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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]

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]

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]

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]

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]

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 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 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]

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]

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]

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]

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]

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]

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]

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]

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]

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

The formula for the area of a rhumb polygon, a polygon whose edges are rhumb lines on an ellipsoid of revolution, is derived and a method is given for computing the area accurately. This paper also points out that standard methods for computing rhumb lines give inaccurate results for nearly east- or west-going lines; this problem is remedied by the systematic use of divided differences. [ABSTRACT FROM AUTHOR]

The Virtual Element Method (VEM) is a promising approach developed in recent years. Within the energy-based framework of the VEM, we can directly compute the consistency and stabilization terms by automatically differentiating the separation of the energy functional. Surface integrals are involved in computing the consistency matrix, and we find that the results depend on the element-partitioning schemes for these integrals. Partitioning scheme that introduces facial centroids works well, but it can only be applied to star-convex polygons. Other partitioning schemes without introducing centroids cause significant errors since the integration weights are not computed from all nodes of the surface. In this paper, we provide a detailed derivation to illustrate these discrepancies. To overcome this challenge, we implement an integration scheme without element-partitioning, which converting the surface integrals into line integrals using a recursive procedure. We employ four examples to demonstrate the effectiveness of the proposed method for both convex and concave polyhedra. [ABSTRACT FROM AUTHOR]

Mechanical perturbation of thin layers of pastes on a substrate results in anisotropic cracking during desiccation which allows for a controlled generation of crack patterns with a high relevance for industrial applications. In the present paper we propose a modeling framework to capture the anisotropy induced by initial perturbations which imprint directional dependent material features into the layer. Our approach is based on a random tessellation of the layer using convex polygons which are made anisotropic by elongating or compressing them along the direction of perturbation. Physical properties of the particles' cohesive contacts are determined by the geometrical characteristics of polygons giving rise to a well controlled anisotropy in the mechanical response of the material and in the structure of the fracture pattern emerging during shrinking. The main advantage of the approach is that both the structural disorder of the layer and the strength of anisotropy are controlled by a single parameter. Based on computer simulations we demonstrate that varying its two parameters the model reproduces the qualitative features of experimental findings and provides advantages over existing modeling approaches. • A discrete element model is proposed for anisotropic cracking of shrinking layers. • Anisotropy is introduced by a geometrical transformation of an isotropic lattice. • Disorder has a single source which affects also the mechanical features of the layer. • The model provides a reasonable agreement with experimental findings. [ABSTRACT FROM AUTHOR]

In this paper, an enhanced prismatic-element scaled boundary perfectly matched layer (SBPML) is developed, which is a novel time-domain artificial boundary method for 3D infinite wave problems. The SBPML permits the utilization of an artificial boundary with general geometry and can consider planar physical surfaces and interfaces extending to infinity. Moreover, this enhancement enables the seamless integration with surface meshes of arbitrary polygonal faces (such as triangles, quadrangles, pentagons, heptagons, octagons, etc.) at the truncation boundaries. The interior domain can be modelled by polyhedrons formulated by the scaled boundary finite element method (SBFEM). The geometric shape of the SBPML elements in the proposed method, within the local coordinate system, takes the form of a regular prism with regular polygonal top and bottom faces. Consequently, it is referred to as the Prismatic-element SBPML. Local scaled boundary coordinates based on the polygonal shape function are firstly introduced into the truncated infinite domain to describe its general geometry properties and provide the coupled capabilities with arbitrary polygonal face meshes at the truncation boundaries of numerical models. Then, a complex stretching function from PML is applied to radial direction of the local scaled boundary coordinates to map the physical space onto the complex space, resulting in a SBPML domain. Upon spatial discretization, the proposed method is formulated using a 2nd-order mixed displacement-stress unsplit-field formulation, allowing seamless coupling with FEM and SBFEM originally designed for the interior computational domain. Finally, three benchmark examples of wave propagation problems in unbounded domains and two soil-structure interaction problems of engineering problems are presented to demonstrate the accuracy and robustness of the Prismatic-element SBPML. [ABSTRACT FROM AUTHOR]

In this paper, we survey some properties, encoding, and bijections involving combinatorial maps, double occurrence words, and chord diagrams. We particularly study quasi-trees from a purely combinatorial point of view and derive a topological representation of maps with a given spanning quasi-tree using two fundamental polygons, which extends the representation of planar maps based on the equivalence with bipartite circle graphs. Then, we focus on Depth-First Search trees and their connection with a poset we define on the spanning quasi-trees of a map. We apply the bijections obtained in the first section to the problem of enumerating loopless rooted maps. Finally, we return to the planar case and discuss a decomposition of planar rooted loopless maps and its consequences on planar rooted loopless map enumeration. [ABSTRACT FROM AUTHOR]

An ellipse can be uniquely determined by five tangents. Given a convex quadrilateral, there are infinitely many ellipses inscribed in it, but the one with maximum area is unique. In this paper, we give a concise and effective solution of this problem. Our solution is composed of three steps: First, we transform the problem from the maximal ellipse construction problem into the minimal quadrilateral construction problem by an affine transformation. And then, we convert the construction problem into a conditional extremum problem by analyzing the key angles. At last, we derive the solution of the conditional extremum problem with Lagrangian multiplier. Based on the conclusion, we designed an algorithm to achieve the construction. The numerical experiment shows that the ellipse constructed by our algorithm has the maximum area. It is interesting and surprising that our constructions only need to solve quadratic equations, which means the geometric information of the ellipse can even be derived with ruler and compass constructions. The solution of this problem means all the construction problems of conic with extremum area from given pure tangents are solved, which is a necessary step to solve more problems of constructing ellipses with extremum areas. Our work also provides a useful conclusion to solve the maximal inscribed ellipse problem for an arbitrary polygon in Computational Geometry. • We find a method to transform the maximal inscribed ellipse for an arbitrary quadrilateral to a conditional extremum problem. • We derive the solution for the conditional extremum problem with elementary operations. • We design the construction algorithm for constructing the maximal ellipse inscribed in an arbitrary quadrilateral. [ABSTRACT FROM AUTHOR]

The generalization of barycentric coordinates to arbitrary simple polygons with more than three vertices has been a subject of study for a long time. Among the different constructions proposed, mean value coordinates have emerged as a popular choice, particularly due to their suitability for the non-convex setting. Since their introduction, they have found applications in numerous fields, and several equivalent formulas for their evaluation have been presented in the literature. However, so far, there has been no study regarding their numerical stability. In this paper, we aim to investigate the numerical stability of the algorithms that compute mean value coordinates. We show that all the known methods exhibit instability in some regions of the domain. To address this problem, we introduce a new formula for computing mean value coordinates, explain how to implement it, and formally prove that our new algorithm provides a stable evaluation of mean value coordinates. We validate our results through numerical experiments. • A new algorithm for evaluating mean value coordinates. • A formal proof that this algorithm is numerically stable. • A comparison of the numerical stability of all known algorithms for mean value coordinates. [ABSTRACT FROM AUTHOR]

A framework, which is a (possibly infinite) graph with a realization of its vertices in the plane, is called flexible if it can be continuously deformed while preserving the edge lengths. We focus on flexibility of frameworks in which 4-cycles form parallelograms. For the class of frameworks considered in this paper (allowing triangles), we prove that the following are equivalent: flexibility, infinitesimal flexibility, the existence of at least two classes of an equivalence relation based on 3- and 4-cycles and being a non-trivial subgraph of the Cartesian product of graphs. We study the algorithmic aspects and the rotationally symmetric version of the problem. The results are illustrated on frameworks obtained from tessellations by regular polygons. [ABSTRACT FROM AUTHOR]

In this paper, we explore the macroscopic elastic behavior of the aperiodic but hyperuniform ein stein (single tile) tiling, as recently reported in Smith et al. (2023a, 2023b). The first step involves assigning mechanical properties to the geometric pattern. The simplest approach includes near-neighbor type (NN) interactions (springs) along the edges of the geometric pattern. To eliminate zero modes, we also incorporate angular interactions using the simplest quadratic approximation—the Kirkwood–Keating three-body potential. We compute the macroscopic elastic response on circular domains for various realizations across an increasing sequence of length scales. As the domain radius increases, the set average at a fixed length scale of the obtained macroscopic Hooke tensor approaches the hyperplane of two-dimensional isotropic Hooke tensors. In a closer-to-experiment scenario, we also discuss a second mechanical setting where edges of all polygons in the considered domain are treated as one-dimensional continua (straight beams). These continua are endowed with both extension/compression, shear and bending energy within the framework of the Timoshenko theory. The macroscopic response of the hat tiling aligns with that of an isotropic elastic continuum. • We endow the edges of the recently discovered ein stein aperiodic tilling of the plane with mechanical interactions and we explore the macroscopic properties of the resulting discrete continuum at the macroscale. • Since the aperiodic tiling with a single tile is hyperuniform by construction, we conjecture that the macroscopic material behavior is that of an isotropic continuum. • In both cases, the numerical experiments performed on different realizations at various length scales indicate that, as expected, the macroscopic elastic law tends towards that of an isotropic material. [ABSTRACT FROM AUTHOR]

• An improved surface morphology method is proposed to describe the bit-rock interaction. • The new method makes the drilling model more accurate and efficient. • The new method avoids some unreasonable behavior in the drilling model. • The blades' shape and cutters' arrangement have effects on the dynamic characteristics. • The cutting error is proposed to evaluate the design of the bit and the bit-rock interaction model. In this paper, the surface morphology method is improved and the bit-rock interaction model between the rock and the PDC bit is studied, considering the influence of blades' shape and the cutters' arrangement. Based on this model, the dynamic model of the deep drilling system equipped with arbitrary shape PDC bit is established, and the stability prediction method is proposed. Based on the improved surface morphology method, the dynamic characteristics of the drilling system equipped with four different types of bit (an actual bit, a Tian bit, a Tian bit with blades, and a traditional bit) are studied, comparing with the Tian model based on polygon clipping. The results indicate that the influence of blades' shape and cutters' arrangement of PDC bit on the nonlinear vibration of the drilling system cannot be ignored. By applying the proposed method, the dynamic model of the drilling system equipped with an actual PDC bit presents excellent computational efficiency, accuracy, and avoids unreasonable behavior, which provides a reliable theory for the dynamic analysis of the drilling system. In addition, the proposed uncut volume error and repeated cutting volume error can be used to evaluate the shape design of the drill bit and the accuracy of the bit-rock interaction model. [Display omitted] [ABSTRACT FROM AUTHOR]

• The formation principles of the equal-width polygon groove-tooth wheel were thoroughly developed and analyzed. • The influence of the equal width polygon groove-tooth wheel on seed population disturbance ability, filling and feeding performance, was compared by DEM simulation. • Notable differences were observed in the average kinetic energy and the disturbance frequency of the seed population. This work aimed to investigate the influence of the equal width polygon groove-tooth wheel on both filling and feeding performance to enhance the performance of the seed feeding device of the air-assisted centralized metering device for wheat. This paper conducted an analysis of the equal width polygon groove-tooth wheel formation principles. The EDEM simulation compared and analyzed the influence of the structural types, including equal width triangular, equal width pentagon, equal width heptagon, equal width nonagon, and circular groove-tooth wheel, on seed population disturbance ability, filling, and feeding performance. The simulation results showed that when the rotating speed of the feeding shaft was 20r/min, the average kinetic energy of the seed population decreased gradually, and the ability to disturb the average seed population decreased in turn, but the interval between the adjacent average kinetic energy peaks decreased, and the frequency of disturbed seed population increased with the increase of the number of sides of the equal width polygon groove-tooth wheel. The average kinetic energy of the seed population was significantly higher under the disturbance of the equal width triangular, pentagon, heptagon, and nonagon groove-tooth wheels, increasing by 371.32 %, 209.23 %, 91.02 %, and 53.37 %, respectively, compared to that of the circular groove-tooth wheel. Within the feeding shaft rotating speed range of 10 ∼ 60r/min, the average kinetic energy of the seed population gradually increased in the equal width polygon groove-tooth wheel type feeding device gradually with the increase of the feeding shaft rotating speed. Using the quality rate of seed filling (2 ± 1 seeds/hole) and the missing rate of seed filling (0 seed/hole) as benchmarks, the equal width heptagon groove-tooth wheel exhibited the quality rate of seed filling of 93.98 % and the missing rate of seed filling of 2.78 %. The seed feeding quantity and seed feeding time had high linearity, and the seed filling and seed feeding performance were better. In the range of 10 ∼ 60r/min for the rotating speed of the equal width polygon groove-tooth wheel, there was a positive correlation observed between the average kinetic energy of the seed population and the increment of the feeding shaft rotating speed. The variation in feeding shaft rotating speed exhibited minimal influence on the feeding performance. The bench test verified the performance of the seed feeding device equipped with the equal width heptagon groove-tooth wheel in terms of both seed feeding and seeding efficiency. The results demonstrated that within the feeding shaft rotating speed range of 10 ∼ 60r/min, the coefficient of variation（CV） for seed feeding rate stability did not exceed 2.14 %. Additionally, the damage rate during the feeding process remained below 0.2 %. These outcomes aligned with the requirements of stable wheat seed feeding. [ABSTRACT FROM AUTHOR]

• Develop an electromechanical model to couple vehicle-track and traction drive systems. • Wheel polygon wear evolution analysis is performed including effects of interharmonic. • The pulsating torque with specific interharmonic excites the wheelset resonance at 70 km/h. The formation and development of wheel polygon wear (WPW) present significant challenges to the safety of railway vehicle operations. It is widely accepted that WPW occurs due to the mechanical vibration within the wheel-rail system. The output torque produced by the traction motor acts as the primary power source for the vehicle's movement. The pulsating torque, resulting from interharmonics can directly impact the wheelset's dynamic behavior, contributing to the occurrence of WPW. However, the precise mechanism by which interharmonics affect WPW remains incompletely understood. Therefore, this paper aims to establish the link between interharmonics originating from the electric traction drive system and the wheelset vibration. A comprehensive WPW evolution model that incorporates an electromechanical coupling model and a wheel wear calculation model is developed. This model allows to elucidate the intricate mechanism by which interharmonics impact WPW. The analysis indicates that when the locomotive is running at 70 km/h, the pulsating torque generated by the | 5 f − 7 f 0 | interharmonic can excite the 1st-order bending mode of the wheelset. This excitation triggers wheelset resonance, which ultimately leads to the development of 20th-order WPW. [ABSTRACT FROM AUTHOR]