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We demonstrate experimentally and numerically the control of magnetization circulation of the closure domain state in thin permalloy squares divided diagonally into four dipolar-coupled isosceles triangles. The control over the magnetization circulation was achieved by the mutual application of a suitable external in-plane magnetic field and interaction with a magnetic force microscopy (MFM) tip. Numerical simulations confirm that the proposed switching protocol is valid for the broad range of MFM tips. Experiments performed on four diagonally cut squares show that this method is robust against slight imperfections in the fabrication process. Moreover, by varying the polarity of the applied field and the direction of the MFM scanning, proper symmetry response of the switching method is observed. [ABSTRACT FROM AUTHOR]

We use P t and C t to denote a path and a cycle on t vertices, respectively. A diamond is a graph obtained from two triangles that share exactly one edge. A kite is a graph that consists of a diamond and another vertex adjacent to a vertex of degree 2 of the diamond. A gem is a graph that consists of a P 4 plus a vertex adjacent to all vertices of the P 4. Choudum et al. (2021) proved that every ( P 7 , C 7 , C 4 , gem)-free graph has either a clique cutset or a vertex whose neighbors is the union of two cliques unless it is a clique blowup of the Petersen graph, and every non-Petersen ( P 7 , C 7 , C 4 , diamond)-free graph has either a clique cutset or has a vertex of degree at most max { 2 , ω (G) − 1 }. In this paper, we extend these two results respectively to ( P 7 , C 4 , gem)-free graphs and ( P 7 , C 4 , diamond)-free graphs. We also prove that if G is a ( P 7 , C 4 , kite)-free graph with δ (G) ≤ ω (G) and without clique cutsets, then there is an integer ℓ ≥ 0 such that G = K ℓ + H , where H is either the Petersen graph or a well-defined graph F. This implies a polynomial algorithm which can determine the chromatic number of ( P 7 , C 4 , diamond)-free graphs, color ( P 7 , C 4 , kite)-free graphs G with (ω (G) + 1) colors, and color ( P 7 , C 4 , gem)-free graphs G with (2 ω (G) − 1) colors. [ABSTRACT FROM AUTHOR]

ABSTRACT With shortening development cycles, digital development rises in importance. Therefore, we present an approach to segment digital carlines based on their corrosion risk using digital data sets. A typical carline is described via STL‐Files that accumulate to approximately 25 million triangles. While physics‐based simulation models exist capable of predicting the onset of corrosion in local geometry settings, an application to a complex surface mesh would be prohibitively expensive to compute. This calls for data reduction techniques that reduce the number of triangles by identifying areas prone to corrosion, as well as those that are protected by measures such as adhesives. Therefore, the implementation of corrosion‐prone areas of a body‐in‐white part, as presented in Waibel et al., is extended by implementing a design measure segmentation for adhesive pipes. This work introduces a method to predict corrosion‐protected areas for flanges with adhesives by implementing a feature extraction and a transfer learning‐based workflow for data‐efficient training. The results of the implementation are the validated prediction on body‐in‐white parts with high precision. [ABSTRACT FROM AUTHOR]

In plane geometry, it is well-known that one can inscribe a circle in any triangle. A less trivial yet natural question could be asked about inscribing circles in topologically embedded triangles. This paper presents a proof for the following general theorem: Given any boundary of an n-simplex topologically embedded in Euclidean k-space, with n ≤ k , there exists a (k − 1) -sphere whose interior is disjoint from the embedded boundary of the simplex and which touches each (n − 1) -dimensional face on the boundary of the simplex at least once. We say this sphere is inscribed in the embedding. This paper also shows that ellipsoids, cubes, and octahedrons can be inscribed in any topologically embedded boundary of the n-simplex. The uniqueness of inscribed spheres is also discussed. [ABSTRACT FROM AUTHOR]

The article highlights the significance of pattern and feature recognition in K-8 mathematics, emphasizing how these skills enhance critical thinking and connections between mathematical concepts, while introducing Google's Teachable Machine as a supportive educational tool. Topics include the application of Teachable Machine in the classroom; and the benefits of using this technology for diverse grade levels.

Summary: This paper provides a review of proofs of Heron's formula for the area of a plane triangle. We groups proofs together into analogy/experimental evidence, geometry (incircle radius multiplied by semi-perimeter), algebra (equivalence with the Pythagorean Theorem), trigonometric proofs (often via the cosine rule), and specialization from a more general theorem. Lastly, we discuss a neglected paper by Baker of 1885 that lists 110 distinct formulae for the area of a plane triangle and relate some of these formulae to those used in proving Heron's formula. [ABSTRACT FROM AUTHOR]

Background: Surgical approaches to the cavernous sinus (CS) and middle cranial fossa (MCF) can be challenging, particularly for young neurosurgeons. The anteromedial (Mullan's) triangle is a triangle by the side of the CS and constitutes part of the floor of the MCF. The contents include the sphenoid sinus, superior ophthalmic vein, and sixth cranial nerve. The literature contains very little research that has precisely defined and measured the anteromedial triangle while considering anatomical variances minimally. Methodology: The present study was conducted on the skulls of 25 adult human cadavers which were dissected to expose the anteromedial (Mullan's) triangle on both sides. After precisely defining the triangle on each side, measurements of the three borders were taken, and using Heron's formula, the area of each triangle was calculated. Results: On average, the length of the medial border was 12.5 (+ 3.1 mm); the length of the lateral border was 9.9 (+ 3.1 mm); the length of the base was 10.75 (+ 2.4 mm) and the area of the anteromedial triangle was 43.9 (+ 15.06 mm2). Conclusion: Precise anatomical knowledge of the Mullan's triangle enables the treatment of disorders in often deformed anatomy or difficult-to-access structures. That is the reason it is important to gain a thorough understanding of the surgical anatomy and to adopt a safe procedure. [ABSTRACT FROM AUTHOR]

We give asymptotically optimal constructions in generalized Ramsey theory using results about conflict-free hypergraph matchings. For example, we present an edge-coloring of K_{n,n} with 2n/3 + o(n) colors such that each 4-cycle receives at least three colors on its edges. This answers a question of Axenovich, Füredi and the second author [J. Combin. Theory Ser B 79 (2000), pp. 66–86]. We also exhibit an edge-coloring of K_n with 5n/6+o(n) colors that assigns each copy of K_4 at least five colors. This gives an alternative very short solution to an old question of Erdős and Gyárfás [Combinatorica 17 (1997), pp. 459–467] that was recently answered by Bennett, Cushman, Dudek, and Prałat [J. Combin. Theory Ser. B 169 (2024), pp. 253–297] by analyzing a colored modification of the triangle removal process. [ABSTRACT FROM AUTHOR]

The space‐time generalized Riemann problems method allows to obtain numerical schemes of arbitrary high order that can be used with very large time steps for systems of linear hyperbolic conservation laws with source term. They have been introduced in Berthon et al. (J. Sci. Comput. 55 (2013), 268–308.) in 1D and on 2D unstructured meshes made of triangles. The objective of this article is to complement them in order to answer some important questions arising when they are involved. The formulation is described in detail on quadrangle meshes, the choice of approximation basis is discussed and Legendre polynomials are used in practical cases. The addition of a limiter to preserve certain properties without compromising accuracy is also considered. Finally, the asymptotic behavior of the scheme in the diffusion regime is studied. [ABSTRACT FROM AUTHOR]

Existing image analysis algorithms cannot achieve consistency with human visual classification results when classifying particles based on angular levels. To address this issue, this paper proposes an image analysis method based on triangle side ratio to quantify particle angularity, referred to as a TSR method. The proposed method utilizes a primary parameter, Mean Angularity, to assess the mean angularity level, and employs three auxiliary parameters to offer insights into the Sharpest Angularity, the Flat Proportion, and the Number of Angularity. When quantifying the angularity, the method further provides the count of convex angles. Each parameter can reflect different characteristic information of the angularity. When using the mean angularity level to order particles, the TSR method achieves the same results as visual classification, and furthermore introduces a range of values for the main parameter corresponding to the different angularity levels. The TSR method is simpler and more stable, since the particle parameters can be calculated directly without contour smoothing, and consistent results are achieved for different shapes with the same degree of angular sharpness. The results of the study on lunar soil, volcanic rock, mechanism stone, and stream stone, show that the TSR method can objectively and comprehensively analyze and quantify the particle angularity. [ABSTRACT FROM AUTHOR]

A (4 , 5) -coloring of K n is an edge-coloring of K n where every 4-clique spans at least five colors. We show that there exist (4 , 5) -colorings of K n using 5 6 n + o (n) colors. This settles a disagreement between Erdős and Gyárfás reported in their 1997 paper. Our construction uses a randomized process which we analyze using the so-called differential equation method to establish dynamic concentration. In particular, our coloring process uses random triangle removal, a process first introduced by Bollobás and Erdős, and analyzed by Bohman, Frieze and Lubetzky. [ABSTRACT FROM AUTHOR]

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In this paper the effect of particle shape and size on the mechanic behavior of glass-particle reinforcing Nylon matrix composite is evaluated. There are many parameters effecting on the composite such as types, size, position and shape of reinforcement material. Particle morphology has received enormous interest in particulate composite; small particles adhere strongly to the polymer, which leads to a strong reinforcing effect. The objective of this research is to investigate the influence of reinforcement shape and size with different volume fractions and for three packing (square, hexagonal and random) using finite element method. The composite is subject to unidirectional tensile loading. Numerical results are presented for a variety of particle shapes and size including circular, triangular, square and two distributions grouping these shapes together. The results show that the variation of the Von Mises and longitudinal stresses consistently increased when the shape of the reinforcement changed from triangle to square, distribution 1 and distribution 2 to circular in this order. [ABSTRACT FROM AUTHOR]

Progress in the field of aircraft construction is usually determined by the functional capabilities of those materials that are used for the creation of aircraft equipment. Compositions of celsian-slawsonite ceramics belonging to the BaO-SO-Al2O3-SiO2 oxide system are promising for use in the aircraft industry due to their high operational properties. In this regard, a theoretical study of its subsolidus structure is of great interest. The paper presents the results of calculating the Gibbs free energy for coupled reactions in the selected multicomponent system. The temperature intervals of the coexistence of phase combinations are given and the formed elementary tetrahedra are constructed. According to the results of theoretical studies, it is established that the considered multicomponent system is divided into 28 elementary tetrahedra. The existence of a combination of SrAl2SiO-2SrSiO3-BaSiO3 phases in the temperature range of 300–700 K is confirmed, a «filled triangle» being formed between them. Based on the results of thermodynamic calculations, it is established that the reaction between Ba2SiO, та SrAl2Si2O8 phases does not occur at temperatures up to 1200 K, and it is thermodynamically possible in the range of 1200–1700 K with the formation of a combination of BaSiO3-BaAl2O4-Sr2Al2SiO7 phases, which form a «filled circuit». [ABSTRACT FROM AUTHOR]

This document is a correction notice for an article titled "Stochastic inexact augmented Lagrangian method for nonconvex expectation constrained optimization." The correction addresses a mistake in the proof of Theorem 3 in the original article. The authors acknowledge the error and provide a modified version of Theorem 1 that corrects the mistake. The correction ensures a smaller overall oracle complexity for the algorithm described in the article. The given text presents a mathematical equation and its implications. The equation involves the "dist" function, which measures the distance between two points in a mathematical space. The equation shows that the distance between two points can be expressed in terms of other variables and functions. The text also mentions assumptions and inequalities related to the equation. Overall, the text provides a mathematical analysis of the "dist" function and its properties. The given text contains a series of mathematical equations and inequalities. It discusses the relationship between certain mathematical functions and their bounds. The text applies various mathematical techniques, such as Young's inequality, to derive these bounds. The information provided is aimed at researchers and mathematicians interested in these specific topics. The given text discusses a mathematical theorem and its application in Algorithm 1. The theorem, referred to as Theorem 2, presents the overall complexity of Algorithm 1 in a research paper. The algorithm aims to find an optimal solution to a problem described in Equation (1) of the paper. The theorem provides conditions and constraints for the algorithm to produce an optimal solution, and it [Extracted from the article]

The contour error evaluation in computer-aided manufacturing (CAM) systems is critical for high-precision measurement and manufacture of free-form surfaces. The existing methods of evaluating the contour error for free-form surfaces have been developed based on various models. Nevertheless, there still exist some of the shortcomings when applied practically, including the inaccurate distance model between the measured point cloud (MPC) and ideal surface, the numerical search required for the correspondence, possible local optimum of the solution, the inapplicability to all the surfaces, the huge point cloud required, and heavy computational load in the registration iteration, and so forth. To overcome those shortcomings, this work proposes the registration method based on a spherically-approximated patch (SAP) for the contour error evaluation of free-form surfaces. The significance of registration between the MPC and the SAP set is theoretically analyzed. Then the spherically approximable condition is provided, and the SAP set for the ideal surface is efficiently generated based on the centroid-midpoint search of parametric triangle subdivision (CMSPTS), and the meshes disobeying the spherically approximable condition are refined to planar patches under prescribed tolerance. The tactics of efficiently computing the projection of the MPC on the SAP set based on a local spreading search function (LSSF) are provided. Finally, the scheme of evaluating contour error based on the Point-To-SAP (PTSAP) registration is developed. The comparative simulation and experiment demonstrate that the proposed method raises the registration level between the MPC and the ideal surface, produces the evaluation result closer to the theoretical one with acceptable efficiency, and reduces the size of the ideal point cloud compared with the existing methods. [ABSTRACT FROM AUTHOR]

ABSTRACT Geofence is a virtual boundary that helps in location‐based services. Unmanned aerial vehicles (UAVs) fly without a pilot, and geofence helps in regulating these vehicles and constraints a UAV to enter the boundary of other UAV. However, these have to go through many safety and security features like avoiding collision, complying within the restricted area, and organizing airspaces with high traffic. The UAVs need to be within the boundaries to avoid various damages to property and loss of lives. Geofences help to detect boundary violation of UAVs to a great extent. This study uses various mathematical algorithms to create geofences and detect boundary violation of UAVs. The existing methods, Delaunay triangulation and triangle weight characterization with adjacency, are implemented. The existing methods have high execution time as well as high standard deviation and low accuracy. Further, a novel method for boundary violation detection is proposed using a convex hull to create geofences. The results show that the proposed method improves the execution time by 59.23% and accuracy by 1.35%. [ABSTRACT FROM AUTHOR]

For each commutative ring R we associate a simple graph Γ1∗(R). We investigate the properties of an Artinian ring R when Γ1∗(R) is a star. [ABSTRACT FROM AUTHOR]

AbstractIn this study, we explore the application of process mining techniques on assessment log data to explore problem-solving strategies in Algebra. By analyzing sequences of student activities, we demonstrate the significant potential of process mining in identifying problem-solving strategies that lead to successful and unsuccessful outcomes. Our findings reveal that students who successfully solve the problem tend to follow one of three structured strategies, displaying a systematic process in filling the boxes of a Pascal’s triangle. Conversely, those who falter often start with a correct strategy but deviate by inserting incorrect values, especially in central boxes. Further analysis provides insights into the strategies and potential misconceptions among students from various disability groups. Notably, autistic students exhibit unique patterns, such as initiating the solution from the triangle’s right side, contrary to the common left-to-right strategy, and consistently applying this approach even when errors occur. [ABSTRACT FROM AUTHOR]

For animals to locate resources and stay safe, navigation is an essential cognitive skill. Blind people use different navigational strategies to encode the environment. Path integration significantly influences spatial navigation, which is the ongoing update of position and orientation during self-motion. This study examines two separate things: (i) how guided and non-guided strategies affect blind individuals in encoding and mentally representing a trajectory and (ii) the sensory preferences for potential navigational aids through questionnaire-based research. This study first highlights the significant role that the absence of vision plays in understanding body centered and proprioceptive cues. Furthermore, it also underscores the urgent need to develop navigation-assistive technologies customized to meet the specific needs of users. [ABSTRACT FROM AUTHOR]

ABSTRACT Most existing remote sensing image watermarking algorithms concentrate on excavation of particular embedding templates, image features, or geometric invariant domains, which present challenges in terms of resistance to desynchronization attacks, embedding domain repetition, and insufficient algorithm versatility. To address these issues, this study proposes a watermarking algorithm that is robust to desynchronization attacks and can adapt to different types of remote sensing images using the geometric invariant domain and hybrid frequency domain. The algorithm uses the multi‐scale SIFT to identify feature points in remote sensing images, then creates a Delaunay triangulation network (DTN) based on these feature points, extracts the tangent circles of triangles, and normalizes these tangent circles using image moment and affine transformation, and the feature domains with geometric invariance are constructed. On this basis, the discrete wavelet transform (DWT) transforms the feature domain to the frequency decomposition state, and the singular value decomposition (SVD) further mines the watermark embedding domain, ensuring the stability of the watermark transforming back and forth in the embedding domain and improving the overall invisibility of the watermarking algorithm. The experimental results indicate that, compared to related algorithms, the proposed watermarking algorithm not only adapts better to remote sensing images with different bands and bit depths but also provides superior invisibility and demonstrates strong robustness against various desynchronization attacks such as splicing, panning, rotating, as well as image processing like noise addition, filtering, and compression. [ABSTRACT FROM AUTHOR]

In this paper we introduce the galaxy of Coxeter groups (of finite rank) – an infinite dimensional, locally finite, ranked simplicial complex which captures isomorphisms between finite rank Coxeter systems. In doing so, we would like to suggest a new framework to study the isomorphism problem for Coxeter groups. We prove some structural results about this space, provide a full characterization in small ranks and propose many questions. In addition we survey known tools, results and conjectures. Along the way we show profinite rigidity of triangle Coxeter groups – a result which is possibly of independent interest. [ABSTRACT FROM AUTHOR]

In Evans and Francis (2022) and Hendel (2021) the authors investigated resistance distance in triangular grid graphs and observed several types of asymptotic behavior. This paper extends their work by studying the initial, non-asymptotic, behavior found when equivalent circuit transformations are performed, reducing the rows in the triangular grid graph one row at a time. The main conjecture characterizes, after reducing an arbitrary number of times an initial triangular grid all of whose edge resistances are identically one, when edge resistance values are less than, equal to, or greater than one. A special case of the conjecture is proven. The main theorem identifies patterns of repeating edge resistances arising in diagonals of a triangular grid reduced s times provided the original grid has at least 4 s rows of triangles. This paper also improves upon the notation, concepts, and proof techniques introduced by the authors previously. [ABSTRACT FROM AUTHOR]

The conventional Ant Colony Optimization (ACO) algorithm, applied to logistics robot path planning in a two-dimensional grid environment, encounters several challenges: slow convergence rate, susceptibility to local optima, and an excessive number of turning points in the planned paths. To address these limitations, an improved ant colony algorithm has been developed. First, the heuristic function is enhanced by incorporating artificial potential field (APF) attraction, which introduces the influence of the target point's attraction on the heuristic function. This modification accelerates convergence and improves the optimization performance of the algorithm. Second, an additional pheromone increment, calculated from the difference in pheromone levels between the best and worst paths of the previous generation, is introduced during the pheromone update process. This adjustment adaptively enhances the path length optimality. Lastly, a triangle pruning method is applied to eliminate unnecessary turning points, reducing the number of turns the logistics robot must execute and ensuring a more direct and efficient path. To validate the effectiveness of the improved algorithm, extensive simulation experiments were conducted in two grid-based environments of varying complexity. Several performance indicators were utilized to compare the conventional ACO algorithm, a previously improved version, and the newly proposed algorithm. MATLAB simulation results demonstrated that the improved ant colony algorithm significantly outperforms the other methods in terms of path length, number of iterations, and the reduction of inflection points, confirming its superiority in logistics robot path planning. [ABSTRACT FROM AUTHOR]

A new flexible bis(amidine) ligand featuring two additional N-donor groups incorporates a defined [CuIMesCu2I]2+ fragment from mesitylcopper into a triangular cluster with a reactive organometallic coordination site. Subtle changes to the reaction protocol result in the formation of an intertwined tetracuprous arrangement that adopts the shape of a Möbius strip. [ABSTRACT FROM AUTHOR]

Photoelectron spectroscopy and theoretical calculations have identified the global minimum structure of the 16‐valence electron Si3Cu3− cluster, which features a planar tetracoordinate silicon (ptSi) in a rhombic arrangement. The Si3 and Cu3 triangles are interconnected by a Si2/Cu2 edge, forming an ordered chain‐like structure. Besides the conventional 2c–2e σ‐bond connecting Si3 and Cu3, the stability of this cluster is reinforced by a delocalized 3c–2e σ‐bond in Cu3 and a π‐bond in Si3. Our study provides experimental confirmation of a planar hypercoordinate heavier Group 14 element, opening possibilities for exploring similar structures in two‐dimensional materials. [ABSTRACT FROM AUTHOR]

The first aqueous silver borate nitrate Ag3B4O6(OH)2(NO3) has a non-centrosymmetric crystal structure (P43212, a = 10.0196(3) Å, c = 9.2594(2) Å). The compound is prepared by soft hydrothermal synthesis inside an evacuated quartz ampoule. The structure of the resulting 1D–5B-borate compound is formed by [B4O6(OH)2]2– chains consisting of two triborate rings with a common (2Δ3□:(‹Δ2□›–‹Δ2□›–)∞ tetrahedron and NO3 triangles with silver atoms between the rings. The relationship between the structures of Ag3B4O6(OH)2(NO3), Tl2B4O6(OH)2·2H2O and the kernite Na2B4O6(OH)2·3H2O (also formed by [B4O6 (OH)2]2– chains) is discussed. [ABSTRACT FROM AUTHOR]

This paper proposes a fluid–cloth simulation method based on dynamic time-stepping and a two-level collision detection algorithm to enhance the realism and real-time interaction of fluid and cloth. The method addresses the slow collision detection speed and poor culling efficiency between flexible cloth and complex fluid models. First, an adaptive k-means algorithm based on smoothed particle hydrodynamics is introduced, which dynamically divides fluid particles into two independent clusters according to velocity differences and applies a dual-scale dynamic time stepping (DDTS) scheme. Second, an extended position-based dynamics method is employed to simulate the cloth, integrating the small time steps derived from the fluid simulation to ensure the numerical stability during the cloth constraint solving process. Finally, a bounding box collision detection algorithm based on spatial hashing is constructed for the purpose of high-level culling between particles and triangles. At the low-level culling stage, a new filter, the dynamic motion filter (DMF), is proposed as a means of further improving culling efficiency. In comparison with the combination of continuous collision detection and position-based dynamics collision constraints, the average frame rate of the model building process with the introduction of the DDTS method is increased by 120%, resulting in higher-quality visual effects. Furthermore, the DMF increases the culling rate for vertex-face collision pairs by 3% to 23% and reduces overall collision detection time by 2% to 8% compared with other collision culling methods, enhancing the culling efficiency of collision. [ABSTRACT FROM AUTHOR]

In an error estimation of finite element solutions to the Poisson equation, we usually impose the shape regularity assumption on the meshes to be used. In this paper, we show that even if the shape regularity condition is violated, the standard error estimation can be obtained if "bad" elements that violate the shape regularity or maximum angle condition are covered virtually by simplices that satisfy the minimum angle condition. A numerical experiment illustrates the theoretical result. [ABSTRACT FROM AUTHOR]

To tackle problems such as the destruction of topological structures and the loss of detailed features in the simplification of 3D building models, we propose a 3D building model simplification algorithm that considers detailed features and topological structures. Based on the edge collapse algorithm, the method defines the region formed by the first-order neighboring triangles of the endpoints of the edge to be collapsed as the simplification unit. It incorporates the centroid displacement of the simplification unit, significance level, and approximate curvature of the edge as influencing factors for the collapse cost to control the edge collapse sequence and preserve model details. Additionally, considering the unique properties of 3D building models, boundary edge detection and face overlay are added as constraints to maintain the model's topological structure. The experimental results show that the algorithm is superior to the classic QEM algorithm in terms of preserving the topological structure and detailed features of the model. Compared to the QEM algorithm and the other two comparison algorithms selected in this paper, the simplified model resulting from this algorithm exhibit a reduction in Hausdorff distance, mean error, and mean square error to varying degrees. Moreover, the advantages of this algorithm become more pronounced as the simplification rate increases. The research findings can be applied to the simplification of 3D building models. [ABSTRACT FROM AUTHOR]

In this paper, we obtain improvements of two known inequalities in an acute triangle. We also propose three conjectured inequalities as open problems. [ABSTRACT FROM AUTHOR]

There are two regular polygons Pn-(R1) and Pn-(R2) which correspond to the given polygonal distances {d1, d2, ..., dn. It is proved that the radii R1, R2 of the circumcircles of the regular polygons and the n distances are connected by the system: ... where m = 1, ..., n - 1. The solutions for the equilateral triangles and the squares are investigated. The two-parametric families of the solutions are given in both cases, when three distances are rational numbers. [ABSTRACT FROM AUTHOR]

The purpose of this paper is to explore the potential and development of the drama triangle, the winner's triangle, and the trauma quadrangle in trauma work and to offer a tool to help identify the roles/patterns replayed when caught in drama. This fosters moving into a more active (winner's) space and retaining a more integrated position. The authors draw on trauma-informed practices and transactional analysis (TA). They propose that the bystander position in response to trauma can be both an internal and external process. As a tool for working with developmental trauma, the authors offer the contactful quad, which through empowerment and witness aims to support the process of decontamination and deconfusion, to aid the healing process, and to support integration of the unresolved somatic experience of trauma responses. Using clinical vignettes, the authors demonstrate the power of the Actively Courageous position of the contactful quad as an antidote to the Bystander position. [ABSTRACT FROM AUTHOR]

The combinatorial problem in this paper is motivated by a variant of the famous traveling salesman problem where the salesman must return to the starting point after each delivery. The total length of a delivery route is related to a metric known as closeness centrality. The closeness centrality of a vertex v in a graph G was defined in 1950 by Bavelas to be C C (v) = | V (G) | − 1 S D (v) , where S D (v) is the sum of the distances from v to each of the other vertices (which is one-half of the total distance in the delivery route). We provide a real-world example involving the Metro Atlanta Rapid Transit Authority rail network and identify stations whose S D values are nearly identical, meaning they have a similar proximity to other stations in the network. We then consider theoretical aspects involving asymmetric trees. For integer values of k, we considered the asymmetric tree with paths of lengths k , 2 k , ... , n k that are incident to a center vertex. We investigated trees with different values of k, and for k = 1 and k = 2 , we established necessary and sufficient conditions for the existence of two vertices with identical S D values, which has a surprising connection to the triangular numbers. Additionally, we investigated asymmetric trees with paths incident to two vertices and found a sufficient condition for vertices to have equal S D values. This leads to new combinatorial proofs of identities arising from Pascal's triangle. [ABSTRACT FROM AUTHOR]

Recent advancements in manufacturing technologies have facilitated the efficient and cost-effective production of functionally graded (FG) sandwich members, offering exceptional structural performances. However, designing these members is challenging due to the difficulties in accurately determining their cross-sectional properties, especially within routine designs that rely on beam-column elements. To this end, this paper introduces a robust finite-element-based cross-section analysis method for FG sandwich sections considering arbitrary shapes and material gradients. A novel nonhomogeneous plane triangle (NPT) element is formulated to enhance the computational efficiency of the developed cross-section analysis approach. A new structural analysis software with a graphic user interface, namely MSASect2, is developed based on the proposed cross-section analysis approach for its convenient implementation in practice. Four examples are displayed to verify the correctness of the present study and MSASect2, indicating their potential to promote the broader application of FG sandwich sections. [ABSTRACT FROM AUTHOR]

In 1966, Zalgaller completed the task of determining all edge-to-edge tilings of the sphere by regular spherical polygons, proving that the only possibilities are the Platonic tilings, the Archimedean tilings, the Johnson tilings, the prism and anti-prism tilings and two tiles sharing their boundaries. We determine all non-edge-to-edge tilings of the sphere by regular spherical polygons of three or more sides, showing there are five continuous families of kaleidoscope tilings, fifteen continuous families of 2-hemisphere tilings, four lunar tilings, five sporadic tilings, five composed tilings and one magic triangle tiling. [ABSTRACT FROM AUTHOR]

Overdetermined boundary value problems and the minimal operators generated by them are extremely important in the description of regular boundary value problems for differential equations, and are also widely used in the study of local properties of solutions. In addition, for inverse problems of mathematical physics arising from applications, when determining unknown data, it is necessary to study problems with overdetermined boundary conditions, which is reflected in the study of problems, including for hyperbolic equations and systems, arising in physics, geophysics, seismic tomography, geoelectrics, electrodynamics, medicine, ecology, economics and many other practical areas. Thus, the study of overdetermined boundary value problems is of both theoretical and applied interest. In this paper, a criterion for the regular solvability of the overdetermined Cauchy problem for the Gellerstedt equation and the minimal differential operator generated by it in a hyperbolic domain is established, as which both the case of a characteristic triangle and the case of a more general domain with fairly general assumptions about the boundary of the domain are considered. Due to overdetermined boundary conditions, the problem under consideration will be ill-posed in the general case, therefore, for its regular solvability, additional conditions must be imposed on the initial data. In other words, we have considered the inverse problem: to determine what requirements the initial data of the problem, in particular the right part of the Gellerstedt equation, should meet, in question, so that the overdetermined Cauchy problem is regularly solvable. The proof is based on the Gellerstedt potential, the properties of solutions of the Goursat problem in the characteristic triangle, and the properties of special functions. [ABSTRACT FROM AUTHOR]

Let $\Gamma =\langle I_{1}, I_{2}, I_{3}\rangle $ be the complex hyperbolic $(4,4,\infty)$ triangle group with $I_1I_3I_2I_3$ being unipotent. We show that the limit set of $\Gamma $ is connected and the closure of a countable union of $\mathbb {R}$ -circles. [ABSTRACT FROM AUTHOR]

For power-generalized Hartogs triangles in $ \mathbb {C}^3 $ C 3 , the Bergman projection satisfies a weak-type estimate at the upper endpoint of $ L^p $ L p boundedness but not at the lower endpoint. Our work complements related results obtained recently for rational Hartogs triangles in $ \mathbb {C}^2 $ C 2 and the punctured unit ball in $ \mathbb {R}^3 $ R 3 . [ABSTRACT FROM AUTHOR]

We apply the parqueting-reflection technique for a triangle domain to achieve the points of the complex plane. Then the Cauchy–Schwarz representation formula is construed by the Cauchy–Pompeiu formula and an explicit solution for the Schwarz boundary value problem (BVP) for the non-homogeneous Cauchy–Riemann equation on the triangle is presented. We also discuss the condition of solvability and by using the Schwarz BVP, the inhomogeneous Dirichlet and homogeneous Neumann BVPs are investigated. [ABSTRACT FROM AUTHOR]

Given a bounded open connected set Ω ⊂ ℝ 2 {\Omega\subset\mathbb{R}^{2}} with Lipschitz boundary, we consider the class of piecewise constant maps u taking three fixed values α , β , γ ∈ ℝ 2 {\alpha,\beta,\gamma\in\mathbb{R}^{2}} , vertices of an equilateral triangle; for any u in this class, using a weak notion of Jacobian determinant valid for BV functions, we give a precise description of Det ⁡ ( ∇ ⁡ u ) {\operatorname{Det}(\nabla\kern 0.569055ptu)} and show that the relaxed graph area of u is bounded from above by a quantity related to the flat norm of Det ⁡ ( ∇ ⁡ u ) {\operatorname{Det}(\nabla\kern 0.569055ptu)} . The provided upper bound allows to show the validity of a De Giorgi conjecture regarding the relaxed area functional when one restricts to this class of piecewise constant functions. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to investigate Sobolev orthogonality and families of orthogonal polynomials on the triangle as a generalization of the results in Xu, Y. Constr. Approx. 46, 349-434 (2017). We define inner products or pseudo-inner products that contain derivatives up to third-order in the Sobolev space on a triangle and study associated orthogonal polynomials that are analogs of the Jacobi polynomials J k , n α , β , γ but with α , β , γ being - 1 , - 2 , or - 3 , and two other families derived under simultaneous permutations of x , y , 1 - x - y and α , β , γ . [ABSTRACT FROM AUTHOR]

The first aqueous silver borate nitrate Ag3B4O6(OH)2(NO3) has a non-centrosymmetric crystal structure (P43212, a = 10.0196(3) Å, c = 9.2594(2) Å). The compound is prepared by soft hydrothermal synthesis inside an evacuated quartz ampoule. The structure of the resulting 1D–5B-borate compound is formed by [B4O6(OH)2]2– chains consisting of two triborate rings with a common (2Δ3□:(‹Δ2□›–‹Δ2□›–)∞ tetrahedron and NO3 triangles with silver atoms between the rings. The relationship between the structures of Ag3B4O6(OH)2(NO3), Tl2B4O6(OH)2·2H2O and the kernite Na2B4O6(OH)2·3H2O (also formed by [B4O6 (OH)2]2– chains) is discussed. [ABSTRACT FROM AUTHOR]

The Scott–Vogelius finite element pair for the numerical discretization of the stationary Stokes equation in 2D is a popular element which is based on a continuous velocity approximation of polynomial order k and a discontinuous pressure approximation of order k - 1 . It employs a "singular distance" (measured by some geometric mesh quantity Θ z ≥ 0 for triangle vertices z ) and imposes a local side condition on the pressure space associated to vertices z with Θ z = 0 . The method is inf-sup stable for any fixed regular triangulation and k ≥ 4 . However, the inf-sup constant deteriorates if the triangulation contains nearly singular vertices 0 < Θ z ≪ 1 . In this paper, we introduce a very simple parameter-dependent modification of the Scott–Vogelius element with a mesh-robust inf-sup constant. To this end, we provide sharp two-sided bounds for the inf-sup constant with an optimal dependence on the "singular distance". We characterise the critical pressures to guarantee that the effect on the divergence-free condition for the discrete velocity is negligibly small, for which we provide numerical evidence. [ABSTRACT FROM AUTHOR]

Denote by C ℓ - the 3-uniform hypergraph obtained by removing one hyperedge from the tight cycle on ℓ vertices. It is conjectured that the Turán density of C 5 - is 1/4. In this paper, we make progress toward this conjecture by proving that the Turán density of C ℓ - is 1/4, for every sufficiently large ℓ not divisible by 3. One of the main ingredients of our proof is a forbidden-subhypergraph characterization of the hypergraphs, for which there exists a tournament on the same vertex set such that every hyperedge is a cyclic triangle in this tournament. A byproduct of our method is a human-checkable proof for the upper bound on the maximum number of almost similar triangles in a planar point set, which was recently proved using the method of flag algebras by Balogh, Clemen, and Lidický. [ABSTRACT FROM AUTHOR]

This note introduces an innovative refinement of the Garfunkel--Bankoff inequality, further improving upon the Finsler--Hadwiger inequality and the Lukarevski-- Marinescu inequality within a triangle. [ABSTRACT FROM AUTHOR]