Problems In Geometry Research Articles

A metric description of flexible octahedra

A new description of flexible Bricard octahedra is obtained using conditions in terms of edge lengths. It is suitable for the study of a number of problems in the metric geometry of octahedra and, in particular, for searching for a proof of the conjecture of Sabitov on the vanishing of all but the leading coefficients of the polynomial for the volume of a type $3$ octahedron. Bibliography: 17 titles.

ON ONE PROBLEM OF INTEGRAL GEOMETRY

The paper is devoted to a problem of integral geometry on a two-dimensional plane. In the band of the two-dimensional space , a two-parameter family of curves is given. Each of these curves consists of two arcs , which are adjacent to each other at the vertex of the curve at an angle , and the other ends rest on the axis. With respect to the branches of the curve , certain smoothness conditions are assumed. The family of curves can be parametrized using the coordinates of the vertices of the curves, that is, each curve from this family can be put in an unambiguous correspondence with the coordinates of its vertex. For each point , there is only one curve belonging to the specified family and having the point as its vertex. It is required to determine the function from the following equation for a given function where – region bounded curves and the axis . The theorem (uniqueness of the solution) is proved. If , then the solution of equation (*) is unique in the class of continuous finite functions with a carrier in . In proving the theorem, the invariance of the family of curves L(ξ,n) and weight functions to shifts along the abscissa axis is significantly used. Further, using the Fourier transform from the function with respect to the variable and the permissible transformations, a family of Volterra integral equations of the second kind is obtained with respect to the Fourier image from the function with respect to the first variable. The paper also provides an estimate of the conditional stability of the considered integral geometry problem

Дифференцирование линейных алгебр с единицей над полем

Linear algebras over a given field arise when studying various problems of algebra, analysis and geometry. The operation of differentiation, which originated in mathematical analysis, was transferred to the theory of linear algebras over a field, as well as to the theory of rings. The set of all differentiations of a linear algebra themselves form a linear algebra. This algebra is called the algebra of differentiations. At the same time, this algebra admits the structure of a Lie algebra. If the algebra whose differentiations are considered is finite-dimensional, then its Lie algebra of differentiations will also be finite-dimensional. Therefore, the­re is a natural problem of determining the dimension of the Lie algebras of the differentiations of the linear algebra under consideration or to obtain an estimate from above of the dimension of the algebra of differentiations. To solve these problems, a system of linear homogeneous equations is obtained, which is satisfied by the components of arbitrary differen­tia­tion. Evaluation of the rank of this system allows us to obtain an estimate from below of the rank of the matrix under consideration.

The quality of mathematical proficiency in solving geometry problem: difference cognitive independence and motivation

The quality of mathematical proficiency in solving geometry problem: difference cognitive independence and motivation

Distance geometry for word representations and applications

Many machine learning methods used for the treatment of sequential data often rely on the construction of vector representations of unitary entities (e.g. words in natural language processing, or k-mers in bioinformatics). Traditionally, these representations are constructed with optimization formulations arising from co-occurrence based models. In this work, we propose a new method to embed these entities based on the Distance Geometry Problem: find object positions based on a subset of their pairwise distances or inner products. Considering the empirical Pointwise Mutual Information as a surrogate for the inner product, we discuss two Distance Geometry based algorithms to obtain word vector representations. The main advantage of such algorithms is their significantly lower computational complexity in comparison with state-of-the-art word embedding methods, which allows us to obtain word vector representations much faster. Furthermore, numerical experiments indicate that our word vectors behave quite well on text classification tasks in natural language processing as well as regression tasks in bioinformatics.

Analisis Kecerdasan Spasial Siswa dalam Menyelesaikan Soal-Soal Geometri Berdasarkan Tingkat Berpikir Van Hiele

The purpose of this research is to describe, identify and analyze students' spatial intelligence in working on geometry problems according to Van Hiele's theory. The method used in this research is descriptive research with a qualitative approach. This research was conducted at SMP Negeri 2 Tarakan. The research instrument used was the Van Hiele geometric thinking level test and spatial ability test, interview guides and documentation. Prior to use, the test, an interview guide, was first validated. The data analysis technique will be used in 4 stages, namely collecting, reducing, presenting, and concluding all the data obtained. The results of this study are that students with high spatial intelligence (KST) are at the level of informal deductive thinking (level 3) where at this ability students are able to know the relationship between shapes. While students with moderate spatial intelligence (SCC) are at the level of analytical thinking (level 2), namely the ability to know and mention the properties of flat shapes. Students with low spatial intelligence (KSR) are at the level of visualization thinking (level 1). In conclusion, it is important for teachers to know students' abilities in learning geometry by paying attention to the level of geometric thinking, being able to use the Van Hiele Geometry learning model, increasing spatial intelligence and van Hiele levels can be done through training involving geometric media or software (geogebra).
 
 Keywords: spatial intelligence, geometry, level of thinking, Van Hiele

Improving Self-Regulated Learning and Creativity of Students through GeoGebra in Solving Spatial Geometry Problems

This study aims to determine the implementation of GeoGebra in solving problems to increase students' independence and creativity. The research subjects were 6 grade 12 students at SMA IPA HSPG Bekasi Odd Semester 2022/2023 Academic Year. This research method is qualitative, with data collection in task-based interviews. The main instrument in this study is the researcher and the auxiliary device, namely the problem-solving test (PST). Data analysis techniques in this study were carried out by (1) grouping data into independent learning and creativity categories, (2) presenting data in the form of narrative text, and (3) conducting analysis to conclude. There was an increase in learning independence in high-category students. Students were more enthusiastic about using GeoGebra in solving questions, and students' creativity has also increased by finding one new way. There was an increase in learning independence in moderate-category students. Students were increasingly enthusiastic about using GeoGebra in solving questions even though students were still unsure. Student creativity also increased by finding one new way, but it could have been better. There was an increase in learning independence in low-category students. Students started trying GeoGebra even though students could only make geometric shapes. There was no increase in student creativity.

Analysis of students’ anxiety based on Van Hiele’s levels in solving geometry problems

[English]: Mathematics anxiety emerges due to discomfort when dealing with mathematical problems, including problems in geometry. This qualitative research aims to analyze how students' anxiety in solving the problems referring to Van Hiele’s levels. The participants were sixty grade 7 students. Data was collected through a test, questionnaire, observations, and semi-structured interviews. It was then analyzed following the stages of condensation, presentation, and drawing and verifying conclusions. The research found that students at the visualization level have moderate anxiety and panic levels, students at the analysis level have moderate anxiety, and students with informal deduction levels have low anxiety. The high level of students' geometric thinking does not necessarily result in lower anxiety or vice versa. Math anxiety is able to encourage students, but at a certain level, it could be detrimental to students. [Bahasa]: Kecemasan matematika timbul salah satunya karena rasa tidak nyaman ketika berhadapan dengan masalah matematika termasuk dalam menyelesaikan masalah geometri. Penelitian kualitatif ini bertujuan untuk menganalisis kecemasan siswa dalam menyelesaikan soal geometri ditinjau dari level Van Hiele. Enam puluh siswa SMP kelas 7 dilibatkan dalam penelitian. Data penelitian dikumpulkan melalui tes, angket, observasi, dan wawancara semi terstruktur. Analisis data dilakukan melalui tiga tahap yaitu kondensasi data, penyajian data, serta penarikan dan verifikasi kesimpulan. Hasil penelitian menunjukkan bahwa siswa yang berada di level visualisasi memiliki tingkat kecemasan sedang dan panik, siswa pada level analisis memiliki kecemasan sedang, dan siswa pada level deduksi informal memiliki kecemasan rendah. Tingginya tingkat berpikir geometris siswa belum tentu mengakibatkan kecemasan yang rendah atau sebaliknya. Kecemasan matematika dapat mendukung siswa, akan tetapi pada tingkat kecemasan tertentu, dapat merugikan siswa.

Gender Specific Study of Students' Spatial Inference in Solving Geometry Problem

Pemikiran spasial sangat penting untuk akuisisi dan konstruksi. Pemikiran spasial dapat membantu siswa memahami konsep geometris. Siswa mulai memahami konsep geometri pada tahap prilaku. Pemikiran spasial siswa pada tahap perilaku konkret tidak sedikit mampu melakukan eksplorasi. Penelitian ini mengkaji kemampuan siswa menggunakan penalaran spasial untuk menyelesaikan masalah geometri berbasis gender. Jenis penelitian ini adalah deskriptif kualitatif. Melalui lembar kerja, siswa mampu melakukan penalaran spasial. Selain lembar kerja, pengumpulan data pendukung diperoleh melalui wawancara mendalam. Tes dan wawancara digunakan untuk mengumpulkan data. Sampel yang menjadi target pada penelitian ini adalah siswa sekolah dasar kelas lima, responden pria dan wanita. Hasil penelitian menunjukkan bahwa siswa laki-laki memiliki keunggulan penalaran dibandingkan siswa perempuan, khususnya dalam visualisasi spasial.

Students’ difficulties in solving geometry problems

This study is based on research findings that examines students' difficulties utilizing Polya's problem to solve geometry problems -solving stages. Thirty-one students were assigned to work on questions about the material in order to calculate the distance in space. Researchers examined each student’s work to determine which stage of Polya they struggled with. In-depth interviews were also conducted to clarify answers and investigate students' difficulties in solving math problems. The results showed that students who were able to answer completely and correctly, based on Polya’s stages in understanding the problem, compiling or thinking about a solution plan, solving the problem according to the plan and re- examining the procedure and the results of solving the answer, had a very low success rate. Scaffolding is given to students who have difficulty and recognition of the accuracy of correct answers for those who are successful. Keywords: Geometry, Polya’s problem-solving stages, students difficulties.

Integration of character education in android-based m-learning media on problem-solving ability

This study aims to produce M-learning media that is effective, practical, effective by including character education and the ability to solve problems in the media in cube material and building blocks. This study adopted the ADDIE development design model which included five main stages, namely analysis, design, development, implementation and evaluation. The subject of this study was a grade VIII student of SMP Negeri 10 Sungai Kakap, totaling 15 students. The instruments used in this study were validation sheets, questionnaires, and pretest and posttest questions. The results showed that the average level of verification of mobile learning media by the three experts was 85.56%, meets the very valid criteria. The cumulative score of teacher and student questionnaires reaches 88.66%, very practical. The results of inferential statistical tests from pretest and posttest using the t test, it was found that T_Hitung -t_table with significant α = 5% which means there is a significant difference in the ability to solve problems between before and after using mobile learning media so that it can be said that the media is classified as effective. Therefore, it can be concluded that the use of mobile learning media combined with character education on the ability to solve the geometry problem of junior high school students is valid, practical and effective.

Harnessing low dimensionality to visualize the antibody–virus landscape for influenza

Antibodies constitute a key line of defense against the diverse pathogens we encounter in our lives. Although the interactions between a single antibody and a single virus are routinely characterized in exquisite detail, the inherent tradeoffs between attributes such as potency and breadth remain unclear. Moreover, there is a wide gap between the discrete interactions of single antibodies and the collective behavior of antibody mixtures. Here we develop a form of antigenic cartography called a ‘neutralization landscape’ that visualizes and quantifies antibody–virus interactions for antibodies targeting the influenza hemagglutinin stem. This landscape transforms the potency–breadth tradeoff into a readily solvable geometry problem. With it, we decompose the collective neutralization from multiple antibodies to characterize the composition and functional properties of the stem antibodies within. Looking forward, this framework can leverage the serological assays routinely performed for influenza surveillance to analyze how an individual’s antibody repertoire evolves after vaccination or infection.

The smooth extension embedding method with Chebyshev polynomials

AbstractWe propose an implementation of the smooth extension embedding method (SEEM), first described by Agress and Guidotti's study, in the setting of Chebyshev polynomials. SEEM is a hybrid fictitious domain/collocation method which solves general boundary value problems in complex domains by recasting them as constrained optimization problems in a simple encompassing set. Previously, SEEM was introduced and implemented using a periodic box (read a torus) using Fourier series; here, it is implemented on a (non‐periodic) rectangle using Chebyshev polynomial expansions. This implementation has faster convergence on smaller grids. Numerical experiments will demonstrate that the method provides a simple, robust, efficient, and high order fictitious domain method which can solve elliptic and parabolic problems in complex geometries, with non‐constant coefficients, and for general boundary conditions. We consider applications to two and three dimensional boundary value problems as well as an initial boundary value problem via a genuinely space–time discretization.

Water content variation of railway track sub-ballast layer in seasonal frost area: A case study from Finland

Many previous international studies have found a clear relationship between railway track water content and track geometry problems. The proper drainage of substructures is important in ensuring the safe and efficient operation of tracks. This importance will be emphasized in the future with the increasing frequency of extreme weather phenomena and the raising axle weights of train traffic. In many studies, excessive water content has been found to cause excessive geometry deterioration and mud pumping. These problems are usually related to thin substructure layers. However, in this case study, Finland, track substructures are thick due to frost protection requirements and behave differently to thin substructures. In this study, the long-term water content of three instrumented test sites was monitored to study the seasonal and freeze–thaw effects, as well as the effects of drainage improvement made at one site. In a previously reported part of the same research project, the measured water contents were used in an extensive cyclic loading triaxial test series evaluating the loading resistance of substructure materials taken from the same sites. The drainage improvement resulted in a marked decrease in the average water content at the drainage improvement test site. Simultaneously, high momentary water content values also disappeared. The role of capillary water was observed to be less than expected on Finnish railways.

Elementary School Students’ Ways of Thinking in Geometry Through the Lens of Geometric Habits of Mind

Geometric habits of mind include several ways of thinking that can be used in the learning and teaching of geometry as well as the reasoning ability in the process of solving the geometry problem. Based on this framework, revealing students' geometric reasoning process can shed light to researchers and teachers in the selection and design of methods and materials. This study is aimed to investigate the geometric problem-solving processes of elementary school students based on the GHoM. In this qualitative study case study design was adopted. The participants of the study consisted of six elementary school students. In the interviews the students can easily use reasoning with the relationship GHoM in all tasks. Generalizing and investigating invariants habits can be seen in some of the tasks. However balancing exploration and reflection habits were appeared rare when compared to other GHoMs. It can be suggested that to reveal the ways of reasoning of the students, non-routine and open-ended geometry tasks similar to those designed for this research can be utilized in mathematics classes.

Reduced order models for thermal radiative transfer problems based on moment equations and data-driven approximations of the Eddington tensor

A new group of structure and asymptotic preserving reduced-order models (ROMs) for multidimensional nonlinear thermal radiative transfer (TRT) problems is presented. They are formulated by means of the nonlinear projective approach and data compression techniques. The nonlinear projection is applied to the Boltzmann transport equation (BTE) to derive a hierarchy of low-order moment equations. Approximation of the Eddington tensor that provides exact closure for the system of moment equations is found with projection-based data-driven methodologies. These include the (i) proper orthogonal decomposition (POD), (ii) dynamic mode decomposition (DMD) and (iii) a variant of the DMD. A parameterization is derived for this ROM for the temperature of radiation incoming to the problem domain (the radiation drive temperature). This parameterization is informed from results of a dimensionless study of the TRT problem. Analysis of the ROMs is performed on the classical Fleck-Cummings TRT multigroup test problem in 2D geometry with a radiation-driven Marshak wave. Numerical results are presented to demonstrate the performance of these ROMs for the simulation of evolving radiation and heat waves. Results show these models to be sufficiently accurate for practical computations with rather low-rank representations of the Eddington tensor. As the rank of the approximation is increased, the errors of solutions generated by the ROMs gradually decreases.

THE AREA METHOD AT THE PRIMARY LEVEL WITH MEDIAN LINE PROBLEMS

When solving any problem or geometry problem, we often have thoughts and find the other solutions or how to discover new problems from the original problem. For geometry problems, in order to have a new problem we often change or add sumptions/conclusions and etc. In this work, it comes from solvingthe median line problem in primary level. We will develop andexpand it to include new problems on the basis of changing theratio of each dividing point on the two sides of the triangle and/oradding appropriate assumptions.

Problem-Solving and Students' Use of Metacognitive Skills

We examined how high school students use their cognitive and metacognitive skills in the geometry problem-solving process. This research employed a mixed-methods descriptive sequential design. Data were collected in the 2019–2020 academic year at secondary education institutions in the central districts of Adana, Türkiye. Using stratified sampling, 313 students participated in the quantitative component, of 313 students 149 are girls and 164 were boys and they were all 15 years olds. Then, using extreme case sampling, 18 students were selected as participant of the qualitative component. Fourteen of them were boys and four were girls. Measures included the metacognitive skills scale, geometry problem test, thinking-aloud protocol, and an observation form. Descriptive statistics and content analysis were applied for data analysis. Results showed that students with high metacognitive skills used metacognitive skills more when solving geometry problems than students with low metacognitive skills. As the implication of the result it is suggested that attention should be paid to the development of students' metacognitive skills in schools. In this context, it would be beneficial to train teachers to develop metacognitive skills. In addition, there is a need to investigate the effect of metacognitive skills on learning in different learning areas.

Latest technologies in production, automated design systems, computer modelling, innovative teaching methods, graphic disciplines

The relevance of the study is due to the continuous development of automated design systems and global trends in the digital transformation of production at all levels of the product life cycle. Since one of the main principles of education is integration with science and production, this integration is the main condition for its further development and a factor in ensuring its quality. The product life cycle begins with project design, which students of technical specialities begin to master by studying descriptive geometry, engineering, and computer graphics. In this course, the skills of working in automated design systems are pawned, so its teaching should take place using updated technologies that the present needs. The purpose of the study is to analyse the experience of teaching graphic disciplines in higher educational institutions and to substantiate the methods of studying the discipline “Engineering and computer graphics” based on the author’s experience, considering trends in the development of digital production. The research work was conducted using general scientific methods by analysing scientific-technical information on the subjects covered and using the method of pedagogical experiment. The study suggests using computer-aided design systems starting with the study of descriptive geometry, provides practical ways to solve classical problems of descriptive geometry by means of automated design systems, defines the minimum skills within the course to ensure further assimilation of the capabilities of automated design systems in special courses, during the implementation of course projects and to reduce the period of adaptation of the future specialist to professional activities. The practical importance of the research is in the development of a methodology for the transition from conventional methods of studying descriptive geometry, which involve the use of drawing tools, to modern ones using computer-aided design systems, which contributes to their accelerated mastering by students to further apply them in the study of technical disciplines

Distinct angles in general position

The Erdős distinct distance problem is a ubiquitous problem in discrete geometry. Somewhat less well known is the Erdős distinct angle problem, the problem of finding the minimum number of distinct angles between n non-collinear points in the plane. Recent work has introduced bounds on a wide array of variants of this problem, inspired by analogous questions in the distance setting.In this short note, we improve the best known upper bound for the minimum number of distinct angles formed by n points in general position from O(nlog2⁡(7)) to O(n2). We consider a point-set to be in general position if no three points lie on a common line and no four lie on a common circle. Before this work, similar bounds relied on projections onto a generic plane from higher dimensional space. In this paper, we introduce a construction employing the geometric properties of a logarithmic spiral, sidestepping the need for a projection.We also apply this configuration to reduce the upper bound on the largest integer such that any set of n points in general position has a subset of that size with all distinct angles. This bound is decreased from O(nlog2⁡(7)/3) to O(n1/2).