Problems In Geometry Research Articles

Simultaneous identification of multi-parameter for power hardening elastoplastic problems in three-dimensional geometries

PurposeThe purpose of this study is to simultaneously determine the constitutive parameters and boundary conditions by solving inverse mechanical problems of power hardening elastoplastic materials in three-dimensional geometries.Design/methodology/approachThe power hardening elastoplastic problem is solved by the complex variable finite element method in software ABAQUS, based on a three-dimensional complex stress element using user-defined element subroutine. The complex-variable-differentiation method is introduced and used to accurately calculate the sensitivity coefficients in the multiple parameters identification method, and the Levenberg–Marquardt algorithm is applied to carry out the inversion.FindingsNumerical results indicate that the complex variable finite element method has good performance for solving elastoplastic problems of three-dimensional geometries. The inversion method is effective and accurate for simultaneously identifying multi-parameters of power hardening elastoplastic problems in three-dimensional geometries, which could be employed for solving inverse elastoplastic problems in engineering applications.Originality/valueThe constitutive parameters and boundary conditions are simultaneously identified for power hardening elastoplastic problems in three-dimensional geometries, which is much challenging in practical applications. The numerical results show that the inversion method has high accuracy, good stability, and fast convergence speed.

A sangaku problem involving two circles and an equilateral triangle

Summary We generalize a problem in Wasan geometry involving two circles and an equilateral triangle in terms of oriented circles and oriented lines, and get an invariant property of the figure consisting of an isosceles triangle of fixed height and a line passing through the apex of the triangle. Several results for the incircle and the excircles of the isosceles triangle and the right triangle are obtained as special cases of the generalization.

Uniform distribution and geometric incidence theory

A celebrated unit distance conjecture due to Erdős says that the unit distances cannot arise more than Cϵn1+ϵ times (for any ϵ>0) among n points in the Euclidean plane (see e.g. [8] and the references contained therein). In three dimensions, the conjectured bound is Cn43 (see e.g. [6] and [11]). In dimensions four and higher, this problem, in its general formulation, loses meaning because the Lens example shows that one can construct a set of n points in dimension 4 and higher where the unit distance arises ≈n2 times (see e.g. [1]). However, the Lens example is one-dimension in nature, which raises the possibility that the unit distance conjecture is still quite interesting in higher dimensions under additional structural assumptions on the point set. This point of view was explored in [4], [5], [7] and has led to some interesting connections between the unit distance problem and its continuous counterparts, especially the Falconer distance conjecture ([3]). In this paper, we study the unit distance problem and its variants under the assumption that the underlying family of point sets is uniformly distributed. We prove several incidence bounds in this setting and clarify some key properties of uniformly distributed sequences in the context of incidence problems in combinatorial geometry.

Meshless Monte Carlo radiation transfer method for curved geometries using signed distance functions.

.Significance: Monte Carlo radiation transfer (MCRT) is the gold standard for modeling light transport in turbid media. Typical MCRT models use voxels or meshes to approximate experimental geometry. A voxel-based geometry does not allow for the precise modeling of smooth curved surfaces, such as may be found in biological systems or food and drink packaging. Mesh-based geometry allows arbitrary complex shapes with smooth curved surfaces to be modeled. However, mesh-based models also suffer from issues such as the computational cost of generating meshes and inaccuracies in how meshes handle reflections and refractions.Aim: We present our algorithm, which we term signedMCRT (sMCRT), a geometry-based method that uses signed distance functions (SDF) to represent the geometry of the model. SDFs are capable of modeling smooth curved surfaces precisely while also modeling complex geometries.Approach: We show that using SDFs to represent the problem’s geometry is more precise than voxel and mesh-based methods.Results: sMCRT is validated against theoretical expressions, and voxel and mesh-based MCRT codes. We show that sMCRT can precisely model arbitrary complex geometries such as microvascular vessel network using SDFs. In comparison with the current state-of-the-art in MCRT methods specifically for curved surfaces, sMCRT is more precise for cases where the geometry can be defined using combinations of shapes.Conclusions: We believe that SDF-based MCRT models are a complementary method to voxel and mesh models in terms of being able to model complex geometries and accurately treat curved surfaces, with a focus on precise simulation of reflections and refractions. sMCRT is publicly available at https://github.com/lewisfish/signedMCRT.

Teacher-student interaction supporting students’ creative mathematical reasoning during problem solving using Scratch

ABSTRACT Studies have shown that learning mathematics through programming can be complex and that the programming itself might even hamper students’ learning. However, few studies have focused on the role of the teacher and the teacher-student interaction that aims to support students’ learning when using programming. The present study examines a didactic design in which a teacher uses well-prepared questions and suggestions that focus on students’ reasoning to solve mathematical problems using Scratch, a block-based programming environment. Forty students, 10–11 years old, solved a geometry problem using Scratch supported by their teacher. The students’ screen activities and teacher-student interactions were recorded. The results indicate that well-prepared general and task-specific questions targeting students’ creative reasoning can help to overcome some of the complexities of learning mathematics while programming. The study furthermore shows that when students have sufficient programming skills, Scratch could even have the potential to support their reasoning and at the same time guide the teacher to give timely feedback.

Ray Statement of the Acoustic Tomography Problem

The ray statement of the inverse problem of determining three unknown variable coefficients in the linear acoustic equation is studied. These coefficients are assumed to differ from given constants only inside some bounded domain. There are point pulse sources and acoustic receivers on the boundary of this domain. Acoustic signals are measured by a receiver near the moment of time at which the signal from a source arrives at the receiver. It is shown that this information makes it possible to uniquely determine all the three desired coefficients. Algorithmically, the original inverse problem splits into three subproblems solved successively. One of them is a well-known inverse kinematic problem (of determining the speed of sound), while the other two lead to the same integral geometry problem for a family of geodesic lines determined by the speed of sound.

Kemampuan Penalaran Matematis dalam Menyelesaikan Soal Geometri ditinjau dari Persepsi Siswa

Mathematical reasoning ability is a person's ability to think logically in drawing a conclusion from problem solving in the mathematics learning process. By thinking logically, students can have confidence that the mathematics they learn is reasonable so that it affects the mathematical information and knowledge that students receive. It will also lead to different perceptions of each student in participating in mathematics learning. To determine the ability of mathematical reasoning when viewed from the perception of students, it can be done by giving geometry questions. The purpose of this study was to describe the mathematical reasoning ability of students with positive and negative perceptions in working on geometry problems. This research uses a qualitative approach with exploratory descriptive research. This research was conducted at SMPN 2 Balongbendo with the research subjects of two students of class VIII E, with each of them being a student with positive perceptions and a student with negative perceptions. Data collection techniques used are tests and interviews, while to determine the validity of the data using time triangulation. The results of this study indicate that students with positive perceptions can achieve every indicator of mathematical reasoning ability. While students with negative perceptions have not been able to fulfill every indicator of mathematical reasoning ability.

Exploring the relationship between commognition and the Van Hiele theory for studying problem-solving discourse in Euclidean geometry education

This article is an advanced theoretical study as a result of a chapter from the first author’s PhD study. The aim of the article is to discuss the relationship between commognition and the Van Hiele theory for studying discourse during Euclidean geometry problem-solving. Commognition is a theoretical framework that can be used in mathematics education to explain mathematical thinking through one’s discourse during problem-solving. Commognition uses four elements that characterise mathematical discourse and the difference between ritualistic and explorative discourses to explain how one displays mastery of mathematical problem-solving. On the other hand, the Van Hiele theory characterises five levels of geometrical thinking during one’s geometry learning and development. These five levels are fixed and mastery of one level leads to the next, and there is no success in the next level without mastering the previous level. However, for the purpose of the Curriculum and Assessment Policy Statement (CAPS) we only focused on the first four Van Hiele levels. Findings from this theoretical review revealed that progress in the Van Hiele levels of geometrical thinking depends mainly on the discourse participation of the preservice teachers when solving geometry problems. In particular, an explorative discourse is required for the development in these four levels of geometrical thinking as compared to a ritualistic discourse participation.

A parabolic approach to the Calabi–Yau problem in HKT geometry

We consider the natural generalization of the parabolic Monge–Ampère equation to HKT geometry. We prove that in the compact case the equation has always a short-time solution and when the hypercomplex structure is locally flat and admits a compatible hyperkähler metric, then the equation has a long-time solution whose normalization converges to a solution of the quaternionic Monge–Ampère equation first introduced in Alekser and Verbitsky (Isr J Math 176:109–138, 2010). The result gives an alternative proof of a theorem of Alesker (Adv Math 241:192–219, 2013).

Rigid fibers of integrable systems on cotangent bundles

(Non-)displaceability of fibers of integrable systems has been an important problem in symplectic geometry. In this paper, for a large class of classical Liouville integrable systems containing the Lagrangian top, the Kovalevskaya top and the C. Neumann problem, we find a non-displaceable fiber for each of them. Moreover, we show that the non-displaceable fiber which we detect is the unique fiber which is non-displaceable from the zero-section. As a special case of this result, we also show the existence of a singular level set of a convex Hamiltonian, which is non-displaceable from the zero-section. To prove these results, we use the notion of superheaviness introduced by Entov and Polterovich.

DEVELOPMENT OF ELECTRONIC MODULE WITH AUGMENTED REALITY ON PYRAMID FOR VIII GRADE

In order to create a joyful learning environment, especially during a pandemic that requires learning to be carried out online, an innovation is needed such as the use of interactive learning media for mutual learning which can be accessed online. Electronic modules with Augmented Reality are packaged in the form of learning media that can be used independently everytime and everywhere. This study aims to develop an electronic module with android-based Augmented Reality named "AR LIMAS" as a media to improve students' abilities on pyramid geometry problem. The selection of pyramid material in the electronic module was based on several previous studies regarding the misconceptions experienced by students on pyramid. The development of this electronic module used the ADDIE research method by taking into account the scores ​​of validity, practicality, and effectiveness for the creation of good learning media. This electronic module was tested on 3 students of VII grade who have low, medium, and high mathematical ability. The categorization of students' mathematical ability was based on the final examination scores in the previous semester. Based on the assessment of experts, this electronic module was declared to have a good category with a validity percentage of 86.59%. Then the results of the questionnaire using the electronic module obtained a practicality percentage value of 98.35% with a very good category. The results of pretest with an average score of 15 increased in posttest with an average score of 82.5, so the module was considered effective in improving students' understanding on pyramid.
 Keywords: electronic module, android, augmented reality, pyramid, development.

Smooth solutions to the Gauss image problem

In this paper we study the the Gauss image problem, which is a generalization of the Aleksandrov problem in convex geometry. By considering a geometric flow involving Gauss curvature and functions of normal vectors and radial vectors, we obtain the existence of smooth solutions to this problem.

A thermalized electrokinetics model including stochastic reactions suitable for multiscale simulations of reaction–advection–diffusion systems

We introduce a scheme to simulate the spatial and temporal evolution of the densities of charged species, taking into account diffusion, thermal fluctuations, coupling to a carrier fluid, and chemical reactions. To this end, the diffusive fluxes in the electrokinetic model by Capuani et al. (2004) are supplemented with thermal fluctuations. Chemical reactions are included via an additional source term in the mass balance equation. The diffusion–reaction model is then coupled to a solver for fluctuating hydrodynamics based on the lattice Boltzmann method. This combination is particularly useful for soft matter simulations, due to the ability to couple particles to the lattice-Boltzmann fluid. These could, e.g., be charged colloids or polymers, which then interact with an ion distribution. We describe one implementations based on the automatic code generation tools pystencils and lbmpy, and another one that is contained in the molecular dynamics package ESPResSo and that allows for an easy coupling of particles to the density fields. We validate our implementations by comparing to several known analytic results. Our method can be applied to coarse-grained catalysis problems as well as to many other multi-scale problems that require the coupling of explicit-particle simulations to flow fields, diffusion, and reaction problems in arbitrary geometries.

Examining the effect of dynamic geometry software on supporting geometric habits of mind: A qualitative inquiry

Geometric habits of mind (GHoM) are the approaches that individuals use to solve geometry problems. This study aimed to determine the effect of dynamic geometry software (DGS) in activating the GHoM of pre-service mathematics teachers. The study was conducted with six teacher candidates who were enrolled in a semester-long course titled "Computer Assisted Mathematics Teaching" that entailed a focus on GHoM. In order to identify the impact of the DGS on their GHoM, clinical interviews were conducted with the participants before and after the course wherein they were asked questions relating to each habit. The preliminary interviews showed that, prior to the intervention, the pre-service teachers primarily used the software in the context of the habits of exploration and reflection and visualizing geometric shapes to find the solution of a given problem. On the other hand, the post-intervention interviews revealed that the DGS supported the habits of investigating invariants, reasoning with relationships and considering specific cases and generalizing geometric ideas. This paper discusses how the DGS supported each habit.

Conditioning problems for invariant sets of expanding piecewise affine mappings: application to loss of ergodicity in globally coupled maps

We propose a systematic approach to the construction of invariant union of polytopes (IUP) in expanding piecewise affine mappings whose linear components are isotropic scalings. The approach relies on using empirical information embedded in trajectories in order to infer, and then to solve, a so-called conditioning problem for some generating collection of polytopes. A conditioning problem consists of a series of requirements on the polytopes’ localisation and on the dynamical transitions between these elements. The core element of the approach is a reformulation of the problem as a set of piecewise linear inequalities for some matrices which encapsulate geometric constraints. In that way, the original topological puzzle is converted into a standard problem in computational geometry. This transformation involves an optimisation procedure that ensures that both problems are equivalent. As a proof of concept, the approach is applied to the study of the loss of ergodicity in basic examples of globally coupled maps. The study explains, completes and substantially extends previous achievements about asymmetric IUP in these systems. Comparison with the numerics reveals sharp existence conditions depending on the map parameters, and accurate fits of the empirical ergodic components. In addition, this application also reveals unanticipated features about conditioning problem solutions, especially as the dependence on the set of admissible face directions is concerned.

Secondary students’ spatial thinking in solving the minimum competency assessment (MCA) on geometry

Reflecting on PISA and TIMSS, international mathematical, reading, and science literacy assessments, the Indonesian government began implementing the Minimum Competency Assessment (MCA) for primary, secondary, and high school students. MCA demands the ability to solve daily life problems, one of which is geometry problems that require spatial ability. This research focused on describing spatial thinking ability used in numeracy competencies on geometry content. This research is descriptive with a qualitative approach. Thirty junior high school students in Malang, East Java, Indonesia, took the MCA preparation test and assessed their spatial thinking skills. The research was conducted in three stages: preparation, implementation, and analysis, and showed that three aspects of spatial ability, namely visualization, orientation, and relations, had not yet fully emerged. Some students cannot arrange the expected pattern. There are even students who ignore the geometric shapes requested. Students simplify drawing a hexagon with a quadrilateral, which are two different geometric objects, which leads to fatal errors. This study suggests that teachers emphasize learning mathematics that can train students' spatial thinking skills because it can lead to better mathematical understanding and performance.

A Minimax Model for Generalized Penetration Distance Between Convex Sets by Directed Hausdorff Distance

The penetration distance, which characterizes the depth of intersection between two sets, is a fundamental computational geometry problem with a wide variety of applications in image processing, robotics, and circuit design. However, the penetration distance (even in Euclidean metric) is typically challenging to handle because it is essentially a nonconvex nonlinear optimization involving the projection onto the complement of a convex set. In this letter, we explore the penetration distance in non-Euclidean metrics (e.g., Manhattan, Chebyshev, and hexagon distances) by deploying the gauge function in convex analysis, and reformulate it as a nonlinear equation by virtue of the directed Hausdorff distance. More precisely, the generalized penetration distance amounts to seeking the largest zero point of a nondecreasing convex “black-box” function, whose function values can be obtained by the recent algorithmic advances in minimax optimization. We develop a solver for calculating generalized penetration distance between compact convex sets (possibly non-polyhedron) in arbitrary <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$n$</tex-math></inline-formula> dimension. Numerical experiments on variant types of compact convex sets demonstrate that the proposed solver is compelling to render solutions with high accuracy.

A quantum approach to the discretizable molecular distance geometry problem

The Discretizable Molecular Distance Geometry Problem (DMDGP) aims to determine the three-dimensional protein structure using distance information from nuclear magnetic resonance experiments. The DMDGP has a finite number of candidate solutions and can be solved by combinatorial methods. We describe a quantum approach to the DMDGP by using Grover's algorithm with an appropriate oracle function, which is more efficient than classical methods that use brute force. We show computational results by implementing our scheme on IBM quantum computers with a small number of noisy qubits.

Some integral geometry problems for wave equations

We consider the Cauchy problem and the source problem for normally hyperbolic operators on the Minkowski spacetime, and study the determination of solutions from their integrals along light-like geodesics. For the Cauchy problem, we give a new proof of the stable determination result obtained by Vasy and Wang (2021 Commun. Math. Phys. 384 503–32). For the source problem, we obtain stable determination for sources with space-like singularities. Our proof is based on the microlocal analysis of the normal operator of the light ray transform composed with the parametrix for strictly hyperbolic operators.

A Global Fundamental Matrix Estimation Method of Planar Motion Based on Inlier Updating.

A fundamental matrix estimation based on matching points is a critical problem in epipolar geometry. In this paper, a global fundamental matrix estimation method based on inlier updating is proposed. Firstly, the coplanar constraint was incorporated into the solution of the fundamental matrix to reduce the number of parameters to be solved. Subsequently, an inlier updating matrix was introduced according to the threshold of the epipolar geometry distance to eliminate the potential outliers and obtain a reliable initial value of the fundamental matrix. On this basis, we employed a four-point iterative method to estimate the fundamental matrix and make it satisfy the rank constraint at the same time. Finally, the epipolar geometry in binocular vision was extended to triple-view, and the fundamental matrix obtained in the previous step was globally optimized by minimizing the coordinate deviation between the intersection point and feature point in each group of images. The experiments show that the proposed fundamental matrix estimation method is robust to noise and outliers. In the attitude measurement, the maximum static error was 0.104° and dynamic measurement error was superior to 0.273°, which improved the reconstruction accuracy of feature points. Indoor images were further used to test the method, and the mean rotation angle error was 0.362°. The results demonstrate that the estimation method proposed in this paper has a good practical application prospect in multi-view 3D reconstruction and visual localization.