching sessions in the mathematics teaching method course and two lessons in the student teaching). Participants' TPACK levels were examined based on the development model proposed by Niess, M. L., Sadri, P., & Lee, K. (2007, April). Dynamic spreadsheets as learning technology tools: Developing teachers' technology pedagogical content knowledge (TPCK). Paper presented at the meeting of the American Educational Research Association Annual Conference, Chicago, IL. Results showed that participants had not used technology effectively and efficiently in their microteaching sessions and the first lessons in their student teaching. After evaluating their first lessons in schools, PSTs improved their teaching considerably. We suggest giving more opportunities for PSTs to teach with technology in classrooms and to assess their teaching practices reflectively. [ABSTRACT FROM AUTHOR]

In this paper, we introduce an m-fold illumination number I m (K) of a convex body K in Euclidean space E d , which is the smallest number of directions required to m-fold illuminate K, i.e., each point on the boundary of K is illuminated by at least m directions. We get a lower bound of I m (K) for any d-dimensional convex body K, and get an upper bound of I m (K) for any d-dimensional convex body K with smooth boundary. We also prove that I m (K) = 2 m + 1 , for a 2-dimensional smooth convex body K. Furthermore, we obtain some results related to the m-fold illumination numbers of convex polygons and cap bodies of a d-dimensional unit ball B d in small dimensions. In particular, we show that , for a regular convex n-sided polygon P. [ABSTRACT FROM AUTHOR]

The paper is devoted to some extremal problems, related to convex polygons in the Euclidean plane and their perimeters. We present a number of results that have simple formulations, but rather intricate proofs. Related and still unsolved problems are also discussed. [ABSTRACT FROM AUTHOR]

A 2006 conjecture of Antunes and Freitas is addressed connecting the scaling-invariant polygonal isoperimetric and principal frequency deficits for triangles. This yields a quantitative polygonal Faber–Krahn inequality for triangles with an explicit constant. Furthermore, a problem mentioned in the 1951 book "Isoperimetric Inequalities In Mathematical Physics" by Pólya and Szegö is addressed: a formula is given for the principal frequency of a triangle. Moreover, a space of polygons is constructed for the classical Pólya and Szegö problem: in 1947, Pólya proved that if n = 3, 4 the regular polygon Pn minimizes the principal frequency of an n-gon with given area α > 0 and suggested that the same holds when n ≥ 5. In 1951, Pólya and Szegö discussed the possibility of counterexamples. This paper constructs explicit (2n − 4)–dimensional polygonal manifolds M (n , α) and proves the existence of a computable N ≥ 5 such that for all n ≥ N, the admissible n-gons are given via M (n , α) and there exists an explicit set A n (α) ⊂ M (n , α) such that Pn has the smallest principal frequency among n-gons in A n (α). Inter-alia when n ≥ 3, a formula is proved for the principal frequency of a convex P ∈ M (n , α) in terms of an equilateral n-gon with the same area; and, the set of equilateral polygons is proved to be an (n − 3)–dimensional submanifold of the (2n − 4)–dimensional manifold M (n , α) near Pn. The techniques involve a partial symmetrization, tensor calculus, the spectral theory of circulant matrices, and W2,p/BMO estimates. Last, an application is given in the context of electron bubbles. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to investigate the multivariate frequency polygon as a density estimator for stationary random fields indexed by multidimensional lattice points space. Optimal cell widths that asymptotically minimize integrated mean square error (IMSE) are derived. Under weak conditions, the IMSE of frequency polygons achieves the same rate of convergence to zero as that of kernel estimators. The frequency polygon can also attain the optimal uniform rate of convergence and the almost sure convergence under general conditions. Finally, a result of L 1 convergence is given. Frequency polygons thus appear to be very good density estimators with respect to the criteria of IMSE, of uniform convergence, of almost sure convergence and of L 1 convergence. We apply our results to simulated data and real data. [ABSTRACT FROM AUTHOR]

In order to study the wheel polygon wear problems in high-speed trains, an iterative model of wheel wear was established based on $ 1) $ 1) the train-track rigid-flexible coupling dynamics model, $ 2) $ 2) Kik-Piotrowiski + Fastsim wheel-rail contact model, and $ 3) $ 3) Archard wear model to predict the developing process of wheel polygon. Taking a certain type of high-speed train in China as an example, the developing process of the 20-order wheel polygon is simulated. Compared with the Hertz model, the results of the Kik-Piotrowiski model are more consistent with the actual situation, and the prediction of the wheel polygon is more accurate. The results show that with the increase of vehicle running speed, the dominant order of the final wheel polygon decreases gradually, while the excitation frequency always locates between 550 and 600 Hz, which verifies the 'frequency fixed' mechanism. By comparing the development of wheel polygon wear results under three working conditions $ 1) $ 1) flexible wheelset + flexible rail, $ 2) $ 2) rigid wheelset + flexible rail, and $ 3) $ 3) flexible wheelset + rigid rail), and by analysing the harmonic response of rail, it is concluded that the third-order bending modal vibration of rail plays an important role in the forming of wheel polygon.. [ABSTRACT FROM AUTHOR]

Automated extraction of polygonal building contours from high-resolution remote sensing images is important for various applications. However, it remains a difficult task to achieve automated extraction of polygonal buildings at the level of human delineation due to diverse building structures and imperfect image conditions. In this paper, we propose Line2Poly, an end-to-end approach that uses feature lines as geometric primitives to achieve polygonal building extraction by recovering topological relationships among these lines within an individual building. To extract building feature lines with precision, we adopt a two-stage strategy that combines Convolutional Neural Network (CNN) and transformer architectures. A CNN-based module extracts preliminary feature lines, which serve as positional priors for initializing positional queries in the subsequent transformer-based module. For polygonal building contour reconstruction, we devise a learnable polygon topology reconstruction module that predicts adjacency relationships among discrete lines, and integrates lines into building polygons. The resultant building polygons, based on feature lines, exhibit inherent regularity that aligns with manual labeling standards. Extensive experiments on the Vectorizing World Buildings dataset, the WHU aerial building dataset and the WHU-Mix (vector) dataset validate Line2Poly's impressive performance in building feature line extraction and instance-level building detection. Moreover, Line2Poly's predictions exhibit the highest level of concurrence with manual delineations, with over 83% agreement on the WHU aerial building test set and 68.7/59.7% on the WHU-Mix (vector) test set I and II, respectively. [ABSTRACT FROM AUTHOR]

The ArtGallery problem is a fundamental visibility problem in Computational Geometry. The input consists of a simple polygon P, (possibly infinite) sets G and C of points within P, and an integer k; the task is to decide if at most k guards can be placed on points in G so that every point in C is visible to at least one guard. In the classic formulation of ArtGallery, G and C consist of all the points within P. Other well-known variants restrict G and C to consist either of all the points on the boundary of P or of all the vertices of P. Recently, three new important discoveries were made: the above mentioned variants of ArtGallery are all W[1]-hard with respect to k [Bonnet and Miltzow in 24th Annual European Symposium on Algorithms (Aarhus 2016)], the classic variant has an O (log k) -approximation algorithm [Bonnet and Miltzow in 33rd International Symposium on Computational Geometry (Brisbane 2017)], and it may require irrational guards [Abrahamsen et al. in 33rd International Symposium on Computational Geometry (Brisbane 2017)]. Building upon the third result, the classic variant and the case where G consists only of all the points on the boundary of P were both shown to be ∃ R -complete [Abrahamsen et al. in 50th Annual ACM SIGACT Symposium on Theory of Computing (Los Angeles 2018)]. Even when both G and C consist only of all the points on the boundary of P, the problem is not known to be in NP. Given the first discovery, the following question was posed by Giannopoulos [Lorentz Workshop on Fixed-Parameter Computational Geometry (Leiden 2016)]: Is ArtGallery FPT with respect to r, the number of reflex vertices? In light of the developments above, we focus on the variant where G and C consist of all the vertices of P, called Vertex-Vertex Art Gallery. Apart from being a variant of ArtGallery, this case can also be viewed as the classic DominatingSet problem in the visibility graph of a polygon. In this article, we show that the answer to the question by Giannopoulos is positive: Vertex-VertexArtGallery is solvable in time r O (r 2) · n O (1) . Furthermore, our approach extends to assert that Vertex-BoundaryArtGallery and Boundary-VertexArtGallery are both FPT as well. To this end, we utilize structural properties of "almost convex polygons" to present a two-stage reduction from Vertex-VertexArtGallery to a new constraint satisfaction problem (whose solution is also provided in this paper) where constraints have arity 2 and involve monotone functions. [ABSTRACT FROM AUTHOR]

Nowadays, deep learning allows to automatically learn features from data. Buildings are one of the most important objects in urban environments. They are used in applications such as inputs to building reconstruction, disaster monitoring, city planing and environment modelling for autonomous driving. However, it is not enough to represent them in raster format, since applications require buildings as polygons. We use an existing, learning based approach to extract building footprints from ortho imagery and digital surface model (DSM) and propose a pipeline for building polygon extraction, which we call primary orientation learning (POL). The first step is to extract initial polygons, that contain a vertex for each pixel in the boundary of the footprint. Afterwards, the two primary orientation angles are regressed continuously. Using these orientation, we insert vertices such that all consecutive edges are perpendicular. To the best of our knowledge, our approach is the first to predict a continuous orientation angle for building boundary regularization. Furthermore, the proposed method is highly efficient with an average processing time of 2.879 ms for a single building. [ABSTRACT FROM AUTHOR]

Stationary Poisson processes of lines in the plane are studied, whose directional distributions are concentrated on $k\geq 3$ equally spread directions. The random lines of such processes decompose the plane into a collection of random polygons, which form a so-called Poisson line tessellation. The focus of this paper is to determine the proportion of triangles in such tessellations, or equivalently, the probability that the typical cell is a triangle. As a by-product, a new deviation of Miles's classical result for the isotropic case is obtained by an approximation argument. [ABSTRACT FROM AUTHOR]

The idea of using the generalised inverse of a singular matrix A to solve the matrix equation A x = b has been discussed in the earlier papers [1, 2, 3, 4] in the Gazette. Here we discuss three simple geometric questions which are of interest in their own right, and which illustrate the use of the generalised inverse of a matrix. The three questions are about polygons and circles in the Euclidean plane. We need not assume that a polygon is a simple closed curve, nor that it is convex: indeed, abstractly, a polygon is just a finite sequence (v 1, ..., v n ) of its distinct, consecutive, vertices. It is convenient to let v n + 1 = v 1 and (later) C n + 1 = C 1. [ABSTRACT FROM AUTHOR]

Point-in-spherical-polygon tests are a fundamental problem in computational geometry. For such tests, efficiency is much required for many global information processing matters, especially their real-time processing. Though many efforts have been made, it is still challenging due to the constraints from the non-Euclidean space of the sphere. Recently, a method is proposed to construct an Adaptive Hexagonal Hierarchical Grid (AHHG) to manage spherical polygon edges, by which the non-Euclidean space constraint of the sphere can be well handled to improve point-in-spherical-polygon tests. However, this method is very expensive in the construction of AHHGs, which prevents its use in practice. In this paper, we present novel measures to divide the task of AHHG construction into several subtasks to solve the problems that arise in the parallel construction of incongruent adaptive hierarchies, so we can exploit the parallel computing potential of GPUs for acceleration. We also develop measures to adaptively optimize the hierarchical levels of an AHHG for high efficiency. Experimental results show that we can answer 1,000,000 query points against dynamically varying spherical polygons in over 100,000 edges in real time on a personnel computer, where AHHGs for spherical polygons are individually constructed. This is much superior to the existing methods. [ABSTRACT FROM AUTHOR]

In this paper, we study the Newton polygon of the L -function of a generalized Kloosterman polynomial with two variables over finite fields. We give the explicit form of the monomial basis of the top dimensional cohomology space of the p -adic complex associated to the L -function. One consequence of this result is a concrete method for computing the Hodge polygon. Using decomposition theorems of Wan, we determine when a generalized Kloosterman polynomial is ordinary and when its Newton polytope is generically ordinary. [ABSTRACT FROM AUTHOR]

In this article, we continue the study of a certain family of 2-Calabi–Yau tilted algebras, called dimer tree algebras. The terminology comes from the fact that these algebras can also be realized as quotients of dimer algebras on a disk. They are defined by a quiver with potential whose dual graph is a tree, and they are generally of wild representation type. Given such an algebra |$B$|⁠ , we construct a polygon |$\mathcal {S}$| with a checkerboard pattern in its interior, which defines a category |$\text {Diag}(\mathcal {S})$|⁠. The indecomposable objects of |$\text {Diag}(\mathcal {S})$| are the 2-diagonals in |$\mathcal {S}$|⁠ , and its morphisms are certain pivoting moves between the 2-diagonals. We prove that the category |$\text {Diag}(\mathcal {S})$| is equivalent to the stable syzygy category of the algebra |$B$|⁠. This result was conjectured by the authors in an earlier paper, where it was proved in the special case where every chordless cycle is of length three. As a consequence, we conclude that the number of indecomposable syzygies is finite, and moreover the syzygy category is equivalent to the 2-cluster category of type |$\mathbb {A}$|⁠. In addition, we obtain an explicit description of the projective resolutions, which are periodic. Finally, the number of vertices of the polygon |$\mathcal {S}$| is a derived invariant and a singular invariant for dimer tree algebras, which can be easily computed form the quiver. [ABSTRACT FROM AUTHOR]

In this paper we study uniformly random lozenge tilings of strip domains. Under the assumption that the limiting arctic boundary has at most one cusp, we prove a nearly optimal concentration estimate for the tiling height functions and arctic boundaries on such domains: with overwhelming probability the tiling height function is within n δ of its limit shape, and the tiling arctic boundary is within n 1 / 3 + δ to its limit shape, for arbitrarily small δ > 0 . This concentration result will be used in Aggarwal and Huang (Edge statistics for lozenge tilings of polygons, II: Airy line ensemble, 2021. arXiv:2108.12874) to prove that the edge statistics of simply-connected polygonal domains, subject to a technical assumption on their limit shape, converge to the Airy line ensemble. [ABSTRACT FROM AUTHOR]

The Land Use/Cover Area frame Survey (LUCAS) of the European Union (EU) presents a rich resource for detailed understanding of land cover and use, making it invaluable for Earth Observation (EO) applications. This manuscript discusses the recent advancements and improvements in the LUCAS Copernicus module, particularly the data collection process of 2022, its protocol simplifications, and geometry definitions compared to the 2018 survey and data. With approximately 150,000 polygons collected in 2022, an increase from 60,000 in 2018, the LUCAS Copernicus 2022 data provides a unique and comprehensive in-situ dataset for EO applications. The protocol simplification also facilitates a faster and more efficient data collection process. In 2022, there are 137,966 polygons generated, out of the original 149,408 LUCAS Copernicus points, which means 92.3% of the points were actually surveyed. The data holds 82 land cover classes for the Copernicus module LUCAS level 3 legend (88 classes). For land use the data holds 40 classes, along with 18 classes of land use types. The dataset is available here for download (PID: http://data.europa.eu/89h/e3fe3cd0-44db-470e-8769-172a8b9e8874). The paper further elaborates on the implications of these enhancements and the need for continuous harmonisation to ensure semantic consistency and temporal usability of data across different periods. Moreover, it calls for additional studies exploring the potential of the collected data, especially in the context of remote sensing and computer vision. The manuscript ends with a discussion on future data usage and dissemination strategies. [ABSTRACT FROM AUTHOR]

Photocatalytic hydrogen evolution using suitable photocatalysts in water is a sustainable way of energy production. In this paper, Ca2+ doped TiO 2 nano-sized polygon plates with oxygen vacancies were synthesized. The incorporation of Ca2+ induces the loss of oxygen and the formation of oxygen vacancies in the TiO 2 lattice, confirmed by XPS and EPR. TiO 2 with 0.040% Ca adding amount shows the best photocatalytic hydrogen production activity (4485 μmol g−1 h−1) with the apparent quantum efficiency of 46.4% at 370 nm, higher than undoped TiO 2 (2182 μmol g−1 h−1). The successful incorporation of Ca2+ and the formation of oxygen vacancies effectively inhibit the recombination of carriers in the reaction process and enhance the photocatalytic hydrogen production activity. • Ca2+ doped TiO 2 nano-sized polygon plates with oxygen vacancies were synthesized. • The incorporation of Ca2+ induces the formation of oxygen vacancies. • TiO 2 with 0.040% Ca shows the best photocatalytic H 2 production activity. • The incorporation of Ca2+ inhibits the recombination of electron-hole pairs. [ABSTRACT FROM AUTHOR]