Problems In Geometry Research Articles

A uniqueness result for the inverse problem of identifying boundaries from weighted Radon transform

Abstract We study the problem of the integral geometry, in which the functions are integrated over hyperplanes in the n-dimensional Euclidean space, n = 2 ⁢ m + 1 {n=2m+1} . The integrand is the product of a function of n variables called the density and weight function depending on 2 ⁢ n {2n} variables. Such an integration is called here the weighted Radon transform, which coincides with the classical one if the weight function is equal to one. It is proved the uniqueness for the problem of determination of the surface on which the integrand is discontinuous.

A finite volume method based multi-group neutron diffusion model for space-time nuclear reactor kinetics problems in RZ geometry

The transient analysis of Fast Breeder Reactor (FBR) requires the development of high fidelity space-time nuclear reactor kinetics codes for the realistic prediction of spatial and temporal behavior of neutron flux and fission power along with the feedback effects. So, a space-time nuclear reactor kinetics model has been developed for solving the coupled time dependent multi-group neutron diffusion equations and the delayed neutron precursor equations in axi-symmetric cylindrical (r-z) geometry. The numerical model employs an orthogonal mesh finite volume method for spatial discretization and implicit method for time discretization. A computer code named MANDIR-KRZ (Multi-group Analysis of Neutron DIffusion in Reactor for Kinetics problems in RZ geometry) has been developed and employed to reconstruct the keff, transient neutron flux and power results of different benchmark problems. Mathematical formulation of the numerical model and results of the code for Dodds and FBR transient benchmark problems are discussed in this paper.

Physics-constrained neural network for solving discontinuous interface K-eigenvalue problem with application to reactor physics

Machine learning-based modeling of reactor physics problems has attracted increasing interest in recent years. Despite some progress in one-dimensional problems, there is still a paucity of benchmark studies that are easy to solve using traditional numerical methods albeit still challenging using neural networks for a wide range of practical problems. We present two networks, namely the Generalized Inverse Power Method Neural Network (GIPMNN) and Physics-Constrained GIPMNN (PC-GIPIMNN) to solve K-eigenvalue problems in neutron diffusion theory. GIPMNN follows the main idea of the inverse power method and determines the lowest eigenvalue using an iterative method. The PC-GIPMNN additionally enforces conservative interface conditions for the neutron flux. Meanwhile, Deep Ritz Method (DRM) directly solves the smallest eigenvalue by minimizing the eigenvalue in Rayleigh quotient form. A comprehensive study was conducted using GIPMNN, PC-GIPMNN, and DRM to solve problems of complex spatial geometry with variant material domains from the field of nuclear reactor physics. The methods were compared with the standard finite element method. The applicability and accuracy of the methods are reported and indicate that PC-GIPMNN outperforms GIPMNN and DRM.

Problems of Extrinsic Geometry of Foliations

Problems of Extrinsic Geometry of Foliations

Geometric Algebra for Optimal Control With Applications in Manipulation Tasks

Many problems in robotics are fundamentally problems of geometry, which have led to an increased research effort in geometric methods for robotics in recent years. The results were algorithms using the various frameworks of screw theory, Lie algebra, and dual quaternions. A unification and generalization of these popular formalisms can be found in geometric algebra. The aim of this article is to showcase the capabilities of geometric algebra when applied to robot manipulation tasks. In particular, the modeling of cost functions for optimal control can be done uniformly across different geometric primitives leading to a low symbolic complexity of the resulting expressions and a geometric intuitiveness. We demonstrate the usefulness, simplicity, and computational efficiency of geometric algebra in several experiments using a Franka Emika robot. The presented algorithms were implemented in c++20 and resulted in the publicly available library <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">gafro</i> . The benchmark shows faster computation of the kinematics than state-of-the-art robotics libraries.

A numerical-and-computational study on the impact of using quaternions in the branch-and-prune algorithm for exact discretizable distance geometry problems

A numerical-and-computational study on the impact of using quaternions in the branch-and-prune algorithm for exact discretizable distance geometry problems

Proving geometry theorems: Student prospective teachers’ perseverance and mathematical reasoning

Proof of geometry is a topic that involves mathematical reasoning abilities and relates to perseverance involving hard work, the spirit of achievement, and self-confidence. The current important problem that occurs at this time is that students who are future teachers of mathematics still experience difficulties in compiling proofs, especially those who are not challenged to work hard. This qualitative research explores mathematics teacher candidates' reasoning abilities and perseverance in proving geometric theorems. Therefore, the research design used a case study. There were three participants in this study, and they were student prospective mathematics teachers' s taking geometry courses. Data were collected through working documents, open questionnaires, and semi-structured interviews and were analyzed using iterative techniques consisting of data condensation, data exposure, and verification. The study's results showed that students' prospective teachers did not prioritize proof in solving geometry problems, even though they worked hard to solve the problems independently until they were finished. The students' perseverance also impacts their mathematical reasoning in proving geometric theorems. Students with more hard work values tend to have more reasoning values. The results of this study have implications that there needs to be an effort from the teacher to get used to giving proof questions to support students' perseverance and mathematical reasoning abilities.

Almost sharp weighted Sobolev trace inequalities in the unit ball under constraints

In this paper, we establish some improved weighted Sobolev trace inequalities [Formula: see text] under the zero higher order moments constraint via the concentration compactness principle, where [Formula: see text] is a defining function of [Formula: see text] and [Formula: see text]. This relates to the fractional (conformal) Laplacians and related problems in conformal geometry. We construct some test functions and show that the inequality is almost optimal when [Formula: see text].

Prescribed curvature measure problem in hyperbolic space

AbstractThe problem of the prescribed curvature measure is one of the important problems in differential geometry and nonlinear partial differential equations. In this paper, we consider the prescribed curvature measure problem in the hyperbolic space. We obtain the existence of star‐shaped k‐convex bodies with prescribed (n‐k)‐th curvature measures by establishing crucial C2 regularity estimates for solutions to the corresponding fully nonlinear PDE in the hyperbolic space.

Communication costs in a geometric communication network

We represent a communication network as a graph in which each node has only local information about the graph, except for an upper bound on the number of nodes, and nodes communicate by passing messages along its edges. Here, we consider a geometric communication network where the nodes also occupy points in space and the distance between points is the Euclidean distance. Our goal is to understand the communication cost needed to solve several fundamental geometry problems, including Farthest Pair, Convex Hull, Closest Pair, and approximations of these problems, in the asynchronous CONGEST KT1 model, where each node knows its ID and those of its neighbors. This extends the 2011 result of Rajsbaum and Urrutia for finding a convex hull of a planar geometric communication network to networks of arbitrary topology.We define a new model where each node has a position on the plane and nodes can communicate to each other if and only if there is an edge between them. We motivate the model and study a number of geometric problems in this model. We prove lower bounds on the communication complexity of the problems in this new model and present approximation algorithms for them. We prove lower bounds on the number of expected bits required for any randomized algorithm to solve the problems.

An explicit structural optimization method for temperature-sensitive hydrogel actuation devices

Hydrogel actuation devices are commonly composed of responsive and passive materials, and their actuation effect depends on their unique structure. In this paper, an explicit optimization method is proposed for the design of temperature-sensitive hydrogel-based soft devices involving swelling and shrinking to achieve specific actuation effects. In this method, the outline of configuration and the responsive layer are explicitly described by the ordered B-spline curve and movable hydrogel patch, respectively. The parametric modeling method is proposed to construct the geometric model with local features, which can transform the complex patch stacking problem into the fusion problem of three-dimensional geometry. Based on the large deformation theory of temperature-sensitive hydrogel, the deformation behavior of soft device is accurately simulated. The adaptive mesh division method is applied to numerical simulation of complex structures to avoid mesh distortion. Furthermore, the optimization problem is effectively solved by combining the gradient optimization algorithm. Several numerical examples are presented to verify the accuracy of the method, and interesting structures such as the Miura-ori structure, switch and bionic structures are designed. The method can significantly reduce the optimization design variables, and the explicit optimization results can be used as the basis for the additive manufacturing of soft devices.

Differential inclusions for the Schouten tensor and nonlinear eigenvalue problems in conformal geometry

Let g0 be a smooth Riemannian metric on a closed manifold Mn of dimension n≥3. We study the existence of a smooth metric g conformal to g0 whose Schouten tensor Ag satisfies the differential inclusion λ(g−1Ag)∈Γ on Mn, where Γ⊂Rn is a cone satisfying standard assumptions. Inclusions of this type are often assumed in the existence theory for fully nonlinear elliptic equations in conformal geometry. We assume the existence of a continuous metric g1 conformal to g0 satisfying λ(g1−1Ag1)∈Γ′‾ in the viscosity sense on Mn, together with a nondegenerate ellipticity condition, where Γ′=Γ or Γ′ is a cone slightly smaller than Γ. In fact, we prove not only the existence of metrics satisfying such differential inclusions, but also existence and uniqueness results for fully nonlinear eigenvalue problems for the Schouten tensor. We also give a number of geometric applications of our results. We show that the sign of a nonlinear eigenvalue for the σ2 operator is a conformal invariant in three dimensions. We also give a generalisation of a theorem of Aubin & Ehrlick on pinching of the Ricci curvature, and an application in the study of Green's functions for fully nonlinear Yamabe problems.

Errors and Misconceptions in Euclidean Geometry Problem Solving Questions: The Case of Grade 12 Learners

Euclidean geometry provides an opportunity for learners to learn argumentation and develop inductive and deductive reasoning. Despite the significance of Euclidean geometry for developing these skills, learner performance in mathematics, particularly geometry, remains a concern in many countries. Thus, the current study examined the nature of learners’ errors in Euclidean geometry problem-solving, particularly regarding the theorem for angle at the centre and its applications. Van Heile’s theory of geometric thinking and teacher knowledge of error analysis were used as conceptual frameworks to make sense of the nature of learners’ errors and misconceptions. Using a participatory action research approach, the study was operationalised by five mathematics teachers from four secondary schools in Motheo district in the Free State Province of South Africa and three academics from two local universities. The study analysed 50 sampled midyear examination scripts of Grade 12 learners from four schools. The findings of this study revealed that most learner errors resulted from concepts on Van Heile’s operating Levels 0 and 1, while the questions mainly required Level 3 thinking. The study recommends that teachers determine their learners’ level of geometric thinking and integrate this knowledge in their lesson preparations and material development.

Geometry Problems in Formalizing the Creation of Machining Processes on Machine Tools

Introduction: The main problems hindering the creation of fully automatic systems of design of parts and technologies of their production are considered. The search for and formalization of regularities and relationships between the design parameters of parts and the technology of their production is an urgent task. At the same time, it has not yet been possible to create a unified general theory of geometric construction and transformation of objects at the macro level. Purpose: To show and formulate the key problems of formalizing the design process and identify possible ways to solve them. Methods: The study uses a wide range of methods of algebra, geometry, and discrete mathematics, as well as the main provisions of the geometry of non-ideal objects. Results: The need for fundamental laws of generation of geometric configurations and their elements is a key formalization problem. The geometry of non-ideal objects allows us to formally solve these problems. A geometric object is represented in six-dimensional space (three linear and three angular dimensions) with binary non-uniform metrics. The considered geometry has a principal possibility to formally generate geometric configurations of objects and their elements. The geometrical structure of a part is unambiguously connected with a set of technological schemes providing the mutual arrangement of surfaces during machining. Rules for the formal generation of a complete set of technological structures that are guaranteed to provide the specified parameters during the machining of parts. Conclusion: Sets of surfaces and relations between them define the individual structure of geometric configuration for each specific object. This individuality can be effectively used for the formal recognition of geometric objects. The geometry of non-ideal objects can serve as a fundamental basis for the formal theory of the creation of original technological processes of their processing.

Level of visual geometry skill towards learning style Kolb in junior high school

This study aims to conduct an in-depth analysis of the visual thinking level of junior high school students with the learning style of assimilators, converges, accommodators, and divergers in solving geometry problems. The type of research used is qualitative research with a grounded theory and case study design. The subjects studied were junior high school students consisting of 6 of 56 students. Data were collected through a learning style inventory (LSI) test given to 56 students to group participants based on the learning style of the Kolb model, then a geometry problem-solving test and interviews were given to 6 students, namely two assimilator students, one converges, one accommodator, and two diverger students. The analysis is based on data from written test results and interviews. Then, time triangulation is carried out to obtain valid research data. The analysis was conducted based on data from written test results and interview results paired with video recordings. Then, triangulation of time is carried out to obtain valid research data. The results of the analysis showed that assimilator students and converger students were able to achieve at the global visual level, namely being able to carry out visual thinking activities well in solving problems, illustrate the problem correctly in geometric drawings/objects, represent problems in mathematical symbols precisely and can express relationships between images well. While accommodator and diverger students can only reach the local visual level, they have yet to be able to show every visual thinking activity well in solving geometry problems, illustrating problems in geometry drawings that could be more precise, and solving rudimentary geometry problems.

ANALISIS KESULITAN PENYELESAIAN SOAL GEOMETRI PADA SD ISLAM AL AZHAR 34 MAKASSAR

&#x0D; Geometry at the elementary school level brings various benefits to students. Firstly, it is a critical stage in the development of spatial abilities. Geometry helps students understand the world around them in different ways. This study aims to analyze the difficulties in solving geometry problems among elementary school students. The research design used is quantitative research with data collection techniques using tests and documentation. The research subjects are fifth-grade students of SD Islam Al Azhar 34 Makassar, consisting of 30 individuals. The research sample consists of 5 students. The results of the study indicate that visual-spatial understanding, perception, interpretation, and mathematical communication skills play a crucial role in students' understanding of geometric concepts. Student A needs to be introduced to the concept of classifying plane figures to improve their spatial understanding, while student B needs a deeper understanding of the types of plane figures. Student C needs to develop mathematical communication skills and pay attention to the instructions of the problems in order to accurately transfer their understanding of concepts. Student D has adequate conceptual understanding but needs to improve their accuracy in solving the final steps. Student E has good conceptual understanding of solving geometry problems but needs to be reminded of the importance of completing all solution steps accurately. An approach that includes introducing the concept of classifying plane figures, deepening understanding of the types of plane figures, developing mathematical communication skills, and paying attention to problem instructions needs to be implemented to enhance students' understanding of geometric concepts and their ability to solve problems accurately.

The Impact of Written Feedback in Geometry Problem Solving through a Gallery Walk

Traditionally, students experience difficulties solving problems involving geometric shapes and their properties. In line with the current curricular guidelines, it’s important to reflect about the use of active learning strategies, which directly engage students in meaningful mathematical activity, that contribute to reverse this situation. This paper refers to a study that aims to understand the influence of peer written feedback on students' performance in a Gallery Walk context, in the scope of problem solving involving areas of plane figures. We followed a qualitative, interpretative approach and collected data in a 5th grade class, with 19 students, through participant observation, a questionnaire, interviews, written productions and photographic records. Results show that, when the sender of feedback is a student, different types of feedback are used, however not all of them promote the improvement of the receiver’s performance. It was noticed that the most effective feedback is focused mainly on the emerging mathematical content, commenting on the student's performance in terms of task solution, process and results. It was also found that the feedback must be clear and simple so that the receiver clearly understands what the sender wants to convey.

ANALYSIS OF STUDENT’S MATHEMATICAL REASONING ABILITY IN LEARNING INDEPENDENCE ON GEOMETRY MATERIALS

One of the standards of competence included in the default development process is the ability to reason, which is one of the skills to be developed. In resolving potentially challenging problems, students need to develop the ability of a student's mathematical reasoning. It is, therefore essential to improve students' mathematical reasoning ability. The purpose of this study is to describe how junior high students use mathematical reasoning to solve geometry problems with their learning independence. This study used a descriptive qualitative research methodology using 30 class VIII junior high school students in Bekasi as research subjects. The methods used to obtain information include written tests for mathematical reasoning abilities, independent learning questionnaires, and interviews to find out whether the answers given are correct as they are and to strengthen the results of the previous written test. The results showed that students with the highest mathematical reasoning abilities also had a high level of learning independence from all subjects in this study. However, students who have not fully met the indicators of mathematical reasoning ability have good learning independence.

Combinatorial Gray Codes — an Updated Survey

A combinatorial Gray code for a class of objects is a listing that contains each object from the class exactly once such that any two consecutive objects in the list differ only by a `small change'. Such listings are known for many different combinatorial objects, including bitstrings, combinations, permutations, partitions, triangulations, but also for objects defined with respect to a fixed graph, such as spanning trees, perfect matchings or vertex colorings. This survey provides a comprehensive picture of the state-of-the-art of the research on combinatorial Gray codes. In particular, it gives an update on Savage's influential survey [C. D. Savage. A survey of combinatorial Gray codes. SIAM Rev., 39(4):605--629, 1997.], incorporating many more recent developments. We also elaborate on the connections to closely related problems in graph theory, algebra, order theory, geometry and algorithms, which embeds this research area into a broader context. Lastly, we collect and propose a number of challenging research problems, thus stimulating new research endeavors.

Optimal Geometry and Motion Coordination for Multisensor Target Tracking with Bearings-Only Measurements.

This paper focuses on the optimal geometry and motion coordination problem of mobile bearings-only sensors for improving target tracking performance. A general optimal sensor-target geometry is derived with uniform sensor-target distance using D-optimality for arbitrary n (n≥2) bearings-only sensors. The optimal geometry is characterized by the partition cases dividing n into the sum of integers no less than two. Then, a motion coordination method is developed to steer the sensors to reach the circular radius orbit (CRO) around the target with a minimum sensor-target distance and move with a circular formation. The sensors are first driven to approach the target directly when outside the CRO. When the sensor reaches the CRO, they are then allocated to different subsets according to the partition cases through matching the optimal geometry. The sensor motion is optimized under constraints to achieve the matched optimal geometry by minimizing the sum of the distance traveled by the sensors. Finally, two illustrative examples are used to demonstrate the effectiveness of the proposed approach.