Problems In Geometry Research Articles

Level of Critical Thinking Abilities of Madrasah Aliyah Negeri 1 Jombang Students in Solving Geometry Problems: A Review Based on Cognitive Style

Level of Critical Thinking Abilities of Madrasah Aliyah Negeri 1 Jombang Students in Solving Geometry Problems: A Review Based on Cognitive Style

A Probabilistic Approach in the Search Space of the Molecular Distance Geometry Problem.

The discovery of the three-dimensional shape of protein molecules using interatomic distance information from nuclear magnetic resonance (NMR) can be modeled as a discretizable molecular distance geometry problem (DMDGP). Due to its combinatorial characteristics, the problem is conventionally solved in the literature as a depth-first search in a binary tree. In this work, we introduce a new search strategy, which we call frequency-based search (FBS), that for the first time utilizes geometric information contained in the protein data bank (PDB). We encode the geometric configurations of 14,382 molecules derived from NMR experiments present in the PDB into binary strings. The obtained results show that the sample space of the binary strings extracted from the PDB does not follow a uniform distribution. Furthermore, we compare the runtime of the symmetry-based build-Up (SBBU) algorithm (the most efficient method in the literature to solve the DMDGP) combined with FBS and the depth-first search (DFS) in finding a solution, ascertaining that FBS performs better in about 70% of the cases.

Optimal Algorithm for the Planar Two-Center Problem

We study a fundamental problem in Computational Geometry, the planar two-center problem. In this problem, the input is a set $S$ of $n$ points in the plane and the goal is to find two smallest congruent disks whose union contains all points of $S$. A longstanding open problem has been to obtain an $O(n\log n)$-time algorithm for planar two-center, matching the $\Omega(n\log n)$ lower bound given by Eppstein [SODA'97]. Towards this, researchers have made a lot of efforts over decades. The previous best algorithm, given by Wang [SoCG'20], solves the problem in $O(n\log^2 n)$ time. In this paper, we present an $O(n\log n)$-time (deterministic) algorithm for planar two-center, which completely resolves this open problem.

Some Numerical Results of the Albedo Problem in Spherical Geometry with Linearly Anisotropic Scattering

We examine the effect of linear anisotropic scattering on the solution of spherical geometry transport problems. We consider the albedo problem of a solid homogeneous sphere with linearly anisotropic scattering and write two matrix equations to solve the albedo problem with the H N and SVD methods in pseudo slab geometry. These methods are based on Case method of singular eigenfunctions. The numerical results of both methods are in agreement with each other.

Elliptic geometry-based kernel matrix for improved biological sequence classification

Protein sequence classification plays a pivotal role in bioinformatics as it enables the comprehension of protein functions and their involvement in diverse biological processes. While numerous machine learning models have been proposed to tackle this challenge, traditional approaches face limitations in capturing the intricate relationships and hierarchical structures inherent in genomic sequences. These limitations stem from operating within high-dimensional non-Euclidean spaces. To address this issue, we introduce the application of the elliptic geometry-based approach for protein sequence classification. First, we transform the problem in elliptic geometry and integrate it with the Gaussian kernel to map the problem into the Mercer kernel. The Gaussian-Elliptic approach allows for the implicit mapping of data into a higher-dimensional feature space, enabling the capture of complex nonlinear relationships. This feature becomes particularly advantageous when dealing with hierarchical or tree-like structures commonly encountered in biological sequences. Experimental results highlight the effectiveness of the proposed model in protein sequence classification, showcasing the advantages of utilizing elliptic geometry in bioinformatics analyses. It outperforms state-of-the-art methods by achieving 76% and 84% accuracies for DNA and Protein datasets, respectively. Furthermore, we provide theoretical justifications for the proposed model. This study contributes to the burgeoning field of geometric deep learning, offering insights into the potential applications of elliptic representations in the analysis of biological data.

Mathematics anxiety of junior high school students in solving geometry problems

Mathematics anxiety of junior high school students in solving geometry problems

Critical thinking of masculine students with impulsive cognitive style in solving geometry problems

The problem and the aim of the study. Critical thinking has a vital role in education because it is one of the essential skills that students must possess to face future challenges. By thinking critically, students can learn how to identify problems, analyze information from various sources, evaluate alternatives, and make logical decisions. In solving mathematical problems, it is common to find masculine students who are highly capable and impulsive (fast in solving problems). Therefore, this study aims to explore the critical thinking of masculine students with an impulsive cognitive style in solving geometry problems. Research methods. The study involved 70 junior high school students sitting in grade 9. The selection of research subjects was carried out in three public schools in Pringsewu Regency, Lampung province in Indonesia, namely Public Junior High Schools in Pringsewu Regency-Lampung, namely: SMP Negeri 2 Pringsewu, SMP N 4 Pringsewu and SMP N 1 Ambarawa. The research subjects were masculine students with moderate mathematics ability and above who had an impulsive cognitive style. In recruiting the research subjects, the gender questionnaire, cognitive style questionnaire, and math ability test were used. Moreover, the research is a descriptive qualitative study. The primary data was collected through geometry problem-solving test results and in-depth interviews. Several procedures were also set for analyzing critical thinking data to achieve these goals. Lastly, data analysis was done through data reduction, presentation, interpretation, and conclusion drawing. Results. After 70 students were given a gender questionnaire, cognitive style and math ability test with a minimum score of 75 found as many as 7 students with masculine gender with impulsive cognitive style with high math ability. With scores (mean score masculine: 4.2 ≥ 4: median score masculine) and (mean score feminine: 4.6 ≤ 5 median score feminine). However, one student (S1) was recommended by the math teacher at the school where the research was conducted to be used as a research subject. After in-depth analysis and interviewing S1, it was found that there were 5 aspects of critical thinking in solving geometry problems on the circle theme, namely aspects of clarity, relevance, depth, logicality, and significance. In conclusion, from the qualitative data analysis it was found that during the problem solving process this study has shown aspects of critical thinking namely aspects of clarity, relevance, depth, logicality, and significance.

Boundary Value Problems in Ellipsoidal Geometry and Applications

Boundary Value Problems in Ellipsoidal Geometry and Applications

Investigating different answers to one Euclidean geometry problem: a case study of grade 12 mathematics examinations in South African secondary schools

This article investigates the tendency for distinctly different answers to a problem in Euclidean Geometry as shown in some grade 12 examination memos for secondary schools in South Africa. The study is based on a qualitative analysis of some examples in the final grade 12 examinations where solutions to specific problems are given in distinctly different forms. Both primary and secondary data were applied to the investigation. The secondary data was the literature review of studies on the problems students come across when solving Euclidean geometry problems, and the primary data was acquired from the example of a memorandum where distinctly different answers were given to the solution of a problem. The approach to the problem-solution presentation using distinctly different answers is appreciated as it broadens the understanding of problem-solving. However, does this approach not leave the markers ignorant of other solution options, thus disadvantaging the learners and denying them a pass?

Analytical Study on the Impact of Teaching-Learning of ‘Transformation Geometry’ in Secondary Schools in General using Van Hiele’s Model

An analysis of the methods involved in the teaching-learning of the topics related to Transformation Geometry’ is one of the topics in mathematics in which learners have not been performing very well. The purpose of the study is aimed to identify the difficulties of the learners encounter when learning transformation geometry using Van Hiele’s model. Data was analysed through a process of coding, categorizing, clustering and performance indicators corresponding to the Van Hiele’s model were used in the analysis of data. The result of the study indicated that learners, experienced difficulties to solve problems involving enlargement transformation geometry problems. Learners struggle to grasp the fundamental concepts of enlargement, particularly with the concept of the centre and scale factor. The study findings revealed the challenges faced by the students in comprehending geometrical figures that were transformed to a larger dimension. The Learners inability to understand the steps of transformation geometry because they had insufficient background knowledge on the topic is taken to relevance in this paper writing. The study focuses on the limitations, challenges and remedies being intended for better learning prospects.

공학 도구 기반의 보조 구성 도형 분석을 통한 논증 기하 학습 방법 탐색

This paper proposes proof methods for synthetic geometry problem by dividing it into a process of observing geometric properties of auxiliary constructions(auxiliary lines and auxiliary circles) drawn using technology tools without logical proof and a process of logically proving the observes geometric properties. Through this method, students can approach problems more easily and the utilization of technology tools is maximized, so each student can solve geometry problems that were difficult to solve using traditional methods. In addition, it can be expected that this method will play a meaningful role in improving students' logical thinking abilities by being able to solve problems with only basic geometry knowledge without using difficult geometric theorems.

Conjectural Criteria for the Most Singular Points of the Hilbert Schemes of Points

We provide conjectural necessary and (separately) sufficient conditions for the Hilbert scheme of points of a given length to have the maximum dimension tangent space at a point. The sufficient condition is claimed for 3D and reduces the original problem to a problem in convex geometry. Proving either of the two conjectural statements will in particular resolve a long-standing conjecture by Briançon and Iarrobino back in the ’70s for the case of the powers of the maximal ideal. Furthermore, for specific classes of lengths, we conjecturally classify points satisfying the conjectural sufficient conditions. This in particular (conjecturally) provides many new explicit families of examples of maximum dimension tangent space at a point of the Hilbert schemes of points of lengths strictly between two consecutive tetrahedral numbers ( 3 + k 3 ) .

Geometric thinking program to enhance problem-solving skills of primary school teacher candidates

Geometry is a mathematics subject that has been widely considered hard. Several problems in learning geometry lead to geometric problem-solving skills. After exploring various causes, theories, and processes of geometric thinking, there should be an evaluation program to optimize problem-solving skills. This study aims to (1) develop a geometric thinking evaluation program and (2) enhance geometric problem-solving skills observed from the students’ thinking level. The research and development adopted the Analysis, Design, Development, Implementation, and Evaluation (ADDIE) model, which started from the analysis stage involving four lecturers and 306 of students. The research was conducted at Primary School Teacher Education Department, Universitas Negeri Semarang. The process continued to the design stage, at which the evaluation program was drafted, followed by the development stage to test the validity. The last two phases included the trials in which the t-test and n-gain were applied to measure the program's effectivity and the evaluation phase aimed at assessing its practicality and the student's achievement in geometry. The established evaluation program covers an instrument of 39 multiple-choice items as an early assessment to determine the thinking level and learning trajectories as a scaffolding for improving geometric thinking skills. The trajectories, in the form of worksheets, were adjusted to the students' thinking level and were effective in assisting them in elevating to the informal reasoning level as well as refining their problem-solving skills.

Elastic-Degenerate String Matching with 1 Error or Mismatch

An elastic-degenerate (ED) string is a sequence of n finite sets of strings of total length N, introduced to represent a set of related DNA sequences, also known as a pangenome. The ED string matching (EDSM) problem consists in reporting all occurrences of a pattern of length m in an ED text. The EDSM problem has recently received some attention by the combinatorial pattern matching community, culminating in an O~(nmω-1)+O(N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\ ilde{O}}(nm^{\\omega -1})+\\mathcal {O}(N)$$\\end{document}-time algorithm [Bernardini et al., SIAM J. Comput. 2022], where ω\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\omega $$\\end{document} denotes the matrix multiplication exponent and the O~(·)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {\ ilde{O}}(\\cdot )$$\\end{document} notation suppresses polylog factors. In the k-EDSM problem, the approximate version of EDSM, we are asked to report all pattern occurrences with at most k errors. k-EDSM can be solved in O(k2mG+kN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(k^2mG+kN)$$\\end{document} time, under edit distance, or O(kmG+kN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(kmG+kN)$$\\end{document} time, under Hamming distance, where G denotes the total number of strings in the ED text [Bernardini et al., Theor. Comput. Sci. 2020]. Unfortunately, G is only bounded by N, and so even for k=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$k=1$$\\end{document}, the existing algorithms run in Ω(mN)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varOmega (mN)$$\\end{document} time in the worst case. In this paper we make progress in this direction. We show that 1-EDSM can be solved in O((nm2+N)logm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}((nm^2 + N)\\log m)$$\\end{document} or O(nm3+N)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(nm^3 + N)$$\\end{document} time under edit distance. For the decision version of the problem, we present a faster O(nm2logm+Nloglogm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(nm^2\\sqrt{\\log m} + N\\log \\log m)$$\\end{document}-time algorithm. We also show that 1-EDSM can be solved in O(nm2+Nlogm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {O}(nm^2 + N\\log m)$$\\end{document} time under Hamming distance. Our algorithms for edit distance rely on non-trivial reductions from 1-EDSM to special instances of classic computational geometry problems (2d rectangle stabbing or 2d range emptiness), which we show how to solve efficiently. In order to obtain an even faster algorithm for Hamming distance, we rely on employing and adapting the k-errata trees for indexing with errors [Cole et al., STOC 2004]. This is an extended version of a paper presented at LATIN 2022.

Self-efficacy of mathematics education students in solving geometry problems

The problem and the aim of the study. Teachers as key holders of student success must have selfefficacy so that the learning process goes well and gets optimal results. In other words, teachers who have good self-efficacy will be able to foster students' self-efficacy in working on math problems. Researchers propose to analyze the influence of mathematics education students' self-efficiency in solving geometric problems. Participants and Research methods. This research uses a combination of quantitative and qualitative methods, which aims to analyze the self-efficacy of mathematics education students in solving geometry problems. This research was conducted on 43 students STKIP YPUP Makassar, which consisted of 7 boys students and 36 girl students. Two instruments were used in this study, namely the Self-Efficacy Questionnaire (SEQ) and The Geometry Problem Solving Test (GPST). Forty-three students in the research sample. were asked to complete SEQ and GPST-1 with a time duration of 30 minutes and 80 minutes respectively, the implementation of which was directly supervised by the researcher and assisted by lecturers who taught geometry courses. The results were grouped into low, medium and high mathematics ability groups. Then one student from each group was selected to be the research subject. Results. Data analysis for the influence test obtained: (i) F count 56,863 with a significance of 0.001 < 0.05, meaning that students' self-efficacy and Geometry Values simultaneously have a significant influence on the geometry problem solving, and (ii) For SEQ, t count was 3,480. with a significance of 0.001 < 0.05, meaning that the self-efficacy partially has a significant influence on the geometry problem solving, while for GV, the t count is 8.775 with a significance of 0.001 < 0.05, meaning that Geometry Values partially has a significant influence on the geometry problem solving. LS can complete the understanding of aspects of the problem, MS is able to solve problems by understanding the problem and formulating a plan, and HS is able to solve problems on all indicators. Conclusion. Quantitatif: Self-efficacy and geometry values has a significant influence on students' geometry problem-solving. Qualitative: Each subject in each group has a different geometry problem solving process at each stage of problem solving.

Soft cells and the geometry of seashells.

A central problem of geometry is the tiling of space with simple structures. The classical solutions, such as triangles, squares, and hexagons in the plane and cubes and other polyhedra in three-dimensional space are built with sharp corners and flat faces. However, many tilings in Nature are characterized by shapes with curved edges, nonflat faces, and few, if any, sharp corners. An important question is then to relate prototypical sharp tilings to softer natural shapes. Here, we solve this problem by introducing a new class of shapes, the soft cells, minimizing the number of sharp corners and filling space as soft tilings. We prove that an infinite class of polyhedral tilings can be smoothly deformed into soft tilings and we construct the soft versions of all Dirichlet-Voronoi cells associated with point lattices in two and three dimensions. Remarkably, these ideal soft shapes, born out of geometry, are found abundantly in nature, from cells to shells.

Process Errors Committed By Senior Secondary School Students In Solving Problems In Geometry In Cross River State, Nigeria

Process Errors Committed By Senior Secondary School Students In Solving Problems In Geometry In Cross River State, Nigeria

Profile Of Students' Mathematical Literacy Skills Private High Schools In Yogyakarta In Differentiated Learning Reviewed In Terms Of Process Differentiation

This study aims to describe the implementation of differentiated learning in terms of differentiation of processes held in class XII MIPA in a high school in Yogyakarta. Differentiated learning is a systematic approach to designing curriculum and learning instruction for students who have a variety of abilities, interests, and learning needs. The research will also describe students' mathematical literacy skills in solving numeracy literacy problems in the fields of geometry and measurement, algebra, and data and uncertainty. This type of research is qualitative descriptive with quantitative assistance. The subjects of this study are 40 students in grade XII MIPA. Research data includes observation sheets and student worksheets. Observation data was processed by calculating the implementation of differentiated learning, while student worksheets were described, and categorized based on the same type of work into the level of PISA standard mathematical literacy ability. The first result of the study is that differentiated learning has been 82% well carried out and the second is the result based on the results of student work, there are 87.5% of students have level 2 mathematical literacy skills and 5% of students at level 3 in answering geometry and measurement problems. There were 2.5% of students with level 1 mathematical literacy skills and 97.5% of students with level 2 mathematical literacy skills in solving geometry and second type measurement problems. On the topic of algebra, there are 5% of students who have level 1 mathematical literacy skills, 80% of students at level 2, and 5% of students at level 3. On the topic of data and uncertainty, there are 87.5% of students who have level 3 mathematical literacy skills.

The classifications of o-monomials and of 2-to-1 binomials are equivalent

AbstractWe observe that on the binary finite fields the classification of 2-to-1 binomials is equivalent to the classification of o-monomials, which is a well-studied and elusive problem in finite geometry. This connection implies a complete classification of 2-to-1 binomials $$b=x^d+ux^e$$ b = x d + u x e for a large set of values of (d, e). Further, we show that a number of the known infinite families of 2-to-1 maps can be traced back to o-polynomials or to difference maps of APN maps. We also provide some connections between 2-to-1 maps and hyperovals in non-desarguesian planes.

Finite element solution of heat conduction in complex 3D geometries

This paper presents the use of the finite element method (FEM) to solve heat conduction problems in complex 3-dimensional geometries not amenable to analytical solutions. Heat conduction is important across engineering domains, but closed-form solutions only exist for basic shapes. For intricate real-world component geometries, numerical techniques like FEM must be applied. The paper outlines the mathematical formulation of FEM, starting from the heat conduction governing equations. The domain is discretized into a mesh of interconnected finite elements. Element equations are derived and assembled into a global matrix system relating nodal temperatures. Boundary conditions are imposed and the matrix equations solved to find the temperature distribution. An example problem analyzes steady state conduction in an L-shaped block with 90-degree corners and surface convection. Results show FEM can capture localized gradients and discontinuities difficult to model otherwise. Detailed temperature contours provide insight. FEM enables robust thermal simulation of complex 3D geometries with localized effects, expanding analysis capabilities beyond basic analytical shapes. Proper application of FEM is critical for accurate results.