Geometric Problems Research Articles

Applications of a q-Integral Operator to a Certain Class of Analytic Functions Associated with a Symmetric Domain

In this article, our objective is to define and study a new subclass of analytic functions associated with the q-analogue of the sine function, operating in conjunction with a convolution operator. By manipulating the parameter q, we observe that the image of the unit disc under the q-sine function exhibits a visually appealing resemblance to a figure-eight shape that is symmetric about the real axis. Additionally, we investigate some important geometrical problems like necessary and sufficient conditions, coefficient bounds, Fekete-Szegö inequality, and partial sum results for the functions belonging to this newly defined subclass.

Method for prediction magnetic silencing of uncertain energy-saturated extended technical objects in prolate spheroidal coordinate system

Aim. Development of method for prediction by energy-saturated extended technical objects magnetic silencing based on magnetostatics geometric inverse problems solution and magnetic field spatial spheroidal harmonics calculated in prolate spheroidal coordinate system taking into account of technical objects magnetic characteristics uncertainties. Methodology. Spatial prolate spheroidal harmonics of extended technical objects magnetic field model calculated as magnetostatics geometric inverse problems solution in the form of nonlinear minimax optimization problem based on near field measurements for prediction far extended technical objects magnetic field magnitude. Nonlinear objective function calculated as the weighted sum of squared residuals between the measured and predicted magnetic field COMSOL Multiphysics software package used. Nonlinear minimax optimization problems solutions calculated based on particle swarm nonlinear optimization algorithms. Results. Results of prediction extended technical objects far magnetic field magnitude based on extended technical objects spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system using near field measurements with consideration of extended technical objects magnetic characteristics uncertainty. Originality. The method for prediction by extended technical objects magnetic cleanliness based on spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of magnetic characteristics uncertainty is developed. Practical value. The important practical problem of prediction extended technical objects magnetic silencing based on the spatial prolate spheroidal harmonics of the magnetic field model in the prolate spheroidal coordinate system with consideration of extended technical objects magnetic characteristics uncertainty solved. References 48, figures 2.

Reconstruction of the shape of a flaw in ferromagnetic plate by solving inverse problem of magnetostatics and series of direct problems

The article presents a verification technique for solving the inverse geometric problem of magnetostatics in a soft magnetic ferromagnet plate. The technique involves solving a number of direct problems, in which the shape of the defect obtained by solving the inverse geometric problem of magnetostatics is used as a first approximation, and then increasing or decreasing the depth of the defect without changing the shape of the boundary surface — comparing the topographies of the magnetic field components obtained during measurements above the plate surface and calculated (as a result of solving the direct problem) at the same points of the components of the magnetic stray field from the reconstructed three-dimensional defect. As a result of applying the technique, the geometric parameters of the defect under study can also be refined. Obtaining the initial conditions for solving the inverse problem and solving direct problems of magnetostatics is carried out using the finite element method in the ELMER program. The technique works with one-sided access to any surface of the plate (a defect-free surface or a surface with a defect).

Development of a flexible phased array electromagnetic acoustic testing system with array pickups

The thermal barrier coatings (TBCs) used to protect blades of heavy-duty gas turbines is prone to debonding defects during fabrication and service. It is difficult to detect the debonding defects in the TBC system non-destructively and in situ due to the curved shape of blade and the narrow space between the blades setting in the gas turbine. In this paper, a flexible phased array electromagnetic acoustic (FPA-EMA) testing system with array pickups is developed, which is capable to drive a flexible phased array electromagnetic acoustic transducer (FPA-EMAT) to generate and receive ultrasonic wave directly in the metallic substrate of TBCs, and is potential to be applied to the inspection of debonding defects in the TBC system of gas turbine blades. A phased array excitation unit, a multi-channel signal receiving unit and a new signal post-processing algorithm are developed for the new EMA testing system to enhance the surface acoustic wave in the layered structure and to improve the low conversion efficiency of the conventional EMA testing system. In addition, a bias magnetic field coil fed with long pulse current of large amplitude is adopted to make the new EMAT thin and flexible in order to be applied in a narrow space. The interfering noise and geometric size problems brought by the multiple excitation and pickup channels are well addressed in the new system. The good performances of the developed FPA-EMA testing system were verified experimentally and show the system of good potential to be applied to NDT of practical curved structures.

Structure Prediction of Large RNAs with AlphaFold3 Highlights its Capabilities and Limitations

DeepMind’s AlphaFold3 webserver offers exciting new opportunities to make structural predictions of heterogeneous macromolecular systems. Here we attempt to apply AlphaFold3 to large RNA molecules whose 3D atomic structures are unknown but whose physical dimensions have been studied experimentally. One difficulty that we encounter is that models returned by AlphaFold3 often contain severe steric clashes and, less frequently, clear breaks in the phosphodiester backbone, with the probability of both events increasing with the length of the RNA. Restricting attention to those RNAs for which non-clashing models can be obtained, we find that hydrodynamic radii computed from the AlphaFold3 models are much larger than those reported experimentally under low salt conditions but are in better agreement with those reported in the presence of polyvalent cations. For two RNAs whose shapes have been imaged experimentally, the computed anisotropies of the AlphaFold3-predicted structures are too low, indicating that they are excessively spherical; extending this analysis to larger RNAs shows that they become progressively more spherical with increasing length. Overall, the results suggest that AlphaFold3 is capable of producing plausible models for RNAs up to ∼2000 nucleotides in length, but that thousands of predictions may be required to obtain models free of geometric problems.

Analysis of the Work of Foreign Researchers on the Prevention of Typical Errors made by Students

The article reviews the theoretical literature on the issue of the occurrence and causes of errors in learning, identifies the types of errors. The author analyzes studies on typical errors made by students when mastering geometric problems. This article investigates how foreign researchers, including Schleppenbach and others, examine student errors in classroom settings. A key comparison is made between Chinese and American teachers, revealing different approaches: American teachers often hide or avoid discussing student mistakes, while Chinese teachers encourage students to reflect on their errors conceptually. This reflective process is essential for students to develop a deeper understanding of the material. Ball emphasizes that teachers should use mistakes as a learning tool, delving beyond simple "right or wrong" analyses. Newman's error analysis also plays a crucial role, highlighting stages where students encounter difficulties—reading, comprehension, transformation, and processing. The study also emphasizes spatial intelligence, including skills like spatial perception, mental rotation, and visualization, to assess how students solve geometric problems. Understanding the types of errors, such as procedural or encoding mistakes, is essential for improving teaching methods. Furthermore, researchers like Jacobs and Ambrose suggest that teachers can guide students to think critically about their mistakes, allowing them to correct misconceptions.

A Cognitive Load Approach to Molecular Geometries: Augmented Reality Technology and Visuospatial Abilities in Chemistry

Within chemistry education, methods for effectively teaching students the three-dimensional spatial arrangements of matter at the molecular level remains a topical issue. As a form of geometric problem solving, it requires learners to apply mental rotation abilities as an evolved visuospatial skill to obtain subject-specific content knowledge. Recent research into the use of Cognitive Load Theory (CLT) as a framework for instructional design in conjunction with augmented reality (AR) technology as a learning tool has begun to show promise in reducing unnecessary cognitive activity to improve learning. Yet, broader conclusions remain inconclusive, especially within the context of a learner’s mental rotation abilities. This study investigated the relationship between these factors by collecting data using a 2 × 3 experimental design that divided a sample of Year 10 students (n = 42) into two groups. The intervention group (n = 24) used mobile devices utilising AR technology with instructional 3D molecular geometry content featuring design principles based on CLT to encourage hand movements to rotate three-dimensional molecular structures. The non-AR-based control group (n = 18) was taught using traditional methods. Analysis of the data revealed participants using AR technology that featured CLT design principles experienced less cognitive load and improved achievement in post-testing compared to those taught using traditional methods, suggesting under certain conditions, the use of hand movement applied to AR design material improves learning.

Local geometric properties of conductive transmission eigenfunctions and applications

Abstract The purpose of the paper is twofold. First, we show that partial-data transmission eigenfunctions associated with a conductive boundary condition vanish locally around a polyhedral or conic corner in $\mathbb{R}^n$ , $n=2,3$ . Second, we apply the spectral property to the geometrical inverse scattering problem of determining the shape as well as its boundary impedance parameter of a conductive scatterer, independent of its medium content, by a single far-field measurement. We establish several new unique recovery results. The results extend the relevant ones in [26] in two directions: first, we consider a more general geometric setup where both polyhedral and conic corners are investigated, whereas in [26] only polyhedral corners are concerned; second, we significantly relax the regularity assumptions in [26] which is particularly useful for the geometrical inverse problem mentioned above. We develop novel technical strategies to achieve these new results.

Optimization of high-speed reducer in electric vehicle based on analysis of lubrication

PurposeThe purposes of this paper are optimization of high speed reducer in electric vehicles based on the analysis of lubrication and verification of simulation accuracy and optimization results.Design/methodology/approachThe traditional CFD method presents poor applicability to complex geometric problems due to grid deformity. Therefore, moving particle semi-implicit (MPS) method is applied in this study to simulate lubrication of the reducer and analyze the influence of input speed and lubrication system design on the distribution. According to the results, the reducer is optimized. Meanwhile, the experiments for lubrication and churning power loss is carried out to verify the accuracy of simulation and optimization effects.FindingsThe flow field of lubricant inside the reducer is obtained. The lubrication system of reducer needs to be improved. Simulation and experiment show that the optimization is sufficient and efficient.Originality/valueAccording to the simulation of lubrication, the reducer is optimized. The lubrication experimental setup is established. The conclusion of paper can provide the method and tool for reducer in electric vehicle.Peer reviewThe peer review history for this article is available at: https://publons.com/publon/10.1108/ILT-04-2024-0123/

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.

PPLICATION OF KINEMATIC APPROACHES IN GEOMETRIC PROBLEMS

This work is devoted to the use of kinematic methods in the process of solving geometric problems. The first part of the work is devoted to solving geometric problems by considering the movements of some image points. The minimum knowledge required to apply the kinematic method in elementary geometry is briefly outlined. Then a solution to a generalized version of a well-known geometric problem is presented. The importance of studying elements of operator calculus from an interdisciplinary perspective is also presented. The next part of the article examines problems selected according to the principle of gradual complication, from high school to bachelor’s degree. In these problems, the radius of curvature at various points is determined by kinematic way. Using the result of solving a kinematic problem about a body thrown in a horizontal direction, the expression for the radius of curvature of a parabola is determined. One physical method for determining the radius of curvature of a cycloid is presented. The problem of motion, which generates the equations of an ellipse and a hyperbola, is considered, and a kinematic version of determining the radii of curvature at the points of intersection of these curves and axes of symmetry is presented. The main methodological novelty of the work lies in the development, generalization and systematic presentation of kinematic problems that bring geometric results. Some methodological guidelines and arguments are also given to substantiate the importance of the discussed kinematic approaches to solving geometric problems.

The contribution of working memory and spatial perception to the ability to solve geometric problems

Geometry is a branch of mathematics that deals with the properties of space, including distance, shape, size, and the relative position of figures. It is one of the oldest branches of mathematics and has applications in various fields such as science, art, architecture, and even in areas seemingly unrelated to mathematics. Studies show that working memory and spatial perception contribute to students' geometry performance. This paper presents multiple studies demonstrating the brain regions activated when solving geometric problems. Interestingly, the brain areas activated when solving algebraic problems are different from those activated when solving geometric problems. Finally, multiple studies are presented that indicate students with learning difficulties lag in geometry, as solving geometric problems requires good reading and arithmetic skills.

Morrey regularity theory of Rivière’s equation

This note is devoted to developing Morrey regularity theory for the following system of Rivière − Δ u = Ω ⋅ ∇ u + f in B 2 , \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f f belongs to some Morrey space. Our results extend the L p L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24].

МЕКТЕП ГЕОМЕТРИЯСЫНДАГЫ ФИГУРАЛАРДЫН КЛАССИФИКАЦИЯСЫ

This work proposes a classification of geometric figures studied in a systematic course in school geometry. Classification of geometric shapes is said to be an important aspect of teaching geometry to enable students to understand and analyze different shapes; identify similarities and differences between figures; recognize common features and differences between figures belonging to the same class or type; systematize and organize knowledge; reveal patterns and relationships between figures with similar properties, which provides the basis for subsequent discoveries; derive new properties and theorems; solve spatial geometric problems. New areas of spatial perception (virtual space, multidimensional space, digital technologies, including cloud, etc.), technologies and industries are increasingly emerging. In such changes, the relevance of the science of geometry does not disappear, but rather increases. This is due to the fact that geometric knowledge and geometric thinking give a person the basis for solving various problems in theoretical (geometric) and practical (physical) spaces. However, despite this, geometry is considered a difficult subject to perceive and understand.

Streaming Algorithms for Geometric Steiner Forest

We consider a generalization of the Steiner tree problem, the Steiner forest problem , in the Euclidean plane: the input is a multiset \(X\subseteq{\mathbb{R}}^{2}\) , partitioned into \(k\) color classes \(C_{1},\ldots,C_{k}\subseteq X\) . The goal is to find a minimum-cost Euclidean graph \(G\) such that every color class \(C_{i}\) is connected in \(G\) . We study this Steiner forest problem in the streaming setting, where the stream consists of insertions and deletions of points to \(X\) . Each input point \(x {\in} X\) arrives with its color \(\mathsf{color}(x) {\in} [k]\) , and as usual for dynamic geometric streams, the input is restricted to the discrete grid \(\{1,\ldots,\Delta\}^{2}\) . We design a single-pass streaming algorithm that uses \(\operatorname{poly}(k\cdot\log\Delta)\) space and time, and estimates the cost of an optimal Steiner forest solution within ratio arbitrarily close to the famous Euclidean Steiner ratio \(\alpha_{2}\) (currently \(1.1547\leq\alpha_{2}\leq 1.214\) ). This approximation guarantee matches the state-of-the-art bound for streaming Steiner tree, i.e., when \(k=1\) , and it is a major open question to improve the ratio to \(1+\varepsilon\) even for this special case. Our approach relies on a novel combination of streaming techniques, like sampling and linear sketching, with the classical Arora-style dynamic-programming framework for geometric optimization problems, which usually requires large memory and so far has not been applied in the streaming setting. We complement our streaming algorithm for the Steiner forest problem with simple arguments showing that any finite multiplicative approximation requires \(\Omega(k)\) bits of space.

Applying matrix diagonalisation in the classroom with GeoGebra: parametrising the intersection of a sphere and plane

In this paper, we explore how the typical second year undergraduate topic of matrix diagonalisation can be applied to solve a geometric problem which arises frequently in Vector Calculus: to find the curve of intersection of a plane and a surface. We use GeoGebra, which offers a cost-free and dynamic interface, to explore a specific variant of this problem, namely the parametrisation of the intersection of a plane and sphere. GeoGebra not only allows students to confirm their calculations, but it can also provide motivation to create connections between prior learning experiences and new knowledge.

On some geometrical eigenvalue inverse problems involving the p-Laplacian operator

On some geometrical eigenvalue inverse problems involving the p-Laplacian operator

Applying a Fuzzy Ordering Approach in Transportation Problems with Decagonal Intuitionistic Fuzzy Numbers

In this study, the paper delves into precision challenges within traditional transportation problem solutions, which rigidly define cost, supply, and demand. Acknowledging the inherent vagueness in real world contexts, the research explores the efficacy of intuitive fuzzy sets as a potent tool. Organized into four distinct sections, this work utilizes decagonal intuitionistic fuzzy numbers for managing supply and demand, while upholding conventional approaches for cost considerations. Employing a fuzzy ordering method, optimal solutions are derived by adjusting the configuration of decagonal intuitive fuzzy numbers across each segment. Through a comparative analysis, the Study identifies the most effective solution, with initial sections addressing balanced geometric intuitionistic fuzzy transportation problems and the final part focusing on unbalanced scenarios, specifically emphasizing supply and demand complexities.

Application of meshless generalized finite difference method (GFDM) in single-phase coupled heat and mass transfer problem in three-dimensional porous media

This paper achieves effective and precise meshless modeling of three-dimensional (3D) single-phase coupled heat and mass transfer problems based on the generalized finite difference method (GFDM). It utilizes the Taylor formula and the weighted least squares method in the node influence domains to derive a generalized finite difference scheme for spatial derivatives of pressure and temperature. Consequently, a sequential coupled discrete scheme for the pressure diffusion equation and heat convection–conduction equation is formulated, resulting in the determination of pressure and temperature. An example conducts sensitivity analysis with different schemes of node collocation and different radius of influence domains. The calculation results demonstrate that this method exhibits good convergence. Two 3D model examples with regular and irregular boundaries illustrate the advantages of the GFDM in handling complex geometric problems within the computational domain, showcasing its superior flexibility and simplicity. This paper demonstrates the significant potential of GFDM in addressing complex geometric multi-physics field coupling challenges, offering innovative ideas for geothermal resource development, groundwater management, and thermal recovery in oil and gas reservoirs.

Analysıs Of Students' Problem-Solvıng Abılıtıes On Hıgher Order Thınkıng Skıll (Hots) Geometry Problems In Terms Of Spatıal Reasonıng

Geometric concepts in mathematics possess abstract properties that require a spatial reasoning process. Evaluating problem-solving abilities through HOTS (Higher-Order Thinking Skills) questions focusing on spatial reasoning is crucial for enhancing students' skills. The aims of this research are: (1) To understand the process of students' problem-solving skills in geometry HOTS based on spatial reasoning; (2) To analyze and pinpoint the challenges faced by students in solving geometry problems through HOTS questions that involve spatial reasoning. This study employs a qualitative approach with data analysis following the Miles and Huberman model, comprising data collection, data reduction, data presentation, and conclusion. The data gathered includes test outcomes, interviews, and documentation. The participants consist of 2 fifth-grade teachers and 3 students from SD N Lulut 05 and SD N Lulut 03. The findings indicate the necessity of regular training for students to effectively tackle HOTS questions and avoid difficulties. The test results showed that the spatial reasoning of the two schools reached 32% in the high category, while the HOTS geometry problem solving reached 30% in the high category. Students with strong spatial reasoning skills excel in solving HOTS problems, whereas those with moderate to low abilities encounter challenges in geometry problem-solving. Therefore, the spatial reasoning skills of students with different proficiency levels vary in addressing geometric problems compared to those with high mathematical abilities. A tailored learning strategy is essential to cater to the diverse needs of these students.