Geometric Problems Research Articles

Geometric problem solving mediated by Minecraft Education Edition in fifth grade elementary school children with ADHD

En este trabajo se describe una actividad la cual contiene tres problemas de nociones geométricas, se valoran las fases de la resolución de problemas consideradas en el marco TRU de Schoenfeld y el desempeño matemático de los estudiantes por cada problema propuesto, teniendo en cuenta los derechos básicos de aprendizaje (DBA 5) y (DBA 7) para grado quinto en el marco referencia colombiano. Fue aplicada en estudiantes neurotípicos y con diagnóstico TDAH de la institución educativa Mauricio Nelson Visbal en el grado 5-02, de forma presencial, con la guía del docente, se analiza la resolución de problemas en los pensamientos métrico y espacial; además participan en la actividad, dos estudiantes con TDAH en el mismo grado de escolaridad de la institución educativa Docente de Turbaco, los cuales la realizan con la mediación de la plataforma Minecraft Education de Microsoft, ambas instituciones se localizan en el norte del departamento de Bolívar. El estudio adopta un enfoque cualitativo mediante un diseño de investigación de acción participativa. Se logró que los estudiantes que desarrollaron la actividad con el uso del recurso gamificado Minecraft Education Edition y con las consideraciones y estrategias metodológicas del modelo inclusivo TDR se destacaran en el desempeño matemático y asimismo en el domino de las fases de la resolución de problemas en comparación con aquellos estudiantes que no usaron el recurso tecnológico.

Preparing students for solving geometric problems of the EIA in mathematics by means of GeoGebra

Preparing students for solving geometric problems of the EIA in mathematics by means of GeoGebra

Narrowing the Pareto set on the basis of fuzzy information about the ratio of ODA preferences

A multi-criteria choice model is considered, which includes a set of possible options, a numerical vector criterion and a fuzzy binary preference relation of the decision-maker person (DPR). The task of multi-criteria selection is to choose one or more "best" options from the Pareto set, that is, to narrow this set taking into account the information about the preference ratio of the DPR. Narrowing is carried out according to the axiomatic approach. The paper considers the algorithm for narrowing the Pareto set based on an arbitrary finite set of "quanta" of fuzzy information about the preference ratio of the DPR. Accordingly, an algorithm is proposed that allows, on the basis of an arbitrary set of clear information, to construct an estimate from above for an unknown set of options to be chosen, that is, to narrow the Pareto set. The purpose of this paper is to extend this algorithm to the case of a fuzzy preference relation, where the DPR assigns different degrees of certainty to its reasoning. In the fuzzy case under consideration, the set of options to be chosen and the top estimate constructed for it are also fuzzy. In the first section of the article, the statement of the problem of multi-criteria selection is presented and the basic axioms are formulated. The second chapter is devoted to the description of the reduction of this task to the geometric problem of constructing a fuzzy double cone. In the third section, a generalization of the algorithm is given, which makes it possible to construct the generators of a fuzzy double cone. Based on these constituents, a new vector criterion is constructed, the Pareto set with respect to which is the desired narrowing of the original Pareto set. An illustrative example is considered in the fourth chapter.

An intelligent decision support system for solving optimized geometric design problems

An integrated intelligent approach for solving geometric design problems is studied. A general optimization placement problem of arbitrary shaped objects in a bounded container is formulated as a mathematical programming problem in terms of the phi-function technique. Various technological requirements (geometric and mechanical) are considered, including continuous translations and rotations of the objects, allowable distances between objects, prohibited zones in the container, balancing conditions, mechanical strength constraints. Solution strategies, methods, and algorithms to solve different variants of the model are discussed and illustrated with examples.

Random Access in Persistent Strings and Segment Selection

We consider compact representations of collections of similar strings that support random access queries. The collection of strings is given by a rooted tree where edges are labeled by an edit operation (inserting, deleting, or replacing a character) and a node represents the string obtained by applying the sequence of edit operations on the path from the root to the node. The goal is to compactly represent the entire collection while supporting fast random access to any part of a string in the collection. This problem captures natural scenarios such as representing the past history of an edited document or representing highly-repetitive collections. Given a tree with n nodes, we show how to represent the corresponding collection in O(n) space and \(O(\log n/ \log \log n)\) query time. This improves the previous time-space trade-offs for the problem. Additionally, we show a lower bound proving that the query time is optimal for any solution using near-linear space. To achieve our bounds for random access in persistent strings we show how to reduce the problem to the following natural geometric selection problem on line segments. Consider a set of horizontal line segments in the plane. Given parameters i and j, a segment selection query returns the j th smallest segment (the segment with the j th smallest y-coordinate) among the segments crossing the vertical line through x-coordinate i. The segment selection problem is to preprocess a set of horizontal line segments into a compact data structure that supports fast segment selection queries. We present a solution that uses O(n) space and support segment selection queries in \(O(\log n/ \log \log n)\) time, where n is the number of segments. Furthermore, we prove that that this query time is also optimal for any solution using near-linear space.

ОЗНАЧЕННЯ, ПОЗНАЧЕННЯ КОМПОЗИЦІЙНИХ МАТРИЦЬ ТА ЇХ АНАЛІЗ

The existing algebraic theory of matrices is created for the abbreviated recording of solutions and solutions to problems of linear algebra, that is, it "serves" the performance of operations with linear forms. Because of this, in our study we will call these traditional matrices - "algebraic", in contrast to composite matrices (compomatrices), which are introduced and developed by us. By numerical field we mean any set of numbers within which are always defined and can unambiguously perform four operations: addition, subtraction, multiplication and division. Since algebraic matrices are formed over the field K, to determine their elements over the field K must first formulate a problem, determine the initial conditions for it and make the appropriate linear algebraic equations, which determine the elements of algebraic matrices. It follows that the elements of algebraic matrices can’t be arbitrarily selected from the set of the field K. All of them are determined by certain algorithms, as a result, the elements of the algebraic matrix are always combination values, ie such that changing the value of any one of them, entails a change in the values of all its other elements. In other words, the linear transformation problem uniquely defines an algebraic matrix. Conversely, any algebraic matrix uniquely defines a linear transformation. Composite matrices (compomatrices) are completely different in nature and purpose. If algebraic matrices are designed for abbreviated writing and compact solution of problems presented by linear algebra in matrix form, computer matrices are designed for analytical formalization of geometric figures by compositional geometry and abbreviated writing and compact solving of geometric problems in analytical form. However, compositional geometry is radically different from analytical geometry in that the equations of geometric objects are formed relative to the basis points of the original GF, for which the problem is solved. Conversely, in analytical geometry, the equations are formed relative to the coordinate system in which the original GF is located. Keywords: basis points, geometric figure, compositional geometry, compomatrices, compositional geometric modeling, point polynomial.

On Embeddability of Unit Disk Graphs Onto Straight Lines

Unit disk graphs are the intersection graphs of unit radius disks in the Euclidean plane. Deciding whether there exists an embedding of a given unit disk graph, i.e. unit disk graph recognition, is an important geometric problem, and has many application areas. In general, this problem is known to be \(\exists \mathbb {R}\)-complete. In some applications, the objects that correspond to unit disks, have predefined (geometrical) structures to be placed on. Hence, many researchers attacked this problem by restricting the domain of the disk centers. Following the same line, we also describe a polynomial-time reduction which shows that deciding whether a graph can be realized as unit disks onto given straight lines is NP-hard, when the given lines are parallel to either x-axis or y-axis. Adjusting the reduction, we also show that this problem is NP-complete when the given lines are only parallel to x-axis. We obtain those results using the idea of the logic engine introduced by Bhatt and Cosmadakis in 1987.

Minimum Scan Cover and Variants: Theory and Experiments

We consider a spectrum of geometric optimization problems motivated by contexts such as satellite communication and astrophysics. In the problem Minimum Scan Cover with Angular Costs , we are given a graph G that is embedded in Euclidean space. The edges of G need to be scanned, i.e., probed from both of their vertices. To scan their edge, two vertices need to face each other; changing the heading of a vertex incurs some cost in terms of energy or rotation time that is proportional to the corresponding rotation angle. Our goal is to compute schedules that minimize the following objective functions: (i) in Minimum Makespan Scan Cover (MSC-MS), this is the time until all edges are scanned; (ii) in Minimum Total Energy Scan Cover (MSC-TE), the sum of all rotation angles; and (iii) in Minimum Bottleneck Energy Scan Cover (MSC-BE), the maximum total rotation angle at one vertex. Previous theoretical work on MSC-MS revealed a close connection to graph coloring and the cut cover problem, leading to hardness and approximability results. In this article, we present polynomial-time algorithms for one-dimensional (1D) instances of MSC-TE and MSC-BE but NP-hardness proofs for bipartite 2D instances. For bipartite graphs in 2D, we also give 2-approximation algorithms for both MSC-TE and MSC-BE. Most importantly, we provide a comprehensive study of practical methods for all three problems. We compare three different mixed-integer programming and two constraint programming approaches and show how to compute provably optimal solutions for geometric instances with up to 300 edges. Additionally, we compare the performance of different meta-heuristics for even larger instances.

An efficient moving pseudo-boundary MFS for void detection

We consider the method of fundamental solutions (MFS) for the determination of the boundary of a void. In the proposed formulation the location of the pseudo-boundary is not fixed. The MFS discretization of the corresponding inverse geometric boundary value problem yields a system of nonlinear equations in which the coefficients in the MFS approximation, the discrete radii in the polar parametrization, the coordinates of the centre of the void and the expansion and dilation coefficients of the pseudo-boundaries are unknown. For the minimization of the resulting functional we employ the nonlinear least squares minimization routine lsqnonlin from the MATLAB® optimization toolbox . In contrast to previous studies, we exploit the option which enables the user to provide the analytical expression for the Jacobian of the system, and show that, although tedious, this leads to spectacular savings in computational time. The case of multiple voids is also addressed.

Computational Thinking Situations in Mathematics classes: an analysis of Elementary School textbook collections

This study aims to analyze how textbook collections, approved by the Programa Nacional do Livro Didático, address the algorithms construction for solving geometric problems. This is a qualitative research, based on assumptions of Content Analysis. The analysis was based on situations related to Computational Thinking and geometric concepts, contained in eight collections of Elementary School textbooks, showing skills of the Base Nacional Comum Curricular. We found that of the skills associated with Computational Thinking, EF06MA23 contained the largest number of situations involving algorithms construction. As for the representations proposed for the organization of the algorithm, approximately 46% of the mobilizations carried out were approached by exploring natural language and 41% in the flowchart/schema representation. Furthermore, only 9 situations explore different types of representation in the algorithm construction. The algorithm represented in programming language was identified in only two collections, which suggest the use of visual programming environments/software.

On the Computation of Geometric Features of Spectra of Linear Operators on Hilbert Spaces

Computing spectra is a central problem in computational mathematics with an abundance of applications throughout the sciences. However, in many applications gaining an approximation of the spectrum is not enough. Often it is vital to determine geometric features of spectra such as Lebesgue measure, capacity or fractal dimensions, different types of spectral radii and numerical ranges, or to detect gaps in essential spectra and the corresponding failure of the finite section method. Despite new results on computing spectra and the substantial interest in these geometric problems, there remain no general methods able to compute such geometric features of spectra of infinite-dimensional operators. We provide the first algorithms for the computation of many of these long-standing problems (including the above). As demonstrated with computational examples, the new algorithms yield a library of new methods. Recent progress in computational spectral problems in infinite dimensions has led to the solvability complexity index (SCI) hierarchy, which classifies the difficulty of computational problems. These results reveal that infinite-dimensional spectral problems yield an intricate infinite classification theory determining which spectral problems can be solved and with which type of algorithm. This is very much related to S. Smale’s comprehensive program on the foundations of computational mathematics initiated in the 1980s. We classify the computation of geometric features of spectra in the SCI hierarchy, allowing us to precisely determine the boundaries of what computers can achieve (in any model of computation) and prove that our algorithms are optimal. We also provide a new universal technique for establishing lower bounds in the SCI hierarchy, which both greatly simplifies previous SCI arguments and allows new, formerly unattainable, classifications.

Features of organizing an elective course on non-standard problems for pupils of grades 5-6

Currently, one of the relevant topics is the inclusion of elective courses in the school curriculum. The education of creative potential, the development of logical thinking, a certain contribution to hardening will lead to a significant result. This article will focus on the elective course program for solving non-standard tasks for grades 5-6 of secondary school. The introduction provides some information about the logical problem. At the same time, the definition of a non-standard report is given. Information about the elective course has been announced. In addition, approaches to the organization of an elective course are provided. About twenty topics have been selected for the organization of this elective course. The topics listed are listed according to the number of hours. We are talking about the logical sequence of the received topics. We are talking about the fact that it is compiled in a logical sequence, based on the shedule plan of the secondary school and comparison on the topics covered. In particular, importance is also attached to the sequential arrangement of algebraic and geometric problems. In order to reveal the meaning of the chosen topic, reports on each topic are provided. The solution of all problems is shown. In the final part, we are talking about preparing pupils for the release of a series of reports on these topics. In the future, it is planned to issue a specific methodological manual for the elective course. Keywords: non-standard problems, elective course, logic, secondary school, methodical manual,  problem book.

Differential geometric formulation of hanging membranes: Shell membrane theory and variational principle

In this paper, we present differential geometric formulation of hanging membrane forms based on the equilibrium equations in the shell membrane theory and the geometric variational problems. We also present the equations of hanging membranes in isothermic coordinates which will be an effective representation for numerical analysis in our forthcoming paper.

Revisiting Geometrical Preservation Properties Using Proof Without Words and Interactive Technology

The complexity of the art of teaching and learning, especially in the field of mathematics, calls for a broader approach than the traditional status quote. Proof without words in geometry coupled with the dynamic and interactive GeoGebra applet have the capacity to be used in all levels of mathematics education from grade schools to higher education. It is the aim of this article to present a few open-ended geometrical problems where proofs without words and dynamic interactive GeoGebra applets can be used in mathematics education.

Method for Solving Inverse Geometric Problem of Magnetostatics for Surface Defects of a Soft Magnetic Ferromagnet

Method for Solving Inverse Geometric Problem of Magnetostatics for Surface Defects of a Soft Magnetic Ferromagnet

Analysis of critical thinking and communication skills in geometric transformation problem solving using TransGeo application

The research aims to analyze narratively critical thinking and communication skills using the Android-based TransGeo application on geometry transformation in junior high school. This study uses a qualitative with a descriptive method obtained from tests and interviews. The test contains contextual problems in the android application, while the interviews are based on student test answers. Then, test results are adjusted descriptively by presenting the achievement of each student to be categorized based on critical thinking and communication scoring. Next, one student was randomly selected from each category as the interview subject. The indicator of critical thinking skills is the most difficult to achieve in the fourth indicator. Students are less precise in writing conclusions related to the initial problem; the indicator that many students succeed is the first indicator where students have written what is known and asked in difficulties. The indicator of communication skills is the most difficult to achieve in the second indicator. Students did not describe or write mathematical sentences to present an explanation of the transformation type or final position area in the problem. Many students achieve the first indicator, where they have written what they know and are asked about.

Triangles, Fractals and Spaghetti

The well-known Broken Spaghetti Problem is a geometric problem which can be stated as: A stick of spaghetti breaks into three parts and all points of the stick have the same probability to be a breaking point. What is the probability that the three sticks, putting together, form a triangle?
 
 In this note, we describe a hidden geometric pattern behind the symmetric version of this problem, namely a fractal that parametrizes the sample space of this problem. Using that fractal, we address the question about the probability to obtain a δ-equilateral triangle.

Partial Differential Equations

The workshop covered topics in nonlinear elliptic and parabolic Partial Differential Equations as well as topics in Geometric Measure Theory, touching topics such as geometric variational problems and minimal surfaces, geometric flows, free boundaries and the structure of nodal sets of eigenfunctions as well as real and complex Monge–Ampère equations.

Axial Symmetry in Non-standard Problems with Connection to Musical Art

In the paper, we will focus on the relationship between the teaching subjects of mathematics and musical art from the point of view of solving non-standard geometric problems. We designed several problems in order to point out the connection between theoretical knowledge and practical life. We present interdisciplinary relations as one of the possible innovative forms of teaching in secondary school.

Identifikasi Penalaran Kreatif Siswa Madrasah Ibtidaiyah dalam Memecahkan Masalah Bangun Ruang

The reasoning is one of the many skills that must be mastered by students when working on math problems. One type of reasoning in mathematics is creative reasoning. Creative reasoning is classified into Local Creative Reasoning (LCR) and Global Creative Reasoning (GCR). The reasoning is called Local Creative Reasoning (LCR) if, in the problem-solving process, the steps are still memorizing or imitating, and only a small part uses creative reasoning. Meanwhile, the reasoning is called the Global Creative Reasoning (GCR) type if the problem-solving is not based on an algorithm and, as a whole, requires creative reasoning. This research aims to identify the types of creative reasoning of Islamic elementary school students in solving geometrical problems. This study used the descriptive qualitative method. The subjects of this study were fifth-grade students of MI Nurul Ulum Moropelang, consisting of 2 students with high mathematical abilities, 2 students with moderate mathematical abilities, and 2 students with low mathematical abilities. Data was collected through tests of mathematical ability, creative reasoning tests and interviews. The results showed that students who had high mathematical abilities were currently using Local Creative Reasoning (LCR) type creative reasoning. In contrast, those with low mathematical abilities did not use both types of creative reasoning in solving geometrical problems.