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Mathematics is a difficult subject for elementary school students. One of the causes is errors in mathematical concepts and principles made by students. For this reason, efforts need to be made to improve the concept and principles of quadrilaterals. The aim of this research is to design student learning trajectories in finding the perimeter and area of a rectangle in the context of a fishing pond. This is research with a design research approach. The subjects of this research were 50 students divided into two groups, 25 people each. Data collection techniques through observation, interviews and documentation. The data analysis technique used is qualitative and is developed through certain relationship patterns. The results of this research show that the mathematics learning trajectory design was implemented very well. It improves the ability of understanding about quadrilaterals. The learning path is activity-1: students create a representation of the context of a fishing pond and a rectangular image. Activity-2: students determine the perimeter of a rectangular image by placing match sticks around the rectangle in sequence, holding them tightly together. Activity-3: students determine the circumference of the fishing pond. Activity-4: students determine how to calculate the circumference of the pond by drawing a rectangle using matchsticks. Activity-5: students discover the principle of perimeter of a rectangle. The conclusion of this research is that the mathematics learning trajectory design is valid, practical, and increases the ability to understand concepts and principles from quadrilaterals. [ABSTRACT FROM AUTHOR]

Edge-to-edge tilings of the sphere by congruent a2bc-quadrilaterals are classified as 3 classes: (1) A 1-parameter family of quadrilateral subdivisions of the octahedron with 24 tiles, and a flip modification for one special parameter; (2) a 2-parameter family of 2-layer earth map tilings with 2n tiles for each n ≥ 3; (3) a 3-layer earth map tiling with 8n tiles for each n ≥ 2, and two flip modifications for each odd n. The authors also describe the moduli of parameterized tilings and provide the full geometric data for all tilings. [ABSTRACT FROM AUTHOR]

Edge-to-edge tilings of the sphere by congruent quadrilaterals are completely classified in a series of three papers. This last one classifies the case of a 3 ⁢ b -quadrilaterals with some irrational angle: there are a sequence of 1-parameter families of quadrilaterals admitting 2-layer earth map tilings together with their basic flip modifications under extra condition, and 5 sporadic quadrilaterals each admitting a special tiling. A summary of the full classification is presented in the end. [ABSTRACT FROM AUTHOR]

In this research, based on the historical statistics of the deposits of a private bank (Ansar bank before merging) into four different categories (funds), we are looking for the results and prediction of bank deposits in each of the categories, using machine learning prediction methods for the first time in internal studies. In this forecasting method, the superiority of the machine learning method is confirmed due to the examination of all the time changes of the data in a large volume (from the beginning until now). Forecasting the above deposit statistics for bank decision makers in the future according to the economic conditions can be the basis of macro banking decisions. The results show that partly stable long-term deposit and stable deposit will take a downward trend in the future in the Ansar bank, and unstable and interest-sensitive deposit will also take a downward trend. Qarzol-hasane deposit has an upward trend, although it has always fluctuations similar to sin-cos fluctuations. It should be noted that this machine learning prediction method is a stable method without sensitivity analysis and has high reliability. According to this research, paying attention to long-term and stable deposits as well as Qarzol-ha-sa-ne is of great importance in the bank's budget. [ABSTRACT FROM AUTHOR]

The inverse finite element method (iFEM) is a powerful tool for shape sensing and structural health monitoring and has several advantages with respect to some other existing approaches. In this study, a two-dimensional four-node quadrilateral inverse finite element formulation is presented. The element is suitable for thin structures under in-plane loading conditions. To validate the accuracy and demonstrate the capability of the inverse element, four different numerical cases are considered for different loading and boundary conditions. iFEM analysis results are compared with regular finite element analysis results as the reference solution, and very good agreement is observed between the two solutions demonstrating the capability of iFEM approach. [ABSTRACT FROM AUTHOR]

Given a planar graph family F ${\rm{ {\mathcal F} }}$, let exP(n,F) $e{x}_{{\mathscr{P}}}(n,{\mathscr{F}})$ and spexP(n,F) $spe{x}_{{\mathscr{P}}}(n,{\mathscr{F}})$ be the maximum size and maximum spectral radius over all n $n$‐vertex F ${\rm{ {\mathcal F} }}$‐free planar graphs, respectively. Let tCℓ $t{C}_{\ell }$ be the disjoint union of t $t$ copies of ℓ $\ell $‐cycles, and tC $t{\mathscr{C}}$ be the family of t $t$ vertex‐disjoint cycles without length restriction. Tait and Tobin determined that K2+Pn−2 ${K}_{2}+{P}_{n-2}$ is the extremal spectral graph among all planar graphs with sufficiently large order n $n$, which implies the extremal graphs of both spexP(n,tCℓ) $spe{x}_{{\mathscr{P}}}(n,t{C}_{\ell })$ and spexP(n,tC) $spe{x}_{{\mathscr{P}}}(n,t{\mathscr{C}})$ for t≥3 $t\ge 3$ are K2+Pn−2 ${K}_{2}+{P}_{n-2}$. In this paper, we first determine spexP(n,tCℓ) $spe{x}_{{\mathscr{P}}}(n,t{C}_{\ell })$ and spexP(n,tC) $spe{x}_{{\mathscr{P}}}(n,t{\mathscr{C}})$ and characterize the unique extremal graph for 1≤t≤2 $1\le t\le 2$, ℓ≥3 $\ell \ge 3$ and sufficiently large n $n$. Second, we obtain the exact values of exP(n,2C4) $e{x}_{{\mathscr{P}}}(n,2{C}_{4})$ and exP(n,2C) $e{x}_{{\mathscr{P}}}(n,2{\mathscr{C}})$, which solve a conjecture of Li for n≥2661 $n\ge 2661$. [ABSTRACT FROM AUTHOR]

We apply Diophantine analysis to classify edge-to-edge tilings of the sphere by congruent almost equilateral quadrilaterals (i.e., edge combination a 3 b ). Parallel to a complete classification by Cheung, Luk, and Yan, the method implemented here is more systematic and applicable to other related tiling problems. We also provide detailed geometric data for the tilings. [ABSTRACT FROM AUTHOR]

In this article, using the formulas derived in (Nenkov 2019) and (Nenkov 2022) for the distances from some remarkable points in the quadrilateral to its vertices and to the midpoints of its sides and diagonals, formulas for the distances from these remarkable points to the intersection of the diagonals and to the intersections of the pairs of opposite sides were obtained. Formulas are also derived for the distances from another remarkable point to the vertices of the quadrilateral, as well as to the remarkable points discussed in (Nenkov 2019). Using the obtained formulas for these distances, a number of interesting inequalities in the quadrilateral are proved. [ABSTRACT FROM AUTHOR]

Objective: To identify the differences between trilateral and quadrilateral retinoblastoma based on radiologic imaging. Case: We found three cases of retinoblastoma at Sanglah General Hospital. The retinoblastoma patients we found were boys under the age of 5. The first patient presented with swelling, pain, and impaired vision in the right eye. The second and third patients presented with eye protrusion and leukocoria. Computed tomography (CT) imaging revealed a calcified mass in the intraorbital region that extended to the cerebral hemispheres. It may be accompanied by hair-on-end periosteal reaction. In the first patient, the mass extended to the pineal gland. In the second and third patients, the mass extended to the suprasellar and pineal regions. Trilateral retinoblastoma consists of a primordial midline neuroectodermal tumor that originates in the pineal or suprasellar region. It is an uncommon combination of unilateral or bilateral retinoblastomas. Quadrilateral retinoblastoma, on the other hand, presents with bilateral retinoblastoma and tumors in both the suprasellar and pineal regions. The histopathology of trilateral and quadrilateral retinoblastoma shows a different pattern from retinoblastoma. Conclusion: Retinoblastoma poses one of the most challenging problems due to its distinct patterns of growth, extension, and recurrence. Intracranial metastasis of trilateral and quadrilateral retinoblastoma exhibits a different pattern. The prognosis and diagnosis of retinoblastoma are significantly influenced by clinical examination, radiologic imaging, and histopathology.

In this paper, we introduce new generalized barycentric coordinates (coined as moment coordinates) on convex and nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with $ n $ vertices (nodes) in $ \mathbb{R}^2 $, the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank $ n $, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties.

We prove 30 necessary and sufficient conditions for when a convex quadrilateral has a pair of opposite sides of equal length. [ABSTRACT FROM AUTHOR]

In this paper, we introduce new generalized barycentric coordinates (coined as moment coordinates) on convex and nonconvex quadrilaterals and convex hexahedra with planar faces. This work draws on recent advances in constructing interpolants to describe the motion of the Filippov sliding vector field in nonsmooth dynamical systems, in which nonnegative solutions of signed matrices based on (partial) distances are studied. For a finite element with n vertices (nodes) in ℝ², the constant and linear reproducing conditions are supplemented with additional linear moment equations to set up a linear system of equations of full rank n, whose solution results in the nonnegative shape functions. On a simple (convex or nonconvex) quadrilateral, moment coordinates using signed distances are identical to mean value coordinates. For signed weights that are based on the product of distances to edges that are incident to a vertex and their edge lengths, we recover Wachspress coordinates on a convex quadrilateral. Moment coordinates are also constructed on a convex hexahedra with planar faces. We present proofs in support of the construction and plots of the shape functions that affirm its properties. [ABSTRACT FROM AUTHOR]

This study uses three turbulence model variations, i.e., S - A, k - ε, and k – ω turbulence models. In addition, there are two variations of cell shape and three variations of cell number. The number of cells is 500, 5000, 50000, and 100000. Verification is carried out in the mesh refinement study and validated by aerodynamic performances. Based on the mesh refinement study, quadrilateral cells with the k - ε are in the asymptotic convergence range. Based on the Cl, it can be concluded that the quadrilateral mesh with 50000 and 100000 cells simulated using the k-ε turbulence model shows very low errors, namely 4.1151% and 3.8643%, respectively. It shows consistency based on the quadrilaterals Cd mesh data with the k-ε and k-ω turbulence models. However, k-ε shows the lowest error with the number of cells 50000 and 100000, i.e., 127.7682% and 110.4175%, respectively. However, choosing mesh 50000 cells are advisable because it only takes 23 minutes 48 seconds in computation, while mesh 100000 cells take 1 hour 17 minutes 21 seconds. Only Cm from quadrilateral mesh with the turbulence model k-ω shows consistency. An error of mesh 50000 cells is 22.0717%, and the error value for 100000 cells is 18.1630%. By considering computation time, mesh 50000 cells are preferable because it only takes 27 minutes 16 seconds, which is faster 43 minutes 14 seconds than 100000 cells.

The inverse finite element method (iFEM) is a powerful tool for shape sensing and structural health monitoring and has several advantages with respect to some other existing approaches. In this study, a two-dimensional eight-node quadrilateral inverse finite element formulation is presented. The element is suitable for thin structures under in-plane loading conditions. To validate the accuracy and demonstrate the capability of the inverse element, four different numerical cases are considered for different loading and boundary conditions. iFEM analysis results are compared with regular finite element analysis results as the reference solution and very good agreement is observed between the two solutions, demonstrating the capability of the iFEM approach. [ABSTRACT FROM AUTHOR]

We consider the approximation properties of primal discontinuous Petrov-Galerkin (DPG) method on quadrilateral meshes. We show how the previous convergence results as well as the Aubin-Nitsche type duality arguments can be extended to cover arbitrary convex quadrilateral elements with bilinear isomorphisms. The arguments are based on the approximation theory of quadrilateral vector finite element spaces associated to the numerical flux variable of the DPG approximation. The theoretical results are validated by a numerical experiment that features also a comparison between the primal DPG method and a conventional least squares finite element method with the same number of degrees of freedom. [ABSTRACT FROM AUTHOR]

This study uses three turbulence model variations, i.e., S - A, k - ε, and k - ω turbulence models. In addition, there are two variations of cell shape and three variations of cell number. The number of cells is 500, 5000, 50000, and 100000. Verification is carried out in the mesh refinement study and validated by aerodynamic performances. Based on the mesh refinement study, quadrilateral cells with the k - ε are in the asymptotic convergence range. Based on the Cl, it can be concluded that the quadrilateral mesh with 50000 and 100000 cells simulated using the k-ε turbulence model shows very low errors, namely 4.1151% and 3.8643%, respectively. It shows consistency based on the quadrilaterals Cd mesh data with the k-ε and k-ω turbulence models. However, k-ε shows the lowest error with the number of cells 50000 and 100000, i.e., 127.7682% and 110.4175%, respectively. However, choosing mesh 50000 cells are advisable because it only takes 23 minutes 48 seconds in computation, while mesh 100000 cells take 1 hour 17 minutes 21 seconds. Only Cm from quadrilateral mesh with the turbulence model k-ω shows consistency. An error of mesh 50000 cells is 22.0717%, and the error value for 100000 cells is 18.1630%. By considering computation time, mesh 50000 cells are preferable because it only takes 27 minutes 16 seconds, which is faster 43 minutes 14 seconds than 100000 cells. [ABSTRACT FROM AUTHOR]

Understanding quadrilateral and triangle material is still a problem for students. These problems occur because teachers did not design learning trajectories before they tough in the classroom, or they still did not apply innovative approaches and media in the learning process. Designing and developing a learning trajectory for the quadrilateral and triangle material as an activities series based on science, technology, engineering, and mathematics (STEM) using rubber-powered car projects through lesson study is the aim of this study. This design is referred to as a hypothetical learning trajectory in teaching quadrilateral and triangle material. This study used design research as a research method, which consists of three stages: preliminary design, design experiment, and retrospective analysis. The subject of this research was seventh-grade students at a state junior high school in Semarang City. Data were collected from various sources: pre- and post-test, student worksheets, interviews, documentation of photos, and video recording of the learning process. Data analyses through qualitative research about validity and reliability refer to the quality of the observation. This study result described the learning trajectory consisting of three activities, namely creating a rubber-powered car project, identifying the property and relation among quadrilaterals, and solving a contextual problem in the area and perimeter of quadrilaterals. This research indicated that a series of activities STEM-based designed through lesson study can make students understand material actively, joyfully, and meaningfully. [ABSTRACT FROM AUTHOR]

Eld-independent cognitive styles in solving problems on rectangular shape material. This type of research is descriptive qualitative research, namely by interpreting data with the stages of image analysis, interviews, observation, and documentation. The subject of this study was taken by purposive sampling after carrying out the GEFT test in class V where 4 samples were taken based on field dependent and field independent cognitive styles. The instruments prepared in this study were GEFT test questions and mathematical representation questions. From the results of this study it shows that: 1) students who have a field dependent cognitive style carry out mathematical representations visually students are not always able to make pictures and symbols of a problem and in mathematical expression students can make mathematical models and can perform calculations according to what has been taught by the teacher accurately or always think globally without making new discoveries and in written expression students are less able to make solutions in logical and systematic written words. 2) students who have a field independent cognitive style make mathematical representations in solving problems with visualization stages can understand the problem and change it in the form of images and symbols in accordance with the problems that are understood and in mathematical expression students and can make plans for solving with mathematical models and can carry out calculations in their own way in accordance with their experience which is considered fast average in written expression students are able to express solutions in written words in a systematic and logical manner.

We demonstrate that the Schwarz-Cristoffel mapping, which maps the given quadrilateral to the upper half-plane, is a solution to the general Heun equation. Furthermore, we provide an explicit formula for the accessory parameter of the Heun equation as a function of the singular point. [ABSTRACT FROM AUTHOR]

Mathematical knowledge for teaching (MKT) is one of the factors that affect the competence of prospective teachers in learning. This study aims to describe the MKT of prospective teachers in terms of differences in ability levels in the scope of quadrilateral. The MKT studied in this study includes common content knowledge, and knowledge of the mathematical horizon. The research subjects consisted of three prospective math teachers, namely high-skill, intermediate-skill and low-skill. This research is qualitative research using vignettes. Data in this study was collected through vignettes and interviews. Validity of data using time triangulation. The results of this study showed that the ability of prospective teachers of intermediate-skilled and low-skill teachers still have difficulty regarding quadrilateral convex and quadrilateral concaves and still have difficulty understanding the relationships of squares, rectangles, rhombus and parallelogram. The MKT of intermediate and low-skill subjects of being produced is still limited compared to high-skill subjects. This difference occurs because it has not been able to eliminate the rectangular prototype image. To develop their abilities, prospective teachers need to be equipped with quadrilateral exploration tasks in education and training.

Lately, the foundations of a complete geometry of the quadrilateral were laid, built in analogy of the classical geometry of the triangle. Several articles were devoted to it, the PhD-dissertation work of one of the authors of this article (Stefanov 2020), and a book encompassing it in its entirety is soon to be published. This publication aims to convince the readers in the benefits of studying this geometry. During the last years the rubrics „Contest problems“ and „Problems М+“ of the journals „Mathematics and Informatics“ and „Mathematics Plus“ included difficult problems, whose solutions are complex for somebody who doesn‘t know and doesn‘t use elements of the quadrilateral’s geometry. Knowing the properties of the remarkable points, lines and circles, as well as the mappings in the quadrilateral, which are subjects of its geometry, not only provides ideas for solving this problem, but also serves for their implementation (to carry out the solution). This was shown once in (Nenkov, Stefanov & Haimov 2020). We will confirm it here again, with new interesting examples. [ABSTRACT FROM AUTHOR]

In this paper, we focus on studying the distortion of the exterior conformal modulus of a quadriilateral of a sufficiently arbitrary form under the stretching map along the abscissa axis with coefficient . By using the properties of quasiconformal transformations and taking into account some facts from the theory of elliptic integrals, we confirm that the asymptotic behavior of this modulus does not depend on the shape of the boundary of the quadrilateral. Especially, it is equivalent to as . Therefore, we give a solution to the Vuorinen problem for the exterior conformal modulus of a sufficiently arbitrary quadrilateral. [ABSTRACT FROM AUTHOR]

In 1972, the United Methodist Church's Theological Study Commission for Doctrine and Doctrinal Standards received near-unanimous approval from the General Conference for its formulation of a quadrilateral framework of scripture, tradition, reason, and experience for understanding John Wesley's norms of theology and Methodist doctrine. Despite the overwhelming support of the Conference, the quadrilateral would go on to become one of the most contentious expressions of Methodist doctrine in ensuing years. In this discussion, however, Outler's original intent for the quadrilateral has often been obscured. This article examines the formation of the quadrilateral in the Theological Study Commission and argues that the intended function of the quadrilateral was educational and ecumenical. It was intended to help Methodists discover the "wellsprings" of their own theological heritage, as well as offer opportunities for ecumenical engagement through processes of theological reflection and discourse. [ABSTRACT FROM AUTHOR]

In this work, we study the distortion of the exterior conformal modulus of a symmetric quadrilateral, when stretched in the direction of the abscissa axis with the coefficient . By using some facts from the theory of elliptic integrals, we confirm that the asymptotic behavior of this modulus does not depend on the shape of the boundary of the quadrilateral; moreover, it is equivalent to as . [ABSTRACT FROM AUTHOR]

Some research about identifying the differences between male and female students’ written communication skills has been done before. However, the previous research only focused on qualitative studies. Therefore, there is still a lack of empirical proof of the result. This study aims to test whether a significant difference exists between male and female students' written mathematical communication skills and identify their performance difficulties. The subjects were 14 male and 14 female students at the junior high school. The instrument was a test of mathematical communication skills on quadrilateral topics. The research method used was a mixed method with a sequential explanatory design. The data were analyzed using Mann-Whitney U to measure the difference between male and female students’ written mathematical communication skills, and document analysis was conducted to analyze their difficulties in performing written mathematical communication. The results showed that even though there is no significant difference between written mathematical communication skills between male and female students, there is a significant difference when viewed from each indicator. In addition, the difficulties experienced by male and female students are difficulties expressing mathematical ideas using mathematical symbols, using facts and concepts to solve problems, and operating algebraic forms.

Background and Objectives: To investigate the digital measurement method for the plate trajectory of dynamic anterior plate-screw system for quadrilateral plate (DAPSQ), and then design a third-generation DAPSQ plate that conforms to the needs of the Chinese population through collating a large sample anatomical data. Materials and Methods: Firstly, the length of the pubic region, quadrilateral region, iliac region, and the total length of the DAPSQ trajectory were measured by a digital measurement approach in 22 complete pelvic specimens. Then, the results were compared with the direct measurement of pelvic specimens to verify the reliability of the digital measurement method. Secondly, 504 cases (834 hemilateral pelvis) of adult pelvic CT images were collected from four medical centers in China. The four DAPSQ trajectory parameters were obtained with the digital measurement method. Finally, the third-generation DAPSQ plate was designed, and its applicability was verified. Results: There was no statistically significant difference in the four trajectory parameters when comparing the direct measurement method with the digital measurement method (p > 0.05). The average lengths of the pubic region, quadrilateral region, iliac region, and the total length in Chinese population were (60.96 ± 5.39) mm, (69.11 ± 5.28) mm, (84.40 ± 6.41) mm, and (214.46 ± 10.15) mm, respectively. Based on the measurement results, six models of the DAPSQ plate including small size (A1,A2), medium size (B1,B2), and the large size (C1,C2) were designed. The verification experiment showed that all these six type plates could meet the requirement of 94.36% cases. Conclusions: A reliable computerized method for measuring irregular pelvic structure was proposed, which not only provided an anatomical basis for the design of the third-generation DAPSQ plate, but also provided a reference for the design of other pelvic fixation devices. [ABSTRACT FROM AUTHOR]

А new formula for the area of an arbitrary quadrilateral is applied to derive formulae for the areas of some kinds of polygons with any number of sides; these formulae are further applied to find specific relationships for concrete number of sides. [ABSTRACT FROM AUTHOR]

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The inverse finite element method (iFEM) is a powerful tool for shape sensing and structural health monitoring and has several advantages with respect to some other existing approaches. In this study, a two-dimensional eight-node quadrilateral inverse finite element formulation is presented. The element is suitable for thin structures under in-plane loading conditions. To validate the accuracy and demonstrate the capability of the inverse element, four different numerical cases are considered for different loading and boundary conditions. iFEM analysis results are compared with regular finite element analysis results as the reference solution and very good agreement is observed between the two solutions, demonstrating the capability of the iFEM approach.

We study the mixed Dirichlet–Neumann problem for the Laplace equation in the complement of a bounded convex polygonal quadrilateral in the extended complex plane. The Dirichlet / Neumann conditions at opposite pairs of sides are { 0 , 1 } and { 0 , 0 } , resp. The solution to this problem is a harmonic function in the unbounded complement of the polygon known as the potential function of the quadrilateral. We compute the values of the potential function u including its value at infinity. The main result of this paper is Theorem 4.3 which yields a formula for u (∞) expressed in terms of the angles of the polygonal given quadrilateral and the well-known special functions. We use two independent numerical methods to illustrate our result. The first method is a Mathematica program and the second one is based on using the MATLAB toolbox PlgCirMap. The case of a quadrilateral, which is the exterior of the unit disc with four fixed points on its boundary, is considered as well. [ABSTRACT FROM AUTHOR]

We describe the asymptotic behavior of the conformal modulus of an unbounded doubly-connected domain, non-symmetric with respect to the coordinate axes, when stretched in the direction of the abscissa axis with coefficient . Therefore, we give a partial answer to a problem, suggested by M. Vourinen, for the case of an arbitrary unbounded domain. [ABSTRACT FROM AUTHOR]

This article presents research activity by outstanding high school students aimed at discovering interesting geometric conservation properties in the context of Van Aubel's theorem, specifically in the case where the quadrilateral first degenerates into a triangle and later into a straight line. For some properties, proofs are presented with explanations of every stage. This activity was accompanied by computer software that facilitates dynamic research into various forms. [ABSTRACT FROM AUTHOR]

A convex quadrilateral, Q, is called a midpoint diagonal quadrilateral if the intersection point of the diagonals of Q coincides with the midpoint of at least one of the diagonals of Q. A parallelogram, Þ, is a special case of a midpoint diagonal quadrilateral since the diagonals of Þ bisect one another. We prove two results about ellipses inscribed in midpoint diagonal quadrilaterals, which generalize properties of ellipses inscribed in parallelograms. First, Q is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in Q has tangency chords which are parallel to one of the diagonals of Q. Second, Q is a midpoint diagonal quadrilateral if and only if each ellipse inscribed in Q has a pair of conjugate diameters parallel to the diagonals of Q. Finally, we show that there is a unique ellipse, EI, of minimal eccentricity incribed in a midpoint diagonal quadrilateral, Q, and also that the unique pair of conjugate diameters parallel to the diagonals of Q are the equal conjugate diameters of EI. [ABSTRACT FROM AUTHOR]

For given graphs G 1 , G 2 , ⋯ , G k , k ≥ 2 , the k-color Ramsey number, denoted by R (G 1 , G 2 , … , G k) , is the smallest integer N such that if we arbitrarily color the edges of a complete graph of order N with k colors, then it always contains a monochromatic copy of G i in color i, for some 1 ≤ i ≤ k . Let C m be a cycle of length m and W n a wheel of order n + 1 . In this paper, we show that R (C 4 , C 4 , W n) ≤ n + 4 n + 5 + 3 for n = 42 , 48 , 49 , 50 , 51 , 52 or n ≥ 56 . Furthermore, we prove that R (C 4 , C 4 , W ℓ 2 - ℓ) ≤ ℓ 2 + ℓ + 2 for ℓ ≥ 9 , and if ℓ is a prime power, then the equality holds. [ABSTRACT FROM AUTHOR]

In this article we have studied some interesting properties of a Quadrilateral inscribed in a circle. We have shown using co-ordinate geometry that the point of intersection of the diagonals of such a quadrilateral and the point . of intersection of any two opposite sides (when they are not parallel) of that quadrilateral when joined by a straight line, the straight line meets the circle at a point at which the line joining this point and the point of intersection of the other two opposite sides (when they too are not parallel) of the quadrilateral is tangent to the circle. [ABSTRACT FROM AUTHOR]

We show that up to Euclidean motions, there are 4 infinite families of lattice equable kites given by corresponding Pell or Pell-like equations, exactly 5 lattice equable trapezoids (2 isosceles, 2 right, 1 singular) and 4 lattice equable cyclic quadrilaterals. We also show that with one exception, the interior diagonals of lattice equable quadrilaterals are irrational. [ABSTRACT FROM AUTHOR]

Prospective teachers are required to develop their creative thinking abilities. The creative thinking abilities can help prospective teachers in designing the learning process, one of which is in designing non-routine problems as an evaluation of learning. This study aims to describe the creative thinking abilities of six sixth-semester students of Mathematics Education at Sanata Dharma University in solving and posing open-ended problems on the topic of quadrilateral. Data collection methods used are written tests to solve and pose open-ended problems and interviews. Data analysis adopted the framework of qualitative data analysis namely data reduction, data display, and conclusion drawing. The results showed that (1) the students' creative thinking abilities in solving open-ended quadrilateral problems were categorized into four categories namely very poor (1 student), poor (3 students), fair (2 students) and good (1 student), (2) the students' creative thinking abilities in posing open-ended quadrilateral problems were categorized into three categories of creative thinking abilities namely very poor (2 students), poor (1 student) and very good (3 students).

Tujuan penelitian ini adalah untuk mendeskripsikan pedagogical content knowledge (PCK) guru matematika SMP di kabupaten Merauke pada materi segiempat yang ditinjau dari segi gender. Jenis penelitian ini adalah deskriptif kualitatif. Subjek terdiri atas dua guru perempuan dan dua guru laki-laki. Teknik pengumpulan data yang digunakan dalam penelitian ini yaitu penyebaran kuesioner PCK dan wawancara. Teknik analisis data yang digunakan yaitu triangulasi teknik yang meliputi reduksi data, penyajian data, dan menarik kesimpulan atau verifikasi. Hasil penelitian ini menunjukkan pengetahuan konten guru perempuan lebih baik daripada guru laki-laki. Untuk pengetahuan pedagogik keempat guru menunjukkan penguasaan tentang kurikulum dan kemampuan siswanya dengan sangat baik. Namun terkait rencana pembelajaran masing-masing guru memiliki rencana yang berbeda, ada yang menyusun RPP berdasarkan anjuran Kemendikbud 2020 dan ada pula yang menyusun RPP berdasarkan Kurikulum 2013, namun inti pembelajarannya sama, menggunakan metode pembelajaran diskusi dan tanya jawab untuk mengeksplorasi pengetahuan dan menemukan penyelesaian. Untuk pelaksanaan pembelajaran baik guru laki-laki maupun guru perempuan sama-sama representasi berupa benda-benda di sekitar siswa dengan harapan siswa dapat lebih mudah memahami materi bangun segiempat. Untuk pengetahuan terkait teknik evaluasi pembelajaran, baik guru laki-laki maupun guru perempuan memiliki pengetahuan dalam penilaian proses dan hasil belajar yang baik dengan mempertimbangkan tingkat pemahaman dan kemampuan siswa. The purpose of this paper is to describe the pedagogical content knowledge (PCK) of junior high school mathematics teachers in Merauke district on quadrilateral in terms of gender. This type of research is descriptive qualitative. Subjects consisted of one female teacher and one male teacher. Data collection techniques used in this study were distributing PCK questionnaires and interviews. The data analysis technique used is the triangulation technique which includes data reduction, data presentation, and drawing conclusions or verification. The results of this study indicate that female teachers' content knowledge is better than male teachers. For pedagogic knowledge, both female and male teachers demonstrated excellent mastery of the curriculum and the abilities of their students. However, regarding the learning plan, each teacher has a different plan, there are those who prepare lesson plans based on recommendations from the Ministry of Education and Culture 2020 and some are compiling lesson plans based on the 2013 Curriculum, but the core of learning is the same, using discussion and question and answer learning methods to explore knowledge and find solutions. For the implementation of learning, both male and female teachers are representations of objects around students with the hope that students can more easily understand the material of rectangular shapes. For knowledge related to learning evaluation techniques, both male and female teachers have good knowledge in assessing learning processes and outcomes by considering the level of understanding and ability of students

Abstract Background Surgical treatment of acetabular fractures involving a quadrilateral plate is a challenge to orthopedic surgeons. We have developed a novel fixation technique using a specially shaped reconstruction plate combined with several buttress screws of a quadrilateral plate which was also called a dynamic anterior plate-screw system for quadrilateral plate (DAPSQ) to treat acetabular fractures involving quadrilateral plate since 2005 (RP group). And the long-term follow-up results have confirmed the effectiveness and safety of this technique. After 2016, standardized titanium plate (STP group) of DAPSQ have been designed and applied. The aim of the study was to compare the clinical efficacy of anatomical plate and the reconstruction plate of DAPSQ in the treatment of quadrilateral plate fractures. Methods We led a propensity-matched cohort study of quadrilateral plate fractures. Twenty-two patients were included in the STP group during the inclusion period (2016–2018) and were matched to 22 cases in our database of the RP group (2008–2016). The primary outcome measures were the quality of reduction and functional outcomes. Intraoperative conditions were also compared. Results Of these 22 consecutive patients in the STP group, the mean age was 46.7 years and the most common fracture pattern was a both-column fracture (12 cases, 54.5%) according to Letournel-Judet classification. The mean follow-up period was 23.1 months (range 12–37). There were no significant differences between the two groups with regard to the quality of reduction using the Matta radiological criteria and functional outcomes evaluated by the modified Merle d’Aubigné score (P > 0.05). Compared with the RP group, the STP group had a shorter operation time (245.1 min vs. 286.8 min, P = 0.020), less intraoperative blood loss (1136.4 mL vs. 1777.3 mL, P = 0.014), and transfusion (780.9 vs. 1256.8 mL, P = 0.035). The complication rate was 18.2% in the STP group, and there was no significant difference compared with the RP group (36.4%) (P > 0.05). None of the cases in the two groups had quadrilateral screws entering the hip or implant failure. Conclusions The fixation of standardized titanium plate in quadrilateral plate fractures showed a similar result to the reconstruction plate, in terms of quality of reduction and functional outcome. The standardized titanium plate of DAPSQ has the advantages of a short operation time, less intraoperative bleeding, and blood transfusion, and it is worth further promotion and research.

Objectives To evaluate the efficacy and safety of a novel fixation technique referred to as the dynamic anterior plate–screw system for quadrilateral plate (DAPSQ) for complex acetabular fractures with quadrilateral plate involvement through the single ilioinguinal approach. Methods A total of 32 patients with acetabular fractures, selected between January 2009 and March 2016, were managed by DAPSQ with “quadrilateral screws” through single ilioinguinal approach. The primary outcomes measure was the reduction quality evaluated according to the Matta radiological criteria, and the functional outcomes were evaluated by the modified Merle d'Aubigné score at the last follow‐up. Secondary outcomes were postoperative complications and intraoperative conditions included, for instance, operation time and blood loss. Results Of these 32 consecutive patients, 19, 9, and 4 were classified as both columns, anterior column posterior hemi‐transverse, and T‐shaped fractures, respectively, and with an average of 47 months’ follow‐up. Anatomic reduction was obtained in 19 cases (59%), imperfect reduction in 9 cases (28%), and poor reduction in 4 cases (13%). The modified Merle d'Aubigné scores were excellent in 15 cases (47%), good in 13 cases (41%), fair in 2 cases (6%), and poor in 2 cases (6%). Three cases sustained temporary lateral femoral cutaneous nerve injuries. One patient had a superficial wound infection that resolved after debridement. Five patients had posttraumatic arthritis; one of them underwent total hip arthroplasty at 46 months. No cases had quadrilateral screws entering the hip joint. Conclusion The use of DAPSQ with quadrilateral screws is an effective and safe choice for complex acetabular fractures with quadrilateral plate involvement through the single ilioinguinal approach.

Background and Objectives: To investigate the digital measurement method for the plate trajectory of dynamic anterior plate-screw system for quadrilateral plate (DAPSQ), and then design a third-generation DAPSQ plate that conforms to the needs of the Chinese population through collating a large sample anatomical data. Materials and Methods: Firstly, the length of the pubic region, quadrilateral region, iliac region, and the total length of the DAPSQ trajectory were measured by a digital measurement approach in 22 complete pelvic specimens. Then, the results were compared with the direct measurement of pelvic specimens to verify the reliability of the digital measurement method. Secondly, 504 cases (834 hemilateral pelvis) of adult pelvic CT images were collected from four medical centers in China. The four DAPSQ trajectory parameters were obtained with the digital measurement method. Finally, the third-generation DAPSQ plate was designed, and its applicability was verified. Results: There was no statistically significant difference in the four trajectory parameters when comparing the direct measurement method with the digital measurement method (p > 0.05). The average lengths of the pubic region, quadrilateral region, iliac region, and the total length in Chinese population were (60.96 ± 5.39) mm, (69.11 ± 5.28) mm, (84.40 ± 6.41) mm, and (214.46 ± 10.15) mm, respectively. Based on the measurement results, six models of the DAPSQ plate including small size (A1,A2), medium size (B1,B2), and the large size (C1,C2) were designed. The verification experiment showed that all these six type plates could meet the requirement of 94.36% cases. Conclusions: A reliable computerized method for measuring irregular pelvic structure was proposed, which not only provided an anatomical basis for the design of the third-generation DAPSQ plate, but also provided a reference for the design of other pelvic fixation devices.

We describe the asymptotic behavior of the conformal modulus of an unbounded doubly connected planar domain, symmetric with respect to the coordinate axes, when stretched in the direction of the abscissa axis with coefficient tending to infinity. Therefore, we give a partial answer to a problem, suggested by M. Vourinen, for the case of unbounded domains. We also use classical theorems by Ahlfors and Warshavskii to investigate the asymptotical behavior of the conformal modulus of a quadrilateral under stretching. [ABSTRACT FROM AUTHOR]