Geometric Classification Research Articles

Polynomial Bounds on Torsion From a Fixed Geometric Isogeny Class of Elliptic Curves

Polynomial Bounds on Torsion From a Fixed Geometric Isogeny Class of Elliptic Curves

A geometric classification of rod complements in the 3-torus

Rod packings are used in crystallography to describe crystal structures with linear or zigzag chains of particles, and each rod packing can be topologically viewed as a collection of disjoint geodesics in the 3 3 -torus. Hui and Purcell developed a method to study the complements of rods in the 3 3 -torus with the use of 3 3 -dimensional geometry and tools from the 3 3 -sphere, and they partially classified the geometry of some families of rod complements in the 3 3 -torus. In this paper, we provide a complete classification of the geometry of all rod complements in the 3 3 -torus using topological arguments.

A geometric classification of the holomorphic vertex operator algebras of central charge 24

A geometric classification of the holomorphic vertex operator algebras of central charge 24

Eye movements in conceptual and non-conceptual thinking

AbstractHow to influence and assess whether students engage in conceptual thinking are longstanding methodological problems in mathematics education. Recently, eye-tracking technology has fueled a discussion on whether eye movement analysis can support valid inferences about mathematical thinking. This study investigates whether eye movement analysis can distinguish between conceptual and non-conceptual thinking in a geometric classification task where both modes of thinking lead to identical responses. Participants were asked to classify geometric shapes while we tracked their eye movements and to report their thinking verbally. Our findings indicate that self-reported conceptual thinking is characterised by fewer eye movements between task shapes and response shapes, and that self-reported non-conceptual thinking involves comparing the shapes’ similarity directly. A logistic regression model correctly classified the self-reported ways of thinking in 80.3% of the cases. We conclude that eye-tracking can contribute to making inferences about mathematical thought processes and facilitate research on how to engage students in conceptual thinking and development.

The Determination of Criticality for Ice Shapes Based on CCAR-25

Determining the criticality of ice shapes is a necessary condition for verifying compliance with icing airworthiness regulations. However, the clear, concise, and applicable criterion based on the geometric characteristics of ice shapes has not been clearly given out by current advisory circulars. To address this problem, this paper summarizes aerodynamic performance items and recommended ice shapes the latest version of CCAR-25 and corresponding advisory circulars for a variety of flight phases, including takeoff, holding, en route, DTO, etc., instead of the single phase of holding in the previous research. Based on the geometric classification of the ice shapes, the dominant parameters of various ice shapes are clarified by the correlation between the geometric parameters and aerodynamic effects. The geometric parameters to determine the criticality of specific ice shapes are defined as the roughness height and range for the roughness ice and the total projection height in the direction of lift for the horn ice. On this basis, the detailed determination criterion of critical ice shape geometries corresponding to different flight phases and aircraft components is formulated, which will provide an operational selection methodology for determining the geometries of critical ice shapes at the airworthiness certification stage.

On the Complexity of Target Set Selection in Simple Geometric Networks

We study the following model of disease spread in a social network. At first, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbors are already infected. We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs. We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph classes, such as interval graphs and grid graphs.

Geometric nonlinear classification of Hilbert $$C^*$$-modules based on strong Birkhoff–James orthogonality

Geometric nonlinear classification of Hilbert $$C^*$$-modules based on strong Birkhoff–James orthogonality

Interactive effects of elevation difference, slope variation, and terrain formation on road traffic crashes occurrences using triangular irregular network

AbstractNumerous factors contribute to traffic crashes, which result in fatalities and injuries. Road traffic crashes and their severity level are likely caused by variations in elevation, slope, and terrain formation. As far as our knowledge is concerned, the triangular irregular network (TIN) was not used to analyze dangerous zones of the road network with respect to traffic crashes' location. The main purpose of this study was to investigate a dangerous road network that was affected by traffic crashes using TIN. To address the goal, this study used elevation differences slope variation, and terrain between road traffic crash locations. The study used the intersection zone road network coordinates of traffic crashes in the city of Budapest that were recorded from 2017 to 2021. Using tools like ArcGIS and MS Excel, the study organized, modeled, and analyzed the data. Geometrical interval classification was used to classify the data in accordance with the characteristics of elevation deference and slope variation. Both inferential and descriptive statistics were engaged to explain, analyze, and summarize the findings. The results of this study demonstrated that the majority of traffic crashes occurred in locations with lower elevations and their differences, steeper/cliff slopes between points, and flat terrain. The finding was in contrast to what was observed in other studies. This study advocates that TIN is the optimal method and substitute for analyzing the risky zone of the road network at macroscopic level.

Asymmetric lesion detection with geometric patterns and CNN-SVM classification

In dermoscopic images, which allow visualization of surface skin structures not visible to the naked eye, lesion shape offers vital insights into skin diseases. In clinically practiced methods, asymmetric lesion shape is one of the criteria for diagnosing Melanoma. Initially, we labeled data for a non-annotated dataset with symmetrical information based on clinical assessments. Subsequently, we propose a supporting technique—a supervised learning image processing algorithm—to analyze the geometrical pattern of lesion shape, aiding non-experts in understanding the criteria of an asymmetric lesion. We then utilize a pre-trained convolutional neural network (CNN) to extract shape, color, and texture features from dermoscopic images for training a multiclass support vector machine (SVM) classifier, outperforming state-of-the-art methods from the literature. In the geometry-based experiment, we achieved a 99.00 % detection rate for dermatological asymmetric lesions. In the CNN-based experiment, the best performance is found 94 % Kappa Score, 95 % Macro F1-score, and 97 % weighted F1-score for classifying lesion shapes (Asymmetric, Half-Symmetric, and Symmetric).

CARBO: Clustering and rotation based oversampling for class imbalance learning

Class imbalance of a data set is a crucial problem in machine learning where one class significantly outnumbers others. In such a data set, classification is a troublesome task for the standard classification algorithms, leading to bias towards the majority class. Different methods have been developed so far, such as oversampling, undersampling, and cost-sensitive learning, to deal with class imbalance circumstances. Among these techniques, oversampling technique does not suffer from the information loss and critical cost selection challenges. However, appropriate synthetic sample generation can be challenging and vulnerable to privacy leakage. This research proposed an oversampling technique, called CARBO, using threshold-based geometric rotation and majority class influenced clustering. Unlike the existing resampling approaches to class imbalance problem, we contribute to consider the data privacy and optimal sample generation together for effective oversampling. The performance of CARBO is evaluated using 44 benchmark imbalanced data set. The empirical analysis elucidates that CARBO can make boosting-based C4.5 ensemble classifiers perform higher for 73% of the data set than six state-of-the-art approaches. In addition, the theoretical compatibility analysis of CARBO demonstrates its robustness.

Disconnected gauge groups in the infrared

Gauging a discrete 0-form symmetry of a QFT is a procedure that changes the global form of the gauge group but not its perturbative dynamics. In this work, we study the Seiberg-Witten solution of theories resulting from the gauging of charge conjugation in 4d N\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\mathcal{N} $$\\end{document} = 2 theories with SU(N) gauge group and fundamental hypermultiplets. The basic idea of our procedure is to identify the ℤ2 action at the level of the SW curve and perform the quotient, and it should also be applicable to non-lagrangian theories. We study dynamical aspects of these theories such as their moduli space singularities and the corresponding physics; in particular, we explore the complex structure singularity arising from the quotient procedure. We also discuss some implications of our work in regards to three problems: the geometric classification of 4d SCFTs, the study of non-invertible symmetries from the SW geometry, and the String Theory engineering of theories with disconnected gauge groups.

Technology and geometry: Fostering young children's geometrical concepts through a research-based robotic coding program

AbstractGeometric concepts are fundamental to early geometry education, and developmentally appropriate practices are crucial for teaching them to young children. Robotic coding is an effective tool in many areas of early mathematics and has significant potential in teaching geometric concepts. This study aimed to test the impact of a research-based robotic coding program on young children's understanding of geometric concepts. A quasi-experimental design was used with two intervention groups. The study involved 52 5-year-old children from a public kindergarten. The intervention-1 group (n = 18) received a research-based robotic coding program, while the intervention-2 group (n = 16) received a research-based no-coding program. The non-intervention group (n = 18) attended their regular program. Data on participants' understanding of geometrical concepts was collected through individual interviews using a geometric shapes classification test. Hierarchical linear modeling (HLM) was utilized to assess the effectiveness of two interventions. The results showed that both interventions had a significant effect with the robotic program being more effective. Additionally, follow-up tests indicated that both interventions had a lasting effect on the children's understanding of geometrical concepts. The study highlighted the potential of incorporating robotic coding and relevant research in fostering young children's geometrical development.

Macrobenthic fauna and the ecological status of a shellfish farm in the Mediterranean Sea (Algeria)

A faunistic and ecological study on the benthic macrofauna was carried out in the shellfish farm in Ain Chorb (Ain Taya), central Algeria. This study aims to characterise the composition, structure, and functioning of the benthic community to establish the reference state and ecological quality of the shellfish farm and to monitor its evolution through an adapted sampling and observation methodology adapted and used by the scientific community at the international level. Benthic fauna samples were collected in two different periods (warm and cold season) between February 2020 and June 2021; six grab samples were carried out at three sampling stations. Different indices were evaluated based on the benthic community characteristics (species richness, abundance, density, ecological and trophic groups, and the biotic index) and indicator species. An inventory of the macrobenthic fauna of the farm allowed us to identify 6 phyla, 10 classes, 48 families, 106 genera, and 138 species, with a total of 45960 ind/m2. The benthic assemblage is characterised by the dominance of the species Abra alba, Salvatoria clavata, Caecum spp., and Bittium spp. The results of the four benthic indices (the Shannon-Weaver diversity index [H′], the AZTI Marine Biotic Index [AMBI], the multivariate AZTI Marine Biotic Index [M-AMBI], and the BENTIX) indicated that that the sampling stations have a moderate and good ecological status. These results were confirmed by the abundance/biomass comparison curves and geometric abundance class methods. This study provides the first inventory and represents the reference state of the soft-bottom communities of the aquaculture farm. The findings also indicate that the macrobenthic assemblage is excellent indicator of the ecological status.

Method for solving system of linear equations mxn, m=1, 2, 3; Based on its Geometrical Interpretations

In this paper, by using the geometrical classification of the types of linear equations or systems of two equations, we determine a method for solving systems of linear equations. The analysis specifically refers to systems of linear equations that reduce to a quadratic system of the second or third order in which all determinants of the system are zero. The methodological significance of the method is also discussed.

Research on the application of knowledge mapping and knowledge structure construction based on adaptive learning model

This project has developed a geometry learning software that integrates multiple computer technologies to address the challenges of deep analysis of knowledge points and establishing connections in learning software. The software combines Long Short-Term Memory (LSTM) and Residual Neural Network (ResNet101) to encode text and image features. A self-attention mechanism is used to fuse information from both modalities, enabling decoding of geometric models and classification of corresponding knowledge points.This project uses LSTM and ResNet101 models to extract text and visual features for problem-solving using the Multi Mode Thinking Chain (CoT) method. Classification labels are utilized to generate text responses for problem-solving ideas. Furthermore, a recommendation module is proposed, which combines knowledge tracking and neural collaborative filtering algorithms to capture student behavior and knowledge point vectors. Implicit factors representing students' mastery of different knowledge points are used as inputs in neural collaborative filtering for personalized recommendations. The results demonstrate improvements in accuracy using the ResNet + LSTM multimodal algorithm, achieving a 13 % increase compared to single-modal classification. The multimodal CoT approach also outperforms language models like GPT3.5 and VisualBert by 10 %. Additionally, the combined algorithm of knowledge tracking and neural collaborative filtering shows a 13.3 % higher F1 value compared to ordinary algorithms, confirming the superiority of the adopted method in this project.

Detection and Diagnostics of Bearing and Gear Fault under Variable Speed and Load Conditions Using Heterogeneous Signals

In industrial applications, rotating machines operate under real-time variable speed and load regimes. In the presence of faults, the degradation of critical components is accelerated significantly. Therefore, robust monitoring algorithms able to identify these faults become crucial. In the literature, it is hard to find comprehensive monitoring systems that include variable speed and load regimes with combined gearbox faults using electrical and vibration signals. For this purpose, a novel signal processing methodology including a geometric classification technique is proposed. This methodology is based on using different types of sensors such as current, voltage and vibration sensors with a regime normalization, which allows the grouping of different regimes belonging to the same health state. It consists of reducing dispersion between the class observations and separating other classes representing different health states including the variation in speed and load. Then, a peripheral threshold is proposed in our classifier to diagnose new health states. To verify the effectiveness of the methodology, current, voltage and vibration data from a gearbox system are collected under variable speed and load levels.

Convolutional Neural Network (CNN) Model for the Classification of Varieties of Date Palm Fruits (Phoenix dactylifera L.).

The popularity and demand for high-quality date palm fruits (Phoenix dactylifera L.) have been growing, and their quality largely depends on the type of handling, storage, and processing methods. The current methods of geometric evaluation and classification of date palm fruits are characterised by high labour intensity and are usually performed mechanically, which may cause additional damage and reduce the quality and value of the product. Therefore, non-contact methods are being sought based on image analysis, with digital solutions controlling the evaluation and classification processes. The main objective of this paper is to develop an automatic classification model for varieties of date palm fruits using a convolutional neural network (CNN) based on two fundamental criteria, i.e., colour difference and evaluation of geometric parameters of dates. A CNN with a fixed architecture was built, marked as DateNET, consisting of a system of five alternating Conv2D, MaxPooling2D, and Dropout classes. The validation accuracy of the model presented in this study depended on the selection of classification criteria. It was 85.24% for fruit colour-based classification and 87.62% for the geometric parameters only; however, it increased considerably to 93.41% when both the colour and geometry of dates were considered.

Active hydraulics laws from frustration principles

Viscous flows are laminar and deterministic. Robust linear laws accurately predict their streamlines in geometries as complex as blood vessels, porous media and pipe networks. However, biological and synthetic active fluids defy these fundamental laws. Irrespective of their microscopic origin, confined active flows are intrinsically bistable, making it challenging to predict flows in active fluid networks. Although early theories attempted to tackle this problem, quantitative experiments to validate their relevance to active hydraulics are lacking. Here we present a series of laws that accurately predict the geometry of active flows in trivalent networks. Experiments with colloidal rollers reveal that active hydraulic flows realize dynamical spin ices: they are frustrated, non-deterministic and yield degenerate streamline patterns. These patterns split into two geometric classes of self-similar loops, which reflect the fractionalization of topological defects at subchannel scales. Informed by our measurements, we formulate the laws of active hydraulics in trivalent networks as a double-spin model. We then use these laws to predict the random geometry of degenerate streamlines. We expect our fundamental understanding to provide robust design rules for active microfluidic devices and to offer avenues to investigate the motion of living cells and organisms in complex habitats.

Partial and Simultaneous Transitive Orientations via Modular Decompositions

A natural generalization of the recognition problem for a geometric graph class is the problem of extending a representation of a subgraph to a representation of the whole graph. A related problem is to find representations for multiple input graphs that coincide on subgraphs shared by the input graphs. A common restriction is the sunflower case where the shared graph is the same for each pair of input graphs. These problems translate to the setting of comparability graphs where the representations correspond to transitive orientations of their edges. We use modular decompositions to improve the runtime for the orientation extension problem and the sunflower orientation problem to linear time. We apply these results to improve the runtime for the partial representation problem and the sunflower case of the simultaneous representation problem for permutation graphs to linear time. We also give the first efficient algorithms for these problems on circular permutation graphs.

The algebraic and geometric classification of nilpotent binary and mono Leibniz algebras

This paper is devoted to the complete algebraic and geometric classification of complex 5-dimensional nilpotent binary Leibniz and 4-dimensional nilpotent mono Leibniz algebras. As a corollary, we have the complete algebraic and geometric classification of complex 4-dimensional nilpotent algebras of nil-index 3.