Concepts Of Geometry Research Articles

Mathematical foundation of High-Dimensional Data Analysis: Leveraging Topology and Geometry for Enhanced Model Interpretability in AI

One of the most important challenges for modern AI and machine learning is the analysis of high-dimensional data. Traditional methods face serious complications in such cases due to high complexity of datasets: the curse of dimensionality, overfitting, and lack of transparency of model behavior. In this paper, we adopt a novel approach to analyze high-dimensional data; topological and geometric techniques will be exploited, taking advantage of better model interpretability and deeper insights into the structure. Precisely, we discuss Topological Data Analysis, mainly Persistent Homology (Edelsbrunner et al., 2002), which allows the extraction of topological features-like loops and connected components that enable the extracting knowledge about the global structure of data. We also see how some concepts of differential geometry and Riemannian geometry (Do Carmo, 1976) can be used to cast light on manifold data structure lying at the heart of any attempt at modeling intrinsic patterns in high-dimensional spaces. We will review how these mathematical pillars, combined with state-of-the-art techniques for dimensionality reduction like t-SNE, UMAP, Principal Component Analysis, are able to provide interpretable and low-dimensional representations of high-dimensional data that can be used to understand models and make decisions. Case studies are also included, which explain the practical working of these methods in AI systems and show how much complex models can be made transparent using these, especially in domains that are very critical, such as healthcare (Caruana et al., 2015), finance (Chen et al., 2018), and autonomous systems ( Wang et al., 2019). We also discuss some of the difficulties in using these methods for practical applications: computational complexity; the need for large-scale data processing (Bengio et al., 2007); and integration of topological and geometric intuition with the rest of the machine learning pipeline (Zhu et al., 2020). We conclude with possible future directions of research toward fine-tuning these methods and exploring their broader applicability to AI in its quest for more robust, interpretable, and reliable AI models. Given this work, we focus on how linking topology, geometry, and AI bears great promise for solving one of today's critical challenges: model interpretability in high-dimensional data analysis.

Ekplorasi Etnomatematika Kain Tenun Kampung Bokuna di Kecamatan Wewewa

In general, the North Wewewa Wewewa Woven Fabric of Bokuna Village is passed down from the ancestors and followed until now this Woven Fabric has beautiful motifs made based on daily life and also from nature. Ethnomathematics is a bridge between culture and mathematics, ethmomathematics is adapted by cultural groups to see that the culture contains mathematical aspects. This research aims to find out the concept of geometry in Woven Fabrics. The data collected is in the form of documentation, interviews, and observation results. The type of research used in qualitative descriptive is to describe the mathematical elements contained in woven fabrics. Data collection uses observation and interviews. Data analysis is carried out qualitatively with the stages of data reduction, data presentation, and conclusion drawing. The results of the analysis show that Woven Fabrics are sacred crafts that are inherited from ancestors or from generation to generation. In the study, the researcher found several one-dimensional and two-dimensional buildings. The geometric shapes found by the researcher are: rectangular shape on the entire shape of the woven fabric, an equilateral triangle shape on the mamuli motif, an isosceles triangle shape on the flower motif, a bele ketupat shape on the star motif, a rectangular shape on the entire woven fabric, a square shape on the butterfly motif

Role of Linear Diophantine Equations in RSA Encryption

Abstract. Diophantine equations are mathematical equations that include only integer solutions and two or more unknown variables. To illustrate, the most elementary form of Diophantine equations are linear Diophantine equations when two different one-degree monomials added up to a constant value. Such equations were given the name of the extraordinary Greek mathematician Diophantus who made great contribution to algebra and number theory. In addition, they help in defining concepts in algebraic geometry and contribute to the development of algorithms for cryptography. They work in the same way as public keys do in cryptocurrencies within blockchain technology to curb cyber threats and scams targeting public systems. They serve as a means of securing transactions on a computer network when using cryptocurrencies such as bitcoins. These encryption techniques also permit cryptographers and mathematicians to create new ideas in relation to these number systems, thus making it possible for further cryptographic techniques to be put into place.

Symmetry Preservation in Hamiltonian Systems: Simulation and Learning

This work presents a general geometric framework for simulating and learning the dynamics of Hamiltonian systems that are invariant under a Lie group of transformations. This means that a group of symmetries is known to act on the system respecting its dynamics and, as a consequence, Noether’s theorem, conserved quantities are observed. We propose to simulate and learn the mappings of interest through the construction of G-invariant Lagrangian submanifolds, which are pivotal objects in symplectic geometry. A notable property of our constructions is that the simulated/learned dynamics also preserves the same conserved quantities as the original system, resulting in a more faithful surrogate of the original dynamics than non-symmetry aware methods, and in a more accurate predictor of non-observed trajectories. Furthermore, our setting is able to simulate/learn not only Hamiltonian flows, but any Lie group-equivariant symplectic transformation. Our designs leverage pivotal techniques and concepts in symplectic geometry and geometric mechanics: reduction theory, Noether’s theorem, Lagrangian submanifolds, momentum mappings, and coisotropic reduction among others. We also present methods to learn Poisson transformations while preserving the underlying geometry and how to endow non-geometric integrators with geometric properties. Thus, this work presents a novel attempt to harness the power of symplectic and Poisson geometry toward simulating and learning problems.

Penggunaan Teknologi Augmented Reality pada Aplikasi Bangun Ruang Sederhana Berbasis Unity dan Vuforia Engine

Advances in augmented reality (AR) technology have opened up new possibilities in learning, such as help with mathematics. The aim of this research is to develop a simple interactive learning application for building rooms based on AR technology using Unity and Vuforia Engine. This application is designed to help students understand spatial geometry concepts visually and interactively so as to increase interest and understanding of the subject matter. This application development uses Unity as a development platform and Vuforia Engine as an SDK to facilitate AR integration. This application creates a more interesting learning experience and supports the visualization of geometric objects such as cones, cubes, tubes, balls and pyramids. AR technology allows students to interact with these objects in a 3D environment, which can be very helpful in clarifying concepts and formulas that are difficult to understand using traditional learning media. It is hoped that this research can provide innovative solutions to improve the quality of geometry learning in schools.

Poisson’s ratio-LambdaRho rock-physics templates and a study on sensitivity of different fluid indicators

Seismic fluid indicators are highly accurate when calibrated, but when the data is scarce, an interpreter must choose between different indicators. We investigate the data from multiple basins to understand the most effective fluid indicators for use. Unlike previous studies, the data comprises a large, equal, and statistically meaningful number of hydrocarbon and brine sands from different basins. We show that Poisson’s ratio-LambdaRho templates can be used to separate different facies in synthetic and real data and can be used as a useful crossplot interpretation tool. We develop a fluid indicator, as LambdaRho Poisson’s ratio multiplication [Formula: see text], which is highly sensitive to saturation and can be potentially used as a fizz-gas discriminator. We compare the sensitivity of different fluid indicators to hydrocarbon saturation by using Bhattacharyya distance, which provides a quantitative metric for how well different indicators separate hydrocarbon and brine sands, and it is a nonparametric measure unlike other fluid indicator scores developed in similar studies. Absolute properties derived from seismic by inversion are rarely available in regional studies, whereas relative elastic properties can be easily obtained and used. Following the concepts of elastic reflectivity vectors and geometry of intercept-gradient crossplots, we show how the reflectivity of different fluid indicators can be approximated from amplitude variation with offset (AVO) parameters at various chi ([Formula: see text]) angles, enabling an interpreter to use them even when inversion products are not available. Finally, we compare the effectiveness of relative elastic parameters on different AVO classes and show that no single attribute works best across all classes but for general screening purposes fluid factor and [Formula: see text] can be good choices. The findings of this study can help better characterization of fluids in exploration, appraisal, and development of hydrocarbons, and in other areas where monitoring produced and injected fluids is important like 4D seismic or carbon capture storage.

Exploring geometry and measurement in the Perjuangan Subkoss Garuda Sriwijaya Museum: An ethnomathematics study

The Museum of Perjuangan Subkoss Garuda Sriwijaya is a museum that preserves the history of South Sumatra’s struglle against the Dutch and Japanese colonialists. This research aims to obtain the results of an ethnomathematics study from the Museum of Perjuangan Subkoss Garuda Sriwijaya regarding historical, philosophical, and mathematical aspects and implement it in the Merdeka Curriculum in mathematics for elementary schools. This research employs an ethnographic approach with a qualitative descriptive approach, utilizing observation, interviews, documentation, field notes, and data triangulation. The analysis techniques include data reduction, data presentation, verification, and conclusions drawings. The study involved three informants who had direct interaction with the Museum of Perjuangan Subkoss Garuda Sriwijaya. The results of this research shows that ethnomathematics studies offer several insights: the historical aspect introduces local history and culture through artifacts that supported the struggle during the physical revolution; the philosophical aspectteaches the values ​​of caring, cooperation, creativity, and diversity within society; and the mathematical concepts reveals the concept of geometry, including area, perimeter, volume of both two-dimensional figures and three-dimensional figures, and length measurement.

Log-Cholesky filtering of diffusion tensor fields: Impact on noise reduction

Diffusion tensor imaging (DTI) is a powerful neuroimaging technique that provides valuable insights into the microstructure and connectivity of the brain. By measuring the diffusion of water molecules along neuronal fibers, DTI allows the visualization and study of intricate networks of neural pathways.DTI is a noise-sensitive method, where a low signal-to-noise ratio (SNR) results in significant errors in the estimated tensor field. Tensor field regularization is an effective solution for noise reduction.Diffusion tensors are represented by symmetric positive-definite (SPD) matrices. The space of SPD matrices may be viewed as a Riemannian manifold after defining a suitable metric on its tangent bundle. The Log-Cholesky metric is a recently developed concept with advantages over previously defined Riemannian metrics, such as the affine-invariant and Log-Euclidean metrics. The utility of the Log-Cholesky metric for tensor field regularization and noise reduction has not been investigated in detail.This manuscript provides a quantitative investigation of the impact of Log-Cholesky filtering on noise reduction in DTI. It also provides sufficient details of the linear algebra and abstract differential geometry concepts necessary to implement this technique as a simple and effective solution to filtering diffusion tensor fields.

Penerapan Kearifan Lokal Samosir dalam Mengeksplorasi Bentuk-Bentuk Geometri

The purpose of this study is to investigate how the Batak people in Samosir use their local knowledge to understand geometric shapes. The literature review research method, which is a data collection strategy focused on the evaluation of relevant and current scientific literature related to the research problem, is used in this publication. The data sources in the study came from Google Scholar, Scopus, and SINTA. Data analysis was carried out qualitatively consisting of data reduction, data presentation, and conclusion drawing. The results of the data analysis show that the way local knowledge is applied to the study of geometric forms shows how Batak culture has used geometric principles in religious ceremonies as well as art and construction. In addition to having a practical and aesthetic purpose, these geometric shapes have a deep philosophical feel that enhances the cultural identity and bond of the community with nature and its ancestors. Through the local wisdom of the Batak people in Samosir, geometric ideas have been successfully identified and applied to various parts of their lives, including traditional house buildings, traditional artifacts, and ulos weaving crafts. Overall, this study shows how the concept of geometry is used organically by Batak local wisdom, which is essential for preserving community peace, fostering cultural identity, and fostering spiritual ties with nature and ancestors.

Rogramming the Atoms & Compounds and the Unification of Physics and Chemistry

One of the most important concepts in Geometry is, distance, which is the Quanta in geometry, while in Material-Geometry the composition of Opposite, the Material-Point which is the Quanta in Chemistry and Physics. As in Algebra Zero ,0, is the Master-key number for all Positive and Negative numbers and this because their sum and multiplication become zero, and the same on any coordinate-System where ± axes pass from zero, The Rolling of Positive ⊕, constituent on the Negative ⊝, constituent, creates the Neutral Material Point Which Equilibrium. Angular-Momentum is identical with Spin and consists the First-Discrete-Energy-Monad which occupies, Discrete Value and Direction, in contradiction to the Point which is Nothing, Dimensionless and without any Direction. Quaternion [(+) ↻↺ (-)] ≡ Box which carries the Principal stress σ between A (+), B (-) which σ, as Centripetal-acceleration is the minimum Energy becoming from the in-storage AB acceleration and is equal to the Gravity g. Because of the two different motions, Revolving and Periodic, acceleration of the Gravity g ≡ ± σ exists as the First Energy-Box- ,while in the Second is followed the Local-Extreme-case this acceleration of Gravity g ≡ ± σ, is altered Locally by changing the Principal-stress σ with a Local-uniform-Pressure → ≡ g k = g. [ Force/Area] = G, i.e. The minimum Local - Energy acceleration is the known, Universal Gravitational-constant G = g k = g = σ, for Macrocosm and Microcosm, Obeying Newton`s Laws of motion. It was proved that, Constant G, is the mechanism for the First-kick-Start on the Granular-Energy-monad, g, which Acts in the lightest and less-mass Particle and which is the Hydrogen. The Electron-Nutation-Energy due to g, effect is the minimum frequency ≡ = 2 ,8398447. , and which so exists in all Atoms. This Energy in Hydrogen-Cave as E-M, Conductor ≡ The Pin of Atom-Plug Into their Sockets, which are the Orbit – Bracket – Hooks ≡ The Hands of Atoms, i.e. → The Atoms Plug with their Pins into the other Atoms-Drains = Holes, and so are Bonded. This is the Resonance frequency between all Atoms, and because Hydrogen is Common to all Atoms, so Bond to Molecules and Crystals, and all other Compounds in this Cosmos.

Students’ Knowledge of Integration of GeoGebra in Mathematics Classes in Uganda

GeoGebra, an open-source Dynamic Mathematics Software (DMS), has emerged as a transformative tool in mathematics education, allowing students and teachers to visualize, explore, and manipulate geometric concepts in dynamic virtual environments. This study investigated the students’ knowledge of integration of GeoGebra in mathematics classes in lower secondary schools in Rwampara District, Southwestern Uganda. The objectives of this study were to identify the most abstract geometrical concepts posing challenges to students, examine the impact of GeoGebra training on the identified abstract concepts, and assess the relationship between students' attitudes and the integration of GeoGebra. This study utilized a quantitative approach, employing quasi-experimental and cross-sectional survey designs. Data were collected from 103 students using questionnaires and pre-and post-tests to evaluate the impact of GeoGebra software intervention. Independent samples t-test, Pearson's linear correlation, and descriptive statistics such as means, standard deviations, and frequencies were employed as data analysis methods. The study found that the students identified rotational transformation as the most abstract geometrical concept. The pre-test results indicated that respondents had similar baseline knowledge of rotational transformation. However, after the GeoGebra intervention, the post-test results showed that the difference in mean scores between the experimental and control groups was statistically significant. Additionally, the experimental group demonstrated a positive attitude towards the software. The study concluded that while rotational transformation posed a significant challenge for students, GeoGebra significantly improved their conceptual understanding and performance in this geometry concept. The study recommended that teachers should strongly consider integrating GeoGebra into their ICT-based teaching and learning resources to help improve students' conceptual understanding, motivation, attitude, and achievement in rotational transformations and other geometric concepts

Differential geometry and general relativity with algebraifolds

It is often noted that many of the basic concepts of differential geometry, such as the definition of connection, are purely algebraic in nature. Here, we review and extend existing work on fully algebraic formulations of differential geometry which eliminate the need for an underlying manifold. While the literature contains various independent approaches to this, we focus on one particular approach that we argue to be the most natural one based on the definition of algebraifold, by which we mean a commutative algebra A for which the module of derivations of A is finitely generated projective. Over R as the base ring, this class of algebras includes the algebra C∞(M) of smooth functions on a manifold M, and similarly for analytic functions. An importantly different example is the Colombeau algebra of generalized functions on M, which makes distributional differential geometry an instance of our formalism. Another instance is a fibred version of smooth differential geometry, since any smooth submersion M→N makes C∞(M) into an algebraifold with C∞(N) as the base ring. Over any field k of characteristic zero, examples include the algebra of regular functions on a smooth affine variety as well as any function field.Our development of differential geometry in terms of algebraifolds comprises tensors, connections, curvature, geodesics and we briefly consider general relativity.

Edge morphology attention mechanism and optimal geometric matching connection model for vascular segmentation

Over the past decades, medical image segmentation methods have been intensively developed, but there are still a number of unsolved challenging problems in vascular image segmentation, including broken vessels, insufficient vessel branches, and missing small vessels. In order to optimize the topology and accuracy of segmented vessels, we propose a novel Edge Morphology Attention Network (EMA-Net) for the segmentation of vessel-like structures, and an Optimal Geometric Matching Connection (OGMC) model to connect the broken vessel fragments. The EMA-Net has an edge attention module that is able to improve segmented performances of edges and small branches by morphology operation extracting boundary voxels on multi-scales. The OGMC model uses the geometry concept of curve touching to filter out fragmented vessel endpoints, and then employs minimal surfaces to determine the optimal connection order between blood vessels. Finally, we adopt geodesics to repair missing vessels under a Riemannian metric. Our method achieves superior or competitive results compared to state-of-the-art methods on four datasets of 3D vascular segmentation tasks, both effectively reducing vessel broken and increasing vessel branch richness, yielding blood vessels with more precise topological structures.

EKSPLORASI ETNOMATEMATIKA PADA BENTUK JAJANAN PASAR DI DAERAH JEPARA UNTUK PEMBELAJARAN MATEMATIKA SMP

In the cultural context, traditional market snacks in the Jepara area have the potential to integrate ethnomathematics into mathematics learning. Traditional market snacks in Jepara offer various forms of snacks in which there is a geometric concept. The purpose of this study is to find out the concept of geometry in traditional market snacks in the Jepara area. In this study, qualitative methods are used with an ethnographic approach. The objects studied by the researcher were 5 traditional market snacks, with 3 respondent subjects, namely traditional market snack sellers found in Jepara market, Pecangaan market, and Tegalsambi market. Several techniques for collecting data are used in this study, including observation techniques, interviews, and documentation. Based on the research conducted by the researcher, the results were obtained that there are geometric concepts and cultural elements in 5 traditional market snacks in the Jepara area, including a ball-shaped klepon, a rectangular layer, a ball-shaped onde-onde, a triangular prism-shaped banana bongko, and a solid half-ball shaped kintelan. The results of this research are expected to function as an effective alternative learning resource to introduce geometry concepts to students.

BUDAYA TELLASEN TOPAK: DARI KETUPAT KE IDE AKTIVITAS SPATIAL SKILLS DALAM PEMBELAJARAN MATEMATIKA

Tellasen topak, which is identical to ketupat, is a culture of choosing, weaving, and processing janur into a medium for ketupat food. This culture is full of spatial skills activities, because it turns two-dimensional objects into space objects (three-dimensional). The purpose of this study is to explore the idea of spatial skills activities from ketupat in mathematics learning. The research method is qualitative research. Data collection techniques are observation, documentation, and internet search. Data analysis is in the form of data reduction, data presentation, and conclusion. The results concluded that in choosing, weaving, and processing janur into a medium for ketupat food, there are spatial skills activity ideas in the form of spatial visualization, spatial orientation and spatial rotation. In math learning, these ideas play a role in explaining the concepts of geometry, measurement and number. The results of this study contribute to mathematics learning, namely in developing culture-based teaching modules.

Electromagnetic forces and their finite element computation

AbstractThis paper introduces the concepts of differential geometry that are necessary to establish a systematic definition for electromagnetic forces by means of a natural thermodynamic approach. It is shown that standard electromagnetic force formulae used in finite element computational electromagnetism are particular instances of that general approach. Finally, the paper offers a complete conceptual framework, as well as a mathematical toolbox, to derive electromagnetic force formulae for multi‐physics finite element models with complex materials.

On the role of geometric phase in the dynamics of elastic waveguides.

The geometric phase provides important mathematical insights to understand the fundamental nature and evolution of the dynamic response in a wide spectrum of systems ranging from quantum to classical mechanics. While the concept of geometric phase, which is an additional phase factor occurring in dynamical systems, holds the same meaning across different fields of application, its use and interpretation can acquire important nuances specific to the system of interest. In recent years, the development of quantum topological materials and its extension to classical mechanical systems have renewed the interest in the concept of geometric phase. This review revisits the concept of geometric phase and discusses, by means of either established or original results, its critical role in the design and dynamic behaviour of elastic waveguides. Concepts of differential geometry and topology are put forward to provide a theoretical understanding of the geometric phase and its connection to the physical properties of the system. Then, the concept of geometric phase is applied to different types of elastic waveguides to explain how either topologically trivial or non-trivial behaviour can emerge based on the geometric features of the waveguide. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 2)'.

Exploring the Role of Geometry Understanding and Spatial Ability in Academic Achievement: The Moderating Effect of Self-Confidence

This study investigates the impact of understanding geometry concepts, spatial ability, and self-confidence on student learning outcomes. Understanding geometry enhances mathematical logic, spatial ability strengthens spatial comprehension, and self-confidence boosts learning motivation. Collectively, these variables are crucial for improving student learning outcomes. The research employed an explanatory design, conducted in two elementary schools within the same cluster. A random sampling method selected samples from one class of an Elementary School in Panyikokkang with 21 students and another class with 23 students, resulting in a total of 44 students. Data collection instruments included a validated Self-Confidence questionnaire, Geometry Concept Understanding test, Spatial Ability test, and a Student Learning Outcomes test derived from daily quizzes on geometry topics. Data were analyzed using Inferential Statistical Analysis and Moderated Regression Analysis (MRA). The Moderated Regression Analysis (MRA) revealed that self-confidence acts as a quasi-moderator in the relationship between understanding geometry concepts and learning outcomes. Specifically, students with higher self-confidence showed a better understanding of geometry concepts, positively influencing their learning outcomes. However, self-confidence did not moderate the effect of spatial ability on learning outcomes. These findings highlight the importance of understanding geometry concepts in enhancing students' comprehension of geometry and developing spatial abilities at the elementary level. The results suggest that self-confidence plays a significant role in mediating the relationship between understanding geometry concepts and learning outcomes but not between spatial ability and learning outcomes. The study underscores the necessity for educators to focus on understanding geometry concepts and fostering self-confidence to design effective learning strategies that improve student achievement in mathematics, particularly in geometry.

Mathematical Concepts and Cultural Values in Sonaf Tamkesi: Ethnomathematics Study in a Traditional Village on Timor Island

Mathematics is a universal science, and it is crucial for everyone to master it, both in terms of its applications and its cognitive patterns. The purpose of this research is to explore the concepts of mathematics and cultural values found in the artifacts of Sonaf Tamkesi Manufui. The study was conducted in the traditional village of Tamkesi, south Biboki District, North Central Timor Regency, in July 2023. The subjects of this study were three community leaders. The type of data collected for this research was primary data obtained through documentation and interviews between the researcher and the respondents. Data analysis was conducted through stages of domain analysis, taxonomy, componential analysis, and cultural themes. The research findings indicate that there are concepts of plane geometry in the artifacts such as trapeziums, rectangles, squares, rhombuses, triangles, circles, cylinders, hexagons, and cubes. Meanwhile, the cultural values of the artifacts in Sonaf Tamkesi are the roof symbolizes harmony, the door symbolizes greatness and respect for ancestors, the pillars symbolize vertical relations with ancestors, the betel-nut container symbolizes service, the clay pot symbolizes a gift from ancestors to the living, the hearth symbolizes balance, and harmony in achieving goals.

Student Teachers’ Knowledge of School-level Geometry: Implications for Teaching and Learning

<p style="text-align:justify">This study aimed to assess the geometric knowledge of student teachers from a university in the Eastern Cape province of South Africa. The study used a sample of 225 first-year student teachers who completed school mathematics baseline assessments on a computer- aided mathematics instruction (CAMI) software. The study adopted a descriptive cross-sectional research design, using quantitative data to measure student teachers’ geometry achievement level, and qualitative data to explain the challenges encountered. The results show that student teachers exhibited a low level of understanding of school-level geometry. The low achievement levels were linked to various factors, such as insufficient grasp of geometry concepts in their secondary school education, difficulty in remembering what was done years ago, low self-confidence, and lack of Information and Communications Technology (ICT) skills along with the limited time for the baseline tests. These results suggest that appropriate measures should be taken to ensure that student teachers acquire the necessary subject-matter knowledge to teach effectively in their future classrooms.</p>